The Physical Processes and Observing Techniques of Radio Astronomy: An Introduction [1st ed.] 9783319169811, 9783319169828

Citation preview

Undergraduate Lecture Notes in Physics

Thomas G. Pannuti

The Physical Processes and Observing Techniques of Radio Astronomy An Introduction

Undergraduate Lecture Notes in Physics Series Editors Neil Ashby, University of Colorado, Boulder, CO, USA William Brantley, Department of Physics, Furman University, Greenville, SC, USA Matthew Deady, Physics Program, Bard College, Annandale-on-Hudson, NY, USA Michael Fowler, Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen, Department of Physics, University of Oslo, Oslo, Norway Michael Inglis, Department of Physical Sciences, SUNY Suffolk County Community College, Selden, NY, USA

Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading. ULNP titles must provide at least one of the following: • An exceptionally clear and concise treatment of a standard undergraduate subject. • A solid undergraduate-level introduction to a graduate, advanced, or nonstandard subject. • A novel perspective or an unusual approach to teaching a subject. ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level. The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career.

More information about this series at http://www.springer.com/series/8917

Thomas G. Pannuti

The Physical Processes and Observing Techniques of Radio Astronomy An Introduction

Thomas G. Pannuti Department of Physics, Earth Science and Space Systems Engineering Morehead State University Morehead, KY, USA

ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notes in Physics ISBN 978-3-319-16981-1 ISBN 978-3-319-16982-8 (eBook) https://doi.org/10.1007/978-3-319-16982-8 © Springer International Publishing Switzerland 2020 This work is subject to copyright. All rights are reserved 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 International Publishing Switzerland The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For Angie and Michael—my wife and my son, my loves and my world. For my mother Mary, who nurtured and encouraged my scientific curiosity since my earliest years. This book is dedicated to her loving memory. For all of my family members and their boundless love and support over the years. “You can observe a lot by just watching.”—Yogi Berra

Preface

The content of this book was first gathered as the substance for a course on radio astronomy that is taught once an academic year at Morehead State University. The students who typically enroll in this course are physics majors pursuing a concentration in astrophysics as well as space systems engineering majors. Nominally, both types of students enrolled in this class are juniors and seniors and the class itself is a graduation requirement for each type of student. The structure of the course includes lectures as well as the remote observation of prominent radio sources with the 21 Meter Space Tracking Antenna that is adjacent to campus and operated by the Space Science Center of the Department of Physics, Earth Sciences and Space Systems Engineering of Morehead State University. I have personally taught this course since the Fall 2010 semester and guided its development: as I taught the class, I found myself unable to find a textbook with sufficiently detailed descriptions of natural phenomena and their accompanying physical processes along with examples and problems that illustrate these concepts as well as facilitate understanding by students of these phenomena. To help fill this void, I prepared my own lectures, exams, and homework assignments for the course: these prepared materials form the basis of this text. Radio astronomy is an extremely vast subject spanning many fields of physics, engineering, and computer science: thus, no single textbook—especially a textbook intended for a single-semester undergraduate-level course—could do the entire field as a whole justice. In recognition of this reality coupled with the need to reconcile the scope of the textbook with a semester-long course, I have first provided a background review of some pertinent basic concepts that is relevant to the text that follows, such as angles and angular measurement (which segues into a discussion about sky coordinate systems), gravity, orbits, and orbital motion (with a discussion and review of Kepler’s three laws of planetary motion), light (with treatment of both wave- and particle-like properties, the polarization of light—which is essential knowledge to master in radio astronomy—along with the Doppler Effect), and finally the Bohr model of the atom and relativity. The instructor of the course and the students may proceed with this chapter as a whole or in part—or omit it in whole or in part—based on the level of preparation of the students and the scope vii

viii

Preface

of the course. After the review of background physics concepts, I have chosen to concentrate on the most commonly encountered physical processes encountered in radio astronomy (namely, synchrotron radiation, thermal bremsstrahlung emission, and molecular transitions), thus laying the foundation when astronomical sources that emit copious amounts of radiation through these processes are encountered later in the text. Following the discussions about the background physical processes, I next discuss the principles and techniques of radio astronomy, namely the observing strategies of single dish radio telescopes and interferometers. As part of this discussion, I emphasize presenting some commonly encountered equations that will be useful for computing such crucial properties as the sensitivity of the radio telescope or interferometer. In this context, I also provide a description of Fourier transforms: my motivation for providing treatment of Fourier transforms here is twofold. Firstly, Fourier transforms are clearly relevant in the creation of astronomical sources at all wavelengths and not solely at the radio. Secondly, because Fourier transforms will be encountered by both types of students whom have typically enrolled in my radio astronomy class (specifically, in higher-level physics and mathematics courses for the physics majors and higher-level engineering courses for the space science majors), I feel that an exposure to Fourier transforms in this context is timely and helpful to the students. For similar reasons, I have presented a discussion in this chapter about Bessel functions as well. In this chapter (and, for that matter, in this textbook), I do not provide a very rigorous treatment about the hardware involved in radio astronomy: I have taken this approach for two reasons. Firstly, the available literature on hardware for radio astronomy is already quite plentiful. Secondly, to keep the scope of the material that may be covered by this book within the time frame of a semester, I have elected not to provide much discussion on this topic. In the remaining chapters, discussions are presented in turn about radio observations of sources in the Solar System, the Galaxy, external galaxies, and cosmology itself. It is clearly beyond the scope of any text to provide very detailed descriptions of radio observations (and the insights revealed by these observations) of all types of astronomical sources, so these chapters are best viewed as an attempt to cover in a broad manner the types of astronomical sources that are currently the subjects of extensive study and coverage in the modern research literature on radio astronomy. I hope this textbook proves useful for both instructors and students who are exploring radio astronomy as part of a broader study of astrophysics or space science. I welcome feedback and comments from all readers to help improve the quality of this work. Morehead, KY, USA July 2020

