Radiative Transfer: An Introduction to Exact and Asymptotic Methods 3030952460, 9783030952464

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Hélène Frisch

Radiative Transfer An Introduction to Exact and Asymptotic Methods

Radiative Transfer

Hélène Frisch

Radiative Transfer An Introduction to Exact and Asymptotic Methods

Hélène Frisch Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange Université Côte d’Azur Nice Cedex 04, France

ISBN 978-3-030-95246-4 ISBN 978-3-030-95247-1 (eBook) https://doi.org/10.1007/978-3-030-95247-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Uriel, Anne, and Thomas

Preface

Transport of particles such as photons, neutrons, electrons, and molecules through a host medium usually takes place by multiple (repeated) scatterings. This book presents methods for constructing exact solutions of stationary radiative transfer problems related to the transport of photons in a stellar or planetary atmosphere or any foggy medium. When the scatterings do not modify the physical properties of the host medium, the equations describing the transport of particles are linear. In spite of this, neither the construction of exact solutions nor numerical solutions are straightforward, because of the nonlocal character of multiple scattering. Exact solutions provide trustful descriptions of the physical phenomenon and can serve as testing ground for numerical methods of solutions. Exact solutions, that is solutions which can be expressed as known functions or quadratures, can be constructed for a few, but widely used, types of scattering processes and geometries of the host medium. Scattering processes allowing exact solutions discussed in this book are monochromatic scattering, complete frequency redistribution, and Rayleigh scattering. Monochromatic scattering and Rayleigh scattering (which accounts for the linear polarization and blue color of the sky) govern the formation of continuous spectra. Photons change directions at each scattering but keep the same frequency. Complete frequency redistribution (an idiomatic expression of the astrophysical literature) governs the formation of most atomic spectral lines. Photons do undergo changes of direction, but also of frequencies, in an uncorrelated way, within the frequency range of the spectral line. Concerning the host medium, the spreading of a source of light by a large number of scatterings can be studied by assuming a medium of infinite extension. A standard Fourier transformation can then provide the relevant solution. In astrophysics, and also in nuclear reactors, a very important question is how a radiation field will be reflected and transmitted through a scattering medium, such as a stellar or planetary atmosphere. A closely related question is how photons created inside these atmospheres will escape the medium, and what will be their observable distribution in directions and frequencies. A reasonably good model for a stellar atmosphere is a semi-infinite medium. Finding exact solutions is possible but rather hard. The first exact solution of a transport problem in a semi-infinite plane-parallel medium was vii

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Preface

constructed by (Wiener and Hopf 1931) for the Milne problem, which describes the temperature profile of a stellar atmosphere in radiative equilibrium. Their method of solution, which now carries their name, is one the greatest achievements in mathematical physics in the first half of this century. It relies on the properties of analytic functions in the complex plane. In contrast, when the question is posed to find the distribution of photons that have been reflected or transmitted through a scattering medium of finite extension, a slab for example, then there is no explicit expression for the radiation field, inside or outside the medium. However, there are many interesting exact relations and integral equations, which can be solved numerically. This topic is only briefly mentioned in this book. Following the work of Wiener and Hopf, several methods have been developed to tackle semi-infinite medium problems. One of the best known method in astrophysics is the construction of a nonlinear integral equation for the famous H function, a sort of special function for half-space transport problems. Proposed for the first time by Ambartsumian (1942), it largely avoids complex plane analysis and relies on invariance properties of the radiative transfer process. Plasma physicists became strongly interested in the subject in relation to the transport of neutrons in nuclear reactors. New methods were developed, such as the Case singular eigenfunction expansion method (Case 1960), which leads to linear singular integral equations with Cauchy-type kernels. Methods of solutions for these equations, also based on complex plane analysis, rely largely on a technique introduced by Carlemann (1922) and developed by Muskhelishvili (1953). These methods lead to boundary value problems in the complex plane, known as Riemann–Hilbert problems, and provide explicit expressions for the H -function, as does the Wiener– Hopf method. A key motivation for writing this book was to present in the same volume several of the methods leading to exact results for semi-infinite media, usually presented in separate books or articles, for both scalar and polarized radiation fields, and also connections, that are not always clearly apparent, between the various methods. When the number of scatterings in the host medium is very large, asymptotic techniques may be employed to analyze the large-scale behavior of the random walk of the photons and to explain, for example, why monochromatic scattering has all the characteristics of an ordinary diffusion process, while complete frequency redistribution of spectral lines has in contrast those of a Lévy walk. This asymptotic approach is presented in Part III of the book, while exact methods are presented in Parts I and II. Part I deals with unpolarized radiation and Part II with monochromatic radiation, polarized by Rayleigh scattering, and spectral lines, polarized by resonance scattering and the Hanle effect, which is a modification of resonance polarization by a weak magnetic field. Details on the organization of each part are given in Chap. 1. The intended readership for the book ranges from first-year graduate students to professional scientists. Astrophysicists can find in this book exact methods of solutions used in radiative transfer, but also applied in distant fields, such as financial mathematics. A major effort has been made in the organization of the material to give a synthetic view on exact and asymptotic methods in radiative transfer.

Preface

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Although oriented towards methods of solutions, this book also provides exact expressions for the radiation field intensity and polarization for a number of standard problems. I am deeply grateful to many colleagues for their help and encouragement in this undertaking. My sincere thanks go particularly to V. Bommier, V. V. Ivanov, B. Rutily, and P. Zweifel, who have read significant parts of the book and have taken the time to share with me their intimate knowledge of the field. I have adopted many of their suggestions. My thanks go also to my family, especially to my mother D. Piron-Lévy, for lasting encouragements; to my husband, Uriel, for reading the full manuscript and playing the role of the non-specialist; and to L. Anusha, E. Lega, and M. Sampoorna for their very generous help in the preparation of the figures. This book would not exist without the continuous support of Springer teams in Europe and India. I owe them a thousands thanks. Nice Cedex 04, France

Hélène Frisch

References Ambartsumian, V.A.: Light scattering by planetary atmospheres. Astron. Zhurnal 19, 30–41 (1942) Carleman, T.: Sur la résolution de certaines équations intégrales. Ark. Mat. Astr. Fys. 16, 1–19 (1922) Case, K.M.: Elementary solutions of the transport equation and their applications. Ann. Phys. (New York) 9, 1–23 (1960) Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen (Based on the Second Russian Edition Published in 1946) (1953); Dover Publications (1991) Wiener, N., Hopf, D.: Über eine Klasse singulärer Integralgleichungen, Sitzungsberichte der Preussichen Akademie, Mathematisch-Physikalische Klasse 3. Dez. 1931, vol. 31, pp. 696–706 (ausgegeben 28. Januar 1932) (1931); English translation: In: Paley, R.C., Wiener, N., Fourier transforms in the Complex Domain. Am. Math. Soc. Coll. Publ., vol. XIX, pp. 49–58 (1934)

Contents

1

An Overview of the Content .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Part I: Scalar Radiative Transfer Equations . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Part II: Scattering Polarization .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Part III: Asymptotic Properties of Multiple Scattering . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part I

1 1 4 6 8

Scalar Radiative Transfer Equations

2

Radiative Transfer Equations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 The Integro-Differential Radiative Transfer Equations .. . . . . . . . . . . . 2.1.1 Monochromatic Scattering . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 Complete Frequency Redistribution . . .. . . . . . . . . . . . . . . . . . . . 2.1.3 The Diffuse Radiation Field. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Integral Equations for the Source Function . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Monochromatic Scattering . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 The Milne Problem . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 Complete Frequency Redistribution . . .. . . . . . . . . . . . . . . . . . . . 2.3 Neumann Series Expansion .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 The Green Function and Associated Functions .. . . . . . . . . . . . . . . . . . . . 2.4.1 Some Definitions . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.2 A Lemma on Wiener–Hopf Equations.. . . . . . . . . . . . . . . . . . . . 2.4.3 The Source Function and the Resolvent Function . . . . . . . . 2.4.4 Some Properties of the Green Function . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

11 12 17 18 22 23 23 25 27 30 32 32 34 35 36 37

3

Exact Methods of Solution: A Brief Survey . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 The Infinite Medium and the Dispersion Function . . . . . . . . . . . . . . . . . 3.2 Exact Methods for a Semi-Infinite Medium . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 The Wiener–Hopf Method . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Traditional Real-Space Methods .. . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 The Singular Integral Equation Approach .. . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

41 42 43 43 44 45 47 xi

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4

Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 The Inverse Laplace Transformation . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 The Half-Space Case . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 The Full-Space Case. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 The Direct Laplace Transformation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 The Hilbert Transform Method of Solution .. . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 A Special Case . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 Application to Radiative Transfer.. . . . .. . . . . . . . . . . . . . . . . . . . Appendix A: Hilbert Transforms .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1 Definition and Properties.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2 The Plemelj Formulae .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.3 The Plemelj Formulae for a Dirac Distribution .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

51 52 52 54 55 57 58 59 61 63 63 64 66 67

5

The Scattering Kernel and Associated Auxiliary Functions . . . . . . . . . . . 5.1 The Kernel and Its Inverse Laplace Transform .. . . . . . . . . . . . . . . . . . . . 5.2 The Dispersion Function .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 Symmetries and Analyticity Properties .. . . . . . . . . . . . . . . . . . . 5.2.2 The Zeroes of the Dispersion Function .. . . . . . . . . . . . . . . . . . . 5.2.3 The Index κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 The Half-Space Auxiliary Function X(z) . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Some Properties of X(z) . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.1 Complete Frequency Redistribution . . .. . . . . . . . . . . . . . . . . . . . 5.4.2 Monochromatic Scattering . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4.3 Some Properties of the H -Function . . .. . . . . . . . . . . . . . . . . . . . Appendix B: Properties of the Half-Space Auxiliary Function .. . . . . . . . . . . B.1 Factorization Relation .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.2 Identities for the Boundary Values of X(z) . . . . .. . . . . . . . . . . . . . . . . . . . B.3 Integral Equations for X(z) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.4 Values of X(z) and H (z) . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . B.5 Moments of X(z) and H (z) . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

69 69 73 73 78 80 80 83 83 84 85 88 89 89 91 92 93 94

6

The Surface Green Function and the Resolvent Function . . . . . . . . . . . . . 6.1 Infinite Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.1 The Fourier Transform Method .. . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.2 The Inverse Laplace Transform Method .. . . . . . . . . . . . . . . . . . 6.2 Semi-infinite Medium. The Inverse Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 Complete Frequency Redistribution . . .. . . . . . . . . . . . . . . . . . . . 6.2.2 Monochromatic Scattering . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 The H -Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Appendix C: The Direct Laplace Transform Method . .. . . . . . . . . . . . . . . . . . . . C.1 Complete Frequency Redistribution .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.2 Monochromatic Scattering .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

97 98 98 101 102 103 105 108 110 111 113 114

Contents

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7

The Emergent Intensity and the Source Function . .. . . . . . . . . . . . . . . . . . . . 7.1 The Emergent Intensity . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 The Source Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 Complete Frequency Redistribution . . .. . . . . . . . . . . . . . . . . . . . 7.2.2 Monochromatic Scattering. A Fourier Inversion .. . . . . . . . . 7.3 Some Standard Results . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.1 Uniform Primary Source Term .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.2 Exponential Primary Source Term . . . . .. . . . . . . . . . . . . . . . . . . . 7.3.3 Diffuse Reflection . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Appendix D: The Monochromatic Source Function . . .. . . . . . . . . . . . . . . . . . . . D.1 The Inverse Laplace Transform Method . . . . . . . .. . . . . . . . . . . . . . . . . . . . D.2 The Direct Laplace Transform Method . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

115 115 118 119 123 126 126 128 130 134 135 139 142

8

Spectral Line with Continuous Absorption . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 The Radiative Transfer Equation . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Properties of the Auxiliary Functions . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 The Dispersion Function . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.2 The Half-Space Auxiliary Function . . .. . . . . . . . . . . . . . . . . . . . 8.3 Some Exact Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.1 The Resolvent Function . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.2 The Line Source Function .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.3 The Emergent Intensity.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Appendix E: Kernels for Spectral Lines with a Continuous Absorption .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

143 144 147 147 149 149 150 151 154

Conservative Scattering: The Milne Problem . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 The Monochromatic Milne Problem . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.1 The Auxiliary Functions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.2 The Source Function and the Emergent Intensity .. . . . . . . . 9.1.3 The Hopf Function . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 The Milne Problem for a Spectral Line . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Appendix F: The Conservative Auxiliary Functions. Complete Frequency Redistribution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . F.1 Asymptotic Behavior of the Kernel . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . F.2 The Dispersion Function and the Half-Space Auxiliary Function Near the Origin . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Appendix G: The Milne Problem with the Inverse Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

159 160 162 163 165 167

10 The Case Eigenfunction Expansion Method. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 The Eigenfunctions and Eigenvalues .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.1 Monochromatic Scattering . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1.2 Complete Frequency Redistribution . . .. . . . . . . . . . . . . . . . . . . .

179 180 180 182

9

155 158

169 170 171 174 176

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10.2 The Diffuse Reflection Problem .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.1 Monochromatic Scattering . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2.2 Complete Frequency Redistribution . . .. . . . . . . . . . . . . . . . . . . . 10.3 Further Applications of the Case Method .. . . . . .. . . . . . . . . . . . . . . . . . . . 10.3.1 A Complete Redistribution Full-Space Problem .. . . . . . . . . 10.3.2 A Half-Space Problem with Internal Sources . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . √ 11 The -law √and the Nonlinear H -Equation . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 The -law .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1.1 A Quadratic Approach . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1.2 A Green Function Approach . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 Construction of the Nonlinear H -Equation . . . . .. . . . . . . . . . . . . . . . . . . . 11.2.1 Scattering Coefficient Method . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2.2 Exponential Primary Source Method . .. . . . . . . . . . . . . . . . . . . . 11.3 Some Properties of the H -Equation .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3.1 A Factorization Relation . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3.2 Uniqueness and Alternative Nonlinear H -Equations . . . . . Appendix H: An Elementary Construction of the H -Function . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

195 195 197 198 200 200 204 206 206 207 211 213

12 The Wiener–Hopf Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.1 An Overview of the Wiener–Hopf Method . . . . .. . . . . . . . . . . . . . . . . . . . 12.2 Full Description of the Wiener–Hopf Method . .. . . . . . . . . . . . . . . . . . . . 12.2.1 Decomposition Formula .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.2.2 Analyticity of Complex Fourier Transforms . . . . . . . . . . . . . . 12.2.3 Properties of the Dispersion Function V (z) . . . . . . . . . . . . . . . 12.2.4 Factorization of the Dispersion Function .. . . . . . . . . . . . . . . . . 12.2.5 Decomposition of the Inhomogeneous Term .. . . . . . . . . . . . . 12.2.6 Fourier Transform of the Source Function . . . . . . . . . . . . . . . . 12.3 The Resolvent Function .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.4 The Milne Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.5 The Wiener–Hopf Method for Spectral Lines . .. . . . . . . . . . . . . . . . . . . . 12.5.1 Factorization of V (z) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.5.2 The Fourier Transform of the Source Function . . . . . . . . . . . 12.5.3 The Emergent Intensity.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12.6 The Sommerfeld Half-Plane Diffraction Problem . . . . . . . . . . . . . . . . . . Appendix I: The Auxiliary Function Vu (z) . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . I.1 Explicit Expression of Vu (z). . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . I.2 The Auxiliary Functions Vu (z) and X(z) . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

215 216 220 220 222 224 224 228 229 231 233 237 238 239 240 241 246 247 248 250

Part II

184 184 188 190 190 192 194

Scattering Polarization

13 The Scattering of Polarized Radiation . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 255 13.1 Description of a Polarized Radiation .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 256

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13.2 The Rayleigh Scattering Phase Matrix . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2.1 The Scattering of a Collimated Radiation Beam .. . . . . . . . . 13.2.2 The Rayleigh Phase Matrix in a Fixed Reference Frame .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3 Resonance Scattering of Spectral Lines . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3.1 Populations and Coherences . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3.2 The Scattering Phase Matrix . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4 The Hanle Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4.1 Some General Properties . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4.2 The Hanle Phase Matrix .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Appendix J: Spectral Details of Resonance Scattering .. . . . . . . . . . . . . . . . . . . . Appendix K: Spectral Details of the Hanle Effect.. . . . .. . . . . . . . . . . . . . . . . . . . K.1 Magnetic Field Reference Frame . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . K.2 Atmospheric Reference Frame . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . K.3 Approximations for the Redistribution Matrix ... . . . . . . . . . . . . . . . . . . . Appendix L: The Hanle Effect with the Classical Harmonic Oscillator Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . L.1 The Harmonic Oscillator Model .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . L.2 Polarization Direction and Polarization Rate . . .. . . . . . . . . . . . . . . . . . . . L.3 Frequency Redistribution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

261 262

14 Polarized Radiative Transfer Equations . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.1 Rayleigh Scattering. The Radiative Transfer Equation .. . . . . . . . . . . . 14.2 Conservative Rayleigh Scattering . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2.1 The Polarized Milne Problem .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2.2 The Diffuse Reflection Problem . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3 Rayleigh Scattering. A (KQ) Expansion of the Stokes Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3.1 Cylindrically Symmetric Radiation Field . . . . . . . . . . . . . . . . . 14.3.2 Azimuthally Dependent Radiation Field . . . . . . . . . . . . . . . . . . 14.4 Resonance Polarization of Spectral Lines . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.4.1 One-Dimensional Medium .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.4.2 Multi-Dimensional Medium . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5 The Hanle Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5.1 The Hanle Redistribution Matrix . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5.2 One-Dimensional Medium .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5.3 Multi-Dimensional Medium . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . √ 15 The -Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.1 The Source Vector: Its Derivative and Matrix Representation . . . . . √ 15.2 The -Law .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.2.1 Rayleigh Scattering and Resonance Polarization . . . . . . . . . 15.2.2 The Hanle Effect .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

305 307 308 309 311

264 271 272 274 277 277 279 282 285 286 289 289 290 290 293 295 300

312 312 316 321 323 327 330 331 334 338 339 341 342 344 345 347

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15.3 The Green Matrix and the Resolvent Matrix .. . .. . . . . . . . . . . . . . . . . . . . 15.3.1 Rayleigh Scattering and Resonance Polarization . . . . . . . . . 15.3.2 The Hanle Effect .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.4 Construction of the H-Equation . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.4.1 Rayleigh Scattering and Resonance Polarization . . . . . . . . . 15.4.2 The Hanle Effect .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5 Alternative Definitions of the H-Matrix .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5.1 Exponential Primary Source . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5.2 Uniform Primary Source . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.6 Singular Integral Equations .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.6.1 Rayleigh Scattering and Resonance Polarization . . . . . . . . . 15.6.2 The Hanle Effect .. . . . . .√ . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.7 Factorization Relations and the -Law .. . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

349 349 351 352 353 355 356 357 357 359 360 361 362 364

16 Conservative Rayleigh Scattering: Exact Solutions.. . . . . . . . . . . . . . . . . . . . 16.1 Radiative Transfer Equations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.2 The Dispersion Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3 Construction of the Auxiliary X-matrix . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.4 The Resolvent Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.5 The H-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.6 The Polarized Milne Problem.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.6.1 Some Physical Properties . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.6.2 The Source Vector . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.6.3 The Generalized Hopf Functions . . . . . .. . . . . . . . . . . . . . . . . . . . 16.6.4 The Emergent Radiation Field . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.6.5 The Stokes Parameters I and Q . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.7 The Diffuse Reflection Problem .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.7.1 The Emergent Radiation Field . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.7.2 The Source Vector and the Radiation Field at Infinity .. . . Appendix M: Properties of the Auxiliary Functions . . .. . . . . . . . . . . . . . . . . . . . M.1 The Functions L1 (z) and L2 (z) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . M.2 Some Properties of the Matrix L(ν) . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . M.3 The Auxiliary Functions X1 (z) and X2 (z) . . . . . .. . . . . . . . . . . . . . . . . . . . M.3.1 Construction of X1 (z) and X2 (z) . . . . . .. . . . . . . . . . . . . . . . . . . . M.3.2 Factorizations Relations . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . M.3.3 Nonlinear Integral Equations.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . M.3.4 The Functions Hl (z) and Hr (z) . . . . . . . .. . . . . . . . . . . . . . . . . . . . M.3.5 Moments of Hl (μ) and Hr (μ) . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Appendix N: The Milne Integral Equation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

365 366 369 370 373 377 378 379 381 382 385 386 389 390 392 394 395 397 398 398 399 400 401 402 404 409

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.1 Non-Conservative Rayleigh Scattering.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.1.1 The Dispersion Matrix . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.1.2 The Dispersion Function . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

411 412 413 414

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17.1.3 The Auxiliary X-Matrix . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.1.4 A Factorization of the Dispersion Matrix . . . . . . . . . . . . . . . . . 17.2 Resonance Polarization . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.2.1 The Dispersion Matrix . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.2.2 The Auxiliary X-Matrix . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.3 The Hanle Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

417 419 421 422 423 424 426

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-Matrix, and the I-Matrix... . . . . . . . . . . . . . . . . . . . 18.1 Non-Conservative Rayleigh Scattering.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.1.1 The Resolvent Matrix . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.1.2 The H-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.1.3 The H-Equation in the Complex Plane . . . . . . . . . . . . . . . . . . . . 18.1.4 Uniqueness of the Solution of the H-Equation .. . . . . . . . . . . 18.1.5 An Alternative H-Equation .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.1.6 The Emergent Radiation Field and the I-Matrix . . . . . . . . . . 18.2 Resonance Polarization . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.2.1 The Resolvent Matrix . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.2.2 The H-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.2.3 Nonlinear Integral Equations.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.3 The Hanle Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.3.1 The H-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18.3.2 Nonlinear Integral Equations.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . ˜ Appendix O: The Singular Integral Equation for G(p) .................... References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

427 428 428 432 434 436 437 439 442 443 444 445 447 448 449 451 455

Part III

Asymptotic Properties of Multiple Scattering

19 Asymptotic Properties of the Scattering Kernel K(τ ) .. . . . . . . . . . . . . . . . . 19.1 The Monochromatic Kernel . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 19.2 The Complete Frequency Redistribution Kernel.. . . . . . . . . . . . . . . . . . . 19.2.1 Asymptotic Behavior at Large Optical Depths .. . . . . . . . . . . 19.2.2 The Inverse Laplace Transform Near the Origin .. . . . . . . . . 19.2.3 The Fourier Transform Near the Origin . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

459 461 461 462 463 464 466

20 Large Scale Radiative Transfer Equations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.1 Asymptotic Analysis of the Source Function . . .. . . . . . . . . . . . . . . . . . . . 20.1.1 Monochromatic Scattering . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.1.2 Complete Frequency Redistribution . . .. . . . . . . . . . . . . . . . . . . . 20.1.3 Higher Order Terms . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.2 Asymptotic Analysis in Fourier Space . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.3 The Thermalization Length .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 20.4 Ordinary and Anomalous Diffusion .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

467 468 470 470 472 473 474 477 478

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21 The Photon Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 21.1 The Mean Displacement After n Steps . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 21.1.1 Mean Displacement: Method I . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 21.1.2 Mean Displacement: Method II . . . . . . . .. . . . . . . . . . . . . . . . . . . . 21.1.3 Application to Normal Diffusion and Lévy Walks . . . . . . . . 21.2 The Mean Positive Maximum After n Steps . . . .. . . . . . . . . . . . . . . . . . . . 21.2.1 A Wiener–Hopf Integral Equation .. . . .. . . . . . . . . . . . . . . . . . . . 21.2.2 A Precise Asymptotics √ for the Mean Maximum . . . . . . . . . . 21.3 A Probabilistic Proof of the -Law.. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Appendix P: Universality of Escape from a Half-Space . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

479 481 481 486 487 491 491 494 499 502 505

22 Asymptotic Behavior of the Resolvent Function . . . .. . . . . . . . . . . . . . . . . . . . 22.1 The Infinite Medium Resolvent Function . . . . . . .. . . . . . . . . . . . . . . . . . . . 22.1.1 Infinite Medium with a Plane Source ... . . . . . . . . . . . . . . . . . . . 22.1.2 Infinite Medium with a Point Source . .. . . . . . . . . . . . . . . . . . . . 22.2 Semi-Infinite Medium .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Reference .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

507 507 508 511 513 514

23 The Asymptotics of the Diffusion Approximation . .. . . . . . . . . . . . . . . . . . . . 23.1 The Rescaled Equation .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 23.2 The Interior Radiation Field . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 23.3 An Improved Diffusion Equation .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 23.4 The Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 23.4.1 Boundary Conditions for the Interior Solution .. . . . . . . . . . . 23.4.2 The Emergent Intensity.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 23.5 The Eddington Approximations . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 23.6 Determination of the Critical Size . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

515 516 517 520 522 524 525 526 529 531

24 The Diffusion Approximation for Rayleigh Scattering .. . . . . . . . . . . . . . . . 24.1 The Radiative Transfer Equation . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 24.2 The Interior Radiation Field Expansion .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 24.3 The Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 24.3.1 The Boundary Layer as a Diffuse Reflection Problem.. . . 24.3.2 Matching of the Interior and Boundary Layer Fields . . . . . 24.4 The Emergent Radiation Field . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 24.4.1 The Polarization Rate. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

533 534 536 539 539 541 543 544 548

25 Anomalous Diffusion for Spectral Lines . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 25.1 Interior and Boundary Layer Expansions . . . . . . .. . . . . . . . . . . . . . . . . . . . 25.2 The Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 25.3 Resonance Polarization . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 25.4 Scaling Laws for a Slab .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 25.5 Mean Number of Scatterings and Mean Path Length . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

549 550 552 553 556 558 562

Contents

26 Asymptotic Results for Partial Frequency Redistribution . . . . . . . . . . . . . 26.1 RI Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 26.2 RIII Asymptotic Behavior .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 26.3 RV Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 26.4 RII Asymptotic Behavior.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 26.4.1 Diffusion Equation for the Source Function . . . . . . . . . . . . . . 26.4.2 Diffusion Equation for the Radiation Field.. . . . . . . . . . . . . . . 26.4.3 RII Scaling Laws .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

xix

563 565 568 569 572 574 576 578 580

Author Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 583 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 587

List of Figures

Fig. 2.1

Fig. 2.2

Fig. 2.3 Fig. A.1

Fig. 5.1

Fig. 5.2

Left panel: the space variable τ and the inclination angle θ of a ray with direction n. The axis z is normal to the surface of the medium. Right panel: the heliocentric interpretation of the angle θ . The Sun is viewed from the right. Observations at disk center correspond to θ = 0 and those at the limb to θ = π/2 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Doppler profile, and several Voigt profiles with different values of the parameter a. The left panel shows ϕ(x) and the right panel log ϕ(x). In the right panel, the Lorentzian wing regime characterized by ϕ(x) ∼ 1/x 2 can be observed for |x| larger than (− ln a)1/2 .. . . . . . . . . . . . . . . . . . . . Complete frequency redistribution. The integration domains (ξ, x) and (x, ξ ), with ξ = μ/ϕ(x) . . .. . . . . . . . . . . . . . . . . . . . Integration contour for the Plemelj formulae. The contribution from the singular point is obtained by letting ρ → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The function g(ξ ) for the Doppler profile and the Lorentz profile, 1/(π(1 + x 2 )), in linear scales in the upper panel and in log-log scales in the lower panel. The value of g(ξ ) is constant for ξ ≤ 1/ϕ(0). The algebraic behavior of g(ξ ) for ξ → ∞ given in Eqs. (5.8) and (5.9) can be observed in the lower panel. The Voigt profile has the same algebraic behavior as the Lorentz profile .. . . . . . . . . . . . . . . . . . . . The function k(ν) for the Doppler profile and the Lorentz profile, 1/(π(1 + x 2 )), in linear scales in the upper panel and in log-log scales in the lower panel. The angular point is located at ν = ϕ(0). For the Doppler and Lorentz profiles, k(ν) tends to zero as 1/ν for ν → ∞. For ν → 0, one can observe in the lower panel the algebraic behaviors given in Eqs. (5.8) and (5.9). The Voigt profile has the same algebraic behaviors as the Lorentz profile . . . . . . . . . . .

14

21 29

65

71

72 xxi

xxii

Fig. 5.3

Fig. 5.4

Fig. 5.5

Fig. 5.6

Fig. 5.7

Fig. 5.8

Fig. B.1

Fig. 6.1

Fig. 6.2

List of Figures

Branch cuts of the monochromatic dispersion function. The solid black lines on the real axis show the branch cuts of L(z) . For complete frequency redistribution, L(z) is singular along the full real axis, except at the origin . . . . . . . . . . . . . . . 74 Monochromatic scattering. The function λ(ν), real part of L± (ν), defined in Eq. (5.28), is shown here for  = 1/8 = 0.125. It becomes infinite for ν = ±1, takes the value  at ν = 0 and has two zeroes in the interval [−1, +1] at ±ν0 . They coalesce into a double zero at the origin for  = 0. The zeroes at ±ν0 correspond to the two zeroes of L(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 76 Monochromatic scattering. Phase diagrams of L+ (ν) and of its complex conjugate L− (ν) for ν between 1 and +∞. The change of sign of [L+ (ν)] corresponds to the change of sign of λ(ν) in the interval ν ∈ [1, ∞[. The angle θ is the argument of L+ (ν). It varies between −π and 0 for ν ∈ [1, ∞[. The destruction probability  has the value 0.125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 77 Complete frequency redistribution. Sketch of the phase diagrams of L+ (ν) and L− (ν) for a Lorentz profile and  = 0. The phase diagram lies entirely to the right of the imaginary axis since [L+ (ν)] = λ(ν) has no zero .. . . . . . . . . . . . . . . 77 Contour for the determination of the number of zeroes of the dispersion function L(z). For monochromatic scattering, the end point of the cuts are at ν = ±1. For complete frequency redistribution, they are at ν = 0 .. . . . . . . . . . . . . . 78 The monochromatic function H (μ) for isotropic scattering, with different values of . For √  = 0, H (μ) = 3μ as μ → ∞. For  = 0, √ H (μ) = 1/  as μ → ∞. For the chosen values of , √ 1/  = 6.3, 3.2, 2.2, 1.6. The numerical data are from Chandrasekhar (1960, p. 125) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 87 Complete frequency redistribution. Closed contour for the construction of the nonlinear integral equation satisfied by X(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 91 Contours for the determination of the infinite medium Green function by application of the Fourier inversion formula. Left panel: complete frequency redistribution; right panel: monochromatic scattering . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 ˜ Contours for the calculation of G(p), the Laplace transform of the surface Green function G(τ ). Left panel: complete frequency redistribution. Right panel: monochromatic scattering . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 108

List of Figures

Integration contours for the application of the Fourier inversion formula. Left panel: calculation of the resolvent function (τ ) for complete frequency redistribution. Right panel: calculation of the source function S(τ ) for monochromatic scattering . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 7.1 The source function for a uniform primary source Q∗ =  and several choices of . The absorption profile is a Voigt −2 profile with parameter √ a = 10 . The surface value S(0) follows the exact -law . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 7.2 The source function for an exponential primary source Q∗ (τ ) = exp(−10−4 τ ) and  = 10−6 , for a Voigt profile and a Doppler profile. For the latter, the straight line shows the asymptotic behavior of the boundary layer solution (see also Fig. 9.1) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. E.1 The function kβ (ν, β) for the Doppler profile and different values of the parameter β, ratio of the continuous absorption coefficient to the line absorption coefficient. The function kβ (ν, β) is zero up to ν = β. It is defined in Eq. (8.17) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 9.1 Large τ asymptotic behavior of the source function for conservative scattering. The function S(τ ) is calculated with a uniform primary source  = 10−6 for a Doppler profile and a Voigt profile. The straight lines show the corresponding asymptotic behaviors, τ 1/2 (ln τ )1/4 and τ 1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. F.1 Complete frequency redistribution. Sketch of the phase diagrams of L+ (ν) and L− (ν) for  = 0. At the point ν = 0, L± (0) = L(0) = 0 and the phase diagrams are tangent to the vertical axis in the case of the Doppler profile and and make an π/4 angle with this axis for a Voigt profile. For ν → ∞, L± (∞) = L(∞) = 1 and the phase diagrams are tangent to the vertical axis .. . . . . . . . . . . . . . . . . . . . Fig. 12.1 Integration contour for the decomposition formula. The vertical lines with an arrow-head indicate that f + (z) is analytic in an upper half-plane and that f − (z) is analytic in a lower half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.2 The analyticity domains of V (z), Su (z), the Fourier transform of S + (τ ), and of Sl (z), the Fourier transform of S − (τ ). The dashed line a has been placed arbitrarily below the line +1. The vertical lines with two arrows indicate a strip of analyticity, those with a simple arrow indicate a half-plane analyticity. The functions V (z) and Su (z) have a common strip of analyticity defined by a < (z) < 1. The function V (z) is analytic in −1 < (z), +1 and has two zeroes at ±i ν0 . . . .. . . . . . . . . . . . . . . . . . . .

xxiii

Fig. C.1

112

127

129

157

169

173

221

223

xxiv

List of Figures

Fig. 12.3 The figure shows the strips of analyticity of V (z), the dispersion function, and of V ∗ (z) defined in Eq. (12.34). It shows also the analyticity half-planes of Vu (z) and Vl (z), which satisfy V (z) = Vu (z)/Vl (z), and of Vu∗ (z) and Vl∗ (z), which satisfy V ∗ (z) = Vu∗ (z)/Vl∗ (z) . . . . . . . . . . . . . . . . . . Fig. 12.4 Analyticity half-planes of the inhomogeneous term Qˆ ∗ (z) and of the functions Gu (z) and Gl (z) defined in Eq. (12.47). The thick solid line indicates the branch cut of Vl (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.5 The analyticity half-planes of the functions in the left and right hand-sides of Eq. (12.16) and their common analyticity strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.6 The Wiener–Hopf method for the Milne problem. The half-plane analyticity domains of the functions in the left and right hand-sides of Eq. (12.73) and their common strip of analyticity. The function Vu (z) has a double zero at the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.7 The Sommerfeld diffraction problem: (a) the geometry; (b) the boundary conditions for the field ϕ(x, y) and for ϕ (x, 0± ) = ∂ϕ(x, 0± )/∂y, the normal derivative of ϕ(x, y) on the axis y = 0. The notation 0± means that y → 0 by positive or negative values. The continuity across the axis y = 0 is indicated by a vertical line with two arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 12.8 The Sommerfeld diffraction problem: the factorization of γ = (k − λ)1/2 (k + λ)1/2 . The principal determination of (k + λ)1/2 is analytic in (k) > 0 and that of (k − λ)1/2 in (k) < 0 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. I.1 Monochromatic scattering. (a) Contours for obtaining an explicit expression of Vu (z) in terms of the logarithm of the dispersion function. The point z lies in the upper half-plane (z) > δ > −ν0 . (b) Contour for obtaining the relation between Vu (z) and X(z). The point z lies in the upper half-plane (z) > −1 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.1 Ellipse described by the tip of the electric field vector in the plane orthogonal to the direction of propagation. The angle  (0 ≤  ≤ π) specifies the orientation of the ellipse and the angle χ (−π/4 ≤ χ ≤ π/4) the ellipticity and the sense in which the ellipse is being described . . . . . . . . . . . . . . Fig. 13.2 Rotation of the reference direction by an angle α . . . . . . . . . . . . . . . . . . Fig. 13.3 Geometry of a single scattering. The components Ir and Il are respectively perpendicular and parallel to the scattering plane (,  ) . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

225

229

230

235

241

245

247

257 259

262

List of Figures

Fig. 13.4 The atmospheric reference frame. The axis Oz is along the direction normal to the surface. A direction  is defined by its colatitude θ and azimuth χ. In the plane perpendicular to , ea is the reference direction and eb is perpendicular to ea . γ is the angle between the tangent to the meridian plane and ea . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 13.5 Sketch of the Zeeman displacement of the magnetic sublevels for a normal Zeeman triplet. The horizontal axis is the magnetic field intensity B. The Hanle effect can be observed as long as the Zeeman sublevels are overlapping each other. The magnetic field intensity Btyp corresponds to a Hanle factor H of order unity . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. L.1 The spherical unit vectors e0 , e+1 , and e+1 . The vector e0 is aligned with the magnetic field vector B . . . . .. . . . . . . . . . . . . . . . . . . . Fig. L.2 Motion of the damped classical oscillator in the plane perpendicular to the direction of the magnetic field. The parametric plot shows the time evolution of the component xy (t) versus the component xx (t), defined in Eq. (L.14). The left panel shows a rosette with a Hanle parameter H = 1 and the right one a daisy with H = 9. The curves have been calculated with γ = 1, 2πν0 = 50, and a maximum value t = 2.5 for the rosette and t = 1.5 for the daisy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 14.1 The elements of the kernel matrix K(τ ) for the Doppler profile, with the depolarization parameter set equal to 1. The left panel shows the elements Kij (τ ) and the right panel shows log K11 (τ ) (left vertical axis) and the ratios K12 (τ )/K11 (τ ) and K22 (τ )/K11 (τ ) (right vertical axis) . . . . . . . . . . . . Fig. 16.1 The functions Hl (μ) and Hr (μ) for Rayleigh scattering, and the function H (μ) for conservative monochromatic scattering. For μ → ∞, √ √ Hl (μ) increases at infinity as 5μ and H (μ) as 3μ, while Hr (μ) tends to the constant value 0.1414. The linear growth of Hl (μ) and H (μ) is already installed for μ around one. The data are from Chandrasekhar (1960, p. 125 and p. 248) . . . . . . . . . . . . . . . . . . . . . Fig. 20.1 The thermalization length for monochromatic scattering and complete frequency redistribution. The figure shows the source function S(τ ) for a uniform primary source  with  = 10−6 . The thermalization length, distance from the surface at which the source√ function S(τ ) reaches its saturation value, is of order 1/  for monochromatic scattering, 1/ for the Doppler profile, and a/( 2) for the Voigt profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

xxv

264

277 292

294

327

373

475

xxvi

List of Figures

Fig. 20.2 Ordinary diffusion and anomalous diffusion. The left panel, shows a diffusive-type random walk. All the random steps have the same constant value. The right panel shows an anomalous random walk. The random steps are calculated with a probability distribution having the form 1/(1 + |x|β+1 ), with β = 1, mimicking the Doppler frequency redistribution . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 21.1 A typical discrete random walk on a line. The vertical axis represents the position on the line of a random walker starting at 0. After n steps, the random walker has the position xn .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 21.2 Examples of Lévy walks. The probability distribution has the form 1/(1 + |x|β+1). In the left panel, β = 2 and in the right panel β = 1.5, which is a model for a Holtsmark distribution. The case β = 1 is shown in the right panel of Fig. 20.2. The particles are starting at the origin (0,0). Each graphic shows 104 steps . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 21.3 A typical random walk on a line, for a random walker starting at the point x. Mn indicates the maximum position reached while performing n steps. The first step brings the random walker from x to x1 , x1 ≤ y . . . . . . . . . . . . . . . . . . . . Fig. 22.1 Geometry for establishing the relation between of the point source and the plane source resolvent functions . . . . . . . . . . . . . Fig. 23.1 At each point r b of the surface, the boundary layer is a plane-parallel, semi-infinite medium extending from s = 0 to s = −∞. The surface of the boundary layer is tangential to the surface of the medium at r b . . .. . . . . . . . . . . . . . . . . . . . Fig. 24.1 The surface polarization rate (in percent). A comparison between the asymptotic predictions of Eq. (24.60) (the dashed lines) and numerical solutions of the polarized transfer equation, for different values of  (the solid lines). The √ model is a slab with an optical thickness larger than , an unpolarized and uniform primary source and no incident radiation. From top to bottom,  = 10−5 , 10−3 , 10−2 , 10−1 . The asymptotic result is not plotted for  = 10−1. The Chandrasekhar limit is graphically indistinguishable from the  = 10−5 results . . . . . . . . . . . Fig. 26.1 The RI partial redistribution function. This figure show the ratio r¯I (x, x )/ϕ(x) as a function of x for different values of x; x and x are respectively the incident and scattered dimensionless frequencies . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

476

480

490

492 512

520

546

566

List of Figures

xxvii

Fig. 26.2 The RV partial redistribution function. This figure shows the partial redistribution function rV (ξ, ξ ) in the atomic rest frame given in Eq. (26.22) for the damping parameter γ1 = 0.05, the incident frequency ξ = 2, and three different values of the ratio γ2 /γ1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 569 Fig. 26.3 The partial RII redistribution function. This figure shows the ratio r¯II (x, x )/ϕ(x) as a function of x for different values of the frequency x; x and x are respectively the incident and scattered dimensionless frequencies . . . . . . . . . . . . . . . . . . 573

List of Tables

Table 5.1

Table B.1

Table 6.1

Table 6.2

Monochromatic scattering and complete frequency redistribution. Real part, imaginary part, and argument of L+ (ν) for  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 76 The functions X(z), X∗ (z) = (ν0 − z)X(z), and H (z) for  = 0: values at z = 0 and behavior at infinity. For monochromatic scattering, ν0 ∈ [0, 1] is the positive root of the √ dispersion function L(z). For small values of , ν0  3. The behavior at infinity of H (z) for  = 0 is discussed in Chap. 9. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 Infinite medium. G∞ (τ ) is the infinite medium Green function and ∞ (τ ) the resolvent function. Columns (2) and (3) correspond to the Fourier transform and inverse Laplace transform of the functions in column (1). Column (4) gives the primary source term for the convolution integral equation satisfied by the functions in column (1). Columns (5) and (6) give the Fourier and inverse Laplace transforms of the functions in column (4). The notation—stands for undefined functions . . . . . . . . . . . . . . . 98 Semi-infinite medium. G(τ ) is the surface Green function, and (τ ) the resolvent function. Columns (2) and (3) give the direct and inverse Laplace transforms of the functions in column (1). Column (4) gives the primary source term for the convolution integral equation satisfied by the functions in column (1). Columns (5) and (6) give the direct and inverse Laplace transforms of the functions in column (4). The notation—stands for nonexistent functions . . .. . . . . . . . . . . . . . . . . . . . 103

xxix

xxx

List of Tables

Table 7.1

Semi-infinite medium. S(τ ) denotes the source function, Q∗ (τ ) the primary source term and K(τ ) the kernel of the Wiener–Hopf integral equation for S(τ ). Columns (2) and (3) correspond to the direct and inverse Laplace transforms of the functions in column (1). Column (4) lists examples of primary source terms and columns (5) and (6) their direct and inverse Laplace transforms . . . . . . . . . . . . . . . Table 8.1 The line source function at the surface and at infinity. The effects of a continuous absorption and of a continuous emission. β is the ratio of the continuous to line absorption coefficient, ¯ the effective destruction probability, and ρB the continuous emission. The parameters β, , ρ and B are constants . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table F.1 Complete frequency redistribution. Real part, imaginary part, and argument of L+ (ν) for  = 0 . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 15.1 Notation for inverse Laplace transforms of vectors and matrices depending on the optical depth τ . . . .. . . . . . . . . . . . . . . . . . . . Table 16.1 The Rayleigh Milne problem. The surface value and behavior at infinity of the two components of the source vector for different representation of the polarized radiation field. Sl and Sr correspond to the (Il , Ir ) representation and SI and SQ correspond to the (I, Q) one. For comparison the first line shows the source function S(τ ) for the scalar Milne problem. For the scalar √ case, s1 = 3F /(4π) and for Rayleigh scattering, s1R = 6F /(8π) with F the total flux, defined for Rayleigh scattering in Eq. (16.77). We recall that c = β/α = 0.873 and q = 2/α = 0.690 . . . . . .. . . . . . . . . . . . . . . . . . . . Table M.1 Real parts, imaginary parts, and arguments of L+ 1 (ν) and L+ (ν) along the cut [1, ∞[ . . . . . . . . . . . . . . .. . .................. 2 Table M.2 The functions L1,2 (z), X1,2 (z) and Hl,r (z) at three special points z = 0, 1, ∞. We recall that zX1 (−1/z) = Hl (z) and X2 (−1/z) = Hr (z). The values of Hl,r (1) are from Chandrasekhar. The values of X1,2 (1) are derived from the values of Hl,r (1) and the factorization relations in Eq. (M.24) . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table M.3 Moments of Hl (μ) and Hr (μ) and some numerical constants. The two first moments of Hl (μ) and Hr (μ), the constants q and c, and the values of Hl (1) and Hr (1) are from (Chandrasekhar 1960, p. 248), rounded to 3 significant digits. This reference provides also tables for Hl (μ) and Hr (μ), μ ∈ [0, 1], which we have used to calculate numerically the second and third-moments . . . . . . . . . . . . .

119

153 172 359

388 396

399

402

List of Tables

˜ ˜ R (z), and HR (z) at the origin Table 18.1 The matrices G(z), H(z), G and at infinity. The matrix R = GR (∞) is a rotation matrix, which can be calculated by solving numerically ˜ R (z) or HR (z). The the nonlinear integral equations for G matrix E = diag[, Q ] is equal to L(0). The I-matrix is defined by I(0, z) = HR (z) . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 19.1 The asymptotic behaviors of K(τ ) for τ → ∞, g(ξ ) for ˆ ξ → ∞, k(ν) for ν → 0, and K(k) for k → 0. The kernel is an even function of τ . Here τ , ξ , and ν are positive. For monochromatic scattering, k(ν) is zero in the interval [0, 1[ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 19.2 The index α for the Doppler and Voigt profiles and the asymptotic expressions of f (y) and f (y) for y large. The function f (y) is the inverse function of the profile ϕ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 22.1 Infinite medium. Behavior at large optical depths of the resolvent function for small values of , the destruction probability per scattering. The constant a is the Voigt parameter of the line. The nearly conservative zone corresponds to 1 τ < τeff and the strong absorption zone to τ > τeff . For monochromatic scattering τeff ∼ ν0 . . . . . . . . Table 22.2 Semi-infinite medium. Asymptotic behavior of the resolvent function in the nearly conservative and strong absorption zones. The thermalization lengths τeff are given in Table 22.1. For monochromatic scattering τeff ∼ ν0 . For  = 0, the monochromatic resolvent √ function has exactly the value 3 at infinity . .. . . . . . . . . . . . . . . . . . . . Table 24.1 Moments of Hl (μ) and Hr (μ) and other numerical constants. The two first moments of Hl (μ) and Hr (μ), the constants q and c, and the values of Hl (1) and Hr (1) are from Chandrasekhar (1960, p. 248). This reference provides also tables for Hl (μ) and Hr (μ), μ ∈ [0, 1], used to calculate numerically the third and fourth order moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 25.1 Scaling laws for complete frequency redistribution with a Doppler profile. In the first column, the small expansion parameter; in row 1: β = 0 and T = ∞, in row 2:  = 0 and T = ∞, and in row 3: β =  = 0. The four subsequent columns show τeff , the characteristic scale of variation of the radiation field, xc , the characteristic frequency defined by τeff ϕ(xc ) ∼ 1, the mean number of scatterings N, and the mean path length L .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

xxxi

439

465

466

509

513

544

559

xxxii

List of Tables

Table 25.2 Scaling laws for complete frequency redistribution with a Voigt profile. a Voigt parameter of the line. Same presentation as in Table 25.1 . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 559 Table 26.1 Scaling laws for the RII partial frequency redistribution. a parameter of the line profile. The expansion parameter is  for an infinite medium and 1/T for a slab with total optical thickness T . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 579

Chapter 1

An Overview of the Content

The book is organized in three parts: Part I, is devoted to the presentation of exact methods of solution for scalar radiative transfer problems, and Part II to their generalization to vector and matrix problems for linearly polarized radiation fields. In Part I we deal only with the radiation field intensity, whereas in Part II we also consider the linear polarization created by the scattering of photons. In Part III, we use asymptotic methods to examine the structure of polarized and unpolarized radiation fields when photons undergo a very large number of scatterings.

1.1 Part I: Scalar Radiative Transfer Equations In Part I we consider two different scattering processes. One of them is monochromatic scattering. At each scattering, photons undergo a change of direction but keep the same frequency (energy). This process is relevant to the formation of continuous spectra. The other process is relevant to the formation of spectral lines. The photons undergo, at each scattering, a change of direction and also of frequency, within the width of the spectral line. The frequency of the absorbed and scattered photons are assumed to be fully uncorrelated, a situation known as complete frequency redistribution or noncoherent scattering in the astronomical literature. It is a frequently valid assumption. The general case, with frequency correlations between the absorbed and scattered photons is known as partial frequency redistribution. For monochromatic scattering and complete frequency redistribution, we assume that the scattering in direction is isotropic. Although fairly different, monochromatic scattering and complete frequency redistribution have a common property, namely that the local emission of photons, which is the sum of a scattering term and of a primary source of photons, depends only on the position inside the medium. This local emission term is known in the astronomical literature as the source function, an expression used throughout this book.

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For a one-dimension medium, the source function, satisfies a convolution integral equation, the kernel of which contains the scattering properties of the transport process. For monochromatic scattering, the kernel has an exponential tail and for complete frequency redistribution an algebraic one, due to changes in frequency. In all the examples that are considered, the kernel is translation invariant, otherwise no exact solutions would not exist. For semi-infinite media, these integral equations are often referred to as Wiener–Hopf1 integral equations, a custom, which we follow. Wiener–Hopf integral equations for the source function and for its associated Green function are starting points for many exact methods of solution for radiative transfer problems. Scattering processes are also characterized by a destruction probability per scattering, henceforth denoted . When  = 0, all the absorbed photons are reemitted and the single scattering albedo, (1 − ), is equal to one. It will be said that the scattering, or the medium, are conservative. The term non-conservative is employed when  = 0; the single scattering albedo is then less than one. Part I is organized as follows. In Chap. 2 we introduce the radiative transfer equations for monochromatic scattering and complete frequency redistribution, the source function, the Green2 function, and the resolvent function, which is the regular part of the surface value of the Green function. We also introduce the convolution integral equations satisfied by these functions and discuss some of their properties. We present in Chap. 3 a summary of each of the exact methods of solution considered in the subsequent chapters. Their presentation follows, more or less, a chronological order, while their detailed description and application follows, also more or less, an inverse chronological order, ending with the Wiener–Hopf method. A common property of exact methods of solution for radiative transfer problems in semi-infinite media is that exact solutions, when they exist, ultimately require the solution of a nonlinear problem, although the starting point is a linear radiative transfer equations or a convolution integral equation. Examples will be found in the subsequent chapters. Exact methods of solutions often start with the transformation of a radiative transfer equation, or a Wiener–Hopf integral equation, into another linear equation, to which can be applied more or less standard mathematical tools. In Chap. 4, we show how to transform a Wiener–Hopf integral equation into a singular integral equation with a Cauchy-type3 kernel, by a direct or an inverse Laplace transform. We then describe the main lines of a Hilbert transform method of solution, which transforms singular integral equations into boundary layer problems in the complex plane, known as Riemann–Hilbert problems.4

1

Norbert Wiener: 1894–1964 (Columbia, USA-Stockholm); Eberhard Hopf: 1902–1983 (Salzburg-Bloomington, USA). 2 George Green: 1793–1841 (Sneiton-Nottingham, GB). 3 Augustin-Louis Cauchy: 1789–1857 (Sceaux-Paris). 4 Bernhard Riemann: 1826–1866 (Jameln, Germany-Verbenia, Italy); David Hilbert: 1862–1943 (Königsberg-Göttingen).

1.1 Part I: Scalar Radiative Transfer Equations

3

In Chap. 5, we show how to solve Riemann–Hilbert problems arising from radiative transfer. We show in particular that for semi-infinite media, the solution requires the introduction of an auxiliary function, solution of a homogeneous Riemann–Hilbert problem. The latter is a nonlinear problem of the type mentioned above. The auxiliary function is essentially the function, known nowadays as the H -function. The singular integral equation approach, associated to a Hilbert transform method is employed in Chap. 6 to construct an exact expression of the resolvent function for non-conservative scattering. The resolvent function is chosen as first example for the application of an exact method of solution. It depends exclusively on the scattering process and has a simple expression in terms of the H -function. The relation between the resolvent function and the H -function can be generalized to polarized radiation fields, as will be shown in Part II. In the three subsequent chapters, we give other examples of the same singular integral equation method. In Chap. 7 we construct exact expressions for the source function and the emergent intensity, for non-conservative scattering, considering both monochromatic scattering and complete frequency redistribution. In Chap. 8, we treat the case of a spectral line embedded in a continuum, and in Chap. 9 conservative scattering. In this latter chapter, we consider the Milne5 problem, the diffusion reflection problem, and their generalized versions for complete frequency redistribution. We then describe three alternative methods of solution: the Case eigenfunction expansion method, the traditional resolvent method, and, last but not least, the Wiener–Hopf method. The Case expansion method is presented in Chap. 10. Its starting point is the radiative transfer equation for the radiation field intensity. An eigenfunction expansion of the radiation field leads to singular integral equations with Cauchy-type kernels similar to the singular integral equations introduced in Chap. 4. In Chap. 11, we present a traditional approach, sometimes referred to as the resolvent method. Based mainly on invariance properties, it is a real-space method, which does not require complex plane analysis. It leads to a nonlinear integral equation for the H -function, in brief referred to as the H-equation. The resolvent method is another example of how a linear radiative transfer problem is transformed into a nonlinear one. We briefly discuss the properties of H -equations, in particular the non-uniqueness of the solution for monochromatic scattering. In √ the same chapter we give a simple algebraic proof of the famous exact -law, applicable in a semi-infinite medium with a uniform primary source. The Wiener–Hopf method is presented in Chap. 12. It is a generalization to convolution integral equations on a semi-infinite line of the standard Fourier transform method for convolution integral equations on the full line. It makes a systematic use of the analyticity properties of Fourier transforms in the complex plane. We describe it for monochromatic scattering and then for complete frequency redistribution. In the latter case, it becomes a Riemann–Hilbert problem of the type

5

Edward Arthur Milne: 1896–1950 (Kingston upon Hull-Dublin).

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1 An Overview of the Content

solved in Chap. 5. One of the key ingredient of the Wiener–Hopf method makes its applicable to partial differential equations. We give, with a wave diffraction problem, an example in Sect. 12.6.

1.2 Part II: Scattering Polarization The second Part is devoted to radiative transfer problems for fields linearly polarized by scattering processes sharing the same Rayleigh polarization phase matrix, namely the Rayleigh scattering of monochromatic radiation, the resonance polarization of spectral lines, formed with complete frequency redistribution, and the associated Hanle6 effect, which takes into account the effects of a weak magnetic field. An important feature of the Rayleigh polarization phase matrix is that it can be factorized as the product of two matrices depending separately on the directions of the incident and scattered beams. We make use of this property to introduce a representation of the polarized radiation field based on the irreducible spherical tensors for polarization introduced by Landi Degl’Innocenti (1984). With this representation, which can be introduced for one-dimensional and multi-dimensional media, the scattering term (the integral over the directions of the incident beams weighted by a polarization matrix) in the radiative transfer equation depends only on the position inside the medium. The radiative transfer equations have thus the same structure as those of Part I, except that the radiation field and the source term are now vectors. For a one-dimensional medium, the convolution equation for the source term has also the same structure as in the scalar case, but its kernel is a matrix. The Green function, the resolvent function, the H -function also become matrices. The scalar Riemann–Hilbert problem becomes a matrix problem and the nonlinear H -equation becomes a nonlinear matrix equation. The matrix structure makes the situation significantly more complicated and it is in general impossible to construct explicit solutions for these equations. In some very special cases, homogeneous matrix Riemann–Hilbert problems have an explicit solution, conservative monochromatic Rayleigh scattering is one of them. The reason is that the corresponding matrix problem can actually be transformed into two scalar ones. This property was first demonstrated by Chandrasekhar7 (1946), by a detailed analysis of the properties of the matrix kernel and has allowed him to construct exact solutions for the Milne and the diffuse reflection problems in terms of two functions Hl and Hr with exact expressions. There is no such decomposition for non-conservative Rayleigh scattering, however an exact solution was obtained by Siewert et al. (1981, see also Aoki and Cercignani 1985) with very advanced mathematics, out of the scope of this book. It is a

6 7

Wilhem Hanle: 1901–1993 (Mannheim-Giessen). Subramayan Chandrasekhar: 1910–1995 (Lahore-Chicago).

1.2 Part II: Scattering Polarization

5

real breakthrough but the solution that has been obtained has a very complicated form, not easily amenable to a numerical solution. Here we treat non-conservative Rayleigh scattering in the same way as the resonance polarization of spectral lines and the Hanle effect. For spectral lines, there is no exact solution, even when are formed with complete frequency redistribution and the scattering is conservative. We show, by different methods, how to construct a nonlinear matrix H-equation, which can be solved numerically. Part II is organized as follows. Chapter 13 is devoted to various aspects of scattering physics. After recalling the classical electromagnetic description of a polarized beam of light, we present polarization matrices for the Rayleigh scattering,8 the resonance polarization of spectral lines and the Hanle effect. For spectral lines, the polarization matrix depends on the frequencies of the incident and scattered photons and will be referred to as the redistribution matrix. Wandering from our main subject, we described in some details, for a simple-two level atom, the spectral profile of the scattered radiation. The purpose is to place the assumption of complete frequency redistribution in a broader context and also to offer a brief introduction to the literature on the scattering polarization of spectral lines. In Chap. 14, we show how the radiative transfer equations for the Stokes parameters of a field linearly polarized by one of the three mechanisms mentioned above, can be transformed into new radiative transfer equations, in which the source term depends only on the position in the medium, as in the scalar case. For a onedimensional medium, these new equations can then be recast as vector or matrix Wiener–Hopf integral equations and used as starting points to generalize some of the methods of solution described in the scalar case. In Chap. 15 we generalize to a polarized radiation √ the real-space methods described in Chap. 11 for the scalar case. We construct -laws for Rayleigh scattering, resonance polarization, and the Hanle effect, and also the corresponding nonlinear matrix H-equations. We also introduce matrix singular integral equations that will be used in the subsequent chapters. In the remaining three chapters we show how vector and matrix singular integral equations with Cauchy-type kernels can be treated with Hilbert transform methods. For conservative Rayleigh scattering, we explain in Chap. 16 how the matrix Riemann–Hilbert problem can be transformed into two scalar problems and how the polarized Milne and the diffuse reflection problems can be solved exactly in terms of the two scalar functions Hl and Hr . For polarization problems with no exact solution (non-conservative Rayleigh scattering, resonance polarization, and Hanle effect), we establish in Chap. 17 homogeneous matrix Riemann–Hilbert problems and discuss in detail the asymptotic behavior of their solutions at infinity (in the complex plane). Although they contain undetermined constants, these solutions can be used, as we show in Chap. 18, to recover the nonlinear H-equations, established in Chap. 15 with a real-space method. This second method has the advantage of proving that the existence of solutions to these nonlinear equations. We also discuss in the same

8

The expression Rayleigh scattering always implies that the scattering is monochromatic.

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1 An Overview of the Content

chapter a non-uniqueness problem for the monochromatic nonlinear H-equation and present alternative nonlinear integral equations, well suited to numerical solutions. We present in particular the equation for an I-matrix, constructed with the emergent radiation field, which has been introduced by Ivanov (1996).

1.3 Part III: Asymptotic Properties of Multiple Scattering We describe in Part III asymptotic methods based on the existence of small or large parameters such as , the destruction probability per scattering, or T the optical thickness of the medium. They are applicable when photons undergo a very large number of scatterings and greatly contribute to the physical understanding of the formation of a radiation field by multiple scatterings. For example, it will be shown that for monochromatic scattering photons perform an ordinary diffusion random walk. In particular, that the averaged distance travelled after n scatterings (n large) goes as n1/2 . For complete frequency redistribution, the random walk of photons has an anomalous behavior, performing a so-called Lévy9 walk, because of the changes of frequency within the spectral line. The averaged distance travelled by photons varies as n1/α with α smaller than 2. The difference between ordinary and anomalous diffusion has its origin in the asymptotic behavior of the kernel. These scaling laws are derived in Part III with an asymptotic analysis of the integral equation for the source function and also with a statistical analysis of the random walk of the photons. Asymptotic methods can address radiative transfer problems, which have no exact solution and are usually solved numerically. They provide equations that are simpler than the original equations. Their solutions are approximations to the original problem, but their domain of validity and accuracy are controlled by the small expansion parameter. We give examples with non-conservative Rayleigh scattering and partial frequency redistribution of spectral lines. Another strength of asymptotic methods is that they provide spatial and frequency scales of variation for the radiation field, which describe the spreading of photons in space and frequency (in the case of spectral lines). For example the spreading in space, which is known in the astronomical literature as thermalization length, henceforth denoted τeff , can be identified with the characteristic scale for the spatial variations of the radiation field. Part III is organized as follows. The somewhat technical Chap. 19 contains a detailed analysis of the asymptotic behavior of the kernel K(τ ) for large values of τ ˆ and of its Fourier transform K(k) for k → 0. In Chap. 20 we show how to perform a large scale asymptotic analysis of the integral equation for the source function in an infinite medium, by a proper rescaling of the optical depth. This analysis provides scaling laws for the thermalization length τeff for monochromatic scattering and

9

Paul Pierre Lévy: 1886–1971 (Paris-Paris).

1.3 Part III: Asymptotic Properties of Multiple Scattering

7

complete frequency redistribution. We show in particular that the ordinary diffusion is encountered when the second order moment of the kernel is finite, that is when 

+∞ −∞

τ 2 K(τ ) dτ < ∞.

(1.1)

In the following Chap. 21 we present a statistical analysis of a random walk of photons, progressing with random steps on an infinitely long line. We examine the mean displacement and the mean maximum displacement after n steps. We show how they are related to the thermalization length, how the discrete description is related to the continuous description with radiation fields, and how anomalous diffusion is related to stable probability distributions. For complete frequency redistribution, the analysis of the random walk of the photons is further developed in Chap. 22, where we analyze the propagation of photons away from a source, using exact expressions of the Green function in an infinite medium established in Part I. The two subsequent chapters are devoted to monochromatic scattering in a medium with boundaries, the scalar case in Chap. 23 and the Rayleigh scattering in Chap. 24. Starting from the radiative transfer equation, we show that the radiation field can be decomposed into an interior field, which satisfies a diffusion equation, and a boundary layer field, solution of a semi-infinite conservative radiative transfer equation. The two fields are described by equations, which are simpler than the original transfer equation, and which can be asymptotically matched to obtain a solution valid in the full medium. For the Rayleigh scattering, we obtain corrections to the Chandrasekhar’s polarization laws for conservative Rayleigh scattering. Chapter 25 is devoted to an asymptotic analysis of spectral lines formed with complete frequency redistribution. We show that the radiation field can also be decomposed into an interior field and a boundary layer field and that they can be matched together to construct a solution valid in the whole medium. The asymptotic expansion is performed on the integral equation for the source function as in Chap. 20, the anomalous diffusion of the photons preventing an asymptotic analysis of the radiative transfer equation itself. We also examine the case of resonance polarization and present various scaling laws concerning the mean number of scatterings and the mean pathlength, an accumulated distance travelled by photons. Finally, we present in Chap. 26 some asymptotic properties of spectral lines formed with partial frequency redistribution. We consider the four elementary partial redistribution functions introduced in Hummer (1962). We show that three of them (RI , RIII , RV ) are behaving as complete frequency redistribution and that RII , which describes a coherent scattering in the atomic frame, is of the ordinary diffusion type at large frequencies.

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References Aoki, K., Cercignani, C.: On the matrix Riemann–Hilbert problem relevant to Rayleigh scattering. J. Appl. Math. Phys. (ZAMP) 36, 61–69 (1985) Chandrasekhar, S.: On the radiative equilibrium of a stellar atmosphere. X. Astrophys. J. 103, 351–370 (1946) Hummer, D.G.: Non-coherent scattering I. The redistribution functions with Doppler broadening. Mon. Not. R. Astr. Soc. 125, 21–37 (1962) Ivanov, V.V.: Generalized Rayleigh scattering. III. Theory of I-matrices, Astron. Astrophys. 307, 319–331 (1996) Landi Degl’Innocenti, E.: Polarization in Spectral lines III : Resonance polarization in the nonmagnetic, collisionless regime. Sol. Phys. 91, 1–26 (1984) Siewert, C.E., Kelley, C.T., Garcia, R.D.M.: An analytical expression for the H-Matrix relevant to Rayleigh scattering. J. Math. Anal. Appl. 84, 509–518 (1981)

Part I

Scalar Radiative Transfer Equations

Chapter 2

Radiative Transfer Equations

In this chapter, we present the scalar radiative transfer equations used in Part I to illustrate exact method of solutions for radiative transfer equations in semi-infinite media. We also present different types of integral equations that can be derived from the integro-differential equations. All the equations are time-independent and one-dimensional. They describe the formation of continuous spectra and spectral lines. For continuous spectra, there is no change in frequency at each scattering. We refer to it as monochromatic scattering. For a spectral line, photons change their frequency at each scattering. They can be fully decorrelated, a situation known as complete frequency redistribution, or partially correlated. One says that there is partial frequency redistribution. An example of partial frequency redistribution is presented in Appendix J of Chap. 13 and some asymptotic results on the large scale behavior of the radiation field in Chap. 26. Otherwise, we consider only spectral lines formed with complete frequency redistribution (also known as noncoherent scattering) for which exact results can be obtained. The integro-differential radiative transfer equations are presented in Sect. 2.1 for monochromatic scattering and complete frequency redistribution. Section 2.2 is devoted to the source function, sum of the scattered radiation and of a primary source of photons. With our assumptions on the frequency redistribution, it depends only on the position in the medium. We show that it satisfies an integral equation of the convolution type, with a translation invariant kernel. Section 2.4 is devoted to the Green function and some associated functions, such as the resolvent function, regular part of the surface Green function. These functions, which characterize the scattering process and the geometry of the medium, independently of the primary sources, obey integral equations identical to those of the source function, except for inhomogeneous terms. In this same Sect. 2.4, we prove some fundamental properties of the Green function and of convolution type integral equations for semi-infinite media.

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2.1 The Integro-Differential Radiative Transfer Equations Radiative transfer equations are linear Boltzmann equations describing the propagation of an ensemble of photons inside a host medium and their interactions with the particles of the medium. Each photon is treated as a particle of zero mass, moving with the velocity of light c. At an instant t, photons are characterized by a position r, a direction of motion n, and an energy E = hν, with h the Planck constant and ν the frequency. A systematic derivation of radiative transfer equations from the Maxwell equations is outside the scope of this book. The electro-magnetic description of the radiation field is introduced briefly in Part II when dealing with polarized radiation fields. A systematic electro-magnetic description of multiple scattering of light can be found in, e.g., Mishchenko et al. (2006) and Mishchenko (2008) (see also Landi Degl’Innocenti and Landolfi 2004, Chap. 5). In this book we consider exclusively interaction processes relevant to the formation of spectral lines and continuous spectra. These processes are (i) creation, (ii) absorption followed by a destruction and (iii) absorption followed by a reemission. The third type of interaction is referred to as a scattering process. Creations correspond to a radiative decay of a level (bounded or unbounded) that has been first excited by a collision, say, with electrons. Destruction is the opposite process: a level excited by photon absorption is deexcited by collision with some particles of the medium. These three processes, together with the photons entering or escaping the medium participate in the formation of the radiation field. The time evolution of an ensemble of photons can be described by a distribution function f (t, r, n, ν) dV d dν, which is the number of photons at time t in an infinitesimal volume dV around r, a solid angle d around n and a frequency interval dν around ν. In Astronomy, the custom is to work with the specific intensity, I (t, r, n, ν), related to the photon distribution function by I (t, r, n, ν) = c h νf (t, r, n, ν).

(2.1)

The specific intensity is proportional to the density of energy of the radiation field. In the traditional radiometric point of view, it is defined in terms of the amount of radiant energy dE in a frequency interval dν around ν, transported across an element of area dσ in an infinitesimal solid angle d around a direction n, during a time dt. By definition dE = I (t, r, n, ν, t) cos θ dν dσ d dt,

(2.2)

where θ is the angle between the direction n and the outward normal to dσ . The specific intensity is a positive quantity. The description of the radiation field with the specific intensity, or with a scalar distribution function, is sufficient as long as the polarization of the radiation is ignored. In Astronomy, polarized radiation is usually described by its four Stokes parameters (see Chap. 13).

2.1 The Integro-Differential Radiative Transfer Equations

13

A radiative transfer equation has the general form: 1 ∂I (t, r, n, ν) + n · ∇I (t, r, n, ν) = c ∂t − [χd (t, r, n, ν) + χs (t, r, n, ν)]I (t, r, n, ν)  ∞ d

dν (t, r, n, ν, n , ν )I (t, r, n , ν ) + 4π 0 + Q∗ (t, r, n, ν),

(2.3)

with 

∞

χs (t, r, n, ν) = 0

(t, r, n, ν, n , ν )

d

dν . 4π

(2.4)

The right hand side in Eq. (2.3) describes the sinks and sources of photons. The coefficient χd corresponds to an absorption followed by a destruction and the coefficient and χs to an absorption followed by a reemission. The integral term describes the scattering of an incident beam of radiation with direction n and frequency ν into a scattered beam with direction n and frequency ν. The integration is over all possible directions and frequencies of the incident beam. The last term Q∗ (t, r, n, ν) represents a source of primary photons. An analysis of the interaction between the radiation field and the matter of the host medium is needed to determine the coefficients of the radiative transfer equation. For problems relevant to Astronomy and more specifically to the formation of stellar spectra, this subject is developed in several text books such as Aller (1963), Jefferies (1968), Mihalas (1978), Ivanov (1973), Landi Degl’Innocenti and Landolfi (2004), and Hubeny and Mihalas (2015). The equations considered in this book are much simpler than Eq. (2.3) but are not mathematical oddities. In particular they can describe the formation of spectral lines and other spectral features such as atomic continua in a stellar atmosphere. The simplifications come from several assumptions: (i) The radiation field is assumed to be independent of time. The time dependence plays a role only for very fast evolving phenomena, namely those with a characteristic time of variation t such that t  l/c, with l the scale of variation in the physical space. (ii) The medium is assumed to be one-dimensional. This assumption holds for atmospheres with a thickness small compared to the radius of the star, such as the solar atmosphere. The one-dimensional hypothesis implies that the physical properties of the atmosphere (temperature, density, chemical composition, and so on) have no horizontal variations. The one-dimensional variable, here denoted z, is measured along the normal to the medium (see Fig. 2.1). This one-dimensional assumption is raised for some problems considered in Parts II and III.

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2 Radiative Transfer Equations

z

z

+∞

θ n

n θ

z n

θ τ +∞

z

n

−∞

Fig. 2.1 Left panel: the space variable τ and the inclination angle θ of a ray with direction n. The axis z is normal to the surface of the medium. Right panel: the heliocentric interpretation of the angle θ. The Sun is viewed from the right. Observations at disk center correspond to θ = 0 and those at the limb to θ = π/2

(iii) In Part I, we also assume that the radiation field has a cylindrical symmetry. The direction of the field is fully defined by the colatitude θ ∈ [0, π], i.e. inclination of the ray with respect to the z-axis (see Fig. 2.1) or with μ = cos θ , μ ∈ [−1, +1]. For solar observations, n being the direction of the line-ofsight, then θ is the so-called heliocentric angle. Observations at disk center correspond to θ = 0 (μ = 1) and those at the limb to θ = π/2 (μ = 1) (see Fig. 2.1). Radiation fields without a cylindrical symmetry are considered in Part II, when we discuss the polarization of radiation fields. (iv) The absorption coefficient is assumed to be independent of the direction n. It may then be written as χ(t, r, n, ν) = kd (r)f (ν).

(2.5)

This form of coefficient holds for continuous spectra and spectral lines. The coefficient kd has the dimension of the inverse of a length. It is proportional to the number of scattering atoms per unit volume multiplied by a cross-section. The function f (ν) gives the probability density that a photon is absorbed in a frequency interval [ν, ν + dν]. (v) Concerning the scattering process, we consider monochromatic scattering and complete frequency redistribution. For monochromatic scattering (t, r, n, ν, n , ν ) = ks (r)δ(ν − ν )f (ν),

(2.6)

where δ(ν − ν ) is the Dirac distribution. This coefficient describes scatterings of photons without change of frequency and a spherical (isotropic) indicatrix. It applies to the formation of continuous spectra.

2.1 The Integro-Differential Radiative Transfer Equations

15

For complete frequency redistribution, (t, r, n, ν, n , ν ) = ks (r)f (ν )f (ν).

(2.7)

This coefficient is encountered in the formation of spectral lines. The atomic transitions are occurring between bounded atomic levels, broadened by radiative decay (natural broadening) and by collisions with particles such as electrons or neutral hydrogen atoms. Because of the broadening of the levels, absorbed and scattered photons have in general different frequencies. When there is a high rate of collisions, these frequencies can be fully uncorrelated and the scattering coefficient may be written as above. Complete frequency redistribution is the term used for this situation. In this book, we consider only isolated spectral line. The function f (ν), is referred to as the absorption profile and following a well establish custom, it will be henceforth denoted ϕ(ν). It takes its maximum value at the line center frequency νc and typically has the form of a Gaussian or of a Voigt function, which is the convolution product of a Gaussian and a Lorentzian (see Sect. 2.1.2). All the absorption profiles, which will be considered, are normalized to unity, that is, 

∞

ϕ(ν) dν = 1.

(2.8)

0

Equation (2.4), which expresses the detail balance between the absorptions and emissions takes then the form χs (t, r, n, ν) = ks (r)ϕ(ν).

(2.9)

(vi) For spectral lines and continuous spectra, the primary source of photons often comes from the radiative decay of an atomic level (bound or free) that has been excited by a collision. When the colliding particles have a Maxwellian velocity distribution, the primary source has the form Q∗ (r, n, ν) = kd (r)ϕ(ν)Q∗ν (r),

(2.10)

where Q∗ν (r) stands for the Planck function at the point r and at the frequency ν. With the assumptions listed above, Eq. (2.3) becomes ∂ I (z, ν, μ) = −κ(z)ϕ(ν)I (z, ν, μ) ∂z   1 +1 ∞ R(ν, ν )I (z, ν , μ ) dν dμ + kd (z)ϕ(ν)Q∗ν (z), + ks (z) 2 −1 0 μ

(2.11)

16

2 Radiative Transfer Equations

where κ(z) ≡ kd (z) + ks (z).

(2.12)

In the scattering integral, R(ν, ν ) = δ(ν − ν )ϕ(ν ),

or

R(ν, ν ) = ϕ(ν)ϕ(ν ),

(2.13)

respectively for monochromatic scattering and complete frequency redistribution. We introduce the optical depth τ , defined by dτ = −κ(z) dz.

(2.14)

The use of the optical depth as space variable is quite popular in radiative transfer because it provides a scale for the extinction of the radiation field. For a semi-infinite medium, it is positive inside the medium with τ = 0 at the surface (see Fig. 2.1). Equation (2.11) can thus be rewritten as ∂ I (τ, ν, μ) = ϕ(ν)I (τ, ν, μ) ∂τ   1 +1 ∞ − [1 − (τ )] R(ν, ν )I (τ, ν , μ ) dν dμ − (τ )ϕ(ν)Q∗ν (τ ), (2.15) 2 −1 0 μ

where (τ ) =

kd (τ ) kd (τ ) = . κ(τ ) kd (τ ) + ks (τ )

(2.16)

The coefficient [1 − (τ )] gives the probability that a photon that has been absorbed will be reemitted and (τ ) that it will be destroyed (its energy turned into thermal energy). The notation  for the destruction probability per scattering can already by found in Eddington (1929) and has been continuously in use (e.g. Thomas 1957). For continuous spectra (monochromatic scattering), kd (τ ) is due to bound-free and free-free transitions and ks (τ ), the scattering coefficient, is due to, e.g., Rayleigh or Thomson scattering. For a spectral line, kd (τ ) is the absorption coefficient from the lower level of the transition and ks (τ ) is related to the spontaneous radiative deexcitation from the upper level of the transition. For a two-level atom, the destruction coefficient may be written as (τ ) =

C21 (τ ) , A21 + C21 (τ )

(2.17)

where A21 is the Einstein coefficient for spontaneous emission, typically around 107 − 108 s−1 , and C21 (τ ) a rate of de-population of the excited level by inelastic

2.1 The Integro-Differential Radiative Transfer Equations

17

collisions, mainly with electrons. It depends on local physical conditions (temperature and density). For resonance lines1  can be very small, of order 10−3 or less. In the following we assume that (τ ) is a constant independent of τ , simply denoted . Exact solutions exist only under this condition. When  = 0, all the absorbed photons are scattered. The scattering (or the medium) is said to be conservative. When  = 1, there is no scattering, hence no long range coupling between different regions. All the sources and sinks of photons are local, and the radiative transfer equation becomes an ordinary differential equation. In an infinite medium, the mean number of scattering events that a photon can suffer before its energy is returned to thermal energy is of order 1/. Radiative transfer equations are often written in terms of the single scattering albedo 1 − , usually denoted λ or  or a. We now rewrite the transfer equation in Eq. (2.15) separately for monochromatic scattering and complete frequency redistribution, assuming, as everywhere in this book, that  is a constant.

2.1.1 Monochromatic Scattering Monochromatic scattering was first formulated by Schwarzschild (1906) as a model for the radiative equilibrium of a stellar atmosphere with a grey absorption coefficient, i.e., independent of frequency (see Sect. 2.2.2). Monochromatic scattering is also a reasonable approximation for the transport of fast and slow neutrons (Kourganoff 1963; Case and Zweifel 1967; Ivanov 1973). The radiative transfer equation can be treated frequency by frequency, since R(ν, ν ) = δ(ν − ν )ϕ(ν ). Introducing the monochromatic optical depth τν , defined by dτν = −ϕ(ν)κ(z) dz, henceforth simply denoted τ , and the notation I (τ, μ) for the radiation field, Eq. (2.15) may be written as μ

1 ∂I (τ, μ) = I (τ, μ) − (1 − ) ∂τ 2



+1 −1

I (τ, μ ) dμ − Q∗ (τ ).

(2.18)

Here Q∗ (τ ) stands for Q∗ν (τ ). Equation (2.18) with  = 0 describes the radiative equilibrium of a grey stellar atmosphere (see Sect. 2.2.2). It is convenient to rewrite the radiative transfer equation as μ

1

∂I (τ, μ) = I (τ, μ) − S(τ ), ∂τ

For a resonance line, the lower level is the atomic fundamental level.

(2.19)

18

2 Radiative Transfer Equations

where the so-called source function S(τ ) is defined by 1 S(τ ) ≡ (1 − ) 2



+1 −1

I (τ, μ) dμ + Q∗ (τ ).

(2.20)

The first term expresses the contribution of the photons that have been scattered at least once and the second term yields the primary source of photons. The source function depends only on the space variable and satisfies an integral equation of the convolution type (see Sect. 2.2). For this reason, it plays a fundamental role in the solution of radiative transfer equations.

2.1.2 Complete Frequency Redistribution As pointed out by Eddington2 (1929), monochromatic scattering, or coherent scattering as it used to be called, cannot be employed for the study of spectral lines because it ignores the natural damping of the energy levels. It also ignores broadening by collisions with electrons and other species and the thermal movements of the atoms (see Woolley and Stibbs 1953; Hummer 1962, for a history of the subject). The opposite assumption, namely complete frequency redistribution, was found to be adequate for a large majority of spectral lines formed in stellar atmospheres, the main cause for the absence of correlation being the perturbation of the excited levels of the transitions by collisions with passing atoms. The first investigations on multiple scattering with complete frequency were made independently by Holstein (1947) and Biberman (1947). For very strong lines, this assumption will break down. Because of the very short mean life time of the excited level, photons can be reemitted without changing their frequency and there is, as one says partial frequency redistribution. In this case, radiative transfer problems have in general no exact solutions. Attempts have been made by Hemsh and Ferziger (1972) and by Yengibarian and Khachatrian (1991) at finding exact solutions for the resonant scattering of gamma-ray quanta by Mösbauer nuclei, a process which can be described by a linear combination of monochromatic scattering and complete frequency redistribution. The authors could show that the problem can be reduced to the construction of a function depending only on the optical depth. When dealing with one single isolated spectral line, as done here, it is convenient to introduce a dimensionless frequency variable, x=

2

ν − νc , νD

Arthur Eddington (Sir): 1882–1944 (Kendal-Cambridge, UK).

(2.21)

2.1 The Integro-Differential Radiative Transfer Equations

19

where νc is the frequency of the line center and νD , the Doppler width. The Doppler width measures the broadening of the absorption line profile by small scale velocity fields, thermal and turbulent. It has the form νD =

νc 2 2 )1/2 , (v + vturb c th

(2.22)

with vturb the turbulent velocity and vth = (2kT /M)1/2, the thermal velocity. We recall that k is the Boltzmann constant, T the temperature of the medium, M the mass of the scattering atom and c the speed of light. The Doppler width varies in general slowly inside a stellar atmosphere. Here we take it as a constant. The frequency νc being very large for spectral lines in the optical domain (νc  1015 − 1016 Hz), we make the usual assumption that x varies from minus infinity to plus infinity, with x = 0 at line center. Introducing now R(ν, ν ) = ϕ(ν)ϕ(ν ) into Eq. (2.15) and the frequency variable x, we can write the radiative transfer equation as ∂I (τ, x, μ) = ϕ(x)I (τ, x, μ) ∂τ  +∞ +1 1 −(1 − ) ϕ(x) ϕ(x )I (τ, x , μ ) dμ dx − ϕ(x)Q∗x (τ ). 2 −∞ −1

μ

(2.23)

Over the width of a spectral line, the frequency variation of the Planck function is in general very small and the value of Q∗x (τ ) can be taken at the line center frequency. Here we assume that Q∗x (τ ) is independent of frequency and denote it Q∗ (τ ). For the standard two-level atom model (see e.g. Mihalas 1978),  is given by Eq. (2.16) and may take very small values, say, 10−4 or less. The normalization of the absorption profile in Eq. (2.8) becomes 

+∞

−∞

ϕ(x) dx = 1.

(2.24)

The optical depth τ is known as the frequency integrated or frequency averaged line optical depth. For spectral lines in the optical range, the absorption profile is in general a Voigt function. A Voigt function is the convolution of a Lorentz function and of a Doppler function, namely a U (x, a) = √ π π



+∞ −∞

e−y dy. a 2 + (x − y)2 2

(2.25)

20

2 Radiative Transfer Equations

√ The function H (x, a) = π U (x, a) is the so-called Voigt function. In contrast to U (x, a) it is not normalized to unity. Here we refer to U (x, a) as the Voigt function. The Lorentz function, ϕL (x) =

a 1 , 2 π a + x2

(2.26)

takes into account the natural width of the energy levels and their broadening by collisions with electrons and other species. The parameter a is a positive constant, called the Voigt parameter of the line, defined by a = /4πνD , where , measured in s −1 , is the line damping rate (see, e.g., Landi Degl’Innocenti and Landolfi 2004, p. 590). In stellar atmospheres, typical values for a are 10−1 –10−3 . The Doppler function, 1 2 ϕD (x) = √ e−x , π

(2.27)

takes care of the velocity distribution of the atoms, assumed to be a Maxwellian distribution. A Voigt function has a Gaussian core and Lorentzian wings for frequencies greater than (− ln a)1/2. Several approximations for the Voigt function can be found in the literature (see, e.g., Hummer 1965, Hubeny and Mihalas 2015, p. 320, Landi Degl’Innocenti and Landolfi 2004, p. 167). A simple one valid for small values of a and x = 0 is a 3 1 2 [1 + 2 + . . .]. U (x, a)  √ e−x + πx 2 2x π

(2.28)

Voigt profiles for three different values of the parameter a smaller than one are shown in Fig. 2.2. The Doppler profile corresponds to a = 0.3 The exact methods of solutions presented in this book hold if ϕ(x) is an even function of x, is decreasing to zero monotonously as |x| → ±∞ and is integrable, that is, satisfies some normalization condition such as Eq. (2.24). For complete frequency redistribution, Eq. (2.23) can also be written as μ

∂I (τ, x, μ) = ϕ(x)[I (τ, x, μ) − S(τ )]. ∂τ

(2.29)

The source function S(τ ), which contains the primary sources of photons and the contribution from the scattered photons, is defined by 1 S(τ ) ≡ (1 − ) 2

3



+1  +∞ −1

−∞

ϕ(x)I (τ, x, μ) dμ dx + Q∗ (τ ).

(2.30)

Voigt profiles for a larger than one, can be found in https://en.wikipedia.org/wiki/Voigt_profile.

2.1 The Integro-Differential Radiative Transfer Equations

21

Fig. 2.2 The Doppler profile, and several Voigt profiles with different values of the parameter a. The left panel shows ϕ(x) and the right panel log ϕ(x). In the right panel, the Lorentzian wing regime characterized by ϕ(x) ∼ 1/x 2 can be observed for |x| larger than (− ln a)1/2

Because of the assumption of complete frequency redistribution S(τ ) depends only on the optical depth. The definition of the source function, given in Eq. (2.30) for a spectral line formed with complete frequency redistribution and in Eq. (2.20) for monochromatic scattering, is independent of the description of the atomic structure of the radiating atom. From an atomic point of view, the source function for a simple two-level atom model, assuming complete frequency redistribution, and neglecting stimulated emission, has the form S(τ ) =

2hνc3 gl nu , c2 gu nl

(2.31)

where νc is the frequency at line center, h the Planck constant, c the speed of light, gl , nl and gu , nu the statistical weights and the populations of the lower and upper levels of the transition. The populations of the levels satisfy statistical equilibrium equations, which describe the balance between population and depopulation by radiative and collisonal processes. When collisions dominate over radiative processes, the populations of the levels follow the Saha–Boltzmann law and the source function reduces to the Planck function. This situation is known in Astronomy as Local Thermodynamic Equilibrium (LTE). In contrast, when radiative processes have a dominant role, which is the case for strong resonance lines, the source function will depend on the radiation field. For simple atomic models it will take the form written in Eq. (2.30), where Q∗ (τ ) is the Planck function at the frequency of the line center. This physical picture, known as Nonlocal Thermodynamic Equilibrium (NLTE) was stressed in the late fifties by J. T. Jefferies and R. N. Thomas (for historical references, see Hubeny and Mihalas 2015, p. 13). For spectral lines formed in NLTE, the determination of the radiation field demands that statistical equilibrium equations and radiative transfer equations be solved simultaneously (see, e.g., Landi Degl’Innocenti and Landolfi 2004; Hubeny and Mihalas 2015).

22

2 Radiative Transfer Equations

2.1.3 The Diffuse Radiation Field Primary photons can be created inside the medium, but can also come from outside. It is a typical situation for planetary atmospheres, for which the main and usually only source of energy comes from an outside illumination. The radiation field inside the atmosphere and the reemitted fraction, known as the albedo, can be derived from the radiative transfer equation. There are two different ways of handling the incident radiation: as a boundary condition for the radiative transfer equation, or by introducing a diffuse field. We describe this method, for a one-dimensional medium, say a semi-infinite one, first for monochromatic scattering and then for complete frequency redistribution. Diffuse fields are used in several chapters, for unpolarized and polarized radiation. We assume that a field I inc (μ) is incident on the surface at τ = 0. The idea of the method is to separate the field in two parts: a direct field and a diffuse field. The diffuse field denoted I d (τ, μ) is defined by I d (τ, μ) ≡ I (τ, μ) − I inc (μ) e−τ/|μ| , I d (τ, μ) ≡ I (τ, μ),

μ < 0,

μ > 0.

(2.32)

With our convention, μ < 0 for the radiation incident on the surface and μ > 0 for the radiation emerging from the medium. The diffuse field is zero at the surface for incoming directions. It is equal to the total field for the outgoing directions and at τ = 0 yields the emerging radiation. Introducing Eq. (2.32) into Eq. (2.18), one obtains for I d (τ, μ) the equation μ

∂I d 1 (τ, μ) = I d (τ, μ) − (1 − ) ∂τ 2



+1 −1

I d (τ, μ ) dμ − Q∗t (τ ),

(2.33)

with Q∗t (τ )

1 = Q (τ ) + (1 − ) 2 ∗



1

I inc (−μ)e−τ/μ dμ.

(2.34)

0

The second term in the right-hand side describes the scattering of the incident field inside the medium. The important point is that it depends only on τ . The source function for the diffuse field is S(τ ) = (1 − )

1 2



+1 −1

I d (τ, μ) dμ + Q∗t (τ ).

(2.35)

For complete frequency redistribution, one introduces in a similar way a diffuse field I d (τ, x, μ) defined by I d (τ, x, μ) = I (τ, x, μ) − I inc (x, μ)e−τ ϕ(x)/|μ| ,

μ < 0,

(2.36)

2.2 Integral Equations for the Source Function

I d (τ, x, μ) = I (τ, x, μ),

23

μ > 0.

(2.37)

It satisfies Eq. (2.23) with the internal primary source Q∗ (τ ) replaced by Q∗t (τ ) = Q∗ (τ ) + (1 − )

1 2



1  +∞ 0

−∞

ϕ(x)I inc (x, −μ) e−τ ϕ(x)/μ dx dμ. (2.38)

The second term in the right-hand side describes the scattering of the incident photons and depends only on τ , just as with monochromatic scattering. Radiative transfer problems with no internal source but an incident radiation are usually referred to as diffuse reflection problems. Thanks to the introduction of a diffuse field, any exact method of solution applicable to a problem with an internal primary source, can also be employed when the the medium is illuminated from outside. With the Case singular eigenfunction expansion method described in Chap. 10, it is actually preferable to work with the original radiation field and the associated boundary condition.

2.2 Integral Equations for the Source Function We now construct integral equations of the convolution type for the monochromatic and for the complete redistribution source functions. The method is quite simple. It suffices to introduce the solution of the transfer equations, expressed in terms of S(τ ), into the defining equations for S(τ ). Monochromatic scattering and complete frequency redistribution lead to similar integral equations.

2.2.1 Monochromatic Scattering Solving Eq. (2.19) as if S(τ ) were a known function, we can write I (τ, μ) = I (T , μ) exp(−

T −τ )+ μ



T τ

S(τ ) exp(−

τ − τ dτ

) , μ μ

(2.39)

where T and τ are two points inside the medium. When T > τ , Eq. (2.39) holds for μ > 0 and when T < τ , it holds for μ < 0. To calculate the radiation field we need boundary conditions. They depend on the geometry of the medium. For an infinite medium, physically meaning full solutions of the radiative transfer equation should grow less rapidly than an exponential at plus and minus infinity. This condition, applied at T = +∞ for μ > 0 and at T = −∞ for μ < 0, implies that the first term in the right-hand side of Eq. (2.39) is zero. Hence, in an infinite

24

2 Radiative Transfer Equations

medium 

±∞

I (τ, μ) =

S(τ ) exp(−

τ

τ − τ dτ

) , μ μ

(2.40)

where +∞ corresponds to μ > 0 and −∞ to μ < 0. Inserting Eq. (2.40) into the definition of S(τ ) given in Eq. (2.20), we find that S(τ ) satisfies the convolution type integral equation,  S(τ ) = (1 − )

+∞ −∞

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

(2.41)

with K(τ ) ≡

1 2



1

exp(− 0

1 |τ | dμ ) = E1 (|τ |). μ μ 2

(2.42)

Here E1 (τ ) is the first integro-exponential function (see e.g. Abramovitz and Stegun 1964). Equation (2.41) can have a finite solution only when the integral of q ∗ (τ ) over ] − ∞, +∞[ is finite. Once S(τ ) has been calculated, the radiation field can be derived from Eq. (2.40). The kernel contains all the physical properties of the scattering process. Equation (2.42) shows that it diverges logarithmically for |τ | → 0 and decreases exponentially for |τ | → ∞. It shows also that the kernel is a superposition of exponentials, which means that it can be written as a Laplace transform, namely as  ∞ K(τ ) = k(ν) e−ντ dν, (2.43) 0

with  k(ν) =

0 0 ≤ ν < 1, 1/2ν ν ≥ 1.

(2.44)

The function k(ν), which is the inverse Laplace transform of the kernel, plays a central role in almost all the exact methods of solution presented in the following chapters. Finally we note that the kernel, as we have defined it, is normalized to unity, that is 

+∞ −∞



∞

K(τ ) dτ = 2 0

K(τ ) dτ = 1.

(2.45)

2.2 Integral Equations for the Source Function

25

This implies 

+∞ −∞

k(ν)

dν =2 ν



∞

k(ν) 0

dν = 1. ν

(2.46)

For a semi-infinite medium, τ ∈ [0, ∞[.4 At plus infinity, the boundary condition is the same as for the infinite medium. At τ = 0, an incident radiation field corresponding to the directions μ < 0 must be prescribed. For outgoing directions (μ > 0), 

+∞

I (τ, μ) =

S(τ ) exp(−

τ

τ − τ dτ

) , μ μ

μ ∈ [0, 1].

For incoming directions (μ < 0),  τ τ − τ dτ

inc −τ/|μ| I (τ, μ) = I (μ)e ) , + S(τ ) exp(− |μ| |μ| 0

(2.47)

μ ∈ [−1, 0], (2.48)

where I inc (μ), μ < 0 is the incident field at the surface. Inserting the expressions of I (τ, μ) for μ > 0 and μ < 0 given in Eqs. (2.47) and (2.48) into Eq. (2.20), we find  S(τ ) = (1 − ) 0

∞

K(τ − τ )S(τ ) dτ + Q∗t (τ ),

τ ≥ 0,

(2.49)

where Q∗t (τ ) is the total primary source given in Eq. (2.34). It reduces to Q∗ (τ ) when the incident field is zero. Equation (2.49) can also be derived from the radiative transfer equation for the diffuse field I d (τ, μ) introduced in Sect. 2.1.3. Because radiation can escape freely through the surface boundary, Eq. (2.49) may have a finite solution even when the integral of Q∗ (τ ) over [0, +∞[ is infinite. Halfspace convolution equations such as Eq. (2.49) are known as Wiener–Hopf integral equations, from the Wiener and Hopf (1931) work, in which a method for obtaining exact closed form solutions was proposed for the first time.

2.2.2 The Milne Problem The Milne problem, as originally introduced by Schwarzschild (1906), describes the radiative equilibrium of a semi-infinite plane-parallel atmosphere in which the opacity coefficient is independent of frequency and the emission of photons depends 4 The notation [ means that the lower limit of an interval is included and the notation ] that it is excluded. For an upper limit the corresponding notation is ] and [.

26

2 Radiative Transfer Equations

on the local temperature, according to the Stefan’s law. It is moreover assumed that there is no radiation field impinging on the surface. The transfer equation for the radiation field may then be written as μ

∂I (τ, μ) = I (τ, μ) − S(τ ), ∂τ

(2.50)

where S(τ ) = σ T 4 (τ )/π. Here T is the thermodynamic temperature and σ the Stefan–Boltzmann constant. The boundary condition at the surface is I (0, μ) = 0 for μ < 0. To satisfy the condition that the atmosphere is in radiative equilibrium, the radiative flux, defined by  F (τ ) ≡ 2π

+1

−1

I (τ, μ)μ dμ,

(2.51)

should be constant and hence its derivative should be zero. Integrating Eq. (2.50) over μ, one immediately sees that this constraint leads to 1 2

S(τ ) =



+1 −1

I (τ, μ) dμ.

(2.52)

The transfer equation thus becomes μ

1 ∂I (τ, μ) = I (τ, μ) − ∂τ 2



+1

−1

I (τ, μ) dμ.

(2.53)

It describes a conservative and isotropic scattering problem. The formal solution of Eq. (2.50) combined with Eq. (2.52) leads to the homogeneous Wiener–Hopf integral equation 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ ,

(2.54)

0

solved in Wiener and Hopf (1931). The kernel K(τ ) is given in Eq. (2.42). The solution of Eq. (2.54) is defined within a multiplicative constant, which, for the Milne problem, is chosen to be the value of the flux F . We show in Chap. 9 that its solution may be written as S(τ ) =

3 F [τ + q(τ )], 4π

(2.55)

where √ q(τ ), the Hopf function, varies monotonically from the value q(0) = 1/ √ 3 = 0.5773 to the value 0.7104 at infinity. Hence, at the surface S(0) = 3F /(4π).

2.2 Integral Equations for the Source Function

27

For a star with a luminosity L and a radius R, the radiative flux per unit area 4 , with T is F = L/(4πR 2 ) = σ Teff eff the effective temperature. Hence, for an atmosphere in radiative equilibrium, the temperature varies as T 4 (τ ) =

3 4 T [τ + q(τ )]. 4 eff

(2.56)

The important point is that the fourth power of the temperature increases linearly with the optical depth. This famous law is employed, e.g., in Guillot (2010) to analyze the interplay between radiative transfer and advection in exoplanetary atmospheres.

2.2.3 Complete Frequency Redistribution The construction of a convolution integral for S(τ ) goes for complete frequency redistribution exactly as for monochromatic scattering. It suffices to replace μ by μ/ϕ(x) and to perform the integrations over the frequency x. For an infinite medium, S(τ ) satisfies Eq. (2.41) and for a semi infinite medium it satisfies Eq. (2.49) with Q∗t (τ ) given by Eq. (2.38). The kernel for complete frequency redistribution is significantly more complicated and also more interesting. It may be written as K(τ ) ≡

1 2



+∞  1 −∞

0

  ϕ(x) dμ dx. ϕ 2 (x) exp −|τ | μ μ

(2.57)

The monochromatic kernel is recovered when ϕ(x) is a rectangular profile with ϕ(x) = 1 for |x| ≤ 1/2 and ϕ(x) = 0 for |x| > 1/2. Taking into account the normalization of ϕ(x) (see Eq. (2.24)), it is easy to verify that K(τ ) satisfies the normalization to unity written in Eq. (2.45). The kernel for complete frequency redistribution is known in the Russian literature as the Biberman–Holstein kernel. It was introduced simultaneously by Holstein (1947) and Biberman (1947) to investigate the imprisonment of resonance radiation in enclosures such as light bulbs. Nowadays, in addition to Astrophysics, it is used to analyze plasmas in tokomaks (Kukushkin and Sdvizhenskii 2014; Kukushkin et al. 2018). There are differences and similarities between the monochromatic and complete redistribution kernels. For |τ | → 0, they both have a logarithmic divergence, but they have different asymptotic behaviors at infinity. For |τ | → ∞, the monochromatic kernel decreases exponentially and the complete frequency redistribution one algebraically, because of the integration over frequency. This difference has strong implications on exact methods of solutions and on asymptotic properties of the scattering process (see Part III). For complete frequency redistribution, the law of decrease at infinity √ depends on the absorption profile ϕ(x). For the Doppler profile K(τ ) ∼ 1/(τ 2 ln τ ) and for the Voigt profile K(τ ) ∼ 1/τ 3/2 (see, e.g., Eq. (5.10)).

28

2 Radiative Transfer Equations

To handle the formation of spectral lines with complete frequency redistribution, Sobolev5 (1949) has suggested to introduce the new variable ξ = μ/ϕ(x).

(2.58)

All the expressions involving an integration over frequency and direction can then be written as simple integrals over ξ . For example, the kernel K(τ ) takes the form 

∞

K(τ ) =

g(ξ ) exp(− 0

|τ | dξ ) , ξ ξ

(2.59)

where  g(ξ ) =

∞

ϕ 2 (u) du,

(2.60)

y(ξ )

and  y(ξ ) =

0

0 < ξ ≤ 1/ϕ(0),

ϕ (1/ξ )

ξ ≥ 1/ϕ(0).

−1

(2.61)

−1

The notation ϕ denotes the inverse function of ϕ. The transformation from the (x, μ) variables to the (x, ξ ) variables is meaningful only when ϕ(x) is an even function of x and is monotonously decreasing with |x|. These conditions are usually satisfied for spectral lines, when there is no systematic velocity field in the medium. The transformation is carried out by expressing μ in terms of ξ and then changing the order of integration from (ξ, x) to (x, ξ ) as shown in Fig. 2.3. The functions g(ξ ) and y(ξ ) are positive and even. The function g(ξ ) has a constant value for 0 ≤ ξ ≤ 1/ϕ(0) and tends to zero as ξ → ∞. The normalization of K(τ ) implies 

+∞ −∞



∞

g(ξ ) dξ = 2

g(ξ ) dξ = 1.

(2.62)

0

Equation (2.59) shows that the kernel can also be written as 

∞

K(τ ) =

k(ν) dν, 0

5

Viktor Victorovich Sobolev: 1915–1999 (Petrograd-St Petersburg).

(2.63)

2.2 Integral Equations for the Source Function

29

ξ 1/ϕ(x) 1/ϕ(0)

x 0 1/ϕ(0)

y(ξ)

Fig. 2.3 Complete frequency redistribution. The integration domains (ξ, x) and (x, ξ ), with ξ = μ/ϕ(x)

with k(ν) =

1 1 g( ), ν ν

ν ∈ [0, ∞[,

(2.64)

and the normalization 

∞

k(ν) 0

1 dν = . ν 2

(2.65)

The functions g(ξ ) and k(ν) are plotted in Figs. 5.1 and 5.2. In contrast to monochromatic scattering there is no interval in ν in which k(ν) is identically zero. This has strong implications for exact methods of solutions. The properties of k(ν) for complete frequency redistribution, in particular its behavior for ν → 0 and ν → ∞ are discussed in detail in Chap. 5 and in Appendix F of Chap. 9. The introduction of the variable ξ = μ/ϕ(x) simplifies many expressions involving complete frequency redistribution and also highlights the analogies and differences between monochromatic scattering and complete frequency redistribution. For example, for monochromatic scattering, the intensity emerging from a semi-infinite medium may be written as 

∞

I (0, μ) = 0

τ dτ , S(τ ) exp(− ) μ μ

μ ∈ [0, 1],

(2.66)

ξ ∈ [0, ∞].

(2.67)

and for complete frequency redistribution as 

∞

I (0, ξ ) = 0

τ dτ , S(τ ) exp(− ) ξ ξ

30

2 Radiative Transfer Equations

One easily sees that in both cases the emergent intensity is the Laplace transform of the source function. The only difference lies in the domains of definition of the Laplace variable. These simple expressions lead to the so-called Eddington–Barbier relations (Kourganoff 1963): I (0, μ)  S(μ);

I (0, x, μ)  S(

μ ), ϕ(x)

μ ∈ [0, 1].

(2.68)

These approximate relations become exact when S(τ ) is a linear function of τ , that is when S(τ ) = S0 + S1 τ . For monochromatic scattering, the Eddington–Barbier relation explains the phenomenon of limb-darkening observed at the surface of the Sun. The source function for the visible continuum, given by the Planck function, decreases from the interior to the surface as it follows the run of the temperature in the photosphere. The continuum radiation at the limb, coming from a region closer to the surface, will thus be darker than the continuum at disk center coming from a deeper region. The Eddington–Barbier relation approximation also explains the formation of absorption lines. In a stellar atmosphere, the source function of a line formed by multiple scattering is decreasing towards the surface, because of the escape of photons. The variation of I (0, x, μ) away from the line center simply follows the increase of the source function towards the interior. By this mechanism, an absorption line can be formed even in a medium with a constant temperature.

2.3 Neumann Series Expansion We have shown in Sects. 2.2.1 and 2.2.3 that the source function S(τ ), for monochromatic scattering and complete frequency redistribution, satisfies a convolution integral equation, having the form ∗

S(τ ) = Q (τ ) + (1 − )

 D

K(τ − τ )S(τ ) dτ ,

(2.69)

where D is the integration domain. The kernel K(τ ) contains the physical properties of the scattering process. It is defined in such a way that its integral from −∞ to +∞ is equal to 1. The parameter  is thus the destruction probability per scattering and (1 − ) the single scattering albedo. It will be seen in Part II that similar equations arise for polarized radiative transfer. The kernel becomes a matrix and the source function a vector or a matrix. The solution of this integral equation can be written as a series expansion, known as a Neumann series.6 Also known as the method of successive substitutions, it was

6

Karl Gottfried Neumann: 1832–1925 (Königsberg-Leipzig).

2.3 Neumann Series Expansion

31

introduced by Liouville (1837). It is based on the recursion relation, Sn (τ ) = Q∗ (τ ) + (1 − )

 D

K(τ − τ )Sn−1 (τ ) dτ .

(2.70)

We start for n = 0 with S0 (τ ) = Q∗ (τ ).

(2.71)

Inserting S0 (τ ) into Eq. (2.69), we obtain S1 (τ ) = Q∗ (τ ) + (1 − )

 D

K(τ − τ )S0 (τ ) dτ .

(2.72)

Repeating this process, we find at order n, Sn (τ ) = Q∗ (τ ) +

   n  i (1 − ) ... i=1

D D

D

K(τi − τi−1 )K(τi−1 − τi−2 ) . . . K(τ1 − τ )Q∗ (τi )dτ1 dτ2 . . . dτi .

(2.73)

In the limit n → ∞, this series converges to the solution of the integral equation, when (1 − ) is strictly smaller than one. The first term in the right-hand side yields the local creation rate Q∗ (τ ). The first term in the series contains the contribution, at the point τ , of the photons that have been created everywhere in the medium and scattered only one time before reaching the point τ . The second one term takes into account the photons that have been scattered twice, and so on. In a random walk approach to scattering problems, the source function is a generating function, an essential concept in probability theory, and the scattering kernel K(τ ) a density probability probability. The link between the integral equation for the source function and generating functions is discussed in Chap. 21. When the summation is carried out to infinity, a Neumann series expansion can be used to established exact results. For example, the exact expression of the surface √ value of the source function for the Milne problem, S(0) = 3F /4π, was derived by Hopf (1930) with this method. The capacity of Neumann series expansion to establish exact results in plane-parallel media is amply illustrated in a series of articles by Rutily and Bergeat (1987). Limited to a finite number of terms, a Neumann series expansion is a powerful tool for numerical solutions of radiative transfer problems. It is the basic ingredient of a numerical method, known as -iteration. This method has a poor convergence rate when  is very small and the integration domain has a large optical depth, but is otherwise efficient and simple to implement numerically. Special -accelerated iterations methods have been developed to obviate this shortcoming. They are widely used for scalar and polarization problems (Hubeny and Mihalas 2015;

32

2 Radiative Transfer Equations

Nagendra 2019). In the following chapters, we use Neumann series expansions to establish some symmetry properties, in particular in Sects. 2.4.1 and 15.3.

2.4 The Green Function and Associated Functions A standard method for solving linear differential equations or linear integral equations with an inhomogeneous term is to determine the Green function of the problem. The solution of the full problem can then be obtained by integrating the Green function weighted by the inhomogeneous term. This general method works also in radiative transfer. For half-space problems it was introduced by Sobolev (1963) and systematically developed by the Sobolev’s school in Leningrad/St. Petersburg. It is in general referred to as the resolvent method. Whereas the source function takes into account the scattering process and also the effect of the primary source, the Green function describes only the scattering process. This is an advantage, but has also a drawback. Whereas the source function depends on only one variable, the position in the medium, the Green function depends on two variables, since it describes the reaction of the medium to a point source at a given location. When the medium is translation invariant, the Green function can be expressed in terms of the so called resolvent function, a function of only one variable. We now present some properties of the Green and resolvent functions used in the following chapters.

2.4.1 Some Definitions The Green function G(τ, τ0 ) and the source function S(τ ) are related by  S(τ ) =

D

G(τ, τ0 )Q∗ (τ0 ) dτ0 .

(2.74)

Equation (2.69) shows that the Green function satisfies the convolution integral equation  G(τ, τ0 ) = (1 − )

D

K(τ − τ )G(τ , τ0 ) dτ + δ(τ − τ0 ),

τ0 , τ ∈ D.

(2.75)

For all the problems considered in Part I, the kernel K(τ ) is an even function of τ . Hence, G(τ, τ0 ) is symmetric with respect to τ and τ0 : G(τ, τ0 ) = G(τ0 , τ ).

(2.76)

This result can be demonstrated by writing the solution of (2.75) as a Neumann series expansion. The Green function G(τ, τ0 ) is actually a distribution, as it

2.4 The Green Function and Associated Functions

33

contains a Dirac distribution at τ = τ0 . Its regular part (τ, τ0 ) = G(τ, τ0 ) − δ(τ − τ0 ), known as the resolvent, or as the resolvent kernel, satisfies the integral equation  (τ, τ0 ) = (1 − )

D

K(τ − τ )(τ , τ0 ) dτ + (1 − )K(τ − τ0 ),

τ, τ0 ∈ D. (2.77)

For an infinite medium, the Green function is an even function of one variable, denoted here G∞ (τ ). Its domain of definition is ] − ∞, +∞[, also denoted R. The source function is given by  S(τ ) =

+∞ −∞

G∞ (τ − τ )Q∗ (τ ) dτ .

(2.78)

The function G∞ (τ ) satisfies the integral equation  G∞ (τ ) = (1 − )

+∞ −∞

K(τ − τ )G∞ (τ ) dτ + δ(τ ),

τ ∈ R.7

(2.79)

Its regular part, the infinite medium resolvent function ∞ (τ ) ≡ G∞ (τ ) − δ(τ ), is solution of the integral equation  ∞ (τ ) = (1 − )

+∞

−∞

K(τ − τ )∞ (τ ) dτ + (1 − )K(τ ),

τ ∈ R.

(2.80)

For a semi-infinite medium, the domain of integration D is [0, ∞[. A function playing an important role in the construction of exact solutions for half-space problems is G(τ ) ≡ G(0, τ ) = G(τ, 0).

(2.81)

The function G(τ ) can be interpreted as the source function of a semi-infinite medium with a plane isotropic source uniformly distributed on the surface τ = 0 and emitting 4π of energy per unit area. Here we refer to G(τ ) as to the surface Green function. The function G(τ ) and its regular part (τ ), (τ ) ≡ G(τ ) − δ(τ ),

(2.82)

known as the resolvent function, satisfy the Wiener–Hopf integral equations  G(τ ) = (1 − )

∞

K(τ − τ )G(τ ) dτ + δ(τ ),

τ ∈ [0, ∞[,

0

7

The symbol IR stands for the full real line, that is IR =] − ∞, +∞[.

(2.83)

34

2 Radiative Transfer Equations

and  (τ ) = (1 − )

∞

K(τ − τ )(τ ) dτ + (1 − )K(τ ),

τ ∈ [0, ∞[.

(2.84)

0

The functions G(τ ) and (τ ) play an important role in the solution of semiinfinite problems. There are several reasons. As will be shown in Sect. 2.4.4, the knowledge of G(τ ) is sufficient to determine G(τ, τ0 ). For some special primary sources, the source function S(τ ) has a simple expression in terms of (τ ). It can be derived from an important lemma established below.

2.4.2 A Lemma on Wiener–Hopf Equations Let f (τ ) satisfy the Wiener–Hopf integral equation 

∞

f (τ ) = (1 − )

K(τ − τ )f (τ ) dτ + f ∗ (τ ),

τ ∈ [0, ∞[,

(2.85)

0

with K(τ ) an even function of τ , and f ∗ (τ ) a given inhomogeneous term. The lemma states that the derivative of f (τ ) satisfies a Wiener–Hopf integral equation with the same kernel, namely d f (τ ) = (1 − ) dτ

 0

∞

K(τ − τ )

d f (τ )

d f ∗ (τ ) . dτ + f (0)(1 − )K(τ ) + dτ

dτ (2.86)

The simplest way to establish Eq. (2.86) (Hopf 1934 p. 34; Ivanov 1994) is to proceed directly from the definition of the derivative: d f (τ ) 1 ≡ lim [f (τ + τ ) − f (τ )]. τ →0 τ dτ

(2.87)

Applying this definition to Eq. (2.85), we obtain  ∞ d f (τ ) 1− K(τ − u)[f (u + τ ) − f (u)] du = lim τ →0 τ dτ 0   0 d f ∗ (τ ) . K(τ − u)f (u + τ ) du + + dτ −τ

(2.88)

The limit τ → 0 leads to Eq. (2.86). Other proofs of this result can be found in Kourganoff (1963, §13), Sobolev (1963, Section 4 of Chapter 3), Ivanov (1973, p. 199). We present now two applications of this lemma.

2.4 The Green Function and Associated Functions

35

2.4.3 The Source Function and the Resolvent Function For some special primary sources, the source function S(τ ) can be expressed in terms of the resolvent function (τ ) and of the surface value S(0). These expressions are no substitute for a solution of the Wiener–Hopf integral equation for S(τ ), but are quite useful and can be readily established with the above lemma. Let us assume that Q∗ (τ ), the primary source in the integral equation for S(τ ) (see Eq. (2.69)) has a constant value Q∗ . The above lemma leads to d S(τ ) = (1 − ) dτ



∞

K(τ − τ )

0

d S(τ )

dτ + (1 − )K(τ )S(0). dτ

(2.89)

Comparing this equation with the integral equation for (τ ) written in Eq. (2.84), we find d S(τ ) = (τ )S(0), dτ

(2.90)

hence 

τ

S(τ ) = S(0)[1 +

(τ ) dτ ].

(2.91)

0

This simple relation holds for a uniform source term and also for the Milne problem for which Q∗ = 0 (see Eq. (9.20)). The knowledge of (τ ) is of course not sufficient to determine S(τ ), since S(0) has still to be determined. For an exponential source term Q∗ (τ ) = Q∗ e−ατ , α > 0, the source function, denoted Sα (τ ), can also be directly expressed in terms of the resolvent function. According to the above lemma, the derivative of Sα (τ ) satisfies the Wiener–Hopf integral equation dSα (τ ) = dτ  ∞ dSα (τ )

(1 − ) K(τ − τ ) dτ + (1 − )K(τ )Sα (0) − αQ∗ e−ατ . dτ

0

(2.92)

A comparison with the equation for (τ ) leads to dSα (τ ) = (τ )Sα (0) − αSα (τ ), dτ

(2.93)

and the integration of Eq. (2.93) to   Sα (τ ) = Sα (0) e−ατ +

τ 0



e−α(τ −τ ) (τ ) dτ .

(2.94)

36

2 Radiative Transfer Equations

This expression is very useful for solving diffuse reflection problems. Of course, when α = 0, one recovers Eq. (2.91).

2.4.4 Some Properties of the Green Function We show here how the knowledge of the surface Green function G(τ ) allows one to determine the Green function G(τ, τ0 ). The relation between the two functions may be written as  τ G(τ, τ0 ) = G(τ − t)G(τ0 − t) dt, τ = min(τ0 , τ ). (2.95) 0

This relation has a Laplace transform version, ˜ ˜ G(p) G(ν) ˜˜ , G(ν, p) = p+ν

(2.96)

˜˜ where G(ν, p) is the double Laplace transform of G(τ, τ0 ), defined by ˜˜ G(p, ν) =



∞

e−pτ



0

∞

e−ντ0 G(τ, τ0 ) dτ0 dτ,

(2.97)

0

˜ and G(p) is the Laplace transform of G(τ ) defined by ˜ G(p) =



∞

e−pτ G(τ ) dτ.

(2.98)

0

An elegant proof of Eq. (2.95) based on probabilistic arguments is given in Ivanov (1973, p. 206, also 1994). Here we reproduce the purely algebraic proof of Sobolev (1963), based on the lemma written in Eq. (2.86). The first proof of Eq. (2.96) can be found in Hopf (1934, p. 59). The key ingredients are the symmetry and translational invariance of the kernel and the semi-infinite character of the medium. We consider the integral equation for the resolvent,  (τ, τ0 ) = (1 − )

∞

K(τ − τ )(τ , τ0 ) dτ + (1 − )K(τ − τ0 ),

τ, τ0 ∈ [0, ∞[,

0

(2.99) and apply the differential operator D=

∂ ∂ + . ∂τ ∂τ0

(2.100)

References

37

Using DK(τ − τ0 ) = 0 and (τ, τ0 ) = (τ0 , τ ), which follow from the evenness of K(τ ), the lemma in Eq. (2.86), and (τ0 ) = (0, τ0 ), we obtain  D(τ, τ0 ) = (1−)

∞

K(τ −τ )D(τ , τ0 ) dτ +(1−)K(τ )(τ0).

(2.101)

0

Comparing Eq. (2.101) with Eq. (2.84), we see that D(τ, τ0 ) and (τ ) satisfy the same integral equation except for the factor (τ0 ) which multiplies the inhomogeneous term in Eq. (2.101). The linearity of these two integral equations implies D(τ, τ0 ) = (τ )(τ0 ).

(2.102)

The integration of this partial differential equation, with the boundary condition (0, τ0 ) = (τ0 ), yields  (τ, τ0 ) = (τ − τ ) +

τ

(t)(t + τ − τ ) dt,

(2.103)

0

with τ ≡ min(τ, τ0 ) and τ ≡ max(τ, τ0 ). Expressing (τ, τ0 ) in terms of G(τ, τ0 ), (τ ) in terms of G(τ ), and using the symmetry G(τ, τ0 ) = G(τ0 , τ ), one recovers Eq. (2.95). Equation (2.96) can also be derived from Eq. (2.102). Taking its Laplace transform with respect to τ and with respect to τ0 , we obtain ˜˜ ˜ ˜ ˜ (p + ν)(p, ν) = (p)(ν) + (0, ν) + (p, 0).

(2.104)

Equation (2.96) is recovered after some simple algebra. It is used in the following chapters to calculate the emergent radiation field and to construct nonlinear integral equations for the H -function. Among the many techniques that exist for establishing Eqs. (2.95) and (2.96), Sobolev’s method, which requires only the fundamental physical properties of the scattering process, is particularly elegant and efficient.

References Abramovitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC (1964) Aller, L.H.: Astrophysics. The Atmospheres of the Sun and Stars. Ronald Press, New York (1963) Biberman, L. M.: On the theory of resonance radiation. Zhurn. Eksperim. Theor. Phys. 17, 416 (1947) (in Russian) Case, K.M., Zweifel, P.: Linear Transport Theory. Addison-Wesley, Reading, MA (1967) Eddington, A.S.: The formation of absorption lines. Mon. Not. R. Astr. Soc. 89, 620–636 (1929) Guillot, T.: On the radiative equilibrium of irradiated planetary atmospheres. Astron. Astrophys. 520, A27 (13 pp.) (2010). https://doi.org/10.1051/0004-6361/200913396

38

2 Radiative Transfer Equations

Hemsh, M.J., Ferziger, J.H.: Radiative transfer with partially coherent scattering. J. Quant. Spectrosc. Radiat. Transf. 12, 1029–1046 (1972) Holstein, T.: Imprisonment of radiation in gases. Phys. Rev. 72, 1212–1233 (1947) Hopf, E.: Remarks on the Schwarzschild–Milne model of the outer layers of a star. Mon. Not. R. Astr. Soc. 90, 287–293 (1930) Hopf. E.: Mathematical Problems of Radiative Equilibrium. Cambridge University Press, London (1934) Hubeny, I., Mihalas, D.: Theory of Stellar Atmospheres. Princeton University Press, Princeton (2015) Hummer, D.G.: Non-coherent scattering I. The redistribution functions with Doppler broadening. Mon. Not. R. Astr. Soc. 125, 21–37 (1962) Hummer, D.G.: The Voigt-function: an eight-significant-figure table and generating procedure. Mon. Not. R. Astr. soc. 70, 1–32 (1965) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ., vol. 385, translation by D.G. Hummer and E. Weppner (1973); Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Ivanov, V.V.: Resolvent method: exact solutions of half-space transport problems by elementary means. Astron. Astrophys. 286, 328–337 (1994) Jefferies, J.T.: Spectral Line Formation. Blaisdell Publishing Company, Waltham, MA (1968) Kourganoff, V.: Basic Methods in Transfer Problems. Radiative Equilibrium and Neutron Diffusion. Dover Publications, New York (1963); First edition: Oxford University Press, London (1952) Kukushkin, A.B., Sdvizhenskii, P.A.: Scaling laws for non-stationary Biberman–Holstein radiative transfer in plasmas. In: 41st EPS Conference on Plasma Physics, vol. P4.133 (2014) Kukushkin, A.B., Nerenov, V.S., Sdvizhenskii, P.A., Voloshinov, V.V.: Automodel solutions of Biberman–Holstein equation for stark broadening of spectral Lines. MDPI, Atoms 6, 43 (15 pp.) (2018) Landi Degl’Innocenti, E., Landolfi, M.: Polarization in Spectral Lines. Kluwer Academic Publisher, Dordrecht (2004) Liouville, J.: Sur le développement des fonctions ou parties de fonctions en séries dont les termes sont assujettis à satisfaire à une même équation différentielle du second ordre contenant un paramètre variable. J. Math. Pures Appl. 2, 16–55 (1837) Mihalas, D.: Stellar Atmospheres, 2nd edn. W.H. Freeman and Company, San Francisco, CA (1978) Mishchenko, M.I.: Multiple scattering, radiative transfer, and weak localization in discrete random media: unified microphysical approach. Rev. Geophys. 46, RG2003 (33 pp.) (2008). https://doi. org/10.10029/2007/RG000230 Mishchenko, M.I., Travis, L.D., Lacis, A.A.: Multiple Scattering of Light by Particles. Radiation Transfer and Backscattering. Cambridge University Press, Cambridge (2006) Nagendra, K.N.: Polarized line formation: Methods and solutions. In: Belluzzi, L., Casini, R., Romolli, M., Trujillo Bueno, J. (eds.) Solar Polarization 8. ASP Conference Series, vol. 526, pp. 99–118 (2019) Rutily, B., Bergeat, J.: The Neumann solution of the multiple scattering problem in a plane parallel medium. J. Quant. Spectrosc. Radiat. Transfer 38, IIa (1987). Semi-infinite space and the Hfunction, pp. 47–60, IIb. The resolvent function in a semi-infinite space, pp. 61–70, IIc. The specific intensity in a semi-infinite space, pp. 71–78 Schwarzschild, K.: Über das Gleichgewicht der Sonnenatmosphäre. Nachr. von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Math.-phys. Klasse, 195, 41–53 (1906) Sobolev, V.V.: A Treatise on Radiative Transfer, Von Nostrand Company, Princeton, New Jersey, transl. by S.I. Gaposchkin (1963); Russian original, Transport of Radiant Energy in Stellar and Planetary Atmospheres, Gostechizdat, Moscow (1956) Thomas, R.N.: The source function in a non-equilibrium atmosphere I. The resonance lines. Astrophys. J. 125, 260–274 (1957)

References

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Wiener, N., Hopf, D.: Über eine Klasse singulärer Integralgleichungen. Sitzungsberichte der Preussichen Akademie, Mathematisch-Physikalische Klasse 3. Dez. 1931, vol. 31, pp. 696–706 (ausgegeben 28. Januar 1932) (1931); English translation: In: Paley, R.C., Wiener, N., Fourier transforms in the Complex Domain. Am. Math. Soc. Coll. Publ., vol. XIX, pp. 49–58 (1934) Woolley, R., Stibbs, D.: The Outer Layers of a Star. Clarendon Press, Oxford (1953) Yengibarian, N.B., Khachatrian, A.K.: On a problem of multiple resonance scattering of gammaray quanta. J. Quant. Spectrosc. Radiat. Transf. 46, 565–575 (1991)

Chapter 3

Exact Methods of Solution: A Brief Survey

The preceding chapter contains a presentation of several equations, differential and integral ones, describing multiple scatterings of photons. In this chapter we present, more or less in a chronological order, a brief summary of exact methods of solution. They are described in detail in the next chapters, following an inverse chronological order, recent methods allowing us to introduce tools and concepts that make it easier to understand the first exact solution of the Milne problem by Wiener and Hopf. Several of the methods described here for scalar equations are applied in Part II to vector equations describing linearly polarized radiation fields. We show in particular how to construct the exact solution of the Milne problem for the Rayleigh scattering, first obtained by Chandrasekhar (1946b,c). Analytical work in radiative transfer has started roughly at the end of the nineteenth century with the works of Lommel (1889) and Chwolson (1890). They were the first to set the foundations for a mathematical theory of the propagation of light by repeated scatterings through a diffusive medium and to show that the problem could be formulated as an integral equation. A detailed account of this pioneering work is described by Ivanov (1991a). O. D. Chwolson, who was a chemical-physicist, had particularly in mind the propagation of light in milk (opal) glass. E. Lommel was interested in the determination of the albedo of a diffuse medium. The work of these two authors remained largely unnoticed, except in the Russian literature. The sustained interest in the problem came with studies on the transfer of light through stellar and planetary atmospheres. Schuster (1905) stressed that the effect of scatterings could be of primary importance in interpreting the nature of stellar atmospheres. In a famous paper Schwarzschild (1906) showed that the transfer of light through a grey stellar atmosphere in radiative equilibrium is equivalent to a transfer problem in a diffusive conservative, plane-parallel, semi-infinite medium and that it could be formulated as an integro-differential equation, where the integral term describes the scattering process. Subsequently Schwarzschild (1914) and Milne (1921) have shown that the integro-differential equation can be expressed as an integral equation of the convolution type, rediscovering thus the integral formulation first proposed by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_3

41

42

3 Exact Methods of Solution: A Brief Survey

E. Lommel and O. D. Chowlson. The problem stated by Schwarzschild, known as the Schwarzschild–Milne or simply Milne problem, is described in Sect. 2.2.2, where we show that it leads to the integral equation 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ ,

(3.1)

0

where S(τ ) and K(τ ) are the source and kernel functions. Several fairly good approximate solutions, properly catching the linear asymptotic behavior at infinity (see Eq. (2.55)), were proposed in the twenties (see e.g. Hopf 1934 for references) but the exact analytical solution was constructed only in 1931 by Wiener and Hopf (1931). Because the range of integration is from zero to infinity and not from minus infinity to plus infinity, the solution cannot be constructed by a simple Fourier transformation. The method introduced by Wiener and Hopf involves an auxiliary function, which expresses the fact that the scattering occurs in a semi-infinite medium. In any half-space transport problem, will appear a half-space auxiliary function. For a scalar transport equation, this function is a scalar. For vector transfer problems arising in polarized transfer, this function becomes a matrix (see Part II). In the astrophysical literature, this auxiliary function is nowadays usually called the H -function, a notation introduced by Chandrasekhar (1946a). Each transfer problem has its own H -function and in each problem the H -function plays a fundamental role. For example, for the Milne problem, the angular variation of the emergent radiation field is given by I

em

√ 3 F H (cos θ ), (cos θ ) = 4π

(3.2)

where θ is the angle between the normal to the surface and the direction of the emergent radiation and F is the constant radiative flux inside the atmosphere. This important result was established by Hopf (1934). For this reason the H -function is also referred to as the limb-darkening function (Mihalas 1978). The basic ingredient entering the construction of this half-space auxiliary function is not the kernel K(τ ) itself, but the so-called dispersion function, directly related to the kernel. It appears naturally when solving radiative transfer problems in an infinite medium. We introduce this dispersion function before presenting exact methods of solutions for radiative transfer equations in a semi-infinite medium.

3.1 The Infinite Medium and the Dispersion Function The convolution integral equation for the source function has the form  S(τ ) = (1 − )

D

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

(3.3)

3.2 Exact Methods for a Semi-Infinite Medium

43

where Q∗ (τ ) is a given primary source and D an integration domain (see Sect. 2.2). This equation holds for monochromatic scattering as well as complete frequency redistribution. For an infinite medium (D =] − ∞, +∞[), Eq. (3.3) can be solved by a Fourier transformation. We define the real Fourier transform fˆ(k) of a function f (τ ) with fˆ(k) ≡



+∞ −∞

f (τ )ei kτ dτ,

k

real.

(3.4)

The Fourier transform of Eq. (3.3) is ˆ S(k)V (k) = Qˆ ∗ (k),

(3.5)

ˆ where S(k) and Qˆ ∗ (k) are the Fourier transforms of S(τ ) and Q∗ (τ ), and V (k), known as the dispersion function, is defined by ˆ V (k) ≡ 1 − (1 − )K(k).

(3.6)

ˆ Here K(k) is the Fourier transform of K(τ ). Equation (3.5) provides an explicit expression of the Fourier transform of S(τ ). An explicit (closed form) expression of S(τ ) can then be established with the Fourier inversion formula, namely 1 S(τ ) = 2π



+∞ −∞

ˆ Q(k) e−i kτ dk. V (k)

(3.7)

We show how to perform this inversion with various examples.

3.2 Exact Methods for a Semi-Infinite Medium For a semi-infinite medium, the integral equation for the source function is given by Eq. (3.3), the integration domain being now [0, ∞[. We present here the main ideas of methods that will be described in the following chapters.

3.2.1 The Wiener–Hopf Method The method of solution invented by Wiener and Hopf (1931) to solve the Milne problem, which now carries their name, is a major achievement in Mathematical Physics, in the first half of the twentieth century. It makes a systematic use of the analyticity properties of Fourier transforms in the complex plane. In the Fourier transforms introduced above, k is now a complex variable. In the Milne problem, the scattering is monochromatic scattering and it follows that the kernel K(τ )

44

3 Exact Methods of Solution: A Brief Survey

decreases exponentially as τ tends to infinity. As a result, the dispersion function V (z), z complex, is analytic in a strip in the complex plane, say a horizontal strip. A fundamental step in the Wiener–Hopf method is to write the dispersion function as V (z) = Vu (z)/Vl (z),

(3.8)

where Vu (z) is analytic in an upper half-plane and Vl (z) is analytic in a lower halfplane, with the two half-planes having a common strip of analyticity included in the analyticity strip of V (z). We show in Chap. 12 that these functions are related to the H -function by Vu (z) = 1/H (−i /z) and Vl (z) = 1/H (i /z). For some problems, a factorization of V (z), of the type shown in Eq. (3.8), can be made upon inspection. For radiative transfer problems, it requires a detailed analysis of the behavior of the dispersion function in the complex plane. The Wiener–Hopf method was developed for exponentially decreasing kernels, but it can be extended to algebraically decreasing kernels encountered with spectral lines formed with complete frequency redistribution. In that case, the analyticity strip becomes a line. The Wiener–Hopf method has found a wide range of applications in such fields as diffraction of electromagnetic waves, propagation of acoustic waves, elasticity theory, each time a semi-infinite geometry is involved. Books by Noble (1958), Carrier et al. (1966), Roos (1969), Duderstadt and Martin (1979), Ablowitz and Fokas (1997) present a large variety of such problems. We show in Sect. 12.6 how to apply the Wiener–Hopf method to a diffraction problem.

3.2.2 Traditional Real-Space Methods For the standard radiative transfer problems described in Chap. 2, the kernel is a linear superposition of exponentials. In brief, it has the form of a Laplace transform and can be written as  ∞ K(τ ) = k(ν)e−τ ν dν, τ ≥ 0. (3.9) 0

For example, for monochromatic scattering, Eq. (2.42) shows that  k(ν) =

0 1/2ν

0 ≤ ν < 1, ν ≥ 1.

(3.10)

For complete frequency redistribution k(ν) is given in Sect. 2.2.3. Ambartsumian1 (1942) was the first to take advantage of the representation of the kernel as a Laplace transform. A detailed description of the method can be found

1

Viktor Amazaspovich Ambartsumian: 1908–1996 (Tbilisi-Buryakan).

3.2 Exact Methods for a Semi-Infinite Medium

45

in Kourganoff (1963). It is based on manipulations of the integral equations for the source function and for the Green function. One of its marking result is the construction of a nonlinear integral equation, λ ψ(x) = 1 + xψ(x) 2



1 0

ψ(x ) dx , x + x

(3.11)

for a half-space auxiliary function, denoted ψ(x), which turns out in modern notation to be the H -function. The influence of Ambartsumian (1942) paper has been quite large. We describe in Chap. 11 a method à la Ambartsumian leading to Eq. (3.11). Belonging to what we have for simplicity referred to as traditional real-space methods are the very fundamental theoretical developments by V. V. Sobolev (Sobolev 1963, also Ivanov 1973), on radiative transfer for spectral lines formed with complete frequency redistribution. They include the construction of an H -function for half-space problems and of a nonlinear integral equation similar to Eq. (3.11). A comparison between Ambartsumian’s and Sobolev’s methods of solution for Eq. (3.3), which holds for monochromatic scattering as well as for complete frequency redistribution, is presented in Rutily and Bergeat (1994) for the infinite, semi-infinite, and finite space cases (see also Rutily 1992, p. 117). The nonlinear integral equation in Eq. (3.11), which holds for an isotropic scattering, and similar ones, for anisotropic scattering and spectral lines, can be solved exactly. Most methods of solution are factorization methods à la Wiener– Hopf, which make use of complex plane analysis (see e.g. Chandrasekhar 1960, Busbridge 1960, Kourganoff 1963, Sobolev 1963, Ivanov 1973). We will not describe them here, as we are mainly interested in exact methods of solution of Eq. (3.3), but we describe in Chap. 11 Ivanov’s (1994) method of solution, which involves only real-space valued functions. For polarized radiative transfer treated in Part II, the H -function becomes an Hmatrix, but nonlinear integral equations similar to Eq. (3.11) can be constructed with the methods developed in the scalar case. The emergent polarized radiation field has an expression similar to Eq. (3.2). The main difference with the scalar case is that the nonlinear integral equations are matrix equations, which have in general no exact explicit solutions. They have to be solved numerically.

3.2.3 The Singular Integral Equation Approach For kernels which can be written as a Laplace transform, the radiative transfer equation can be transformed into a linear singular integral equation with a Cauchy-type kernel. For these equations, a powerful method of solution based on complex plane

46

3 Exact Methods of Solution: A Brief Survey

analysis was introduced by Carleman2 (1922) and developed by Muskhelishvili3 (1953). We present two methods, leading to singular integral equations. They can both be viewed as generalizations of the discrete ordinates method of Chandrasekhar (1960). One is based on an expansion of the radiation field and the other one on an expansion of the source function. In Chap. 10 we present the eigenfunction expansion method introduced by Case (1960). It amounts to writing the radiation field as an expansion having the form  I (τ, ξ ) =

a(ν)(ν, ξ )e−ντ dν,

(3.12)

where a(ν) are the coefficients of the expansion and (ν, ξ ) the eigenfunctions. For monochromatic scattering ξ = μ and for complete frequency redistribution ξ = μ/ϕ(x). The eigenfunctions are deduced from of the homogeneous version of the radiative transfer equation. For half-space or infinite space problems, the expansion coefficient a(ν) satisfies a singular integral equation with a Cauchy-type kernel, which may be written as  α(ν)a(ν) + β(ν)

a(ν ) dν = γ (ν).

Lν −ν

(3.13)

The notation means that the integral should be taken in Cauchy Principal Value and L is an interval (or an ensemble of intervals) on the real line, which may be infinite. One can observe that the integral operator is universal, in the sense that it does not depend on the properties of the scattering mechanism. The latter are contained in the coefficients α(ν) and β(ν). Singular integral equations with Cauchy-type kernels can also be derived directly from convolution integral equations such as Eq. (3.1). One can either take their Laplace transforms (Halpern et al. 1938), or their inverse Laplace transforms (Frisch and Frisch 1982). The latter method amounts to assume that the source function can be written as  ∞ S(τ ) = s(ν) e−ντ dν. (3.14) 0

Equation (3.14) is simply an integrated version of Eq. (3.12). It leads to a singular integral equation for s(ν), similar to Eq. (3.13). Singular integral equations such as Eq. (3.13) have been extensively studied in the mathematical literature, in particular by Carleman (1922), who was the first to show that these equations can be transformed into boundary value problems in the complex plane by a Hilbert transform.

2 3

Torsten Carleman: 1892–1949 (Osby-Stockholm). Nikolai Ivanovich Muskhelishvili: 1891–1976 (Tbilisi-Tbilisi).

References

47

For full-space problems, the solution of singular integral equations of the type written in Eq. (3.13) can be expressed in terms of the dispersion function. In contrast, half-space problems require the construction of a half-space auxiliary function, which we denote X(z). It must be analytic in the complex plane and, along some interval LX , included in L, satisfy a boundary value problem, which has the form X+ (ν)/X− (ν) = W (ν),

ν ∈ LX .

(3.15)

Here X+ (ν) and X− (ν) are the limiting values of X(z) on each side of the interval LX and W (ν) is a function depending only on the dispersion function. The construction of X(z) is equivalent to the factorization problem of the Wiener–Hopf method. The functions X(z), V + (z) and V − (z) are very simply connected and are of course directly related to the H -function. Equation (3.15) is a boundary value problem in the complex plane, known under the name of Riemann–Hilbert problem. Riemann–Hilbert problems can be homogeneous as in Eq. (3.15) or inhomogeneous. The Riemann–Hilbert problem was introduced by Riemann (1851) and generalized by Hilbert (1900, 1912). Some historical context can be found in Bothner (2021).4 We show in Chap. 4 how to transform Wiener–Hopf integral equations into singular integral equations, by direct or inverse Laplace transforms, and how the latter can be transformed into boundary value problems in the complex plane by a Hilbert transform. The construction of the auxiliary function X(z) is described in Chap. 5. Several applications of this method are presented in Chaps. 6–10. The transformation of Wiener–Hopf integral equations into singular integral equations with Cauchy-type kernels and their subsequent transformation into boundary value problems is less general than the Wiener-Hopf method because it is restricted to kernels that can be written as a superposition of exponentials, but technically speaking, the boundary values problems in the complex plane, are simpler to solve and can be generalized to vectorial problems, such as those encountered in scattering polarization (see Part II).

References Ablowitz, M.J., Fokas, A.S.: Complex Variables, Introduction and Applications. Cambridge University Press, Cambridge (1997) Ambartsumian, V.A.: Light scattering by planetary atmospheres. Astron. Zhurnal 19, 30–41 (1942) Bothner, T.: On the origins of Riemann–Hilbert problems in mathematics. Nonlinearity 34, R1– R73 (2021) Busbridge, I.W.: The Mathematics of Radiative Transfer. Cambridge University Press, London (1960) Carleman, T.: Sur la résolution de certaines équations intégrales. Ark. Mat. Astr. Fys. 16, 1–19 (1922)

4

See also the https://en.wikipedia.org/wiki/Riemann-Hilbert_problem, and https://encyclopediaofmath.org/wiki/Riemann-Hilbert_problem.

48

3 Exact Methods of Solution: A Brief Survey

Carrier, G.F., Krook, M., Pearson, C.E.: Functions of a Complex Variable. McGraw-Hill Book Company, New York (1966) Case, K.M.: Elementary solutions of the transport equation and their applications. Ann. Phys. (New York) 9, 1–23 (1960) Chandrasekhar, S.: On the radiative equilibrium of a stellar atmosphere. IX. Astrophys. J. 103, 165–192 (1946a) Chandrasekhar, S.: On the radiative equilibrium of a stellar atmosphere. X. Astrophys. J. 103, 351–370 (1946b) Chandrasekhar, S.: On the radiative equilibrium of a stellar atmosphere. XI. Astrophys. J. 104, 110–132 (1946c) Chandrasekhar, S.: Radiative Transfer, Dover Publications, New York (1960); First edition, Oxford University Press, Oxford (1950) Chwolson, O.D.: Grundzüge einer Mathematischen Theorie der Inneren Diffusion des Lichtes. Bull. Acad. Imp. Sci. St. Pétersbourg 33, 221–256 (1890), also in Mélanges Phys. et Chim. XIII, 83–118 (1889) Duderstadt, J.J., Martin, W.R.: Transport Theory. Wiley, New York (1979) Frisch, H., Frisch, U.: A method of Cauchy integral equation for non-coherent transfer in halfspace. J. Quant. Spectrosc. Radiat. Transf. 28, 361–375 (1982) Halpern, O., Lueneburg, R., Clark, O.: On multiple scattering of neutrons I. Theory of the albedo of a plane boundary. Phys. Rev. 53, 173–183 (1938) Hilbert, D.: Mathematische probleme. Göttinger Nachrichten 3, 253–297 (1900) Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. ed. Teubner, Leipzig (1912) Hopf, E.: Mathematical Problems of Radiative Equilibrium. Cambridge University Press, London (1934) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office. NBS Spec. Publ. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Ivanov, V.V.: Resolvent method: exact solutions of half-space transport problems by elementary means. Astron. Astrophys. 286, 328–337 (1994) Ivanov, V.V.: A hundred year of the radiative transfer integral equation, in “Centennial of the Integral Transport Equation: Symposium in Leningrad”. Transport Theory Stat. Phys. 20, 525– 539 (1991a). ed. T.A. Germogenova Kourganoff, V.: Basic Methods in Transfer Problems. Radiative Equilibrium and Neutron Diffusion. Dover Publications, New York (1963); first edition: Oxford University Press, London (1952) Lommel, E.: Die Photometrie der diffusen Zurückwerfung. Ann. Phys. und Chemie (Wiedemann Annalen) 36, 473–502 (1889) Mihalas, D.: Stellar Atmospheres, 2nd edn. W.H. Freeman and Company, San Francisco, CA (1978) Milne, E.A.: Radiative equilibrium in the outer layers of a star : the temperature distribution and the law of darkening. Mon. Not. R. astr. Soc. 81, 361–375 (1921) Muskhelishvili, N.I.: Singular Integral Equations, Noordhoff, Groningen (based on the second Russian edition published in 1946) (1953); Dover Publications (1991) Noble, B.: Methods Based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, New York (1958) Riemann, B.: Grundlagen für eine allgememeine Theorie der Funktionen einer verändlichen komplexen Grösse Inauguraldisserationn Göttingen 1851 (1851); zweiter unveränderte Andruck, Göttingen (1867); in Werke, Leipzig pp. 3–43 (1876); in collected Works, Dover (1953) Roos, B.W.: Analytic Functions and Distributions in Physics and Engineering. Wiley, New York (1969) Rutily, B.: Solutions exactes de l’équation de transfert et applications astrophysiques. In: Thèse de Doctorat d’État. Université Claude Bernard, Lyon I, N◦ , 92.07 (1992)

References

49

Rutily, B., Bergeat, J.: The solution of the Schwarzschild-Milne integral equation in an homogeneous isotropically scattering plane-parallel medium. J. Spectrosc. Radiat. Transf. 51, 823–847 (1994) Schuster, A.: Radiation through a foggy atmosphere. Astrophys. J. 21, 1–22 (1905) Schwarzschild, K.: Über das Gleichgewicht der Sonnenatmosphäre. Nachr. von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Math.-phys. Klasse 195, 41–53 (1906) Schwarzschild, K.: Über Diffusion und Absorption in der Sonnenatmosphäre. Sitzungber. Acad. Wissen. Berlin 47, 1183–1200 (1914) Sobolev, V.V.: A Treatise on Radiative Transfer. Von Nostrand Company, Princeton, NJ (1963), transl. by S.I. Gaposchkin; Russian original, Transport of Radiant Energy in Stellar and Planetary Atmospheres, Gostechizdat, Moscow (1956) Wiener, N., Hopf, D.: Über eine Klasse singulärer Integralgleichungen. Sitzungsberichte der Preussichen Akademie, Mathematisch-Physikalische Klasse 3. Dez. 1931 31, 696–706 (1931) (ausgegeben 28. Januar 1932); English translation: in Fourier transforms in the Complex Domain, Paley, R.C., Wiener, N.: Am. Math. Soc. Coll. Publ. XIX, 49–58 (1934)

Chapter 4

Singular Integral Equations

The construction of exact solutions for radiative transfer equations is always based on the transformation of these equations into new ones, for which exact methods of solutions have been developed, in other branches of physics or in mathematical physics. For infinite media, as shown in Sect. 3.1, the integral equation for the source function can be transformed into an algebraic equation by performing a Fourier transform. For radiative transfer problems in a semi-infinite medium, the situation is not as simple, but they can be transformed into singular integral equations with Cauchy-type kernels for which powerful methods of solutions have been developed, such as the Hilbert transform method introduced by Carleman (1922). In this Chapter, we show how to transform a Wiener–Hopf integral equation into a singular integral equation. We present two different methods, using the equation for S(τ ) as example. We then present the main lines of the Hilbert transform method of solution, which will be applied to scalar as well as to polarized radiative transfer problems. A third method leading to a singular integral equation is described in Chap. 10. It is based on a singular eigenfunction expansion of the radiation field, introduced in Case (1960). In Sect. 4.1 we present the inverse Laplace transform method introduced by Frisch and Frisch (1982). As pointed out in Sect. 2.2, the kernel K(τ ) is a superposition of exponentials. The idea, already mentioned in Sect. 3.2.3, is to assume that the source function has a similar structure, that is can be written as 

∞

S(τ ) =

s(ν) e−ντ dν,

ν ≥ 0.

(4.1)

0

The new unknown s(ν) can be a regular function or a distribution. The same assumption is made for the primary source Q∗ (τ ). Once s(ν) has been determined, the source function itself is simply given by Eq. (4.1). The other method presented in Sect. 4.2 is more widely known (see e.g. Kourganoff 1963). Already employed in Halpern et al. (1938), it amounts to taking the direct Laplace transform of the Wiener–Hopf integral equation S(τ ). It provides © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_4

51

52

4 Singular Integral Equations

a Cauchy-type singular integral equation for s˜ (p), the Laplace transform of S(τ ). Once s˜(p) has been determined, S(τ ) can be calculated with the Fourier inversion formula. It will in general require an integration in the complex plane. We show also in this Section that s˜ (p) can be derived from the inverse Laplace transform s(ν). The advantage is that singular integral equations derived by inverse Laplace transforms are in general easier to solve than the equations obtained by a direct Laplace transform. For a full-space problem, the direct Laplace transform is equivalent to a Fourier transform. We then present in Sect. 4.3 and in Appendix A of this chapter the main lines of the Hilbert transform method of solution for singular integral equations with a Cauchy-type kernel. The properties of the dispersion function and the construction of the half-space auxiliary function X(z) are described in Chap. 5.

4.1 The Inverse Laplace Transformation Introduced in Frisch and Frisch (1982) for half-space transfer problems with complete frequency redistribution, it was then extended to monochromatic scattering in Frisch (1988b). It draws its inspiration from Sobolev’s and Ivanov’s writing of the kernel as a superposition of exponentials. It is applicable to half-space as well as full-space problems.

4.1.1 The Half-Space Case We start from the half-space integral equation 

∞

S(τ ) = (1 − )

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

τ ∈ [0, ∞[,

(4.2)

0

where the inhomogeneous term Q∗ (τ ) is a given primary sources of photons. The only restriction on Q∗ (τ ) is that the solution of Eq. (4.2) is bounded at the surface and grows less rapidly than an exponential at infinity. We now assume that all the functions in Eq. (4.2) can be represented as Laplace transforms. Thus we write 

∞

S(τ ) =

s(ν)e−τ ν dν,

τ ∈ [0, ∞[,

(4.3)

0

with s(ν) a regular or generalized function. For instance, for monochromatic scattering, s(ν) is the sum a regular function and of a Dirac distribution. For complete redistribution, distributions may also appear, but they are not a generic feature of the solution.

4.1 The Inverse Laplace Transformation

53

For Q∗ (τ ) we make a similar assumption, writing it as 

Q∗ (τ ) =

∞

q ∗ (ν)e−τ ν dν,

τ ∈ [0, ∞[.

(4.4)

0

Here again, q ∗ (ν) may involve distributions. For example, for Q∗ (τ ) = Q∗ with Q∗ a constant, one has q ∗ (ν) = Q∗ δ(ν). As for the kernel K(τ ), we recall that it may be written as 

∞

K(τ ) =

k(ν)e−|τ |ν dν,

(4.5)

0

for monochromatic scattering and complete frequency redistribution. We now introduce Eqs. (4.3) and (4.5) into Eq. (4.2) and perform the integration over τ , having first separated the integral in two parts, one spanning the range 0 ≤ τ < τ and the other one the range τ ≤ τ < ∞. We thus obtain  S(τ ) = (1 − ) + (1 − ) 0

∞

k(ν)

0 ∞  ∞



 s(ν ) −τ (ν −ν)

[1 − e e−τ ν dν ] dν

−ν ν 0  k(ν)

dν s(ν )e−τ ν dν + Q∗ (τ ), τ ∈ [0, ∞[.

ν +ν 

∞

0

(4.6)

The idea now is to rewrite the right-hand side in such a way that each term becomes a Laplace transform. The second integral has already the right form. The first integral can be separated into two integrals, leading to 



∞

∞

k(ν) 0

0

  ∞ s(ν )

−τ ν dν e dν + ν − ν 0

∞ 0

 k(ν)

dν s(ν )e−τ ν dν .

ν−ν (4.7)

These two terms have now the form of a Laplace transform. Because the integrands are singular for ν = ν , the integrals have to be defined in Cauchy Principal Value as indicated by the notation for the integral sign.1 Expressing now in Eq. (4.6) S(τ ) and Q∗ (τ ) in terms of their inverse Laplace transforms, we obtain the singular integral equation  λ(ν)s(ν) + η(ν)

∞ 0

s(ν ) dν = q ∗ (ν), ν − ν

ν ∈ [0, ∞[,

(4.8)

Cauchy Principal Value means that an infinitesimal interval [−t, +t], t → 0, centered on the singularity at ν = ν is excluded from the integral.

1

54

4 Singular Integral Equations

where  λ(ν) = 1 +

∞

η(ν )(

0

1 1 +

) dν , ν − ν ν +ν

(4.9)

and η(ν) = −(1 − )k(ν).

(4.10)

Equation (4.8) is of the type announced in Sect. 3.2.3. The Cauchy-type kernel is independent of the scattering process, which is described by the coefficients λ(ν) and η(ν). We show in Sect. 5.2 how they are related to the dispersion function, introduced in Sect. 3.1 as V (k).

4.1.2 The Full-Space Case The full-space version of Eq. (4.2) is  S(τ ) = (1 − )

+∞ −∞

K(τ − τ )S(τ ) dτ + Q(τ ),

τ ∈ R.

(4.11)

We assume that the source function can be represented by 

∞

S(τ ) = 0

S(τ ) = −



s(ν)e−τ ν dν, 0

−∞

s(ν)e−τ ν dν,

τ ∈ [0, ∞], τ ∈] − ∞, 0].

(4.12) (4.13)

A similar representation is assumed for the primary source term Q∗ (τ ). To ensure that Eq. (4.11) has a solution, the integral of Q∗ (τ ) over the full real axis must be finite. Considering the positive and negative values of τ separately and proceeding as described in Sect. 4.1, we find  λ(ν)s(ν) + η(ν)

+∞

s(ν ) dν = q(ν),

−∞ ν − ν

ν ∈ R.

(4.14)

The solution of this equation involves only the dispersion function. An example is given in Sect. 6.1.2. For infinite-space problems, one can of course use the classical Fourier transform method. The inverse Laplace transform is alternative technique, which avoids the Fourier inversion. It may be preferred.

4.2 The Direct Laplace Transformation

55

4.2 The Direct Laplace Transformation The idea of considering the Laplace transform of the integral equation for the source function S(τ ) can be traced back to Halpern et al. (1938). The determination of the source function by a Laplace transformation is a natural approach, since the Laplace transform of the source function yields the emergent intensity at the surface of a semi-infinite medium (see Chap. 2). The Laplace transform of the integral equation for S(τ ) is a singular integral equation with a Cauchy-type kernel, as we now show. In subsequent chapters we solve these type of equations by the Hilbert transform method, described below in Sect. 4.3. In Halpern et al. (1938), it was solved by a factorization method à la Wiener–Hopf. We introduce the Laplace transforms 

∞

s˜ (p) =

S(τ ) e−pτ dτ,

q˜ ∗ (p) =

0



∞

Q∗ (τ ) e−pτ dτ,

(4.15)

0

with p real and positive. For K(τ ), we use the representation as an inverse Laplace transform written in Eq. (4.5) with the lower bound of the integral set to νl , with νl = 0 for complete frequency redistribution and νl = 1 for monochromatic scattering. The Laplace transform of Eq. (4.2) may be written as  s˜ (p) = (1 − )

∞ ∞ ∞

0

0



k(ν)S(τ ) e−ν|τ −τ | e−pτ dν dτ dτ + q(p). ˜

(4.16)

νl

Following a standard method (Kourganoff 1963, p. 67), we separate the integral over τ in two intervals, [0, τ ] and [τ , ∞[. We thus obtain 

∞

s˜ (p) = (1 − )

× e





∞

S(τ ) 0

k(ν) νl

−ντ 1 − e

−(p−ν)τ

p−ν

+e

ντ e

−(p+ν)τ

p+ν

 dν dτ + q˜ ∗ (p).

(4.17)

We separate the first term in the square bracket in two parts and introduce Cauchy Principal Values to avoid divergences at ν = p. We thus obtain 

∞ ∞

k(ν)

S(τ )e−ντ dτ dν p − ν νl 0  ∞  ∞ 1 1

− ] dν + q˜ ∗ (p). + (1 − ) S(τ )e−pτ dτ

k(ν)[ p + ν p − ν 0 νl s˜ (p) = (1 − )

(4.18)

56

4 Singular Integral Equations

Introducing η(ν) = −(1 − )k(ν), we see that Eq. (4.18) may be written as  λ(p)˜s (p) −

∞ η(ν)˜ s (ν) νl

ν −p

dν = q(p), ˜

p ≥ 0.

(4.19)

Equation (4.19) is a singular integral equation with a Cauchy-type kernel. The integral operator is adjoint to the integral operator in Eq. (4.8). In Sect. 4.3 we describe a Hilbert transform method of solution for Eq. (4.8). It can be applied to Eq. (4.19) as well. Examples are given in the following chapters. Once s˜ (p) has been determined, a Laplace inversion is needed to obtain the source function S(τ ) itself. The Laplace inversion formula is 1 ω→∞ 2iπ



c+iω

S(τ ) = lim

s˜ (p) epτ dp,

(4.20)

c−iω

where c and ω are real. We can take c = 0, since s˜(p) is defined for all p > 0. To calculate S(τ ), the integration along the imaginary axis is included into a closed contour in the complex plane. For τ ∈ [0, ∞[, the contour must lie in the halfplane (p) < 0, to insure the convergence of the integral. It must turn around the singularities of s˜ (p). The form of S(τ ) is determined by the position and nature of these singularities. For example, when s˜ (p) has a simple pole at p = −ν0 , S(τ ) will contain a term proportional to e−ν0 τ . A branch cut on the negative real axis will yield a real Laplace integral. In fine, one obtains a real Laplace integral, similar to Eq. (4.3), but involving the Laplace transform q˜ ∗ (p) of the primary source. Examples of Laplace inversions are described in Sects. 6.1.1.1 and 7.2.2. We have seen that the inverse Laplace transform and the direct Laplace transform of a same Wiener–Hopf integral equation lead to two singular integral equations with adjoint integral operators. One may expect that their solutions will be closely connected. Indeed, combining the definitions of s˜ (p) and of s(ν) given in Eqs. (4.15) and (4.3), one immediately finds  s˜ (p) = 0

∞

s(ν) dν. ν+p

(4.21)

This relation provides s˜ (p) in terms of the inverse Laplace transform q ∗ (ν) of the ˜ primary source. It is used in Sect. 6.3, to determine G(p), the Laplace transform of the surface Green function. Indeed, it is simpler to solve the singular integral equation for (ν), the inverse Laplace transform of the resolvent function (τ ), ˜ than the singular equation for G(p).

4.3 The Hilbert Transform Method of Solution

57

4.3 The Hilbert Transform Method of Solution We present here the Hilbert transform method introduced by Carleman (1922) for the solution of singular integral equations with Cauchy-type kernels. A general mathematical presentation of this family of equations can be found in, e.g., Muskhelishvili (1953), Gakhov (1966), Estrada and Kanwal (1987). Here we restrict our attention to the equations arising in radiative transfer, or other transport problems. They are simpler than the most general equations considered in Muskhelishvili (1953), but the concepts that are introduced to construct their solution are quite general. The main idea of the method is to transform a singular integral equation with a Cauchy-type kernel into a boundary value problem in the complex plane, also known as a Riemann-Hilbert problem. For half-space problems, the solution will involve an auxiliary function, solution of a boundary problem, introduced in Eq. (3.15). Several other methods of solution have been introduced for these singular integral equations. For example, Halpern et al. (1938) apply a Wiener–Hopf factorization method. Van der Mee and Zweifel (1990) have introduced a method based on orthogonality relations, which is somewhat similar to that of Mc Cormick and Kušˇcer (1973). This method avoids the introduction of Hilbert transforms, but the auxiliary function is still introduced as the solution of a Riemann–Hilbert problem. In Sects. 4.1 and 4.2 we have shown how to derive a singular integral equation from a Wiener–Hopf integral equation. An inverse Laplace transform leads to an equation having the form  α(ν)f (ν) + β(ν)

f (ν ) dν = q(ν),

Lν −ν

ν ∈ L,

(4.22)

ν ∈ L.

(4.23)

and a direct Laplace transform to  α(ν)f (ν) −

β(ν )f (ν )

dν = h(ν),

L ν −ν

The unknown is the function f (ν), the integration contour L is an interval on the real axis, which can be of infinite length. The coefficients α(ν), β(ν) are are functions of the kernel K(τ ), and q(ν) and h(ν) are Laplace transforms (inverse or direct) of the primary source. Equations (4.23) and (4.22) are adjoints to each other. We now describe the transformation of Eq. (4.22) into a boundary value problem in the complex plane by a Hilbert transform method. With minor changes it can be applied to Eq. (4.23) (see Appendix C in Chap. 6). The method is based on the introduction of the Hilbert transform of f (ν) : 1 F (z) ≡ 2i π

 L

f (ν) dν, ν−z

(4.24)

58

4 Singular Integral Equations

| . This Hilbert transform is a where z belongs to the complex plane, denoted here C Cauchy integral, the properties of which are described in Appendix A of this chapter. We summarize them here :

• (i) F (z) is analytic in the complex plane cut along the contour L, that is in | \L. shorthand notation, analytic in C • (ii) F (z) satisfies the Plemelj formulae : F + (ν) − F − (ν) = f (ν),  f (ν ) 1 dν , F + (ν) + F − (ν) = i π L ν − ν

(4.25) (4.26)

where F ± (ν) are the limiting values of F (z) on each side of the cut L (see the definition in Eq. (A.4)). Equations (4.25) and (4.26) will be referred to as the first and second Plemelj formula respectively. • (iii) F (z) tends to zero at infinity. • (iv) F (z) has only mild (integrable) singularities near the end points of L. Using the Plemelj formulae, Eq. (4.22) can be recast as F + (ν)[α(ν) + i πβ(ν)] − F − (ν)[α(ν) − i πβ(ν)] = q(ν),

ν ∈ L.

(4.27)

This equation describes a boundary value problem in the complex plane. An important feature of this equation, which determines the method of solution, is that it resembles the first Plemelj formula, given in Eq. (4.25). When Eq. (4.27) is a Plemelj formula, one readily finds F (z) and then f (ν) by using Eq. (4.25). This is a special case, which is encountered when the transfer occurs in an infinite medium. When Eq. (4.27) is not a Plemelj formula, it can be transformed into a Plemelj formula by the introduction of an auxiliary function, namely the function X(z) introduced in Eq. (3.15). This is the general situation for all half-space problems. We now show how to construct F (z) in the special and general cases.

4.3.1 A Special Case The functions [α(ν) + i πβ(ν)] and [α(ν) − i πβ(ν)] are the limiting values  + (ν) | \L. Introducing and  − (ν) above and below the cut L of a function (z) analytic C the Hilbert transform  q(ν) 1 dν, (4.28) Q(z) ≡ 2i π L ν − z | \L, we can rewrite Eq. (4.27) as which is analytic in C

[F + (ν) + (ν) − Q+ (ν)] − [F − (ν) − (ν) − Q− (ν)] = 0.

(4.29)

4.3 The Hilbert Transform Method of Solution

59

The general solution of this equation has the form F (z)(z) − Q(z) = P(z),

(4.30)

where P(z) is an entire function. Indeed. the left-hand side is by construction | \L and, according to Eq. (4.29), its jump across L is zero. We recall analytic in C that an entire function is an analytic function of the complex plane with no poles, nor singularities, nor cuts. It takes the form of a polynomial. According to the Liouville theorem (Carrier et al. 1966), the behavior at infinity of an entire function is sufficient to determine the function.2 The behavior of P(z) at infinity is in general not difficult to find. Many examples are given in the following chapters. Knowing P(z), Q(z) and (z), one can then calculate F (z). The first Plemelj formula yields f (ν) = F + (ν) − F − (ν). A full-space transport problem belonging to this special case is treated in Sect. 6.1.

4.3.2 The General Case The functions [α(ν) + i πβ(ν)] and [α(ν) − i πβ(ν)] are not the limiting values of a | \L. The trick is to transform Eq. (4.27) into a Plemelj formula. function analytic in C The first step is a division of Eq. (4.27) by [α(ν) − i πβ(ν)], assumed to be nonzero. When α(ν) and β(ν) are real functions of the real variable ν, this assumption is equivalent to the statement that α(ν) and β(ν) have no common zeroes for ν ∈ L. The form taken by Eq. (4.27) after this division suggests to introduce an auxiliary | \L and satisfying the jump condition function X(z), analytic in C X+ (ν) = W (ν)X− (ν),

ν ∈ L,

(4.31)

with W (ν) =

α(ν) + i πβ(ν) . α(ν) − i πβ(ν)

(4.32)

The function W (ν) is a known function, which for radiative transfer depends only on the dispersion function, and X+ (ν) and X− (ν) are the limiting values of X(z) along the cut. Some additional conditions described in Sect. 5.3 have to be imposed on X(z). Equation (4.31) is the homogeneous Riemann–Hilbert problem associated

2

The Liouville theorem states that a function, which is analytic and bounded in the full complex plane, is a constant. Thus, when an analytic function is bounded and vanishes at infinity it vanishes in the full complex plane.

60

4 Singular Integral Equations

to Eq. (4.27). It is the defining relation for X(z). Equation (4.27) can now be written as F + (ν)X+ (ν) − F − (ν)X− (ν) = γ (ν),

ν ∈ L,

(4.33)

where γ (ν) =

q(ν)X− (ν) , α(ν) − i πβ(ν)

ν ∈ L.

(4.34)

The left-hand side is now in the form of the jump across L of the function F (z)X(z). The idea now is to write the right-hand side in the same form. The first Plemelj formula shows that it suffices to introduce the Hilbert transform  1 γ (ν) G(z) = dν, (4.35) 2i π L ν − z | \L. Equation (4.33) becomes which is analytic in C

[F + (ν)X+ (ν) − G + (ν)] − [F − (ν)X− (ν) − G − (ν)] = 0,

ν ∈ L.

(4.36)

Its general solution has the form F (z)X(z) − G(z) = P(z),

(4.37)

| \L and where P(z) is an entire function. Indeed the left-hand side is analytic in C according to Eq. (4.36) its jump across L is zero. The condition that F (z) tends to zero as 1/z as z → ∞, combined with the known behaviors of X(z) and G(z) for z → ∞, allows one to determine the behavior at infinity of P(z) and hence P(z) itself by the Liouville theorem. The solution of Eq. (4.22) is then given by f (ν) = F + (ν) − F − (ν). The construction method sketched above is applied in the following chapters to several singular integral equations such as Eqs. (4.22) and (4.23). For some problems (an example is given in Sect. D.1), the function q(ν) contains a free parameter. The existence of a solution with the correct behavior at infinity may require some conditions on G(z), known as solvability conditions, which provide a full determination of q(ν). In the general theory of singular integral equations with Cauchy-type kernels (see e.g. Muskhelishvili 1953; Gakhov 1966), ν and ν are complex variables belonging to a contour L of the complex plane and the coefficients α(ν), β(ν) and q(ν) are Hölder continuous functions.3 The solution f (ν) is sought in the same functional space. The Hölder continuity condition can be weakened, in particular for the inhomogeneous term q(ν), some integrability condition being sufficient to ensure

3

Hölder continuity implies continuity but not necessarily differentiability.

4.3 The Hilbert Transform Method of Solution

61

the existence of solutions (see Case and Zweifel 1967, Estrada and Kanwal 1987). We show in Sect. A.3 that it is possible to define a Hilbert transform for a δ-function.

4.3.3 Application to Radiative Transfer In Sect. 4.1, we show that s(ν), the inverse Laplace transform of the source function S(τ ), satisfies  λ(ν)s(ν) + η(ν)

s(ν ) dν = q ∗ (ν),

Lν − ν

ν ∈ [0, ∞[.

(4.38)

For an infinite medium, L =] − ∞, +∞[ and for a semi-infinite one L = [0, ∞[. The coefficients of the equation are 

∞

λ(ν) = 1 +

η(ν )(

0

1 1 +

) dν , ν − ν ν +ν

(4.39)

and η(ν) = −(1 − )k(ν).

(4.40)

The properties of the scattering process are contained in the coefficients λ(ν) and η(ν), themselves depending only on the dispersion function, as we now show. ˜ The dispersion function is introduced in Eq. (3.6) as V (k) = 1 − (1 − )K(k), ˜ where K(k) is the Fourier transform of the kernel. Expressing K(τ ) in terms of its inverse Laplace transform k(ν), and setting k = z, z complex, we obtain 

∞

V (z) = 1 − (1 − )

k(ν)( 0

1 1 + ) dν. ν + iz ν − iz

(4.41)

Changing i z to z, we can also write the dispersion function as 

∞

L(z) ≡ 1 − (1 − )

k(ν)( 0

1 1 + ) dν. ν−z ν +z

(4.42)

Comparing the expression of L(z) with that of λ(ν), we see that L(z) is the analytic continuation in the complex plane of λ(ν). We use V (z) in Chap. 12 on the Wiener– Hopf method, but elsewhere we use L(z). Equation (4.42) shows that L(z) is a sum of two Hilbert transforms. The integrals with 1/(ν − z) and 1/(ν + z) being singular along the positive real axis and the negative real axis, respectively, L(z) is analytic in the complex plane, cut along the full real axis. Actually this statement is valid for complete frequency redistribution, only. For monochromatic scattering, k(ν) = 0 for ν ∈ [0, 1]. The lower bound of

62

4 Singular Integral Equations

the integral in the expression of L(z) is now 1. As a consequence, L(z) is analytic in the complex plane cut along the semi-infinite lines ] − ∞, −1] and [1, +∞[. Monochromatic scattering is considered at the end of this section. We now pursue the discussion with complete frequency redistribution. For ν ∈ [0, ∞[ , the Plemelj formulae (see Eqs. (4.25) and (4.26)) lead to L+ (ν) − L− (ν) = −2i π(1 − )k(ν),   ∞ + −

k(ν ) L (ν) + L (ν) = 2 1 − (1 − ) 0

1 1 +

ν − ν ν +ν



(4.43)

dν , (4.44)

where L± (ν) are the limiting values of L(z) above and below the cut [0, ∞[. Similar formulae hold for negative values of ν (see Eq. (5.24)). We thus obtain: L± (ν) = λ(ν) ± i πη(ν),

ν ∈ [0, ∞[.

(4.45)

The comparison of Eqs. (4.22) and (4.38) shows that α(ν) ± i πβ(ν) = λ(ν) ± i πη(ν).

(4.46)

Therefore, L(z) is the function introduced as (z) in Sect. 4.3.1. To solve Eq. (4.38) for s(ν), we introduce the Hilbert transform S(z) ≡

1 2i π

 L

s(ν) dν, ν −z

(4.47)

| \L). analytic in the complex plane cut along the contour L (analytic in C When L is the full real axis, S(z) and L(z) have the same analyticity domains and s(ν) can be constructed with the method described in Sect. 4.3.1. There is no need to introduce an auxiliary function. | \] − ∞, +∞[ while S(z) When L = [0, ∞[, the function L(z) is analytic in C | is analytic in C\[0, ∞[. Their domains of analyticity being different, we are in the general case considered in Sect. 4.3.2. An auxiliary function has to be introduced. It can be derived from the Riemann-Hilbert problem in Eq. (4.31), provided α(ν) and β(ν), that is λ(ν) and η(ν), have no common zeroes on the cut [0, ∞[. For complete frequency redistribution, this condition is satisfied (see Sect. 5.2) and we can define the auxiliary function X(z) as in Eq. (4.31), with

W (ν) =

L+ (ν) , L− (ν)

ν ∈ [0, ∞[.

(4.48)

The construction of X(z) is described in Sect. 5.3. The Riemann–Hilbert problem was introduced by Riemann with W (ν), ratio of two imaginary conjugate function as in Eqs. (4.32) and (4.48). This restriction was raised by Hilbert (Pandey 1996, p. 57). All the Riemann–Hilbert problems encountered in this book arise from singular

A.1 Definition and Properties

63

integral equations with real coefficients such as Eq. (4.22), therefore W (ν) is always the ratio of two imaginary conjugate functions. For monochromatic scattering, k(ν) = 0, for ν ∈ [0, 1[, and λ(ν) has two zeroes at ν = ±ν0 , ν0 ∈ [0, 1[ (see Sect. 5.2.2). Therefore λ(ν) and η(ν) have a common zero at ν = ν0 . Equation (4.38) can nonetheless be solved by the Hilbert transform method. It suffices, as shown in Sect. 6.2.2, to consider separately the intervals ν ∈ [0, 1] and ν ∈ [1, ∞[. The integration in Eq. (4.38) being along [0, ∞[, the two solutions are coupled by a parameter, which is determined by a solvability condition (see the discussion in Sect. 4.3.2). To obtain the full solution, it suffices to add the contributions from the two intervals. The contribution from the interval ν ∈ [0, 1] contains distributions, such as the Dirac delta function. Distributional solutions are also encountered in the singular eigenfunction expansion method presented in Chap. 10. Other examples of distributional solutions for singular integral equations with Cauchy-type kernels can be found in, e.g., Estrada and Kanwal (1987), Frisch (1988b), Frisch et al. (1990).

Appendix A: Hilbert Transforms A.1 Definition and Properties The Hilbert transform F (z) of a function f (u) is defined by F (z) ≡

1 2i π

 L

f (u) du, u−z

(A.1)

where L is a line in the complex plane, z a point in the complex plane, not belonging to L and f (u) a complex-valued function defined on L. The line L can be an arc or a closed contour. The integral in Eq. (A.1) is a Cauchy integral. These integrals are well known to mathematicians and their properties have been studied in great detail by Muskhelishvili (1953), when f (u) is a Hölder continuous function. Cauchy integrals are treated in detail in many textbooks dealing with functions of a complex variable, such as Gakhov (1966), Pogorzelski (1966), Carrier et al. (1966), Roos (1969), Dautray and Lions (1984, vol. 6), Ablowitz and Fokas (1997) and also in books devoted to transport theory, such as Noble (1958), Case and Zweifel (1967), Duderstadt and Martin (1979), to name only a few. We recall here some properties that are in frequent use in this book. • F (z) is analytic5 in the complex plane cut along the line L. The existence of a cut implies that F (z) has different values on each side of the cut, say F + (u) and

Analytic in a domain D of the complex plane means differentiable for all points z in D . An extended definition leaves open the possibility of a finite number of singular points.

5

64

4 Singular Integral Equations

F − (u) and to go from one side of the cut to the other, it is necessary to circulate around the cut. • The limiting values, F + (u) and F − (u) satisfy the Plemelj formulae (also called the Sokhotskii–Plemelj formulae). They state that F + (u) − F − (u) = f (u),  f (u ) 1 du , F + (u) + F − (u) = i π L u − u

(A.2) (A.3)

where means that the integral is taken in Cauchy Principal Value. Here, L will in general be an interval on the real axis, oriented from −∞ to +∞. Then F + (u) is the limit from above, while F − (u) is the limit from below, that is F ± (u) ≡ lim F (u ± i η), η→0

u and η real,

η > 0.

(A.4)

For an arbitrary arc L, the limiting values are defined with respect to the outside and the inside of a closed oriented contour including the arc L. A simplified derivation of Plemelj formulae is presented in Sect. A.2 and a verification that they hold also when f (u) is a Dirac distribution in Sect. A.3. • F (z) tends to zero at infinity. How fast it tends to zero depends on L and f (u). For instance, when f (u) is integrable, F (z) tends to zero as 1/z. When the integral of f (u) is zero, then F (z) tends to zero as 1/z2 . • F (z) has a mild (integrable) singularity at the end point(s) of L. This means that for a point z in the neighborhood of the extremity a of the curve L, but not on L, we have |F (z)| ≤ C/|z − a|α ,

0 ≤ α < 1,

(A.5)

where C is a constant. When f (a) has a non zero value, then F (z) has a logarithmic singularity, which does satisfy this condition.

A.2 The Plemelj Formulae Rigorous proofs can be found in most of the references listed above. Carrier et al. (1966) is recommended. Here we give only the main lines of the proof. Also to simplify we assume that L is an interval [a, b] on the real axis and that f (u) is real.

A.2 The Plemelj Formulae

L a

65

F +(u) u

z u ρ

b

L a

u F − (u )

ρ

u

b

z Fig. A.1 Integration contour for the Plemelj formulae. The contribution from the singular point is obtained by letting ρ → 0

We consider a point z above the real axis approaching the point u on the line L. We assume that f (z) is analytic at the point u and deform the integration line L, as shown in Fig. A.1. We denote by ρ the radius of the half-circle centered on u, lying below the real axis. Along this contour, Eq. (A.1) becomes  u−ρ  b 1 f (u ) f (u )

du du

+ F (z) =

2i π a u − z u+ρ u − z   2π f (u + ρei θ ) i θ + (A.6) ρe i dθ . u + ρei θ − z π Letting z → u and ρ → 0, replacing f (u + ρei θ ) by f (u) in the last integral, and performing the integration over θ , we obtain F + (u) =

1 1 f (u) + 2 2i π



 f (u )

du .

au −u b

(A.7)

We stress that the integral over [a, b] is a Principal Value integral. We now consider a point z lying below L and deform the integration line as shown in Fig. A.1. The half-circle lies above the real axis. Proceeding exactly as above, we obtain 1 1 F − (u) = − f (u) + 2 2i π



 f (u )

du .

au −u b

(A.8)

The negative sign in the first term comes from the variation of θ from π to 0. Combining Eqs. (A.7) and (A.8), we obtain the Plemelj formulae (A.2) and (A.3).

66

4 Singular Integral Equations

A.3 The Plemelj Formulae for a Dirac Distribution Although the Dirac distribution δ is not a Hölder continuous function, one can still define a Hilbert transform and establish the Plemelj formulae. The subject of Hilbert transform for Schwartz distributions in treated in Pandey (1996). Here we give the example of the Dirac distribution. We consider F (z) =

1 2i π



b a

δ(u) du. u−z

(A.9)

We assume that the interval [a, b] contains the origin. We consider a point z = u + i ,  > 0 (located above [a, b]). We have F (u + i ) =

1 2i π



b a

1 1 δ(u ) du = − .

u − (u + i ) 2i π u + i 

(A.10)

Similarly, for a point z = u − i , located below the interval [a, b] we have F (u − i ) =

1 2iπ



b a

1 δ(u ) 1 du = − .

u − (u − i ) 2i π u − i 

(A.11)

Thus 1  2 π u + 2 u 1 . F (u + i ) + F (u − i ) = − 2 i π u + 2

F (u + i ) − F (u − i ) =

(A.12) (A.13)

In the limit  → 0, the right-hand side in Eq. (A.12) tends to the Dirac distribution and the right-hand side in Eq. (A.13) to −(1/i π)(P /u), where P stands for the Cauchy Principal Value. Hence, we obtain F + (u) − F − (u) = δ(u) F + (u) + F − (u) = −

1 P . iπ u

(A.14) (A.15)

We can rewrite these equations as F + (u) =

1 P 1 [δ(u) − ] ≡ δ − (u), 2 iπ u

− F − (u) =

1 P 1 [δ(u) + ] ≡ δ + (u). 2 iπ u

(A.16) (A.17)

References

67

The distributions δ + and δ − are defined in such a way that δ + + δ − = δ. They are of frequent use in Quantum Mechanics. They are related to the Fourier transforms (k) and Y − (k) of the Heaviside function Y (x) and Y− (x) ≡ Y (−x). Using Y lim Yˆ (k + i ) =

→0



∞

ei (k+i )x dx,

k real,

 > 0,

(A.18)

0

− (k), we obtain and a similar definition for Y (k) = 2πδ − (k), Y

− (k) = 2πδ + (k). Y

(A.19)

References Ablowitz, M.J., Fokas, A.S.: Complex Variables, Introduction and Applications. Cambridge University Press, Cambridge (1997) Carleman, T.: Sur la résolution de certaines équations intégrales. Ark. Mat. Astr. Fys. 16, 1–19 (1922) Carrier, G.F., Krook, M., Pearson, C.E.: Functions of a Complex Variable. McGraw-Hill Book Company, New York (1966) Case, K.M.: Elementary solutions of the transport equation and their applications. Ann. Phys. (New York) 9, 1–23 (1960) Case, K.M., Zweifel, P.: Linear Transport Theory. Addison-Wesley, Reading, MA (1967) Dautray, R., Lions, J.-L.: Analyse Mathématique et calcul numérique pour les sciences et les techniques. Masson, Paris (1984); 1984–1985 (3 Vols.); 1988 (9 Vols.); English edition: Mathematical Analysis and Numerical Methods for Science and Technology (6 Vols.). Springer, Berlin (1990) Duderstadt, J.J., Martin, W.R.: Transport Theory. Wiley, New York (1979) Estrada, R., Kanwal, R.P.: The Carleman type singular integral equations. SIAM Rev. 29, 263–290 (1987) Frisch, H.: A Cauchy integral equation method for analytic solutions of half-space convolution equations. J. Quant. Spectrosc. Radiat. Transf. 39, 149–162 (1988b) Frisch, H., Frisch, U.: A method of Cauchy integral equation for non-coherent transfer in halfspace. J. Quant. Spectrosc. Radiat. Transf. 28, 361–375 (1982) Frisch, H., van der Mee, C.V., Zweifel, P.: Distributional solutions of singular integral equations. J. Integral Equa. Appl. 2, 205–210 (1990) Gakhov, F.D.: Boundary Value Problems. Pergamon Press, London (1966); translation by I.N. Sneddon of the 2nd Russian edition (1963). Fizmatgiz, Moscow Halpern, O., Lueneburg, R., Clark, O.: On multiple scattering of neutrons I. Theory of the albedo of a plane boundary. Phys. Rev. 53, 173–183 (1938) Kourganoff, V.: Basic Methods in Transfer Problems. Radiative Equilibrium and Neutron Diffusion. Dover Publications, New York (1963); First edition: Oxford University Press, London (1952) Mc Cormick, N.J., Kušˇcer, I.: Singular eigenfunction expansions in neutron transport theory. In: Henley, E.J., Lewins, J. (eds.) Advances in Nuclear Science and Technology, vol. 7, pp. 181– 282. Academic Press, New York (1973) Muskhelishvili, N.I.: Singular Integral Equations, Noordhoff, Groningen (based on the second Russian edition published in 1946) (1953); Dover Publications (1991) Noble, B.: Methods Based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, New York (1958)

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Pandey, J.N.: The Hilbert Transform of Schwartz Distributions and Applications. Wiley Inc., New York (1996) Pogorzelski, W.: Integral Equations and their Applications. Pergamon Press, New York (1966) Roos, B.W.: Analytic Functions and Distributions in Physics and Engineering. Wiley, New York (1969) Van der Mee, C.V.M., Zweifel, P.F.: Application of orthogonality relations to singular integral equations. J. Integral Equa. Appl. 2, 185–203 (1990)

Chapter 5

The Scattering Kernel and Associated Auxiliary Functions

In the preceding chapters, we have introduced the scattering kernel K(τ ), the dispersion function, defined as V (k) or L(z), and a half-space auxiliary function X(z), needed to solve radiative transfer equations in a semi-infinite medium. In this chapter we analyze in detail the properties of these functions, which all play a role in the build up of a radiation field by multiple scatterings. Section 5.1 is devoted to the scattering kernel K(τ ) and Sect. 5.2 to the dispersion function L(z). In Sect. 5.3, we show how to construct the function X(z). Its main properties and its relation with the H -function are described in Sect. 5.4 and in Appendix B of this chapter. In each section we consider monochromatic scattering and complete frequency redistribution. Some additional properties of these functions, specific to the case  = 0, are established in Chap. 9, devoted to conservative scattering.

5.1 The Kernel and Its Inverse Laplace Transform The kernel K(τ ), which characterizes the Wiener–Hopf integral equations introduced in Chap. 2, may be written as 1 2



1

dμ , μ

(5.1)

  ϕ(x) dμ dx, ϕ 2 (x) exp −|τ | μ μ

(5.2)

K(τ ) ≡

0

e−|τ |/μ

for monochromatic scattering, and as K(τ ) ≡

1 2



+∞  1 −∞

0

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_5

69

70

5 The Scattering Kernel and Associated Auxiliary Functions

for complete frequency redistribution. In both cases the kernel can be written as 

∞

K(τ ) =

k(ν)e−|τ |ν dν.

(5.3)

0 ≤ ν < 1, ν ≥ 1.

(5.4)

ν ∈ [0, ∞[.

(5.5)

0

For monochromatic scattering,  k(ν) =

0 1/2ν

For complete frequency redistribution, k(ν) =

1 1 g( ), ν ν

The function g(ξ ) (ξ = 1/ν) is defined by  g(ξ ) =

∞

ϕ 2 (v) dv,

(5.6)

y(ξ )

where  y(ξ ) =

0

0 < ξ ≤ 1/ϕ(0),

ϕ (1/ξ )

ξ ≥ 1/ϕ(0).

−1

(5.7)

The variations of k(ν) and g(ξ ) are plotted in Figs. 5.1 and 5.2. Although, k(ν) is introduced for positive values of ν, Eqs. (5.4) and (5.5) show that k(ν) can also be defined for negative values of ν and that it is an odd function of ν. For ν → ∞, k(ν)  1/ν for monochromatic scattering and also for complete frequency redistribution. For monochromatic scattering, this behavior is readily deduced from Eq. (5.4). The same behavior holds for complete frequency redistribution, since g(ξ ) tends to a constant as ξ → 0 (see Fig. 5.1). More interesting and more important is the behavior of k(ν) for complete frequency redistribution as ν → 0, because it controls the large-τ behavior of K(τ ). The asymptotic behaviors of k(ν) for ν → 0 and g(ξ ) for ξ → ∞ and also K(τ ) for τ → ∞ are established in Appendix F.1 of Chap. 9. At leading order, gD (ξ ) 

1 1 , √ 4ξ 2 ln ξ

ξ → ∞,

kD (ν) 

1 ν √ , 4 − ln ν

ν → 0,

(5.8)

and 1 a 1/2 gV (ξ )  √ , 3 π ξ 3/2

ξ → ∞,

1 kV (ν)  √ a 1/2ν 1/2 , 3 π

ν → 0,

(5.9)

5.1 The Kernel and Its Inverse Laplace Transform

71

Fig. 5.1 The function g(ξ ) for the Doppler profile and the Lorentz profile, 1/(π(1+x 2 )), in linear scales in the upper panel and in log-log scales in the lower panel. The value of g(ξ ) is constant for ξ ≤ 1/ϕ(0). The algebraic behavior of g(ξ ) for ξ → ∞ given in Eqs. (5.8) and (5.9) can be observed in the lower panel. The Voigt profile has the same algebraic behavior as the Lorentz profile

where a is the Voigt parameter of the line (see Eq. (2.26)). The subscript D stands for Doppler and V for Voigt. One can then easily infer that K(τ ) decreases algebraically at infinity as KD (τ ) 

1 √

4τ 2 ln τ

,

KV (τ ) 

a 1/2 , 6τ 3/2

τ → ∞.

(5.10)

72

5 The Scattering Kernel and Associated Auxiliary Functions

Fig. 5.2 The function k(ν) for the Doppler profile and the Lorentz profile, 1/(π(1 + x 2 )), in linear scales in the upper panel and in log-log scales in the lower panel. The angular point is located at ν = ϕ(0). For the Doppler and Lorentz profiles, k(ν) tends to zero as 1/ν for ν → ∞. For ν → 0, one can observe in the lower panel the algebraic behaviors given in Eqs. (5.8) and (5.9). The Voigt profile has the same algebraic behaviors as the Lorentz profile

We show in Figs. 5.1 and 5.2, g(ξ ) and k(ν) in linear scales and log-log scales, where the asymptotic behaviors are easily observed. The large τ algebraic behavior of the elements of KD (τ ) can be observed in Fig. 14.1. Finally we recall that the kernel is normalized to unity, that is 

+∞ −∞

K(τ ) dτ = 1.

(5.11)

5.2 The Dispersion Function

73

This normalization implies 

∞

k(ν) 0

1 dν = . ν 2

(5.12)

5.2 The Dispersion Function We have given in Sect. 4.3.3 an expression of the dispersion function, well adapted to the solution of singular integral equations with Cauchy-type kernels, namely  L(z) ≡ 1 − (1 − )

∞

k(ν)( νl

1 1 + ) dν. ν−z ν +z

(5.13)

For monochromatic scattering νl = 1 and for complete frequency redistribution νl = 0. We analyze here the properties of L(z). Those of V (z) = L(i z) are straightforwardly obtained with a rotation of π/2. For monochromatic scattering, L(z) has an explicit expression. Using νl = 1 and k(ν) = 1/2ν, Eq. (5.13) can be written as L(z) = 1 −

1− 2z



∞

( 1

1 1 − ) dν, ν−z ν+z

(5.14)

and, after integration over ν, as L(z) = 1 −

1− 1+z ln . 2z 1−z

(5.15)

In Astronomy, for monochromatic scattering, and in the neutron transport literature, the dispersion function is in general written in terms of 1/z. For z real, this amounts to write the dispersion function in terms of the direction variable μ, with μ ∈ [−1, +1]. We first discuss the analyticity and symmetry properties, which are easily derived from the definition. We then discuss other important properties, not so easy to establish, namely the number of zeroes and the so-called index.

5.2.1 Symmetries and Analyticity Properties • The dispersion function is an even function of z, i.e. L(z) = L(−z).

(5.16)

74

5 The Scattering Kernel and Associated Auxiliary Functions

• For z = 0, L(0) = .

(5.17)

This result is readily deduced from the normalization of k(ν) given in Eq. (5.12). The case  = 0 is discussed in Sect. 5.2.2. • At infinity, lim L(z) = 1.

(5.18)

z→∞

• The analyticity properties of L(z) were already considered in Sect. 4.3.3. We recall that L(z)

| \ ] − ∞, −ν ] ∪ [+ν , +∞[. analytic in C l l

(5.19)

The notation ∪ is the logic notation for union. The branch cuts of L(z) are shown in Fig. 5.3. • The limiting values of L(z) along the positive branch cut, ν ∈ [νl , ∞[, are defined by L± (ν) ≡ lim L(ν ± i ξ ),

ξ real,

ξ →0

ξ > 0.

(5.20)

(See Fig. 5.3.) • The Plemelj formulae lead to L± (ν) = λ(ν) ± i πη(ν),

ν ∈ [νl , ∞[,

(5.21)

ξ

L+(ν) = lim L(ν + i ξ) ξ→0

−1

ν

+1 −

L (ν) = lim L(ν − i ξ) ξ→0

Fig. 5.3 Branch cuts of the monochromatic dispersion function. The solid black lines on the real axis show the branch cuts of L(z) . For complete frequency redistribution, L(z) is singular along the full real axis, except at the origin

5.2 The Dispersion Function

75

with η(ν) = −(1 − )k(ν),

(5.22)

and  λ(ν) = 1 +

∞ νl

η(ν )(

1 1 +

) dν . ν − ν ν +ν

(5.23)

• For negative values of ν, L± (−ν) = L∓ (ν),

ν ∈ [νl , ∞[.

(5.24)

Indeed, L(z) being an even function of z, one has L(−x − i y) = L(x + i y), x and y real. Taking the limits y → 0+ and y → 0− , we obtain Eq. (5.24) (the symbols 0+ and 0− mean that y tends to zero keeping positive and negative values respectively). • The limiting values L+ (ν) and L− (ν) are complex conjugate, a consequence of the fact that λ(ν) and η(ν) are real. They can be written as L+ (ν) = |L+ (ν)| exp[i θ (ν)],

L− (ν) = |L+ (ν)| exp[−i θ (ν)].

(5.25)

We choose θ (ν) = 0 at infinity, for both monochromatic scattering and complete frequency redistribution. We recall that L(z) = 1 for z → ∞. According to Eq. (5.21), [L+ (ν)] = λ(ν) and [L+ (ν)] = πη(ν), hence we can write  θ (ν) = arctan

 πη(ν) . λ(ν)

(5.26)

We choose the determination of arctan such that − π ≤ θ (ν) ≤ 0.

(5.27)

• For monochromatic scattering, λ(ν) is shown in Fig. 5.4. It has the expression λ(ν) = 1 −

1− ν +1 ln , 2ν |ν − 1|

ν ∈ [0, ∞[.

(5.28)

One can observe that λ(ν) becomes infinite for ν = ±1 and that it has two symmetric zeroes in the interval [−1, +1]. They give rise to the zeroes of L(z) discussed in Sect. 5.2.2. For ν → ∞, λ(ν) tends to 1. An asymptotic expansion of Eq. (5.28) yields λ(ν)  1 − (1 − )/ν 2 .

76

5 The Scattering Kernel and Associated Auxiliary Functions

Fig. 5.4 Monochromatic scattering. The function λ(ν), real part of L± (ν), defined in Eq. (5.28), is shown here for  = 1/8 = 0.125. It becomes infinite for ν = ±1, takes the value  at ν = 0 and has two zeroes in the interval [−1, +1] at ±ν0 . They coalesce into a double zero at the origin for  = 0. The zeroes at ±ν0 correspond to the two zeroes of L(z)

• For monochromatic scattering, η(ν) = −(1 − )πk(ν) is zero in the interval [−1, +1], hence L(z) is analytic in the strip defined by |(z)| < 1, and we have L(ν) = λ(ν) = L+ (ν) = L− (ν),

ν ∈] − 1, +1[.

(5.29)

• For complete frequency redistribution, there is no interval in which k(ν) is zero (see Fig. 5.2). The function λ(ν) has a simple variation, increasing monotonically from λ(0) =  to λ(∞) = 1. An asymptotic expansion for ν → ∞ shows that limν→∞ (λ(ν) − 1) = O(1/ν 2 ). The question arises whether z = 0 belongs to the cut. When  = 0, the origin does not belong to the cut since L(0) = . When  = 0, the dispersion function tends to zero and the origin does not belong to the cut either. The zero of the dispersion function at z = 0 has a rather complicated nature discussed in Sect. F.2. We summarize in Table 5.1, the values of λ(ν), πη(ν) and θ (ν) at the end points of the cuts. The variation of θ (ν) between the end points of cuts plays an important role in the construction of the half-space auxiliary function. In Figs. 5.5 and 5.6, we show for monochromatic scattering and complete frequency redistribution, respectively, the variations of the real and imaginary parts of L+ (ν) and L− (ν) for ν ∈ [νl , ∞[. The imaginary part of L+ (ν) is negative and that of L− (ν) is positive. At the point L± (∞) = (1, 0), the tangent to the phase diagrams is perpendicular to the real axis (L+ (ν)). Its direction is given by the ratio t (ν) = −πk (ν)/λ (ν) for ν → ∞ (the prime indicates a derivative). Using k(ν) ∼ 1/ν and (λ(ν) − 1) ∼ 1/ν 2 , we find t (ν) ∼ ν, hence a vertical tangent. For Table 5.1 Monochromatic scattering and complete frequency redistribution. Real part, imaginary part, and argument of L+ (ν) for  = 0

ν [L+ (ν)] = λ(ν)

[L+ (ν)] = πη(ν)

θ(ν)

Monochromatic 1 ∞ −∞ 1 −(1 − )π/2 0 −π 0

Redistribution 0 ∞  1 0 0 0 0

5.2 The Dispersion Function

0.4

77

(L±)

L−(1) (1−ε)π/2 0.2

L−(ν) L±(∞)

−0.2

0.2

1.0

θ

(L±)

L+(ν)

−0.2

−(1−ε)π/2 +

L (1) −0.4

Fig. 5.5 Monochromatic scattering. Phase diagrams of L+ (ν) and of its complex conjugate L− (ν) for ν between 1 and +∞. The change of sign of [L+ (ν)] corresponds to the change of sign of λ(ν) in the interval ν ∈ [1, ∞[. The angle θ is the argument of L+ (ν). It varies between −π and 0 for ν ∈ [1, ∞[. The destruction probability  has the value 0.125

(L±)

L−(ν) L±(∞)

L±(0) = θ

1.0

(L±)

L+(ν)

Fig. 5.6 Complete frequency redistribution. Sketch of the phase diagrams of L+ (ν) and L− (ν) for a Lorentz profile and  = 0. The phase diagram lies entirely to the right of the imaginary axis since [L+ (ν)] = λ(ν) has no zero

78

5 The Scattering Kernel and Associated Auxiliary Functions

complete frequency redistribution, the phase diagram of L+ (ν) at the point (, 0), corresponding to ν = 0, makes an angle −π/4 with the real axis for the Lorentz profile and −π/2 for the Doppler profile. These angles are determined in Sect. F.2, where we treat conservative scattering.

5.2.2 The Zeroes of the Dispersion Function In the overview on the Hilbert transform method in Sect. 4.3.3, we indicate how a zero of the dispersion function will modify the method of solution. Finding the zeroes of the dispersion function is a prerequisite for any exact method of solution for convolution-type integral equations arising in radiative transfer. Knowing their number and position is needed for the Hilbert transform method as well as for the Wiener–Hopf method. We apply here a general method for finding the zeroes of analytic functions, following Case and Zweifel (1967). According to a classical property of analytic functions (e.g. Carrier et al. 1966), the number of zeroes of L(z) is equal to the change in the argument of L(z) divided by 2π as z varies along a closed contour in the cut plane. We show in Fig. 5.7 a close contour appropriate for monochromatic scattering. For complete redistribution, the contour is similar, but the end points of the cuts are at the origin. To calculate the change of the argument, the outer curves are sent to infinity. They do not contribute to this variation, since L(z) tends to one as z → ∞. The total variation of θ (ν) is thus determined by its variation around the cuts. As shown above, L+ (ν) and L− (ν) are complex conjugate

(z)

ν

−1

+1

Fig. 5.7 Contour for the determination of the number of zeroes of the dispersion function L(z). For monochromatic scattering, the end point of the cuts are at ν = ±1. For complete frequency redistribution, they are at ν = 0

5.2 The Dispersion Function

79

and L± (−ν) = L∓ (ν). Hence, the change in the argument of L(z) is four times the change in the argument of L+ (ν) as ν varies from ∞ to νl . The number of zeroes is thus given by N=

4 [θ (∞) − θ (νl )], 2π

(5.30)

with νl = 1 for monochromatic scattering and νl = 0 for complete frequency redistribution. The variation of θ (ν) is shown in Table 5.1. For monochromatic scattering, θ (ν) varies monotonically, from zero to −π, as ν varies from ∞ to ν = 1 (see also Fig. 5.5). Equation (5.30) yields N = 2. This number is independent of the value of . The function L(z) being even and satisfying L(¯z) = L(z) (here the bar denotes the complex conjugate), the zeroes lie either on the real or on the imaginary axis. For  > 0 they are real and are located in the interval ν ∈] −1, +1[ at ν = ±ν0 . For  < 0, they lie on the imaginary axis. The single scattering albedo is then larger than one. We give an example in Sect. 23.6, where we discuss a criticality problem for neutron transport. The equation L(z) = 0, which provides the zeroes of the dispersion function, is usually referred to as the characteristic equation and ±ν0 are the two roots of this equation. When  is small, the position of the zeroes can be found by expanding L(z) about z = 0. A Taylor expansion of Eq. (5.14) yields L(z)   − (1 − )

z2 + O(z4 ), 3

z → 0.

(5.31)

Hence, for small , ν0 

√ 3.

(5.32)

When  = 0, Eq. (5.31) reduces to L(z)  −z2 /3. The two simple zeroes coalesce into a double zero at the origin. The value of ν0 approaches 1 when  increases. Values of ν0 for several values of  can be found in Chandrasekhar (1960, p. 19). We note that the zeroes of λ(ν), which can be observed in Fig. 5.4 for |ν| larger than 1, are not zeroes of L(z), since η(ν) = 0 in this interval. For complete frequency redistribution, Table 5.1 shows that the variation of θ (ν), when ν varies between zero and infinity is zero. This property can also be observed by following the variation θ (ν) on the phase diagrams of L+ (ν) and L− (ν) shown in Fig. 5.6. In contrast to monochromatic scattering, these diagrams lie entirely to the right of the imaginary axis. For complete frequency redistribution, L(z) has no zero for  = 0. For  = 0, there is a zero at the origin, the nature of which is fairly complicated and depends on the absorption profile ϕ(x) (see Sect. F.2). It is not an isolated zero and it does not play the same role as the zeroes of the monochromatic dispersion function.

80

5 The Scattering Kernel and Associated Auxiliary Functions

5.2.3 The Index κ The structure of the solutions of a singular integral equation with a Cauchy-type kernel can be classified according to the sign of a number known as the index of the singular integral equations. It is actually a property of the homogeneous Riemann-Hilbert problem defining the auxiliary function X(z). The general theory is developed in Muskhelishvili (1953) (see also Gakhov 1966; Case and Zweifel 1967; Duderstadt and Martin 1979). Here we determine the value of this index, denoted κ, for monochromatic scattering and complete frequency redistribution. How it affects the construction of the function X(z) is described in Sect. 5.3. We define the index κ as κ≡

1 [arg[ (∞)] − arg[ (νl )]], 2π

(5.33)

where  (ν) = L+ (ν)/L− (ν) = exp[2i θ (ν)].

(5.34)

Thus κ=

1 N [θ (∞) − θ (νl )] = . π 2

(5.35)

For monochromatic scattering κ = 1 and for complete redistribution κ = 0. It is obvious, but we want to stress it again, that the differences in the values of N and κ, between monochromatic scattering and complete redistribution, are directly related to the fact that the kernel K(τ ) has different asymptotic behaviors at large optical depths: it decreases exponentially for monochromatic scattering and algebraically for complete redistribution. In the latter case, the algebraic behavior of K(τ ) at infinity depends on the law of decrease of the profile ϕ(x) for large |x|, but the value of κ and the number of zeroes are independent of the detail of this law.

5.3 The Half-Space Auxiliary Function X(z) In the overview about the Hilbert transform method in Sects. 4.3.2 and 4.3.3, we explained that an auxiliary function X(z) is needed to solve half-space problems, and that this function is solution of a homogeneous Riemann–Hilbert problem defined by X+ (ν)/X− (ν) = L+ (ν)/L− (ν),

ν ∈ [νl , ∞[,

(5.36)

5.3 The Half-Space Auxiliary Function X(z)

81

where L± (ν) are the limiting values of the dispersion function L(z) along the real axis. We recall that νl = 0 for complete redistribution and νl = 1 for monochromatic scattering. In addition to Eq. (5.36), other conditions have to be imposed on X(z) to guarantee that the functions F (z) or S(z), which serve to solve the singular integral equations such as Eqs. (4.22) and (4.38), are Hilbert transforms. All together, the conditions that have to be satisfied by X(z) are: • • • •

| \[ν , ∞[, (i) analyticity in C l + − (ii) X (ν)/X (ν) = L+ (ν)/L− (ν), (iii) limz→νl (νl − z)/X(z) = 0, (iv) finite degree at infinity.

ν ∈ [νl , ∞[,

We note that any function of the form X(z)P(z), with P(z) an entire function with no zero at νl , satisfies these four conditions. We impose the additional constraint that X(z) has the lowest possible degree at infinity. The solution which satisfies this additional condition is called the fundamental solution. Condition (iii) means that 1/X(z) can have a mild (integrable) singularity at the end point. It stems from the requirement that F (z) should be integrable at the end point of a branch cut (see condition (iv) on F (z)). A similar integrability condition appears in orthogonality method of solution (Van der Mee and Zweifel 1990). Condition (iv) means that X(z) should have an algebraic behavior at infinity as opposed to an exponential one. An exponential behavior would be inconsistent with the fact that F (z) is a Hilbert transform. To construct X(z), we start from the condition (ii). Using Eq. (5.25), we can rewrite Eq. (5.36) as X+ (ν)/X− (ν) = exp[2i θ (ν)],

ν ∈ [νl , ∞[,

(5.37)

where θ (ν) is the argument of L+ (ν) . The determination of θ (ν) is chosen in such a way that −π ≤ θ (ν) ≤ 0 and θ (∞) = 0 (see Sect. 5.2.1). We now take the logarithm of Eq. (5.37). It becomes, ln[X+ (ν)] − ln[X− (ν)] = 2i θ (ν),

ν ∈ [νl , ∞[.

(5.38)

This equation is a Plemelj formula for the function Y0 (z) ≡

1 π



∞

θ (ν) νl

dν . ν−z

(5.39)

By construction this function is analytic in the complex plane cut along the line [νl , ∞[. The function X0 (z) = exp[Y0 (z)],

(5.40)

82

5 The Scattering Kernel and Associated Auxiliary Functions

is thus a solution of Eq. (5.37), but we must examine whether it satisfies the four conditions listed above. Since θ (ν) → 0 as ν → ∞, the integral over ν in Eq. (5.39) is convergent. The conditions (i) are (ii) are trivially satisfied. We now consider condition (iii). When z is near the end point νl of the cut, θ (νl ) ln(νl − z), π

Y0 (z)  −

(5.41)

hence X0 (z) ∼ (νl − z)−θ(νl )/π .

(5.42)

Complete frequency redistribution and monochromatic scattering are now considered separately. For complete frequency redistribution, the end point of the cut is at νl = 0. For  = 0, as shown in Table 5.1, θ (ν) → 0 as ν → 0. Hence, X0 (z) has a finite value at the end point of the cut. Condition (iii) on the integrability of 1/X(z) is satisfied and we can choose X(z) ≡ X0 (z).

(5.43)

For  = 0, although θ (ν) does not tend to zero as ν → 0 (see Sect. F.2), 1/X0 (z) is still integrable and one can also choose X(z) = X0 (z). For monochromatic scattering, the end point of the cut is at νl = 1. As shown in Sect. 5.2.1, θ (ν) → −π as ν → 1 for any value of . Hence, X0 (z) ∼ (1 − z) as z → 1. Condition (iii) is not satisfied. We define X(z) as X(z) ≡

1 X0 (z). 1−z

(5.44)

Now X(z) satisfies the conditions (i) to (iii). Finally we examine the condition (iv) concerning the algebraic behavior at infinity. For z → ∞ in the cut plane, Y0 (z) → 0,

hence X0 (z) → 1.

(5.45)

Thus X(z) → 1 X(z)  −1/z

for complete frequency redistribution

(5.46)

for monochromatic scattering.

(5.47)

So, depending on whether we are dealing with complete redistribution or monochromatic scattering, X(z) is given by Eq. (5.43) or Eq. (5.44), with X0 (z) defined by Eqs. (5.40). Of course, the functions Y0 (z) and X0 (z) are different for

5.4 Some Properties of X(z)

83

complete frequency redistribution and monochromatic scattering, since they involve an integration of θ (ν). It is clear that the solutions given in Eqs. (5.43) and (5.44) are those with the lowest degree at infinity. The algebraic factor, which prevents the appearance of a singularity of 1/X(z) at the end point of the branch cut, is related to the value of the index κ introduced in Eq. (5.33). For complete redistribution κ = 0, no algebraic factor is needed, while for monochromatic scattering κ = +1. When the index κ is positive, X(z) always contains an algebraic factor raised to the power −κ (see Muskhelishvili 1953, Gakhov 1966, Duderstadt and Martin 1979). The value of the index κ, because it determines the behavior of X(z) for z → ∞, also controls the solutions of singular integral equations with Cauchy-type kernels. When κ = 0, the solution is unique, when κ is negative, the solution is not unique, and when κ is positive, the existence of a unique solution requires that the inhomogeneous term satisfies κ conditions. For complete redistribution there is always uniqueness of solution since κ = 0. For monochromatic scattering the existence of a solution imposes one solvability condition. An example is given in Sect. D.1.

5.4 Some Properties of X(z) We present here the main properties of X(z), treating first complete redistribution and then monochromatic scattering. We give in particular an important factorization relation between X(z) and the dispersion function L(z). Proofs of the results and some additional properties can be found in Appendix B of this chapter. The results are presented in this section are for non-conservative scattering, namely  = 0. Some changes appear for conservative scattering. They are treated in Chap. 9.

5.4.1 Complete Frequency Redistribution • The function X(z) satisfies the factorization relation X(−z)X(z) = L(z),

(5.48)

| \R , where R which holds in C + + is the real axis without the origin. This factorization relation is at the heart of many exact results given in this book. Along the real axis, it becomes

X(−ν)X± (ν) = L± (ν), ±

∓

X (−ν)X(ν) = L (ν),

ν ∈ [0, ∞[

(5.49)

ν ∈] − ∞, 0].

(5.50)

The functions X± (ν) satisfy several very useful relations given in Appendix B of this chapter. All of them are easily derived from Eqs. (5.21) and (5.36).

84

5 The Scattering Kernel and Associated Auxiliary Functions

• The factorization relation readily yields the key result, √ .

X(0) =

(5.51)

It suffices to use L(0) =  and the positivity of X(z) for z on the negative real axis (see Eq. (5.40)). For  = 0, X(z) goes algebraically to zero at the origin, with a power law depending on the absorption profile (see Table F.1). • X(z) obeys the nonlinear integral equation  X(z) = 1 − (1 − )

∞

0

dν k(ν) . X(−ν) ν − z

(5.52)

The proof is given in Appendix B of this chapter. We note that the integral involves only values of X(z) for z on the negative part of the real axis, where X(z) is analytic and takes real values.

5.4.2 Monochromatic Scattering We show in Appendix B of this chapter, that the results given above for complete frequency redistribution hold for monochromatic scattering, provided νl = 0 is replaced by νl = 1 and X(z) is replaced by X∗ (z), defined by X∗ (z) = (ν0 − z)X(z),

(5.53)

where ν0 is the positive zero of L(z) (0 ≤ ν0 < 1). • The factorization relation (5.48), which is now X(−z)X(z) = L(z)/(ν02 − z2 ),

(5.54)

can also be written as X∗ (−z)X∗ (z) = L(z),

| \ ] − ∞, −1] ∪ [1, +∞[. C

(5.55)

In Eq. (5.54), the denominator (ν02 − z2 ) cancels the zeroes of L(z). Along the real axis, Eq. (5.54) becomes X(−ν)X± (ν) =

L± (ν) , ν02 − ν 2

ν ∈ [1, ∞[,

X± (−ν)X(ν) =

L∓ (ν) , ν02 − ν 2

ν ∈] − ∞, −1].

Other relations involving X± (ν) are given in Appendix B of this chapter.

(5.56)

5.4 Some Properties of X(z)

85

• At the origin, ν0 X(0) = X∗ (0) =

√ .

(5.57)

For  = 0, as shown by Eq. (5.31), L(z)  −z2 /3 as z → 0. Equation (5.54) yields √ X(0) = 1/ 3.

(5.58)

√ This result can also be obtained by setting ν0  3, √ the asymptotic behavior of ν0 for  → 0 in Eq. (5.57). The value X(0) = 1/ 3 plays an important role in the solution of Milne problem. • The nonlinear integral equation for X∗ (z) takes the form X∗ (z) = 1 −

1− 2

 1

∞

1 νX∗ (−ν)

dν . ν −z

(5.59)

It can be obtained by replacing in Eq. (5.52) X(z) by X∗ (z) and k(ν) by 1/(2ν). A proof is given in Appendix B of this chapter.

5.4.3 Some Properties of the H -Function The auxiliary function for half-space problems, known in Astronomy under the name of the H -function since Chandrasekhar’s very complete investigation (Chandrasekhar 1960), has been the subject of numerous publications. This function first appeared, at least implicitly, in the the famous Wiener and Hopf (1931) article. It was popularized with the work of Ambartsumian (1942), Chandrasekhar (1960), Sobolev (1963), as the solution of a nonlinear integral equation or the solution of a singular linear integral equation with a Cauchy-type kernel. There exist many definitions of the H -function, such as the surface value of the source function for an exponential primary source term, as the center-to-limb variation of the emergent intensity for a uniform primary source, or as the Laplace transform of the surface Green function G(τ ). References to fundamental articles can be found in books by, e.g., Chandrasekhar (1960), Kourganoff (1963), Sobolev (1963), Case and Zweifel (1967), Ivanov (1973), van de Hulst (1980). The H -function will be encountered in several of the following chapters, in particular in Chap. 11. In this Section, we derive the relation between the H function and X(z) by comparing the nonlinear integral equation for X(z) with the nonlinear integral equation for the H -function given for monochromatic scattering in Chandrasekhar (1960, p. 97) and for complete frequency redistribution in Ivanov (1973, p. 212). We also present some properties of H -function and different exact expressions.

86

5 The Scattering Kernel and Associated Auxiliary Functions

For monochromatic scattering, with an isotropic phase function, the nonlinear integral equation satisfied by the H -function can be written as 1− μH (μ) H (μ) = 1 + 2



1

0

H (μ ) dμ , μ + μ

μ ∈ [0, 1].

(5.60)

For complete frequency redistribution, it has the form 

∞

H (ξ ) = 1 + (1 − )ξ H (ξ ) 0

H (ν) 1 1 k( ) dν, ξ +ν ν ν

ξ ∈ [0, ∞[.

(5.61)

These equations can be analytically continued to the complex plane. A comparison with the nonlinear integral equations for X(z) and X∗ (z) given in Eqs. (5.52) and (5.59) leads to 1 H (z) = 1/X(− ), z

for complete frequency redistribution,

1 H (z) = 1/X∗ (− ), z

for monochromatic scattering,

and to the expressions    z ∞ θ (ν) H (z) = exp − dν , π 0 1 + νz

(5.62) (5.63)

for complete frequency redistribution (5.64)

and

   z ∞ θ (ν) 1+z exp − dν , H (z) = 1 + ν0 z π 1 1 + νz

for monochromatic scattering. (5.65)

Here, θ (ν) is the argument of L+ (ν) introduced in Eq. (5.25). The analyticity properties of H (z) are easily deduced from those of X(z). For monochromatic scattering, H (z) has a branch cut along the interval [−1, 0] and a pole at z = −1/ν0 . For complete frequency redistribution, H (z) has a cut along the negative part of the real axis and no pole. The factorization relations given for X(z) in Eqs. (5.48) and (5.54) lead to a single relation, 1 H (z)H (−z)L( ) = 1. z

(5.66)

The introduction of the H -function often unifies monochromatic and frequency redistribution expressions. For both scattering processes and for conservative and non-conservative scattering, the factorization relation leads to H (0) = 1.

(5.67)

5.4 Some Properties of X(z)

87

Fig. 5.8 The monochromatic function H (μ) for isotropic scattering, with different values of √ . For  = 0, ∞. H (μ) = 3μ as μ →√ For  = 0, H (μ) = 1/  as μ → ∞. For the chosen values √ of , 1/  = 6.3, 3.2, 2.2, 1.6. The numerical data are from Chandrasekhar (1960, p. 125)

For z → ∞, Eq. (5.66) yields √ lim H (z) = 1/ ,

z→∞

(5.68)

for both scattering processes, when  = 0. When  = 0, that is for conservative scattering, H (z) tends√to infinity. For monochromatic scattering, Eq. (5.63) combined with X(0) = 1/ 3 and X∗ (z) = −zX(z) leads to √ H (z)  z 3.

(5.69)

For monochromatic scattering, we show H (μ) in Fig. 5.8, for different values of . The linear behavior for  = 0 is easy to detect. The numerical calculation of H (μ) is not straightforward. For example, the solution of Eq. (5.60) raises a uniqueness problem discussed in Chap. 11. For complete frequency redistribution, the behavior at infinity is controlled by the large frequency dependence of the absorption profile ϕ(x). As shown in Appendix F of Chap. 9, HD (z) ∼ z1/2 (ln |z|)1/4,

HV (z) ∼ z1/4 ,

(5.70)

with D standing for the Doppler profile and V for the Voigt profile. In Eq. (5.65) the H -function is given in terms of the argument of L+ (ν). Historically, the first closed-form expressions were obtained with the Wiener–Hopf factorization method (e.g. Hopf 1934; Halpern et al. 1938) and were expressed in terms of the dispersion function itself. We show in Chap. 12, that the Wiener–Hopf factorization leads to    +∞ 1 1 dk H ( ) = exp − , (5.71) ln V (k) z 2i π −∞ k − iz

88

5 The Scattering Kernel and Associated Auxiliary Functions

˜ ˆ where V (k) = 1 − (1 − )K(k). We recall that K(k) is the Fourier transform of the kernel and that V (k) = L(i k) . Using the evenness of V (k), we can write    +∞ z 1 dk H ( ) = exp − ln V (k) 2 . z 2π −∞ k + z2

(5.72)

Equation (5.71) holds for monochromatic scattering and complete frequency redistribution (see e.g., Ivanov 1973, p. 216). We remark here that Eq. (5.72) can provide the values of H (z) at zero and at infinity. The √ value H (0) = 1 is obtained by letting z tends to infinity. For  = 0, the value 1/  at infinity is obtained by setting k = zu and V (k) = [1 − (1 − )(1 − ˆ ˆ K(k))/]. It suffices then to let z → 0, and to remember that K(0) = 1 and that 1/(1 + u2 ) is the derivative of arctan u. An alternative form of Eq. (5.72), proposed by Chandrasekhar (1960, p. 115) (see also Kourganoff 1963), is  H (z) = exp

z 2i π



+i ∞

−i ∞

 ln T (ω) dω ,  2 − z2

(5.73)

where T (ω) = V (1/i ω). Equation (5.73) is easily derived from Eq. (5.72) by setting k = 1/(i ω) and changing z into 1/(i z). Several other closed form expressions for the monochromatic H -function can be found in the literature (see e.g. Nagirner 1968; Siewert 1975; Kawabata and Limaye 2011). In Appendix I of Chap. 12, we show how Eq. (5.71) can be transformed into the expressions given in Eqs. (5.64) and (5.65).

Appendix B: Properties of the Half-Space Auxiliary Function In this Appendix we prove some results presented in Sect. 5.4 and discuss some additional properties of the X-function. The corresponding proofs and results for the H -function are easily deduced from the relations between H (z) and X(z) given in Eqs. (5.62) and (5.63). The proofs are established for complete redistribution, because the algebra is simpler than for monochromatic scattering, but are inspired by proofs given by Case and Zweifel (1967) for monochromatic scattering. Most of the expressions established for complete redistribution remain valid for monochromatic scattering, when X(z) is replaced by X∗ (z) = (ν0 − z)X(z), but one should be careful about domains of validity.

B.2 Identities for the Boundary Values of X(z)

89

B.1 Factorization Relation We introduce the function A(z) ≡

L(z) . X(z)X(−z)

(B.1)

It follows from the properties of L(z) and X(z) that A(z) is an even function of z, analytic in the complex plan cut along the full real axis. We now consider the ratio A+ (ν)/A− (ν) for ν ∈ [0, ∞[. It can be written as A+ (ν) L+ (ν) X− (ν) X− (−ν) = . A− (ν) L− (ν) X+ (ν) X+ (−ν)

(B.2)

Using the continuity of X(z) across the negative real axis, which states that X− (−ν) = X+ (−ν) and the defining relation L+ (ν) X+ (ν) = − , − X (ν) L (ν)

ν ∈ [0, ∞[,

(B.3)

we obtain A+ (ν) = 1, A− (ν)

ν ∈ [0, ∞[.

(B.4)

This equation shows that the jump of A(z) across the real positive axis is zero. Since A(z) is an even function of z, Eq. (B.4) holds also for ν ∈] − ∞, 0]. Hence, the jump of A(z) across the full real axis is zero and we may conclude that A(z) is an entire function. At infinity L(z) and X(z) tend to unity, hence A(z) goes also to unity. Using the Liouville theorem, we find A(z) = 1 for all z. We thus obtain the factorization relation X(z)X(−z) = L(z),

(B.5)

given in Eq. (5.48). For monochromatic scattering, similar arguments lead to Eq. (5.55). For the proof, the function A(z) should be defined as in Eq. (B.1) with the right-hand side divided by (ν02 − z2 ).

B.2 Identities for the Boundary Values of X(z) The boundary values of X(z) satisfy simple but important relations, which can be derived from the defining relation in Eq. (5.36) and the Plemelj formulae for the

90

5 The Scattering Kernel and Associated Auxiliary Functions

dispersion function, namely L+ (ν) − L− (ν) = 2i πη(ν),

L+ (ν) + L− (ν) = 2λ(ν)

ν ∈ [νl , ∞[,

(B.6)

with νl = 0 for complete frequency redistribution and νl = 1 for monochromatic scattering. For complete redistribution, they are X+ (ν) − X− (ν) =

2i πη(ν) , X(−ν)

1 + λ(ν) [X (ν) + X− (ν)] = , 2 X(−ν) 1 X+ (ν)

−

1 X− (ν)

= −X(−ν)

(B.7) (B.8)

2i πη(ν) , R(ν)

1 1 λ(ν) 1 [ + + − ] = X(−ν) . 2 X (ν) X (ν) R(ν)

(B.9) (B.10)

For monochromatic scattering, they are 2i πη(ν) , − ν 2 )X(−ν)

(B.11)

1 + λ(ν) [X (ν) + X− (ν)] = 2 , 2 (ν0 − ν 2 )X(−ν)

(B.12)

X+ (ν) − X− (ν) =

(ν02

1 1 2i πη(ν) − = −(ν02 − ν 2 )X(−ν) , X+ (ν) X− (ν) R(ν) 1 1 1 λ(ν) [ + + − ] = (ν02 − ν 2 )X(−ν) . 2 X (ν) X (ν) R(ν)

(B.13) (B.14)

The function R(ν) is given by R(ν) ≡ L+ (ν)L− (ν) = λ2 (ν) + π2 η2 (ν).

(B.15)

For complete redistribution, these relations hold for ν ∈ [0, ∞[ and for monochromatic scattering for ν ∈ [1, ∞[. Written in terms of the function X∗ (z), Eqs. (B.11) to (B.14) become identical to Eqs. (B.7) to (B.10), but the domain of ν is [1, ∞[ instead of [0, ∞[. Equations (B.7) to (B.10) hold also when a continuous absorption is added to the line absorption coefficient (see Chap. 8).

B.3 Integral Equations for X(z)

91

B.3 Integral Equations for X(z) Here we give the proofs of the integral equations written in Eqs. (5.52) and (5.59) for complete redistribution and monochromatic scattering respectively. The proofs do not make use of the explicit expressions of X(z), only of its definition, namely that it satisfies the jump condition in Eq. (5.36). Following Case and Zweifel (1967), we write X(z) as a Cauchy integral, i.e. as X(z) =

1 2iπ

 C

X(ξ ) dξ, ξ −z

(B.16)

where C is a contour turning around the branch cut of X(z). For complete redistribution the contour C is shown in Fig. B.1. Since X(z) → 1 for z → ∞ in the cut complex plane, there is a contribution from the outer curve, which can be calculated by assuming that the outer curve is a circle of radius R, and letting R tend to infinity. The contribution from the horizontal lines above and below the cut can be expressed in terms of the identity given in Eq. (B.7): X+ (ν) − X− (ν) =

2iπη(ν) , X(−ν)

ν ∈ [0, ∞[.

(B.17)

√ There is no contribution from the endpoint of the cut. Indeed, X(0) =  for  = 0 and X(z) → 0 as z → 0 when  = 0. Summing the contributions, we obtain the nonlinear integral equation 

∞

X(z) = 1 − (1 − ) 0

k(ν) dν . X(−ν) ν − z

(B.18)

For the H -function it leads to Eq. (5.61). For monochromatic scattering, the contour C is similar to the contour in Fig. B.1 with the endpoint of the cut at ν = 1. There is no contribution from the outer curve since X(z)  −1/z at infinity. The contribution from the two lines above and below the cut is deduced from Eq. (B.11). There is no contribution from Fig. B.1 Complete frequency redistribution. Closed contour for the construction of the nonlinear integral equation satisfied by X(z)

(z) 0

92

5 The Scattering Kernel and Associated Auxiliary Functions

the endpoint of the cut since X(1) has a finite value. We thus obtain the nonlinear integral equation X(z) = −

1− 2



∞ 1

dν 1 . 2 2 X(−ν)(νo − ν ) ν(ν − z)

(B.19)

A very useful identity can be derived from this equation. Taking the limit z → ∞ and using the asymptotic behavior X(z) → −1/z for z → ∞, we find the identity 1− 2

 1

∞

1 dν = −1. 2 2 X(−ν)(νo − ν ) ν

(B.20)

The equation for X(z) can be transformed into an equation for X∗ (z) = (ν0 − z)X(z). The transformation requires a multiplication by (ν0 − z), a decomposition of [(ν0 − ν)(ν − z)]−1 into simple fractions and then the use of Eq. (B.20). One obtains  dν 1 1− ∞ . (B.21) X∗ (z) = 1 − ∗ 2 νX (−ν) ν − z 1 It leads to the nonlinear integral equation for H (z) written in Eq. (5.60). The nonlinear integral equations for X∗ (z) and H (z) do not have a unique solution. The correct solution must satisfy the identity in Eq. (B.20). The question of the non-uniqueness is discussed in Sect. 11.3.2, where we also introduce an alternative H -equation better suited to a numerical solution. Finally we point out that all the nonlinear integral equations given here hold also for conservative scattering. It suffices to set  = 0.

B.4 Values of X(z) and H (z) The integral equations for X(z) and H (z) and all the results established in the following chapters show that numerical values of these functions are needed only for real values of z, inside the analyticity domain. For X(z) this means z = ν ∈ ] − ∞, −νl ], with νl = 0 for complete frequency redistribution and νl = 1 for monochromatic scattering. For H (z) this means z = μ ∈ [0, 1/νl ]. We show in Table B.1, for complete frequency redistribution and monochromatic scattering, the values of X(z), X∗ (z) = (ν0 − z)X(z) and H (z) at z = 0 and infinity for  = 0. We recall that H (z) = 1/X(−1/z) for complete frequency redistribution and H (z) = 1/X∗ (−1/z) for monochromatic scattering. For  = 0, the behavior of these functions is discussed in Chap. 9. For monochromatic scattering, it is √ shown that X(0) = 1/ 3 and H (z) ∼ z for z → ∞. For complete frequency redistribution, one has X(z) → 0 as z → 0, with XD (z) ∼ z1/2 and XV (z) ∼ z1/4 and for z → ∞, H D (z) ∼ z1/2 and H V (z) ∼ z1/4.

B.5 Moments of X(z) and H (z)

93

Table B.1 The functions X(z), X ∗ (z) = (ν0 − z)X(z), and H (z) for  = 0: values at z = 0 and behavior at infinity. For monochromatic scattering, ν0 ∈ [0, 1] is the positive root of the dispersion √ function L(z). For small values of , ν0  3. The behavior at infinity of H (z) for  = 0 is discussed in Chap. 9. z 0 ∞

Redistribution X(z) H (z) √  1 √ 1 1/ 

Monochromatic X(z) √ /ν0 ∼ −1/z

X ∗ (z) √  1

H (z) 1 √ 1/ 

For monochromatic scattering, a detailed historical account of the methods employed to calculate the H -function is presented in Bosma and de Rooij (1983). The methods are based on exact integral representations (e.g. Ivanov and Nagirner 1965; Case and Zweifel 1967; Kawabata and Limaye 2011) or on nonlinear integral equations (see e.g. Chandrasekhar 1960, Bosma and de Rooij (1983)). We briefly discuss this second approach in Sect. 11.3.2. Tables of H (μ), for μ ∈ [0, 1] and different values of , can be found in, e.g., Placzek (1947), Stibbs and Weir (1959), Chandrasekhar (1960), Ivanov (1973, p. 129), Bosma and de Rooij (1983), and Case and Zweifel (1967). In the latter reference, the table is for the function ν0 1 1 Xcz (z) ≡ − √ X( ). z z

(B.22)

√ This function Xcz (z) remains finite in the limit  → 0 since ν0  3. For complete frequency redistribution, tables of H (μ) for different profiles and different values of  can be found in Ivanov (1973, p. 232), for the Doppler profile (see also Ivanov and Nagirner 1965). Other tables have also been published in Warming (1970). Our definition and that in Ivanov’s book are related by H (μ) = Hi [μϕ(0)],

(B.23)

where the subscript i stands for Ivanov and ϕ(0) is the value of the absorption profile at line center.

B.5 Moments of X(z) and H (z) The moments of the H -function appear in some exact expressions. We give here definitions and explicit expressions, when they exist. Monochromatic Scattering Following Chandrasekhar (1960), we introduce the moments 

1

αn = 0



∞

H (μ)μ dμ = n

1

1 X∗ (−ν)

dν , ν n+2

(B.24)

94

5 The Scattering Kernel and Associated Auxiliary Functions

where n is an integer. They are well defined for all values of n ≥ 0, but only the zeroth-moment, and the first-moment for  = 0, have explicit expressions. Setting √ z = 0 in the integral equation for X∗ (z) (see Eq. (B.21)) and using X∗ (0) = , we immediately obtain α0 = 2

√ 1−  . 1−

(B.25)

For conservative scattering, α0 = 2. To calculate α1 for the conservative case, we set  = 0, ν0√= 0 and z = 0 in the integral equation for X(z) (see Eq. (B.19)). Using X(0) = 1/ 3, we obtain 

∞

α1 = 1

1 X∗ (−ν)

dν 2 = 2X(0) = √ . 3 ν 3

(B.26)

Numerical values of αn for n up to four can be found in Bosma and de Rooij (1983) for isotropic and anisotropic scattering processes with  = 0 and  = 0. Complete Frequency Redistribution As suggested by Ivanov (1973), the moments of the H -function for complete frequency redistribution can be defined by  αn = 2 0

∞

 g(ν)H (ν)ν n dν = 2 0

∞

k(ν) dν . X(−ν) ν n+1

(B.27)

Setting z = 0 in To calculate α0 , one can proceed as for monochromatic scattering. √ the integral equation for X(z) (see Eq. (B.18)) and using X(0) = , we find that α0 is also√given by Eq. (B.25). This moment depends only on X(0), or X∗ (0), both equal to . Higher moments are divergent. The behavior of k(ν) for ν → 0 discussed in Chap. 9 shows that k(ν)/ν n+1 is non integrable at ν = 0, for n > 0. The fundamental reason is that the kernel K(τ ) is decreasing algebraically at large optical depths.

References Ambartsumian, V.A.: Light scattering by planetary atmospheres. Astron. Zhurnal 19, 30–41 (1942) Bosma, P.B., de Rooij, W.A.: Efficient methods to calculate Chandrasekhar’s H-functions. Astron. Astrophys. 126, 283–292 (1983) Carrier, G.F., Krook, M., Pearson, C.E.: Functions of a Complex Variable. McGraw-Hill Book Company, New York (1966) Case, K.M., Zweifel, P.: Linear Transport Theory. Addison-Wesley, Reading, MA (1967) Chandrasekhar, S.: Radiative Transfer. Dover Publications, New York (1960); First edition, Oxford University Press, Oxford (1950) Duderstadt, J.J., Martin, W.R.: Transport Theory. Wiley, New York (1979)

References

95

Gakhov, F.D.: Boundary Value Problems. Pergamon Press, London (1966); translation by I.N. Sneddon of the 2nd Russian edition (1963), Fizmatgiz, Moscow Halpern, O., Lueneburg, R., Clark, O.: On multiple scattering of neutrons I. Theory of the albedo of a plane boundary. Phys. Rev. 53, 173–183 (1938) Hopf. E.: Mathematical Problems of Radiative Equilibrium. Cambridge University Press, London (1934) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Ivanov, V.V, Nagirner, D.I.: H-functions in the theory of resonance radiation transfer. Astrophysics 1, 86–101 (1965); translation from Astrofizika 1, 103–146 (1965) Kawabata, K., Limaye, S.S.: Rational approximation for Chandrasekhar’s H-function for isotropic scattering. Astrophys. Space Sci. 332, 365–371 (2011) Kourganoff, V.: Basic Methods in Transfer Problems. Radiative Equilibrium and Neutron Diffusion. Dover Publications, New York (1963); First edition: Oxford University Press, London (1952) Muskhelishvili, N.I.: Singular Integral Equations, Noordhoff, Groningen (based on the second Russian edition published in 1946) (1953); Dover Publications (1991) Nagirner, D.I.: Multiple light scattering in a semi-infinite atmosphere. Uch. Zap. Leningr. Univ. 337, 3 (1968) Placzek, G.: The angular distribution of neutrons emerging from a plane surface. Phys. Rev. 72, 556–558 (1947) Siewert, C.E.: On a problem of uniqueness regarding H -function calculations. J. Quant. Spectrosc. Radiat. Transf. 15, 385–387 (1975) Sobolev, V.V.: A Treatise on Radiative Transfer, Von Nostrand Company, Princeton, NJ (1963), transl. by S.I. Gaposchkin; Russian original, Transport of Radiant Energy in Stellar and Planetary Atmospheres, Gostechizdat, Moscow (1956) Stibbs, D.W.N., Weir, R.E.: On the H-functions for isotropic scattering. Mon. Not. Roy. Astr. Soc. 119, 512–525 (1959) van de Hulst, H.C.: Multiple Light Scattering Tables, Formulas and Applications (2 Vols.). Academic Press, New York (1980) Van der Mee, C.V.M., Zweifel, P.F.: Application of orthogonality relations to singular integral equations. J. Integral Equa. Appl. 2, 185–203 (1990) Warming, R.F.: The direct calculation of the H -function for completely noncoherent scattering. Astrophys. J. 159, 593–604 (1970) Wiener, N., Hopf, D.: Über eine Klasse singulärer Integralgleichungen. Sitzungsberichte der Preussichen Akademie, Mathematisch-Physikalische Klasse 3. Dez. 1931 31, 696–706 (1931) (ausgegeben 28. January 1932); English translation: in Fourier transforms in the Complex Domain, Paley, R.C., Wiener, N.: Am. Math. Soc. Coll. Publ. XIX, 49–58 (1934)

Chapter 6

The Surface Green Function and the Resolvent Function

We have shown in Sect. 2.4 that the surface Green function G(τ ) = G(τ, 0) = G(0, τ0 ) and its regular part the resolvent function (τ ) = G(τ ) − δ(τ ) can be considered as building blocks for solutions of radiative transfer problems. The convolution integral equations satisfied by G(τ ) and (τ ) have fairly simple inhomogeneous terms, for G(τ ) it is δ(τ ) and for (τ ) it is (1 − )K(τ ) (see e.g. Eqs. (2.79) and (2.80)). These two functions appear appropriate for a presentation of exact methods of solution. Some exact methods of solution for convolution equations are presented in Chap. 3: the Fourier transform method for to full space problems, the inverse Laplace transform, applicable to both full-space and half-space equations, and the direct Laplace transform applicable to half-space problems only. The Fourier and direct Laplace methods require a Fourier inversion to obtain the solution in the physical space. The Laplace transforms can be obtained as solutions of a singular integral equation or by integrating an inverse Laplace transform. We use here (τ ) and G(τ ) to illustrate these methods, for complete frequency redistribution and monochromatic scattering. This chapter is organized as follows. Section 6.1 is devoted to the infinite medium case. We show how to apply the Fourier transform method and the inverse Laplace transform to obtain an exact expression of the infinite medium resolvent function ∞ (τ ). We also show how to transform a Fourier type integral into into a Laplace type one. The semi-infinite medium case is considered in Sect. 6.2, where we apply the inverse Laplace transform method to the determination of φ(ν), the inverse Laplace transform of (τ ). Section 6.3 is devoted to G(τ ). We first use ˜ the expression of φ(ν) obtained in Sect. 6.2 to derive its Laplace transform, G(p), ˜ and then show that G(p) provides a definition of the H -function. An alternative ˜ determination of G(p) with the direct Laplace transform is presented in Appendix C of this chapter.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_6

97

98

6 The Surface Green Function and the Resolvent Function

6.1 Infinite Medium The infinite medium Green function G∞ (τ ) and its regular part, the infinite medium resolvent function ∞ (τ ) = G∞ (τ ) − δ(τ ), satisfy convolution integral equations written in Eqs. (2.79) and (2.80). The equation for ∞ (τ ) is  ∞ (τ ) = (1 − )

+∞ −∞

K(τ − τ )∞ (τ ) dτ + (1 − )K(τ ).

(6.1)

The Green function G∞ (τ ) satisfies a similar equation, but the inhomogeneous term is a Dirac distribution δ(τ ). The singularity of G∞ (τ ) at τ = 0 prevents the application of the Fourier transform method and of the inverse Laplace transform method to the equation for G∞ (τ ). These two methods are thus presented here for ∞ (τ ), the Fourier method in Sect. 6.1.1 for complete frequency redistribution and monochromatic scattering and the inverse Laplace transform method in Sect. 6.1.2 for complete frequency redistribution only. Some of the notations used in this Chapter and throughout the book are listed in Table 6.1.

6.1.1 The Fourier Transform Method ˆ ∞ (k), the Fourier transform Taking the Fourier transform of Eq. (6.1), we find that  of ∞ (τ ), satisfies the algebraic equation ˆ ∞ (k) = 



 1 −1 , V (k)

(6.2)

where V (k) is the dispersion function. It is defined by ˆ V (k) = 1 − (1 − )K(k),

(6.3)

Table 6.1 Infinite medium. G∞ (τ ) is the infinite medium Green function and ∞ (τ ) the resolvent function. Columns (2) and (3) correspond to the Fourier transform and inverse Laplace transform of the functions in column (1). Column (4) gives the primary source term for the convolution integral equation satisfied by the functions in column (1). Columns (5) and (6) give the Fourier and inverse Laplace transforms of the functions in column (4). The notation—stands for undefined functions (1) G∞ (τ ) ∞ (τ )

(2) Fourier ˆ ∞ (k) G ˆ ∞ (k)

(3) inv. Laplace – φ∞ (ν)

(4) source δ(τ ) (1 − )K(τ )

(5) Fourier 1 ˆ (1 − )K(k)

(6) inv. Laplace – (1 − )k(ν)

6.1 Infinite Medium

99

ˆ where K(k) is the Fourier transform of K(τ ). The function ∞ (τ ) can then be calculated by the classical Fourier inversion formula: ∞ (τ ) =

1 2π



+∞ −∞

e−i kτ [

1 − 1] dk. V (k)

(6.4)

The solution depends only on V (k). This Fourier type integral can be transformed into a Laplace type integral by a proper deformation of the path of integration. Because V (k) has different analyticity properties for complete frequency redistribution and monochromatic scattering, we consider the two cases separately.

6.1.1.1 Complete Frequency Redistribution We assume τ ≥ 0. The path of integration is shown in the left panel of Fig. 6.1. It lies in the lower half of the complex plane to ensure that the argument of the exponential has a negative real part, hence that the exponential goes to zero at infinity. The contour takes into account the analytical structure of V (z), namely its cut along along the negative imaginary axis, which does not include the point z = 0. The horizontal line is along the real axis. Since V (z) → 1 for z → ∞, there is no contribution from the outer curve, when it is sent to infinity. This statement can be verified by transforming the outer curve into a half-circle of radius R, and letting R go to infinity. The Jordan’s lemma (Carrier et al. 1966, p. 81) is needed to prove that the contribution from this outer circle is zero. There is no contribution from the end point of the cut since V (z) has a finite value at z = 0. The only contributions to the

-R

+R

0

(z)

-R

+R

(z)

−i ν0 −i

ξ

ξ

Fig. 6.1 Contours for the determination of the infinite medium Green function by application of the Fourier inversion formula. Left panel: complete frequency redistribution; right panel: monochromatic scattering

100

6 The Surface Green Function and the Resolvent Function

contour integral come from the real axis and the two vertical lines on each side of the cut. We thus obtain    0 1 1 1 −i ξ τ − ∞ (τ ) = lim e dξ, η > 0, real. η→0 2π −i ∞ V (ξ − η) V (ξ + η) (6.5) Making the change of variable ξ = −i ν and using V (z) = L(i z), we obtain  ∞ (τ ) =

∞ 0

φ∞ (ν)e−ντ dν,

τ ∈ [0, ∞[,

(6.6)

with   1 1 1 − . φ∞ (ν) = 2i π L+ (ν) L− (ν)

(6.7)

The Plemelj formulae for L(z) given in Eq. (5.21) yield φ∞ (ν) =

(1 − )k(ν) , λ2 (ν) + π2 (1 − )2 k 2 (ν)

ν ∈ [0, ∞[.

This equation can be found in (Ivanov 1973, , p. 155). The infinite medium Green function can then be written as  ∞ G∞ (τ ) = δ(τ ) + φ∞ (ν)e−ντ dν, τ ∈ [0, ∞[.

(6.8)

(6.9)

0

Since G∞ (τ ) and ∞ (τ ) are even functions of τ , Eqs. (6.6) and (6.9) hold also for τ < 0, provided τ is changed to |τ |. As already mentioned, the Fourier transform method cannot be applied directly to G∞ (τ ) because it has a singularity at τ = 0. The behavior of ∞ (τ ) for large values of τ is examined in Chap. 22. It is shown that G∞ (τ ) decreases algebraically at large optical depths, the power law depending on the large frequency behavior of the absorption profile ϕ(x).

6.1.1.2 Monochromatic Scattering The infinite medium resolvent function ∞ (τ ) is also given by Eq. (6.4). The dispersion function V (z) has two zeroes at z = ±i ν0 , ν0 ∈ [0, 1[, and is analytic in the complex plane cut along ] − i ∞, −i ] ∪ [−i , i ∞[. To calculate the integral in Eq. (6.4) for τ positive, we choose the integration contour shown in the right panel of Fig. 6.1. It is similar to the contour shown in the left panel of the same figure, except that it includes a simple pole of the integrand at k = −i ν0 . The contribution of the vertical lines along the cut can be written as in Eq. (6.5) with the upper bound of the integral set to −i . The contribution of the outer curves also tends to zero

6.1 Infinite Medium

101

as they are sent to infinity. The contribution of the pole can be calculated with the Residue Theorem, or by modifying the integration contour to turn around the pole at 0 exp(−ν τ ), where φ 0 can be derived −i ν0 . The pole yields a term of the form φ∞ 0 ∞ from the Taylor expansion of V (z) around −i ν0 . Using V (z) = L(i z), we obtain 0 =− φ∞

1 , L (ν0 )

(6.10)

where L (ν0 ) is the derivative of L(z) at ν0 . This derivative is well defined since L(z) is analytic in the strip |(z)| < 1 containing ν0 . Using Eq. (5.15) we obtain 0 = φ∞

ν0 (1 − ν02 )

(6.11)

. ν02 −  √ √ 0  3/(2√). When  tends to zero, ν0  3, and φ∞ We thus obtain for the infinite medium Green function, G∞ (τ ) =

0 −ν0 |τ | δ(τ ) + φ∞ e

 +

∞

1

φ∞ (ν)e−ν|τ | dν,

(6.12)

where φ∞ (ν) =

(1 − )k(ν) , λ2 (ν) + π2 (1 − )2 k 2 (ν)

ν ∈ [1, ∞[.

(6.13)

Not surprisingly, φ∞ (ν) has the same expression for monochromatic scattering and complete frequency redistribution. For monochromatic scattering, k(ν) = 1/(2ν) for ν ∈ [1, ∞[ and λ(ν) has an explicit expression, given in Eq. (5.28) and shown in Fig. 5.4. For |τ | → ∞, G∞ (τ ) decreases exponentially as e−ν0 τ , while it has an algebraic behavior for complete frequency redistribution.

6.1.2 The Inverse Laplace Transform Method We consider the infinite medium resolvent function for complete frequency redistribution. The application of an inverse Laplace transform to Eq. (6.1) leads to the singular integral equation  λ(ν)φ∞ (ν) + η(ν)

+∞ φ −∞

∞ (ν ) ν − ν

dν = −η(ν),

ν ∈ R.

(6.14)

We solve it here with the Hilbert transform method described in Sect. 4.3.1. Introducing 1 F (z) ≡ 2iπ



+∞ −∞

φ∞ (ν) dν, ν−z

(6.15)

102

6 The Surface Green Function and the Resolvent Function

and using the Plemelj formulae for F (z) and L(z), we can rewrite Eq. (6.14) as L+ (ν)[F + (ν) +

1 1 ] − L− (ν)[F − (ν) + ] = 0, 2iπ 2iπ

ν ∈ R.

(6.16)

To obtain Eq. (6.16), the right-hand side −η(ν) has been expressed in terms of the limiting values L± (ν). This kind of trick should be used whenever possible because it simplifies the algebra. Equation (6.16) is of the type described in Sect. 4.3.1, because L(z) and F (z) have the same domain of analyticity. The general solution of Eq. (6.16) takes the form L(z)[F (z) +

1 ] = P (z), 2iπ

(6.17)

where P (z) is an entire function. At infinity P (z) → 1/(2i π) since F (z) → 0 and L(z) → 1. So, by the Liouville theorem,1 P (z) = 1/2i π, for all values of z. We thus find F (z) =

1 1 [ − 1]. 2iπ L(z)

(6.18)

Combining Eq. (6.18) with φ∞ (ν) = F + (ν) − F − (ν) (the first Plemelj formula for F (z)), we recover the result given in Eq. (6.7). The inverse Laplace transform method avoids the Fourier inversion step, as it directly provides (τ ) as a Laplace type integral, but it requires the introduction of the singular integral equation for φ∞ (ν).

6.2 Semi-infinite Medium. The Inverse Laplace Transform Method We apply here the inverse Laplace transform method to construct an exact expression for the resolvent function (τ ). The surface Green G(τ ) is then simply given by G(τ ) = (τ ) + δ(τ ). Complete frequency redistribution is treated in Sect. 6.2.1 and monochromatic scattering in Sect. 6.2.2. Notations for the direct and inverse Laplace transforms of the Green and resolvent functions are given in Table 6.2.

1

When P (z) is a polynomial tending to a constant at infinity, it is easy to conceive that it has only one term equal to its value at infinity. Similarly, when P (z) increases linearly at infinity, then it will necessary be of the form P (z) = p0 + p1 z, where the coefficients p0 and p1 are constants.

6.2 Semi-infinite Medium. The Inverse Laplace Transform Method

103

Table 6.2 Semi-infinite medium. G(τ ) is the surface Green function, and (τ ) the resolvent function. Columns (2) and (3) give the direct and inverse Laplace transforms of the functions in column (1). Column (4) gives the primary source term for the convolution integral equation satisfied by the functions in column (1). Columns (5) and (6) give the direct and inverse Laplace transforms of the functions in column (4). The notation—stands for nonexistent functions (1) G(τ ) (τ )

(2) laplace ˜ G(p) ˜ φ(p)

(3) inv. laplace – φ(ν)

(4) source δ(τ ) (1 − )K(τ )

(5) laplace 1 –

(6) inv. laplace – (1 − )k(ν)

6.2.1 Complete Frequency Redistribution For a semi-infinite medium, the Wiener–Hopf integral equation for the resolvent function (τ ) may be written as 

∞

(τ ) = (1 − )

K(τ − τ )(τ ) dτ + (1 − )K(τ ).

(6.19)

0

Compared to Eq. (6.1) the integration is now on [0, ∞[. Applied to Eq. (6.19), the inverse Laplace transform method described in Sect. 4.1.1, leads to a singular integral equation for φ(ν), the inverse Laplace transform of (τ ). This equation may be written as  λ(ν)φ(ν) + η(ν)

∞ 0

φ(ν ) dν = −η(ν), ν − ν

ν ∈ [0, ∞[.

(6.20)

Following the Hilbert transform method described in Sect. 4.3, we rewrite Eq. (6.20) as L+ (ν)[F + (ν) +

1 1 ] − L− (ν)[F − (ν) + ] = 0, 2iπ 2iπ

ν ∈ [0, ∞[,

(6.21)

where L± (ν) are the boundary values of L(z) and F ± (ν) the boundary values of 1 F (z) ≡ 2iπ



∞ 0

φ(ν) dν, ν −z

ν ∈ [0, ∞[.

(6.22)

The functions L(z) and F (z) do not have the same domain of analyticity since | cut along the full real axis (the point z = 0 being outside L(z) is analytic in C | cut along the positive real axis. Proceeding as the cut) and F (z) is analytic in C described in Sect. 4.3.2, we divide by L− (ν), introduce the auxiliary function X(z), which satisfies X+ (ν)/X− (ν) = L+ (ν)/L− (ν), and multiply by X− (ν). These

104

6 The Surface Green Function and the Resolvent Function

operations are legitimate because L± (ν) and X± (ν) have no zeroes for ν ∈ [0, ∞[. Equation (6.21) can thus be written as X+ (ν)[F + (ν) +

1 1 ] − X− (ν)[F − (ν) + ] = 0, 2iπ 2iπ

ν ∈ [0, ∞[.

(6.23)

The general solution of Eq. (6.23) has the form X(z)[F (z) +

1 ] = P (z), 2i π

(6.24)

where P (z) is an entire function which should be chosen in such a way that F (z) → 0 as z → ∞. This condition is satisfied when P (z) → 1/(2i π) as z → ∞, since X(z) → 1 for z → ∞. The Liouville theorem then tells us that P (z) = 1/(2i π) for all z. We thus obtain F (z) =

1 1 [ − 1]. 2i π X(z)

(6.25)

The Plemelj formulae in Eqs. (4.25) and (4.26) yield 1 1 1 [ + − − ], 2i π X (ν) X (ν)

φ(ν) = 

∞ 0

ν ∈ [0, ∞[,

1 1 φ(ν ) 1 dν = [ + + − ] − 1.

ν −ν 2 X (ν) X (ν)

(6.26)

(6.27)

Combining Eq. (6.26) with Eq. (6.7) and using the factorization relation in Eq. (5.48), we find φ(ν) = X(−ν)φ∞ (ν),

ν ∈ [0, ∞[,

(6.28)

where φ∞ (ν) is given in Eq. (6.8). The exact expression for the surface Green function may thus be written as  G(τ ) = δ(τ ) + 0

∞

X(−ν)φ∞ (ν)e−ντ dν,

τ ∈ [0, ∞[.

(6.29)

At infinity G(τ ) goes algebraically to zero. The scaling depends on the behavior of the line absorption profile ϕ(x) at large values of |x| (see Chap. 22).

6.2 Semi-infinite Medium. The Inverse Laplace Transform Method

105

6.2.2 Monochromatic Scattering For monochromatic scattering, φ(ν) satisfies the singular integral equation  λ(ν)φ(ν) + η(ν)

∞ 0

φ(ν ) dν = −η(ν), ν − ν

ν ∈ [0, ∞[.

(6.30)

The difference with Eq. (6.20), written above for complete frequency redistribution, lies in the coefficients λ(ν) and η(ν). For monochromatic scattering, η(ν) = 0 for ν ∈ [0, 1] and λ(ν) has a zero at ν0 ∈ [0, 1] as shown in Sect. 5.2.2. The condition α(ν) − i πβ(ν) = 0 stated in Sect. 4.3.2 as being necessary for the application of the Hilbert transform method is not satisfied for monochromatic scattering because of the zero of λ(ν) − i πη(ν) at ν = ν0 . The method for solving Eq. (6.30) is to consider separately the two intervals ν ∈ [0, 1[ and ν ∈ [1, ∞[. Interval ν ∈ [0, 1[ In this interval, Eq. (6.30) reduces to λ(ν)φ(ν) = 0.

(6.31)

Because λ(ν) has a zero at ν0 , the solution of Eq. (6.31) has the form φ(ν) = φ0 δ(ν − νo ),

ν ∈ [0, 1[,

(6.32)

where φ0 is a constant, undetermined at this stage. Here we use the property that the Dirac distribution δ(x) satisfies xδ(x) = 0. This property is easily verified by applying xδ(x) to a test function (Schwartz 1961). Interval ν ∈ [1, ∞[ Introducing Eq. (6.32) into Eq. (6.30), we obtain  λ(ν)φc (ν) + η(ν)

∞φ 1

c (ν )

ν −ν

dν = −η(ν)[1 −

φ0 ], ν − ν0

ν ∈ [1, ∞[.

(6.33)

The function φc (ν) is defined by φc (ν) ≡ φ(ν)

for ν ∈ [1, ∞[,

φc (ν) = 0

for ν ∈ [0, 1].

(6.34)

The equation for φc (ν) satisfies the conditions for an application of the Hilbert transform method since η(ν) = 0 for ν ∈ [1, ∞[. Proceeding now as above, we introduce the Hilbert transform 1 F (z) ≡ 2i π



∞ 1

φc (ν) dν, ν −z

ν ∈ [1, ∞[.

(6.35)

106

6 The Surface Green Function and the Resolvent Function

Expressing λ(ν) and η(ν), in both the left-hand side and right-hand side, in terms of the limiting values of L(z) and using the Plemelj formulae for F (z), we obtain the boundary value equation   φ0 1 ( − 1) L+ (ν) F + (ν) − 2i π ν − ν0   φ0 1 − − ( − L (ν) F (ν) − − 1) = 0, 2i π ν − ν0

ν ∈ [1, ∞[.

(6.36)

Multiplying by (ν − ν0 ) and replacing L+ (ν)/L− (ν) by X+ (ν)/X− (ν), Eq. (6.36) becomes   1 [φ0 − (ν − ν0 )] X+ (ν) (ν − ν0 )F + (ν) − 2i π   1 [φ0 − (ν − ν0 )] = 0, ν ∈ [1, ∞[. (6.37) − X− (ν) (ν − νo )F − (ν) − 2i π The general solution of Eq. (6.37) has the form   1 [φ0 − (z − ν0 )] = P (z), X(z) (z − ν0 )F (z) − 2i π

(6.38)

where P (z) is an entire function. Knowing that X(z)  −1/z as z → ∞, the condition that F (z) tends to zero as z → ∞ is satisfied when P (z) tends to −1/(2i π) at infinity. The Liouville theorem leads to P (z) = −1/(2i π),

(6.39)

for all z. Being a Hilbert transform, F (z) should be free of singularities, except for the cut along [1, ∞[. This condition is satisfied by choosing φ0 =

1 . X(ν0 )

(6.40)

The condition on φ0 corresponds to the solvability condition discussed in Sect. 4.3.2. We thus obtain F (z) =

1 1 1 1 1 [ − ]− . 2i π ν0 − z X(z) X(ν0 ) 2i π

(6.41)

The first Plemelj formula applied to F (z) yields φc (ν) =

1 1 1 1 [ − ], 2i π ν0 − ν X+ (ν) X− (ν)

ν ∈ [1, ∞[.

(6.42)

6.2 Semi-infinite Medium. The Inverse Laplace Transform Method

107

Using Eq. (B.13) to express the square bracket in terms of X(ν), and introducing the function X∗ (z) = (ν0 − z)X(z), we can rewrite φc (ν) as φc (ν) = φ∞ (ν)X∗ (−ν),

ν ∈ [1, ∞[,

(6.43)

where φ∞ (ν) is given in Eq. (6.13). We recover for φc (ν) the complete frequency redistribution result given in Eq. (6.28), provided we replace X(z) by X∗ (z). Adding the contributions from the intervals [0, 1] and [1, ∞[, we thus obtain φ(ν) =

1 δ(ν − ν0 ) + φc (ν). X(ν0 )

(6.44)

We note that the second Plemelj formula yields 

∞φ 1

c (ν )

ν −ν

dν =

1 1 1 1 1 1 [ + ]−1− , 2 ν0 − ν X + (ν) X − (ν) νo − ν X(ν0 )

ν ∈ [1, ∞[.

(6.45) Using φ0 = 1/X(ν0 ) and introducing X∗ (z), we can rewrite Eq. (6.45) as 

∞ 0

  1 φ(ν ) 1 1

dν = −1 + ∗ ν − ν 2 [X∗ (ν)]+ [X (ν)]− = X∗ (−ν)

λ(ν) − 1, R(ν)

ν ∈ [1, ∞[.

Our final result for the resolvent function is thus  ∞ 1 e−ν0 τ + φ∞ (ν)X∗ (−ν)e−ντ dτ, (τ ) = X(ν0 ) 1

τ ∈ [0, ∞[.

(6.46)

(6.47)

It can also be given an expression similar to that of complete frequency redistribution, namely 

∞

(τ ) =

φ(ν)e−ντ dν,

(6.48)

0

where φ(ν) is given by Eq. (6.44). For monochromatic scattering, the inverse Laplace transform φ(ν) contains a distribution. The inverse Laplace transform of the source function S(τ ) also contains a distribution (see Appendix D.1 in Chap. 7). Equation (6.47) shows that (τ ) and G(τ ) = (τ ) + δ(τ ) vary as e−ν0 τ for large values of τ . Hence, for  = 0, they tend exponentially to √ zero as τ → ∞. For  = 0, since ν0 = 0, they tend to the constant 1/X(0) = 3. This result is also demonstrated in Chap. 9 devoted to the Milne problem.

108

6 The Surface Green Function and the Resolvent Function

6.3 The H -Function ˜ We show here that the H -function is directly related to the Laplace transform G(p) of the surface Green function G(τ ). A simple method described in this Section for ˜ obtaining G(p) is to derive it from the expression of φ(ν), already established in Sect. 6.2. Another method, described in Appendix C of this chapter, is to solve the ˜ singular integral equation satisfied by G(p). As has been pointed out in Sect. 4.2, direct and inverse Laplace transforms are simply related (see Eq. (4.21)). Combining the definitions of the direct and inverse Laplace transforms of (τ ) and using G(τ ) = (τ ) + δ(τ ), we obtain ˜ G(p) =



∞

G(τ )e−pτ dτ =

0



∞

0

φ(ν) dν + 1. ν+p

(6.49)

In Sect. 6.2, we give expressions of φ(ν) in terms of the jump of the function 1/X(z) across its singular line. For complete frequency redistribution, this expression is given in Eq. (6.26). For monochromatic scattering, it is given in Eqs. (6.44) and (6.42). These expressions suggest to introduce a contour integration for the ˜ calculation of G(p). For complete frequency redistribution, we introduce the contour integral 1 2i π

 C

1 1 dξ, p + ξ X(ξ )

p ∈ [0, ∞[,

| , ξ ∈C

(6.50)

where the contour C is shown in the left panel of Fig. 6.2. The contribution from the outer curve and from the two lines above and below the cut yield the right-hand side in Eq. (6.49). We recall that X(ξ ) → 1 as ξ → ∞. The contribution of the outer curve is calculated by considering that this curve is a circle of radius R, and letting R go to infinity. The residue of the simple pole at ξ = −p gives a term −1/X(−p). (ξ)

(ξ)

(ξ)

(ξ) −p

0

−p

0 ν 0

1

˜ Fig. 6.2 Contours for the calculation of G(p), the Laplace transform of the surface Green function G(τ ). Left panel: complete frequency redistribution. Right panel: monochromatic scattering

6.3 The H -Function

109

One thus obtains ˜ G(p) =

1 , X(−p)

p ∈ [0, ∞[.

(6.51)

For monochromatic scattering, Eq. (6.44) leads to ˜ G(p) =1+ +

1 2i π



∞ 1

1 X(ν0 )(ν0 + p)

1 1 1 1 [ − ] dν, ν + p ν0 − ν X+ (ν) X− (ν)

p ∈ [0, ∞[.

(6.52)

To calculate the integral, we consider the contour integral 1 2i π

 C

1 1 1 dξ, ξ + p ν0 − ξ X(ξ )

| , ξ ∈C

(6.53)

where C is the contour shown in the right panel of Fig. 6.2. Using the Residue Theorem to calculate the contributions from the simple poles at −p and ν0 and remembering that X(ξ ) ∼ −1/ξ for ξ → ∞, we obtain ˜ G(p) =

1 1 1 = ∗ . ν0 + p X(−p) X (−p)

(6.54)

We recover the expression given for complete frequency redistribution, with X∗ (−p) instead of X(−p). These relations can be analytically continued to the complex plane. We have observed in Sect. 5.4, that that H (z) = 1/X(−1/z) for complete frequency redistribution and H (z) = 1/X∗ (−1/z) for monochromatic scattering ˜ (see Eq. (5.62)). The expressions of G(p) obtained above show that 1 ˜ G(p) = H ( ), p

p ∈ [0, ∞[,

(6.55)

for both complete frequency redistribution and monochromatic scattering. Hence, the H -function can be defined as the Laplace transform of the surface Green function. This definition can be generalized to Rayleigh scattering and resonance lines polarization (see Part II). For p = 0, Eq. (6.55) leads to 1 ˜ G(0) = H (∞) = √ , 

(6.56)

110

6 The Surface Green Function and the Resolvent Function

and to the normalization  ∞



∞

G(τ ) dτ =

0

0

1 (τ ) dτ + 1 = √ . 

(6.57)

The integral of G(τ ) provides the mean number of scatterings of photons starting √ their random walk at the boundary of a semi-infinite medium. The value 1/  comes from the symmetry of the random walk, as demonstrated in Sect. 21.3. ˜ In Appendix C of this chapter we show how to determine G(p) with the ˜ direct Laplace transform method. The equation for G(p) is also a singular integral equation, but the method of solution has some differences with the method described in Sect. 6.2 for φ(ν). The reason is that direct and inverse Laplace transforms lead to singular integral equations, which are adjoint to each other (see Sect. 4.3).

Appendix C: The Direct Laplace Transform Method ˜ We give here an alternative proof of the result G(p) = H (1/p). We derive it ˜ from the singular integral equation satisfied by G(p). This equation is similar to the singular integral equation for φ(ν), but its integral operator is the adjoint of the operator in the equation for φ(ν) (see Eq. (6.20)). We also apply the Hilbert transform method of solution, but there are some differences with the method described in Sect. 6.2 to solve Eq. (6.20), which are worth presenting. The surface Green function G(τ ) satisfies the Wiener–Hopf integral equation  G(τ ) = (1 − )

∞

K(τ − τ )G(τ ) dτ + δ(τ ).

(C.1)

0

Applying a Laplace transform and proceeding as shown in Sect. 4.2, we obtain the singular integral equation ˜ λ(p)G(p) −



∞ η(p )G(p ˜ ) 0

p − p

dp = 1,

p ∈ [0, ∞[.

(C.2)

It is of the type introduced in Eq. (4.23) and, as we now show, can be solved by Hilbert transform method, which leads to a boundary value problem in the complex plane. The method is applicable even when η(p) is not different from zero everywhere. The only requirement is that λ(p) and η(p) do not have a common zero. This means that, at some point, it will become necessary to distinguish between complete frequency redistribution and monochromatic scattering. We introduce the Hilbert transform 1 F (z) ≡ 2i π



∞ 0

˜ η(p)G(p) dp, p−z

p ∈ [0, ∞[.

(C.3)

C.1 Complete Frequency Redistribution

111

Its limiting values along the interval [0, ∞[ satisfy the Plemelj formulae

1 iπ



˜ η(p)G(p) = F + (p) − F − (p), ∞ η(p )G(p ˜ )

p − p

0

dp = F + (p) + F − (p),

(C.4) p ∈ [0, ∞[.

(C.5)

We now follow the method suggested in Muskhelishvili (1953, p. 128). Using Eq. (C.5), we can rewrite Eq. (C.2) as ˜ λ(p)G(p) = i π[F + (p) + F − (p)] + 1.

(C.6)

Equation (C.4) can be rewritten as ˜ i πη(p)G(p) = i π[F + (p) − F − (p)].

(C.7)

Adding and subtracting these two equations, and using the Plemelj formulae for L(z) written in Eq. (5.21), we can write ˜ = 2i πF + (p) + 1, L+ (p)G(p) ˜ L− (p)G(p) = 2i πF + (p) − 1,

(C.8)

p ∈ [0, ∞[.

(C.9)

˜ This operation, which leads Dividing by L+ (p) and L− (p), we can eliminate G(p). to a boundary value problem, requires that L+ (p) and L− (p) have no zero for p ∈ [0, ∞[. This condition is satisfied for complete frequency redistribution but not for monochromatic scattering. This leads us to treat the two cases separately.

C.1 Complete Frequency Redistribution Combining Eqs. (C.8) and (C.9), we obtain L− (p)[F + (p) +

1 1 ] − L+ (p)[F − (p) + ] = 0, 2i π 2i π

p ∈ [0, ∞[.

(C.10)

Since we are considering a half-space problem, L(z) and F (z) have different domains of analyticity. We therefore introduce the half-space auxiliary function X(z). Using L+ (p)/L− (p) = X+ (p)/X− (p), we readily obtain 1 1 1 1 [F + (p) + ]− − [F − (p) + ] = 0, X+ (p) 2i π X (p) 2i π

p ∈ [0, ∞[.

(C.11)

112

6 The Surface Green Function and the Resolvent Function

Proceeding then exactly as in Sect. 6.2.1, we find F (z) =

1 [X(z) − 1]. 2i π

(C.12)

The Plemelj formula in Eq. (C.4) leads to ˜ η(p)G(p) =

1 [X+ (p) − X− (p)], 2i π

p ∈ [0, ∞[.

(C.13)

Replacing the square bracket by the expression in Eq. (B.7), we recover Eq. (6.51), namely ˜ G(p) =

1 , X(−p)

p ∈ [0, ∞[.

(C.14)

The expression of G(τ ) given in Eq. (6.29) can be recovered by applying the ˜ Fourier inversion formula to G(p)−1, the Laplace transform of (τ ). The inversion formula may be written as 1 (τ ) = lim R→∞ 2i π



c+i R



c−i R

 1 − 1 eξ τ dξ, X(−ξ )

| , ξ ∈C

(C.15)

˜ where c and R are real. We can choose c = 0 since G(p) = is defined for all p real positive. To calculate (τ ), we integrate along the contour shown in the left panel of Fig. C.1. Since τ ≥ 0, the contour must lie in the half-plane (ξ ) < 0 to ensure the convergence of the integral. There is no contribution from the outside curves when they are sent to infinity, since [1/X(ξ ) − 1] tends to zero as ξ → ∞, nor from the (ξ)

(ξ)

c+iR

c+iR

0

(ξ)

(ξ)

−1 ν0

0

c−iR

c−iR

Fig. C.1 Integration contours for the application of the Fourier inversion formula. Left panel: calculation of the resolvent function (τ ) for complete frequency redistribution. Right panel: calculation of the source function S(τ ) for monochromatic scattering

C.2 Monochromatic Scattering

113

end point of the cut because 1/X(ξ ) has a finite value at ξ = 0 for non-conservative scattering ( = 0). The only non-zero contribution comes from the integration of 1/X(−ξ ) along the horizontal lines below and above the cut at a distance |y| from the cut. We thus obtain   ∞ e+i |y|τ 1 e−i |y|τ − (τ ) = (C.16) e−ντ dν. 2i π 0 X(ν + i |y|) X(ν − i |y|) Taking now the limit y → 0, we recognize in the square bracket the expression of φ(ν) given in Eq. (6.26).

C.2 Monochromatic Scattering ˜ The Laplace transform G(p) satisfies Eq. (C.2), with the lower bound of the singular operator set to one, since η(p) = 0 for p ∈ [0, 1]. We define F (z) as in Eq. (C.3), with the lower bound set to 1. The Plemelj formulae in Eqs. (C.4) and (C.5) hold for p ∈ [1, ∞[. Equation (C.6) holds for p ∈ [0, ∞[ and Eq. (C.7) for p ∈ [1, ∞[. We treat the two intervals p ∈ [1, ∞[ and p ∈ [0, 1[ separately. Interval p ∈ [1, ∞[ In this interval, we can proceed exactly as described in Sect. C.1. We obtain Eq. (C.11) with p ∈ [1, ∞[. It solution has the form F (z) = X(z)P (z) −

1 , 2i π

(C.17)

where P (z) is a polynomial in z. For monochromatic scattering, X(z)  −1/z as z → ∞. To ensure that F (z) → 0 for z → ∞, one must have P (z)  −z/(2i π) for z → ∞. This implies P (z) = −

z + P0 , 2i π

(C.18)

where P0 is a constant, still to be determined. Interval p ∈ [0, 1[ Since F (z) is analytic for z ∈ [0, 1[, Eq. (C.2) reduces to ˜ λ(p)G(p) − 2i πF (p) = 1,

p ∈ [0, 1[.

(C.19)

Inserting the expression of F (z) given in Eq. (C.17), we obtain ˜ λ(p)G(p) = 2i πX(p)P (p),

p ∈ [0, 1[.

(C.20)

114

6 The Surface Green Function and the Resolvent Function

The right-hand side contains the undetermined constant P0 . It can be determined by setting p = ν0 . For p = ν0 , one has λ(ν0 ) = 0, while X(ν0 ) has a finite value and ˜ 0 ) also, otherwise G(τ ) would tend to infinity as eν0 τ . Therefore Eq. (C.20) can G(ν be satisfied for p = ν0 , if and only if P (ν0 ) = 0. This constraint leads to P (z) =

1 (ν0 − z). 2i π

(C.21)

The function F (z) can thus be written as F (z) =

1 [X∗ (z) − 1]. 2i π

(C.22)

We recover the expression of F (z) given in Eq. (C.12) for complete frequency redistribution, with X(z) replaced by X∗ (z). ˜ The function G(p) can now be determined for all values of p. For the interval p ∈ [1, ∞[, it suffices to apply the first Plemelj formula to Eq. (C.22) and to use Eq. (B.11). For the interval p ∈ [0, 1], it suffices to replace in Eq. (C.20) λ(p) by L(p) and then to use the factorization relation X(p)X(−p) = L(p)/(ν02 − p2 ). One thus obtains, ˜ G(p) =

1 1 1 = ∗ . ν0 + p X(−p) X (−p)

p ∈ [0, ∞[.

(C.23)

We recover the expression given in Eq. (6.54), derived with an inverse Laplace transform method. The function (τ ) can then be obtained by a Fourier inversion. We give in Sect. D.2 another example of application of the Hilbert transform method to a singular integral equation similar to Eq. (C.2).

References Carrier, G.F., Krook, M., Pearson, C.E.: Functions of a Complex Variable. McGraw-Hill Book Company, New York (1966) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Muskhelishvili, N.I.: Singular Integral Equations, Noordhoff, Groningen (Based on the Second Russian Edition Published in 1946) (1953); Dover Publications (1991) Schwartz, L.: Méthodes Mathématiques pour les Sciences Physiques, Hermann, Paris (1961); English translation: Mathematical Methods for the Physical Sciences. Dover, New York (2008)

Chapter 7

The Emergent Intensity and the Source Function

In this chapter we present exact results for the emergent intensity and the source function, for complete frequency redistribution, relevant to the formation of spectral lines, and for monochromatic scattering, relevant to the formation of continuous spectra. We make use of results obtained in the preceding chapters for the Green and resolvent functions and apply the inverse and direct Laplace transforms methods to solve Wiener–Hopf integral equations for the source function. The Chapter is organized as follows. In Sect. 7.1 we make use of results on the Green function established in Chap. 2 to construct a simple explicit expression of the emergent intensity. It involves the H -function and q ∗ (ν), the inverse Laplace transform of the primary source Q∗ (τ ). Section 7.2 is devoted to the source function S(τ ). For complete frequency redistribution, we describe the inverse and direct Laplace transform methods. These methods are algebraically more difficult to apply for monochromatic scattering. They are presented in Appendix D of this chapter. For monochromatic scattering, we use the expression of the emergent intensity established in Sect. 7.1 to give an exact expression of S(τ ). Depending on the method, S(τ ) is obtained in terms of q ∗ (ν) or q˜ ∗ (p), the inverse and direct Laplace transforms of the primary source. Finally, in Sect. 7.3, the general expressions obtained for the emergent intensity and source function are applied to the standard problems of a uniform and an exponential primary source, and to the diffuse reflection problem.

7.1 The Emergent Intensity In a semi-infinite medium, the emergent intensity for monochromatic scattering and complete frequency redistribution can be written as  Iξ (0, ξ ) = 0

∞

S(τ )e−τ/ξ

1 dτ, ξ

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_7

(7.1)

115

116

7 The Emergent Intensity and the Source Function

where ξ = μ, μ ∈ [0, 1], for monochromatic scattering and ξ = μ/ϕ(x), ξ ∈ [0, ∞[, for complete frequency redistribution. For monochromatic scattering Iξ (0, ξ ) = I (0, μ) and for complete frequency redistribution, Iξ (0, ξ ) = I (0, x, μ) (see Eqs. (2.66) and (2.67)). Introducing p = 1/ξ , we can rewrite the emergent intensity in terms of the Laplace transform of the source function, namely 

∞

I (0, p) = p

S(τ ) e−pτ dτ = ps˜ (p).

(7.2)

0

One can with this equation construct a simple expression of the emergent intensity in terms of the H -function. Expressing S(τ ) in terms of the Green function G(τ, τ0 ), we can write 

∞ ∞

I (0, p) = p 0

G(τ, τ0 ) Q∗ (τ0 ) e−pτ dτ0 dτ,

(7.3)

0

where Q∗ (τ0 ) is the primary source term. Introducing q ∗ (ν), the inverse Laplace transform of Q∗ (τ0 ), we can rewrite Eq. (7.3) as 

∞

I (0, p) = p

˜˜ G(p, ν)q ∗ (ν) dν,

(7.4)

0

˜˜ where G(p, ν) is the double Laplace transform of G(τ, τ0 ) defined by ˜˜ G(p, ν) ≡



∞ ∞ 0

G(τ, τ0 ) e−pτ e−ντ0 dτ0 dτ.

(7.5)

0

It is shown in Eq. (2.96) that ˜ ˜ G(p) G(ν) ˜˜ , G(ν, p) = ν+p

(7.6)

˜ where G(p) is the Laplace transform of the surface Green function G(τ ), and in Eq. (6.55) that 1 ˜ G(p) = H ( ). p

(7.7)

We thus obtain 1 I (0, p) = H ( ) p



∞ 0

p 1 q ∗ (ν) dν. H( ) ν p+ν

(7.8)

7.1 The Emergent Intensity

117

For monochromatic scattering Eq. (7.8) leads to 

∞

I (0, μ) = H (μ) 0

1 1 q ∗ (ν)H ( ) dν, 1 + νμ ν

μ ∈ [0, 1].

(7.9)

and for complete frequency redistribution to μ ) I (0, x, μ) = H ( ϕ(x)



∞ 0

ϕ(x) 1 q ∗ (ν)H ( ) dν, ϕ(x) + νμ ν

μ ∈ [0, 1].

(7.10)

These expressions involve only the inverse Laplace transform of the primary source term and the auxiliary function for half-space problems. They are applied to three standard problems in Sect. 7.3. With the expressions obtained for the emergent intensity we can derive some properties of the source function. Combining Eqs. (7.2) and (7.8), we can write the Laplace transform of the source function as 1 s˜ (p) = H ( ) p



∞ 0

1 1 q ∗ (ν) dν. H( ) ν p+ν

(7.11)

We show in Sect. 7.2.2 how to derive from this equation an exact expression of S(τ ). The surface value of S(τ ) is simply obtained by setting μ = 0 in the expressions of the emergent intensity given in Eqs. (7.10) and (7.9). One thus obtains 

∞

S(0) = 0

1 H ( )q ∗ (ν) dν, ν

(7.12)

for both complete frequency redistribution and monochromatic scattering. The dependence on the scattering process and on the value of the destruction parameter ∗  are accounted for by the H -function. For a uniform √ primary source, Q (τ ) = B, ∗ we have q (ν) = Bδ(ν). Using H (∞) = 1/  we readily recover the famous √ -law, S(0) =

√ B.

(7.13)

This law holds for any symmetric (even) kernel. It is a fundamental property of symmetric random walks in a semi-infinite medium. Many proofs of this fundamental result, involving only the symmetry of the kernel and the semi-infinite √ geometry of the medium have been proposed. We return to the subject of the -law in Sects. 11.1 and 21.3.

118

7 The Emergent Intensity and the Source Function

7.2 The Source Function In Chaps. 2 and 6, we show that the source function can be written as 

∞

S(τ ) =

G(τ, τ0 ) Q∗ (τ0 ) dτ0 ,

(7.14)

0

where G(τ, τ0 ) is the Green function. We also show that G(τ, τ0 ) can be expressed in terms of the surface Green function G(τ ), and construct, for the latter, exact expressions given in Eqs. (6.29) and (6.47), for complete frequency redistribution and monochromatic scattering, respectively. This approach provides S(τ ) in terms of the primary source term Q∗ (τ ) itself. Here, we use the inverse and direct Laplace transform methods described in the preceding chapters to solve the Wiener–Hopf integral equation  S(τ ) = (1 − )

∞

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

τ ∈ [0, ∞[.

(7.15)

0

The inverse Laplace transform method yields 

∞

S(τ ) =

s(ν) e−ντ dν,

(7.16)

0

and the direct Laplace transform method, S(τ ) =

1 2i π



+i ∞ −i ∞

s˜ (p) epτ dp,

τ ∈ [0, ∞[.

(7.17)

Table 7.1 recapitulates the notations for the direct and inverse Laplace transforms of the functions in Eq. (7.15). The function s(ν) is obtained in terms of q ∗ (ν) in Sect. 7.2.1.1 for complete frequency redistribution and in Appendix D.1 of this chapter for monochromatic scattering. The function s˜ (p) is obtained in terms of q˜ ∗ (p) in Sect. 7.2.1.2 for complete frequency redistribution and in Appendix D.2 of this chapter for monochromatic scattering. We also show in Sect. 7.2.2 for monochromatic scattering how to obtain s˜ (p) in terms of q ∗ (ν). The choice between different methods can be based on the structure of the direct and inverse Laplace transforms of the primary source. For example, for the standard problems considered in Sect. 7.3, q ∗ (ν) has a simpler structure than q˜ ∗ (p). The inverse Laplace transform method is simpler to apply. Whether one uses the inverse or direct Laplace transform method, the Wiener– Hopf integral equation for S(τ ) is easier to solve for complete frequency redistribution than for monochromatic scattering because the dispersion function has no zero. We start in Sect. 7.2.1 with complete frequency redistribution and consider monochromatic scattering in Sect. 7.2.2.

7.2 The Source Function

119

Table 7.1 Semi-infinite medium. S(τ ) denotes the source function, Q∗ (τ ) the primary source term and K(τ ) the kernel of the Wiener–Hopf integral equation for S(τ ). Columns (2) and (3) correspond to the direct and inverse Laplace transforms of the functions in column (1). Column (4) lists examples of primary source terms and columns (5) and (6) their direct and inverse Laplace transforms (1) S(τ ) Q∗ (τ ) K(τ )

(2) laplace s˜ (p) q˜ ∗ (p) ˜ k(p)

(3) inv. laplace s(ν) q ∗ (ν) k(ν)

(4) source δ(τ ) Q∗ Q∗ e−ατ

(5) laplace 1 Q∗ /p Q∗ /(p + α)

(6) inv. laplace – Q∗ δ(ν) Q∗ δ(ν − α)

7.2.1 Complete Frequency Redistribution The results obtained here with the inverse or direct Laplace transform methods are expressed in terms of the auxiliary function X(z). They can be written in terms of the H -function by using the relation H (z) = 1/X(−1/z).

7.2.1.1 Inverse Laplace Transform Method The inverse Laplace transform method applied to Eq. (7.15) leads to the singular integral equation  λ(ν)s(ν) + η(ν)

∞ 0

s(ν ) dν = q ∗ (ν), ν − ν

ν ∈ [0, ∞[,

(7.18)

already presented in Eq. (4.8). We solve it now with the Hilbert transform method described in Sect. 4.3 and applied to the calculation of the resolvent function (τ ) in Sect. 6.2.1. The only difference is that we deal here with an arbitrary inhomogeneous term q ∗ (ν). The Hilbert transform method leads to the boundary value equation : S + (ν)X+ (ν) − S − (ν)X− (ν) = G + (ν) − G − (ν),

ν ∈ [0, ∞[.

(7.19)

Here S ± (ν) and G ± (ν) are the limiting values of the functions 1 S(z) ≡ 2i π



∞ 0

s(ν) dν, ν −z

G(z) ≡

1 2i π



∞ 0

γ (ν) dν, ν −z

(7.20)

where γ (ν) ≡

q ∗ (ν)X− (ν) q ∗ (ν) = , L− (ν) X(−ν)

ν ∈ [0, ∞[.

(7.21)

120

7 The Emergent Intensity and the Source Function

The factorization relation (5.49) is used to express γ (ν) in terms of X(−ν). The general solution of Eq. (7.19) has the form S(z)X(z) − G(z) = P (z),

(7.22)

where P (z) is an entire function. Since X(z) → 1 and G(z) → 0 for z → ∞, the condition that S(z) tends to zero as z → ∞ is satisfied when P (z) → 0 for z → ∞. Hence, P (z) = 0 for all z. One obtains S(z) =

G(z) , X(z)

(7.23)

and s(ν) = S + (ν) − S − (ν) =

G − (ν) G + (ν) − , X+ (ν) X− (ν)

ν ∈ [0, ∞[.

(7.24)

The Plemelj formulae applied to G(z) (see Eqs. (A.7) and (A.8)) lead to 1 q ∗ (ν) 1 1 [ + ] 2 X(−ν) X+ (ν) X− (ν)  ∞ ∗

1 1 1 q (ν ) dν

+ [ + − − ] .



2i π X (ν) X (ν) 0 X(−ν ) ν − ν

s(ν) =

(7.25)

Using the relations (B.10) and (B.9), we can rewrite this result as   1 ∗ s(ν) = λ(ν)q (ν) − η(ν)X(−ν) R(ν)

∞ 0

 q ∗ (ν ) dν

, X(−ν ) ν − ν

ν ∈ [0, ∞[ (7.26)

with R(ν) = λ2 (ν) + π2 η2 (ν). This expression involves X(−ν) and the functions η(ν) and λ(ν) defined in Eqs. (5.22) and (5.23). Thus to calculate s(ν) and then S(τ ) by taking the Laplace transform of s(ν), one needs only the values of X(z) on the negative part of the real axis. It is also possible to express s(ν), in terms of φ(ν), the inverse Laplace transform of the resolvent function (τ ). Combining Eq. (7.25) with the Plemelj formulae for φ(ν) written in Eqs. (6.26) and (6.27), we obtain s(ν) =

  q ∗ (ν) 1+ X(−ν)

∞ 0

  φ(ν )

dν + φ(ν) ν − ν

∞ 0

q ∗ (ν ) dν

, X(−ν ) ν − ν

ν ∈ [0, ∞[. (7.27)

7.2 The Source Function

121

This equation is used in Sect. 7.3, where we consider some standard problems with simple primary sources.

7.2.1.2 Direct Laplace Transform Method The direct Laplace transform method described in Sect. 4.2 leads to  λ(p)˜s (p) −

∞ η(p )˜ s (p )

p − p

0

dp = q˜ ∗ (p),

p ∈ [0, ∞[.

(7.28)

˜ When q˜ ∗ (p) = 1, this equation reduces to the singular integral equation for G(p) (see Appendix C in Chap. 6). For q˜ ∗ (p) = 1/(ν + p), it yields the singular integral ˜˜ ˜˜ for G(p, ν), the solution of which is given in Eq. (7.6). We recall that G(p, ν) is the double Laplace transform of G(τ, τ0 ). Returning to the general case, we introduce a function S(z) ≡

1 2i π



∞ 0

η(p)˜s (p) dp. p−z

(7.29)

Proceeding as described in Appendix C in Chap. 6, we can rewrite Eq. (7.28) as S − (p) S + (p) − = G + (p) − G − (p), X+ (p) X− (p)

p ∈ [0, ∞[,

(7.30)

where G + (p) and G − (p) are the limiting values of the function 1 G(z) = 2i π



∞ 0

t (p)q˜ ∗ (p)

dp , p−z

(7.31)

with t (p) =

η(p) . L+ (p)X− (p)

(7.32)

The function t (p) is simply related to the inverse Laplace transform of the resolvent function (τ ). Using the first Plemelj formula for L(z), the defining relation for X(z) given in Eq. (5.36), and the expression of φ(ν) given in Eq. (6.26), we find t (p) = −

  1 1 1 − = −φ(p), 2i π X+ (p) X− (p)

(7.33)

For p → ∞, as shown in Sect. 6.2.1, φ(p)  1/p. Thus G(z) will tend to zero at infinity as 1/z, provided that q ∗ (p)/p is integrable, a condition which is usually satisfied.

122

7 The Emergent Intensity and the Source Function

The general solution of Eq. (7.30) has the form S(z) − G(z) = P (z), X(z)

(7.34)

where P (z) is an entire function of z. At infinity, X(z) → 1 and G(z) → 0. The condition S(z) → 0 for z → ∞ will be satisfied, when P (z) tends to zero at infinity. Since P (z) is a polynomial, this condition implies that P (z) = 0 for all z. One thus has the solution S(z) = X(z)G(z).

(7.35)

The first Plemelj formula applied to S(z) yields η(p)˜s (p) = X+ (p)G + (p) − X− (p)G − (p).

(7.36)

Making use of the Plemelj formulae for G(z), we obtain 1 [X+ (p) − X− (p)] η(p)˜s (p) = − 2i π



∞

φ(p )q˜ ∗ (p )

0

1 − [X+ (p) + X− (p)]φ(p)q˜ ∗ (p). 2

dp

p − p (7.37)

Expressing the sum and the difference of X± (p) with Eqs. (B.7) and (B.8) and φ(p) with the expression given in Eq. (6.28), namely φ(p) = φ∞ (p)X(−p) = −

η(p) X(−p), R(p)

(7.38)

where R(p) = L+ (p)L− (p), we can write s˜ (p) =

λ(p) ∗ 1 q˜ (p) − R(p) X(−p)



∞ 0

φ(p )q˜ ∗ (p )

dp

. −p

p

(7.39)

The first term in the right-hand side depends only on the dispersion function. The second one takes the half-space geometry into account. One can verify that this ˜ ˜ equation with q˜ ∗ (p) = 1, yields G(p) = 1/X(−p), with G(p) the Laplace transform of G(τ ) (hint : use the singular integral equation for φ(ν) in Eq. (6.20)). A Fourier inversion is still needed to write S(τ ) in the form of a Laplace type integral. An example is given in Sect. 7.2.2 for monochromatic scattering.

7.2 The Source Function

123

7.2.2 Monochromatic Scattering. A Fourier Inversion In Appendix D.1 of this chapter and in Appendix D.2 of this chapter, we apply to monochromatic scattering the inverse and direct Laplace transform methods described for complete frequency redistribution. Here we show that the expression of s(ν) that will be obtain in Appendix D.1 of this chapter can also be derived by a Fourier inversion from the expression of s˜ (p) given in Eq. (7.11). First we rewrite Eq. (7.11) as s˜ (p) =

1 1 ν0 + p X(−p)



∞ 0

1 q ∗ (ν) 1 dν, ν0 + ν X(−ν) p + ν

p, ν ∈ [0, ∞[.

(7.40)

The H -function has been replaced by its expression in terms of the X-function, in order to exhibit the hidden poles. The function X(z) for monochromatic scattering is analytic in the complex plane cut along [1, +∞[, free of poles and of zeroes and behaves as −1/z for z → ∞. The Fourier inversion formula may be written as S(τ ) =

1 2i π



+i ∞

−i ∞

s˜(ξ )eξ τ dξ,

τ ∈ [0, ∞[.

(7.41)

To carry out the integration, we include the imaginary axis in a contour in the complex plane. For τ ≥ 0, the contour must lie in the half-plane (ξ ) < 0 to ensure the convergence of the integral. To calculate the integral, it is convenient to write s˜(ξ ) as s˜ (ξ ) =

1 1 2i πG(−ξ ), ν0 + ξ X(−ξ )

(7.42)

with 1 G(ξ ) = 2i π

 0

∞

q ∗ (ν) dν 1 . ν0 + ν X(−ν) ν − ξ

(7.43)

The choice of the integration contour depends on the position of the poles and analyticity domain of s˜(ξ ). The function X(−ξ ) has a cut along ] − ∞, −1], the function G(−ξ ) has a cut along ] − ∞, 0]. In addition there is a pole at ξ = −ν0 embedded into the cut of G(−ξ ), since ν0 ∈ [0, 1[. The contour chosen for the integration is shown in the right panel of Fig. C.1. The contribution of the outside curve is zero because ξ X(−ξ ) → 1 and G(−ξ ) → 0 as 1/ξ , as ξ → ∞. For convenience we write S(τ ) = S0 (τ ) + S1 (τ ) + S2 (τ ),

(7.44)

124

7 The Emergent Intensity and the Source Function

with S0 (τ ) the contribution from the pole at −ν0 , S1 (τ ) the contribution from the two straight lines with (ξ ) ∈] − ∞, −1], and S2 (τ ) the contribution from the two intervals with (ξ ) ∈] − 1, 0]. First we consider the contribution of the pole at −ν0 . We set ξ = −ν0 + ρei θ with ρ → 0 and take into account the fact that G(−ξ ) has different values above and below the cut. We may then write  S0 (τ ) = lim

y→0 0

 +

π

2π π

G(ν0 + i y) −ν0 τ i dθ e X(ν0 )

G(ν0 − i y) −ν0 τ e i dθ, X(ν0 )

y > 0.

(7.45)

Taking the limit y → 0, we obtain S0 (τ ) = i π

 1  + G (ν0 ) + G − (ν0 ) e−ν0 τ . X(ν0 )

(7.46)

The second Plemelj formula applied to G(ξ ) yields S0 (τ ) = s0 e−ν0 τ

(7.47)

with s0 =

1 X(ν0 )



∞ 0

1 dν q ∗ (ν) . X(−ν) ν0 + ν ν − ν0

(7.48)

We now calculate the term S1 (τ ). The contributions of the two lines below and above the cut yield  S1 (τ ) = lim  +

−1

y→0 −∞

−∞

−1

G(−ν + i y) e(ν−i y)τ dν (ν0 + ν − i y)X(−ν + i y)

G(−ν − i y) e(ν+i y)τ dν, (ν0 + ν + i y)X(−ν − i y)

y > 0.

(7.49)

Making the change of variable ν → −ν and letting y → 0, we obtain 

∞

S1 (τ ) =

s1 (ν) e−ντ dν,

(7.50)

1

with s1 (ν) =

1 ν0 − ν



 G + (ν) G − (ν) − , X+ (ν) X− (ν)

ν ∈ [1, ∞[.

(7.51)

7.2 The Source Function

125

Using the Plemelj formulae for G(z) and Eqs. (B.13) and (B.14), we obtain s1 (ν) =

  1 λ(ν)q ∗ (ν) − η(ν)X∗ (−ν) R(ν)

∞ 0

 q ∗ (ν ) dν

, ν ∈ [1, ∞[. X∗ (−ν ) ν − ν (7.52)

We recall that X∗ (z) = (ν0 − z)X(z). The expression of s1 (ν) is identical to the expression of s(ν) given in Eq. (7.24) for complete redistribution, except that X(z) is replaced by X∗ (z). Written with the H -function, they would be identical. We finally consider the interval ] − 1, 0], ignoring the pole at ξ = −ν0 , which has already been taken into account. In this interval X(ξ ) is analytic but G(ξ ) has a branch cut. Proceeding as above, we obtain  S2 (τ ) = lim

y→0 0

1

  G − (ν) 1 G + (ν) − e−ντ dν, X(ν) ν0 − ν − i y ν0 − ν + i y

y > 0. (7.53)

We can let y → 0, provided we take the integral in Principal Value. Using the first Plemelj formula for G(ξ ), the factorization X(−z)X(z)(ν02 − z2 ) = L(z) and L(ν) = λ(ν) for ν ∈ [0, 1[, we obtain 

1

S2 (τ ) =

s2 (ν) e−ντ dν,

ν ∈ [0, 1],

(7.54)

0

with s2 (ν) = P

q ∗ (ν) , λ(ν)

(7.55)

where P stands for Principal Value. The value of s2 (ν) is actually equal to s1 (ν), when the expression given in Eq. (7.52) is taken in the interval [0, 1]. Indeed, in this interval η(ν) = 0 and R(ν) = λ2 (ν). Summing the contributions to s(ν) coming from the intervals [0, 1[, [1, ∞[, and from the pole, we obtain S(τ ) = s0 e−ν0 τ +



∞

s1 (ν)e −ντ dν,

(7.56)

0

with s0 given by Eq. (7.48) and s1 (ν) by Eq. (7.52). This result is also establish in the Appendix D.1 of this chapter with the inverse Laplace transform method applied to the Wiener–Hopf integral equation for S(τ ). The derivation presented here is somewhat simpler. For complete frequency redistribution, Eq. (7.27) shows that s(ν) can be expressed in terms of the resolvent function. The same can be done for monochromatic scattering. Using the results obtained for the inverse Laplace transform φ(ν)

126

7 The Emergent Intensity and the Source Function

in Sect. 6.2.2 and regrouping the two terms in Eq. (7.56), we can write 

∞

S(τ ) =

s(ν)e−ντ dν,

(7.57)

0

with s(ν) =

    ∞ φ(ν ) q ∗ (ν)

dν + φ(ν) 1 + X∗ (−ν) ν − ν 0

∞ 0

dν

q ∗ (ν ) , ν ∈ [0, ∞[. ∗



X (−ν ) ν − ν (7.58)

We recover the expression given in Eq. (7.27) with X(z) replaced by X∗ (z). We recall that φ(ν), for monochromatic scattering, contains a Dirac distribution δ(ν − ν0 ) (see Eq. (6.44)).

7.3 Some Standard Results We now apply the general results obtained for the source function and the emergent intensity to three standard problems : a uniform primary source, an exponential primary source, and a diffuse reflection problem.

7.3.1 Uniform Primary Source Term For Q∗ (τ ) = Q∗ , with Q∗ a constant, q ∗ (ν) = Q∗ δ(ν). For complete frequency redistribution, Eq. (7.27) leads to S(τ ) =

   ∞ φ(ν) Q∗ 1+ (1 − e−ντ ) dν . X(0) ν 0

(7.59)

For monochromatic scattering Eq. (7.58) leads to the same equation, √ with X(0) replaced by X∗ (0). We recall that X(0) and X∗ (0) are both equal to . In terms of the resolvent function (τ ), Eq. (7.59) becomes    τ Q∗



(τ ) dτ . S(τ ) = √ 1 +  0

(7.60)

7.3 Some Standard Results

127

This expression is valid for both complete frequency redistribution and monochromatic scattering. When the normalization of (τ ) given in Eq. (6.57) is taken into account, we can write    ∞ Q∗ 1 S(τ ) = √ √ − (7.61) (τ ) dτ .   τ For τ = 0, Eq. (7.60) leads to Q∗ S(0) = √ . 

(7.62)

√ For Q∗ = B, we recover the -law already given in Eq. (7.13). Equation (7.60) has already been given in Sect. 2.4.3, but with S(0) still unknown. Equation (7.61) shows that S(τ ) goes at infinity to the constant Q∗ /. A result, which can also be deduced from the Wiener–Hopf integral equation for S(τ ). It suffices to take S(τ ) out of the integral and use the fact that the kernel is normalized to unity (see Sect. 5.1). We show in Fig. 7.1 S(τ ) in log-log scales for Q∗ = , the Voigt profile with a = 10−2 , and several values of . The effect of the Lorentzian wings appears for values of τ such that aτ  1. The distance from the surface at which S(τ ) reaches its constant value Q∗ / = 1 is known as the thermalization length. For the Voigt profile it varies with  as a/ 2 . The concept of thermalization length is discussed in detail in Part III of this book, in particular in Chap. 20.

Fig. 7.1 The source function for a uniform primary source Q∗ =  and several choices of . The −2 absorption √ profile is a Voigt profile with parameter a = 10 . The surface value S(0) follows the exact -law

128

7 The Emergent Intensity and the Source Function

Because S(τ ) has an exact and simple expression at the surface and also at infinity, the case of a uniform source term is very often used to test numerical methods in radiative transfer and verify scaling laws on the behavior of the source function at large optical depths (see e.g. Avrett and Hummer 1965). Exact expressions of the emergent intensity are given in Sect. 7.1 for complete frequency redistribution and monochromatic scattering. For a primary source Q∗ = B, Eq. (7.10) leads to I (0, x, μ) =

√

BH (

μ ), ϕ(x)

μ ∈ [0, 1].

(7.63)

For a given μ, Eq. (7.63) describes an absorption line. The intensity has its minimum value at line center and tends to B in the line wings. For monochromatic scattering, Eq. (7.9) leads to I (0, μ) =

√ BH (μ),

μ ∈ [0, 1].

(7.64)

The center-to-limb variation of the emergent intensity follows the H -function. For this reason, the H -function is also referred to as the limb-darkening function (e.g. Mihalas 1978). This limb-darkening law can also be used to test numerical codes.

7.3.2 Exponential Primary Source Term We now assume Q∗ (τ ) = Q∗ e−ατ , with α a positive constant. Its inverse Laplace transform is q ∗ (ν) = Q∗ δ(ν − α). Exponential primary sources can serve to model stellar atmospheres with temperature dependent primary sources (Ivanov 1973, p. 274) and, as we show in Sect. 7.3.3, they play an important role in half-space diffuse reflection problems. We denote the source function by Sα (τ ). We first examine the surface value Sα (0). For complete frequency redistribution and monochromatic scattering, Eq. (7.12) shows that it takes the simple form 1 Sα (0) = Q∗ H ( ). α

(7.65)

This explains why the surface value of the source function for an exponential primary source can be used to define the H -function. To obtain Sα (τ ), we insert q ∗ (ν) = Q∗ δ(ν − α) into Eqs. (7.27) and (7.58). For complete frequency redistribution, we obtain    ∞ −ατ e − e−ντ Q∗ −ατ φ(ν) dν , Sα (τ ) = + e X(−α) ν−α 0

(7.66)

7.3 Some Standard Results

129

and for monochromatic scattering, Sα (τ ) =

   ∞ φc (ν) −ατ Q∗ e−ατ − e−ν0 τ −ατ −ντ + (e + − e ) dν . e X∗ (−α) (ν0 − α)X(ν0 ) ν−α 1 (7.67)

Using 1/X(−α) = H (1/α) for complete frequency redistribution and 1/X∗ (−α) = H (1/α) for monochromatic scattering, both equations can written as   Sα (τ ) = Sα (0) e−ατ +

τ



(τ )e−α(τ −τ ) dτ .

(7.68)

0

It was shown in Sect. 2.4.3 that this expression can be derived directly from the Wiener–Hopf integral equations for Sα (τ ) and (τ ), but with Sα (0) still undetermined. We show in Fig. 7.2, logS(τ ) versus logτ for an exponential source with α = 10−4 , a destruction parameter  = 10−6 , a Doppler profile, and a Voigt profile with a = 10−2 . The results are derived from a numerical solution of the radiative transfer equation. Starting from the value Sα (0) at τ = 0, the source function goes through a maximum and then decreases to zero as τ tends to infinity. Equation (7.68) shows it decreases as (τ ). The physical reason, pointed out in Ivanov (1973), is that the source function behaves at large optical depths as if all the primary sources were concentrated at the surface. In Chap. 22 we show that (τ ) decreases as 1/τ 2 for the Doppler profile and as 1/τ 3/4 for a Voigt profile and relate these asymptotic

Fig. 7.2 The source function for an exponential primary source Q∗ (τ ) = exp(−10−4 τ ) and  = 10−6 , for a Voigt profile and a Doppler profile. For the latter, the straight line shows the asymptotic behavior of the boundary layer solution (see also Fig. 9.1)

130

7 The Emergent Intensity and the Source Function

behaviors to the shape of the profiles at large frequencies. The first numerical values of the source function for an exponential source term were derived by Avrett and Hummer (1965) from the integral equation for S(τ ). The results have been used for testing the accuracy of approximate solutions of transfer equations (Frisch and Frisch 1975; Ivanov and Serbin 1984). For complete frequency redistribution, the general expression for the emergent intensity given in Eq. (7.10) leads to I (0, x, μ) = Sα (0)

μ ϕ(x) H( ), ϕ(x) + αμ ϕ(x)

μ ∈ [0, 1].

(7.69)

For a given μ, Eq. (7.69) describes an emission line. It may have a central reversal depending on the value of α. A detailed study of the shape of I (0, x, μ) as function of α, and also as a function of μ, can be found in Ivanov (1973, p. 270). For monochromatic scattering, Eq. (7.9) leads to I (0, μ) = Sα (0)

H (μ) , 1 + αμ

μ ∈ [0, 1].

(7.70)

Because of the concentration of the primary sources near the surface, the variation between the center and the limb is flattened by a factor 1/(1 + αμ), compared to the uniform primary source case.

7.3.3 Diffuse Reflection We now assume that a given radiation field is incident on the surface at τ = 0 and that there is no primary source term inside the medium. As shown in Chap. 2, the radiation field inside the medium can be decomposed into a directly transmitted field and a diffuse field. The emergent radiation of the diffuse field is equal to the emergent radiation of the total field. For the diffuse field, the incident radiation is zero, but there is a primary source term, which depends on the given incident field. For monochromatic scattering it has the form Q∗ (τ ) =

1− 2



1

I inc (−μ)e−τ/μ dμ,

(7.71)

0

and for complete frequency redistribution Q∗ (τ ) =

1− 2

1  +∞

 0

−∞

  ϕ(x) I inc (x, −μ) exp −τ ϕ(x) dx dμ. μ

(7.72)

7.3 Some Standard Results

131

In both cases we can write 

∗

∞

Q (τ ) =

q ∗ (ν) e−ντ dν.

(7.73)

0

For monochromatic scattering, the general formula for the emergent intensity established in Sect. 7.1 is  ∞ 1 1 I (0, μ) = H (μ) q ∗ (ν)H ( ) dν, μ ∈ [0, 1]. (7.74) 1 + νμ ν 0 and the inverse Laplace transform of Q∗ (τ ) is q ∗ (ν) =

1 −  1 inc 1 I (− ) 2 ν2 ν

ν ∈ [1, ∞[,

q ∗ (ν) = 0,

ν ∈ [0, 1].

(7.75)

Inserting Eq. (7.75) into Eq. (7.74), we recover the well known expression (e.g. Chandrasekhar 1960) : I (0, μ) =

1− H (μ) 2

 0

1

μ H (μ ) inc I (−μ ) dμ , μ + μ

μ ∈ [0, 1].

(7.76)

When the incident radiation is beam of radiation I inc (−μ) = I inc (μ0 )δ(μ − μ0 ), μ, μ0 ≥ 0, the emergent intensity is given by I (0, μ) =

1− μ0 H (μ)H (μ0) I inc (μ0 ). 2 μ + μ0

(7.77)

This expression, which holds for  = 0 and  = 0, is in Hopf (1934, p. 59) for  = 0 and in Halpern et al. (1938) for  = 0 (see Ivanov 1973, p. 141 for an historical account). When the incident radiation is isotropic, that is I inc (μ) = I inc , the emergent intensity is given by I (0, μ) = I inc [1 −

√ H (μ)],

μ ∈ [0, 1].

(7.78)

To derive this expression we use the nonlinear integral equation for H (μ) and the zeroth-moment of H (μ) : 

1

α0 = 0

H (μ) dμ = 2

√ 1−  . 1−

(7.79)

Thus for  = 0, that is for conservative scattering, the emerging radiation is equal to the incident one.

132

7 The Emergent Intensity and the Source Function

We now introduce the albedo. Denoted A, it is the ratio F em /F inc , where F em and F inc , the emergent and incident flux, are defined by 

1

F em = 2π

 I em (0, μ)μ dμ,

F inc = 2π

0

1

I inc (−μ)μ dμ.

(7.80)

0

The albedo characterizes the reflecting properties of a medium to a given incident radiation. For a semi-infinite medium, the albedo can be derived from Eq. (7.76). Using the nonlinear integral for H (μ) and the expression of α0 , simple algebra leads to √ F em A = inc = 1 −  F

1 0

H (μ)I inc (−μ)μ dμ . 1 inc 0 I (−μ)μ dμ

(7.81)

When  = 0, the albedo is equal to one. This means that the medium is fully reflecting. The albedo has a very simple expression for the two incident fields considered above. For an isotropic incident radiation, √ A=1−2 



1

H (μ)μ dμ,

(7.82)

0

and for a pencil of radiation coming from the direction μ0 , A=1−

√ H (μ0 ).

(7.83)

This expression shows that the albedo is larger for a grazing incident beam (with a small μ0 ) than for beam normal to the surface (μ0 around one). This has to do with the fact that a beam normal to the surface can penetrate deeper inside the medium. We recall that H (μ) is an increasing function of μ. We now briefly comment on the source function S(τ ). Setting μ = 0 in Eq. (7.74), we see that the surface value is given by 

∞

S(0) = 0

1 q ∗ (ν)H ( ) dν. ν

(7.84)

As shown in Sect. 7.2.2, S(τ ) may be written as S(τ ) = s0 e

−ν0 τ

 +

∞

s1 (ν)e−ντ dν,

(7.85)

1

where s0 is given in Eq. (7.48) and s1 (ν) in Eq. (7.52). There is no contribution from ν ∈ [0, 1] since q ∗ (ν) is zero in this interval. We recall that ν0 , the positive zero of

7.3 Some Standard Results

133

the dispersion function, is zero for  = 0 and lies in the interval ]0, 1] for  = 0. Inserting the expression of q ∗ (ν) given in Eq. (7.75) into Eq. (7.48), we obtain 1− 1 s0 = 2 X(ν0 )



1

I inc (−μ)H (μ)

0

μ dμ. 1 − μν0

(7.86)

Equation (7.85) shows that S(τ ) decreases exponentially at infinity as s0 e−ν0 τ when  = 0 and that it tends to a constant at infinity when  = 0. At infinity the radiation field becomes isotropic√ and equal to the source function. Setting ν0 = 0 in Eq. (7.86), and using X(0) = 1/ 3, we obtain √  1 3 S(∞) = I (∞, μ) = I inc (−μ)H (μ)μ dμ. 2 0

(7.87)

This expression seems to have been first given by Bensoussan et al. (1979) (see also Frisch and Bardos 1981). We use it in Chap. 23, where we discuss the diffusion approximation for monochromatic scattering. We also derive it in Chap. 9. For complete frequency redistribution, the emergent intensity may be written as  ∞ ϕ(x) μ 1 ) q ∗ (ν)H ( ) dν, μ ∈ [0, 1]. (7.88) I (0, x, μ) = H ( ϕ(x) 0 ϕ(x) + νμ ν For a pencil of radiation around the frequency x0 and the direction −μ0 , the emergent intensity is given by I (0, x, μ) =

1− H 2





μ0 ϕ(x) μ μ0 H ϕ(x0 ) I inc (x0 , −μ0 ). ϕ(x) ϕ(x0 ) μ0 ϕ(x) + μϕ(x0 )

(7.89) The shape of the emergent profile in the x0 , μ0 parameter space is discussed in detail in Ivanov (1973, p. 269). Many interesting properties are pointed out. For example, when μ0 is close to zero, the emergent radiation is an emission line, with I (0, x, μ) ∝ H

μ ϕ(x)

ϕ(x) . μ

(7.90)

The reason is that the source of photons coming from the incident radiation is concentrated at the boundary. Now, when x0 is large, meaning that the incident radiation is in the far wing, the emerging radiation is an absorption line with I (0, x, μ) ∝ H

μ . ϕ(x)

(7.91)

134

7 The Emergent Intensity and the Source Function

The reason is that incident photons can now penetrate deep inside the medium. Another type of diffusion problem, discussed in detail in Ivanov (1973, p. 278), is the illumination by a continuum radiation I inc , isotropic and independent of frequency in the spectral range of the line. In this case I (0, x, μ) = I inc [1 −

√ μ )], H ( ϕ(x)

μ ∈ [0, 1].

(7.92)

The source function has also a simple expression, namely S(τ ) =

√ inc I



∞

(τ ) dτ ,

(7.93)

τ

or, after taking the normalization of (τ ) into account, S(τ ) = I inc [1 −

√

 (1 +

τ

(τ ) dτ )].

(7.94)

0

Several applications are discussed in Ivanov (1973). We mention also that illumination by a continuum radiation has been used by Faurobert et al. (1988) to analyze the solar hydrogen spectrum predicted by Skumanich and Lites (1986). The diffuse reflection problem is also investigated in Chap. 10, where it is shown that the Case eigenfunction expansion method is particularly well adapted to radiative transfer problems with an incident radiation field.

Appendix D: The Monochromatic Source Function In this Appendix we construct exact expressions for the monochromatic source function, with the inverse and the direct Laplace transform methods applied to the Wiener–Hopf integral equation  S(τ ) = (1 − )

∞

K(τ − τ )S(τ ) dτ + Q(τ ),

τ ∈ [0, ∞[.

(D.1)

0

The inverse Laplace transform method is an alternative way for obtaining the expression of s(ν) in terms of q ∗ (ν) given in Eq. (7.58). The direct Laplace transform method yields s˜ (p), the direct Laplace transform of S(τ ), in terms of q˜ ∗ (p), the direct Laplace transform of Q∗ (τ ).

D.1 The Inverse Laplace Transform Method

135

D.1 The Inverse Laplace Transform Method As shown in Chap. 4, s(ν), defined by 

∞

S(τ ) =

s(ν)e−ντ dν,

(D.2)

0

satisfies the singular integral equation  λ(ν)s(ν) + η(ν)

∞ 0

s(ν ) dν = q ∗ (ν), ν − ν

ν ∈ [0, ∞[.

(D.3)

Because of the zeroes of the dispersion function, this equation is not a standard singular integral equation of the Cauchy-type. One has to consider separately the intervals ν ∈ [0, 1[ and ν ∈ [0, ∞[. (see e.g. Sect. 6.2.2). Interval ν ∈ [0, 1[ Since η(ν) = 0 for ν ∈ [0, 1[, Eq. (D.3) reduces to λ(ν)s(ν) = q ∗ (ν),

ν ∈ [0, 1[.

(D.4)

Because λ(ν) has a zero at ν = ν0 , ν0 ∈ [0, 1[, the general solution of this equation takes the form s(ν) = s0 δ(ν − ν0 ) + P

q ∗ (ν) , λ(ν)

ν0 ∈ [0, 1[,

(D.5)

where s0 is a constant, which cannot be determined at this stage, and P stands for Cauchy Principal Value. The Cauchy Principal Value is needed because of the zero of λ(ν) at ν0 . Interval ν ∈ [1, ∞[ Inserting Eq. (D.5) into Eq. (D.3) we obtain  λ(ν)s(ν) + η(ν)

∞ 1

s(ν ) dν = γ (ν), ν − ν

ν ∈ [1, ∞[.

(D.6)

The inhomogeneous term γ (ν) can be written as γ (ν) = qt∗ (ν) + η(ν)

s0 , ν − ν0

(D.7)

with qt∗ (ν)

∗

= q (ν) − η(ν)



1 q ∗ (ν ) 0

dν

, L(ν ) ν − ν

ν ∈ [1, ∞[.

(D.8)

136

7 The Emergent Intensity and the Source Function

Now we can apply to Eq. (D.6) the standard Hilbert transform method of solution. First we solve Eq. (D.6) with the expression of γ (ν) given in Eq. (D.7) (Step 1). To obtain the final result (Step 2), qt∗ (ν) is replaced by the expression given in Eq. (D.8). Step 1 We introduce the Hilbert transform S(z) =



1 2i π

∞

1

s(ν) dν. ν−z

(D.9)

Proceeding as in Sect. 6.2.2, we write η(ν) = [L+ (ν)−L− (ν)]/(2i π) and introduce the X-function. Equation (D.6) becomes X+ (ν)[S + (ν)(ν − ν0 ) −

s0 s0 ] − X− (ν)[S − (ν)(ν − ν0 ) − ] 2i π 2i π

= Ut+ (ν) − Ut− (ν),

(D.10)

where Ut+ (ν) and Ut− (ν) are the limiting values of the Cauchy integral 1 Ut (z) = 2i π



∞ 1

X− (ν) ∗ ν − ν0 q (ν) dν. L− (ν) t ν −z

(D.11)

Using the factorization relation X(z)X(−z) = L(z)/(ν02 − z2 ) (see Sect. B.1), and X∗ (z) ≡ (ν0 − ν)X(z), we can rewrite Ut (z) as Ut (z) = −

1 2i π



∞

1

qt∗ (ν) dν . X∗ (−ν) ν − z

(D.12)

The general solution of Eq. (D.10) satisfies X(z)[(z − ν0 )S(z) −

s0 ] − Ut (z) = P (z), 2i π

(D.13)

where P (z) is an entire function. All the functions in the left-hand side of Eq. (D.13) tend to zero at infinity as 1/z, hence P (z) goes also to zero at infinity and by the Liouville theorem P (z) = 0 for all z. The solution of Eq. (D.13) has thus the form  S(z) =

 s0 1 Ut (z) Ut (z) 1 s0 + + =− ∗ . X(z) 2i π z − ν0 X (z) 2i π z − ν0

(D.14)

with s0 = −2i π

Ut(ν0 ) . X(ν0)

(D.15)

D.1 The Inverse Laplace Transform Method

137

The expression of s0 is obtained by setting z = ν0 in Eq. (D.13). We remark that z = ν0 is not a pole of S(z) since the square bracket is zero for z = ν0 . We can easily verify that S(z) has all the properties of a Hilbert transform. The solution of the Cauchy integral equation (D.6) may thus be written as 

 Ut− (ν) Ut+ (ν) − s(ν) = − , X∗+ (ν) X∗− (ν)

ν ∈ [1, ∞[.