Thomas G. Pannuti

Acknowledgements

Significant efforts like the creation of a textbook over such an expansive topic as radio astronomy are never created in a vacuum, and the present work is no exception. I would like to express my most heartfelt thanks to the following people, without whom the dream of this textbook would never have been realized. Thanks to everyone at Springer Publishing for shepherding this effort to completion. I would like to thank Hannah Kaufman, Silembarasan Panneerselvam, Mario Gabriele and Jeffrey Taub for guiding me through the process of writing this book, making all of the necessary arrangements, fielding all of my questions, and providing boundless encouragement along the way. Thanks also to Nora Rawn from Springer Publishing for also assisting in the development of the manuscript as well. Thanks to Morehead State University for granting me a sabbatical during the 2014–2015 academic year to concentrate on the writing of this textbook. Thanks to the Director of the Space Science Center and my former department chair— Dr. Benjamin Malphrus—for encouraging the development of astronomy courses and a research program in astrophysics centered on undergraduate students within the Space Science Center at the Department of Physics, Earth Sciences, and Space Systems Engineering (formerly the Department of Earth and Space Sciences) at Morehead State University. Thanks to Space Science Center staff engineers Mike Combs, Bob Kroll, and Jeff Kruth for keeping the 21 Meter Space Tracking Antenna at the Space Science Center in working order and capable of making sensitive astronomical observations. Thanks also to Jeff Kruth for his willingness to give guest lectures in my courses on the hardware of radio astronomy. Special thanks to my colleagues in the Department of Physics, Earth Sciences, and Space Systems Engineering at Morehead State University: to Professor Eric Jerde for his encouragement of this work as department chairperson and Professor Dirk Grupe for helping to “field test” lectures, homework assignments, and examinations that I had prepared for the course during the Fall 2014 semester while I was on sabbatical. Thanks as well to Dr. Chuck Conner for useful discussions about Fourier Transforms.

ix

x

Acknowledgements

Thanks also to all of the Morehead State University students who have taken this course over the years and provided useful feedback on all of the course content that formed the basis of this book. Among the many students who provided feedback, I’d like to specifically thank Kristen Ammons, Megan Conley, Ethan Palmer and Katerina Winters. Special thanks to Mikey Awbrey, Matthew Hezelstine, and Marina Vitatoe—all graduates of Morehead State University—for preparing many of the figures seen in the book. Thanks also to Evan van Daniker of the Craft Academy for also providing comments and feedback on the text. Thanks to Dr. Elias Brinks who taught me the fundamentals of radio astronomy in a graduate and advanced undergraduate level course on the subject when I was a bright-eyed first-year graduate student at the Department of Physics and Astronomy at the University of New Mexico. The content presented by Dr. Brinks in that course helped form many of my ideas for teaching radio astronomy, and those ideas are fleshed out in the present textbook. Thanks also to my dissertation advisor Dr. Nebojsa Duric who taught me so much about radio astronomy as I worked to complete my dissertation topic. I am very grateful to the National Radio Astronomy Observatory (NRAO) who provided financial support and wonderful hospitality for my sabbatical visit to the Science Operations Center (SOC) at Socorro during the Fall 2014 semester. Thanks to SOC Director Dr. Dale Frail for arranging the funding for my sabbatical visit, providing a welcoming atmosphere, offering feedback on an earlier version of this text, and hosting wine tasting parties at his home in Socorro. Thanks to SOC Assistant Director Dr. Claire Chandler for serving as my host for my visit, making arrangements for my stay at the SOC, and for providing comments on the manuscript. There were many very kind people at Socorro who did so much to make sure my visit was very pleasant and productive. The SOC is a vibrant and welcoming facility and there were so many staff members, staff engineers, postdocs, and staff astronomers who were always welcoming, friendly, and encouraging, whether assisting me with the writing of this manuscript, inviting me out to weekly social dinners, or inviting me to join the SOC’s soccer team. Thanks to Skip Lagoyda, Allen Lewis, Lori Appel, Terry Lopez, and Bernadette Lucero for helping me to become settled in Socorro and in the SOC and for answering all of my questions, including ones about the copy machine. The engineers, postdocs, and astronomers were all very helpful and encouraging in my efforts: many thanks to Brent Avery, Hichem Ben Frej, Dr. Barry Clark, Dr. Mark Claussen, Dr. Paul Demorest, Dr. Vivek Dhawan, Steve Durand, Dave Finley, Brian and Marie Glendenning, Dr. Miller Goss, Dr. Eric Griesen, Dr. Chris Hales, Dr. Hubertus Intema, Preshanth Jagannathan, Minnie Mao, Drew and Heidi Medlin, Dr. Betsy Mills, Dr. Amy Mioduszewski, Dr. Steve Meyers, Dr. Juergen Ott, Dr. Frazer Owen, Peggy and Dr. Rick Perley, Dr. Lorant Sjouwerman, Meri Stanley, Stephan Witz, and Dr. Joan Wrobel for all that they did during my visit. Thanks to Judy Stanley and Yvonne Magener—also in the SOC—for kindly inviting me to participate in public outreach events at the Very Large Array during