(D.16)

Step 2 The goal is to express Ut (z) in terms of q ∗ (ν). We present a method requiring very little algebraic work but a careful use of the properties of analytic functions with branch cuts. It follows from Eqs. (D.8) and (D.12) that the jump of Ut (z) may be written as Ut+ (ν) − Ut− (ν) = −

η(ν) q ∗ (ν) + X∗ (−ν) X∗ (−ν)



1 q ∗ (ν ) 0

L(ν )

dν

, −ν

ν

ν ∈ [1, ∞[. (D.17)

The first term in the right-side suggests to introducing the function A1 (z) ≡ −

1 2i π



∞ 1

q ∗ (ν) dν . X∗ (−ν) ν − z

(D.18)

Concerning, the second term, we remark, using Eq. (B.11), that η(ν) 1 = [X∗+ (ν) − X∗− (ν)], X∗ (−ν) 2i π

ν ∈ [1, ∞[.

(D.19)

This suggest to introducing the function A2 (z) ≡ X∗ (z)

1 2i π



1 q ∗ (ν) 0

dν . L(ν) ν − z

(D.20)

The integral is taken in Principal Value because L(ν) has a zero in the interval [0, 1[. The function A2 (z) has a branch cut along the interval [1, ∞[ because of the factor X∗ (z), but also along the interval [0, 1[ because of the integral over [0, 1]. The corresponding jump may be written − A+ 2 (ν) − A2 (ν) = (ν0 − ν)X(ν)

q ∗ (ν) q ∗ (ν) = , L(ν) (ν0 + ν)X(−ν)

ν ∈ [0, 1]. (D.21)

138

7 The Emergent Intensity and the Source Function

This jump can be cancelled by redefining A1 (z) with an integration over [0, ∞[. The function Ut (z), analytic in the complex plane cut along the interval [1, ∞[ and satisfying the jump condition in Eq. (D.17), that we are looking for, is thus 1 Ut (z) = − 2i π



∞ 0

q ∗ (ν) dν 1 + X∗ (z) X∗ (−ν) ν − z 2i π



1 q ∗ (ν) 0

dν . L(ν) ν − z

(D.22)

We observe that it also has the proper behavior at infinity, namely behaves as 1/z. Final Result We can now obtain an explicit expression for s0 and for s(ν) in terms of q ∗ (ν). Combining Eqs. (D.15) and (D.22) we find 1 s0 = X(ν0 )



∞ 0

dν q ∗ (ν) . X∗ (−ν) ν − ν0

(D.23)

The second term in Eq. (D.22) does not contribute to s0 since X∗ (ν0 ) = 0. Equations (D.16) and (D.22) lead to 1 1 1 q ∗ (ν) [ ∗+ + ∗− ] ∗ 2 X (−ν) X (ν) X (ν)  1 1 1 ∞ q ∗ (ν ) dν

1 − ] . + [ ∗+ 2 X (ν) X∗− (ν) i π 0 X∗ (−ν ) ν − ν

s(ν) =

(D.24)

This equation is similar to Eq. (7.25) with X± (ν) replaced by X∗± (ν). It can thus also be written as Eq. (7.26) with X(−ν) replaced by X∗ (−ν), namely as   1 ∗ ∗ λ(ν)q (ν) − η(ν)X (−ν) s(ν) = R(ν)

∞ 0

 q ∗ (ν ) dν

, X∗ (−ν ) ν − ν

ν ∈ [1, ∞[. (D.25)

Summing the contributions to s(ν) coming from the intervals [0, 1[ and [1, ∞[, we obtain S(τ ) = s0 e−ν0 τ +



1 q ∗ (ν) 0

λ(ν)

e −ντ dν +



∞

s(ν)e −ντ dν,

(D.26)

1

with s0 given by Eq. (D.23) and s(ν) by Eq. (D.25). We recover the result obtained in Sect. 7.2.2 by a Fourier inversion. The expression of s0 is identical to the expression given in Eq. (7.48), the second term is identical to s2 (ν) in Eq. (7.55) and s(ν) is identical to the expression given for s1 (ν) in Eq. (7.52).

D.2 The Direct Laplace Transform Method

139

D.2 The Direct Laplace Transform Method We determine here s˜ (p), the Laplace transform of the source function, in terms of q˜ ∗ (p), the Laplace transform of the primary source term. It satisfies the singular integral equation  λ(p)˜s (p) −

∞ η(p )˜ s (p ) 1

dp = q˜ ∗ (p),

p − p

p ∈ [0, ∞[.

(D.27)

We follow the method of solution applied in the Appendix C.2 of Chap. 6 to calculate the Laplace transform of the surface Green function, in particular we consider separately the intervals p ∈ [0, 1[ and p ∈ [1, ∞[. Interval p ∈ [1, ∞[ We introduce the Hilbert transform S(z) =

1 2i π



∞ 1

η(p)˜s (p) dp. p−z

(D.28)

Using the Plemelj formulae for S(z) and L(z) and introducing the function X(z), we can rewrite Eq. (D.27) as S + (p) S − (p) − = G + (p) − G − (p), X+ (p) X− (p)

p ∈ [1, ∞[,

(D.29)

dp , p−z

(D.30)

.

(D.31)

where 1 G(z) = 2i π



∞

t (p)q˜ ∗ (p)

1

with t (p) ≡

η(p) L+ (p)X− (p)

Using the first Plemelj formula for L(z), the defining relation for X(z) given in Eq. (5.36), and introducing the expression of φc (p), the inverse Laplace transform of the resolvent function (τ ), given in Eq. (6.42), we can write t (p) = −

  1 1 1 − = −(ν0 − p)φc (p). 2i π X+ (p) X− (p)

(D.32)

To continue the derivation of S(z) we must examine the behavior of G(z) for z → ∞. As pointed out in Sect. 6.2.2, φc (p) ∼ 1/p for p → ∞, hence t (p) ∼ 1. To ensure the converge of the integral, q˜ ∗ (p) should decrease sufficiently fast at infinity. For example, when Q∗ (τ ) is a constant, say Q∗ (τ ) = 1, then q˜ ∗ (p) = 1/p, the

140

7 The Emergent Intensity and the Source Function

integral converges and G(z) tends to zero at infinity. In the Appendix C.2 of Chap. 6, ˜ where we calculate G(p), the Laplace transform of the surface Green, q˜ ∗ (p) = 1, but the right-hand side in Eq. (D.29) is equal to ([X− (p)]−1 − [X+ (p)]−1 )/(2i π) and can be incorporated into the left-hand side of the equation. The solution of Eq. (D.27) has the form S(z) = X(z)[G(z) + P (z)],

(D.33)

where P (z) is an entire function of z. At infinity G(z) tends to zero and S(z)/X(z) tends to a constant. Hence P (z) tends to a constant and by the Liouville theorem P (z) = P0 with P0 a constant, to be determined as we consider the interval p ∈ [0, 1[. Interval p ∈ [0, 1[ Equation (D.27) reduces to λ(p)˜s (p) − 2i πS(p) = q˜ ∗ (p),

p ∈ [0, 1[.

(D.34)

In this interval, λ(p) has a zero at p = ν0 . The function s˜ (p) should not have a pole at p = ν0 , otherwise S(τ ) would be increasing exponentially at infinity. This condition is satisfied if 2i πS(ν0 ) + q˜ ∗ (ν0 ) = 0.

(D.35)

Using the expression of S(z) in Eq. (D.33), we find P0 = −

1 q˜ ∗ (ν0 ) − G(ν0 ). 2i π X(ν0 )

(D.36)

Final Result for S(z) The combination of Eqs. (D.33) and (D.36) with Eq. (D.32) leads to   1 q˜ ∗ (ν0 ) S(z) = X(z) − + G(z) − G(ν0 ) , 2i π X(ν0 )

(D.37)

where G(z) − G(ν0 ) =

z − ν0 2i π



∞

φc (p)q˜ ∗ (p)

1

dp . p−z

(D.38)

Regrouping the contributions from the zero at ν0 and from the cut along [1, ∞[ and introducing also X∗ (z) ≡ (ν0 − z)X(z), we can rewrite S(z) as 1 S(z) = −X (z) 2i π ∗



∞ 0

φ(p)q˜ ∗ (p)

dp . p−z

(D.39)

D.2 The Direct Laplace Transform Method

141

Determination of s˜(p) To determine s˜ (p) for p in the interval [0, 1], we rewrite Eq. (D.34) as s˜ (p) =

1 [q˜ ∗ (p) + 2i πS(p)], λ(p)

p ∈ [0, 1[,

(D.40)

where S(p) is given by Eq. (D.39) with z = p. By construction the square bracket in Eq. (D.40) is zero for p = ν0 . Using L(z) = X∗ (z)X∗ (−z) and L(p) = λ(p) for p ∈ [0, 1], we obtain   λ(p) 1 ∗ q˜ (p) − ∗ s˜ (p) = λ(p) X (−p)

∞ 0

 dp

, φ(p )q˜ (p )

p −p

∗



p ∈ [0, 1]. (D.41)

To determine s˜ (p) in the interval [1, ∞[, we apply the first Plemelj formula to S(z). Using Eqs. (A.7) and (A.8) to calculate the limiting values of the Cauchytype integral in the right-hand side, Eqs. (B.11) and (B.12) to express the sum and the difference of X± (−p), and φc (p) = φ∞ (p)X∗ (−p) = −

η(p) ∗ X (−p), R(p)

p ∈ [1, ∞[,

(D.42)

(see Eq. (6.43)), we obtain s˜ (p) =

1 λ(p) ∗ q˜ (p) − ∗ R(p) X (−p)



∞ 0

φ(p )q˜ ∗ (p )

dp

, −p

p

p ∈ [1, ∞[. (D.43)

This expression is identical to the expression given in Eq. (7.39) for complete frequency redistribution, provided X∗ (−p) is replaced by X(−p). We remark also that it holds for p ∈ [0, 1[. Indeed, for p ∈ [0, 1[, the ratio λ(p)/R(p) reduces to 1/λ(p). The expression of s˜ (p) given in Eq. (D.43) can be considered as the final ˜ result, holding for p ∈ [0, ∞[. For q˜ ∗ (p) = 1, we recover G(p) = 1/X∗ (−p), ˜ where G(p) is the Laplace transform of G(τ ). For monochromatic scattering, the derivation of s˜ (p) is simpler that the derivation of the inverse Laplace transform s(ν), but to obtain a closed-form expression for S(τ ), an inverse Fourier transformed has still to be applied to s˜ (p). The exponentially decreasing term s0 e−ν0 τ will come from the distribution δ(ν0 − p ) in φ(p ).

142

7 The Emergent Intensity and the Source Function

References Avrett, E.H., Hummer, D.G.: Non-coherent scattering II: Line formation with a frequency independent source function. Mon. Not. R. Astr. Soc. 130, 295–331 (1965) Bensoussan, A., Lions, J.L., Papanicolaou, G.: Boundary layers and homogenization of transport processes. Publ. RIMS Kyoto Univ. 15, 53–157 (1979) Chandrasekhar, S.: Radiative Transfer. 1st edn. Oxford University Press, Oxford (1950). Dover Publications, New York (1960) Faurobert, M., Frisch, H., Skumanich, A.: A model for the penetration of Lyman-α in the solar chromosphere. Astrophys. J. 328, 856–859 (1988) Frisch, H., Bardos, C.: Diffusion approximations for the scattering of resonance-line photons. Interior and boundary layer solutions. J.√ Quant. Spectrosc. Radiat. Transf. 26, 119–134 (1981) Frisch, U., Frisch, H.: Non-LTE transfer.  revisited. Mon. Not. R. Astr. Soc. 173, 167–182 (1975) Halpern, O., Lueneburg, R., Clark, O.: On multiple scattering of neutrons I. Theory of the albedo of a plane boundary. Phys. Rev. 53, 173–183 (1938) Hopf. E.: Mathematical Problems of Radiative Equilibrium. Cambridge University Press, London (1934) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ. 385, (1973) translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Ivanov, V.V, Serbin, V.M.: The transfer of line radiation. I. General analysis of approximate solutions; II. Approximate solutions for semi-infinite atmospheres. Soviet Astron. 28, 405– 409 (1984) and pp. 524–531; translation from Astron. Zhurnal 61, 691–699 and pp. 900–912 (1984) Mihalas, D.: Stellar Atmospheres, 2nd edn. W.H. Freeman and Company, San Francisco, California (1978) Skumanich, A., Lites, B.W.: Radiative transfer diagnostics - understanding multilevel transfer calculations. I. Analysis of the full statistical equilibrium equations. Astrophys. J. 310, 419– 431 (1986)

Chapter 8

Spectral Line with Continuous Absorption

Spectral lines are in general embedded in a continuous background created by photoionizations and recombinations, and free-free emission. In cool stars such as the Sun the dominant source of continuous absorption is the negative hydrogen ion H− , an ion with a single bound state of low binding energy. In this Chapter we show how a continuous absorption and a continuous emission modify the results presented in Chap. 7 for spectral lines formed with complete frequency redistribution. Exact solutions can be constructed for semi-infinite media when the continuous absorption is independent of frequency over the width of the line and the ratio of the continuous to the line absorption coefficient is independent of the optical depth. The first exact solution for the formation of a spectral with a continuous absorption is by Abramov et al. (1967) with the Wiener–Hopf method applied to the Wiener–Hopf integral equation for the resolvent function. Here we show how to apply the inverse Laplace transform method to the integral equations for the resolvent function and for the line source function. We obtain singular integral equations with a Cauchy-type kernel very similar to the equations obtained in the absence of continuous absorption. We also employ the method based on the Laplace transform of the Green function, described in Sect. 7.1, to calculate the emergent intensity. The formation of spectral lines with a continuous absorption has been thoroughly investigated in articles by Hummer (1968) and Nagirner (1968) and in Ivanov’s book (1973, Chapter VII). Most of the discussions and results of this chapter can be found in these documents. Hummer (1968) presents numerical solutions of the radiative transfer equation obtained with a discrete ordinate method à la Chandrasekhar. In Ivanov’s book, a strong attention is given to exact and asymptotic results. This Chapter is organized as follows. In Sect. 8.1 we present the radiative transfer equation. Following Hummer (1968), we introduce an effective destruction probability and an effective scattering kernel, which make it is possible to construct, for the source function, an integral equation almost similar to the integral equation for the source function in the absence of continuous absorption. The properties of the effective kernel are presented in Appendix E of this chapter. Section 8.2 is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_8

143

144

8 Spectral Line with Continuous Absorption

devoted to the dispersion function and the half-space auxiliary function. It is shown in particular that a continuous absorption does not modify in any essential way the properties of the half-space auxiliary function. In Sect. 8.3 we present exact results concerning the resolvent function, the source function, and the emergent intensity. The resolvent function and the source function are derived with the Hilbert transform method applied the corresponding singular integral equations. Exact results are used to discuss the effects of a continuous absorption and to distinguish between weak and strong continuous absorption regimes.

8.1 The Radiative Transfer Equation We consider a spectral line formed with complete frequency redistribution and assume that in the frequency range of the line there is a continuous absorption and a continuous emission, both with a negligible frequency dependence and that the continuous emission is purely thermal. The radiation field satisfies the transfer equation ∂ I (z, x, μ) = −[κl (z)ϕ(x) + κc (z)]I (z, x, μ) ∂z   1 +1 +∞ ϕ(x )I (z, x , μ ) dx dμ

+ [1 − (z)]kl (z)ϕ(x) 2 −1 −∞

μ

+ κl (z)ϕ(x)Q∗l (z) + κc (z)Q∗c (z).

(8.1)

Here ϕ(x) is the line absorption profile, with x the frequency in Doppler width unit and x = 0 at line center, κl (z) is the line absorption (extinction) coefficient, κc (z) the continuous absorption coefficient, (z) the destruction probability per scattering. Finally, Q∗l (z) and Q∗c (z) are the primary sources for the line and the continuum. We now introduce the line frequency integrated optical depth dτ = −κl (z) dz and the dimensionless parameter β = κc (z)/κl (z), which measures the ratio between the continuous and line absorption coefficients. For spectral lines in stellar atmospheres, β is in general much smaller than unity. We assume that β and  are independent of the depth in the medium. These assumptions are necessary for the existence of closed form solutions. The radiative transfer equation may then be written as μ

∂ I (τ, x, μ) = [ϕ(x) + β]I (τ, x, μ) − ϕ(x)Sl (τ ) − βQ∗c (τ ), ∂τ

(8.2)

where Sl (τ ), referred to as the line source function, is given by Sl (τ ) ≡ (1 − )

1 2



+1  +∞ −1

−∞

ϕ(x)I (τ, x, μ) dx dμ + Q∗l (τ ).

(8.3)

8.1 The Radiative Transfer Equation

145

We assume that there is no incident radiation on the surface of the medium. Incorporating the formal solution of Eq. (8.2) into the definition of Sl (τ ), we find, after the usual algebra, that it satisfies the Wiener–Hopf integral equation  Sl (τ ) = (1 − )

∞

K(τ − τ , β)Sl (τ ) dτ + Q∗t (τ, β),

0

τ ∈ [0, ∞[.

(8.4)

The primary source Sl (τ ) depends on β, a dependence, which we have not indicated explicitly to simplify the notation. The kernel may be written as K(τ, β) ≡

1 2



1  +∞ −∞

0

  dμ ϕ(x) + β dx . ϕ 2 (x) exp −|τ | μ μ

(8.5)

The primary source term is given by Q∗t (τ, β)

≡

Q∗l (τ ) + (1 − )βF (β)



∞

0

L1 (τ − τ , β)Q∗c (τ ) dτ ,

(8.6)

where L1 (τ, β) ≡

1 2F (β)

1  +∞



−∞

0

  dμ ϕ(x) + β dx . ϕ(x) exp −|τ | μ μ

(8.7)

The notation L1 (τ ) comes from Hummer (1968). The first term in the right-hand side of Eq. (8.6) is the thermal creation rate of line photons (denoted Q∗ (τ ) in preceding chapters). The second one comes from the scattering by the line of the continuous emission Qc (τ ). The new kernel L1 (τ, β) has a structure similar to that of K(τ, β), with one of the factor ϕ(x) in the integrand replaced by β. Because of the destruction of photons by continuous absorption, the integral of the kernel, denoted n(β), is smaller than unity. Indeed, the integration over τ leads to  n(β) ≡

+∞ −∞

 K(τ, β) dτ =

+∞

−∞

ϕ 2 (x) dx. ϕ(x) + β

(8.8)

As suggested by Hummer (1968), the effects of the continuous absorption are ¯ conveniently explained by introducing a new kernel K(τ, β), normalized to unity, ¯ K(τ, β) ≡

1 K(τ, β), n(β)

(8.9)

and an effective destruction probability ¯ defined by ¯ ≡  + (1 − )βF (β),

(8.10)

146

8 Spectral Line with Continuous Absorption

with  F (β) ≡

+∞

−∞

ϕ(x) dx. ϕ(x) + β

(8.11)

When β → 0, the function F (β) tends to infinity, as is clear from Eq. (8.11), but the product βF (β) in Eq. (8.6) tends to zero. An asymptotic expansion of F (β) for β → 0 can be obtained with the method described in Sect. F.1 for the construction of an asymptotic expansion of the kernel for τ → ∞. The idea of the method is to choose ϕ(x) as independent variable. The leading terms are given by  FD (β)  2 − ln β[1 + O(

1 )], − ln β

 FV (β) 

πa β

1/2 [1 + O(β)],

(8.12)

for the Doppler and Voigt profiles. Explicit expressions for the higher order terms can be found in (Ivanov 1973, p. 316). Whereas the destruction of photons by inelastic collisions is happening during a scattering event, their destruction by a continuous absorption may be viewed as occurring in flight (Hummer 1968). It is controlled by the factor βF (β), larger than β, because of the amplification by multiple scatterings. Continuous absorption contributes to the destruction of photons but it also contributes to the emission term Q∗t (τ, β) (see Eq. (8.6)). The order of magnitude of this emission can be evaluated by making the assumption that Q∗c (τ ) is a constant, say, Q∗c (τ ) = 1. This emission term is then given by  (1 − )βF (β)

∞

L1 (τ − τ , β) dτ = (1 − )βF (β)[1 − L2 (τ, β)],

(8.13)

0

with L2 (τ, β) =

1 2F (β)

 0

1  +∞ −∞

  ϕ(x) + β ϕ(x) exp −τ dx dμ. ϕ(x) + β μ

(8.14)

The function L2 (τ, β) tends to zero for τ → ∞ and is equal to one-half for τ = 0. Hence, the emission term coming from the continuous absorption is of order βF (β) and increases from βF (β)/2 and βF (β), when τ increases from zero to infinity, a mere change of a factor two. In terms of the renormalized kernel and destruction probability, the integral equation for Sl (τ ) becomes 

∞

Sl (τ ) = (1 − ¯ ) 0

¯ − τ , β)Sl (τ ) dτ + Q∗t (τ, β), K(τ

τ ∈ [0, ∞[.

(8.15)

8.2 Properties of the Auxiliary Functions

147

Thanks to the introduction of this new kernel, the combined effects of inelastic collisions and continuous absorption can be expressed in much the same way as the effects of inelastic collisions alone and general features of Sl (τ ) can be derived with the arguments used for β = 0.

8.2 Properties of the Auxiliary Functions The presence of a continuous absorption does not modify the structure of the kernel, which remains a superposition of exponentials. We show in the Appendix E of this chapter that it can be written as ¯ K(τ, β) =



∞

kβ (ν, β)e−ν|τ | dν,

(8.16)

0

with kβ (ν, β) =

β 1 (1 − )k(ν − β), n(β) ν kβ (ν, β) = 0,

ν ∈ [β, ∞[,

ν ∈ [0, β].

(8.17) (8.18)

For β = 0, one recovers k(ν), the inverse Laplace transform of the kernel K(τ ). ¯ Equation (8.5) and also Eq. (E.13) show that K(τ, β) decreases exponentially at infinity as e−βτ . This exponential behavior holds for τ > 1/β. For 1 τ < 1/β, ¯ K(τ, β) has the same algebraic behavior as K(τ ). Several figures showing these two regimes for different values of β can be found in Ivanov (1973) for the Doppler and the Voigt profiles (see also Hummer 1968, Faurobert and Frisch 1985). We now discuss the main properties of the dispersion function L(z) and of the half-space auxiliary function X(z). As shown below, they are essentially similar to those of the corresponding functions with β = 0. We keep the notation L(z) and X(z) although these functions depend on β.

8.2.1 The Dispersion Function The dispersion function takes the form  L(z) ≡ 1 − (1 − ¯ )

∞

kβ (ν, β)( β

1 1 + ) dν. ν −z ν +z

(8.19)

148

8 Spectral Line with Continuous Absorption

We see that L(z) is analytic in the complex plane cut along ]−∞, −β] and [β, +∞[. ¯ At infinity L(z) tends to 1. The normalization of the kernel K(τ, β) leads to 

∞

kβ (ν, β) β

dν 1 = . ν 2

(8.20)

Setting z = 0 in Eq. (8.19) and making use of Eq. (8.20), one readily obtains L(0) = ¯ .

(8.21)

We now discuss the somewhat controversial subject of the zeroes of L(z). Cormick and Siewert (1970) applied the Case eigenfunction expansion method to investigate the effects of a continuous absorption, treating the problem as if L(z) had two zeroes, as for monochromatic scattering. Actually as shown by Abramov et al. (1967), L(z) has no zeroes, unless both β and  are zero. We now give a proof that L(z) is free of zeroes, employing the method presented in Sect. 5.2.2 based on the variation of the argument of L+ (ν) along the positive branch of the cut. The Plemelj formulae for L(z) yield L+ (ν) = λβ (ν, β) − i π(1 − ¯ )kβ (ν, β),

ν ∈ [β, ∞[,

(8.22)

where kβ (ν, β) is given by Eq. (8.17) and  λβ (ν, β) = 1 − (1 − ) ¯

∞

kβ (ν , β)(

β

ν

1 1 +

) dν . −ν ν +ν

(8.23)

The function kβ (ν, β) is positive and tends to zero for ν → β and ν → ∞. This implies that L+ (ν) lies in the lower half of the complex plane and tends to the real axis as ν → β and ν → ∞. Equation (8.23) shows that λβ (ν, β) tends to one as ν → ∞. We now examine the value of λβ (ν, β) for ν = β. Expressing kβ (ν, β) in terms of k(ν) (see Eq. (8.17)) and introducing the variable t = ν − β, we can write  λβ (β, β) = 1 − (1 − ) 0

∞

2k(t) dt. t + 2β

(8.24)

The normalization of k(t) to 1/2 (see Eq. (8.20) with β = 0), implies that the integral is smaller than one. Thus, even when  = 0, λβ (β, β) is positive. To summarize, for ν = β and ν at infinity, L+ (ν) is real and has positive values, λ(β, β) and 1, respectively. For all intermediate values of ν, L+ (ν) remains below the real axis. From these results we can conclude that the change in the argument of L+ (ν) is zero along the cut [β, ∞[, hence that L(z) has no zero. Another proof showing that L(z) is free of zeroes is presented in (Ivanov 1973, p. 311). It goes essentially as above and amounts to show that λβ (ν, β) has no zeroes for ν ∈ [0, β].

8.3 Some Exact Results

149

8.2.2 The Half-Space Auxiliary Function Since L(z) has no zeroes, the construction of the function X(z) goes exactly as described in Sect. 5.3 for complete frequency redistribution for β = 0. We can thus choose   ∞  1 dν X(z) = X0 (z) = exp θ (ν) , (8.25) π β ν −z where θ (ν) is the argument of L+ (ν). At infinity, X(z) → 1, for all values of β. The properties of X(z) are similar to those described in Sect. 5.4, except that the lower limit of the cut is β instead of zero. Thus X(z) is analytic in the complex plane cut along [β, ∞[, satisfies the integral equation, 

∞

X(z) = 1 − (1 − ¯ ) β

kβ (ν, β) dν , X(−ν) ν − z

(8.26)

and obeys the factorization relation X(−z)X(z) = L(z),

z ∈ C /] − ∞, −β] ∪ [β, ∞[.

(8.27)

We recall that the notation z ∈ C /] − ∞, −β] ∪ [β, ∞[ means that z belongs to the complex plane cut along ] − ∞, −β] and [β, ∞[. The factorization relation leads to X(0) =

√ ¯ .

(8.28)

The identities given in Eqs. (B.7)–(B.15) concerning the limiting values X± (ν) hold for ν ∈ [β, ∞[. The H -function can be defined by H (z) ≡ 1/X(−1/z).

(8.29)

Numerical values H (z), z real, calculated for the Voigt profile and for different values of β can be found in (Ivanov 1973, p. 332). The effect of the continuous absorption can be observed when βz becomes larger than unity.

8.3 Some Exact Results We examine here the effects of a continuous absorption on the resolvent function, the line source function and the emergent intensity. For the resolvent function, the effects are easily incorporated into the renormalized kernel and effective destruction probability. For the line source function and the emergent intensity, the contribution of the continuous emission to the line source function plays an important role.

150

8 Spectral Line with Continuous Absorption

8.3.1 The Resolvent Function In Sect. 6.2.1, we show how to determine the resolvent function for complete frequency redistribution with no continuous absorption. The results are straightforwardly generalized to include a continuous absorption. It suffices to replace  by the effective destruction probability ¯ , the kernel K(τ ) by Kβ (τ, β), hence k(ν) by kβ (ν, β) and λ(ν) by λβ (ν, β). There are two reasons why the generalization is so simple : the resolvent function describes only the propagation of photons inside the medium and the dispersion function has no zero. Although there are only very minor differences, we repeat here for β = 0 the argument developed in Sect. 6.2.1 for β = 0. The resolvent function (τ ) satisfies the Wiener–Hopf integral equation 

∞

(τ ) = (1−) ¯

¯ ), ¯ −τ , β)(τ ) dτ +(1−) ¯ K(τ K(τ

τ ∈ [0, ∞[,

(8.30)

0

and its inverse Laplace transform, φ(ν), satisfies the singular integral equation  λβ (ν, β)φ(ν) + ηβ (ν, β)

∞ 0

φ(ν ) dν = −ηβ (ν, β), ν − ν

ν ∈ [0, ∞[,

(8.31)

where ηβ (ν, β) = −(1 − ¯ )kβ (ν, β).

(8.32)

In the interval ν ∈ [0, β], ηβ (ν, β) = 0. Since λβ (ν, β) has no zero, the solution of Eq. (8.31) is simply φ(ν) = 0. In contrast, for monochromatic scattering λ(ν) has a zero in the interval ν ∈ [0, 1] in which η(ν) = 0. In the interval, ν ∈ [β, ∞[, φ(ν) is solution of Eq. (8.31), where the lower bound of the Principal Value integral is now β. It can be solved with the Hilbert transform method of solution described in Sect. 6.2.1, the cut of X(z) being now along [β, ∞[. One thus obtains  ∞ (τ ) = φ(ν)e−ν|τ | dν, (8.33) β

with φ(ν) =

  1 1 1 − = φ∞ (ν)X(−ν), 2i π X+ (ν) X− (ν)

ν ∈ [β, ∞[.

(8.34)

Here φ∞ (ν) is the resolvent function for the infinite medium. It is given by Eq. (6.8) with k(ν) and λ(ν) replaced by kβ (ν, β) and λβ (ν, β). Because β > 0, (τ ) decreases exponentially at infinity for τ > β. For 1 τ < β, the influence of the continuous absorption is negligible. A brief discussion on the asymptotic behavior

8.3 Some Exact Results

151

of (τ ) for β = 0 is presented in Sect. 22.2. A more thorough investigation can be found in Ivanov (1973). ˜ The function G(p), Laplace transform of the surface Green function G(τ ) can calculated as described in Sect. 6.3 for β = 0.√It suffices to set the lower bound of the integral in Eq. (6.49) to β and use X(0) = ¯ . One thus obtains ˜ G(p) =

1 1 = H ( ), X(−p) p

p ∈ [0, ∞[.

(8.35)

1 (τ ) dτ = √ . ¯

(8.36)

The normalization of G(τ ) becomes 

∞

¯ G(τ ) dτ = G(0) =1+



0

∞ 0

This integral provides the mean number of scatterings inside that medium, of photons that have been √ at the surface. The continuous absorption produces √ created a decreases from 1/  to 1/ ¯ .

8.3.2 The Line Source Function The line source function Sl (τ ) satisfies the Wiener–Hopf integral equation written in Eq. (8.15). It can be calculated with the inverse Laplace transform method, exactly as described in Sect. 7.2.1. We write  ∞ Sl (τ ) = sl (ν) e−ντ dν, (8.37) 0

and introduce the inverse Laplace transform of the primary source term, qt∗ (ν, β), defined by Q∗t (τ, β) =



∞ 0

qt∗ (ν, β) e−ντ dν.

(8.38)

The function sl (ν) is given by   1 ∗ λβ (ν, β)qt (ν, β) − ηβ (ν, β)X(−ν) sl (ν) = R(ν, β)

∞ q ∗ (ν , β) t

0 X(−ν )

 dν

, ν − ν (8.39)

where R(ν, β) = λ2β (ν, β) + π2 ηβ2 (ν, β). This expression holds for all values of ν ∈ [0, ∞[. For ν ∈ [0, β], it reduces to the first term in the square bracket. A detailed analysis of Sl (τ ) has been performed by Hummer (1968) for Q∗l (τ ) = B and Q∗c (τ ) = ρB, with B a constant and ρ ∈ [0, 1]. We present here some results

152

8 Spectral Line with Continuous Absorption

for the same choice of primary sources. With these assumptions, the total primary source is given by Q∗t (τ, β) = [ + (1 − )βF (β)ρ]B − (1 − )βF (β)L2 (τ, β)ρB,

(8.40)

and its inverse Laplace transform is qt∗ (ν, β) = [ + (1 − )βF (β)ρ]Bδ(ν) − (1 − )βk2(ν, β)ρB,

(8.41)

where k2 (ν, β) is the inverse Laplace transform of F (β)L2 (τ, β) (see Eq. (E.23)). An explicit expression of Sl (τ ) can be obtained by inserting Eq. (8.41) into Eq. (8.39) and then taking the Laplace transform of the result. The full explicit expression is a bit cumbersome. Here we discuss only a few general features. The behavior of Sl (τ ) at infinity can be derived directly from the integral equation in Eq. (8.15). Taking Sl (τ ) out of the integral, we can write lim Sl (τ ) =

τ →∞

1 [ + (1 − )βF (β)ρ]B. ¯

(8.42)

A part from the factor 1/¯ , the right-hand side is the factor which multiplies δ(ν) in Eq. (8.41). For β = 0, one recovers Sl (∞) = B. For β = 0 and ρ = 0 (continuous absorption but no continuous emission), the source function is smaller than B by the factor /¯ . The continuous emission contributes to increase the value of Sl (τ ) at large optical depths The surface value Sl (0) is not as easy to evaluate. The simplest method is to introduce the Laplace transform  s˜l (p) =

∞

Sl (τ )e−pτ dτ,

(8.43)

0

and to use Sl (0) = lim ps˜l (p). p→∞

(8.44)

We now express Sl (τ ) in terms of the Green function G(τ, τ0 ), the primary source in terms of its inverse Laplace transform qt∗ (ν), and introduce ˜ ˜ G(p) G(ν) ˜˜ , G(p, ν) = p+ν

p, ν ∈ [0, ∞[.

(8.45)

8.3 Some Exact Results

153

It is easy to verify that Eq. (8.45), which has been established in Sect. 2.4.4 for β = 0, holds also for β = 0. We thus find  ˜ ˜ ps˜l (p) = G(p) G(0)[ + (1 − )βF (β)ρ]   ∞ p ˜ dν B. G(ν)k −(1 − )βρ (ν) 2 p+ν 0

(8.46)

˜ Using the expression of k2 (ν, β) given in Eq. (E.23), G(ν) = H (1/ν), and introducing ξ = 1/ν, we can write 

∞

˜ G(ν)k 2 (ν)

0

p dν = p+ν





1/β

H (ξ )g0 0

ξ 1 − βξ

[1−

1 ] dξ, 1 + pξ

(8.47)

where g0 (ξ ) is defined in Eq. (E.18). When the limit p → ∞ is taken in Eq. (8.47), one recovers the function denoted W (z) in (Ivanov 1973, p. 329). √ We now take the ˜ ˜ limit p → ∞ in Eq. (8.46). Using G(∞) = 1 and G(0) = 1/ ¯ , we obtain  Sl (0) =

 1 √ [ + (1 − )βF (β)ρ] − (1 − )βρα00 B, ¯

(8.48)

where 



1/β

α00 =

H (ξ )g0 0

ξ 1 − βξ

dξ.

(8.49)

Numerical values of α00 are given in (Ivanov 1973, Table 35, p. 333) for different values of the Voigt parameter. We show in Table 8.1 the surface value and the value at infinity of the line source function for different choices of the parameters characterizing the continuous absorption and emission. The first line corresponds to a line with a continuous absorption such that  and βF (β) are of the same order, and for which there is no continuous emission at the frequency of the line (ρ = 0). The primary source of photons is B. The continuous absorption produces a global decrease of the source

Table 8.1 The line source function at the surface and at infinity. The effects of a continuous absorption and of a continuous emission. β is the ratio of the continuous to line absorption coefficient, ¯ the effective destruction probability, and ρB the continuous emission. The parameters β, , ρ and B are constants βF (β) ∼  , ρ = 0 βF (β)  < 1  βF (β) < 1

Sl (0) √ ¯ (/¯ )B √ B [βF (β)]1/2 ρB

Sl (∞) (/¯ )B B ρB

154

8 Spectral Line with Continuous Absorption

function by a factor about /¯ . The line source function and the resolvent function are related by      ∞  ∞ B 1 dν B 1 Sl (τ ) = √ √ − = √ √ − φ(ν)e−ντ (τ ) dτ . ν ¯ ¯ ¯ ¯ β τ (8.50) Setting β = 0, we recover the expression of the source function in a semi-infinite medium with a uniform primary source B. In the second line, we assume βF (β)  < 1. The continuous absorption and emission are negligible compared to the collisional processes contributing to the line formation. The source function behaves essentially as with β = 0. The third line corresponds to the opposite situation, with the continuous absorption and emission dominating the collisional processes. There is a √ very strong increase in the surface value of the source function with respect to the -law. These different regimes are very well illustrated in Hummer (1968).

8.3.3 The Emergent Intensity The radiative transfer equation for the radiation field, given in Eq. (8.2), shows that the emergent intensity may be written as 

∞

I (0, x, μ) = 0

[ϕ(x)Sl (τ ) + βQ∗c (τ )]e−τ (ϕ(x)+β)/μ

dτ . μ

(8.51)

Introducing q˜c∗ (p), the Laplace transform of Q∗c (τ ), the continuous emission, we can write the emergent intensity as I (0, x, μ) =

ϕ(x) β p s˜l (p) + p q˜c (p), ϕ(x) + β ϕ(x) + β

(8.52)

where s˜l (p), is the Laplace transform of Sl (τ ), and p = (ϕ(x) + β)/μ. We assume as above that the primary sources for the line and the continuum are uniform, given by Q∗l (τ ) = B and Q∗c (τ ) = ρB. Replacing ps˜l (p) by Eq. (8.46) and using Eq. (8.47), we obtain  ϕ(x)B μ  + (1 − )βF (β)ρ βρB + H( ) I (0, x, μ) = √ β + ϕ(x) ϕ(x) + β ϕ(x) + β ¯    1/β 1 ξ ) dξ . (8.53) H (ξ )g0 ( − (1 − )βρ α00 − 1 − βξ 1 + pξ 0 The integral over ξ in the square bracket is tabulated in (Ivanov 1973, Chapter VII) for  = 0, several values of β, and of the Voigt parameter.

Appendix E: Kernels for Spectral Lines with a Continuous Absorption

155

Equation (8.53) represent an absorption lines, which tends to ρB in the line wings since ϕ(x) → 0. At the line center, the dominant term is the first term in the curly braces and the intensity has roughly the same value as Sl (0). The variation with μ is controlled by the H -function. This equation was first derived by Sobolev in a series of papers listed in (Ivanov 1973, p. 344). Their content can be found in Sobolev (1963).

Appendix E: Kernels for Spectral Lines with a Continuous Absorption In this Appendix we calculate the inverse Laplace transforms kβ (ν, β), k1 (ν, β) and ¯ k2 (ν, β) of the kernel K(τ, β) and of the kernels L1 (τ, β) and L2 (τ, β). The kernels are defined by ¯ K(τ, β) ≡

1 1 n(β) 2

L1 (τ, β) ≡ 1 F (β)

L2 (τ, β) ≡

1 F (β) 

∞



+∞

ϕ 2 (x)E1 [τ (ϕ(x) + β)] dx,

(E.1)

ϕ(x)E1[−τ (ϕ(x) + β)] dx,

(E.2)

−∞



∞ 0

ϕ(x) E2 [−τ (ϕ(x) + β)] dx, ϕ(x) + β

0

(E.3)

and their inverse Laplace transforms by ¯ K(τ, β) =



∞

kβ (ν, β) e

−ν|τ |



∞

dν =

0

gβ (ξ, β) e−|τ |/ξ

0



∞

F (β)Ln (τ, β) =

kn (ν, β) e

−ν|τ |



∞

dν =

0

dξ , ξ

gn (ξ, β) e−|τ |/ξ

0

dξ , ξ

(E.4)

(E.5)

with n = 1, 2. We recall that E1 and E2 are the first and second exponential integrals. The functions n(β) and F (β) are defined in Eqs. (8.8) and (8.11). For ¯ simplicity we assume that τ is positive. For β = 0, K(τ, β) reduces to K(τ ) =

1 2



+∞ −∞

ϕ 2 (x)E1 [τ ϕ(x)] dx.

(E.6)

As shown in Sect. 2.2.3, it can be written as 

∞

K(τ ) = 0

|τ | dξ = g(ξ ) exp(− ) ξ ξ



∞ 0

k(ν)e−τ ν dν,

(E.7)

156

8 Spectral Line with Continuous Absorption

where 

∞

g(ξ ) =

ϕ 2 (u) du,

(E.8)

y(ξ )

and  y(ξ ) =

0

0 < ξ ≤ 1/ϕ(0),

ϕ (1/ξ )

ξ ≥ 1/ϕ(0).

−1

(E.9)

−1

Here, the notation ϕ denotes the inverse function of ϕ. The functions k(ν) and g(ξ ) are plotted in Figs. 5.1 and 5.2 and are related by k(ν) =

1 1 g( ), ν ν

ν ∈ [0, ∞[.

(E.10)

We now follow the method described in (Ivanov 1973, Chapter VII) to obtain an expression of kβ (ν, β) and of the other Laplace transforms, in terms of k(ν) or g(ξ ). ¯ We start with kβ (ν, β). The idea is to write the derivative of K(τ, β) with respect to τ in terms of the derivative of K(τ ). Using the definition 

∞

E1 (t) ≡ t



e−t dt , t

(E.11)

and e−at dE1 (at) =− , dt t

a constant,

(E.12)

we can write  ∞ ¯ d K(τ, β) e−βτ dK(τ ) 1 ν k(ν) e−(ν+β)τ dν. = =− dτ n(β) dτ n(β) 0

(E.13)

An integration over τ leads to ¯ K(τ, β) =

1 n(β)

 0

∞

ν k(ν) e−(ν+β)τ dν. ν+β

(E.14)

Introducing the variable ν = ν + β, we obtain kβ (ν, β) =

β 1 (1 − )k(ν − β), ν ∈ [β, ∞[, n(β) ν

gβ (ξ, β) =

ξ 1 g( ), n(β) 1 − βξ

ξ ∈ [0, 1/β].

(E.15) (E.16)

Appendix E: Kernels for Spectral Lines with a Continuous Absorption

157

Fig. E.1 The function kβ (ν, β) for the Doppler profile and different values of the parameter β, ratio of the continuous absorption coefficient to the line absorption coefficient. The function kβ (ν, β) is zero up to ν = β. It is defined in Eq. (8.17)

The function kβ (ν, β) is shifted by a quantity β. It is zero for ν < β and for ν > β, it is smaller than k(ν) by a factor (1 − β/ν)/n(β). The function gβ (ξ, β) is zero for ξ > 1/β and has a constant value for ξ ∈ [0, 1/(φ(0) + β)]. The function kβ (ν, β) is drawn in Fig. E.1 for the Doppler profile and different values of β. The same method can be used to calculate k1 (ν, β), the inverse Laplace transform of L1 (τ, β). We introduce 1 L0 (τ ) ≡ 2



+∞  1 −∞

ϕ(x)e 0

−τ ϕ(x)/μ

dμ dx = μ



∞

ϕ(x)E1 [τ ϕ(x)] dx.

(E.17)

0

This leads to the following exponential representation, 

∞

L0 (τ ) = 0

k0 (ν)e−ντ dν =



∞

g0 (ξ )e−τ/ξ

0

dξ , ξ

(E.18)

where g0 (ξ ) is defined almost as in Eq. (E.8), except that ϕ 2 (x) is replaced by ϕ(x). The derivatives with respect to τ of L1 (τ, β) and L0 (τ, β) are related by e−βτ dL0 (τ ) dL1 (τ, β) = . dτ F (β) dτ

(E.19)

158

8 Spectral Line with Continuous Absorption

The integration of Eq. (E.19) over τ leads to k1 (ν, β) = (1 − g1 (ξ, β) = g0 (

β )k0 (ν − β), ν ξ ), 1 − βξ

ν ≥ β,

ν ≤ 1/β.

(E.20) (E.21)

Equations (E.20) and (E.21) are identical to Eqs. (8.17) and (E.16), except for the factor 1/n(β). The functions k1 (ν, β) and g1 (ξ, β) are zero for ν < β and ξ > 1/β, respectively, and are related to k0 (ν) and g0 (ξ ) exactly as kβ (ν, β) and gβ (ξ, β) are related to k(ν) and g(ξ ). The function k2 (ν, β) is easily derived from L2 (τ, β) = −L1 (τ, β), dτ

τ ≥ 0.

(E.22)

One obtains k2 (ν, β) =

β 1 1 k1 (ν, β) = (1 − )k0 (ν − β), ν ν ν

g2 (ξ, β) = ξg1 (ξ, β) = ξg0 (

ξ ), 1 − βξ

ν ≥ β,

(E.23)

ξ ∈ [0, 1/β].

(E.24)

These functions are zero for ν < β and ξ > 1/β, respectively.

References Abramov, Yu.Yu., Dykhne, A.M., Napartovich, A.P.: Transfer of resonance radiation in a halfspace. Astrophysics 3, 215–223 (1967); translation from Astrofizika 3, 459–479 (1967) Faurobert, M., Frisch, H.: Line transfer with complete frequency redistribution in an absorbing medium. Scaling laws and approximations. Astron. Astrophys. 149, 372–382 (1985) Hummer, D.G.: Non-coherent scattering-III. The effect of continuous absorption on the formation of spectral lines. Mon. Not. R. Astr. Soc. 133, 73–108 (1968) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ., vol. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Mc Cormick, N.J., Siewert, C.E.: Spectral line formation by noncoherent scattering. Astrophys. J. 162, 633–647 (1970) Nagirner, D.I.: Multiple light scattering in a semi-infinite atmosphere. Uch. Zap. Leningr. Univ. 337, 3 (1968) Sobolev, V.V.: A Treatise on Radiative Transfer. Von Nostrand Company, Princeton, New Jersey (1963); transl. by S.I. Gaposchkin; Russian original, Transport of Radiant Energy in Stellar and Planetary Atmospheres, Gostechizdat, Moscow (1956)

Chapter 9

Conservative Scattering: The Milne Problem

We show in this chapter how to construct an exact solution for the Milne problem and for its generalized complete frequency redistribution version. For both problems the scattering process is conservative ( = 0) and the source function satisfies the homogeneous Wiener–Hopf integral equation, 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ .

(9.1)

0

The Milne problem, introduced in Sect. 2.2.2, describes the radiative equilibrium of a grey plane-parallel semi-infinite atmosphere. It was formulated by Schwarzschild (1906) and its integral formulation was established by Schwarzschild (1914) and Milne (1921). It is the first half-space convolution type integral equation with an exact solution (Wiener and Hopf 1931). The Milne problem has received a very large attention in the literature. References can be found in books by, e.g., Chandrasekhar (1960), Kourganoff (1963), Ivanov (1973), Landi Degl’Innocenti and Landolfi (2004). For spectral lines with complete frequency redistribution, the Milne problem was introduced by Ivanov (1962) (see also Hummer and Stewart 1966; Ivanov 1973). Referred to by V.V. Ivanov, and also here, as the generalized Milne problem, it describes the behavior of the source function in a plane-parallel boundary layer at the surface of a medium. When the destruction probability  is very small, the radiation field can be described at large optical depths by a so called interior field, which depends on the primary source of photons, and a boundary layer field, the variation of which has a universal behavior described by Eq. (9.1). This universal behavior can be observed by comparing the optical depth variation of the source function for a uniform primary source and for an exponentially decreasing primary source, plotted in Figs. 7.1 and 7.2, respectively. The decomposition into an interior field and boundary layer field is discussed in detail in Part III, Chap. 23. Equation (9.1) has of course the trivial solution S(τ ) = 0, but it also has a one-parameter family of solutions, which tends to infinity as τ → ∞. For © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_9

159

160

9 Conservative Scattering: The Milne Problem

monochromatic scattering, it grows linearly at infinity, a result already suggested in Schwarzschild (1914) and confirmed by Eddington (1926). For complete frequency redistribution the τ -variation depends on the absorption profile: τ 1/4 for the Voigt profile and τ 1/2(ln τ )1/4 for the Doppler profile (Ivanov 1973, p. 231). The solution of Eq. (9.1) is defined up to a multiplicative constant. Because of its physical meaning, the free parameter for the monochromatic Milne problem is usually chosen to be the constant radiative flux F of the radiation field inside the atmosphere. For the generalized Milne problem, the usual choice for the free parameter is the surface value of the source function, because the radiative flux may be infinite. The primary source being zero in Eq. (9.1), the source function S(τ ) and the resolvent function (τ ) are related by   S(τ ) = S(0) 1 +

τ

 (τ ) dτ ,

(9.2)

0

as shown in Sect. 7.3.1. This relation holds for monochromatic scattering treated in Sect. 9.1 as well as for complete frequency redistribution treated in Sect. 9.2. In each section, we examine the properties of the conservative dispersion function L(z), of the half-space auxiliary function X(z), of the conservative resolvent function (τ ), and discuss in detail the growth at infinity of the source function. For monochromatic scattering we introduce the Hopf function and show in Appendix G of this chapter how to derive S(τ ) with the inverse Laplace transform method. For complete frequency redistribution, we show how the growth at infinity is controlled by the behavior of X(z) about the origin. The latter is determined in Appendix F of this chapter.

9.1 The Monochromatic Milne Problem The equations for the monochromatic Milne problem have been presented in Sect. 2.2.2. We recall them here. The radiative transfer equation may be written μ

1 ∂I (τ, μ) = I (τ, μ) − ∂τ 2



+1

−1

I (τ, μ) dμ.

(9.3)

The surface boundary condition for the radiation field is I (0, μ) = 0,

μ ∈ [−1, 0[.

(9.4)

The source function is equal to the direction averaged intensity J (τ ), namely S(τ ) = J (τ ) ≡

1 2



+1 −1

I (τ, μ) dμ.

(9.5)

9.1 The Monochromatic Milne Problem

161

It satisfies the homogeneous Wiener–Hopf integral equation written in Eq. (9.1), the solution of which is uniquely defined by the value of the constant radiative flux F . Some important partial results can be derived from this integral equation with a simple heuristic approach, in particular those concerning the behavior at infinity of the source function. By taking moments of the radiative transfer equation, Eddington (1926) has shown that the Milne integral equation has a non trivial solution increasing linearly with τ as τ → ∞. The method is the following. We introduce the radiative flux F (τ ) and the second-moment J (2)(τ ) defined by  F (τ ) ≡ 2π

+1

−1

I (τ, μ)μ dμ,

J

(2)

1 (τ ) ≡ 2



+1

−1

I (τ, μ)μ2 dμ.

(9.6)

The integration of Eq. (9.3) over μ yields dF (τ ) = 0, dτ

hence F (τ ) = F.

(9.7)

The multiplication of Eq. (9.3) by μ, followed by an integration over μ yields F dJ (2) (τ ) = , dτ 4π

hence J (2) (τ ) =

F (τ + C), 4π

(9.8)

where C is a constant. Its value, given in Eq. (9.27), requires the exact solution of the problem. As pointed out by Eddington, I (τ, μ) becomes isotropic and tends to S(τ ) far from boundaries, as their influence becomes negligible. The isotropization of the radiation field allows one to write lim J (2)(τ ) 

τ →∞

1 lim S(τ ). 3 τ →∞

(9.9)

This approximation, combined with Eq. (9.8) leads to lim S(τ ) 

τ →∞

3 F. 4π

(9.10)

The first non trivial exact result concerning the Milne problem is the Hopf– Bronstein relation, established independently by Bronstein (1929) and Hopf (1930) (see Chandrasekhar 1960, for a detailed historical account). This relation states that √ 3 S(0) = F. 4π

(9.11)

It provides an exact expression of the angle-averaged intensity emerging from a gray stellar atmosphere in terms of the star radiative flux. There is no elementary proof for the Hopf–Bronstein relation. Those of Bronstein (1929) and Hopf (1930) are

162

9 Conservative Scattering: The Milne Problem

fairly involved. Hopf’s proof relies on a Neumann series expansion of the source function. Simpler proofs have been developed in the course of time (see e.g. Landi Degl’Innocenti and Landolfi 2004, Appendix 15). In Sect. 9.1.2 we construct an exact solution, which automatically provides the Hopf–Bronstein relation.

9.1.1 The Auxiliary Functions To construct an exact solution we must first determine the properties of the dispersion function L(z) and of the half-space auxiliary function X(z). The properties of these auxiliary functions are derived in Chap. 5 for  = 0. We give them here for  = 0. For  = 0, the dispersion function may be written as L(z) = 1 −

1+z 1 ln . 2z 1 − z

(9.12)

When  = 0, L(z) has two zeroes at ±ν0 , with ν0 ∈ [0, 1[ (see Sect. 5.2.2). They coalesce into a double zero at the origin (see Fig. 5.4) when  = 0. A Taylor expansion of L(z) around zero leads to L(z)  −

z2 + O(z4 ), 3

z → 0.

(9.13)

The limiting values L± (ν) can be defined as in Eq. (5.21) and vary as shown in Fig. 5.5, except that the imaginary part of L± (1) take the values ±π/2. The imaginary and real parts are given by Eqs. (5.22) and (5.23) with  = 0. The function X(z) can be defined and constructed exactly as described in Sect. 5.3. The factorization relation given in Eq. (5.54), which is a consequence of the defining relation, becomes X(z)X(−z) = −

1 L(z). z2

(9.14)

Combined with Eq. (9.13), it leads to 1 X(0) = √ . 3

(9.15)

This value can also be obtained by taken the limit  → 0 in Eq. (5.57). The behavior of X(z) for z → ∞ is not modified. One still has X(z)  −1/z (see Eq. (5.47)). The function X∗ (z) becomes X∗ (z) = −zX(z),

(9.16)

9.1 The Monochromatic Milne Problem

163

and the relation between the functions X(z) and H (z) is 1 1 H (z) = 1/X∗ (− ) = z/X(− ). z z

(9.17)

Thus H (0) = 1,

√ lim H (z)  z 3.

(9.18)

z→∞

√ For non-conservative scattering, H (z) tends to the limit 1/  as z → ∞. When  = 0, it tends to infinity, increasing linearly with z. In contrast, the value of H (0) is independent of the value of . The zeroth and first-moment are α0 = 2,

√ α1 = 2/ 3.

(9.19)

To obtain α0 , it suffices to set  = 0 in Eq. (B.25). How to calculate α1 is explained in Sect. B.5.

9.1.2 The Source Function and the Emergent Intensity A few preliminaries remarks concerning the resolvent function and its inverse Laplace transform φ(ν) are needed before we make use of the relation   S(τ ) = S(0) 1 +

τ



(τ ) dτ



 .

(9.20)

0

First of all, we remark that (τ ) tends to a constant at infinity. This property can be derived from Eq. (9.20) itself and the knowledge that S(τ ) grows linearly as τ → ∞. It can also be derived from Eq. (6.47) when ν0 is set to zero. Actually, all the results concerning φ(ν) given in Sect. 6.2.2 for  = 0 hold also for  = 0, provided one uses the conservative version of X(z) and sets ν0 = 0. The reason is that the solution of the singular integral equation for φ(ν) in the region ν ∈ [0, 1] (see Eq. (6.31)), compatible with ν0 = 0 and the constraint that (τ ) tends to a constant at infinity, is φ0 δ(ν). Thus Eq. (6.40) becomes φ0 = and Eq. (6.47) becomes  √ (τ ) = 3 + 1

∞

√ 1 = 3, X(0)

φ∞ (ν)νX(−ν)e−ντ dν,

(9.21)

τ ∈ [0, ∞[,

(9.22)

164

9 Conservative Scattering: The Milne Problem

φ∞ (ν) being given by Eq. (6.13) with  = 0. It depends only on the dispersion function. Inserting Eq. (9.22) into Eq. (9.20) we obtain   √ S(τ ) = S(0) 1 + 3τ +

∞ 1

 φ∞ (ν)X(−ν)(1 − e−ντ ) dν .

(9.23)

We show Appendix G in this chapter, where we apply the inverse Laplace transform method to S(τ ), that the linear growth at infinity is caused by the double zero of the dispersion function. It remains to determine S(0). This we can do by considering the asymptotic behavior at infinity of the source function. For τ → ∞, Eq. (9.23) leads to √ S(τ )  S(0) 3τ and the Eddington approach to S(τ )  (3/4π)F τ . Equating these two expressions, we obtain the Hopf–Bronstein relation, √ S(0) = ( 3/4π)F.

(9.24)

A proof of this relation, based on the exact solution of the Milne integral equation by an inverse Laplace transform, is given in Appendix G of this chapter. To determine the emergent intensity, we follow the remark of Ivanov (1973) that the Milne problem, and also the generalized Milne problem for complete frequency redistribution, are the limit for  → 0 of the non-conservative inhomogeneous equation 

∞

S(τ ) = (1 − )

K(τ − τ )S(τ ) dτ +

√ S(0).

(9.25)

0

√ In the limit  → 0, one recovers a homogeneous equation and the -law ensures that the source function takes the value S(0) at the surface. For a uniform primary source term, the emergent intensity is simply given by I (0, μ) = S(0)H (μ) (Sect. 7.3.1). For the Milne problem, one thus obtains I (0, μ) = S(0)H (μ) =

√ 3 F H (μ), 4π

μ ∈ [0, 1].

(9.26)

We are now in the position to determine the constant C appearing in the expression of J (2)(τ ). Inserting Eq. (9.26) into Eq. (9.8), we obtain √  1 3 C= H (μ)μ2 dμ. 2 0

(9.27)

This constant equals q(∞), the value at infinity of the Hopf function, as we show in Eq. (9.31).

9.1 The Monochromatic Milne Problem

165

9.1.3 The Hopf Function Following Hopf (1930), the source function for the Milne problem is usually written as S(τ ) =

3 F [τ + q(τ )], 4π

(9.28)

where q(τ ) is known as the Hopf function. The properties of this function have been studied in great detail (for references see e.g. Chandrasekhar 1960, Kourganoff 1963, Ivanov 1973, Landi Degl’Innocenti and Landolfi 2004). Equations (9.17) and (9.23) show that q(τ ) may be written as

  ∞ 1 1 − e−τ ν dν q(τ ) = √ 1 + φ∞ (ν) . H ( ν1 ) ν 3 1

(9.29)

This expression was established by Mark (1947) with the Wiener–Hopf method. Equation (G.13) shows that it can also be written as q(τ ) =

1 X (0) −√ X(0) 3



∞ 1

φ∞ (ν)νX(ν)e−τ ν

dν . ν

(9.30)

√ The Hopf function is monotonic. At the surface q(0) = 1/ 3 = 0.5773. At infinity, according to Eq. (G.15), it can be written as √  1 X (0) 3 q(∞) = = H (μ)μ2 dμ = C, X(0) 2 0

(9.31)

where C is the constant in the expression of J (2)(τ ). Numerically, q(∞)  0.7104, value given by Hopf (1934) with four significant digits. More accurate determinations of q(∞) have been obtained subsequently (see Ivanov 1973, p. 140 for references). Note the small amplitude of variation of q(τ ) as τ varies between 0 and ∞. The value q(∞) is often referred to as the extrapolation length or the extrapolation endpoint. It is denoted τe or τ0 depending on the authors. It is the distance above τ = 0 at which the linear approximation, S(τ ) 

3F [τ + q(∞)], 4π

(9.32)

extrapolates to zero. The values of q(τ ) at zero and infinity can be expressed in terms of the moments αn of the H -function. The values of α0 and α1 given in Eq. (9.19), combined with the definition of α2 in Eq. (9.31), lead to q(0) =

α1 , α0

q(∞) =

α2 . α1

(9.33)

166

9 Conservative Scattering: The Milne Problem

We also note that q(τ ) satisfies the inhomogeneous Wiener–Hopf integral equation 

∞

q(τ ) = 0

1 K(τ − τ )q(τ ) dτ + E3 (τ ), 2

(9.34)

where E3 (τ ) is the third exponential integral function. This equation can be obtained by combining the definition of q(τ ) in Eq. (9.28) with the integral equation for S(τ ) in Eq. (9.1). A numerical algorithm for solving Eq. (9.34) is described in Landi Degl’Innocenti and Landolfi (2004, Appendix 16). An interesting physical interpretation proposed by Ivanov (1973, p. 289) explains why the values of q(τ ) lie in the interval [0, 1]. This interpretation is based on the expression of the primary source, 1 1 E3 (τ ) = 2 2



1

e−t /μ μ dμ,

(9.35)

p(τ, μ)μ dμ,

(9.36)

0

which shows that q(τ ) can be written as  q(τ ) = 2π

1

0

where p(τ, μ) is the solution of p(τ, μ) =

1 2



∞

E1 (τ − τ )p(τ , μ) dτ +

0

1 −τ/μ e . 4π

(9.37)

We recall that K(τ ) = E1 (τ )/2. The integration of Eq. (9.37) over μ shows that  2π

1

p(τ, μ) dμ = 1.

(9.38)

0

The Hopf function can thus be written as 1

p(τ, μ)μ dμ q(τ ) = 0 1 . 0 p(τ, μ) dμ

(9.39)

A Neumann series expansion of the integral equation for p(τ, μ) shows, as pointed out by Ivanov (1973, pp. 205, 290), that p(τ, μ)dω is the probability for a photon created at depth τ to escape from the medium within a solid angle dω around the direction defined by μ. The Hopf function can thus be interpreted as the mathematical expectation (mean) cosine of the angle of emergence of photons that began their random walk at a depth τ . The concept of the Hopf function as defined by Eq. (9.39) can be generalized to spectral lines (Ivanov 1973, p. 293).

9.2 The Milne Problem for a Spectral Line

167

9.2 The Milne Problem for a Spectral Line The radiative transfer equation corresponding to the generalized Milne problem may be written as     1 +∞ +1 ∂I μ (τ, x, μ) = ϕ(x) I (τ, x, μ) − I (τ, x, μ)ϕ(x) dμ dx . ∂τ 2 −∞ −1 (9.40) Here x is the frequency in Doppler width unit with x = 0 at line center. It is assumed that there is no incident field on the surface at τ = 0. The source function S(τ ) is defined by 1 S(τ ) ≡ 2



+∞  +1 −∞

−1

I (τ, x, μ)ϕ(x) dμ dx,

(9.41)

and satifies the Wiener–Hopf integral equation in Eq. (9.1). The source function can be written in the same way as the monochromatic source function, namely as   S(τ ) = S(0) 1 +

τ

 (τ ) dτ ,

(9.42)

0

with (τ ) the resolvent function. The free parameter is S(0). For complete frequency redistribution, the radiative flux, given by the integral of I (τ, x, μ)μ over direction and frequency, may become infinite, while its derivative remains zero. We now use Eq. (9.42) to determine the behavior of S(τ ) for τ → ∞. First we must determine the behavior at infinity of (τ ) for  = 0. In Sect. 6.2.1 we show that (τ ) can be written as 

∞

(τ ) =

φ(ν)e−ντ dν,

(9.43)

0

with   1 1 1 − φ(ν) = = φ∞ (ν)X(−ν), 2i π X+ (ν) X− (ν)

ν ∈ [0, ∞[.

(9.44)

Because X+ (ν) and X− (ν) are complex conjugate (see Sect. B.2), Eq. (9.44) can also be written as   1 1 . (9.45) φ(ν) =  π X+ (ν) All these equations still hold for  = 0, but the behavior of φ(ν) for ν → 0, needed to find the behavior of (τ ) at infinity, depends on the value of . We show in

168

9 Conservative Scattering: The Milne Problem

Appendix F of this chapter that π + XD (ν)  ( )1/2 ν 1/2 (− ln ν)−1/4 e−i π/2, ν → 0, 4 √ 2π 1/2 1/4 1/4 −i π/4 + ) a ν e XV (ν)  ( , ν → 0. 3

(9.46) (9.47)

Taking the imaginary parts, we find after a few lines of algebra: D (τ ) 

1 4 1/2 1 1/2 ( ) ( ) (ln τ )1/4 π π τ



∞ 0

3 1 1 1 V (τ )  √ ( √ )1/2 ( )1/4 ( )3/4 a τ π 2 2π

u−1/2 e−u du, 

∞

u−1/4 e−u du.

(9.48) (9.49)

0

The integrals in these expressions can be expressed in terms of the -function, 

∞

(x) =

ux−1 e−u du.

(9.50)

0

Using ( 12 ) =

√ √ π and ( 34 ) = π 2/ ( 14 ), we obtain 2 1 1/2 ( ) (ln τ )1/4 , π τ 1 1 3 1 V (τ )  ( √ )1/2 1 ( )1/4( )3/4 . a τ ( 4 ) 2π

D (τ ) 

(9.51) (9.52)

The order of the next term is given in Ivanov (1973, p. 227). In contrast to monochromatic scattering, the resolvent function for complete frequency redistribution in a conservative semi-infinite medium tends to zero at infinity. This difference is of course due to the frequency redistribution, which allows photons to escape more easily because they can be scattered towards the line wings. The integration of (τ ) over τ , leads to the asymptotic behavior 4 SD (τ )  S(0) τ 1/2 (ln τ )1/4 , π √ 27/4 a −1/4τ 1/4 . SV (τ )  S(0) 3 π1/4 ( 14 )

(9.53) (9.54)

In Fig. 9.1 we compare these scaling laws with the source function calculated for  = 10−6 , with a Doppler profile and a Voigt profile with a Voigt parameter a = 10−2 . One can observe the onset of the Lorentz regime for τ larger than 1/a. The scaling laws in Eq. (9.54) can be found in Ivanov (1973, p. 231). These scaling laws are used in Part III for some asymptotic results on finite slabs.

Appendix F: The Conservative Auxiliary Functions. Complete Frequency. . .

169

Fig. 9.1 Large τ asymptotic behavior of the source function for conservative scattering. The function S(τ ) is calculated with a uniform primary source  = 10−6 for a Doppler profile and a Voigt profile. The straight lines show the corresponding asymptotic behaviors, τ 1/2 (ln τ )1/4 and τ 1/4

Appendix F: The Conservative Auxiliary Functions. Complete Frequency Redistribution For conservative scattering, that is for  = 0, the dispersion function L(z) is zero at the origin. For monochromatic scattering, a simple Taylor expansion of Eq. (9.12) leads to the expansion in Eq. (9.13). In this Appendix, we deal with complete frequency redistribution and the algebra is slightly more difficult. The dispersion function was introduced in Sect. 5.2 as  L(z) ≡ 1 − (1 − )

∞

k(ν)( 0

1 1 + ) dν, ν −z ν+z

(F.1)

where k(ν), the inverse Laplace transform of the kernel K(τ ), is defined by 

∞

K(τ ) =

k(ν)e−|τ |ν dν.

(F.2)

0

The kernel is normalized to unity, that is 

+∞

∞

k(ν) 0

−∞

K(τ ) dτ = 1. This implies

1 dν = . ν 2

(F.3)

Equation (F.1) shows that the behavior of L(z) for z → 0 is controlled by the behavior of k(ν) for ν → 0 and that the latter, as shown by Eq. (F.2) controls the

170

9 Conservative Scattering: The Milne Problem

behavior of K(τ ) for large optical depths. In Sect. F.1 we determine the asymptotic behaviors of k(ν) and K(τ ) and in Sect. F.2, those of L(z) and X(z).

F.1 Asymptotic Behavior of the Kernel In Sect. 5.1, k(ν) is written as k(ν) =

1 1 g( ), ν ν

(F.4)

where 

∞

g(ξ ) =

ϕ 2 (v) dv,

(F.5)

y(ξ ) −1

y(ξ ) = 0 for 0 < |ξ | ≤ 1/ϕ(0) and y(ξ ) = ϕ (1/ξ ) for |ξ | ≥ 1/ϕ(0). The behavior of k(ν) for ν → 0 is thus controlled by the behavior of g(ξ ) for ξ → ∞, with ξ defined by ξ = μ/ϕ(x). Large values of ξ thus correspond to large values of the frequency. The variations of gD,V (ξ ) and kD,V(ν) are plotted in Figs. 5.1 and 5.2. Using ϕ(v) as independent variable and replacing y(ξ ) by its expression for ξ ≥ ϕ(0), we obtain  g(ξ ) =

0 1 ξ

−1

u2 d( ϕ (u)).

(F.6)

2 √ For the Doppler profile, ϕ(u) = e−u / π, thus



1 ξ

gD (ξ ) = 0

u  √ du. 2 − ln(u π)

(F.7)

Introducing the variable w = ξ u, we can write this integral as gD (ξ ) 

1 √ 2 2ξ ln ξ



1 0

[1 −

√ ln w π −1/2 ] w dw. ln ξ

(F.8)

Expanding the square bracket in the limit ξ → ∞, we obtain to leading order gD (ξ ) 

1 1 , √ 4ξ 2 ln ξ

ξ → ∞,

kD (ν) 

1 ν √ , 4 − ln ν

ν → 0.

(F.9)

F.2 The Dispersion Function and the Half-Space Auxiliary Function Near the. . .

171

For the Voigt profile, when the parameter a is small, the asymptotic behavior for large values of ξ may be written as ϕ(x)  a/(πx 2). Thus 1 gV (ξ )  √ a 1/2 2 π



1 ξ

√ u du.

(F.10)

0

Performing the integration, we find, at leading order, 1 a 1/2 gV (ξ )  √ , 3 π ξ 3/2

ξ → ∞,

1 kV (ν)  √ a 1/2ν 1/2 , 3 π

ν → 0.

(F.11)

√ Thus for the Doppler profile k(ν) tends to zero as ν/ − ln ν and for the Voigt profile as ν 1/2 . These expressions lead to KD (τ ) 

1 1 √ , 4τ 2 ln τ

KV (τ ) 

a 1/2 , 6τ 3/2

(F.12)

The asymptotic behavior of K(τ ) for large values of τ is discussed in detail in Sect. 19.2. We show how to construct higher order terms and introduce a general formula, valid for both profiles.

F.2 The Dispersion Function and the Half-Space Auxiliary Function Near the Origin Taking into account the normalization of k(ν) given in Eq. (F.3), the dispersion function can be written as  ∞ dν k(ν) . (F.13) L(z) = −2z2 (ν − z)(ν + z) ν 0 Replacing k(ν) by its asymptotic form for ν → 0, setting z = ρei θ and ν = uρ (ρ real), and considering the limit ρ → 0, we find after some algebra π π z(− ln |z|)−1/2 exp(−i ), 4 2 √ 2π 1/2 1/2 π a z exp(−i ), z → 0. LV (z)  3 4 LD (z) 

(F.14)

These asymptotic behaviors can also be derived from the limiting values L± (ν). For conservative scattering, L± (ν) = λc (ν) ∓ i πk(ν),

(F.15)

172

9 Conservative Scattering: The Milne Problem

Table F.1 Complete frequency redistribution. Real part, imaginary part, and argument of L+ (ν) for  = 0

Profile ν [L+ (ν)] [L+ (ν)]

θ(ν)

Doppler 0 ν ∼ (− ln ν)3/2 ν ∼− (− ln ν)1/2 π − 2

∞

Voigt 0

∞

1

∼ ν 1/2

1

0

∼ −ν 1/2

0

0

−

π 4

0

with  λc (ν) = −2ν

2

∞ 0

k(ν ) dν . ν (ν 2 − ν 2 )

(F.16)

Making the change of variable ν = ρν, letting ν → 0, and introducing the asymptotic behavior of k(ν) for ν → 0 given in Eqs. (F.9) and (F.11), we obtain λD c (ν)

ν π  −( )2 , 4 (− ln ν)3/2

λV c (ν)

√ π 1/2 1/2 a ν .  3

(F.17)

For the Doppler profile, the factor (− ln ν)3/2 comes from the expansion of (− ln ρν) and the remark that the Principal Value integral, from zero to infinity, of 1/(1 − ρ 2 ) is zero. The constant factors in Eq. (F.17) can be obtained by a contour integration along the upper right quadrant of a circle indented at ρ = 1 on the real axis. Details on the calculations can be found in Ivanov (1973, p. 84), Frisch and Frisch (1982), Ivanov et al. (1997). The leading terms of the real and imaginary parts of L+ (ν) are listed in Table F.1. The ratio t (ν) = −πk (ν)/λ c (ν), giving the slope of the phase diagram of L+ (ν) for each value of ν, has thus the values t D (ν) 

4 (− ln ν), π2

t V (ν)  −1,

ν → 0.

(F.18)

θV (ν) → −π/4,

ν → 0,

(F.19)

This implies θD (ν) → −π/2;

where θ (ν) is the argument of L+ (ν). For  = 0, the modulus of L+ (ν) is given |L+ (ν)| = [λ2c (ν) + π2 k 2 (ν)]1/2.

(F.20)

F.2 The Dispersion Function and the Half-Space Auxiliary Function Near the. . .

173

(L±) L− D (ν) L− V (ν) L(0)

L(∞) 1.0

(L±)

L+ V (ν) L+ D (ν)

Fig. F.1 Complete frequency redistribution. Sketch of the phase diagrams of L+ (ν) and L− (ν) for  = 0. At the point ν = 0, L± (0) = L(0) = 0 and the phase diagrams are tangent to the vertical axis in the case of the Doppler profile and and make an π/4 angle with this axis for a Voigt profile. For ν → ∞, L± (∞) = L(∞) = 1 and the phase diagrams are tangent to the vertical axis

Replacing λc (ν) and k(ν) by their asymptotic values, we find π π ν(− ln ν)−1/2 exp(−i ), 4 2 √ 2π 1/2 1/2 π a ν exp(−i ), ν → 0, L+ V (ν)  3 4

L+ D (ν) 

ν > 0.

(F.21)

The phase diagram of L+ (ν) about ν = 0 and for ν → ∞ is sketched in Fig. F.1. At infinity, L+ (ν) tends to unity, independently of the value of . Equation (F.21) can also be derived from Eq. (F.14) by setting z = ν. To obtain L− (ν), for ν > 0, one should set z = νei π . The asymptotic behavior of X(z) for z → 0 can now be deduced from the relation X(z)X(−z) = L(z). Equation (F.14) leads to π π XD (z)  ( )1/2z1/2 (− ln |z|)−1/4 exp(−i ), 4 2 √ 2π 1/2 1/4 1/4 π ) a z exp(−i ), z → 0. XV (z)  ( 3 4

(F.22)

The relation H (z) = 1/X(−1/z) provides the H -function at infinity. Ignoring constant factors, we obtain HD (z) ∼ z1/2(ln |z|)1/4,

HV (z) ∼ a −1/4z1/4 ,

z → ∞.

(F.23)

174

9 Conservative Scattering: The Milne Problem

Appendix G: The Milne Problem with the Inverse Laplace Transform Method Here we show how to solve the Milne integral equation for monochromatic scattering by the inverse Laplace transform method. It was applied for monochromatic, non-conservative scattering in Sect. 6.2.2 and in Appendix D.1 of Chap. 7, for the resolvent function and the source function. The Milne integral equation is simpler because it has no inhomogeneous term, but there is a new feature because the dispersion function has a double zero at the origin. The inverse Laplace transform of Eq. (9.1) leads to the singular integral equation  λ(ν)s(ν) + η(ν)

∞ 0

s(ν ) dν = 0, ν − ν

ν ∈ [0, ∞[,

(G.1)

where s(ν) is the inverse Laplace transform of S(τ ). In the interval ν ∈ [0, 1[, Eq. (G.1) reduces to λ(ν)s(ν) = 0,

(G.2)

since η(ν) = −k(ν) = 0. Because λ(ν) has a double zero at the origin, the solution of Eq. (G.2) has the form (Frisch 1988), s(ν) = s1 δ (ν) + s0 δ(ν),

ν ∈ [0, 1[,

(G.3)

where δ (ν) is the derivative of the Dirac distribution and s1 and s0 are two constants to be determined. The term proportional to δ (ν), which comes from the double zero of at the origin of the dispersion function, is responsible for the linear behavior of S(τ ) at infinity. In the interval ν ∈ [1, ∞[, Eq. (G.1) may be written as  λ(ν)s(ν) + η(ν)

∞ 1

s(ν ) s1 s0 dν = η(ν)[− 2 + ], ν − ν ν ν

ν ∈ [1, ∞[.

(G.4)

Introducing S(z) =

1 2i π



∞ 1

s(ν) dν, ν −z

(G.5)

using η(ν) = [L+ (ν) − L− (ν)]/(2i π) and X+ (ν)/X− (ν) = L+ (ν)/L− (ν), we can rewrite Eq. (G.4) as X+ (ν)[2i πν 2 S + (ν) − s0 ν + s1 ] − X− (ν)[2i πν 2 S − (ν) − s0 ν + s1 ] = 0.

(G.6)

Appendix G: The Milne Problem with the Inverse Laplace Transform Method

175

The general solution of this equation has the form X(z)[2i πz2S(z) − s0 z + s1 ] = P (z),

(G.7)

where P (z) is an entire function. For z → ∞, X(z) and S(z) tend to zero as 1/z, hence P (z) tends to a constant and by the Liouville theorem P (z) = P0 with P0 a constant. To determine P0 , it suffices to set z = 0 in Eq. (G.7). One finds P0 = s1 X(0).

(G.8)

  1 X(0) − 1) + s0 z . S(z) = s1 ( 2i πz2 X(z)

(G.9)

We can thus write

Since S(z) is a Hilbert transform, there should be no singularity at z = 0. To satisfy this condition, the constant term and the term of order z in the square bracket must be zero for z = 0. Taylor expanding X(z) around z = 0 to first order, we see that this condition is satisfied for s0 = s1

X (0) . X(0)

(G.10)

The function s(ν) is given by the jump of S(z) across the branch cut [1, ∞[. We thus obtain   1 X(0) 1 − s(ν) = s1 , ν ∈ [1, ∞[. (G.11) 2i πν 2 X+ (ν) X− (ν) This result can also be expressed in terms of φ(ν), the inverse Laplace transform of the resolvent function (τ ). Letting ν0 = 0 in Eq. (6.42), we obtain s(ν) = −s1 X(0)

φc (ν) , ν

ν ∈ [1, ∞[,

(G.12)

√ where φc (ν) = φ∞ (ν)νX(−ν) and X(0) = 1/ 3. We recall that φc (ν) = φ(ν) for ν ∈ [1, ∞[. Adding the contributions from the regions ν ∈ [0, 1] and ν ∈ [1, ∞[, we find that the source function may be written as    ∞ 1 dν X (0) −√ φc (ν)e−ντ S(τ ) = s1 τ + . X(0) ν 3 1

(G.13)

The second and third term in the square bracket provide the Hopf function q(τ ) introduced in Sect. 9.1.3.

176

9 Conservative Scattering: The Milne Problem

Finding the expression of s1 is very easy. The Eddington approximate solution leads to S(τ )  (3F /4π)τ for τ → ∞ (see Eq. (9.10). Hence, s1 =

3F . 4π

(G.14)

There are different ways of writing X (0). We give two different expressions, one in terms of the second order moment of the H -function and the other one in terms of φ(ν). We consider the nonlinear integral equation for X(z) in Eq. (B.19). Taking its derivative and setting  = ν0 = 0, we immediately obtain 1 X (0) = 2



∞ 1

1 dν 1 1 = X(−ν) ν 4 ν 2



1

H (μ)μ2 dμ.

(G.15)

0

To express X (0) in terms of the resolvent function, we consider F (z), the Hilbert transform of φc (ν), defined in Eq. (6.35). Setting z = 0 in Eq. (6.35) and taking the limits z → 0 and then ν0 → 0 in Eq. (6.41), we obtain X (0) =1+ X2 (0)



∞ 1

φc (ν) dν. ν

The source function can then be written as     ∞ 3F φc (ν) −ντ (1 − e ) dν . S(τ ) = τ + X(0) 1 + 4π ν 1

(G.16)

(G.17)

√ Setting X(0) = 1/ 3 and φc (ν) = φ∞ (ν)νX(−ν) = φ∞ (ν)/H (1/ν), we recover the expression of q(τ ) in Eq. (9.29).

References Bronstein, M.: Über das Verhältnis der effektiven Temperatur der Sterne zur Temperature ihrer Oberfläche. Z. Phys. 59, 144–148 (1929) Chandrasekhar, S.: Radiative Transfer, 1st edn. Dover Publications, New York (1960); Oxford University Press (1950) Eddington, A.S.: The Internal Constitution of Stars, p. 322. Dover, New York (1926) Frisch, H.: A Cauchy integral equation method for analytic solutions of half-space convolution equations. J. Quant. Spectrosc. Radiat. Transf. 39, 149–162 (1988) Frisch, H., Frisch, U.: method of Cauchy integral equation for non-coherent transfer in half-space. J. Quant. Spectrosc. Radiat. Transf. 28, 361–375 (1982) Hopf. E.: Mathematical Problems of Radiative Equilibrium. Cambridge University Press, London (1934) Hummer, D.G., Stewart, J.C.: Thermalization lengths and the homogeneous transfer equation in line formation. Astrophys. J. 146, 290–294 (1966) Hopf, E.: Remarks on the Schwarzschild–Milne model of the outer layers of a star. Mon. Not. R. Astr. Soc. 90, 287–293 (1930)

References

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Ivanov, V.V.: Diffusion of resonance radiation in stellar atmospheres and nebulae. I. Semiinfinite medium. Sov. Astron. 6, 783–801 (1962); translation from Astron. Zhurnal 39, 1020–1032 (1962) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ., vol. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Ivanov, V.V, Grachev, S.I., Loskutov, V.M.: Polarized line formation by resonance scattering II. Conservative case. Astron. Astrophys. 321, 968–984 (1997) Kourganoff, V.: Basic Methods in Transfer Problems. Radiative Equilibrium and Neutron Diffusion, 1st edn. Dover Publications, New York (1963); Oxford University Press, London (1952) Landi Degl’Innocenti, E., Landolfi, M.: Polarization in Spectral Lines. Kluwer Academic Publisher, Dordrecht (2004) Mark, C.: The neutron density near a plane surface. Phys. Rev. 72, 558–564 (1947) Milne, E.A.: Radiative equilibrium in the outer layers of a star: the temperature distribution and the law of darkening. Mon. Not. R. astr. Soc. 81, 361–375 (1921) Schwarzschild, K.: Über das Gleichgewicht der Sonnenatmosphäre. Nachr. von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Math.-phys. Klasse 195, 41–53 (1906) Schwarzschild, K.: Über Diffusion und Absorption in der Sonnenatmosphäre. Sitzungber. Acad. Wissen. Berlin 47, 1183–1200 (1914) Wiener, N., Hopf, D.: Über eine Klasse singulärer Integralgleichungen. Sitzungsberichte der Preussichen Akademie, Mathematisch-Physikalische Klasse 3. Dez. 1931 31 696–706 1931 (ausgegeben 28. Januar 1932); English translation: in Fourier transforms in the Complex Domain, Paley, R. C. & Wiener, N., Am. Math. Soc. Coll. Publ. XIX, 49–58 (1934)

Chapter 10

The Case Eigenfunction Expansion Method

As nicely summarized by Mc Cormick and Kušˇcer (1973), The method of singular eigenfunction expansions is modeled after the Fourier approach to partial differential equations. The basic idea is to construct a complete set of eigenmodes, i.e., solutions of the homogeneous transport equation with separated variables. The eigenmodes are used to expand arbitrary solutions, and the main task consists of the determination of the expansion coefficients.

The method of singular eigenfunction expansions has been introduced by Case (1960) for one-speed neutron transport. Here, we referred to it as the Case expansion method. One-speed, also known as monokinetic neutron transport, corresponds to monochromatic scattering in the radiative transfer terminology. The method has been described in many books and review articles (e.g. Case and Zweifel 1967; Roos 1969; Mc Cormick and Kušˇcer 1973; Duderstadt and Martin 1979). The Case method was subsequently extended to energy dependent scattering problems with so-called separable kernels resembling radiative transfer problems with complete frequency redistribution (see the review by Mc Cormick and Kušˇcer 1973). In this chapter we describe the Case expansion method for monochromatic scattering and also a complete frequency redistribution generalization. Our presentation closely follows the original article by Case (1960). The radiation field for monochromatic scattering and complete frequency redistribution has the form I (τ, ξ ) with ξ = μ for monochromatic scattering and ξ = μ/ϕ(x) for complete frequency redistribution (see e.g. Chap. 2). The idea of the Case expansion method is to look for elementary solutions of radiative transfer equations of the form Iν (τ, ξ ) = (ν, ξ ) e−ντ ,

(10.1)

and to expand the radiation field in terms of these solutions. The form of the elementary solution is dictated by the translational invariance of the homogeneous radiative transfer equation (the radiative transfer equation without the primary source). In analogy with ordinary terminology, one says that the function (ν, ζ ) is the eigenfunction corresponding to the eigenvalue ν. Once the eigenfunctions have © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_10

179

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10 The Case Eigenfunction Expansion Method

been determined, as solutions of the homogeneous radiative transfer equation, the solution of a given radiative transfer problem with its boundary conditions is sought in the form  (10.2) I (τ, ξ ) = a(ν)(ν, ξ )e−ντ dν. The domain of variation of ν depends on the problem at hand. The expansion coefficient a(ν) may be an ordinary function or a distribution. Inserting Eq. (10.2) into a radiative transfer equation, we obtain for a(ν) a singular integral equation with a Cauchy-type kernel, similar to the integral equations stemming from the direct or inverse Laplace transform methods. The relation between the Case method and the discrete ordinate method of Chandrasekhar introduced for monochromatic scattering, it is quite obvious. Chandrasekhar starts also with the ansatz in Eq. (10.1), but with a variable ν taking only a finite number of discrete values. Hence, the equation corresponding to Eq. (10.2) contains a summation instead of an integral. The two methods become identical when the number of terms in the summation becomes infinite. Many exact results in Chandrasekhar (1960) are obtained by taking this limit. The Case eigenfunction expansion method can be used for full and half-space radiative transfer problems with an incident radiation and/or internal sources, for monochromatic scattering and complete frequency redistribution. In Sect. 10.1 we determine the eigenfunctions for monochromatic scattering and complete frequency redistribution. In Sect. 10.2 we determine the expansion coefficients a(ν) for the diffuse reflection problem, treating both monochromatic scattering and complete frequency redistribution. To illustrate the method somewhat further, we show in Sect. 10.3, for complete frequency redistribution, two other examples of application: a full-space problem with a point source at the origin and a half-space problem with internal sources.

10.1 The Eigenfunctions and Eigenvalues The eigenfunctions (ν, ξ ) are solution of the homogeneous radiative transfer equation. As already mentioned above, they can be regular functions or distributions. We determine them first for monochromatic scattering and then for complete frequency redistribution.

10.1.1 Monochromatic Scattering We consider the homogeneous radiative transfer equation  ∂I 1 +1 μ (τ, μ) = I (τ, μ) − (1 − ) I (τ, μ ) dμ . ∂τ 2 −1

(10.3)

10.1 The Eigenfunctions and Eigenvalues

181

We use here the same convention regarding notation as in the preceding chapters. The optical depth τ and the direction cosine μ are chosen in such a way that the absorption term in the right-hand side appears with a plus sign. In Case and Zweifel (1967) and in the literature on the Case method, the absorption term usually appears with a minus sign and the variable here denoted ν is usually denoted 1/η or 1/ξ . The original notation has been changed to keep contact with the inverse Laplace transform method of solution described in preceding chapters. We look for elementary solutions of the form Iν (τ, μ) = (ν, μ) e−ντ ,

(10.4)

with ν real. Inserting Eq. (10.4) into Eq. (10.3), we obtain the eigenvalue equation (1 + νμ)(ν, μ) =

1 (1 − )n(ν), 2

(10.5)

with  n(ν) ≡

+1 −1

(ν, μ) dμ.

(10.6)

Since the homogeneous transfer equation is linear, the normalization of (ν, μ) is arbitrary. Following a standard custom, we choose n(ν) = 1.

(10.7)

The eigenvalue equation in Eq. (10.5) holds for any value of destruction probability . Negative values of  are encountered, for example in the determination of criticality condition (see e.g. Case and Zweifel 1967; Sect. 23.6)). Two different regions must be distinguished, depending on whether the factor (1 + νμ) may become zero or not. 10.1.1.1 Discrete Spectrum For ν ∈]−1, +1[, the factor (1+νμ) is never zero since μ ∈ [−1, +1]. The solution of Eq. (10.5) has the form 0 (ν, μ) =

1 1 (1 − ) . 2 1 + νμ

(10.8)

When this expression is inserted into Eq. (10.6), the normalization condition n(ν) = 1 leads to 1−

1− 1+ν ln = 0, 2ν 1−ν

ν ∈] − 1, +1[.

(10.9)

We recognize in the left-hand side the dispersion function for monochromatic scattering L(z), for z = ν ∈] − 1, +1[ (see Eq. (5.15)). In this interval L(z) is analytic, and we can set z = ν. Equation (10.9) shows that in the interval ] − 1, +1[

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10 The Case Eigenfunction Expansion Method

there will be a finite number of allowed values of ν, given by the zeroes of the dispersion function. These zeroes are commonly referred to as the eigenvalues. It is shown in Sect. 5.2.2, that the dispersion function has two zeroes located at ν = ±ν0 , ν0 ∈]0, 1], when  > 0, The discrete eigenfunctions are given by Eq. (10.8) with ν = ±ν0 . For  = 0, the two zeroes coalesce into a double zero at the origin and when  is negative, L(z) has two zeroes, located on the imaginary axis, at ±i ν0 . 10.1.1.2 Continuous Spectrum For |ν| ∈ [1, ∞[, the factor (1 + μν) may become zero. Therefore Eq. (10.5) has solutions of the form c (ν, μ) =

P 1 1 (1 − ) + c(ν)δ(μ + ), 2 1 + μν ν

(10.10)

where c(ν) is an arbitrary function and P stands for Cauchy Principal Value. The presence of the Dirac distribution comes from the fact that xδ(x) = 0. The combination of Eq. (10.10) with the normalization n(ν) = 1, leads to c(ν) = 1 −

1+ν 1− ln | |, 2ν 1−ν

|ν| ∈ [1, ∞[.

(10.11)

We recognize in the right-hand side the function λ(ν) = [L+ (ν) + L− (ν)]/2 with L± (ν) the limiting values of L(z) along the cut |ν| ∈ [1, ∞[ (see Eq. (5.21)). To summarize, the transport operator in Eq. (10.3) has a discrete spectrum with two eigenvalues values at ν = ±ν0 and two eigenfunctions 0 (±ν0 , μ) given by Eq. (10.8) with ν = ±ν0 . It also has a continuous spectrum formed by two intervals on the real axis defined by |ν| ∈ [1, ∞[ and a continuous set of eigenfunctions c (ν, μ) defined by Eq. (10.10) with c(ν) = λ(ν). The continuous eigenfunctions are distributions, but the solution of the transfer equation are regular functions, as they involve only integrals over these distributions. Singular eigenfunctions were introduced by van Kampen (1955) to investigate plasma oscillations.

10.1.2 Complete Frequency Redistribution For complete frequency redistribution, the homogeneous radiative transfer equation may be written as ∂I (τ, x, μ) = ϕ(x)I (τ, x, μ) ∂τ  +∞  − (1 − )ϕ(x) ϕ(x ) dx

μ

−∞

+1 −1

I (τ, x , μ )

dμ

. 2

(10.12)

10.1 The Eigenfunctions and Eigenvalues

183

Actually I (τ, x, μ) is a function of τ and ξ = μ/ϕ(x), with ξ ∈] − ∞, +∞[. The transfer equation can thus be rewritten as1  +∞ ∂I ξ (τ, ξ ) = I (τ, ξ ) − (1 − ) I (τ, ξ )g(ξ ) dξ, (10.13) ∂τ −∞ where g(ξ ) has been defined in Eq. (2.60). We now look for elementary solutions of the form (ν, ξ )e−ντ ,

(10.14)

where ν is real. The eigenvalue equation is (1 + νξ )(ν, ξ ) = (1 − )n(ν),

(10.15)

with  n(ν) =

+∞ −∞

g(ξ )(ν, ξ ) dξ.

(10.16)

We choose the same normalization as for monochromatic scattering, namely n(ν) = 1.

(10.17)

Since ξ and ν can take any real values, there is no interval in ν over which the factor (1 + νξ ) can remain free of zeroes. Therefore, there is only a continuous spectrum and the continuous eigenfunctions are distributions. They have the form (ν, ξ ) = (1 − )

c(ν) 1 P + δ(ξ + ), 1 + νξ ν ν

(10.18)

with ν ∈] − ∞, +∞[. The coefficient c(ν) can be determined by imposing the normalization condition n(ν) = 1. Using k(ν) =

1 1 g( ), ν ν

(10.19)

with k(ν), the inverse Laplace transform of the kernel, we find c(ν) =

λ(ν) , k(ν)

(10.20)

1 The notation I is kept for the radiation field when introducing the variable ξ = μ/ϕ(x), contrary to the rule, but there no risk of error here since there is no derivative with respect to μ and x in the radiative transfer equation.

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10 The Case Eigenfunction Expansion Method

with  λ(ν) = 1 − (1 − )

∞

k(ν )(

0

1 1 +

) dν . ν − ν ν +ν

(10.21)

As for monochromatic scattering, λ(ν) = [L+ (ν) + L− (ν)]/2, with L± (ν) the limiting values along the real axis of the dispersion function L(z). In contrast to monochromatic scattering, the transport operator for complete redistribution has only a continuous spectrum covering the positive and negative parts of the real axis.

10.2 The Diffuse Reflection Problem We now show how to apply the Case expansion method to solve a diffuse reflection problem for non-conservative scattering. We first treat monochromatic scattering and then complete frequency redistribution. We consider a semi-infinite medium illuminated from outside by an incident field and free of internal sources. We also assume that  > 0. This same problem is solved in Sect. 7.3.3, where the incident radiation is transformed into a primary source, through the introduction of a diffuse field.

10.2.1 Monochromatic Scattering The radiation field for τ ≥ 0 obeys the set of equations 1 ∂I μ (τ, μ) = I (τ, μ) − (1 − ) ∂τ 2



+1 −1

I (τ, μ ) dμ ,

μ ∈ [−1, 0],

I (0, μ) = I inc (μ),

I (τ, μ) → 0 for τ → ∞.

(10.22) (10.23) (10.24)

We assume that it has the form I (τ, μ) = a0 0 (ν0 , μ)e−ν0 τ +



∞

a(ν)c (ν, μ)e−ντ dν,

(10.25)

1

with 0 (ν, μ) and c (ν, μ) given by Eqs. (10.8) and (10.10). Note that we have retained only positive values of ν so as to satisfy the condition at infinity.

10.2 The Diffuse Reflection Problem

185

Integrating over μ and taking into account the normalization of the eigenfunctions, we see that the source function  1 −  +1 I (τ, μ ) dμ , (10.26) S(τ ) ≡ 2 −1 has the form S(τ ) =

   ∞ 1− a0 e−ν0 τ + a(ν)e−ντ dν . 2 1

(10.27)

We recognize here the expansion written for the calculation of the source function by an inverse Laplace transform. This shows how close the two methods are and that it suffices to determine the coefficients a0 and a(ν) to construct the source function and the radiation field. We now show how to determine these coefficients with the Case eigenfunction expansion.

10.2.1.1 The Expansion Coefficients The expansion (10.25) should be valid for all values of τ and μ. This means in particular that it should hold for the boundary condition in Eq. (10.23), hence that it should satisfy  ∞ a(ν)c (ν, μ) dν, μ ∈ [−1, 0]. (10.28) I inc (μ) = a0 0 (ν0 , μ) + 1

This is the determining equation for a0 and a(ν). The calculation of these coefficients amounts to prove a half-range completeness theorem (Case and Zweifel 1967), namely that any function of μ has an expansion of the form written in Eq. (10.28). Replacing in Eq. (10.28) the eigenfunctions by their expressions (see Eqs. (10.8) and (10.10)), we find after some simple algebra that a(ν) satisfies the singular integral equation  λ(ν)a(ν) + η(ν)

∞ 1

a(ν ) dν = χ(ν), ν − ν

ν ∈ [1, ∞[,

(10.29)

where, as in the other chapters, η(ν) = −(1 − )k(ν).

(10.30)

For monochromatic scattering, k(ν) = 1/2ν. The inhomogeneous term is given by χ(ν) =

a0 1 inc 1 I (− ) + η(ν) . ν2 ν ν − ν0

(10.31)

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10 The Case Eigenfunction Expansion Method

Equation (10.29) is a singular integral equation of the type encountered with monochromatic scattering problems in the domain ν ∈ [1, ∞[ (see for example Eqs. (6.33) or (D.6)). One can apply the method developed in Sect. D.1, Step 1, or the method usually described in the literature on the Case expansion method, which relies on a so-called solvability condition. We refer to these two approaches as Method (i) and Method (ii)). Method (i) We regroup in Eq. (10.29) the two terms containing the factor η(ν) and apply the Hilbert transform method, exactly as described in Sect. D.1. Introducing the function 1 G(z) ≡ 2i π



∞

1 1 inc 1 dν 1 , I (− ) ν0 + ν X(−ν) ν 2 ν ν −z

1

(10.32)

which corresponds to the function denoted Ut (z) in Sect. D.1, we find a0 = 2i π

G(ν0 ) , X(ν0 )

(10.33)

and a(ν) =

1 ν0 − ν



 G − (ν) G + (ν) − , X+ (ν) X− (ν)

ν ∈ [1, ∞[,

(10.34)

where G ± (ν) are the limiting values of G(z) along [1, ∞[. The coefficient a(ν) can be expressed in terms of X(−ν) and of the coefficients λ(ν), η(ν) (see for example Sect. 7.2.1.1), however the expression in Eq. (10.34) is more useful, for example, to determine the emergent intensity. Method (ii) We introduce, the Hilbert transform F (z) =

1 2i π



∞ 1

a(ν) dν. ν−z

(10.35)

Proceeding in the usual way, we obtain the boundary value equation X+ (ν)F + (ν) − X− (ν)F − (ν) = Gt+ (ν) − Gt− (ν),

ν ∈ [1, ∞[,

(10.36)

where Gt± (ν) are the limiting values of the function Gt (z) ≡

1 2i π



∞ 1

1 1 dν χ(ν) . X(−ν) ν0 + ν ν0 − ν ν − z

Here χ(ν), which is defined in Eq. (10.31), contains the parameter a0 .

(10.37)

10.2 The Diffuse Reflection Problem

187

The general solution of Eq. (10.35) has the form X(z)F (z) − Gt (z) = P (z),

(10.38)

where P (z) is an entire function. The function F (z) should be analytic in the complex cut along [1, ∞[ and tend to zero at infinity as 1/z. The first condition is clearly satisfied. We know that X(z) ∼ 1/z for z → ∞. The second condition will be satisfied by choosing P (z) = 0 and by imposing that Gt (z) behaves as 1/z2 for z → ∞. Expanding 1 ν 1  − [1 + + . . .], ν −z z z we see that the second condition requires  ∞ χ(ν) 1 1 dν = 0. X(−ν) ν + ν ν 0 0−ν 1

(10.39)

(10.40)

This solvability condition allows one to determine the unknown parameter a0 contained in Gt (z). Inserting the expression of χ(ν) given in Eq. (10.31) into Eq. (10.40), we obtain for a0 the determining equation  ∞ 1 η(ν) 1 dν = −a0 ν − ν X(−ν)(ν + ν) (ν 0 0 0 − ν) 1  ∞ 1 1 inc 1 1 (10.41) I (− ) dν. X(−ν)(ν0 + ν) (ν0 − ν) ν 2 ν 1 To calculate the integral in the left-hand side, we express X(−ν) in terms of the difference X+ (ν) − X− (ν) (see Eq. (B.11)) and introduce a contour turning around the cut [1, ∞[ and the pole at ν = ν0 . The integral in the right-hand side is, up to a constant, simply the function G(z) defined in Eq. (10.32) taken at z = ν0 . We thus recover the expression of a0 given in Eq. (10.33). The presentation of the two methods shows that the algebra is always simpler when the coefficient η(ν) is expressed, whenever it appears, in terms on the difference L+ (ν) − L− (ν). This procedure is used for example to determine the monochromatic resolvent function with Eq. (6.33). Having obtained explicit expressions for the expansion coefficients a0 and a(ν), we can calculate the emergent intensity.

10.2.1.2 The Emergent Intensity For τ = 0 and positive values of μ, Eq. (10.28) becomes    ∞ a0 1− a(ν) dν , μ ∈ [0, 1], I (0, μ) = + 2 1 + μν0 1 + μν 1

(10.42)

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10 The Case Eigenfunction Expansion Method

with a0 given by Eq. (10.33) and a(ν) by Eq. (10.34). There is no contribution from the Dirac distribution in c (μ, ν), since ν and μ are both positive. The Principal Value is not needed either. To calculate the integral, which is the contribution from the continuum spectrum, we consider the Cauchy integral 

G(ζ ) 1 dζ , X(ζ ) ζ − ν0 1 + μζ

C

ζ ∈C.

(10.43)

The integrand in Eq. (10.43) has a pole at ζ = ν0 , another pole at ζ = −1/μ and a cut along the interval [1, ∞[. Performing the integral over a closed contour C turning around these singularities, we obtain 

∞ 1

2i π a(ν) dν = 1 + μν 1 + μν0



 G(ν0 ) . − X(− μ1 ) X(ν0 ) G(− μ1 )

(10.44)

The first term in the square bracket of Eq. (10.42) is cancelled by the contribution from the discrete eigenvalue in Eq. (10.44). This cancellation between the discrete and continuous spectrum occurs because we are at τ = 0. We thus obtain I (0, μ) =

1− 1 1 1 2 ν0 + μ X(− μ1 )  ∞ 1 1 inc 1 dν 1 , × I (− ) 2 ν + ν X(−ν) ν ν 1 + μν 0 1

μ ∈ [0, 1].

(10.45)

Introducing 1 1 H (z) = [(ν0 + )X(− )]−1 , z z

(10.46)

we recover the famous result, already written in Eq. (7.76), namely 1− H (μ) I (0, μ) = 2

 0

1

H (μ )μ inc I (−μ ) dμ , μ + μ

μ ∈ [0, 1].

(10.47)

The discrete eigenvalue does not appear explicitly in the expression of the emergent intensity, once the H -function is introduced.

10.2.2 Complete Frequency Redistribution The calculation of the emergent intensity for complete frequency redistribution goes essentially as described for monochromatic scattering. It is actually simpler because there no discrete spectrum.

10.2 The Diffuse Reflection Problem

189

We assume an incident field I (0, x, −μ) = I inc (−ξ ),

μ ∈ [0, 1],

(10.48)

with ξ = μ/ϕ(x). The radiation field is then a function of τ and ξ and satisfies the radiative transfer equation for I (τ, ξ ) in Eq. (10.12). When the incident field is not a function ξ , the radiation field can be decomposed into a directly transmitted field and a diffuse field. The primary source for the diffuse field takes into account the incident radiation (see the discussion in Sect. 7.3.3). We assume an expansion of the form 

∞

I (τ, ξ ) =

a(ν)(ν, ξ )e−ντ dν,

(10.49)

0

where the eigenfunction (ν, ξ ) has the form λ(ν) 1 P + δ(ξ + ). 1 + νξ k(ν) ν

(ν, ξ ) = (1 − )

(10.50)

The combination of Eq. (10.50) with Eq. (10.48) leads to the singular integral equation,  λ(ν)a(ν) + η(ν)

∞ 0

a(ν ) dν = γ (ν), ν − ν

ν ∈ [0, ∞[,

(10.51)

where γ (ν) =

k(ν) inc 1 I (− ). ν ν

(10.52)

The Hilbert transform method of solution readily leads to a(ν) =

G − (ν) G + (ν) − − , + X (ν) X (ν)

(10.53)

where X± (ν) are the limiting values of the auxiliary function X(z) along [0, ∞[ and G ± (ν) those of the function 1 2i π

G(z) =



∞ 0

γ (ν) dν . X(−ν) ν − z

(10.54)

The emergent intensity is given by  I (0, x, μ) = (1 − ) 0

∞

a(ν) dν, 1 + νξ

ξ ∈ [0, ∞[.

(10.55)

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10 The Case Eigenfunction Expansion Method

Since ν and ξ have the same sign, the term with the Dirac distribution in the expression of (ν, ξ ) does not contribute to the emergent intensity. The Principal Value is not needed either. To calculate the integral, we proceed as above. We introduce the contour integral  C

G(ζ ) 1 dζ, X(ζ ) 1 + zζ

ζ ∈C,

(10.56)

where C is a closed contour in the complex plane turning around the positive real axis and the pole at ζ = −1/ξ . We obtain I (0, x, μ) =

1− X(− ξ1 )

 0

∞

γ (ν) dν . X(−ν) 1 + νξ

(10.57)

Expressing the X-function in terms of the H -function and using k(ν) = g(1/ν)/ν, we can write

 ∞ μ νϕ(x) I (0, ξ ) = (1 − )H dν, ξ ≥ 0. H (ν)I inc (−ν)g(ν) ϕ(x) 0 νϕ(x) + μ (10.58) For a pencil of radiation around the frequency x0 and in the direction −μ0 , we recover Eq. (7.88). We refer the reader to Sect. 7.3.3 for additional comments on the diffuse reflection problem.

10.3 Further Applications of the Case Method The singular eigenfunctions expansion method can also be used to solve scattering problems with internal sources in an infinite or a semi-infinite plane parallel medium. We give two examples. To simplify the presentation we restrict ourselves to complete frequency redistribution. The general idea for solving problems with internal sources is to recast them as boundary value problems similar to the diffuse reflection problem treated in Sect. 10.2.

10.3.1 A Complete Redistribution Full-Space Problem We consider an infinite plane parallel medium with a plane isotropic source at the origin. The radiation field, denoted I∞ (τ, ξ ), satisfies the radiative transfer equation ∂I∞ = I∞ (τ, ξ ) − (1 − ) ξ ∂τ



+∞ −∞

I∞ (τ, ξ )g(ξ ) dξ − δ(τ ).

(10.59)

10.3 Further Applications of the Case Method

191

The boundary condition at infinity is I∞ (τ, ξ ) → 0,

as

τ → ±∞.

(10.60)

The source term S(τ ), sum of the scattering term and of the Dirac distribution, is actually the infinite medium Green function G∞ (τ ) investigated in Sect. 6.1 and the scattering term alone is the resolvent function ∞ (τ ) = G∞ (τ ) − δ(τ ). Following the method presented in Case and Zweifel (1967, Chapter 5) for monochromatic scattering, we integrate Eq. (10.59) over an infinitesimal interval around the origin. This is a very standard procedure to treat point source inhomogeneous term. Now I∞ (τ, ξ ) satisfies the homogeneous version of Eq. (10.59) with the jump condition ξ [I∞ (0+ , ξ ) − I∞ (0− , ξ )] = −1,

(10.61)

where I∞ (0+ , ξ ) and I∞ (0− , ξ ) are the values of I∞ (τ, ξ ) on each side of the origin, at an infinitesimal distance from it. For the function I∞ (τ, ξ ) we assume an expansion of the form  I∞ (τ, ξ ) =

∞ 0



I∞ (τ, ξ ) = −

a(ν)(ν, ξ )e−τ ν dν, 0

−∞

a(ν)(ν, ξ )e−τ ν dν,

τ ∈ [0, ∞[, τ ∈] − ∞, 0].

(10.62) (10.63)

This expansion guarantees the vanishing of I∞ (τ, ξ ) at plus and minus infinity. The minus sign in the expansion valid for τ < 0 is there for convenience. Substituting this expansion into the jump condition (10.61), we obtain the integral equation  ξ

+∞

−∞

a(ν)(ν, ξ ) dν = −1.

(10.64)

Replacing now (ν, ξ ) by Eq. (10.18) and using Eqs. (10.19) and (10.21), we find that a(ν) satisfies the singular integral equation  λ(ν)a(ν) + η(ν)

+∞

a(ν ) dν = k(ν),

−∞ ν − ν

ν ∈] − ∞, +∞[.

(10.65)

When k(ν) is replaced by (1 − )k(ν) we recover Eq. (6.14), satisfied by φ∞ (ν). Thus, a(ν) =

k(ν) φ∞ (ν) = , 1− R(ν)

a(−ν) = −a(ν),

ν ∈ [0, ∞[.

(10.66)

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10 The Case Eigenfunction Expansion Method

We recall that R(ν) = λ2 (ν)+π2 (1−)2 k 2 (ν). Using the normalization of (ν, ξ ), we find  ∞ a(ν)e−ν|τ | dν, τ ∈ [0, ∞[. (10.67) S(τ ) = G∞ (τ ) = δ(τ ) + (1 − ) 0

We recover the expression given in Eq. (6.9), obtained with other techniques (Fourier transform or inverse Laplace transform). Since we are dealing with an infinite medium, a(ν) involves only the dispersion function, for reasons explained in Chap. 4.

10.3.2 A Half-Space Problem with Internal Sources Half-space problems with internal sources can also be solved with the Case expansion method. We present here for complete frequency redistribution the main lines of a proof given in Case and Zweifel (1967, Chapter 5, p. 115) for monochromatic scattering. We assume that Eq. (10.12) contains a primary source term ϕ(x)Q∗ (τ ) and that no radiation is incident on the surface. The radiation field is thus a function of τ and ξ = μ/ϕ(x). It satisfies the equation ∂I ξ (τ, ξ ) = I (τ, ξ ) − (1 − ) ∂τ



+∞ −∞

I (τ, ξ )g(ξ ) dξ − Q∗ (τ ),

(10.68)

with the surface boundary condition I (0, ξ ) = 0 for ξ ∈] − ∞, 0]. The solution is sought in the form 

∞

I (τ, ξ ) =

G(τ, τ0 , ξ )Q∗ (τ0 ) dτ0 ,

τ ∈ [0, ∞[,

(10.69)

0

where G(τ, τ0 , ξ ) is a Green function, depending also on frequency and direction. In the half-space τ ∈ [0, ∞[, G(τ, τ0 , ξ ) is solution of ξ

∂G (τ, τ0 , ξ ) = G(τ, τ0 , ξ ) − (1 − ) ∂τ



+∞

−∞

G(τ, τ0 , ξ )g(ξ ) dξ − δ(τ − τ0 ), (10.70)

with the boundary condition G(0, τ0 , ξ ) = 0,

ξ ∈] − ∞, 0].

(10.71)

The solution of Eq. (10.70) is constructed in two successive steps: one first determines the full-space Green function with a plane source at τ = τ0 , and then

10.3 Further Applications of the Case Method

193

determines a half-space Green function with a known boundary condition at τ = 0. More explicitly, the solution is sought in the form G(τ, τ0 , ξ ) = G∞ (τ, τ0 , ξ ) + G> (τ, τ0 , ξ ),

τ ∈ [0, ∞[.

(10.72)

We now introduce this decomposition into Eq. (10.70) and into the boundary condition in Eq. (10.71). Equation (10.70) is then decomposed into two equations, one for G∞ (τ, τ0 , ξ ) and another one for G> (τ, τ0 , ξ ). The equation for G∞ (τ, τ0 , ξ ) is Eq. (10.70), to which we associate the jump condition ξ [G∞ (τ0+ , τ0 , ξ ) − G∞ (τ0− , τ0 , ξ )] = −1,

(10.73)

where τ0+ and τ0− are on each side of τ0 , at an infinitesimal distance of it. The point τ0 takes any value in ]−∞, +∞[. The equation for G> (τ, τ0 , ξ ) is the homogeneous radiative transfer equation in Eq. (10.13), with a boundary condition at τ = 0, derived from Eq. (10.71), namely G> (0, τ0 , ξ ) = −G∞ (0, τ0 , ξ ),

ξ ∈] − ∞, 0].

(10.74)

The functions G∞ and G> are sought in the form  G∞ (τ, τ0 , ξ ) =

∞ 0

G∞ (τ, τ0 , ξ ) = − 



a(ν)(ν, ξ ) e−ν(τ −τ0 ) dν, 0

−∞ ∞

a(ν)(ν, ξ ) e−ν(τ −τ0 ) dν,

b(ν)(ν, ξ ) e−ντ dν,

G> (τ, τ0 , ξ ) =

τ ∈]τ0 , ∞[, τ ∈] − ∞, τ0 [,

τ ∈ [0, ∞[,

(10.75)

0

where (ν, ξ ) is the eigenfunction given in Eq. (10.50). The coefficient a(ν) is determined in Sect. 10.3.1 for τ0 = 0. The solution given in Eq. (10.66) holds also for τ0 = 0, because of the translation invariance of the problem. The coefficient b(ν) satisfies the singular integral equation  λ(ν)b(ν) + η(ν)

∞ 0

b(ν ) k(ν) 1 dν = − G∞ (0, τ0 , − ), ν − ν ν ν

ν ∈ [0, ∞[, (10.76)

which can be solved by a Hilbert transform method. The final result, somewhat cumbersome, yields G(τ, τ0 , ξ ) and hence I (τ, ξ ), in terms of the primary source term Q∗ (τ ). Other applications of the Case expansion method can be found in Case and Zweifel (1967, Chapter 5), for example to two adjacent half-spaces with different single scattering albedos. They are all given for monochromatic scattering but can

194

10 The Case Eigenfunction Expansion Method

be applied to complete frequency redistribution. The algebra is somewhat simpler since there is no discrete spectrum. The applications of the singular eigenfunction expansion method described in this chapter show that this method is particularly well adapted to boundary value problems and that it can also be used for problems with internal sources, provided they are first recast as boundary values problems. It can also be applied to Rayleigh scattering problems and leads to vector singular integral equations with Cauchy-type kernels. Methods of solutions for these equations are presented in Part II, where we consider the Rayleigh scattering of polarized radiation.

References Case, K.M.: Elementary solutions of the transport equation and their applications. Ann. Phys. (New York) 9, 1–23 (1960) Case, K.M., Zweifel, P.: Linear Transport Theory. Addison-Wesley, Reading (1967) Chandrasekhar, S.: Radiative Transfer, 1st edn. Dover Publications, New York (1960); Oxford University Press (1950) Duderstadt, J.J., Martin, W.R.: Transport Theory. Wiley, New York (1979) Mc Cormick, N.J., Kušˇcer, I.: Singular eigenfunction expansions in neutron transport theory. In: Henley, E.J., Lewins, J. (eds.) Advances in Nuclear Science and Technology, vol. 7, pp. 181– 282. Academic Press, New York (1973) Roos, B.W.: Analytic Functions and Distributions in Physics and Engineering. Wiley, New York (1969) van Kampen, N.G.: On the theory of stationnary waves in plasma. Physica 21, 949–963 (1955)

Chapter √ 11

The -law and the Nonlinear H -Equation

We show in this chapter how some exact results concerning radiative transfer in a semi-infinite medium can be derived from the radiative transfer equation and from the Wiener–Hopf integral equation for the source function itself, with elementary methods, to quote Ivanov (1994). They are elementary in the sense that they do not make use of complex plane analysis, whether they are really elementary, is a matter of appreciation. A certain advantage of these methods is that there is in general no need to distinguish between monochromatic scattering and complete frequency redistribution. √ We address two different topics, the construction of the exact -law √ and the construction of the nonlinear integral equation for the H -function. In the -law,  is a destruction probability per scattering with values in the interval ]0, 1[. The methods described in this chapter can be generalized to scattering polarization, as will be seen in Part II, Chap. 15. The√chapter is organized as follows. In Sect. 11.1 we present two different proofs of the -law. One is based on the integral equation for the source function and the other one on properties of the Green function. In Sect. 11.2 we describe two different methods leading to the nonlinear H -equation. A solution of this equation by Ivanov (1994), involving only real-space functions, is described in Appendix H of this chapter. Some properties of the nonlinear H -equation are investigated in Sect. 11.3. We discuss in particular the problem of the non-uniqueness of its solution for monochromatic scattering and present an alternative H -equation, used for numerical solutions. We also recover with simple algebra an important factorization relation derived in Chap. 5 from the exact expression of the H -function.

11.1 The

√ -law

√ The -law is an exact result. It holds for any value of , strictly positive and smaller than one. It states that in a semi-infinite medium with a uniform primary source term © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_11

195

196

11 The

√ -law and the Nonlinear H -Equation

Q∗ = B the surface value of the source function is given by S(0) =

√ B,

(11.1)

for any scattering process for which the kernel K(τ ) is positive, symmetrical (an even function of τ ) and normalized to unity. This law was brought to the attention of the astronomical community by the work of Avrett and Hummer (1965) on the solution of the integral equation for S(τ ). The authors gave an algebraic proof based on the infinite limit of the discrete ordinate method of Chandrasekhar (1960). A numerical confirmation and other important properties of the source function were established in the same article by careful numerical calculations. It must be pointed out that this law is implicitly contained in some prior exact results by Ambartsumian (1942), Chandrasekhar (1960, §84.4) and Sobolev (1963, p. 143) on the emergent radiation from a semi-infinite atmosphere. The Ambartsumian’s and Sobolev’s approaches make use of the nonlinear integral equation for the H -function. Chandrasekhar’s method is based on the discrete ordinate method. Even earlier, Hopf (1930) and Bronstein (1929) √ established the socalled Hopf–Bronstein relation, a conservative version of the -law for the Milne problem (see Sect. 9.1). √ After 1965, several proofs for -law√have been proposed in the astronomical literature. In Ivanov (1973, p. 251) the -law is derived from properties of the resolvent function (τ ), based themselves on an analysis of the trajectories of the photons in a semi-infinite medium. In Frisch and Frisch (1975), the introduction of a quadratic integral leads to a very simple proof described in Sect. 11.1.1. It uses only the symmetry of the kernel and √ the semi-infinite geometry of the medium. In Landi Degl’Innocenti (1979), the -law is derived from a Neumann series expansion of the integral equation for the source function, followed by a resummation of all the terms. For τ = 0 and a√constant primary source, the summation can be done explicitly and provides the -law. Hubeny (1987a,b) has √ proposed probabilistic interpretations based on invariance principles, for the -law itself and for a √ generalized -law valid at all depths introduced by Rybicki (1977). More recently Ivanov (1994) has proposed a proof based on an integral relation between the infinite medium Green function and the surface Green function of a semi-infinite medium. The method introduced in Frisch and Frisch (1975) has been generalized to polarized radiative transfer for resonance scattering of spectral lines Ivanov (1990, 1995) and to the Hanle effect and Zeeman effects (Landi Degl’Innocenti √ and Bommier 1994; Frisch 1999). The history of the -law is described in detail in Ivanov (2021). √ In this book, the -law has already been established in Sects. 7.1 and 7.3.1, for complete frequency redistribution and monochromatic scattering. It is derived from the exact value at the origin of the auxiliary function X(z), or equivalently, from the behavior at infinity of H -function. The origin of this law actually lies in a universality property of symmetrical random walks in a half-space. A probabilistic √ proof of the -law and of the universality property of symmetric random walks is described in Sect. 21.3, where we reproduce a proof by Frisch and Frisch (1995).

11.1 The

√

197

-law

In Sect. 11.1.1 we reproduce the very simple proof of Frisch and Frisch (1975), based on the introduction of a quadratic quantity and in Sect. 11.1.2 that of Ivanov (1994), based on properties of the Green function. None of them makes use of complex plane analysis.

11.1.1 A Quadratic Approach We present here a somewhat simplified version of the method introduced in Frisch and Frisch (1975). We consider the Wiener–Hopf integral equation 

∞

S(τ ) = (1 − )

K(τ − τ )S(τ ) dτ + Q∗ ,

(11.2)

0

where Q∗ is a uniform primary source. At infinity, the solution of this equation tends to a constant, denoted S(∞). Taking the limit τ → ∞ in Eq. (11.2) and remembering that the kernel is normalized to unity (see Eq. (5.11)) simple algebra leads to S(∞) = Q∗ /.

(11.3)

We introduce the constant 

∞

F =

S(τ ) 0

dS(τ ) dτ. dτ

(11.4)

As shown in Sect. 2.4.3, the derivative of S(τ ) satisfies the integral equation dS(τ ) = (1 − ) dτ



∞

K(τ − τ )S (τ ) dτ + (1 − )K(τ )S(0),

(11.5)

0

where S (τ ) = dS/dτ . It is convenient to write F = F1 + F2 ,

(11.6)

S(τ )K(τ − τ )S (τ ) dτ dτ,

(11.7)

with 

∞ ∞

F1 = (1 − ) 0

0

and 

∞

F2 = (1 − )

 S(τ )K(τ ) dτ S(0).

0

(11.8)

198

11 The

√ -law and the Nonlinear H -Equation

The kernel being an even function of τ , we can rewrite F1 as  ∞ ∞ S(τ )K(τ − τ )S (τ ) dτ dτ . F1 = (1 − ) 0

(11.9)

0

Exchanging the order of the integration, a procedure justified in Frisch and Frisch (1975), and making use of Eq. (11.2), we obtain  ∞ F1 = [S(τ ) − Q∗ ]S (τ ) dτ, (11.10) 0

and, after expanding the integrand,  ∞ F1 = F − Q∗ S (τ ) dτ = F − Q∗ [S(∞) − S(0)].

(11.11)

0

To calculate F2 , we combine Eq. (11.2), in which we set τ = 0, with Eq. (11.8). This leads to F2 = [S(0) − Q]S(0).

(11.12)

It suffices now to perform the sum F = F1 + F2 . Using S(∞) = Q∗ /, we obtain S 2 (0) = Q∗ S(∞) = [Q∗ ]2 /.

(11.13) √ This is the -law written in Eq. (11.1) for Q∗ = B. In Sect. 15.2 we show how this quadratic approach can be generalized to Rayleigh scattering, resonance polarization, and the Hanle effect.

11.1.2 A Green Function Approach The method described now, proposed by Ivanov (1994), is based on integral properties of the Green function G(τ, τ0 ) and associated functions, in particular on the normalization of the resolvent function (τ ). First we establish a nonlinear relation between G∞ (τ ) and G(τ ), the infinite medium and surface Green functions, respectively. In Sect. 2.4.4, it is shown that  τ G(τ − t)G(τ0 − t) dt, τ = min(τ0 , τ ). (11.14) G(τ, τ0 ) = 0

The proof is based on the translation invariance of the kernel and the semi-infinite character of the medium. As pointed out in Ivanov (1994), G∞ (τ ), for τ ∈ [0, ∞[, can be defined as G∞ (τ ) = lim G(τ + T , T ). T →∞

(11.15)

11.1 The

√

199

-law

It satisfies the convolution integral equation written in Eq. (2.79), namely  G∞ (τ ) = (1 − )

+∞

−∞

K(τ − τ )G∞ (τ ) dτ + δ(τ ).

(11.16)

Replacing in Eq. (11.14) τ by τ + T , τ0 by T , and t by T − t and then taking the limit T → ∞, we obtain  G∞ (τ ) =

∞

G(τ + t )G(t ) dt ,

τ ∈ [0, ∞[.

(11.17)

0

This relation can be viewed as the real space counterpart of factorization relations between the dispersion function L(z) and the auxiliary function X(z), given in Eqs. (5.48) and (5.55) for complete frequency redistribution and monochromatic scattering. We now consider the integral of G∞ (τ ) over the domain τ ∈] − ∞, +∞[. The integration of Eq. (11.16) yields the normalization 

+∞

−∞

G∞ (τ ) dτ =

1 , 

(11.18)

and the integration of Eq. (11.17) leads to 

+∞ −∞

 G∞ (τ ) dτ = 2



∞

∞

G(t) dt 0

 G(u) du =

t

2

∞

G(t) dt

.

(11.19)

0

The proof make uses of G∞ (−τ ) = G∞ (τ ) and of the change of variable u = τ +t. Equating the results in Eqs. (11.18) and (11.19), we obtain 

∞

√ G(τ ) dτ = 1/ .

(11.20)

0

The sign ambiguity, when taking the square-root, is raised by considering the integral equation for G(τ ), namely, 

∞

G(τ ) = (1 − )

K(τ − τ )G(τ ) dτ + δ(τ ).

(11.21)

0

Indeed, for  = 1, the integral of G(τ ) over [0, ∞[ is positive, equal to one. The normalization of G(τ ) leads to 

∞

√ (τ ) dτ + 1 = 1/ ,

0

where (τ ) = G(τ ) − δ(τ ) is the resolvent function.

(11.22)

200

11 The

We now have all the elements to establish the for a uniform primary source,   S(τ ) = S(0) 1 +

τ

√ -law and the Nonlinear H -Equation

√ -law. As shown in Sect. 7.3.1,

 (τ ) dτ .

(11.23)

0

Letting τ → ∞, using the normalization of (τ ) and knowing that S(τ ) tends to Q∗ / for √ τ → ∞, we recover Eq. (11.1). The -law holds of course only for non-conservative scattering. For monochromatic scattering, it has a conservative version, namely the Hopf–Bronstein relation √ S(0) = ( 3/4π)F , with F the radiative flux (see Chap. 9).

11.2 Construction of the Nonlinear H -Equation Following the purely mathematical methods introduced in Wiener and Hopf (1931) and Halpern et al. (1938) to construct of exact solutions of half-space radiative transfer problem, physicists and astronomers started to develop in the early forties various methods based on invariance principles. They are described, together with detailed historical accounts, in books by, e.g., Chandrasekhar (1960), Ambartsumian (1958), Sobolev (1963), Ivanov (1973). The methods based on invariance principle lead in general to a nonlinear integral equation for the H -function. Complex plane analysis is then employed to construct an exact expression of the H -function, although it is possible to do without it, as shown by Ivanov’s proof (1994), reproduced in the Appendix H of this chapter. We describe two different methods of construction for the nonlinear integral equation satisfies by the half-space auxiliary function H (μ). The first one, in Sect. 11.2.1, is based on the calculation of the reflection coefficient for a diffuse reflection problem, and the second one, in Sect. 11.2.2, on the determination of the surface value of the source function for a medium with an exponential primary source term.

11.2.1 Scattering Coefficient Method We follow here the method described in Chandrasekhar (1960, p. 96) for a monochromatic scattering diffuse reflection problem. For simplicity, we assume an isotropic phase function. The general case of an arbitrary phase function is treated in Chandrasekhar’s book. The radiative transfer equation for the diffuse reflection problem is μ

1 ∂I (τ, μ) = I (τ, μ) − (1 − ) ∂τ 2



+1 −1

I (τ, μ ) dμ .

(11.24)

11.2 Construction of the Nonlinear H -Equation

201

The notation is the same as in the other chapters, with τ ∈ [0, ∞[ denoting the optical depth in the medium and μ ∈ [−1, +1] the direction cosine of a radiation beam. Outward and inward going beams correspond respectively to positive and negative values of μ. To emphasize the direction of the beams, we denote by I + (τ, μ) and I − (τ, μ) the outward and inward going beams. At the surface τ = 0, we assume a parallel beam of radiation in the direction −μ0 , μ0 positive. The boundary condition is written as I − (0, −μ) = I inc (μ0 )δ(μ − μ0 ), μ

μ, μ0 ∈ [0, 1].

(11.25)

Following Chandrasekhar, we introduce a scattering function Ssc (μ, μ ), where and μ are the directions of the incident and scattered beams. It is defined by I + (τ, μ) =

1 2μ



1

Ssc (μ, μ )I − (τ, −μ ) dμ .

(11.26)

0

This function Ssc (μ, μ ) is identical to the Chandrasekhar’s function S(μ, μ ). The notation has been changed to avoid a confusion with the source function S(τ ). The existence of a function Ssc (μ, μ ), independent of the position in the medium, is made possible by the fact that the medium is semi-infinite and translation invariant. Thus at any depth τ , the region [τ, ∞[ behaves as a semi-infinite medium, the region [0, τ ] providing the incident field at the depth τ . We show first that Ssc (μ, μ ) satisfies a quadratic integral equation, from which one can then derive the nonlinear integral equation for H (μ) (in brief the H -equation). It is convenient for the construction of the integral equation for Ssc (μ, μ ) to introduce a diffuse radiation field I˜(τ, μ), defined, as in Sect. 2.1.3, by I˜− (τ, μ) ≡ I − (τ, μ) − I inc (μ0 )e−τ/μ0 , ˜+

+

I (τ, μ) ≡ I (τ, μ),

μ ∈ [−1, 0],

μ ∈ [0, 1].

(11.27)

For μ ∈ [0, 1], the diffuse field is equal to the total field. The radiative transfer equation for the diffuse field is μ

∂ I˜ (τ, μ) = I˜(τ, μ) − S(τ ), ∂τ

(11.28)

and the source function S(τ ) is defined by 1− S(τ ) ≡ 2



+1

−1

1 −  inc I (μ0 )e−τ/μ0 . I˜(τ, μ ) dμ + 2

(11.29)

At the surface τ = 0, the boundary condition for the diffuse field is I˜− (0, μ) = 0,

μ ∈ [−1, 0].

(11.30)

202

11 The

√ -law and the Nonlinear H -Equation

In terms of the diffuse field, Eq. (11.26) becomes    1 1 + inc −τ/μ0

˜−



˜ I (μ0 )Ssc (μ, μ0 )e + Ssc (μ, μ )I (τ, −μ ) dμ . I (τ, μ) = 2μ 0 (11.31) At the surface, the emergent diffuse field is equal to the total emergent field, hence 1 inc I˜+ (0, μ) = I (μ0 )Ssc (μ, μ0 ), 2μ

μ ∈ [0, 1].

(11.32)

This expression can also be derived from Eq. (11.31) by setting I˜− (0, −μ ) = 0 inside the integral. Following Chandrasekhar, we now calculate the derivative of Eq. (11.31) at τ = 0. It may be written as ∂ I˜+ (τ, μ)|τ =0 ∂τ 

 1 ˜− 1 Ssc (μ, μ0 ) inc

∂I



−I (μ0 ) (τ, −μ )|τ =0 dμ . = + Ssc (μ, μ ) 2μ μ0 ∂τ 0 (11.33) We now use the radiative transfer equation to calculate the derivatives. Taking the derivative of Eq. (11.28) and using Eq. (11.32), we can write   ∂ I˜+ 1 ˜+ 1 1 inc (τ, μ)|τ =0 = [I (0, μ) − S(0)] = I (μ0 )Ssc (μ, μ0 ) − S(0) , ∂τ μ μ 2μ (11.34) and ∂ I˜− 1 (τ, μ)|τ =0 = − S(0). ∂τ μ

(11.35)

Inserting these expressions into Eq. (11.33), we obtain  

 1 1 1 1 1 1 inc

dμ I (μ0 )Ssc (μ, μ0 )( + ) = S(0) 1 + Ssc (μ, μ ) . 2μ μ μ0 μ 2 0 μ (11.36)

11.2 Construction of the Nonlinear H -Equation

203

The surface value S(0) can be derived from Eq. (11.29), in which we insert I˜− (0, μ) = 0, μ ∈ [−1, 0] and the expression of I˜+ (0, μ) = 0, μ ∈ [0, 1] given in Eq. (11.32). We thus obtain    1 1 dμ

1 −  inc

I (μ0 ) 1 + Ssc (μ , μ0 ) . S(0) = 2 2 0 μ

(11.37)

Inserting Eq. (11.37) into Eq. (11.36) and eliminating the factor I inc (μ0 )/(2μ), we obtain the quadratic integral equation, 1 1 + )= μ μ0      dμ

dμ

1 1 1 1 Ssc (μ , μ0 )

Ssc (μ, μ ) . (1 − ) 1 + 1+ 2 0 μ 2 0 μ Ssc (μ, μ0 )(

(11.38)

A proof of this equation is established in Sobolev (1963, p. 60) with the method of addition of layers, which states that the reflecting property of a medium with an infinitely large optical thickness does not change by adding of a small layer of thickness τ . Equation (11.38) is now used to introduce the H -function and to show that it is solution of a nonlinear integral equation. The function Ssc (μ, μ0 ) is symmetric, that is Ssc (μ0 , μ) = Ssc (μ, μ0 ).

(11.39)

This property can be derived from the structure of Eq. (11.38). The latter can thus be written as Ssc (μ, μ0 )(

1 1 + ) = (1 − )H (μ)H (μ0), μ μ0

(11.40)

with 1 H (μ) ≡ 1 + 2



1

Ssc (μ, μ )

0

dμ

. μ

(11.41)

Inserting Eq. (11.40) into Eq. (11.41), we obtain the nonlinear integral equation 1− H (μ)μ H (μ) = 1 + 2

 0

1

H (μ ) dμ . μ + μ

(11.42)

We recover here the nonlinear integral equation derived in Sect. 5.4.3 with the Cauchy integral formula.

204

11 The

√ -law and the Nonlinear H -Equation

We can also recover the expression of the emergent intensity given in Eq. (7.77). The combination of Eq. (11.32) with Eq. (11.40) yields I + (0, μ) =

1 −  inc μ0 I (μ0 ) H (μ0)H (μ). 2 μ + μ0

(11.43)

As for the surface value of the source function, it is simply given by S(0) =

1 −  inc I (μ0 )H (μ0 ). 2

(11.44)

We show in the next section how this expression can be used to define the H function. For monochromatic scattering, the general case of an anisotropic phase function depending on the polar angles of the incident and scattered beams is treated in Chandrasekhar (1960, Chapter IV). A few examples are worked out in detail. The corresponding H -functions satisfy nonlinear integral equations of the form  H (μ) = 1 + (1 − )H (μ)μ 0

1

(μ )

H (μ ) dμ , μ + μ

(11.45)

where (μ) is known as the characteristic function. For isotropic scattering, (μ) = 1/2. The scattering coefficient method can also be applied for complete frequency redistribution and leads to Eq. (11.54).

11.2.2 Exponential Primary Source Method We now use, as originally proposed in Ambartsumian (1942), the surface value of the source function in a semi-infinite medium with an exponential source term, here written e−pτ , to define the H -function and construct the nonlinear H -equation. The Wiener–Hopf integral equation for the source function, denoted Sp (τ ) is 

∞

Sp (τ ) = (1 − )

K(τ − τ )Sp (τ ) dτ + e−pτ ,

(11.46)

0

with p ∈ [0, ∞[. We write its solution as 

∞

Sp (τ ) =

G(τ, τ0 )e−pτ0 dτ0 ,

(11.47)

0

with G(τ, τ0 ) the Green function. Equation (11.47) shows that ˜ Sp (τ ) = G(τ, p)

˜ and Sp (0) = G(p),

(11.48)

11.2 Construction of the Nonlinear H -Equation

205

˜ ˜ where G(τ, p) is the Laplace transform of G(τ, τ0 ) with respect to τ0 and G(p) the Laplace transform of G(τ ) ≡ G(τ, 0) = G(0, τ ). For monochromatic scattering and complete frequency redistribution, the kernel K(τ ) can be represented by a superposition of exponentials, namely as 

∞

K(τ ) =

k(ν)e−ντ dν,

(11.49)

νl

with νl = 1 for monochromatic scattering and νl = 0 for complete frequency redistribution. The expression of k(ν) for monochromatic scattering and complete frequency redistribution can be found in Sect. 5.1. Setting τ = 0 in Eqs. (11.46) and (11.47), we obtain ˜ Sp (0) = G(p) = 1 + (1 − )



∞

˜˜ G(ν, p) k(ν) dν,

(11.50)

νl

˜˜ where G(ν, p) is the double Laplace transform of the Green function. As shown in Sect. 2.4.4, with a proof based on the translation invariance of the kernel, ˜ ˜ G(p) G(ν) ˜˜ . G(ν, p) = p+ν

(11.51)

˜ We thus obtain for G(p), hence for Sp (0), the nonlinear integral equation ˜ ˜ G(p) = 1 + (1 − )G(p)



∞ νl

˜ G(ν) k(ν) dν. ν +p

(11.52)

p ∈ [0, ∞[,

(11.53)

By choosing 1 ˜ H ( ) ≡ G(p), p

we obtain a nonlinear integral equation for the H -function. For monochromatic scattering k(ν) = 1/(2ν). We recover Eq. (11.42). For complete frequency redistribution, the H -equation takes the form 

∞

H (p) = 1 + (1 − )pH (p) 0

H (ν) 1 1 k( ) dν. p+ν ν ν

(11.54)

The method described here for the construction of the nonlinear H -equation can be extended to the scattering of linearly polarized radiation (see Sect. 15.4). The H -function becomes a matrix. It also satisfies a nonlinear integral equation, very similar to the scalar equation, but in contrast to the scalar case, it has in general no exact solution. The only exception is conservative ( = 0) Rayleigh scattering.

206

11 The

√ -law and the Nonlinear H -Equation

11.3 Some Properties of the H -Equation In Chap. 5, we use the definition of the half-space auxiliary function X(z) to show that the H -function satisfies the factorization 1/[H (z)H (−z)] = L(1/z), where L(z) is the dispersion function. This factorization is the starting point of the Wiener–Hopf method, as we have seen Sect. 3.2.1. Following Chandrasekhar (1960, p. 116), we now show that it can be derived from the nonlinear H -equation. In Chandrasekhar (1960, p. 117), the factorization relation is then used to construct an exact expression of the H -function. Two different approaches have been used for obtaining numerical values of the H -function: numerical solution of an explicit expression or solution of the nonlinear integral equation for the H -function. The task is never straightforward, especially when  is small. For this reason there has also been a significant effort for finding simple and accurate approximations (see, e.g., Kawabata and Limaye 2011) A review of the many methods that have been used for monochromatic scattering can be found in Rutily (1992, p. 288). In this case the nonlinear H -equation has two different solutions. We explain in detail in Sect. 11.3.2 that the origin of this non-uniqueness problem lies in the two zeroes of the dispersion function. For complete frequency redistribution there is no uniqueness problem since the dispersion function is free of zeroes. In this same Sect. 11.3.2, we also introduce alternative nonlinear H -equations, well suited to numerical solutions, especially when  is small.

11.3.1 A Factorization Relation We consider the integral equation for H (μ) given in Eq. (11.42). For the sake of generality we continue it to the complex plane, but for simplicity assume an isotropic phase function. The calculation presented below can be generalized to the H -equation for non-isotropic scattering (see Eq. (11.45)) and to complete frequency redistribution. First we rewrite Eq. (11.42) as 1 1 = (1 − ) 1− H (z) 2



1

H (μ) 0

z dμ. z+μ

(11.55)

Following Chandrasekhar (1960, p. 116), we consider the product 1 (1 − )2 1 ][1 − ]= [1 − H (z) H (−z) 4

1 1



H (μ) 0

0

z z H (μ ) dμ dμ . z+μ z − μ

(11.56)

11.3 Some Properties of the H -Equation

207

Inserting into Eq. (11.56) the simple fraction decomposition, z z z = z + μ z − μ

μ + μ



μ μ

+ z + μ z − μ

(11.57)

,

we can rewrite Eq. (11.56) as  1 1 (1 − )2 1 1 1 ][1 − ]= z [1 − ] dμ H (μ) H (z) H (−z) 4 z + μ H (μ) 0  1 (1 − )2 1 1 + H (μ ) [1 − (11.58) z ] dμ .

4 z−μ H (μ ) 0

[1 −

Using again Eq. (11.55), we find 1 1 = 1 − (1 − )z H (z)H (−z) 2

1

 0

1 1 − z+μ z−μ

 dμ.

(11.59)

The right-hand side is the dispersion function. Indeed changing z to 1/z and μ to 1/ν, we readily find 1 1 = 1 − (1 − ) H (1/z)H (−1/z) 2z

∞

 1

 1 1 − dν = L(z). ν−z ν +z

(11.60)

We recover the factorization relation for H (z) given in Eq. (5.66).

11.3.2 Uniqueness and Alternative Nonlinear H -Equations The nonlinear H -equation for monochromatic scattering does not have a unique solution. It is a well known property (see e.g., Chandrasekhar 1960, p. 123). This phenomenon, which is a consequence of the fact that the dispersion function has a pair of real zeroes at z = ±ν0 , is explained here with the integral equations for the functions X(z) and X∗ (z). We show in Sect. B.3 that X(z) satisfies the nonlinear integral equation X(z) = −

1− 2



∞ 1

1 dν , X(−ν)(νo2 − ν 2 ) ν(ν − z)

(11.61)

and that its solution should satisfy the identity 1− 2



∞ 1

dν 1 = −1, 2 2 X(−ν)(νo − ν ) ν

(11.62)

208

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√ -law and the Nonlinear H -Equation

to ensure that X(z)  −1/z as z → ∞. The integral equation for X∗ (z) = (ν0 − z)X(z) is X∗ (z) = 1 −

1− 2



∞ 1

1

dν , ν(ν − z)

X∗ (−ν)

(11.63)

and that for H (z) is H (z) = 1 +

1− zH (z) 2

 0

1

H (μ) dμ. z+μ

(11.64)

We recall that H (z) = 1/X∗ (−1/z).

(11.65)

We introduce the function X1∗ (z) ≡ −(ν0 + z)X(z). We now show that this function also satisfies Eq. (11.63). To prove it, we use the method employed in Sect. B.3 to construct the integral equation for X∗ (z). We multiply Eq. (11.61) by −(ν0 + z) and make a decomposition of the factor [1/(ν0 + ν)(ν − z)] into simple fractions. One finds that H1 (z) ≡ 1/X1∗ (−1/z),

(11.66)

satisfies the same nonlinear equation as H (z). The relation between the two solutions is H1 (z) =

1 + ν0 z H (z). 1 − ν0 z

(11.67)

Whereas H (z) has a pole at z = −1/ν0 , the pole of H1 (z) is located at z = 1/ν0 and leads √ to an unphysical solution. Equation (11.67) shows also that √ H1 (z) tends to −1/  at infinity, instead of 1/ . We can also observe that H1 (z)H1 (−z) = H (z)H (−z), hence and that H1 (z) satisfies the factorization relation H1 (z)H1 (−z)L(1/z) = 1. We see that the latter is insufficient to uniquely define the H (z). The proof that H1 (z) is a solution of the nonlinear H -equation can also be established by algebraic manipulations of the H -equation itself (see e.g. Chandrasekhar 1960, p. 153). Since the H (z) is not uniquely defined by the H -equation, one needs an additional constraint to find the physically correct solution. A simple way to establish this constraint is to write the H -equation as    1 H (μ) 1− z dμ = 1. H (z) 1 − 2 0 z−μ

(11.68)

11.3 Some Properties of the H -Equation

209

To ensure that H (z) becomes infinite for z = −1/ν0 , the square bracket should be equal to zero for z = −1/ν0 . This leads to the linear constraint 1− 2



1

0

H (μ) dμ = 1. 1 − ν0 μ

(11.69)

Expressing in Eq. (11.69) H (μ) in terms of the X-function, we recover the constraint given for X(z) in Eq. (11.62). For conservative scattering  = 0, hence 1 ν0 = 0. The linear constraint reduces to 0 H (μ) dμ = 2, which is the value of the zeroth-order moment of H (μ) for conservative scattering (see Eq. (B.25)). We now discuss an important problem linked to the numerical solution of the H -equation. It is pointed out in Chandrasekhar (1960) that the numerical solution of the H -equation by an iterative method,1 which is the normal approach, presents some difficulties. One of them is of course the non-uniqueness of the solution, the other one is the poor rate of convergence of iterative methods applied to the standard form of the H -equation in Eq. (11.42). An alternative form, having also two different solutions, but leading to faster convergence rates, was actually used by him and Mrs. Breen to obtain numerical values of H (μ), μ ∈ [0, 1] (Chandrasekhar 1960, p. 125). This equation is √ 1 1− = + H (μ) 2



1 0

H (μ )μ

dμ . μ + μ

(11.70)

It is obtained by subtracting from Eq. (11.55) the identity √ 1−  =1− 2



1

H (μ) dμ,

(11.71)

0

√ obtained by taking the limit μ → ∞ in Eq. (11.42) and using H (∞) = 1/ . Whereas the standard Eq. (11.42) yields H (μ) = 1√for μ = 0, it does not directly predict that the limit of H (μ) for μ → ∞ is 1/ . The alternative Eq. (11.70) includes the correct behavior at infinity and also at the origin. Equation (11.70) has also two solutions. They can be both derived from Eq. (11.61). First we multiply Eq. (11.61) by (ν0 − √z) and then subtract from this new equation its value for z = 0. Using ν0 X(0) = , we obtain √ 1− z X (z) =  − 2 ∗



∞ 1

1 1 dν. X∗ (−ν) ν 2 (ν − z)

(11.72)

To find the second solution, we multiply Eq. (11.61) by (ν0 + z) and proceed in the same way. The function X2∗ (z) = (ν0 + z)X(z) = −X1∗ (z) also satisfies Eq. (11.72).

1

Knowing an iterate H (n) (μ) for all relevant values of μ, the next iterate H (n+1) (μ) is calculated by introducing H (n) (μ) into the H -equation.

210

11 The

√ -law and the Nonlinear H -Equation

Hence the function H2 (z) = −H1 (z) = −

1 + ν0 z H (z) 1 − ν0 z

(11.73)

is also solution of Eq. (11.70). A detailed analysis of iterative methods of solution of the standard H -equation and of its alternative form is presented in Bosma and de Rooij (1983). We summarize here some of their remarks and conclusions. The two solutions of the standard H -equation, H (μ) and H1 (μ) are both positive for μ ∈ [0, 1], but those of the alternative equation, H (μ) and H2 (μ), have opposite signs. Thus, with the alternative equation, when the starting function of the iterative process is positive, all the iterates of the correct solution should remain positive, and if the procedure converges, it will lead to the physically correct solution. Another advantage of Eq. (11.70) is its behavior upon iteration: the iterates oscillate about the converged solution, whereas they go monotonically to the converged solution with Eq. (11.64). Methods to speed up the convergence are described in Bosma and de Rooij (1983). We note that Eq. (11.70) holds also in the conservative case ( = 0), where it reduces to 1 1 = H (μ) 2



1 0

H (μ )μ

dμ . μ + μ

(11.74)

This equation correctly predicts the values of H (μ) at the origin and at infinity given in Eq. (9.18). At the origin, one has 1 1 = H (0) 2



1

1 α0 = 1. 2

(11.75)

1 1 α1 = √ . 2μ 3μ

(11.76)

H (μ ) dμ =

0

At infinity, 1 1 = lim μ→∞ H (μ) 2μ



1 0

H (μ )μ dμ =

√ We recall that α0 = 2 and α1 = 2/ 3 for conservative scattering (see Sect. B.5). Equation (11.70) is not the only alternative H -equation proposed to speed up the convergence of iterative methods of solution for monochromatic scattering. Ivanov (1998) has proposed an “albedo shifting” method based on the remark by Domke (1988) that the Wiener–Hopf integral equation for the source function is a member of a one-parameter family of integral equations of the same type and that it suffices to obtain the solution of one of the equation to be able to calculate the solution of all the others. The method is called albedo shifting because it amounts to solve the H -equation with a modified albedo (1 − ).

Appendix H: An Elementary Construction of the H -Function

211

For complete frequency redistribution, there is also an alternative nonlinear integral equation, namely √ 1 =  + (1 − ) H (ξ )



∞ 0

H (ν) 1 k( ) dν. ξ +ν ν

(11.77)

It can be obtained by the same procedure as Eq. (11.70), namely by taking the limit √ ξ → ∞ in Eq. (5.61) and using H (∞) = 1/ . Tables with numerical values of H (ξ ) for different absorption profiles and different values of  can be found in Ivanov and Nagirner (1965) (see also Ivanov 1973, p. 235–250). They have been calculated with Eq. (11.77) and also with an explicit expression of H (ξ ).

Appendix H: An Elementary Construction of the H -Function In Sect. 5.4.3 we present several exact expressions of the H -function. In particular,    +∞ p 1 dk , H ( ) = exp − ln[V (k)] 2 p 2π −∞ k + p2

p ∈ [0.∞[,

(H.1)

where V (k) = L(i k) is the dispersion function, defined by ˆ V (k) = 1 −  K(k).

(H.2)

ˆ We recall that K(k) is the Fourier transform of the kernel K(τ ) and  = 1 − . In Chap. 12, we construct Eq. (H.1) with the Wiener–Hopf factorization method, in which the properties of analytic functions play an important role. In this Appendix, we describe a method proposed by Ivanov (1994) to construct Eq. (H.1) by elementary means. The first remark made in Ivanov (1994) is that Eq. (H.1) is equivalent to 

˜ ∂ ln G(p) ˜ ∞ (p), = ∂

(H.3)

˜ ∞ (p) is the Laplace transform of the infinite space resolvent function where  ˜ ∞ (τ ). To obtain Eq. (H.3), we first take the logarithm of Eq. (H.1). Using, G(p) = H (1/p), we can write p ˜ G(p) =− 2π



+∞ −∞

ln[V (k)]

k2

dk . + p2

(H.4)

212

11 The

√ -law and the Nonlinear H -Equation

˜ ∞ (p) can be written as Now comes the remark that  ˜ ∞ (p) = 

p π



+∞ −∞

ˆ ∞ (k) 

k2

dk , + p2

(H.5)

ˆ ∞ (k) is the Fourier transform of ∞ (τ ). This Fourier transform, according where  to Eq. (6.2), is given by ˆ ˆ ∞ (k) = [ 1 − 1] =  K(k) = − ∂ ln V (k) .  ˆ V (k) ∂ 1 −  K(k)

(H.6)

The combination of Eqs. (H.5), (H.6) and Eq. (H.4) leads to Eq. (H.3). The problem is thus reduced to the proof of Eq. (H.3), first rewritten as ˜ ∂ G(p) ˜ ˜ ∞ (p). = G(p)  ∂



(H.7)

The starting point is the Wiener–Hopf integral equation for the surface Green function G(τ ), namely  G(τ ) = 

∞

K(τ − τ )G(τ ) dτ + δ(τ ).

(H.8)

0

We consider its derivative with respect to  , ∂G(τ ) = ∂



∞

0

∂G(τ )

dτ + K(τ − τ ) ∂



∞

K(τ − τ )G(τ ) dτ .

(H.9)

0

The combination of Eqs. (H.8) and (H.9) leads to 

∂G(τ ) = ∂



∞

K(τ − τ )

0

∂G(τ )

dτ + G(τ ) − δ(τ ). ∂

(H.10)

This equation shows that  ∂G(τ )/∂ satisfies the same integral equation as G(τ ), with [G(τ )−δ(τ )] as primary source term. Introducing the Green function G(τ, τ0 ), we can thus write the solution of Eq. (H.10) as 

∂G(τ ) = ∂



∞

G(τ, τ0 )[G(τ0 ) − δ(τ0 )] dτ0.

(H.11)

0

The Laplace transform of this equation with respect to τ is 

˜ ∂ G(p) = ∂



∞ 0

˜ ˜ G(p, τ0 )G(τ0 ) dτ0 − G(p).

(H.12)

References

213

It is shown in Ivanov (1994) that 

∞ 0

˜ ˜ ˜ ∞ (p). G(p, τ0 )G(τ0 ) dτ0 = G(p) G

(H.13)

This relation is obtained by taking the Laplace transform with respect to τ of the nonlinear integral equation 

τ

G(τ, τ0 ) =

G(τ0 − t)G(τ − t) dt,

τ = min(τ, τ0 ),

(H.14)

0

˜ ∞ (p) − 1 =  ˜ ∞ (p), The combination of Eqs. (H.12), (H.13), and the remark that G then lead to Eq. (H.7), hence to Eq. (H.1). Whether this method of construction of the H -function would have been invented if the result had not be known a priori is a question that may be raised, but it shows that complex plane analysis is not a must for the construction of an explicit expression of the H -function. Complex plane analysis however shows, more clearly than elementary algebra, why it is possible to construct an exact expression for the half-space auxiliary function.