Acknowledgements

xi

my sabbatical visit to Socorro and encouraging me to help describe the Very Large Array to members of the general public. Thanks also to faculty and students at New Mexico Institute of Mining and Technology at Socorro—namely, Crystal Anderson, Dr. Jean Eilek, and Dr. Dave Westpfahl—who also provided feedback and suggestions on the manuscript. Thanks also to all of the astronomers at other institutions who provided input and suggestions on the content of the book. To Dr. Tracey DeLaney (West Virginia Wesleyan College), Dr. Mark Morris (University of California at Los Angeles), Dr. Ylva Pihlstrom (University of New Mexico), Dr. Thomas Troland (University of Kentucky), and Dr. Rosa Murphy Williams (Columbus State University), many thanks for taking the time to review and provide commentary on the text. Thanks to John Stoke for providing me with many images on behalf of NRAO for inclusion in this book and for providing feedback on the images as well. Thanks to Marsha Bishop (Observatory Librarian, NRAO) and Lance Utley (Database Administrator, NRAO) for helping me obtain resources needed for writing this book. Thanks as well to Dr. Yuri Kovalev and Dr. Kardashev Nikolay for furnishing permission to reproduce the figure of the RadioASTRON satellite that appears in this book. Thanks to Dr. Christina Lacey and Dr. Fabian Walter for granting permission for figures from their papers to be included in this book. Thanks to my family for all of their boundless love and support with all of my efforts over the years. All of my successes in my life would not be possible without them. My mother Mary always encouraged my interest in science and in particular loved visiting the Very Large Array—this book is dedicated in part to her loving memory. To my father Carl, my brothers Paul and Carl Jr., and my sister Joan— thanks for all of the encouragement that you provided to your brother as he moved across the country multiple times during his professional career as a scientist. Last but certainly not least, thanks to Angela Brown Pannuti who one night accepted my marriage proposal under a canopy of stars and brings light to all of my days. And to our son Michael Pannuti, who always views the world and the sky in wonder and fascination: he always inspires us and fills our world with love. The research for this book has made use of NASA’s Astrophysics Data System.

Contents

1

2

Introduction: Why Make Astronomical Observations at Radio Frequencies? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 15

Angles, Gravity, Light, the Bohr Model of the Atom and Relativity . . . 2.1 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Trigonometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Sky Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Orbits and Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Kepler’s Laws of Planetary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Kepler’s First Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Kepler’s Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Kepler’s Third Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Wave-Like Properties of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Particle-Like Properties of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Wave-Particle Duality for Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Polarization of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Bohr Model of the Atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 17 20 23 31 34 35 35 37 39 43 43 47 48 48 53 54 59 63 65 67

xiii

xiv

3

4

5

Contents

Emission Mechanisms: Blackbody Radiation, An Introduction to Radiative Transfer, Synchrotron Radiation, Thermal Bremsstrahlung, and Molecular Rotational Transitions . . . . . . . . . . . . . . . . 3.1 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 An Introduction to Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Thermal Bremsstrahlung. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Molecular Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 79 84 102 106 111 114

Radio Observations: An Introduction to Fourier Transforms, Convolution, Observing Through Earth’s Atmosphere, Single Dish Telescopes, and Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 An Introduction to Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Angular Resolution and Observing Through Earth’s Atmosphere . . . 4.4 Single Dish Radio Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Two-Element Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Multi-Element Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 A Discussion about Interferometer Observations . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 121 123 128 137 138 146 153 158 161 163

Solar System Radio Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Radio Emission from the Solar System: An Overview. . . . . . . . . . . . . . . . 5.2 Radio Continuum and Radar Observations of Astronomical Sources in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Continuum Radio Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Radar Observations of Solar System Objects . . . . . . . . . . . . . . . . . 5.3 Radio Observations of the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Radio Observations of the Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Moon and The Terrestrial Planets . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 The Giant Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Radar and Radio Observations of the Minor Bodies in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Radio Searches for Exoplanets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 166 166 169 171 176 176 178 183 185 186 189

Contents

6

7

Galactic Radio Astronomy: Galactic Structure, HII Regions, Supernova Remnants, Neutron Stars and Pulsars . . . . . . . . . . . . . . . . . . . . . . . 6.1 Overview of Radio Observations of the Milky Way Galaxy . . . . . . . . . . 6.2 Radio Observations of the Diffuse Component of the Milky Way Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Galactic Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 HI Emission: The 21-cm Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Galactic Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Motion in the Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Magnetic Fields: The Zeeman Effect and Zeeman Splitting. . 6.2.6 Propagation Through the Interstellar Medium and Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Radio Observations of Star Formation Sites in the Milky Way Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Molecular Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Masers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 HII Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Maser Emission from Evolved Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Microquasars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Supernova Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Neutron Stars and Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Rotation Measure and Dispersion Measure . . . . . . . . . . . . . . . . . . . 6.4.6 Galactic Center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extragalactic Radio Astronomy: Galaxy Classification, Active Galactic Nuclei, Superluminal Motion, Galaxy Clusters, and the Cosmic Microwave Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Extragalactic Radio Astronomy and Galaxy Classification . . . . . . . . . . . 7.1.1 Cosmology and Hubble’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Normal Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 HI Line Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Tully–Fisher Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Radio Continuum Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Radio Galaxies and Active Galactic Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Blazars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Classification System of Radio Galaxies . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Radio Jets from Active Galactic Nuclei . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Superluminal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Radio Lobes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Spectral Energy Distributions of Radio Galaxies . . . . . . . . . . . . . 7.3.7 Doppler Boosting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

191 191 191 191 193 199 206 208 210 211 212 216 216 218 225 225 227 228 244 254 257 262 267