References Ambartsumian, V.A.: Light scattering by planetary atmospheres. Astron. Zhurnal 19, 30–41 (1942) Ambartsumian, V.A.: Theoretical Astrophysics. Pergamon Press, New York (1958), translation by J.B. Sykes; Russian original, Teoreticheskaia Astrofizika, Gostechizdat, Moscow (1952) Avrett, E.H., Hummer, D.G.: Non-coherent scattering II: Line formation with a frequency independent source function. Mon. Not. R. astr. Soc. 130, 295–331 (1965) Bosma, P.B., de Rooij, W.A.: Efficient methods to calculate Chandrasekhar’s H -functions. Astron. Astrophys. 126, 283–292 (1983) Bronstein, M: Über das Verhältnis der effektiven Temperatur der Sterne zur Temperature ihrer Oberfläche. Z. Phys. 59, 144–148 (1929) Chandrasekhar, S.: Radiative Transfer, 1st edn. Dover Publications, New York (1960); Oxford University Press (1950) Domke, H.: An equivalence theorem for Chandraskhar’s H -function and its application for accelerating convergence. J. Quant. Spectrosc. Radiat. Transf. 39, 283–286 (1988) Frisch, H.: Resonance polarization and the Hanle effect; the integral equation formulation and some applications. In: Nagendra, K.N., Stenflo, J.O. (eds.) Solar Polarization, 2nd Solar Polarization Workshop, Kluwer, Dordrecht, pp. 97–113 (1999) √ Frisch, U., Frisch, H.: Non-LTE transfer.  revisited. Mon. Not. R. astr. Soc. 173, 167–182 (1975) Frisch, U., Frisch, H.: Universality of escape from a half-space for symmetrical random walks. In: Shlesinger, M., Zaslavsky, G., Frisch, U. (eds.) Lévy Flights and Related Topics in Physics. Lectures Notes in Physics, vol. 450, pp. 262–268. Springer, Berlin (1995) Halpern, O., Lueneburg, R., Clark, O.: On multiple scattering of neutrons I. Theory of the albedo of a plane boundary. Phys. Rev. 53, 173–183 (1938) Hopf, E.: Remarks on the Schwarzschild–Milne model of the outer layers of a star. Mon. Not. R. astr. Soc. 90, 287–293 (1930)

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√ -law and the Nonlinear H -Equation

√ Hubeny, I.: Probabilistic interpretation of radiative transfer. I. The -law. Astron. Astrophys. 185, 332–335 (1987a) Hubeny, I.: Probabilistic interpretation of radiative transfer. II. Rybicki equation. Astron. Astrophys. 185, 336–342 (1987b) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ., vol. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Ivanov, V.V.: Nonmagnetic polarization of the Doppler core of strong Fraunhofer lines. Sov. Astron. 34, 621–625 (1990); translation from Astron. Zhurnal 67, 1233–1242 (1990) Ivanov, V.V.: Resolvent method: exact solutions of half-space transport problems by elementary means. Astron. Astrophys. 286, 328–337 (1994) Ivanov, V.V.: Generalized Rayleigh scattering I. Basic theory. Astron. Astrophys. 303, 609–620 (1995) Ivanov, V.V.: Albedo shifting: a new method in radiative transfer theory. Astron. Rep. 42, 89–98 (1998); translation from Astron. Zhurnal 75, 102–112 (1998) Ivanov, V.V, Nagirner, D.I.: H-functions in the theory of resonance radiation transfer. Astrophysics 1, 86–101 √ (1965); translation from Astrofizika 1, 103–146 (1965) Ivanov, V.V.: -law: Centennial of the first exact result of classical radiative transfer theory. In: Kokhanovsky, A. (ed.) Springer Series in Light Scattering, vol. 6, pp. 1–52 (2021) Kawabata, K., Limaye, S.S.: Rational approximation for Chandrasekhar’s H-function for isotropic scattering. Astrophys. Space Sci. 332, 365–371 (2011) Landi Degl’Innocenti, E.: Non-LTE transfer. An alternative derivation for sqrt. Mon. Not. R. astr. Soc. 186, 369–375 (1979) Landi Degl’Innocenti, E., Bommier, V.: Resonance line√polarization for arbitrary magnetic fields in optically thick media III. A generalization of the -law. Astron. Astrophys. 284, 865–873 (1994) Rutily, B.: Solutions exactes de l’équation de transfert et applications astrophysiques. Thèse de Doctorat d’État, Université Claude Bernard, Lyon I, N◦ : 92.07 (1992) Rybicki, G.: Integrals of the transfer equation. I. Quadratic integrals for monochromatic, isotropic scattering. Astrophys. J. 213, 165–176 (1977) Sobolev, V.V.: A Treatise on Radiative Transfer. Von Nostrand Company, Princeton (1963). Transl. by S.I. Gaposchkin; Russian original, Transport of Radiant Energy in Stellar and Planetary Atmospheres, Gostechizdat, Moscow (1956) Wiener, N., Hopf, D.: Über eine Klasse singulärer Integralgleichungen. Sitzungsberichte der Preussichen Akademie, Mathematisch-Physikalische Klasse 3. Dez. 1931 31, 696–706 (1931) (ausgegeben 28. Januar 1932); English translation: In Fourier transforms in the Complex Domain, Paley, R.C., Wiener, N. Am. Math. Soc. Coll. Publ., vol. XIX, pp. 49–58, (1934)

Chapter 12

The Wiener–Hopf Method

The first method leading to an exact solution for a convolution integral equation on a half-line was presented in the now famous Wiener and Hopf (1931) article, where the authors propose an explicit solution for the integral equation 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ ,

τ ≥ 0,

(12.1)

0

which describes the monochromatic Milne problem (see Chap. 9). N. Wiener and E. Hopf, were both mathematicians, but E. Hopf has had a lasting interest in the radiative equilibrium of stellar atmospheres (Hopf 1930, 1934). The original article is in German, but appeared in English in Paley and Wiener (1934) and Hopf (1934). Outside Astronomy, the Wiener–Hopf method was within a few years applied to integral equations describing the scattering of neutrons (Halpern et al. 1938) and then in the late fourties to the diffraction of waves on an obstacle (see references in, e.g., Karp 1950, Jones 1952, Noble 1958, Lawrie and Abrahams 2007). The application by Jones (1952) of the Wiener–Hopf method, directly to the partial differential equations arising in wave-diffraction problems, contributed to the popularization of the method. The first step of the Wiener–Hopf method is a Fourier transform, by which a convolution integral equation, or a partial differential equation, is recast into an algebraic or differential equation for Fourier transforms. The analyticity properties of complex Fourier transforms are then used to obtained a functional equation in the complex plane involving analytic functions, each function being analytic in a specified region. This functional equation, sometimes referred to as a functional Wiener–Hopf equation, corresponds to the boundary-value problems in the complex plane (Riemann–Hilbert problems) derived from singular integral equations with a Cauchy-type kernel by Hilbert transforms. The relationship between a Wiener–Hopf problem and a line Riemann–Hilbert problem is examined in, e.g., Kisil (2015) and Kisil et al. (2021). We also discuss it in this Chapter. Descriptions of the original

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_12

215

216

12 The Wiener–Hopf Method

Wiener–Hopf method can be found in, e.g.,Tichmarsh (1937), Morse and Feshbach (1953), Busbridge (1960), Roos (1969), and Duderstadt and Martin (1979). The Wiener–Hopf method, in its original form, requires the factorization of a function analytic in a strip of the complex plane into two functions analytic into two analytic overlapping half-planes (see Sect. 3.2.1). As will be shown, this means that it was restricted to convolution equations with exponentially decreasing kernels, for example radiative transfer with monochromatic scattering. It was extended by Krein (1962) to convolution equations with algebraically decreasing kernels. The functional Wiener–Hopf equation becomes then identical to a Riemann–Hilbert boundary value problem. As we have seen in preceding chapters, algebraically decreasing kernels are found in the formation of spectral lines with complete frequency redistribution (no correlation between the frequencies of the absorbed and scattered photons), but they are also found in the analysis of time series (Wiener 1949) and in the kinetic theory of gases (e.g. Latyshev et al. 1985). Applied to differential equations with half-space geometries, the Wiener–Hopf method does also lead to Riemann–Hilbert problems. Many examples are proposed in, e.g., Noble (1958), Carrier et al. (1966), Dautray and Lions (1984), Ablowitz and Fokas (1997). Lawrie and Abrahams (2007) present a brief historical perspective of the WienerHopf technique with several references to its application to partial differential equations. We treat in Sect. 12.6 the specific case of the Sommerfeld half-plane diffraction problem. This chapter is organized as follows. Section 12.1 is devoted to an overview of the Wiener–Hopf method (henceforth abbreviated as WH method). The details of the method are described in Sect. 12.2 and in Appendix I of this chapter for monochromatic non-conservative scattering, that is for an exponentially decreasing kernel and  = 0. Applications of the WH method are then presented in the subsequent sections: in Sect. 12.3 to the determination of the resolvent function for monochromatic non-conservative scattering, in Sect. 12.4 for the solution of the monochromatic Milne problem ( = 0), and in Sect. 12.5 for spectral lines formed with complete frequency redistribution. Finally in Sect. 12.6, we show how the WH method can be employed to solve a partial differential equation describing a wave diffraction problem.

12.1 An Overview of the Wiener–Hopf Method For the sake of generality, we present the main ideas of the WH method for the source function S(τ ). We assume monochromatic scattering and  = 0 (nonconservative scattering). We recall that  ∈ [0, 1] is a destruction probability per scattering. Some modifications are of course needed for conservative scattering or for complete frequency redistribution, but the main steps remain the same.

12.1 An Overview of the Wiener–Hopf Method

217

In a semi-infinite medium, the source function S(τ ) satisfies the integral equation:  ∞ S(τ ) = (1 − ) K(τ − τ )S(τ ) dτ + Q∗ (τ ), τ ≥ 0. (12.2) 0

For a full-space problem, the integral equation for S(τ ) can be solved by a standard Fourier transformation (see, e.g., Sect. 3.1). In Eq. (12.2), the integral is from 0 to ∞ and the unknown S(τ ) sought for τ ∈ [0, +∞[. As pointed out in Wiener and Hopf (1931), this equation can be transformed into a new integral equation where the integration is from −∞ to +∞ by introducing a new unknown function S − (τ ), defined by S − (τ ) ≡ (1 − )



∞

K(τ − τ )S + (τ ) dτ , for τ < 0,

and S − (τ ) ≡ 0

for τ ≥ 0.

0

(12.3) The function S + (τ ) is defined by S + (τ ) ≡ S(τ )

S + (τ ) ≡ 0

for τ ≥ 0,

for τ < 0.

(12.4)

In a similar way, we introduce Q+ (τ ) = Q∗ (τ )

Q+ (τ ) = 0 for τ < 0,

for τ ≥ 0,

(12.5)

Equation (12.2) can then be written as S + (τ ) + S − (τ ) = (1 − )



+∞ −∞

K(τ − τ )S + (τ ) dτ + Q+ (τ ),

τ ∈] − ∞, +∞[.

(12.6) It is valid on the full real axis, but involves two unknown functions S + (τ ) and S − (τ ), coupled by Eq. (12.3). This key idea of adding a new unknown function, defined for negative values of τ , is not restricted to convolution integral equations. We show in Sect. 12.6 how it can be carried out for a diffraction problem. A complex Fourier transform is then applied to Eq. (12.6). The Fourier transforms of the kernel K(τ ) and of the other functions appearing in Eq. (12.6) are defined by ˆ K(z) ≡ 

+∞

Su (z) ≡

+

S (τ )e 0



+∞ −∞

K(τ )ei zτ dτ, 

i zτ

dτ,

Sl (z) ≡

0 −∞

S − (τ )ei zτ dτ,

(12.7)

(12.8)

218

12 The Wiener–Hopf Method

Qˆ ∗ (z) ≡



+∞

Q+ (τ )ei zτ dτ,

(12.9)

0

where z is a complex variable. Complex Fourier transforms have very specific analyticity properties, which depend on the integration range. We discuss them in Sect. 12.2.2, where we show that Su (z) and Qˆ ∗ (z) are analytic in an upper halfplane, Sl (z) is analytic in a lower half-plane, whereas V (z) is analytic in a horizontal strip (see Figs. 12.3, 12.4 and 12.5). The original WH method has been designed with the double sided Laplace transform (z instead of i z). The Fourier version was introduced by Tichmarsh (1937) and is now commonly used. The difference between the two formulations amounts to a rotation of π/2 in the complex plane. We consider now the complex Fourier transform of Eq. (12.6). It may be written as Su (z)V (z) = −Sl (z) + Qˆ ∗ (z),

(12.10)

ˆ V (z) ≡ 1 − (1 − )K(z),

(12.11)

where

is the dispersion function, already introduced in Eq. (3.6). Although Eq. (12.10) has two unknowns, Su (z) and Sl (z), it can be solved separately for these two functions by making use of their analyticity properties and of a very clever factorization of the dispersion function. A second key idea of the WH method, already presented in Sect. 3.2.1, is the factorization of V (z) as V (z) = Vu (z)/Vl (z),

(12.12)

where Vu (z) is analytic in an upper half-plane and Vl (z) is analytic in a lower halfplane (see Fig. 12.3). For monochromatic scattering, V (z) is analytic in (z) ∈ [−1, +1] because the kernel decreases exponentially at infinity. For complete frequency redistribution, the analyticity strip is actually reduced to the real axis because the kernel decreases algebraically at infinity, but the decomposition remains possible (see Sect. 12.5). The factorization of V (z) makes use of a general formula for the decomposition of a function analytic in a strip, applied to ln V (z). The general decomposition formula is presented in Sect. 12.2.1. Equation (12.10) can now be written as Su (z)Vu (z) = −Sl (z)Vl (z) + Qˆ ∗ (z)Vl (z),

(12.13)

where Su (z)Vu (z) is analytic in an upper half-plane, while Sl (z)Vl (z) is analytic in a lower half-plane (see Fig. 12.5). The function ˆ ∗ (z)Vl (z), G(z) ≡ Q

(12.14)

12.1 An Overview of the Wiener–Hopf Method

219

which is analytic in an horizontal strip (see Fig. 12.4), can be decomposed as G(z) = Gu (z) − Gl (z),

(12.15)

where Gu (z) is analytic in an upper half-plane and Gl (z) is analytic in a lower one (see Fig. 12.4). With the introduction of the functions Vu (z), Vl (z), Gu (z), and Gl (z), Eq. (12.10) has been transformed into Su (z)Vu (z) − Gu (z) = −Sl (z)Vl (z) − Gl (z),

(12.16)

where the left-hand side is analytic in an upper half-plane and the right-hand side in a lower half-plane. Let us assume that these two half-planes have a common strip of analyticity (see Fig. 12.5). According to Eq. (12.16), in this strip, the functions in the right-hand side and in the left-hand side are equal. There is a theorem, usually referred to as the identity theorem (e.g. Carrier et al. 1966), from which one can infer that these two functions are two different representations of a single function, say P (z), analytic in the full complex plane. Equation (12.16) thus leads to Su (z) = [Gu (z) + P (z)]/Vu (z),

(12.17)

Sl (z) = −[Gl (z) + P (z)]/Vl (z).

(12.18)

The function P (z), because it is analytic in the full complex plane, which means that it is an entire function, can be derived from its behavior at infinity with help of the Liouville theorem (see Sect. 4.3.1). How to determine P (z) is shown explicitly with the determination of the resolvent function in Sect. 12.3 and the solution of the Milne problem in Sect. 12.4. The expressions given in Eqs. (12.17) and (12.18) hold in the domains where Su (z) and Sl (z) are analytic, but, it is an important point, also in the full complex plane by analytic continuation. Outside their analyticity half-planes, these functions can have poles and branch cuts. They have to be considered to finally determine S(τ ) by a Fourier inversion of Su (z). We note here that for spectral lines formed with complete frequency redistribution the analyticity strip is reduced to a line (the real axis), but the identity theorem is still applicable to an equation similar to Eq. (12.16) and hence permits the construction of an entire function P (z). The factorization of V (z) becomes a homogeneous Riemann–Hilbert problem of the type discussed in Sect. 5.3 (see also Sect. 12.5). Although we are somewhat spoiling the suspense of how the WH method really works for a radiative transfer equation, we make here a contact with the Hilbert transform method presented in the preceding chapters. It is clear that the factorization in Eq. (12.12) corresponds to the factorization of the dispersion function L(z) in terms of the auxiliary function X(z) and that the construction of

220

12 The Wiener–Hopf Method

Vu (z) and Vl (z) is the counterpart of the construction of the function X(z). The decomposition of G(z) also has its counterpart in the Hilbert transform method. The relations between the auxiliary functions Vu (z), Vl (z), and X(z) can be deduced from their analyticity properties, their behavior at infinity, and the factorization relations L(z) = X(z)X(−z)(ν02 − z2 ) and V (z) = Vu (z)/Vl (z). For monochromatic non-conservative scattering ( = 0), 1 , H (−1/i z)

(12.19)

1 1 1 = ∗ = H ( ). (ν0 + i z)X(−i z) X (−i z) iz

(12.20)

Vu (z) = (ν0 − i z)X(i z) = X∗ (i z) = Vl (z) =

Similar relations exist for the Milne problem ( = 0) and for complete frequency redistribution (see Eq. (12.90)). In Appendix I of this chapter, we derive explicit expressions of Vu (z) and Vl (z) in terms of ln V (z) and then show how they lead to Eqs. (12.19) and (12.20). The WH method and the Hilbert transform presented in the preceding chapters have a somewhat different philosophy. Whereas the determination of X(z) is based on the solution of a homogeneous Riemann–Hilbert problem along branch cuts of the dispersion function L(z), in the WH method, it is the existence of an analyticity strip (possibly reduced to a line), which permits the factorization of V (z) in terms of two functions Vu (z) and Vl (z), analytic in overlapping half-planes. In a detailed analysis of the relationship between a strip Wiener–Hopf problem and a line Riemann–Hilbert problem, Kisil (2015) shows that more regularity is assumed for the Wiener–Hopf method than for the Riemann–Hilbert problem. It is also stressed, as can be observed by perusing the literature, that there has been little interaction between the communities using the Wiener–Hopf and Riemann–Hilbert methods.

12.2 Full Description of the Wiener–Hopf Method We now carry out the steps of the WH method presented above.

12.2.1 Decomposition Formula A fundamental ingredient of the Wiener–Hopf method is a decomposition formula for functions analytic in a strip. Its formulation is fairly simple, but it has wide implications. Its proof relies only on the Cauchy’s integral formula for analytic functions,  f (ζ ) 1 dζ, (12.21) f (z) = 2i π C ζ − z

12.2 Full Description of the Wiener–Hopf Method Fig. 12.1 Integration contour for the decomposition formula. The vertical lines with an arrow-head indicate that f + (z) is analytic in an upper half-plane and that f − (z) is analytic in a lower half-plane

221

(z) f +(z)

δ−

α

δ+ β +R

−R f −(z)

where C is a closed contour in the analyticity domain of f (z). Assuming that the analyticity domain is a strip α < (z) < β and that f (z) → 0 as z → ∞ in the strip, Cauchy’s integral formula, applied to the rectangle in Fig. 12.1, allows us to write f (z) = f + (z) − f − (z),

(12.22)

with f + (z) ≡ f − (z) ≡

1 2i π 1 2i π

 

+∞+i δ+ −∞+i δ+ +∞+i δ− −∞+i δ−

f (η) dη, η−z f (η) dη. η−z

(12.23)

Indeed, the contributions from the two vertical lines with abscissae ±R tend to zero as R → ∞, since f (z) → 0 as z → ∞. The functions f + (z) and f − (z) are Hilbert transforms of the type introduced in Sect. 4.3.3. The function f + (z) defines a function, which is analytic in the upper half-plane (z) > δ+ and f − (z) a function, which is analytic in the lower half-plane (z) < δ− . The function f (z), analytic in the strip α < (z) < β, is thus decomposed into two functions analytic in overlapping half-planes. We note that this decomposition is not unique, since a same entire function can be added to f + (z) and f − (z). Proofs of this decomposition can be found in textbooks describing the WH factorization method (see e.g. Carrier et al. 1966, p. 382; Roos 1969; Dautray and Lions 1984.

222

12 The Wiener–Hopf Method

12.2.2 Analyticity of Complex Fourier Transforms An essential property of a complex Fourier transform is that it defines an analytic function in the domain of the complex plane where the integral is convergent. A second important property is that it tends to zero at infinity in this domain. In Sect. 12.1, we consider several Fourier transforms having the form fˆ(z) =



∞

f (τ ) ei zτ dτ.

(12.24)

f (τ ) e−yτ ei xτ dτ.

(12.25)

0

Setting z = x + i y, we can write fˆ(z) =



∞ 0

This expression shows that the domain of convergence depends on the behavior of f (τ ) at infinity. Let us assume that f (τ )e−γ τ → 0 as

τ → ∞,

(12.26)

with γ real, positive or negative. Equation (12.25) shows that in the upper half-plane (z) = y > γ , fˆ(z) exists, is analytic, and tends to zero at infinity. For a function f (z), being analytic in a region of the complex plane implies that the derivative with respect to z exists at each point in the region. Equation (12.10) involves the Fourier transforms Su (z) and Qˆ ∗ (z), defined in Eq. (12.24). Their analyticity domains can be found with the rule stated above. Assuming that S + (τ )e−aτ → 0 and Q∗ (τ )e−cτ → 0 as τ → ∞, their Fourier transforms are analytic in the upper half-planes (z) > a and (z) > c, respectively. Equation (12.10) also involves the Fourier transform,  Sl (z) ≡

0 −∞

S − (τ )ei zτ dτ,

(12.27)

with S − (τ ) defined by Eq. (12.3). For monochromatic scattering, the kernel K(τ ) decreases exponentially to zero for τ → ±∞. Thus S − (τ )  eτ for τ → −∞. Inserting this behavior into Eq. (12.27), we see that Sl (z) is analytic in the lower half-plane y = (z) < 1 and tends to zero at infinity in this half-plane. For monochromatic scattering, the kernel K(τ ) is defined by K(τ ) =

1 2



1

exp(− 0

|τ | dμ ) . μ μ

(12.28)

12.2 Full Description of the Wiener–Hopf Method

223

At infinity: K(τ ) = O(e−|τ | ),

τ → ±∞.

(12.29)

A more precise determination of the asymptotic behavior leads to K(τ ) ∼ e−|τ | /|τ |, τ → ±∞ (Abramovitz and Stegun 1964), but the subdominant algebraic factor plays no role in the analysis of the Fourier transform. The Fourier transform of K(τ ) is ˆ K(z) =



+∞ −∞

K(τ )e−yτ ei xτ dτ.

(12.30)

The contribution from the integral over [0, +∞[ is analytic in an upper half-plane y = (z) > −1, whereas the contribution from the integral over ] − ∞, 0] is ˆ analytic in a lower half-plane y = (z) < 1. Hence K(z) is analytic in the strip −1 < (z) < 1 and tends to zero at infinity. The dispersion function V (z) = ˆ 1 − (1 − )K(z) has thus the same analyticity strip, −1 < (z) < 1, and tends to 1 as z → ∞. The horizontal analyticity strip defined by (z) ∈] − 1, +1[ is henceforth denoted S. We show in Fig. 12.2 the domain of analyticity of V (z), Su (z), and Sl (z). We can observe that the condition a < 1 is necessary, for the existence of a strip of analyticity common to these functions. This discussion shows that the Wiener–Hopf integral equation has no solution growing at infinity more rapidly than exp(τ ).

(z)

Su(z)

+1

common strip +ν0

a 0

V (z)

−ν0

−1 zero

Sl (z)

Fig. 12.2 The analyticity domains of V (z), Su (z), the Fourier transform of S + (τ ), and of Sl (z), the Fourier transform of S − (τ ). The dashed line a has been placed arbitrarily below the line +1. The vertical lines with two arrows indicate a strip of analyticity, those with a simple arrow indicate a half-plane analyticity. The functions V (z) and Su (z) have a common strip of analyticity defined by a < (z) < 1. The function V (z) is analytic in −1 < (z), +1 and has two zeroes at ±i ν0

224

12 The Wiener–Hopf Method

12.2.3 Properties of the Dispersion Function V (z) ˆ The calculation of K(z), Fourier transform of the kernel K(τ ), leads to 1− V (z) = 1 − 2



∞ 1

1 dν 1 + ) = 1 − (1 − ) ( ν − iz ν + iz ν

 1

∞

dν . ν 2 + z2 (12.31)

This explicit expression shows that V (0) = L(0) =  and that limz→∞ V (z) = 1. It also shows that outside the analyticity strip S, V (z) has two branch cuts along the imaginary axis, one above z = i and the other one below z = −i . It also shows that V (z) has logarithmic singularities at z ± i . In Sect. 5.2, we have studied in detail the properties of the dispersion function L(z) related to V (z) by L(i z) = V (z). It was shown that L(z) has two zeroes at ±ν0 , ν0 ∈ [0, 1[. Hence, V (z) has two zeroes on the imaginary axis at ±i ν0 , inside the strip S. For √  = 0, the two zeroes coalesce into a double zero at z = 0. We recall that ν0  3, for small values of . We also recall that for L(z), the analyticity strip is vertical, defined by −1 < (z) < +1. The π/2 rotation comes from the definitions of V (z) and L(z), based on Fourier and a Laplace transforms, respectively. The existence of the analyticity strip S permits in principle the application of the decomposition formula written in Eq. (12.22). Indeed, taking the logarithm of Eq. (12.12), we can write ln V (z) = ln Vu (z) − ln Vl (z).

(12.32)

However, because of the zeroes of V (z), ln V (z) has singularities at ±i ν0 inside the analyticity strip S, hence, it does not directly obey the conditions of applicability of the decomposition theorem described in Sect. 12.2.1, which requires that the function f (z) be analytic in a strip. The presence of the zeroes of V (z) inside the strip S significantly complicates the decomposition. We describe how to overcome the difficulty: in Sect. 12.2.4 for  = 0 and in Sect. 12.4 for  = 0. The general theory for the factorization of a function which has isolated singularities inside the analyticity strip is described in most of the references on the WH method indicated at the beginning of this chapter.

12.2.4 Factorization of the Dispersion Function The goal is to construct two functions Vu (z) and Vl (z) such that, V (z) = Vu (z)/Vl (z),

(12.33)

12.2 Full Description of the Wiener–Hopf Method

225

(z) Vu∗(z)

Vu(z)

z−

+1

+ν0 z

0 V (z)

V ∗(z) −ν0

zero

z+

−1 Vl∗(z)

pole

Vl (z)

Fig. 12.3 The figure shows the strips of analyticity of V (z), the dispersion function, and of V ∗ (z) defined in Eq. (12.34). It shows also the analyticity half-planes of Vu (z) and Vl (z), which satisfy V (z) = Vu (z)/Vl (z), and of Vu∗ (z) and Vl∗ (z), which satisfy V ∗ (z) = Vu∗ (z)/Vl∗ (z)

with Vu (z) analytic in an upper half-plane and Vl (z) analytic in a lower half-plane, the two half-planes having a common strip of analyticity. To avoid the singularities of ln V (z) at ±i ν0 , we introduce a new function V ∗ (z), free of zeroes, defined by V ∗ (z) ≡ V (z)

(z − z− )(z − z+ ) , z2 + ν02

(12.34)

with z− = i and z+ = −i (see Fig. 12.3). It is an even function of z (V ∗ (−z) = V ∗ (z)). The factor (z2 + ν02 ) cancels the zeroes of V (z). The factor (z − z− )(z − z+ ) plays several roles. It cancels the logarithmic singularities of V (z) at z = ±i and ensures that V ∗ (z) tends to one at infinity. To satisfy the condition f (z) → 0 as z → ∞ (see Sect. 12.2.1), one must verify that the variation of the argument of ln V ∗ (z) is zero, as z moves along an infinite horizontal line inside S. Because V (z)/(ν02 + z2 ) is analytic and free of zeroes in the strip S, the variation of its argument is zero on a closed contour. It is zero on the real axis, because V (z) is real, hence it is zero on any horizontal line, inside the strip. The variation of the argument of (z − z− )(z − z+ ) is also zero, since z− and z+ are on opposite sides of the analyticity strip. If they had been on the same side, the argument of (z − z− )(z − z+ ) would change by a factor 2π. The decomposition formula applied to ln V ∗ (z) leads to V ∗ (z) = Vu∗ (z)/Vl∗ (z),

(12.35)

226

12 The Wiener–Hopf Method

where Vu∗ (z) = exp[vu (z)],

Vl∗ (z) = exp[vl (z)].

(12.36)

The functions vu (z) and vl (z) are given by vu (z) =

1 2i π



+∞−i δ −∞−i δ

ln V ∗ (η) dη, η−z

(z) > −δ,

(12.37)

with δ = 1 − ζ , and ζ positive and as small as needed. The function vl (z) is defined by 1 vl (z) = 2i π



+∞+i δ −∞+i δ

ln V ∗ (η) dη, η−z

(z) < δ.

(12.38)

Changing η into −η and z into −z and using V ∗ (z) = V ∗ (−z), we find Vl∗ (z) = 1/Vu∗ (−z),

(12.39)

and the factorization relations V ∗ (z) = Vu∗ (−z)Vu∗ (z) = 1/[Vl∗ (−z)Vl∗ (z)].

(12.40)

We see here the analogy with the factorization relations for the auxiliary functions X(z) or the H -function (see Chap. 5). To summarize, by construction, the functions Vu∗ (z) and Vl∗ (z) are analytic, free of zeroes and tend to unity at infinity in their domain of analyticity, which are the upper half-plane (z) > −1 for Vu∗ (z) and the lower half-plane (z) < +1 for V˜l (z) (see Fig. 12.3). We now give an expression of Vu (z) in terms of ln V (z), which will be used in the Appendix I.2 of this chapter to establish the expression of H (1/z) given in Eq. (5.72) (see also Eq. (12.19)). Replacing V ∗ (z) by Vu∗ (z)/Vl∗ (z) in Eq. (12.34), we can write V (z) =

V ∗ (z)(z + i ν0 )(z − i ν0 ) Vu (z) = u∗ . Vl (z) Vl (z)(z − z− )(z − z+ )

(12.41)

To construct Vu (z) and Vl (z), we must decide how to associate the factors (z ± i ν0 ), (z − z− ), and (z − z+ ) to Vu∗ (z) and Vl∗ (z). To make the choice, we impose the condition that Vu (z) is analytic in an upper half-plane and that Vl (z) is analytic in a lower half-plane, with the two planes overlapping, that Vu (z) and Vl (z) tend to one at infinity, and that Su (z) and Sl (z), which depend on Vu (z) and Vl (z) as shown

12.2 Full Description of the Wiener–Hopf Method

227

in Eqs. (12.17) and (12.18), are also free of poles in their analyticity domains. The only choice compatible with these constraints is Vu (z) = Vu∗ (z)

z + i ν0 , z − z+

(12.42)

Vl (z) = Vl∗ (z)

z − z− . z − i ν0

(12.43)

and

Here we treat the non-conservative case ( = 0) with ν0 = 0. A different choice is made for  = 0 (see Eqs. (12.69) and (12.70)). We recall that z+ = −i and z− = i . The dependence on z+ and z− disappears in the final expression given in Eq. (12.45). We show in Fig. 12.3 the domains of analyticity of Vu (z) and Vl (z), their zeroes and also their branch cuts outside the analyticity domains. The function Vl (z) is analytic in the lower half-plane (z) < ν0 . The function Vu (z) is analytic in the upper half-plane (z) > −1 and has a zero at z = −i ν0 . For Vu (z), the zero at z = −i ν0 and the branch cut intervene in final Fourier inversions. As shown in Sect. 12.3 devoted to the resolvent function (τ ), the zero at −iν0 produces a term behaving as e−ν0 τ . Using the expression of Vu (z) given in Eq. (12.42), we show in the rather technical Appendix I.1 in this chapter that Vu (z) can be written as  Vu (z) = exp



1 2i π

 dη , η−z

(12.44)

 dη . η 2 − z2

(12.45)

+∞

−∞

ln V (η)

or, using V (−z) = V (z), as  Vu (z) = exp

z iπ



+∞

−∞

ln V (η)

This relation between the auxiliary function Vu (z) and the dispersion function V (z) readily provides the value of Vu (0). Changing z to i z in Eq. (12.45) and proceeding as in the Sect. 5.4.3 on the properties of the H -function, we obtain Vu (0) =

√ ,

√ Vl (0) = 1/ .

(12.46)

Other properties of Vu (z), such as the position of the zero, are somewhat hidden in Eq. (12.45). They are more apparent when Vu (z) is expressed in terms of X(z). The relations between Vu (z), Vl (z), X(z), and H (z) announced in Eq. (12.19) are established in Appendix I.2 of this chapter, where we also recover the expressions of the H -function given in Eqs. (5.65) and (5.71).

228

12 The Wiener–Hopf Method

This section has hopefully clarified the remark made at the end of Sect. 12.1 that in the WH method, it is the existence of an analyticity strip which permits the factorization of V (z) in terms of two functions Vu (z) and Vl (z), analytic in overlapping half-planes, whereas the determination of X(z) is based on the solution of a homogeneous Riemann–Hilbert problem along branch cuts of the dispersion function L(z), which are actually those of V (i z). The factorization required by the WH method is not always as involved as described here for monochromatic scattering. Examples of factorization by inspection can be found in Carrier et al. (1966) or Kisil (2015).

12.2.5 Decomposition of the Inhomogeneous Term We now examine the decomposition of Qˆ ∗ (z)Vl (z), the inhomogeneous source term in Eq. (12.13), into Qˆ ∗ (z)Vl (z) = Gu (z) − Gl (z),

(12.47)

where Gu (z) and Gl (z) are analytic in two overlapping half-planes, an upper and a lower one. To perform this decomposition, we must identify a strip where Qˆ ∗ (z)Vl (z) is analytic and examine its behavior at infinity. Assuming that Q∗ (τ )e−cτ → 0 for τ → ∞, Qˆ ∗ (z) will be analytic in an upper half-plane (z) > c and tend to zero at infinity (see Sect. 12.2.2). The function Vl (z) is analytic, free of zeroes in the lower half-plane (z) < ν0 and tends to unity at infinity. Thus, provided c < ν0 , there is a strip where Qˆ ∗ (z)Vl (z) is analytic and tends to zero at infinity. Hence, we can apply the decomposition formula given in Eq. (12.23) and obtain two functions Gu (z) and Gl (z), with Gu (z) analytic in the upper half-plane (z) > c, c < ν0 , and Gl (z) analytic in the lower half-plane (z) < ν0 (see Fig. 12.4). By Eq. (12.23), it follows that the function Gu (z) may be written as 1 Gu (z) = 2i π



+∞+iγ −∞+iγ

dη , Vl (η)Qˆ ∗ (η) η−z

(z) > c,

γ = c + ζ,

(12.48)

with ζ positive as small as needed. The function Gl (z) has a similar expression with γ = ν0 − ζ . We can observe that the decomposition of the inhomogeneous term in the WH method is very similar to the decomposition of the inhomogeneous term in the inhomogeneous Riemann–Hilbert problem considered in Sect. 4.3.2. We also remark that this decomposition provides a constraint on the growth at infinity of the primary source term Q∗ (τ ) in the Wiener–Hopf integral equation for S(τ ). The condition c < ν0 < 1 implies that there will be no solution to this equation unless Q∗ (τ ) grows less rapidly than eν0 τ .

12.2 Full Description of the Wiener–Hopf Method

(z)

ˆ ∗(z) Q

229

Gu(z)

+1 +ν0

common strip c 0

−ν0

−1

pole Vl (z)

Gl (z)

Fig. 12.4 Analyticity half-planes of the inhomogeneous term Qˆ ∗ (z) and of the functions Gu (z) and Gl (z) defined in Eq. (12.47). The thick solid line indicates the branch cut of Vl (z)

12.2.6 Fourier Transform of the Source Function Our problem is now of the form Su (z)Vu (z) − Gu (z) = −Sl (z)Vl (z) − Gl (z).

(12.49)

We have to verify that there is a strip of analyticity common to the left-hand side and right-hand side of this equation. The domains of analyticity of these functions are shown in Fig. 12.5. In the right-hand side, Sl (z), Vl (z) and Gl (z) are analytic in lower half-planes, the upper boundaries being defined by (z) = 1 for Sl (z) and (z) = ν0 for Vl (z) and Gl (z). Hence the right-hand side is analytic in the lower half-plane (z) < ν0 . In the left-hand side, Su (z), Vu (z) and Gu (z) are analytic in upper half-planes. Figure 12.5 shows that there can be a strip of analyticity common to left and right-hand sides of Eq. (12.49), provided a and c are smaller than ν0 . The constant c depends only on the given primary source Q∗ (τ ). For the constant a, as it depends on S + (τ ), the condition that it satisfies a < ν0 must be check a posteriori, once S + (τ ) has been determined. Figure 12.5 is drawn with a and c positive, but they can take negative values. We note also that the zero of Vu (z) at z = −i ν0 should be outside the common analyticity strip, otherwise it would induce a singularity in Su (z) (see Eq. (12.50)). The analyticity strip common to the left hand-side and righthand side of Eq. (12.49) is thus included inside the strip ] − ν0 , +ν0 [. Having identified a strip of analyticity common to the left hand-side and righthand sides of Eq. (12.49), we can apply the identity theorem, already mentioned in Sect. 12.1, from which one can conclude that the two sides of Eq. (12.49) are two

230

12 The Wiener–Hopf Method

left−hand side Gu(z) Vu(z)

Su(z)

(z)

+1 +ν0

common strip

a c

−ν0

−1

zero pole

Gl (z)

Vl (z)

Sl (z)

right−hand side Fig. 12.5 The analyticity half-planes of the functions in the left and right hand-sides of Eq. (12.16) and their common analyticity strip

different representations of single function P (z), analytic in the full complex plane, hence an entire function (a polynomial), which can determined by considering its behavior at infinity. One thus finally obtains Su (z) = [Gu (z) + P (z)]/Vu (z).

(12.50)

We give two examples of the determination of P (z): for the resolvent function in Sect. 12.3 and for the Milne problem in Sect. 12.4. In both cases P (z) tends to a constant. These examples are somewhat simpler than the general scheme described above. For the Milne problem, the primary source term Q∗ (τ ) is zero. For the resolvent function, Qˆ ∗ (z) = 1 − V (z). A simple inspection shows that Gu (z) = −Vu (z) and Gl (z) = −Vl (z). The determination of Su (z) is sufficient to determine the emergent intensity. Indeed, as shown in preceding chapters, I (0, μ) = ps˜(p), p = 1/μ, μ ∈ [0, 1], where s˜ (p) is the Laplace transform of S + (τ ). The Fourier transform of S + (τ ) and its Laplace transform are related by Su (z) = s˜ (−i z). Hence, we can write I (0, μ) =

1 i Su ( ), μ μ

μ ∈ [0, 1].

(12.51)

12.3 The Resolvent Function

231

With this formula we can recover the exact expressions of the emergent intensity in terms of the H -function derived by other methods in preceding chapters. We apply it in Sect. 12.4 for the monochromatic Milne problem and in Sect. 12.5.2 for complete frequency redistribution. The source function S(τ ) = S + (τ ) can be calculated with the inversion formula, S(τ ) =

1 2π



+∞+i γ −∞+i γ

Su (z)e−i zτ dz,

τ ≥ 0,

(12.52)

where Su (z) is given by Eq. (12.50). The inversion line must be chosen inside the analyticity domain of Su (z). We indicate in Sect. 12.3 how to perform the Fourier inversion for the resolvent function. A remarkable feature of the WH method is that it is sufficient to identify a strip of analyticity for obtaining the Fourier transform Su (z) and hence prove the existence of a solution to the Wiener–Hopf integral equation for S(τ ).

12.3 The Resolvent Function An explicit expression of the resolvent function (τ ) is derived with the singular integral equation method in Chap. 6. We recover it here with the WH method. The resolvent function (τ ) satisfies the Wiener–Hopf equation 

∞

(τ ) = (1 − )

K(τ − τ )(τ ) dτ + Q∗ (τ ),

τ ≥ 0,

(12.53)

0

with Q∗ (τ ) = (1 − )K(τ ). We introduce the functions − (τ ), defined as in ˆ u (z) and Eq. (12.3), + (τ ), defined as in Eqs. (12.4), and their Fourier transforms  ∗ ˆ l (z), defined as in Eqs. (12.24) and (12.27). The Fourier transform of Q (τ ) is ˆ = 1 − V (z). Qˆ ∗ (z) = (1 − )K(z)

(12.54)

The Fourier transform of Eq. (12.53) is thus ˆ u (z) + 1] = − ˆ l (z) + 1. V (z)[

(12.55)

The factorization of V (z) allows us to rewrite it as ˆ u (z) + 1] = Vl (z)[1 −  ˆ l (z)]. Vu (z)[

(12.56)

We now look for a strip of analyticity common to the left-hand and right-hand side. The situation is similar to that shown in Fig. 12.5, provided we ignore Gu (z) and ˆ u (z) and  ˆ l (z), and assume that + (τ )e−aτ → Gl (z), replace Su (z) and Sl (z) by  0 as τ → ∞. The right-hand side is analytic in the lower half-plane (z) < ν0

232

12 The Wiener–Hopf Method

and the left-hand side is analytic in the half-plane (z) > a. Therefore, the left and right-hand side can have a common strip of analyticity defined by max(−ν0 , a) < ˆ u (z) and (z) < ν0 . In this strip, Vu (z) and Vl (z) tend to one at infinity, while  ˆ l (z) tend to zero. The entire function P (z), defined by either side of Eq. (12.56), tends thus to one at infinity. Hence, P (z) = 1 for all z and we find ˆ u (z) = 

1 − 1. Vu (z)

(12.57)

The function (τ ) can be calculated with the Fourier inversion formula, 1 (τ ) = 2π



+∞+i γ −∞+i γ

ˆ u (z)e−i zτ dz, τ ≥ 0. 

(12.58)

Proceeding as in Chap. 6, this integral can be transformed into a Laplace type integral by introducing an integration contour in the complex plane extending to infinity, similar to the contour shown in the right panel of Fig. 6.1. Setting z = x + i y, we see that the convergence of the integral for z → ∞ requires a contour, which lies in the lower-half complex plane. The horizontal line can ˆ u (z). This means that γ be chosen anywhere inside the analyticity domain of  should satisfy max(−ν0 , a) < γ < ν0 . The contour turns around the singularity ˆ u (z) along the negative real axis and encircles the pole of  ˆ u (z) at −i ν0 . The of  Fourier inversion leads to the expression of (τ ) obtained with the inverse Laplace transform method in Sect. 6.2.2, namely 1 e−ν0 τ + (τ ) = X(ν0 )



∞ 1

φ∞ (ν)(ν0 + ν)X(−ν)e−ντ dν.

(12.59)

Here φ∞ (ν) is the inverse Laplace transform of the infinite space resolvent function (see Eq. (6.13)). The properties of Vu (z) that matter for the inversion are the zero at z = −i ν0 and the branch cut along the imaginary axis for (z) < −1. The existence of this branch cut is easily derived from the relation Vu (z) = (ν0 − i z)X(i z) = X∗ (i z) given in Eq. (12.19). Here are some details about the the calculation of (τ ). We use the expression u (z) =

1 − 1, (ν0 − i z)X(i z)

(12.60)

and integrate over the contour shown in the right panel of Fig. 6.1. The contribution from the outer curve is zero because the integrand tends to zero at infinity. There is a logarithmic singularity at the end point of the cut, but it is incorporated into the definition of the X-function.

12.4 The Milne Problem

233

The contribution from the pole at z = −i ν0 is 1 e−ν0 τ . X(ν0 )

(12.61)

To calculate the contribution from the integrals along the lines L1 and L2 , on each side of the cut, we introduce ξ = x − i y, with y > 1. We can write 1 2π

 L1 +L2

(

1 i − 1)e−i ξ τ dξ = X ∗ (i ξ ) 2π



∞ 1

e−yτ [

1 1 − ] dy. X ∗ (y − i x) X ∗ (y + i x)

(12.62) In the limit x → 0, the contribution from the lines L1 and L2 becomes 1 2i π



∞ 1

1 1 1 [ + − − ] e−yτ dy, ν0 − y X (y) X (y)

(12.63)

where X∗+ (y) and X∗− (y) are the limiting values of X∗ (z), above and below its real branch cut along the interval [1, ∞[. In this integral, we recognize φ(ν), the inverse Laplace transform of (τ ), given by φ(ν) =

1 1 1 1 [ − ] = φ∞ (ν)X∗ (−ν), 2i π ν0 − ν X+ (ν) X− (ν)

(12.64)

(see Sect. 6.2.2). Regrouping the contributions from the pole and from the branch cut, we recover Eq. (12.59). With this example, we see how the WH method directly provides an algebraic equation in the complex plane for the Fourier transform of the solution. This is clearly an advantage, compared to method described in Sect. 6.2.2. In the latter, the Wiener–Hopf integral equation is first transformed into a singular integral equation with a Cauchy-type kernel, before it can be transformed into a boundary layer problem in the complex plane. The construction of the two auxiliary functions Vu (z) and Vl (z), the search for an analyticity strip, and the final Fourier inversion are however somewhat more delicate than the construction of a single auxiliary function X(z) and the solution of a boundary value equation.

12.4 The Milne Problem We now apply the Wiener–Hopf method to the Milne problem. The source function S(τ ) satisfies Eq. (12.1). A Fourier transformation leads to Su (z)V (z) = −Sl (z),

(12.65)

234

12 The Wiener–Hopf Method

where Su (z) and Sl (z) are defined as in Eq. (12.8), and the factorization of V (z) as V (z) = Vu (z)/Vl (z) to Su (z)Vu (z) = −Sl (z)Vl (z).

(12.66)

For the Milne problem, the destruction probability per scattering, , is zero. The two simple zeroes of V (z) at ±i ν0 coalesce into a double zero at the origin, however the construction of Vu (z) and Vl (z) can be carried out as described in Sect. 12.2.4. The function V ∗ (z) is now defined by V ∗ (z) = V (z)

(z − z+ )(z − z− ) , z2

(12.67)

with z+ = −i and z− = +i . The function V ∗ (z) is free of zeroes in the strip |(z)| < 1 and can be factorized as in Eq. (12.35) with Vu∗ (z) and Vl∗ (z) analytic and free of zeroes in the half-planes (z) > −1 and (z) < 1, respectively. We can thus write V (z) =

Vu∗ (z) z2 . ∗ Vl (z) (z − z+ )(z − z− )

(12.68)

Now comes a difference with the case  = 0. With the choice of Vu (z) and Vl (z) made in Eqs. (12.42) and (12.43), Vu (z) and Vl (z) would tend to one at infinity, the two sides in Eq. (12.66) would then tend to zero at infinity and the construction of Su (z) would be impossible. But there is another choice, which provides a solution to the Milne problem, namely Vu (z) = Vu∗ (z)

z2 , z − z+

(12.69)

and Vl (z) = Vl∗ (z)(z − z− ).

(12.70)

With this choice Vu (z) and Vl (z) increase as z at infinity, Vu (z) has a double zero at the origin and is analytic and free of zero in upper half-plane (z) > 0, as for Vl (z), it is analytic in a lower half-plane (z) < 1 (see Fig. 12.6). The relations between Vu (z), Vl (z), and X(z) given in Eqs. (12.19) and (12.20) for  = 0 are replaced by Vu (z) = −i z2 X(i z) = zX∗ (i z) = z/H (1/(−i z)),

(12.71)

Vl (z) = −i /X(−i z) = z/X∗ (−i z) = zH (1/(−i z)).

(12.72)

12.4 The Milne Problem

235

left−hand side Vu(z)

Su(z)

(z)

+1 common strip

a 0

−1

Vl (z)

Sl (z)

right−hand side Fig. 12.6 The Wiener–Hopf method for the Milne problem. The half-plane analyticity domains of the functions in the left and right hand-sides of Eq. (12.73) and their common strip of analyticity. The function Vu (z) has a double zero at the origin

We must now check that the two sides of the equation, Su (z)Vu (z) = −Sl (z)Vl (z),

(12.73)

have a common strip of analyticity and determine their behavior at infinity. The analyticity domains of the functions in Eq. (12.73) are shown in Fig. 12.6. The right-hand side is analytic in the lower half-plane (z) < 1. With the assumption S(τ )e−aτ → 0 for τ → ∞, Su (z) is analytic in the upper half-plane (z) > a. The horizontal strip max(a, 0) < (z) < 1 is thus common to both sides of the equation and each side is a representation of a entire function P (z). For z → ∞, Su (z) and Sl (z) → 0, whereas Vu (z) and Vl (z) → z. Since P (z) is a polynomial, this implies that P (z) is of degree smaller than one and hence that it tends to a constant at infinity. We thus find Su (z) = A/Vu (z),

(12.74)

where A is a constant, still to be determined. We know that the solution of the Milne equation is a one parameter family. The free parameter is in general the radiative flux F . Another possible choice is the surface value of the √ source function S(0), related to the flux by the Hopf–Bronstein relation S(0) = 3F /4π (see Eq. (9.11)).

236

12 The Wiener–Hopf Method

We now show how to determine the constant A in terms of S(0). First we remark that Su (z) and s(ν), the inverse Laplace transform of S + (τ ) defined by S + (τ ) =



∞

s(ν)e−ντ dν,

(12.75)

0

are related by 

∞

Su (z) = 0

s(ν) dν. ν − iz

(12.76)

Taking the limit z → ∞ in Eq. (12.76) and using Vu (z)  z for z → ∞, we find 

∞

A = lim [Vu (z)Su (z)] = lim [zSu (z)] = i z→∞

z→∞

s(ν) dν = i S(0).

(12.77)

0

The solution of Eq. (12.73) can thus be written as Su (z) = i S(0)

1 . Vu (z)

(12.78)

The source function S(τ ) can be calculated with the inverse Fourier transform formula. Using Vu (z) = −i z2 X(i z), it may be written as S(τ ) = −

1 S(0) 2π



+∞+i γ −∞+i γ

1 e−i zτ dξ, τ ≥ 0. z2 X(i z)

(12.79)

The integration line should be chosen inside the analyticity domain of Su (z), and the integral can be calculated with the contour shown in the right panel of Fig. 6.1. One thus recovers the expression given in Eq. (9.23), namely   √ S(τ ) = S(0) 1 + 3τ +

∞ 1

 φ∞ (ν)X(−ν)(1 − e−ντ dν .

(12.80)

The linear growth at infinity comes from the double pole at z = 0. Its contribution is obtained by Taylor expanding e−i zτ /X(i z) to first order around z = 0. Details of the calculation can be found in Appendix G of Chap. 9, where we solve the Milne problem with the inverse Laplace transform method. The emergent radiation field is given by I (0, μ) =

1 i Su ( ). μ μ

(12.81)

Using Vu (z) = zX∗ (i z) = z/H (1/(−i z)), we readily recover I (0, μ) = S(0)H (μ) (see e.g. Sect. 9.1.2). The calculation of the radiative flux leads to S(0) =

12.5 The Wiener–Hopf Method for Spectral Lines

237

√ 3F /(4π). We recall √ that the first moment of H (μ) for conservative scattering has the value α1 = 2/ 3 (see Eq. (B.26)).

12.5 The Wiener–Hopf Method for Spectral Lines For spectral lines formed with complete frequency redistribution, the source function S(τ ) also satisfies Eq. (12.2), but with a kernel K(τ ), which decreases algebraically as τ → ∞. We recall that complete frequency redistribution means that there is no correlation between the frequencies of the absorbed and scattered photons. For complete frequency redistribution, the dispersion function is given by 

∞

V (z) = 1 − (1 − )

k(ν)( 0

1 1 + ) dν, ν − iz ν + iz

(12.82)

where k(ν) is the inverse Laplace transform of K(τ ) (see e.g. Eq. (5.3)). Setting z = x + i y, we see that V (z) has two branch cuts along the imaginary axis, one in the lower half-plane for y ∈] − i ∞, 0[ and the other in the upper half-plane for y ∈]0, +i ∞[. For y = 0, the integrand has no singularity. The analyticity strip of the dispersion function V (z) is thus reduced to the real axis. The properties of V (z) are straightforwardly derived from those of L(z) = V (i z). In this section we assume that  = 0, hence we will have V (0) = . One way of applying the WH method to complete frequency redistribution is a regularization technique introduced by Abramov et al. (1967). As seen in Chap. 8, in the presence of a continuous absorption, K(τ ) has an exponential tail, varying as e−β|τ | , β being the ratio of the continuous to the line opacity. The Fourier transform of K(τ ), and hence the dispersion function, are analytic in a strip |(z)| < β. The factorization of V (z) can be carried out as described above for monochromatic scattering, and there is no need to introduce an intermediate function V ∗ (z), since V (z) has no zeroes as shown in Chap. 8. The standard WH method can then be safely applied and at the end it suffices to take the limit β → 0. We now show that the WH method for complete frequency redistribution actually leads to a boundary layer problem in the complex plane, in other terms to a Riemann–Hilbert problem. Another example of a WH method leading to a Riemann–Hilbert problem is presented in Sect. 12.6, where we discuss a wave diffraction problem. When the analyticity trip of V (z) is reduced to the real axis, the Fourier transform of Eq. (12.2) becomes, Su (x)V (x) = −Sl (x) + Qˆ ∗ (x),

x ∈] − ∞, +∞[,

(12.83)

with x real in ] − ∞, +∞[. All the functions in Eq. (12.83) are real Fourier transforms. They are well defined and tend to zero at infinity, provided the initial functions can be bounded by a function with an algebraic behavior at infinity

238

12 The Wiener–Hopf Method

(Schwartz 1961). Real Fourier transforms may be distributions. For example, when Q∗ (τ ) has the constant value Q∗ for τ ∈ [0, ∞[, then Qˆ ∗ (x) = Q∗ [πδ(x) − P /(i x)], where P stands for Cauchy Principal Value (see Appendix A.3 in Chap. 4). Equation (12.83) can be interpreted as a boundary value problem in the complex plane, as encountered in Chap. 4. It suffices to consider that Su (x) and −Sl (x) are the limiting values on the real axis of a function S(z) defined by S(z) ≡ Su (z) for (z) > 0,

S(z) ≡ −Sl (z) for (z) < 0.

(12.84)

The minus sign is here for convenience only. By analogy with the notation used for boundary layer problems in preceding chapters, we introduce S + (x) ≡ Su (x) and S − (x) ≡ Sl (x). They are defined by S + (x) ≡ lim S(x + iy), y→0

S − (x) ≡ lim S(x − iy), y→0

y > 0.

(12.85)

With this notation, Eq. (12.83) becomes S + (x)V (x) = S − (x) + Qˆ ∗ (x),

x ∈] − ∞, +∞[.

(12.86)

It can be solved, as we now show, by introducing a homogeneous Riemann–Hilbert problem.

12.5.1 Factorization of V (z) To solve Eq. (12.86), we need two functions Vu (z) and Vl (z), respectively analytic in the half-planes (z) > 0 and (z) < 0, and such that their limiting values on the real axis, Vu+ (x) and Vl− (x), satisfy the jump condition Vu+ (x)/Vl− (x) = V (x),

x ∈] − ∞, +∞[.

(12.87)

Equation (12.82) shows that V (x) is a real function, which tends to one at infinity, and is free of zero, since  = 0. This implies that ln V (x) is also real, has no singularities on the real axis and tends to zero at plus and minus infinity. We can thus proceed exactly as in Sect. 5.3, where we show how to construct the auxiliary function X(z). We take the logarithm of Eq. (12.87) and consider the resulting equation as a Plemelj formula. We then introduce 

1 V(z) ≡ exp 2i π



+∞ −∞

 dx . ln V (x) x−z

(12.88)

12.5 The Wiener–Hopf Method for Spectral Lines

239

This function is analytic in the complex plane cut along the real axis, free of zeroes, and tends to one at infinity. The functions Vu (z) and Vl (z) are given by Vu (z) ≡ V(z) for (z) > 0,

Vl (z) ≡ V(z) for (z) < 0.

(12.89)

and satisfy the factorization relation V (z) = Vu (z)/Vl (z). The expression of V(z) is similar to the expression given in Eq. (12.44) for monochromatic scattering. The relations between Vu (z), Vl (z), X(z), and H (z) are given by the complete frequency redistribution version of Eqs. (12.19) and (12.20), namely Vu (z) = X(i z) = 1/H (−1/i z), (z) > 0, Vl (z) = 1/X(−i z) = H (1/i z), (z) < 0.

(12.90)

These relations can also be derived by setting V (x) = L(i x) in Eq. (12.88).

12.5.2 The Fourier Transform of the Source Function The construction of Su (z), the Fourier transform of the source function, can then be carried out exactly as described for monochromatic scattering. The factorization of V (x) allows us to rewrite Eq. (12.86) as ˆ ∗ (x), S + (x)Vu+ (x) = S − (x)Vl− (x) + Vl− (x)Q

x ∈] − ∞, +∞[.

(12.91)

The inhomogeneous term can be written as − Vl− (x)Qˆ ∗ (x) = G+ u (x) − Gl (x),

x ∈] − ∞, +∞[,

(12.92)

− where G+ u (x) and Gl (x) are the limiting values of two functions Gu (z) and Gl (z), analytic in the half-planes (z) > 0 and (z) < 0, respectively, defined by

Gu (z) = G(z) for (z) > 0,

Gl (z) = G(z)

for (z) < 0,

(12.93)

with G(z) ≡

1 2i π



+∞ −∞

ˆ ∗ (x) Vl− (x)Q

dx . x−z

(12.94)

The function G(z) is a Hilbert transform. It is analytic in the complex plane cut along the full real axis and tends to zero at infinity. Equation (12.91) now becomes − − − S + (x)Vu+ (x) − G+ u (x) = S (x)Vl (x) − Gl (x).

(12.95)

240

12 The Wiener–Hopf Method

Applying the identity theorem (see Sects. 12.1 or 12.2.6), we obtain Su (z) = [Gu (z) + P (z)]/Vu (z),

(z) > 0,

(12.96)

where P (z) is an entire function, which can be determined from its behavior at infinity. We recover the expression given in Eq. (12.50) for monochromatic scattering. Since Gu (z) and Su (z) tend to zero at infinity, P (z) = 0 for all z, and thus Su (z) = Gu (z)/Vu (z).

(12.97)

The source function S(τ ) can then be obtained by a Fourier inversion. It amounts to integrate e−i zτ Su (z) over the contour shown in the left panel of Fig. 6.1. The application of the WH method to convolution equations with algebraically decreasing kernels described here is an adaptation of a more general proof given in Dautray and Lions (1984). In the latter it is assumed that one can find b real, b > 0, such that K(τ ) exp(−bτ ) and Q∗ (τ ) exp(−bτ ) are integrable on the real axis and one looks for solutions such that S(τ ) exp(−bτ ) is integrable on the real axis. The proof goes then exactly as above, except that the real line is replaced by the horizontal line (z) = b.

12.5.3 The Emergent Intensity The emergent intensity is given by I (0, ξ ) =

1 i Su ( ), ξ ξ

ξ ≥ 0.

(12.98)

Here ξ is defined by ξ = μ/ϕ(w), with ϕ(w) the absorption line profile. The frequency variable is denoted w. It is shown in, e.g., Sect. 7.1, that the emergent intensity can be written as  I (0, ξ ) = H (ξ ) 0

∞

dν 1 , q ∗ (ν)H ( ) ν 1 + ξν

ξ ≥ 0,

(12.99)

where q ∗ (ν) is the inverse Laplace transform of the primary source term Q∗ (τ ). We now show how Eq. (12.99) can be derived from the expression of G(z) given in Eq. (12.94). The definition of Qˆ ∗ (z) in Eq. (12.9) shows that Qˆ ∗ (z) =



∞ 0

q ∗ (ν)

dν . ν − iz

(12.100)

12.6 The Sommerfeld Half-Plane Diffraction Problem

241

ˆ ∗ (z), for z complex, is a Hilbert transform. To calculate Gu (z), we replace Hence Q in Eq. (12.94) the integral over the real axis by an integration along the contour in the complex plane shown in the left panel of Fig. 6.1. We thus obtain 1 Gu (z) = lim 2i π x→0



∞

ˆ ∗(−iy+x)−Q ˆ ∗(−iy−x)] Vl (−iy)[Q

0

dy . y − iz

(12.101)

There is no contribution from the outside circle since Qˆ ∗ (z)Vl (z) tends to zero at ˆ ∗ (z) across the branch cut, is infinity. The square bracket, which is the jump of Q ∗ equal to 2i π q (y). Using Vl (−i y) = 1/X(−y) (see Eq. (12.90)), we obtain 

∞

Gu (z) = 0

q ∗ (ν) dν . X(−ν) ν − z

(12.102)

Combining Eqs. (12.97), (12.98), (12.102), and Vu (z) = X(i z) = 1/H (i /z), we readily recover Eq. (12.99). A similar calculation can be performed for monochromatic scattering. Slightly more complicated because Vl (z) has pole at i ν0 , it also leads to Eq. (12.99), where the lower bound of the integral should be set to 1.

12.6 The Sommerfeld Half-Plane Diffraction Problem We consider now the so-called Sommerfeld half-plane diffraction problem. It addresses the calculation of the reflected and diffracted acoustic (or electromagnetic) waves generated by a plane wave incident on a semi-infinite plane. The geometry is shown in Fig. 12.7a. This diffraction problem was solved by

y

y

ϕ(x, 0+)

ϕI θ

semi-infinite plate

(a)

x

ϕ(x, 0−)

ϕ (x, 0+)

−

ϕ (x, 0 )

ϕ(x, 0+) ϕ(x, 0−)

ϕ (x, 0+) x ϕ (x, 0−)

(b)

Fig. 12.7 The Sommerfeld diffraction problem: (a) the geometry; (b) the boundary conditions for the field ϕ(x, y) and for ϕ (x, 0± ) = ∂ϕ(x, 0± )/∂y, the normal derivative of ϕ(x, y) on the axis y = 0. The notation 0± means that y → 0 by positive or negative values. The continuity across the axis y = 0 is indicated by a vertical line with two arrows

242

12 The Wiener–Hopf Method

Sommerfeld (1896) with an ingeneous extension of the method of images to Riemann surfaces, a method considered as an inimitable tour-de-force. Here, we show how the Wiener–Hopf approach allows one to transform this wave problem into a boundary value problem in the complex plane, similar to the problem solved in Sect. 12.5 for radiative transfer with complete frequency redistribution. We follow the descriptions by Noble (1958) and by Ablowitz and Fokas (1997). Our notation is that of Ablowitz and Fokas (1997). One considers an incident wave ϕ˜I = ϕI ei ωt with ϕI (x, y) = exp[−i λ(x cos θ + y sin θ )],

(12.103)

coming from a direction θ with θ ∈ [−π/2, +π/2]. The total field, denoted ϕ˜ T , is the sum of the incident field ϕ˜I and of a field ϕ, ˜ created by the presence of the plate. The fields, ϕ˜T = ϕ˜ I + ϕ˜ and ϕ˜ I satisfy the linear wave equation, so it follows that ϕ˜ = ϕei ωt also satisfies this equation and that ϕ satisfies the so-called Helmholtz equation, ∂ 2ϕ ∂ 2ϕ (x, y) + (x, y) + λ2 ϕ(x, y) = 0. ∂x 2 ∂y 2

(12.104)

The solution of this equation depends on the boundary conditions imposed on ϕ(x, y) and on its derivative ∂ϕ(x, y)/∂y, for y = 0 and at infinity. For the socalled Sommerfeld problem the boundary conditions are the following: (a) the normal velocity of the total field, ∂ϕT /∂y, vanishes on the plate, so that ∂ϕ ∂ϕI |y=0 = − |y=0 , ∂y ∂y

x ∈] − ∞, 0].

(12.105)

(b) the normal derivative ∂ϕT (x, y)/∂y and therefore ∂ϕ(x, y)/∂y are continuous on y = 0 for x ∈] − ∞, +∞[, (c) the function ϕT (x, y) and therefore ϕ(x, y) are continuous on y = 0 for x ∈ [0, +∞[, (d) at infinity, there should be outgoing waves only, that is ϕ ∼ c e−i λ r ,

as r → ∞.

(12.106)

The continuity requirements for ϕ(x, y) and for its derivative ∂ϕ(x, y)/∂y are reproduced in Fig. 12.7b. It should be observed that ϕ(x, y) is discontinuous on the plate, that is for x ∈] − ∞, 0]. Following a standard method of solution for wave equations, we take the Fourier transform of Eq. (12.104). The Fourier transform of ϕ(x, y), defined by ˆ (k, y) ≡



+∞ −∞

ϕ(x, y)ei kx dx,

k ∈] − ∞, +∞[,

(12.107)

12.6 The Sommerfeld Half-Plane Diffraction Problem

243

satisfies the second order differential equation, ˆ d 2 ˆ (k, y) − γ 2 (k, y) = 0, dy 2

γ = (k − λ)1/2 (k + λ)1/2 .

(12.108)

Taking into account the condition at infinity, its solution has the form ˆ (k, y) = A1 (k)e−γ y ,

y ∈]0, +∞[;

ˆ (k, y) = B2 (k)eγ y ,

y ∈] − ∞, 0[. (12.109)

The continuity of ∂ϕ(x, 0)/∂y for x ∈] − ∞, +∞[ implies that A1 (k) = −B2 (k). Thus ˆ (k, y) = A(k)e−γ y ,

y ∈]0, +∞[;

ˆ (k, y) = −A(k)eγ y ,

y ∈] − ∞, 0[, (12.110)

with A(k) ≡ A1 (k) = −B2 (k). The problem is reduced to the determination of A(k). ˆ Following the Wiener–Hopf approach, the function (k, y) is decomposed into two new functions ˆ + (k, y) ≡ 



∞

ϕ(x, y)e

i kx

dx,

0

ˆ − (k, y) ≡ 



0 −∞

ϕ(x, y)ei kx dx.

(12.111)

The variable k is now assumed to belong to the complex plane. For k complex, ˆ − (k, y) are analytic in the upper and lower half complex k-plane, ˆ + (k, y) and   ˆ + (k, y)/∂y and ∂  ˆ − (k, y)/∂y, respectively. Their derivatives with respect to y, ∂  have the same analyticity property. We now establish the boundary conditions for these two new functions. We first consider the range x ∈]0, +∞[, outside the plate. The continuity of ϕ(x, y) and of ∂ϕ(x, y)/∂y across the x-axis leads to ˆ + (k, y) = lim  ˆ + (k, y) ≡  ˆ + (k, 0). lim 

y→0+

y→0−

(12.112)

and lim

y→0+

ˆ+ ˆ+ ˆ+ ∂ ∂ ∂ (k, y) = lim (k, y) ≡ (k, 0). ∂y ∂y y→0− ∂y

(12.113)

The notation 0+ and 0− means that y → 0 with positive or negative values, ˆ + (k, y) and its derivative with respect to y are thus respectively. The function  continuous across the x-axis for x positive.

244

12 The Wiener–Hopf Method

In the range x ∈] − ∞, 0[, which coincides with the plate, ϕ(x, y) is discontinuous across the x-axis but its derivative is continuous. Thus, ˆ − (k, y) = lim  ˆ − (k, y). lim 

y→0+

y→0−

(12.114)

and lim

y→0+

ˆ− ˆ− ˆ− ∂ ∂ ∂ (k, y) = lim (k, y) ≡ (k, 0). ∂y ∂y y→0− ∂y

(12.115)

We use now Eqs. (12.112) to (12.115) to establish the boundary conditions for the ˆ ˆ + (k, y)+ ˆ − (k, y) for y → 0± . The combination of Eqs. (12.112) sum (k, y) =  and (12.114) with Eq. (12.110) leads to ˆ ˆ ˆ − (k, y) − lim  ˆ − (k, y) = 2A(k). lim (k, y) − lim (k, y) = lim 

y→0+

y→0−

y→0+

y→0−

(12.116) A second expression for A(k) can be established by considering the limit y → 0 ˆ − are continuous for ˆ ˆ  ˆ + , and  of the derivative ∂ (k, y)/∂y. The derivatives of , y → 0 (see Eqs. (12.110), (12.113) and (12.115)). One can thus write ˆ+ ˆ− ˆ ∂ ∂ ∂ (k, y) = (k, 0) + (k, 0) = −γ A(k). y→0 ∂y ∂y ∂y lim

(12.117)

Equation (12.116) shows that A(k) has the same analyticity domain as Φˆ − (k, y), hence is analytic in the lower half of the complex k-plane, that is for (k) < 0. ˆ − (k, y) across the axis y = 0. The It shows also that 2A(k) is the jump of the  + ˆ function ∂  (k, 0)/∂y is analytic in the upper half complex k-plane, that is for ˆ − (k, 0)/∂y, it plays the role of a given inhomogeneous term. (k) > 0. As for ∂  Indeed, according to Eq. (12.105), ˆ− ∂ (k, 0) = − ∂y



0

−∞

ei kx

∂ϕI (x, 0) dx. ∂y

(12.118)

Equation (12.117) is thus a Riemann–Hilbert boundary value problem for k ∈] − ∞, +∞[. To stress this interpretation and simplify the notation, we introduce  − (k) ≡ A(k),

 + (k) ≡

ˆ+ ∂ (k, 0), ∂y

Q(k) ≡

ˆ− ∂ (k, 0). ∂y

(12.119)

Replacing γ by its full expression given in Eq. (12.108), Eq. (12.117) becomes  + (k) = −(k 2 − λ2 )1/2 − (k) + Q(k),

k ∈] − ∞, +∞[.

(12.120)

12.6 The Sommerfeld Half-Plane Diffraction Problem

245

This equation is similar to the boundary value equation for complete frequency redistribution written in Eq. (12.86). The Wiener–Hopf method tells us that to solve Eq. (12.120), the function γ = (k − λ)1/2 (k + λ)1/2 must be factorized as the product of two functions analytic respectively in a lower and a upper half-plane. Because of the square-root, each term is multivalued. The factor (k + λ)1/2 has a branch cut along −∞ < k < −λ. The branch cut of (k − λ)1/2 is along λ < k < +∞. We define (k + λ)1/2 for k real as the limit of the principal determination of (k + λ)1/2 , as k approaches the negative real axis from above. Similarly, (k − λ)1/2 for k real is defined as the limit of the analytic function (k − λ)1/2 as k approaches the positive real axis from below (see Fig. 12.8). Equation (12.120) can thus be written as  + (k) Q(k) = −(k − λ)1/2  − (k) + , (k + λ)1/2 (k + λ)1/2

(12.121)

The inhomogeneous term can be decomposed as Q(k) = Q+ (k) − Q− (k), (k + λ)1/2

(12.122)

with Q+ (k) and Q− (k) analytic in the upper and lower half complex k-plane, respectively. The functions Q+ (k) and Q− (k) are given in Ablowitz and Fokas (1997). The left hand-side is analytic in an upper half-plane and the right-hand is side analytic in a lower half-plane. All the terms in Eq. (12.121) tend to zero as k → ∞. One thus obtains  + (k) = Q+ (k), (k + λ)1/2

A(k) =  − (k) = −

Q− (k) . (k − λ)1/2

(12.123)

The field ϕ(x, y) can then be obtained by taking the inverse Fourier transform of ˆ ˆ (k, y). The expression of (k, y) given in Eq. (12.110) leads to sgn(y) ϕ(x, y) = 2π Fig. 12.8 The Sommerfeld diffraction problem: the factorization of γ = (k − λ)1/2 (k + λ)1/2 . The principal determination of (k + λ)1/2 is analytic in (k) > 0 and that of (k − λ)1/2 in (k) < 0



+∞ −∞

A(k)e−γ y e−i kx dk.

(12.124)

(k) (k + λ)1/2 −λ

+λ

(k) (k − λ)1/2

246

12 The Wiener–Hopf Method

The final result is (Ablowitz and Fokas 1997, p. 574) sign(y) (λ − λ cos θ )1/2 ϕ(x, y) = − 2π

 L

e−i kx−γ |y| dk, (k − λ)1/2 (k − λ cos θ )

(12.125)

where L =] − ∞, +∞[ is indented underneath the pole at k = λ cos θ . Other partial differential equations, reducible to Riemann–Hilbert problems can be found in Noble (1958), Carrier et al. (1966), Ablowitz and Fokas (1997). The latter book addresses the inverse scattering transform, a very powerful method introduced in the late sixties by Gardner et al. (1967), by which nonlinear differential equations, such as the Korteweg–de Vries equation or the Kadomtsev–Petviashvili equation, can be transformed into linear scattering problems (Ablowitz and Fokas 1997, pp. 609–613). This is done via a time independent Schrödinger equation, ψxx + (q(x) + k 2 )ψ = 0,

−∞ < x < +∞.

(12.126)

The idea is to assume that the potential q(x) evolves in time according to, say, a Korteweg–de Vries equation and then to exploit an important property of the Schrödinger equation, namely that the reconstruction of the potential q(x) from scattering data can be transformed into a matrix Riemann–Hilbert problem.

Appendix I: The Auxiliary Function Vu (z) The Wiener–Hopf method is based on the decomposition of the dispersion function V (z), analytic in a horizontal strip of the complex plane, into two functions Vu (z) and Vl (z) analytic in overlapping upper and lower half-planes, and such that V (z) = Vu (z)/Vl (z). In this Appendix we establish the explicit expression of Vu (z) given in Eq. (12.44), namely ln Vu (z) =

1 2i π



+∞ −∞

ln V (η)

dη , η−z

(I.1)

and the relation between Vu (z), X(z) and H (z) given in Eq. (12.19). The technique is similar in both cases. It relies on integrations in the complex plane on properly chosen contours.

I.1 Explicit Expression of Vu (z)

247

I.1 Explicit Expression of Vu (z) Combining Eqs. (12.34), (12.36), (12.37), and (12.42) we can write 1 ln Vu (z) = 2i π + ln



+∞+i δ −∞+i δ

[ln V (η) + ln

η − z+ η − z− dη + ln ] η + i ν0 η − i ν0 η − z

z + i ν0 , z − z+

(I.2)

for (z) > δ > −ν0 . We recall that z+ = −i and z− = i . The function Vu (z) is analytic and free of zero in the half-plane (z) > −ν0 (see Fig. 12.3). To calculate the second term of the integrand, we consider the contour in the upper half of the complex plane shown in Fig. I.1a, consisting in the integration line and a semi-circle Ru with a radius Ru → ∞. There is no contribution from the semi-circle since the logarithm tends to zero as R → ∞. There is only one pole inside this contour at η = z. We thus obtains 1 2i π



+∞+i δ −∞+i δ

ln

z − z+ η − z+ dη = ln . η + i ν0 η − z z + i ν0

(I.3)

This term is canceled by the last term in Eq. (I.2). To calculate the third term, we close the integration line by a semi-circle Rl in the lower-half complex plane also shown in Fig. I.1a. There is no pole inside the contour and the contribution of the

(z)

(z) z

Ru +1

+ν0 −ν0 Rl

−1

z

+1

(z) δ

V ∗(z)

(z)

−1

R

(a)

(b)

Fig. I.1 Monochromatic scattering. (a) Contours for obtaining an explicit expression of Vu (z) in terms of the logarithm of the dispersion function. The point z lies in the upper half-plane (z) > δ > −ν0 . (b) Contour for obtaining the relation between Vu (z) and X(z). The point z lies in the upper half-plane (z) > −1

248

12 The Wiener–Hopf Method

semi-circle tends to zero as its radius Rl → ∞, hence the integral is zero. We thus obtain the explicit expression 

1 Vu (z) = exp 2i π



+∞+i δ

−∞+i δ

 dη . ln V (η) η−z

(I.4)

Since Vu (z) is analytic in the half-plane (z) > −ν0 , we can shift the integration line inside this domain. Setting δ = 0, we obtain the expression of Vu (z) given in Eq. (12.44).

I.2 The Auxiliary Functions Vu (z) and X(z) We now recover the relation given in Eq. (12.19) between the function Vu (z) and the auxiliary function X(z) of the Riemann–Hilbert approach. It holds in the analyticity domain of Vu (z), that is for (z) > −ν0 . Taking the logarithm of Eq. (I.4), using Eq. (12.42) with z+ = −i to express Vu (z) in terms of Vu∗ (z), and also Eqs. (12.36) and (12.37), we can write ln Vu (z) = I (z) + ln(ν0 − i z) − ln(1 − i z),

(I.5)

where 1 I (z) ≡ 2i π



+∞ −∞

ln V ∗ (η)

dη , η−z

(I.6)

and V ∗ (z) = V (z)

1 + z2 . ν02 + z2

(I.7)

The function V ∗ (z), already introduced in Eq. (12.34), has two branch cuts along the imaginary axis, one above z = i and one below z = −i coming from V (z). Otherwise, it is by construction analytic and free of zeroes in the strip −1 < (z) < 1, and has no singularities at the end points ±i of the branch cuts. To calculate the integral I (z), we can shift the integration line to a line defined by (z) = −1 + ζ , ζ → 0. We include this line into a closed contour, lying in the lower half-plane, and turning around the branch cut as shown in Fig. I.1b. There is no singularity inside

I.2 The Auxiliary Functions Vu (z) and X(z)

249

the contour since (z) > −ν0 and there is no contribution from the circle of radius R, R → ∞, since V ∗ (η) → 1, hence ln V ∗ (η) → 0. We can thus write I (z) =

1 2i π



−1 −∞

ln V ∗ (i y − ζ )

i dy + iy − z



−∞ −1

ln V ∗ (i y + ζ )

 i dy , iy − z (I.8)

where ζ > 0 and ζ → 0. A change of y to −y leads to I (z) =

1 2i π



∞

ln 1

dy V ∗ (i y − ζ ) . V ∗ (i y − ζ ) −y + i z

(I.9)

We now set V (z) = L(i z) in Eq. (I.7). The ratio (1 + z2 )(ν02 + z2 ) having no discontinuity across the branch cut, Eq. (I.9) can be written as 1 I (z) = 2i π



∞

ln 1

dy L− (y) , L+ (y) −y + i z

(I.10)

where L+ (y) and L+ (y) are the limiting values of L(z) below and above the branch cut y ∈ [1, ∞[. Recalling that L+ (y) and L+ (y) are complex conjugate, we obtain I (z) =

1 π



∞

θ (y) 1

dy , y − iz

(I.11)

where θ (y) is defined by L+ (y) = |L+ (y)| exp[i θ (y) (see Eq. (5.25)). The sum of the three terms in Eq. (I.5) leads thus to Vu (z) =

  ∞  ν0 − i z dy 1 θ (y) exp . 1 − iz π 1 y − iz

(I.12)

The comparison with Eq. (5.44), established for X(z) in Sect. 5.3, shows that Vu (z) = (ν0 − i z)X(i z) = X∗ (i z) = 1/H (−1/i z).

(I.13)

This is the result announced in Eq. (12.19). We also recover the expression of the H -function for monochromatic scattering given in Eq. (5.65), namely    z ∞ dy 1+z exp − θ (y) H (z) = . 1 + ν0 z π 1 1 + yz

(I.14)

The expression of Vu (z) involving ln V (z) is more compact and may seem a priori more simple, but is not very convenient to perform Fourier inversions needed to return to the physical τ -space.

250

12 The Wiener–Hopf Method

References Ablowitz, M.J., Fokas, A.S.: Complex Variables, Introduction and Applications. Cambridge University Press, Cambridge (1997) Abramov, Yu.Yu., Dykhne, A.M., Napartovich, A.P.: Transfer of resonance radiation in a halfspace. Astrophysics 3, 215–223 (1967); translation from Astrofizika 3, 459–479 (1967) Abramovitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. (1964) Busbridge, I.W.: The Mathematics of Radiative Transfer. Cambridge University Press, London (1960) Carrier, G.F., Krook, M., Pearson, C.E.: Functions of a Complex Variable. McGraw-Hill Book Company, New York (1966) Dautray, R., Lions, J.-L.: Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris; 1984–1985 (3 Vols.) (1984); 1988 (9 Vols.); English edition: Mathematical Analysis and Numerical Methods for Science and Technology (6 Vols.), Springer Verlag, Berlin (1990) Duderstadt, J.J., Martin, W.R.: Transport Theory. Wiley, New York (1979) Gardner, C.S., Greene, J.M., Kruskal, M.D, Miura, R.M.: Method for solving the Korteveg-de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967) Hopf, E.: Remarks on the Schwarzschild–Milne model of the outer layers of a star. Mon. Not. R. astr. Soc. 90, 287–293 (1930) Hopf. E.: Mathematical Problems of Radiative Equilibrium. Cambridge University Press, London (1934) Halpern, O., Lueneburg, R., Clark, O.: On multiple scattering of neutrons I. Theory of the albedo of a plane boundary. Phys. Rev. 53, 173–183 (1938) Jones, D.S.: A simplifying technique in the solution of a class of diffraction problems. Q. J. Math 3, 189–196 (1952) Karp, S.: Separation of variables and the Wiener–Hopf technique. Research Report New York University No EM-25, (88 p.) (1950) Kisil, A.V.: The relationship between a strip Wiener–Hopf problem and a line Riemann-Hilbert problem. IMA J. Appl. Math. 80, 1569–1581 (2015) Kisil, A.V., Abrahams, D., Mishuris,G., Rogosin, S.V.: The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods. Proc. R. Soc. A 477, 20210533 (2021) Krein, M.G.: Integral Equations on a half-line with kernels depending upon the difference of the arguments. Am. Math. Soc. Transl. Ser. 2 22, 163–288 (1962); Russian original Uspekhi. Mat. Nauk. 13, 3–120 (1958) Latyshev, A.V., Manucharyan, G.A., Yalamov, Yu. L.: Integral equations of convolution type and boundary value problems of the kinetic theory of gases. Soviet Phys. Dokl. 30, 763 (1985) Lawrie, J.B., Abrahams, I.D.: A brief historical perspective of the Wiener–Hopf method. J. Eng. Math. 59, 351–358 (2007) Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill Book Company, New York (1953) Noble, B.: Methods based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, New York (1958) Paley, R.C., Wiener, N.: Fourier Transforms in the Complex Domain. Amer. Math. Soc. Coll. Publ. XIX (1934) Roos, B.W.: Analytic Functions and Distributions in Physics and Engineering. Wiley, New York (1969) Schwartz, L.: Méthodes Mathématiques pour les Sciences Physiques. Hermann, Paris (1961); English translation: Mathematical methods for the Physical Sciences, Dover (2008) Sommerfeld, A.: Mathematische Theorie der Diffraction. Math. Ann. 47, 317–375 (1896) Tichmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1937)

References

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Wiener, N.: Extrapolation, Interpolation and Smoothing of Stationary Time Series, Appendix C by N. Levinson (a heuristic exposition of Wiener’s mathematical theory of prediction and filtering), pp. 149–158. The MIT Press, Cambridge (1949) Wiener, N., Hopf, D.: Über eine Klasse singulärer Integralgleichungen. Sitzungsberichte der Preussichen Akademie, Mathematisch-Physikalische Klasse 3. Dez. 1931 31, 696–706 (1931) (ausgegeben 28. Januar 1932); English translation: in Fourier transforms in the Complex Domain, Paley, R.C., Wiener, N.: Am. Math. Soc. Coll. Publ. XIX, 49–58, (1934)

Part II

Scattering Polarization

Chapter 13

The Scattering of Polarized Radiation

In Part I of this book, we treated scattering problems for monochromatic radiation and spectral lines, ignoring the linear polarization created by the scattering process itself. This scattering polarization accounts for the polarization and the blue color of the sky, as discovered by Rayleigh (1871; see Chandrasekhar 1960 for additional historical references). It accounts for the polarization of the continuum of stellar spectra and also for the linear polarization of spectral lines, observable in the absence of magnetic fields. Polarization triggered by scattering is known as Rayleigh scattering when the scattering is on atoms and molecules with a size smaller than the wavelength of the radiation and as Thomson scattering when the scattering is on free electrons. The names are different but the phase matrix describing the polarization process are identical. For spectral lines, the radiative excitation of an atomic energy level followed by a spontaneous radiative deexcitation can produced a linearly polarized emission when the scattering process leads to an uneven population of the Zeeman atomic sublevels involved in the atomic transition. The polarization created by this mechanism is known as resonance polarization. Systematic observations of the linear non-magnetic polarization of the solar spectrum, which started in 1980s (Stenflo et al. 1980, 1983a,b), have resulted in the discovery of the second solar spectrum, a term coined by Ivanov (1991), which shows dozens of polarizations features, smaller or larger than the polarization of the adjacent continuum. A complete polarimetric survey of this second solar spectrum from 3160 Å to 6995 Å has been completed by Gandorfer (2000, 2002, 2005). Resonance polarization can survive in the presence of a weak magnetic field, as observed by Hanle (1924) with a mercury vapor illuminated anisotropically (for historic references see Trujillo Bueno 2001). The modifications due to the presence of a magnetic field are referred to as the Hanle effect. In general it produces a depolarization and a rotation of the plane of polarization. This second solar spectrum, offers in the terms of Stenflo and Keller (1997), a new window for diagnostics of the Sun. Whether one deals with a continuum spectrum or a spectral line, the polarization is triggered by the anisotropy of the radiation field incident on the particles. In a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_13

255

256

13 The Scattering of Polarized Radiation

stellar or planetary atmosphere, the anisotropy of the radiation field is weak, hence the polarization created by Rayleigh scattering or resonance scattering is weak also, a few per cent at most, and for this reason is often neglected when analyzing the intensity spectrum. In contrast, a collimated beam of unpolarized light may become 100% polarized after a scattering. To give some physical content to the equations that are considered in the following chapters, we give in this chapter a short survey of the main properties of polarized radiation, Rayleigh scattering, resonance scattering, and the Hanle effect. For each scattering process, we make a rapid presentation of the construction of the phase matrix entering in the radiative transfer equation. Our presentation follows closely the books of Stenflo (1994) and Landi Degl’Innocenti and Landolfi (2004, henceforth in this chapter referred to as LL04). In-depth treatments of these subjects can be found in several other books, e.g., Chandrasekhar (1960), Dolginov et al. (1995), Born and Wolf (1999, 7th edition), Landi Degl’Innocenti (2014, Chap. 2) and in lecture notes, e.g., Sahal-Bréchot (1984), Landi Degl’Innocenti (1992), Bommier (2009). This chapter is organized as follows. Section 13.1 is devoted to a brief description of some fundamental properties of a beam of polarized radiation propagating in the vacuum, using the classical electromagnetic wave representation. Section 13.2 is devoted to the Rayleigh scattering phase matrix. We give its expression for a collimated beam and then in a fixed reference frame and also introduce the irreducible spherical tensor representation developed by Landi Degl’Innocenti (1984). The redistribution matrix for the resonance polarization of spectral lines is discussed in Sect. 13.3 and in Appendix J of this chapter. We briefly introduce the density matrix concept, needed to describe the polarization of spectral lines created by a scattering process and also the spectral profile of the scattered radiation, derived from exact Quantum Mechanics calculations (Bommier 1997a), for a two-level atom with unpolarized ground-level. The modifications induced by a weak magnetic field on the resonance polarization properties are described in Sect. 13.4 and in Appendix K of this chapter, devoted to the Hanle effect. We also present in Appendix L of this chapter, a classical harmonic oscillator model proposed by Bommier and Stenflo (1999) for the interpretation of the spectral profile of the resonance polarization and of the Hanle effect.

13.1 Description of a Polarized Radiation According to the Maxwell equations, the electric field vector E(t) associated to a monochromatic beam of radiation propagating in a direction, say ez , can be described at each point along the beam by two components Ex (t) and Ey (t) along two orthogonal axis ex and ey (see Fig. 13.1). It is assumed that ex , ey , ez , in this

13.1 Description of a Polarized Radiation Fig. 13.1 Ellipse described by the tip of the electric field vector in the plane orthogonal to the direction of propagation. The angle  (0 ≤  ≤ π) specifies the orientation of the ellipse and the angle χ (−π/4 ≤ χ ≤ π/4) the ellipticity and the sense in which the ellipse is being described

257

ey

Ψ ex χ

order, form a right-handed orthogonal coordinate system. These components may be written Ex (t) = E1 cos(φ1 − 2i πνt) = [E1 e−2i πνt ], Ey (t) = E2 cos(φ2 − 2i πνt) = [E2 e−2i πνt ],

(13.1)

where E1 ≡ E1 ei φ1 , E2 ≡ E2 ei φ2 . Here ν is the wave frequency,  stands for real part, and E1 , E2 , φ1 and φ2 are positive constants. Actually three constants are sufficient to characterize the field: E1 , E2 and the phase shift δ = φ2 − φ1 . The two components of the electric field perpendicular to the direction of propagation may be regrouped into a single vector, known as the Jones vector (see Eq. (13.47)). A monochromatic wave can also be described by four parameters known as the Stokes parameters, introduced by Stokes (1852). Ignoring a dimensional constant, the Stokes parameters for a monochromatic wave can be defined by Im = E1 E1∗ + E2 E2∗ , Qm = E1 E1∗ − E2 E2∗ = I cos 2 cos 2χ, Um = E1∗ E2 + E2∗ E1 = I sin 2 cos 2χ, Vm = i [E1∗ E2 − E2∗ E1 ] = I sin 2χ,

(13.2)

where ∗ stands for complex conjugate. The angles  and χ are shown in Fig. 13.1. Stokes I is the total intensity of the wave, Stokes V measures the circular polarization and Stokes Q and U the linear polarization. In the real world, a pure monochromatic wave does not exist. A so called monochromatic radiation beam is a statistical mixture of monochromatic waves, with frequencies in a small interval ν, random amplitudes, random phases, and a random directions. The complex amplitudes become functions of time and position

258

13 The Scattering of Polarized Radiation

(to simplify the notation we indicate only the time dependence). The observable Stokes parameters are mean values given by I = E1 (t)E1∗ (t) + E2 (t)E2∗ (t), Q = E1 (t)E1∗ (t) − E2 (t)E2∗ (t), U = E1∗ (t)E2 (t) + E2∗ (t)E1 (t), V = i [ E1∗ (t)E2 (t) − E2∗ (t)E1 (t)].

(13.3)

where . . .  is an average over a time interval much longer than the wave period and a small surface perpendicular to the direction of the wave (justifications for the averaging procedure can be found in e.g. Chandrasekhar (1960), Stenflo (1994), Dolginov et al. (1995), Born and Wolf (1999), p. 356; LL04, p. 8). To illustrate the need for the time averaging, we recall that for a typical spectral line in the optical wavelength range, the period of the wave is around 2 10−15 s (ν ≈ 6 1014 s−1 ), whereas the life-time of a wave packet, given by the inverse of Aul, the Einstein coefficient for spontaneous emission, is typically around 10−7 –10−8 s. The mean Stokes parameters can still be written in terms of a polarization ellipse, that is, as Q = I cos 2 cos 2χ, U = I sin 2 cos 2χ, V = I sin 2χ,

(13.4)

where  and χ are now mean values. The Stokes parameters are measurable quantities, since each of the quadratic quantities Ei (t)Ej∗ (t) appearing in Eq. (13.3) has the dimension of a radiative flux per unit time and per unit surface (Poynting flux). Moreover, as shown by Chandrasekhar (1960), they satisfy a vector radiative transfer equation with a structure similar to that of a scalar radiative transfer equation. These two properties make them essential for any investigations related to polarized light. The measurements of Q and U require the choice of a reference direction, while I and V are independent of the reference system. Indeed, when the system ex , ey is rotated by an angle α as shown in Fig. 13.2, the complex amplitudes along the axes eX and eY become EX = E1 cos α + E2 sin α, EY = −E1 sin α + E2 cos α.

(13.5)

13.1 Description of a Polarized Radiation

259

Fig. 13.2 Rotation of the reference direction by an angle α

ey

α ex

Hence, according to Eq. (13.3), the Stokes parameters in the new reference frame, become Iα = I, Qα = Q cos 2α + U sin 2α, Uα = −Q sin 2α + U cos 2α, Vα = V .

(13.6)

A ±90◦ rotation of the reference frame changes Q and U into −Q and −U , a ±45◦ rotation changes Q into ±U and U into ∓Q, but the sum (Q2 + U 2 ) is invariant under a rotation of the reference frame. In solar observations, the reference direction is usually chosen to be the radial direction from the center of the Sun to the point where the observation is made or the direction parallel to the solar limb (perpendicular to this radius). For a linearly polarized beam, χ = 0. The direction , known as the direction of polarization, can be defined (modulo π/2) by U = tan 2. Q

(13.7)

A better definition (modulo π) is given by cos 2 = 

Q Q2

+ U2

,

sin 2 = 

U Q2

+ U2

.

(13.8)

For a linearly polarized radiation beam, the measurement of the direction of polarization can be done by measuring the intensity transmitted through a polaroid plate rotating around the direction of the beam. Given a reference frame, the direction of polarization is reckoned by the direction of the maximum value of Stokes I .

260

13 The Scattering of Polarized Radiation

An important quantity characterizing a polarized light is the total polarization rate p defined by  p≡

Q2 + U 2 + V 2 . I

(13.9)

It is invariant in a rotation of the reference frame. For a purely monochromatic wave, p = 1, but after the averaging process, p < 1 (see Chandrasekhar 1960, p. 33; LL04, p. 9). The definition of the Stokes vector in Eq. (13.3) shows that the building blocks are the four quadratic quantities Ji,j = Ei∗ (t)Ej (t),

i, j = 1, 2.

(13.10)

They can be used to construct a 2 × 2 Hermitian matrix J, known as the polarization tensor or coherency matrix. When the reference direction is aligned with the unit vector ex J=

  1 I + Q U − iV . 2 U + iV I − Q

(13.11)

The case of an arbitrary reference direction is discussed in LL04, p. 22. We note here that the matrices defined in this chapter and in the following ones on polarized radiation are denoted with uppercase sans serif roman characters. For vectors we use bold italic characters. The propagation of an electric field can be described in a Cartesian coordinate system, but it is often more convenient, to employ spherical vectors (also called cyclic unit vectors), in particular when a rotation of the coordinate system is needed. Slightly different definitions are found in the literature. In Dolginov et al. (1995) and LL04, they are defined as 1 e+1 ≡ √ (−ex + i ey ), 2 e0 ≡ ez , 1 e−1 ≡ √ (ex + i ey ). 2

(13.12)

The vector e0 is along the direction of propagation of the electric field and the vectors e±1 in the plane perpendicular to e0 . In Brink and Satchler (1968) and Stenflo (1994), the vectors e±1 are the complex conjugate of the vectors in Eq. (13.12). The properties of the spherical vectors and their relation with Wigner rotations matrices, describing the rotation of a coordinate system, are presented very

13.2 The Rayleigh Scattering Phase Matrix

261

clearly in the Appendix A of Dolginov et al. (1995). One important property is that [eq ]∗ = (−1)q e−q ,

q = 0, ±1.

(13.13)

The spherical components of the electric field, Eα (t) and Eβ (t) (α, β = ±1), reckoned with respect to the vectors e±1 , can be used to construct a polarization tensor. Its elements are J˜αβ = Eα∗ (t)Eβ (t). In terms of the Stokes parameters,   I + V −Q + i U ˜J = 1 . 2 −Q − i U I − V

(13.14)

The polarization tensor is particularly convenient for theoretical problems. The polarization tensor and the Stokes parameters are related by 1 Si σˆ i , 2 3

J=

i=0

1 J˜ = Si σ¯ i , 2 3

(13.15)

i=0

where the Si are the four Stokes parameters in the order I, Q, U, V and the σˆ i and σ¯ i are Pauli spin matrices (see e.g. LL04, pp. 22 and 199). The relations between the polarization tensors and the Stokes parameters depend on the choice of the reference direction (LL04, p. 23).

13.2 The Rayleigh Scattering Phase Matrix The Rayleigh scattering of a radiation beam can be described by a so-called phase matrix P(,  ) defined by I (s) () d = P(,  )I (0) ( )

d

. 4π

(13.16)

Here I (0) ( ) and I (s) () are vectors constructed with the four Stokes parameters I, Q, U, V or Il , Ir , Q, U, V . The superscript (0) indicates the incident beam, coming from the direction  within a elementary solid angle d , and the superscript (s) the scattered beam, emitted in the direction , in a solid angle d. The phase matrix for Rayleigh scattering depends only on the angle between the incident and scattered beams (see Fig. 13.3). There is no dependence on the frequency of the incident beam. In Sect. 13.2.1 we will consider the scattering of a collimated radiation beam and explain the origin of the linear polarization. In Sect. 13.2.2 we present different expressions of the Rayleigh phase matrix to be used in radiative transfer problems: a standard expression established by Chandrasekhar (1960), a multipole expansion in

262

13 The Scattering of Polarized Radiation

Fig. 13.3 Geometry of a single scattering. The components Ir and Il are respectively perpendicular and parallel to the scattering plane (,  )

ey incident beam

Ir ez

Ω

Ir

Θ

Il

ex

Il

Ω scattered beam

terms of irreducible spherical tensors TQK (i, ) introduced in Landi Degl’Innocenti (1983, 1984), and the Jones matrix approach by Stenflo (1994).

13.2.1 The Scattering of a Collimated Radiation Beam We consider a monochromatic electric field travelling in the direction  along ez and scattered in a direction  making an angle with the direction of the incident field (see Fig. 13.3). We introduce the unit vectors ex and ey , parallel and perpendicular to the scattering plane containing  and . The components of the incident electric field along ex and ey may be written as Ex(0) (t) = [E1(0) e−2i πνt ], Ey(0) (t) = [E2 e−2i πνt ]. (0)

(13.17)

Sufficiently far away from the particle, i.e. a few wavelengths away, the scattered electric field can be treated as a plane wave. Ignoring a constant factor, depending on the exact physical nature of the scattering process, the components Ex(s)(t) and (s) Ey (t) of the scattered field have the form Ex(s) (t) = [E1 e−2i πνt ] cos (0)

Ey(s) (t) = [E2 e−2i πνt ]. (0)

(s)

, (13.18)

The component Ey (t) along ey , perpendicular to the scattering plane (plane of the incident and scattered beams), is unaltered, because it remains perpendicular to the (s) direction of the scattered beam (see Fig. 13.3). The component Ex (t) along ex is ◦ multiplied by a factor cos . For cos = 90 , this component becomes zero and after the scattering the monochromatic wave is linearly polarized in the direction perpendicular to the scattering plane.

13.2 The Rayleigh Scattering Phase Matrix

263

A polarized radiation field can also be described by its components Il , Ir , U , and V , with Il and Ir and respectively parallel and perpendicular to the scattering plane (see Fig. 13.3). This description is employed in e.g., Chandrasekhar (1960). Hence, Il = E1 (t)E1∗ (t) and Ir = E2 (t)E2∗ (t). They are related to the Stokes parameters by I = Il + Ir and Q = Il − Ir . Combining Eqs. (13.3) and (13.18), we obtain (s)

Il

∗(0)

(0)

= E1 (t)E1

(t) cos2

(0)

= Il

cos2

,

Ir(s) = E2(0) (t)E2∗(0) (t) = Ir(0) , U (s) = [ E1∗(0) (t)E2(0) (t) + E2∗(0) (t)E1(0) ] cos ∗(0)

V (s) = i [ E1

∗(0)

(0)

(t)E2 (t) − E2

(0)

(t)E1 ] cos

= U (0) cos = V (0) cos (0)

, .

(13.19) (0)

When the incident beam is composed of natural light, that is, when Il = Ir , the component Il(s) becomes zero for = 90◦ . The scattered beam is polarized in the direction perpendicular to the scattering plane and its intensity is divided by two. In matrix form, Eq. (13.19) can be written as (see, e.g., Chandrasekhar 1960, p. 37) I (s)() d = σ R( )I (0) ( )

d

, 4π

(13.20)

with ⎡

cos2 3⎢ 0 R( ) = ⎢ 2⎣ 0 0

0 0 1 0 0 cos 0 0

0 0 0 cos

⎤ ⎥ ⎥. ⎦

(13.21)

Here the vectors I (0) and I (s) are constructed with the components Il , Ir , U , and V . The factor 3/2 is a normalizing factor, which can be deduced from the conservation of total energy. The factor σ in Eq. (13.20) depends on the scattering process. For example, for Thomson scattering σ = (8π/3)(e02/mc2)2 with e0 and m the charge and mass of the electron and c the speed of light. This expression is for cgs units.1

1

To obtain the expression of the Thomson scattering cross-section in SI units, mc2 must be multiplied by 4π0 , with 0 the permittivity of free space. Books treating of the polarization of radiation use either the cgs unit system, as in LL04, or the S.I. system as in Stenflo (1994). In this book we use the cgs unit system.

264

13 The Scattering of Polarized Radiation

For the Stokes parameters I ,Q,U ,V , the matrix R( ) may be written as (LL04, p. 518) ⎡

1 + cos2 3 ⎢ sin2 R( ) = σ ⎢ 0 4 ⎣ 0

sin2 1 + cos2 0 0

0 0 2 cos 0

0 0 0 2 cos

⎤ ⎥ ⎥. ⎦

(13.22)

To obtain this matrix, Stokes Q must be defined as Q = Ir − Il .

13.2.2 The Rayleigh Phase Matrix in a Fixed Reference Frame The expression given in Eq. (13.21) holds for the simple geometry of Fig. 13.3. To treat the transport of polarized radiation in, say, a stellar atmosphere, the directions of the incident and scattered beams must be reckoned with respect to a reference frame attached to the atmosphere. For a plane parallel atmosphere, it is convenient to choose an orthogonal reference frame (see Fig. 13.4). A direction  is defined by its colatitude θ and azimuthal angle χ. The construction of a phase matrix, describing the scattering of an beam with incident direction  = (θ , χ ) and outgoing direction  = (θ, χ) is thus essentially a problem of geometry.

z plane perpendicular to Ω

θ 0 Ω

eb

γ

ea y

χ x

Fig. 13.4 The atmospheric reference frame. The axis Oz is along the direction normal to the surface. A direction  is defined by its colatitude θ and azimuth χ. In the plane perpendicular to , ea is the reference direction and eb is perpendicular to ea . γ is the angle between the tangent to the meridian plane and ea

13.2 The Rayleigh Scattering Phase Matrix

265

There are different methods to construct this phase matrix, henceforth denoted PR (,  ), and the final expression depends on the representation chosen for the radiation field. In Eq. (13.23) we give a standard expression directly derived from the phase matrix R( ), in Eq. (13.32) an expression in terms of the irreducible spherical tensors for polarization TQK (i, ) introduced by Landi Degl’Innocenti (1984), and in Eq. (13.53) the result of the coherency matrix approach (Stenflo 1994). We limit the presentation of these methods to a few general remarks. The two last methods have in common that the starting point is the construction of a phase matrix for the polarization tensor J, introduced in Eq. (13.10).

13.2.2.1 A Standard Expression To construct of the matrix PR (,  ) = PR (θ, χ, θ , χ ), two rotation operators defined by their Euler angles have to be applied to R( ). The procedure is described in detail in e.g. Chandrasekhar (1960), Stenflo (1994), LL04 (Chap. 5). It leads to the following expression: PR (,  ) = PR (θ, θ ) + PR (θ, θ , χ − χ) + PR (θ, θ , 2(χ − χ)). (0)

(1)

(2)

(13.23)

The matrix PR is cylindrically symmetric, PR contains terms linear in cos(χ −χ)



and sin(χ −χ), and P(2) R terms linear in cos 2(χ −χ) and sin 2(χ −χ). The relation (0)

cos

(1)

= cos θ cos θ + sin θ sin θ cos(χ − χ ).

(13.24)

explains that PR (,  ) contains terms up to the second order only. Equation (13.23) is actually an azimuthal Fourier series expansion, namely PR (θ, χ, θ , χ ) =

l=+2 

P˜ (l) (θ, θ )ei l(χ−χ ) .

(13.25)

l=−2

The radiation scattered in a solid angle d around a direction  = (θ, χ) at a point r has thus the form  d

, (13.26) S sc (r, θ, χ) = PR (θ, χ, θ , χ )I (r, θ , χ ) 4π where I (r, θ , χ ) is the incoming Stokes vector at a point r, in the direction (θ , χ ), per solid angle. The integration is over all the possible directions of the incident radiation field. The vector S sc (r, θ, χ) is in general multiplied by a single scattering albedo representing the fraction of the absorbed photons that are being reemitted. An important property of the matrix PR (,  ) is that it is reducible with respect to the Stokes parameter V . In other terms, Stokes V is decoupled from the other Stokes parameters. Hence, when the incident beam contains no circular polarization,

266

13 The Scattering of Polarized Radiation

the same is true of the scattered beam. This property is related to the fact that Stokes V remains invariant in a rotation of the reference direction. When the circular polarization is zero, the transfer of linear polarization can be described by a 3×3 phase matrix. Moreover, when the radiation field is independent of χ, i.e. has a cylindrical symmetry with respect to the axis Oz, the scattering process is described by the cylindrically symmetric component P(0) (θ, θ ) and the radiation field is fully defined by Il and Ir , or by the Stokes parameters I and Q. For the (Il , Ir ) representation, the Rayleigh phase matrix, properly normalized, may be written as PR (μ, μ ) = (0)

  3 2(1 − μ2 )(1 − μ 2 ) + μ2 μ 2 μ2 , μ 2 1 4

(13.27)

where μ = cos θ and μ = cos θ . This matrix can be factorized as PR (μ, μ ) = AR (μ)ATR (μ ), (0)

(13.28)

where T stands for transposed and √  √  3 μ2 2(1 − μ2 ) AR (μ) = . 2 1 0

(13.29)

This decomposition, suggested by Sekara (1963), is also employed in Siewert and Fraley (1967), Siewert and Burniston (1972). It is well adapted to the construction of exact solutions of conservative Rayleigh scattering problems and is used in Chap. 16. The factorization is not unique. One can always insert a term I = UUT , with U an orthogonal matrix and I the identity matrix (van de Hulst 1980).

For the Stokes (I, Q) representation, the matrix P(0) R (μ, μ ), may be written as (0) PR (μ, μ )

3 = 8



8 3

 + 13 (1 − 3μ2 )(1 − 3μ 2 ) (1 − 3μ2 )(1 − μ 2 ) . (1 − μ2 )(1 − 3μ 2 ) 3(1 − μ2 )(1 − μ 2 )

(13.30)

It can be factorized as in Eq. (13.28) with ⎡

⎤ 1 2 (3μ − 1) 1 √ ⎢ 2 2 ⎥ ⎥. AR (μ) = ⎢ ⎣ ⎦ 3 2 0 − √ (1 − μ ) 2 2

(13.31)

The factorization of the Rayleigh phase matrix plays a very important role in the analytical methods presented in subsequent chapters and also in numerical methods of solution of radiative transfer equations. It is of course independent of the geometry of the scattering medium, since it is a local property, depending only on the phase matrix.

13.2 The Rayleigh Scattering Phase Matrix

267

13.2.2.2 The Spherical Tensors Representation As shown in LL04 (Chap. 5), when the radiation field is represented by its Stokes parameters, the elements of the full 4 × 4 Rayleigh phase matrix can be written as, [PR (,  )]i,j =

 K (−1)Q TQK (i, )T−Q (j,  ), K

i, j = 0, . . . , 3,

Q

(13.32) where the indices i and j refer to the Stokes parameters with i = 0, 1, 2, 3 corresponding to I , Q, U , V , in this order. The irreducible spherical tensors for polarimetry, TQK (i, ), were introduced by Landi Degl’Innocenti (1984) to describe the polarization created by scattering processes. They are of the same family as the J rotation operators DMN (R) (LL04, Chapters 2, 3, 5; Brink and Satchler 1968). They may be written as TQK (i, ) = ei Qχ T˜QK (i, θ ),

(13.33)

and satisfy the conjugation property K (i, ). [TQK (i, )]∗ = (−1)Q T−Q

(13.34)

Here θ and χ are the colatitude and azimuth of . The tensors T˜QK (i, θ ) depend also on the reference direction in the plane perpendicular to  (see Fig. 13.4). They are trigonometric functions of θ involving cos(Kθ ) and sin(Kθ ) and are in general complex numbers. The index K takes the values 0, 1, 2. For each value of K, Q takes 2K+1 values, −K ≤ Q ≤ +K. The index K is related to the transfer of polarization. The value K = 0, which implies Q = 0, corresponds to the scalar case (zero polarization). T00 (0, ) = 1 while all the others T00 (i, ) are equal to zero. The index K = 1 corresponds to circular polarization, hence it is associated to Stokes V and all the tensors with K = 1 are zero except TQ1 (3, ), Q = −1, 0, 1. The index K = 2 is associated to linear polarization, that is to Stokes Q and U . Hence all the TQ2 (3, ) associated to Stokes V are zero. Explicit expressions of these spherical tensors can be found in LL04, Table 5.6, p. 211. For Q = 0, one obviously has T0K (i, ) = T˜0K (i, θ ). In terms of the variable μ = cos θ , 1 3 T˜00 (0, μ) = 1, T˜02 (0, μ) = √ (3μ2 − 1), T˜02 (1, μ) = − √ cos 2γ (1 − μ2 ), 2 2 2 2 (13.35) where the angle γ defines the reference direction in the plane perpendicular to  (see Fig. 13.4).

268

13 The Scattering of Polarized Radiation

The proof of Eq. (13.32) is described in detail in LL04, Chapter 5. The first step is the construction of a phase matrix for the polarization tensor J. The relation between the Stokes parameters and the polarization tensor (see Eq. (13.15)) leads to an expression of the Rayleigh phase matrix in terms of tensors Tqq (i, ), q, q = 0, ±1, defined in LL04, Eq. (5.131) (see also Eq. (L.35)). The tensors Tqq (i, ) are then expressed in terms of the tensors TQK (i, ). The relation between these two family of tensors can be found in LL04 (p. 201, 208). In the subsequent chapters, we consider mainly cylindrically symmetric problems for which the Rayleigh phase matrix reduces to its cylindrically symmetric

term P(0) R (μ, μ ). The summation in Eq. (13.32) contains only the value Q = 0 and readily leads to

P(0) R (μ, μ )



 T˜00 (0, μ)T˜00 (0, μ ) + T˜02 (0, μ)T˜02 (0, μ ) T˜02 (0, μ)T˜02 (1, μ ) = . T˜02 (1, μ)T˜02 (0, μ ) T˜02 (1, μ)T˜02 (1, μ ) (13.36)

This matrix can be factorized as  0  0  0 T˜0 (0, μ) T˜02 (0, μ) T˜0 (0, μ ) (0)

PR (μ, μ ) = . T˜02 (0, μ ) T˜02 (1, μ ) 0 T˜02 (1, μ)

(13.37)

Setting γ = 0, in Eq. (13.35), one readily recovers the matrix AR (μ) in Eq. (13.31). It is clear that this decomposition is not unique. Choosing the opposite sign for the second column of the first matrix and for the second line of the second matrix, we recover the decomposition used in Ivanov (1995) and Nagendra et al. (1998). For a non-cylindrically symmetric radiation field, it remains possible to factorize the phase matrix as the product of matrices depending respectively on  and  (see e.g., Frisch 2007). In Chap. 14 we show how to take advantage of this decomposition to simplify the radiative transfer equations, whenever the radiation field is noncylindrically symmetric. The summation over the indices K and Q in Eq. (13.32) shows that the Rayleigh phase matrix can be written as a multipole expansion, that is as PR (,  ) =



PK (,  ),

K = 0, 1, 2,

(13.38)

K

where PK (,  ) is the multipole components of order K. Its elements are [PK (,  )]ij =

 Q

K (−1)Q TQK (i, )T−Q (j,  ).

(13.39)

13.2 The Rayleigh Scattering Phase Matrix

269

The comparison of the multipole expansion in Eq. (13.39) with the azimuthal Fourier expansion in Eq. (13.25) shows that ˜ (l) (θ, θ )]ij = [P



K (−1)l T˜l K (i, θ )T˜−l (j, θ ).

(13.40)

K

The factorization of the Rayleigh phase matrix can be generalized to linear combination of the multipole components PK (,  ). For simplicity let us consider a cylindrically symmetric phase matrix having the form P(μ, μ ) =



aK PK (μ, μ ),

K = 0, 2,

(13.41)

i, j = 0, 1.

(13.42)

K

with aK a positive constant, and [PK (μ, μ )]ij = T˜0K (i, μ)T˜0K (j, μ ),

The multipole component corresponding to K = 0, is the so-called isotropic matrix Pis : 

 10 Pis = . 00

(13.43)

Introducing w = a2 /a0 , Eq. (13.41) can be written as P(μ, μ ) = a0 A(μ)AT (μ ),

(13.44)

√ ⎤ w 2 √ (3μ − 1) 1 ⎢ 2 2 ⎥ ⎥. √ A(μ) = ⎢ ⎣ ⎦ w 2 0 − √ 3(1 − μ ) 2 2

(13.45)

with ⎡

Linear combinations similar to Eq. (13.41) are encountered in the resonance polarization of spectral lines (see Sect. 13.3.2) and for Rayleigh scattering by a random mixture of anisotropic particles (Silant’ev et al. 2014). They are often written as a linear combination of isotropic and Rayleigh scattering, that is as,

P(μ, μ ) = αP(0) R (μ, μ ) + βPis .

(13.46)

The factorization takes the form written in Eq. (13.44) with a0 = α + β and w = α/(α + β).

270

13 The Scattering of Polarized Radiation

13.2.2.3 The Coherency Matrix We follow here the presentation and the notation used in Stenflo (1994). First we introduce the Jones vector, J = e−2i πνt



E1 E2

,

(13.47)

where E1 and E2 are the complex amplitudes of the electric field introduced in Eq. (13.1). We also introduce the Jones matrix w(,  ) describing the interaction of a radiation beam with the medium. It is defined by J (s) () = w(,  )J (0) ( ),

(13.48)

where J (0) ( ) and J (s) () are the Jones vector of the incident and scattered beams. The Jones vector describes only a fully polarized monochromatic radiation. To treat partially polarized light, it is necessary to take statistical averages, namely averages over a time much longer than the period of the electric oscillation (see Sect. 13.1). This averaging can be carried out on the coherency matrix defined by D = J J †,

(13.49)

where J † is the adjoint (transposed and complex conjugate) of J . It is easy to verify that

D = J,

(13.50)

where  stands for the statistical average and J is the polarization tensor, the elements of which are defined in Eq. (13.10). Now comes a step which is proper to the coherency matrix method. The 2 × 2 matrix D is transformed into a 4-dimensional vector D v . The vector D v , defined by D v = [ D11 , D12 , D21 , D22 ]T , where T stands for transposed, is transformed by the scattering process according to  (0)  D (s) v () = W(,  )D v ( ),

(13.51)

W = w ⊗ w∗ .

(13.52)

where

The symbol ⊗ denotes a tensor product. The final step is the construction of the 4 × 4 Rayleigh phase matrix PR (,  ) for the Stokes parameters. In Stenflo (1994) this matrix is denoted M(,  ) and referred to as the Mueller matrix, in analogy with the scattering of a light beam by an

13.3 Resonance Scattering of Spectral Lines

271

optical device. The relation between the elements of the polarization (=coherency) matrix and the Stokes parameters written in Eq. (13.15) leads to PR (,  ) = M(,  ) = T W(,  ) T−1 .

(13.53)

The matrix T is a combination of Pauli matrices and contains only 0, ±1 and ±i . An explicit expression of T can be found in Stenflo (1994, Eq. (2.49)). One should be aware that the expression of T depends on the definition of the Stokes parameters. We add here a few indications on the construction of the matrix W. We first consider the Jones matrix w(,  ). In a component notation, Eq. (13.48) may be written as  (0) Eα(s) () = wαβ (,  )Eβ ( ), α, β = 1, 2, (13.54) β

where α and β refer to the two components of the electric field perpendicular to the directions  and  . The elements of the Jones matrix w(,  ) may be written as wαβ (,  ) =

 [qα ( )]∗ qβ (),

q = 0, ±1,

(13.55)

q

where qα ( ) and q () are the director cosines of the components Eα and Eβ with respect to three unit vectors of the atmospheric reference frame, referred to by the index q. The elements of the 4×4 matrix W have thus the form β

wαβ (,  )[wα β (,  )]∗ =



β

[qα ( )]∗ qβ ()qα ( )][q ()]∗ .

(13.56)

qq

It is possible to include into the matrix elements wαβ some physical property related to the scattering mechanism, derived for example from the classical harmonic oscillator model (see the Appendix L in this chapter) or from the KramersHeisenberg scattering formula of Quantum Mechanics (see references in Nagendra 2019).

13.3 Resonance Scattering of Spectral Lines The term resonance scattering, means the excitation an atomic level by a beam of radiation, followed by a spontaneous radiative decay onto the same level (a decay to a different level is known as Raman scattering). The radiative transitions between the atomic levels are in general modified by inelastic and elastic collisions. Inelastic collisions participate in the excitation and deexcitation of the atomic levels and elastic collisions trigger transitions between magnetic energy sublevels, without modifying the energy of the levels. For atomic transitions, with an appropriate

272

13 The Scattering of Polarized Radiation

quantum structure, the scattered radiation will be linearly polarized. A typical example is a Zeeman triplet. For sufficiently simple atomic transitions, the scattering process can be described by a redistribution matrix R(ν, , ν ,  ), with ν and  the frequency and direction of the scattered beam. The prime variables are those of the incident beam. The redistribution matrix is a generalization of the Rayleigh phase matrix, which, for spectral lines, takes into account the frequency changes at each scattering. All the exact results presented in subsequent chapters are obtained with the assumption that the scattering process can be described by a redistribution matrix. The first Quantum Mechanics calculation of resonance polarization taking collisions into account was published by Omont et al. (1972) (see Faurobert-Scholl et al. (1995) for a summary of the method). Redistribution matrices, which could be incorporated into radiative transfer equations, were then constructed by Domke and Hubeny (1988). A fully Quantum Electrodynamics calculation was applied to resonance polarization by Bommier (1997a) and was extended in Bommier (1997b) to take the effects of magnetic fields into account. Our presentation of the redistribution matrix for resonance polarization in Sect. 13.3.2 and in Appendix J of this chapter is based on Bommier’s results for a two-level atom with unpolarized ground level.

13.3.1 Populations and Coherences A description of the atomic system by the populations of the levels and magnetic sublevels is not sufficient to take polarization phenomena into account. It must be replaced by the more sophisticated concept of density operator, which we now briefly introduce. A complete presentation of the density operator can be found in LL04 (Chapter 3). Usually denoted ρ, it is characterized by its matrix elements ρnm = un |ρ|um ,

(13.57)

where un and um are members of a complete orthogonal basis {|un } of unit vectors for the atomic system, in general chosen to be the eigenvectors of the total angular momentum. The matrix elements are then denoted ρ(J M, J M ) (other quantum numbers may be included the list of arguments), where J and M and J

and M are the total angular momentum and magnetic quantum numbers of the levels involved in a transition. The diagonal density matrix elements ρ(J M, J M) give the population of the magnetic sublevels J, M and the nondiagonal terms the couplings, usually called coherences, between these sublevels (see e.g. LL04, Chapter 3). The radiation emitted by the spontaneous deexcitation of a level is polarized when the magnetic sublevels are unevenly populated. Uneven populations can be created when the atomic levels are excited by an anisotropic radiation field or by an anisotropic distribution of particles. The linear polarization induced by these excitation mechanisms are referred to as resonance polarization and impact polarization, respectively (see, e.g., Percival and Seaton 1958).

13.3 Resonance Scattering of Spectral Lines

273

The density matrix can also be described by its so called multipole moments, also K (J, J ), (−K ≤ Q ≤ +K) (see, e.g., LL04, called spherical statistical tensors, ρQ K (J, J ) is that they obey a simple pp. 122–130). The advantage of the tensors ρQ transformation law in a rotation R of the reference system, namely K (J, J )]new = [ρQ

 K

K ∗ [ρQ

(J, J )]old [DQ Q (R)] ,

(13.58)

Q

K (R) are the matrix elements of the rotation operator R and the star where DQ

Q stands for complex conjugate. K (J, J ) may be written as The relations between the ρ(J M, J M ) and ρQ K (J, J ) = ρQ



√ (−1)(J −M) 2K + 1

KQ

ρ(J M, J M ) =

 KQ



J J K M −M Q

√ (−1)(J −M) 2K + 1



ρ(J M, J M ),

J J K M −M Q

K (J, J ). ρQ

(13.59)

(13.60)

The parenthesis is a 3 − j symbol. Hence, |J − J | ≤ K ≤ J + J , M − M + Q = 0 and −K ≤ Q ≤ +K. The index K is always an integer. Explicit of expressions of K (J, J ) in terms of the ρ(J M, J M ) can be found in, e.g., LL04, Tables 3.6 the ρQ and 3.7, pp. 125–127. K (J, J ) or ρ(J M, J M ), satisfy conservation The density matrix elements, ρQ laws, usually called statistical equilibrium equations, which describe the balance between creations and destructions due to radiative and collisional processes. The radiative transitions rates include spontaneous deexcitation rates, excitation rates coming from radiative absorption, and the often negligible stimulated deexcitation rate. The collisional processes include inelastic collisions, inducing transitions between the two J -levels of a spectral line, and elastic collisions inducing transitions between magnetic sublevels belonging to the same J -level. These elastic collisions, often due to hydrogen atoms, may contribute strongly to decreasing the polarization created by the scattering process (see the Appendices J and L). K (J, J ), J = J are negligible when the Bohr frequency corresponding The ρQ to the energy separation between two atomic levels J and J is much larger than the inverse life time of the upper level. They are not negligible for a spectral line belonging to a multiplet. For an isolated spectral line, modelled by a two-level atom, K (J ) for the upper each level can be described by its density matrix elements, ρQ u K (J ), for the lower one, with J and J the total angular momentum of level and ρQ l u l the upper and lower level, respectively. The total population of a given J -level is K (J ) is sufficient to describe the atomic system, given by ρ00 (J ). The upper level ρQ u when the lower level population ρ00 (Jl ) is known and its magnetic sublevels evenly populated. The elements with K = 1 are known as the orientation components and

274

13 The Scattering of Polarized Radiation

those with K = 2 as the alignment components. A geometrical interpretation is given in LL04, p. 128. K (J ), the spontaneous deexcitation rate is In the statistical equation for ρQ u K proportional to Aul ρQ (J ), where Aul is the Einstein coefficient for spontaneous deexcitation. When the lower level is unpolarized, the radiative excitation rate is proportional to Blu J¯QK (r)ρ00 (Jl ), where Blu is the Einstein coefficient for radiative excitation, and J¯QK (r) are moments of the radiation field, often referred to as the spherical moments of the radiation field. Here r is the position inside the medium. For resonance scattering, J¯QK (r) ≡

j =3    j =0

K (j, )ϕ(ν)Ij (r, ν, ) (−1)Q T−Q

d dν, 4π

(13.61)

where ν is the frequency variable, ϕ(ν) the line absorption profile and Ij (r, ν, ) the j -component of the Stokes vector. In the presence of a magnetic field, these moments become J¯QK (r) ≡

j =3     d

K

dν. (−1)Q KK Q (ν)T−Q (j, )Ij (r, ν, ) 4π

j =0

(13.62)

K



The index K takes the values 0, 1, 2 and |Q| ≤ min(K, K ). The KK Q (ν) were introduced in Landi Degl’Innocenti et al. (1991) with the name generalized profiles. They are linear combinations of the Zeeman components profiles. A brief presentation is given in the Appendix K of this chapter. K (J ) the radiation field asymmetries, The moments J¯QK (r) are transferring to ρQ also called the degree of order of the radiation field (Trujillo Bueno 2001). When the radiation field is isotropic, the only non-zero component is J¯00 (r), which represents the direction and frequency averaged Stokes I .

13.3.2 The Scattering Phase Matrix The statistical equilibrium equations for the density matrix elements are coupled to the radiation field by the radiative excitation terms. The statistical equilibrium equations and the transfer equation for the radiation field have thus to be solved simultaneously. When the scattering process can be described by a redistribution matrix R(ν, , ν ,  ), the statistical equilibrium equations and transport equation for the radiation field can be combined into a single radiative transfer equation, describing simultaneously the transport and the scattering process. This situation is typical of strong resonance lines (lines originating from the ground level), which can be represented by a two-level atomic model. For lines belonging to a multiplet,

13.3 Resonance Scattering of Spectral Lines

275

such as the NaI D1 and D2 lines, the introduction of a redistribution matrix is not possible. Two equations have to be solved simultaneously, a statistical equilibrium equation and a radiative transfer equation (see, e.g., Bommier 2016a,b, 2017). Using Bommier (2017), Alsina Ballester et al. (2021) have solved the paradox of the D1 line polarization that has challenged solar physicists for many years. When the ratio J20 /J00 , which measures the anisotropy of the radiation field, is small, the construction of a redistribution matrix may become possible, even for an atomic system, for which it was a priori ruled out. This so-called weak anisotropy approximation is described in detail in LL04, p. 513. In the solar photosphere and in most stellar atmospheres, the anisotropy of the radiation field is weak and this approximation could be used for the interpretation of polarization measurements. The introduction of the statistical equilibrium equations for multi-level and also multi-term atoms can be avoided by making use of the Kramers–Heisenberg formula (see e.g., Stenflo 1994; Nagendra 2014, 2019). Designed to handle radiative scattering processes, it has been extended to account for elastic collisions (Sampoorna et al. 2007a,b). We present in Appendix J of this chapter the redistribution matrix R(ν, , ν ,  ) established by Bommier (1997a,b) for a two-level atom with unpolarized lower level. Here we comment on the frequency integrated redistribution matrix, PRS (,  ) ≡



R(ν, , ν ,  ) dν dν ,

(13.63)

often referred as the scattering phase matrix. The integration of Eq. (J.1) over frequencies leads to PRS (,  ) =

 K

WK PK (,  ), 1 + I + δ (K)

K = 0, 2.

(13.64)

Here PK (,  ) is the multipole component of the Rayleigh phase matrix corresponding to the index K, introduced in Eq. (13.38). Its elements are defined by [PK (,  )]ij =



K (−1)Q TQK (i, )T−Q (j,  ),

(13.65)

Q

(see Eq. (13.39)). For K = 0, the matrix P0 (,  ) is the isotropic matrix. The parameter WK is defined for all atomic dipole transitions. It depends on the angular momentum of the upper and lower levels (LL04 Table 10.1 p. 515). Its largest value is one. One always has W0 = 1. For a normal Zeeman triplet, W2 = 1 (a normal Zeeman triplet has Jl = 0, Ju = 1). The parameter I ≡ I / R ,

(13.66)

276

13 The Scattering of Polarized Radiation

takes into account the effect of inelastic collisions. Here R and I , in s−1 , are the spontaneous radiative deexcitation rate and inelastic collisional deexcitation rate of the upper level, respectively. R is the Einstein coefficient Aul . The parameter δ (K) is defined by δ (K) ≡ D (K) / R ,

(13.67)

K (J ) by elastic where D (K) is a rate of destruction of the multipole component ρQ collisions. D (2) is often referred to as a depolarization rate, as it contributes to the destruction of the alignment and therefore to the destruction of the linear polarization. It is in general much larger than I . Elastic collisions do not modify the population of a J -level and therefore D (0) = δ (0) = 0. When elastic collisions are ignored (δ (K) = 0), Eq. (13.64) may be written as

PRS (,  ) =

1 [(1 − W2 )Pis + W2 PR (,  )]. 1 + I

(13.68)

For a normal Zeeman triplet, W2 = 1, hence the square bracket reduces to the Rayleigh phase matrix. An expression almost identical to Eq. (13.68) has been established by Hamilton (1947) (see also Chandrasekhar 1960), in the framework of Quantum Mechanics, namely P( ) = (1 − W )Pis + W PR ( ),

(13.69)

where is the scattering angle and W is actually the constant W2 . Frequency redistribution and elastic collisions are ignored in Hamilton’s approach, but it is, as far as we know, the first Quantum Mechanics calculation for resonance scattering. In the subsequent chapters, to obtain exact results for resonance polarization, we assume that the redistribution matrix has the form R(ν, , ν ,  ) = ϕ(ν)ϕ(ν )PRS (,  ),

(13.70)

where ϕ(ν) is the line absorption profile. The frequency redistribution is treated with the same complete frequency redistribution assumption as in the scalar case. It is justified for spectral lines without extended line wings, such as the Sr I line at 4607 Å. It is also a consequence of the flat spectrum assumption employed in LL04 (see e.g., p. 257). In Appendix J of this chapter we present the frequency dependence of the redistribution matrix for a two-level atom with unpolarized ground level, as established by Bommier (1997a). It will be seen that it contains a coherent frequency redistribution term and an incoherent one. The interpretation of these two terms by Bommier and Stenflo (1999) with a classical harmonic oscillator is presented in Appendix L of this chapter.

13.4 The Hanle Effect

277

13.4 The Hanle Effect We present here, very qualitatively, the changes brought to resonance polarization by a weak magnetic field. In a magnetized atmosphere, what we have referred to as the absorption coefficient becomes a matrix, known as the propagation matrix (see e.g. Landi Degl’Innocenti and Landolfi 2004), but when the magnetic field is weak enough, in the sense discussed below, the nondiagonal terms can be neglected and the absorption term remains the same as in the scalar case. In contrast, the coherences (nondiagonal elements of the density matrix elements) are partially destroyed, leading to changes in the linear polarization described below.

13.4.1 Some General Properties The shear presence of the magnetic field, in an otherwise cylindrically symmetrical medium, breaks the symmetry of the medium, except when the magnetic field is in the direction of the symmetry axis, or consists in an isotropic turbulent field. The three Stokes parameters I , Q, and U are thus needed to characterize the linear polarization. The Hanle effect is observable for weak magnetic fields only. The sketch in Fig. 13.5 can help to understand the reason. It shows the Zeeman displacement of the magnetic Zeeman sublevels for a two-level atom with Jl = 0 and Ju = 1 (normal Zeeman triplet). The Hanle effect is present as long as the three magnetic sublevels remain coupled, that is, as long as the ratio H =

Fig. 13.5 Sketch of the Zeeman displacement of the magnetic sublevels for a normal Zeeman triplet. The horizontal axis is the magnetic field intensity B. The Hanle effect can be observed as long as the Zeeman sublevels are overlapping each other. The magnetic field intensity Btyp corresponds to a Hanle factor H of order unity

2πνL gu , Aul

(13.71)

Energy

J=1 natural width

J=0 Btyp

B

278

13 The Scattering of Polarized Radiation

stays smaller or around of unity. This ratio is usually referred to as the Hanle factor. In Eq. (13.71), νL is the Larmor frequency, which measures the Zeeman splitting, 1/Aul the life time of the upper level, and gu the Landé2 factor, equal to one for Ju = 1. We recall that the Larmor frequency is defined by νL = (e0 /(4πmc))B, with B the magnetic field strength in Gauss, e0 and m the charge and mass of the electron, and c the speed of light.3 The condition H  1 defines a typical magnetic field value Btyp . In the cgs system νL ≈ 1.4106 B. For a permitted transition Aul ≈ 107 − 108 s−1 . Thus for gu = 1, H = 0.879B/Aul, with B in Gauss and Aul in 107 s−1 . This means that the Hanle effect is sensitive to weak fields, order of a few Gauss. For larger field values, the sublevels are decoupled and one enters the realm of the Zeeman effect. There is an intermediate regime, known as Hanle– Zeeman where both effects have to be taken into account (Sampoorna et al. 2017 and references therein). A detailed discussion on the conditions to be satisfied for the Hanle effect to exist can be found in Landi Degl’Innocenti (1992), Stenflo (1994), LL04 (Chapter 5). There is an additional condition for the presence of the Hanle effect, namely a Larmor frequency smaller than the Doppler width. When it is satisfied, the full Zeeman propagation matrix can be approximated by the scalar absorption profile ϕ(ν) and the magnetic field effects are contained exclusively in the redistribution matrix. In general, the Hanle effect leads to a depolarization and a rotation of the direction of the polarization (see the sketches in Landi Degl’Innocenti 1992; Trujillo Bueno 2001 (Figure 1), LL04, p. 216). However, this is not always the case. For observations near the solar disk center, the resonance polarization is essentially zero for symmetry reason, but in a magnetized region close to the disk center, some linear polarization created by the Hanle effect may become observable (Anusha et al. 2011). The Hanle effect is usually detected near the solar limb, where the resonance polarization takes its largest values. It is more sensitive to horizontal magnetic fields than to vertical ones, the latter preserving the cylindrical symmetry of the atmosphere. The Hanle effect does not disappear in the case of a turbulent magnetic field, in contrast to the longitudinal Zeeman effect, which becomes zero when magnetic fields with opposite directions and equal strength fall in the same field of view. This property was stressed by Stenflo (1982), and from that time on the Hanle effect has been extensively used to determine turbulent magnetic fields in the solar photosphere, with observational techniques (Stenflo et al. 1998) or numerical solutions of radiative transfer equations (see, e.g., Faurobert-Scholl 1993).

2

Alfred Landé, 1888–1976, (Elberfeld, Ge.-Colombus, Ohio). The definition of the Larmor frequency depends on the system of units. In the cgs system νL = (eo /(4πmc))B with B in Gauss and eo = 4.8 10−10 in esu. In the S.I. units νL = (eo /(4πm))B with B in Tesla and eo = 1.6 10−19 Coulomb.

3

13.4 The Hanle Effect

279

Finally let us mention that the Hanle effect is observable only in the center of spectral lines. Typically it disappears about three Doppler widths away from the line center. This property can be established by making proper asymptotic expansions in the limit (ν − ν0 ) → ∞ of the frequency dependence of the redistribution matrix (Stenflo 1994, p. 83; Stenflo 1998; LL04, p. 525; see also Eq. (K.12)).

13.4.2 The Hanle Phase Matrix The conditions needed to define a redistribution matrix are those discussed in Sect. 13.3.2 for resonance scattering, but its construction is more complicated because one has to take into account the Zeeman splitting and the symmetry breaking by the magnetic field. The Hanle effect was discussed with a Quantum Mechanics approach first in Omont et al. (1973), then by Bommier (1997b), who gave an explicit expression of the redistribution matrix in the presence of a magnetic field for a two-level atom with unpolarized ground level. A weak field limit yields the Hanle redistribution matrix. The latter is first constructed in the magnetic reference frame, i.e, with the quantization axis in the direction of the magnetic field. A rotation has then to be applied to this magnetic field redistribution matrix to obtain a redistribution matrix that can be used in a reference frame attached to the medium. In Appendix K of this chapter, we present the redistribution matrix for the Hanle effect, derived in Bommier (1997b) for a two-level atom with unpolarized ground level. This matrix, denoted RH (ν, , ν ,  , B), depends on the frequency and direction of the incident and scattered beams and on the magnetic field vector B. In this section we consider only the frequency integrated version of RH (ν, , ν ,  , B), usually referred to as the Hanle phase matrix, denoted here by PH (,  , B). For lines formed with complete frequency redistribution, the redistribution matrix is then simply RH (ν, , ν ,  , B) = ϕ(ν)ϕ(ν )PH (,  , B).

(13.72)

We first present the Hanle phase matrix in the magnetic reference frame and use it to explain the rotation of the direction of polarization and the decrease of the linear polarization induced by the magnetic field. We then give the Hanle phase matrix in the atmospheric reference frame and discuss the effect of a turbulent magnetic field.

13.4.2.1 Magnetic Reference Frame In the magnetic reference frame, the Hanle phase matrix, depends only on the strength B of the magnetic field. Here denoted PH (,  , B), it is given by 

PH (,  , B) ≡



RH (ν, , ν ,  , B) dν dν ,

(13.73)

280

13 The Scattering of Polarized Radiation

where RH (ν, , ν ,  , B) is the redistribution matrix in the magnetic frame. The integration over frequencies in Eq. (K.1) yields [PH (,  , B)]ij =

 KQ

WK K (−1)Q TQK (i, )T−Q (j,  ). 1 + I + δ (K) + i QH (13.74)

For a non-cylindrically symmetric radiation field, both Stokes Q and U are needed to describe the linear polarization. Therefore the indices i and j take the values 0, 1, 2. The indices K and Q take the values K = 0, Q = 0, K = 2, Q = −2, . . . + 2. The parameters WK , I , and δ (K), which are independent of the magnetic field, are defined in Sect. 13.3.2. The directions  and  are reckoned with respect to the z-axis, which coincides with the direction of the magnetic field (see Fig. 13.4). The magnetic field appears in the term i QH , where H is the Hanle factor introduced in Eq. (13.71). In the calculation of the scattered radiation, this imaginary term is always associated to its complex conjugate, so that the Stokes parameters remain real quantities. The terms i QH , with Q = 0, are responsible for a rotation of the polarization direction and for a decrease in the polarization rate. To understand the origin of these effects, we rewrite the first factor as K 1 1 1 K −i αQ = cos αQ e , 1 + I + δ (K) 1 + i QHK

1 + I + δ (K)

(13.75)

where HK =

H , 1 + I + δ (K)

(13.76)

and K tan αQ ≡ QHK .

(13.77)

2 are often referred to as the Hanle mixing angles (Stenflo 1994, p. 91). The angles αQ K , K = 2, are responsible for the decrease of the polarization rate The factors cos αQ

and the phase factors e−i αQ for the rotation of the direction of the polarization. In the classical oscillator model described in the Appendix L of this chapter, term i QH appears as a phase factor in the electric field emitted by the oscillator. The Hanle phase matrix, in the magnetic reference frame, was first given by Stenflo (1978). K

13.4 The Hanle Effect

281

13.4.2.2 Atmospheric Reference Frame The redistribution matrix in the atmospheric reference frame, PH (,  , B) depends also on the direction of the magnetic field. As explained in e.g., LL04, Chapter 5, it can be obtained by applying a rotation R(0, −θB , −χB ) to the magnetic frame redistribution matrix, where θB and χB are the polar angles of the magnetic field in the atmospheric reference frame (see Fig. 13.4). Applied to Eq. (13.74), this rotation leads to [PH (,  , B)]ij =

 KQ

 WK K K Q K

T (i, ) NQQ

(B)(−1) T−Q (j,  ). Q 1 + I + δ (K)

Q

(13.78) K (), which contains the dependence on the magnetic field strength The factor NQQ

and direction, may be written as

K i (Q −Q)χB NQQ

(B) = e



K K dQQ

(θB )dQ

Q (−θB )

Q

1 , 1 + i Q

HK

(13.79)

K (θ ) are the reduced rotation where HK is defined in Eq. (13.76). The matrices dQQ

B matrices. They can be found in LL04 (Table 2.1, p. 57) and in any book on K (B) reduces to angular momentum theory. When the magnetic field is zero, NQQ

the Kronecker symbol δQQ . One recovers the Rayleigh phase matrix written in Eq. (13.32). K (B) must be replaced In the presence of a turbulent magnetic field, the term NQQ

by its average over the magnetic field distribution. For an isotropic distribution, or cylindrically symmetric magnetic field distribution, with a constant magnetic field strength B, this average may be written as K

NQQ

(B) = μK (B)δQQ .

(13.80)

For an isotropic distribution (LL04, p. 215), μ0 (B) = 1,

μ2 (B) =

1 (1 + 2 cos2 α1 + 2 cos2 α2 ). 5

(13.81)

The average of Eq. (13.78) over the magnetic field distribution reduces to the resonance scattering phase matrix in Eq. (13.64), except for the multiplication by a factor μK (B) of the terms in the right-hand side. Smaller than unity, μ2 (B) contains the depolarizing effect of a turbulent magnetic field. As pointed out in Stenflo (1994, p. 229), its value depends rather weakly on the angular distribution of the magnetic field. In the solar atmosphere, the mean intensity of a turbulent magnetic field may be obtained by comparing observed polarization rates with numerical predictions

282

13 The Scattering of Polarized Radiation

of resonance polarization rates, calculated with a given atmospheric model. The values obtained by this technique depend on the model and also on the assumed magnetic field angular distribution (Frisch et al. 2009; Anusha et al. 2010). Special observational techniques have been developed to avoid the dependence on the atmospheric model (Stenflo et al. 1998; Kleint et al. 2011). In Chaps. 15 and 18, we establish some exact results for the Hanle effect with the redistribution matrix RH (ν, , ν ,  , B) = ϕ(ν)ϕ(ν )PH (,  , B),

(13.82)

where the elements of PH (,  , B) are those given in Eq. (13.78). The frequency redistribution in Eq. (13.82) is simply complete frequency redistribution. This approximation is justified for weak lines without extended wings, such as the Sr I 4607 Å line. In-depth analysis of the linear polarization of this line have been performed to determine the small scale turbulent magnetic field in the solar photosphere (Faurobert-Scholl 1993; Faurobert-Scholl et al. 1995; Trujillo Bueno et al. 2004; Bommier et al. 2005). They all point to a turbulent field around fifty Gauss. For a two-level atom with unpolarized ground level, the frequency dependence of the Hanle redistribution matrix, in the atomic rest frame, contains a coherent frequency redistribution term and an incoherent one (see Appendix K in this chapter). In this respect, it is similar to that of resonance scattering (see Appendix J in this chapter). There is however a major difference between the Hanle effect and resonance scattering. For the Hanle effect, the frequency redistribution cannot be factored out of the redistribution matrix, as it can be for resonance polarization. The reason for this difference is explained in Appendix K of this chapter.

Appendix J: Spectral Details of Resonance Scattering For a two-level atom with an unpolarized and infinitely thin lower level, the redistribution matrix in the atomic rest frame may be written as Ra (ν, , ν ,  ) =



K  WK RK a (ν, ν )P (,  ),

(J.1)



(K) − α) ϕa (ν )ϕa (ν). RK a (ν, ν ) = α δ(ν − ν )ϕa (ν) + (β

(J.2)

K

where

The coefficients α and (β (K) −α) are the branching ratios for the frequency coherent and incoherent terms. The variables ν and  are the frequency and direction of the scattered beam. The prime variables are those of the incident beam. The first term corresponds to coherent scattering and the second one to complete frequency

Appendix J: Spectral Details of Resonance Scattering

283

redistribution. Here ϕa (ν) is the line absorption profile in the atomic rest frame. It is defined in Eq. (J.4). The elements of the matrix PK (,  ) are [PK (,  )]ij =



K (−1)Q TQK (i, )T−Q (j,  ),

i, j = 0, . . . , 3.

(J.3)

Q

The decomposition of the Rayleigh phase matrix in multipole components PK (,  ) is introduced in Eq. (13.38). The value K = 0 corresponds to the unpolarized scattering problem. Linear polarization is described with K = 0 and K = 2. The constant WK , which depends on the angular momentum of the upper and lower level of the transition, is defined in Sect. 13.3.2. For a transition Jl → Ju → Jl with Jl = 0 and Ju = 1 (called a normal Zeeman triplet), WK = 1 for K = 0 and K = 2. In Bommier (1997a), Eq. (J.1) is obtained by an expansion of the Schrödinger equation for the density matrix in terms of the interaction potential between the atom and the radiation, both being treated with Quantum Mechanics. Odd order terms do not contribute to the solution. The second order term yields the incoherent term proportional to β (K) , whereas the fourth order term yields two terms proportional to α, namely the coherent one and an incoherent one. Higher order terms can be included in the second and fourth order terms. The coherent term corresponds to photons that are reemitted before the upper level could be perturbed by elastic collisions, whereas the incoherent term takes into account the perturbations by elastic collisions. The coherent and the incoherent scattering terms can be interpreted in terms of a classical harmonic oscillator model proposed by Bommier and Stenflo (1999) and presented in Appendix L of this chapter. The profile ϕa (ν) is normalized to unity when ν is integrated from −∞ to +∞. Therefore, when Eq. (J.1) is integrated over frequency, the contribution of the fourth order term disappears since the two terms proportional to α cancel each other. One recovers the phase matrix PRS (,  ) given in Eq. (13.64). The line absorption profile ϕa (ν) may be written as ϕa (ν) =

 γl 1  γl φ (ν) + [φ γl (ν)]∗ = , 2 2 (γl /2) + (2π(ν0 − ν))2

(J.4)

where φ γl (ν), the complex absorption profile, is defined by φ γl (ν) ≡

2 . 2i π(ν0 − ν) + γl /2

(J.5)

Here ν0 is the line center frequency and γl = R + I + E ,

(J.6)

284

13 The Scattering of Polarized Radiation

is the line broadening coefficient. The coefficients α and β (K) are defined by α≡ β (K) ≡

R 1 = , R + I + E 1 + I + E

(J.7)

R 1 = . R + I + D (K) 1 + I + δ (K)

(J.8)

The parameters R and I are deexcitation rates of the upper level by radiative transitions and inelastic collisions, respectively. The effects of elastic collisions are described by the parameters E and D (K) . They are zero when their is no elastic collisions. In this case α = β (K) and therefore the frequency redistribution reduces to the coherent term. The parameters E and D (K) play different roles. As pointed out in Sect. 13.3.2, D (K) is a rate of destruction by elastic collisions of the K (J ). The total population being independent of elastic density matrix elements ρQ u collisions, one has D (0) = 0. The coefficient D (2) being the rate of destruction of 2 (J ), it participates in the destruction of the linear polarization. The the alignment ρQ u (2) coefficient D contributes to the width of the upper level, given by R + I + D (2) . The coefficient E accounts for two of the effects induced by elastic collisions: broadening of levels and interferences between levels. There is no simple relation between D (2) and E , but E can be expected to be larger than D (2) . The different roles played by D (2) and E and the role of E have been derived from Quantum Mechanics calculations (Omont et al. 1972; Sahal-Bréchot and Bommier 2019, and references therein) and can be physically understood with a classical analogue of elastic collisions developed by Stenflo (1994). A usual approximation is to assume that D (2) = E /2 (see Appendix L in this chapter). The redistribution matrix Ra (ν, , ν ,  ) in Eq. (J.1) is frequently written in terms of the Rayleigh and isotropic phase matrices PR (,  ) and Pis (see e.g., Faurobert-Scholl et al. 1995). Using P0 (,  ) = Pis ,

P2 (,  ) = PR (,  ) − Pis ,

(J.9)

and Eq. (J.2), the redistribution matrix for a normal Zeeman triplet (WK = 1) may be written as, Ra (ν, , ν ,  ) = [α δ(ν − ν )ϕa (ν) + (β (2) − α)ϕa (ν)ϕa (ν )]PR (,  ) + (β (0) − β (2))ϕa (ν)ϕa (ν )Pis .

(J.10)

When the summation over K in Eq. (J.1) is limited to the term with K = 0, one recovers the partial redistribution function for the scalar case. Introducing c = (1 + I )(1 + I + E )−1 , it can be written R0a (ν, ν ) =

1 [c δ(ν − ν )ϕa (ν) + (1 − c) ϕa (ν )ϕa (ν)]. 1 + I

(J.11)

Appendix K: Spectral Details of the Hanle Effect

285

This expression is for an atom a rest. In the absence of elastic collisions, c = 1. The redistribution function contains only the coherent term. The ratio 1/(1 + I ) is the fraction of the absorbed photons that are being reemitted. In a laboratory experiment or in a stellar atmosphere, atoms are in general not at rest and the Doppler effect due to thermal velocities has to be taken into account. To calculate the Doppler effect, also referred to as Doppler redistribution, one generally assumes that the atomic velocity distribution is a Maxwellian (see however Bommier (2016a,b), where the velocity distribution depends on the scattering process). The Doppler effect modifies only the frequency dependent terms

RK a (ν, ν ). The so-called laboratory or atmospheric frame redistribution matrix is then given by R(ν, , ν,  ) =



WK RK (ν, , ν ,  )PK (,  ),

(J.12)

K

where RK (ν, , ν ,  ) = α rII (ν, , ν ,  ) + (β (K) − α) rIII (ν, , ν ,  ).

(J.13)

Here rII (ν, , ν ,  ) and rIII (ν, , ν ,  ) are the scalar redistribution functions in the laboratory frame, first written by Hummer (1962). The frequency dependent term RK (ν, , ν ,  ) depends now also on the directions  and  . The Doppler effect induces a coupling between the polarization phase matrix and the frequency redistribution functions. This coupling disappears when the redistribution functions are replaced by their angle-averages r¯II (ν, ν ) and r¯III (ν, ν ), also given in Hummer (1962). This approximation is nearly always justified for the calculation of the radiation field intensity (Stokes I ). For the polarization profiles, the validity of the approximation depends on the total optical thickness of the line. Justified for strong lines, because of the averaging effect of a very large number of scatterings, it can lead to incorrect predictions for weak lines (see e.g., Sampoorna et al. 2011 and references therein).

Appendix K: Spectral Details of the Hanle Effect The first explicit expression of the frequency dependent redistribution matrix for the Hanle effect has been established by Bommier (1997b). It is valid for a two level atom with an unpolarized and infinitely thin lower level. The construction method is a generalization of the perturbation expansion method developed for resonance scattering in Bommier (1997a). The expression given in Bommier (1997b) holds for magnetic fields of any strength and therefore in particular for the Hanle effect, which covers a regime of magnetic field strength from the milligauss to a few tens of Gauss. We consider the redistribution matrix first in the magnetic field reference frame (quantization axis along the magnetic field), then in the atmospheric reference frame.

286

13 The Scattering of Polarized Radiation

K.1 Magnetic Field Reference Frame The geometry of the scattering process is shown in Fig. 13.4. The direction of the magnetic field is along the axis Oz. The elements of the redistribution matrix depend only on the magnetic field strength, denoted B, and have a structure similar to those of the redistribution matrix for resonance polarization. For an atom at rest, they may be written as 



Q K

TQK (i, )RKK [RH (ν, , ν ,  , B)]ij = Q (ν, ν , B)(−1) T−Q (j,  ), KK Q

(K.1) where



RKK Q (ν, ν , B) ≡





(KQ) KK

− α (Q) ]KK α (Q) wJl ,Ju δ(ν − ν )KK Q (ν) + [β Q (ν)Q (ν ). (K)

(K.2)

The magnetic field strength enters in the coefficients α (Q) , β (KQ), and in the

generalized profile KK defined below. Since we consider linear polarization only, Q the indices K and Q take the values K = 0, K = 2, −K ≤ Q ≤ +K. The index K takes the values 0,1,2. The coefficients α (Q) and β (K,Q) are given by α (Q) = β (KQ) =

R , R + I + E + 2i πνL Q

(K.3)

R . R + I + D (K) + 2i πνL Q

(K.4)

The effect of the magnetic field is contained in the term 2i πνL Q, where νL is the Larmor frequency, which must be multiplied by the Landé factor of the upper level. Here, it is taken equal to one. For Q = 0, one recovers the parameters α and β (K) of the resonance scattering given in Eqs. (J.7) and (J.8). The factor 2i πνL Q produces, among other effects, a rotation of the polarization plane. Its origin is explained with the classical oscillator model in the Appendix L of this chapter. For the Hanle effect, the magnetic field couples the frequency dependent terms and the direction dependent terms, since they both depend on the indices K and Q. For resonance polarization, as can be observed on Eq. (J.1) there is no such coupling.

The frequency dependence is contained in the functions KK Q (ν), a simpli

fied notation for the generalized profiles K,K (Jl , Ju , ν) introduced in Landi Q Degl’Innocenti et al. (1991) (see also LL04, Eq. (A13.1), p. 525). These general-

K.1 Magnetic Field Reference Frame

287

ized profiles are linear combinations of the Zeeman components absorption and dispersion profiles. They have the form

(Jl , Ju , ν) = K,K Q

 qq

C(Jl , Ju , K, K , Q, q, q ) φqql (ν). γ

(K.5)

γ

The frequency dependent function φqql (ν) is defined by γ

φqql (ν) ≡

 1  γl γ φq (ν) + [φq l (ν)]∗ , 2

q, q = 0, ±1,

(K.6)

γ

where φq l (ν), the complex absorption profile, is given by4 γ

φq l (ν) ≡

2 . 2i π(ν0 + qνL − ν) + γl /2

(K.7)

For νL = 0, one recovers the absorption profile defined in Eq. (J.5). The line broadening parameter γl is defined by γl = R + I + E , as in Eq. (J.6). The coefficient C(Jl , Ju , K, K , Q, q, q ) can be calculated explicitly. It involves the product of several 3 − j symbols and a summation over the magnetic sublevels of the lower and upper levels given. The index K can take the values 0, 1, 2, only. The indices q, q = 0, ±1 refer to a reference coordinate system defined by its spherical unit vectors, with e0 align with the direction of the magnetic field (see, e.g., Eqs. (13.12), (L.5), and (L.6)). Explicit expressions of the generalized profiles for a Zeeman triplet are given in LL04, p. 833. The properties of the generalized profiles are described in LL04, Appendix A13. Here we want to mention two important features. In the limit of zero magnetic field

(K)

lim K,K (Jl , Ju , ν) = δKK wJl ,Ju ϕa (ν), Q

(K.8)

νL →0

where ϕa (ν) is the absorption profile in the atomic reference frame defined in (K) Eq. (J.4). The coefficient wJl ,Ju depends on the angular momentum of the lower values can be found in LL04 and upper level of the transition. A table of wJ(K) l ,Ju (K)

(Table 10.1, p. 515). The square of wJl ,Ju is the depolarization coefficient WK . For (K)

Jl = 0, Ju = 1 (normal Zeeman triplet), wJl ,Ju = (−1)K , hence WK = 1 for K = 0, 2.

4

γ

a Other definitions of the complex profile can be found in the literature, such as φ−q (ν) or γa ∗ [φq (ν)] . They all lead to the same redistribution matrix.

288

13 The Scattering of Polarized Radiation

The integral over frequency is: 

+∞

−∞



K,K (Jl , Ju , ν) dν = δKK wJ(K) . Q l ,Ju

(K.9)

with δKK the Kronecker symbol. The integration of Eq. (K.1) over the frequencies ν and ν yields the Hanle phase matrix written in Eq. (13.74), namely [PH (,  , B)]ij =



K WK β (KQ)(−1)Q TQK (i, )T−Q (j, ).

(K.10)

KQ

When the magnetic field is zero, one recovers the resonance polarization phase matrix in Eq. (13.64). An important property of the Hanle effect, listed in Sect. 13.4, is that it disappears in the spectral line wings. This means that away from the line core, one recovers the resonance polarization. The wings, observable only for strong lines, are essentially



formed by coherent scattering described by the term α (Q) KK Q (ν)δ(ν − ν ), with

α (Q) given by Eq. (K.3) and KK Q (ν) by Eq. (K.5). The disappearance of the Hanle

effect can be explained by considering the product α (Q) KK Q (ν) for νo − ν  νL . γl First one remarks that the profile φqq (ν), defined in Eqs. (K.6) and (K.7), can also be written as γ

φqql (ν) =

γl + i (q − q )2πνL . [2i π(ν0 + qνL − ν) + γl /2][−2i π(ν0 + q νL − ν) + γl /2]

(K.11)

In the limit νo − ν  νL , γ

φqql (ν)  ϕa (ν)

1 [γl + i (q − q )2πνL ]. γl

(K.12)

As pointed out in LL04 (p. 526), the summation over q and q in Eq. (K.5) can be carried out explicitly when the width of the lower level is negligible compared to that of the upper level, in particular when the lower level is infinitely sharp, as assumed here. The summation leads to



KK Q (ν)  (1 + i QH )ϕa (ν)δKK ,

(K.13)

where H = 2πνL /γl and Q = q − q . Rewriting the parameter α (Q) as α (Q) =

1 R , γl 1 + i QH

(K.14)

we can observe that the magnetic field dependent term (1 + i QH ) disappears in

the product α (Q) KK Q (ν). We recover the coherent scattering term of the resonance polarization (see Eq. (J.1)).

K.3 Approximations for the Redistribution Matrix

289

K.2 Atmospheric Reference Frame The redistribution matrix in the atmospheric reference frame is obtained by application of a rotation R(0, −θB , χB ), to the expression given in Eq. (K.1) (see Fig. 5.9 in LL04, p. 185). The angles θB and χB are the polar angles of the magnetic field. After the rotation, the elements of the redistribution matrix may be written as [RH (ν, , ν ,  , B)]ij =



TQK (i, )

KK Q

 Q





KK

Q K

NQQ

(ν, ν , B)(−1) T−Q (j,  ),

(K.15) where



KK

i (Q−Q )χB NQQ

(ν, ν , B) = e





K K KK

dQQ

(θB )dQ

Q (−θB )RQ

(ν, ν , B).

Q

(K.16)



The frequency dependent term RKK Q

(ν, ν , B) is given in Eq. (K.2) for an atom at rest. Resonance polarization is recovered when the magnetic field is zero. Indeed, KK (ν, ν , B) = δ K

K

when B = 0, one has NQQ

QQ δKK Ra (ν, ν ), where Ra (ν, ν ) is the frequency dependent term in Eq. (J.1). When the Doppler effect is taken into account, the frequency redistribution term



RKK Q

(ν, ν , B) depends also on the direction of the incident and scattered beams. But, even when this term is replaced by its angle-averaged version, the frequency redistribution cannot be factored out of the redistribution matrix. Radiative transfer calculations taking into account the Hanle effect are therefore extremely demanding in both time and memory resources, unless some approximation is being made.

K.3 Approximations for the Redistribution Matrix Several approximations are proposed in Bommier (1997b) to treat the resonance polarization in the presence of a magnetic field. They have been very systematically used for calculating the Hanle effect. They are based on approximations of the

Doppler broadened generalized profiles KK Q (ν). A first remark is that only the diagonal terms KK Q (ν) contribute in any significant way to the summation in Eqs. (K.1) and (K.15), when the magnetic field is weak. Then a detailed asymptotic analysis is made of the frequency dependence of the broadened generalized profiles

KK Q (ν). For each coherent and incoherent term, the frequency domain (ν, ν ) is decomposed into different regions, the shape of which depends also on the angle between the directions  and  . In a given domain, the redistribution matrix can then be written as a product of a scalar partial frequency redistribution function, rII (ν, , ν ,  ) or rIII (ν, , ν ,  ), times a linear combination of Hanle

290

13 The Scattering of Polarized Radiation

phase matrices with Hanle coefficients H depending on the frequency domain. An angle-averaged version of this decomposition has also been introduced in Bommier (1997b). The different regions depend then only on ν and ν and the redistribution functions are replaced by their angle-averaged versions. The approximations introduced in Bommier (1997b) were first employed in Nagendra et al. (1998). Tests on their validity were performed in Nagendra and Sampoorna (2011), using for comparison a scattering expansion method. Additional references describing the use of these approximations can be found in Nagendra and Sampoorna (2009) and Nagendra (2014).

Appendix L: The Hanle Effect with the Classical Harmonic Oscillator Model The classical harmonic oscillator model assumes that an atom can be represented by an electron (with a negative electric charge −e0 ), which performs a damped harmonic oscillation around a positive charge (the nucleus). This model, introduced by Henrik A. Lorentz to interpret the Zeeman effect has been abundantly described (see e.g. Mitchell and Zemansky 1934; Bommier 1987; Landi Degl’Innocenti 1992; Stenflo 1994; Trujillo Bueno 2001, LL04; Sampoorna 2008). We show here how it can be applied to the Hanle effect. In Sect. L.1, we recall the main features of this model and in Sect. L.2 use it, in the context of the Hanle effect, to explain the rotation of the direction of polarization and the decrease of the polarization rate. Then in Sect. L.3, we present the method developed in Bommier and Stenflo (1999) to explain with the classical oscillator model the mixture of coherent and incoherent frequency redistribution produced by resonance scattering.

L.1 The Harmonic Oscillator Model In the presence of a constant magnetic field B and of an oscillating electric field E e−2i πνt , the position x(t) of the electron satisfies a second order differential equation, which may be written as dx eo dx e0 d 2x ( × B) + γ + 4π2 ν02 x = − Ee−2i πνt , + 2 dt mc dt dt m

(L.1)

where ν0 is the frequency of the oscillator, γ a damping constant, e0 and m the charge and mass of the electron, and c the speed of light. The damping coefficient γ corresponds to the radiative deexcitation rate R . The oscillating electron emits an electric field, which, far from the source, can be treated as a plane wave. The laws of

L.1 The Harmonic Oscillator Model

291

classical electrodynamics tell that at a distance r from the source, it may be written as E(r, , t) =

k 2 ei kr p ⊥ (t), r

(L.2)

where p ⊥ (t) is the component of the electric dipole p = −e0 x(t) in the plane perpendicular to the direction . The solution of Eq. (L.1) allows one to calculate the elements Ei (t)Ej∗ (t) of the polarization tensor, hence the Stokes parameters of the radiation emitted by a collection of such oscillators. We note here that the factor ei kr disappears when one performs the product Ei (t)Ej∗ (t). The factor 1/r disappears in the definition of the radiation emitted per solid angle and per unit time (see LL04, Eq. 5.75). When the magnetic field is zero, there is no preferred direction in medium. One can choose any orthogonal reference system ex , ey and ez and decompose the vector x(t) as x(t) = xx (t)ex + xy (t)ey + xz (t)ez ,

(L.3)

and similarly for E. It is easy to verify that Eq. (L.1) can be rewritten as 3 independent scalar equations, each components of x(t), let us say xx (t), satisfying dxx e0 d 2 xx + 4π2ν02 xx = − Ex e−2πνt , +γ dt 2 dt m

(L.4)

where Ex is the component of E in the ex direction. The motion of the electron can thus be represented by three independent linear harmonic oscillators vibrating along the axes ex , ey and ez . Let us assume that the electron is excited by an incident electric field travelling in the direction ez (see Fig. 13.3). Its components Ex and Ey perpendicular to ez will excite respectively the linear oscillators along ex and ey . For example, when the incident beam is unpolarized and the observation made in the direction ey , i.e. making an angle of 90◦ with the incident beam, the observed radiation is linearly polarized in the direction ex perpendicular to the scattering plane. When the incident and scattered beams make an angle , one recovers the scattering law written in Eq. (13.18). In the presence of a magnetic field, the situation is not so simple. When for example the vector ez is chosen along the magnetic field direction, the equations for xx (t) and xy (t) are coupled by the presence of the term (dx/dt) × B in Eq. (L.1). The introduction of a new reference system allows one to describe the motion of the electron with three uncoupled oscillators. The new unit vectors are: e0 = ez ,

(L.5)

292

13 The Scattering of Polarized Radiation

Fig. L.1 The spherical unit vectors e0 , e+1 , and e+1 . The vector e0 is aligned with the magnetic field vector B

e+1

ex e−1 e0

B

ez

ey

in the direction of the magnetic field (see Fig. L.1) and two spherical unit vectors defined as in Eq. (13.12), namely √ e±1 = (∓ex + i ey )/ 2.

(L.6)

The unit vectors satisfy eq × e0 = −i qeq ,

q = 0, ±1.

(L.7)

The expansion of x(t) on the basis of the vectors eq takes the form  x(t) = xq (t)eq , q = 0, ±1.

(L.8)

q

Equation (L.1) shows that the three oscillators xq (t) are solutions of dxq dxq d 2 xq e0 +γ +4π2ν02 xq = − Eq e−2i πνt , −q4πi νL 2 dt dt m dt

q = 0, ±1,

(L.9)

where Eq are the components of the electric vector E in the new basis and νL = eo B/(4πmc) is the Larmor frequency. With this decomposition, the motion of the electron is represented by three uncoupled oscillators, a linear one, in the direction of the magnetic field, and two counter-rotating circular ones in the plane perpendicular to the magnetic field direction. The frequency νq of these oscillators can be determined by looking for solutions of the homogeneous version of Eq. (L.9) having the form xq (t) = Aq e−2i πνq t ,

(L.10)

where Aq is a constant. The frequency νq is solution of a quadratic equation. For spectral lines in the optical wavelength range and magnetic field strengths encountered in stellar atmospheres, ν0  γ and ν0  νL , hence νq  ±ν0 − qνL − i

γ . 4π

(L.11)

L.2 Polarization Direction and Polarization Rate

293

To be in agreement with the definition given in Eq. (13.1), it is the plus sign in front of ν0 , which should be kept. One thus obtains xq (t) = Aq e−γ t /2e−2i π(ν0 −qνL )t .

(L.12)

The two rotating oscillators corresponding to q = ±1 have thus different frequencies, ν0 +νL and ν0 −νL . In the absence of magnetic field, the two circular oscillators are vibrating at the same frequency ν0 with a phase relationship constant with time. In the presence of a magnetic field, the circular oscillators are vibrating at different frequencies, hence the initial phase relationship is lost little by little during the emitting process. In the density matrix formalism, the effect of the magnetic field is to partially remove the coherences. Many features of resonance polarization, of the Zeeman effect, and of the Hanle effect, can be derived from the properties of x(t), for example the refractive index in a magnetized gas, hence the propagation matrix for the Zeeman effect.

L.2 Polarization Direction and Polarization Rate In Sect. 13.4 it is pointed out that a weak magnetic field will in general decrease the polarization created by resonance scattering and produce a rotation of the direction of polarization. These effects can be explained with the oscillator model. We consider the simple geometry of Fig. L.1, namely a beam of radiation propagating in the direction ez aligned with the direction of the magnetic field. According to Eqs. (L.6) and (L.8), the motion of the electron in the plane ex , ey may be written as  1  x(t) = √ [−x+1 (t) + x−1 (t)]ex + i [x+1 (t) + x−1 (t)]ey . 2

(L.13)

We assume that at time t = 0 the radiation is linearly polarized in the direction ex . The combination of Eqs. (L.12) and (L.13) leads then to √ xx (t) = a 2e−γ t /2 cos(2πνL t)[e−2i πν0 t ], √ xy (t) = a 2e−γ t /2 sin(2πνL t)[e−2i πν0 t ],

(L.14)

where a is a constant. Because of the factor e−γ t /2, the amplitude of the oscillation tends to zero as e−γ t . The trajectory of the electron describes a rosette (Mitchell and Zemansky 1934; Stenflo 1994; LL04, p. 219). Its shape depends on the value of the parameter H ≡

2πνL . γ

(L.15)

294

13 The Scattering of Polarized Radiation

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-0.5

0

1

0.5

-1

-0.5

-1

0

1

0.5

Fig. L.2 Motion of the damped classical oscillator in the plane perpendicular to the direction of the magnetic field. The parametric plot shows the time evolution of the component xy (t) versus the component xx (t), defined in Eq. (L.14). The left panel shows a rosette with a Hanle parameter H = 1 and the right one a daisy with H = 9. The curves have been calculated with γ = 1, 2πν0 = 50, and a maximum value t = 2.5 for the rosette and t = 1.5 for the daisy

We show in Fig. L.2 the shape of the rosette for H = 1 and H = 9. For H = 0, the tip of the electric field oscillates along the horizontal axis. For H = 1, the rosette is, as one says, incomplete. The time-averaged direction is now tilted with respect to the horizontal axis. For H = 9, the movement of the electron looks like a daisy and the polarization is destroyed. We stress that these figures hold for the special geometry of Fig. L.1, in which the magnetic field is align with the direction of observation. We now derive from Eq. (L.14) the Stokes parameters of the radiation field emitted in the direction ez , using the definition given in Eq. (13.3). In the plane (ex , ey ) perpendicular to ez , the components of the electric field are proportional to the components of p ⊥ (t) (see Eq. (L.2)), hence to xx (t) and xy (t). Setting E1 (t) = e−γ t /2 cos 2πνL t and E2 (t) = e−γ t /2 sin 2πνL t in Eq. (13.3), we obtain I ∝ e−γ t , Q ∝ e−γ t cos 4πνL t, U ∝ e−γ t sin 4πνL t, V = 0,

(L.16)

where  stands for a time average. Integrating over time from 0 to ∞, we find I ∝ 1,

Q∝

1 , 1 + 4H 2

U∝

2H , 1 + 4H 2

(L.17)

with H given in Eq. (L.15). We observe that the polarization tends to zero as H → ∞, that is when the incomplete rosette becomes a daisy.

L.3 Frequency Redistribution

295

For the linear polarization rate p, defined by 

Q2 + U 2 , I

(L.18)

1 p= √ . 1 + 4H 2

(L.19)

p= Equation (L.16) leads to

The √polarization rate, chosen to be equal to one at time t = 0, is reduced by a factor 1/ 1 + 4H 2 under the action of the magnetic field. The polarization direction , can be defined, modulo  π/2, by tan 2 = U/Q (see Eq. (13.7)), or modulo π, by cos 2 = Q/ Q2 + U 2 , sin 2 =  2 2 U/ Q + U ) (see Eq. (13.8)). By construction,  = 0 at t = 0. Equation (L.17) leads to tan 2 = 2H,

1 cos 2 = √ , 1 + 4H 2

sin 2 = √

2H 1 + 4H 2

.

(L.20)

These relations explain the rotation of the polarization under the action of the magnetic field. When H is small, the rotation is of order H . We note that a rotation of 180◦ of the magnetic field direction amounts to replace νL by −νL in Eq. (L.14), hence to change the sign of U . This amounts also to make a symmetry of the rosette with respect to the axis ex (see Fig. L.2). When summing two such rosettes,  becomes zero but Q is not changed. This behavior is radically different from that of the Zeeman effect. In this section, we have assumed for simplicity that the magnetic field is aligned with the direction of the observation. Additional properties of the Hanle effect can be analyzed with a harmonic oscillator model when this assumption is raised (see e.g. Landi Degl’Innocenti 1992; Trujillo Bueno 2001; LL04, Chapter 5).

L.3 Frequency Redistribution The redistribution matrices for resonance scattering and the Hanle effect, presented in Appendices J and K show that the redistribution matrix can be written as the sum of two terms, describing respectively a frequency coherent scattering and a frequency incoherent one. These two types of frequency redistribution having no direct relation with the polarization, the arguments presented here apply also when the polarization is ignored.

296

13 The Scattering of Polarized Radiation

As pointed out in Bommier and Stenflo (1999), it is possible to explain the coherent and incoherent terms with the classical harmonic oscillator equation. For an ordinary Zeeman triplet (Jl = 0, Ju = 1), the classical oscillator model is able to reproduce the results of a full Quantum Mechanics calculation (Sampoorna et al. 2007a,b). For an arbitrary transition Jl → Ju → Jl , the classical model can also reproduce some of the full Quantum Mechanics calculations, provided the lower level is unpolarized (Sampoorna 2011). We present here the main of the classical approach. Details can be found in the above references. Equation (L.9) is an inhomogeneous second order differential equation with constant coefficients. Its solution is the sum of a solution of the homogeneous equation and of a particular solution of the inhomogeneous equation. The latter can be constructed by the method of variation of parameters (see, e.g., Bender and Orszag 1978). The full solution may be written as  xq (t) = Cq c1 e

−2i πνq+ t

+ c2 e

−2i πνq− t



e−2i πν t + 2i π



1 1 + −

ν − νq ν − νq−

 . (L.21)

In the right-hand side, the two first terms are solution of the homogeneous equation and c1 and c2 are undetermined constants. These solutions oscillate at the frequencies νq± ≡ ±ν0 − νL q − i

γ . 4π

(L.22)

The imaginary term in Eq. (L.22) shows that they tend to zero as e−γ t /2 as t → ∞. The third term, particular solution of the inhomogeneous equation, takes into account the inhomogeneous term −(e0 /m)Eq e−2i πνt . The particular solution is locked at the frequency of the incident field, henceforth denoted ν . The multiplicative constant Cq is given by Cq = −

e0 Eq . m

(L.23)

Following Bommier and Stenflo (1999), we refer to the damped solution as the transitory solution, since it tends to zero as t → ∞. To be in agreement with Eq. (13.1), we keep only the damped oscillation with frequency νq+ = ν0 − νL q − i γ /4π. The transitory solution has thus the form xqtrans(t) ∝ e−γ t /2 e2i πqνL t e−2i πν0 t .

(L.24)

It provides the incoherent frequency redistribution term of the redistribution matrix. In the oscillator model, the frequency of the incident field should be close to the frequency ν0 of the oscillator. The particular solution contains a resonant term corresponding to 1/(ν − νq+ ), and a non-resonant one, corresponding 1/(ν − νq− ),

L.3 Frequency Redistribution

297

which is of order 1/2ν0 , and can be neglected. The resonant term is referred to in Bommier and Stenflo (1999) as the stationary solution. It has the form

xqstat(t)

e−2i πν t ∝

. ν − νq+

(L.25)

This stationary solution, provides the frequency coherent part of the redistribution matrix. It is time to make a contact with the Quantum Mechanics results. The inelastic collisions deexcitation rate can be incorporated into this classical model by choosing for the damping parameter γ = R + I .

(L.26)

Elastic collisions are not so easily incorporated, since they modify the line broadening, described by the parameter γ , but also the polarization rate, which depends on the parameter D (K) (see the discussions in the Appendices J and K). Elastic collisions have to be introduced in a heuristic way, directly into the solution of the oscillator equation. We describe below a method suggested by Stenflo (1994, p. 211), employed in Bommier and Stenflo (1999). A somewhat different method is presented in LL04 (p. 220). They are both quite general and can be applied to polarized or unpolarized scattering problems. Random collisions will scramble the phases of the oscillators. The phase of the stationary solution, which is driven by the electromagnetic wave, is not affected, but that of the transitory solution is. At the time of a collision, there is an abrupt phase change. Without these collisions, there would be no incoherent term in the redistribution matrix, since the transitory solution would tend exponentially to zero. Random collisions are also destroying correlations between the stationary and the transitory solutions. The spectrum of the coherent and incoherent solutions can thus be calculated independently and then added to obtain the total spectrum of emitted radiation. The two spectrum are given by trans Eqq ˜qtrans(ν)[x˜qtrans (ν)]∗ ,

(ν) = x

stat ∗ Eqq ˜qstat(ν)[x˜qstat

(ν) = x

(ν)] ,

(L.27)

where x˜q (ν) is the Fourier transforms of xq (t). Mathematically, a collision corresponds to a truncation of the Fourier transform of xqtrans(t). Instead of extending it from −∞ to +∞, it is calculated over a finite interval [0, tc ] during which the transitory solution is unperturbed by a collision. Since the collisions occur randomly, the time tc can be assumed to obey a Poisson statistics with a probability density e−tc /τc /τc , where τc is the mean time between collisions. The spectrum is then obtained by averaging the tc -dependent product x˜qtrans(ν)[x˜qtrans (ν)]∗ over the Poisson distribution. The inverse of τc can serve to

298

13 The Scattering of Polarized Radiation

introduce a damping constant γc , which is defined by γc /2 = 1/τc . The averaging procedure leads to trans Eqq

(ν) ∝

1 2i π(q

− q )νL

+γ +

γc 2

γ +γ

φqq c (ν),

(L.28)

γ +γ

where φqq c (ν) is the generalized profile defined in Eq. (K.6). Concerning the stationary solution, Eq. (L.25) shows that the time dependence of xqstat(t) is locked to that of the incident field. Therefore the spectrum of the stationary solution is independent of q and q and has the form E stat(ν) ∝ δ(ν − ν ),

(L.29)

with ν the frequency of the incident electric field. We now consider the redistribution matrix itself. Equations (L.28) and (L.29) provide the spectrum of the radiation emitted at a frequency ν after absorption of a photon with a frequency ν . This spectrum must be multiplied by an absorption term describing the probability that a photon has been absorbed at a frequency ν . This absorption term has the form (Bommier and Stenflo 1999) γ +γc

abs

fqq

(ν ) ∝ R φq γ +γc

Using the expression of φq

abs

fqq

(ν ) ∝

γ +γc

(ν )[φq

(ν )]∗ .

(L.30)

(ν) given in Eq. (K.7), it may be rewritten as

4R γ +γ φq c (ν ). γ + 2i π(q − q)νL + γc

(L.31)

Up to now, we have not taken into account the fact that the incident radiation at the frequency ν has a direction  and the emitted radiation at the frequency ν a direction . Taking this geometry into account and multiplying the stationary and transitory spectra by the absorption term, the redistribution matrix for the classical oscillator may be written as (Bommier, 1999, unpublished):



[Rcl H (ν, , ν ,  , B)]ij =



x˜qq (ν, ν )Tqq (i, )[Tqq (j,  )]∗ ,

(L.32)

qq

where   bqq

R γ +γc

γ +γc

x˜qq (ν, ν ) = aqq

φ (ν) . φ (ν ) × δ(ν − ν ) + γqq + γc qq γqq + γc /2 qq (L.33)

L.3 Frequency Redistribution

299

To simplify the notation, we have introduced γqq ≡ γ + 2i π(q − q)νL = R + I + 2i π(q − q)νL .

(L.34)

In Eq. (L.33), aqq and bqq are constants, undetermined at this stage. The tensors Tqq (i, ) in Eq. (L.32) are defined in LL04 (Eq. (5.131)). They are written as Tqq (i, ) ≡

1 α,β

2

(σ¯ i )α,β Eq,q (β, α, ),

α, β = ±1; q, q = 0, ±1,

(L.35)

where (σ¯ i ) are the Pauli matrices relating the Stokes parameters to the polarization matrix J (see Eq. (13.15)) and Eq,q (α, β, ) = (cα )q [(cβ )q ]∗ . Here (cα )q and (cβ )q are the director cosines of the two spherical components of electric field with respect to a reference coordinate system, defined by its spherical unit vectors (LL04, p. 196). The indices α and β correspond to the two components of the electric field perpendicular to the direction  and the indices q, q to the reference coordinate system. Explicit expressions for the tensors Tqq (i, ) can be found in LL04, Table 5.3, p. 206. We note that the tensors Tqq (i, ) are the starting point for the construction of the spherical irreducible tensors TQK (i, ) (see LL04, Chapter 5, p. 208). In a rotation of the reference system, the transformation law for Tqq (i, ) involves two rotation operators (see, e.g. LL04 p. 207). The tensors TQK (i, ) are defined in such a way that the transformation law can be described by a single rotation operator. K (J, J ) in a rotation of the reference system (see This same operator transforms ρQ Eq. (13.58)). Moreover, the irreducible tensor representation makes it possible to take into account the isotropy of the collision process (LL04, p. 342) and hence to explicitly describe elastic collisions with the parameter D (K) .



The comparison of x˜qq (ν, ν ) with RKK Q (ν, ν , B), the frequency redistribution function for the Hanle effect given in Eq. (K.2), leads to the identifications γc = E ,

1 γc = D (2) , 2

(L.36)

and bqq = γc − γc /2. The treatment of elastic collisions in Bommier and Stenflo (1999) correctly predicts that the depolarization rate γc /2 is smaller than E . We stress again that no simple relation exists between E and D (2) . The identification of γc /2 with D (2) can also be verified by calculating the scattering phase matrix. It is obtained by integrating Eq. (L.32) over the frequencies γ +γ ν and ν . Taking into account the normalization to unity of φqq c (ν) and using bqq = γc − γc /2, the integration over frequency leads to  [Pcl H (,  , B)]ij =  R Tqq (i, )[Tqq (j,  )]∗ . aqq

 +  + 2i πν R I L (q − q ) + γc /2

qq

(L.37)

300

13 The Scattering of Polarized Radiation

This expression should reduce to the Rayleigh phase matrix, when I , γc and νL are set to zero. As shown in LL04 (p. 200), the Rayleigh phase matrix can also be written as  [PR (,  )]ij = 3 Tqq (i, )[Tqq (j,  )]∗ . (L.38) qq

Hence aqq = 3. The identification of γc /2 with D (2) can also be verified by comparing Eq. (L.37) with the phase matrix for the Hanle effect written in Eq. (13.74), namely [PH (,  , B)]ij =

 KQ

R K (−1)Q TQK (i, )T−Q (j, ). R + I + D (K) + 2i πνL Q (L.39)

Although it cannot replace a Quantum Mechanics approach, the classical oscillator model provides a simple interpretation for the coherent and incoherent frequency redistribution mechanisms at play in resonance scattering and the Hanle effect. It also shows how the presence of a magnetic field modifies the phase of the transitory solution by a term 2i πνL q, which then appears as an additional deexcitation coefficient 2i πνL (q − q ) in the line broadening coefficient and in the width of the upper level. In the Quantum Mechanics approach, the corresponding term, 2i πνLQ, K (J ) (see, e.g., Bommier is included in the statistical equilibrium equations for ρQ u 1997b, Eq. (7), LL04 (p. 284)).

References Alsina Ballester, E., Belluzzi, L., Trujillo Bueno, J.: 2021, Solving the paradox of the Solar Sodium D1 line polarization. Phys. Rev. Let. 127, 081101, (5 pp.) (2021) Anusha, L.S., Sampoorna, Frisch, H., Nagendra, K.N.: The Hanle effect as diagnostic tool for turbulent magnetic fields. In: Hasan, S.S., Rutten, R.J. (eds.) Magnetic coupling between the interior and the atmosphere of the Sun, Astrophys. & Space Sci. Proceedings, pp. 390–394 (2010) Anusha, L.S., Nagendra, K.N., Bianda, M., Stenflo, J.O., Holzreuter, R., Sampoorna, M., Frisch, H., Ramelli, R., Smitha, H.N.: Analysis of the forward-scattering Hanle effect in the Ca I 4227 Åline. Astrophys. J. 737, 95(17pp) (2011) Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York (1978) Bommier, V.: Détermination du vecteur champ magnétique et de la densité électronique des protubérances solaires par l’interprétation de l’effet Hanle et de la dépolarisation collisionnelle. Thèse de Doctorat d’ État ès Sciences Physques, Université de Paris VII (1987) Bommier, V.: Master equation applied to the redistribution of polarized radiation, in the weak radiation field limit. I. Zero magnetic field case. Astron. Astrophys. 328, 706–725 (1997a) Bommier, V.: Master equation applied to the redistribution of polarized radiation, in the weak radiation field limit. II. Arbitrary magnetic field case. Astron. Astrophys. 328, 726–751 (1997b)

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Chapter 14

Polarized Radiative Transfer Equations

The equations of radiative transfer for a field polarized by a scattering process were formulated by Chandrasekhar (1946) and Sobolev (1949) and are presented in several books by, e.g., Chandrasekhar (1960), Sobolev (1963), Stenflo (1994), Landi Degl’Innocenti and Landolfi (2004). In this chapter, we present a few linearly polarized radiative transfer equations describing monochromatic Rayleigh scattering, resonance polarization, and the Hanle effect. They are used, at least part of them, in the following chapters to construct exact and asymptotic solutions. As we have seen in Chap. 13, a polarized radiation field is described by a vector, constructed, for example, with the Stokes parameters. It can also be described by a matrix (see Chap. 15). For polarized radiative transfer, the scattering term, which contains the polarization phase matrix, depends on the position in the medium but also on the direction of the scattered beam (see Eq. (13.26)). We show in this chapter that it is possible to represent the radiation field in such a way that it satisfies a radiative transfer equation in which the scattering term is independent of the direction of the beam. These new radiative transfer equations can be constructed for Rayleigh scattering, resonance polarization of spectral lines, and even the Hanle effect, for one-dimensional or three dimensional media. They lead, under appropriate conditions (one-dimensional medium, complete frequency redistribution) to convolution integral equations similar to scalar convolutions equations encountered in Part I. We recall that they have the form.  S(τ ) = (1 − ) K(τ − τ )S(τ ) dτ + Q∗ (τ ). (14.1) D

The convolution integral equations for polarized radiation fields can be treated by more or less the same methods as their scalar counterparts. The only difference is that the unknown is a vector, or a matrix, and the kernel a matrix. For multidimensional media, the new radiative transfer equations do not lead to convolution integral equations, but are simpler to solve numerically than the original ones.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_14

305

306

14 Polarized Radiative Transfer Equations

The construction of the new equations is based on an important property of the Rayleigh phase matrix, already presented in Chap. 13, namely that it can be factorized as PR (,  ) = A()AT ( ),

(14.2)

where  and  are the directions of the incident and scattered beams and T stands for transposed. This factorization makes it possible to extract from the scattering term the dependence on the direction . A multiplication of the radiative transfer equation for the Stokes vector I = (I, Q, U ) by the matrix A−1 () leads to a new transfer equation for a vector I ≡ A−1 ()I ,

(14.3)

in which the scattering term depends only on the position in the medium. We have seen in Sect. 13.2.2.2 that the elements of A() can be expressed in terms of the irreducible spherical tensors TQK (i, ) (i is the index of the Stokes parameter). The relation I = A()I becomes in component notation Ii =

 K

K TQK (i, )IQ ,

(14.4)

Q

K the components of the where Ii are the component of the Stokes vector and IQ vector I. The solution of the transfer equation for I is, as we said, simpler that the K one can then easily construct the Stokes vector. equation for I . Knowing IQ For a cylindrically symmetric radiation field, which can be represented by the two Stokes parameters I and Q, the vector I has two components, corresponding to K = Q = 0 and K = 2, Q = 0. For a radiation field without cylindrical symmetry, the three Stokes parameters I , Q, U are needed to represent the intensity and polarization of the field. The vector I has six components corresponding to K will be referred to as K = Q = 0 and K = 2, −2 ≤ Q ≤ +2. The components IQ the (KQ) components of the radiation field and the expansion in Eq. (14.4) as the (KQ) representation of the radiation field. This terminology is more explicit than “reduced radiation field” introduced in earlier work (e.g. Frisch 2007). The construction of the new radiative transfer equations is carried out in Sects. 14.1 to 14.3 for several Rayleigh scattering problems in a one-dimensional medium, namely conservative and non-conservative Rayleigh scattering for radiation fields with symmetrical symmetry and without it. We show how to construct the new fields I and convolution integral equations such as Eq. (14.1). Resonance polarization is considered in Sect. 14.4. We use the direction averaged version of the two-level atom redistribution matrix R(x, , x ,  ) introduced in Chap. 13 and Appendix J in Chap. 13. The (KQ) decomposition technique is applied to radiation fields with cylindrical symmetry and without it, and for onedimensional and multi-dimensional geometries. For a one-dimensional medium

14.1 Rayleigh Scattering. The Radiative Transfer Equation

307

and complete frequency redistribution, we construct a convolution equation similar to Eq. (14.1). We recall that complete frequency redistribution means a total decoupling between the frequencies of the incident and scattered beams. The Hanle effect is considered in Sect. 14.5. We recall that the Hanle effect describes resonance polarization modified by the presence of a uniform weak magnetic field. The three Stokes parameters I , Q, U are needed to represent the radiation field, even in a one-dimensional medium, since the magnetic field is breaking the cylindrical symmetry of the medium. We consider only complete frequency redistribution and construct for a one-dimensional medium, a convolution integral equation similar to Eq. (14.1), in which the integral is multiplied by a term depending on the magnetic field, while the kernel remains unchanged.

14.1 Rayleigh Scattering. The Radiative Transfer Equation In a plane-parallel atmosphere, the vector radiative transfer equation for a radiation field with a cylindrical symmetry, may be written as ∂I λ (z, μ) = −[σc (λ, z) + κc (λ, z)]I λ (z, μ) ∂z  1 +1 P(μ, μ )I λ (z, μ ) dμ + κc (λ, z)Q∗λ (z). + σc (λ, z) 2 −1 μ

(14.5)

Here I λ (z, μ) is the polarized radiation field, λ is the frequency, z is the space variable in the direction of the normal to the atmosphere and μ is the cosine of the direction of the ray with respect to the outward normal. The polarized field is a twocomponent vector and P(μ, μ ), the 2 × 2, cylindrically symmetric Rayleigh phase matrix. The coefficient κc (λ, z) is due to bound-free and free-free transitions and σc (λ, z) is the Rayleigh scattering the coefficient. It varies with frequency as 1/λ4 . Finally Q∗λ (z) is a given primary source term. We introduce the monochromatic optical depth, dτλ = −[σc (λ, z) + κc (λ, z)] dz,

(14.6)

and the parameter λ (z) =

κc (λ, z) , σc (λ, z) + κc (λ, z)

which represents the destruction probability per scattering.

(14.7)

308

14 Polarized Radiative Transfer Equations

The radiative transfer equation can then be written as ∂I λ (τλ , μ) = I λ (τλ , μ) ∂τλ  1 +1 − [1 − λ (τλ )] P(μ, μ )I λ (τλ , μ ) dμ − λ (τλ )Q∗λ (z). 2 −1 μ

(14.8)

The parameter λ (τλ ) depends on the density in the medium and on the frequency. The smaller , the larger the polarization. For example, one observes in the solar spectrum that the continuum polarization, which is created by Rayleigh scattering, is a strongly decreasing function of the wavelength (Fluri and Stenflo 1999). This can be explained by the 1/λ4 behavior of the Rayleigh scattering. An increase of the wavelength produces a decrease in σc (λ, z), hence an increase of , and consequently a decrease in the polarization. An increase in density, will also decrease the polarization. Here for the construction of exact solutions and also for the perturbation analysis carried out in Part III, we assume a uniform . Since there is no coupling between frequencies, the wavelength λ can be omitted. Henceforth, to treat Rayleigh scattering problems, we use the radiative transfer equation 1 ∂I μ (τ, μ) = I (τ, μ) − (1 − ) ∂τ 2



+1 −1

P(μ, μ )I (τ, μ ) dμ − Q∗ (τ ),

(14.9)

with τ = 0 at the surface and τ → ∞ in the interior. The parameter  plays a very important role in the scattering term but not in the primary source. The equation being linear, it is just a scaling factor. We have set it to one.

14.2 Conservative Rayleigh Scattering Two conservative Rayleigh scattering problems will be considered: the polarized Milne problem and the diffuse reflection problem. In both cases, the primary source Q∗ (τ ) is zero. The radiation field has a cylindrical symmetry and can be represented by the Stokes parameters I and Q or by the components Il and Ir . For conservative scattering, it is algebraically more convenient to use the components Il and Ir , in brief the (lr) representation. The two-component radiation field I (τ, μ) satisfies the radiative transfer equation 1 ∂I μ (τ, μ) = I (τ, μ) − ∂τ 2



+1 −1

P(μ, μ )I (τ, μ ) dμ ,

(14.10)

14.2 Conservative Rayleigh Scattering

309

where P(μ, μ ) is the cylindrically symmetric part of the Rayleigh phase matrix, (0) also denoted PR . The boundary condition at τ = 0 are I (0, μ) = 0,

Milne problem,

I (0, μ) = I inc (μ),

μ ∈ [−1, 0],

diffuse reflection,

(14.11)

μ ∈ [−1, 0].

(14.12)

For the Milne problem, the total radiative flux is a given positive constant F . For the (lr) representation, the phase matrix P(μ, μ ) takes the form 3 P(μ, μ ) = 4



2(1 − μ2 )(1 − μ 2 ) + μ2 μ 2 μ2 μ 2

 .

(14.13)

1

It can be factorized as P(μ, μ ) = A(μ)AT (μ ),

(14.14)

where √  √  3 μ2 2(1 − μ2 ) A(μ) = . 2 1 0

(14.15)

This matrix A(μ) has already been introduced in Eq. (13.29). We now use this factorization to solve the Milne problem and diffuse reflection problem.

14.2.1 The Polarized Milne Problem Although there is no incident flux and no primary source, the condition that the total radiative flux is a given positive constant ensures that there is a solution, which is not identically zero and, as will be shown, increases linearly at infinity just as in the scalar case. Since the phase matrix P(μ, μ ) depends on the directions μ and μ of the incident and scattered radiation, the source function vector, 1 S(τ, μ) ≡ 2



+1 −1

P(μ, μ )I (τ, μ ) dμ ,

(14.16)

depends on the direction μ. This is in contrast with scalar problems treated in Part I, in which the source function depends only on the optical depth. To take advantage

310

14 Polarized Radiative Transfer Equations

of the factorization of the phase matrix, we introduce a new radiation field I(τ, μ), with components I1 (τ, μ) and I2 (τ, μ), defined by I(τ, μ) ≡ A−1 (μ)I (τ, μ).

(14.17)

The idea of introducing this new field is already in Stenflo and Stenholm (1976) and more explicitly in Rees (1978). In component notation, Eq. (14.17) may be written as √ √ 3 2 [μ I1 (τ, μ) + 2(1 − μ2 )I2 (τ, μ)], Il (τ, μ) = 2 √ 3 Ir (τ, μ) = I1 (τ, μ). (14.18) 2 For μ = 1, the radiation field should be unpolarized. One easily checks that for μ = 1, the components Il (τ, μ) and Ir (τ, μ) are equal. The field I(τ, μ) satisfies the equation μ

∂I (τ, μ) = I(τ, μ) − S(τ ), ∂τ

(14.19)

with the surface boundary condition I(0, μ) = 0,

μ ∈ [−1, 0].

(14.20)

The source function vector may be written as S(τ ) ≡

1 2



+1 −1

(μ )I(τ, μ ) dμ ,

(14.21)

where the matrix (μ) is defined by ⎤ √ 2 2 2 μ (1 − μ ) 3 ⎦. (μ) ≡ AT (μ)A(μ) = ⎣ √ 4 2 μ2 (1 − μ2 ) 2(1 − μ2 )2 ⎡

μ4 + 1

(14.22)

For the new field, the source function vector depends only on the optical depth. The introduction of the formal solution of Eq. (14.19) into the definition of S(τ ) readily leads to 

∞

S(τ ) = 0

K(τ − τ )S(τ ) dτ ,

(14.23)

14.2 Conservative Rayleigh Scattering

311

where the kernel K(τ ) is defined by 1 K(τ ) ≡ 2



1

(μ) exp(− 0

|τ | dμ ) . μ μ

(14.24)

The matrix K(τ ) is an even function of τ and a symmetric matrix, since (μ) is symmetric by construction. As in the scalar case, it decreases exponentially at infinity and has the structure of a Laplace transform, a property fully exploited in Chap. 15. An exact solution of Eq. (14.23) is constructed in Chap. 16.

14.2.2 The Diffuse Reflection Problem We now assume that there is an incident radiation on the medium. The problem is to calculate the field inside the medium and the emergent field. To construct the new field I(τ, μ), it is convenient to introduce a diffuse field, defined in such a way that the medium contains a primary source term but has no incident radiation. Proceeding as for the scalar diffuse reflection problem, we introduce a diffuse field I d (τ, μ) defined by I d (τ, μ) ≡ I (τ, μ) − I inc (μ) eτ/μ , d

I (τ, μ) ≡ I (τ, μ),

μ ∈ [−1, 0],

μ ∈ [0, 1],

(14.25)

where I inc (μ), μ ∈ [−1, 0] is the incident radiation at τ = 0. The radiative transfer equation for the diffuse field is  ∂I d 1 +1 (τ, μ) = I d (τ, μ) − P(μ, μ )I d (τ, μ ) dμ

∂τ 2 −1  1 1

− P(μ, μ )I inc (−μ ) e−τ/μ dμ , 2 0

μ

(14.26)

with the boundary condition I d (0, μ) = 0,

μ ∈ [−1, 0].

(14.27)

We now use Eq. (14.17), with I (τ, μ) replaced by I d (τ, μ), to introduce a new field I(τ, μ). It satisfies the radiative transfer equation in Eq. (14.19). The source vector is given by S(τ ) =

1 2



+1 −1

(μ )I(τ, μ ) dμ + Q∗ (τ ),

(14.28)

312

14 Polarized Radiative Transfer Equations

where Q∗ (τ ) =

1 2



1



AT (μ )I inc (−μ ) e−τ/μ dμ .

(14.29)

0

By construction, the boundary condition at τ = 0 for I(τ, μ) is I(τ, μ) = 0, μ ∈ [0, −1]. The integral equation for the source vector S(τ ) is 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

(14.30)

0

where K(τ ) is defined in Eq. (14.24) and Q∗ (τ ) in Eq. (14.29). An exact solution of Eq. (14.30) is constructed in Chap. 16.

14.3 Rayleigh Scattering. A (KQ) Expansion of the Stokes Parameters We introduce in this section a spherical tensor decomposition of the Stokes vector, for non-conservative scattering Rayleigh in a one-dimensional medium. We first treat the case of a cylindrically symmetric radiation field, represented by its Stokes parameters I and Q (Sect. 14.3.1) and then of a non-cylindrically symmetric field represented by I , Q, and U (Sect. 14.3.2). For non-conservative Rayleigh scattering, there is no advantage to use the (lr) representation of the radiation field. We shall always use the Stokes parameters representation. For non-conservative scattering, a fraction of the photons disappearing at each scattering (turned into thermal energy), there is no possible stationary radiation field, when the loss of photons is not compensated by an incident radiation or a primary source inside the medium. We now assume that the medium contains an arbitrary primary source but that there is no incident radiation, the case of an incident radiation having been treated in Sect. 14.2.2.

14.3.1 Cylindrically Symmetric Radiation Field A cylindrically symmetric radiation field in a one-dimensional medium can be represented by the two-component Stokes vector I (τ, μ) = (I (τ, μ), Q(τ, μ)). It satisfies the radiative transfer equation ∂I (τ, μ) = I (τ, μ) − ∂τ  1 +1 (1 − ) P(μ, μ )I (τ, μ ) dμ − Q∗ (τ, μ), 2 −1

μ

(14.31)

14.3 Rayleigh Scattering. A (KQ) Expansion of the Stokes Parameters

313

where  is the destruction probability per scattering and Q∗ (τ, μ) is the primary source. Here, for the sake of generality, we consider a mixture of Rayleigh and isotropic scattering. The phase matrix is written as P(μ, μ ) = W PR (μ, μ ) + (1 − W )Pis ,

(14.32)

where PR (μ, μ ) is the Rayleigh phase matrix and Pis the isotropic matrix. We recognize here the phase matrix written in Eq. (13.46). It suffices to set α = W and β = (1 − W ). The depolarization parameter W can be given any value in [0, ∞[ (see Ivanov 1995). Here we assume that W ∈ [0, 1]. A combination of Rayleigh and isotropic scattering is encountered in, e.g., media with a random distribution of anisotropic particles (Silant’ev et al. 2015) and for the resonance polarization of spectral lines, when elastic collisions are perturbing the Zeeman sublevels (see Eq. (J.10)). When the Stokes vector is represented by its components I and Q (in brief (IQ) representation), ⎤ ⎡8 1 2

2 2

2 + (1 − 3μ )(1 − 3μ ) (1 − 3μ )(1 − μ ) 3 ⎦. PR (μ, μ ) = ⎣ 3 3 8 2 2 (1 − μ2 )(1 − 3μ ) 3(1 − μ2 )(1 − μ )

(14.33)

and  Pis =

 10 . 00

(14.34)

It is explained in Sect. 13.2.2.2 how the decomposition of the Rayleigh phase matrix in terms of the irreducible spherical tensors TQK (i, ) naturally leads to the factorization PR (μ, μ ) = A(μ)AT (μ ) with ⎡ A(μ) = ⎣

T˜00 (0, μ)

T˜02 (0, μ)

⎤

⎡

⎢1 ⎦=⎢ ⎣ T˜00 (1, μ) T˜02 (1, μ) 0

⎤ 1 2 √ (3μ − 1) ⎥ 2 2 ⎥. ⎦ 1 2 − √ 3(1 − μ ) 2 2

(14.35)

We recall that TQK (i, ) = T˜QK (i, θ )ei Qχ (see Eq. (13.33)). Here Q = 0, hence T˜0K (i, μ) = T0K (i, ). It is easy to verify that the matrix P(μ, μ ) can be factorized as P(μ, μ ) = AW (μ)ATW (μ )

(14.36)

314

14 Polarized Radiative Transfer Equations

with √ AW (μ) = A(μ) W,

(14.37)

W = diag[1, W ].

(14.38)

and W the diagonal matrix,

In expanded form √

⎤ W 2 ⎢ 1 2√2 (3μ − 1) ⎥ ⎥. √ AW (μ) = ⎢ ⎦ ⎣ W 2 0 − √ 3(1 − μ ) 2 2 ⎡

(14.39)

An arbitrary source cannot be factorized as a product AW (μ)Q∗ (τ ), a necessary condition to construct a radiative transfer equation with a source term depending only on the optical depth. As pointed out by Ivanov et al. (1997) (see also Frisch 2007), the solution of Eq. (14.31) can always be written as I (τ, μ) = I d (τ, μ) + I p (τ, μ),

(14.40)

where I p (τ, μ) is a direct field, created by the primary source, and I d (τ, μ) a diffuse field. The direct field is solution of the equation μ

∂I p (τ, μ) = I p (τ, μ) − Q∗ (τ, μ), ∂τ

(14.41)

and the diffuse field is solution of the equation  ∂I d 1 +1 d (τ, μ) = I (τ, μ) − (1 − ) μ P(μ, μ )I d (τ, μ ) dμ

∂τ 2 −1  1 1 −(1 − ) P(μ, μ )I p (τ, μ ) dμ . (14.42) 2 −1 The primary source has now the required form. The boundary conditions for the direct and diffuse field are I p (0, μ) = I d (0, μ) = 0,

μ ∈ [−1, 0].

(14.43)

We now consider exclusively the diffuse radiation field I d (τ, μ) and drop the superscript d. Proceeding as above, we introduce a new radiation field I(τ, μ) ≡ A−1 W (μ)I (τ, μ).

(14.44)

14.3 Rayleigh Scattering. A (KQ) Expansion of the Stokes Parameters

315

with components I00 (τ, μ) and I02 (τ, μ). The Stokes parameters may be written √ W I (τ, μ) = I00 (τ, μ) + √ (3μ2 − 1)I02 (τ, μ), 2 2 √ 3 W Q(τ, μ) = − √ (1 − μ2 )I02 (τ, μ), 2 2

(14.45)

or in more compact form, Ii (τ, μ) =

 WK T˜0K (i, μ)I0K (τ, μ),

i = 0, 1,

K = 0, 2,

(14.46)

K

with W0 = 1 and W2 = W . The indices i = 0, 1 correspond respectively to Stokes I and Stokes Q. Similar decompositions are established for resonance polarization in a one-dimensional medium, in a multi-dimensional medium, and for the Hanle effect (see Sects. 14.4.1, 14.4.2, 14.5.2, and 14.5.3). For a multi-dimensional medium and for the Hanle effect, the irreducible spherical tensors TQK (i, ), with Q = 0 are also involved in the decomposition. The field I(τ, μ) satisfies the radiative transfer equation written in Eq. (14.19), in which the source term is given by S(τ ) ≡ (1 − )

1 2



+1 −1

W (μ )I(τ, μ ) dμ + Q∗ (τ ).

(14.47)

√ √ W(μ) W,

(14.48)

Here W (μ) = ATW (μ)AW (μ) = with ⎡ 1

1 √ (1 − 3μ2 ) 2 2

⎤

⎥ ⎢ ⎥. (μ) = ⎢ ⎦ ⎣ 1 1 √ (1 − 3μ2 ) (5 − 12μ2 + 9μ4 ) 4 2 2

(14.49)

In expanded notation ⎡

√

W √ (1 − 3μ2 ) 2 2

⎤

1 ⎥ ⎢ ⎥. √ W (μ) = ⎢ ⎦ ⎣ W 2 W 2 4 (5 − 12μ + 9μ ) √ (1 − 3μ ) 4 2 2

(14.50)

316

14 Polarized Radiative Transfer Equations

The primary source term is 1 Q (τ ) = (1 − ) 2 ∗



1 −1

ATW (μ )I p (τ, μ ) dμ .

(14.51)

The introduction of a direct field I p (τ, μ) to handle an arbitrary primary source term Q∗ (τ, μ) is not needed when the thermal source is unpolarized, of thermal origin for example. In this case Q∗ (τ ) = Q∗ (τ ). Indeed, for an unpolarized vector V (τ, μ) = (V1 (τ, μ), V2 (τ, μ)), the second component V2 (τ, μ) is zero, then AW (μ)V (τ, μ) = V (τ, μ).

(14.52)

The source vector S(τ ) satisfies a convolution integral equation similar to Eq. (14.30), namely 

∞

S(τ ) = (1 − )

K(τ − τ )S(τ ) dτ + Q∗ (τ ).

(14.53)

0

The kernel K(τ ) is defined as in Eq. (14.24), the matrix (μ) being replaced by the matrix W (μ) given in Eq. (14.50). The factor (1 − ) is not as innocent as it looks, since it actually prevents the construction of exact solutions (see however Siewert et al. 1981).

14.3.2 Azimuthally Dependent Radiation Field When the linearly polarized radiation field has no cylindrical symmetry, whatever the reason, the three Stokes parameters I , Q, and U are needed for its description. We show here, for Rayleigh scattering, how to generalize the (KQ) expansion introduced in Sect. 14.3.1 to a non-cylindrically symmetric field. The same type of expansion is carried out for resonance polarization in a multi-dimensional medium (Sect. 14.4.2) and for the Hanle effect (Sect. 14.5). The starting point is always the decomposition of the polarization phase matrix in terms of the irreducible spherical tensors TQK (i, ). As shown in Faurobert-Scholl (1991), for the Hanle effect, an azimuthal Fourier expansion of the polarization phase matrix reaches essentially the same goal as a (KQ) decomposition, but a (KQ) decomposition is significantly simpler than an azimuthal Fourier decomposition. For simplicity, we assume a one dimensional medium and pure Rayleigh scattering (W = 1). The Rayleigh phase matrix, PR (,  ), now a 3 × 3 matrix, contains azimuthally dependent terms. It can be written as in Eq. (13.23) or, as

14.3 Rayleigh Scattering. A (KQ) Expansion of the Stokes Parameters

317

explained in Sect. 13.2.2.2, in terms of the irreducible spherical tensor TQK (i, ). Each element of the matrix may be written as [PR (,  )]i,j =

 K

K (−1)Q TQK (i, )T−Q (i,  ),

i, j = 0, 1, 2,

(14.54)

Q

with K = 0, 2, −K ≤ Q ≤ +K,  and , the directions of the incident and scattered beams. The properties of the tensors TQK (i, ) are described in Sect. 13.2.2.2. For a cylindrically symmetric radiation field, the Stokes parameters I and Q can be expanded in terms of T˜0K (i, μ) (see Eq. (14.46)). For a noncylindrically symmetric radiation field the (KQ) expansion will involve all the TQK (i, ) for K = 0, 2 and −K ≤ Q ≤ +K. In a one-dimensional medium, the 3-dimension Stokes vector I (τ, ) constructed with I , Q, U satisfies the radiative transfer equation μ

∂I (τ, ) = I (τ, ) − S(τ, ), ∂τ

(14.55)

where the source vector S(τ, ) is given by  S(τ, ) = (1 − )

PR (,  )I (τ,  )

d

+ Q∗ (τ ). 4π

(14.56)

The components of the source vector may be written as Si (τ, ) = (1 − )

2    j =0

K TQK (i, )(−1)Q T−Q (j,  )Ij (τ,  )

K,Q

d

+ Q∗i (τ ). 4π (14.57)

For simplicity, but without loss of generality, we assume that the primary source of photons Q∗ (τ ) depends only on the position in the medium and is unpolarized. More general primary sources can be handled with the introduction of a direct field as explained in Sect. 14.3.1. The components of the primary source can thus also be written as  Q∗i (τ ) = TQK (i, )QK i = 0, 1, 2. (14.58) Q (τ ), K

Q

When Q∗ (τ ) is unpolarized, Q00 (τ ) is the only non zero component. The components of the source vector can thus be written as  K TQK (i, )SQ (τ ), (14.59) Si (τ, ) = K,Q

318

14 Polarized Radiative Transfer Equations

where K SQ (τ )

= (1 − )

2   j =0

K (j,  )Ij (τ,  ) (−1)Q T−Q

d

+ QK Q (τ ). 4π

(14.60)

K (τ ) depends only on the The important point is that the expansion coefficient SQ optical depth τ . The direction dependence of Si (τ, ) is entirely described by the spherical tensors. The radiative transfer equation for the components of the Stokes vector may thus be written as

μ

 ∂Ii K (τ, ) = Ii (τ, ) − TQK (i, )SQ (τ ). ∂τ

(14.61)

KQ

To solve this equation we need boundary conditions. We assume that there is no incident radiation on the surface at τ = 0 and for simplicity, but without loss of generality, that the medium is semi-infinite. For μ ∈ [0, 1], the solution of this equation is Ii (τ, ) =



 TQK (i, )

KQ

∞

e−(τ

−τ )/μ

τ

K

SQ (τ )

dτ

. μ

(14.62)

There is a similar equation for μ ∈ [−1, 0]. These equations show that the radiation field components can be written as Ii (τ, ) =



K TQK (i, )IQ (τ, μ).

(14.63)

KQ K depend only on μ, their azimuthal The important point is that the components IQ K dependence being included in the tensor TQ (i, ). The reason for this simplification is of course that the source term depends only on the optical depth. The components K (τ, μ) are solution of the radiative transfer equation IQ

μ

K ∂IQ

∂τ

K K (τ, μ) = IQ (τ, μ) − SQ (τ ).

(14.64)

K (τ ) in terms of I K (τ, μ). Inserting Eq. (14.63) into The last step is to express SQ Q Eq. (14.57), we obtain

K SQ (τ )

= (1 − )

2   j =0

K (j, ) (−1)Q T−Q

 K Q





K TQK (j, )IQ

(τ, μ)

d + QK Q (τ ). 4π (14.65)

14.3 Rayleigh Scattering. A (KQ) Expansion of the Stokes Parameters

319

Using now TQK (j, ) = ei Qχ T˜QK (j, θ ), we see that product of the two spherical

tensors contains a factor ei (Q −Q)χ . Upon integration over the solid angle d, the contribution of this term is zero unless Q = Q. This leads to K SQ (τ )

= (1 − )

 K

+∞ −∞

1 2



+1 −1





KK K Q (μ)IQ (τ, μ) dμ + QK Q (τ ),

(14.66)

with

KK Q (μ) =

2 

K (−1)Q T˜−Q (j, μ)T˜QK (j, μ),

K, K = 0, 2.

(14.67)

j =0

In vector notation, the radiative transfer equation for the 6-dimension vector K (τ, μ) is I(τ, μ), constructed with the components IQ μ

∂I (τ, μ) = I(τ, μ) − S(τ ). ∂τ

(14.68)

K (τ ) is given by The source term S(τ ) with components SQ

S(τ ) = (1 − )

1 2



+1 −1

(μ)I(τ, μ) dμ + Q∗ (τ ).

(14.69)

It is evident that the radiative transfer equation for I(τ, μ) is significantly simpler than the transfer equation for the Stokes vector in Eq. (14.55). In Eq. (14.69), Q∗ (τ ) = [Q00 (τ ), 0, 0, 0, 0, 0] with Q00 (τ ), the first component of the unpolarized primary source Q∗ (τ ). The 6 × 6 matrix (μ) has a simple block structure: ⎤ ⎡ 00 02 0 0 0 0 0 0 ⎢  20  22 0 0 0 0 ⎥ ⎥ ⎢ 0 ⎢ 0 00  22 0 0 0 ⎥ ⎥ ⎢ −1 (μ) = ⎢ (14.70) ⎥, 22 0 ⎢ 0 0 0 +1 0 ⎥ ⎥ ⎢ 22 0 ⎦ ⎣ 0 0 0 0 −2 22 0 0 0 0 0 +2 where 000 = 1,

1 002 = 020 = √ (3μ2 − 1), 2 2

022 =

1 (5 − 12μ2 + 12μ4 ). 4 (14.71)

320

14 Polarized Radiative Transfer Equations

The other diagonal elements are 22 22 −1 = +1 =

3 (1 − μ2 )(1 + 2μ2 ), 4

22 22 −2 = +2 =

3 (1 + μ2 )2 . 8

(14.72)

The 2 × 2 block corresponds to the cylindrically symmetric part of the radiation field. Using the expressions of the tensors TQK (i, ) given in Landi Degl’Innocenti and Landolfi (2004, p. 211)1, Eq. (14.63), becomes in expanded form: 1 I0 (τ, ) = I (τ, μ, χ) = I00 (τ, μ) + √ (3μ2 − 1)I02 (τ, μ) 2 2 √ 3 2 (1 − μ2 )1/2 μ[I12 (τ, μ) ei χ − I−1 (τ, μ) e−i χ ] − 2 √ 3 2 (1 − μ2 )[I22 (τ, μ) e2i χ + I−2 (τ, μ) e−2i χ ], + 4 3 I1 (τ, ) = Q(τ, μ, χ) = − √ (1 − μ2 )I02 (τ, μ) 2 2 √ 3 2 (1 − μ2 )1/2 μ[I12 (τ, μ) ei χ − I−1 (τ, μ) e−i χ ] − 2 √ 3 2 − (1 − μ2 )[I22 (τ, μ) e2i χ + I−2 (τ, μ) e−2i χ ], 4

(14.73)

(14.74)

√ 3 2 i (1 − μ2 )1/2 μ[I12 (τ, μ) ei χ − I−1 (τ, μ) e−i χ ] I2 (τ, ) = U (τ, μ, χ) = − 2 √ 3 2 2i μ[I22 (τ, μ) e2i χ + I−2 (τ, μ) e−2i χ ]. (14.75) − 4 We note that the azimuthal dependence of I0 (τ, ) given in Eq. (14.73) is different from the azimuthal dependence of the radiation field intensity obtained with the scalar Rayleigh scattering phase function (3/4)(1 + cos2 ). K (τ, μ) are complex numbers. Actually they are real for Q = The components IQ K (τ, μ) satisfies 0 and complex for Q = 0. Since the Stokes parameters are real, IQ K the same conjugation relation as TQ (i, ), namely K K [IQ (τ, μ)]∗ = (−1)Q I−Q (τ, μ).

1

They dependent on an angle γ shown in Fig. 13.4, which is set here to zero.

(14.76)

14.4 Resonance Polarization of Spectral Lines

321

Equations (14.73) to (14.75) can also be written in terms of the real quantities K )= (IQ

1 K K [I + (−1)Q I−Q ], 2 Q

i K K K (IQ ) = − [IQ − (−1)Q I−Q ], 2

Q ≥ 0. (14.77)

The azimuthal dependence is then described by the trigonometric functions sin χ, sin 2χ, cos χ, and cos 2χ. The source vector S(τ ) in the (KQ) representation also satisfies a convolution integral equation, obtained by inserting the formal solution of the transfer equation for I(τ, μ) into the definition of S(τ ) given in Eq. (14.69). The convolution equation for S(τ ) may be written as  ∞ K(τ − τ )S(τ ) dτ + Q∗ (τ ), (14.78) S(τ ) = (1 − ) 0

where 1 K(τ ) ≡ 2

 0

1

  |τ | dμ . (μ) exp − μ μ

(14.79)

The kernel has the same block diagonal structure as the matrix (μ). This implies K (τ ), Q = ±1, ±2, which control the azimuthal dependence that the components SQ of the radiation field, can be calculated independently of the cylindrically symmetric components. This well known property of Rayleigh scattering can be derived from the standard azimuthal decomposition of the phase matrix given in Sect. 13.2.2.1 (see e.g. Chandrasekhar 1960) or from the representation of the Rayleigh phase matrix introduced by Domke (1971). The TQK (i, ) decomposition employed here provides a simpler proof.

14.4 Resonance Polarization of Spectral Lines The mechanism of the linear polarization of spectral lines by resonance scattering is presented in Chap. 13 and the redistribution matrix, in the atom rest frame, for a two-level atom with an unpolarized and infinitely thin ground level in Appendix J of Chap. 13. In the atom rest frame, the frequency redistribution and the Rayleigh polarization phase matrix are decoupled (see Eq. (J.2)). They become coupled, in the laboratory frame, when the Doppler effects due to the atomic velocities are taken into account. Indeed, the elements of the redistribution matrix in the laboratory frame become Rij (x, , x ,  ) =   WK rK (x, , x ,  )TQK (i, )TQK (j,  ), i, j = 0, 1, 2, (14.80) K=0,2 Q

322

14 Polarized Radiative Transfer Equations

with rK (x, , x ,  ) = αrII (x, , x ,  ) + (β (K) − α)rIII (x, , x ,  ).

(14.81)

Here x and  are the frequency and direction of the scattered beam. The prime variables correspond to the incident beam. The frequencies are measured in Doppler width unit, with zero at line center. The coefficients WK depend on the angular momentum of the atomic levels of the transition (see Appendix J in Chap. 13). For a normal Zeeman triplet W0 = W2 = 1. The scalar redistribution functions are the standard RII and RIII (Hummer 1962) scalar redistribution functions. The coefficients α and β (K) are α=

1 , 1 + I + E

β (K) =

1 , 1 + I + δ (K)

(14.82)

where I = R / I , E = R / E , δ (K) = R /D (K) . The coefficients R , I , E are deexcitation rates by radiative decay, inelastic and elastic collisions, respectively. The coefficient D (2) is the destruction rate of the alignment by elastic collisions and D (0) = 0 (see Appendix J in Chap. 13). The dependence of the frequency redistribution function rK (x, , x ,  ) on the directions  and  prevents a separation between the frequency redistribution and the polarization matrix. Therefore it is a priori impossible to construct a source function vector, which is independent of the direction of the beam. The decoupling is achieved by replacing the direction-dependent redistribution functions by their direction-averages, also known as angle-averages. Most numerical investigations of resonance polarization employ angle-averaged redistribution functions. It has been shown to have negligible effects on the intensity of the radiation field. On the polarization profile, the effects depend on the optical thickness of the line. They may be significant for weak lines, but disappear for strong lines because of the averaging effect over a very large number of scatterings (see e.g., Sampoorna et al. 2011 and references within). The direction-averaged redistribution functions are defined by 1 r¯II, III (x, x ) = 2



π

rII, III (x, x , ) sin

d ,

(14.83)

0

where is the angle between  and  . The elements of the angle-averaged ¯ redistribution matrix R(x, , x ,  ) may thus be written ¯ ij (x, , x ,  ) = R



WK r¯K (x, x )T0K (i, )T0K (j,  ), i, j = 0, 1,

K=0,2

(14.84)

14.4 Resonance Polarization of Spectral Lines

323

with r¯K (x, x ) = α r¯II (x, x ) + (β (K) − α)¯rIII (x, x ).

(14.85)

With this redistribution matrix, we now establish a (KQ) decomposition of the radiation field, first for a one-dimensional medium with a cylindrically symmetric radiation field, then for a multi-dimensional medium. For a one-dimensional medium, the method is essentially identical to the method described in Sect. 14.3.1 for Rayleigh scattering. For a multi-dimensional medium, the radiation field has no cylindrical symmetry, hence three Stokes parameters I , Q, U , or the six (KQ) components, are needed to describe the radiation field. Moreover, the cylindrically symmetric and non-cylindrically symmetric (KQ) components are coupled. The 6 × 6 matrix (μ) becomes a full matrix.

14.4.1 One-Dimensional Medium We consider a one dimensional cylindrically symmetric medium in which the radiation field is also cylindrically symmetric, which means that it can be described by the two Stokes parameters I and Q. The elements of the redistribution matrix may be written as ¯ ij (x, μ, x , μ ) = R



WK r¯K (x, x )T˜0K (i, μ)T˜0K (j, μ ), i, j = 0, 1.

K=0,2

(14.86) In matrix notation, √ √ ¯ R(x, μ, x , μ ) = A(μ) WT(x, x ) WAT (μ ),

(14.87)

where W = diag[W0 , W2 ] = diag[1, W2 ],

(14.88)

and T(x, x ) =



 α r¯II + (β (0) − α)¯rIII 0 . 0 α r¯II + (β (2) − α)¯rIII

(14.89)

324

14 Polarized Radiative Transfer Equations

The matrix A(μ) is defined in Eq. (14.35). We introduce √ ⎤ W2 2 √ ⎢ 1 2√2 (3μ − 1) ⎥ ⎥. √ AW (μ) = A(μ) W = ⎢ ⎣ ⎦ W2 2 0 − √ 3(1 − μ ) 2 2 ⎡

(14.90)

The parameter W2 is the depolarization parameter. We now follow the method described for Rayleigh scattering. For simplicity we assume that there is no incident radiation on the medium and that the primary source term is unpolarized and depends only on the position inside the medium. When these assumptions are not satisfied, it is possible to introduce a diffuse field, as explained in Sects. 14.2.2 and 14.3.1. All the results given below will hold for the diffuse field. The Stokes vector I (τ, x, μ) satisfies the radiative transfer equation μ

∂I (τ, x, μ) = ϕ(x)[I (τ, x, μ) − S(τ, x, μ)], ∂τ

(14.91)

where τ is the frequency integrated optical depth, ϕ(x) the line absorption profile. the source function vector S(τ, x, μ) is defined by S(τ, x, μ) ≡

1 2



+∞ +1 −∞

−1

1 ¯ R(x, μ, x , μ )I (τ, x , μ ) dμ dx + Q∗ (τ ), ϕ(x) (14.92)

where Q∗ (τ ) is the unpolarized primary source. We introduce the two-component vector I(τ, x, μ) defined by I(τ, x, μ) ≡ A−1 W (μ)I (τ, x, μ).

(14.93)

The components of I and I are related as shown in Eq. (14.45), with W replaced by W2 . Expressing in Eq. (14.92) the Stokes vector I (τ, x , μ ) in terms of the field I(τ, x, μ), we obtain for I(τ, x, μ) the radiative transfer equation μ

∂I (τ, x, μ) = ϕ(x)[I(τ, x, μ) − S(τ, x)], ∂τ

(14.94)

where S(τ, x) =

1 2



+∞ +1 −∞

−1

1 T(x, x )W (μ)I(τ, x , μ) dμ dx + Q∗ (τ ), ϕ(x) (14.95)

14.4 Resonance Polarization of Spectral Lines

325

and W (μ) =

√ √ W (μ) W.

(14.96)

The elements of the matrix (μ) are given in Eq. (14.49) and those of W (μ) in Eq. (14.50) with W = W2 . Since Q∗ (τ ) is assumed to be unpolarized, Q∗ (τ ) = Q∗ (τ ). Some simplifying assumptions are often made in the treatment of resonance polarization. One of them consists in keeping only the coherent scattering term. This approximation holds when the elastic collisions responsible for the parameters E and δ (2) are negligible compared to I . It is an approximation which holds for strong resonance lines with extended wings. In this case the matrix T(x, x ) reduces to the scalar αrII (x, x ). The opposite assumption amounts to keep only the term β (K)r¯III (x, x ). In addition one usually replaces r¯III (x, x ) by the product ϕ(x)ϕ(x ). This complete frequency redistribution limit is justified for lines which are devoid of strong line wings, such as the Sr I 4607 Å resonance line (Faurobert-Scholl 1993; Bommier et al. 2005), since the rII term plays an important role only in the wings of strong lines. As for replacing the rIII term by complete frequency redistribution, it is a safe approximation, as shown in e.g., Sampoorna et al. (2017). We recall that complete frequency redistribution is also a consequence of the flat-spectrum approximation made in Landi Degl’Innocenti and Landolfi (2004, p. 257). For complete frequency redistribution, the elements of the redistribution matrix become  ¯ ij (x, μ, x , μ ) = R WK β (K) ϕ(x)ϕ(x )T˜0K (i, μ)T˜0K (j, μ ), i, j = 0, 1. K=0,2

(14.97) The integration over frequencies yields the scattering phase matrix in Eq. (13.64). The source vector S(τ, x) becomes independent of x and satisfies a convolution integral equation, established below. First, to emphasize the non-conservative properties of the redistribution matrix, we introduce the parameter K = (1 + I )β (K)WK = W

1 + I WK , 1 + I + δ (K)

(14.98)

and the diagonal matrix  = diag[W 2 ], 0 , W W

0 = 1, W

2 = W

1 + I W2 . 1 + I + δ (2)

(14.99)

326

14 Polarized Radiative Transfer Equations

We also introduce =

I . 1 + I

(14.100)

The source vector S(τ ) may be then be written as  S(τ ) = (1 − )

+∞

−∞

1 2



+1 −1

∗ W  (μ)ϕ(x)I(τ, x, μ) dμ dx + Q (τ ),

(14.101)

    The elements of   (μ) are given by Eq. (14.50) with W W(μ) W.  (μ) = W 2 . with W replaced by W For a semi-infinite medium, S(τ ) satisfies the vector Wiener–Hopf integral equation 

∞

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

(14.102)

  |τ |ϕ(x) dμ dx. W  (μ)ϕ (x) exp − μ μ

(14.103)

S(τ ) = (1 − ) 0

with the kernel 1 K(τ ) ≡ 2



+∞  1 −∞

0

2

The matrix K(τ ) is symmetric. Its elements have the normalization 

+∞ −∞

K11 (τ ) dτ = 1,



+∞ −∞

K12 (τ ) dτ = 0,



+∞ −∞

K22 (τ ) dτ =

7  W2 . 10 (14.104)

2 is the depolarization parameter defined in Eq. (14.99). The same normalHere W ization holds for a mixture of Rayleigh and isotropic scattering (see Eq. (14.32)), 2 replaced by W . The element K11 (τ ) is equal to the complete frequency with W 2 = redistribution kernel K(τ ). We show in Fig. 14.1, for the Doppler profile and W 1, the elements of the matrix K(τ ). The left panel shows K11 (τ ), K12 (τ ) = K21 (τ ), and K22 (τ ). The right panel shows the same curves as in Ivanov et al. (1997), namely the logarithm of K11 (τ ) (left vertical axis) and the ratios K12 (τ )/K11 (τ ) and K22 (τ )/K11 (τ ) (right vertical axis). One can observe that the ratio K22 (τ )/K11 (τ ) is smaller than one, except for values of τ smaller than 10−1 , which have a negligible contribution to the integral of K22 (τ ) and that K12 (τ ) is positive for τ small and negative for τ large. One can also observe that K12 (τ )/K11 (τ ) and K22 (τ )/K11 (τ ) are constant for log τ larger than 1, which means that K12 (τ ) and K22 (τ ) have the same large τ algebraic asymptotic behavior as K11 (τ ). The latter has the same behavior as the complete frequency √ redistribution kernel K(τ ), which, for the Doppler profile, is K(τ ) ∼ 1/(τ 2 ln τ ). The τ -dependence of the elements of K(τ )

14.4 Resonance Polarization of Spectral Lines

327

Fig. 14.1 The elements of the kernel matrix K(τ ) for the Doppler profile, with the depolarization parameter set equal to 1. The left panel shows the elements Kij (τ ) and the right panel shows log K11 (τ ) (left vertical axis) and the ratios K12 (τ )/K11 (τ ) and K22 (τ )/K11 (τ ) (right vertical axis)

is also shown for the Doppler profile in, e.g., Faurobert-Scholl and Frisch (1989) and Faurobert-Scholl et al. (1997). The matrix kernel can also be written as a Laplace transform. Combining the frequency variable x and the direction variable μ into a single variable ξ = μ/ϕ(x) as in Sect. 2.2.3, we can write  K(τ ) =

∞

ˆ )e−|τ |/ξ g(ξ

0

dξ = ξ



∞

M(ν)e−ντ dν,

(14.105)

0

with ˆ )= g(ξ



∞ y(ξ )

2 W  [ξ ϕ(u)]ϕ (u) du,

M(ν) =

1 1 ˆ ), g( ν ν

(14.106)

and  y(ξ ) =

0

0 < ξ ≤ 1/ϕ(0),

ϕ (1/ξ )

ξ ≥ 1/ϕ(0).

−1

(14.107)

The inverse Laplace transform M(ν), now a matrix, plays an important role in the following chapters, where we discuss methods of solution for vector Wiener–Hopf integral equations.

14.4.2 Multi-Dimensional Medium The (KQ) decomposition of the radiation field for a multi-dimensional medium was developed in Anusha and Nagendra (2011a) for resonance polarization and

328

14 Polarized Radiative Transfer Equations

in Anusha and Nagendra (2011b) for the Hanle effect. It has been applied to the polarization of rotating disks in Mili´c and Faurobert (2014). The radiative transfer equation for three-component Stokes vector I (r, x, ), with components I , Q, U , may be written as .∇I (r, x, ) = −σ (r)ϕ(x)[I (r, x, ) − S(r, x, )].

(14.108)

Here r is the position in the medium,  the beam direction, σ (r) an absorption coefficient per unit length, and ϕ(x) the line absorption profile, normalized to unity. The source vector S(r, x, ) has the form S(r, x, ) = S sc (r, x, ) + Q∗ (r),

(14.109)

where  sc

S (r, x, ) ≡

+∞ −∞

d

1 ¯ R(x, , x ,  )I (r, x ,  ) dx . ϕ(x) 4π

(14.110)

We assume for simplicity that the primary source term Q∗ (r) depends only on the position in the medium and that there is no incident radiation on the medium. It is shown in Sects. 14.2.2 and 14.3.1 how the introduction of a diffuse field allows one to treat more complex problems. The elements of the redistribution matrix may be written as in Eq. (14.84). The construction of a (KQ) expansion of the source vector S(r, x, ) in terms K (r, x), independent of the direction, goes as described in of six functions SQ Sect. 14.3.2 for Rayleigh scattering. First one observes that the components of the scattering term S sc (r, x, ) may be written as   Sisc (r, x, ) = WK TQK (i, )JQK (r, x), (14.111) K

Q

where JQK (r, x)

=



 WK

+∞ r¯ (x, x ) K −∞

ϕ(x)

2   j =0

[TQK (j,  )]∗ Ij (r, x ,  )

d

dx . 4π (14.112)

K (j,  ) = [T K (j,  )]∗ , where ∗ stands for complex We have used (−1)Q T−Q Q √ √ √ conjugate. The factor WK has been written WK × WK with one factor WK K (j,  ) by analogy with the oneassociate to TQK (i, ) and the other one to T−Q dimensional case.

14.4 Resonance Polarization of Spectral Lines

329

Expanding the components Q∗i (r) of the primary source term as   WK TQK (i, )QK Q (r),

Q∗i (r) =

K

i = 0, 1, 2.

(14.113)

Q

we can write S i (r, x, ) =



K WK TQK (i, )SQ (r, x),

i = 0, 1, 2,

(14.114)

KQ

with K SQ (r, x) = JQK (r, x) + QK Q (r).

(14.115)

The three-component source vector S(r, x, ) has been replaced by a sixK (r, x). component vector S(r, x), independent of the direction, with components SQ It remains to construct an expansion for the radiation field. Inserting the expansion of the source term into the radiative transfer equation for the Stokes vector I (r, x, ) and using the formal solution of the equation, it can be shown that the components of the Stokes vector can be written as Ii (r, x, ) =

 K WK TQK (i, )IQ (r, x, ),

i = 0, 1, 2.

(14.116)

KQ

The details of the proof can be found in Anusha and Nagendra (2011a). The final step is the construction of a radiative transfer equation for the K (r, x, ). Introducing Eq. (14.116) into Eq. (14.112), we can write components IQ JQK (r, x) ×

2  

=



 WK

[TQK (j,  )]∗

+∞ −∞

r¯K (x, x ) ϕ(x)





K Q

j =0



K



WK TQK (j,  )IQ

(r, x ,  )

d

dx . (14.117) 4π

In matrix notation, this equation becomes  J (r, x) =

+∞ −∞

T(x, x ) ϕ(x)

 √

√ d

dx , W M ( ) W I(r, x ,  ) 4π (14.118)

330

14 Polarized Radiative Transfer Equations

where J (r, x) and I(r, x ,  ) are six-component vectors with components K (r, x, ), respectively. The matrix  () is a 6 × 6 matrix JQK (r, x) and IQ M with elements

KK  QQ

( ) =

j =2 



[TQK (j,  )]∗ TQK (j,  ).

(14.119)

j =0

KK (), known as the multipole coupling coefficients, have been The coefficients QQ

introduced in Landi Degl’Innocenti et al. (1990), where they are denoted GKQ,K Q . Explicit expressions can be found in this reference and in Landi Degl’Innocenti and Landolfi (2004, Appendix A.20). The matrix M () is a full matrix, in contrast with the block diagonal matrix of the one-dimensional case. This is the main difference between a one-dimensional and a multi-dimensional geometry. The other matrices are defined as follows:

W = diag[W0 , W2 , W2 , W2 , W2 , W2 ], ˆ rIII (x, x ), T(x, x ) = αˆ r¯II (x, x ) + (βˆ − α)¯ αˆ = αI, βˆ = diag[β (0), β (2) , β (2) , β (2), β (2) , β (2) ].

(14.120) (14.121) (14.122) (14.123)

Here I is the 6 × 6 unity matrix. The radiative transfer equation in the (KQ) representation has the same form as the radiative transfer equation for the Stokes vector, namely .∇I(r, x, ) = −σ (r)ϕ(x)[I(r, x, ) − S(r, x)],

(14.124)

where I(r, x, ) and S(r, x) are 6-component vectors constructed the corresponding (KQ) components. What has been gained with this decomposition is that the source term is independent of the direction  of the beam. The boundary K (r, x, ) are easily derived from the boundary conditions for conditions for IQ the components Ii (r, x, ). The source vector becomes independent of frequency when the coherent scattering term αˆ r¯II (x, x ) is neglected and rIII (x, x ) replaced by ϕ(x)ϕ(x ). Numerical solutions of Eq. (14.124) are carried out in Anusha and K ) and (I K ) defined in Eq. (14.77). Nagendra (2011a) with the real variables, (IQ Q

14.5 The Hanle Effect The main effects of a weak magnetic field on resonance polarization, known as the Hanle effect, are described in Sect. 13.4.1 and some detail on the redistribution matrix for a two-level atom with unpolarized ground level in the Appendix K of

14.5 The Hanle Effect

331

Chap. 13. The three Stokes parameters I , Q, and U are needed to describe the linear polarization of the radiation field since the presence of the magnetic field will break the cylindrical symmetry of the radiation field, in an otherwise cylindrically symmetric medium. For resonance polarization, we show in Sect. 14.4.2 how the source function vector for the three Stokes parameters I , Q, U can be expressed as a six-component vector independent of the direction of the beam. This decomposition is possible when the frequency dependent terms are replaced by their direction averages. For the Hanle effect, the same type of decomposition is not possible, even if the frequency dependent terms are replaced by their direction averages, because the frequency dependent terms involve the generalized profiles, which depend on the magnetic field (see the expression of the redistribution matrix given in Eqs. (K.15), and (K.2)). Bommier (1997) has proposed approximations for the redistribution matrix based on an asymptotic analysis of the generalized profiles in different frequency domains (see Appendix K in Chap. 13). They provide, for each domain, an explicit expression of the redistribution matrix in which the frequency redistribution terms are separated from the scattering phase matrix. When, moreover, the frequency redistribution terms are replaced by their direction averages, it is possible to perform in each domain a (KQ) decomposition. Here, for simplicity, we make the complete frequency redistribution assumption. It automatically separates the frequency redistribution from the scattering phase matrix but is valid only for lines without extended wings.

14.5.1 The Hanle Redistribution Matrix The Hanle redistribution matrix is introduced in Sect. 13.4 and in the Appendix K of Chap. 13. For a two-level atom with an unpolarized ground level, the elements of the redistribution matrix, in the magnetic reference frame and for an atom at rest, have the form 



Q K

TQK (i, )RKK [RH (ν, , ν ,  , B)]ij = Q (ν, ν , B)(−1) T−Q (j,  ). KK Q

(14.125)



The expression of RKK Q (ν, ν , B), which describes the frequency redistribution is given in Eq. (K.2). Here we make a number a simplifying assumptions: we neglect

the coherent scattering contribution and replace the generalized profiles KK Q (ν) √ by WK ϕ(ν), their limit for a zero magnetic field (see Eq. (K.8)). With these assumptions, the frequency redistribution matrix, in the magnetic reference frame, becomes

RKK Q (x, x , B) = WK

ϕ(x)ϕ(x ) . 1 + I + δ (K) + i H Q

(14.126)

332

14 Polarized Radiative Transfer Equations

The absorption profile ϕ(x) takes into account the Doppler effects. The dependence on the magnetic field strength B is contained in the Hanle factor H = 2i πνL / R . Typically, it has a value of few unity. Except for Q = 0, the product H Q is much larger than the destruction coefficients I and δ (K). To single out the role of the Hanle effect, it is convenient to write Eq. (14.126) as

RKK Q (x, x , B) =

WK

ϕ(x)ϕ(x ), 1 + i QHK

(14.127)

where WK =

WK , 1 + I + δ (K)

HK =

H . 1 + I + δ (K)

(14.128)

In the laboratory frame, the redistribution matrix may then be written as RH (x, , x ,  , B) = ϕ(x)ϕ(x )PH (,  , B),

(14.129)

where the elements of the phase matrix PH (,  , B) are given by [PH (,  , B)]ij =



WK

K



TQK (i, )

Q

 Q



K Q K NQQ T−Q (j,  ).

(B)(−1)

(14.130) K (B), which can be written as The magnetic field vector B appears in NQQ



K i (Q−Q )χB NQQ

(B) = e



K K dQQ

(θB )dQ

Q (−θB )

Q

1 , 1 + i Q

HK

(14.131)

with χB and θB the azimuthal and polar angles of the magnetic field. K (B) can serve to construct a 6 × 6 matrix N(B), which can The elements NQQ

be written as N(B) = R(χB )MH (θB )R(−χB ),

(14.132)

with  −1 D(−θB ). MH (θB ) = D(θB )[I + i H2 Q]

(14.133)

ˆ and R(χB ) are diagonal, The matrices Q  = diag[0, 0, −1, +1, −2, +2], Q R(χB ) = diag[1, 1, e−i χB , e+i χB , e−2i χB , e+2i χB ].

(14.134) (14.135)

14.5 The Hanle Effect

333

The matrix N(B) reduces to the matrix unity when the magnetic field is zero. The matrix D(θB ) has the block structure  D(θB ) =

 1 0 , 0 D(2) (θB )

(14.136)

where D(2) (θB ) is a full 5 × 5 matrix. Its elements are the reduced rotation matrices 2 (θ ) (see e.g. LL04, p. 54). They are real numbers and satisfy a number of dQQ

B symmetry relations, such as 2 2 (β) = (−1)M−N d−M−N (β), dMN 2 (β), = (−1)M−N dNM 2 (−β). = (−1)M−N dMN

(14.137)

D(2) (−θB ) = [D(2)]T (θB ) = [D(2) ]−1 (θB ),

(14.138)

They lead to

and D(2)(−θB )D(2)(θB ) = I.

(14.139)

The same relations hold for the matrix D(θB ) itself. We note here that M−1 H (θB ) has a simple expression. Combining Eqs. (14.133) and (14.138), we can write



 M−1 H (θB ) = D(θB )[I + i H2 Q]D(−θB ) = I + i H2 O(θB ),

(14.140)

where   B ) ≡ D(θB )QD(−θ O(θ B ).

(14.141)

 B ) has the block structure shown in Eq. (14.136). The 5 × 5 lower The matrix O(θ (2) (θB ), may be written as block denoted, O 

⎡ 0

3 2

 sin θB

⎢ ⎢ 3 ⎢ 2 sin θB − cos θB  (2) (θB ) = ⎢ O ⎢ 3 0 ⎢ 2 sin θB ⎢ ⎣ 0 sin θB 0 0

3 2

⎤ sin θB

0

0

0

sin θB

0

cos θB

0

sin θB

0 sin θB

−2 cos θB 0 0 2 cos θB

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(14.142)

334

14 Polarized Radiative Transfer Equations

 The matrices MH (θB ), M−1 H (θB ), and O(θB ) are symmetric. This is important for the discussions in Chaps. 17 and 18. We consider first the one-dimensional case and then the multi-dimensional one. The first case resemble very much Rayleigh scattering in a one-dimensional medium with a non-cylindrically symmetric radiation field and the second one, resonance polarization in a multi-dimensional medium. In the first case, thanks to the assumption of complete frequency redistribution, it is possible to construct a Wiener–Hopf type integral equation for the source vector.

14.5.2 One-Dimensional Medium In a one-dimensional medium, the radiative transfer equation for the Stokes vector I = (I, Q, U ) may be written as μ

∂I (τ, x, ) = ϕ(x)[I (τ, x, ) − S(τ, )]. ∂τ

(14.143)

The assumption of complete frequency redistribution makes the source vector independent of the frequency. It can be written as S(τ, ) = S sc (τ, ) + Q∗ (τ ).

(14.144)

The primary source term Q∗ (τ ) is assumed to depend only on the position in the medium and, for simplicity, to be unpolarized. The scattering term is given by  S sc (τ, ) ≡

+∞ −∞

PH (,  , B)ϕ(x )I (τ, x ,  )

d

dx . 4π

(14.145)

Although it is not indicated explicitly, all the quantities involving the radiation field depend on the magnetic field B. We now apply the (KQ) decomposition method described in Sect. 14.4.2. Using the expression of the Hanle phase matrix given in Eq. (14.130), the components of the scattering term may be written as Sisc (τ, ) =

 K

WK



TQK (i, )



K K NQQ

(B)JQ (τ ),

i = 0, 1, 2,

Q

Q

(14.146) with JQK (τ ) =



WK



2  +∞  −∞ j =0



K





(−1)Q T−Q

(j,  )ϕ(x )Ij (r, x ,  )

d

dx . 4π (14.147)

14.5 The Hanle Effect

335

Adding the primary source term and expanding it as in Eq. (14.113), we can write Si (τ, ) =



K WK TQK (i, )SQ (τ ),

i = 0, 1, 2,

(14.148)

KQ

with K SQ (τ ) =



K K K NQQ

(B)JQ (τ ) + QQ (τ ).

(14.149)

Q

One can then proceed exactly as in Sect. 14.4.2. The construction of the formal solution of Eq. (14.143) shows that the components the radiation field can be written as  K Ii (τ, x, ) = WK TQK (i, )IQ (τ, x, ), i = 0, 1, 2. (14.150) KQ K (τ, x, ) satisfy the transfer equation The components IQ

μ

K ∂IQ

∂τ

K K (τ, x, ) = ϕ(x)[IQ (τ, x, ) − SQ (τ )].

(14.151)

K (τ ) depend only on τ , the components I K (τ, x, ) Because the components SQ Q depend only on μ. The azimuthal dependence of the Stokes vector I (τ, x, ) is contained in the spherical tensor TQK (i, ). Combining now Eqs. (14.147) and (14.150), we can write

JQK (τ )

=



×

K Q

2  

+∞ 

j =0 −∞



K WK (−1)Q T−Q (j, )

K WK TQK (j, )ϕ(x)IQ

(τ, x, μ)

d dx. 4π

(14.152)

As pointed out in Sect. 14.3.2, the product of the two spherical tensors contains a

factor ei (Q −Q)χ . Upon integration over the solid angle d, the contribution of this term is zero unless Q = Q. We thus finally obtain JQK (τ ) =

 K

+∞

−∞

1 2



+1  −1

 KK

K

WK Q (μ) WK ϕ(x)IQ (τ, x, μ) dμ dx, (14.153)

336

14 Polarized Radiative Transfer Equations

with

KK Q (μ) =

2 

K (−1)Q T˜−Q (j, μ)T˜QK (j, μ),

K, K = 0, 2.

(14.154)

j =0

In vector notation, Eq. (14.153) can be written as  J (τ ) =

+∞

−∞



1 2

+1 −1

W (μ)ϕ(x)I(τ, x, μ) dμ dx,

(14.155)

√ √ W (μ) W .

(14.156)

where W (μ) =

The 6 × 6 matrix (μ) is given in Eq. (14.70) and W = diag[

1 , W2 , W2 , W2 , W2 , W2 ], 1 + I

(14.157)

with W2 = 1/(1 + I + δ (2) ). To summarize, the 6-dimension vector I(τ, x, μ) with K (τ, x, μ) satisfies the radiative transfer equation components IQ μ

∂I (τ, x, μ) = ϕ(x)[I(τ, x, μ) − S(τ )]. ∂τ

(14.158)

K (τ ) may be written The source term S(τ ) with components SQ

 S(τ ) = N(B)

+∞

−∞

1 2



+1 −1

W (μ)ϕ(x)I(τ, x, μ) dμ dx + Q∗ (τ ).

(14.159)

where Q∗ (τ ) is the primary source with components QK Q (τ ). The matrix N(B), which describes the effects of the magnetic field, is defined in Eq. (14.132). The source vector S(τ ) satisfies a convolution integral equation easy to obtain by the standard method. Inserting the formal solution of Eq. (14.158) into Eq. (14.159), assuming that the medium is semi-infinite, that there is no incident radiation, and introducing the parameter  = I /(1 + I ), we obtain the Wiener–Hopf type integral equation 

∞

S(τ ) = (1 − )N(B)

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

(14.160)

0

for the 6-dimension vector S(τ ). An identical integral equation has been given in Faurobert-Scholl (1991, Eq. (39)) for a 6-dimension vector !(τ ). The equation for !(τ ) is constructed with an azimuthal Fourier expansion of the polarization

14.5 The Hanle Effect

337

phase matrix. Except for the multiplication by the matrix N(B), Eq. (14.160) has the same structure as the Wiener–Hopf integral equation for resonance scattering in Eq. (14.102).  K(τ ) is a 6 × 6 matrix, also given by Eq. (14.103), where  The kernel   The matrix (μ) has the block structure in Eq. (14.70) W W(μ) W. (μ) =  and  = diag[1, W 2 , W 2 , W 2 , W 2 , W 2 , W 2 ]. W

(14.161)

2 is defined in Eq. (14.99). The depolarization parameter W Although it is not explicitly written, the vector S(τ ) depends on the direction of the magnetic field vector, that is on the angles χB and θB . Since it is assumed that there is no incident radiation and that the primary source term Q∗ (τ ) is isotropic, the cylindrical symmetry of the problem is broken exclusively by the magnetic field. As a consequence, the vector S(τ ) has a simple dependence on χB . The expression of N(B) given in Eq. (14.132) suggests the Ansatz S(τ ) = R(χB )S 0 (τ ),

(14.162)

where S 0 (τ ) depends only on θB . Introducing Eq. (14.162) into Eq. (14.160) and using Eq. (14.132), we see that S 0 (τ ) satisfies the integral equation  S 0 (τ ) = (1 − )M(θB )R(−χB )

∞

K(τ − τ )R(χB )S 0 (τ ) dτ + R(−χB )Q∗ (τ ).

0

(14.163) It is easy to verify, using Eq. (14.135) and remembering that K(τ ) has the same block diagonal structure than (μ), that R(−χB )K(τ )R(χB ) = K(τ ).

(14.164)

The assumption that Q∗ (τ ) depends only on τ and is unpolarized implies that Q∗ (τ ) has only one non-zero component, namely Q00 (τ ). Hence R(−χB )Q∗ (τ ) = Q∗ (τ ).

(14.165)

This relation holds for any cylindrically symmetric primary source, since the only non-zero components will be Q00 (τ ) and Q20 (τ ). The integral equation for S 0 (τ ) reduces thus to 

∞

S 0 (τ ) = (1 − )M(θB )

K(τ − τ )S 0 (τ ) dτ + Q∗ (τ ).

(14.166)

0

The dependence of S(τ ) on χB is simply given by Eq. (14.162). The Hanle effect and resonance scattering leading to similar integral equations, they can be treated with the same numerical techniques. The only important difference is the

338

14 Polarized Radiative Transfer Equations

multiplication by MH (θB ). A detailed numerical investigation of Eq. (14.166) is presented in Nagendra et al. (1998).

14.5.3 Multi-Dimensional Medium It is shown in Sect. 14.4.2, for resonance polarization, how to apply a (KQ) decomposition for a multi-dimensional medium and in Sect. 14.5.2 how to treat the one-dimensional Hanle effect. It suffices to combine the results of these two sections to obtain a (KQ) decomposition of the radiation field for the Hanle effect in a multi-dimensional medium. As in Sect. 14.5.2, we assume complete frequency redistribution and an unpolarized primary source depending only on the position in the medium. The radiative transfer equation for the radiation field is .∇I (r, x, ) = −σ (r)ϕ(x)[I (r, x, ) − S(r, )].

(14.167)

The notation is the same as in Sect. 14.4.2. The source term is independent of the frequency because of the assumption of complete frequency redistribution. Its components can be written as in Eq. (14.148) and (14.149) with τ replaced by r. The solution of the radiative transfer equation shows that the Stokes parameters can be written as in Eq. (14.116), namely as Ii (r, x, ) =

 K WK TQK (i, )IQ (r, x, ),

i = 0, 1, 2.

(14.168)

KQ K (r, x, ) satisfies the The 6-dimension vector I(r, x, ) with components IQ radiative transfer equation

.∇I(r, x, ) = −σ (r)ϕ(x)[I(r, x, ) − S(r)].

(14.169)

The 6-component source vector has the form S(r) = N(B)J (r) + Q∗ (r).

(14.170)

The matrix N(B), which describes the effects of the magnetic field, is defined in Eq. (14.132) and  J (r) =

+∞  −∞

√ √ d

dx . W M ( ) W ϕ(x )I(r, x ,  ) 4π

(14.171)

References

339

The matrix W is diagonal. It has the form given in Eq. (14.120) with W0 = W0 = 1/(1 + I ) and W2 = W2 = 1/(1 + I + δ (2) ). The elements of the 6 × 6 matrix M (), introduced in Eq. (14.119), are

KK  QQ

( ) =

j =2 



[TQK (j,  )]∗ TQK (j,  ).

(14.172)

j =0

The matrix M () is a full matrix in contrast to the matrix (μ) of the onedimension medium. This (KQ) decomposition of the radiation field for a multi-dimensional medium, presented here for complete frequency redistribution, is implemented in Anusha and Nagendra (2011b) for lines formed with partial frequency redistribution, with the approximation proposed in Bommier (1997), whereby the frequency domain is decomposed into several regions and in each region the frequency redistribution is decoupled from the polarization phase matrix.

References Anusha, L.S., Nagendra, K.N.: Polarized Line formation in multidimensional media. I. Decomposition of the Stokes parameters in arbitrary geometries. Astrophys. J. 726, 6–19 (2011a) Anusha, L.S., Nagendra, K.N.: Polarized Line formation in multidimensional media. III. Hanle effect with partial frequency redistribution. Astrophys. J. 738, 116 (19pp) (2011b) Bommier, V.: Master equation applied to the redistribution of polarized radiation, in the weak radiation field limit. II. Arbitrary magnetic field case. Astron. Astrophys. 328, 726–751 (1997) Bommier, V., Derouich, M., Landi DeglInnocenti, E., Molodij, G., Sahal-Bréchot, S.: Interpretation of the second solar spectrum observations of the Sr I 4607 Å line in a quiet region : Turbulent magnetic field strength determination. Astron. Astrophys. 432, 295–305 (2005) Chandrasekhar, S.: On the radiative equilibrium of a stellar atmosphere. XI. Astrophys. J. 104, 110–132 (1946) Chandrasekhar, S.: Radiative Transfer. Dover Publications, New York (1960); First edition, Oxford University Press (1950) Domke, H.: Radiation transport with Rayleigh scattering. I. Semiinfinite atmosphere. Soviet Astron. 15, 266–275 (1971); translation from Astron. Zhurnal 50, pp. 126–136 (1971) Faurobert-Scholl, M.: Hanle effect with partial frequency redistribution. I. Numerical methods and first applications. Astron. Astrophys. 246, 469–480 (1991) Faurobert-Scholl, M.: Investigation of microturbulent magnetic fields in the solar photosphere by their Hanle effect in the Sr I 4607 Å line. Astron. Astrophys. 268, 765–774 (1993) Faurobert-Scholl, M., Frisch, H.: Asymptotic analysis of resonance polarization and escape probability approximations. Astron. Astrophys. 219, 338–351 (1989) Faurobert-Scholl, M., Frisch, H., Nagendra, K.N.: An operator perturbation method for polarized line transfer I. Non-magnetic regime in 1D media. Astron. Astrophys. 322, 896–910 (1997) Fluri, D., Stenflo J.O.: Continuum polarization in the solar spectrum. Astron. Astrophys. 341, 902–911 (1999) Frisch, H.: The Hanle effect. Decomposition of the Stokes parameters into irreducible components. Astron. Astrophys. 476, 665–674 (2007) Hummer, D.G.: Non-coherent scattering I. The redistribution functions with Doppler broadening. Mon. Not. R. Astron. Soc. 125, 21–37 (1962)

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14 Polarized Radiative Transfer Equations

Ivanov, V.V.: Generalized Rayleigh scattering I. Basic theory. Astron. Astrophys. 303, 609–620 (1995) Ivanov, V.V, Grachev, S.I., Loskutov, V.M.: Polarized line formation by resonance scattering I. Basic formalism. Astron. Astrophys. 318, 315–326 (1997a) Landi Degl’Innocenti, E., Landolfi, M.: Polarization in Spectral Lines. Kluwer Academic Publisher, Dordrecht (2004) Landi Degl’Innocenti, E., Bommier, V., Sahal-Bréchot, S.: Resonance line polarization and the Hanle effect in optically thick media I. Formulation for the two-level atom. Astron. Astrophys. 235, 459–471 (1990) Mili´c, I., Faurobert, M.: Scattering line polarization in rotating, optically thick disks. Astron. Astrophys. 571, A79, 14 pp (2014) Nagendra, K.N., Frisch, H., Faurobert-Scholl,M.: An operator perturbation method for polarized line transfer III. Applications to the Hanle effect in 1D media. Astron. Astrophys. 332, 610–628 (1998) Rees, D.E.: Non-LTE resonance line polarization in the absence of magnetic fields. Publ. Astron. Soc. Japan 30, 455–466 (1978) Sampoorna, M., Nagendra, K. N., Frisch, H.: Spectral line polarization with angle-dependent partial frequency redistribution II. Accelerated lambda iteration and scattering expansion methods for Rayleigh scattering. Astron. Astrophys. 527, A89 (15p) (2011) Sampoorna, M., Nagendra, K.N., Senflo, J.O.: Polarized line formation in arbitrary strength magnetic fields. Angle-averaged and angle-dependent partial frequency redistribution. Astrophys. J. 844, 97 (10 pp.) (2017) Siewert, C.E., Kelley, C.T., Garcia, R.D.M.: An analytical expression for the H-Matrix relevant to Rayleigh scattering. J. Math. Anal. Appl. 84, 509–518 (1981) Silant’ev, N.A., Alekseeva, G.A., Novikov, V.V.: Depolarization of multiple scattered light in atmospheres due to anisotropy of small grains and molecules. II. The problems with sources. Astrophys. Space Sci. 357, 53–62 (2015) Sobolev, V.V.: Non-coherent light scattering in stellar atmospheres. Astron. Z. 26, 129–137 (1949) Sobolev, V.V.: A Treatise on Radiative Transfer. Von Nostrand Company, Princeton, New Jersey (1963), transl. by S.I. Gaposchkin; Russian original, Transport of Radiant Energy in Stellar and Planetary Atmospheres, Gostechizdat, Moscow (1956) Stenflo, J.O.: Solar Magnetic Fields: Polarized Radiation Diagnotics. Kluwer Academic Publisher, Dordrecht (1994) Stenflo, J.O., Stenholm, L.: Resonance-line polarization. II. Calculations of linear polarization in solar UV emission lines. Astron. Astrophys. 46, 69–79 (1976)

Chapter √ 15

The -Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations

For scalar scattering problems, we showed in Part I that many exact results for semi-infinite media can be derived with the so-called resolvent method, based on convolution-type integral equations satisfied by the Green and the resolvent functions. For a polarized radiation field these functions become matrices or vectors but, as we show in this chapter, they satisfy similar convolution integral equations to which can be applied the same methods of solution as in the scalar case. For a polarized radiation field, as we show in Chap. 14, it is possible with a (KQ) representation of the radiation field to obtain convolution-type integral equations for a source vector S(τ ). They exist for monochromatic scattering and spectral lines, provided they are formed with complete frequency redistribution, whether a magnetic field is present or not. These√equations are the starting points of the results given in this chapter, in particular -laws and nonlinear integral equations for H-matrices. The chapter is organized as follows. Section 15.1 serves as introductory section. We recall the Wiener–Hopf integral equation satisfied by the source term S(τ ) and show that the derivative of S(τ ) also satisfies a Wiener–Hopf integral equation, exactly as in the scalar case. We also introduce a matrix representation of the radiation field suggested in Ivanov (1995), by which the vectors representing the KQ-components of the radiation field √ and of the source term can be replaced by matrices. We use them to construct -laws. √ Section 15.2 is devoted to the construction of -laws for monochromatic scattering, resonance polarization and the Hanle effect. We stress again that  is not a small expansion parameter, but a destruction probability per scattering with values in [0, 1]. The proof is based on a quadratic expression involving the source term and its derivative. It is a generalization of the proof given in Sect. 11.1.1 for the scalar case. Section 15.3 is devoted to the Green matrix G(τ, τ0 ) and associated matrices such as the surface value of the Green matrix G(τ, 0) and its regular part the resolvent (τ ). We discuss symmetry properties and introduce the associated Wiener–Hopf integral equations. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_15

341

342

15 The

√

-Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations

The two subsequent sections are devoted to the H-matrix. It is introduced in Sect. 15.4, where it is defined as the Laplace transform of the surface Green matrix G(τ, 0). In the same section, we construct nonlinear H-equations for monochromatic scattering, resonance polarization and the Hanle effect. In Sect. 15.5, we introduce alternative definitions of the H-matrix based on the emergent radiation field for a uniform primary source and on the surface value of the source matrix for an exponential primary source. All the proofs are direct generalizations to scattering polarization of the proofs given in Chap. 11 for the scalar case. Finally, we show in Sect. 15.6, that for a semi-infinite medium, the convolutiontype integral equations for the source vector and resolvent matrix can be transformed into matrix or vector linear singular integral equations. We also introduce the dispersion matrix and examine some of its properties. The linear equations are used in subsequent chapters to obtain exact results.

15.1 The Source Vector: Its Derivative and Matrix Representation In Chap. 14, we have considered several scattering processes: conservative and nonconservative Rayleigh scattering, resonance polarization and the Hanle effect. For each process we show how a factorization of the polarization matrix allows one to simplify the radiative transfer equations in such a way that the source term becomes independent of the direction of the beam and, for a one dimensional medium, satisfies a convolution integral equation. For a semi-infinite medium, except for the Hanle effect, this equation can be written as  S(τ ) = (1 − )

∞

K(τ − τ )S(τ ) dτ + Q∗ (τ ).

(15.1)

0

For conservative Rayleigh scattering this equation is established in Sect. 14.2.2 and for non-conservative Rayleigh scattering in Sect. 14.3.1. Actually, for conservative scattering  = 0, and for the Milne problem Q∗ (τ ) = 0. For resonance polarization, this Wiener–Hopf integral equation is established in Sect. 14.4.1. It holds for spectral lines formed with complete frequency redistribution. For the Hanle effect, Eq. (15.1) becomes  S(τ ) = (1 − )M(θB )

∞

K(τ − τ )S(τ ) dτ + Q∗ (τ ).

(15.2)

0

It is established in Sect. 14.5.2. The angle θB is the colatitude of the direction of the magnetic field. The matrix MH (θB ) is defined in Eq. (14.133). For the Hanle effect, S(τ ) is a 6-dimension vector, constructed with the (KQ) representation of the radiation field. For Rayleigh scattering and resonance polarization, S(τ ) is a 2-dimension vector.

15.1 The Source Vector: Its Derivative and Matrix Representation

343

√ In Sect. 15.2 of the present chapter, we establish a -law for non-conservative Rayleigh scattering, the resonance polarization, and the Hanle effect. The proof is based on a quadratic quantity involving S(τ ) and its derivative S (τ ) = dS/dτ . For Rayleigh scattering and resonance polarization, S (τ ) satisfies the integral equation dS(τ ) = (1−) dτ



∞

K(τ −τ )S (τ ) dτ +(1−)K(τ )S(0)+

0

dQ∗ (τ ) . dτ

(15.3)

The kernel Kτ ) is defined in Eq. (14.24) for monochromatic scattering and in Eq. (14.103) for resonance polarization. The simplest way to find this equation is, as in the scalar case, to consider the derivative of S(τ ) as the limit of [S(τ +h)−S(τ )] for h → 0 (see Eq. (2.89)). For the Hanle effect, S (τ ) satisfies Eq. (15.3), the two first terms being multiplied by MH (θB ). √ The proof of -laws that will be given in Sect. 15.2 also makes use of Eq. (14.104), giving the norm (integral over the full space) of the elements of K(τ ). This equation allows one to write  (1 − )

+∞

−∞

K(τ ) dτ = I − E,

(15.4)

where I is the matrix unity and E a diagonal matrix. For Rayleigh scattering and resonance polarization, E is a 2 × 2 matrix:  E=

  0 , 0 Q

(15.5)

where Q ≡ 1 − (1 − )

7 Wp . 10

(15.6)

Here Wp is the depolarization parameter smaller than unity, denoted W for Rayleigh 2 for resonance polarization and the Hanle effect. scattering and W The constant Q plays for the polarization a role similar to that of the constant  for the intensity. In contrast to , the value of Q is larger than 3/10. This difference has its importance for numerical methods of solutions. In contrast to Stokes I , the polarization can be calculated by a standard -iteration (Nagendra and Sampoorna 2009). The limit Q tending to zero has been investigated in detail under the name of Generalized Rayleigh Scattering, in a series of articles (Ivanov 1995; Ivanov et al. 1995; Ivanov 1996; Ivanov et al. 1996).

344

15 The

√

-Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations

For the Hanle effect, the kernel K(τ ) is given by Eq. (14.103) with the matrix W  (μ) given in Eq. (14.70). The normalization of the kernel can also be written as in Eq. (15.4). Now E is a diagonal 6 × 6 matrix:  E=

  0 , 0 EQ

(15.7)

where EQ is a 5 × 5 diagonal matrix, all the elements of which are also equal to Q . As pointed out in Ivanov (1995), it is possible and often more convenient to represent a polarized radiation field by a matrix instead of a vector. To make this transformation, it suffices to represent the primary source vector by a diagonal matrix. For Rayleigh scattering or resonance polarization, the two-component primary source vector Q∗ (τ ) = (Q∗1 (τ ), Q∗2 (τ )) serves to define the diagonal matrix Q∗ (τ ) = diag[Q∗1 (τ ), Q∗2 (τ )].

(15.8)

The vector S(τ ) is related to its matrix representation S(τ ) by   1 S(τ ) = S(τ ) . 1

(15.9)

The components S1 (τ ) and S2 (τ ) of the vector S(τ ) are thus related to the elements of the matrix S(τ ) by S1 (τ ) = S11 (τ ) + S12 (τ ), S2 (τ ) = S21 (τ ) + S22 (τ ).

(15.10)

The vector radiation fields I(τ, μ) and I(τ, x, μ) are related to their matrix representations I(τ, μ) and I(τ, x, μ) as in Eq. (15.9). For the Hanle effect, the sixdimension source vector S(τ ) can in the same way be represented by a 6 ×6 matrix.

15.2 The

√ -Law

√ In Sect. 11.1, we have established the -law, for the scalar case, essentially reproducing a proof given in Frisch and Frisch (1975), which is based on the introduction of a properly chosen quadratic quantity. We recall that this law provides the exact expression for the surface value of the source function for a semiinfinite medium with a uniform primary source and that it is a consequence of the symmetry property of the kernel. √ The proof given in Frisch and Frisch (1995) is summarized in Sect. 21.3. The -law has been generalized to polarization problems, by Ivanov (1990) for Rayleigh scattering and resonance polarization,

15.2 The

√

345

-Law

and by Landi Degl’Innocenti and Bommier (1994) for resonance scattering in the presence of a magnetic field of arbitrary strength, in particular for the Hanle effect. The proofs by Ivanov (1990) and by Landi Degl’Innocenti and Bommier (1994) are also based on the introduction of quadratic quantities. The same approach is employed in Ivanov (1995) to construct a somewhat more general formula for non-symmetric kernels and depth-dependent primary sources, referred to as the Rybicki–Hopf–Bronstein general formula. In this section we show how the quadratic approach can be employed to construct √ a -law for Rayleigh scattering, resonance polarization and the Hanle effect. We use the matrix representation of the source function introduced in Eq. (15.9), somewhat better adapted than the vector representation.

15.2.1 Rayleigh Scattering and Resonance Polarization The Rayleigh scattering and the resonance polarization can be treated together, since they lead to the same Wiener–Hopf integral equation namely, 

∞

S(τ ) = (1 − )

K(τ − τ )S(τ ) dτ + Q∗ ,

(15.11)

0

where Q∗ is a constant matrix. For the proof given below, it does not have to be diagonal. The solution of this equation tends to a constant, denoted S(∞), at infinity. Taking the limit τ → ∞ in Eq. (15.11) and using the normalization of the kernel in Eq. (15.4), simple algebra leads to S(∞) = E −1 Q∗ ,

(15.12)

where E = diag[, Q ]. We now introduce the quadratic quantity 

∞

F= 0

ST (τ )

dS(τ ) dτ. dτ

(15.13)

The matrix ST (τ ) is solution of  S (τ ) = (1 − ) T

∞

ST (τ )K(τ − τ ) dτ + [Q∗ ]T .

(15.14)

0

The symmetry of K(τ ) has been used to write Eq. (15.14). As shown above, the derivative of S(τ ) satisfies the integral equation dS(τ ) = (1 − ) dτ



∞ 0

K(τ − τ )S (τ ) dτ + (1 − )K(τ )S(0),

(15.15)

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15 The

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-Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations

where S (τ ) = dS/dτ . It is convenient to write the constant matrix F as F = F1 + F2 ,

(15.16)

with 

∞ ∞

F1 = (1 − ) 0

ST (τ )K(τ − τ )S (τ ) dτ dτ,

(15.17)

0

and 

∞

F2 = (1 − )

 T

S (τ )K(τ ) dτ S(0).

(15.18)

0

The kernel being an even function of τ , we can rewrite F1 as 

∞ ∞

F1 = (1 − ) 0

ST (τ )K(τ − τ )S (τ ) dτ dτ .

(15.19)

0

An exchange of the order of integration, a procedure justified in Frisch and Frisch (1975), and the use of Eq. (15.14) leads to 

∞

 ST (τ ) − [Q∗ ]T S (τ ) dτ.

(15.20)

S (τ ) dτ = F − [Q∗ ]T [S(∞) − S(0)].

(15.21)

F1 = 0

We thus obtain F1 = F − [Q∗ ]T



∞ 0

To calculate F2 , we set τ = 0 in Eq. (15.11). This leads to F2 = [ST (0) − [Q∗ ]T ]S(0).

(15.22)

It suffices now to perform the sum F = F1 + F2 to obtain ST (0)S(0) = [Q∗ ]T S(∞). Inserting the expression of S(∞) given in Eq. (15.12), we obtain the Rayleigh scattering and resonance polarization, namely ST (0)S(0) = [Q∗ ]T E −1 Q∗ .

(15.23) √ -law for

(15.24)

15.2 The

√

347

-Law

√ √ This -law for the matrix S(0) can be transformed into a -law for the vector S(0). Assuming that Q∗ is a diagonal matrix and using S(τ ) = S(τ )e = eT ST (τ ), where e = [1, 1]T, we obtain S(0) · S(0) = Q∗ · E −1 Q∗ ,

(15.25)

where Q∗ is the constant primary source vector and the dot stands for scalar product. In terms of the components S1 (τ ) and S2 (τ ), Eq. (15.25) becomes [S1 (0)]2 + [S2 (0)]2 =

[Q∗ ]2 [Q∗1 ]2 + 2 .  Q

(15.26)

Equation (15.26) can also be constructed with the integral equation for S(τ ) and the quadratic quantity 

∞

F = 0

S(τ ) ·

dS(τ ) dτ. dτ

(15.27)

√ We stress again that the -law is a consequence of the semi-infinite geometry of the medium and of the symmetry properties of the kernel. When the primary source is unpolarized, that is when Q∗2 = 0, the component S1 (0) is much larger than the component S2 (0), which describes the polarization. Equation (15.26) is then not very useful for testing numerical codes. To take advantage of this equation, one must introduce in the numerical code a polarized primary source term (see e.g. Ambartsumian 1942, also Frisch 1999).

15.2.2 The Hanle Effect For the Hanle effect, the integral equation for the source function matrix is  S(τ ) = (1 − )M(θB )

∞

K(τ − τ )S(τ ) dτ + Q∗ .

(15.28)

0

It is identical to Eq. (15.11), except for the multiplication by the matrix M(θB ) defined in Eq. (14.133). We consider the quadratic quantity 

∞

F= 0

ST (τ )M−1 (θB )

dS(τ ) dτ. dτ

(15.29)

Proceeding exactly as described in Sect. 15.2.1, we obtain ∗ T −1 ST (0)M−1 H (θB )S(0) = [Q ] MH (θB )S(∞).

(15.30)

348

15 The

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-Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations

Equation (15.28), taken in the limit τ → ∞, leads to −1 ∗ S(∞) = L−1 H (0)MH (θB )Q ,

(15.31)

LH (0) = M−1 H (θB ) − I + E.

(15.32)

with

It will be shown in Sect. 15.6.2 that LH (0) is the value at zero of the dispersion matrix. The 6 × 6 matrix E is defined in Eq. (15.7). We thus obtain −1 −1 ∗ T −1 ∗ ST (0)M−1 H (θB )S(0) = [Q ] MH (θB )LH (0)MH (θB )Q .

(15.33)

−1 We now introduce √ in the right-hand side the expression of MH (θB ) given in Eq. (14.133). The -law for the Hanle effect finally takes the form ∗ T −1 ∗ ST (0)M−1 H (θB )S(0) = [Q ] D(θB )[EH ] D(−θB )Q .

(15.34)

The elements of the matrix D(θB ), defined in Eq. (14.136), are the reduced rotation 2 (θ ). The 6 × 6 diagonal matrix [E ]−1 may be written as matrices dQQ

B H ˆ ˆ −1 [I + i H Q]. ˆ [EH ]−1 ≡ [I + i H2 Q][E + i H2 Q] 2 Its elements are 

1 1 (1 − i H2 )2 (1 + i H2 )2 (1 − 2i H2 )2 (1 + 2i H2 )2 . , , , , ,  Q Q − i H2 Q + i H2 Q − 2i H2 Q + 2i H2

(15.35)

(15.36)

The constant H2 , given by HK for K = 2, contains the effect of the magnetic field and of the depolarizing elastic collisions (see Eq. (14.128)). When the primary source term is unpolarized,√the matrix Q∗ is diagonal and has the form Q∗ = diag[Q∗ , 0, 0, 0, 0, 0]. The -law reduces to ST (0)M−1 H (θB )S(0) =

1 ∗2 [Q ] . 

In terms of the six dimension vectors S(τ ) and Q∗ , the

(15.37) √ -law takes the form

∗ −1 ∗ S(0) · M−1 H (θB )S(0) = Q · D(θB )[EH ] D(−θB )Q ,

(15.38)

15.3 The Green Matrix and the Resolvent Matrix

349

where the dot indicates a scalar product. It can be derived directly from the integral equation for the vector S(τ ) and the quadratic quantity 

∞

F =

S(τ ) · M−1 (θB )

0

For an unpolarized primary source, the

√

dS(τ ) dτ. dτ

(15.39)

-law reduces to

S(0) · M−1 H (θB )S(0) =

[Q∗ ]2 . 

(15.40)

K (τ ) describing The component S00 (τ ) being much larger than the components SQ the polarization, Eq. (15.40) is not very useful for testing numerical codes, unless a polarized primary source term is introduced in the numerical code.

15.3 The Green Matrix and the Resolvent Matrix In the scalar case, the Green function G(τ, τ0 ), its surface value G(0, τ ) = G(τ, 0), and the resolvent function (τ ), regular part of the latter, satisfy Wiener–Hopf integral equations, which can be used to construct solutions of radiative transfer problems. These functions become matrices for polarized radiation, but they also satisfy Wiener–Hopf integral equations, which can be used to generalize many of the relations established for the scalar case. In the present section we introduce the Green matrix and the resolvent matrix, treating Rayleigh scattering and resonance polarization in Sect. 15.3.1 and the Hanle effect in Sect. 15.3.2.

15.3.1 Rayleigh Scattering and Resonance Polarization The solution of the integral equation for S(τ ), given in Eq. (15.1), can be written as 

∞

S(τ ) =

G(τ, τ0 )Q∗ (τ0 ) dτ0 ,

(15.41)

0

where G(τ, τ0 ) is the Green function, here actually a 2 × 2 matrix. It satisfies the Wiener–Hopf integral equation  G(τ, τ0 ) = (1 − )

∞

K(τ − τ )G(τ , τ0 ) dτ + δ(τ − τ0 )I,

(15.42)

0

where δ denotes the Dirac distribution and I the identity matrix. For the Milne problem G(τ, τ0 ) is defined as the solution of Eq. (15.42) since Q∗ (τ0 ) = 0.

350

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-Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations

We now discuss the relation between G(τ, τ0 ) and G(τ0 , τ ). In the scalar case G(τ, τ0 ) = G(τ0 , τ ). When the Green function is a matrix, there is in general no simple relation between G(τ, τ0 ) and G(τ0 , τ ). Here, because the kernel is an even function of τ and a symmetric matrix, GT (τ0 , τ ) = G(τ, τ0 ).

(15.43)

To prove this result, we first remark that G(τ, τ0 ) is also solution of the integral equation 

∞

G(τ, τ0 ) = (1 − )

G(τ, τ )K(τ − τ0 ) dτ + δ(τ − τ0 )I.

(15.44)

0

The simplest method to prove this relation is to write the solution of the two integral equations for G(τ, τ0 ) as Neumann series expansions (that is as series expansions in powers of the transport operator). It is then easy to verify that Eqs. (15.42) and (15.44) lead to the same expansion. We now transpose Eq. (15.44). It leads to 

∞

T

G (τ, τ0 ) = (1 − )

KT (τ − τ0 )GT (τ, τ ) dτ + δ(τ − τ0 )I.

(15.45)

0

Making use of the fact that the kernel is a symmetric matrix and an even function of τ , we obtain, after a change of notation (τ → τ0 and then τ0 → τ ), 

∞

GT (τ0 , τ ) = (1 − )

K(τ − τ )GT (τ0 , τ ) dτ + δ(τ − τ0 )I.

(15.46)

0

The comparison of Eq. (15.46) with Eq. (15.42) leads to Eq. (15.43). The regular part of the Green matrix, defined by (τ, τ0 ) ≡ G(τ, τ0 ) − δ(τ − τ0 )I,

(15.47)

satisfies the integral equation 

∞

(τ, τ0 ) = (1 − )

K(τ − τ )(τ , τ0 ) dτ + (1 − )K(τ − τ0 ).

(15.48)

0

We now introduce two useful new matrices, G(τ ) ≡ G(τ, 0),

(τ ) ≡ (τ, 0),

(15.49)

related by (τ ) = G(τ ) − δ(τ )I.

(15.50)

15.3 The Green Matrix and the Resolvent Matrix

351

Henceforth, we referred to (τ ) as the resolvent matrix and to G(τ ) as the surface value of the Green matrix. Equation (15.43) leads to G(0, τ ) = GT (τ, 0) = GT (τ ),

(0, τ ) =  T (τ, 0) = T (τ ).

(15.51)

In the scalar case, one simply has G(0, τ ) = G(τ, 0) = G(τ ). The matrices G(τ ) and (τ ) satisfy the integral equations 

∞

K(τ − τ )G(τ ) dτ + δ(τ )I,

(15.52)

K(τ − τ )(τ ) dτ + (1 − )K(τ ).

(15.53)

G(τ ) = (1 − ) 0

and  (τ ) = (1 − )

∞

0

All the integral equations given here are generalizations of similar equations established in the scalar case and will be employed for the same purposes, such as the construction of a nonlinear integral equation for a H-matrix.

15.3.2 The Hanle Effect The Green matrix satisfies the Wiener–Hopf integral equation  G(τ, τ0 ) = (1 − )M(θB )

∞

K(τ − τ )G(τ , τ0 ) dτ + δ(τ − τ0 )I.

(15.54)

0

For the Hanle effect, the simple relation G(τ, τ0 ) = GT (τ0 , τ ) does not hold because the matrix M(θB )K(τ ) is not symmetric (since the product of two symmetric matrices is in general not symmetric). To find the relation between G(τ, τ0 ) and G(τ0 , τ ), we follow the approach described above. We first remark that G(τ, τ0 ) is also solution of  ∞ G(τ, τ0 ) = (1 − ) G(τ, τ )MH (θB )K(τ − τ0 ) dτ + δ(τ − τ0 )I. (15.55) 0

The equivalence of the two equations can be established by performing on each of them a Neumann series expansion. The equation obtained by a transposition of Eq. (15.55) is  G (τ, τ0 ) = (1 − ) T

0

∞

K(τ − τ0 )MH (θB )GT (τ, τ ) dτ + δ(τ − τ0 )I.

(15.56)

352

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After a change of notation, it becomes  GT (τ0 , τ ) = (1 − )

∞

K(τ − τ )MH (θB )GT (τ0 , τ ) dτ + δ(τ − τ0 )I.

(15.57)

0

The comparison between Eqs. (15.55) and (15.57) shows that GT (τ0 , τ ) = M−1 H (θB )G(τ, τ0 )MH (θB ).

(15.58)

This relation can be verified by replacing in Eq. (15.57) GT (τ0 , τ ) by the above expression. The matrix (τ, τ0 ) satisfies the integral equation 

∞

(τ, τ0 ) = (1 − )M(θB )

K(τ − τ )(τ , τ0 ) dτ + (1 − )M(θB )K(τ − τ0 ),

0

(15.59) and its surface value (τ ) the integral equation 

∞

(τ ) = (1 − )M(θB )

K(τ − τ )(τ ) dτ + (1 − )M(θB )K(τ ).

(15.60)

0

All the equations given for the Hanle effect reduce to the equations for resonance scattering when the magnetic field is zero, since M(θB ) reduces to the unity matrix.

15.4 Construction of the H-Equation In the scalar case, the H -function can be defined as 1 ˜ H ( ) ≡ G(p), p

p ∈ [0, ∞[,

(15.61)

˜ where G(p) is the Laplace transform of G(τ ), the surface value of the Green function G(τ, τ0 ). It is defined by ˜ G(p) =



∞

G(τ ) e−pτ dτ,

p ∈ [0, ∞[.

(15.62)

0

All the Laplace transforms are denoted with a tilde. Two other definitions of the H -function are commonly used: surface value of the source function for a semi-infinite medium with an exponential primary source Q∗ (τ ) = exp(−pτ ) and center-to-limb variation of the emergent intensity for a uniform primary source. The definition in Eq. (15.61) has the advantage

15.4 Construction of the H-Equation

353

of depending exclusively on the scattering mechanism. The construction of the nonlinear integral equation for the H -function makes use of the fundamental relation, ˜˜ G(ν, p) =

1 ˜ ˜ G(p)G(ν), ν +p

ν, p ∈ [0, ∞[,

(15.63)

˜˜ where G(ν, p) is the double Laplace transform of G(τ, τ0 ). Here, we define the H-matrix with the polarized version of Eq. (15.61), namely 1 ˜ H( ) ≡ G(p), p

(15.64)

˜ where G(p) is the Laplace transform of G(τ ) = G(τ, 0). In the polarized case, the other definitions are not fully equivalent. We discuss them in Sect. 15.5. To construct the nonlinear H-equation we follow closely the method presented in Chap. 11 for the scalar case. The Rayleigh scattering and the resonance polarization are treated together since they lead to identical Wiener–Hopf equations.

15.4.1 Rayleigh Scattering and Resonance Polarization First we show how to generalize Eq. (15.63). The proof goes essentially as in the scalar case (see Sect. 2.4.4). The starting point is the integral equation for the resolvent matrix 

∞

(τ, τ0 ) = (1 − )

K(τ − τ )(τ , τ0 ) dτ

0

+ (1 − )K(τ − τ0 ),

τ, τ0 ∈ [0, ∞[.

(15.65)

We apply the differential operator D=

∂ ∂ + . ∂τ ∂τ0

(15.66)

We now take the derivative with respect to τ and τ0 , as shown in Sect. 15.1. Using DK(τ − τ0 ) = 0 and (0, τ0 ) =  T (τ0 , 0) = T (τ0 ), we obtain  D(τ, τ0 ) = (1 −) 0

∞

K(τ −τ )D(τ, τ ) dτ +(1 −)K(τ )T (τ0 ).

(15.67)

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The comparison of this equation, with the integral equation for (τ ), namely  (τ ) = (1 − )

∞

K(τ − τ )(τ ) dτ + (1 − )K(τ ),

(15.68)

0

leads to D(τ, τ0 ) = (τ )T (τ0 ).

(15.69)

We now take the Laplace transform of this equation with respect to τ and then with respect to τ0 . These transformations lead to ˜˜ T ˜ ˜ ˜ (p + ν)(p, ν) = (p) (ν) + (0, ν) + (p, 0),

(15.70)

and after simple algebra to ˜˜ G(p, ν) =

1 ˜ ˜ T (ν), G(p)G ν +p

p, ν ∈ [0, ∞[.

(15.71)

We can now construct a nonlinear integral equation for the H-matrix, proceeding exactly as described in Sect. 11.2.2. For τ = 0, the Wiener–Hopf integral equation for G(τ, τ0 ), written in Eq. (15.42), becomes 

∞

G(0, τ0 ) = (1 − )

K(τ )G(τ , τ0 ) dτ + δ(τ0 )I.

(15.72)

0

We now express K(τ ) in terms of its inverse Laplace transform M(ν), defined by 

∞

K(τ ) =

M(ν)e−ν|τ | dν.

(15.73)

0

Equation (15.72) becomes 

∞

G(0, τ0 ) = (1 − )

˜ M(ν)G(ν, τ0 ) dν + δ(τ0 )I.

(15.74)

0

We now take the Laplace transform with respect to τ0 . Using the expression of ˜˜ G(p, ν) given in Eq. (15.70), G(0, τ0 ) = GT (τ0 ) and transposing the result, we find ˜ ˜ G(p) = I + (1 − )G(p)

 0

∞

˜ T (ν)M(ν) dν . G p+ν

(15.75)

15.4 Construction of the H-Equation

355

For Rayleigh scattering,  M(ν) =

 for ν ∈ [0, 1[, , for ν ∈ [1, ∞[

0 W (1/ν)/2ν

(15.76)

˜ (see where W (μ) is given in Eq. (14.50). Using the definition H(p) ≡ G(1/p) Eq. (15.61)), we recover the standard form of the nonlinear H-equation, namely H(μ) = I +

1− H(μ)μ 2



1

HT (μ )W (μ )

0

dμ

. μ + μ

(15.77)

This equation can be found in many references, usually for W = 1 (see e.g. Fymat 1967; Pahor 1968; Bond and Siewert 1971; de Rooij et al. 1989; Ivanov 1996). Numerical solutions were proposed by several authors (see e.g. Lenoble 1970; Abhyankar and Fymat 1970, 1971; Kriese and Siewert 1971; de Rooij et al. 1989; Ivanov et al. 1996). Problems raised by the numerical solution of the H-equation for Rayleigh scattering are discussed in Chap. 18. For resonance polarization, M(ν) is given  ∞ 1 1 2 ˆ )= ˆ ), g(ξ W (15.78) M(ν) = g(  [ξ ϕ(u)]ϕ (u) du, ν ν y(ξ ) with  y(ξ ) =

0

0 < ξ ≤ 1/ϕ(0),

ϕ (1/ξ )

ξ ≥ 1/ϕ(0).

−1

(15.79)

2 (see The elements of the matrix W  (μ) are given in Eq. (14.96) with W = W Eq. (14.99)). The same simple algebraic manipulations lead to  H(μ) = I + (1 − )H(μ) μ 0

∞

ˆ HT (p)g(p)

dp . μ+p

(15.80)

The numerical solution of this equation is discussed in Sect. 18.2.3. The nonlinear H-equation are constructed in the present section for real values of μ, μ ∈ [0, 1] for Rayleigh scattering and μ ∈ [0, ∞[ for resonance polarization. In both case the domain of definition can be extended to complex values of μ (see Chap. 18).

15.4.2 The Hanle Effect The method of construction described above being easily generalized to the Hanle effect, we indicate only the main steps.

356

15 The

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˜˜ We define the H-matrix, as in Eq. (15.64). The double Laplace transform G(p, ν) is derived from the integral equations for (τ, τ0 ) and (τ ) given in Eqs. (15.59) and (15.60). The application of the differential operator D = ∂/∂τ + ∂/∂τ0 to Eq. (15.59) leads to ˜˜ G(p, ν) =

1 ˜ ˜ T (ν)M−1 (θB ). G(p)MH (θB )G H p+ν

(15.81)

One then proceeds with the integral equation for G(τ, τ0 ) given in Eq. (15.54) exactly as describes above. Using the relation between GT (τ, τ0 ) and G(τ, τ0 ) in Eq. (15.58), and the symmetry property of MH (θB ), we obtain ˜ ˜ G(p) = I + (1 − )G(p)M H (θB )

 0

∞

˜ T (ν)M(ν) dν . G p+ν

(15.82)

The expression of M(ν) given in Eq. (15.78), holds also Hanle effect, the  for the    matrix W  (μ) is now a 6×6 matrix given by W  (μ) = W(μ) W, where (μ)  = diag[1, W 2 , W 2 , W 2 , W 2 , W 2 ]. Equation (15.82) is given by Eq. (14.70) and W can be rewritten as  ∞ dp ˆ . (15.83) H(μ) = I + (1 − )H(μ)MH (θB ) μ HT (p)g(p) μ+p 0 Although it is not explicitly written, the H-matrix depends of on the angle θB . We also note that H(μ) is a 6 × 6 matrix. The method of construction of H-equations presented in this section requires, as we have shown, only elementary steps, but the existence of solutions has to be proven, a not so easy task (Siewert and Burniston 1972). We present in Chap. 18, an alternative method based on the matrix singular integral equation introduced in Eq. (15.110). It requires some complex plane analysis, but automatically proves the existence of solutions.

15.5 Alternative Definitions of the H-Matrix Ambartsumian (1942) pointed out that the H -function for the scalar case can be defined as the surface value of the source function for a semi-infinite medium with an exponential primary source term. In the scalar case, the function H (μ) can also be defined as the center-to-limb variation of the emergent intensity for a medium with a uniform primary source term. We now examine these definitions for the polarization processes discussed above, using the matrix representation of source term and of the radiation field.

15.5 Alternative Definitions of the H-Matrix

357

15.5.1 Exponential Primary Source We assume that the primary source Q∗ (τ ) behaves as an decreasing exponential, that is has the form Q∗ (τ ) = e−pτ I,

(15.84)

with p positive and I the identity matrix. For Rayleigh scattering and resonance polarization, the integral equation for the source function matrix Sp (τ ) is 

∞

Sp (τ ) = (1 − )

K(τ − τ )Sp (τ ) dτ + e−pτ I.

(15.85)

0

The solution of this equation may be written as  Sp (τ ) =

∞

˜ G(τ, τ0 ) e−pτ0 dτ0 = G(τ, p).

(15.86)

0

Setting τ = 0, using G(τ, 0) ≡ G(τ ), G(τ, τ0 ) = GT (τ0 , τ ) and the definition of the H-matrix in Eq. (15.64), we obtain ˜ ˜ T (p) = HT ( 1 ). Sp (0) = G(0, p) = G p

(15.87)

The surface value of the source matrix provides the transposed of the H-matrix, not the H-matrix itself. Of course, when the primary source has the form e−τ/p I, then Sp (0) = HT (p). For the Hanle effect, the relation between G(τ, τ0 ) and G(τ0 , τ ) given in Eq. (15.58) leads to 1 Sp (0) = MH (θB )HT ( )M−1 (θB ). p H

(15.88)

Properly multiplied by M(θB ), the surface value Sp (0) can serve to define the Hmatrix.

15.5.2 Uniform Primary Source We now examine the definition based on the emergent radiation field. For Rayleigh scattering, the emergent radiation field is given by 

∞

I(0, μ) = 0

S(τ ) e−τ/μ

dτ . μ

(15.89)

358

15 The

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Writing the primary source term as 

Q∗ (τ ) =

∞

e−ντ q∗ (ν) dν,

(15.90)

0

expressing S(τ ) in terms of the Green matrix (see Eq. (15.41)), and using the fundamental relation in Eq. (15.71), we can then write ˜ 1) I(0, μ) = G( μ



∞

˜ T (ν)q∗ (ν) G

0

dν . 1 + μν

(15.91)

For a uniform source, Q∗ (τ ) = Q∗ and q∗ (ν) = Q∗ δ(ν). Hence, ˜ T (0)Q∗ = H(μ)G ˜ 1 )G ˜ T (0)Q∗ = H(μ)S(0). I(0, μ) = G( μ

(15.92)

with ˜ T (0)Q∗ . S(0) = G

(15.93)

For resonance polarization, the emergent intensity is given by 

∞

I(0, x, μ) =

S(τ )e−τ ϕ(x)/μ

0

ϕ(x) dτ. μ

(15.94)

The field Iξ (0, ξ ) = I(0, x, μ), with ξ = μ/ϕ(x), has the form 

∞

Iξ (0, ξ ) =

S(τ )e−τ/ξ

0

dτ . ξ

(15.95)

Equation (15.92) with μ replaced by ξ leads to Iξ (0, ξ ) = H(ξ )S(0).

(15.96)

For the Hanle effect, the emergent intensity may also be written as in Eqs. (15.94) and (15.95). Expressing the source function in terms of the Green function G(τ, τ0 ), the primary source term in terms of its inverse Laplace transform q∗ (ν), and using ˜˜ the expression of G(p, ν) given in Eq. (15.81), we observe that Iξ (0, ξ ) can be written as in Eq. (15.96), with ˜ T (0)M−1 (θB )Q∗ . S(0) = MH (θB )G

(15.97)

Although the center-to-limb variation of the emergent radiation field matrix follows the variation of the H-matrix, this definition is not as easy to use as in the scalar case. In Sect. 18.3.2, we show that the matrix representing the emergent radiation

15.6 Singular Integral Equations

359

field satisfies a nonlinear integral equation, which provides an interesting alternative to the H-equation, having better numerical properties. Introduced in Ivanov (1996), it is known as the I-matrix.

15.6 Singular Integral Equations For scalar problems, we show in Chap. 4 how the Wiener–Hopf integral equations can be transformed into singular integral equations by an inverse or direct Laplace transform. These equations are then transformed into boundary value problems in the complex plane by a Hilbert transform method and solved in terms of an auxiliary function X(z), which has an exact expression. The same approach can be used for polarized transfer. The function X(z) becomes a matrix X(z). An exact expression can be obtained for conservative Rayleigh scattering ( = 0). This means that closed form solutions can be obtained for the Milne problem and the diffuse reflection problem, as we show in Chap. 16. For the other scattering mechanisms considered in this book, the matrix X(z) has no explicit expression, but can be used to establish exact relations, to construct the H-equation and to analyze the properties of the Hmatrix. In the present section we present some singular integral equations studied in detail in following chapters. We recall that the inverse Laplace transform approach amounts to assume that S(τ ), Q∗ (τ ), K(τ ), and (τ ) can be written as Laplace transforms. The inverse Laplace transforms of the kernel and of the primary source are introduced in Eqs. (15.73) and (15.90), those of the source function vector and resolvent matrix are defined by 

∞

S(τ ) =

s(ν) e−ντ dν,

0



∞

(τ ) =

(ν) e−ντ dν.

(15.98)

0

We give in Table 15.1 the names of the inverse Laplace transforms. The same symbol is used for the matrix (τ ) and for its transform (ν), but the independent variable being always explicitly indicated, there should be no confusion. We examine first together the Rayleigh scattering and the resonance polarization and then the Hanle effect. The presence of the matrix MH (θB ) modifies the properties of the scattering term. Table 15.1 Notation for inverse Laplace transforms of vectors and matrices depending on the optical depth τ

Function S (τ ) Q∗ (τ ) K(τ ) (τ )

Inverse Laplace s(ν) q ∗ (ν) M(ν) (ν)

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15 The

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15.6.1 Rayleigh Scattering and Resonance Polarization The inverse Laplace transform applied to the integral equations for S(τ ) and (τ ) leads to the singular integral equations  L(ν)s(ν) − (1 − )M(ν)

∞ 0

s(ν ) dν = q ∗ (ν), ν − ν

ν ∈ [0, ∞[,

(15.99)

and 

∞

L(ν)(ν)−(1−)M(ν) 0

(ν )

dν = (1−)M(ν), ν − ν

ν ∈ [0, ∞[,

(15.100)

where  L(ν) ≡ I − (1 − )

∞

M(ν)( 0

1 1 +

) dν . ν + ν ν −ν

(15.101)

The matrix M(ν) is given in Eq. (15.76) for Rayleigh scattering and in Eq. (15.78) for resonance scattering. The integrals over ν are taken in Principal Value. The properties of the kernel K(τ ) are contained in the matrices M(ν) and L(ν). By changing the real variable ν into a complex variable z, L(ν) can be analytically continued into a matrix L(z) known as the dispersion matrix. It can be defined, as in the scalar case, by taking the Fourier transform of the infinite medium version of Eq. (15.1). This transformation leads to 

 ˆ ˆ ˆ ∗ (k). I − (1 − )K(k) S(k) =Q

(15.102)

Here k is the Fourier variable and the hat indicates a Fourier transform. The square bracket defines the dispersion matrix. With k = i z and the kernel expressed in terms of M(ν), it can be written as in Eq. (15.101) with ν = z. The properties of L(z) are described in detail in Chap. 17. Equation (15.4) shows that   0 , L(0) = E = 0 Q 

(15.103)

7 with Q ≡ 1 − (1 − ) 10 Wp . In Eq. (15.99), the unknown is a vector, while it is a matrix in Eq. (15.100). The Hilbert transform method can be applied to both types of equations but some differences appear when solving the corresponding boundary value equations. The equations for s(ν) and (ν) have exact solutions for Rayleigh scattering when  = 0, as we show in Chap. 16. For non-conservative Rayleigh scattering and for resonance polarization (whatever the value of ), there is no exact solution. The

15.6 Singular Integral Equations

361

existence or nonexistence of exact solutions depends on the properties of the matrix L(z). A direct Laplace transform of a Wiener–Hopf integral equation, also leads to a singular integral equation. The Cauchy-type integral operator is the adjoint of the integral operator deriving from an inverse Laplace transform, but the singular integral equation can be solved by similar methods. For example, a direct Laplace transform of Eq. (15.52), the Wiener–Hopf integral equation for G(τ ), yields the singular integral equation ˜ L(p)G(p) + (1 − )



∞

M(p )

νl

˜ ) G(p dp = I, p − p

p ∈ [0, ∞[,

(15.104)

where νl = 1 for Rayleigh scattering, νl = 0 for resonance polarization, and ˜ G(p) =



∞

G(τ ) e−pτ dτ,

p ∈ [0, ∞[.

(15.105)

0

˜ The equations for G(p) and (ν) differ by the position of the matrix M(ν).

15.6.2 The Hanle Effect In the Wiener–Hopf integral equations describing the Hanle effect, the integral term is multiplied by the matrix M−1 (θB ). As a consequence, a dispersion function defined as in Eq. (15.102) would not be symmetric, a property needed to apply the Hilbert transform method of solution. A symmetric dispersion matrix for the Hanle effect is easy to construct. We consider the infinite medium version of the integral equation for S 0 (τ ) written in Eq. (15.2) and multiply it by M−1 (θB ) before taking the Fourier transform. We obtain   −1 ˆ ˆ ˜∗ (15.106) M−1 H (θB ) − (1 − )K(k) S(k) = MH (θB )Q (k). The matrix M−1 H (θB ) is symmetric (see Eq. (14.140)), hence the matrix inside the square bracket also and we can use it to define a dispersion matrix for the Hanle effect, henceforth denoted LH (z). Setting k = i z and expressing the kernel in terms of M(ν), we can write LH (z) ≡ M−1 H (θB ) − (1 − )



∞

M(ν)( 0

1 1 + ) dν. ν +z ν −z

(15.107)

362

15 The

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Except for the behavior at infinity, LH (z) has the same properties as the dispersion matrix for resonance polarization. Its value at zero is LH (0) = M−1 H (θB ) − I − E,

(15.108)

where E is the 6 × 6 matrix defined in Eq. (15.7). The singular integral equations for the inverse Laplace transforms of (τ ) and of S 0 (τ ) are given by Eqs. (15.100) and (15.99), where L(ν) is replaced by LH (ν). In the case of S 0 (τ ), the primary source term is multiplied by M−1 H (θB ).

15.7 Factorization Relations and the

√ -Law

In the scalar case, it is shown in Sect. 5.4.3 that the dispersion function L(z) and the H -function are related by H (z)H (−z)L(1/z) = 1. The proof is based on the definition of the auxiliary function X(z), as being a solution of a homogeneous Riemann–Hilbert problem. As shown by Chandrasekhar (1960, p. 116), this factorization relation can be derived from the nonlinear H -equation by considering the product [1 − 1/H (p))][1 − 1/H (p))]. The proof is reproduced in Sect. 11.3.1. Another method, suggested in Ivanov et al. (1997) for resonance polarization, is ˜ to combine the nonlinear H-equation and the singular integral equation for G(p) = 1/H(1/p). We now present this latter method and then use the resulting factorization √ relation to recover the -laws established in Sect. 15.2. For the Rayleigh scattering and the resonance polarization, the nonlinear Hequation can be written as shown in Eq. (15.75), namely ˜ ˜ G(p) = I + (1 − )G(p)



∞ νl

˜ T (ν)M(ν) dν . G p+ν

(15.109)

The linear equation is ˜ L(p)G(p) + (1 − )



∞ νl

M(p )

˜ ) G(p dp = I, p − p

p ∈ [0, ∞[,

(15.110)

where νl = 1 for the Rayleigh scattering and νl = 0 for resonance polarization. −1 on the right, using the symmetry and ˜ Multiplying Eq. (15.110) by [G(p)] evenness properties of L(p), we can write   ˜ T (p)]−1 I − (1 − ) L(p) = [G

 ˜ T (ν)M(p ) dp . G p − p νl ∞

(15.111)

15.7 Factorization Relations and the

√

363

-Law

Comparing the bracket with Eq. (15.109), we obtain the factorization relation −1 ˜ T (p)]−1 [G(−p)] ˜ L(p) = [G ,

(15.112)

which, for the H-matrix, becomes 1 HT (p)L( )H(−p) = I. p

(15.113)

For the Hanle effect, the nonlinear integral equation is ˜ ˜ G(p) = I + (1 − )G(p)M H (θB )



∞ 0

˜ T (ν)M(ν) dν , G p+ν

(15.114)

and the linear one is ˜ LH (p)G(p) + (1 − )



∞

M(p )

0

˜ ) G(p dp = M−1 H (θB ), p − p

p ∈ [0, ∞[. (15.115)

Proceeding as above, we find the factorization relation −1 −1 ˜ T (p)]−1 M−1 (θB )[G(−p)] ˜ ˜ T (p)]−1 [G(−p)] ˜ LH (p) = [G = [M(θB )G , (15.116)

which, for the H-matrix, becomes 1 MH (θB )HT (p)LH ( )H(−p) = I. p

(15.117)

The √ factorization relations in Eqs. (15.112) and (15.117) can be used to recover the -law derived in Sect. 15.2 by considering a quadratic quantity involving the source vector and its derivative. For the Rayleigh scattering and the resonance polarization, it is shown in Sect. 15.5.2 that the surface value of the source function matrix for a medium with a uniform primary source may be written as ˜ T (0)Q∗ . S(0) = G

(15.118)

Using the factorization √ relation in Eq. (15.112) and L(0) = E, we recover Eq. (15.24), hence the -law. For the Hanle effect, we have ˜ T (0)M−1 (θB )Q∗ . S(0) = MH (θB )G

(15.119)

(see Eq. (15.97)). The factorization relation in Eq. (15.116) readily leads to the law written in Eq. (15.34).

√ -

364

15 The

√

-Law, the Nonlinear H-Equation, and Matrix Singular Integral Equations

References Abhyankar, K.D., Fymat, A.L.: Imperfect Rayleigh scattering in a semi-infinite atmosphere. Astron. Astrophys. 4, 101–110 (1970) Abhyankar, K.D., Fymat, A.L.: Tables of auxiliary functions for the nonconservative Rayleigh phase matrix in semi-infinite atmospheres. Astrophys. J. Suppl. Ser. 195, 35–101 (1971) Ambartsumian, V.A.: Light scattering by√planetary atmospheres. Astron. Zhurnal 19, 30–41 (1942) Bommier, V., Štˇepán, J.: A generalized -law. The role of unphysical source terms in resonance line polarization transfer and its importance as an additional test of NLTE radiative transfer. Astron. Astrophys. 468, 797–801 (2007) Bond, G.R., Siewert, C.E.: On the nonconservative equation of transfer for a combination of Rayleigh and isotropic scattering. Atrophys. J. 164, 97–110 (1971) de Rooij, W.A., Bosma, P.B., van Hooff, J.P.C.: 1989, A simple method for calculating the H-matrix for molecular scattering. Astron. Astrophys. 226, 347–356 (1989) Frisch, H.: Resonance polarization and the Hanle effect; the integral equation formulation and some applications. In: Nagendra, K.N., Stenflo, J.O. (eds.) Solar Polarization, 2nd Solar Polarization Workshop, Kluwer, Dordrecht, pp. 97–113 √ (1999) Frisch, U., Frisch, H.: Non-LTE transfer.  revisited. Mon. Not. R. Astron. Soc. 173, 167–182 (1975) Frisch, U., Frisch, H.: Universality of escape from a half-space for symmetrical random walks. In: Lévy Flights and Related Topics in Physics, eds. M. Shlesinger, G. Zaslavsky, U. Frisch, Lectures Notes in Physics, vol. 450, pp. 262–268. Springer Verlag, Berlin (1995) Fymat, A.L.: Theory of radiative transfer in atmospheres exhibiting polarized resonance fluorescence, Ph.D. thesis, Los Angeles, UCLA Ivanov, V.V.: Nonmagnetic polarization of the Doppler core of strong Fraunhofer lines. Sov. Astron. 34, pp. 621–625 (1990); translation from Astron. Zhurnal 67, pp. 1233–1242 (1990) Ivanov, V.V.: Generalized Rayleigh scattering I. Basic theory. Astron. Astrophys. 303, 609–620 (1995) Ivanov, V.V.: Generalized Rayleigh scattering III. Theory of I-matrices. Astron. Astrophys. 307, 319–331 (1996) Ivanov, V.V., Kasaurov, A.M., Loskutov, V.M., Viik, T.: Generalized Rayleigh scattering II. Matrix source functions. Astron. Astrophys. 303, 621–634 (1995) Ivanov, V.V., Kasaurov, A.M., Loskutov, V.M.: Generalized Rayleigh scattering IV. Emergent radiation. Astron. Astrophys. 307, 332–346 (1996) Ivanov, V.V, Grachev, S.I., Loskutov, V.M.: Polarized line formation by resonance scattering I. Basic formalism. Astron. Astrophys. 318, 315–326 (1997) Kriese, J.T., Siewert, C.E.: An expedient method for calculating H -matrices. Astrophys. J. 164, 389–391 (1971) Landi Degl’Innocenti, E., Bommier, V.: Resonance line√polarization for arbitrary magnetic fields in optically thick media III. A generalization of the -law. Astron. Astrophys. 284, 865–873 (1994) Lenoble, J.: Importance de la polarisation dans le rayonnement diffusé par une atmosphère planétaire. J. Quant. Spectrosc. Radiat. Transf. 10, 533–556 (1970) Nagendra, K.N., Sampoorna, M.: Numerical methods in polarized line formation theory. In: Solar Polarization 5, S. Berdyugina, K.N. Nagendra, R. Ramelli (eds), ASP Conference Series 405, 261–273 (2009) Pahor, S. : Albedo and Milne ’s problem for thermal neutrons. Nuclear Sci. Eng. 31, 110–116 (1968) Siewert, C.E., Burniston, E.E.: On existence and uniqueness theorems concerning the H-Matrix of radiative transfer. Astrophys. J. 174, 629–641 (1972)

Chapter 16

Conservative Rayleigh Scattering: Exact Solutions

It is known since the work of Chandrasekhar (1946a,1946b, see also 1960) that the Milne problem, which describes conservative Rayleigh scattering in a plane-parallel semi-infinite medium with a given total radiative flux F has an exact solution. Chandrasekhar could thus establish that the emergent polarized radiation field has an explicit expression in terms of two auxiliary functions Hl (μ) and Hr (μ), and that these auxiliary functions satisfy nonlinear integral equations, similar to the nonlinear H -equation, which can be solved explicitly. Following the work of Chandrasekhar based on the discrete ordinate method, (Siewert and Fraley 1967) applied to the Milne problem the Case singular eigenfunction expansion method presented in Chap. 10 for the scalar case. This approach leads to a matrix homogeneous Riemann–Hilbert problem, characterized by the dispersion matrix L(z) introduced in Sect. 15.6.1. For conservative Rayleigh scattering this matrix can be diagonalized and the matrix Riemann–Hilbert problem reduced to two scalar Riemann–Hilbert problems with an exact solution. Other methods based on the exact solution of the nonlinear integral equation for the H-matrix have also been developed (see references in e.g. Van de Hulst 1980, Chap. 16, Ivanov et al. 1997a). In the present Chapter we use a method similar to that of Siewert and Fraley (1967) to construct exact solutions for the Milne problem and for the diffuse reflection problem. Instead of the Case eigenfunction expansion method, we apply the inverse Laplace transform method described in Sect. 15.6.1. We arrive of course at the same homogeneous matrix Riemann–Hilbert problem. We solve it for a halfspace auxiliary matrix X(z) and then use a Hilbert transform method of solution to solve the singular integral equations for the matrix (ν) and the vector s(ν), inverse Laplace transforms of the resolvent matrix (τ ) and of the source vector S(τ ), respectively. The Chapter is organized as follows. In Sect. 16.1 we recall the radiative transfer equations for conservative scattering in the (lr) representation and the results of Sect. 14.2 concerning the factorization of the Rayleigh phase matrix and the construction of the new radiation field I(τ, μ). The properties of the dispersion © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_16

365

366

16 Conservative Rayleigh Scattering: Exact Solutions

matrix, in particular its diagonalization, are discussed in Sect. 16.2 and in Appendix M in this chapter. Section 16.3 is devoted to the construction of a half-space auxiliary matrix X(z). We show that it has an exact expression in terms of two scalar functions X1 (z) and X2 (z), simply related to the functions Hl (μ) and Hr (μ). The exact expression of X(z) is then used in Sect. 16.4 to solve the matrix singular integral equation for (ν). The matrix (ν) is then used in Sect. 16.5 to introduce the H-matrix and to give an explicit expression in terms of the functions Hl (μ) and Hr (μ). In Sects. 16.6 and 16.7 we use the exact expression of (τ ) to construct exact solutions of the Milne problem, and of the diffuse reflection problem. In both cases we calculate the emergent radiation field and discuss in detail the behavior of the field at infinity. In Appendix N of this chapter we present an alternative method of solution for the Milne problem, based on the solution of the vector singular integral equation for s(ν).

16.1 Radiative Transfer Equations Only conservative Rayleigh scattering being considered in this Chapter, we describe the polarized radiation field as in Sect. 14.2 by its two components Il (τ, μ) and Ir (τ, μ). We recall that μ = cos θ with θ the colatitude of the ray and τ the optical depth with τ = 0 at the surface and τ → ∞ in the interior. For μ = 1 (ray direction perpendicular to the surface), the radiation field is unpolarized, that is Il (τ, μ) = Ir (τ, μ), because of the cylindrical symmetry. The radiative transfer equations for conservative Rayleigh scattering are presented in Sect. 14.2. We recall those that are most needed in this Chapter. In the (Il , Ir ) representation, the cylindrical part of the Rayleigh phase matrix may be written as

2

2 2 2 2  3 2(1 − μ )(1 − μ ) + μ μ μ

P(μ, μ ) = , (16.1) 4 μ 2 1 where μ and μ are the direction cosines of the scattered and incident beams. The matrix P(μ, μ ), can be factorized as P(μ, μ ) = A(μ)AT (μ ), with

⎤ √ ⎡ 2√ 2(1 − μ2 ) 3 μ ⎣ ⎦. A(μ) = 2 1 0

(16.2)

(16.3)

16.1 Radiative Transfer Equations

367

Exactly as in Sect. 14.2.1, we introduce a field I(τ, μ), with components I1 (τ, μ) and I2 (τ, μ), defined by I(τ, μ) ≡ A−1 (μ)I (τ, μ),

(16.4)

where I (τ, μ) is the vector with components Il (τ, μ) and Il (τ, μ). For the diffuse reflection problem, I (τ, μ) is the diffuse part of the field. In explicit notation, √

√ 3 2 [μ I1 (τ, μ) + 2(1 − μ2 )I2 (τ, μ)], 2 √ 3 Ir (τ, μ) = I1 (τ, μ). 2

Il (τ, μ) =

(16.5)

The vector I(τ, μ) satisfies the radiative transfer equation 1 ∂I(τ, μ) = I(τ, μ) − μ ∂τ 2



+1 −1

(μ )I(τ, μ ) dμ − Q∗ (τ ),

(16.6)

with ⎡

√

1 + μ4

3⎣ √ 2 4 2μ (1 − μ2 )

(μ) ≡ AT (μ)A(μ) =

2μ2 (1 − μ2 )

2(1 − μ2 )2

⎤ ⎦.

(16.7)

The primary source term is Q∗ (τ ) = 0, 1 Q (τ ) = 2 ∗



1

Milne,

AT (−μ)I inc (−μ) e−τ/μ dμ,

diffuse reflection.

(16.8) (16.9)

0

For both problems, the boundary condition at τ = 0 is I(0, μ) = 0,

μ ∈ [−1, 0].

(16.10)

The source function vector S(τ ), defined by 1 S(τ ) ≡ 2



+1 −1

(μ)I(τ, μ) dμ + Q∗ (τ ),

(16.11)

satisfies the Wiener–Hopf integral equation, 

∞

S(τ ) = 0

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

(16.12)

368

16 Conservative Rayleigh Scattering: Exact Solutions

with the kernel K(τ ) given by 

1 K(τ ) ≡ 2

|τ | dμ ) . μ μ

1

(μ) exp(− 0

(16.13)

In the scalar case, the source function for the Milne problem and for the diffuse reflection problem have very simple expressions in terms of the resolvent function. The same relations holds for Rayleigh scattering. For the Milne problem, we have   S(τ ) = I +

τ

 (τ ) dτ S(0).