269 269 271 275 275 279 281 285 287 288 288 293 298 302 308

xvi

Contents

7.4 Galaxy Clusters and Associated Diffuse Radio Emission. . . . . . . . . . . . . 7.5 The Cosmic Microwave Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The Sunyaev–Zeldovich Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309 313 316 317 320

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Acronyms

AGN ALMA ATCA AU AUI CGS DEC DM DRAO EDT FAST FR-I FR-II FSRQ GMRT HBL HI HII JVLA LBL LHA LST MKS NRAO NSF OVV RA RM UT

Active Galactic nuclei Atacama Large Millimeter/Submillimeter Array Australia Telescope Compact Array Astronomical unit Associated Universities, Inc. Centimeter–grams–seconds unit system Declination Dispersion measure Dominion Radio Astrophysical Observatory Eastern daylight time Five hundred meter Aperture Spherical Telescope Fanaroff–Riley class I radio galaxy Fanaroff–Riley class II radio galaxy Flat spectrum radio-loud quasars Giant Metrewave Radio Telescope High-frequency peaked blazars Neutral (atomic) hydrogen Ionized hydrogen Karl G. Jansky Very Large Array Low-frequency peaked blazars Local hour angle Local sidereal time Meter–kilograms–seconds unit system National Radio Astronomy Observatory National Science Foundation Optically violent variable blazars Right ascension Rotation measure Universal time

xvii

xviii

VLBA VLBI WMAP WSRT

Acronyms

Very Long Baseline Array Very long baseline interferometry Wilkinson Microwave Anisotropy Probe Westerbork Synthesis Radio Telescope

Chapter 1

Introduction: Why Make Astronomical Observations at Radio Frequencies?

1.1 Introduction The title of this chapter—“Why Make Astronomical Observations at Radio Frequencies?”—provides a starting point for the present text and motivates a description of radio astronomy itself in the context of observational astronomy as conducted at all wavelength bands. At the outset of human history, astronomical observations were conducted solely at optical wavelengths, first with the naked eye and then—commencing roughly in the seventeenth century by Galileo Galilei and his contemporaries—with the telescope. Radio astronomy may trace its origins to the 1930s, but it did not flourish as a robust and formal branch of astronomy in its own right until after World War II. Like many fields of astronomy that operate outside of optical wavelengths, astronomers were initially reluctant to believe the results presented by observers conducting observations at radio wavelengths. In fact, the community of professional astronomers initially greeted descriptions of radio observations of astronomical sources with skepticism that these sources would produce any appreciable amounts of emission at radio wavelengths. However, in time the community came to accept observations made at radio wavelengths of astronomical sources as accurate, meaningful, and crucial in helping to advance our understanding of the properties of astronomical sources. Today, astronomers routinely draw upon radio observations along with observations made at other wavelengths (such as optical and infrared) when making multi-wavelength studies of objects as they develop the most thorough understanding possible of these objects. In fact, professional astronomers often receive training in conducting and analyzing observations made at not just optical wavelengths but at other wavelengths too (including radio) during their undergraduate, graduate, and postdoctoral studies. This multi-wavelength training illustrates the direction of modern astronomers to develop the most thorough understanding possible of the properties of astronomical

© Springer International Publishing Switzerland 2020 T. G. Pannuti, The Physical Processes and Observing Techniques of Radio Astronomy, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-16982-8_1

1

2

1 Introduction

sources. Within the developments of these understandings, radio observations often play a very crucial role. A detailed history about radio astronomy is beyond the scope of the current text: thorough treatments of this subject have been presented elsewhere [1–5] and only a very brief summary is presented here. Like all forms of observational astronomy, radio engineers and astronomers design hardware to collect light (electromagnetic radiation) from astronomical sources.1 It is important to provide an initial context for the range of the wavelengths of light (the distance from one peak of a wave to the next) that will be considered in this text to correspond to radio astronomy. While there is no firmly defined definition of the range of wavelengths for the radio domain of the electromagnetic spectrum, a workable definition would include radiation with wavelengths of approximately one millimeter (10−3 m) or greater. Published refereed papers on radio observations will describe observations that have been conducted at a particular wavelength: in a complementary manner, an observation may also be said to have been conducted at a particular frequency (the time required for the wave to complete one cycle). Either convention is appropriate, given the complementary nature of wavelengths and frequencies for light: therefore, a complementary definition of radio astronomy based on frequency of the observed radiation would be for light with frequencies of 1011 cycles per second—also known as 1011 Hertz (Hz)—or greater.2 Like all forms of electromagnetic radiation, the study of radio waves emitted from astronomical sources must contend with the effects of the media through which the waves propagate: these media include both the atmosphere of the Earth and the space between the Earth and the astronomical source of interest. The effects upon electromagnetic radiation by the medium through which the radiation propagates are illustrated with the well-known phenomenon of stars appearing to “twinkle” at night due to turbulence in the atmosphere. At radio wavelengths, astronomers must consider such effects as the opacity of the Earth’s ionosphere (which absorbs light waves with frequencies of approximately 9 MHz or less and prevents such light from reaching the surface of the Earth) and absorption by molecules in the Earth’s atmosphere at particular frequencies. These considerations necessitate the placement of radio observatories that conduct observations at those wavelengths at high altitudes (such as the tops of mountains). At such altitudes, a significant portion (by mass) of the atmosphere by mass may lie below the height of the observatory, thereby reducing the column of the atmosphere (and the potential amount of turbulence) through which the observatory will detect radiation from the source of interest. A more detailed discussion about the challenges of making observations with radio telescopes is presented in Sect. 4.3. In modern times, radio observations must also contend with interference from 1A

detailed treatment of the properties of light is presented in Sect. 2.4.1. unit is named in tribute to German physicist Heinrich Hertz (1857–1894), who was the first scientist to conclusively prove the existence of electromagnetic waves. In radio astronomy, the typical frequency ranges at which astronomical observations are measured are usually expressed in terms of Megahertz (106 Hertz—abbreviated as “MHz”) and Gigahertz (109 Hertz—abbreviated as “GHz”).

2 This

1.1 Introduction

3

Fig. 1.1 Karl Guthe Jansky with the antenna he constructed in Holmdel, New Jersey, with which the discovery of the extraterrestrial radio emission (namely, emission from the center of the Milky Way Galaxy) was made. Image Credit: NRAO/AUI/NSF

artificial sources, such as cellular phone towers and wireless Internet devices, which have rapidly proliferated. These devices transmit extensive stray radiation at radio frequencies, thus threatening the environments of radio observatories that were formerly quite pristine. The history of radio astronomy may best be described as a story of serendipity. The birth of radio astronomy is typically attributed to Karl Guthe Jansky (1905– 1950). Jansky earned a bachelor’s degree in physics from the University of Wisconsin and joined the staff of the Bell Telephone Laboratories in Holmdel, New Jersey, in 1928. Jansky was tasked with identifying sources of radio static that affected trans-Atlantic radio communications: to conduct his analysis, he constructed an antenna with a diameter of approximately 30 m and a height of approximately 6 m. The antenna was designed to detect radio waves with a wavelength of 14.6 m (corresponding to a frequency of 20.5 MHz) and rotated on a turntable so that it could point in any direction (see Fig. 1.1). Rotating the antenna also allowed Jansky to pinpoint the origin of sources of radio emission. As expected prior to the start of his observations, Jansky detected radio emission from several terrestrial sources (namely thunderstorms) but also detected an additional source of emission from an uncertain origin. Noticing that the intensity of this

4

1 Introduction

peculiar source increased and decreased over the course of a day, Jansky initially believed that the source of this emission may be the Sun. However, more careful analysis of this emission revealed that the periodicity of the emission was 23 h and 56 min, which corresponds to the sidereal day (the period of time required for the Earth to complete one rotation relative to the reference frame of distant stars) rather than a solar day (which lasts 24 h and corresponds to the amount of time required for the Sun to complete one whole path across the sky).3 Comparing optical maps of the sky with his observations, Jansky realized that the source of the radio emission corresponded to the Milky Way Galaxy—the home galaxy to which the Sun belongs—and that this form of emission was strongest in the direction of the constellation of Sagittarius. Modern astronomers now know that the supermassive black hole that is located at the center of our Galaxy is coincident with this peak of strongest emission identified by Jansky: this radio source is now known as Sagittarius A∗ (pronounced “Sagittarius A-star”).4 The discovery of this unexpected source of radio emission and its remarkable origin intrigued Jansky: his work represented the first detection of radio emission from an extraterrestrial source. In 1933, he published his discovery in an engineering journal as a paper with the somewhat prosaic title “Electrical Disturbances Apparently of Extraterrestrial Origin” [6]. The title was prosaic in the sense that it belied the remarkable discovery of radio emission from far beyond the Earth! Jansky would go on to propose to Bell Laboratories the construction of a dish antenna with a diameter of 30 m with the purpose of conducting formal astronomical observations. However, Bell Labs rejected this proposal: with the origin of the source of noise determined and because this noise did not dramatically affect commercial communications, the company had no further interest in this phenomenon and funding its investigation. Furthermore, the United States was still in the midst of the Great Depression and funding for ambitious new research apparatus such as the one Jansky proposed was extremely scarce. The onset of World War II further delayed the development of radio astronomy on an international scale: in the meantime, Jansky was re-assigned to another research project for Bell Laboratories and he never conducted any further work on radio astronomy. Jansky’s crucial discovery attracted little attention from the community of professional astronomers (or from many scientists or engineers at all), but it did pique the interest of Grote Reber (1911–2002) (see Fig. 1.2). Reber had earned a degree in electrical engineering and was an amateur radio operator: he read about Jansky’s work in 1933 and decided that he wanted to pursue professionally the study of radio emission from astronomical sources. Reber actually applied for a job at the Bell Laboratories (the same company that employed Jansky), but unfortunately the United States was still in the midst of the Great Depression and jobs were not plentiful at that time. Undeterred, Reber built a radio telescope in a side yard of his mother’s home in Wheaton, Illinois (see Fig. 1.3). This telescope consisted of a

3 The

concept of sidereal time is discussed more fully in Sect. 2.1.3.2. of the center of the Milky Galaxy are discussed in detail in Sect. 6.4.6.