(16.14)

0

For the diffuse reflection of a parallel beam with direction −μ0 , μ0 ∈ [0, 1], the primary source term varies as e−τ/μ0 . We now have   −τ/μ0 I + S(τ ) = e

τ

e

−(τ −τ )/μ0



(τ ) dτ



 S(0).

(16.15)

0

In these equations, (τ ) is the resolvent matrix introduced in Eq. (15.50). The relations between S(τ ) and (τ ) can be established by comparing the integral equation for (τ ) given in Eq. (15.53) with the integral equation for the derivative of the source vector given in Eq. (15.3). The resolvent matrix (τ ) satisfies the Wiener–Hopf integral equation, 

∞

(τ ) =

K(τ − τ )(τ ) dτ + K(τ ),

(16.16)

0

and its inverse Laplace transform (ν), the singular integral equation given in Sect. 15.6, namely  L(ν)(ν) − M(ν)

∞ 0

(ν )

dν = M(ν), ν − ν

ν ∈ [0, ∞[.

(16.17)

This a matrix singular integral equation. We show in the subsequent sections how to solve it by a Hilbert transform method. The source function S(τ ) can also be derived from the singular integral equation for its inverse Laplace transform s(ν) given in Sect. 15.6.1, namely  L(ν)s(ν) − M(ν)

∞ 0

s(ν ) dν = q ∗ (ν), ν − ν

ν ∈ [0, ∞[.

(16.18)

This is a vector singular integral equation. Its solution for the Milne problem (q ∗ (ν) = 0) is described in Appendix N of this chapter.

16.2 The Dispersion Matrix

369

The matrix M(ν) is the inverse Laplace transform of the kernel K(τ ). It follows from Eq. (16.13) that  M(ν) =

0 (1/ν)/2ν

for ν ∈ [0, 1[, for ν ∈ [1, ∞[.

(16.19)

The matrix L(ν) is defined by  L(ν) ≡ I −

∞

M(ν)( 0

1 1 +

) dν . ν + ν ν −ν

(16.20)

The inhomogeneous term q ∗ (ν) is zero for the Milne problem. It is given in Eq. (16.141) for the diffuse reflection of a parallel beam of radiation. The singular integral equations for (ν) and s(ν) are solved by a Hilbert transform method, in Sect. 16.4 for (ν) and Appendix N in this chapter in for s(ν). To carry out the Hilbert transform method, we first analyze the properties of the dispersion matrix L(z), continuation of L(ν) in the complex plane, and then construct a half-space auxiliary matrix X(z), solution of a homogeneous RiemannHilbert problem.

16.2 The Dispersion Matrix The dispersion matrix L(z), already introduced in Sect. 15.6.1, is defined by 

∞

L(z) ≡ I −

M(ν)( 1

1 1 + ) dν. ν +z ν −z

(16.21)

The matrices L(ν) and L(z) are the matrix counterparts of the functions λ(ν) and L(z) of the scalar case (see Sect. 4.3.3). In the following, boldface sans serif is used for matrices defined in the complex plane. Equation (16.21) shows that L(z) is analytic in the complex plane cut along the intervals ] − ∞, −1] and [1, +∞[ that is (in C /] − ∞, −1] ∪ [1, +∞[). Saying that a matrix is analytic in some domain, means that all its elements have the same analyticity property. A very important property of the matrix L(z), first pointed out by Mullikin (1969) (see also Siewert and Burniston 1972), is that it can be factorized as L(z) =

1 1 1 1 (1 − 2 )−1 AT ( )D(z)A( ), 3 z z z

(16.22)

where D(z) is a diagonal matrix:  D(z) =

 L1 (z) 0 . 0 L2 (z)

(16.23)

370

16 Conservative Rayleigh Scattering: Exact Solutions

The functions Ln (z), n = 1, 2 are defined by Ln (z) ≡ (−1)n + 3(1 −

1 ) L(z), z2

n = 1, 2.

(16.24)

where 1 L(z) ≡ 1 − 2



∞

( 1

1 dν 1 1+z 1 + ) =1− ln , ν +z ν−z ν 2z 1 − z

(16.25)

is the dispersion function for scalar monochromatic scattering. For nonconservative Rayleigh scattering, it is still possible to diagonalize L(z), however the properties of the diagonalizing matrices make the construction of exact solutions extremely difficult (Siewert et al. 1981). The properties of Ln (z) are described in Appendix M.1 of this chapter and those of the matrices L(z) and L(ν) in Appendix M.2 of this chapter.

16.3 Construction of the Auxiliary X-matrix In this section we describe the construction of a matrix X(z), analytic in C /[1, ∞[ and solution of the matrix homogeneous Riemann–Hilbert problem L+ (ν)X+ (ν) = L− (ν)X− (ν),

ν ∈ [1, ∞[,

(16.26)

where L± (ν) and X± (ν) are the limiting values of L(z) and X(z) along the cut ν ∈ [1, ∞[. In addition we require X(z) to be nonsingular (det X(z) = 0), its elements to be free of zeroes and singularities and to have an algebraic behavior at infinity. We note that there is not a unique choice for the jump condition. The positions of the matrices of L+ (ν) and L− (ν) can be exchanged (see for example Sect. 17.1.3 where we consider nonconservative Rayleigh scattering). Homogeneous matrix Riemann–Hilbert problems of this type have usually no closed form expressions. Conservative Rayleigh scattering is an exception because Eq. (16.26) can be reduced to two scalar homogeneous Riemann–Hilbert problems for which closed form expressions can be found. A key ingredient is the diagonalization of L(z) written in Eq. (16.22) and the fact that the matrices involved in this diagonalization are analytic functions of z, with poles only. Using the factorization of L(z) given in Eq. (16.22), we can rewrite Eq. (16.26) as 1 1 A( )X+ (ν) = [D+ (ν)]−1 D− (ν)A( )X− (ν), ν ν

ν ∈ [1, ∞[.

(16.27)

We now introduce the matrix 1 Z(z) ≡ A( )X(z). z

(16.28)

16.3 Construction of the Auxiliary X-matrix

371

Equations (16.22), (16.23), and (16.26) show that its boundary values satisfy Z+ (ν) = [D+ ]−1 (ν)D− (ν)Z− (ν).

(16.29)

Since D(z) = diag[L1 (z), L2 (z)] is a diagonal matrix, we can write Z(z) as Z(z) ≡ diag[X1 (z), X2 (z)],

(16.30)

with the functions X1 (z) and X2 (z) satisfying Xn+ (ν) =

L− n (ν) − Xn (ν), L+ n (ν)

ν ∈ [1, ∞[,

n = 1, 2.

(16.31)

The functions X1 (z) and X2 (z) are thus solutions of homogeneous Riemann–Hilbert problems of the type discussed in Sect. 5.3. We require them to be analytic functions in C /[1, ∞[, to be free of zeroes, and to have an integrable singularity at z = 1. The construction of X1 (z) and X2 (z) is described in the Appendix M.3 of this chapter. They may be written as

1 X1 (z) = (1 − z) exp 2i π





∞

ln 1

L− 1 (ν) L+ 1 (ν)



 dν , ν −z

(16.32)

and

1 X2 (z) = exp 2i π





∞

ln 1

L− 2 (ν) L+ 2 (ν)



 dν . ν −z

(16.33)

The factor (1 − z) appearing in X1 (z) is here to prevent a singularity at z = 1. For z → ∞, X1 (z)  −z and X2 (z)  1. Other properties are presented in the Appendix M.3 of this chapter. We note that [X1 (z)]−1 and [X2 (z)]−1 have the same structure as, respectively, the monochromatic and complete frequency redistribution X-functions. The solution of the homogeneous Riemann–Hilbert problem in Eq. (16.26) can thus be written as   1 X1 (z) 0 X(z) = A−1 ( ) R(z), (16.34) 0 X2 (z) z where the elements of R(z) are rational functions of z (their jump across the cut [1, ∞[ is zero). In addition, the matrix R(z) has to be chosen in such a way that that the elements of X(z) are free of zeroes and of singularities and such that the determinant of X(z), given by √ 2 2 z2 det X(z) = √ X1 (z)X2 (z) det R(z), 3 1 − z2

(16.35)

372

16 Conservative Rayleigh Scattering: Exact Solutions

is free of zeroes. A suitable choice for the matrix R(z) is  R(z) =

z−2 (α + βz) 1

√ −2 2  2z (z − 1) , 0

(16.36)

where α and β are constant parameters still to be determined. The matrix X(z) has thus the form ⎡ ⎤ (z) 0 X 2 2 ⎦. X(z) = √ ⎣ √ (16.37) 3 [ 2(z2 − 1)]−1 [(α + βz)X1 (z) − X2 (z)] X1 (z) The determinant of X(z) is now free of zeroes and singularities. To satisfy the condition that the elements of X(z) are free of singularities, one must have (α + βz)X1 (z) − X2 (z) = 0,

for z = ±1.

(16.38)

The factorization relations for X1 (z) and X2 (z) given in Eq. (M.24), used together with L1 (1) = −1 and L2 (1) = 1, show that X1 (±1) and X2 (±2) have finite values. Equation (16.38) then leads to α = [X2 (1)X1 (−1) + X1 (1)X2 (−1)]/4, β = [X2 (1)X1 (−1) − X1 (1)X2 (−1)]/4.

(16.39)

The parameters α and β satisfy α 2 − β 2 = 2.

(16.40)

This relation is easily derived from Eq. (16.38) and from the factorization relations in Eq. (M.24). The correspondence between our notation and the Chandrasekhar’s one is 1 μX1 (− ) = Hl (μ), μ q=

2 , α

1 X2 (− ) = Hr (μ), μ c=

β , α

μ ∈ [0, 1],

q 2 + 2c2 = 2.

(16.41)

(16.42)

The numerical values of the constants c and q are c = 0.873 and q = 0.690 (see Table M.3). The functions Hl (μ) and Hr (μ) for μ ∈ [0, 1] are shown in Fig. 16.1. The correspondence with the auxiliary functions introduced in Siewert and Fraley (1967), denoted X(z) and Y (z), is √ 1 5 , − zX1 ( ) = z X(z)

√ 1 2 X2 ( ) = , z Y (z)

(16.43)

16.4 The Resolvent Matrix

373

Fig. 16.1 The functions Hl (μ) and Hr (μ) for Rayleigh scattering, and the function H √ (μ) for conservative √ monochromatic scattering. For μ → ∞, Hl (μ) increases at infinity as 5μ and H (μ) as 3μ, while Hr (μ) tends to the constant value 0.1414. The linear growth of Hl (μ) and H (μ) is already installed for μ around one. The data are from Chandrasekhar (1960, p. 125 and p. 248)

In Siewert and Burniston (1972) the functions X(z) and Y (z) are denoted F1 (z) and F2 (z), respectively.

16.4 The Resolvent Matrix The half-space auxiliary matrix X(z) having been determined, we can now solve the singular integral equation for the matrix (ν), namely  L(ν)(ν) − M(ν)

∞ 0

(ν )

dν = M(ν), ν − ν

ν ∈ [0, ∞[,

(16.44)

already given in Eq. (16.17). The method of solution of Eq. (16.44) presented here is largely inspired by Siewert and Burniston (1972), where similar matrix singular integral equations are treated. It is also quite close to the method described in Sect. 6.2.2 for the determination of the resolvent function in the scalar case. In the interval ν ∈ [0, 1], Eq. (16.44) reduces to L(ν)(ν) = 0.

(16.45)

We show in Appendix M.2 of this chapter that det L(ν) =

1 λ1 (ν)λ2 (ν), 8

(16.46)

374

16 Conservative Rayleigh Scattering: Exact Solutions

and that det L(ν) has a double zero at the origin. Hence Eq. (16.45) has solutions which are linear combinations of a Dirac distribution and of its derivative. For the Milne problem the source vector increases linearly at infinity (see Eq. (16.88)). One can thus deduce from the relation between S(τ ) and the resolvent matrix (τ ) given in Eq. (16.14), that (τ ) goes at infinity to a constant matrix, here denoted 0 . The solution of Eq. (16.45) compatible with this property has the form (ν) = 0 δ(ν),

(16.47)

L(0) 0 = 0.

(16.48)

and 0 satisfies the condition

0 will be determined when we consider the interval ν ∈ [1, ∞]. In the interval ν ∈ [1, ∞], Eq. (16.44) becomes  L(ν)(ν) − M(ν)

∞ 1

0 (ν )

dν = M(ν)[I − ],

ν −ν ν

ν ∈ [1, ∞[.

(16.49)

We now transform this equation into a boundary value problem in the complex plane. We introduce the matrix Hilbert transform 1 F(z) = 2i π



∞ 1

(ν) dν. ν −z

(16.50)

Using the Plemelj formulae for F(z), namely F+ (ν) − F− (ν) = (ν),  1 ∞ (ν )

+ − dν , F (ν) + F (ν) = i π 1 ν − ν

ν ∈ [1, ∞[,

(16.51)

those for L(z), L+ (ν) − L− (ν) = −2i πM(ν), L+ (ν) + L− (ν) = 2L(ν),

ν ∈ [1, ∞[,

(16.52)

and multiplying by ν, we can rewrite Eq. (16.49) as     1 1 + − − [νI + 0 ] − L (ν) νF (ν) + [νI + 0 ] = 0. L (ν) νF (ν) + 2i π 2i π (16.53) +

16.4 The Resolvent Matrix

375

Using the jump condition in Eq. (16.26), we obtain the new boundary value equation   1 [νI − 0 ] [X+ (ν)]−1 νF+ (ν) + 2i π   1 − −1 − [νI − 0 ] = 0. − [X (ν)] νF (ν) + 2i π

(16.54)

It has solutions of the form F(z) =

1 1 1 [X(z)P(z) + 0 ] − I, z 2i π 2i π

(16.55)

where P(z) is a matrix of polynomials. Defined as a Hilbert transform, F(z) should be free of singularities and tend to zero for z → ∞. These two conditions allow us to determine the matrices P(z) and 0 . The condition that F(z) tends to zero at infinity is satisfied when 1 X(z)P(z) → I, z

z → ∞.

(16.56)

For z → ∞, X1 (z)  −z and X2 (z)  1. The expression of X(z) given in Eq. (16.37) leads to ⎤ ⎡ 1 0 2 ⎦, X(z)  √ ⎣ β 3 − √ −z 2

z → ∞.

(16.57)

Equation (16.56) is thus satisfied when P(z) has the form √ ⎡p − z p ⎤ 1 3⎣ 0 ⎦, β P(z) = − √ 1 2 2

(16.58)

where p0 and p1 are constants still to be determined. The condition that F(z) has no singularity at z = 0 leads to X(0)P(0) + 0 = 0.

(16.59)

We know that L(0)0 = 0. Multiplying by L(0) we obtain the condition L(0)X(0)P(0) = 0.

(16.60)

376

16 Conservative Rayleigh Scattering: Exact Solutions

The matrices L(0) and X(0) being known, this condition allows us to determine P(0), hence the constants p0 and p1 . Using the expression of L(0) given in Eq. (M.16), namely, L(0) = and X2 (0) =

√   1 1 − 2 √ , 10 − 2 2

(16.61)

√ 2, we find p0 = β/α,

p1 =

√ 2/α.

(16.62)

The matrix P(z) is now fully determined, ⎡

√ ⎤ β 2 √ ⎢ −z ⎥ α α ⎢ ⎥ 3⎢ ⎥, P(z) = − ⎢ ⎥ 2 ⎣ β ⎦ √ 1 2

(16.63)

and Eq. (16.59) leads to ⎡

√ ⎤ ⎡ 2 c ⎢ ⎥ α ⎥ √ ⎢ √ ⎢ ⎢ ⎥ = 2⎢ 0 = 2 ⎢ ⎢ ⎥ ⎣ c ⎣ 1 β 1 ⎦ √ √ 2 2α α β α

q ⎤ √ 2⎥ ⎥ ⎥, q ⎦

(16.64)

2

where c and q are the Chandrasekhar parameters defined in Eq. (16.42). We now return to the matrix (ν). The first Plemelj formula applied to Eq. (16.55) leads to (ν) =

1 1 [X+ (ν) − X− (ν)]P(ν), ν 2i π

ν ∈ [1, ∞[.

(16.65)

The jump of the matrix X(z) can be expressed in terms of the functions X1 (−ν) and X2 (−ν), but actually the expression given in Eq. (16.65) turns out to be more useful, in particular to construct an H-matrix. To summarize, the resolvent matrix may be written as  ∞ (τ ) = 0 + (ν)e−ντ dν, (16.66) 1

where 0 is given in Eq. (16.64) and (ν) in Eq. (16.65).

16.5 The H-matrix

377

16.5 The H-matrix In Sect. 15.4 we have defined the H-matrix as 1 ˜ H( ) ≡ G(p), p

(16.67)

˜ where G(p) is the Laplace transform of the surface Green matrix G(τ ). We now use the exact expression of (ν) given in Eq. (16.65) to construct an exact expression of H(p). Taking the Laplace transform of the relation G(τ ) = (τ ) − I and using the exact expression of (τ ) given in Eq. (16.66), we obtain 1 ˜ G(p) = I + 0 + p



∞ 1

(ν) dν. ν +p

(16.68)

To calculate the integral, we replace (ν) by the expression given in Eq. (16.65) and consider the contour integral 1 2i π

 X(ξ )P(ξ ) C

1 1 dξ, ξ ξ +p

(16.69)

where the contour C turns around the poles at ξ = 0 and ξ = −p, around the cut along [1, ∞[ and is closed by a circle of radius R, R → ∞. We thus obtain 1 ˜ G(p) = − X(−p)P(−p), p

p ∈ [0, ∞[.

(16.70)

This expression can be continued to the complex plane, leading to 1 ˜ G(z) = − X(−z)P(−z). z

(16.71)

˜ This equation shows that G(z) has the same analyticity properties than X(z), namely ˜ that it is is analytic in C /] − ∞, −1]. It is also easy to verify that G(z) → I for ˜ z → ∞ and that G(z)  0 /z, z → 0, the pole at z = 0 being a consequence of the fact that G(τ ) tends to the constant matrix 0 at infinity. The H-matrix can be defined as ˜ 1 ). H(z) ≡ G( z

(16.72)

378

16 Conservative Rayleigh Scattering: Exact Solutions

Combining Eq. (16.63) with Eq. (16.71), we obtain the explicit expression H(z) = ⎡

⎤ q √ z(1 − z2 ) Hr (z) 1 ⎢ 2 ⎥ ⎦ . (16.73) ⎣ z2 q q 3 1 − z2 √ [ Hl (z) − (1 + cz)Hr (z)] [(1 − cz)Hl (z) − z Hr (z)] 2 2 z (1 − z2 )(1 + cz)Hr (z)

The matrix H(z) is analytic in C /]−1, 0]. The limits of H(z) for z → 0 and z → ∞ ˜ are easily derived from the properties of G(z). For z → ∞, H(z)  z0 and at the origin H(0) = I. These expressions can be compared√to those of the scalar Milne problem, H (0) = 1 and limz→∞ H (z)  zX(0) = z/ 3. For conservative Rayleigh scattering, there is a priori no need to look for a nonlinear H-equation such as Eq. (15.77), since H(z) has an explicit expressions in terms of the functions Hl (z) and Hr (z). We give it nonetheless. It may be written as 1 H(z) = I + zH(z) 2



1

HT (μ)(μ)

0

dμ . μ+z

(16.74)

It can be constructed with the method described in Sect. 15.4.1 or with the method that will be described in Chap. 18 for non-conservative Rayleigh scattering. It suffices then to set  = 0. When this nonlinear equation is used to define H(z), as in Siewert and Burniston (1972), the existence of a solution has to be proved. The work is essentially equivalent to that of the construction of the matrix X(z).

16.6 The Polarized Milne Problem The polarized Milne problem describes conservative Rayleigh scattering in a semiinfinite plane-parallel medium with no incident radiation, no internal source, but a given radiative flux. The two-component radiation field, I (τ, μ) = (Il (τ, μ), Ir (τ, μ)), satisfies the radiative transfer equation 1 ∂I μ (τ, μ) = I (τ, μ) − A(μ) ∂τ 2



+1 −1

AT (μ )I (τ, μ ) dμ ,

(16.75)

with the boundary condition I (0, μ) = 0,

μ ∈ [−1, 0].

(16.76)

16.6 The Polarized Milne Problem

379

The total radiative flux is defined by  F (τ ) = Fl (τ ) + Fr (τ ) = 2π

+1

−1

[Il (τ, μ) + Ir (τ, μ)]μ dμ.

(16.77)

This Section is organized as follows. In Sect. 16.6.1 we describe some simple physical properties, easily derived from the radiative transfer equation itself. In Sect. 16.6.2 we use the relation between the source vector S(τ ) and the resolvent matrix (τ ) given in Eq. (16.14) to establish an exact expression for S(τ ). We then show in Sect. 16.6.3 how to express S(τ ) in terms of generalized Hopf functions. In Sect. 16.6.4, devoted to the emergent radiation field, we present the exact expressions of Il (0, μ) and Ir (0, μ), μ > 0, in terms of the functions Hl (μ) and Hr (μ). Some of the main results are rewritten in Sect. 16.6.5 for the Stokes parameters I and Q.

16.6.1 Some Physical Properties Some interesting physical properties can be derived from the moments of the radiative transfer equation. We introduce the moments J (τ ), F (τ ), and J (2) (τ ) defined by 1 J (τ ) ≡ 2



−1

 F (τ ) ≡ 2π

J

(2)

1 (τ ) ≡ 2

+1

+1

−1



+1 −1

I (τ, μ) dμ,

(16.78)

I (τ, μ)μ dμ,

(16.79)

I (τ, μ)μ2 dμ.

(16.80)

(2) Their components are denoted Jl,r (τ ), Fl,r (τ ), Jl,r (τ ). Integrating the radiative transfer equation over μ from −1 to +1, we obtain

1 d 2π dτ



Fl (τ ) Fr (τ )



⎡ ⎤ (2) 3Jl (τ ) − Jr (τ ) 1 ⎦. = ⎣ 2 −3J (2)(τ ) + J (τ ) l

(16.81)

r

The sum of the two components leads to d [Fl (τ ) + Fr (τ )] = 0. dτ

(16.82)

380

16 Conservative Rayleigh Scattering: Exact Solutions

Multiplying the radiative transfer equation by μ before integrating it over μ, we obtain ⎤ ⎡ 

(2) d ⎣ Jl (τ ) ⎦ 1 Fl (τ ) = . (16.83) dτ J (2) (τ ) 4π Fr (τ ) r According to Eq. (16.82), the total radiative flux F (τ ), defined in Eq. (16.77) is a constant, henceforth denoted F . For the Milne problem, F is strictly positive. Its value specifies the solution among the family of all possible solutions of the Milne problem. The sum Il +Ir represents the total energy of the electromagnetic radiation field. We recall that Stokes I is given by I = Il + Ir . Equation (16.83) shows that d (2) 1 [Jl (τ ) + Jr(2)(τ )] = F. dτ 4π

(16.84)

Hence, Jl(2)(τ ) + Jr(2)(τ ) =

1 F (τ + C p ), 4π

(16.85)

where Cp =

4π (2) [J (0) + Jr(2)(0)]. F l

(16.86)

An exact expression of the constant C p is given in Eq. (16.131). It involves the functions Hl (μ) and Hr (μ). At infinity, the radiation field becomes isotropic (i.e. independent of μ) and unpolarized. A rigorous proof of this result, obvious on physical grounds, is given in Sect. 16.6.2, where we calculate S(τ ). For τ → ∞, one thus has (2)

Jl,r (τ ) 

1 Jl,r (τ ), 3

Jl (τ )  Jr (τ ),

Sl,r (τ, μ)  Jl,r (τ ),

τ → ∞. (16.87)

The combination of Eqs. (16.85) and (16.87) leads to the asymptotic behavior Sl (τ, μ)  Sr (τ, μ) 

3 F τ. 8π

(16.88)

In the scalar case, the intensity at large distances from the surface becomes isotropic and increases linearly with the optical depth τ . For Rayleigh scattering, the same holds true for the components Il and Ir and also for Stokes I , whereas Stokes Q, which describes the net polarization, tends to zero at infinity. Because the source vector is increasing indefinitely at infinity, the radiative flux may remain constant

16.6 The Polarized Milne Problem

381

at infinity, although the radiation field becomes isotropic and unpolarized. It is sometimes convenient to view the Milne problem as having a source of infinite strength at infinity.

16.6.2 The Source Vector The relation between the source vector S(τ ) and the resolvent matrix (τ ), written in Eq. (16.14) is   S(τ ) = I +

τ

 (τ ) dτ S(0).

(16.89)

0

An exact expression of (τ ) is given in Eq. (16.66). It remains to find S(0). This step is fairly simple. Remembering that (τ ) tends to a constant matrix 0 at infinity, we infer from Eq. (16.89) that S(τ )  0 S(0) τ,

τ → ∞.

(16.90)

To find S(0), we first determine the expression of S(τ ) for τ → ∞. Applying the matrix A−1 (μ) to Eq. (16.88), we obtain √ S(τ ) 

6 F j 0 τ, 8π

τ → ∞,

(16.91)

where j 0 = A−1 (μ)

   √   √ 1 2 1 2 0 2(1 − μ2 ) ≡ √ √ . ≡ 1 −μ2 1 3 2(1 − μ2 ) 1

(16.92)

Combining now Eqs. (16.90) and (16.91) with the expression of 0 given in Eq. (16.64), we find ⎡√ β ⎤ √ √  3 ⎢ 2α ⎥ 3 2c S(0) = F⎣ F . ⎦= 2 8π 8π q α √

(16.93)

For the scalar Milne problem, the surface value of the source function is given by the Hopf–Bronstein relation, √ 3 F. S(0) = 4π

(16.94)

382

16 Conservative Rayleigh Scattering: Exact Solutions

For the polarized Milne problem, Eq. (16.93) combined with q 2 + 2c2 = 2, leads to the generalized Hopf–Bronstein relation S12 (0) + S22 (0)

1 = 2

√ 3F 4π

2

.

(16.95)

The variation of S(τ ) with the optical depth can be deduced from Eq. (16.89). Inserting the expression of (τ ) given by Eq. (16.66), we obtain   S(τ ) = 0 τ + I +

∞ 1

 ∞

 (ν) (ν) −ντ dν − e dν S(0). ν ν 1 (16.96)

We see that the source function vector contains a term increasing linearly with optical depth, giving the behavior of S(τ ) at infinity, a constant term, and a τ dependent term, decreasing exponentially. In Appendix N of this chapter, devoted to the solution of the Wiener–Hopf integral equation for S(τ ), we obtain a solution having the form  S(τ ) = [s1 τ + s0 ]j 0 + where s(ν) is given in Eq. (N.21), s1 =  s0 = s1

√

∞

s(ν)e−ντ dν,

(16.97)

1

6F /(8π), and

 1 β − √ X2 (0) . α 2

(16.98)

Here X2 (0) is the derivative of X2 (z) taken at z = 0. An expression of X2 (0) in terms of the function Hr (μ) is given in Eq. (M.47). It is easy to verify that the τ dependent terms in Eqs. (16.96) and (16.97) are identical. It is slightly more difficult to prove the equality of the constant terms. Some contour integrations involving the elements of the matrix X(z) are needed.

16.6.3 The Generalized Hopf Functions The source function for the scalar Milne problem can be written as √ 3F S(τ ) = [τ + q(τ )], 4π

(16.99)

where q(τ ) is the famous Hopf function (see Sect. 9.1.2). Generalized Hopf functions can be defined for Rayleigh scattering (see, e.g., Domke 1971). The exact

16.6 The Polarized Milne Problem

383

expression of S(τ ) given in Eq. (16.96), or Eq. (16.97), shows that the components S1 (τ ) and S2 (τ ) can be written as Si (τ ) = ci [τ + qi (τ )],

i = 1, 2,

(16.100)

where c1 and c2 are constants. The functions q1 (τ ) and q2 (τ ) play for Rayleigh scattering the role played by the Hopf function q(τ ) for the scalar Milne problem. According to Eq. (16.97), we can write √

3 F [τ + q1 (τ )], 4π √ 3 F S2 (τ ) = √ [τ + q2 (τ )], 4π 2 S1 (τ ) =

(16.101)

(16.102)

where q1 (τ ) =

p q∞

1 + √ s1 2 p



∞

s1 (ν) e−ντ dν,

(16.103)

1

q2 (τ ) = q∞ +

1 s1



p

∞

s2 (ν) e−ντ dν,

(16.104)

1

q∞ ≡

s0 . s1

(16.105)

Here s1 (ν) and s2 (ν) are the two components of s(ν). The ratio s0 /s1 is given in Eq. (16.98). In Eq. (16.131) we give an expression of this ratio in terms of moments of the functions Hl (μ) and Hr (μ). Explicit expressions for the generalized Hopf functions can be deduced from the expression of s(ν) given in Eq. (N.21). Using Eq. (M.26) to calculate the jump of X(z), we find √  p q1 (τ ) = q∞ + 3 2 q2 (τ ) =

p q∞

1 0

(μ − c)(1 − μ2 ) −τ/μ e dμ, Hr (μ)R2 (1/μ)

(16.106)

11

√  −3 2 0

 1 (μ − c)μ2 + e−τ/μ dμ, α Hl (μ)R1 (1/μ) Hr (μ)R2 (1/μ) (16.107)

where − R1,2 (ν) = L+ 1,2 (ν)L1,2 (ν),

ν ∈ [1, ∞[.

(16.108)

384

16 Conservative Rayleigh Scattering: Exact Solutions

The limiting values of L1 (z) and L2 (z) are complex numbers (see Eq. (M.4)), but their limiting values above and below the cut are complex conjugate. The functions R1,2 (ν) are thus real. The values of L1 (z) and L2 (z) given in Sect. 16.2, for z = 1 and z → ∞ show that R1 (1/μ) and R2 (1/μ) keep finite values for μ ∈ [0, 1]. One can note that the expressions of q1 (τ ) and q2 (τ ) are somewhat similar to the expression of q(τ ) given in Eq. (9.29) for the scalar Milne problem. For τ = 0, the value of S(0) in Eq. (16.93) leads to, β 1 c q1 (0) = √ = √ = 0.61726, α 2 2

√ q 2 = √ = 0.48783. q2 (0) = α 2 (16.109) p

For τ → ∞, the functions q1 (τ ) and q2 (τ ) tend to the same constant value q∞ . Numerical values have been calculated by several authors. Bond and Siewert (1967) p give q∞ = 0.712109761. A value with 17 significant digits can be found in Viik p (1990). The value of q∞ is slightly larger than q(∞), the value at infinity of the scalar Hopf function, namely 0.7121 instead of 0.7104 (see Sect. 9.1.2). The functions q1 (τ ) and q2 (τ ) have, as the Hopf function q(τ ) itself, a rather small amplitude of variation. Numerical values of q1 (τ ) and q2 (τ ) can be found in Domke (1971) for τ ∈ [0, 3.8] (see also Viik (1990)). They show that q1 (τ ) and q2 (τ ) have almost converged to their asymptotic value at τ = 3.8, indeed q1 (3.8) = 0.71208 and q2 (3.8) = 0.71069. It should be noted that our definition of q1 (τ ) corresponds to Domke’s q2 (τ ) and conversely. In the scalar case, the Hopf function satisfies an inhomogeneous Wiener–Hopf integral equation (see Eq. (9.34)). For Rayleigh scattering, there is a corresponding set of integral equations. Introducing Eqs. (16.101) and (16.102) into the integral equation for S(τ ), we find 

∞

q1 (τ ) = 0



∞

q2 (τ ) = 0

1 [K11 (τ −τ )q1 (τ )+ √ K12 (τ −τ )q2 (τ )] dτ +q1∗ (τ ), 2

(16.110)

√ [ 2K21 (τ − τ )q1 (τ ) + K22 (τ − τ )q2 (τ )] dτ + q2∗ (τ ),

(16.111)

where 3 q1∗ (τ ) = √ [E3 (τ ) + E5 (τ )], 4 2

q2∗ (τ ) =

3 [E3 (τ ) − E5 (τ )]. 4

(16.112)

Here E3 (τ ) and E5 (τ ) are the third and fifth exponential integral functions. The functions Kij (τ ) are the elements of the kernel matrix K(τ ) (see Eq. (16.13)). We recall that K12 (τ ) = K21 (τ ). This set of equations can be used to calculate the numerical values of q1 (τ ) and q2 (τ ). Because the integral of K12 (τ ) is zero, the coupling terms are smaller than the diagonal terms. In Domke (1971) the generalized Hopf functions are calculated with the expressions given in Eqs. (16.106) and

16.6 The Polarized Milne Problem

385

(16.107), using accurate tables of Hl (μ) and Hr (μ) calculated by Bond and Siewert (1967). Viik (1990) makes use of the Chandrasekhar discrete ordinate method. In the generalized theory of Rayleigh scattering developed by Ivanov and coworkers (Ivanov 1995, 1996; Ivanov et al. 1995, 1996), the matrix representation of the radiation field and of the source function leads to a matrix representation of the generalized Hopf functions.

16.6.4 The Emergent Radiation Field The emergent radiation field I(τ, μ) may be written as 

∞

I(0, μ) =

S(τ )e−τ/μ

0

dτ , μ

μ ∈ [0, 1],

(16.113)

p = 1/μ,

(16.114)

or equivalently as ˜ I(0, μ) = pS(p),

˜ where S(p) is the Laplace transform of S(τ ). Taking the Laplace transform of the derivative of Eq. (16.14), we can rewrite the emergent field as ˜ ˜ I(0, μ) = [I + (p)]S(0) = G(p)S(0),

(16.115)

˜ or, using the expression of G(p) given in Eq. (16.70), as 1 1 I(0, μ) = −μX(− )P(− )S(0). μ μ

(16.116)

Multiplying by the matrix A(μ) (see Eq. (16.3)), using the expressions of X(z), P(z), and S(0) given in Eqs. (16.37), (16.63), and (16.93), and also α 2 −β 2 = 2, we obtain 3 F 2 1 μX1 (− ), Il (0, μ) = √ 8π α μ 2 β 3 F 1 Ir (0, μ) = √ (μ + )X2 (− ). α μ 2 8π

(16.117)

In terms of the functions Hl (μ) and Hr (μ) defined in Eq. (16.41), we recover the Chandrasekhar’s result 3F q √ Hl (μ), 8π 2 3F μ + c Ir (0, μ) = √ Hr (μ). 8π 2 Il (0, μ) =

(16.118)

386

16 Conservative Rayleigh Scattering: Exact Solutions

Setting μ = 0 in Eq. (16.118), applying the matrix A−1 (μ) with μ = 0, and using Hl (0) = Hr (0) = 1, we recover the surface value of S(0) given in Eq. (16.93). We recall that for the scalar Milne problem, the emergent intensity is given by I (0, μ) = √ 3/(4π)F H (μ). The coefficients α and β (or q and c) are defined by the two conditions given in Eq. (16.38). These conditions are actually a direct consequence of the cylindrical symmetry of the problem. Imposing Il (0, ±1) = Ir (0, ±1) in Eq. (16.117), we can write (α + β)X2 (−1) = 2X1 (−1),

(α − β)X2 (1) = 2X1 (1).

(16.119)

Making use of the factorization relations X1 (1)X1 (−1) = 2 and X2 (1)X2 (−1) = 4 (see Eq. (M.24)), Eq. (16.119) may also be written as (α ± β)X1 (±1) = X2 (±1).

(16.120)

We recover the conditions on α and β given in Eq. (16.38). We give in Eq. (M.37) the corresponding relations for the matrices Hl (μ) and Hr (μ). Following Chandrasekhar, we define the polarization rate of the emergent radiation as p(μ) =

(μ + c)Hr (μ) − qHl(μ) Ir (0, μ) − Il (0, μ) = . Ir (0, μ) + Ir (0, μ) (μ + c)Hr (μ) + qHl(μ)

(16.121)

The polarization, which is zero at disk center since Il (0, 1) = Ir (0, 1), takes its maximum value at the limb for μ = 0. Using Hl (0) = Hr (0) = 1, we recover the famous result, first obtained by Chandrasekhar, namely p(0) = (c − q)/(q + c) = 0.11713 (see e.g. Chandrasekhar 1960, Table XXIV, p. 248).

16.6.5 The Stokes Parameters I and Q Observations and numerical work on polarized radiative transfer are usually presented in terms of the Stokes parameters I and Q. Here we use the expressions obtained for S1 (τ ) and S2 (τ ), the components of S(τ ), to discuss some properties of the Stokes parameters I = Il + Ir and Q = Il − Ir and of the corresponding source functions SI (τ, μ) and SQ (τ, μ). Using I (τ, μ) = A(μ)I(τ, μ) (see Eq. (16.4)), we can write 

  √    2 √ 3 1 1 I (τ, μ) 2(1 − μ2 ) I2 (τ, μ) μ . = Q(τ, μ) 1 0 I1 (τ, μ) 2 1 −1

(16.122)

16.6 The Polarized Milne Problem

387

The same relation holds for the components SI (τ, μ) and SQ (τ, μ) of the source vector associated to the (I, Q) representation. It leads to √

√ 3 [(1 + μ2 )S1 (τ ) + 2(1 − μ2 )S2 (τ )], 2 √ √ 3 (1 − μ2 )[S1 (τ ) − 2S2 (τ )]. SQ (τ, μ) = − 2

SI (τ, μ) =

(16.123) (16.124)

It is obvious from Eq. (16.124) that the polarization is zero in the direction normal to the surface corresponding to μ = 1. In terms of the generalized Hopf functions, we can write   3F μ2 1 [q1 (τ ) − q2 (τ )] , SI (τ, μ) = τ + [q1 (τ ) + q2 (τ )] + 4π 2 2 SQ (τ, μ) = −

3F (1 − μ2 )[q1 (τ ) − q2 (τ )]. 8π

(16.125)

Introducing the explicit expressions of q1 (τ ) and q2 (τ ) in Eqs. (16.106) and (16.107), we can write SQ (τ, μ) as 9 F SQ (τ, μ) = − √ (1 − μ2 ) 2 4π

 1 0

 (μ − c)μ2 1 q + e−τ/μ dμ. 2 Hl (μ)R1 (1/μ) Hr (μ)R2 (1/μ) (16.126)

This expression shows that SQ (τ, μ) decreases exponentially at infinity. Actually the polarization can be considered to be zero at optical depths larger than, say, 5. For Stokes I , Eq. (16.125) shows that SI (τ, μ) 

3F p (τ + q∞ ), 4π

τ → ∞.

(16.127)

We can observe that SI (τ, μ) behaves essentially as the source function of the p scalar Milne problem. The constant q∞ plays the role of an extrapolation length (or extrapolation endpoint), which is defined as the distance above the surface at which the linear approximation of the source function tends to zero (see Sect. 9.1.2). We summarize in Table 16.1 the surface values and behavior at infinity of the components of the source vector for different representations of the radiation field. It is shown in Sect. 16.6.1 that JI(2) (τ ) ≡ Jl(2) (τ ) + Jr(2) (τ ) =

F (τ + C p ), 4π

(16.128)

388

16 Conservative Rayleigh Scattering: Exact Solutions

Table 16.1 The Rayleigh Milne problem. The surface value and behavior at infinity of the two components of the source vector for different representation of the polarized radiation field. Sl and Sr correspond to the (Il , Ir ) representation and SI and SQ correspond to the (I, Q) one. For comparison the first line shows the source function S(τ ) for the √ scalar Milne problem. For the scalar case, s1 = 3F /(4π) and for Rayleigh scattering, s1R = 6F /(8π) with F the total flux, defined for Rayleigh scattering in Eq. (16.77). We recall that c = β/α = 0.873 and q = 2/α = 0.690 τ =0 1 √ s1 3

Source function S(τ )

τ →∞ s1 τ √

s1R c

S1 (τ )

q s1R √ 2 √ 3 R s [(1 − μ2 )q + μ2 c] 2 1 √ 3 R s c 2 1 √ 3 R s [(1 + μ2 )c + (1 − μ2 )q] 2 1 √ 3 R − s (1 − μ2 )(c − q) 2 1

S2 (τ )

Sl (τ, μ) Sr (τ, μ) SI (τ, μ) SQ (τ, μ)

(2)

2s1R τ

s1R τ √

6 R s τ 2 1 √ 6 R s τ 2 1 √ R 6s1 τ 0

(2)

where Jl (τ ) and Jr (τ ) are the components of the vector J 2 (τ ) defined in Eq. (16.80) and C p is a constant. Using JI(2)(τ ) 

1 F p SI (τ, μ)  (τ + q∞ ), 3 4π

τ → ∞,

(16.129)

we readily find p

C p = q∞ =

4π (2) J (0). F I

(16.130)

The expression of the emergent radiation field given in Eq. (16.118) leads to 3 p q∞ = √ 4 2

 0

1

3 [qHl(μ) + (μ + c)Hr (μ)]μ2 dμ = √ [qα2l + cα2r + α3r ]. 4 2 (16.131)

Numerical values of the moments of Hl (μ) and Hl (μ) are given in Table M.3. Other p expressions for q∞ can be found in the literature, e.g. in Bond and Siewert (1967). For τ = 0, the expression of S(0) given in Eq. (16.93) leads to 3 F [(1 + μ2 ) c + (1 − μ2 ) q], SI (0, μ) = √ 2 8π

(16.132)

16.7 The Diffuse Reflection Problem

389

3 F (1 − μ2 )(c − q). SQ (0, μ) = − √ 8π 2

(16.133)

Setting μ = 0, we recover the limb polarization rate p(0) = |Q(0, 0)|/I (0, 0) = |SQ (0, 0)|/SI (0, 0) = (c − q)/(c + q) = 0.11713. For Stokes I and Stokes Q, exact expressions of the surface values I (0, μ) and Q(0, μ), μ ∈ [0, 1], are readily deduced from the expressions of Il (0, μ) and Ir (0, μ) given in Eq. (16.118). The polarized Milne problem can of course be solved directly for I and Q, but it is clear from Eq. (16.118) that the algebra is simpler with Il and Ir . At infinity, Stokes Q tends to zero and Stokes I becomes isotropic, while increasing linearly with optical depth. A detailed analysis of the behavior of I and Q at infinity is carried out by V.V. Ivanov and co-workers in a series of four papers (Ivanov 1995; Ivanov et al. 1995, 1996) dealing with both conservative and nonconservative Rayleigh scattering.

16.7 The Diffuse Reflection Problem The radiative transfer equation for the radiation field I (τ, μ) is the same for diffuse reflection and for the Milne problem. It is given in Eq. (16.75). For diffusion reflection the source of photons comes from the illumination by an external radiation field. For simplicity we assume that the external radiation is a parallel beam in the direction −μ0 , μ0 > 0. It can be written as I inc (−μ) = I inc δ(μ − μ0 ),

μ, μ0 > 0.

(16.134)

The two-component vector 2πI inc represents the flux of the incident beam, per unit area, in the direction perpendicular to the beam. As observed in Sect. 14.2.2, the radiation field can be decomposed into a direct field, describing the ballistic penetration of the incident field, and a diffuse field, I d (τ, μ). For the diffuse field, the radiation field incident on the surface is zero, but there is a source of photons inside the medium due to the scattering of the external radiation. In the following we consider only the diffuse field. By application of the matrix A−1 (μ) (see Eq. (16.4)), we transform it into a field I(τ, μ), which satisfies the radiative transfer equation μ

∂I (τ, μ) = I(τ, μ) − S(τ ), ∂τ

(16.135)

with I(0, μ) = 0, μ ∈ [−1, 0]. The source vector S(τ ) satisfies a Wiener–Hopf integral equation (see Eq. (16.12)) in which the primary source is Q∗ (τ ) =

1 T A (−μ0 )I inc e−τ/μ0 . 2

(16.136)

390

16 Conservative Rayleigh Scattering: Exact Solutions

For a primary source term varying as e−τ/μ0 , the source vector S(τ ) and the resolvent matrix (τ ) are related by   −τ/μ0 I+ S(τ ) = e

τ

e

−(τ −τ )/μ0



(τ ) dτ



 S(0),

(16.137)

0

(see Eq. (16.15)). The resolvent matrix is determined in Sect. 16.4, where we show in particular that (τ ) tends to a constant value 0 for τ → ∞. To determine S(τ ), one still needs S(0). The integration over τ shows that S(τ ) goes at infinity to the constant value S(τ )  μ0 0 S(0),

τ → ∞.

(16.138)

In contrast to the Milne problem, there is no simple physical argument allowing one to deduce the value of S(0) from the behavior of S(τ ) at infinity. The only thing that we know is that at infinity S(τ ) tends to a constant, and that the radiation field will become unpolarized and isotropic and also tend to a constant value. A word about the total radiative flux F (τ ) defined in Eq. (16.77). Equation (16.82) shows that it has a constant value. The radiation field tending to a constant at infinity, Eq. (16.83) shows that the flux tends to zero at infinity, and since it has a constant value, it is zero everywhere in the medium. We now determine the emergent radiation field at the surface I(0, μ), μ ∈ [0, 1]. We recover for I (0, μ) the standard Chandrasekhar result. We then derive an exact expression of S(0) from the relation S(0) = I(0, 0) and used it in Sect. 16.7.2 to find exact expressions of S(τ ) and of the radiation field at infinity.

16.7.1 The Emergent Radiation Field The field I(τ, μ) satisfies Eq. (16.135), hence at the surface, for the outgoing directions, 

∞

I(0, μ) = 0

e−τ/μ S(τ )

dτ =p μ



∞

e−pτ S(τ ) dτ,

p = 1/μ,

μ ∈ [0, 1].

0

(16.139) Expressing S(τ ) in terms of the Green matrix G(τ, τ0 ) (see Eq. (15.41)) and the primary source term Q∗ (τ ) in terms of its inverse Laplace transform q ∗ (ν), we can write  ∞ ˜˜ I(0, μ) = p G(p, ν)q ∗ (ν) dν, (16.140) 0

16.7 The Diffuse Reflection Problem

391

˜˜ where G(p, ν) is the double Laplace transform of G(τ, τ0 ). For a beam with direction −μ0 , ⎧ ⎨ 0, ν ∈ [0, 1] 1 1 T 1 inc q ∗ (ν) = (16.141) ), ν ∈ [1, ∞[. ⎩ 2 A (− )I δ(ν − 2ν ν μ0 For the scalar case, it is shown in Sect. (11.2.2) that ˜˜ G(p, ν) =

1 ˜ ˜ G(p)G(ν). ν +p

(16.142)

For Rayleigh scattering, there is a similar result established in Sect. 15.4.1, namely ˜˜ G(p, ν) =

1 ˜ ˜ T (ν). G(p)G ν +p

(16.143)

Hence, we can write ˜ I(0, μ) = pG(p)



∞

˜ T (ν)q ∗ (ν) dν. G

(16.144)

0

˜ Using G(1/μ) = H(μ) and the expression of q ∗ (ν) given in Eq. (16.141), we obtain I(0, μ) =

1 μ0 H(μ)HT (μ0 )AT (−μ0 )I inc . 2 μ + μ0

(16.145)

For the positive values of μ, the diffuse field is equal to radiation field, hence to obtain the emergent components Il (0, μ) and Ir (0, μ), it suffices to apply the matrix ˜ A(μ) to I(0, μ). Using the expression of G(p) given in Eq. (16.70), we obtain I (0, μ) =

3 μ0 S(μ, μ0 )I inc , 8 μ + μ0

(16.146)

where the four elements of the matrix S(μ, μ0 ) may be written as Sl,l (μ, μ0 ) = Hl (μ)2[1 − c(μ + μ0 ) + μμ0 ]Hl (μ0 ), Sl,r (μ, μ0 ) = Hl (μ)q(μ + μ0 )Hr (μ0 ), Sr,l (μ, μ0 ) = Hr (μ)q(μ + μ0 )Hl (μ0 ), Sr,r (μ, μ0 ) = Hr (μ)[1 + c(μ + μ0 ) + μμ0 ]Hr (μ0 ).

(16.147)

We recover the result given in Chandrasekhar (1960, pp. 254–256) by setting I inc = (Ilinc , Irinc ) = (Fl , Fr )/2 and S(μ, μ0 ) = S (0) (μ, μ0 )(μ + μ0 )/(μμ0 ), where S (0)(μ, μ0 ) is the Chandrasekhar scattering matrix.

392

16 Conservative Rayleigh Scattering: Exact Solutions

The expression obtained for a Dirac distribution can be generalized to an incident field I inc (μ) = (Ilinc (μ), Irinc (μ)), μ < 0. It suffices to integrate over the direction of the incident field. We thus obtain 

μ  2[1 − c(μ + μ ) + μμ ]Hl(μ )Ilinc (−μ )

0 μ+μ  + q(μ + μ )Hr (μ )Irinc (−μ ) dμ , (16.148)

Il (0, μ) =

3 Hl (μ) 8



1

μ  [1 + c(μ + μ ) + μμ ]Hr (μ )Irinc (−μ )

0 μ+μ  + q(μ + μ )Hl (μ )Ilinc (−μ ) dμ . (16.149)

3 Ir (0, μ) = Hr (μ) 8

1

Equations (16.147), (16.148), and (16.149) satisfy the cylindrical symmetry of the problem, that is Il (0, 1) = Ir (0, 1) and the condition that the radiation is unpolarized, that is Il (0, μ) = Ir (0, μ), when the incident radiation is in the direction of the symmetry axis, i.e. μ0 = 1. These properties are easily verified with the help of Eq. (M.37). When the incident field is unpolarized and independent of μ, that is Ilinc (μ) = Irinc (μ) = I inc , μ ∈ [−1, 0], then I (τ, μ) = (I inc , I inc ) is solution of the radiative transfer equation, the emergent field is equal to the incident field and inside the medium, the field is everywhere uniform, unpolarized and isotropic. We are now finally able to calculate S(0). Using S(0) = I(0, 0) and the expression of I(0, μ) given in Eq. (16.145), we find ⎤ √ √ ⎡ √ 2qμ0 Hl (μ0 ) 2(1 + cμ0 )Hr (μ0 ) 3 ⎣ ⎦ I inc . S(0) = √ 4 2 2(1 − cμ0 )Hl (μ0 ) qμ0 Hr (μ0 )

(16.150)

When the incident field is polarized and depends on μ, it suffices to replace I inc by the vector (Ilinc (−μ), Irinc (−μ)), μ > 0, and integrate over μ, μ ∈ [0, 1].

16.7.2 The Source Vector and the Radiation Field at Infinity The behavior of the radiation field at infinity for the diffuse reflection problem is in general not considered in the literature (see however Domke 1973), but it plays an important role in the analysis of the diffusion limit for Rayleigh scattering treated in Chap. 24. We use it to perform an asymptotic matching between the interior and boundary layer solutions. As shown by Eq. (16.138), the source function S(τ ) tends to a constant vector at infinity. Different possibilities exist to find this vector and hence the radiation field at infinity.

16.7 The Diffuse Reflection Problem

393

As shown in Eq. (16.138), limτ →∞ S(τ ) = μ0 0 S(0). Explicit expressions for 0 and for S(0) are given in Eqs. (16.64) and (16.150). Simple algebra leads to lim S(τ ) = s0 j 0 ,

τ →∞

(16.151)

√ where j 0 = ( 2, 1) and √ 3 μ0 [qHl(μ0 )Ilinc + (μ0 + c)Hr (μ0 )Irinc ], s0 = 4

(16.152)

for an incident beam of radiation in the direction −μ0 , μ0 ∈ [0, 1]. For an incident field I inc (−μ) = (Ilinc (−μ), Irinc (−μ)), μ > 0, √  1 3 s0 = [qHl(μ)Ilinc (−μ) + (μ + c)Hr (μ)Irinc (−μ)]μ dμ. 4 0

(16.153)

At infinity, the radiation field I(τ, μ) tends to the constant value of S(τ ). Using I (τ, μ) = A(μ)I(τ, μ), we obtain $   3 1 I (τ, μ)  s0 , 2 1

τ → ∞,

(16.154)

hence, 3 Il∞ = Il∞ = √ 4 2



1 0

[qHl (μ)Ilinc (−μ) + (μ + c)Hr (μ)Irinc (−μ)]μ dμ. (16.155)

Equation (16.155) is a generalization to Rayleigh scattering of the value obtained in the scalar case, namely S∞ = I ∞ =

√  1 3 H (μ)I inc (−μ)μ dμ. 2 0

(16.156)

Another method, for calculating the radiation field at infinity makes use of the expression of the emergent radiation field. For the diffuse reflection problem, we (2) have seen that F = 0. Equation (16.84) with F = 0 shows that the sum Jl (τ ) + (2) Jr (τ ) is a constant. We thus have Jl(2)(0) + Jr(2) (0) =

1 ∞ (I + Ir∞ ). 3 l

(16.157)

394

16 Conservative Rayleigh Scattering: Exact Solutions

At the surface, we know the given incident radiation and the emergent radiation from Eqs. (16.148) and (16.149). An integration over μ, μ ∈ [−1, +1], will provide after some algebra the radiation field at infinity. The expression given in Eq. (16.155) is rather simple and suggests that it could be derived from the solution of the Milne problem. We consider the definition of the flux given in Eq. (16.77) and the emergent radiation field for the Milne problem given in Eq. (16.118). Since the flux is constant in the medium and the incident radiation at the surface is zero, we have the identity 3 F = √ 4 2



1

[qHl(μ)F + (c + μ)Hr (μ)F ]μ dμ.

(16.158)

0

We now consider the diffuse reflection problem. First we assume that the incident radiation is unpolarized and independent of μ, that is Ilinc (0, μ) = Irinc (0, μ) = I inc , μ ∈ [−1, 0]. In this case, as mentioned above, the radiation field is isotropic, unpolarized, independent of the optical depth and equal to (I inc , I inc ) everywhere inside the medium. Since Eq. (16.158) is an identity, we can set F = I inc = Ilinc (0, −μ) = Irinc (0, −μ) = Il∞ = Ir∞ . Thus, when the incident radiation is independent of μ, we can write 3 Il∞ = Ir∞ = √ 4 2  1  qHl (μ)Ilinc (0, −μ) + (μ + c)Hr (μ)Irinc (0, −μ) μ dμ. ×

(16.159)

0

Equations (16.148) and (16.149) show that the functions Hl (μ) and Hr (μ) should be associated respectively to the l and r components of the radiation field. The two terms in the integral describe the transformation of the two components of the incident radiation by the scattering process inside the semi-infinite medium. This process does not depend on the incident field. Hence, Eq. (16.159) holds for any incident radiation field and leads to Eq. (16.155). The argument given here for Rayleigh scattering holds also in the scalar case and leads to Eq. (16.156).

Appendix M: Properties of the Auxiliary Functions We discuss in this Appendix some properties of the functions L1 (z), L2 (z), X1 (z), X2 (z), and of the matrices L(ν), L(z), and X(z). These functions are introduced in Sects. 16.2 and 16.3.

M.1 The Functions L1 (z) and L2 (z)

395

M.1 The Functions L1 (z) and L2 (z) The functions L1 (z) and L2 (z) are defined by Ln (z) ≡ (−1)n + 3(1 −

1 ) L(z), z2

n = 1, 2,

(M.1)

where L(z) ≡ 1 −

1 2



∞

( 1

1 dν 1 1+z 1 + ) =1− ln , ν+z ν −z ν 2z 1 − z

(M.2)

is the dispersion function for the scalar case. The definition of Ln (z) shows that its properties are determined by those of L(z), which have been studied in Sect. 5.2. We recall here the main properties of L(z) for  = 0. Equation (M.2) shows that L(z)  −

3 z2 (1 + z2 ) + O(z6 ), 3 5

z → 0,

(M.3)

and that L(z) → 1 for z → ∞. It also shows that L(z) is analytic in C /]−∞, −1]∪ [1, +∞[. Its limiting values along the cut may be written as L± (ν) = λ(ν) ∓ i π

1 , 2ν

|ν| ∈ [1, ∞[,

(M.4)

where λ(ν) is given by Eq. (M.2), with z replaced by ν and 1 − z by |1 − ν| for |ν| ∈ [1, ∞[. The variation of λ(ν) with ν is shown in Fig. 5.4 for  = 0. When  = 0, λ(ν) has double zero at the origin, behaving as shown in Eq. (M.3), with z replaced by ν. Otherwise λ(ν) behaves as shown in Fig. 5.4, namely is negative for |ν| ∈ [0, 1[, has a logarithmic divergence at ν = ±1, and tends to 1 for |ν| → ∞. One deduces from the definition of Ln (z) and the properties of L(z) that L1 (z)  2,

L2 (z)  4,

2 L1 (z)  − z2 , 5

for z → ∞,

L2 (z)  2,

for z → 0,

(M.5) (M.6)

and Ln (±1) = (−1)n , These values are displayed in Table M.2.

n = 1, 2.

(M.7)

396

16 Conservative Rayleigh Scattering: Exact Solutions

One can also observe that the functions Ln (z) have the same analyticity properties as L(z). Hence, these two functions are analytic in C /]−∞, −1]∪[1, ∞[ and their limiting values along the cut are given by L± n (ν) = λn (ν) ∓ i π3(1 −

1 1 ) , ν 2 2ν

ν ∈] − ∞, −1] ∪ [1, ∞[,

(M.8)

1 )λ(ν), ν2

(M.9)

with λn (ν) = (−1)n + 3(1 −

n = 1, 2.

Actually, the points ν = ±1 do not belong to the cut of Ln (z) since Ln (±1) = − (−1)n . The functions L+ n (ν) and Ln (ν) are complex conjugate. To construct the auxiliary functions X1 (z) and X2 (z) with the method described in Sect. M.3, we need the variation of θn (ν), the argument of L+ n (ν), n = 1, 2, along the interval [1, ∞[. We write + i θn (ν) L+ , n (ν) = |Ln (ν)|e

ν ∈]1, ∞[,

n = 1, 2.

(M.10)

Then θn (ν) is given by θn (ν) = arctan

[L+ n (ν)] , [L+ n (ν)]

−π ≤ θn (ν) ≤ 0.

(M.11)

+ Equation (M.8) shows that the imaginary parts of L+ 1 (ν) and L2 (ν) are equal, that they are negative for ν ∈]1, ∞[, and that they are zero at ν = 1 and at infinity. The real part of L+ 1 (ν) changes its sign, varying from −1 to 2, when ν varies from 1 to ∞. Hence the argument θ1 (ν) has a variation of π along [1, ∞[. The real part of L+ 2 (ν) is always positive, varying between 1 and 4, when ν varies from 1 to ∞. Hence the variation of θ2 (ν) is zero. These properties are summarized in Table M.1.

Table M.1 Real parts, imaginary parts, and arguments of L+ 1 (ν) and L+ (ν) along the cut [1, ∞[ 2

1

∞

ν ∈ [1, ∞[

−1

2

change of sign

0

0

negative

1

4

positive

θ1 (ν)

−π

0

θ2 (ν)

0

0

ν [L+ 1 (ν)] = [L+ 1,2 (ν)] [L+ 2 (ν)]

λ1 (ν)

= λ2 (ν)

M.2 Some Properties of the Matrix L(ν)

397

M.2 Some Properties of the Matrix L(ν) The matrix L(ν) is defined by  L(ν) ≡ I −

∞

M(ν)( 0

1 1 +

) dν , ν + ν ν −ν

(M.12)

where M(ν), the inverse Laplace transform of the kernel K(τ ), is given in Eq. (16.19). The matrix L(ν) is self-adjoint and an even function of ν. Its four elements may be written as   1 1 1 1 + 3(1 + 4 )λ(ν) + 2 , 4 ν ν √   1 2 1 L12 (ν) = L21 (ν) = −1 + 3(1 − )λ(ν) , 4 ν2 ν2   1 1 1 L22 (ν) = −1 + 3(1 − 2 )2 λ(ν) + 2 . 2 ν ν

L11 (ν) =

(M.13)

The solution of the singular integral equation for (ν) (see Eq. (16.44)) depends on the existence and position of possible zeroes of the determinant of L(ν) for ν ∈ [0, 1]. This determinant is usually referred to as the dispersion function. One can show, using for example (16.22), that det L(ν) =

1 λ1 (ν)λ2 (ν), 8

(M.14)

where λn (ν), n = 1, 2, are defined in Eq. (M.9). The factorization of the determinant of L(ν) was first demonstrated by Chandrasekhar (1946b) (see also Schnatz and Siewert 1971). We now show that det L(ν) has a double zero at the origin, coming from a double zero of λ1 (ν). We consider the interval ]0, 1[. In this interval the product 3(1 − 1/ν 2 )λ(ν) is positive, hence λ2 (ν) is positive, and has the values λ2 (0) = 2 and λ2 (1) = 1 at the end points of the interval. In contrast, λ1 (ν) is negative, with λ1 (ν)  − 25 ν 2 for ν → 0 and λ1 (1) = −1. Therefore λ1 (ν) and hence det L(ν) have a double zero at the origin. We now consider L(ν) for ν = 0. To find the limit of Lij (ν) for ν = 0, we replace λ(ν) by the asymptotic expansion λ(ν)  −ν 2 (1 +3ν 2 /5)/3 given in Eq. (M.3). We thus obtain √ 2 1 1 2 1 L11 (ν)  − ν ; L12 (ν) = L21 (ν)  − (1+3ν 2); L22 (ν)  (1+2ν 2), 10 4 10 5 (M.15)

398

16 Conservative Rayleigh Scattering: Exact Solutions

and ⎡ √ ⎤ 1 ⎣ 1 − 2⎦ . L(0) = √ 10 − 2 2

(M.16)

For ν → ∞, L11 (ν) and L22 (ν) tend to one and L12 (ν) = L21 (ν) to zero.

M.3 The Auxiliary Functions X1 (z) and X2 (z) M.3.1 Construction of X1 (z) and X2 (z) For the construction of X1 (z) and X2 (z), we proceed essentially as described in Sect. 5.3. For the jump condition, we choose Xn+ (ν) =

L− n (ν) − Xn (ν), L+ n (ν)

ν ∈ [1, ∞[,

n = 1, 2.

(M.17)

+ The ratio L− n (ν)/Ln (ν) is the inverse of the ratio used in Sect. 5.3, but this change does not modify the method of construction. We introduce the functions  −1 ∞ dν , (M.18) Yn (z) ≡ θn (ν) π 1 ν−z

where θn (ν) is the argument of L+ n (ν) and, as in the scalar case, consider exp[Yn (z)]. It behaves as exp[Yn (z)] ∼ (1 − z)θn (ν)/π

(M.19)

when z is near ν = 1, the end point of the cut. For n = 1, θ1 (ν) → −π as ν → 1, therefore the right-hand side behaves as 1/(1 − z). To avoid a singularity at z = 1 (see condition (iii) in Sect. 5.3) we define X1 (z) as X1 (z) ≡ (1 − z) exp[Y1 (z)].

(M.20)

For n = 2, θ2 (ν) → 0 as ν → 1, therefore the right-hand side in Eq. (M.19) tends to a constant and we can define X2 (z) as X2 (z) ≡ exp[Y2 (z)].

(M.21)

M.3 The Auxiliary Functions X1 (z) and X2 (z)

399

Table M.2 The functions L1,2 (z), X1,2 (z) and Hl,r (z) at three special points z = 0, 1, ∞. We recall that zX1 (−1/z) = Hl (z) and X2 (−1/z) = Hr (z). The values of Hl,r (1) are from Chandrasekhar. The values of X1,2 (1) are derived from the values of Hl,r (1) and the factorization relations in Eq. (M.24) z 0 +1 ∞

L1 (z)

L2 (z)

−2z2 /5 −1 2

2 +1 4

X1 (z) √ 5 0.58 −z

X2 (z) √ 2 3.13 1

Hl (z) 1 3.47 √ 5z

Hr (z) 1 1.28 √ 2

For z → ∞, X1 (z) → −z,

X2 (z) → 1.

(M.22)

Table M.2 displays the values of the functions L1,2 (z), X1,2 (z) and Hl,r (z) for z = 0, 1, ∞. The explicit expressions of X1 (z) and X2 (z) are given in Eqs. (16.32) and (16.33).

M.3.2 Factorizations Relations Following the method described for the scalar case in the Appendix B of Chap. 5, we introduce the functions f1 (z) ≡ X1 (z)X1 (−z)L1 (z),

f2 (z) ≡ X2 (z)X2 (−z)L2 (z).

(M.23)

According to the jump condition written in Eq. (M.17), f1 (z) and f2 (z) are analytic functions of z. We deduce from the behavior at infinity of Xn (z) and Ln (z), n = 1, 2, that X1 (z)X1 (−z) = −

2z2 , L1 (z)

X2 (z)X2 (−z) =

4 , L2 (z)

z ∈ C /]1, ∞[. (M.24)

We note that these relations hold for z = ±1, since these points do not belong to the cut of Ln (z) (see Sect. 16.2). Setting z = 0 and using the values of L1,2 (0) given in Table M.2, we obtain X1 (0) =

√

5,

X2 (0) =

√ 2.

(M.25)

400

16 Conservative Rayleigh Scattering: Exact Solutions

Combining Eq. (M.24) with the jump condition in Eq. (M.17), we can calculate the jumps of X1 (z) and X2 (z) across the cut ν ∈]1, ∞[ and the jumps of their inverses. They may be written as X1+ (ν) − X1− (ν) = −2i π

1 1 3 2ν 2 (1 − 2 ) , − 2ν ν X1 (−ν) L+ (ν)L 1 1 (ν)

1 3 4 1 (1 − 2 ) , − 2ν ν X2 (−ν) L+ (ν)L 2 2 (ν)

X2+ (ν) − X2− (ν) = 2i π

(M.26)

1 1 1 1 3 − = 2i π (1 − 2 ) 3 X1 (−ν), 4 ν ν X1+ (ν) X1− (ν) 1 1 1 3 1 − = −2i π (1 − 2 ) X2 (−ν). 8 ν ν X2+ (ν) X2− (ν)

(M.27)

The limiting values L± n (ν) are given in Eq. (M.8).

M.3.3 Nonlinear Integral Equations The functions X1 (z) and X2 (z) satisfy nonlinear integral equations, which can be constructed with the method described in Sect. B.3 for the scalar case. The idea is to write 1/Xn (z) as a Cauchy integral, namely: 1 1 = Xn (z) 2i π

 C

dξ 1 , Xn (ξ ) ξ − z

n = 1, 2,

(M.28)

where C is a contour in the complex plane turning around the branch cut of Xn (z), n = 1, 2. Using X1 (z)  −z and X2 (z)  1 for z → ∞ and Eq. (M.27), we obtain the nonlinear integral equations 3 1 = X1 (z) 4



∞

(1 −

1

1 1 dν , ) 3 X1 (−ν) 2 ν ν ν−z

(M.29)

and 3 1 =1− X2 (z) 8



∞ 1

(1 −

1 1 dν . ) X2 (−ν) 2 ν ν ν−z

(M.30)

M.3 The Auxiliary Functions X1 (z) and X2 (z)

401

Changing z → −1/z, ν → 1/μ and using the behavior at infinity of X1 (z) and X2 (z) given in Eq. (M.22), we see that these equations can also be written as 1 1 3 zX1 (− ) = 1 + zX1 (− ) z z z 4 1 1 3 X2 (− ) = 1 + X2 (− ) z z z 8



1

1 dμ , μX1 (− )(1 − μ2 ) μ μ+z

(M.31)

1 dμ . X2 (− )(1 − μ2 ) μ μ+z

(M.32)

0



1 0

M.3.4 The Functions Hl (z) and Hr (z) Most results for Rayleigh scattering are given in the literature in terms of Hl (μ) and Hr (μ). We establish here the correspondence between the functions X1 (z) and X2 (z) and the functions Hl (μ) and Hr (μ). Following (Chandrasekhar 1960, p. 246), the nonlinear equations for Hl (μ) and Hr (μ) are usually written in the form  H (μ) = 1 + μH (μ) 0

1

ψ(μ ) H (μ ) dμ , μ + μ

(M.33)

where the characteristic function ψ(μ) is given by ψl (μ) =

3 (1 − μ2 ), 4

ψr (μ) =

3 (1 − μ2 ). 8

(M.34)

A comparison between the equations for X1,2 (z) given in Eqs. (M.31) and (M.32), with the equation for Hl,r (μ) leads to 1 zX1 (− ) = Hl (z), z

1 X2 (− ) = Hr (z). z

(M.35)

The functions Hl (z) and Hr (z) are analytic in the complex plane cut along the interval ] − 1, 0]. Numerical values of Hl (μ) and Hr (μ) for μ ∈ [0, 1] can be found in Chandrasekhar (1960, p. 248, Table XXII). Both functions are increasing monotonically with μ, starting from Hl (0) = Hr (0) = 1. For μ = 1, Hl (1) = 3.4695 √ and Hr (1) = 1.27797. For z √ → ∞, one has Hr (z) = X2 (−1/z) → 2 and Hl (z) = zX1 (−1/z) → 5z. The numerical values given in Chandrasekhar (1960, p. 123) are calculated by an iterative method based on an alternate form of Eq. (M.33). More accurate values have been calculated subsequently (see, e.g., Bond and Siewert 1967).

402

16 Conservative Rayleigh Scattering: Exact Solutions

The factorization relations in Eq. (M.24) become 1 Hl (z)Hl (−z) = 2/L1 ( ), z

1 Hr (z)Hr (−z) = 4/L2 ( ), z

(M.36)

and the relations in Eq. (16.119) become qHr (1) = 2(1 − c)Hl(1),

qHl (1) = (1 + c)Hr (1).

(M.37)

M.3.5 Moments of Hl (μ) and Hr (μ) Several relations between the moments of Hl (μ) and Hr (μ) can be derived from their respective nonlinear integral equations and from properties of the Milne and of the diffuse reflection problem. Some of these relations are used in Chap. 24 dealing with the diffusion approximation for Rayleigh scattering. Following Chandrasekhar (1960), we introduce the moments  αnl

≡

1

 n

μ Hl (μ) dμ, 0

αnr

1

≡

μn Hr (μ) dμ,

(M.38)

0

where n is a positive integer. They can also be written as  αnl ≡

∞ 1

1 dν X1 (−ν) 3 , n ν ν

 αnr ≡

∞ 1

1 dν X2 (−ν) 2 . n ν ν

(M.39)

Numerical values of the two first moments of Hl (μ) and Hl (μ) and of the constants q and c can be found in Chandrasekhar (1960, p. 248). They are reproduced in Table M.3. Values of q and c, more accurate than those of Chandrasekhar, can be found in Bond and Siewert (1967) (see also Siewert 1968). Because of the factor μn , the moments of Hl (μ) and Hr (μ) are decreasing functions of n as can be observed in Table M.3. Table M.3 Moments of Hl (μ) and Hr (μ) and some numerical constants. The two first moments of Hl (μ) and Hr (μ), the constants q and c, and the values of Hl (1) and Hr (1) are from (Chandrasekhar 1960, p. 248), rounded to 3 significant digits. This reference provides also tables for Hl (μ) and Hr (μ), μ ∈ [0, 1], which we have used to calculate numerically the second and third-moments α0l = 2.297

α0r = 1.197

q = 0.690

α1l

= 1.349

α1r

= 0.617

c = 0.873

α2l

= 0.964

α2r

= 0.416

Hl (1) = 3.4695

α3l

= 0.752

α3r

= 0.314

Hr (1) = 1.2780

M.3 The Auxiliary Functions X1 (z) and X2 (z)

403

Letting z → ∞ in Eq. (M.29) and√setting z = 0 in Eq. (M.30), while using X1 (z)  −z for z → ∞, and X2 (0) = 2, we find 3 r 1 (α0 − α2r ) = 1 − √ . 8 2

3 l (α − α2l ) = 1, 4 0

(M.40)

As shown in Chandrasekhar (1960), for any characteristic function of ψ(μ ) = a + bμ 2 , the integration over μ of the nonlinear integral equation in Eq. (M.33) yields 1 α0 = 1 + [a(α0 )2 + b(α1 )2 ]. 2

(M.41)

3 α0l = 1 + [(α0l )2 − (α1l )2 ], 8

(M.42)

3 [(α r )2 − (α1r )2 ]. 16 0

(M.43)

Hence,

α0r = 1 +

The emergent radiation for the Milne problem yields another relation. Using the constancy of the total flux F = Fl (τ ) + Fr (τ ) and the expression of the emergent radiation field given in Eq. (16.118), we can write 3 √ 4 2



1

[qHl(μ) + (μ + c)Hr (μ)]μ dμ = 1,

(M.44)

0

hence, qα1l

+ α2r

+ cα1r

√ 4 2 . = 3

(M.45)

For the diffuse reflection problem, when the incident radiation in isotropic and unpolarized, the emergent radiation is equal to the incident radiation (see Sect. 16.7.1). Using Eqs. (16.148) and (16.149) with μ = 0, we find 3 (2α0l − 2cα1l + qα1r ) = 1, 8

3 r (α + cα1r + qα1l ) = 1. 8 0

(M.46)

A relation involving the third-moment α3r can be derived from two different expressions of X2 (0). Taking the derivative of Eq. (M.30) with respect to z, setting √ z = 0, and using X2 (0) = 2, we can write X2 (0) =

3 4



1 0

3 1 (1 − μ2 )μX2 (− ) dμ = μ 4



1 0

(1 − μ2 )μHr (μ) dμ.

(M.47)

404

16 Conservative Rayleigh Scattering: Exact Solutions p

Equation (16.98) together with q∞ = s0 /s1 shows that X2 (0) =

√ p 2(c − q∞ ).

(M.48)

Hence, √ 3 r p (α1 − α3r ) = 2(c − q∞ ). 4

(M.49) p

The values of c and α1r given in Table M.3, combined with q∞ = 0.712, lead to α3r = 0.314. We have found no similar relation for α3l . Finally we consider the constants α and β defined in Eq. (16.39). Expressing the functions X1 (z) and X2 (z) in terms of Hl (z) and Hr (z) (see Eq. (M.35)), we can write 1 [Hl (1)Hr (−1) − Hl (−1)Hr(1)], 4 1 β = [Hl (1)Hr (−1) + Hl (−1)Hr (1)]. 4

α=

(M.50)

The nonlinear integral equation in Eq. (M.33) yields 3 Hl (±1)[1 − (α0l ∓ α1l )] = 1, 4 3 Hr (±1)[1 − (α0r ∓ α1r )] = 1. 8

(M.51)

The combination of Eqs. (M.50) with (M.51) leads to 3 9 α = − (α1r − 2α1l ) + (α0l α1r − α1l α0r ), 2 8

(M.52)

3 9 β = −4 + (α0r + 2α0l ) − (α0l α0r − α1l α1r ). 2 8

(M.53)

Additional relations can be found in Chandrasekhar (1960, Chapter X).

Appendix N: The Milne Integral Equation In this Appendix we show how to solve a vector singular integral equation with a Cauchy-type kernel. Chosen for its simplicity, the presentation is carried out with the Milne problem. We recover the results obtained in Sect. 16.6.

Appendix N: The Milne Integral Equation

405

The primary source being zero, the source vector S(τ ) satisfies the homogeneous Wiener–Hopf integral equation 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ ,

(N.1)

0

and its inverse Laplace transform s(ν) satisfies the singular integral equation  L(ν)s(ν) − M(ν)

∞ 0

s(ν ) dν = 0, ν − ν

ν ∈ [0, ∞[.

(N.2)

The matrices M(ν) and L(ν) are defined in Eqs. (16.19) and (M.12). We now apply to this equation the Hilbert transform method of solution. In the interval ν ∈ [0, 1[, Eq. (N.2) reduces to L(ν)s(ν) = 0,

ν ∈ [0, 1[.

(N.3)

As shown in the Appendix M.2 of this chapter, det L(ν) has a double zero at the origin. Hence, this equation has solutions having the general form s(ν) = [s1 δ (ν) + s0 δ(ν)]j 0 ,

(N.4)

where s1 and s0 are constants, δ(ν) is the Dirac distribution, and δ (ν) its derivative. This type of solution, involving a combination of δ(ν) and δ (ν), is encountered in the scalar Milne problem (see Eq. (G.3)). The constant vector j 0 is solution of L(0)j 0 = 0,

(N.5)

where L(0) is given in Eq. (M.16). The solution is √  2 j0 = . 1

(N.6)

The solution of Eq. (N.5) is actually defined within a multiplicative constant, set here to one. The vector j 0 corresponds to an unpolarized field, in the (lr) representation. The coefficient s0 in Eq. (N.4) will be obtained together with the expression of s(ν) when we consider the region ν ∈ [1, ∞[ and the coefficient s1 will be deduced from the behavior of the source function at infinity. The term s1 δ (ν)j 0 indicates that S(τ )  s1 τ j 0 for τ → ∞. In the region ν ∈ [1, ∞[, Eq. (N.2) becomes  L(ν)s(ν) − M(ν)

∞ 1

s(ν ) 1 s1 dν = ( − s0 )M(ν)j 0 , ν − ν ν ν

ν ∈ [1, ∞[.

(N.7)

406

16 Conservative Rayleigh Scattering: Exact Solutions

We introduce the Hilbert transform 1 F (z) ≡ 2i π



∞ 1

s(ν) dν. ν −z

(N.8)

By construction, the vector F (z) is analytic in C /[1, ∞[, behaves as 1/z for z → ∞, and satisfies the Plemelj formulae, F + (ν) − F − (ν) = s(ν),  1 + − F (ν) + F (ν) = iπ

∞ 1

s(ν ) dν , ν − ν

ν ∈ [1, ∞[.

(N.9)

When stating that the vector F (z) has these properties, we mean that they hold for its two components. Using the Plemelj formulae for F (z), those for L(z) (see Eq. (16.52)), and multiplying by ν 2 , we can rewrite Eq. (N.7) as 1 (s1 − s0 ν)j 0 ] 2i π 1 − L− (ν)[ν 2 F − (ν) + (s1 − s0 ν)j 0 ] = 0, 2i π

L+ (ν)[ν 2 F + (ν) +

ν ∈ [1, ∞[.

(N.10)

We now express the boundary values of L(z) in terms of those of X(z). Using Eq. (16.26) we obtain 1 (s1 − s0 ν)j 0 ] 2i π 1 (s1 − s0 ν)j 0 ] = 0, − [X− (ν)]−1 [ν 2 F − (ν) + 2i π

[X+ (ν)]−1 [ν 2 F + (ν) +

ν ∈ [1, ∞[. (N.11)

The transformation of Eq. (N.10) into Eq. (N.11) requires that detX(z) = 0, a condition imposed for the construction of X(z) in Sect. 16.3. The solution of Eq. (N.11) has the form F (z) =

1 1 (s1 − s0 z)j 0 ], [X(z)P (z) − 2 z 2i π

(N.12)

where P (z) is a two-component vector, each component being a polynomial function of z. The matrix X(z) is given in Eq. (16.37). The vector F (z) has the correct analyticity properties, but in addition must behave as 1/z for z → ∞ and be free of singularities. These constraints will be satisfied by a convenient choice of P (z).

Appendix N: The Milne Integral Equation

407

We assume that   1 γ F (z)  , z δ

z → ∞,

(N.13)

where γ and δ are some constants. Inserting into Eq. (N.12) the assumed asymptotic expression of F (z) and that of X(z) given in Eq. (16.57), we see that when P (z) has the form   p0 + p1 z , (N.14) P (z) = p2 F (z) will have the proper behavior at infinity. There are five constants to be determined: the three constants p0 , p1 , p2 and s0 and s1 . Actually, at this stage one cannot find both s1 and s0 , because s1 depends on the value of the total radiative flux, which has not been entered into the previous equations. To ensure that F (z) is free of singularities, the two first coefficients of the Taylor expansion √ of the square bracket in Eq. (N.12) around z = 0 should be zero. Using X2 (0) = 2, we find that this condition is satisfied provided √

3 s1 p0 = , 2 2i π

√ 3 s1 β p1 = − , 2 2i π α

√ 3 s1 α p2 = √ , 2 2i π 2

(N.15)

and  s0 = s1

 1 β − √ X2 (0) . α 2

(N.16)

Here X2 (0) is the derivative of X2 (z) taken at z = 0. An explicit expression of X2 (0) is given in Eq. (M.47). A numerical value and an interpretation of the ratio s0 /s1 is given in Sect. 16.6.3. The constant s1 can be deduced from the behavior of the source vector for τ → ∞, namely S(τ )  s1 τ j 0 .

(N.17)

Applying the matrix A(μ) to Eq. (N.17) and comparing the result with lim Sl (τ, μ) = lim Sr (τ, μ) =

τ →∞

τ →∞

3 F τ, 8π

(N.18)

(see Eq. (16.88)), we obtain √ 6 s1 = F. 8π

(N.19)

408

16 Conservative Rayleigh Scattering: Exact Solutions

To summarize, the solution of the Wiener–Hopf integral equation (16.12) may be written as  ∞ S(τ ) = [s1 τ + s0 ]j 0 + s(ν)e−ντ dν, (N.20) 1

with s(ν) = F + (ν) − F − (ν) =

1 + [X (ν) − X− (ν)]P (ν), ν2

ν ∈ [1, ∞[,

(N.21)

and ⎡ β ⎤ √ ν 1 − 3 s1 ⎢ α ⎥ P (ν) = ⎣ α ⎦. 2 2i π √ 2

(N.22)

The elements of the matrix [X+ (ν) − X− (ν)] can be calculated by combining Eq. (16.37) with Eq. (M.26). They can be expressed in terms of the functions − Xn (−ν) and the products L+ n (ν)Ln (ν), n = 1, 2. The surface value S(0) can be deduced from Eq. (N.20) by setting τ = 0 and performing the integration over s(ν). One recovers √

3 F S(0) = 8π

√  2c , q

(N.23)

the expression given in Eq. (16.93). The emergent radiation field is now easily derived from 

∞

I(0, μ) = 0

S(τ )e−τ/μ

dτ , μ

μ ∈ [0, 1].

(N.24)

Using the expression of S(τ ) given in Eqs. (N.20) and (N.21) and performing a contour integration to calculate the contribution from the interval ν ∈ [1, ∞[, we obtain 1 1 I(0, μ) = (s1 μ + s0 )j 0 + 2i π F (− ), μ μ

μ ∈ [0, 1].

(N.25)

Using now the expression of F (z) given in Eq. (N.12), we find 1 1 I(0, μ) = 2i πμX(− )P (− ). μ μ

(N.26)

References

409

To recover the expressions of Il (0, μ) and Ir (0, μ) given in Eq. (16.117), it suffices to apply the matrix A(μ) to I(0, μ). For the Milne problem, the method presented here is algebraically simpler than the method presented in Sect. 16.6, which requires the determination of the resolvent matrix. Of course, the same method can be applied to the diffuse reflection problem. The Wiener–Hopf integral equation for the source vector is 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

(N.27)

0

where 1 Q (τ ) = 2 ∗



1

AT (−μ)I inc (−μ) e−τ/μ dμ.

(N.28)

0

The singular integral equation for s(ν) is identical to Eq. (N.2), except that the righthand side contains an inhomogeneous term q ∗ (ν) =



0 AT (−1/ν)I inc (−1/ν)/(2ν 2 )

for ν ∈ [0, 1[ for ν ∈ [1, ∞[.

(N.29)

A main difference with the Milne problem is that the equation L(ν)s(ν) = 0,

ν ∈ [0, 1],

(N.30)

has the solution s(ν) = s0 δ(ν)j 0 ,

ν ∈ [0, 1],

(N.31)

where the constant s0 has still to be determined. This constant yields the constant value of the source vector at infinity. Because of the presence of the inhomogeneous term, the algebra is not as simple as with the Milne problem and a direct calculation of the source vector S(τ ) has no real advantage over the method based on the determination of the resolvent matrix.

References Bond, G.R., Siewert, C.E.: On a Rayleigh scattering problem in stellar atmospheres. Astrophys. J. 150, 357–359 (1967) Chandrasekhar, S.: On the radiative equilibrium of a stellar atmosphere. X. Astrophys. J. 103, 351–370 (1946a) Chandrasekhar, S.: On the radiative equilibrium of a stellar atmosphere. XI. Astrophys. J. 104, 110–132 (1946b)

410

16 Conservative Rayleigh Scattering: Exact Solutions

Chandrasekhar, S.: Radiative Transfer. Dover Publications, New York (1960). 1st edn. Oxford University, Oxford (1950) Domke, H.: Radiation transport with Rayleigh scattering. I. Semiinfinite atmosphere. Soviet Astron. 15, 266–275 (1971); translation from Astron. Zhurnal 50, 126–136 (1971) Domke, H.: Multiple scattering of polarized light in a semiinfinite atmosphere with small true absorption. Soviet Astron. 17, 81–87 (1973); transaltion from Astron. Zhurnal 50, 126–136 (1973) Hulst, H.C.: van de: Multiple Light Scattering Tables, Formulas and Applications, vol. 2. Academic Press, New York (1980) Ivanov, V.V.: Generalized Rayleigh scattering I. Basic theory. Astron. Astrophys. 303, 609–620 (1995) Ivanov, V.V.: Generalized Rayleigh scattering III. Theory of I-matrices. Astron. Astrophys. 307, 319–331 (1996) Ivanov, V.V., Kasaurov, A.M., Loskutov, V.M., Viik, T.: Generalized Rayleigh scattering II. Matrix source functions. Astron. Astrophys. 303, 621–634 (1995) Ivanov, V.V., Kasaurov, A.M., Loskutov, V.M.: Generalized Rayleigh scattering IV. Emergent radiation. Astron. Astrophys. 307, 332–346 (1996) Ivanov, V.V, Grachev, S.I., Loskutov, V.M.: Polarized line formation by resonance scattering I. Basic formalism. Astron. Astrophys. 318, 315–326 (1997a) Mullikin, T.W.: Multiple scattering of partially polarized light. In: Transport Theory SIAM-AMS Proceedings, vol. 1, pp. 3–16 (1969) Schnatz, T.W., Siewert, C.E.: On the transfer of polarized light in Rayleigh-scattering half spaces with true absorption. Mon. Not. R. Astr. Soc. 152, 491–508 (1971) Siewert, C.E.: A new approach to Chandrasekhar’s scattering matrix for a semi-infinite Rayleighscattering atmosphere. Astrophys. J. 152, 835–840 (1968) Siewert, C.E., Burniston, E.E.: On existence and uniqueness theorems concerning the H-Matrix of radiative transfer. Astrophys. J. 174, 629–641 (1972) Siewert, C.E., Fraley, S.K.: Radiative transfer in a free-electron atmosphere. Ann. Phys. (New York) 43, 338–359 (1967) Siewert, C.E., Kelley, C.T., Garcia, R.D.M.: An analytical expression for the H-Matrix relevant to Rayleigh scattering. J. Math. Anal. Appl. 84, 509–518 (1981) Viik, T: Rayleigh-Cabannes scattering in planetary atmospheres III. The Milne problem in conservative atmospheres. Earth, Moon, and Planets 49, 163–175 (1990)

Chapter 17

Scattering Problems with No Exact Solution I: The Auxiliary Matrices

In this Chapter and the following one, we consider the following scattering mechanisms: the Rayleigh scattering with true absorption (non-conservative), the resonance polarization, and the Hanle effect. For these processes, there is no explicit solution for radiative transfer problems in semi-infinite plane parallel media. They have to be solved numerically, directly from the radiative transfer equations, or by means of a nonlinear H-equation. The fundamental reason for the absence of exact solutions is that the dispersion matrix L(z) has no diagonalization similar to Eq. (16.22). In Chap. 15 we have shown how to construct a nonlinear H-equation by algebraic manipulations of convolution type integral equations in the real τ -space. In this Chapter and the following one, we show that this nonlinear integral equation can be constructed with the method described in Chap. 16 for conservative Rayleigh scattering. There is no explicit expression for the matrix X(z), but it is possible to find its behavior for z → ∞ and this is sufficient to construct a nonlinear integral equation. The construction presented in this Chapter and the following one, requires more advanced tools and concepts related to matrix Riemann–Hilbert problems. We believe that it is worth the effort, as it clarifies some properties of radiative transfer in semi-infinite media not so easily caught with the elementary approach. The present chapter is devoted to the dispersion matrix and the half-space auxiliary matrix X(z), for non-conservative Rayleigh scattering, the resonance polarization and the Hanle effect. The construction of the nonlinear integral equation is carried out in Chap. 18 for the above three scattering mechanisms. Section 17.1 is devoted to non-conservative Rayleigh scattering, Sect. 17.2 to resonance polarization, and Sect. 17.3 to the Hanle effect. Complete frequency redistribution is assumed for the resonance polarization and the Hanle effect. The radiation field is represented by its Stokes parameters I, Q, U . This chapter and the following one are largely inspired by a work of Siewert and Burniston (1972). The authors assume the existence of a nonlinear integral equation for the H-matrix and then prove the existence of solutions. We proceed in the opposite way: we introduce a half-space auxiliary matrix, solution of a matrix © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_17

411

412

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices

Riemann–Hilbert problem, and then derive the nonlinear integral equation for the H-matrix.

17.1 Non-Conservative Rayleigh Scattering Non-conservative Rayleigh scattering is introduced in Sect. 14.3.1. We consider here the same phase matrix P(μ, μ ) = W PR (μ, μ ) + (1 − W )Pis ,

(17.1)

where PR (μ, μ ) is the Rayleigh phase matrix, Pis the isotropic matrix, and W a constant with values in [0, 1]. In Sect. 14.3.1 we show that it can be factorized as P(μ, μ ) = AW (μ)ATW (μ ),

(17.2)

with ⎡ ⎢1 AW (μ) = ⎢ ⎣ 0

√ ⎤ W √ (3μ2 − 1) ⎥ 2√ 2 ⎥. ⎦ W 2 − √ 3(1 − μ ) 2 2

(17.3)

The source vector S(τ ) satisfies the Wiener–Hopf integral equation in Eq. (14.53) in which the kernel K(τ ) has the form 1 K(τ ) ≡ 2



1

W (μ) exp(− 0

|τ | dμ ) , μ μ

(17.4)

with ⎡ 1

√ W √ (1 − 3μ2 ) 2 2

⎤

⎢ ⎥ ⎥. W (μ) = ⎢ ⎣ √W ⎦ 2 W 2 4 √ (1 − 3μ ) (5 − 12μ + 9μ ) 4 2 2

(17.5)

The kernel elements Kij (τ ), i, j = 1, 2 satisfy 

+∞ −∞

K11 (τ ) dτ = 1,



+∞

−∞

Kij (τ ) dτ = 0, i = j,



+∞ −∞

K22 (τ ) dτ =

7 W. 10 (17.6)

17.1 Non-Conservative Rayleigh Scattering

413

The matrix M(ν), inverse Laplace transform of the kernel (see Table 15.1), is given by  M(ν) =

0 W (1/ν)/2ν

for ν ∈ [0, 1[, for ν ∈ [1, ∞[.

(17.7)

17.1.1 The Dispersion Matrix For conservative Rayleigh scattering, the dispersion matrix L(z) has the form shown in Eq. (16.21). For non-conservative scattering it has the same form, except that the integral is multiplied by a factor (1 − ). We now have  L(z) ≡ I − (1 − )

∞

M(ν)( 1

1 1 + ) dν, ν+z ν −z

(17.8)

where M(ν) is defined in Eq. (17.7). The matrix L(z) has the same symmetry and analyticity properties as its conservative version. Its limiting values along the cut ν ∈] − ∞, −1] ∪ [1, +∞[ satisfy the Plemelj formulae L+ (ν) − L− (ν) = −(1 − )2i πM(ν), L+ (ν) + L− (ν) = 2 L(ν),

ν ∈] − ∞, −1] ∪ [1, +∞[,

(17.9)

where  L(ν) ≡ I − (1 − )

∞

M(ν )(

1

ν

1 1 +

) dν . +ν ν −ν

(17.10)

The elements Lij (z), i, j = 1, 2, may be written as 

∞

L11 (z) = L(z) = 1 − (1 − ) 1

$ L12 (z) = L21 (z) = − L22 (z) = 1 −

1 1 1 ( + ) dν, 2ν ν + z ν − z

3 3 W [1 − 2 − (1 − 2 )L(z)], 8 z z

3 12 W 9 W 9 [5 − 2 (1 + 3) + 4 ] + [5 − 2 + 4 ]L(z). 4 z z 4 z z

(17.11)

(17.12)

(17.13)

Here L(z) is the scalar dispersion function for monochromatic scattering: L(z) = 1 − (1 − )

1+z 1 ln . 2z 1 − z

(17.14)

414

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices

The asymptotic behaviors of L(z) for z → ∞ and z → 0 play an important role in the following sections. For z → ∞, Eq. (17.8) shows that L(z) → I. It is easy to verify, using L(z) → 1, that L11 (z)  L22 (z) → 1, and L12 (z) → 0. For z → 0, L11 (z) = L(z)   − (1 − )( $ L12 (z) = L21 (z) 

z2 z4 + ) + O(z6 ), 3 5

(17.15)

2 W (1 − ) z2 + O(z4 ), 2 5

(17.16)

7 + O(z2 ). 10

(17.17)

L22 (z)  1 − (1 − )W For z = 0, L(0) is a diagonal matrix: 

  0 L(0) = , 0 Q

(17.18)

with Q ≡ 1 − (1 − )7W/10. As pointed out in Sect. 15.1, the constant Q plays for Stokes Q the role played by  for Stokes I . The expression of L(0) can also be deduced from the definition of L(z), combined with the normalization of the kernel elements given in Eq. (17.6).

17.1.2 The Dispersion Function The dispersion function, henceforth denoted (z), is defined by (z) ≡ det L(z).

(17.19)

For conservative Rayleigh scattering, this determinant can be factorized as in Eq. (M.14). This is impossible for non-conservative Rayleigh scattering, because of the factor (1 − ). This determinant, as we now show has a pair of real zeroes at z = ±ν0 , ν0 ∈]0, 1]. They are reminiscent of the two zeroes of the scalar dispersion function L(z) for monochromatic scattering and play a somewhat similar role. They coalesce into a double zero at the origin when  = 0. The function (z) is given by (z) = L11 (z)L22 (z) − [L12 (z)]2 .

(17.20)

17.1 Non-Conservative Rayleigh Scattering

415

Inserting the expression of Lij (z), we find (z) =

W 3 L1 (z)L2 (z) + [1 − W (1 − 2 )]L(z), 8 2z

(17.21)

3 1 ) + 3(1 − 2 )L(z), 2 z z

(17.22)

with L1 (z) = −(1 − L2 (z) = (1 −

3 1 ) + 3(1 − 2 )L(z). z2 z

(17.23)

When  = 0 and W = 1, the functions L1 (z) and L2 (z) reduce to the functions λ1 (ν) and λ2 (ν) in Eq. (16.24). But when  = 0, (z) does not have the form of a product and for this reason the construction of the auxiliary matrix X(z) cannot be reduced to two scalar Riemann–Hilbert problems. A number of remarks will help us to find the number and position of the zeroes of (z). Because L(z), L1 (z), and L2 (z) are real functions of the complex variable z, the zeroes are located either on the real or on the imaginary axis and because (z) is an even function of z they come by pair. According to a standard theorem quoted in Sect. 5.2.2, the number of zeroes is equal to the variation of the argument of (z), divided by 2π, as z varies along a closed contour in the cut plane. Equations (17.21), (17.22), and (17.23) show that (z) has the same the analyticity properties as L(z), namely: the analyticity domain is the cut plane C /]−∞, −1]∪[1, +∞[, the limiting values ± (ν) are complex conjugate and satisfy ± (ν) = ∓ (−ν). The change in the argument of (z) along a contour turning around the intervals [1, +∞[ and ] − ∞, −1] and closed by a circle of radius R (see Fig. 5.7) is thus four times the variation of θ (ν), the argument of + (ν), as ν varies from 1 to ∞. The number of zeroes of (z) is thus given by N=

4 [θ (∞) − θ (1)]. 2π

(17.24)

To find the variation of θ (ν), we need the real and imaginary parts of + (ν). Equation (17.21) shows that they depend only on L+ (ν), the limiting value of L(z) analyzed in Chap. 5. Using the same notation, L+ (ν) = λ(ν) + i πη(ν),

ν ∈ [1, ∞[,

(17.25)

416

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices

simple algebra yields [+ (ν)] =

  3 W 1 −(1 − 2 )2 + 9(1 − 2 )[λ2 (ν) − π2 η2 (ν)] 8 ν ν

+λ(ν)[1 − W (1 − [+ (ν)] =

3 )], ν2

(17.26)

1 W 3 πλ(ν)η(ν)(1 − 2 )2 + πη(ν)[1 − W (1 − 2 )]. 4 ν 2ν

(17.27)

We know from Chap. 5 (or by applying the Plemelj formulae to Eq. (17.11)), that η(ν) = −(1 − )/2ν, that λ(ν) → 1 as ν → ∞, and that λ(ν) has a logarithmic divergence for ν → 1. We thus obtains lim [+ (ν)] ∼ λ(ν) ∼ −∞,

ν→1

lim [+ (ν)]  1,

ν→∞

lim [+ (ν)]  −π

ν→1

3 1− [1 − W (1 − )], 2 2 (17.28)

lim [+ (ν)]  0.

ν→∞

(17.29)

Hence, θ (1) = −π,

θ (∞) = 0,

(17.30)

and according to Eq. (17.24) N = 2. We thus find that (z) has two zeroes, symmetric with respect to z = 0 on the real or the imaginary axis. Knowing that the two zeroes coalesce into a double zero at the origin, their position for  small can be derived from a Taylor expansion of L1 (z) and L2 (z) around the origin. The expansion of L(z) in Eq. (17.15) combined with (17.22) and (17.23) leads to

L2 (z)  −

2 L1 (z)  2 − (1 − ) z2 + O(z4 ), 5

(17.31)

6 2 + 2(1 + ) − (1 − ) z2 + O(z4 ). 2 z 5

(17.32)

Thus for z → 0 and  → 0, (z)  (1 −

7 z2 7 W ) − (1 − W ) + O( 2 ) + O(z2 ). 10 3 10

(17.33)

Assuming  > 0, Eq. (17.33) shows that the zeroes of (z) are located at ±ν0  √ 3. At leading order, they are independent of W . Thus, for small values of , the position of the zeroes of (z) are close to the positions of the zeroes of scalar dispersion function L(z), as one could expect.

17.1 Non-Conservative Rayleigh Scattering

417

17.1.3 The Auxiliary X-Matrix The auxiliary matrix X(z) is defined in Chap. 16 as the solution of a homogeneous Riemann–Hilbert problem. The same definition is adopted here, namely we look for a matrix X(z) analytic in C /[1, +∞[, nonsingular (its determinant should be free of zeroes) and solution of the equation X+ (ν) = W(ν)X− (ν),

ν ∈ [1, ∞[,

(17.34)

where W(ν) ≡ L+ (ν)[L− (ν)]−1 .

(17.35)

In Chap. 16, devoted to the conservative Rayleigh scattering, the matrix W(ν) is defined as W(ν) = [L+ (ν)]−1 L− (ν). The definition chosen here is more standard and is used in most theoretical work on homogeneous Riemann–Hilbert appearing in transport theory, in particular in the fundamental work of Muskhelishvili (1953) and Vekua (1967). Equation (17.34) for  = 0 has no exact solutions. Some of their main properties are listed below. They are used in following chapters, in particular to construct an H-equation. The first simple remark is that Eq. (17.34) has an infinite number of solutions. Indeed, if X(z) is a solution then X(z)P(z), where P(z) is a matrix of polynomials, is also a solution. There exist solutions of the Riemann-Hilbert problem, which for z → ∞ satisfy

X(z)

zκ1 0 0 zκ2





 a∞ b∞  , c∞ d ∞

a∞ d∞ = b∞ c∞ ,

(17.36)

where a∞ , b∞ , c∞ , and d∞ are constants and κ1 and κ2 integer numbers. Such solutions are known as fundamental or canonical solutions. A precise definition of fundamental solutions and of their method of construction can be found in (Ablowitz and Fokas 1997, p. 580) (see also Siewert and Burniston 1972). For the problem at hand, W−1 (ν) = W∗ (ν), where the superscript ∗ stands for complex conjugate. This property implies that X∗ (z∗ ) is also solution of Eq. (17.34) and moreover that X∗ (z∗ ) = X(z). As a consequence, the constants a∞ , b∞ , c∞ , and d∞ are real. and X(ν) is also real for ν real in ] − ∞, 1[. The constants κ1 and κ2 , are known as the partial indices or individuals indices (Ablowitz and Fokas 1997, p. 580). They satisfy κ1 + κ2 = κ, where 2πκ, is the variation of the argument of the determinant of W(ν), as ν varies from 1 to ∞. The

418

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices

index κ is known as the index of the homogeneous Riemann–Hilbert problem. Since L+ (ν) and L− (ν) are complex conjugate, we can write κ=

1 [2θ (∞) − 2θ (1)], 2π

(17.37)

where is θ (ν) is the argument of + (ν) given in in Eq. (17.30). Using θ (∞) = 0 and θ (1) = −π, we obtain κ = 1 and κ1 +κ2 = 1. There is no formula enabling one to calculate the values of the partial indices, but following (Burniston and Siewert 1970), one can show that no index can be negative. Let us suppose that one of the partial indices, say κ1 is negative (since κ1 + κ2 = 1 only one of them can be negative), then the elements of the first column of X(z) for z → ∞ becomes infinite at infinity (see Eq. (17.36)). Recalling that L(z) tends to a constant matrix at infinity, we see that this behavior is incompatible with the relation between L(z) and X(z) that will be written in Eq. (17.54) and which is a consequence of the definition of X(z) as solution of Eq. (17.34). Thus the only possible values are κ1 = 0, κ2 = 1, or κ1 = 1, κ2 = 0. Here we have chosen κ1 = 0 and κ2 = 1. Hence for z → ∞, ⎤ b∞ + ... ⎥ ⎢ z X(z)  ⎣ ⎦, d∞ + ... c∞ + . . . z ⎡

a∞ + . . .

z → ∞,

(17.38)

where the dots stands for terms going to zero as z → ∞. A matrix playing an important role in the following is the leading term, X∞ (z), defined by ⎡ ⎢ X∞ (z) ≡ ⎣

a∞ c∞

b∞ ⎤ z ⎥. d∞ ⎦ z

(17.39)

Its inverse, [X∞ (z)]

−1

  1 d∞ −b∞ = ,  −c∞ z a∞ z

 = a ∞ d ∞ − b ∞ c∞ ,

(17.40)

is a matrix of polynomials. There are no explicit expressions for the parameters a∞ , b∞ , c∞ , and d∞ but they satisfy some relations given in Eq. (17.43). We make here a contact with the scalar case. For complete frequency redistribution X(z) tends to a constant at infinity because the index κ is zero and for monochromatic scattering X(z) tends to zero as 1/z at infinity because κ = 1. Loosely speaking, the first column of the matrix X∞ (z) behaves as complete frequency redistribution because it corresponds to the index κ1 = 0 and second column, corresponding to the index κ2 = 1, behaves as monochromatic scattering.

17.1 Non-Conservative Rayleigh Scattering

419

We note that the asymptotic behavior at infinity of the auxiliary matrix Xc (z) given in Eq. (16.57) for conservative scattering does not follow Eq. (17.38) (the subscript c is here to indicate conservative scattering). The reason is that Xc (z) has T −1 − been introduced with the jump condition X+ c (ν) = [W (ν)] Xc (ν). Taking the inverse and transposing the expression of Xc (∞) given in Eq. (16.57), we recover an asymptotic behavior similar to Eq. (17.38). Detailed mathematical proofs regarding the existence of solutions to the homogeneous Riemann–Hilbert problem stated in Eq. (17.34) can be found in books by Muskhelishvili (1953), Vekua (1967), Ablowitz and Fokas (1997) and in articles by Burniston and Siewert (1970) and Siewert and Burniston (1972).

17.1.4 A Factorization of the Dispersion Matrix For conservative Rayleigh scattering, the functions X1 (z) and L1 (z), on the one hand and X2 (z) and L2 (z), one the other hand, satisfy simple factorization relations given in Eq. (M.24). Here we establish a factorization involving the matrices L(z) and X(z). This factorization holds in the cut plane C /] − ∞, −1] ∪ [1, ∞[ and may be written as L(z) = X(z)D(z)DT (−z)XT (−z),

(17.41)

where D(z) is the diagonal matrix  D(z) =

 1 0 . 0 ν0 − z

(17.42)

We first verify that this relation is consistent with the properties of L(z). The matrix X(z) is analytic in C /[1, ∞[ and non singular, hence the right-hand side is analytic in C /] − ∞, −1] ∪ [1, ∞[. Equation (17.41) also satisfies L(z) = L(−z) = LT (z). We know that the determinant of L(z) has a pair of simple zeroes at z = ±ν0 . In the right-hand side of Eq. (17.41), the zeroes are contained in the diagonal matrices D(z) and DT (−z). The matrices X(z) and D(z) are not independent, in the sense that the choice of a specific matrix D(z) imposes conditions on X(z). With D(z) defined by Eq. (17.42), and knowing that L(z) → I as z → ∞, the constants a∞ , b∞ , c∞ , and d∞ satisfy 2 2 2 2 + b∞ = c∞ + d∞ = 1, a∞

a∞ c∞ + b∞ d∞ = 0.

(17.43)

These relations imply 2 = 1 and 1 X∞ (z)XT∞ ( ) = I. z

(17.44)

420

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices

Equation (17.43) and 2 = 1 can be satisfied with, e.g., a∞ = ±1, d∞ = ±1, c∞ = b∞ = 0. The proof of Eq. (17.41) given below is strongly inspired by the proof given in Burniston and Siewert (1970). The only ingredients are the symmetry properties of L(z) and the defining relation for X(z), X+ (ν) = L+ (ν)[L− (ν)]−1 X− (ν),

ν ∈ [1, ∞[,

(17.45)

to which we also refer to as the jump condition. We introduce the matrix R(z) ≡ L(z)[XT (−z)]−1 .

(17.46)

We now show that R(z) satisfies the same jump condition as X(z) and that it has the same analyticity properties. To prove the first part of the statement, we consider the interval ν ∈ [1, ∞[. To prove the second part, we show that the jump of R(z) along the interval ] − ∞, −1] is zero. In the interval [−1, +1], the jumps of L(z) and X(z) are zero, hence the jump of R(z) is zero also. For z ∈ [1, ∞[, the jump of XT (−z) is zero, hence R+ (ν) = L+ (ν) and R− (ν) = − L (ν). Using Eq. (17.45), we find R+ (ν) = X+ (ν)[X− (ν)]−1 R− (ν),

ν ∈ [1, ∞[.

(17.47)

Comparing Eq. (17.47) with Eq. (17.45), one can indeed conclude that R(z) satisfies the same jump condition as X(z). For z ∈]−∞, −1] some more work is required. We have seen that L(z) = L(−z), implies L+ (ν) = L− (−ν) and L− (ν) = L+ (−ν), ν > 0. Hence, R− (−ν) = L− (−ν)[[X+ (ν)]T ]−1 = L+ (ν)[[X+ (ν)]T ]−1 ,

(17.48)

R+ (−ν) = L+ (−ν)[[X− (ν)]T ]−1 = L− (ν)[[X− (ν)]T ]−1 .

(17.49)

We recall that R− (−ν) = lim R(−ν − i y), y→0+

R+ (−ν) = lim R(−ν − i y). y→0−

(17.50)

Multiplying Eq. (17.45) by [L+ (ν)]−1 on the left, then taking its inverse and its transposed, and finally using L(z) = LT (z), we obtain [L+ (ν)][X+T (ν)]−1 = L− (ν)][X−T (ν)]−1 ,

ν ∈ [1, ∞[.

(17.51)

Referring now to Eqs. (17.48) and (17.49), we find R− (−ν) = R+ (−ν),

ν ∈ [1, ∞[.

(17.52)

17.2 Resonance Polarization

421

Hence the jump of R(z) is zero for ν ∈ [−∞, 1]. We have shown that R(z) has the same analyticity properties as X(z) and satisfies the same jump condition, therefore, it is possible to write it as R(z) = X(z)T(z),

(17.53)

where T(z) is a matrix of polynomials. This result combined with Eq. (17.46) shows that one may write L(z) = X(z)T(z)XT (−z).

(17.54)

Equation (17.54) imposes a number of conditions on T(z). They are dictated by the properties of L(z) and X(z). The relation L(−z) = LT (z) implies T(−z) = TT (z). The determinant of L(z) has a pair of zeroes at z = ±ν0 . Since X(z) is nonsingular, the determinant of T(z) must have only two zeroes, at z = ±ν0 . The choice T(z) = D(z)DT (−z),

(17.55)

with D(z) the diagonal matrix given in Eq. (17.42) satisfies these two conditions. Needless to say that other factorizations of T(z) could have been chosen. For z = 0, Eq. (17.41) becomes L(0) = X(0)D2 (0)XT (0).

(17.56)

This equation is a generalization of the scalar factorization L(0) = ν02 X2 (0) =  (see Eq. (5.54)).

17.2 Resonance Polarization The radiative transfer equations for resonance polarization are presented in Sect. 14.4. For complete frequency redistribution, the source vector for the (KQ) representation of the radiation field satisfies the Wiener–Hopf integral equation given in Eq. (14.102) with the kernel K(τ ) ≡

1 2



+∞  1 −∞

0

  |τ |ϕ(x) dμ 2 dx, W (μ)ϕ (x) exp −  μ μ

(17.57)

where ϕ(x) is the line absorption profile, normalized to unity. The matrix W  (μ) 2 defined in Eq. (14.99). is given in Eq. (17.5), with W replaced by the constant W The elements of the kernel are normalized as shown in Eq. (17.6), with W replaced 2 . The inverse Laplace transform M(ν) is defined in Eqs. (14.105), (14.106), by W and (14.107). In contrast to Rayleigh scattering, there is no interval in which the matrix M(ν) is zero. The same difference is observed in the scalar case: for complete

422

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices

frequency redistribution there is no interval in which k(ν), the inverse Laplace transform of the kernel, is zero, while for monochromatic scattering k(ν) is zero for ν ∈ [0, 1]. For resonance polarization the construction of the matrix X(z) will be much simpler than for Rayleigh scattering. First we must examine the properties of the dispersion matrix L(z).

17.2.1 The Dispersion Matrix The definition of the dispersion matrix L(z), given in Eq. (17.8) for Rayleigh scattering, holds also for resonance polarization, but the lower bound of the integral should be set to zero. We now have  ∞ 1 1 + ) dν. (17.58) M(ν)( L(z) ≡ I − (1 − ) ν + z ν − z 0 The matrix L(z) is symmetric, it is an even function of z, and at infinity it tends to the identity matrix. There is no explicit expression for the elements Lij (z). Using the normalization of the kernel in Eq. (17.6), one readily finds that  L(0) =

  0 , 0 Q

(17.59)

2 (7/10). For z → ∞, we have L(z) → I. where Q = 1 − (1 − )W Because the integration over ν is from 0 to infinity, L(z) is analytic in the complex plane cut along ] − ∞, +∞[, while the point z = 0 is not part of the cut. Another important consequence is that (z), the determinant of L(z) has no zero. Indeed, the number of zeroes is given by N=

4 [θ (∞) − θ (0)], 2π

(17.60)

where θ is the argument of + (ν). At the origin + (0) = Q and at infinity + (∞) = 1. Therefore θ (∞) = θ (0) = 0 and N = 0. In the scalar case there is a similar difference between complete frequency redistribution and monochromatic scattering. In the first case L(z) has no zero and in the second case it has a pair of real zeroes (see Sect. 5.2.2). A consequence of N = 0 is that κ, the index defined in Eq. (17.37), is also zero.

17.2 Resonance Polarization

423

17.2.2 The Auxiliary X-Matrix We now proceed with the construction of a matrix X(z), analytic in the complex plane cut along the positive real axis, with an algebraic behavior at infinity, and solution of the homogeneous Riemann–Hilbert problem X+ (ν) = W(ν)X− (ν),

ν ∈ [0, ∞[,

(17.61)

where W(ν) ≡ L+ (ν)[L− (ν)]−1 .

(17.62)

In addition, the elements of X(z) should be free of singularities and its determinant non-zero. Equation (17.61) has no explicit solution, but as explained in Sect. 17.1.3, it has a so called canonical solution, which obeys

lim X(z)

z→∞

zκ1 0 0 zκ2





 a∞ b∞  , c∞ d ∞

a∞ d∞ = b∞ c∞ ,

(17.63)

where a∞ , b∞ , c∞ , and d∞ are constants and κ1 and κ2 integer numbers. The argument developed in Sect. 17.1.3 concerning the determination of the partial indices κ1 and κ2 can be applied here. We have seen that the sum κ = κ1 + κ2 = 0. Since there can be no partial index with a negative value, the only choice compatible with κ = 0 is κ1 = κ2 = 0. Thus for resonance scattering with complete frequency redistribution the matrix X(z) goes at infinity to a constant matrix  a∞ b∞ . X(∞) ≡ c∞ d ∞ 

(17.64)

The elements of X(∞) satisfy the relations 2 2 2 2 a∞ + b∞ = c∞ + d∞ = 1,

a∞ c∞ + b∞ d∞ = 0.

(17.65)

They are identical to the relations given in Eq. (17.43) for Rayleigh scattering and are derived from the factorization relation given in Eq. (17.70). We now show that the matrix X(z) satisfies the factorization relation L(z) = X(z)XT (−z).

(17.66)

The proof goes exactly as in Sect. 17.1.4. One introduces a matrix R(z) ≡ L(z)[XT (−z)]−1 .

(17.67)

424

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices

Using the jump condition in Eq. (17.61), the symmetry of L(z), and L(z) = L(−z), one can show that R(z) has the same analyticity properties as X(z), hence that it has the form R(z) = X(z)T(z),

(17.68)

where T(z) is a matrix of polynomials. We thus obtain L(z) = X(z)T(z)XT (−z).

(17.69)

All the properties of L(z), in particular that its determinant is free of zero can be satisfied by choosing for T(z) the identity matrix, hence Eq. (17.66). The limit z → ∞ leads to I = X(∞)XT (∞),

(17.70)

hence to the relations between the constants a∞ , b∞ , c∞ , d∞ given in Eq. (17.65).

17.3 The Hanle Effect As shown in Sects. 14.5.2 and 15.6.2, the 6×6 dispersion matrix for the Hanle effect can be defined as  ∞ 1 1 −1 + ) dν, (17.71) LH (z) ≡ MH (θB ) − (1 − ) M(ν)( ν + z ν − z 0 with

ˆ M−1 H (θB ) = D(θB )[I + i H2 Q]D(−θB ).

(17.72)

We recall that H2 = 1/(1 + I + δ (2) ) and ˆ = diag[0, 0, −1, +1, −2, +2]. Q

(17.73)

The elements of D(θB ) are the reduced rotation matrices. The matrix L(z) tends to M−1 H (θB ) at infinity, otherwise it has the same properties as the dispersion matrix for resonance polarization: it is symmetric, an even function of z, and analytic in the complex plane cut along the real positive axis. The point z = 0 does not belong to the cut. The normalization of the elements of the kernel matrix leads to L(0) = M−1 H (θB ) − I + E,

(17.74)

17.3 The Hanle Effect

425

where 

  0 E= . 0 EQ

(17.75)

All the elements of the 5 × 5 diagonal matrix EQ are equal to Q . When the magnetic field is zero, one recovers the matrix L(0) for resonance scattering (see Eq. (17.59)). Using D(θB )D(−θB ) = I and Eq. (17.72), it is possible to write Eq. (17.74) in the more symmetric form,   ˆ D(−θB ). L(0) = D(θB ) E + i H2 Q

(17.76)

To determine the number of zeroes of (z), determinant of L(z), we can use the general formula given in Eq. (17.60). At the origin, as shown by Eq. (17.76), ˆ = Q [ 2 + (H )2 ][ 2 + 4(H )2 ]. (0) = det[E + i H2 Q] Q 2 Q 2

(17.77)

At infinity L(∞) = M−1 H (θB ), hence, according to Eq. (17.72), ˆ = [1 + (H )2 ][1 + 4(H )2 ]. (∞) = det[I + i H2 Q] 2 2

(17.78)

At both ends of the cut [0, ∞[, (z) is real and positive. Hence the variation of the argument θ (ν) of + (ν) is zero and (z) is free of zero, as for resonance polarization. We see that the number of zeroes depends only on the properties of the kernel K(τ ), in particular on its behavior for large values of τ . For complete frequency redistribution K(τ ) decreases algebraically to zero and there is no zero, for monochromatic scattering K(τ ) decreases exponentially and there is a pair of real zeroes. An auxiliary matrix XH (z) for the Hanle effect can be constructed as described in Sect. 17.2.2 for resonance polarization. It tends to a constant matrix at infinity, henceforth denoted XH (∞), which satisfies the factorization relation LH (z) = XH (z)XTH (−z).

(17.79)

At infinity, the latter becomes T M−1 H (θB ) = XH (∞)XH (∞).

(17.80)

Although there is no explicit expression for the half-space auxiliary matrix introduced in this chapter, the behavior at infinity and factorization relations such as Eq. (17.79) are sufficient for constructing solutions of matrix and vector singular integral equations. It will be shown in Chap. 18 that these solutions, although not explicit, can be used to recover the nonlinear integral equations established in Chap. 15, with an entirely different method.

426

17 Scattering Problems with No Exact Solution I: The Auxiliary Matrices

References Ablowitz, M.J., Fokas, A.S.: Complex Variables, Introduction and Applications. Cambridge University, Cambridge (1997) Burniston, E.E., Siewert, C.E.: Half-range expansion theorems in studies of polarized light. J. Math. Phys. 11, 3416–3420 (1970) Siewert, C.E., Burniston, E.E.: On existence and uniqueness theorems concerning the H-Matrix of radiative transfer. Astrophys. J. 174, 629–641 (1972) Muskhelishvili, N.I.: Singular Integral Equations, Noordhoff, Groningen (based on the second Russian edition published in 1946) (1953); Dover Publications, New York (1991) Vekua, N.P.: Systems of Singular Integral Equations. Noordhoff, Groningen (1967)

Chapter 18

Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-Matrix, and the I-Matrix

For problems with no exact solution, it is still possible to define a half-space auxiliary matrix X(z), as we have shown in the preceding chapter. We show in the present chapter, that it is also possible to solve vector or matrix Wiener– Hopf integral equations with the method based on the solution of singular integral equations, described for the polarized Milne problem in Appendix N of Chap. 16. Neither the Wiener–Hopf integral equations nor the singular integral equations have explicit solutions, since there is no exact expression for the matrix X(z), however the construction of a nonlinear integral equation for a H-matrix remains possible. In Chap. 15 we have shown that this nonlinear integral equation can be constructed by an algebraic approach in the physical τ -space. The method described in the present chapter requires some complex plane analysis, but has the advantage of proving the existence of solutions. The Chapter is organized as follows. We consider non-conservative Rayleigh scattering in Sect. 18.1, resonance polarization of spectral lines formed with complete frequency redistribution in Sect. 18.2, and the associated Hanle effect in Sect. 18.3. In each section we solve the singular integral equation for (ν), the inverse Laplace transform of the resolvent matrix (τ ), calculate the Laplace ˜ transform G(p) of the surface Green matrix by a contour integration, and then ˜ determine the nonlinear integral equation for G(p). The nonlinear H-equation is ˜ then easily derived from the relation H(p) = G(1/p). In Appendix O of this chapter we show that the same equation, can be derived directly from the solution ˜ of singular integral equation for G(p). For Rayleigh scattering, we also discuss the non-uniqueness of the solution of the H-equation. The problem arises because the determinant of the dispersion matrix has two zeroes, as explained in Chap. 17. For all three scattering processes, we introduce alternative nonlinear integral equations, in particular the nonlinear equation for the I-matrix, proposed by Ivanov (1996), which is better suited to numerical solutions than the H-equation itself.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_18

427

428

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

18.1 Non-Conservative Rayleigh Scattering The treatment of non-conservative Rayleigh scattering presented in this section has many similarities with the treatment of conservative Rayleigh scattering in Chap. 16. In particular we determine the resolvent matrix and the H-matrix as in Sects. 16.4 and 16.5. Concerning nonuniqueness of solution of the nonlinear H-equation and the construction of alternative equations the similarities are with the discussion carried out in Sect. 11.3.2 for the scalar case.

18.1.1 The Resolvent Matrix For non-conservative scattering, the singular integral equation for (ν), the inverse Laplace transform of the resolvent matrix (τ ), may be written as 

∞ 0

(ν )

dν = (1 − )M(ν), ν − ν



∞

L(ν)(ν) − (1 − )M(ν)

ν ∈ [0, ∞[,

(18.1)

dν .

(18.2)

with L(ν) = I − (1 − )

0



1 1 +

M(ν )

ν +ν ν −ν



The matrix M(ν) is defined by  M(ν) =

0 W (1/ν)/2ν

 for ν ∈ [0, 1[, , for ν ∈ [1, ∞[.

(18.3)

and W (μ) by Eq. (17.5). We recall that it is an even function of μ and a symmetric matrix. Equation (18.1) is similar to Eq. (16.44), except for the factor (1 − ), which multiplies the integral. In Sect. 17.1.2, it was shown that (z), the determinant of L(z), has a pair of zeroes at ±ν0 , ν0 ∈ [0, 1]. Since M(ν) = 0 in this interval, we must proceed as in Sect. 16.4, namely consider separately the intervals ν ∈ [0, 1] and ν ∈ [1, ∞]. In the interval ν ∈ [0, 1], Eq. (18.1) reduces to L(ν)(ν) = 0,

ν ∈ [0, 1].

(18.4)

Since (ν0 ) = 0, this equation has a non trivial solution having the form (ν) = 0 δ(ν − ν0 ),

(18.5)

18.1 Non-Conservative Rayleigh Scattering

429

with 0 a constant matrix, still to be determined, and δ(ν) the Dirac distribution. The matrix 0 satisfies L(ν0 )0 = 0.

(18.6)

In the interval ν ∈ [1, ∞[, Eq. (18.1) has the form  L(ν)(ν) − (1 − )M(ν)

∞ 1

(ν )

0 dν = (1 − )M(ν)[I + ],

ν −ν ν0 − ν

ν ∈ [1, ∞[.

(18.7) Proceeding as in Sect. 16.4, we introduce the Hilbert transform 1 F(z) ≡ 2i π



∞ 1

(ν) dν. ν −z

(18.8)

Introducing in Eq. (18.7), the Plemelj formulae for L(z) and for F(z), namely L+ (ν) − L− (ν) = −(1 − )2i πM(ν), L+ (ν) + L− (ν) = 2L(ν),

ν ∈] − ∞, −1] ∪ [1, +∞[,

(18.9)

and F+ (ν) − F− (ν) = (ν),  1 ∞ (ν )

dν . F+ (ν) + F− (ν) = i π 1 ν − ν

ν ∈ [1, +∞[,

(18.10)

we obtain the inhomogeneous Riemann–Hilbert problem    0 0 1 1 − − L (ν) F (ν) + [I + ] − L (ν) F (ν) + [I + ] = 0. 2i π ν0 − ν 2i π ν0 − ν (18.11) +



+

At this stage, we must introduce the auxiliary matrix X(z) defined in Sect. 17.1.3. It satisfies the jump condition X+ (ν) = L+ (ν)[L− (ν)]−1 X− (ν),

ν ∈ [1, ∞[,

(18.12)

and the factorization relation L(z) = X(z)D(z)DT (−z)XT (−z), where D(z) = diag(1, ν0 − z).

(18.13)

430

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

Proceeding as usual, we introduce Eq. (18.12) into Eq. (18.11) and thus obtain the boundary value equation   1 [(ν0 − ν)I + 0 ] [X+ (ν)]T (ν0 − ν)F+ (ν) + 2i π   1 − T − [(ν0 − ν)I + 0 ] = 0, ν ∈ [1, ∞[. −[X (ν)] (ν0 − ν)F (ν) + 2i π (18.14) Note that we have multiplied by ν0 − ν. Equation (18.14) has solutions having the form   I 1 0 T −1 [X (z)] P(z) − − , (18.15) F(z) = ν0 − z 2i π 2i π where P(z) is a matrix of polynomials, still to be determined. To obtain the matrices P(z) and 0 , we use the following properties: (i) F(z) → 0 as z → ∞, (ii) F(z) should not be singular at z = ν0 , and (iii) L(ν0 )0 = 0. To satisfy the condition (i), the first term in the right-hand side should cancel the constant term 1/(2i π). This leads to P(z)  −

z T X (z), 2i π ∞

z → ∞.

(18.16)

In Sect. 17.1.3 we show that X(z)  X∞ (z), for z → ∞ with ⎡ ⎢ X∞ (z) ≡ ⎣

a∞ c∞

b∞ ⎤ z ⎥, d∞ ⎦ z

[X∞ (z)]−1 =

  1 d∞ −b∞ ,  −c∞ z a∞ z

(18.17)

and  = a∞ d∞ − b∞ c∞ . The constants a∞ , b∞ , c∞ , d∞ satisfy the relation given in Eq. (17.43). As shown in Sect. 17.1.4, at infinity we also have 1 X∞ (z)XT∞ ( ) = I. z

(18.18)

We now choose for P(z) a matrix with the form P(z) =

1 [C − zXT∞ (z)], 2i π

(18.19)

where C is a constant matrix. With this choice, we ensure that P(z) has the proper behavior at infinity and that it is a matrix of polynomials. The contribution of C to the behavior of F(z) at infinity should be zero. Multiplying Eq. (18.19) by

18.1 Non-Conservative Rayleigh Scattering

431

[XT∞ (z)]−1 , we see that this constraint is satisfied when  C=

 C11 C12 , 0 0

(18.20)

where the constants C11 and C12 are still to be determined. The condition (ii) leads to 0 = [XT (ν0 )]−1 P(ν0 ). 2i π

(18.21)

Therefore the condition (iii) is satisfied for L(ν0 )[XT (ν0 )]−1 P(ν0 ) = 0.

(18.22)

Using the expression of X∞ (z) given in Eq. (18.17), the factorization relation in Eq. (18.13), and L(ν0 ) = L(−ν0 ), we find 

 a ∞ c∞ . 0 0

(18.23)

  1 a∞ (ν0 − z) c∞ (ν0 − z) , −b∞ −d∞ 2i π

(18.24)

C = ν0 Hence, P(z) = and

 0 0 . −b∞ −d∞

(18.25)

1 (ν0 − z)XT∞ (z − ν0 ). 2i π

(18.26)

0 = [X (ν0 )] T

−1



A more useful form of P(z) is P(z) = We thus finally obtain F(z) =

 1  T I [X (z)]−1 P(z) − [XT (ν0 )]−1 P(ν0 ) − . ν0 − z 2i π

(18.27)

The first Plemelj formula applied to F(z) yields (ν) =

% 1  +T [[X ] (ν)]−1 − [[X− ]T (ν)]−1 XT∞ (ν − ν0 ). 2i π

(18.28)

432

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

This expression, as we now show, can be expressed in terms of the matrix X(−ν). Using the defining relation for X(z) in Eq. (18.12) and the factorization relation in (18.13), we can write (ν) = (1 − )[L+ (ν)]−1 M(ν)[L− (ν)]−1 X(−ν)D(−ν)DT (ν)XT∞ (ν − ν0 ). (18.29) Now we remark that D(−ν)DT (ν)XT∞ (ν − ν0 ) = XT∞ (−

1 ). ν0 + ν

(18.30)

We finally obtain (ν) = (1 − )[L+ (ν)]−1 M(ν)[L− (ν)]−1 X(−ν)XT∞ (−

1 ), ν0 + ν

ν ∈ [1, ∞[. (18.31)

It is interesting to compare the expressions of 0 , given in Eq. (18.25), and (ν), given in Eq. (18.31), with the corresponding scalar functions φ0 and φ(ν) established in Sect. 6.2.2 for monochromatic scattering. In the scalar case φ0 = 1/X(ν0 ) and φ(ν) = (1 − )

k(ν) L+ (ν)L− (ν)

X(−ν)(ν0 + ν).

(18.32)

The analogy is easy to make. The matrix M(ν) corresponds to k(ν), the matrices L± (ν) to L± (ν), and the two last terms in Eq. (18.31) correspond to the scalar auxiliary function X∗ (−ν) = X(−ν)(ν0 + ν). In the physical space the resolvent matrix may be written as (τ ) = 0 e

−ν0 τ



∞

+

(ν)e−ντ dν.

(18.33)

1 −ν0 τ as τ → ∞ The first term √ shows that (τ ) goes exponentially to zero as e with ν0  3 for small values of  (see Sect. 17.1.2). For conservative Rayleigh scattering ν0 = 0 and the resolvent matrix tends to a constant at infinity (see Sect. 16.4).

18.1.2 The H-Matrix For conservative scattering, we have defined the matrix H(z) as ˜ 1 ), H(z) ≡ G( z

(18.34)

18.1 Non-Conservative Rayleigh Scattering

433

˜ where G(z) is the Laplace transform of the surface Green matrix G(τ ) = G(τ, 0). The same definition is adopted here. In the scalar case, we show in Sect. 6.3 how to ˜ deduce the Laplace transform G(z) from φ(ν), the inverse Laplace transform of the resolvent function (τ ). We use the same method here. The surface Green matrix G(τ ) is related to the resolvent matrix (τ ), by G(τ ) = (τ ) + δ(τ )I.

(18.35)

Taking the Laplace transform of this equation and expressing (τ ) in terms of (ν) and 0 , one readily obtains ˜ G(p) =I+

 0

∞

0 (ν) dν = I + + ν+p ν0 + p



∞ 1

(ν) dν, ν+p

p ∈ [0, ∞[. (18.36)

To calculate the integral over [1, ∞[, we replace (ν) by the expression given in Eq. (18.28) and consider the contour integral 

[XT (ξ )]−1 C

1 1 P(ξ ) dξ, ξ + p ν0 − ξ

(18.37)

where the integration contour turns around the poles at −p and ν0 , around the cut along [1, ∞[ and is closed by a circle of radius R, R → ∞. The contribution from the circle of radius R cancels the identity matrix in the right-hand side of Eq. (18.36) and the pole at ν0 cancels the contribution from 0 . Using Eq. (18.26), we obtain ˜ G(p) = [XT (−p)]−1 XT∞ [−(ν0 + p)],

(18.38)

1 ), ν0 + p

(18.39)

or ˜ G(p) = [XT (−p)]−1 X−1 ∞ (−

when the factorization in Eq. (18.18) is employed. These expressions can be continued to the complex plane, leading to1 ˜ G(z) = [XT (−z)]−1 XT∞ [−(ν0 + z)] = [XT (−z)]−1 X−1 ∞ (−

1 ). ν0 + z

(18.40)

For further use, we note that T

˜ (−z)]−1 = X(z)X−1 (−(ν0 − z)) = X(z)XT (− [G ∞ ∞

1

1 ), ν0 − z

(18.41)

As a general rule we use boldface sanserif capitals for matrices defined in the complex plane.

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

434

with XT∞ (−

  1 a∞ c∞ )= . −b∞ (ν0 − z) d∞ (ν0 − z) ν0 − z

(18.42)

We give in Appendix O of this chapter a different derivation of Eq. (18.40), based ˜ on the solution of the singular integral equation for G(p). Simple algebra shows that the factorization of L(z) in Eq. (18.13) may be written as ˜ −1 (−z), ˜ T (z)]−1 G L(z) = [G

(18.43)

1 H(z)L( )HT (−z) = I. z

(18.44)

or as

This factorization relation, combined with the analyticity properties of L(z), shows ˜ that G(z) is analytic in C /] − ∞, −1], that it has a pole at z = −ν0 , and that ˜ G(z) → I as z → ∞. The presence of a pole at z = −ν0 is consistent with the fact that the surface Green matrix decreases as e−ν0 τ for τ → ∞. As for H(z), it is analytic in C /[−1, 0]. It has a pole at z = −1/ν0 and H(0) = I.

18.1.3 The H-Equation in the Complex Plane In the scalar case, the H -equation is easily deduced from the Cauchy integral formula applied to the function X(z) (see Sect. B.3). In the present section we show that the H-equation can be deduced from the Cauchy integral formula for the matrix ˜ T (−z)]−1 . The reason for choosing this matrix is that it tends to the matrix unity [G as z → ∞, that it has no singularity, and that its jump across the interval [1, ∞[ can be expressed in terms of the jump of X(z). Indeed, as shown by Eq. (18.41), +

−

˜ (−z)]T ]−1 − [[G ˜ (−z)]T ]−1 = [X+ (ν) − X− (ν)]XT (− [[G ∞

1 ). ν0 − ν

(18.45)

To calculate the jump of X(z), we combine Eqs. (18.12), (18.13), and (18.9), giving the definition of X(z), the factorization of L(z) and the first Plemejl formula for L(z). We thus obtain 1 [X+ (ν)−X− (ν)] = −(1−)M(ν)[XT (−ν)]−1 D−1 (−ν)D−1 (ν), 2i π

ν ∈ [1, ∞[. (18.46)

18.1 Non-Conservative Rayleigh Scattering

435

˜ In the right-hand side, we use Eq. (18.38) to express [XT (−ν)]−1 in terms of G(ν). We thus obtain 1 T −1 −1 −1 ˜ [X+ (ν) − X− (ν)] = −(1 − )M(ν)G(ν)[X ∞ (−(ν0 + ν))] D (−ν)D (ν). 2i π (18.47) The right hand-side can be simplified by introducing Eq. (18.30). Equation (18.47) becomes thus 1 1 ˜ [X+ (ν) − X− (ν)]XT∞ (− ) = −(1 − )M(ν)G(ν). 2i π ν0 − ν

(18.48)

Comparing this equation with Eq. (18.45), we see that the sought for jump is given by ˜ + (−z)]T ]−1 − [[G ˜ − (−z)]T ]−1 = −(1 − )M(ν)G(ν). ˜ [[G

(18.49)

It suffices now to consider the Cauchy integral formula, T

˜ (−z)]−1 = [G

1 2i π



T

C

˜ (−ξ )]−1 [G dξ, ξ −z

(18.50)

where the contour C turns around the cut [1, ∞[ and is closed by a circle of radius R, R → ∞. Using Eq. (18.49) to calculate the contribution from the cut along [1, ∞[, we obtain  ∞ dν ˜ ˜ T (−z)]−1 = I − (1 − ) [G . (18.51) M(ν)G(ν) ν−z 1 To find the H-equation, we use M(ν) = W (1/ν)/2ν (see Eq. (18.3)), change z to ˜ −1/z, set ν = 1/μ and G(1/z) = H(z). We thus obtain    1 ˜ 1) I − 1 −  z ˜ T ( 1 )W (μ) dμ = I, G( G z 2 μ μ+z 0

(18.52)

and 1− H(z) = I + H(z)z 2



1 0

HT (μ)W (μ)

dμ . μ+z

(18.53)

The H-equation shows that H(0) = I and that H(z) is analytic in the complex plane cut along [−1, 0]. That it has a singularity at z = −1/ν0 is not so easy to see, but this property is easily deduced from the factorization relation in Eq. (18.44). The H-equation for conservative scattering is simply recovered by setting  = 0. In

436

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

Appendix O of this chapter we show how to derive Eq. (18.53) from the solution of ˜ the singular integral equation for G(p).

18.1.4 Uniqueness of the Solution of the H-Equation ˜ The nonlinear equations for the matrices G(z) and H(z) have actually two solutions, one with a pole at −ν0 (respectively 1/(−νo )), which is the physically correct solution, and one with a pole at ν0 (respectively 1/(νo )). The reason for the existence of these two solutions is that the construction of the nonlinear H-equation makes use of the factorization relation H(z)L(1/z)HT (−z) = I, which is satisfied by both ˜ solutions. The construction of the matrices G(z) and H(z) in Sect. 18.1.2 shows that ˜ the physically correct solution has a pole at −ν0 for G(z) and 1/(−νo ) for H(z). A detailed discussion of this non-uniqueness problem can be found in de Rooij et al. (1989). The authors show that the physically correct solution and the other one are related by det H1 (z) =

1 + zν0 det H0 (z), 1 − zν0

(18.54)

where the subscript 0 refers to the correct solution and 1 to the other solution. The non-uniqueness problem encountered with Rayleigh scattering is similar to the nonuniqueness problem for the scalar H -equation (see Sect. 11.3.2). To guarantee that a numerical solution of Eq. (18.53) leads to the correct solution, one must combined it with a so called linear constraint. A very simple method for establishing the linear ˜ constraint is presented in de Rooij et al. (1989). We apply it here to G(z). As shown in Sect. 17.1.2, (z), the dispersion function, which is the determinant of L(z), has two zeroes, located at ν = ±ν0 . Therefore, it is possible to construct a nontrivial (nonzero) vector j (ν0 ), solution of L(ν0 )j (ν0 ) = 0.

(18.55)

j (ν0 ) = [L22 (ν0 ), −L12 (ν0 )]T .

(18.56)

One such solution is the vector

We now consider the factorization relation T

˜ (z)]−1 G ˜ L(z) = [G

−1

(−z),

(18.57)

18.1 Non-Conservative Rayleigh Scattering

437

˜ written in Eq. (18.43). If G(z) is the correct solution with a pole at z = −ν0 , then ˜ for z = ν0 , G(ν0 ) has a finite value and Eq. (18.55) is satisfied if, and only if, ˜ −1 (−ν0 )j (ν0 ) = 0. G

(18.58)

˜ Equation (18.51) combined with G(z) = H(1/z), shows that the linear constraint to be associated with Eq. (18.53) is 

1− I − η0 2



1 0

 dμ j (ν0 ) = 0, H (μ)W (μ) η0 − μ T

(18.59)

where η0 = 1/ν0 , η0 ∈ [1, ∞[. As pointed out in de Rooij et al. (1989) there is also a linear constraint associated to the unphysical solution H1 (z). It can be established with the same kind of arguments as above and is given by Eq. (18.59) with η0 changed into −η0 . The linear constraint serves to check that the numerical solution of the H-equation is the correct one. Numerical values of the four elements of H(z) for μ ∈ [0, 1] are given in de Rooij et al. (1989) for  = 0, 10−3 , 10−2 , 10−1 .

18.1.5 An Alternative H-Equation In the scalar case, the standard H -equation is not the best choice for a numerical calculation of the H -function by an iterative method, and it is much preferable to use an alternative equation given in Eq. (11.70). This remark is already in Chandrasekhar (1960) and the reasons for the needed change are clearly explained in Bosma and de Rooij (1983) (see also Sect. 11.3.2). For Rayleigh scattering, the situation is similar. De Rooij et al. (1989) and Ivanov et al. (1996) have proposed an alternative form of the usual H-equation, better suited for the numerical calculation of the elements of the H-matrix. This alternative equation, written for a new matrix closely related to the H-matrix, is a generalization to Rayleigh scattering of the scalar Eq. (11.70). We show now how to derive the alternative equation proposed in de Rooij et al. (1989). In the scalar case, the standard integral equation explicitly contains the value of the H -matrix at zero, whereas the alternative √ equation involves its value at infinity. We recall that H (0) = 1 and H (∞) = 1/ . The values of H (z) being known both at zero and at infinity, it is not difficult to derive the alternative integral equation from the standard one and conversely. For Rayleigh scattering the situation is different. ˜ One knows that G(z) tends to the identity matrix for z → ∞, but for z = 0, one knows only that it satisfies the factorization relation T

˜ (0)]−1 G ˜ [G

−1

(0) = L(0),

(18.60)

438

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

where L(0) = diag[, Q ]. To construct an alternative integral equation, one needs ˜ R (z), with a known value at z = 0. a new matrix, here denoted G First we rewrite Eq. (18.51) as ˜ −1 (z) = G ˜ −1 (0) + (1 − ) G



∞

˜ T (ν)M(ν) G

1

z dν, ν(ν + z)

(18.61)

where ˜ −1 (0) = I − (1 − ) G



∞

1

˜ T (ν)M(ν) dν . G ν

(18.62)

It is convenient to rewrite Eq. (18.61) as   ˜ ˜ −1 (0) + (1 − ) G(z) G

∞

˜ T (ν)M(ν) G

1

 z dν = I. ν(ν + z)

(18.63)

˜ R (z) defined by We now introduce a new matrix G ˜ R (z) ≡ G(z) ˜ G R,

(18.64)

˜ R (∞), since where R is a rotation matrix, that is satisfies R RT = I. It is equal to G T ˜ G(z) = I for z → ∞. Using R R = I, we can also write Eq. (18.60) as T

˜ ˜ (0)]−1 R RT G [G

−1

(0) = L(0).

(18.65)

˜ R (z) satisfies the factorization relation Hence, the matrix G −1

T

˜ (0)]−1 G ˜ (0) = L(0). [G R R

(18.66)

˜ R (z) the particular solution, which satisfies We choose for G ˜ −1 (0) = L1/2 (0), G R

(18.67)

  0 . E ≡ L(0) = 0 Q

(18.68)

and introduce the notation 

Combining Eqs. (18.63) with (18.64) and using Eq. (18.66), we readily obtain   1/2 ˜ GR (z) E + (1 − )

∞ 1

˜ T (ν)M(ν) G R

 z dν = I. ν(ν + z)

(18.69)

18.1 Non-Conservative Rayleigh Scattering

439

˜ ˜ R (z), and HR (z) at the origin and at infinity. The matrix Table 18.1 The matrices G(z), H(z), G R = GR (∞) is a rotation matrix, which can be calculated by solving numerically the nonlinear ˜ R (z) or HR (z). The matrix E = diag[, Q ] is equal to L(0). The I-matrix integral equations for G is defined by I(0, z) = HR (z) ∞ I

0 ˜ G H ˜R G HR

E −1/2 R−1

E −1/2 R−1

I E −1/2

R E −1/2

R

This is the equation that we were looking for. The numerical solution of Eq. (18.69) ˜ R (∞) and the rotation matrix R. yields G An equation similar to Eq. (18.69), can be constructed for a new matrix ˜ R (1/z) = G(1/z)R ˜ HR (z) ≡ G = H(z)R.

(18.70)

Expressing M(ν) in terms of (μ), we can rewrite Eq. (18.69) as    1− 1 T μ 1/2 HR (μ)W (μ) HR (z) E + dμ = I. 2 μ+z 0

(18.71)

This equation plays for Rayleigh scattering the role played by Eq. (11.70) for the scalar case. We summarize in Table 18.1 the values at zero and infinity of the various matrices introduced in this Section. The equation for HR (z) also has a uniqueness problem with a physical correct (0) (1) solution, say HR (z), and an unphysical one, say HR (z). As shown in de Rooij (0) et al. (1989) these two solutions are well separated in the sense that det HR (μ) > 0 (1) and det HR (μ) < 0, for μ ∈ [0, 1]. In contrast, the determinants of the solutions of Eq. (18.53) keep the same signs for μ ∈ [0, 1] since the factor (1 + ν0 )/(1 − ν0 ) Eq. (18.54) remains positive. According to de Rooij et al. (1989), experience shows that the correct solution is obtained when the iteration of Eq. (18.71) is started with the identity matrix.

18.1.6 The Emergent Radiation Field and the I-Matrix In the scalar case, the emergent radiation field for a semi-infinite medium with a uniform primary source Q∗ =  1/2 is given by I (0, μ) = H (μ),

μ ∈ [0, 1].

(18.72)

440

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

where H (μ) is the scalar H -function. This result, which serves as one of the definition of the H -function, has been extended to Rayleigh scattering by Ivanov (1996) with the introduction of the I-matrix. We give here the definition of this Imatrix and show that ˜ R ( 1 ) = HR (μ) = H(μ)R, I(0, μ) = G μ

(18.73)

where R is the rotation matrix introduced in Eq. (18.65). In Sect. 15.1, we have introduced the matrix representation of the radiation field. The vectors I(τ, μ) and S(τ ), constructed with the (KQ) components of the diffuse radiation field and source vector are related to their matrix representation by I(τ, μ) = I(τ, μ)[1, 1]T ,

(18.74)

S(τ ) = S(τ )[1, 1]T.

(18.75)

The matrices I(τ, μ) and S(τ ) satisfy the same equations as the vectors I(τ, μ) and S(τ ), namely μ

∂I (τ, μ) = I(τ, μ) − S(τ ), ∂τ

(18.76)

and 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ + Q∗ (τ ),

(18.77)

0

with Q∗ (τ ) = diag[Q∗1 (τ ), Q∗2 (τ )].

(18.78)

Here, Q∗1 (τ ) and Q∗2 (τ ) are the components of the vector Q∗ (τ ). For a semi-infinite medium, the emergent radiation field is given by  I(0, μ) =

∞

S(τ )e−τ/μ

0

dτ . μ

(18.79)

Expressing S(τ ) in terms of the Green matrix G(τ, τ0 ) and the primary source term Q∗ (τ ) in terms of its inverse Laplace transform q∗ (ν), we can write 1 I(0, ) = p p

 0

∞

˜˜ G(p, ν)q∗ (ν) dν,

p = 1/μ.

(18.80)

18.1 Non-Conservative Rayleigh Scattering

441

˜˜ where G(ν, p), the double Laplace transform of the Green matrix, is defined by ˜˜ G(ν, p) =



∞ ∞ 0

G(τ, τ0 ) e−ντ e−pτ0 dτ dτ0 .

(18.81)

0

As shown in Sect. 15.4.1, it satisfies the important relation ˜˜ G(ν, p) =

1 ˜ ˜ T (p), G(ν)G ν+p

p, ν ∈ [0, ∞],

(18.82)

˜ where G(ν) is the Laplace transform of G(τ ) = G(τ, 0). For a uniform primary source, q∗ (ν) = Q∗ δ(ν), with Q∗ a constant. Hence, I(0,

1 ˜˜ ) = pG(p, 0)Q∗ . p

(18.83)

Equation (18.82) leads to I(0,

1 ˜ ˜ T (0)Q∗ = G ˜ R (p)G ˜ T (0)Q∗ . ) = G(p) G R p

(18.84)

˜ R (0) = G ˜ T (0) = E −1/2 , we obtain Using G R ˜ R ( 1 )E −1/2 Q∗ = HR (μ)E −1/2 Q∗ = H(μ)RE −1/2 Q∗ . I(0, μ) = G μ

(18.85)

For μ = 0, Eq. (18.85) combined with H(0) = I, leads to I(0, 0) = RE −1/2 Q∗ .

(18.86)

˜ R (0)E −1/2 Q∗ = E −1 Q∗ . I(0, ∞) = G

(18.87)

For μ → ∞, it leads to,

For Q∗ = E 1/2 , we recover the result stated in Eq. (18.73), namely I(0, μ) = HR (μ) = H(μ)R, and R = I(0, 0) = S(0).

(18.88)

which shows that the matrix R has a simple physical meaning. In the (KQ) representation of the radiation field, it is the surface value of the source matrix for a primary source Q∗ = E 1/2 . A detailed analysis of R in terms of a rotation angle ϕ(, Q ) is presented in Ivanov et al. (1995).

442

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

In the atmospheric model used in Ivanov (1996) to construct the I-matrix the 1/2 primary source is Q∗ = E 1/2 = diag[ 1/2 , Q ]. The corresponding primary source term for the Stokes vector, with components I and Q, is Q∗ (τ, μ) = 1/2 AW (μ)[ 1/2, Q ]T , where AW (μ) is defined in Eq. (17.3). The matrix I(0, μ) can be continued to complex values of μ. The matrix I(0, z) is analytic in the complex plane cut along [−1, 0], satisfies the same nonlinear integral equation as HR (z), namely,    1− 1 T μ 1/2 dμ = I, I (0, μ)W (μ) I(0, z) E + 2 μ+z 0

(18.89)

and the factorization relation 1 L−1 ( ) = I(0, −z)IT (0, z). z

(18.90)

The four elements of I(0, z) are calculated in Ivanov et al. (1996) with an iterative method based on Eq. (18.89), for z = μ, with μ from 0 to 106 and several values of . The matrix I(0, μ) also satisfies a singular linear integral equation with a Cauchytype kernel. This equation can be derived from the Laplace transform of the Wiener– ˜ Hopf integral equation for S(τ ) and the relation I(0, 1/p) = pS(p). For a uniform ∗ primary source Q , it may be written as 1− 1 L( )I(0, μ) − μ 2



1

W (μ )I(0, μ )

0

μ dμ

= Q∗ . μ − μ

(18.91)

For μ = 0, Eq. (18.91) leads to 1− 2



1

W (μ )I(0, μ ) dμ = I(0, 0) − Q∗ ,

(18.92)

0

which can serve to check the consistency of numerical codes. We recall that L(∞) = I. Additional properties of the I-matrix can be found in Ivanov (1996).

18.2 Resonance Polarization For resonance polarization with complete frequency redistribution, we show in Sect. 17.2 that (z), the determinant of the dispersion matrix L(z), has no zero. This makes the determination of the resolvent matrix and of all the associated matrices much easier than for non-conservative Rayleigh scattering. Proceeding as in Sect. 18.1, we first determine (ν), the inverse Laplace transform of the resolvent ˜ matrix, and then the Laplace transform G(z) of the surface Green matrix. Finally we

18.2 Resonance Polarization

443

construct the nonlinear equations for the H and I matrices. These equations have a unique solution.

18.2.1 The Resolvent Matrix The inverse Laplace transform (ν) satisfies the matrix singular integral equation  L(ν)(ν) − (1 − )M(ν)

∞ 0

(ν )

dν = (1 − )M(ν), ν − ν



∞

ν ∈ [0, ∞[,

(18.93)

dν ,

(18.94)

where L(ν) = I − (1 − )

0

M(ν )



1 1 +

ν + ν ν −ν

and M(ν) =

1 1 ˆ ). g( ν ν

(18.95)

ˆ The matrix g(1/ν) is defined in Eq. (15.78). In contrast to Rayleigh scattering, there is no interval on the real axis where M(ν) is identically zero, hence there is no need to distinguish different intervals in ν. We solve Eq. (18.93) with the Hilbert transform method described in Sect. 18.1, using the properties of L(z) and X(z) determined in Sect. 17.2. We introduce the Hilbert transform  ∞ 1 (ν) F(z) ≡ dν. (18.96) 2i π 0 ν − z Using the Plemelj formulae for L(z) and F(z) and the relation defining X(z), given in Eq. (18.12) with now ν ∈ [0, ∞[, we obtain the boundary value problem     I I + − T − [X (ν)] F (ν) + − [X (ν)] F (ν) + = 0, 2i π 2i π +

T

ν ∈ [0, ∞[. (18.97)

The solutions of this equation have the form F(z) = [XT (z)]−1 P(z) −

I , 2i π

(18.98)

444

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

where P(z) is a matrix of polynomials. By construction F(z) → 0 at infinity. This condition is satisfied when lim P(z) =

z→∞

XT (∞) , 2i π

(18.99)

where X(∞) is the constant value of X(z) at infinity, namely 

 a∞ b∞ X(∞) ≡ , c∞ d ∞

a∞ d∞ = b∞ c∞ .

(18.100)

Since P(z) is a matrix of polynomials, Eq. (18.99) leads to P(z) = XT (∞)/(2i π) for all z. We thus obtain  % XT (∞) (ν) = [[X+ (ν)]T ]−1 − [[X− (ν)]T ]−1 . 2i π

(18.101)

It is possible to express (ν) in terms of L± (ν), X(−ν), and XT (∞), as in ˜ Eq. (18.31), but Eq. (18.101) is more useful to determine G(p) by a contour integration of (ν).

18.2.2 The H-Matrix ˜ ˜ We define the H-matrix as in Eq. (18.34), namely H(p) = G(1/p), with G(p), the Laplace transform of G(τ ). As shown in Eq. (18.36), ˜ G(p) =I+



∞ 0

(ν) dν, ν +p

p ∈ [0, ∞[.

(18.102)

To calculate the integral we proceed with a contour integration as in Sect. 18.1.2. The contour turns around the pole at −p, around the cut along [0, ∞[ and is closed by a circle of radius R, R → ∞. The contribution from circle of radius R cancels the identity matrix in the right-hand side of Eq. (18.102). We thus obtain ˜ G(p) = [XT (−p)]−1 XT (∞).

(18.103)

This relation can be continued to the complex plane, allowing us to define ˜ G(z) = [XT (−z)]−1 XT (∞).

(18.104)

18.2 Resonance Polarization

445

˜ We see that G(z) is analytic in the complex plane cut along the negative real axis, is free of singularities, and tends to the identity matrix at infinity. It satisfies the factorization relation −1 ˜ T (z)]−1 [G(−z)] ˜ L(z) = [G ,

(18.105)

which is easily deduced from the factorization relations established in Sect. 17.2.2, namely L(z) = X(z)XT (−z) and I = X(∞)XT (∞). ˜ ˜ We can deduce from the properties of G(z) that H(z) = G(1/z) is also analytic in the complex plane cut along the negative real axis and free of singularities. At z = 0, it is equal to the identity matrix. The factorization relation for H(z) is straightforwardly deduced from Eq. (18.105).

18.2.3 Nonlinear Integral Equations The proofs given for Rayleigh scattering in Sect. 18.1 concerning the construction ˜ of nonlinear integral equations for the matrices G(z), H(z), HR (z) and the the definition of the I-matrix are easily extended to resonance scattering. ˜ The construction of the nonlinear integral equation for G(z) is based on the relation ˜ G(z) = [XT (−z)]−1 XT (∞),

(18.106)

and on the defining relation for X(z) given in Eq. (17.61), namely X+ (ν) = W(ν)X− (ν). Proceeding as in Sect. 18.1.3, we calculate the difference X+ (ν) − ˜ T ]−1 (z). These two steps X− (ν) and then apply the Cauchy integral formula to [G lead to ˜ M(ν)G(ν) =−

1 [X+ (ν) − X− (ν)]XT (∞), 2i π

ν ∈ [0, ∞[,

(18.107)

and to −1 ˜ [G(−z)] = I − (1 − )



∞

T ˜ M(ν)G(ν)[X (∞)]−1 X−1 (∞)

0

dν . ν−z

(18.108)

As shown by Eq. (17.70), the product of the two X-matrices is the identity matrix. We thus obtains the nonlinear integral equation ˜ ˜ G(z) = I + (1 − )G(z)



∞ 0

˜ T (ν)M(ν) dν . G ν+z

(18.109)

446

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

˜ For H(z) = G(1/z), it becomes  H(z) = I + (1 − )H(z) z

∞

ˆ HT (p)g(p)

0

dp . z+p

(18.110)

Because the determinant of L(z) has no zero, this equation has a unique solution. In Appendix O of this chapter we show, for Rayleigh scattering, how to solve ˜ the singular integral equation for G(p) and how to construct the H-equation. The method of Appendix O in this chapter can be applied to resonance scattering and ˜ the Hanle effect. The calculation of G(p) requires about the same algebraic work as that of (ν), but the construction of the H-equation is simpler. In particular there is no need to introduce a Cauchy integral formula. ˜ R (z) = G(z)R ˜ ˜ R (z) = The construction of the alternative equations for G and H ˜ H(z)R, with R an orthogonal matrix, goes exactly as in Sect. 18.1.5. Choosing ˜ −1 (0) = L1/2 (0) and using L(0) = E = diag[, Q ], we obtain G R

  HR (z) E 1/2 + (1 − )

∞

0

ˆ HTR (p)g(p)

 p dp = I. z+p

(18.111)

An I-matrix can be constructed exactly as described in Sect. 18.1.6. The emergent field, given by 

∞

I(0, x, μ) =

S(τ )e−τ ϕ(x)/μ

0

ϕ(x) dτ, μ

(18.112)

depends only the variable ξ = μ/ϕ(x). We introduce Iξ (0, ξ ) = I(0, x, μ), defined by 

∞

Iξ (0, ξ ) = 0

S(τ )e−τ/ξ

dτ . ξ

(18.113)

For a uniform primary source term Q∗ , this expression becomes Iξ (0, ξ ) =

1≈ 1 G( , 0)Q∗ , ξ ξ

(18.114)

as shown for Rayleigh scattering in Sect. 18.1.6. The double Laplace transform satisfies the fundamental relation in Eq. (18.82). Hence Eqs. (18.85)–(18.88) established for Rayleigh scattering hold for Iξ (0, ξ ). In particular, for a uniform primary source term Q∗ = L1/2 (0) = E 1/2 , Iξ (0, ξ ) = HR (ξ ) = H(ξ )S(0),

(18.115)

18.3 The Hanle Effect

447

with ˜ T (0)E 1/2 . S(0) = R = Iξ (0, 0) = G

(18.116)

The expressions at zero and infinity of G(z), GR (z), H(z), and HR (z) given in Table 18.1 for Rayleigh scattering also hold for resonance scattering with complete frequency redistribution. For example HR (0) = R and HR (∞) = L−1/2 (0) = E −1/2 . The nonlinear integral equation for the Iξ (0, z) is identical to the equation for HR (z), namely   Iξ (0, z) E 1/2 + (1 − )

∞

0

ˆ ITξ (0, p)g(p)

 p dp = I. z+p

(18.117)

An iterative method of solution for Eq. (18.111) is described in Ivanov et al. (1997b). Some of the results concerning the polarization of the emergent radiation are presented in Sect. 25.3. The I-matrix also satisfies a linear integral equation of the Cauchy-type. It is similar to Eq. (18.91), namely 1 L( )Iξ (0, p) − (1 − ) p



∞

ˆ g(ν)I ξ (0, ν)

0

νdν = Q∗ . ν −p

(18.118)

For p = 0, this equation provides the identity  (1 − )

∞

∗ ˆ g(ν)I ξ (0, ν) dν = Iξ (0, 0) − Q .

(18.119)

0

It is similar to Eq. (18.92) and can be used to check numerical codes. This identity, with Q∗ = E 1/2 , can also be derived from the nonlinear Eq. (18.117), in which one sets z = 0. A detailed asymptotic analysis of Eq. (18.118) is carried out in Ivanov et al. (1997b) for the conservative case.

18.3 The Hanle Effect The results obtained for resonance scattering concerning the H-matrix and the Hequation are easily extended to the Hanle effect. It often suffices to replace the auxiliary matrix X(z) by XH (z). The fact that the matrices considered in the present section are 6 × 6 matrices, plays essentially no role. The main difference is that the dispersion matrix LH (z) tends at infinity to the constant matrix M−1 (θB ) instead of the unity matrix. After introducing the H-matrix, we construct the nonlinear integral equation for the I-matrix, given for the first time by Grachev (2001a).

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

448

18.3.1 The H-Matrix We define the H-matrix in the usual way, as ˜ H(z) ≡ G(1/z),

(18.120)

˜ where G(z) is the analytic continuation of the Laplace transform of the surface Green matrix G(τ ) = G(τ, 0). Following the method employed for resonance polarization, we first solve the singular integral equation for (ν). For the Hanle effect, it may be written as  LH (ν)(ν) − (1 − )M(ν)

∞ 0

(ν )

dν = (1 − )M(ν), ν − ν

ν ∈ [0, ∞[, (18.121)

with LH (ν) ≡ M−1 H (θB ) − (1 − )



∞ 0

M(ν )(

ν

1 1 +

) dν . +ν ν −ν

(18.122)

(see Sect. 15.6.2). The solution has the form given in Eq. (18.101) with X(z) replaced by XH (z), namely  % T − T −1 T −1 XH (∞) (ν) = [[X+ . (ν)] ] − [[X (ν)] ] H H 2i π

(18.123)

The properties of the dispersion matrix LH (z) and that of XH (z) are described in Sect. 17.3. We recall that L(z) can be factorized as LH (z) = XH (z)XTH (−z),

(18.124)

LH (∞) = M−1 H (θB ),

(18.125)

XH (∞)XTH (∞) = M−1 (θB ).

(18.126)

that we have

hence,

˜ ˜ The matrix G(p) and its continuation to the complex plane, G(z), can be derived from (ν) with the method described in Sect. 18.2.2 for resonance scattering. It leads to ˜ G(z) = [XTH (−z)]−1 XTH (∞),

(18.127)

18.3 The Hanle Effect

449

an expression similar to Eq. (18.104). Using Eqs. (18.124) and (18.126), we see that the factorization of LH (z) in Eq. (18.124) can also be written as −1 ˜ T (z)]−1 M−1 (θB )[G(−z)] ˜ LH (z) = [G . H

(18.128)

˜ The matrices G(z) for resonance polarization and for the Hanle effect have the same analyticity properties: they tend to the matrix unity at infinity, they are analytic in the complex plane cut along the real negative axis and free of singularity. Therefore H(z) is analytic in the complex plane cut along the negative real axis and H(0) = I.

18.3.2 Nonlinear Integral Equations ˜ ˜ To determine the nonlinear integral equations for G(z) and H(z), we proceed as described in Sect. 18.2.3 for resonance polarization. The starting point is the ˜ expression of G(z) given in Eq. (18.106) in which X(z) is replaced by XH (z). Introducing Eq. (18.126) into Eq. (18.108), we obtain ˜ ˜ G(z) = I + (1 − )G(z)M(θ B)



∞ 0

˜ T (ν)M(ν) dν . G ν +z

(18.129)

The equation for H(z) is 

∞

H(z) = I + (1 − )H(z)MH (θB ) z

ˆ HT (p)g(p)

0

dp . z+p

(18.130)

When the magnetic field is zero, MH (θB ) = I and one recovers the nonlinear integral equation for resonance polarization. We now construct a nonlinear integral equation for the I-matrix. We first establish ˜ a relation between the I-matrix and the matrix G(z) and then combine it with the ˜ nonlinear integral equation for G(z). As shown in Sect. 18.2.3, for a uniform primary source term Q∗ , Iξ (0, ξ ) =

1≈ 1 G( , 0)Q∗ , ξ ξ

(18.131)

≈

where G(ν, p) is the double Laplace transform of G(τ, τ0 ). For the Hanle effect, ≈

G(p, ν) =

1 ˜ ˜ T (ν)M−1 (θB ), G(p)MH (θB )G p+ν

(18.132)

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

450

as we show in Sect. 15.4.2. The emergent radiation field can thus be written as ˜ 1 )MH (θB )G ˜ T (0)M−1 (θB )Q∗ , Iξ (0, ξ ) = G( ξ

(18.133)

or as ˜ 1 )S(0) = H(ξ )S(0), Iξ (0, ξ ) = G( ξ

(18.134)

˜ T (0)M−1 (θB )Q∗ . S(0) = MH (θB )G

(18.135)

with

Equation (18.134) shows that the Hanle H-matrix can also be defined with the emergent radiation field when the primary source is uniform. The expression of S(0) becomes much simpler, when it is assumed, as in Grachev (2001a), that Q∗ = MH (θB )E 1/2.

(18.136)

The surface value of the source function is then given by T

˜ (0)E 1/2 . S(0) = MH (θB )G

(18.137)

T

˜ (0)E 1/2 (see Eq. (18.116)). When the magnetic field is zero, one recovers S(0) = G We know use Eqs. (18.134) and (18.135) to transform the nonlinear integral ˜ equation for G(z) written in Eq. (18.129) into a nonlinear integral equation for Iξ (0, ξ ). Proceeding as in Sect. 18.1.5, we can rewrite Eq. (18.129) as   ˜ ˜ −1 (0) + (1 − )M(θB ) G(z) G

∞ 0

˜ T (ν)M(ν) G

 zdν = I. ν(ν + z)

(18.138)

This is the Hanle version of Eq. (18.63). Using Eqs. (18.128) for z = 0, Eq. (18.133), and MT (θB ) = M(θB ), we obtain the nonlinear integral equation  Iξ (0, ξ )[Q∗ ]−1 M(θB )LH (0) I +  ∞ 1 dξ  +(1 − )M(θB )[[Q∗ ]T ]−1 = I. ITξ (0, ξ )M( )

ξ ξ +ξ 0

(18.139)

Equation 18.139) can be continued to the complex plane by setting ξ = z. Expressˆ ) (see Eq. (18.95)), and choosing Q∗ = M(θB )E 1/2 , we ing M(1/ξ ) in terms of g(ξ recover an equation established in Grachev (2001a), namely  ∞    ξ

ˆ ) Iξ (0, z) E −1/2 LH (0)E −1/2 E 1/2 + (1 − ) ITξ (0, ξ )g(ξ dξ = I. z + ξ

0 (18.140)

˜ Appendix O: The Singular Integral Equation for G(p)

451

As shown in Sect. 17.3,

ˆ LH (0) = M−1 H (θB ) − I + E = D(θB )[E + i H2 Q]D(−θB ).

(18.141)

This matrix has a block structure, the first row and first column contain only zeroes, except the first element, which is equal to . Because of this block structure, the first bracket is actually independent of the destruction probability . When the magnetic field is zero, LH (0) = E. We recover Eq. (18.111). Numerical calculations of the surface polarization for the Hanle effect are presented in Grachev (2001b). Although the author recalls the nonlinear integral equation, his numerical results are based on the integral equation for the source matrix S(τ ). It seems that Eq. (18.140) has not yet been solved numerically.

˜ Appendix O: The Singular Integral Equation for G(p) We show here, for non-conservative Rayleigh scattering, how to solve the matrix ˜ singular integral equation for G(p), the Laplace transform of the surface Green matrix G(τ ) = G(τ, 0). The solution is then used to establish a nonlinear integral ˜ ˜ equation for G(p). Using H(p) = G(1/p), it is easily transformed into the nonlinear H-equation. ˜ The matrix G(p) satisfies the singular integral equation ˜ L(p)G(p) −



∞ 1

N(p )

˜ ) G(p dp = I, p − p

p ∈ [0, ∞[.

(O.1)

This equation can be derived by applying a Laplace transform to the Wiener–Hopf integral equation for G(τ ) in Eq. (15.52). We recall that N(p) = −(1 − )M(p),

(O.2)

where M(p) is the inverse Laplace transform of the kernel given in Eq. (17.7). The matrix L(ν) and its analytic continuation, the dispersion matrix L(z), are defined in Sect. 15.6.1. For the scalar case, we describe in Sect. C.2 a Hilbert transform method of solution for the scalar version Eq. (O.1). We apply the same Hilbert transform method to Eq. (O.1), but because we now deal with a matrix equation, the construction of the solution is not quite straightforward. The method we present here is largely inspired by a work of Siewert and Burniston (1972). The goal of the work was to construct a solution for a singular integral equation essentially similar to Eq. (O.1), satisfied by the matrix H(p).

452

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

Because we are dealing with the Rayleigh scattering, the dispersion function (z) = det L(z) has a zero at z = ν0 . We have to consider separately the intervals p ∈ [1, ∞[ and p ∈ [0, 1], as in Sect. C.2. Interval p ∈ [1, ∞[ We introduce the Hilbert transform, 1 F(z) ≡ 2i π



∞

N(p) 1

˜ G(p) dp. p−z

(O.3)

By construction, the matrix F(z) is analytic in C /[1, +∞[, tends to zero for z → ∞ and is nonsingular. The Plemelj formulae for F(z) are ˜ F+ (p) − F− (p) = N(p)G(p),  ∞ ˜ ) 1 G(p dp , F+ (p) + F− (p) = N(p )

iπ 1 p −p

p ∈ [1, +∞[.

(O.4)

Proceeding as described in Appendix C of Chap. 6, we combine Eq. (O.1) with the second Plemelj formula. We now have the two equations, ˜ L(p)G(p) = i π[F+ (p) + F− (p)] + I,

(O.5)

˜ i πN(p)G(p) = i π[F+ (p) − F− (p)].

(O.6)

Adding and subtracting these two equations, using the Plemelj formulae for L(z) given in Eq. (17.9), and remembering that L± (p) are nonsingular for p ∈ [1, ∞[, we obtain the inhomogeneous Riemann–Hilbert equation, [L+ (p)]−1 [F+ (p) +

I I ] − [L− (p)]−1 [F− (p) + ] = 0, 2i π 2i π

p ∈ [1, ∞[. (O.7)

Introducing now the auxiliary matrix X(z), we can write this equation as [X+ (p)]−1 [F+ (p) +

I I ] − [X− (p)]−1 [F− (p) + ] = 0, 2i π 2i π

p ∈ [1, ∞[. (O.8)

Its solution may be written as F(z) =

1 [X(z)P(z) − I], 2i π

where P(z) is a matrix of polynomials, still to be determined.

(O.9)

˜ Appendix O: The Singular Integral Equation for G(p)

453

The condition F(z) → 0 as z → ∞ implies X∞ (z)P(z)  I,

z → ∞.

(O.10)

The expression of [X∞ (z)]−1 given in Eq. (18.17) shows that P(z) has the form P(z) = [X∞ (z)]−1 + C, where C is a constant matrix. We write it as   C11 C12 . C= C21 C22

(O.11)

(O.12)

To satisfy Eq. (O.10), the matrix C must satisfy X∞ (z)C → 0 for z → ∞. Using the expressions of the matrices X∞ (z) and X∞ (z)−1 given in Eq. (18.17), we see that this constraint leads to   a∞ C11 a∞ C12 = 0. (O.13) c∞ C11 c∞ C12 Because  = a∞ d∞ − b∞ c∞ = 0, one cannot have simultaneously a∞ = 0 and c∞ = 0. Thus Eq. (O.13) leads to C11 = C12 = 0. At this stage we have P(z) = [X∞ (z)]

−1



 0 0 + . C21 C22

(O.14)

The two unknown elements are determined when we consider the interval p ∈ [0, 1]. Interval p ∈ [0, 1] In this interval F(z) is analytic, which means that F+ (p) = F− (p) = F(p). Inserting the expression of F(z) given in Eq. (O.9) into Eq. (O.6), we see that Eq. (O.1) may be written as ˜ L(p)G(p) − X(p)P(p) = 0,

p ∈ [0, 1].

(O.15)

Following Siewert and Burniston (1972), we use Eq. (O.15) to determine P(p) for p = ν0 . This is sufficient to determine the constants C21 and C22 , hence to fully determine P(z). It is shown in Sect. 17.1.2 that the determinant of L(z) has two zeroes on the real axis located at z = ±ν0 , ν0 ∈ [0, 1], and that it exists a vector j (ν0 ) such that L(ν0 )j (ν0 ) = 0.

(O.16)

454

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

Transposing Eq. (O.15), using Eq. (O.16), and remembering that L(z) is a symmetric matrix, we obtain PT (ν0 )XT (ν0 )j (ν0 ) = 0.

(O.17)

Using the factorization relation L(z) = X(z)D(z)DT (−z)XT (−z),

(O.18)

where D(z) = diag(1, ν0 − z) and remembering that L(z) is an even function of z, we can rewrite Eq. (O.16) as X(−ν0 )D(ν0 )XT (ν0 )j (ν0 ) = 0.

(O.19)

We now consider the vector XT (ν0 )j (ν0 ), which appears in Eqs. (O.17) and (O.19), and denote a and b its components: XT (ν0 )j (ν0 ) ≡

  a . b

(O.20)

It follows from Eq. (O.19) that   a X(−ν0 ) = 0. 0

(O.21)

Since X(z) is nonsingular, Eq. (O.21) implies a = 0. Hence, Eq. (O.17) reduces to   0 = 0. P (ν0 ) 1 T

(O.22)

Without loss of generality, we have set b = 1. Using now the expression of P(z) given in Eq. (O.14), we can calculate the constants C21 and C22 . We finally obtain P(z) = [X∞ (z − ν0 )]−1 ,

(O.23)

 1  X(z)[X∞ (z − ν0 )]−1 − I . 2i π

(O.24)

and F(z) =

The first Plemelj formula applied to F(z) yields ˜ N(p)G(p) =

1 [X+ (p) − X− (p)][X∞ (p − ν0 )]−1 , 2i π

p ∈ [1, ∞[.

(O.25)

References

455

Replacing the jump of X(z) by the expression given in Eq. (18.46), we finally obtain ˜ G(p) = [XT (−p)]−1 [X∞ (−

1 )]−1 , ν0 + p

p ∈ [1, ∞[.

(O.26)

It can be verified, using the factorization in Eq. (O.18), that Eq. (O.26) holds also in the region p ∈ [0, 1], namely that it satisfies Eq. (O.15). Equation (O.26) can be continued to the complex plane, leading to ˜ G(z) = [XT (−z)]−1 [X∞ (−

1 )]−1 . ν0 + z

(O.27)

We recover the expression given in Eq. (18.40), derived from the singular integral equation for (ν) and a contour integration. ˜ The construction of the nonlinear equation for G(z) is now fairly easy. It suffices ˜ to equate Eq. (O.3) with Eq. (O.24), use Eq. (18.40) to express X(z) in terms of G(z), and change z to −z. One obtains ˜ T (z) + G ˜ T (z) G

 1

∞

˜ N(p)G(p)

dp = U(z), p+z

(O.28)

where U(z) = [XT∞ (z − ν0 )]−1 D−1 (z)D−1 (−z)[X∞ (−z − ν0 )]−1 .

(O.29)

2 + b 2 = d 2 + c 2 = 1 and a c + b d = 0 (see Eq. (17.43)) The relations a∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ lead to U(z) = I. The nonlinear equation for H(z) given in Eq. (18.53) is recovered ˜ by setting G(p) = H(1/p), N(p) = −(1 − )(1/p)/(2p), and μ = 1/p. Compared to the solution of the singular integral equation for (ν) in ˜ Sect. 18.1.1, it is clear that the solution of the equation for G(p) is more complex, but the construction of the nonlinear H-equation much simpler. ˜ The method described here to solve the singular integral equation for G(p) and construct the H-equation can be applied to resonance polarization and the Hanle effect. The algebra is simpler than described here because L(z) has no zeroes and X∞ (z) is a constant matrix.

References Bosma, P.B., de Rooij, W.A.: Efficient methods to calculate Chandrasekhar ’s H -functions. Astron. Astrophys. 126, 283–292 (1983) Chandrasekhar, S.: Radiative Transfer, 1st edn. Dover Publications, New York (1960); Oxford University Press (1950) de Rooij, W.A., Bosma, P.B., van Hooff, J.P.C.: A simple method for calculating the H-matrix for molecular scattering. Astron. Astrophys. 226, 347–356 (1989)

456

18 Scattering Problems with No Exact Solution II: The Resolvent Matrix, the H-. . .

Grachev, S.I.: Transfer of polarized radiation: nonlinear integral equations for I-matrices in the general case and for resonance scattering in a weak magnetic field. Astrophysics 44, 369–381 (2001a); translation from Astrofizika 44, 455–467 (2001) Grachev, S.I.: The Formation of polarized lines: allowance for the Hanle effect. Astronomy Reports 45, 960–966 (2001b); translation from Astronom. Zhurnal 78, 1092–1098 (2001) Ivanov, V.V.: Generalized Rayleigh scattering III. Theory of I-matrices. Astron. Astrophys. 307, 319–331 (1996) Ivanov, V.V., Kasaurov, A.M., Loskutov, V.M., Viik, T.: Generalized Rayleigh scattering II. Matrix source functions. Astron. Astrophys. 303, 621–634 (1995) Ivanov, V.V., Kasaurov, A.M., Loskutov, V.M.: Generalized Rayleigh scattering IV. Emergent radiation. Astron. Astrophys. 307, 332–346 (1996) Ivanov, V.V, Grachev, S.I., Loskutov, V.M.: Polarized line formation by resonance scattering II. Conservative case. Astron. Astrophys. 321, 968–984 (1997b) Siewert, C.E., Burniston, E.E.: On existence and uniqueness theorems concerning the H-Matrix of radiative transfer. Astrophys. J. 174, 629–641 (1972)

Part III

Asymptotic Properties of Multiple Scattering

Chapter 19

Asymptotic Properties of the Scattering Kernel K(τ )

We have shown in Chap. 2, that for a plane parallel medium the source function S(τ ), for monochromatic scattering as well as for complete frequency redistribution, satisfies a convolution type integral equation, which may be written as  S(τ ) = (1 − )

D

K(τ − τ )S(τ ) dτ + Q∗ (τ ).

(19.1)

We recall that monochromatic scattering is relevant to the formation of continuous spectra and complete frequency redistribution to the formation of spectral lines. In Eq. (19.1), D is the integration domain and Q∗ (τ ) a given primary source. In this first chapter of Part III devoted to asymptotic properties of the radiation field for media with a large optical thickness, we consider in some detail the properties of the scattering kernel, in particularly its behavior at large τ . It has been seen in preceding chapters that the kernel K(τ ) for large τ decreases exponentially for monochromatic scattering, while it√decreases algebraically for complete frequency redistribution, behaving as 1/(τ 2 ln τ ) for the Doppler profile and as 1/τ 3/2 for the Voigt profile. As a consequence, the second-moment of the kernel, defined by 

+∞ −∞

τ 2 K(τ ) dτ,

(19.2)

has a finite value for monochromatic scattering and an infinite one for complete frequency redistribution. The first-moment of K(τ ) calculated with |τ | is also infinite for the Doppler and Voigt profiles. The large scale behavior of the radiation field, which we study in subsequent chapters, critically depends on whether this second-moment is finite or not. For monochromatic scattering, which has a finite second-moment, the random walk of the photons after a large number of scatterings has a ordinary diffusion behavior, while complete frequency redistribution, which has an infinite second-moment, has a so-called anomalous diffusion behavior. We explain in Chaps. 20 and 21 what is meant by ordinary and anomalous diffusion.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_19

459

460

19 Asymptotic Properties of the Scattering Kernel K(τ )

In this Chapter on the asymptotic behavior of K(τ ) for large values of τ , we consider monochromatic scattering in Sect. 19.1 and complete frequency redistribution in Sect. 19.2. Actually, most of the chapter is devoted to complete frequency redistribution, the monochromatic kernel having a trivial asymptotic behavior. The behavior of K(τ ) is derived directly from the expression of K(τ ) for |τ | → ∞, but also from the behavior of its inverse Laplace transform k(ν), for ν → 0. We also ˆ examine the behavior of the Fourier transform of the kernel, K(k), for k → 0. It will be shown that this behavior can be characterized, for monochromatic scattering as well as for complete frequency redistribution, by a parameter, denoted α, to which we refer to as the characteristic index, or in short index. This index actually serves in the classification of random processes and is known as the stability parameter, for processes with an infinite second-moment. The reason of it is explained in Sect. 21.1.2. The parameter α is positive and takes values in the interval 0 < α ≤ 2. Ordinary diffusion corresponds to α = 2 and anomalous diffusion to α < 2. We start this Chapter with some definitions. For monochromatic scattering, 1 K(τ ) ≡ 2



1

exp(− 0

1 |τ | dμ ) = E1 (|τ |), μ μ 2

(19.3)

with E1 (|τ |) the first exponential integral function, and for complete frequency redistribution     1 +∞ 1 2 ϕ(x) dμ dx, (19.4) K(τ ) ≡ ϕ (x) exp −|τ | 2 −∞ 0 μ μ where ϕ(x) is the line absorption frequency profile, normalized to unity. In both case, the kernel is normalized to unity, namely 

+∞ −∞



∞

K(τ ) dτ = 2

K(τ ) dτ = 1.

(19.5)

0

The inverse Laplace transform k(ν) is defined by 

∞

K(τ ) =

k(ν)e−ντ dν,

(19.6)

K(τ ) e−i kτ dτ.

(19.7)

0

ˆ and the Fourier transform K(k) by ˆ K(k) ≡



+∞ −∞

19.2 The Complete Frequency Redistribution Kernel

461

The inverse and direct Laplace transforms are related by ˆ K(k) =



∞ 0

2νk(ν) dν. ν 2 + k2

(19.8)

Using the normalization of K(τ ), this relation can also be written as ˆ K(k) = 1 − 2k 2



∞ 0

k(ν) dν, ν(ν 2 + k 2 )

(19.9)

ˆ an expression very convenient to determine the limit of K(k) for k → 0.

19.1 The Monochromatic Kernel For monochromatic scattering, the asymptotic behavior of the kernel is quite simple, since exponential integral functions are decreasing exponentially at infinity. The large τ -expansion of E1 (τ ) (see, e.g., Abramovitz and Stegun 1964) leads to   1 1 −|τ | 1 1 + O( 2 ) . K(τ )  e 1− 2 |τ | |τ | τ

(19.10)

The inverse Laplace transform of K(τ ), already used many times in the preceding chapters, is k(ν) = 1/(2ν),

ν ≥ 1,

k(ν) = 0,

ν ∈ [0, 1].

(19.11)

The Fourier transform of K(τ ), easily derived from Eq. (19.9), is given by ˆ K(k) =



∞ 1

dν 1  1 − k2, ν 2 + k2 3

k → 0.

(19.12)

The power of k is the characteristic index α of the scattering process. For monochromatic scattering α = 2.

19.2 The Complete Frequency Redistribution Kernel The large-τ asymptotic expansion of K(τ ) is presented in Sect. 19.2.1 and the limit ν → 0 of the inverse Laplace transform k(ν) in Sect. 19.2.2. In Sect. 19.2.3, we ˆ determine the limit for k → 0 of the Fourier transform K(k) and show that it has the form given in Eq. (19.12) with a value of α depending on the profile and smaller than 2.

462

19 Asymptotic Properties of the Scattering Kernel K(τ )

19.2.1 Asymptotic Behavior at Large Optical Depths We start from the definition in Eq. (19.4). Since K(τ ) is an even function of τ , we assume τ ≥ 0 to simplify the notation. Proceeding as in Appendix F.1 of Chap. 9, we choose ϕ(x) = u as integration variable. We can then write 

1  ϕ(0)

K(τ ) = 0

−1

u2 e−uτ/μ d[ ϕ (u)]

0

dμ . μ

(19.13)

For the Doppler profile, 1 2 ϕ(x) = √ e−x , π

(19.14)

and Eq. (19.13) becomes KD (τ ) =

1  ϕ(0)



1 2

0

dμ u . e−uτ/μ  √ du μ − ln(u π)

0

(19.15)

Introducing y = uτ/μ, we can write 1 KD (τ ) = 2 2τ





1

τ ϕ(0)/μ

μ 0

e−y 

0

y dy dμ. √ − ln( πyμ/τ )

(19.16)

The main contribution to the integral come from values of y about unity. In the limit τ → ∞, the upper limit τ ϕ(0)/μ can be replaced by infinity. An expansion of the logarithmic term yields, for the leading order term, KD (τ ) 

1 1 √ 2τ 2 ln τ





1

∞

μ dμ 0

e−y y dy 

0

1 1 , √ 4τ 2 ln τ

τ → ∞.

(19.17)

Higher order terms in the expansion of the logarithm provide the higher order terms of the expansion. The full expansion has the form 1 1 KD (τ )  2 √ 4τ ln τ

1+

∞  1

 dn , (ln τ )n

(19.18)

where dn are numerical constants, which can be found in Avrett and Hummer (1965) or Ivanov (1973). For the Voigt profile, it is sufficient to keep ϕ(x) 

a 1 , π x2

|x| → ∞,

(19.19)

19.2 The Complete Frequency Redistribution Kernel

463

for the purpose of the asymptotic analysis. Proceeding as above, we obtain, at leading order, 1 KV (τ )  2

$

a 1 π τ 3/2



1

√



∞

e

μ dμ

0

0

√ a 1 y dy  , 6 τ 3/2

−y √

τ → ∞. (19.20)

√ The integral over y is (3/2) = π/2, with  denoting the Gamma function (Abramovitz and Stegun 1964). Higher order terms can be constructed by taking into account the τ -dependence of the upper limit τ ϕ(0)/μ in the integration over y. The full expansion has the form  √

∞  a vn KV (τ )  3/2 1 + . 6τ τn

(19.21)

1

The coefficients vn can be found in Avrett and Hummer (1965) or Ivanov (1973). We note here that the series in Eqs. (19.18) and (19.21) are asymptotic series. These series were introduced by Poincaré1 (1886). These series may be divergent, but are very useful for numerical evaluations, a few terms being often sufficient to reach a good accuracy. The condition for a series to be asymptotic is that the remainder after N terms (difference between the function and the sum of the first N terms) is smaller than the last term of the sum, which has been retained. For a given accuracy, there is an optimal value of N. Several examples of asymptotic series, a precise mathematical definition, and historical notes can be found in, e.g., Bender and Orszag (1978) and Ramis (2012a,b). All the asymptotic expansions given here hold for τ → −∞, provided τ is replaced by |τ |.

19.2.2 The Inverse Laplace Transform Near the Origin The kernel can be written as in Eq. (19.6), with k(ν) the inverse Laplace transform, or as  ∞ dξ , (19.22) K(τ ) = g(ξ )e−|τ |/ξ ξ 0 where  g(ξ ) =

∞ y(ξ )

1

Henri Poincaré: 1854–1912 (Nancy-Paris).

ϕ 2 (v) dv,

(19.23)

464

19 Asymptotic Properties of the Scattering Kernel K(τ ) −1

y(ξ ) = 0 for 0 < |ξ | ≤ 1/ϕ(0) and y(ξ ) = ϕ (1/ξ ) for |ξ | ≥ 1/ϕ(0). The functions k(ν) and g(ξ ) are related by 1 1 g( ). ν ν

k(ν) =

(19.24)

The function g(ξ ) and k(ν) are plotted in Figs. 5.1 and 5.2. The asymptotic behaviors of k(ν) for ν → 0 and g(ξ ) for ξ → ∞ are determined in Appendix F.1 of Chap. 9. Here we simply recall the leading terms. For the Doppler profile, gD (ξ ) 

1 1 , √ 4ξ 2 ln ξ

ξ → ∞,

kD (ν) 

ν 1 , √ 4 − ln ν

ν → 0,

(19.25)

and for the Voigt profile, 1 a 1/2 gV (ξ )  √ , 3 π ξ 3/2

ξ → ∞,

1 kV (ν)  √ a 1/2ν 1/2 , 3 π

ν → 0.

(19.26)

The functions g(ξ ) for ξ → ∞ and K(τ ) for τ → ∞ have the same scaling laws. Inserting the asymptotic expressions of k(ν) into Eq. (19.6), one readily recovers the asymptotic expansions of K(τ ) in Eqs. (19.15) and (19.20).

19.2.3 The Fourier Transform Near the Origin ˆ We start from the expression of K(k) given in Eq. (19.9). Making the change of variable ν = ρ|k|, we can write ˆ K(k) =1−2

 0

∞

k(ρ|k|) dρ . 1 + ρ2 ρ

(19.27)

Inserting the asymptotic behaviors of k(ν) for ν → 0 given in Eqs. (19.25) and (19.26), we obtain π |k| Kˆ D (k)  1 − √ , 4 − ln |k|

Kˆ V (k)  1 −

√ 2π 1/2 1/2 a |k| . 3

(19.28)

ˆ The asymptotic behavior of K(k) for complete frequency redistribution can thus be written as ˆ K(k)  1 − h(|k|)|k|α ,

k → 0,

(19.29)

19.2 The Complete Frequency Redistribution Kernel

465

where h(|k|) is a constant or a slowly varying function of k for k → 0. Slowly varying implies that h(ck) = 1, k→0 h(k) lim

for any c > 0.

(19.30)

ˆ For monochromatic scattering, Eq. (19.12) shows that K(k) can also be written as in Eq. (19.29). The Fourier transform of the kernel, for both monochromatic scattering and complete frequency redistribution, can be written as in Eq. (19.29), with α taking the following values: ⎧ ⎨ 2 monochromatic scattering, α= 1 Doppler profile, ⎩ 1/2 Voigt profile.

(19.31)

For complete frequency redistribution, α gives also the scaling law for k(ν) as ν → 0. The role of α is examined in detail in Chap. 21 where we discuss the random walk of the photons. We summarize in Table 19.1 the asymptotic behaviors derived above. Before closing this Chapter on the asymptotic behavior of K(τ ) for large-τ , we want to mention the method introduced for complete frequency redistribution by Abramov et al. (1967) and described in Ivanov (1973, p. 81). It provides the asymptotic behavior in the form K(τ ) 

(α + 1) f (τ ) , α+1 τ

(19.32)

−1

where f (y) ≡ ϕ (1/y) and f (y) ≡ df (y)/dy. Asymptotic expressions of f (y) and f (y), for y-large, are given in the Table 19.2. The value of α is deduced from the limit, f (ξ/w) = wα . ξ →∞ f (ξ ) lim

(19.33)

Table 19.1 The asymptotic behaviors of K(τ ) for τ → ∞, g(ξ ) for ξ → ∞, k(ν) for ν → 0, ˆ and K(k) for k → 0. The kernel is an even function of τ . Here τ , ξ , and ν are positive. For monochromatic scattering, k(ν) is zero in the interval [0, 1[ Profile Monochromatic Doppler Voigt

K(τ )

g(ξ )

k(ν)

e−τ

1−

2τ 4τ 2

ˆ K(k)

1 √

ln τ

a 1/2

1 6 τ 3/2

1 √ 4ξ 2 ln ξ

ν √ 4 − ln ν

1 a 1/2 √ 3/2 3 πξ

1 √ a 1/2 ν 1/2 3 π

1 2 k 3

π |k| √ 4 − ln |k| √ 2π 1/2 1/2 1− a |k| 3 1−

466 Table 19.2 The index α for the Doppler and Voigt profiles and the asymptotic expressions of f (y) and f (y) for y large. The function f (y) is the inverse function of the profile ϕ(x)

19 Asymptotic Properties of the Scattering Kernel K(τ )

f (y)

Doppler $ y ln √ π

f (y)



1

2y ln α

1

√y π

Voigt a 1/2 1/2 √ y π a 1/2 −1/2 √ y 2 π 1/2

√ With (2) = 1 and (3/2) = π/2, Eq. (19.32) leads to the asymptotic expressions of K(τ ) in Eqs. (19.17) and (19.20). In Ivanov (1973), the exponent α is denoted 2δ or 2γ . The expansion technique leading to Eq. (19.32) can be applied to generalized kernels defined by Eq. (19.4) with ϕ(x) raised to any positive integer power (Ivanov 1973, p. 81).

References Abramov, Yu.Yu., Dykhne, A.M., Napartovich, A.P.: Transfer of resonance radiation in a halfspace. Astrophysics 3, 215–223 (1967); translation from Astrofizika 3, 459–479 (1967) Abramovitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. (1964) Avrett, E.H., Hummer, D.G.: Non-coherent scattering II: Line formation with a frequency independent source function. Mon. Not. R. astr. Soc. 130, 295–331 (1965) Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York (1978) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Poincaré, H.: Sur les intégrales irrégulières des équations linéaires. Acta Mathematica 8, 295–344 (1886) Ramis, J-P.: Poincaré et les développements asymptotiques (Premi‘ere partie). Gazette de la Société Mathématique de France 133, 33–72 (2012a) Ramis, J-P.: Les développements asymptotiques après Poincaré: continuité et. . . divergences. Gazette de la Société Mathématique de France 134, 17–36 (2012b)

Chapter 20

Large Scale Radiative Transfer Equations

In this Chapter, we use the integral equation for the source function to determine large scale properties of the radiation field. We use it, in particular, to distinguish between ordinary and anomalous diffusion processes, to introduce the thermalization length as a characteristic scale of variation of the radiation field and to introduce new equations describing the large scale behavior of the field. These properties are derived from an asymptotic analysis of the integral equation, made possible by the presence in the integral equation of the small parameter , describing the destruction probability per scattering. A similar asymptotic analysis is applied in Chaps. 23 and 24 to the radiative transfer equations for monochromatic scattering and Rayleigh scattering. For elementary presentations of asymptotic expansions of equations, we can recommend (Cole 1968; Bender and Orszag 1978). The present chapter follows closely the article by Frisch and Frisch (1977). The key idea is to introduce of a new optical depth variable τ˜ defined by τ˜ = τ/τeff (),

(20.1)

where the scaling factor τeff () tends to infinity as  tends to zero. The variable τ˜ , henceforth referred to as the rescaled optical depth, is of order unity when τ is order of τeff (). This scaling factor will be identified to the thermalization length. By analogy with time-dependent phenomena, τ˜ plays the role of the slow variable and τ that of the fast variable. In other terms, the variable τ can describe rapidly changing phenomena, while the variable τ˜ describes slowly changing ones. To understand the meaning of the optical depth rescaling, one can also think of road maps with different scales. A map with a scale 1/106, corresponding to 1 cm per 10 km, shows only the main roads and cities. When the scale increases, say to 1/(5 104), corresponding to 1 cm per 500 m, all the villages and small countryside roads are displayed. Anybody familiar with computerized maps, knows that details disappear when looking on a computer screen at a larger region.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_20

467

468

20 Large Scale Radiative Transfer Equations

Another essential ingredient of the method is the assumption that the source function, denoted S (τ ) in this chapter, has an asymptotic expansion of the form ˜ τ˜ ) + h.o.t.], S (τ ) = a()[S(

(20.2)

where h.o.t. stands for higher order terms going to zero as  → 0 and a() is a ˜ τ˜ ), because it depends only on the scaling factor to be determined. The function S( large scale variable τ˜ , describes the large scale behavior of S (τ ) (main roads and cities). The next step is to replace, in the integral equation, S (τ ) by its asymptotic expansion and take the limit  → 0, that is τeff () → ∞. When τeff () is properly ˜ τ˜ ), usually chosen, the initial integral equation for S (τ ) provides an equation for S( simpler than the equation for S (τ ). This asymptotic equation can be used only far boundaries. In this chapter we assume an infinite medium. Finite geometries are considered in subsequent Chapters. The asymptotic analysis of the integral equation for S (τ ) is carried out in Sect. 20.1, for monochromatic scattering and complete frequency redistribution, in an infinite medium. It provides scaling laws for τeff () and large scale equations ˜ τ˜ ). The Fourier transforms of these equations are established in Sect. 20.2. In for S( Sect. 20.3 we present an alternative derivation of the thermalization length, proposed by Avrett and Hummer (1965) for complete frequency redistribution.

20.1 Asymptotic Analysis of the Source Function In the τ -space, the source function S (τ ) satisfies the integral equation  S (τ ) = (1 − )

+∞

K(τ − τ )S (τ ) dτ + Q∗ (τ ),

−∞

(20.3)

where Q∗ (τ ) is a primary source term. Before we perform the asymptotic analysis, Eq. (20.3) must be given an adequate form. For small values of , the integral term and S (τ ) are almost equal. The first step of the analysis is to subtract these two terms. Using the normalization of K(τ ) to unity (see Eq. (19.5)) we rewrite the integral equation as S (τ ) − Q∗ (τ ) = (1 − )



+∞

−∞

K(τ − τ )[S (τ ) − S (τ )] dτ .

(20.4)

We assume that Q∗ (τ ) is a slowly varying function of τ which can be represented by a function Q˜ ∗ (τ˜ ). The handling of rapid variations of the primary source would

20.1 Asymptotic Analysis of the Source Function

469

require a sophisticated asymptotic analysis, known as homogenization (Bensoussan et al. 1978, 1979). Introducing into Eq. (20.4) the expansion of S (τ ), limited to its leading term, and expressing τ in terms of τ˜ , the rescaled optical depth, we can write  τeff () +∞ Q˜ ∗ (τ˜ ) ˜ ˜ τ˜ ) − S( ˜ τ˜ )] d τ˜ , = (1 − ) S(τ˜ ) − K[(τ˜ − τ˜ )τeff ()][S( a()  −∞ (20.5) or  +∞ ˜ ˜ τ˜ )] dσ, ˜ τ˜ ) − Q(τ˜ ) = (1 − ) ˜ τ˜ + σ ) − S( S( K(σ )[S( a()  τeff () −∞

(20.6)

where σ = τ − τ . In Eq. (20.6), a Taylor expansion of the integrand to second order in σ leads to  +∞ 2˜ ˜ 4 ˜ τ˜ ) − Q(τ˜ ) = 1 −  ∂ S(τ˜ ) K(σ )σ 2 dσ + O(τeff ). S( 2 () ∂ τ˜ 2 a() 2τeff −∞

(20.7)

The contribution from the first order term in the Taylor expansion is zero, since K(τ ) is an even function of τ . The integral over σ is the second-moment of K(τ ) introduced in Eq. (19.2). This moment is finite for monochromatic scattering but infinite for complete frequency redistribution. Hence, for monochromatic scattering, it is possible to take the limit  → 0 in Eq. (20.7). For complete frequency redistribution, Eq. (20.7) does not make sense, but the limit  → 0 can be taken in Eq. (20.5). We already see here the difference between scattering processes for which the kernel has a finite or infinite second-moment. The undetermined parameters a() and τeff () are chosen in such a way that in the limit  → 0 all the terms in Eqs. (20.7) and (20.5) are of order unity. This constraint ensures that the scattering term and the primary creation term are properly ˜ τ˜ ) are of order unity (the scaling factor taken into account and that the solutions S( is contained in a()). Concerning a(), these constraints lead to the scaling a() ∼ 1/,

(20.8)

for both Eqs. (20.5) and (20.7). When Q∗ (τ ) is of order , a typical value for an atomic transition, then a() ∼ 1. To determine τeff () and the large scale equation, we use Eq. (20.7) for monochromatic scattering and Eq. (20.5) for complete frequency redistribution.

470

20 Large Scale Radiative Transfer Equations

20.1.1 Monochromatic Scattering Taking the limit  → 0 in Eq. (20.7), we see that the right-hand side has a finite limit as  → 0 when we choose √ τeff () = 1/ .

(20.9)

˜ τ˜ ) is a standard diffusion equation, The equation for S( ˜ τ˜ ) − Q( ˜ τ˜ ) = D  ˜ S( ˜ τ˜ ), S(

1 D= 2



+∞

K(σ )σ 2 dσ,

−∞

(20.10)

˜ = ∂ 2 /∂ τ˜ 2 . The value of the diffusion coefficient D depends on the where  scattering phase function. For an isotropic scattering phase function, D = 1/3. This value is easily derived from the expression of K(τ ) = E1 (|τ |)/2 (see Eq. (19.3)). By ignoring the small scales, the original nonlocal integral equation is transformed into a more simple ordinary differential equation. The properties of the kernel enter only through the diffusion coefficient D. The solution of Eq. (20.10) requires boundaries conditions. In an infinite ˜ τ˜ ) should tend to zero at infinity. The medium, the rescaled source function S( boundary conditions for a bounded medium are discussed in Chap. 23. We also show in this Chap. 23 that the radiative transfer equation for the radiation field can be transformed into a diffusion equation for the direction-average radiation field.

20.1.2 Complete Frequency Redistribution We now consider the limit  → 0 in Eq. (20.5). Looking for an equation valid for τ˜ ∼ 1, it is justified to replace the kernel by its asymptotic expression at large values of τ . For the Doppler profile √ we use √ the expression given in Eq. (19.17) with τ replaced by τ˜ τeff () and ln τ by ln τeff () (the factor τ˜ inside the logarithm can be neglected since τ˜ ∼ 1). We obtain     ˜ τ˜ ) − Q˜ ∗ (τ˜ ) = τeff () ln τeff () S(

+∞ S( ˜ τ˜ ) −∞

4(τ˜

˜ τ˜ )

− S( d τ˜ . − τ˜ )2

(20.11)

The integral must be taken in Cauchy Principal Value to avoid the singularity at τ˜ = τ˜ . The condition that the left-hand tends to a finite limit leads to 1 D τeff () ∼ √ .  − ln 

(20.12)

20.1 Asymptotic Analysis of the Source Function

471

˜ τ˜ ) may be written as The integral equation for S( ˜ τ˜ ) − Q˜ ∗ (τ˜ ) = S(



+∞ S( ˜ τ˜ ) − S( ˜ τ˜ ) −∞

4(τ˜ − τ˜ )2

d τ˜ .

(20.13)

For the Voigt profile, the asymptotic expansion of K(τ ) for τ → ∞ given in Eq. (19.20) leads to V τeff () ∼

a , 2

(20.14)

and to the large scale equation ˜ τ˜ ) − Q˜ ∗ (τ˜ ) = S(



˜ τ˜ ) ˜ τ˜ ) − S( S( d τ˜ .

3/2 −∞ 6(|τ˜ − τ˜ )|) +∞

(20.15)

Equations (20.13) and (20.15) have solutions which remain finite and tend to zero at infinity, when the integral of Q∗ (τ ) over τ is finite. In contrast to monochromatic scattering, the large scale equations for complete redistribution are still nonlocal since they involve an integration over the depth variable τ˜ . They are however simpler than the integral equation for S(τ ), since their kernel, can be written as a fractional power of a negative Laplacian (see Eq. (20.31)). In the presence of a continuous absorption, the line source function Sl (τ ) satisfies an integral equation quite similar to Eq. (20.3) (see Eq. (8.15)) with a generalized ¯ kernel K(τ, β) and a generalized destruction probability ¯ . Hence, for optical depths less than 1/β, all the scaling laws and asymptotic equations given above also hold in the presence of a continuous absorption. It suffices to replace  by ¯ =  + βF (β),

(20.16)

where β is the ratio of the line to the continuous absorption coefficient. For β small, βF (β) can be replaced by the leading term of its asymptotic expansion given in Eq. (8.12). The large-scale equations provide a valuable information for numerical solutions of the integral equation for the source function S (τ ). For the Doppler profile, Eq. (20.13) shows that the condition for the contribution from the region τ˜ ≈ τ ˜ τ˜ ) has a continuous second order derivative. This implies to remain finite is that S( that a representation of S (τ ), which is at least quadratic, is needed to obtain an accurate numerical solution. For the Voigt profile, Eq. (20.15) shows that a linear representation is sufficient. The accuracy of linear versus quadratic representations has been investigated in Bommier et al. (1991).

472

20 Large Scale Radiative Transfer Equations

20.1.3 Higher Order Terms An evaluation of the terms denoted “h.o.t” (for higher order terms) in the large scale expansion of S (τ ) (see Eq. (20.2)) gives an indication on how well S (τ ) is represented by its leading term. We assume that the source function has an expansion of the form ˜ τ˜ ) + S (τ )  a()[S(

∞ 

ln ()S˜n (τ˜ )],

(20.17)

n=1

where ln () and S˜n (τ˜ ) are functions to be determined. We introduce this expansion and the expansions of the kernels given in Eqs. (19.18) and (19.21) into Eq. (20.4). The coefficients ln () are derive from the constraint that all the Sn (τ˜ ) have finite values as  → 0. For the Doppler profile, one thus finds l1 () ∼ d1 /(− ln ),

(20.18)

where d1 is the coefficient of the first-order term in the asymptotic expansion of K(τ ). The first order term S˜1 (τ˜ ) satisfies the singular integral equation S˜1 (τ˜ ) −



+∞ S˜ (τ˜ ) − S˜ (τ˜ ) 1 1

2 −∞ 4(τ˜ − τ˜ )

d τ˜ =



+∞ S( ˜ τ˜ ) − S( ˜ τ˜ ) −∞

4(τ˜ − τ˜ )2

d τ˜ .

(20.19)

It is easy to see that the nth-order term is of order (− ln )n . For the Voigt profile, l1 () ∼ v1  2 ,

(20.20)

where v1 is has the same meaning as d1 , but for the Voigt profile. The equation for S˜1 (τ˜ ) is similar to Eq. (20.19), with a kernel 1/(6|τ˜ − τ˜ |3/2). The corrections terms manifestly decrease with  much faster for the Voigt ˜ τ˜ ) is profile than for the Doppler profile. This shows that the leading term S( a better approximation of S (τ ) for the Voigt than for the Doppler profile. For √ ˜ τ˜ ) is in power of  (see Chap. 23). monochromatic scattering the expansion of S(

20.2 Asymptotic Analysis in Fourier Space

473

20.2 Asymptotic Analysis in Fourier Space For an infinite medium, the Fourier transform of the integral equation for S (τ ) takes a simple algebraic form, namely Sˆ (k) =

ˆ ∗ (k) Q ˆ 1 − (1 − )K(k)

(20.21)

,

ˆ where Sˆ (k), Qˆ ∗ (k) and K(k) are the Fourier transforms of S (τ ), Q∗ (τ ) and K(τ ). The limit τ → ∞ in the physical space corresponds to k → 0 in the Fourier space. The limit k → 0 taken in Eq. (20.21) gives the asymptotic behavior of Sˆ (k) for k → 0. The asymptotic behavior of S (τ ) for τ → ∞ is then recovered by performing an inverse Fourier transform. This program is now applied to monochromatic scattering and complete frequency redistribution. We introduce the new Fourier variable k˜ = kτeff ().

(20.22)

It is of order unity when k is order of 1/τeff () and satisfies k˜ τ˜ ∼ kτ . We also introduce the expansion ˜ k), ˜ Sˆ (k)  a()S(

(20.23)

retaining only the leading term. ˆ In Sect. 19.2.3 we show that the Fourier transform K(k) for k → 0 has the form ˆ K(k)  1 − h(|k|)|k|α ,

k → 0,

(20.24)

where h(|k|) is a slowly varying function. The values of α, given in Eq. (19.31), are 2, 1, and 1/2 respectively for the monochromatic, Doppler and Voigt scattering processes. Introducing Eq. (20.24) into Eq. (20.21) we obtain

 ˜ |k| ∗ ˜ ˜ ˜ ˜ a()S(k) = Q (k)  + h τeff

˜α |k| α τeff

−1 ,

(20.25)

where and τeff stands for τeff (). The functions h(k) are given in Eqs. (19.12) and (19.28) for monochromatic scattering and complete frequency redistribution. The scaling factor τeff () must be chosen in such a way that the second term in the square bracket is of order . Balancing these to terms, we recover the scaling

474

20 Large Scale Radiative Transfer Equations

laws in Eqs. (20.9), (20.12), and (20.14). Matching the right and left-hand sides of Eq. (20.25), we recover a() ∼ 1/. We thus obtain the three asymptotic equations ˜ k) ˜ − Q˜ ∗ (k) ˜ = − 1 k˜ 2 S( ˜ k) ˜ for monochromatic scattering, S( 3 ˜ S( ˜ k) ˜ for the Doppler profile, ˜ = − π |k| ˜ k) ˜ − Q˜ ∗ (k) S( 4 √ ∗ ˜ 1/2S( ˜ k) ˜ − Q˜ (k) ˜ = − 2π |k| ˜ k) ˜ for the Voigt profile. S( 3

(20.26) (20.27) (20.28)

The numerical constants can be eliminated by a proper choice of τeff (). It is easy to show that a factor −k 2 in the Fourier space corresponds to a Laplacian  in the physical space. By analogy, a factor |k| can be symbolically associated ˜ τ˜ ) can thus be to (−)1/2 and |k|1/2 to (−)1/4. The large-scale equations for S( written in the very compact form as ˜ τ˜ ) for monochromatic scattering, ˜ τ˜ ) − Q˜ ∗ (τ˜ ) = 1  ˜ S( S( 3 ˜ τ˜ ) − Q˜ ∗ (τ˜ ) = − π (−) ˜ τ˜ ) for the Doppler profile, ˜ 1/2 S( S( 4 √ 2π ∗ ˜ τ˜ ) for the Voigt profile. ˜ ˜ ˜ 1/4 S( S(τ˜ ) − Q (τ˜ ) = − (−) 3

(20.29) (20.30) (20.31)

˜ stands for the Laplacian with respect to variable τ˜ . The presence of a The notation  fractional power of a Laplacian such as (−)1/2 or (−)1/4 in a transport equation is always the signature of an anomalous transport (Metzler and Klafter 2000). One can verify that the Fourier transforms of the large scale asymptotic equations given in Eqs. (20.10), (20.13), and (20.15) lead to Eqs. (20.26), (20.27), and (20.28). The task is very simple for monochromatic scattering, knowing that a second order derivative yields a factor −k 2 . It is slightly more difficult for complete frequency redistribution because the large scale equations are integral equations.

20.3 The Thermalization Length We have shown that the characteristic scale of variation of the radiation field for monochromatic scattering, the Doppler, and Voigt profiles scales with  as 1 M () ∼ √ , τeff 

1 D () ∼ √ τeff ,  − ln 

V () ∼ τeff

a 1/2 . 2

(20.32)

20.3 The Thermalization Length

475

D (), these scaling laws have the general Ignoring the logarithmic correction in τeff form

τeff () ∼  −1/α ,

(20.33)

with α = 2, 1, 1/2, respectively. This result is also established in Chap. 21, where we show that the mean displacement after n steps scales as n1/α . In an infinite medium, where the destruction probability per scattering is , the mean number of scatterings is of order 1/. Setting n = 1/, we recover Eq. (20.33). This scaling factor τeff is known in Astronomy as the thermalization length. This name was introduced in Avrett and Hummer (1965) (see also Hummer and Stewart 1966) in connection with the calculation of the source function in a semi-infinite medium with a primary source B, where B, the Planck function, is a constant. The thermalization length is the distance from the boundary at which the source function S(τ ) approaches B. Figure 20.1 illustrates the scaling laws given in Eq. (20.32) (see also Fig. 7.1). For complete frequency redistribution, Avrett and Hummer (1965) have shown, using a Neumann series expansion of S(τ ), that the thermalization length, which we denote τth (), can be derived from the relation K2 (

τth () ) = , 2

(20.34)

Fig. 20.1 The thermalization length for monochromatic scattering and complete frequency redistribution. The figure shows the source function S(τ ) for a uniform primary source  with  = 10−6 . The thermalization length, distance √ from the surface at which the source function S(τ ) reaches its saturation value, is of order 1/  for monochromatic scattering, 1/ for the Doppler profile, and a/( 2 ) for the Voigt profile

476

20 Large Scale Radiative Transfer Equations

where K2 (τ ) is a primitive of K(τ ) defined by  K2 (τ ) = 2

∞



τ

K(u) du = 1 − 2

K(u) du.

(20.35)

0

τ

At leading order, K2D (τ ) 

1 √ , 2τ ln τ

K2V (τ ) 

2 a 1/2 , 3 τ 1/2

τ → ∞.

(20.36)

The Doppler and Voigt scaling laws for τeff () are recovered when Eq. (20.36) is combined with Eq. (20.34). The question is why the thermalization length, which depends on the full trajectory of the photons between their creation and destruction can be estimated with K(τ ), knowing that K(τ ) describes the effects of a single scattering. The justification proposed in Avrett and Hummer (1965) and Rybicki and Hummer (1969) for using Eq. (20.34) is based on the concept of longest flight. A photon with a frequency x, has a mean free path order of 1/ϕ(x). Photons are mostly created near the line center, where ϕ(x) has its largest values, but because of the frequency redistribution, their frequency can be shifted to the line wings. Wingphotons, having large values of x, make steps much larger than core-photons. Wing-photons will eventually return to the line core. The random displacement after a large number of steps is thus controlled by the longest flight. This situation is illustrated in Fig. 20.2, right panel. Let us show the origin of Eq. (20.34), assuming that K(τ ) can be used to describe the probability density of a random displacement τ after a large number of steps.

Fig. 20.2 Ordinary diffusion and anomalous diffusion. The left panel, shows a diffusive-type random walk. All the random steps have the same constant value. The right panel shows an anomalous random walk. The random steps are calculated with a probability distribution having the form 1/(1 + |x|β+1 ), with β = 1, mimicking the Doppler frequency redistribution

20.4 Ordinary and Anomalous Diffusion

477

The probability that τ takes a value smaller than τ is given by  Prob(τ < τ ) =

τ

−∞

K(u) du,

(20.37)

and the probability that it takes a value larger than τ is  Prob(τ > τ ) = 1 −

τ −∞

K(u) du =

1 K2 (τ ), 2

(20.38)

Balancing this probability with , the destruction probability per scattering, we recover Eq. (20.34), within constant factors. In Chap. 21, we give a proof that the asymptotic behavior of K(τ ), for large-τ , can be used to describe the effects of multiple scatterings, when the second-moment of the kernel is infinite. For monochromatic scattering, the situation is quite different. The kernel K(τ ) behaving as an exponential for large-τ , the random steps have about the same length (Fig. 20.2, left panel). The kernel K(τ ) cannot be used to describe the displacement after a large number of steps, a point already made in Avrett and Hummer (1965). Equation (20.34) does not hold and the thermalization varies as √ 1/  (see Eq. (20.32)).

20.4 Ordinary and Anomalous Diffusion The difference between an ordinary and an anomalous diffusion is illustrated with the 2D-plots in Fig. 20.2. In the left panel all the random steps have the same length. The random walk mimics monochromatic scattering. For monochromatic scattering K(τ ) decreases exponentially for large-τ and has a finite second-moment. All the random steps have about the same length and, as shown in Sect. 20.1, the large scale equation for the source function is a diffusion equation. The random walk is of the ordinary diffusion type and each single step is contributing to the mean properties of the random walk. In the right panel, the probability density of the random steps varies as 1/(1+τ 2 ). It mimics complete frequency redistribution with a Doppler profile, for which the kernel K(τ ) decreases algebraically for large τ as 1/τ 2 and has an infinite secondmoment. Algebraic tails are also referred to as heavy tail. The mean displacement of the particles is controlled by the longest step (flight). With a magnifying glass one would observe that the “small steps” also contain a mixture of short and much longer steps, a characteristic feature of a self-similar random walk. As shown in Sect. 20.1, when the second-moment of the kernel is infinite, the large scale equation for the source function is an integral (nonlocal) equation. Anomalous diffusion is common in physics, economy, finance, biology. Well known to astronomers is the Holtsmark distribution (Holtsmark 1919), which describes the electric field of a distribution of ions and also the gravitational effect

478

20 Large Scale Radiative Transfer Equations

of a distribution of stars. A fairly comprehensive list of domains where anomalous diffusion is encountered can be found in Metzler and Klafter (2000) (see also Feller 1971, p. 173 and Shlesinger et al. 1995). Metzler and Klafter (2000) discuss in detail the relations between anomalous diffusion, fractional derivatives, and fractal dimensions. An important difference between ordinary diffusion and anomalous diffusion has to be mentioned already here. One may wonder why the large scale asymptotic expansion is carried out on the integral equation for source function and not on the radiation field itself. Actually this is possible for monochromatic scattering and leads to a diffusion equation, for the direction-average radiation field (see Chap. 23). In contrast, for complete frequency redistribution, an asymptotic analysis of the radiative transfer equation is not possible. The physical reason is that photons can jump from one place to another over long distances. A more precise explanation will be given in Chap. 23. A property common to the large scale equations constructed for ordinary and anomalous diffusion processes is that they hold only far from boundaries. In Chaps. 23 and 25, we show that it is possible to construct a solution valid in the entire medium by asymptotically matching large scale equations with boundary layer solutions.

References Avrett, E.H., Hummer, D.G.: Non-coherent scattering II: Line formation with a frequency independent source function. Mon. Not. R. astr. Soc. 130, 295–331 (1965) Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York (1978) Bensoussan, A., Lions, J-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. NorthHolland, New York (1978). Reprinted in 2011 by AMS Chelsea Publishing Bensoussan, A., Lions, J-L., Papanicolaou, G.: Boundary layers and homogenization of transport processes. Publ. RIMS, Kyoto Univ. 15, 53–157 (1979) Bommier, V., Landi Degl’Innocenti, E., Sahal-Bréchot, S.: Resonance polarization and the Hanle effect in optically thick media. II. Case of a plane-parallel atmosphere. Astron. Astrophys. 244, 383–390 (1991) Cole, J.D.: Perturbation Methods in Applied Mathematics. Blaisdell, Waltham, Mass (1968) Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2. Wiley, New York (1971) Frisch, U., Frisch, H.: Non-LTE transfer III. Asymptotic expansions for small . Mon. Not. R. astr. Soc. 181, 273–280 (1977) Holtsmark, J.: Über die Verbreiterung von Spektrallinien. Ann. Phys. 363, 577–630 (1919) Hummer, D.G., Stewart, J.C.: Thermalization lengths and the homogeneous transfer equation in line formation. Astrophys. J. 146, 290–294 (1966) Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 339, 1–77 (2000) Rybicki, G., Hummer, D.G.: Non-coherent scattering V. Thermalization distances and their distribution function. Mon. Not. R. astr. Soc. 144, 313–323 (1969) Shlesinger, M., Zaslavsky, G., Frisch, U.: Lévy Flights and Related Topics in Physics. In: Lectures Notes in Physics, vol. 450 (1995). Springer, Berlin

Chapter 21

The Photon Random Walk

In this Chapter we show how some exact and asymptotic properties of the radiation field, derived in preceding chapters from the radiative transfer equation, can be recovered and given a new interpretation by considering the random motions of the photons. We assume that the photons are making a discrete-time random walk on a continuous line. Figure 21.1 shows one realization of the walk. It is a popular random walk model, simple and yet very powerful. After n steps, the random position x n of the walker is given by x n = x n−1 + ξ n ,

n ≥ 1,

(21.1)

where the step lengths ξ n ’s are independent and identically distributed random variables with zero mean, each drawn from the same probability law with a probability density f (x). We assume that f (x) is symmetric (an even function of x). Being a probability density, f (x) is normalized to unity. Random variables are denoted with bold face characters. The probability that ξ n is smaller than a given value x, also referred to as the cumulative probability, is  Prob(ξ n ≤ x) =

x −∞

f (y) dy.

(21.2)

The analogy is obvious between the discrete random walk model and the scattering processes considered in the preceding chapters, which are described by a symmetric kernel K(τ ), normalized to unity, with τ the one-dimensional optical depth variable. It suffices to identify K(τ ) and f (x). The discrete random walk model can describe monochromatic scattering, by choosing a probability density f (x) with a finite second-moment, but also complete frequency redistribution, when the moments of f (x) are infinite. This model is particularly useful to understand the longest flight concept, used to determine the thermalization length for complete frequency redistribution (see Sect. 20.3).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_21

479

480

21 The Photon Random Walk

xn

0

1

2

3

n

Fig. 21.1 A typical discrete random walk on a line. The vertical axis represents the position on the line of a random walker starting at 0. After n steps, the random walker has the position xn

The discrete random walk model has applications to many different systems. One example is the random walk of a bacterium, which in search of food, jumps from one position to another at discrete time steps. Montroll and Shlesinger (1984) present a large variety of examples. This Chapter is organized as follows. In Sect. 21.1 we determine the mean displacement after n steps and show that it provides the thermalization length, τeff (), which was introduced in the preceding chapter as a scaling factor. We describe two different methods, one which is applicable to monochromatic scattering only, a second, more general one, applicable to ordinary and anomalous diffusion processes, hence to monochromatic scattering and complete frequency redistribution. Anomalous diffusion means that the photons are performing a Lévytype random walk, from the name of P. Lévy, who established the theory of random walks with infinite second order moments (Lévy 1937). In Sect. 21.2 we consider another characteristic property of the random walk, namely the mean positive maximum after n steps. We show that it can be derived from the solution of a Wiener–Hopf integral equation and that it corresponds to the mean displacement on a semi-infinite line. Finally, we √ present in Sect. 21.3 and Appendix P in this chapter a probabilistic proof of the -law, which shows that it is a√universal property of symmetric random walks in a semi-infinite medium. The -law is an exact result for semi-infinite media established in Chap. 11. The results established in this section are largely inspired by work from Ivanov and Sabashvili (1972), Frisch and Frisch (1995), Comtet and Majumbar (2005), and Majumdar et al. (2006).

21.1 The Mean Displacement After n Steps

481

21.1 The Mean Displacement After n Steps We examine in this Section the mean displacement after n steps, n being large, with two different methods. The first one, described in Comtet and Majumbar (2005), provides an asymptotic expansion of the mean displacement, with an exact expression for the leading term, but is restricted to monochromatic scattering. The second one, inspired by Ivanov and Sabashvili (1972), provides only the n-dependence of the leading term, but has a broader range of application, as it can handle monochromatic scattering, and also Doppler or Voigt frequency redistribution, that is both ordinary and anomalous diffusion processes.

21.1.1 Mean Displacement: Method I We consider the discrete random walk on a continuous line defined above. The random position x n of the particle after n steps is evolving as shown in Eq. (21.1). We are interested in the expected (mean) absolute, end to end, distance covered by the particle after n steps (in short, mean displacement). It is given by  E{|x n |} =

+∞ −∞

Pn (x)|x| dx,

(21.3)

where Pn (x) is the probability density of the particle to be between x and x+dx after n steps, starting from x0 = 0 at n = 0. Mean values are designated in this Chapter by the symbol E{.}, standing for expected, the name generally employed in probability theory to designate a mean value. This definition of the mean displacement holds when the first-moment of Pn (x) is finite. We first show that Pn (x) satisfies the recursion relation  Pn (x) =

+∞ −∞

Pn−1 (y)f (x − y) dy,

(21.4)

with the initial condition P0 (x) = δ(x). We consider the first step, ξ 1 = x 1 − x0 , with x0 = 0. It has the probability density P1 (x1 ) = f (x1 ), since all the steps are drawn from the same probability law with probability density f (x). The next step brings the particle to x 2 and we write x 2 − x0 = (x 2 − x 1 ) + (x 1 − x0 ).

(21.5)

The two steps being independent,  P2 (x2 ) =

+∞ −∞

 f (x2 − x1 )f (x1 ) dx1 =

+∞ −∞

P1 (x1 )f (x2 − x1 ) dx1.

(21.6)

482

21 The Photon Random Walk

In the same way, we write x n − x0 = (x n − x n−1 ) + (x n−1 − x n−2 ) + · · · + (x 1 − x0 ).

(21.7)

Thus  Pn (xn ) =



+∞  +∞ −∞

...

−∞

+∞ −∞

f (xn − xn−1 f (xn−1 − xn−2 ) . . . (21.8)

f (x1 ) dx1 dx2 . . . dxn−1 . This equation can be rewritten as  Pn (xn ) =

+∞ −∞

Pn−1 (xn−1 )f (xn − xn−1 ) dxn−1 .

(21.9)

Setting xn = x and xn−1 = y, we obtain the recursion relation in Eq. (21.4). The proof of this recursion relation does not make use of the symmetry of f (x). The integration over y being over the range ] − ∞, +∞[, the recursion relation can be solved by a Fourier transformation. We introduce the characteristic function associated to f (x), defined by fˆ(k) ≡



+∞ −∞

f (ξ )ei kξ dξ ≡ E{ei kξ }.

(21.10)

The characteristic function of a random variable ξ is the Fourier transform of its density probability f (ξ ) and also the mean value of the random variable ei kξ . Characteristic functions are particularly useful to investigate the properties of sums of random variables as will be seen.1 We also introduce the Fourier transform of Pn (x): Pˆn (k) ≡



+∞

−∞

Pn (x) ei kx dx.

(21.11)

The recursion relation leads to Pˆn (k) = [fˆ(k)]n ,

(21.12)

and after a Fourier inversion to Pn (x) =

1

1 2π



+∞ −∞

[fˆ(k)]n e−i kx dk.

(21.13)

When a random variable does not have a probability density, the characteristic function may be defined by the Fourier transform of the measure corresponding to the random variable.

21.1 The Mean Displacement After n Steps

483

For a given fˆ(k), one can thus calculate Pn (x) and then E{|x n |} with Eq. (21.3). To find the behavior of E{|x n |} at large n, it suffices to determine the behavior of Pn (x) at large n and large x. The latter is controlled by the behavior of f˜(k) for k → 0. Depending on the asymptotic behavior of Pn (x), the integral in Eq. (21.3) may or may not be convergent. Before we discuss this point, let us show how the random walk is related to the macroscopic description of the preceding chapters, in particular to the resolvent function. We introduce the generating function Rg (x, s), defined by Rg (x, s) ≡

∞ 

Pn (x)s n ,

(21.14)

n=0

with 0 < s ≤ 1. Generating functions are common and powerful tools for studying the asymptotic behavior of discrete random walks. The quantity we are interested in, here Pn (x), is the coefficient of the s n -term in an expansion of Rg (x, s) in powers of s. The determination of Pn (x) is thus reduced to the determination of a function of a continuous variable s and its asymptotic behavior at large n is obtained by considering the limit s → 1. Other generating functions will be met in this Chapter. A multiplication of the recursion relation in Eq. (21.9) by s n followed by a summation over n leads to the integral equation  Rg (x, s) = s

+∞ −∞

Rg (x , s)f (x − x ) dx + δ(x).

(21.15)

Setting s = 1 − , we recognize the integral equation for the infinite medium Green function G∞ (τ ) written in Eq. (2.79). We now introduce the generating function of E{|x n |}, Xg (s) ≡

∞ 

E{|x n |}s n .

(21.16)

n=0

Using the definitions of E{|x n |} in Eq. (21.3) and of Rg (x, s) in Eq. (21.14), we can write  +∞ Xg (s) = Rg (x, s)|x| dx. (21.17) −∞

Introducing the resolvent function ∞ (τ ) = G∞ (τ )−δ(τ ), now denoted ∞ (τ, s), we obtain  +∞ Xg (s) = ∞ (τ, s)|τ | dτ. (21.18) −∞

484

21 The Photon Random Walk

The notation x has been changed to τ . Equation (21.18) shows that of Xg (s) can be identified with the first-moment of the resolvent function (calculated with |τ |). The convergence of the integral in Eq. (21.18) depends on the behavior of ∞ (τ, s) for τ → ∞. The latter actually follows the behavior of the kernel K(τ ) (see Chap. 22). Hence, ∞ (τ, s) √decreases exponentially for monochromatic scattering and algebraically as 1/(τ 2 ln τ ) and 1/τ 3/2 for the Doppler and Voigt frequency redistribution, respectively. For lack of convergence, Eqs. (21.18) and (21.3) cannot be used to define the mean displacement for the Doppler and Voigt frequency redistributions but can be used for monochromatic scattering. For monochromatic scattering, E{|x n |} is derived in Comtet and Majumbar (2005) from Eq. (21.17), combined with the expressions of Rg (x, s) and Pn (x) given by Eqs. (21.14) and (21.13). Their result, for a probability density with a finite second-moment, is reproduced in Eq. (21.31). Here we show how to use the exact expression of ∞ (τ, s). As shown in Sect. 6.1.1.2, for monochromatic scattering,  ∞ (τ, s) =

∞ 0

φ∞ (ν, s) e−ντ dτ,

τ ∈ [0, ∞[,

(21.19)

with 0 c δ(ν − ν0 ) + φ∞ (ν, s), φ∞ (ν, s) = φ∞

(21.20)

and ν0 ∈ [0, 1[, the positive root of the dispersion function L(z) = V (i z). To determine the asymptotic behavior of E{|x n |} for n large, it suffices to consider the 0 δ(ν − ν ), which controls the behavior of the resolvent function at discrete term φ∞ 0 infinity. As shown in Eq. (6.11), 0 φ∞ =

ν0 (1 − ν02 ) ν02 − 

.

(21.21)

0 is related to the derivative of the dispersion function at ν .) (The value of φ∞ 0 Equation (21.18) leads thus to

Xg (s) 

2 1 − ν02 . ν0 ν02 − 

(21.22)

For  → 0, that is s → 1, ν0 

√ 3 + O( 3/2 ).

(21.23)

Replacing ν0 by its asymptotic expansion and returning to the variable s = 1 − , we obtain 1 Xg (s) = √ (1 − s)−3/2 + O((1 − s)−1/2 ). 3

(21.24)

21.1 The Mean Displacement After n Steps

485

To calculate E{|x n |}, we must still find the coefficient of s n . It can be derived from the general formula for a negative fractional binomial,2 namely (1 − x)−r =

∞  (r)n n=0

n!

x n,

(r + n) . (r)

(r)n ≡

(21.25)

The expansion coefficient (r)n is known as the Pochammer symbol (Abramovitz and Stegun, p. 256). Here r = 3/2 for the leading term and r = 1/2 for the following one. Using the Stirling formula, (n) 

& n 'n $ 2π e

n

n → ∞,

,

(21.26)

valid for large n, and n! = (n + 1), we find nr−1 (r)n  . n! (r) Using (3/2) =

√

(21.27)

π/2, we finally obtain

2 1 √ 1 n + O( √ ). E{|x n |} = √ √ (21.28) n 3 π √ √ The leading term behaves as n, while the second term tends to√zero as 1/ n. Setting n = 1/, we recover the thermalization length τeff () ∼ 1/ . The method described in Comtet and Majumbar (2005) is based on Eq. (21.13), in which fˆ(k), the characteristic function of the probability density Pn (x), has the form σ 2 2 μ4 4 fˆ(k)  1 − k + k + O(k 6 ), 2 24

k → 0.

(21.29)

The constants σ 2 and μ4 , second-moment and fourth-moment of f (ξ ), are defined by  σ2 =

+∞ −∞

 ξ 2 f (ξ ) dξ,

μ4 =

+∞

−∞

ξ 4 f (ξ ) dξ.

(21.30)

The result of Comtet and Majumbar (2005, Eq. (54)) is $ E{|x n |} = σ

2

1 μ4 − 3σ 4 1 2n 1 − √ √ + O( 3/2 ). π σ3 n n 12 2π

Weisstein, E., https://mathworld.wolfram.com/BinomialSeries.html.

(21.31)

486

21 The Photon Random Walk

For monochromatic scattering, fˆ(k) has the form written in Eq. (21.29) with σ 2 /2 = 1/3 (see Eq. (19.12)). Using σ 2 = 2/3, we can check that the leading terms in Eqs. (21.28) √ and (21.31) are identical. To obtain an exact expression of the term of order 1/ n in Eq. (21.28), the expansion of ν0 in Eq. (21.23) must be pushed to the order  3/2 .

21.1.2 Mean Displacement: Method II We now describe a different method for the calculation of the mean displacement. It is simpler than the method presented above, and is applicable to ordinary and anomalous diffusion processes, but provides only the power-law of the leading term. This is however sufficient to determine thermalization lengths. Keeping the same discrete random walk model, we introduce the normalized random sum Xn = (ξ 1 + ξ 2 + . . . + ξ n )/cn ,

(21.32)

where cn is an unknown positive constant. We now show how to choose cn so that it provides, for large n, a scaling law for the mean displacement. The determination of cn is based on the characteristic function of the sum Xn . Since the ξ i ’s are independent random variables with the same probability density f (ξ ), hence with the same characteristic function fˆ(k) (see Eq. (21.10)), the characteristic function of Xn has the form  n n (k) = fˆ( k ) . F cn

(21.33)

The probability density pn (x) of the sum Xn is the inverse Fourier transform of n (k), namely F 1 pn (x) = 2π



+∞ −∞

n (k)e−i kx dk. F

(21.34)

The large scale behavior of the random walk, which can be observed after a large n (k) for k → 0 number of steps, is thus determined by the asymptotic behavior of F and the latter can be derived from the behavior of fˆ(k) for k → 0, as shown by Eq. (21.33). We assume fˆ(k)  1 − h(|k|)|k|α ,

α > 0,

(21.35)

with h(|k|) a constant or a slowly varying function function of k, behaving as shown in Eq. (19.30). For monochromatic scattering α = 2, for the Doppler

21.1 The Mean Displacement After n Steps

487

frequency redistribution α = 1, and for the Voigt frequency redistribution α = 1/2 (Eq. (19.31)). For density probabilities f (ξ ) with a finite second-moment, α = 2. Henceforth, we assume 0 < α ≤ 2. Replacing fˆ(|k|) by its asymptotic expression for k → 0, we readily obtain n (k)  1 − n h( |k| )|k|α , F cnα cn

k → 0.

(21.36)

For Fˆn (k) to describe the asymptotic behavior of the random walk, the righthand side in Eq. (21.36) should have a finite limit when n → ∞. Assuming for simplicity that h(|k|) is a constant C, actually a frequently encountered situation, this constraint leads to cn ∼ n1/α ,

(21.37)

and hence to n (k)  1 − C|k|α , lim F

n→∞

k → 0.

(21.38)

Replacing n by 1/, we see that cn scales as  −1/α . We recover the scaling laws for τeff (), the large scale of variation of the radiation field, given in Eq. (20.33). The thermalization length and the mean displacement after n = 1/ scatterings are thus one and the same thing. It is physically an obvious result, proved here rigorously.

21.1.3 Application to Normal Diffusion and Lévy Walks We examine now in some detail the consequences of the general results established in Eqs. (21.37) and (21.38), for monochromatic scattering and complete frequency redistribution. The role of f (ξ ) is played by the kernel K(τ ), the Fourier transform of which is determined in Sect. 19.2.3. For monochromatic scattering, σ2 2 1 k = 1 − k2, fˆ(k)  1 − 2 3

k → 0.

(21.39)

Here σ 2 is the variance of the random walk and the second-moment of K(τ ). Thus, α = 2 and cn ∼

√ n.

(21.40)

Setting n = 1/, with  the destruction probability per scattering, we recover the √ thermalization length τeff ∼ 1/ .

488

21 The Photon Random Walk

For monochromatic scattering, the variance σ 2 being finite, it is possible to apply the central limit theorem for the sum of random variables. It states that the sum of n independently√and identically distributed random variables with finite variance, normalized by n, tends to a Gaussian (normal) distribution with variance σ 2 as n → ∞ (e.g., Papoulis 1965). For monochromatic scattering, pn (x), the probability density of the sum Xn , tends thus to a Gaussian for n → ∞. That is x2 1 lim pn (x) = √ exp[− 2 ]. n→∞ 2σ 2πσ

(21.41)

We note that the Gaussian behavior of pn (x) differs from the exponential behavior of K(τ ), the probability density of the individual steps. For complete frequency redistribution with a Voigt profile, fˆ(k)  1 − a

1/2

√ 2π 1/2 |k| , 3

k → 0.

(21.42)

The Voigt parameter a is a constant, but it may take values much smaller than unity. Including it into cn , we obtain cn ∼ an2 ,

(21.43)

and √ n (k) = 1 − lim F

n→∞

2π 1/2 |k| . 3

(21.44)

For the Doppler profile, π 1 fˆ(k)  1 − √ |k|, 4 − ln(|k|)

k → 0.

(21.45)

When the logarithmic term is neglected, Eq. (21.36) leads to α = 1 and cn ∼ n. To take the logarithmic term in Eq. (21.45) into account, we remark that the logarithm is a slowly varying function according to the definition given in Eq. (19.30). Thus for cn → ∞, h(

1 π 1 |k| )  h( )  √ . cn cn 4 ln cn

(21.46)

n n cn ∼ √ ∼ √ , ln cn ln n

(21.47)

We thus obtain

21.1 The Mean Displacement After n Steps

489

and n (k) = 1 − π |k|, lim F 4

n→∞

k → 0.

(21.48)

Setting n = 1/ in the scaling laws for cn we √ recover the thermalization lengths, τeff ∼ a/ 2 for the Voigt profile and τeff ∼ 1/( ln −) for the Voigt profile. For the Doppler and Voigt profiles, the second-moment σ 2 being infinite, the central limit theorem is not applicable. The probability density pn (x) is not a Gaussian, but its asymptotic behavior for n large can be derived from the expression n (k) given in Eq. (21.38). In the limit k → 0, the right-hand side in this equation of F can be considered as the two first terms in an expansion of exp(−C|k|α ) for k → 0. The inverse Fourier transformation leads to pn (x) ∼

1 x (α+1)

,

for n → ∞, x → ∞, 0 < α < 2.

(21.49)

This algebraic behavior of pn (x), with α strictly smaller than 2, is typical of Lévytype random walks. When α = 2, the inverse Fourier transform is a Gaussian. For the Doppler and Voigt frequency redistribution, the individual steps have a probability density given by the kernel K(τ ), which, asymptotically for large τ , behaves as KD (τ ) ∼

1 , τ2

KV (τ ) ∼

1 , τ 3/2

τ → 0.

(21.50)

(See Eqs. (19.18) and (19.21).) Setting α = 1 and α = 1/2 in Eq. (21.49), we see that the probability density pn (x) of the normalized sum Xn has asymptotically, the same power-law as that of the individual steps. This is a general property of random walks with infinite second-moments, known as Lévy-type random walks (see, e.g., Shlesinger et al. 1995). It explains why the knowledge of the asymptotic behavior of K(τ ) is sufficient to determine the thermalization length and why the mean displacement of the photons during their life-time is controlled by a single long step (see the remark made in Sect. 20.3). Figures 20.2 and 21.2 show three Lévy-type random walks with a probability density of the form f (x) =

1 1 . A 1 + |x|β+1

(21.51)

Here A is the integral of f (x) from −∞ to −∞. The values chosen for β are β = 2 and β = 1.5 in Fig. 21.2 and β = 1 in Fig. 20.2. For these values of β, the secondmoment is infinite. Because of the algebraic behavior of f (x), different realizations, for a same value of β, can display very significant differences. Lévy-type random walks belong to the family of stable distributions, introduced by Lévy (1937). A precise definition of a stable distribution can be found in Feller (1971, p. 448). Loosely speaking, a distribution is said to be stable when the sum of two or several random variables with the same probability law P (x) also has the

490

21 The Photon Random Walk

Fig. 21.2 Examples of Lévy walks. The probability distribution has the form 1/(1 + |x|β+1 ). In the left panel, β = 2 and in the right panel β = 1.5, which is a model for a Holtsmark distribution. The case β = 1 is shown in the right panel of Fig. 20.2. The particles are starting at the origin (0,0). Each graphic shows 104 steps

probability law P (x).3 The parameter α is known as the stability parameter. The Gaussian distribution is a special case of stable distribution. It is the only stable distribution with a finite variance. The close relation between photon random walks and stable distributions is stressed in Ivanov and Sabashvili (1972) and in van Trigt (1969), where a proof is given that complete frequency redistribution cannot be described by a Gaussian diffusion process. Stable distributions or Lévy walks appear in many domains of physics, in biology, in financial mathematics, etc.. Several examples of Lévy walks in Physics can be found, e.g., in Bouchaud and Georges (1990) and Shlesinger et al. (1995). In the limit n → ∞, Xn can be considered as a random function of a continuous parameter t, say, the time. Identifying n with the time t and introducing the mean square-distance d 2 (t) covered over a time t, Eq. (21.37) leads to d 2 (t) ∼ t 2/α .

(21.52)

For α = 2, one recovers the ordinary diffusion law, stating that the mean squaredistance traveled by a particle grows linearly with time. For superdiffusion, that is when 0 < α < 2, this distance grows faster than linearly because it is actually controlled by the more rare but longer flights. For the Doppler profile it grows as t 2 and for the Voigt profile as t 4 .

A random variable x is said to be stable (or to have a stable distribution) when the sum ax 1 +bx 2 , where a and b are two strictly positive constants and x 1 and x 2 two independent copies of x, has the same distribution as cx + d, where c and d are constants, with c strictly positive (cf. https://en. wikipedia.org/wiki/Stable-distribution).

3

21.2 The Mean Positive Maximum After n Steps

491

The properties that are described here for a one-dimensional random walk, hold also for a three-dimensional random walk. As shown in, e.g., (Feller 1971, p. 31), the probability density p3 (x) of the length of a random vector and the probability density p1 (x) of the length of its projection on a line are related by p3 (x) = −x

dp1 (x) . dx

(21.53)

Thus when p1 (x) ∼ 1/x β for x → ∞, then p3 (x) also has the asymptotic behavior 1/x β . Similarly, when p1 (x) has a Gaussian distribution, then p3 (x) is also a Gaussian. In a one-dimensional medium (plane-parallel medium with a cylindrically symmetry), the photons are actually making a three-dimensional random walk. Equation (21.53) is another justification for the assumption that they are random walking on a line.

21.2 The Mean Positive Maximum After n Steps We consider the same random walk as in the preceding section, but ask a different question. What is the asymptotic behavior of the mean positive maximum after n steps? It is an important question for financial mathematics, queues and waiting times problems. It is a more difficult question than that of the mean displacement treated in Sect. 21.1.1. As will be shown, it amounts to calculate the mean displacement on a semi-infinite line. Following the treatment by Comtet and Majumbar (2005) and Majumdar et al. (2006), we first show that the determination of the mean maximum leads to a convolution equation on a half-line, that is to a Wiener–Hopf integral equation, and then establish the analogy with the mean displacement of a random walker on a semi-infinite line. We assume that the random walk starts at x0 = 0 and introduce M n , the positive maximum of the random walk up to n steps, defined by M n ≡ max(0, x 1 , x 2 , . . . , x n ).

(21.54)

This maximum is a random variable. The problem, treated in the next subsections, is to find the asymptotic large n behavior of the mean (expected) value of M n , denoted E{M n } (see Fig. 21.3).

21.2.1 A Wiener–Hopf Integral Equation We introduce the probability Qn (x, y), that, starting at x, the maximum of the walk, up to n steps, is less or equal to y (see Fig. 21.3). Given that we are considering a

492

21 The Photon Random Walk

Fig. 21.3 A typical random walk on a line, for a random walker starting at the point x. Mn indicates the maximum position reached while performing n steps. The first step brings the random walker from x to x1 , x1 ≤ y

y Mn x1 x 0

1

2

n

3

random walk starting at x = 0, the cumulative probability of M n is given by Prob[M n ≤ y] = Qn (0, y).

(21.55)

The probability density of M n is given by Q n (0, y) = dQn (0, y)/dy and the mean value E{M n } by  +∞ E{M n } = Q n (0, y) y dy. (21.56) −∞

We now show that Qn (x, y) satisfies a recursion relation, which can then be recast as a Wiener–Hopf integral equation. To determine Qn (x, y), we proceed in much the same way as in Sect. 21.1.1. We assume that the random walk is starting at x. We consider the first step, x 1 − x. The probability density of this first step is f (x1 −x). The definition of Qn (x, y) imposes that x and x1 are smaller than y. Hence,  Q1 (x, y) =

y −∞

f (x1 − x) dx1.

(21.57)

We remark that Q1 (x, y) depends only on the difference y − x. We now consider the two first steps, which bring the particle to x 2 . Making the same decomposition as in Eq. (21.5), and imposing the condition that x1 and x2 be smaller than y, we find  Q2 (x, y) =

y



y

−∞ −∞

 f (x2 − x1 )f (x1 − x) dx1 dx2 =

y

−∞

Q1 (x1 , y)f (x1 − x) dx1 .

(21.58)

21.2 The Mean Positive Maximum After n Steps

493

After n steps, the particle has reached the position x n . A decomposition in elementary steps, as in Eq. (21.7), leads to  Qn (x, y) =



y



y

−∞ −∞

...

y −∞

f (xn − xn−1 f (xn−1 − xn−2 ) . . .

f (x1 − x) dx1 dx2 . . . dxn−1 dxn .

(21.59)

This expression can be written as  Qn (x, y) =

y −∞

Qn−1 (x1 , y)f (x1 − x) dx1,

(21.60)

with the initial condition Q0 (x, y) = Y (y − x), where Y (z) is the Heaviside step function. This recursion relation shows that Qn (x, y) depends only the difference (y − x). Introducing z = y − x, z = y − x1 , and qn (z) ≡ Qn (x, y),

z ≥ 0,

we can write the recursion relation as  ∞ qn−1 (z )f (z − z ) dz , qn (z) =

(21.61)

(21.62)

0

valid for all z ≥ 0 and starting with q0 (z) = Y (z). Equation (21.55) then leads to Prob[M n ≤ y] = Qn (0, y) = qn (y),

y ≥ 0.

(21.63)

We see here that the determination of the mean maximum leads to a semi-infinite medium problem. The mean positive maximum E{M n } is given by 

∞

E{M n } = 0

qn (z)z dz,

(21.64)

where qn (z) = dqn /dz is the probability density of M n . We wish to point out here a difference between the method used to find the mean maximum and the method used to find the mean displacement. To establish the recurrence relation in Eq. (21.60), we have considered the cumulative probability Q(x, y), while the recurrence relation in Eq. (21.4) is established with the density probability Pn (x). Cumulative probabilities are needed when looking at questions of maximum or records.

494

21 The Photon Random Walk

We have announced that the problem of finding the mean expected maximum can be expressed as a Wiener–Hopf integral equation. It may indeed be obtained by introducing the generating function, Yg (z, s) ≡

∞ 

qn (z)s n .

(21.65)

n=0

A multiplication of the recursion relation by s n and a summation over n leads to the inhomogeneous Wiener–Hopf integral equation  Yg (z, s) = s

∞

Yg (z , s)f (z − z ) dz + q0 (z),

(21.66)

0

with q0 (z) = Y (z) = 1. This equation holds for any density probability f (x). It does not have to be symmetric. When f (x) is symmetric, Eq. (21.66) is identical to the integral equation for the source function S(τ ) in Eq. (2.69). Here s plays the part of (1 − ), f (x) that of the kernel K(τ ), and q0 (z) that of the primary source term. Given that q0 (z) = 1, we can identify Yg (z, s) with the source function S(τ ) for a semi-infinite medium with a uniform primary source. It is shown in Sect. 7.3.1, that S(τ ) can then be written as   S(τ ) = S(0) 1 +

τ

 (τ ) dτ ,

(21.67)

0

where (τ ) is the resolvent function for a semi-infinite medium √ and S(0) is the surface value of the source function. It has the exact value 1/  for any kernel normalized to unity and a uniform primary source equal to 1. The interpretation of S(τ ) as a generating function is pointed out in Sect. 2.3. We now make use of Eq. (21.67) to find the large n behavior of E{M n }.

21.2.2 A Precise Asymptotics for the Mean Maximum We introduce the generating function ∞ 

Mg (s) ≡

E{M n }s n .

(21.68)

n=0

Using Eqs. (21.64) and (21.65), we can write  Mg (s) = 0

∞

Yg (z, s)z dz,

(21.69)

21.2 The Mean Positive Maximum After n Steps

495

where Yg (z, s) is the derivative of Yg (z, s) with respect to z. Identifying Yg (z, s) with S(τ ), and using the expression of S(τ ) given in Eq. (21.67), we see that Yg (z, s) can be identified with S(0)(τ ). Changing the notation τ to z and (τ ) to (τ, s) = (z, s), we can write 1 Yg (z, s) = √ (z, s). 1−s

(21.70)

We thus obtain 1 Mg (s) = √ 1−s



∞

(z, s)z dz.

(21.71)

0

We can observe that Mg (s), the generating function of E{M n }, and Xg (s), the generating function of E{|x|n }, are both given by the first-moment of the resolvent function, the half-space first-moment for E{M n } and the full-space one for E{|x|n }. Thus, as announced, the calculation of the mean maximum E{M n }, amounts to determine the mean displacement over a semi-infinite line. The resolvent function (τ, s) for τ -large behaves as K(τ ), as shown in Chap. 22. Therefore, the integral in Eq. (21.71) converges for monochromatic scattering, but not for the Doppler or Voigt complete frequency redistribution kernels. For Lévy walks with 1 < α < 2, an asymptotic expansion of Mg (s) can be constructed with the method of Comtet and Majumbar (2005) described in Sect. 21.2.2.2. We now focus on monochromatic scattering.

21.2.2.1 The Mean Maximum for Monochromatic Scattering For monochromatic scattering, E{M n } can be calculated with the exact expression of (τ, s) (alias (τ )) given in Eq. (6.47), namely 1 e−ν0 τ + (τ ) = X(ν0 )



∞ 1

c φ∞ (ν)(ν0 + ν)X(−ν)e−ντ dτ,

τ ∈ [0, ∞[. (21.72)

The function X(z) is the auxiliary function for a semi-infinite medium defined in c (ν, s) is defined in Eq. (21.20). The behavior of E{M } Chap. 5. The function φ∞ n for large n is controlled by the discrete term. Equation (21.71) thus leads to 1 1 . Mg (s)  √ 1 − s ν02 X(ν0 )

(21.73)

496

21 The Photon Random Walk

In the limit s → 1, that is  → 0, the constant ν0 tends to zero as We can thus replace X(ν0 ) by its Taylor expansion, X(ν0 )  X(0) + ν0

dX(ν) |ν=0 , dν

√ √ 3 = 3(1 − s).

(21.74)

and use for X(0) and its derivative X (0) their values for conservative scattering ( = 0), namely 1 X(0) = √ , 3

X (0) =

1 2



1

H (μ)μ2 dμ.

(21.75)

0

(See Eqs. (9.15) and (G.15).) Here H (μ) is the conservative H -function for monochromatic scattering. Inserting the expansion of X(ν0 ) into Eq. (21.73), we obtain √ 1 3 1 1 1 α2 + O( − ), (21.76) Mg (s)  √ 3/2 2 (1 − s) (1 − s)1/2 3 (1 − s) where 

1

α2 =

H (μ)μ2 dμ.

(21.77)

0

The coefficient of s n in the expansion of Mg (s) can be determined with the binomial formula in Eq. (21.25). We thus obtain √ 3 2 1 √ 1 E{M n )  √ √ α2 + O( √ ). n− 2 n π 3

(21.78)

Comparing this result with the power-law for E{|x n |} given in Eq. (21.28), we can observe that E{|x n |} and E{M n } have the same leading terms, but differ by the first correction, the latter having a constant negative value for E{M n }. The most plausible interpretation for the negative value is that the calculation of E{M n ) amounts to determine the mean displacement on a semi-infinite line. For monochromatic scattering, there is an interesting remark, which can be made. For the Milne problem, as shown in Sect. 9.1.3, the source function may be written as S(τ ) =

3 F [τ + q(τ )], 4π

(21.79)

where F is the radiative flux. The function q(τ ), which is √ known as the Hopf function, varies monotonically with τ between q(0) = 1/ 3 = 0.5773 and √ q(∞) = 3α2 /2 = 0.7104. Except for the minus sign, the constant term in

21.2 The Mean Positive Maximum After n Steps

497

Eq. (21.78) is given by the value of the Hopf function at infinity. A high precision value of q(∞) (up to 10 significant digits) can be calculated with the value of α2 given in Bosma and de Rooij (1983). 21.2.2.2 The Mean Maximum for Lévy Walks A general method to determine E{M n }, applicable to any random walk with a characteristic function of the form fˆ(k) = 1 − (a|k|)α + . . . ,

1 < α ≤ 2,

k → 0,

(21.80)

is described in Comtet and Majumbar (2005). We present here the main lines of their proof. The assumption that α is in the range ]1, 2] ensures that the first-moment of f (x) (calculated with |x|) remains finite. We stress that the results in this subsection are not applicable to the Doppler and Voigt scattering processes, since their values of α are 1 and 1/2, but are applicable to the Holtsmark distribution, for which α = 1.5. We introduce the Laplace transform E{e

−pM n



∞

}≡ 0

e−pz qn (z) dz,

(21.81)

and its generating function Ng (p, s) ≡

∞ 

s E{e n

−pM n

 }= 0

n=0

∞

e−pz Yg (z, s) dz,

(21.82)

where Yg (z, s) is the generating function of qn (z) (see Eq. (21.65)). Taking the derivative of Ng (p, s) with respect to p and then the limit p → 0, we obtain ∂Ng (p, s) =− p→0 ∂p



∞

lim

0

Yg (z, s) z dz = −Mg (s),

(21.83)

where Mg (s) is the generating function of E{M n } (see Eqs. (21.68) and (21.69)). Introducing the surface Green function, denoted G(τ ) in preceding Chapters and here G(z, s), and related to the resolvent function (z, s) by G(z, s) = (z, s) + δ(z), we can rewrite Eq. (21.70) as Yg (z, s) = √

1 1−s

[G(z, s) − δ(z)].

(21.84)

Taking the Laplace transform of this equation and using Eq. (21.82), we obtain Ng (p, s) = √

1 1−s

˜ [G(p, s) − 1],

(21.85)

498

21 The Photon Random Walk

˜ where G(p, s) is the Laplace transform of G(z, s). Equation (21.83) thus leads to ˜ ∂Ng (p, s) 1 ∂ G(p, s) = −√ . lim p→0 ∂p ∂p 1 − s p→0

Mg (s) = − lim

(21.86)

˜ We know from preceding chapters that G(p, s) can serve to define the H -function. ˜ Dropping the dependence on s, the relation is G(p) = H (1/p). The exact expression of the H -function in Eq. (5.72) leads to    p ∞ dk ˜ G(p, s) = exp − , ln V (k) 2 π 0 k + p2

(21.87)

with V (k), the dispersion function, given by V (k) = 1 − s fˆ(k).

(21.88)

˜ We note that Eq. (21.85), with G(p, s) given by Eq. (21.87), is known as the Pollaczek–Spitzer formula. It is based on work by Spitzer (1956, 1957) on Wiener– Hopf integral equations, whose kernel is a probability density, and by Pollaczek (1952) on random walks. Equations (21.68) and (21.86) show that the problem of finding an asymptotic expansion of E{M n } at large n is reduced to the calculation of the derivative of ˜ G(p, s) for s → 1 and p → 0. This double expansion is carried out in Comtet and Majumbar (2005) for the function fˆ(k) defined in Eq. (21.80). The authors show that the large n asymptotic behavior of E{M n } has the form E{M n } 

1 aα n1/α (1 − ) n1/α + γ + O( ), π α n

(21.89)

where 1 γ = π



∞ 0



1 − fˆ(k) ln (ak)α



dk . k2

(21.90)

√ This result can be√applied to monochromatic scattering. Setting α = 2, a = 1/ 3, and (1/2) = π, we recover Eq. (21.78), with the constant term containing explicitly the dispersion function V (k). It is shown in Comtet and Majumbar (2005) that the constant term γ is always negative and precise numerical values of γ are given in this article for several probability densities, but not for the density probability corresponding to monochromatic scattering. Precision is needed for applications in, e.g., algorithmic problems (see references in Comtet and Majumbar 2005). An explanation is suggested in Sect. 21.2.2.1 for the negative value of the first correction.

21.3 A Probabilistic Proof of the

√

499

-Law

We give here some indications about the method used in Comtet and Majumbar ˜ (2005) to determine the asymptotic expansion of the derivative of G(p, s) for p → 0 and s → 1 (see Eq. (21.86)). For small values of p, the main contribution to the integral in Eq. (21.87) comes from small values of k for which fˆ(k)  1 − (ak)α . Thus for small k, V (k)   + (1 − )(ak)α ,

(21.91)

where  = 1 − s, is also a small parameter. This expression can be used to construct the first term of the expansion. To obtain the constant γ , V (k) is rewritten as V (k) = [ + (1 − )(ak)α ]

V (k) ,  + (1 − )(ak)α

(21.92)

the idea being to isolated the most singular term, and the logarithm of V (k) is written as ln V (k) = ln[ + (1 − )(ak)α ] + ln

V (k) .  + (1 − )(ak)α

(21.93)

The second term in the right-hand side provides the constant γ .

21.3 A Probabilistic Proof of the

√ -Law

√ The -law is an exact result for scalar radiative transfer. It states that the surface value of the source function in a semi-infinite medium, with a constant primary source B has the value √ S(0) = B, (21.94) for any even kernel K(τ ) normalized to unity and any positive value of , strictly smaller than unity. We recall that  is the destruction probability per scattering. Several proofs of this result are given in Part I, and generalized versions of it for linear polarization in Part II. They either make use of the exact value at the origin of the auxiliary function X(z) or are derived by algebraic manipulations √ from the scalar or vector integral equations for the source function. Actually, the -law is a fundamental property of symmetric random walks in a semi-infinite medium, as we show in the present section with the proof given in Frisch and Frisch (1995). We consider the same discrete random walk as in the preceding sections and introduce the probability pn ≡ Prob[x 1 > 0, x 2 > 0, . . . , x n > 0], where the x i ’s are defined as in Eq. (21.1) and f (ξ ) is the kernel K(τ ).

(21.95)

500

21 The Photon Random Walk

In Appendix P of this chapter we show that pn is a universal function of n, independent of the functional form of K(τ ). More precisely, we establish that p(λ), the generating function of pn , defined by p(λ) ≡

∞ 

pn λn ,

(21.96)

n=1

with 0 < λ < 1, has the explicit expression 1 p(λ) = √ − 1. 1−λ

(21.97)

The variable λ plays the role of the variable denoted s in the preceding sections, but we prefer to keep here the notation used in Frisch and Frisch (1995). An expansion of p(λ) in powers of λ leads then to pn =

(2n)! 22n (n!)2

.

(21.98)

This expression, √ written as (2n − 1)!!/(2n)!!, can be found in a partly heuristic proof of the -law by Landi Degl’Innocenti (1979) based on a Neumann series expansion of the integral equation for the source function. For n → ∞, the Stirling formula gives 1 1 pn  √ √ . π n

(21.99)

The result that p(λ) is independent of the kernel was discovered by Sparre Andersen (1953) with an “ingenious but extremely complicated proof”, as stated by Feller (1971, p. 413). Simpler proofs, but still needing an important background in probability theory can be found in Feller (1971, Chapter XII, §7; Chapter XVIII). The essentially algebraic proof given in Appendix P of this chapter is significantly simpler than earlier proofs. It is independent of the value of the stability index α, hence valid for ordinary random walks with finite variance and for Lévy walks. The universality law in Eq. (21.97) has implications in radiative transfer but in other fields, such as cosmology, and more specifically on the mass distribution of largescale structures (Vergassola√ et al. 1994). We now show how the -law is related to the exact expression of p(λ). We introduce the probability density Pn (x) defined by Pn (x) dx ≡ Prob[x 1 > 0, x 2 > 0, . . . , x n > 0, x ≤ x n ≤ x + dx].

(21.100)

21.3 A Probabilistic Proof of the

√

501

-Law

By definition, Pn (x) is the probability density of the random walker to be in the interval [x, x + dx] after n steps, without having visited negative locations before. The probability density Pn (x) and the probability pn are related by 

∞

pn =

(21.101)

Pn (x) dx. 0

The probability density Pn (x) satisfies the recursion relation  P1 (x) = K(x),

∞

Pn (x) =

K(x − y)Pn−1 (y) dy,

n > 1.

(21.102)

0

It can be established with the method leading to Eqs. (21.9) and (21.60), described in Sects. 21.1.1 and 21.2.1. Equation (21.102) can then be transformed into the Wiener–Hopf integral equation 

∞

Pg (x, λ) = λK(x) + λ

K(x − y)Pg (y, λ) dy,

(21.103)

0

for the generating function Pg (x, λ) ≡

∞ 

Pn (x)λn .

(21.104)

n=1

The technique is the same as in Sects. 21.1.1 and 21.2.1: we multiply the recursion relation by λn and sum all the terms. We recognize in Eq. (21.103) the integral equation for the resolvent function (τ ) written in Eq. (2.84), with λ replacing the factor (1 − ). For a semi-infinite medium with a constant primary source, as already stated in Eq. (21.67), 

τ

S(τ ) = S(0)[1 +

(τ ) dτ ].

(21.105)

0

For a constant primary source B, the source function at infinity takes the value S(∞) = B. Hence   S(0) = B 1 +

∞



(τ ) dτ



−1

  =B 1+

0

−1

∞

Pg (x, λ) dx

.

(21.106)

0

Using now Eqs. (21.101) and (21.104), we can write 

∞ 0

Pg (x, λ) dx =

∞  n=1



∞

λn 0

Pn (x) dx =

∞  n=1

λn pn = p(λ).

(21.107)

502

21 The Photon Random Walk

The exact expression of p(λ) given in Eq. (21.97) leads to S(0) =

√ √ B = 1 − λ B =  B. 1 + p(λ)

(21.108)

√ The probabilistic proof of the -law given here, clearly shows, that in contrast to what has sometimes √ been claimed, this law has nothing to do with a diffusion process. Generalized -laws hold for Rayleigh scattering and resonance polarization (see Sect. 15.2). How this probabilistic approach can be applied to these scattering processes remains to be investigated.

Appendix P: Universality of Escape from a Half-Space In Sect. 21.3 we consider a discrete random walk, on a half-line [0, ∞[, starting at zero, with random independent steps, all having a same symmetric probability density K(x). We introduce the probability pn , that up to the nth-step, the positions x i of the random walker stay positive, that is such that x i ≥ 0 (see Eq. (21.95)) and the probability density Pn (x) of the random walker to be in the interval [x, x + dx] after n steps, without having visited negative locations before (see Eq. (21.100)). They satisfy 

∞

Pn (x) dx = pn .

(P.1)

0

In this Appendix we give a proof of Eq. (21.97), stating that p(λ), the generating function of the probability pn , defined by p(λ) =

∞ 

pn λn ,

(P.2)

n=1

has the exact expression 1 p(λ) = √ − 1. 1−λ

(P.3)

We now essentially follow the algebraic proof given in Frisch and Frisch (1995). As shown in Sect. 21.3, the probability density Pn (x) satisfies the recursion relation  ∞ P1 (x) = K(x), Pn (x) = K(x − y)Pn−1 (y) dy, n > 1. (P.4) 0

Appendix P: Universality of Escape from a Half-Space

503

In somewhat the same spirit as in Sect. 11.1.1, we introduce a quadratic quantity: Qn ≡

 



∞

x

Ps (x)

Pr (z) dz dx,

r+s=n 0

n ≥ 2,

r, s ≥ 1.

(P.5)

0

The universality result in Eq. (P.3) will be derived from a recurrence relation given in Eq. (P.14), obtained by combining two different expressions of Qn . First we remark that  x  ∞  ∞  ∞ Ps (x) Pr (z) dz dx + s ↔ r = Ps (x) dx Pr (z) dz, (P.6) 0

0

0

0

where s ↔ r denotes the same term with the interchange of s and r. Equation (P.1) then leads to Qn ≡

1  pr ps , 2 r+s=n

n ≥ 2,

r, s ≥ 1.

(P.7)

We now make use of the recursion relation in Eq. (P.4) to obtain a second expression of Qn . The algebra is not as straightforward. First we express Ps (x) in terms of Ps−1 (x). Introducing t = s − 1, we can write Qn =





∞ x



∞

Pr (z) dz

r+t=n−1 0

0

K(x − y)Pt (y) dy dx,

n ≥ 2, r ≥ 1, t ≥ 0.

0

(P.8) To deal with an expression symmetrical in r and t, we take the term t = 0 out of the summation and write Qn = An + Bn ,

(P.9)

with An ≡





r+t =n−1 0



∞ x

∞

Pr (z) dz 0

K(x − y)Pt (y) dy dx,

n ≥ 2,

r, t ≥ 1,

0

(P.10) and 



∞

Bn ≡

K(x) 0

x

Pn−1 (z) dz dx. 0

(P.11)

504

21 The Photon Random Walk

The calculation of An and Bn makes use of two identities written in Eqs. (P.18) and (P.19). They lead to An =

1 2







∞

∞

Pr (x) dx

r+t=n−1 0

Pt (y) dy =

0

1 2



pr pt

n ≥ 2,

r, t ≥ 1,

r+t=n−1

(P.12) and to Bn = pn−1 − pn .

(P.13)

Regrouping Eqs. (P.7), (P.9), (P.12), and (P.13), we obtain the recurrence relation pn = pn−1 +

1 2

 r+t =n−1

pr pt −

1  pr ps , 2 r+s=n

n ≥ 2,

r, s, t ≥ 1,

(P.14)

∞ with p1 = 0 K(x) dx = 1/2. Equation (P.14) determines recursively pn in a way which does not involve the functional form of K(x). Hence, pn is a universal function of n. It can be calculated in explicit form by introducing its generating function, p(λ) =

∞ 

pn λn .

(P.15)

1

The multiplication of Eq. (P.14) by λn followed by a summation over n from 1 to ∞, leads to the quadratic equation: 1 1 p(λ) = λp(λ) + λp1 + λp2 (λ) − p2 (λ), 2 2

(P.16)

which, using p1 = 1/2, can also be written as [p(λ) + 1]2 =

1 . 1−λ

(P.17)

The function p(λ) being positive and λ smaller than one, we take the positive root and thus obtain the result stated in Eq. (P.3). For completeness, we give here the two identities used to establish Eqs. (P.12) and (P.13). They are 



∞ x

∞

f (z) dz 0

0

0

 K(x − y)g(y) dy dx + f ↔ g =



∞

∞

f (x) dx 0

g(y) dy, 0

(P.18)

References

and 

505



∞

K(x) 0

0

x





∞

f (z) dz dx +

∞

f (x) 0



∞

K(z − x) dz dx =

0

f (x) dx. 0

(P.19) They +∞ hold for any even positive K(x), normalized to unity, that is such that −∞ K(x) dx = 1. The functions f (x) and g(y) must be absolutely integrable on [0, ∞[ for Lemma 1 and f (x) integrable for Lemma 2. The proofs, given in Frisch and Frisch (1995), make use of the remark that K(x) is a derivative of the odd function K1 (x), defined by 

x

K1 (x) ≡

K(u) du,

(P.20)

0

and that this function has the finite value 1/2 at infinity. Integrations by parts are used to calculate the multiple integrals.

References Bosma, P.B., de Rooij, W.A.: Efficient methods to calculate Chandrasekhar ’s H -functions. Astron. Astrophys. 126, 283–292 (1983) Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Physics Reports 195, 127–293 (1990) Comtet, A., Majumbar, S.N.: Precise asymptotics for a random walker’s maximum. J. Stat. Mech.: Th. and Exp. P06013, 1–17 (2005) Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2. Wiley, New York (1971) Frisch, U., Frisch, H.: Universality of escape from a half-space for symmetrical random walks. In: Shlesinger, M., Zaslavsky, G., Frisch, U. (eds.) Lévy Flights and Related Topics in Physics. Lectures Notes in Physics, vol. 450, pp. 262–268. Springer, Berlin (1995) Ivanov, V.V, Sabashvili, Sh.A.: Transfer of resonance radiation and photon random walks. Astrophys. Space Sci. 17, 13–22 (1972); Russian original ibidem, pp. 3–12 √ Landi Degl’Innocenti, E.: Non-LTE Transfer. An alternative derivation for . Mon. Not. R. Astr. Soc. 186, 369–375 (1979) Lévy, P.: Théorie de l’addition des variables aléatoires. Paris, Gauthier–Villars (1937) Majumdar, S.N., Comtet, A., Ziff, R.M.: Unified solution of the expected maximum of a discrete time random walk and the discrete flux to a spherical trap. J. Stat. Phys. 122, 833–856 (2006) Montroll, E.W., Shlesinger, M.F.: The wonderful world of random walks. In: Lebowitz, J.L., Montroll, E.W. (eds.) Nonequilibrium phenomena II : From Stochastic to Hydrodynamics, Studies in Statistical Mechanics, vol. 11 (1984). North–Holland, Amsterdam Papoulis, N,: Probability, Random Variables, and Stochastic Processes. MacGraw-Hill, New York (1965) Pollaczek, F.: Fonctions caractéristiques de certaines répartitions définies au moyen de la notion d’ordre. Compt. Rendu. Acad. Sci. Paris 234, 2334–2336 (1952) Shlesinger, M., Zaslavsky, G., Frisch, U.: Lévy Flights and Related Topics in Physics. In: Lectures Notes in Physics, vol. 450 (1995). Springer, Berlin

506

21 The Photon Random Walk

Sparre Andersen, E.: On the fluctuations of the sums of random variables. Math. Scand. 1, 263–285 (1953) Spitzer, F.: A combinatorial lemma and its application to probability theory. Trans. Am. Math. Soc 82, 323–339 (1956) Spitzer, F.: The Wiener–Hopf equation whose kernel is a probability density. Duke Math. J. 24, 327–343 (1957) Trigt, C. van: Analytically solvable problems in radiative transfer. I. Phys. Rev. 181, 97–114 (1969) Vergassola, M., Dubrulle, B., Frisch, U., Noullez, A.: Burgers’s equation, Devil’s staircases and the mass distribution for large-scale structures. Astron. Astrophys. 289, 325–356 (1994)

Chapter 22

Asymptotic Behavior of the Resolvent Function

In Chap. 20, we derive the thermalization length from a large scale analysis of the integral equation for the source function and in Chap. 21 from the behavior of the random walk of photons that have undergone a large number of scatterings. In the present chapter, we derive the thermalization length and other properties of the propagation of the photons by carrying out an asymptotic analysis of exact expressions of the resolvent function, established in Chaps. 6 and 9. We recall that the resolvent function describes the propagation of photons away from a primary source and that it is the regular part of the surface Green function. An asymptotic analysis of exact expressions provides a more accurate description than a large scale analysis of the radiative transfer equations, in particular it makes it possible to investigate the effects of multiple scatterings, according to their distance from the source of primary photons. In this Chapter, we consider monochromatic scattering and complete frequency redistribution. In Sect. 22.1 we consider an infinite medium and in Sect. 22.2 a semiinfinite one. The approach developed in this Chapter is largely inspired by Ivanov (1973, Sections 3.4, 4.7, 5.5). The results presented here are a small part of those that can be found in this reference. They hold when the photons perform a large number of scatterings, which means that , the destruction probability per scattering, is assumed to be small.

22.1 The Infinite Medium Resolvent Function The exact expressions of the infinite medium Green function G∞ (τ ) and of the infinite medium resolvent function ∞ (τ ) given in Sect. 6.1 are established with the assumption that there is a plane primary source of photons at the origin. They therefore describe a one-dimensional propagation. In an infinite medium, one can also investigate a three-dimensional propagation by assuming that there is a primary point source at the origin and that the medium is spherically symmetric. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_22

507

508

22 Asymptotic Behavior of the Resolvent Function

In Sect. 22.1.1 we use the explicit expression of the plane source resolvent function ∞ (τ ) to define different asymptotic regimes. In Sect. 22.1.2 we show how the point source resolvent function can be deduced from the plane source resolvent function and discuss one-dimensional propagation versus a three-dimensional one.

22.1.1 Infinite Medium with a Plane Source The exact expressions of the resolvent function ∞ (τ ) = G∞ (τ )−δ(τ ) are given in Eq. (6.9) for complete frequency redistribution and in Eq. (6.12) for monochromatic scattering. We first consider monochromatic scattering, which is easier to deal with. For monochromatic scattering, 0 −ν0 τ ∞ (τ ) = φ∞ e +



∞ 1

c φ∞ (ν)e−ντ dν.

(22.1)

We recall that ν0 , ν0 ∈ [0, 1], is the positive root of the dispersion function, that 0 φ∞ =

ν0 (1 − ν02 ) ν02 − 

(22.2)

,

and that c φ∞ (ν) =

(1 − )k(ν) , λ2 (ν) + (1 − )2 π2 k 2 (ν)

ν ∈ [1, ∞[.

(22.3)

For monochromatic scattering, k(ν) and λ(ν) have √ simple explicit expression, but they are not needed here. Indeed, for  small, ν0  , and thus for optical depths much larger than unity, the term controlling the large τ behavior of ∞ (τ ) is simply 0 e−ν0 τ . It is almost constant for 1 τ < 1/ν and then decreases exponentially φ∞ 0 fast for τ > 1/ν0 . The optical depth 1/ν0 marks the transition between the region where the destructions of photons have a negligible role and the region where photons are converted back to kinetic energy. It provides the order of √ magnitude of the thermalization length. We thus recover the scaling law τeff ∼ 1/ . The two different regions are referred to in Ivanov (1973, p. 160) as the nearly conservative zone and the√ strong absorption zone. 0 ∼ 1/√. Thus in the nearly conservative For ν0 ∼ , Eq. (22.2) shows that φ∞ √ zone, that is for 1 τ < τeff , ∞ (τ ) is essentially constant and of order 1/ . This constant value increases to infinity as  → 0. In the strong absorption zone, that is for τ > τeff , ∞ (τ ) decreases exponentially with τ , but for a fixed value of √ τ , it also tends to infinity as 1/  as  → 0 (see Table 22.1).

22.1 The Infinite Medium Resolvent Function

509

Table 22.1 Infinite medium. Behavior at large optical depths of the resolvent function for small values of , the destruction probability per scattering. The constant a is the Voigt parameter of the line. The nearly conservative zone corresponds to 1 τ < τeff and the strong absorption zone to τ > τeff . For monochromatic scattering τeff ∼ ν0 Profile

τeff

Plane source,  = 0 1 τ < τeff τ > τeff

Point source  = 0, τ  1

Monochromatic

1 √ 

1 √ e−τ/τeff 

Doppler

1 √  − ln 

√ − ln 

3 4πτ √ ln τ τ2

Voigt

a 2

1 a 1/2 τ 1/2

1 √ e−τ/τeff   2τ 2

1 √

ln τ

a 1/2  2 τ 3/2

1 a 1/2 τ 5/2

For complete frequency redistribution,  ∞ (τ ) =

∞ 0

φ∞ (ν)e−ντ dν,

(22.4)

where φ∞ (ν) is also given by the right-hand side in Eq. (22.3). The lower bound of the integral in Eq. (22.4) being equal to 0, the behavior of ∞ (τ ) for large τ depends on the behavior of φ∞ (ν) for ν → 0. For complete frequency redistribution, it is shown in Sect. 19.2.2 that k(ν), the inverse Laplace transform of K(τ ), behaves for ν → 0, as ν kD (ν)  √ , 4 − ln ν

1 kV (ν)  √ a 1/2ν 1/2 , 3 π

(22.5)

for the Doppler and Voigt profiles, respectively. We must now analyze the behavior of the denominator in Eq. (22.3). First we write λ(ν) =  + λc (ν), where λc (ν) is the value of λ(ν) for conservative scattering. Neglecting  compared to 1, the denominator can be written as d   2 + 2λc (ν) + λ2c (ν) + π2 k 2 (ν).

(22.6)

For ν → 0, as shown in Sect. F.2, λD c (ν) ∼

kD (ν) , − ln ν

λV c (ν) ∼ kV (ν).

(22.7)

Ignoring constant factors and the logarithmic correction for the Doppler case, we can write d ∼ [ + k(ν)]2 .

(22.8)

510

22 Asymptotic Behavior of the Resolvent Function

This expression shows that there is a change in the behavior of φ∞ (ν) as k(ν) crosses the value . The transition at k(ν) ∼  defines a characteristic value νeff , hence a characteristic optical depth τeff ∼ 1/νeff . Using the asymptotic behavior of k(ν) for ν → 0 in Eq. √ (22.5), we recover the scaling laws τeff ∼ a/ 2 for the Voigt profile and τeff ∼ 1/( ln −) for the Doppler profile, respectively. We now use φ∞ (ν) 

k(ν) , [ + k(ν)]2

(22.9)

to determine the asymptotic behavior of ∞ (τ ) for large τ . In the strong absorption zone, which holds for τ > τeff , and where k(ν) < , one simply has φ∞ (ν) ∼ k(ν)/ 2 . Hence ∞ (τ )  K(τ )/ 2 ,

τ > τeff .

(22.10)

Here K(τ ) can be replaced by its asymptotic behavior at large τ . The result is shown in Table 22.1. In the nearly conservative zone, which holds for τ < τeff , the assumption k(ν) > , leads to φ∞ (ν) ∼ 1/k(ν) (see Eq. (22.9)). Using Eq. (22.5), we see that for the Voigt profile, φ∞ (ν) ∼ 1/kV (ν) ∼ a −1/2ν −1/2 . The integration in Eq. (22.4) leads to V ∞ (τ ) ∼

1 aτ

1/2 ,

1 τ < τeff .

(22.11)

For the Doppler profile, Eq. (22.5) shows that 1/k(ν) is not integrable. Setting  = 0 in Eq. (22.9) is too drastic. A more refined asymptotic analysis is needed. We introduce the new variables ν˜ and τ˜ defined by ν = (− ln )1/2 ν˜ and τ = τ˜ /(− ln )1/2, which satisfy ν˜ τ˜ = ντ . We now consider the limit  → 0, while keeping ν˜ and τ˜ of order unity. There is no convergence problem around ν = 0 anymore. We thus obtain, at leading order, D ∞ (τ ) ∼

√

− ln ,

1 τ < τeff .

(22.12)

These results are summarized in Table 22.1. In the nearly conservative zone, (τ < τeff ), the behaviors of ∞ (τ ) for the Voigt and Doppler profiles are quite different. For the Voigt profile, V ∞ (τ ) is, at leading order, independent of  and decreases as τ −1/2 . In contrast, for the Doppler profile, D ∞ (τ ) is essentially constant in τ and increases slowly as  decreases. In the strong absorption zone (τ > τeff ), the resolvent function ∞ (τ ) decreases with τ in the same way as the kernel K(τ ). The expressions in the nearly conservative zone and strong absorption zone become identical when τ is replaced by τeff . Figures in Ivanov (1973, p. 190) illustrate these different regimes. For conservative scattering ( = 0), the asymptotic behavior of ∞ (τ ) at large τ can be derived from that of the nearly conservative zone region by letting τeff tend

22.1 The Infinite Medium Resolvent Function

511

to infinity. There is an interesting marked difference between the Doppler profile and√the Voigt profile: for the Doppler profile ∞ (τ ) is independent of τ and grows as − ln , while for the Voigt profile it decreases as τ −1/2 . How does one explain this difference? For conservative scattering the creation of photons cannot be balanced by their destruction, since  = 0, but there are mechanisms, other than destruction, by which the radiation field may remain finite in a infinite conservative medium. As pointed out in Ivanov (1973), multiple scatterings tend to trap the photons in the vicinity of the primary source, but in the case of complete frequency redistribution, photons can escape this neighborhood by a shift in frequency. The efficiency of frequency redistribution for counterbalancing this trapping depends on the profile, since the mean free-path of a photon with frequency x is proportional to 1/ϕ(x). This escape mechanism is more efficient for the Lorentzian or Voigt profiles with extended wings decreasing as 1/x 2, than for the Doppler profile, with wings falling off as exp(−x 2 ). A quantitative analysis of this phenomenon in terms of an accumulation effect can be found in Ivanov (1973, pp. 114 and 174). The other mechanism for preventing an unlimited growth of the radiation field is the spreading of photons in space. The efficiency of this mechanism depends on the dimension of the medium; it is more efficient in a three-dimensional medium than in a one-dimensional one, as we show in Sect. 22.1.2.

22.1.2 Infinite Medium with a Point Source We now assume that there is a point source at the origin and that  = 0. First we establish a relation between the point source and the plane source resolvent functions. p We denote ∞ (ρ) the resolvent function at a point M, with Cartesian coordinates (τx , τy , τz ) and cylindrical coordinates (ρ, φ, τz ) (see Fig. 22.1) and ∞ (τz ) the resolvent function at the same point M, created by a plane source in the plane τz = 0. We now follow (Ivanov 1973, p. 97) for the construction of a relation between p ∞ (ρ) and ∞ (τz ). First we remark that ∞ (τz ) can be written as  ∞ (τz ) =

∞



∞

−∞ −∞

p

∞ (ρ) dτx dτy .

(22.13)

Transforming from the rectangular coordinates, τx , τy , τz , to the cylindrical coordinates τ1 , φ, τz , we can write  ∞ (τz ) = 2π

0

∞

p

∞ (ρ) τ1 dτ1 .

(22.14)

512

22 Asymptotic Behavior of the Resolvent Function

z

M

τz

ρ

τz y

τ1

φ

τx

τy

x

Fig. 22.1 Geometry for establishing the relation between of the point source and the plane source resolvent functions

Observing now that ρ 2 = τ12 + τz2 , and differentiating this relation, while keeping τz constant, we obtain the new expression  ∞ (τz ) = 2π

∞

|τz |

p

(22.15)

∞ (ρ)ρ dρ.

Taking now the derivative with respect to τz , and changing the notation τz to τ , we finally arrive at p

∞ (τ ) = −

1 d∞ (τ ) . 2πτ dτ

(22.16) p

To determine the large τ behavior of the point source resolvent function ∞ (τ ) for conservative scattering, we replace ∞ (τ ) by the asymptotic expression valid in the nearly conservative zone 1 τ < τeff , and then let  → 0. For monochromatic scattering and  small, we have, as seen above, ∞ (τ )  Using ν0 

ν0 (1 − ν02 ) ν02 − 

e−ν0 τ .

(22.17)

√ 3 and letting then  → 0, we obtain p,M

∞ (τ ) 

3 , 4πτ

τ  1.

(22.18)

Here M stands for monochromatic. Because of the spreading in space, the point source resolvent function has a finite value and tends to zero at infinity. For the

22.2 Semi-Infinite Medium

513

plane source, the resolvent function increases indefinitely as  → 0 as shown in Sect. 22.1.1. For complete frequency redistribution, we replace in Eq. (22.4), φ∞ (ν) by φ∞ (ν) ∼ 1/k(ν), and then k(ν) by its behavior for ν → 0 (see Eqs. (22.5) and (22.9)). The derivative with respect to τ producing a multiplication by a factor ν, there is no convergence problem at ν = 0. We thus obtain p,D ∞ (τ )

√ ln τ ∼ , τ2

p,V

∞ (τ ) ∼

1 , a 1/2τ 5/2

τ  1.

(22.19)

Thus even for the Doppler profile, the point source resolvent function for conservative scattering remains finite and tends to zero at infinity. We can observe in Table 22.1, the efficiency of the spreading of the photons in space by comparing the behaviors of the plane source resolvent function in the region 1 τ < τeff and that of the point source resolvent function for  = 0.

22.2 Semi-Infinite Medium Exact expressions of the resolvent function (τ ) for a semi-infinite medium are established in Chap. 6 for  = 0 and in Chap. 9 for  = 0. We use them here to examine the asymptotic behavior at large τ . The results are summarized in Table 22.2. A detailed analysis of (τ ) for small τ can be found in Ivanov (1973, p. 224). For monochromatic scattering, the semi-infinite resolvent function, has the exact expression 1 e−ν0 τ + (τ ) = X(ν0 )



∞ 1

c φ∞ (ν)(ν0 + ν)X(−ν)e−ντ dτ,

τ ∈ [0, ∞[, (22.20)

Table 22.2 Semi-infinite medium. Asymptotic behavior of the resolvent function in the nearly conservative and strong absorption zones. The thermalization lengths τeff are given in Table 22.1. For monochromatic √ scattering τeff ∼ ν0 . For  = 0, the monochromatic resolvent function has exactly the value 3 at infinity Profile Monochromatic Doppler Voigt

=0

 = 0

1 τ < τeff √ −τ/τ eff 3e

τ > τeff √ −τ/τ eff 3e

(ln τ )1/4 τ 1/2

1 √  3/2 τ 2 ln τ

1

a 1/2

a 1/4 τ 3/4

 3/2 τ 3/2

514

22 Asymptotic Behavior of the Resolvent Function

c (ν) is given by Eq. (22.22). The large τ asymptotic behavior is controlled where φ∞ by √ the first term in the right-hand side. For  = 0, this√term is equal to 1/X(0) = 3. Hence, for conservative scattering, (τ ) tends to 3 at infinity. The constant √ value 3 is the result of a balance between the scattering of photons inside the medium, up to infinity,√ and their escape through the surface. For  small, but non zero, we√have (τ )  3e−τ/τeff . Hence, in the nearly conservative zone, (τ ) is close to 3 and then decreases exponentially in the strong absorption zone. For complete frequency redistribution, we have



∞

(τ ) = 0

φ∞ (ν)X(−ν)e−ντ dτ,

τ ∈ [0, ∞[,

(22.21)

(1 − )k(ν) , + (1 − )2 π2 k 2 (ν)

ν ∈ [0, ∞[.

(22.22)

with φ∞ (ν) =

λ2 (ν)

This expression holds also  = 0. Comparing the exact expressions of the resolvent functions for the infinite medium and semi-infinite one, we see that the only difference is the presence of √X(−ν) inside the integral. When  = 0, this function can simply be replaced by . But when  = 0, the function X(−ν) tends to zero as shown in Sect. F.2. For the Doppler and Voigt profiles, the large τ behavior of (τ ) established in Sect. F.2 is thus D (τ ) ∼

(ln τ )1/4 , τ 1/2

V (τ ) ∼

1

1

a 1/4

τ 3/4

,

τ → ∞.

(22.23)

Corrections to this leading term are given in Ivanov (1973, p. 127). In the nearly conservative zone, as its name indicated, (τ ) behaves essentially as shown in Eq. (22.23). In the strong absorption√zone,  starts playing a role again. We now have φ∞ (ν) ∼ k(ν)/ 2 and X(0) = . One thus obtains (τ )  K(τ )/ 3/2,

τ → ∞,

(22.24)

for both the Doppler and Voigt profile. The resolvent function behaves as the kernel and tends to zero at infinity. In the nearly conservative zone, (τ ) is also a decreasing function of τ , thanks to the frequency redistribution, but it decreases more slowly than the kernel.

Reference Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office. NBS Spec. Publ. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969)

Chapter 23

The Asymptotics of the Diffusion Approximation

In Chap. 20, it was shown that monochromatic scattering, and more generally scattering processes with a finite second order moment, can be described, asymptotically, by an ordinary diffusion process, whenever the photons undergo a very large number of scatterings. The proof was based on the analysis of the onedimensional integral equation for the source function in the limit of a small destruction probability . It was also shown that the scale √ of variation of the source function, and thus of the radiation field, is of order 1/ . In this chapter we perform an asymptotic analysis of the radiative transfer equation itself in a medium with boundaries and drop the one-dimensional assumption. We show that the radiation field far from boundaries can be described by a diffusion equation and that close to boundaries it can be described by a one-dimensional diffuse reflection problem with conservative scattering. In Chap. 24, we show that a similar decomposition is possible for Rayleigh scattering. We explain how the radiation fields of the two regions can be matched to construct a solution valid in the whole medium. The method described in this chapter has been developed by Larsen and Keller (1974) for neutron transport. For radiative transfer problems, it has been applied, e.g., by Larsen et al. (1983), Frisch and Faurobert (1984), Frisch (2019). A mathematically rigorous presentation of the diffusion approximation can be found in Bardos et al. (1984). The tools introduced here for monochromatic scattering in a homogeneous medium have a wide range of applications and can be generalized to heterogeneous media. The technique for handling heterogeneous media is known as homogenization. It provides a global description, while taking into account the heterogeneity of the medium. Homogenization can be applied to transport problems (see e.g. Bensoussan et al. 1978, 1979; Tartar 2008), but also to wave propagation in random media (Fouque et al. 2007). In Sect. 23.1, we introduce a new radiative transfer equation, written in terms of a suitably rescaled space variable and in Sect. 23.2 we derive from the rescaled radiative transfer equation, a diffusion equation valid far from boundaries. A radiative transfer equation describing the behavior of the radiation field near © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_23

515

516

23 The Asymptotics of the Diffusion Approximation

boundaries is introduced in Sect. 23.4, where we also show how the interior and boundary layer solutions can be matched to construct a solution valid in the whole medium. We employ the asymptotic analysis in Sect. 23.5 to evaluate the validity of standard laws such as the Fick’s law. Finally we show in Sect. 23.6 how the asymptotic analysis applies to the transport of neutrons and the determination of the criticality condition for a one dimensional slab.

23.1 The Rescaled Equation We consider the radiative transfer equation 1 n.∇I (r, n) = −I (r, n) + (1 − ) k(r)

 I (r, n)

d + Q∗ (r), 4π

(23.1)

where r is the position in physical space, k(r) the absorption coefficient per unit length, assumed to be of order unity,  the destruction probability per scattering (0 <  < 1) and Q∗ (r) the primary source term. Here d/4π denotes the infinitesimal solid angle. We assume that there is no radiation field incident upon the medium.1 The boundary condition for the transfer equation is thus I (r b , n) = 0,

u.n < 0,

(23.2)

where r b is a point on the surface of the medium (see Fig. 23.1). As shown in Chap. 20, when the primary source is of order unity, then the radiation is of order t () ∼ 1/. Here it is assumed that the primary source is of order  so that t () ∼ 1. For the purpose of the asymptotic analysis we first rewrite the radiative transfer equation as    d 1 n.∇I (r, n) = (1 − ) −I (r, n) + I (r, n) + [Q∗ (r) − I (r, n)]. k(r) 4π (23.3) A similar transformation has been applied to the integral equation for the source function in Sect. 20.1. We now introduce the rescaled variable √ r˜ ≡ r ,

(23.4)

and assume that k(r) and Q∗ (r) have no rapid variations, so that they can be expressed in terms of r˜ . We look for solutions I (˜r , n). For an infinite medium,

1 In mathematically oriented articles, this is known as an absorbing boundary condition. No radiation emerging from the medium is falling back onto it.

23.2 The Interior Radiation Field

517

this large scale solution is valid everywhere, but for a finite medium, it will be valid only in the interior of the medium. Another solution valid in the vicinity of the boundary is constructed in Sect. 23.4. To simplify the notation we introduce the small parameter η defined by η≡

√ .

(23.5)

We also introduce the operator L defined by  L[f (n)] ≡ −f (n) +

f (n)

d . 4π

(23.6)

Here f (n) stands for the radiation field. The rescaled radiative transfer equation may then be written as η n.∇r˜ I (˜r , n) = (1 − η2 )L[I ] + η2 [Q∗ (˜r ) − I (˜r , n)]. k(˜r )

(23.7)

The gradient is now taken with respect to the rescaled variable r˜ . The notation I is introduced to indicate that we are dealing with the rescaled field and that it depends on . The radiative transfer equation has been recast into a singular perturbation problem, recognizable by the fact that the derivative is multiplied by a small parameter (see e.g. Cole 1968; Bender and Orszag 1978). The problem is singular in the sense that when η = 0, the boundary conditions associated to the initial non-rescaled equation may not be satisfied. When that happens, boundary layers solutions may have to be introduced. In the asymptotic analysis presented in this chapter, the destruction probability per scattering  is the small parameter. Actually, transport equations are eligible to an asymptotic analysis whenever the mean free-path l of the particles is much smaller than a characteristic dimension of the medium, D. The small expansion parameter is then the ratio η = l/D (see e.g. Larsen and Keller 1974; Frisch and Bardos 1981).

23.2 The Interior Radiation Field We assume that the solution of the rescaled equation may be written as a perturbation series in power of η. The primary source has been chosen in such a way that the radiation field is of order one. So we may write I (˜r , n) = I0 (˜r , n) + ηI1 (˜r , n) + η2 I2 (˜r , n) + η3 I3 (˜r , n) + O(η4 ).

(23.8)

518

23 The Asymptotics of the Diffusion Approximation

Inserting this expansion into Eq. (23.7) and regrouping all the terms with the same power of η, we obtain a hierarchy of equations: L[I0 ] = 0, L[I1 ] = L[I2 ] =

1 n.∇r˜ I0 (˜r , n), k(˜r )

1 n.∇r˜ I1 (˜r , n) + I0 (˜r , n) − Q∗ (˜r ), k(˜r )

(23.9) (23.10)

(23.11)

where the operator L is defined in Eq. (23.6). We now examine each equation in turn. It is easy to verify that isotropic functions, i.e. independent of n, are solutions of Eq. (23.9). To prove that they are the only ones, one must invoke the Krein– Rutman theorem (1950), which is a generalization of the Perron–Frobenius theorem for matrices. The space of functions which satisfies Eq. (23.9) is known as the null space of the operator L. The solutions of Eq. (23.9) have thus the form I0 (˜r , n) = I0 (˜r ).

(23.12)

Because the operator L has a null space, that is a zero eigenvalue, it is not invertible. Therefore the inhomogeneous Eqs. (23.10) and (23.11) may not have a solution. The definition of L given in Eq. (23.6) shows that the integral of the left hand-side, over all directions n, is zero. For Eqs. (23.10) and Eq. (23.11), the solvability condition is thus that the average over directions of their right-hand side is zero. Equation (23.10) satisfies the solvability condition because I0 is isotropic, as just shown. It has solutions of the form I1 (˜r , n) = −

1 [n.∇r˜ I0 (˜r ) + c1 (˜r )], k(˜r )

(23.13)

where c1 (˜r ) satisfies L[c1 (˜r )] = 0, hence belongs to the null space of L. Replacing in Eq. (23.11) I1 (˜r , n) by the expression just obtained, and then integrating over directions, we find that the solvability condition for this equation takes the form 1 1 [∇r˜ ∇r˜ I0 (˜r )] − I0 (˜r ) + Q∗ (˜r ) = 0. 3k(˜r ) k(˜r )

(23.14)

This is a diffusion equation for I0 (˜r ). For a one-dimensional problem, the optical depth can be used as space variable and one recovers the diffusion equation for the source function written in Eq. (20.10) (see also Eq. (23.63)). The construction of the diffusion equation for I0 (˜r ) is possible because the operator L has a null space, made of isotropic functions, which can be used to

23.2 The Interior Radiation Field

519

construct the radiation field. We know from preceding chapters that scattering with complete frequency redistribution cannot lead to a diffusion equation because the second-moment of the kernel is infinite. Another method for reaching the same conclusion is to introduce an operator L, adapted to complete frequency redistribution, defined by  L[f (x, n)] ≡ −f (x, n) +

+∞  −∞

f (x, n)ϕ(x)

d dx, 4π

(23.15)

where x is the frequency and ϕ(x) a line absorption profile, normalized to one. The null space of this operator is made of functions which are both independent of direction and of frequency. This means that they are flat and isotropic and cannot be used to construct frequency dependent solutions. Another way of stating this property is to remark with Field (1959) that photons starting with a given frequency distribution and which are subject to scatterings only (no destruction, no escape) will never reach a stationary frequency distribution; instead their frequency distribution will flatten out indefinitely. For anisotropic scattering, the construction a diffusion equation for the radiation field is possible and goes essentially as described above. We now assume that the operator L is defined by  L[f (n)] = −f (n) +

g(n, n )f (n )

d

, 4π

(23.16)

where the phase function g(n, n ) is strictly positive, is depending only on the angle between n and n , and satisfies 

g(n, n )

d

= 1. 4π

(23.17)

Rayleigh scattering with g(n, n ) = (3/4)(1 + cos2 (n.n )) is a standard example of anisotropic scattering (see Chandrasekhar 1960, p. 35). It is easy to verify that the null space of L is the space of isotropic functions. That they are the only ones is a consequence of the Krein–Rutman theorem. Therefore the leading term in the asymptotic expansion of the radiation field is isotropic. The term of order η has the form I1 (˜r , n) =

1 [D(n).∇I0 (˜r ) + c1 (˜r )], k(˜r )

(23.18)

where D(n) is solution of L[D(n)] = n.

(23.19)

520

23 The Asymptotics of the Diffusion Approximation

For an isotropic phase function, D(n) = −n. When g(n, n ) is an even function of n.n , then I0 (˜r ) satisfies a diffusion equation identical to Eq. (23.14) except that the numerical coefficient 1/3 in front of the second order derivative is replaced by  d − n.D(n) . (23.20) 4π For Rayleigh scattering this coefficient is also 1/3.

23.3 An Improved Diffusion Equation Equation (23.14) can be solved with the Dirichlet boundary condition I0 (˜r b ) = 0,

(23.21)

where r˜ b is a point on the boundary (see Fig. 23.1). It is consistent with the assumption that there is no radiation field incident on the medium, but implies that the emergent radiation field is zero. An improved diffusion approximation, which takes into account c1 (˜r ), the isotropic part of the first order term, can be constructed by introducing the direction average intensity  J (˜r ) =

I (˜r , n)

d . 4π

(23.22)

u

n

u

rb θ n

boundarylayer

0 s

−∞ Fig. 23.1 At each point r b of the surface, the boundary layer is a plane-parallel, semi-infinite medium extending from s = 0 to s = −∞. The surface of the boundary layer is tangential to the surface of the medium at r b

23.3 An Improved Diffusion Equation

521

Keeping only the zeroth order term I0 (˜r ) and the first order term I1 (˜r ), given by Eq. (23.13), we introduce J (˜r ) = I0 (˜r ) − η

c1 (˜r ) . k(˜r )

(23.23)

We now show that J (˜r ) satisfies a diffusion equation. Equation (23.14) shows that I0 (˜r ) satisfies a diffusion equation. It remains to show that c1 (˜r ) does the same. The proof is simple, but requires that we push the hierarchy of equations to order η3 . At order η3 , the rescaled radiative transfer equation in Eq. (23.7) becomes L[I3 ] = L[I1 ] +

1 n.∇r˜ I2 (˜r , n) + I1 (˜r , n). k(˜r )

(23.24)

Using the definition of L given in Eq. (23.6) and the expression of I1 (˜r , n) given in Eq. (23.13), we can also write L[I3 ] =

1 [−c1 (˜r ) + n.∇r˜ I2 (˜r , n)]. k(˜r )

(23.25)

The solvability condition for this equation is  − c1 (˜r ) +

n.∇r˜ I2 (˜r , n)

d = 0. 4π

(23.26)

Equation (23.11) shows that I2 (˜r , n) has the form I2 (˜r , n) =

1 1 c1 (˜r ) [n.∇r˜ ( n.∇r˜ I0 (˜r )) + n.∇r˜ ( ) + c2 (˜r )], k(˜r ) k(˜r ) k(˜r )

(23.27)

where c2 (˜r ) is an isotropic function (it belongs to the null space of the operator L), which plays the role of an integration constant. The first and third terms in the righthand side of Eq. (23.27) do not contribute to the integral in Eq. (23.26) because they are even functions of n. Thus, the solvability condition leads to the homogeneous diffusion equation 1 ∇r˜ 3



c1 (˜r ) 1 ∇r˜ k(˜r ) k(˜r )

− c1 (˜r ) = 0.

(23.28)

Both I0 (˜r ) and c1 (˜r ) satisfying a diffusion equation, J (˜r ) will satisfy the diffusion equation in Eq. (23.14), up to order η, included. The diffusion equations for I0 (˜r ) and J (˜r ) hold only in the interior of the medium. We now construct an equation for the radiation field in the vicinity of the boundary. It will give us a boundary condition for J (˜r ) and the emergent intensity.

522

23 The Asymptotics of the Diffusion Approximation

23.4 The Boundary Layer To construct an equation valid near the boundary, we use a general technique based on the introduction of a stretched variable (see e.g. Cole 1968; Larsen and Keller 1974; Bender and Orszag 1978). We start from the rescaled equation, that is, from the equation in the variable r˜ , and, at each point r˜ b on the boundary, submit this slow variable to a stretching in the direction perpendicular to the boundary, which amounts to enlarge the region next to the boundary. In the stretched variable, here denoted s, the boundary region becomes a plane-parallel semi-infinite medium (see Fig. 23.1). Thanks to this change of variable, the determination of the radiation field in the boundary region leads to one-dimensional half-space problems for which there exist exact solutions, established in preceding chapters. We note that this method is not applicable when the surface of the medium has rapid variations. To carry out this program, we introduce the boundary layer variable s=

1 u.(˜r − r˜ b ), η

(23.29)

where u is the outward normal to the surface at the point r˜ b . In the limit η → 0, the variable s varies between 0, at the surface, and −∞ in the interior. We assume that the radiation field in the boundary layer can be written as I (r, n) = Iint (˜r , n) + Ib (s, μ),

(23.30)

where Iint (˜r , n) is the interior solution discussed in Sect. 23.2, and Ib (s, μ) a boundary layer contribution to the radiation field, in short boundary layer solution. It depends on the variables s and μ = u.n. The boundary layer solution satisfies the radiative transfer equation μ ∂Ib (s, μ) = −(1 − η2 )L[Ib (s, μ)] − η2 Ib (s, μ). k(˜r b ) ∂s

(23.31)

The primary source does not appear in this equation. It has been incorporated in the equation for the interior. The variations of the absorption coefficient inside the boundary layer are neglected. This is consistent with the assumption that the absorption coefficient is a slowly varying function of space. To solve Eq. (23.31), one needs a boundary condition at the surface. At the surface, the sum of the interior and boundary layer solutions must satisfy the boundary condition of the original problem, namely that no radiation is falling onto the medium. This condition leads to Iint (˜r b , n) + Ib (0, μ) = 0,

u.n = μ < 0.

(23.32)

23.4 The Boundary Layer

523

The boundary condition for the boundary layer solution depends on the yet unknown interior radiation field. For the sum in Eq. (23.30) to represent the radiation field in the interior, the boundary layer solution must tend to zero at infinity. This means that the boundary layer solution must satisfy Ib (s, μ) → 0,

s → −∞.

(23.33)

This constraint couples the interior and the boundary layer solutions and provides, as will be seen, a boundary condition for the interior diffusion equation. We now assume that Ib (s, μ) has an expansion of the form Ib (s, μ) = I0b (s, μ) + ηI1b (s, μ) + O(η2 ).

(23.34)

Inserting the sum in Eq. (23.30) into the transfer equation (23.3) and treating τ˜ and s as independent variable (a standard approach for multi-scale expansion, see e.g. Bender and Orszag 1978), we find that I0b (s, μ) and I1b (s, μ) satisfy the onedimensional radiative transfer equation μ ∂Ikb 1 (s, μ) = −Ikb (s, μ) + k(˜r b ) ∂s 2



+1 −1

Ikb (s, μ) dμ,

k = 0, 1.

(23.35)

The boundary condition and the condition at infinity are Ikint (˜r b , n) + Ikb (0, μ) = 0, μ < 0,

Ikb (s, μ) → 0, s → −∞,

k = 0, 1. (23.36)

Equation (23.35) describe conservative scattering in a semi-infinite medium with a given incident radiation. Here the incident radiation is the surface value of the interior field (actually, its opposite). Under the name of diffuse reflection, it is the problem consider in Sect. 7.3.3, where it has been shown that the radiation field tends to a constant at infinity. This constant, here denoted I b (∞), may be written as √  1 3 I (∞) = H (μ)I inc (−μ)μ dμ, 2 0 b

(23.37)

where H (μ) is the Chandrasekhar H -function and I inc the incident field. The condition that I b (∞) is zero at zeroth order and first order in η, will provide the needed boundary condition for the interior diffusion equations satisfied by I0 (˜r ) and J (˜r ) and also the emergent radiation field.

524

23 The Asymptotics of the Diffusion Approximation

23.4.1 Boundary Conditions for the Interior Solution At zeroth order in η, the incident field in Eq. (23.37) is independent of μ, since I0int (˜r b , n) = I0int (˜r b ) is isotropic, hence can be taken out of the integral. The first√ moment of H (μ) being non-zero (it is equal to 2/ 3), the condition I0b (∞) = 0 leads to I0int (˜r b ) = 0,

and I0b (0, μ) = 0,

μ < 0.

(23.38)

We recover the boundary condition for the interior diffusion equation written in Eq. (23.21). The equation for I0b (s, μ) being homogeneous, Eq. (23.38) implies that I0b (s, μ) = 0 for all values of s. The emergent radiation field is thus also zero at this order. We now consider the first order correction I1b (s, μ). At order η, the incident field is equal to −I1int (˜r b , n). Its expression, given in Eq. (23.13), contains a term n.∇r˜ I0int (˜r b ). On the boundary, the tangential component of ∇r˜ I0int (˜r ) is zero since I0int (˜r ) = 0 for all r˜ b . Hence the incident radiation for I1b (s, μ) may be written as I1inc (μ) = −

1 [μu.∇r˜ I0int (˜r b ) + c1 (˜r b )], k(˜r b )

μ < 0,

(23.39)

where ∇r˜ I0int (˜r b ) stands for the gradient of I0int (˜r ) taken at r˜ b . Inserting this μdependent expression into Eq. (23.37), and imposing I1b (∞) = 0, we find c1 (˜r b ) = Lu.∇r˜ I0int (˜r b ),

(23.40)

where L is a constant given by 1

√  1 3 L = 1 H (μ)μ2 dμ. = 2 0 H (μ)μ dμ 0 0

H (μ)μ2 dμ

(23.41)

We recover here the expression of q(∞), the value of the Hopf function at infinity given in Eq. (9.31). We recall that q(∞)  0.7104. This constant is also known as the extrapolation length for reasons given in Chap. 9. We can now calculate the surface value of the interior field at order η. Inserting the expression of c1 (˜r b ) into Eq. (23.13), we find I1int (˜r b , n) = −

1 (μ + L)u.∇r˜ I0int (˜r b ). k(˜r b )

(23.42)

23.4 The Boundary Layer

525

The surface value of the direction-average intensity J (˜r ), defined in Eq. (23.23), is given by J (˜r b ) = −η

L u.∇r˜ I0int (˜r b ). k(˜r b )

(23.43)

As shown in Sect. 23.3, J (˜r ) satisfies the same diffusion equation as I0 (˜r ) (see Eq. (23.14). Replacing in Eq. (23.43) I0int (˜r b ) by J (˜r b ) (the error is O(η2 )), we obtain, for the diffusion equation satisfied by J (˜r ) the so-called Robin-type, or mixed type, boundary condition, J (˜r b ) + η

L u.∇r˜ J (˜r b ) = 0. k(˜r b )

(23.44)

We note that the factor η disappears when the gradient is expressed in terms of the initial variable r. The one-dimensional version of this boundary condition is used in Sects. 23.5 and 23.6, where we discuss the Eddington approximations and the determination of the critical size for neutron transport.

23.4.2 The Emergent Intensity At zeroth order in η, as we have shown above, the interior and boundary layer field are zero on the surface. The emergent radiation field is thus zero. The first non-zero contribution is given by the sum: Iηem (μ) = η[I1int (˜r b , n) + I1b (0, μ)],

μ ∈ [0, 1],

(23.45)

where I1int (˜r b , n), the contribution from the interior field, is given by Eq. (23.42). We still have to calculate the boundary layer contribution I1b (0, μ). For a conservative semi-infinite plane-parallel medium, the emergent intensity may be written as I em (μ) =

1 H (μ) 2

 0

1

H (μ )μ inc I (−μ ) dμ , μ + μ

μ ∈ [0, 1],

(23.46)

where I inc (μ), μ < 0 is the incident field. This famous result (Chandrasekhar 1960, p. 86) is demonstrated in Sect. 7.3.3. Equation (23.46) provides the boundary layer contribution I1b (0, μ) when the incident field is replaced by −I1int (˜r b , n). Equation (23.42) shows that I1int (˜r b , n) contains an isotropic term −Lu.∇r˜ I0int (˜r b )/k(˜r b ). This isotropic term will not contribute to the emergent intensity. Indeed, for a semi-infinite conservative medium, the emergent radiation is equal to the incident one, when the latter is isotropic. Inserting −I1int (˜r b , n), without

526

23 The Asymptotics of the Diffusion Approximation

its isotropic term, into Eq. (23.46), making use of the nonlinear integral equation for the H -function, namely 1 H (μ) = 1 + H (μ)μ 2



1

0

H (μ ) dμ , μ + μ

(23.47)

and of the exact values 

1



1

H (μ) dμ = 2,

0

0

2 H (μ)μ dμ = √ , 3

(23.48)

we find after some simple algebra H (μ) 1 Iηem (μ)  −η √ u.∇r˜ I0int (˜r b ), k(˜ r ) 3 b

μ ∈ [0, 1].

(23.49)

√ Equation (23.49) shows that the emergent radiation field is of order η = , that the factor η disappears when the gradient is taken with respect to the initial variable r, and that the emergent intensity is determined by the gradient of the interior radiation field in the vicinity of the boundary. We also recover the H (μ)-dependence of the emergent radiation field. For a one-dimensional medium, we can use the optical depth τ as space variable. Assuming that the surface is at τ = 0, that τ increases towards the interior of the medium, and defining the optical depth with dτ = −k(0)ds, the emergent intensity takes the form H (μ) dI0int (τ ) Iηem (μ)  √ |τ =0 , dτ 3

μ ∈ [0, 1].

(23.50)

In Sect. 24.4 on Rayleigh scattering we show how higher order terms can be added to this leading term.

23.5 The Eddington Approximations In this section we compare the predictions of the asymptotic analysis to that of a much simpler and older procedure, known as the Eddington approximation, in which it is assume a priori that the radiation field is almost isotropic everywhere in the medium. The comparison is carried out for a one dimensional slab, The one-dimensional version of the radiative transfer equation (23.1) may be written as μ

1 ∂I (τ, μ) = −I (τ, μ) + (1 − ) ∂τ 2



+1 −1

I (τ, μ) dμ + Q∗ (τ ),

(23.51)

23.5 The Eddington Approximations

527

where τ is the optical depth, τ ∈ [−T , +T ], and μ = u.n, with u the outward normal at the surface τ = T . We assume that there is no incident radiation on the boundaries, i.e. I (+T , μ) = 0 μ < 0,

I (−T , μ) = 0 μ > 0.

(23.52)

The moments J (τ ), F (τ ), and J (2)(τ ) are defined, as in Sect. 9.1, by J (τ ) =

1 2





+1

−1

I (τ, μ) dμ,

F (τ ) = 2π

+1

−1

I (τ, μ)μ dμ.

(23.53)

and J (2)(τ ) =

1 2



+1 −1

I (τ, μ)μ2 dμ.