4 Properties

1.1 Introduction

5

Fig. 1.2 Grote Reber in 1975. Image Credit: NRAO/AUI/NSF

parabolic metal mirror called a “dish” with a diameter of 9 m that focused incident radio waves to a receiver located 8 m above the dish. In this respect, Reber’s radio telescope resembled optical telescopes with reflecting mirrors that feature surfaces with parabolic shapes, which bring light to a focus at a position in front of the mirror—this position is known as the prime focus. With its parabolic shape, Reber’s radio telescope—often considered to be the first radio telescope in the history of astronomy—strongly resembled modern radio telescopes that also employ dishes to collect light as well as smaller dishes used to receive and transmit commercial broadcasts. Reber completed his telescope in 1937: his first receiver operated at a frequency of 3300 MHz but failed to detect any emission from astronomical sources. Observations conducted with a second receiver operating at a frequency of 900 MHz also failed to detect such emission but finally—with a third receiver operating at a frequency of 160 MHz—Reber at last detected emission from astronomical sources and thus confirmed Jansky’s discovery. He published his work in the Astrophysical Journal—a leading international research journal in astronomy and astrophysics— in 1940 [7]. Besides confirming Jansky’s result with the detection of radio emission from the Milky Way Galaxy, Reber’s paper also described the detection of radio emission from the nearby Andromeda Galaxy (also known as M31) and the attempt to detect radio emission from several bright stars, the Sun, the Mars, and the Orion Nebula (no emission was detected from any of these three objects). In 1944 Reber published a second paper in the Astrophysical Journal [8] where he presented a radio map of the northern sky (that is, the portion of the sky visible to his telescope in Illinois) made at a frequency of 160 MHz. This map was the first detailed radio map made of at least a portion of the sky. Not only did his map confirm the presence of

6

1 Introduction

Fig. 1.3 The first dish antenna radio telescope (with a diameter of 9 m) as built by Grote Reber in 1937 in a side yard of his mother’s house in Wheaton, Illinois. Reber used wooden rafters, galvanized sheet metal, and spare parts from a Ford Model T truck to build the telescope. Reber would later himself reconstruct his telescope just outside the NRAO office in Green Bank, West Virginia. The reconstructed telescope serves as a monument to Reber’s pioneering work in conducting the first radio astronomical observations. Image Credit: NRAO/AUI/NSF

the powerful radio source detected toward the constellation of Sagittarius by Jansky, but it also contained the first detections of other luminous radio sources seen in the sky known to modern radio astronomers, such as the supernova remnant Cassiopeia A and the luminous radio galaxy Cygnus A. One of the major results of Reber’s research was the first (albeit crude) revelations about the emission processes associated with astronomical sources detected by radio telescopes. By comparing the amount of light or flux emitted by an object at different frequencies, a basic understanding of the emission process associated with the astronomical source (and thus insights into the true nature of the source) may be gleaned. When radio observations were first conducted, astronomers expected that the detected emission would be thermal in nature (as modeled as a phenomenon known as blackbody radiation), in that the amount of detected flux would decrease with decreasing frequency. Instead, Reber found that while the fluxes from some astronomical sources did indeed decline as expected, for other sources the amount of detected flux would in fact increase with decreasing frequency. This result puzzled astronomers and sources that exhibited such spectral behavior were classified as “non-thermal.” It was not until the 1950s that astronomers realized that the true emission process associated with these latter sources was synchrotron radiation,

1.1 Introduction

7

a form of radiation emitted by high-velocity electrons that are gyrating in magnetic fields.5 With the cessation of hostilities at the end of the World War II, radio astronomy truly flourished as a bonafide branch of observational astronomy. Many of the advances in radio astronomy were spurred by surplus military radar hardware that became available to astronomers once military operations had ended: significant technological advances in the capabilities of this hardware as spurred by the necessities of war proved to be essential to astronomers attempting to detect emission from ever fainter radio sources. A key challenge that faces all radio telescopes (and the astronomers who use them) is attaining both a sensitivity (a description of the ability of the telescope to detect signals from astronomical sources to ever lower intensities) and an angular resolution (the ability of the telescope to resolve fine detail in the angular structure of the source) that facilitates a ready comparison with observations made of the same astronomical sources at other wavelengths. In the case of sensitivity, it must be noted at the outset of this text that the radio emission detected from astronomical sources is exceptionally weak (as discussed in Chap. 4) and therefore one of the main purposes of the hardware of radio astronomy (both the physical structure of the telescope and the hardware required to amplify signals) is to collect this weak signal and amplify it considerably so that the emission from these sources are readily detected. Thus, a considerable fraction of the processing of the radio signals from astronomical sources involves amplification. In regard to angular resolution, the angular size of the smallest structure that can be resolved by a telescope is approximately the ratio of the wavelength at which the observation is conducted divided by the aperture or diameter of the telescope (as discussed in Chap. 4). Because optical wavelengths are approximately a million times smaller than the wavelengths of light in the radio domain, large apertures are required for radio telescopes to attain an angular resolution that approaches the angular resolution that is attained routinely at optical (and other shorter) wavelengths. With the goal of attaining the best sensitivity and angular resolution possible, the decades following World War II saw the construction of radio telescopes of ever larger size with increasingly sophisticated hardware for amplifying signal. Examples of such observatories (including their aperture sizes) are given in Table 1.1: one such observatory—the Parkes radio telescope in Australia—is shown in Fig. 1.4. The nominal maximum aperture of a steerable single radio telescope (subject to constraints such as engineering and cost) is approximately 100 m, though novel telescope construction methods have created instruments with apertures that exceed this value.6 Despite such a large diameter, a radio telescope with such an aperture would still be unable to attain the angular resolution attained even by an optical 5 Blackbody radiation and synchrotron radiation will be discussed in elaborate detail when emission

mechanisms are discussed in Chap. 3. 6 For example, regarding the three largest telescopes given in Table 1.1, the Arecibo radio telescope

and the 500-m Aperture Spherical Telescope (FAST) are both actually contoured out of natural valleys. Regarding the RATAN-600 telescope, this telescope is comprised of a large number of individual reflector units arranged in a circle that direct incoming radiation from astronomical sources to a central receiver. This telescope thus achieves the angular resolution of a telescope

8

1 Introduction

Table 1.1 Examples of prominent modern single dish radio telescopes Name RATAN-600 FAST (Five-Hundred-Meter Aperture Spherical Telescope) Arecibo