(23.54)

The integration of Eq. (23.51) over μ yields 1 dF (τ ) = [J (τ ) − Q∗ (τ )], 4π dτ

(23.55)

and its integration over μ, after a multiplication by μ, yields 1 dJ (2)(τ ) = − F (τ ). dτ 4π

(23.56)

The assumption that the radiation field is almost isotropic, provides a closure relation: J (2) (τ ) 

1 J (τ ). 3

(23.57)

This approximation amounts to keep only the leading term√I0 (τ ) in the asymptotic expansion of the radiation field. The error is thus of order . The combination of the exact relation in Eq. (23.56) with the closure relation in Eq. (23.57) leads to the Fick’s law, namely F (τ ) = −

4π dJ (τ ) . 3 dτ

(23.58)

Fick’s law can be derived from the asymptotic expansion of the radiation field limited to its two first term, namely   ∂I0 (τ˜ ) + c1 (τ˜ ) + O(η2 ). I (τ, μ) = I0 (τ˜ ) − η μ ∂ τ˜

(23.59)

528

23 The Asymptotics of the Diffusion Approximation

√ Here τ˜ = τ/η = τ/ . The corresponding radiative flux, is given by F (τ˜ ) = −η

4π ∂I0 (τ˜ ), 3 ∂ τ˜

(23.60)

and the direction average radiation field by J (τ˜ ) = I0 (τ˜ ) − ηc1 (τ˜ ).

(23.61)

The combination of these two expressions leads to F (τ˜ ) = −η

4π ∂J (τ˜ ) + O(η2 ). 3 ∂ τ˜

(23.62)

Returning to the variable τ , we recover Fick’s law with an error of order . Fick’s law implicitly assumes that the radiation field has no steep gradient. Several modifications have been introduced to avoid unphysical values of the flux in regions where this condition does not hold, in particular near boundaries. A well known and successful one is the flux-limited diffusion theory, in brief FDT, introduced by Levermore and Pomraning (1981). The diffusion coefficient 1/3 is replaced by a depth-dependent coefficient DF (τ ), which depends nonlinearly on dJ (τ )/dτ and tends to 1/3 when dJ (τ )/dτ tends to zero. Fick’s law readily leads to a diffusion equation. Combining the derivative of Fick’s law given in Eq. (23.58) with Eq. (23.55), we obtain 1 d 2 J (τ ) = [J (τ ) − Q∗ (τ )]. 3 dτ 2

(23.63)

√ The one-dimensional version of Eq. (23.14) is recovered by introducing τ˜ = τ . In the FDT theory, the diffusion term has the form d(DF (τ )dJ (τ )/dτ )/dτ . Boundary conditions for Eq. (23.63) have been proposed by Eddington (1926) (see Kourganoff 1963) for a slab with no incident radiation. They are based on the assumption that the emergent radiation field is essentially independent of μ. Applied to the definitions of F (τ ) and J (τ ) in Eq. (23.53), it leads to J (±T )  ±

1 F (±T ), 2π

(23.64)

and, combined with Fick’s law, to J (±T ) ±

2 dJ |τ =±T = 0. 3 dτ

(23.65)

23.6 Determination of the Critical Size

529

It is a mixed-type boundary condition, similar to the boundary condition in Eq. (23.44), which, written in the τ -variable, is J (±T ) ± L

dJ | ˜ = 0. dτ τ =±T

(23.66)

The two boundary conditions differ only by the constant multiplying the derivative: 2/3 = 0.666 for the Eddington approximation, and L  0.710 for the systematic asymptotic analysis. More refined closure approximations, based on spherical harmonics expansions of the radiation field, can bring the value of this constant closer to L but not to L itself. The boundary condition of the flux-limited theory is also of the mixed-type, but is nonlinear, as is the associated diffusion equation (Pomraning 1988). Of course, the Eddington approximation brings no information on the directional dependence of the emergent radiation field. Only a boundary layer analysis, such as performed in Sect. 23.4, can bring the information.

23.6 Determination of the Critical Size Asymptotic expansions of the type described in this chapter were introduced and extensively used in neutron transport theory. For neutron transport in a reactor, when more than one neutron is emitted per collision, the single scattering albedo λ = 1 −  may become larger than unity. This means that  is negative and that solutions of Eq. (23.1) will exist only when the size of the medium and the single scattering albedo satisfy a definite relation known as the criticality condition. The criticality condition is derived from the boundary conditions. When the scattering albedo remains close to unity, the diffusion approximation may be used to calculate the critical size. We show here how to determine the critical size for a slab in which the scattering of neutrons is isotropic and the absorption coefficient is a constant equal to unity. We also assume that there is no exterior source of neutrons and no primary sources of neutrons inside the medium. The neutrons are introduced in the medium only through fission. We assume λ = 1 + γ with γ > 0. The transport equation for the neutron distribution (z, μ) may be written as μ

∂ 1 (z, μ) = −(z, μ) + (1 + γ ) ∂z 2



+1 −1

(z, μ) dμ,

(23.67)

with z ∈ [−Z, +Z]. The associated boundary condition is (+Z, μ) = 0 μ < 0,

(−Z, μ) = 0

μ > 0.

(23.68)

530

23 The Asymptotics of the Diffusion Approximation

Proceeding as above, we rewrite Eq. (23.67) as    1 +1 ∂ (z, μ) = (1 + γ ) −(z, μ) + (z, μ) dμ + γ (z, μ). μ ∂z 2 −1

(23.69)

The problem is to find the value of Z, which allows for a finite solution. For small γ , the asymptotic analysis described in the preceding sections provides the answer. To indicate that γ is small, we write γ = η2 γ˜ , where η is the small parameter of the problem, and the asymptotic analysis is performed in the limit η → 0. We also ˜ z, μ), introduce the rescaled variable z˜ = ηz, the rescaled neutron distribution (˜ ˜ and look for an asymptotic expansion of (˜z, μ) in powers of η, of the form ˜ z, μ) =  ˜ 0 (˜z, μ) + η ˜ 1 (˜z, μ) + . . . . (˜

(23.70)

We present two different methods of treating the problem, described in Bardos et al. (1984). Method (i) γ is replaced by η2 γ˜ in both occurrences of γ . The asymptotic expansion can then be carried out exactly as described in the preceding sections. It leads to a diffusion equation, ˜ z) 1 d 2 (˜ ˜ z) = 0, + γ˜ (˜ 3 d z˜ 2

(23.71)

˜ z) =  ˜ 0 (˜z). The solutions of this equation have the form where (˜   ˜ z) = c0 cos( 3γ˜ z˜ ) = c0 cos( 3γ z), (˜

(23.72)

˜ = 0, leads to where c0 is a constant. The Dirichlet boundary condition, (±Z) 1 π Z= √ . 3γ 2

(23.73)

The improved diffusion approximation introduced in Sect. 23.3 also leads to Eq. (23.71), but with the mixed-type (Robin type) boundary condition ˜ d ˜ (±Z) ±L |z=±Z = 0. dz

(23.74)

The combination of Eqs. (23.73) and (23.74) leads to 1 1 Z = √ arctan( √ ). 3γ L 3γ It is easy to verify that the two formulae coincide for small γ .

(23.75)

References

531

Method (ii) γ is now replaced by η2 γ˜ , but in the term γ (z, μ) only. The asymptotic analysis leads to a different diffusion equation, namely, ˜ z) 1 d 2 (˜ ˜ z) = 0. + γ˜ (˜ 3(1 + γ ) d z˜ 2 The solutions have the form   ˜ z) = c0 cos[ 3(1 + γ )γ˜ z˜ ] = c0 cos[ 3γ (1 + γ )z]. (˜

(23.76)

(23.77)

The Dirichlet boundary condition yields 1 π . Z= √ 3γ (1 + γ ) 2

(23.78)

The Robin boundary condition is similar to Eq. (23.74), with the constant L replaced by L/(1 + γ ). The critical size is now given by

1 1+γ Z= √ arctan √ . 3γ (1 + γ ) L 3γ (1 + γ )

(23.79)

A comparison between values of the critical size obtained by numerical simulations for different values of γ and the predictions of the four expressions given above can be found in Bardos et al. (1984). For each method the Robin boundary condition gives a better result than the Dirichlet one. The Robin boundary condition of Method (ii), leading to Eq. (23.79), provides the best approximation. When γ increases from 0.01 to 1, the error remains less than 7%. Even when γ is larger than 1, Eq. (23.79) predicts a critical size with a correct order of magnitude.

References Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc. 284, 617–649 (1984) Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York (1978) Bensoussan, A., Lions, J-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. NorthHolland, New York (1978). Reprinted in 2011 by AMS Chelsea Publishing Bensoussan, A., Lions, J-L., Papanicolaou, G.: Boundary layers and homogenization of transport processes. Publ. RIMS, Kyoto Univ. 15, 53–157 (1979) Chandrasekhar, S.: Radiative Transfer, 1st edn. Dover Publications, New York (1960); Oxford University Press (1950) Cole, J.D.: Perturbation Methods in Applied Mathematics. Blaisdell, Waltham (1968) Eddington, A.S.: The Internal Constitution of Stars, p. 322. Dover, New York (1926) Field, G.: The time relaxation of a resonance-line profile. Astrophys. J. 129, 551–564 (1959)

532

23 The Asymptotics of the Diffusion Approximation

Fouque, J-P., Garnier, J., Papanicolaou, G., Solna, K.: Wave Propagation and Time reversal in Randomly Layered Media. Springer, Berlin (2007) Frisch, H.: Nonconservative Rayleigh scattering. A perturbation approach. Astron. Astrophys. 625, A125 (2019) Frisch, H., Bardos, C.: Diffusion approximations for the scattering of resonance-line photons. Interior and boundary layer solutions. J. Quant. Spectrosc. Radiat. Transf. 26, 119–134 (1981) Frisch, H., Faurobert, M.: Boundary-layer conditions for the transport of radiation in stars. Astron. Astrophys. 140, 57–66 (1984) Kourganoff, V.: Basic Methods in Transfer Problems. Radiative Equilibrium and Neutron Diffusion, 1st edn. Dover Publications, New York (1963); Oxford University Press, London (1952) Krein, M.G., Rutnam, M.A.: Linear operators leaving invariant a cone in a Banach Space. Am. Math. Soc. Transl. 26, 1–128 (1950); Russian original Uspekhi Mat. Nauk 3, 3–35 (1948) Larsen, E.W., Keller, J.B.: Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15, 75–81 (1974) Larsen, E.W., Pomraning, G.C., Badham, V.C.: Asymptotic analysis of radiative transfer problems. J. Quant. Spectrosc. Radiat. Transf. 29, 285–310 (1983) Levermore, C.D., Pomraning, G.C.: A flux-limited diffusion theory. Astrophy. J. 248, 321–334 (1981) Pomraning, G.C.: Initial and boundary conditions for flux-limited diffusion theory. J. Comput. Phys. 75, 73–85 (1988) Tartar, L.: The General Theory of Homogenization: A Personalized Introduction. Lecture Notes of the Unione Matematica Italiana. Springer, Berlin (2008)

Chapter 24

The Diffusion Approximation for Rayleigh Scattering

In Chap. 16, it is shown that the polarized radiative transfer equation for monochromatic Rayleigh scattering in a conservative semi-infinite medium can be solved exactly. We give in particular exact expressions for the emergent Stokes parameters I and Q in terms of the auxiliary functions Hl (μ) and Hr (μ). When the medium is non-conservative, which means that photons have a non-zero probability  to return to the thermal bath at each scattering, there is no exact expressions for the emergent Stokes parameters. As discussed in Chap. 18, the two scalar functions Hl (μ) and Hr (μ) are replaced by a matrix H(z), which satisfies a nonlinear integral equation. The latter can be solved numerically only. In this chapter, we show how to generalize to Rayleigh scattering the asymptotic expansion of the radiation field described in Chap. 23. The method is presented for a plane parallel slab with an unpolarized internal primary source and no incident radiation. For this geometry, the radiation field is cylindrically symmetric and can be represented by its two Stokes parameters I and Q. One key result of this approach is a scaling law for the depolarization of the emergent radiation by the multiple scattering process. The generalizing of this asymptotic method to a threedimensional medium is feasible, but requires that the radiation field be described by its three Stokes parameters I , Q, U . We follow step by step the method described in Chap. 23. In Sect. 24.1, the vector radiative transfer equation for the vector I(τ, μ), constructed with the (KQ) components I00 (τ, μ) and I02 (τ, √ μ) (see Eq. (14.44)) is written in terms of the rescaled space variable τ˜ = τ/ . The scaling factor is the same as in the scalar case. We then construct in Sect. 24.2 a diffusion equation valid far from boundaries. The boundary layer solution and the asymptotic matching between the interior and the boundary are presented in Sect. 24.3. In the scalar case, the boundary region can be described by a semi-infinite medium with conservative scattering. The same simplification holds for Rayleigh scattering and exact results for the Rayleigh diffuse reflection problem obtained in Sect. 16.7.1 are used to describe the boundary layer field. The coupling is a by-product of the constraint that the boundary layer contribution tends to zero in the interior, exactly as in the scalar case. The emergent © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_24

533

534

24 The Diffusion Approximation for Rayleigh Scattering

radiation and polarization rate are calculated in Sect. 24.4. The material presented in this chapter can be found in Frisch (2019) with additional technical details.

24.1 The Radiative Transfer Equation We recall here the polarized radiative transfer equation for Rayleigh scattering, introduced in Chap. 14. For a plane parallel slab and no incident radiation, the radiation field can be represented by a 2-component vector I (τ, μ). It satisfies the radiative transfer equation μ

1 ∂I (τ, μ) = I (τ, μ) − (1 − ) ∂τ 2



+1 −1

P(μ, μ )I (τ, μ ) dμ −  Q∗ (τ ),

(24.1)

where τ is the monochromatic optical depth, τ ∈ [0, T ], and Q∗ (τ ) an unpolarized primary source. We assume that there is no incident radiation. The boundary condition is I (0, μ) = 0,

μ ∈ [−1, 0];

I (T , μ) = 0,

μ ∈ [0, 1].

(24.2)

The factor  in front of Q∗ (τ ) ensures that Stokes I is of order unity in the interior of the medium. The matrix P(μ, μ ) is the Rayleigh phase matrix. Its expression depends on the choice of the representation of the polarized field. The two most standard representations employ the Stokes parameters I and Q or the components Il and Ir parallel and perpendicular to the scattering plane. They are related by I = Il + Ir , Q = Il − Ir .

(24.3)

The definition of Q is the same as in Sect. 16.6.5. To construct the diffusion approximation, it is algebraically simpler to represent radiation field with its (KQ) components I00 (τ, μ) and I02 (τ, μ), introduced in Sect. 16.6.5. Stokes I and Q can then be written as 1 I (τ, μ) = I00 (τ, μ) + √ (3μ2 − 1)I02 (τ, μ), 2 2 3 Q(τ, μ) = − √ (1 − μ2 )I02 (τ, μ). 2 2

(24.4)

24.1 The Radiative Transfer Equation

535

The polarization is carried by the component I02 (τ, μ). The combination of Eqs. (24.3) and (24.4) leads to  1 0 I + Il = 2 0  1 0 Ir = I + 2 0

 1 2 2 √ (3μ − 2)I0 , 2  1 √ I02 . 2

(24.5)

Along the normal to the medium, that is for μ = 1, the polarization is zero, hence Q = 0 and Il = Ir . This property is obvious in both Eqs. (24.4) and (24.5). The radiative transfer equation for the vector I(τ, μ) = (I00 (τ, μ), I02 (τ, μ)), may be written as μ

1 ∂I(τ, μ) = (1 − ) ∂τ 2



+1 −1

(μ )I(τ, μ ) dμ + Q∗ (τ ),

(24.6)

where 

 q ∗ (τ ) Q (τ ) = . 0 ∗

(24.7)

The phase matrix, already introduced in Eq. (14.49), may be written as ⎡

1 √ (3μ2 − 1) 2 2

⎤

1 ⎥ ⎢ ⎥. (μ) = ⎢ ⎣ 1 1 2 2 4 ⎦ √ (3μ − 1) (5 − 12μ + 9μ ) 4 2 2

(24.8)

The matrix (μ) is symmetric and an even function of μ. We now follow the method described in the scalar case. We rewrite the radiative transfer equation as μ

  ∂I(τ, μ) = (1 − )O[I(τ, μ)] +  I(τ, μ) − Q∗ (τ ) , ∂τ

(24.9)

where the operator O is defined by 1 O[V (μ)] ≡ V (μ) − 2



+1 −1

(μ )V (μ ) dμ ,

(24.10)

with V (μ) = (V1 (μ), V2 (μ)) an arbitrary 2-dimensional vector. For Rayleigh scattering, Stokes Q is entirely under the control of Stokes I (Faurobert-Scholl et al. 1997), and the latter behaves as the intensity of the scalar case. Hence for Rayleigh scattering, the thermalization length is also order of

536

24 The Diffusion Approximation for Rayleigh Scattering

√ 1/ . Here we assume an effectively thick slab, i.e., T  τeff , to ensure that the characteristic scale is controlled by  and not by T . We now introduce the rescaled optical depth τ˜ , and the rescaled optical thickness T˜ defined by τ˜ = ητ,

T˜ = ηT ,

(24.11)

√ .

(24.12)

where η=

Assuming that Q∗ (τ ) depends only on the rescaled variable τ˜ , and that the radiation field inside the medium is a function I  (τ˜ , μ) of the rescaled optical depth, we obtain the rescaled radiative transfer equation ημ

  ∂I  (τ˜ , μ) = (1 − η2 )O[I  (τ˜ , μ)] + η2 I  (τ˜ , μ) − Q∗ (τ˜ ) . ∂ τ˜

(24.13)

This is the starting point for the interior and boundary layer equations. The boundary conditions for this equation are given in Eq. (24.2). The range of variation of τ˜ is [0, T˜ ].

24.2 The Interior Radiation Field Expansion We look for solutions of the form I  (τ˜ , μ) = I 0 (τ˜ , μ) + ηI 1 (τ˜ , μ) + η2 I 2 (τ˜ , μ) + O(η3 ).

(24.14)

Regrouping all the terms with the same power of η, we obtain the hierarchy of equations: O[I 0 (τ˜ , μ)] = 0, O[I 1 (τ˜ , μ)] = μ O[I 2 (τ˜ , μ)] = O[I 0 (τ˜ , μ)] + μ

∂I 0 (τ˜ , μ), ∂ τ˜

  ∂I 1 (τ˜ , μ) − I 0 (τ˜ , μ) − Q∗ (τ˜ ) , ∂ τ˜

(24.15) (24.16) (24.17)

e.t.c. The first task is to determine the kernel of the operator O, that is to determine the vectors I 0 (τ˜ , μ), which satisfy Eq. (24.15). The integral term in Eq. (24.10)

24.2 The Interior Radiation Field Expansion

537

is independent of μ, hence the vector I 0 (τ˜ , μ) should also be independent of μ. Moreover, since ⎡ ⎤ 1 0  +1 1 ⎢ ⎥ (24.18) (μ) dμ = ⎣ 7 ⎦ , 2 −1 0 10 the second component of I 0 (τ˜ , μ) is necessarily zero. The solutions of Eq. (24.15) thus have the form   1 I 0 (τ˜ , μ) = i0 (τ˜ ) , (24.19) 0 where i0 (τ˜ ) is at this stage an arbitrary function of τ˜ . We now consider Eq. (24.16). Since the kernel of the operator O is not empty, the operator O is clearly not invertible. Actually, Eqs. (24.16) and (24.17) require a solvability condition. Known as the Fredholm alternative,1 it states that a necessary condition for a solution to exist is that the right-hand side be orthogonal to the kernel of the adjoint operator. Here the operator O is self-adjoint, therefore it has the same kernel as O. The solvability condition for Eq. (24.16) is 

+1

−1

⎡

⎤  di0 (τ˜ ) 1 dμ = 0, d τ˜ ⎦ · 0 0

⎣μ

(24.20)

where the symbol · stands for the scalar product. The integrand being an odd function of μ, the solvability condition is satisfied. It is easy to verify that the solution of Eq. (24.16) has the form ⎤ di0(τ˜ ) + c1 (τ˜ ) ⎦ μ , I 1 (τ˜ , μ) = ⎣ d τ˜ 0 ⎡

(24.21)

where the function c1 (τ˜ ) is arbitrary at this stage. We now consider Eq. (24.17). We infer from Eq. (24.15) that the first term in the right-hand is zero. Using Eq. (24.21) we can write ⎡ O[I 2 (τ˜ , μ)] = ⎣ μ 0

1

2d

2i

0 (τ˜ ) d τ˜ 2

⎤   dc1 ∗ (τ˜ ) − i0 (τ˜ ) − q (τ˜ ) ⎦ . +μ d τ˜

https://en.wikipedia.org/wiki/Fredholm_alternative.

(24.22)

538

24 The Diffusion Approximation for Rayleigh Scattering

The solvability condition for this equation is 

+1 −1

⎡ ⎣μ 0

2d

2i

0 d τ˜ 2

⎤  dc1 ∗ − i0 (τ˜ ) + q (τ˜ ) ⎦ · 1 dμ = 0. +μ d τ˜ 0

(24.23)

Performing the integration and the scalar product, we find the solvability condition 1 d 2 i0 (τ˜ ) − i0 (τ˜ ) + q ∗ (τ˜ ) = 0. 3 d τ˜ 2

(24.24)

This equation provides a diffusion equation for the isotropic and unpolarized leading term of the interior solution. It is identical to the diffusion equation for the scalar case established in Chap. 23. The boundary conditions for this equation are established in Sect. 24.3.2. The solution of Eq. (24.22) has the form ⎡ I 2 (τ˜ , μ) = ⎣

μ

2d

2i

0 (τ˜ ) d τ˜ 2

p i2 (τ˜ )

⎤   dc1 ∗ (τ˜ ) − i0 (τ˜ ) − q (τ˜ ) + c2 (τ˜ ) ⎦ +μ . d τ˜

(24.25)

The function c2 (τ˜ ) is at this stage an arbitrary function of τ˜ . The presence of the p term i2 (τ˜ ) indicates that the radiation field at order η2 is polarized (the superscript p stands for polarization and the subscript 2 indicates that the polarization appears at order 2). The polarization appearing at order η2 is created by the anisotropic term μ2 (d 2 i0 /d τ˜ 2 ). Inserting Eq. (24.25) into Eq. (24.22) and using the expression and normalization of (μ) (see Eqs. (24.8) and (24.18)), we find p

i2 (τ˜ ) =

√ 2 2 1 d 2 i0 (τ˜ ). 3 3 d τ˜ 2

(24.26)

The expression of Stokes Q given in Eq. (24.4) shows that in the interior of the medium, Q is of order  and can be written as Q(τ˜ , μ)  −(1 − μ2 )

1 d 2 i0 (τ˜ ). 3 d τ˜ 2

(24.27)

The factor  can be absorbed in the second order derivative. In the interior Stokes I is of order unity and given by i0 (τ˜ ). A word now about the function c1 (τ˜ ). To evaluate the effects of  on the polarization of the emergent radiation, that is to measure the deviations from the conservative case ( = 0), it is necessary to determine the term of order η2 in the expansion of the emergent radiation. To find this term, the hierarchy of equations

24.3 The Boundary Layer

539

must be continued up to order η3 . Some algebra, similar to the algebra carried out above, shows that c1 (τ˜ ) satisfies the homogeneous diffusion equation, 1 d 2 c1 (τ˜ ) − c1 (τ˜ ) = 0. 3 d τ˜ 2

(24.28)

The boundary conditions needed to solve this equation are derived in Sect. 24.3.2. An explicit expression of c1 (τ˜ ) for a uniform and unpolarized primary source is given in Eq. (24.66). The asymptotic expansion of the interior solution has been determined in this section, with the (KQ) representation of the radiation field. It could of course have been obtained with the Stokes parameters I and Q, but the algebra would have been significantly more cumbersome.

24.3 The Boundary Layer To pursue the asymptotic analysis, we now consider the boundary layers at τ = 0 and at τ = T . Since they have the same properties, it suffices to consider one of them, say the boundary at τ = 0. We proceed then essentially as in the scalar case. First we stretch the rescaled variable τ˜ in the direction perpendicular to the boundary. Since we are dealing with a one dimensional medium, the stretched variable s is simply s = τ˜ /η,

s ∈ [0, ∞].

(24.29)

For the boundary at τ = T , the variable s should be defined as s = (T˜ − τ˜ )/η. The radiation field in the boundary layer is written as b I(τ, μ) = I int  (τ˜ , μ) + I  (s, μ),

(24.30)

b where I int  (τ˜ , μ) is the interior solution discussed above and I  (s, μ) a boundary layer term. The equations satisfied by the boundary layer term are introduced in Sect. 24.3.1 and the matching between the interior field and boundary layer term carried out in Sect. 24.3.2.

24.3.1 The Boundary Layer as a Diffuse Reflection Problem The construction of the boundary solution can be performed essentially as described in the scalar case. Equations (24.31) to (24.36) are the vector version of the

540

24 The Diffusion Approximation for Rayleigh Scattering

corresponding equations in Sect. 23.4. The boundary layer term satisfies the homogeneous radiative transfer equation μ

∂I b (s, μ) = (1 − η2 )O[I b (s, μ)] + η2 I b (s, μ), ∂s

(24.31)

referred to for simplicity as the boundary layer equation. b Applied to the sum I int  (τ˜ , μ)+I  (s, μ), the condition that the radiation incident on the medium is zero leads to I b (0, μ) = −I int  (0, μ),

μ ∈ [−1, 0],

(24.32)

where I int  (0, μ) is the interior field at the surface in direction of negative μ. At infinity, the boundary layer term must satisfy the condition I b (s, μ) → 0,

s → ∞.

(24.33)

We now assume that I b (s, μ) has an expansion of the form I b (s, μ) = I b0 (s, μ) + ηI b1 (s, μ) + η2 I b2 (s, μ) + O(η3 ).

(24.34)

The boundary conditions at s = 0 and at infinity must be satisfied at all orders in η. Inserting Eq. (24.34) into Eq. (24.31), we see that I bk (s, μ), for k = 0, 1, satisfies the equation μ

∂I bk (s, μ) = O[I bk (s, μ)], ∂s

k = 0, 1,

(24.35)

μ ∈ [−1, 0].

(24.36)

with the surface boundary condition I bk (0, μ) = −I int k (0, μ),

Equations (24.35) and (24.36) also hold for k = 2 because I b0 (s, μ) = 0 (see Eq. (24.44)). For k ≥ 3, the equations for I bk (s, μ) will cease to be homogeneous, making the construction of the solution significantly more complicated. Equation (24.35), with the boundary condition in Eq. (24.36), describes conservative Rayleigh scattering in a semi-infinite medium with a given incident radiation. This problem has been treated in Sect. 16.7, where it is shown that the emergent radiation field has an exact expression in terms of the functions Hl (μ) and Hr (μ). These expressions, given in Eqs. (16.148) and (16.149), are 

μ  2[1 − c(μ + μ ) + μμ ]Hl(μ )Ilinc (−μ )

μ + μ 0 

inc + q(μ + μ )Hr (μ )Ir (−μ ) dμ , (24.37)

Il (0, μ) =

3 Hl (μ) 8

1

24.3 The Boundary Layer

541



μ  [1 + c(μ + μ ) + μμ ]Hr (μ )Irinc (−μ )

0 μ+μ  + q(μ + μ )Hl (μ )Ilinc (−μ ) dμ . (24.38)

Ir (0, μ) =

3 Hr (μ) 8

1

These equations will be employed at order η0 , η and η2 . The diffusion reflection problems for the monochromatic and the Rayleigh scattering processes belong to the same family as the boundary layer problems associated to the Boltzmann equation (Golse et al. 1988). They have produced a large body of rigorous mathematical results, in particular that the boundary layer solution tends to a constant at infinity (Bardos et al. 1984). For the scalar case, this constant is given in Eq. (23.37). For the Rayleigh scattering, the radiation field at infinity becomes isotropic and unpolarized and tends to a constant vector with components Il (∞) and Ir (∞) given by 3 Il (∞) = Ir (∞) = √ 4 2



1 0

 qHl(μ)Ilinc (−μ) + (μ + c)Hr (μ)Irinc (−μ) μ dμ. (24.39)

This result is established in Frisch (2019). When Ilinc (−μ) and Irinc (−μ) are replaced by a same constant, say F , the integral is the total radiative flux, F = Fl + Fr , of Milne problem (see Eq. (16.118)) and Eq. (24.39) leads to the identity given in Eq. (M.44). We recall an additional property of the diffuse reflection problem that will be used in the sequel. When the incident field is unpolarized and independent of μ, then the solution of Eq. (24.35) is isotropic, unpolarized and equal to the incident field everywhere in the medium. This means in particular that the emergent field and the field at infinity are equal to the incident field. It is easy to verify that a radiation field with these properties is solution of Eq. (24.35). We now use Eq. (24.39) to match the boundary layer and interior solutions and, thus doing, establish the boundary conditions needed to solve the diffusion equations satisfied by i0 (τ˜ ) and c1 (τ˜ ) given in Eqs. (24.24) and (24.28)).

24.3.2 Matching of the Interior and Boundary Layer Fields The asymptotic expansion of the interior radiation field has been determined in Sect. 24.2 for the (KQ) components. The components of the interior radiation field which appear in Eq. (24.39) are the (lr) components of the field at the surface, henceforth denoted Ilint (0, μ). The incident field appearing in Eq. (24.39) are thus given by Ilinc (−μ) = −Ilint (0, −μ),

Irinc (−μ) = −Irint (0, −μ).

(24.40)

542

24 The Diffusion Approximation for Rayleigh Scattering

The condition that the right-hand side of Eq. (24.39) is zero is now applied at zeroth and first order in η. This is sufficient to obtain the boundary conditions on the diffusion equations for i0 (τ˜ ) and c1 (τ˜ ) and to determine the emergent radiation up to order η2 , included. At zeroth order, the expression of I 0 (τ˜ , μ) given in Eq. (24.19) leads to [Ilint (0, μ)]0 = [Irint (0, μ)]0 =

1 i0 (0). 2

(24.41)

Since the constants q and c, and the functions Hl (μ) and Hl (μ) are positive, the condition that the integral in Eq. (24.39) is zero leads to i0 (0) = i0 (T˜ ) = 0.

(24.42)

This is the boundary condition for Eq. (24.24), the diffusion equation for i0 (τ˜ ). Since the interior field is zero at the surface, Eq. (24.36) leads to I b0 (0, μ) = 0,

μ ∈ [−1, 0].

(24.43)

Since I b0 (s, μ) satisfies the homogeneous Eq. (24.35), Eq. (24.36) leads to I b0 (s, μ) = 0. At zeroth order the boundary layer term is identically zero. At first order, we deduce from Eqs. (24.21) and (24.5) that   1 di0 |τ˜ =0 + c1 (0) . [Ilint(0, μ)]1 = [Irint (0, μ)]1 = μ 2 d τ˜

(24.44)

(24.45)

The condition that the integral in Eq. (24.39) is zero leads to c1 (0) = Lp

di0 |τ˜ =0 . d τ˜

(24.46)

where Lp is a constant. Equation (24.46) is the boundary condition for the diffusion equation satisfied by c1 (τ ) given in Eq. (24.28). The constant Lp is given by 1

[qHl(μ) + (μ + c)Hr (μ)]μ2 dμ L = 01 . 0 [qHl (μ) + (μ + c)Hr (μ)]μ dμ p

(24.47)

The subscript p stands again for polarization. The constant Lp plays for the Rayleigh scattering the role played by the constant L in the scalar case (see Eq. (23.41)). The combination of Eqs. (16.131) and (M.44) leads to p

Lp = q∞  0.712,

(24.48)

24.4 The Emergent Radiation Field

543

p

where q∞ is the common asymptotic value for τ → ∞ of the generalized Hopf functions q1 (τ ) and q2 (τ ) introduced in Sect. 16.6.3. We now have all the elements to determine the emergent radiation field.

24.4 The Emergent Radiation Field The emergent radiation field is given by the sum b I em (0, μ) = I int  (0, μ) + I  (0, μ),

μ ∈ [0, 1],

(24.49)

b where I int  (0, μ) is the surface value of the interior solution and I  (0, μ) the contribution to the emergent radiation of the boundary layer solution. The latter term is given by Eqs. (24.37) and (24.38) in which one must insert the (lr) components of the interior solution at τ = 0. The calculation of the emergent radiation field is described in Frisch (2019). The asymptotic expansion of the (lr) components may be written as em em em (μ) = η[Il,r (μ)]1 + η2 [Il,r (μ)]2 + O(η3 ), Il,r

(24.50)

where q 1 di0 [Ilem (μ)]1 = Hl (μ) √ |0 , 2 2 d τ˜

(24.51)

μ + c 1 di0 |0 , [Irem (μ)]1 = Hr (μ) √ 2 2 d τ˜

(24.52)

and [Ilem (μ)]2 = Hl (μ) [Irem (μ)]2

  1 q dc1 d 2 i0 |0 + t2l (μ) 2 |0 , √ 2 d τ˜ 2 d τ˜

(24.53)

  1 μ + c dc1 d 2 i0 r |0 + t2 (μ) 2 |0 . = Hr (μ) √ 2 d τ˜ 2 d τ˜

(24.54)

  3 10 (μ − c)(α1l − α3l ) + q(α1r − α3r ) , 8 3

(24.55)

Here, t2l (μ) = t2r (μ)

  3 5 8 r r l l 2 (μ + c)(α1 − α3 ) + q(α1 − α3 ) − √ (1 − μ ) . = 8 3 3 2

(24.56)

544

24 The Diffusion Approximation for Rayleigh Scattering

Table 24.1 Moments of Hl (μ) and Hr (μ) and other numerical constants. The two first moments of Hl (μ) and Hr (μ), the constants q and c, and the values of Hl (1) and Hr (1) are from Chandrasekhar (1960, p. 248). This reference provides also tables for Hl (μ) and Hr (μ), μ ∈ [0, 1], used to calculate numerically the third and fourth order moments α0l = 2.297

α0r = 1.197

q = 0.690

α1l = 1.349

α1r = 0.617

c = 0.873

α2l = 0.964

α2r = 0.416

Hl (1) = 3.4695

α3l = 0.752

α3r = 0.314

Hr (1) = 1.2780

l,r The constants αm are moments of Hl (μ) and Hr (μ) defined by

 l,r = αm

1

Hl,r (μ)μm dμ.

(24.57)

0

Their numerical values for m = 0, 1, 2, 3 are listed in Table 24.1. Equations (24.50) to (24.54) hold at the surface τ = T , the derivatives at τ = 0, being replaced by their values at τ = T . The diffusion equation in Eq. (24.24) shows that d 2 i0 /d τ˜ 2 |0 has the exact expression d 2 i0 /d τ˜ 2 |0 = −3q ∗(0). We recall that i0 (0) = 0. The definition of q ∗ (τ ) is given in Eq. (24.7) for an unpolarized primary source. There are a few comments to be made about the expression of the emergent field in Eq. (24.50). First, there is no term of order η0 . As we have shown, at zeroth order, the boundary layer term is identically zero and the interior field is unpolarized and zero at τ = 0. The first nonzero term is of order η. This is consistent with the well known result that the radiation field at the surface of a semi-infinite or effectively √ thick medium is of order , when the primary source is of order . We also note that the components of the term of order η have the same center-to-limb variation than the emergent radiation of the conservative Rayleigh Milne problem given in Eq. (16.118), the factor 3F /4π, with F the total radiative flux, being replaced by di0 (τ˜ )/d τ˜ |0 . A deviation from the center-to-limb variation of the Milne problem appears at order η2 . We discuss it in the next section on the polarization of the emergent radiation.

24.4.1 The Polarization Rate We define the polarization rate by p(μ) ≡

Irem (μ) − Ilem (μ) Qem (0, μ) = − . em Irem (μ) + Il (μ) I em (0, μ)

(24.58)

24.4 The Emergent Radiation Field

545

When the terms of order η2 are neglected, the polarization rate becomes equal to the polarization rate of the Rayleigh Milne problem, namely p0 (μ) =

(μ + c)Hr (μ) − qHl (μ) . (μ + c)Hr (μ) + qHl (μ)

(24.59)

This expression, to which we refer as Chandrasekhar’s law, provides an upper limit for the polarization rate due to the Rayleigh scattering when  = 0. The maximum value, attained at the limb (μ = 0) is 11.7%. When  increases, the mean number of scatterings decreases and so does the polarization. Quantitatively, the effect is contained in the terms of order η2 present in Eqs. (24.53) and (24.54). When these terms are taken into account, the polarization rate becomes p(μ) =

h− (μ) + η[f (0)h− (μ) + g(0)h− 2 (μ)]

h+ (μ) + η[f (0)h+ (μ) + g(0)h+ 2 (μ)]

+ O(η2 ).

(24.60)

To simplify the notation, we have introduced 1 h± (μ) ≡ √ [(μ + c)Hr (μ) ± qHl(μ)] , 2

(24.61)

r l h± 2 (μ) ≡ t2 (μ)Hr (μ) ± t2 (μ)Hl (μ).

(24.62)

and the functions f (τ˜ ) ≡

dc1 di0 / , d τ˜ d τ˜

g(τ˜ ) ≡

d 2 i0 di0 . / d τ˜ 2 d τ˜

(24.63)

The functions t2l (μ) and t2r (μ) are defined in Eqs. (24.55) and (24.56). The Chandrasekhar’s law is given by p0 (μ) =

h− (μ) . h+ (μ)

(24.64)

Numerical calculations of the asymptotic solution and of the initial radiative transfer equation in Eq. (24.6) have been performed in Frisch (2019) for an unpolarized uniform primary source Q∗ (τ ) = (q ∗ , 0). The diffusion equation for i0 (τ˜ ) in Eq. (24.24), with its associated boundary condition in Eq. (24.42), can be solved by a method of variation of the constant for any source term q ∗ (τ˜ ). For a uniform source q ∗ , the solution may be written as i0 (τ˜ ) =

q∗

√ 1 + e− 3T˜



√

1 − e−

3τ˜

√ 3T˜

+ e−

(1 − e

√ 3τ˜

 ) .

(24.65)

546

24 The Diffusion Approximation for Rayleigh Scattering

Fig. 24.1 The surface polarization rate (in percent). A comparison between the asymptotic predictions of Eq. (24.60) (the dashed lines) and numerical solutions of the polarized transfer equation, for√different values of  (the solid lines). The model is a slab with an optical thickness larger than , an unpolarized and uniform primary source and no incident radiation. From top to bottom,  = 10−5 , 10−3 , 10−2 , 10−1 . The asymptotic result is not plotted for  = 10−1 . The Chandrasekhar limit is graphically indistinguishable from the  = 10−5 results

The function i0 (τ˜ ) increases from zero at the surface to a value close to q ∗ in the middle of the slab and decreases to zero towards the other boundary. The diffusion equation for c1 (τ˜ ) given in Eq. (24.28) and its associated boundary condition in Eq. (24.46), has the solution p di0

c1 (τ˜ ) = L

d τ˜

|0

√ 3τ˜

e−

√

3(T˜ −τ˜ ) √ . e− 3T˜

+ e−

1+

(24.66)

The function c1 (τ˜ ) decreases from the boundaries to the middle of the slab, where it becomes exponentially small. The expressions of i0 (τ˜ ) and c1 (τ˜ ), and of their derivatives can be simplified by taking into account that T˜ = ηT  1. Thus for a uniform primary source √ f (0)  − 3Lp ,

√ g(0)  − 3.

(24.67)

In the general case of a slowly depth-dependent primary source, Eq. (24.63) shows that f (0) and g(0) will dependent weakly only on the primary source variations. Numerical solutions of the radiative transfer have been carried out in Frisch (2019) with a PALI2 code based on Nagendra et al. (1999), for a uniform primary source and values of  in the range [10−10, 10−1 ]. A review of numerical methods for polarized radiative transfer can be found in Nagendra and Sampoorna (2009). Figure 24.1 shows for different values of  the surface polarization rate p(μ),

2

PALI is an ALI (Accelerated Lambda Iteration) method for polarized radiation.

24.4 The Emergent Radiation Field

547

calculated numerically and predicted by the asymptotic expression in Eq. (24.60). For  ≤ 10−4 , the numerical results and the asymptotic expression essentially follow the Chandrasekhar limit p0 (μ). As shown in Fig. 24.1, deviations from p0 (μ) appear for larger values of . For  = 10−3 , the asymptotic expansion is still very close to the Chandrasekhar limit, but for  = 10−2 , the polarization rate is underestimated by roughly 30% for μ is around √ 0.1, the discrepancy increasing with μ. The expansion parameter for  = 10−2 is  = 10−1 . For larger values of , say 10−1 , the expansion parameter is about 0.3. The polarization rate predicted by the asymptotic expansion, which is not plotted in Fig. 24.1, is very far from the numerical result. Clearly the expansion cannot catch the rapid drop in the polarization rate at the limb, occuring between  = 10−2 and  = 10−1 . For the polarization, the domain of applicability of the asymptotic analysis is thus limited to values of , smaller than, say  = 10−3 , to be on the safe side. For the center-to-limb variation of Stokes I , the asymptotic expansion is valid up to  = 10−2 (see Frisch 2019, Fig. 4). An interesting quantity, which can be derived from the asymptotic analysis is the deviation from the Chandrasekhar’s law, given by the difference p0 (μ) − p(μ). It describes the depolarization due to the destruction of photons. Subtracting from Eq. (24.64) the expression of p(μ) given in Eq. (24.60), we obtain √ p0 (μ) − p(μ)  − g(0)

+ h− 2 (μ) − p0 (μ)h2 (μ) √ . h+ (μ) + η[− 3LP h+ (μ) + g(0)h+ 2 (μ)]

(24.68)

A numerical calculation of this expression for a uniform primary source shows that it can be fitted with the simple approximation p(μ)  p0 (μ) −

√

 0.516(1 − μ) + O(),

(24.69)

√ (see Frisch 2019, Fig. 3). The deviation from p0 (μ) scales as  and is, as expected, maximum at μ = 0 and equal to zero for μ = 1, as both p(1) and p0 (1) are zero. The factor 0.516 depends on the value of g(0), hence on the τ -dependence of the primary source. However, even with a τ -dependent primary source, the deviation from the Chandrasekhar’s law will remain essentially linear√in μ, since the factor g(0) in the denominator of Eq. (24.68) is multiplied by η = . The asymptotic results presented here can be used to check the accuracy of radiative transfer numerical codes and can prove very helpful to determine the μ and τ -discretization grids. For example, for very small values of , say 10−6 or less, the emergent values of Stokes I and Stokes Q should follow the μ-dependence of Chandrasekhar’s law in Eqs. (24.51) and (24.52). The constant of proportionality, √ di0 /d τ˜ |0 , should be close to 3q ∗ , when the primary source is unpolarized and uniform. As shown in Frisch (2019), the asymptotic expansion described above can be generalized to handle a polarized and anisotropic primary source and an incident radiation field. In the latter case the asymptotic expansion allows to calculate

548

24 The Diffusion Approximation for Rayleigh Scattering

a correction to the general formula of Chandrasekhar given in Eqs. (24.37) and √ (24.38). The correction is found to be of order , as expected, and confirms a result derived in (Domke 1973, Eq. (71)) with a less systematic approach.

References Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc. 284, 617–649 (1984) Chandrasekhar, S.: Radiative Transfer, 1st edn. Dover Publications, New York (1960); Oxford University Press (1950) Domke, H.: Multiple scattering of polarized light in a semiinfinite atmosphere with small true absorption. Soviet Astron. 17, 81–87 (1973); translation from Astron. Zhurnal 50, 126–136 (1973) Faurobert-Scholl, M., Frisch, H., Nagendra, K.N.: An operator perturbation method for polarized line transfer I. Non-magnetic regime in 1D media. Astron. Astrophys. 322, 896–910 (1997) Frisch, H.: Nonconservative Rayleigh scattering. A perturbation approach. Astron. Astrophys. 625, A125 (2019) Golse, F., Perthame, B., Sulem, C.: On a boundary layer problem for the nonlinear Boltzmann equation. C. Arch. Ration. Mech. Anal. 103, 81–96 (1988) Nagendra, K.N., Sampoorna, M.: Numerical methods in polarized line formation theory, in Solar Polarization 5, ASP Conference Series, vol. 405, pp. 261–273 (2009) Nagendra, K.N., Paletou, F., Frisch, H., Faurobert-Scholl, M.: Astrophys. Space Sci. Lib. 243, 127– 142 (1999), Kluwer Academic Publisher (Second Polarization Workshop, Bangalore, India, 12–16 October 1998)

Chapter 25

Anomalous Diffusion for Spectral Lines

In Chaps. 20, 21, and 22 we have presented some asymptotic results concerning the formation of spectral lines formed with the Doppler and Voigt complete frequency redistribution. We have shown that the random walk of the photons becomes a Lévy walk when photons undergo a large number of scatterings. One of the consequences is that the equation for the source term S(τ ), describing its large scale behavior in the interior of the medium, is an integral equation, involving a fractional Laplacian. We have also explained in Sect. 23.2 why the asymptotic analysis cannot be performed on the radiative transfer equation for the radiation field and discussed the large τ behavior of the resolvent function in infinite and sem-infinite media. In this chapter we show how to treat a finite slab, in particular we show that boundary regions can be treated as conservative semi-infinite plane parallel media, exactly as in the case of monochromatic and Rayleigh scattering. In contrast to these diffusion type scattering processes, the diffuse reflection problem for complete frequency redistribution has no exact solution. In other terms, there is no exact expression for the emergent radiation in terms of an incident radiation. One can nevertheless perform an asymptotic matching of the interior and boundary layer solution, which correctly predicts the magnitude of the emergent radiation and of the interior field. The interior and boundary layer analysis of the integral equation for the source function are presented in Sect. 25.1. The large scale behavior of the radiation field is analyzed in Sect. 25.2 and some results concerning resonance polarization are presented in Sect. 25.3. Scaling laws for a conservative slab of optical thickness T are established in Sect. 25.4, and scaling laws for the mean number of scatterings and the mean path length in Sect. 25.5. More detail on the material of this chapter can be found in Frisch and Frisch (1977) and in Frisch (1982, 1988).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_25

549

550

25 Anomalous Diffusion for Spectral Lines

25.1 Interior and Boundary Layer Expansions We show here how to construct an asymptotic expansion for the source function in a medium with boundaries. For simplicity we choose a semi-infinite medium. The generalization to a slab with an optical thickness much larger than the thermalization length τeff () is straightforward. The integral equation for the source function, given in Eq. (20.3) for an infinite medium, may be written as 

∞

S (τ ) = (1 − )

K(τ − τ )S (τ ) dτ + Q∗ (τ ).

(25.1)

0

Using the the normalization of K(τ ) to unity, it can be rewritten as S (τ ) − Q∗ (τ ) =  ∞ 1− S (τ )K2 (τ ), (25.2) (1 − ) K(τ − τ )[S (τ ) − S (τ )] dτ − 2 0 where K2 (τ ), a primitive of K(τ ), is defined in Eq. (20.35). To obtain a large scale equation valid in the interior, we follow the method described in Sect. 20.1. We introduce an asymptotic expansion, ˜ τ˜ ), S (τ )  a()S(

(25.3)

with τ˜ = τ/τc () and replace the kernels K(τ ) and K2 (τ ) by their leading terms for τ → ∞ (see Eqs. (19.18), (19.21), and (20.36)). We also assume that the primary source is a slowly varying function of τ , denoted Q˜ ∗ (τ˜ ). A limit equation, describing the large scale behavior of the source function in the interior, is obtained by choosing a() ∼ 1/,

√ D () ∼ 1/ − ln , τeff

V τeff () ∼ a/ 2 ,

(25.4)

where D and V stands for the Doppler and Voigt profile. Not surprisingly, the scaling laws are the same as for an infinite medium. For the Doppler profile, the interior equation may be written as ˜ τ˜ ) − Q˜ ∗ (τ˜ ) = S(



∞ S( ˜ τ˜ ) ˜ τ˜ ) − S( 0

4(τ˜

− τ˜ )2

d τ˜ −

˜ τ˜ ) S( . 4τ˜

(25.5)

We have as similar equation for the Voigt profile, with the denominators 4(τ˜ − τ˜ )2 and 4τ˜ replaced by 6(|τ˜ − τ˜ |)3/2 and 3τ˜ 1/2 . Exact method of solutions developed in preceding chapters can be applied to these singular integral equations to determine ˜ τ˜ ) for τ˜ → 0. One can thus show that S( ˜ τ˜ ) tends to zero, as τ˜ 1/2 the behavior of S( 1/4 for the Doppler profile and as τ˜ for the Voigt profile.

25.1 Interior and Boundary Layer Expansions

551

The effect of the surface at τ = 0 is to allow the escape of photons. In the vicinity of the boundary, their escape through the boundary dominates over their creation and destruction by inelastic collisions and the source function satisfies the homogeneous equation 

∞

S(τ ) =

K(τ − τ )S(τ ) dτ .

(25.6)

0

This equation can also be established rigorously by taking the limit  → 0 in Eq. (25.1). Equation (25.6) is the integral equation for conservative scattering studied in Sect. 9.2. We write its solution as S(0)S bl (τ ), with S(0) the surface value of the source function, an undetermined constant at this stage. Its magnitude will be found by matching the interior and boundary layer solutions. Equation (25.6) has solutions which grow to infinity as τ → ∞, whereas the ˜ τ˜ ) has solutions which tend to zero as τ˜ → 0. We now assume interior solution S( that the boundary layer and interior expansion have a common domain of validity. This assumption, commonly found in asymptotic analysis (see e.g. Cole 1968), leads to the matching condition1 ˜ τ˜ )] = 0. lim [S(0)S bl (τ ) − a()S(

→0

(25.7)

˜ τ˜ ) decreases as For the Doppler profile, S bl (τ ) increases as τ 1/2(ln τ )1/4 while S( 1/2 bl 1/4 ˜ τ˜ and for the Voigt profile S (τ ) increases as τ while S(τ˜ ) decreases as τ˜ 1/4 (see Eqs. (9.53) and (9.54)). In both cases, a matching is possible and will allow us to evaluate S(0). The matching technique for monochromatic scattering described in Chap. 23 is somewhat different, but the spirit is the same. ˜ τ˜ ) by their asymptotic behavior, τ by Replacing in Eq. (25.7) S bl (τ ) and S( τ˜ τc (), using a() ∼ 1/, and in the case of the Doppler profile, making the asymptotically valid approximation ln[τ˜ τc ()]  ln τc ()  − ln ,

(25.8)

√ S(0) ∼ 1/ ,

(25.9)

we obtain

for the Doppler and the Voigt profiles. Equation (25.9) provides for S(0) the correct order of magnitude for a primary source of order unity. We recall that for a semiinfinite medium, with a constant primary √ source Q∗ , the surface value of the source ∗ function has the exact value S(0) = Q / . Examples of surface boundary layers A more secure procedure, sometimes needed, is to introduce an intermediate scale t = τ˜ /δ with δ chosen in such a way that δτc () → ∞ as δ and  tend to zero. With this choice, τ˜ → 0 and τ → ∞ when letting δ → 0, when t is of order unity.

1

552

25 Anomalous Diffusion for Spectral Lines

and interior solutions are shown for several values of  and different primary sources in Frisch and Froeschlé (1977) (see also Frisch 1988). The scaling laws obtained in Eq. (25.4) also hold when a continuous absorption is present in the frequency range of the line. It suffices to replace  by the effective destruction probability ¯ defined in Eq. (8.10), namely ¯ =  + (1 − )βF (β).

(25.10)

Here, β is the ratio of the continuous to the line opacity coefficients. The function F (β) is given in Eq. (8.12).

25.2 The Radiation Field In the interior, the radiation field I (τ, x, μ) is also a slowly varying function of τ . It can be described by a function of I˜(τ˜ , x, μ), such that I (τ, x, μ)  I˜(τ˜ , x, μ),

(25.11)

given by I˜(τ˜ , x, μ) 



∞ τ˜



e−(τ˜ −τ˜ )τeff ()ϕ(x)/μ

ϕ(x) ˜ τ˜ ) d τ˜ , τeff ()S( μ

μ > 0.

(25.12)

There is a similar equation for μ < 0. For simplicity, we have assumed that the ˜ τ˜ ) is of order unity in the interior. In primary source term is of order  so that S( the limit  → 0, the exponential becomes essentially a Dirac distribution since τc () → ∞. Hence ˜ τ˜ ). lim I˜(τ˜ , x, μ)  S(

→0

(25.13)

This limit is valid when τeff ()ϕ(x) is significantly larger than unity. Therefore for frequencies about the line center, which satisfy ϕ(x) > 1/τeff (), the photons have a distribution, which is flat in frequency √ and isotropic in direction. For example, for  = 10−4 and ϕ(x)  exp(−x 2 )/ π, the frequency profile of the radiation field will be flat up to, roughly, x = 3, that is up to 3 Doppler widths. To examine the radiation field at large frequencies, such that ϕ(x) < 1/τeff (), ˜ μ) defined by we introduce a rescaled profile ϕ˜ and a new radiation field I˜˜(τ˜ , ϕ, I (τ, x, μ)  I˜˜(τ˜ , ϕ, ˜ μ),

ϕ˜ = ϕτc ().

(25.14)

25.3 Resonance Polarization

553

For  → 0, lim I˜˜(τ˜ , ϕ, ˜ μ) 

→0



∞ τ˜



˜ ˜ τ˜ ) e−(τ˜ −τ˜ )ϕ/μ S(

ϕ˜ d τ˜ , μ

μ > 0.

(25.15)

The right-hand side shows that the radiation field is not isotropic, in other terms, the photons are not thermalized at optical depth of order τeff . This property, already pointed out in Avrett and Hummer (1965), has an important consequence for numerical calculations. In numerical calculations, one always deals with a finite range of optical depths, say [0, τmax ] and a finite range of frequencies, say [−xmax , +xmax ]. The value of xmax should be chosen in such a way that τmax ϕ(xmax ) 1. When this condition is not satisfied, one observes a spurious increase of the radiation field at large optical depths because photons with large frequencies are artificially trapped inside the medium. At the surface τ = 0, the emergent radiation is given by 

∞

I (0, x, μ) =

e−τ ϕ(x)/μ

0

ϕ(x) S(τ ) dτ, μ

μ > 0.

(25.16)

For a given frequency and a given direction μ, the main contribution to I (0, x, μ) comes from optical depth values such that τ ϕ(x)/μ is of order one. This remark leads to the Eddington–Barbier approximation, already introduced in Eq. (2.68), namely I (0, x, μ)  S(

μ ). ϕ(x)

(25.17)

This equation is exact when the source function is a linear function of τ . To analyze the frequency profile of the emergent radiation it is convenient to introduce a characteristic frequency xc , such that ϕ(xc ) ∼ 1/τeff (). Assuming μ = 1, we see that the emergent radiation at frequencies smaller than xc will come from the boundary layer and at frequencies larger than xc , from the interior. For a uniform primary source Q∗ , the full profile is an absorption line, much broader than the absorption profile because of the effects of multiple scattering. At line center, the √ intensity is around . When  decreases, the line becomes deeper and deeper and simultaneously broader and broader, its width following the variation of xc () (see e.g. Ivanov 1973, p. 264).

25.3 Resonance Polarization In Sect. 14.4.1 we consider the resonance polarization of spectral lines formed with complete frequency redistribution. We show that the radiation field can be described

554

25 Anomalous Diffusion for Spectral Lines

by its (KQ) components I0K (τ, x, μ), K = 0, 2, and that the corresponding 2-dimension source vector S(τ ), with components S0K (τ ), K = 0, 2, satisfies a convolution integral equation. For a semi-infinite medium, with a destruction probability , it can be written as 

∞

K(τ − τ )S(τ ) dτ + Q∗ (τ ).

(25.18)

  |τ |ϕ(x) dμ dx, W (μ)ϕ (x) exp − μ μ

(25.19)

S(τ ) = (1 − ) 0

The kernel is defined by 1 K(τ ) ≡ 2

+∞  1



−∞

0

2

and the matrix W (μ) is given by √ W √ (1 − 3μ2 ) 2 2

⎡ 1

⎤

⎢ ⎥ ⎥, W (μ) = ⎢ ⎣ √W ⎦ W √ (1 − 3μ2 ) (5 − 12μ2 + 9μ4 ) 4 2 2

(25.20)

where W is a depolarization parameter, smaller or equal to unity (see Sect. 15.1). The elements Kij have the normalization: 



+∞ −∞

K11 (τ ) dτ = 1,



+∞

−∞

Kij (τ ) dτ = 0, i = j,

+∞ −∞

K22 (τ ) dτ =

7 W. 10 (25.21)

The τ -dependence of Kij (τ ) is shown in Fig. 14.1. Large τ asymptotic expansions of Kij (τ ) can be found in Ivanov et al. (1997) and, for the Doppler profile, also in Faurobert-Scholl and Frisch (1989). In the second reference, the expansions of Kij (τ ) for small and large τ are constructed with Padé approximants.2 At large τ , the leading terms of Kij (τ ) vary with τ as the leading term of the scalar kernel K(τ ) (see Fig. 14.1). When the primary source is unpolarized, the integral equations for the components of S(τ ) are  S00 (τ )

∞

= (1 − ) 0

[K11 (τ − τ



)S00 (τ ) +

K12 (τ − τ



)S02 (τ )] dτ



+ Q∗ (τ ), (25.22)

2

An error in one of the numerical coefficients is pointed out in Ivanov et al. 1997, Section 3.

25.3 Resonance Polarization

 S02 (τ )

∞

= (1 − ) 0

[K21 (τ − τ

555

)S00 (τ ) + K22 (τ

−τ



)S02 (τ )] dτ

 .

(25.23)

The component S00 (τ ) drives the polarization, contained in the component S02 (τ ). The asymptotic analysis can be carried out as described in Sect. 25.1. For a primary source of order , the component S00 (τ ) is of order one in the interior and its leading term satisfies the same large scale equations as the scalar source function S(τ ), while the component S02 (τ ) is of order . In the boundary layer, the source vector is solution of Eqs. (25.22) and (25.23) with  = 0 and Q∗ (τ ) = 0. The boundary layer is studied in great detail in Ivanov et al. (1997). The component S00 (τ ) behaves as the scalar source function S(τ ), increasing at infinity as τ 1/2 (ln τ )1/4 for the Doppler profile and as τ 1/4 a −1/4 for the Voigt profile. The behavior of the component S02 (τ ) is derived in Faurobert-Scholl and Frisch (1989) from a simplified (too simplified) version of Eq. (25.23), which leads to S02 (τ ) ∼ τ −1/2 ln τ −1/4 . A systematic and very non trivial asymptotic analysis based on the nonlinear integral equation for the I-matrix carried out in Ivanov et al. (1997) for the Doppler profile shows that the correct behavior is S02 (τ )

√ W τ −1/2 ln τ −9/4 [1 + O(1/ ln τ )]. ∼ 10 − 7W

(25.24)

For the Voigt profile S02 (τ ) ∼ a 1/4 τ −1/4 (Faurobert-Scholl and Frisch 1989). A very detailed analysis of the rate of polarization of the emergent radiation for the Doppler profile, based on numerical solutions and on an asymptotic analysis of the nonlinear integral equation for the I-matrix (defined in Sect. 18.2.3), is carried out in Ivanov et al. (1997). The polarization at the surface, p(x, μ) = |Q(0, x, μ)|/I (0, x, μ), is calculated for the Doppler profile with several values of  and of the depolarization parameter W . The authors find that the limb polarization at line center, for W = 1, behaves as √ p(0, 0) = (9.443 − 38.05 ) %.

(25.25)

For  = 0, the line center limb polarization is somewhat lower than the conservative Rayleigh scattering value 11.7%. As suggested in Ivanov et al. (1997), the frequency redistribution is lowering the anisotropy of the radiation field, which results in a smaller polarization. The influence of the frequency redistribution appears √ however rather weak, at least, for the Doppler profile. The correction scales as . A similar scaling, established with a systematic asymptotic analysis,√ holds for monochromatic Rayleigh scattering (see Sect. 24.4.1). The reason for the -scaling in Eq. (25.25) is not obvious.

556

25 Anomalous Diffusion for Spectral Lines

25.4 Scaling Laws for a Slab The asymptotic results established for a semi-infinite medium with a small destruction probability  are easily extended to a finite slab, with a total frequencyintegrated optical thickness T , much larger than unity. A slab is said to be effectively thin when T τeff () and effectively thick when, on the opposite, T  τeff (). This very useful classification has been introduced in Avrett and Hummer (1965). In the effectively thick case, the slab behaves exactly as a semi-infinite medium, except that it has two boundary layers, one on each side of the slab. In the effectively thin case, the dominant mechanism for the disappearance of photons from the medium is the escape through the boundaries. The slab behaves essentially as a conservative medium, and the small expansion parameter is 1/T . In this section we consider the effectively thin case, corresponding to  = 0 and T  1, and present a brief summary of the scaling laws established in Frisch (1982) for the source function in the interior of the slab and at the surface. The asymptotic analysis can be carried out essentially as in the case of a semi-infinite medium, with the expansion parameter η = 1/T .

(25.26)

The source function, denoted Sη (τ ), to indicate the dependence on the optical thickness T of the slab, satisfies the integral equation  Sη (τ ) =

T

K(τ − τ )Sη (τ ) dτ + Q∗ (τ ).

(25.27)

0

To treat the interior of the slab, we assume an expansion of the form ˜ τ˜ ) + h.o.t.], Sη (τ )  a(η)[S(

(25.28)

where τ˜ = ητ . Proceeding as in Sect. 25.1, we rewrite the integral equation in a form appropriate for the asymptotic analysis, namely, 

Sη (τ ) − Q∗ (τ ) = T 0

1 K(τ − τ )[Sη (τ ) − Sη (τ )] dτ − Sη (τ ) [K2 (τ ) + K2 (T − τ )]. (25.29) 2

Replacing the kernels and the source function by their asymptotic expansion, we ˜ τ˜ ), similar to Eq. (25.5) and the scaling laws obtain a limit equation for S( a D (η) ∼

1 − ln η, η

a V (η) ∼

1 aη

1/2 ,

(25.30)

25.4 Scaling Laws for a Slab

557

where D and V stand as usual for the Doppler and Voigt profile. The large scale equations for the source function can be found in Frisch (1982). They were also established by (Ivanov 1973, p. 392) with a similar method. Introducing the mean primary source 1

Q  = T ∗



T

Q∗ (τ ) dτ,

(25.31)

0

we obtain D ∼ Q∗ T Smax

√ ln T ,

V Smax ∼ Q∗ 

T 1/2 , a 1/2

(25.32)

for the order of magnitude of the source function inside the slab. Replacing Q∗  by B, we recover scaling laws deduced from escape probability arguments (see e.g. Mihalas 1978, p. 347). The same scaling laws, deduced from a detailed analysis of the mean number of scatterings are also given in (Ivanov 1973, p. 430). The matching of the interior and boundary layers can be performed exactly as described for a semi-infinite medium in Sect. 25.1. The asymptotic behavior of the interior solution for τ˜ → 0 and that of the boundary layer solution for τ → ∞ are those of the semi-infinite medium. Equation (25.7), with  replaced by η leads to S D (0) ∼ Q∗ T 1/2 (ln T )1/4 ,

S V (0) ∼ Q∗ a −1/4T 1/4 .

(25.33)

The same laws hold of course at the other boundary. The surface scaling laws can also be derived from escape probability arguments (Jones and Skummanich 1980; see also Athay and Skumanich 1971). As pointed out in Avery et al. (1969), the scaling laws for the source function in the interior and at the surface hold in a bounded medium of arbitrary shape, provided T is chosen to be the smallest dimension of the medium. It is shown in Ivanov (1973, Chap. VIII; also Ivanov 1991), that the scaling laws for the interior and surface value of the source function can also be written as Smax ∼

Q∗  , K2 (T )

Q∗  S(0) ∼ √ . K2 (T )

(25.34)

To obtain these scaling laws, the integral equation for the source function is rewritten as  Sη (τ ) = (1 − ¯ ) 0

T

KT (τ − τ )Sη (τ ) dτ + Q∗ (τ ),

(25.35)

558

25 Anomalous Diffusion for Spectral Lines

with a renormalized kernel   KT (τ ) = K(τ ) 2

−1

T

K(τ ) dτ

,

(25.36)

0

and a renormalized destruction probability ¯ ≡  + (1 − )K2 (T ).

(25.37)

T We recall that 2 0 K(τ ) dτ = 1 − K2(T ). The new kernel KT (τ ) can be continued by a function, which is zero for |τ | > T , and then the upper limit of the integral set to +∞. For an effectively thin slab, one can set  = 0. The parameter ¯ takes into account the escape of photons through the boundaries. The scaling laws in Eq. (25.34) can then be √ derived from the semi-infinite medium scaling laws, a() ∼ 1/ and S(0) ∼ 1/  (see Eqs. (25.4) and (25.9)). In Chap. 8, a renormalized kernel and a renormalized destruction parameter ¯ are also introduced to describe the effects a continuous absorption. Radiative transfer in a plane layer of finite thickness is investigated in detail in (Ivanov 1973, Chap. VIII). The reader can find the definitions of the X-function and Y -function and many more asymptotic results, concerning in particular the Schuster problem (illumination of a slab by an incident radiation). For this problem, the primary source behaves as K2 (τ ) and√the radiation emerging from the side opposite to the illuminated surface scales as K2 (T ) (Ivanov 1973, p. 399). The Schuster problem does not satisfy the conditions of the asymptotic analysis described above.

25.5 Mean Number of Scatterings and Mean Path Length In Chap. 21, we have considered the mean displacement of photons having undergone a given number n of scatterings. In this section we ask a different question, namely, what is the the mean number of scatterings, N, that photons can undergo, knowing that they have a destruction probability  per scattering, or a probability β to be absorbed by a continuous absorption, or are random walking in a medium with a typical optical depth T . Another interesting quantity is the mean accumulated distance L between creation and destruction (or escape). This mean distance is proportional to the mean time time spent by photons in a medium (when the very small capture time is ignored). We present in this section some scaling laws for N and L. We consider a slab with optical thickness T . We assume that  and β are nonzero and that there is no continuum emission in the frequency range of the line, and no incident radiation. The radiative transfer equation may be written as μ

∂ I (τ, x, μ) = [ϕ(x) + β]I (τ, x, μ) − ϕ(x)S(τ ), ∂τ

(25.38)

25.5 Mean Number of Scatterings and Mean Path Length

559

with  S(τ ) = (1 − )

+∞ −∞

ϕ(x)J (τ, x) dx + Q∗ (τ ).

(25.39)

Here J (τ, x) is the direction-averaged radiation field. We also assume,  1, β 1, and T  1. Several, somewhat different definitions of N can be found in the literature. They all lead the same order of magnitude for N. In Hummer (1964) and Ivanov (1973), N is defined as 



T

N ≡

T

S(τ ) dτ/ 0

Q∗ (τ ) dτ,

(25.40)

0

that is as the ratio of the total number of emissions per unit time in a given volume, to the number of primary photons emitted per unit time in the same volume. In Hummer and Kunasz (1980), S(τ ) is replaced by the scattering term in the radiative transfer equation. Equation (25.40) can be used for an infinite or semi-infinite medium, provided the integrals over τ are replaced by convenient limits. The primary source term being of order one, N will scale as S(τ ). Thus according to the asymptotic analysis of Sect. 25.4)

N ∼ a(¯ ).

(25.41)

where ¯ takes into account the effects of inelastic collisions, continuous absorption, and escape through boundaries (see Eqs. (25.10) and (25.37)). The dependence of

N on , on β, and on the slab thickness T is shown in Tables 25.1 and 25.2. Table 25.1 Scaling laws for complete frequency redistribution with a Doppler profile. In the first column, the small expansion parameter; in row 1: β = 0 and T = ∞, in row 2:  = 0 and T = ∞, and in row 3: β =  = 0. The four subsequent columns show τeff , the characteristic scale of variation of the radiation field, xc , the characteristic frequency defined by τeff ϕ(xc ) ∼ 1, the mean number of scatterings N, and the mean path length L Parameter  β 1/T

τeff √ ( − ln )−1 [β(− ln β)]−1 T

xc √ − ln  √ − ln β √ ln T

N  −1 √ [β − ln β]−1 √ T ln T

L ∞ β −1 T ln T

Table 25.2 Scaling laws for complete frequency redistribution with a Voigt profile. a Voigt parameter of the line. Same presentation as in Table 25.1 Parameter  β 1/T

τeff a −2 β −1 T

xc (a)−1 (a/β)1/2 (aT )1/2

N  −1 (1/aβ)1/2 (T /a)1/2

L ∞ β −1 T

560

25 Anomalous Diffusion for Spectral Lines

The mean path length L can be defined as 

T

L ≡



−∞

0



+∞

T

J (τ, x) dx dτ/

Q∗ (τ ) dτ.

(25.42)

0

The concept was introduced by Ivanov (1970) for complete frequency redistribution in an absorbing medium. Because the mean path length is an accumulated distance,

L is larger or equal to N (see Tables 25.1 and 25.2). In an infinite medium,

L becomes infinite when there is no mechanism to limit the path-length such as continuous absorption or escape from the medium. Inelastic collisions limit the number of scatterings but place no constraint on the individual step length, which may tend to infinity as |x| → ∞. The mean values N and L appear in an interesting energy conservation equation introduced by Hummer and Kunasz (1980). The integration of Eq. (25.38) over optical depth, direction and frequency yields  N + β L + fe = 1,

(25.43)

where fe is the emergent flux normalized by the total primary source, that is  fe =

+∞  +1 −∞

−1

 μ[I (T , x, μ) − I (0, x, μ)] dμ dx/

T

Q∗ (τ )dτ.

(25.44)

0

The first term in Eq. (25.43) is the rate of energy loss in collisional destructions, the second term the rate of energy loss in absorption by the continuous opacity, and the third term the energy loss by the radiation escaping from the boundaries of the slab. When the medium is infinite, fe = 0. Hence, one immediately finds

N = 1/, when photons are destroyed by inelastic collisions only, and L = 1/β when they are destroyed by continuous absorption. These results are independent of the absorption profile ϕ(x) and also hold for partial frequency redistribution. The energy argument shows how N scales with  and L with β, but they do not show how they are related. We describe here a simple method to determine L in terms N. The formal solution of Eq. (25.38) allows us to write 

L =

T





T





T

K11 (τ − τ, β) dτ dτ /

S(τ ) 0



0

Q∗ (τ ) dτ ,

(25.45)

dμ . μ

(25.46)

0

where K11 (τ, β) =

1 2

 0

1  +∞ −∞

ϕ(x)e−|τ |(ϕ(x)+β)/μ dx

The notation K11 follows the rule suggested by (Ivanov 1973, p. 76). We note that K11 (τ, β) = F (β)L1 (τ, β), where L1 (τ, β) and F (β) are introduced in Chap. 8 (see Eqs. (8.7) and (8.11)). The function F (β) tends to infinity as β tends to zero,

25.5 Mean Number of Scatterings and Mean Path Length

561

but βF (β) tends to zero. Performing in Eq. (25.45), the integration of K11 (τ, β) over τ , we obtain    1  +∞  T 1 −ντ

ϕ(x) 1



−ν(T −τ ) 1 − [e dτ S(τ ) dμ dx +e ] ,

L = ϕ(x) + β 2 T Q¯ ∗ 0 0 −∞ (25.47) where ν = (ϕ(x) + β)/μ and T Q¯ ∗ is the integral of the primary source (see Eq. (25.31)). When the effects of the continuous absorption dominate over the escape from the boundaries, that is when βF (β) > K2 (T ), the exponential terms in the square bracket can be neglected. One obtains

L  F (β) N.

(25.48)

√ For the Doppler profile F D (β) ∼ − ln β and for the Voigt profile F D (β) ∼ (a/β)1/2 (see Eq. (8.12)). When the effects of the continuous absorption become negligible, compared to the escape through the boundaries, ϕ(x)/(ϕ(x) + β)  1. To evaluate the integral over frequency we introduce a characteristic √ frequency xc defined by the condition T ϕ(xc )  1. For the Doppler profile xcD ∼ ln T and for the Voigt profile xcV ∼ a 1/2T 1/2. For |x| < xc , the exponential terms in the square bracket can be neglected. One obtains

L ∼ xc N.

(25.49)

For |x| > xc , a Taylor expansion of the exponentials terms shows that the square bracket varies as ϕ(x). The integral over x has a finite value, but its contribution can be neglected when compared to the contribution coming from the values of x in the range |x| < xc . A more sophisticated derivation of the scaling laws for L can be found in (Ivanov 1973, p. 437). It is also interesting to compare L with the characteristic scale τeff . Tables 25.1 and 25.2 show that L scales as τeff for the Voigt profile, but that it is slightly larger than τeff for the Doppler profile. For the Voigt profile, L and τeff are fully controlled by a single long flight. For the Doppler effect, several long flights are contributing to L and τeff . They add up together for L, but not for τeff . The random walks in Fig. 21.2 illustrate this remark. It is also interesting to compare the scaling laws for τeff and N. The random walk analysis√presented in Sect. 21.1.3, shows that cn ∼ an2 for the Voigt profile and cn ∼ n/ ln n for the Doppler profile. Identifying cn with τeff and n with N, we obtain

N D τeff ∼ √ , ln N

V ∼ a N2 . τeff

(25.50)

562

25 Anomalous Diffusion for Spectral Lines

The laws in Tables 25.1 and 25.2 are consistent with these relations. Finally a few words about monochromatic scattering. In this case, the random walk of the photons is of the diffusive type and there is no frequency redistribution. Hence, τeff ∼ N1/2 ,

L ∼ N.

(25.51)

√ For an effectively thick slab τeff ∼ N1/2 ∼ 1/  and for an effectively thin slab τeff ∼ N1/2 ∼ T .

References Athay, R.G., Skumanich, A.: Thermalization length and mean numbers of scatterings for line photons. Astrophys. J. 170, 605–611 (1971) Avery, L.W., House, L.L., Skumanich, A.: Radiative transport in finite homogeneous cylinders by the Monte Carlo technique. J. Quant. Spectrosc. Radiat. Transf. 9, 519–531 (1969) Avrett, E.H., Hummer, D.G.: Non-coherent scattering II: line formation with a frequency independent source function. Mon. Not. R. astr. Soc. 130, 295–331 (1965) Cole, J.D.: Perturbation Methods in Applied Mathematics. Blaisdell, Waltham (1968) Faurobert-Scholl, M., Frisch, H.: Asymptotic analysis of resonance polarization and escape probability approximations. Astron. Astrophys. 219, 338–351 (1989) Frisch, H.: Non-LTE transfer with complete redistribution. Scaling laws for a slab. J. Quant. Spectrosc. Radiat. Transf. 28, 377–381 (1982) Frisch, H.: Radiative transfer with frequency redistribution. In: Chmielewsky, Y., Lanz, T. (eds.) Radiation in Moving Gaseous Media, Saas-Fee 18th Advanced Course. Swiss Society of Astronomy and Astrophysics. Publication de l’Observatoire de Genève, pp. 339–448 (1988) Frisch, H., Froeschlé, Ch.: Non-LTE transfer IV. A rapidly convergent iterative method for the Wiener-Hopf integral equations. Mon. Not. R. astr. Soc. 181, 281–292 (1977) Frisch, U., Frisch, H.: Non-LTE transfer III. Asymptotic expansions for small . Mon. Not. R. astr. Soc. 181, 273–280 (1977) Hummer, D.G.: The mean number of scatterings by a resonance-line photon. Astrophys. J. 140, 276–281 (1964) Hummer, D.G., Kunasz, P.: Energy loss by resonance line photons in an absorbing atmosphere. Astrophys. J. 236, 609–618 (1980) Ivanov, V.V.: The mean free path of a photon in a scattering medium. Astrophysics 5, 355–367 (1970); translation from Astrofizika 6, 643–662 (1970) Ivanov, V.V.: Transfer of Radiation in Spectral Lines, Washington Government Printing Office, NBS Spec. Publ., vol. 385 (1973), translation by D.G. Hummer and E. Weppner; Russian original: Radiative Transfer and the Spectra of Celestial Bodies, Nauka, Moscow (1969) Ivanov, V.V.: A hundred year of the radiative transfer integral equation, in “Centennial of the Integral Transport Equation: Symposium in Leningrad. Transp. Theory Stat. Phys. 20, 525– 539 (1991) Ivanov, V.V, Grachev, S.I., Loskutov, V.M.: Polarized line formation by resonance scattering II. Conservative case. Astron. Astrophys. 321, 968–984 (1997) Jones, H.P., Skummanich, A.: The Physical effects of radiative transfer in multidiemsnional media including models of the solar atmosphere. Astrophys. J. Suppl. Series 42, 221–240 (1980) Mihalas, D.: Stellar Atmospheres, 2nd edn. W.H. Freeman and Company, San Francisco (1978)

Chapter 26

Asymptotic Results for Partial Frequency Redistribution

In Chap. 25, we present some asymptotic results for spectral lines formed with complete frequency redistribution. This expression meaning that the frequencies of the absorbed and scattered photons are uncorrelated. As indicated in Appendix J of Chap. 13, even for a simple two-level atom, when collisions are taken into account, the frequency redistribution will be a combination of coherent (in frequency) scattering and complete frequency redistribution. Complete frequency redistribution is an assumption, which in practice holds only for weak lines. The importance of coherent scattering is already stressed in Spitzer (1944). It was further investigated by Unno (1952) and Woolley and Stibbs (1953). A set of four basic redistribution functions (also called elementary redistribution functions) was introduced by Hummer (1962) for two-level atoms. They are listed with roman numerals from RI to RIV . The latter has been subsequently rename RV (Heinzel 1981), as the expression of RIV used in Hummer (1962) was incorrect. Explicit expressions of these elementary redistribution functions, in the rest frame of the atom, in the laboratory frame, and after a direction-averaging, can be found in the books by Jefferies (1968) and (Hubeny and Mihalas 2015, Chapter 10). The latter presents also generalized redistribution functions for transitions with different initial and final states. The elementary redistributions functions describe the following processes: • RI : monochromatic scattering in the atomic rest frame; infinitely sharp atomic levels; • RII : monochromatic scattering in the atomic frame; infinitely sharp lower level; upper level broadened by radiative damping; • RIII : complete frequency redistribution in the atomic frame; infinitely sharp lower level; broad upper level; • RV : non coherent frequency redistribution in the atomic frame; both levels broadened by radiative damping. For spectral lines, the redistribution function depends on the radiative damping of the atomic levels, but also on elastic collisions with surrounding species. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1_26

563

564

26 Asymptotic Results for Partial Frequency Redistribution

redistribution function is thus usually a combination of these elementary functions. For a two-level atom with a sharp lower level, a simple model for a resonance line, the full redistribution function is a combination of RII and RIII (see Appendix J in Chap. 13 and Sect. 14.4). For subordinate lines, the frequency redistribution can be described by a combination of RIII and RV (Heinzel 1981). The redistribution RI , ignoring the natural damping of the upper level, is actually not appropriate for atomic spectral lines. One question discussed in this Chapter is whether a thermalization length can be defined for the elementary partial frequency redistribution functions, although the source function becomes a function of frequency (see Eq. (14.95)). The answer is yes, and as shown in Frisch (1980) and Frisch and Bardos (1981), RI , RIII , and RV have the same large scale behavior as complete frequency redistribution, whereas RII has a diffusive behavior at large optical depths and large frequencies. In the language of critical phenomena RI , RIII , and RV belong to the same universality class as complete frequency redistribution, whereas RII belongs to the class of diffusive processes. To examine this question, it is sufficient to consider the direction-averaged redistribution function r¯ (x, x ) (see Eq. (14.83)). The changes of directions at each scattering have no influence on the large scale behavior. For a one-dimensional medium, with a destruction probability  and a primary source Q∗ (τ ), the radiative transfer equation may be written as μ

∂I (τ, x, μ) = ϕ(x)[I (τ, x, μ) − S(τ, x)], ∂τ

(26.1)

where S(τ, x) ≡ (1 − )

1 2

1  +∞  +1

 0

−∞

−1

r¯ (x, x ) I (τ, x , μ ) dμ dx + Q∗ (τ ). ϕ(x) (26.2)

The source function satisfies an integral equation, which may be written as S(τ, x) = Q∗ (τ )  1   +∞

dμ

1−



r¯ (x, x )



S(τ , x ) e−|τ −τ |ϕ(x )/μ dx dτ , ϕ(x ) + 2 ϕ(x) μ D −∞ 0

(26.3)

where D is the τ -integration interval. To carry out the asymptotic analysis, this equation is rewritten as [S(τ, x) − Q∗ (τ )]ϕ(x) =

  +∞ 1− ϕ(x )¯r (x, x ) 2 D −∞

×[S(τ , x ) − S(τ, x)]E1 [|τ − τ |ϕ(x )] dx dτ ,

(26.4)

26.1 RI Asymptotic Behavior

565

where E1 is the first exponential integral function. Because the normalization of the redistribution has been taken into account, the functions ϕ(x), r¯ (x, x ) in the integral term can be replaced by their asymptotic form, at large x and x , and S(τ, x) by its asymptotic expansion. To establish scaling laws, it is sufficient to assume an infinite medium. Equation (26.4) is the starting point of the asymptotic analysis for the redistribution functions RI and RIII carried out in Sects. 26.1 and 26.2. For RV , considered in Sect. 26.3, we work with the Fourier transform of Eq. (26.4). For RII , which belongs to the class of diffusive processes, the asymptotic analysis can be carried out with Eq. (26.4), but also directly on the radiative transfer equation. The two methods are described in Sect. 26.4.

26.1 RI Asymptotic Behavior In the rest frame of the atom, the RI redistribution function may be written as rI (ξ, ξ ) = δ(ξ0 − ξ )δ(ξ − ξ ),

(26.5)

where ξ0 is the line center frequency. For isotropic scattering, the angle-averaged RI redistribution, in the laboratory frame, may be written as r¯I (x, x ) =

1 erfc(|x|), ¯ 2

|x| ¯ = max(|x|, |x| ).

(26.6)

The complementary error function erfc(x) is defined by 2 erfc(x) ≡ √ π



∞

e−u du. 2

(26.7)

x

The angle-averaged redistribution function satisfies the reciprocity relation r¯I (x, x ) = r¯I (x , x), and the normalization  +∞ r¯I (x, x ) dx = ϕ(x), −∞



+∞

−∞

r¯I (x , x) dx = ϕ(x ).

(26.8)

(26.9)

This equation holds for all the four elementary redistribution functions RI to RV . For RI , ϕ(x) is the Doppler profile. The asymptotic form of r¯I (x, x ) at large frequencies is 1 e−x¯ . r¯I (x, x )  √ ¯ 2 π |x| 2

(26.10)

566

26 Asymptotic Results for Partial Frequency Redistribution

Fig. 26.1 The RI partial redistribution function. This figure show the ratio r¯I (x, x )/ϕ(x) as a function of x for different values of x; x and x are respectively the incident and scattered dimensionless frequencies

The function r¯I (x, x )/ϕ(x ) is plotted in Fig. 26.1. For RI , photons that have been scattered from the line core to the line wings have a greater probability to return to the line center than to remain in the wings. Hence the situation is very similar to that of complete frequency redistribution. That RI behaves as complete frequency redistribution with a Doppler profile has been confirmed with asymptotic approaches (Ivanov 1970; Ivanov and Shneivais 1976; Frisch 1980) and numerical calculations (see references in Frisch 1980). The asymptotic analysis described now shows that, at large optical depths, as predicted by numerical investigations, the RI source function tends to the Doppler complete frequency redistribution √ source function for frequencies less than characteristic frequency xc () ∼ − ln , whereas it lies below the complete frequency redistribution source function for frequencies larger than xc . The asymptotic behavior of RI , described in Frisch (1980), is based on a perturbation analysis around complete frequency redistribution. The idea is to write the frequency dependent source function as ¯ ) + S1 (τ, x), S(τ, x) = S(τ

(26.11)

with ¯ )≡ S(τ



+∞ −∞

ϕ(x)S(τ, x) dx.

(26.12)

¯ ) and S1 (τ, x) are then determined by a perturbation The leading terms of S(τ method, applicable in the limit of a large number of scatterings.

26.1 RI Asymptotic Behavior

567

We first rewrite the integral equation for S(τ, x) in the symbolic form S(τ, x) = Q∗ (τ ) + (1 − )L[S¯ + S1 ],

(26.13)

where L is the linear operator in Eq. (26.3). Averaging over frequencies as in Eq. (26.12) and using Eq. (26.9), we obtain ¯ S] ¯ 1 ], ¯ + (1 − )L[S ¯ ) = Q∗ (τ ) + (1 − )L[ S(τ

(26.14)

where L¯ is the frequency average of L. It is easy to verify that L¯ is the complete frequency redistribution integral operator. The combination of Eqs. (26.13) and (26.14) yields an integral equation for S1 , which may be written as ¯ S] ¯ 1 ]. ¯ + (1 − )[L − L][S S1 (τ, x, n) = (1 − )[L − L][

(26.15)

The idea of the perturbation method is to assume that in the limit  → 0, the terms containing S1 in the right-hand side of Eqs. (26.14) and (26.15) are negligible compared to the terms with S¯ and then to check the consistency of the assumption. ¯ 1 ] in Eq. (26.14), we recover the integral equation for complete Neglecting L[S frequency redistribution. Its asymptotic behavior at large optical depth is analyzed in Chap. √20. For a Doppler profile, it is shown that the thermalization length is τeff () ∼ 1/( − ln ). Neglecting the term with S1 in the right-hand side of Eq. (26.15), we obtain an ¯ To obtain the frequency behavior of S1 at explicit expression of S1 in terms of S. optical depths of order τeff (), we introduce a rescaled profile ϕ˜ ∼ ϕτc (), such that ϕ˜ τ˜ ∼ ϕτ , an associated characteristic frequency xc (), such that ϕ(xc ()) ∼ 1/τ √ c (), and a rescaled frequency x˜ ∼ x/xc (). For the Doppler profile xc () ∼ − ln . The analysis of S1 carried out in Frisch (1980) shows that the correction term S1 is exponentially small (of order exp[−τ ϕ(x)]) for frequencies smaller than xc (), hence the RI source function essentially equal to the complete redistribution source function. For frequencies larger than xc (), the correction term behaves as ˜ ∼− S1 (τ˜ , x)

− ln τ˜ ϕ˜ ˜ S(τ˜ ), − ln 

τ˜ ϕ˜ < 1.

(26.16)

Hence, as announced, S1 lies below the complete redistribution source function and ¯ This factor justifies neglecting S1 in the rightis a factor 1/(− ln ) smaller than S. hand sides of Eq. (26.14) and Eq. (26.15) for performing the asymptotic analysis. Scaling laws for S1 can also be obtained by considering the integral  Pe (τ, x) =

+∞ −∞

r¯ (x, x ) −τ ϕ(x )

e dx , ϕ(x)

(26.17)

568

26 Asymptotic Results for Partial Frequency Redistribution

with r¯ (x, x ) replaced by its asymptotic form given in given in Eq. (26.10) (see Frisch 1980 for details).

26.2 RIII Asymptotic Behavior For RIII , the redistribution function in the atomic rest frame is rIII (ξ, ξ ) = ϕL (ξ, γ )ϕL (ξ , γ ),

(26.18)

with ϕL (ξ, γ ), the Lorentz function, defined by ϕL (ξ, γ ) =

1 γ . 2 π ξ + γ2

(26.19)

The constant γ is the damping rate of the upper level. At large frequencies, the angle-averaged redistribution in the laboratory frame is simply given by r¯III (x, x )  ϕ(x)ϕ(x ).

(26.20)

Here ϕ(x) is the Voigt function with parameter a = γ /(4πνD ), where νD is the Doppler width. The large scale behavior of RIII is thus trivially the same as that of complete frequency redistribution with a Voigt profile. The asymptotic analysis can be carried ˜ τ˜ ) satisfies the same singular out as described above for RI . The leading term S( integral equation as complete frequency redistribution with a Voigt profile. The thermalization length and characteristic frequency are τc () ∼ a −2 ,

xc () ∼ a −1 .

(26.21)

The asymptotic analysis shows that the correction term S1 is of order . This scaling can also be derived from Eq. (26.17) with r¯III (x, x ) = ϕ(x)ϕ(x ) and the absorption profile replaced by its asymptotic form ϕ(x)  a/(πx 2). We stress again that the asymptotic analysis holds only for large frequencies. At line center, there can be significant differences between the emergent profiles calculated with complete frequency redistribution and RIII partial frequency redistribution (Hummer 1969). However, replacing RIII by complete frequency is a common and usually safe practice in the analysis of spectral lines profiles and resonance polarization (Sampoorna et al. 2017).

26.3 RV Asymptotic Behavior

569

26.3 RV Asymptotic Behavior In the atomic rest frame, the RV redistribution function has the form rV (ξ, ξ ) = [ +

γ1 2 ] ϕL (ξ, γt )ϕL (ξ , γt ) γt

γ2 ϕL (ξ − ξ , 2γ1 )[ϕL (ξ, γt ) + ϕL (ξ , γt )] 2γt

+ 2π

γ2γ1 (γ2 + 2γ1 ) ϕL (ξ − ξ , 2γ1 )ϕL (ξ, γt )ϕL (ξ , γt ). γt2

(26.22)

Here γ1 and γ2 are the damping rates of the lower level and upper level and γt = γ1 + γ2 . The line center is at ξ = 0. The expression in Eq. (26.22) was first established by detailed balance arguments (see Woolley and Stibbs 1953) and later confirmed by quantum mechanical calculations (Omont et al. 1972). The expression given in Hummer (1962), based on Heitler’s results does not possess the correct symmetry in ξ and ξ , but holds for Raman scattering (Heinzel 1981; Hubeny 1982). Equation (26.22) encompasses the three other redistribution functions: RIII by setting γ2 = 0, RII by setting γ1 = 0, and RI by letting γ1 and γ2 tend to zero. The ratio rV (ξ, ξ )/ϕL (ξ , γt ) is plotted in Fig. 26.2 as a function of ξ , for ξ = 2 and three different values of the ratio γ2 /γ1 . It shows two well separated peaks with a Lorentzian behavior, one centered at ξ = 0 and the other one at ξ = ξ . The peak at ξ = 0 corresponds to transitions from the middle of the lower level to the middle of the upper level, followed by a reemission to an arbitrary sublevel of the

Fig. 26.2 The RV partial redistribution function. This figure shows the partial redistribution function rV (ξ, ξ ) in the atomic rest frame given in Eq. (26.22) for the damping parameter γ1 = 0.05, the incident frequency ξ = 2, and three different values of the ratio γ2 /γ1

570

26 Asymptotic Results for Partial Frequency Redistribution

lower level. The peak at ξ = ξ corresponds to transitions starting and ending at the middle of the lower level and going to an arbitrary sublevel of the upper level (Woolley and Stibbs 1953; Mihalas 1978; Hubeny and Mihalas 2015). The existence of the two peaks suggests to rewrite rV (ξ, ξ ) as   rV (ξ, ξ ) = ϕL (ξ, γt ) 1 ϕL (ξ , γt ) + 2 δ(ξ − ξ ) +ϕL (ξ, γt )C(ξ, ξ ),

(26.23)

where 1 ≡

γ1 , γt

2 ≡

γ2 . γt

(26.24)

The first term in the right-hand side describes the two peaks and the second one can be shown to be a correction. Indeed, integrating Eq. (26.23) over ξ and ξ and taking into account the normalization of rV (ξ, ξ ), we find 

+∞ −∞







C(ξ, ξ ) dξ = 0,

+∞ −∞

C(ξ, ξ )ϕL (ξ, γt ) dξ = 0.

(26.25)

Although the RV redistribution function has a coherent component at ξ = ξ , the large scale asymptotic behavior is of the complete frequency redistribution type. This can be understood as follows. Assuming that 2 and 1 are of the same magnitude, at each scattering a wing photon has equal probabilities to be reemitted at the same frequency or in the line core. Once a line photon has been reemitted in the line core, it remains around the central frequency and may suffer a large number of scatterings, before it is destroyed or reemitted into the wings. The chances for a wing photon to remain in the wings for many successive scatterings are thus very small and the physical picture is similar to that of complete frequency redistribution. The coherent component has nonetheless to be taken into account in the analysis, as it affects the frequency dependence of the source function. An asymptotic analysis of RV redistribution is carried out in Frisch (1980) with the atomic frame redistribution function rV (ξ, ξ ) in an infinite medium. The use of the atomic frame redistribution function is justified by the fact that the term contributing to the peak about the line center has a Lorentzian behavior. The infinite medium assumption allows one to performed the asymptotic analysis on the Fourier transform of the source source. The main steps of the analysis are as follows. Using the expression of rV (ξ, ξ ) given in Eq. (26.23), the Fourier transform of the integral equation for the source function may be written as   ∗ ˆ ˆ S(k, ξ ) =  Q (k) + (1 − ) 1 ˆ ξ )A(k, ϕ) + + 2 S(k,



∞ −∞

∞ −∞

ˆ ξ )A(k, ϕ )ϕ(ξ ) dξ

S(k,

 ˆ ξ )C(ξ, ξ )A(k, ϕ ) dξ , S(k,

(26.26)

26.3 RV Asymptotic Behavior

571

ˆ ξ ) and Qˆ ∗ (k) are the Fourier transforms of S(τ, ξ ) and of Q∗ (τ ), ϕ = where S(k, ϕ(ξ, γt ), ϕ = ϕ(ξ , γt ), and 

1

A(k, ϕ) ≡ 0

k ϕ2 ϕ dμ = arctan . 2 2 2 k μ +ϕ k ϕ

(26.27)

Equation (26.26) can be rewritten as (k, ξ ) =  Qˆ ∗ (k) + (1 − )1 

∞



∞ −∞

(k, ξ )D(k, ϕ )ϕ(ξ ) dξ

(k, ξ )C(ξ, ξ )D(k, ϕ ) dξ ,

(26.28)

ˆ ξ )[1 − (1 − )2 A(k, ϕ)]. (k, ξ ) ≡ S(k,

(26.29)

+ (1 − )

−∞

where

and D(k, ϕ) ≡ A(k, ϕ)[1 − (1 − )2 A(k, ϕ)]−1 .

(26.30)

The first integral in Eq. (26.28) behaves essentially as a complete frequency redistribution term. The asymptotic analysis can thus be carried out with the method developed for RI in Sect. 26.1. Writing the function (k, ξ ) as ¯ (k, ξ ) = (k) + 1 (k, ξ ),

(26.31)

¯ where (k) is defined as in Eq. (26.12), introducing the rescaled variables k˜ = ˜ the asymptotic analysis ¯ ˜ k), kτeff () and ϕ˜ = ϕτeff (), and assuming that (k)  ( 2 leads to τeff () ∼ γt / and to ˜ = ˜ k) (

˜ k) ˜ Q( , ˜ 1/2 1 + C|k|

(26.32)

where C is a constant of order of unity. We recover the Fourier transform of the singular integral equation for complete frequency redistribution with a Voigt profile given in Eq. (20.15). It can be shown that the correction term 1 (k, ξ ) is of order  (see Frisch 1980). ¯ The combination of Eq. (26.29) with (k, ξ )  (k) and with Eq. (26.32) ˆ ˜ provides for the leading term of S(k, ϕ): ˜ ˆ k, ˜ ϕ) ˜ − 2 A(k, ˜ ϕ)] ˜ k)[1 S( ˜  ( ˜ −1 .

(26.33)

572

26 Asymptotic Results for Partial Frequency Redistribution

ˆ ξ ) is essentially At optical depths of order τeff (), the frequency profile of S(k, flat over a frequency range |x| < xc , with the characteristic frequency xc given by τeff ()ϕ(xc ) ∼ 1. A similar frequency dependence holds for RI and RIII (see ˆ k, ˜ ϕ) Sect. 26.1). The Fourier inversion of S( ˜ can be performed analytically when the two-stream approximation (μ = ±1) is being made (Frisch 1980). The asymptotic analysis in the Fourier space is not easily transposed to the physical space because the contribution of the coherent wing term cannot be factorized as in Eq. (26.26). For RV , as for RI , and RIII , the thermalization length determined for an infinite medium holds also for a semi-infinite one, since it depends only on the scattering mechanism.

26.4 RII Asymptotic Behavior The effect of coherent scattering on spectral line shapes was investigated by Unno (1955), prior to the very systematic analysis of partial frequency redistribution by Hummer (1969). A fundamental result of Hummer (1969) is that coherent scattering, which leads to the RII redistribution function, produces in a semi-infinite medium a line with two symmetric maxima inside a wider absorption feature. This typical feature is observed in strong solar lines such as the H and K lines of Ca II and the h and k lines of Mg II. One of the first observation was reported in Milkey (1976). Coherent scattering plays also a critical role on the shape of linear polarization profiles. For example the linear polarization profile of the Ca I 4224 Å line has a double peak at line center characteristic of RII redistribution (Nagendra 2019). The second solar spectrum (the linear polarization spectrum) shows many other complex polarization profiles, classified by Belluzzi and Landi Degl’Innocenti (2009), for which no explanation can be found outside RII partial frequency redistribution. In the atomic frame, the RII redistribution function may be written as rII (ξ, ξ ) = ϕL (ξ, γ )δ(ξ − ξ ),

(26.34)

where ϕL (ξ, γ ) is the Lorentz profile defined in Eq. (26.19). When Doppler effects are taken into account  

1 (x − x )2 a 1 x + x



exp − , rII (x, , x ,  ) = U , π sin 4 sin2 ( /2) 2 cos( /2) cos( /2) (26.35) where is the angle between  and  and U is the normalized Voigt function. For a 90◦ scattering, this expression reduces to 1 1 rII (x, , x , ) = exp[− √ (x − x )2 ]U π 2



x + x √ √ , 2a . 2

(26.36)

26.4 RII Asymptotic Behavior

573

The first factor produces a peak at x  x , the amplitude of which is controlled by the second factor. The direction-averaged r¯II (x, x ) has an explicit expression, already given in Unno (1952). Its numerical calculation is somewhat demanding. Different methods of calculation have been proposed, including a direct numerical averaging of Eq. (26.35) over the angle (see Adams et al. 1971; Ayres 1985; Hubeny and Mihalas 2015). To reduce the numerical work, several approximations have been proposed (see Hubeny and Mihalas 2015). Equation (26.36) is one of them (Argyros and Mugglestone 1971). A more realistic approximation has been introduced by Ayres (1985), namely r¯II (x, x ) 

a 1 1 erfc(|x|) ¯ + G( |x − x |).

2 2 π[0.5(x + x )] 2

(26.37)

The first term in the right-hand side is the direction-average RI redistribution

function √ defined in Eq. (26.6). It is negligible for large values of x or x , say larger than − ln a. For the asymptotic analysis in Sect. 26.4.1 we retain only the second term. The function G(x) is the first integral of the error function (Abramovitz and Stegun 1964), defined by  G(x) = ierfc(x) =

∞ |x|

1 2 erfc(t) dt = √ e−x − xerfc(x). π

(26.38)

−x 2 /(2√πx 2 ). At the origin, G(x) takes the value For large values √ of x, G(x)  e √ G(0) = 1/ π and its slope is −2/ π. The ratio r¯II (x, x )/ϕ(x ), calculated with exact expressions of r¯II (x, x ) and ϕ(x ), is shown in Fig. 26.3 for several values

Fig. 26.3 The partial RII redistribution function. This figure shows the ratio r¯II (x, x )/ϕ(x) as a function of x for different values of the frequency x; x and x are respectively the incident and scattered dimensionless frequencies

574

26 Asymptotic Results for Partial Frequency Redistribution

√ of x and a = 10−3 . For x smaller than − ln a = 2.63, r¯II (x, x ) is given by the first term in the right-hand side of Eq. (26.37). One can observe that the ratio r¯II (x, x )/ϕ(x ) for x = 0 and x = 2 is essentially equal to the ratio r¯I (x, x )/ϕ(x ) shown in Fig. 26.1. For large x , the ratio r¯II (x, x )/ϕ(x ) has a peaked shape controlled by the second term in Eq. (26.37). The peaks are positioned√at x = x

and their height is essentially independent of x and equal to G(0) = 1/ π  0.56. For x = 5, one can clearly observe that the left wing is more extended than the right one. This asymmetry produces a frequency shift towards the line center discussed in Sect. 26.4.1 below (see Eq. (26.48)). A consequence of the presence of peak in the redistribution function at x  x

is that RII has a diffuse behavior at large frequencies and large optical depths. The optical depth and also the frequency have to be rescaled by a factor, which tends to infinity as  → 0. We thus now introduce τ˜ = τ/τeff (),

x˜ = x/xc (),

(26.39)

with τeff () and xc () tending to ∞ as  → 0. The asymptotic analysis can be carried out on the source function S(τ, x) as in Frisch (1980), or on the radiative transfer equation as in Frisch and Bardos (1981). We now briefly present the two approaches.

26.4.1 Diffusion Equation for the Source Function The source function satisfies the integral equation written in Eq. (26.4), repeated here for convenience:   +∞ 1− [S(τ, x) − Q∗ (τ )]ϕ(x) = ϕ(x )¯r (x, x ) 2 −∞ D ×[S(τ , x ) − S(τ, x)]E1 [|τ − τ |ϕ(x )] dx dτ .

(26.40)

The expansion follows the scheme described in Sect. 20.1, with two variables: τ and x. We assume that ˜ τ˜ , x) S(τ, x)  S( ˜ + h.o.t.

(26.41)

with h.o.t. stands for higher order terms, and Q∗ (τ )  Q˜ ∗ (τ˜ ). In the left-hand side, the profile ϕ(x) can be replaced by δ(x)/x ˜ c (), since the width of ϕ(x) tends to zero in the variable x. ˜ This choice, suggested by by Harrington (1973), preserves the normalization of the profile and that of the primary source term ϕ(x)Q∗ (τ ). In the right-hand side, ϕ(x ) can be replaced by its asymptotic expression, namely ϕ(x)  a/(πx 2), and the angle-averaged redistribution function can be approximated by Eq. (26.37).

26.4 RII Asymptotic Behavior

575

We now write y t , x˜ + ), τc () xc ()

(26.42)

ϕ(x ) = ϕ(x + y) = ϕ(xx ˜ c () + y),

(26.43)

˜ τ˜ , x˜ ) = S( ˜ τ˜ + S(

r¯II (x, x ) = r¯II (x, x + y) = r¯II (xx ˜ c (), xx ˜ c () + y),

(26.44)

with t = τ − τ,

y = x − x

(26.45)

˜ τ˜ , x˜ ) to second order and of r¯II (x, x ) and and perform Taylor expansions of S(

ϕ(x ) to first order. After some algebra, Eq. (26.40) becomes ˜ τ˜ , x) ˜ τ˜ )] δ(x)[ ˜ S( ˜ − Q(

 1 a 2 ∂ S˜ ∂ 2 S˜ 1 xc3 () πx˜ 2 1 ∂ 2 S˜ ] . + − [ = 2 () a 3 ∂ τ˜ 2  τeff x˜ ∂ x˜ πx˜ 2 2xc3() ∂ x˜ 2

(26.46)

At fixed τ˜ and x, ˜ when  → 0, Eq. (26.46) will have a finite limit provided τeff () ∼  −1 ,

xc () ∼ a 1/3 −1/3 .

(26.47)

The frequency xc () provides a characteristic width of the central absorption part of the emergent line profile. The optical depth τeff () plays the role of a thermalization length. The asymptotic equation may be written as ˜ τ˜ , x) ˜ τ˜ )] = δ(x)[ ˜ S( ˜ − Q(

πx˜ 2 1 ∂ 2 S˜ a ∂ 2 S˜ 2 ∂ S˜ ]. + [ 2 − 2 2 a 3 ∂ τ˜ 2πx˜ ∂ x˜ x˜ ∂ x˜

(26.48)

˜ x˜ comes from the This is a space and frequency diffusion equation. The term ∂ S/∂



asymmetry of r¯II (x, x ) with respect to x = x . The factor −1/x˜ is a mean shift per scattering towards the line center (Osterbrock 1962; Adams 1972). The diffusion equation for the source function can also be written as (Harrington 1973) ˜ τ˜ , σ˜ ) − Q( ˜ τ˜ )] = δ(σ˜ )[S(

∂ 2 S˜ ∂ 2 S˜ + , 2 ∂ τ˜ ∂ σ˜ 2

(26.49)

where $ σ˜ =

2 π 3 x˜ . 3 3a

(26.50)

576

26 Asymptotic Results for Partial Frequency Redistribution

˜ τ˜ , σ˜ ) with respect to τ˜ has the simple form The Fourier transform of S( ˆ ˜ ˜ ˆ k, ˜ σ˜ ) = Q(k) exp[−|k||σ˜ |] , S( ˜ 1 + 2|k|

(26.51)

ˆ k, ˜ σ˜ ) has a flat frequency where k˜ = kτeff () ∼ k/. This expression shows that S( profile for σ˜ < k˜ −1 and reaches the same magnitude as the thermal source for k˜  1. This justifies the interpretation of τeff () as a thermalization length and that of xc () as a thermalization frequency. Equation (26.48) shows that the diffusion in space becomes stronger than the diffusion in frequency when x˜ tends to infinity. In this limit RII behaves as monochromatic scattering, a behavior observed in numerical modelling of the Ca I 4227 Å line by Sampoorna et al. (2009). As explained in Hummer (1969), at each scattering the frequency change is of order x ∼ 1, whatever the frequency of the incoming photon, hence the relative change ϕ/ϕ tends to zero as x → ∞ and so does the relative changes of the mean free path. From a numerical point of view, this property implies that it is sufficient to require that the monochromatic optical at the maximum frequency, ϕ(xmax )τmax , is of order unity (Avery and House 1968; Hummer 1969). In contrast, for complete frequency redistribution, this quantity has to be much smaller than unity (see Sect. 25.2).

26.4.2 Diffusion Equation for the Radiation Field For monochromatic scattering and Rayleigh scattering, we show in Chaps. 23 and 24 how to derive, from the radiative transfer equation, a large scale diffusion equation, valid far from boundaries, and the associated boundary conditions. For RII , the same method has been applied in Frisch and Bardos (1981) for a one-dimensional conservative slab and direction-dependent redistribution function. We present here the main steps of the method for a three-dimensional non-conservative medium, with no incident radiation. The starting point is the radiative transfer equation 1 n.∇I (r, x, n) = −ϕ(x)I (r, x, n) k(r)   +∞ d

+ ϕ(x)Q∗ (r). + (1 − ) I (r, x , n )RII (x, n, x , n ) dx

4π −∞

(26.52)

For the asymptotic analysis, it is particularly convenient to use the double Fourier transform representation suggested by Rybicki (1976, unpublished) for the elementary redistribution functions. For RII it has the form

26.4 RII Asymptotic Behavior

577

1 RII (x, n, x , n ) = (2π)2







+∞ −∞

dω

+∞ −∞

dω e−i ωx ei ωx R˜ II (ω, n, ω , n ),

(26.53) with 1

R˜ II (ω, n, ω , n ) = exp[− (ωn − ω n )2 ]e−a|ω−ω | . 4

(26.54)

For RI , the second factor in Eq. (26.54) is 1 and for RIII , it is e−a|ω| e−a|ω | (Frisch 1988). The corresponding representation for the absorption profile is ϕ(x) =

1 2π



+∞ −∞

e−i ωx e−a|ω| e−ω

2 /4

dω.

(26.55)

Our objective is to construct an asymptotic radiative transfer equation valid at frequencies of order xc () and optical depths of order τeff (). The natural expansion parameter, denoted η, is the ratio of a mean free path 1/ϕ(xc ) to the characteristic scale τeff . Using xc ()   −1/3 , τeff () ∼  −1 , and ϕ(x) ∼ 1/x 2, we find 1 1 ∼  1/3 ∼ . ϕ(xc )τeff xc

η∼

(26.56)

To simplify the notation the line parameter a is set to 1. We assume that the radiation field is a function I (˜r , x, ˜ n) of the rescaled variables r˜ and x, ˜ defined by x˜ = ηx,

r˜ = η3 r.

(26.57)

To handle the frequency dependence of the redistribution function, we perform a Taylor expansion of I (˜r , x˜ , n ) around I (˜r , x, ˜ n ), that is we write I (˜r , x˜ , n ) =

∞  (x˜ − x) ˜ j ∂j Iη (˜r , x, ˜ n ). j! ∂ x˜ j

(26.58)

j =0

Following the method described in Sect. 23.2 to construct the interior solution for monochromatic scattering, we introduce the expansion Iη (˜r , x, ˜ n) = I0 (˜r , x, ˜ n) + ηI1 (˜r , x, ˜ n ) + η2 I2 (˜r , x, ˜ n) + O(η3 ).

(26.59)

The construction of the hierarchy of equations is described in Frisch and Bardos (1981). The leading term I0 (˜r , x, ˜ n) is isotropic and satisfies the diffusion equation 1 πx˜ 2 1 1 a ∂ 2 I0 2 ∂I0 ˜ r )−I0 (˜r , x)] ∇r˜ [ ∇r˜ I0 (˜r , x)]+ ]+δ(x)[ ˜ Q(˜ ˜ [ − ˜ = 0. 3 a k(˜r ) k(˜r ) 2πx˜ 2 ∂ x˜ 2 x˜ ∂ x˜ (26.60)

578

26 Asymptotic Results for Partial Frequency Redistribution

We recover a space and frequency diffusion equation similar to the diffusion equation written in Eq. (26.48). The gradient is taken with respect to the rescaled variable r. ˜ An improved diffusion equation can be constructed for the directionaverage intensity,  J (˜r , x) ˜ ≡

[I0int (˜r , x) ˜ + ηI1int (˜r , x, ˜ n)]

d . 4π

(26.61)

The construction of the boundary condition can be performed exactly as described in Sect. 23.4. It leads to the boundary condition J (˜r b , x) ˜ +η

πx˜ 2 L u.∇r˜ J (˜r b , x) ˜ = 0. a k(˜r b )

(26.62)

Here u is the direction normal to the surface at the point r˜ b on the surface. The coefficient η disappears when the original r and x variables are reintroduced. The constant L, defined in Eq. (23.41), is the ratio of the second and first moment of H (μ). The emergent intensity, at frequencies of order xc , is given by H (μ) πx˜ 2 1 u.∇I0int (˜r b , x). I em (˜r b , x, ˜ μ)  −η √ ˜ 3 a k(r b )

(26.63)

In the original variables, it may be written as H (μ) ∂I0 I em (0, x, μ)  √ |τx =0 , 3 ∂τx

μ ∈ [0, 1],

(26.64)

where τx is the monochromatic optical depth along the normal to the surface at the point r b . The optical depth is defined in such as way that τx ∈ [0, +∞[ with τx = 0 at the surface. We recover the expression of the emergent intensity given in Eq. (23.50) for monochromatic scattering. Equation (26.64) holds only for frequencies of order xc ∼  −1/3 . The direction-average emergent intensity J em (0, x) and the emergent radiative flux F em (0, x) are readily obtained √by taking the zeroth and first moment of H (μ), respectively equal to 2 and 2/ 3 (see Sect. B.5). Comparisons between asymptotic predictions and exact numerical solutions are presented in Frisch and Bardos (1981).

26.4.3 RII Scaling Laws We summarize here scaling laws valid for RII . As can be observed in Table 26.1, they differ significantly from those given for complete frequency redistribution

26.4 RII Asymptotic Behavior

579

Table 26.1 Scaling laws for the RII partial frequency redistribution. a parameter of the line profile. The expansion parameter is  for an infinite medium and 1/T for a slab with total optical thickness T Parameter  1/T

τeff  −1 T

xc a 1/3  −1/3 (aT )1/3

Nc a 2/3  −2/3 (aT )2/3

N  −1 T

L a 1/3  −4/3 a 1/3 T 4/3

√ in Tables 25.1 and 25.2 and from the scaling law τeff ∼ 1/  which holds for monochromatic scattering. We have established with the asymptotic analysis of the integral equation for the source function that τeff () ∼  −1 ,

xc () ∼ a 1/3 −1/3 ,

(26.65)

hence xc ()  [aτeff ()]1/3.

(26.66)

This relation, between τeff () and xc (), is a direct consequence of the space and frequency diffusive character of RII for photons with frequency of order xc (). It can be established, as suggested in Bonilha (1979) (see also Harrington 1973) by introducing the mean number of scatterings undergone by photons with frequencies of order xc (), here denoted Nc . Assuming that x ∼ 1 at each scattering, the diffusion in frequency space allows one to write 1/2

xc () ∼ Nc .

(26.67)

The mean free-path of photons with frequency of order xc is of order 1/ϕ(xc ). The diffusion in space then leads to 1/2

τeff () ∼ Nc /ϕ(xc ).

(26.68)

Using the asymptotic form, ϕ(x)  a/(πx 2), we readily recover Eq. (26.66). The corresponding scaling law for a finite medium, xc  (aT )1/3 , is clearly observed in numerical solutions of the radiative transfer equation by Adams (1972). The definitions given in Eqs. (25.40) and (25.42) for the mean number scatterings

N and the mean path length L are easily generalized to partial frequency redistribution. It suffices to replace S(τ ) by S(τ, x) and to integrate over x. It is easy to verify that the energy conservation equation in Eq. (25.43) holds also for RII . For an infinite medium, it yields

N ∼ 1/ ∼ τeff ().

(26.69)

580

26 Asymptotic Results for Partial Frequency Redistribution

The mean path length for photons with characteristic frequency xc is given by

L ∼

1 Nc ∼ xc4 ∼ xc () N. ϕ(xc ) a

(26.70)

For RII , in contrast to complete frequency redistribution, L can remain finite in an infinite medium, even when there is no continuous absorption, because the frequency scattering in the wings is of the diffusive type (Hummer and Kunasz 1980, also Frisch 1988).

References Abramovitz, M., Stegun, I.A.: Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C. (1964) Adams, R.F.: The escape of resonance-line radiation from extremely opaque media. Astrophys. J. 174, 439–448 (1972) Adams, R.F., Hummer, D.G., Rybicki, G.J.: Numerical evaluation of the redistribution function RII−A (x, x ) and of the associated scattering integral. J. Quant. Spectrosc. Radiat. Transf. 11, 1365–1376 (1971) Argyros, J.D., Mugglestone, D.: The scattering of line radiation -I. A generalized redistribution function. J. Quant. Spectrosc. Radiat. Transf. 11, 1621–1632 (1971) Avery, L.W., House, L.L.: An investigation of resonance-line scattering by the Monte Carlo technique. Astrophys. J. 152, 493–507 (1968) Ayres, T.R.: A physically realistic approximate form for the redistribution function RII−A . Astrophys. J. 294, 153–157 (1985) Belluzzi, L., Landi Degl’Innocenti, E.: A spectroscopic analysis of the most polarizing atomic lines of the second solar spectrum. Astron. Astrophys. 495, 577–586 (2009) Bonilha, J.R., Ferch, R.L., Salpeter, E.E., Slater, G., Noerdlinger, P.D.: Monte Carlo calculations for resonance scattering with absorption or differential expansion. Astrophys. J. 233, 649–660 (1979) Frisch, H.: Non-LTE transfer V. The asymptotics of partial redistribution. Astron. Astrophys. 83, 166–183 (1980) Frisch, H.: Radiative transfer with frequency redistribution. In: Chmielewsky, Y., Lanz, T. (eds.) Radiation in Moving Gaseous Media, Saas-Fee 18h Advanced Course, Swiss Society of Astronomy and Astrophysics. Publication de l’Observatoire de Genève, pp. 339–448 (1988) Frisch, H., Bardos, C.: Diffusion approximations for the scattering of resonance-line photons. Interior and boundary layer solutions. J. Quant. Spectrosc. Radiat. Transf. 26, 119–134 (1981) Harrington, J.P.: The scattering of resonance-line radiation in the limit of large optical depth. Mon. Not. R. astr. Soc. 162, 43–52 (1973) Heinzel, P.: Non-coherent scattering in subordinate lines: a unified approach to redistribution functions. J. Quant. Spectrosc. Radiat. Transf. 25, 483–499 (1981) Hubeny, I.: Non-coherent scattering in subordinate lines: III. Generalized redistribution functions. J. Quant. Spectrosc. Radiat. Transf. 27, 593–609 (1982) Hubeny, I., Mihalas, D.: Theory of Stellar Atmospheres. Princeton University Press, Princeton (2015) Hummer, D.G.: Non-coherent scattering I. The redistribution functions with Doppler broadening. Mon. Not. R. astr. Soc. 125, 21–37 (1962) Hummer, D.G.: Non-coherent scattering-VI. Solutions of the transfer problem with a frequencydependent source function. Mon. Not. R. astr. Soc. 145, 95–120 (1969)

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Hummer, D.G., Kunasz, P.: Energy loss by resonance line photons in an absorbing atmosphere. Astrophys. J. 236, 609–618 (1980) Ivanov, V.V.: The mean free path of a photon in a scattering medium. Astrophysics 5, 355–367 (1970); translation from Astrofizika 6, 643–662 (1970) Ivanov, V.V, Shneivais, A.B.: Frequency relaxation in multiple scattering of line radiation. Astrophysics 12, 152–157 (1976); translated from Astrofisika 12, 245 (1976) Jefferies, J.T.: Spectral Line Formation. Blaisdell Publishing Company, Waltham (1968) Mihalas, D.: Stellar Atmospheres, 2nd edn. W.H. Freeman and Company, San Francisco (1978) Milkey, R.W.: Interpretation of Atmospheric Structure, General Assembly 1976, Ed. Cannon, C. J., Publication of the University of Sydney (1976) Nagendra, K.N., Sampoorna, M.: Numerical methods in polarized line formation theory, in Solar Polarization 5, ASP Conference Series, vol. 405, pp. 261–273 (2009) Nagendra, K.N.: Polarized line formation: methods and solutions. In: Belluzzi, L., Casini, R., Romolli, M., Trujillo Bueno, J. (eds.) ASP Conference Series, vol. 526, pp. 99–118 (2019) Omont, A., Smith, E.W., Cooper, J.: Redistribution of resonance radiation I. The effect of collisions. Astrophys. J. 175, 185–199 (1972) Osterbrock, D.E.: The escape of resonance-line radiation from an optically thick nebula. Astrophys. J. 135, 195–211 (1962) Sampoorna, M., Stenflo, J.O., Nagendra, K.N., Bianda, M., Ramelli, R., Anusha, L.S.: Origin of spatial variations of scattering polarization in the wings of the Ca I 4227 Åline. Astrophys. J. 699, 1650–1659 (2009) Sampoorna, M., Nagendra, K.N., Senflo, J.O.: Polarized line formation in arbitrary strength magnetic fields. Angle-averaged and angle-dependent partial frequency redistribution. Astrophys. J. 844, 97 (2017) Spitzer, L.: Notes on the theory of non-coherent scattering. Astrophys.J. 99, 1–7 (1944) Unno, W.: Note on the Zanstra redistribution in planetary nebulae. Publ. Astron. Soc. Jpn. 4, 100– 102 (1952) Unno, W.: Theoretical line contour of the Lyman alpha radiation of ionized helium and the excitation of Bowen lines in planetary nebulae. Publ. Astron. Soc. Jpn. 7, 81–108 (1955) Woolley, R., Stibbs, D.: The Outer Layers of a Star. Clarendon Press, Oxford (1953)

Author Index

A Abhyankar, K.D., 355 Ablowitz, M.J., 44, 63, 216, 242, 246, 417, 419 Abrahams, I.D., 215, 216 Abramovitz, M., 24, 223, 461, 463, 485, 573 Abramov, Yu.Yu., 143, 148, 237, 465 Adams, R.F., 573, 575, 579 Alekseeva, G.A., 269 Aller, L.H., 13 Alsina Ballester, E., 275 Ambartsumian, V.A., viii, 44, 45, 85, 196, 200, 204, 356 Anusha, L.S., ix, 278, 282, 327–330, 339 Aoki, K.S., 4 Argyros, J.D., 573 Asensio Ramos, A., 255, 290, 295 Athay, R.G., 557 Avery, L.W., 557, 576 Avrett, E.H., 128, 130, 196, 462, 463, 468, 475–477, 553, 556 Ayres, T.R., 573

B Badham, V.C., 515, 517, 522 Bardos, C., 133, 515, 517, 530, 531, 541, 564, 574, 576–578 Baur, T.G., 255 Belluzzi, L., 572 Bender, C.M., 296, 463, 467, 517, 522, 523 Bensoussan, A., 133, 469, 515 Berdyugina, S., 282 Bergeat, J., 31, 45 Bianda, M., 576 Biberman, L.M., 18, 27

Bommier, V., ix, 196, 256, 272, 275, 276, 279, 282–285, 289, 290, 296–300, 325, 331, 339, 345, 347, 471 Bond, G.R., 355, 384, 385, 388, 401, 402 Bonilha, J.R., 579 Born, M., 256, 258 Bosma, P.B., 93, 94, 210, 437, 497 Bothner, T., 47 Bouchaud, J.-P., 490 Brink, D.M., 260, 267 Bronstein, M., 161, 196 Burniston, E.E., 266, 356, 369, 373, 378, 411, 417–420, 451, 453 Busbridge, I.W., 45, 216

C Carleman, T., viii, 46, 51, 57 Carrier, G.F., 44, 59, 63, 64, 78, 99, 216, 219, 221, 228, 246 Case, K.M., viii, 17, 46, 51, 61, 63, 78, 80, 85, 88, 91, 93, 179, 181, 185, 191–193 Cercignani, C., 4 Chandrasekhar, S., 4, 41, 42, 45, 46, 79, 85, 87, 88, 93, 131, 159, 161, 165, 180, 196, 200, 204, 206–209, 255, 256, 258, 260, 261, 263, 265, 276, 305, 321, 362, 365, 373, 386, 391, 397, 401–404, 437, 519, 525, 544 Chwolson, O.D., 41 Clark, O., 46, 51, 55, 57, 87, 131, 200, 215 Cole, J.D., 467, 517, 522, 551 Comtet, A., 480, 481, 484, 485, 491, 495, 497–499 Cooper, J., 272, 279, 284, 569

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1

583

584 D Dautray, R., 63, 216, 221, 240 Derouich, M., 325 Dolginov, A.Z., 256, 258, 260, 261 Domke, H., 210, 272, 321, 382, 384, 392, 548 Dubrulle, B., 500 Duderstadt, J.J., 44, 63, 80, 83, 179, 216 Dykhne, A.M., 143, 148, 237, 465 E Eddington, A.S., 18, 161, 528 Elmore, D.F., 255 Estrada, R., 57, 61, 63 F Faurobert, M., 134, 147, 328, 515 Faurobert-Scholl, M., 272, 278, 282, 284, 316, 325, 327, 336, 535, 554, 555 Feller, W., 478, 489, 491, 500 Ferch, R.L., 579 Ferziger, J.H., 18 Feshbach, H., 216 Field, G., 519 Fluri, D., 308 Fokas, A.S., 44, 63, 216, 242, 246, 417, 419 Fouque, J.-P., 515 Fraley, S.K., 266, 365, 372 Frisch, H., ix, 46, 51, 52, 63, 130, 133, 147, 172, 174, 196–198, 268, 282, 306, 314, 327, 344, 346, 347, 467, 480, 499, 500, 502, 504, 505, 515, 517, 534, 541, 543, 545–547, 549, 552, 554–557, 564, 566–568, 570–572, 574, 576–578, 580 Frisch, U., 46, 51, 52, 130, 172, 196–198, 344, 346, 467, 480, 499, 500, 502, 504, 505, 549, 552 Froeschlé, Ch., 552 Fymat, A.L., 355 G Gakhov, F.D., 57, 60, 63, 80, 83 Gandorfer, A., 255 Garcia, R.D.M., 4, 370 Gardner, C.S., 246 Garnier, J., 515 Georges, A., 490 Gnedin, Yu.N., 256, 258, 260, 261 Golse, F., 541 Grachev, S.I., 447, 450, 451 Greene, J.M., 246 Guillot, T., 27

Author Index H Halpern, O., 46, 51, 55, 57, 87, 131, 200, 215 Hamilton, D.R., 276 Hanle, W., 4, 255 Harrington, J.P., 574, 575, 579 Harvey, J.W., 255 Heinzel, P., 563, 564, 569 Hemsh, M.J., 18 Hilbert, D., 2, 47 Holstein, T., 18, 27 Holtsmark, J., 477 Holzreuter, R., 278 Hooff, J.P.C. van, 355, 436, 437, 439 Hopf, E., 31, 34, 36, 42, 87, 131, 159, 165, 196, 215 House, L.L., 576 Hubeny, I., 13, 20, 21, 31, 196, 272, 563, 569, 570, 573 Hulst, H.C. van de, 266, 365 Hummer, D.G., 7, 18, 20, 128, 130, 143, 145–147, 151, 154, 159, 196, 285, 322, 462, 463, 468, 475–477, 553, 556, 559, 560, 563, 568, 569, 572, 576, 580

I Ivanov, V.V., ix, 6, 13, 17, 34, 36, 41, 45, 46, 52, 85, 88, 93, 94, 100, 128–131, 133, 134, 143, 146–149, 151, 153–156, 159, 160, 164–166, 168, 172, 195–198, 200, 210, 211, 213, 255, 268, 313, 314, 326, 341, 343–345, 355, 359, 362, 365, 385, 389, 427, 437, 440–442, 447, 462, 463, 465, 466, 480, 481, 490, 507, 508, 510, 511, 513, 514, 553–555, 557–561, 566

J Jefferies, J.T., 13, 21, 563 Jones, D.S., 215 Jones, H.P., 557

K Kampen, N,G. van, 182 Kanwal, R.P., 57, 61, 63 Kasaurov, A.M., 341, 343–345, 355, 359, 362, 385, 389, 427, 437, 440–442 Kawabata, K., 88, 93, 206 Keller, C.U., 255 Keller, J.B., 515, 517, 522

Author Index Kelley, C.T., 4, 370 Khachatrian, A.K., 18 Kisil, A.V., 215, 220, 228 Klafter, J., 474, 478 Kleint, L., 282 Kourganoff, V., 30, 34, 45, 51, 55, 85, 88, 159, 165, 528 Krein, M.G., 216, 518, 519 Kriese, J.T., 355 Krook, M., 44, 59, 63, 64, 78, 99, 216, 219, 221, 228, 246 Kruskal, M.D., 246 Kukushkin, A.B., 27 Kušˇcer, I., 57, 179

L Lacis, A.A., 12 Landi Degl’Innocenti, E., 4, 13, 20, 21, 159, 165, 196, 256, 262, 265, 267, 274, 277, 278, 290, 295, 305, 320, 325, 330, 345, 500, 572 Landolfi, M., 12, 13, 20, 21, 159, 162, 165, 166, 256, 277, 305, 320, 325, 330 Larsen, E.W., 515, 517, 522 Latyshev, A.V., 216 Lawrie, J.B., 215, 216 Lenoble, J., 355 Levermore, C.D., 528 Lévy, P., 6, 480, 489 Limaye, S.S., 88, 93, 206 Lions, J.-L., 63, 216, 221, 240 Lites, B.W., 134 Lommel, E., 41, 42 Loskutov, V.M., 172, 341, 343–345, 355, 359, 362, 365, 385, 389, 427, 437, 440–442, 447, 549, 553, 554 Lueneburg, R., 46, 51, 55, 57, 87, 131, 200, 215

M Majumbar, S.N., 480, 481, 484, 485, 491, 495, 497–499 Manucharyan, G.A., 216 Mark, C., 165 Martin, W.R., 44, 63, 80, 83, 179, 216 Mc Cormick, N.J., 57, 179 Mee, C.V.M. van der, 57, 81 Metzler, R., 474, 478 Mihalas, D., 13, 19, 20, 21, 31, 42, 128, 557, 563, 570, 573 Mili´c, I., 328 Milkey, R.W., 572

585 Milne, E.A., 3, 41, 159 Mishchenko, M.I., 12 Mitchell, A.C.J., 290, 293 Miura, R.M., 246 Molodij, G., 325 Montroll, E.W., 480 Morse, P.M., 216 Mugglestone, D., 573 Mullikin, T.W., 369 Muskhelishvili, N.I., viii, 46, 57, 60, 63, 80, 83, 111, 417, 419 N Nagendra, K.N., 32, 268, 271, 275, 290, 327–330, 338, 339, 343, 546, 572 Nagirner, D.I., 88, 93, 143, 211 Napartovich, A.P., 143, 148, 237, 465 Nerenov, V.S., 28 Noble, B., 44, 63, 215, 216, 242, 246 Noerdlinger, P.D., 579 Noullez, A., 500 Novikov, V.V., 269 O Omont, A., 272, 279, 284, 569 Orszag, S.A., 296, 463, 467, 517, 522, 523 Osterbrock, D.E., 575 P Pahor, S., 355 Paletou, F., 546 Paley, R.C., 215 Pandey, J.N., 62, 66 Papanicolaou, G., 133, 469, 515 Papoulis, N., 488 Pearson, C.E., 44, 59, 63, 64, 78, 99, 216, 219, 221, 228, 246 Perthame, B., 541 Pogorzelski, W., 63 Poincaré, H., 463 Pollaczek, F., 498 Pomraning, G.C., 528, 529 R Ramelli, R., 278, 576 Ramis, J.-P., 463 Rayleigh (Lord) (John Strutt), 255 Rees, D.E., 310 Riemann, B., 2, 47 Rooij, W.A. de, 93, 94, 210, 355, 436, 437, 439, 497

586

Author Index

Roos, B.W., 44, 63, 179, 216, 221 Rutily, B., ix, 31, 45, 206 Rutnam, M.A., 518 Rybicki, G.J., 576

Št˘epán, J., 347 Stibbs, D., 18, 563, 569, 570 Stokes, G.G., 257 Sulem, C., 541

S Sabashvili, Sh.A., 480, 481, 490 Sahal-Bréchot, S., 256, 284 Salpeter, E.E., 579 Sampoorna, M., ix, 275, 278, 285, 290, 296, 322, 325, 343, 546, 568, 576 Santos, R., 515, 530, 531, 541 Satchler, G.R., 260, 267 Schchukina, N., 282 Schnatz, T.W., 397 Schuster, A., 41 Schwartz, L., 105, 238 Schwarzschild, K., 17, 25, 41, 159 Sdvizhenskii, P.A., 27 Sekara, Z., 266 Sentis, R., 515, 530, 531, 541 Serbin, V.M., 130 Shapiro, A.I., 282 Shlesinger, M.F., 480 Shneivais, A.B., 565 Siewert, C.E., 4, 88, 148, 266, 316, 355, 356, 365, 369, 370, 372, 373, 378, 384, 385, 388, 397, 401, 402, 411, 417–420, 451, 453 Silant’ev, N.A., 269, 313 Skumanich, A., 134, 557 Slater, G., 579 Smitha, H.N., 278 Smith, E.W., 272, 279, 284, 569 Sobolev, V.V., 28, 32, 34, 36, 37, 45, 52, 85, 155, 196, 200, 203, 305 Solna, K., 515 Sommerfeld, A., 242 Sparre Andersen, E., 500 Spitzer, F., 498 Spitzer, L., 563 Stegun, I.A., 24, 223, 461, 463, 485, 573 Stenflo, J.O., 255, 256, 258, 260, 262, 263, 265, 270, 271, 275, 276, 278–284, 290, 293, 296–299, 305, 308, 310 Stenholm, L., 310

T Tartar, L., 515 Thomas, R.N., 16, 21 Tichmarsh, E.C., 216, 218 Travis, L.D., 12 Trigt, C. van, 490 Trujillo Bueno, J., 255, 274, 282, 290, 295 Twerenbold, D., 255

U Unno, W., 563, 572, 573

V Vekua, N.P., 417, 419 Vergassola, M., 500 Viik, T., 384, 385 Voloshinov, V.V., 27

W Warming, R.F., 93 Weisstein, E.W., 485 Wiener, N., viii, 2, 26, 41–43, 85, 159, 200, 215–217 Wolf, E., 256, 258 Woolley, R., 18, 563, 569, 570

Y Yalamov, Yu.L., 216 Yengibarian, N.B., 18

Z Zaslavsky, G., 198, 478, 490 Zemansky, M.W., 290, 293 Ziff, R.M., 480, 491 Zweifel, P., ix, 17, 57, 61, 63, 78, 80, 85, 88, 91, 93, 179, 181, 185, 191–193

Subject Index

A Absorption coefficient continuous, 90, 143, 144, 153, 157, 471 line, frequency integrated, 144 Absorption profile complex, 283, 287 Doppler, 20, 27, 144, 160, 332 Lorentz, 20 Voigt, 20, 27, 160 Addition of layers, 203 Albedo, 2, 17, 22, 30, 41, 79, 132, 193, 210, 265, 529 Alignment, 274, 276, 284, 322 ALI method, 546 Alternative nonlinear equations, 6, 206, 207–211, 427 Analytic function, viii, 59, 78, 137, 211, 215, 245, 370, 371 Angular momentum, 272, 273, 281, 283, 287 Asymptotic expansion, 75, 76, 146, 279, 397, 461, 463, 464, 467, 468, 471, 481, 484, 495, 499, 519, 527, 529, 530, 539, 541, 543, 547, 550, 554, 556, 565 matching, 392, 533, 549 Auxiliary H-function, 3, 45, 85, 119, 196, 206, 226 Auxiliary X-function, 47, 52, 62, 80–83, 88, 103, 119, 147, 160, 162, 189, 196, 199, 206, 219, 248, 362, 370–373, 417–419, 423–424 B Boltzmann constant, 19, 26 equation, 12, 541

Boundary layer, 2, 7, 129, 159, 233, 237, 238, 392, 516, 520, 522–526, 529, 533, 536, 539–544, 549–553, 555–557 Branch cut, 56, 74, 81, 83, 86, 91, 125, 137, 175, 219, 220, 224, 227–229, 232, 233, 237, 241, 245, 248, 249, 400 Branching ratio, 282 Broadening coefficient, 284, 300

C Case expansion method, 3, 179, 184, 186, 193 Cauchy-type kernel, viii, 2, 3, 5, 45–47, 51, 52, 54–57, 60, 63, 73, 80, 83, 85, 143, 180, 194, 215, 233, 404 Center-to-limb, 85, 128, 352, 356, 358, 544, 547 Chandrasekhar’s law, 545, 547 Characteristic equation, 79 Characteristic scale, 474 Circular polarization, 257, 265–267 Classical harmonic oscillator stationary solution, 297 transitory solution, 296, 297 Coherences, 272–274, 277, 293 Coherency matrix, 260, 265, 270–271 Collisions elastic, 271, 273, 275, 276, 283–285, 297, 299, 313, 322, 325, 348, 563 inelastic, 16–17, 146, 147, 271, 276, 284, 297, 322, 551, 559, 560 Conservative non-conservative, 2–6, 83, 86, 113, 163, 164, 174, 184, 200, 220, 227, 306, 312, 325, 342, 343, 360, 378, 411–421, 427–442, 451, 533, 578

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 H. Frisch, Radiative Transfer, https://doi.org/10.1007/978-3-030-95247-1

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588 Continuous spectrum, 182–184, 188 Criticality condition, 181, 516, 529 D Damping constant, 290, 298, 569 Degree of order, 274 Density matrix, 256, 272–274, 277, 283, 284, 293 Depolarization, 255, 276, 278, 287, 299, 313, 324, 326, 327, 337, 343, 533, 547, 554, 555 Destruction probability effective, 143, 145, 149, 150, 153 Diffraction problem, 44, 216, 217, 237, 241–246 Diffuse radiation field, 22–23, 201, 314, 440 Diffuse reflection polarized, 5, 308 scalar, 311 Diffusion anomalous, 6, 7, 459, 460, 467, 476, 478, 480, 481, 486, 549–562 approximation, 133, 402, 515–531, 533–548 equation, 7, 470, 477, 515, 518–521, 523–525, 528–531, 533, 538, 539, 541, 542, 544–546, 576–580 ordinary, viii, 6, 7, 459, 460, 476–478, 480, 481, 490, 515 Dimension multi-dimension, 4, 305, 315, 316, 323, 327–330, 334, 338–339 one-dimension, 2, 4, 11, 13, 22, 305–307, 312, 315–317, 323–327, 330, 334–338, 342, 479, 491, 507, 508, 511, 515, 518, 522, 525, 526, 528, 539, 564, 576 Dirac distribution, 14, 33, 52, 64, 66–67, 98, 105, 126, 174, 182, 188, 190, 191, 349, 374, 392, 405, 429, 552 Dirichlet boundary condition, 520, 530, 531 Discrete ordinate method, 46, 143, 180, 196, 365, 385 Dispersion function, 42–44, 47, 52, 54, 59, 61, 69, 73–81, 83, 87, 90, 93, 98, 100, 118, 122, 133, 135, 144, 147–148, 150, 160, 162, 164, 169, 171–174, 181, 182, 184, 192, 199, 206, 207, 211, 218, 220, 223–228, 237, 246, 247, 361, 362, 370, 395, 397, 413–416, 436, 452, 484, 498, 508 Dispersion matrix, 342, 360, 361, 365–366, 369–370, 411, 413–414, 419–422, 427, 442, 447, 448, 451

Subject Index Distribution function, 12 Doppler broadening, 19 profile, 20, 21, 71, 72, 78, 87, 93, 129, 157, 160, 168–173, 326, 327, 459, 462, 464, 465, 470–472, 474, 475, 477, 488, 490, 509–511, 513, 550, 551, 554, 555, 557, 559, 561, 565, 567 width, 19, 144, 167, 278, 279, 552, 568

E Eddington approximation, 176, 525–529 Eddington–Barbier relation, 30, 553 Effectively thick, 536, 556, 562 Effectively thin, 556, 558, 562 Effective temperature, 27 Eigenfunctions, viii, 3, 23, 46, 51, 63, 134, 148, 179–194, 365 Eigenvalues, 179–184, 188, 518 Einstein coefficients, 16, 258, 274, 276 Elastic collisions, 271, 273, 275, 276, 283–285, 297, 299, 313, 322, 325, 348, 563 Electromagnetic field, 44, 380 Electron scattering, vii, 255 See also Thomson scattering Elementary methods, 195 Elementary redistribution functions, 7, 565, 567 Emission induced (see Stimulated emission) thermal, 144 Escape probability, 557 Expectation, 166 See also Probability Exponential integrals, 166, 384, 460, 461, 565 Extrapolation length also extrapolation point, 165, 387, 524

F Factorization (Wiener–Hopf), 47, 55, 57, 87, 211, 216, 218, 219, 221, 228 Fick’s law, 516, 527, 528 Finite slab, 168, 549, 556 Flat spectrum approximation, 325 Flux-limited diffusion, 528 Fourier inversion, 43, 52, 54, 97, 99, 102, 112, 114, 123–126, 138, 219, 227, 232, 233, 240, 249, 482, 572 Fourier transform complex, 3, 43, 215, 217, 218, 222–223, 233 real, 43, 237, 238

Subject Index Fractional Laplacian, 471, 474, 549 Fredholm alternative, 537

G Gaussian distribution, 488, 490, 491 Generalized profiles, 274, 286, 287, 289, 298, 331 Generalized Rayleigh scattering, 343 Generating function, 31, 483, 494, 495, 497, 500–502, 504 Green function resolvent function, 2, 4, 11, 32–36, 56, 97–115, 151, 191, 198, 349, 497, 507 surface Green function, 11, 36, 56, 97–114, 116, 118, 139, 151, 196, 198, 377, 497, 507 Green matrix resolvent matrix, 349–352, 427, 433, 434, 440, 441, 443–444, 448, 451

H Hanle effect absorption matrix, 277, 278, 282, 332, 411 characteristic field strength, 278, 281, 285 hanle factor, 277, 278, 280, 332 mixing angle, 280 phase matrix, 256, 279–282, 300, 316, 331, 332, 334, 337, 339 rosette, 294, 295 spectral details, 285 in turbulent magnetic fields, 278, 279, 281, 282 Heaviside function, 67, 493 Heavy tail, 477 H-function alternative equation, 437 nonlinear integral equation, 45, 85, 86, 176, 195, 200, 203–205, 207, 213, 341, 351, 353, 354, 359, 363, 526 singular linear integral equation, 85 uniqueness, 207–211, 436–437 Hilbert transform, 2, 3, 5, 46, 47, 51, 55–67, 78, 80, 81, 101, 103, 105, 106, 110, 114, 119, 136, 137, 139, 144, 150, 175, 186, 189, 193, 215, 219–221, 239, 241, 359–361, 365, 368, 369, 374, 375, 405, 406, 429, 443, 451, 452 H-matrix nonlinear integral equation, 351, 354, 365, 412, 427, 439, 445–447, 449–451

589 Holtsmark distribution, 477, 490, 497 Hopf-Bronstein relation, 161, 162, 164, 196, 200, 235, 381, 382 Hopf function generalized Hopf functions, 382–385, 387

I I-matrix, 6, 359, 427–455, 555 Integral equation convolution, 2, 3, 11, 18, 23–25, 27, 32, 41, 42, 46, 78, 97, 98, 103, 159, 199, 215, 217, 305–307, 316, 321, 325, 336, 341, 342, 411, 459, 554 nonlinear, viii, 6, 37, 45, 84–86, 91–93, 131, 132, 176, 195, 196, 200, 203–205, 207, 213, 341, 351, 353, 354, 359, 363, 365, 400–404, 411, 412, 425, 427, 439, 445–447, 449–451, 526, 533, 555 singular with a Cauchy-type kernel, viii, 3, 5, 46, 47, 51, 55–57, 60, 63, 73, 83, 85, 143, 180, 215, 233, 361, 404 Wiener-Hopf, 2, 5, 25, 26, 33, 34, 47, 51, 56, 57, 69, 103, 110, 115, 118, 119, 125, 127, 129, 134, 143, 145, 150, 151, 159, 160, 166, 167, 195, 197, 204, 210, 212, 228, 231, 233, 326, 337, 341, 342, 345, 349, 351, 354, 359, 361, 367, 368, 382, 405, 408, 409, 412, 421, 427, 442, 480, 491–494, 498, 501 Index characteristic, 460, 461 partial, 423 stability, 500 Interior field, 7, 159, 523–525, 539, 540, 542, 544, 549 Inverse scattering transform, 246 Irreducible, 4, 256, 262, 265, 267, 299, 306, 313, 315–317

J Jones matrix, 262, 270, 271 vector, 257, 270 Jordan lemma, 99

K Kernel, 2, 11, 42, 51, 69, 117, 143, 169, 179, 196, 215, 305, 343, 368, 412, 442, 459, 469, 479, 510, 519, 536, 550

590 Kernel function asymptotic expansion, 556 complete frequency redistribution, 461–466, 495 Fourier transform, 61, 88, 217, 224, 465 inverse Laplace transform, 2, 24, 51, 69–73, 147, 155, 169, 183, 369, 397, 413, 451, 463 Laplace transform, 24, 147, 169, 183, 369, 397, 413, 422, 451 monochromatic, 27, 460, 461 Kernel matrix Hanle effect, 5, 196, 279, 305, 307, 330–339, 343–345, 361, 424–425, 427 monochromatic, 27, 460, 461 resonance polarization, 342–347, 349–351, 421, 422 Korteweg-de Vries equation, 246 Kramers-Heisenberg formula, 271, 275

L Laboratory frame, 285, 321, 332, 563, 565, 568 LA04 Landi Degl’Innocenti and Landolfi 2004, 13, 20, 21, 159, 165, 256, 277, 305, 325 Lambda iteration, 32, 343, 546 Landé factor, 278, 286 Laplace transform direct, 51, 52, 55–57, 97, 110–111, 115, 118, 119, 121–123, 134, 139–141, 359, 361, 461 inverse, 2, 24, 46, 47, 51–57, 61, 69–73, 97, 98, 101–108, 110, 114–121, 123, 125, 128, 128, 131, 134–139, 141, 143, 147, 150–152, 155, 157, 160, 163, 169, 174–176, 180, 181, 183, 185, 192, 232, 233, 236, 237, 240, 327, 354, 358–362, 365, 368, 369, 373, 390, 397, 405, 413, 421, 427, 433, 440, 442, 443, 451, 460, 461, 463–464, 509 Larmor frequency, 278, 286, 292 Lévy walks, 480, 487–491, 495, 497–500 Limb-darkening, 30, 42, 128 Linear polarization, vii, 1, 255, 257, 261, 266, 267, 272, 277, 278, 280, 282–284, 286, 295, 321, 331, 386, 499, 572 Liouville theorem, 59, 60, 89, 102, 104, 106, 136, 140, 175, 219 Local thermodynamic equilibrium (LTE) nonlocal thermodynamic equilibrium (NLTE), 21

Subject Index Longest flight, 476, 477, 479 M Magnetic frame, 280 Magnetic sublevels, 271–273, 277, 287 Maxwellian velocity distribution, 15, 20, 285 Mean number of scatterings, 7, 17, 110, 475, 545, 557–562 Mean pathlength, 7 Micro-turbulent magnetic field, 278, 279, 281, 282 velocity, 19 Milne-Eddington atmosphere, 161 Milne equation, 41, 42, 161, 164, 174, 235, 404–409 Milne problem polarized, 5, 309–311, 378–389, 427 scalar, 378, 381–384, 386–388, 405 Monochromatic scattering, 1, 11, 43, 52, 69, 97, 115, 148, 160, 179, 195, 216, 305, 341, 370, 413, 432, 459, 467, 479, 507, 515, 541, 549, 563 Mueller matrix, 270 Multipole components, 268, 269, 275, 276, 283 coupling coefficients, 330 N Negative hydrogen ion, 143 Net linear polarization, 380 Neumann series, 30–32, 162, 166, 196, 350, 351, 475, 500 Noncoherent scattering, see Complete frequency redistribution Nonlinear $H$-equation matrix, 4, 5, 45 scalar, 353 Null space, 518, 519, 521 O Operator perturbation method, 567 See also ALI method P Partial frequency redistribution approximations, 339 atomic frame, 563, 570, 572 direction-averaged, 564, 573, 578 Pauli spin matrices, 261 Perturbation expansion, 285 Phase diagrams, 77, 78, 79, 173

Subject Index Phase function isotropic, 86, 200, 206, 520 Rayleigh, 320 Phase matrix Hanle, 279–282, 288, 300, 334 Rayleigh, 4, 256, 261–272, 275, 276, 281, 283, 288, 300, 306, 309, 313, 316, 321, 365, 366, 412, 534 resonance scattering, 281 Photon distribution function, 12 mean-free path, 579 scattering, 1, 6 Planck function, 15, 19, 21, 30, 469, 475 Plane-parallel atmosphere infinite, 190 semi-infinite, vii, 25, 154, 365, 378, 520, 522, 525 Plasma, viii, 27, 182 Plemelj formulae, 58–60, 62, 64–67, 74, 81, 89, 100, 102, 104, 106, 107, 111–114, 120–122, 124, 125, 139, 141, 148, 238, 374, 376, 406, 413, 416, 429, 431, 443, 452, 454 Pochammer symbol, 485 Polarization basic spherical tensors, 4, 265, 267, 291, 316 continuum, 255, 308 rate, 280–282, 290, 293–295, 297, 386, 389, 534, 544–547 resonance, 5, 198, 255, 256, 272, 276–278, 282, 286, 288, 289, 293, 305–307, 313, 315, 316, 321–331, 334, 338, 341–347, 349–351, 353–355, 358–362, 411, 421–425, 427, 442–447, 449, 455, 502, 549, 553–556, 568 tensor, 261, 265, 268, 270, 291 Polarization ellipse, 258 Primary source plane source, 192, 508–513 point source, 32, 180, 191, 507, 508, 510–513 thermal, 316 Principle of invariance, 196, 200 Probability characteristic function, 482, 485, 486 cumulative probability, 479, 492, 493 expectation, 166 probability density, 14, 297, 476, 477, 479, 481, 482, 484–486, 488, 489, 491–493, 498, 500–502 Profile, see Absorption profile

591 R Radiation field diffuse field, 22–23, 201, 314, 440 emergent intensity, 154, 525, 526 flux, 390, 403 mean intensity, 281 specific intensity, 12 Radiative equilibrium, viii, 17, 26, 27, 41, 159, 215 Radiative transfer equation boundary conditions, 22, 180, 193, 378, 576 formal solution, 329 Raman scattering, 271, 569 Random walk index, 500 mean displacement, 481, 486, 491 mean maximum, 497 Rayleigh scattering blue color of the sky, vii (KQ) expansion, 312–321 phase matrix, 256, 261–271, 275, 276, 281, 283, 300, 305, 306, 309, 313, 316, 321, 365, 366, 412, 534 Real-space methods, 5, 44–45 Redistribution complete, 23, 52, 78, 80–83, 88, 90, 91, 125, 190–192, 567 partial, 7, 284, 566, 569 Redistribution functions elementary, 7, 563–565 generalized, 563 Redistribution matrix, 5, 256, 272, 274–276, 279–282, 284–287, 289–290, 295–298, 306, 321–323, 325, 328, 331–334 Reference frame laboratory, 285, 321, 332, 563, 565, 568 magnetic, 280 Rescaled optical depth, 467, 469, 536 Rescaling, 6, 467 Resolvent function, 2–4, 11, 32–36, 56, 97–115, 119, 120, 125, 126, 139, 143, 144, 149–151, 154, 160, 163, 167, 168, 174–176, 187, 191, 196, 198, 199, 211, 216, 219, 227, 230–233, 349, 368, 433, 483, 484, 494, 495, 497, 501, 507–514, 549 method, 3, 32, 341 Resonance polarization effect of collisions, 272, 313 spectral details, 282–285 Riemann–Hilbert problem

592 homogeneous, 60, 80, 219, 220, 228, 238, 365, 369–371, 417, 423 inhomogeneous, 228, 429, 452 Robin boundary condition, 531 S Saha–Boltzmann law, 21 Scaling laws, 6, 7, 128, 168, 464, 465, 468, 471, 475, 476, 486, 487, 489, 508, 509, 533, 552, 556–562, 565, 567, 578–580 Scattering conservative, 3, 69, 78, 83, 87, 92, 94, 131, 159–176, 209, 210, 216, 237, 365, 419, 432, 435, 496, 509–512, 514, 515, 533, 551 non-conservative, 3, 5, 83, 86, 163, 174, 184, 200, 216, 220, 312, 412, 413, 428 Scattering kernel, see Kernel Scattering phase matrix Hanle effect, 279–282, 288, 316, 334 Rayleigh, 256, 261–271, 275, 276, 281, 283, 300, 306, 307, 309, 313, 316, 321, 365, 366, 412, 534 resonance polarization, 321 Schrödinger equation, 246, 283 Schuster model, 558 Schwarzschild–Milne equation, 42 Second solar spectrum, 255, 572 Singular integral equations, viii, 2, 3, 5, 45–47, 51–67, 73, 80, 81, 97, 101, 102, 105, 108, 110, 114, 121, 122, 135, 143, 144, 150, 163, 174, 186, 189, 191, 193, 194, 215, 233, 341–363, 365, 366, 368, 369, 373, 397, 404, 405, 409, 425, 427, 428, 434, 436, 443, 448, 451–455, 472, 550, 571 Source function integral equation, 2, 6, 7, 11, 23–30, 42, 43, 45, 115, 143, 151, 159, 195, 204, 210, 347, 357, 467, 468, 471, 478, 494, 499, 500, 507, 515, 550, 556, 564, 574, 579 Specific intensity, 12 Spectral lines Ca I 4224 Å, 572 Ca II H 3968 Å, 572 Ca II k 3933 Å, 572 Mg II h 2803 Å, 572 Mg II k 2796 Å, 572 Na I D1 5896 Å, 275 Na I D2 5890 Å, 275 Sr I 4607 Å, 276, 282, 325, 331

Subject Index Spherical tensors, 256, 262, 267–269, 306, 312, 313, 315, 317, 318, 335 Spherical unit vectors, 287, 292, 299 Spectrum continuous, vii, 11, 12, 14–16, 182–184, 188 line, 255 √ -law, 3, 5, 117, 118, 127, 154, 164, 195–213, 341–363, 480, 499–502 Stable probability distribution stability parameter, 490 Statistical equilibrium, 21, 273–275, 300 Stimulated emission, 21 Stirling formula, 485 Stokes parameters, 12, 257–259, 261, 263–268, 270, 271, 277, 280, 291, 294, 299, 305–308, 312–321, 323, 331, 338, 386, 411, 533, 534, 539 profiles, 274 vector, 260, 265, 274, 306, 312, 313, 317–319, 324, 328–330, 334, 335, 442 T Taylor expansion, 79, 101, 169, 407, 416, 469, 496, 561, 575, 577 Thermalization depth, 575 length, 6, 7, 127, 467, 468, 474–477, 479, 480, 485–487, 489, 507, 508, 513, 535, 550, 564, 567, 568, 572, 575, 576 process, 467 Thomson scattering, 16, 255, 263 Translation invariance, 193, 198, 205 TQK spherical tensors, 262, 267, 306, 313, 315, 317, 335 U Universality class, 564 Uniqueness, 83, 87, 206–211, 436–437, 439 V Voigt function approximate form, 20 profile, 20, 21, 27, 71, 72, 87, 127, 129, 146, 147, 149, 160, 168, 169, 171, 173, 459, 462, 464–466, 471, 472, 474, 475, 488–490, 509–511, 514, 550, 551, 555, 557, 559–561, 568, 571

Subject Index W WK factor, 328 Weak anisotropy approximation, 275 Wiener–Hopf (WH) diffuse reflection, 409 factorization, 47, 55, 57, 87, 211, 216, 218, 219, 221, 228 integral equation, 2, 5, 26, 27, 34–36, 47, 51, 56, 57, 69, 103, 110, 115, 118, 119, 127, 129, 134, 143, 145, 150, 151, 159, 161, 166, 167, 195, 197, 204, 210, 212, 215, 223, 228, 231, 233, 326, 327, 334, 336, 337,

593 341, 342, 345, 349, 351, 354, 359, 361, 367, 368, 382, 384, 389, 405, 408, 409, 412, 421, 427, 451, 480, 491–494, 501 Milne equation, 233, 235 for spectral lines, 237–241 Wiener–Hopf (WH) method, 2–4, 43–44, 47, 57, 78, 143, 165, 206, 215–249

Z Zeeman effect, 196, 278, 290, 293, 295