Diameter (m) 600 500

Country Russia China

300

Green Bank Effelsberg Jodrell Bank Galenki Yevpatoria Parkes Sardinia Usuda Large Millimeter Telescope Algonquin Nobeyama Yebes Torún Medicina Hartebeesthoek Hobart Onsala Sheshan Mopra Morehead Metsähovi Warkworth

110×100 100 76 70 70 64 64 64 50 46 45 40 32 32 26 26 25 25 22 21 14 12

United States of America (Puerto Rico) United States of America Germany United Kingdom Russia Ukraine Australia Italy Japan Mexico Canada Japan Spain Poland Italy South Africa Australia (Tasmania) Sweden China Australia United States of America Finland New Zealand

telescope by a wide margin. To address this shortcoming (in a manner that is cost effective), astronomers construct multiple radio telescopes and combine signals from each individual telescope to emulate the angular resolution capabilities of a single radio telescope with a greater diameter. In fact, the angular resolution attained by the array of individual radio telescopes is equivalent to that achieved by a single radio telescope with a diameter corresponding to the distance between the most widely separate individual radio telescopes. This principle of combining (“interfering”) signals from multiple radio telescopes to attain superior angular resolution is known as interferometry, and arrays of radio telescopes that conduct observations through this principle are known as interferometers.7 The best-known with a diameter of 600 m. Note, however, that neither RATAN-600 nor FAST nor Arecibo are fully steerable telescopes and therefore can only observe limited portions of the sky. 7 Interferometry and interferometers will be discussed in more detail when telescopes and observing techniques at radio wavelengths are discussed in Chap. 4.

1.1 Introduction

9

Fig. 1.4 The Parkes 64-m radio telescope. Image Credit: Stewart Duff

radio observatory that operates as an interferometer is the Karl G. Jansky Very Large Array (JVLA) located in New Mexico, United States of America (see Fig. 1.5). The JVLA conducts observations typically at wavelengths ranging from 0.7 to 400 cm: it is comprised of 27 individual radio telescopes each with a diameter of 25 m. The individual telescopes may be moved along railroad tracks to placement in one of four different main configurations (as driven by different scientific objectives). The maximum separation between individual telescopes is approximately 36 km and thus—in that particular configuration—the JVLA may attain the same angular resolution as a single radio telescope with that diameter. The JVLA is situated on the Plains of San Agustin that was leveled by an ancient lake: the flat expanse over which the antennas are situated makes the array of individual antennas to be roughly co-planar, which simplifies the interference of signals from individual antennas. Other examples of interferometers currently in operation include the Giant Metrewave Radio Telescope (GMRT) near Pune, India; the Australia Telescope Compact Array (ATCA) near Narrabri, Australia; the Westerbork Synthesis Radio Telescope (WSRT) near Westerbork, the Netherlands, and finally the Dominion Radio Astrophysical Observatory (DRAO) Synthesis Telescope near Okanagan Falls, British Columbia, Canada. A listing of well-known radio interferometers is given in Table 1.2. In modern astronomy, perhaps the most remarkable interferometer that has recently commenced operations is the Atacama Large Millimeter Array (ALMA) [9, 10]. Commissioned in 2013 and located in the Atacama desert in northern Chile, ALMA makes observations at millimeter wavelengths and operates

10

1 Introduction

Fig. 1.5 An image of the Karl G. Jansky Very Large Array (JVLA). The array is comprised of 27 antennas, each with a diameter of 25 m. Image Credit: W. Grammer, NRAO/AUI/NSF

at an altitude of 5100 m (see Fig. 1.6). The high altitude combined with the very dry climate of the observatory site serves to minimize the effects of water vapor, which may dramatically hamper astronomical observations made within this wavelength range. The placement of an observatory that operates at millimeter wavelengths at an extremely high altitude helps to illustrate the phenomenon of atmospheric windows: the atmosphere of the Earth is not equally transparent to light at all wavelengths but instead is more opaque at some wavelengths than others.8 For example, the atmosphere is opaque to electromagnetic radiation at very short wavelengths (namely gamma-ray, X-ray, and ultraviolet radiation): for this reason, astronomical observations performed at these wavelengths must be conducted from orbiting observatories placed above the Earth’s atmosphere. In contrast, electromagnetic radiation at optical wavelengths may be conducted from sea level because the Earth’s atmosphere is effectively transparent to this radiation. The transmission characteristics of the Earth’s atmosphere in the radio domain of the spectrum are complex: the ionosphere of the Earth reflects radiation at the longest wavelengths (approximately 1 m and greater) into space, making observations of astronomical sources at these wavelengths impossible. The atmosphere of the Earth is essentially transparent to electromagnetic radiation with wavelengths on the order of a centimeter, so observations may be conducted at these wavelengths at

8 The

concept of atmospheric windows will be discussed more extensively in Sect. 4.3.

1.1 Introduction

11

Table 1.2 Examples of prominent modern radio interferometers Name Australia Telescope Compact Array (ATCA) Dominion Radio Astrophysical Observatory (DRAO) Giant Metrewave Radio Telescope (GMRT) Karl G. Jansky Very Large Array (JVLA) Westerbork Synthesis Radio Telescope (WSRT)

Number of elements × diameter per element (m) 6 × 22

Country Australia

7×9

Canada

30 × 45

India

27 × 25

United States of America

14 × 25

Netherlands

Fig. 1.6 The ALMA Array. Notice that the array is composed of a mixture of antennas, some with diameters of 7-m and some with 12-m. Image Credit: N. Gugliucci, CosmoQuest

approximately sea-level elevations. Lastly, copious amounts of water vapor in the Earth’s atmosphere absorb electromagnetic radiation at millimeter wavelengths: therefore, observatories like ALMA that conduct observations at such wavelengths must be placed at high altitudes that exceed the typical scale height of water vapor, thus reducing its impact on these observations. The challenges in observing astronomical sources through the Earth’s atmosphere are discussed in more detail in Sect. 4.3.

12

1 Introduction

Fig. 1.7 The VLBA Antenna in Owens Valley, CA, USA. This antenna is one of the ten that comprise the VLBA: each one has a diameter of 25 m. Image Credit: NRAO/AUI/NSF

Modern technology enables the measurement of the positions of individual telescopes relative to each other with extremely high precision: such knowledge is absolutely essential for conducting observations with interferometers, particularly over ever-widening baselines. In recent decades, this technology has advanced far enough to facilitate interferometric observations conducted over baselines of thousands of kilometers: the angular resolution attained by such observations is superior to that attained by optical observations by several orders of magnitude. Conducting interferometric observations over such long baselines is known as Very Long Baseline Interferometry (VLBI), and an example of such an interferometer with such long baselines is the Very Long Baseline Array (VLBA). The VLBA is based in the United States and conducts observations chiefly at centimeter wavelengths: it is comprised of ten 25-m antennas distributed over the continental United States, Hawaii, and the Virgin Islands. One of these antennas is shown in Fig. 1.7: the baselines that separate these individual antennas span hundreds to thousands of kilometers, and the largest—ranging from Mauna Kea in Hawaii to St. Croix in the Virgin Islands—spans more than 8600 km, or over one-fifth the circumference of the Earth. Another example of such an interferometer is the Joint Institute for VLBI in Europe (JIVE): this interferometer uses extant radio

1.1 Introduction

13

observatories across Europe (such as those listed in Table 1.1) as well as other telescopes in the continent.9 Conducting interferometric observations over such wide baselines requires not just precise measurements of the distances between the individual antenna elements but also very accurate timing measurements at each antenna location in regard to when the observations were conducted. Typically, the data is recorded onto portable hard drives at each observatory and time-stamped using atomic clocks: the hard drives are then shipped to a central location where the datasets are all processed simultaneously and the interferometric analysis is conducted. This processing addresses known and crucial effects that must be taken into account when such large baselines are considered, such as the rotation of the Earth and tiny shifts in the crust of the Earth over time. Remarkably, VLBI observations can also be harnessed for terrestrial applications such as investigating the motion of tectonic plates in the crust of the Earth: because the positions of the antenna elements relative to each other are determined so precisely, any changes in position (due to antenna elements located on different plates being carried toward or away from each other due to the motions of the plates) can be readily determined. In pursuit of ever longer baselines to attain even better angular resolution, several radio antennae have been launched into orbit and operated in concert with ground-based radio antenna elements to conduct interferometric observations with baselines spanning tens of thousands of kilometers. The first such orbiting radio telescope was the HALCA (Highly Advanced Laboratory for Communications and Astronomy) Observatory [11]. HALCA—which had a diameter of 8 m—was a Japanese observatory launched in 1997: it attained a highly eccentric elliptical orbit with perigee and apogee distances (that is, the nearest and farthest distances, respectively, between the observatory and the Earth in the course of the orbit of HALCA around the Earth)10 of approximately 530 km and 21,000 km, respectively. Such an eccentric orbit facilitated the extremely long baselines that exceeded the baselines typically associated with even the most widely separated antenna elements of terrestrial VLBI facilities. HALCA conducted observations at centimeter wavelengths and routinely imaged the central portions of luminous Galactic nuclei in concert with terrestrial radio telescope networks until it was decommissioned in 2005. More recently, RadioAstron (also known as Spektr-R) is an orbiting radio observatory with an aperture of 10 m that was launched by Russia in 2011 (see Fig. 1.8) [12]. Like HALCA, RadioAstron conducts observations chiefly at centimeter wavelengths. After launch, it attained an elliptical orbit with a high eccentricity: its perigee distance of 10,000 km while its apogee distance is 390,000 km—this distance is comparable to the average distance between the Earth and the Moon! RadioAstron makes very long baseline interferometric observations in concert with

9 For

completeness, note that on occasion these interferometers may draw upon other observatories (such as JIVE incorporating the Arecibo antenna and the VLBA incorporating the Effelsberg antenna) to increase the maximum baseline distance included when the observations are conducted. 10 Orbital mechanics and terminology will be discussed in more detail in Sect. 2.3.

14

1 Introduction

Fig. 1.8 An artist’s conception of RadioAstron, a space-based international Very Long Baseline Interferometry (VLBI) project led by the Astro Space Center of Lebedev Physical Institute in Moscow, Russia. Copyright: Astro Space Center of Lebedev Physical Institute, Russian Academy of Sciences

radio telescopes located across the Earth, and—like HALCA—it has imaged the innermost regions of the radio luminous nuclei of galaxies, amongst other classes of radio sources. The enormous baselines achieved by observations conducted with HALCA and RadioAstron are the largest ever incorporated into interferometric observations and have thus yielded the best angular resolution ever realized in astronomical observations. The advantages of conducting radio observations from above the atmosphere of the Earth include a reduction in the magnitude of confusing radio emission from terrestrial sources as well as the potential to collect radiation emitted by astronomical sources at long wavelengths that cannot penetrate the ionosophere of the Earth (and therefore cannot be detected by ground-based radio observatories). The main disadvantage of space-based radio observations is the determination of the distances from the orbiting antenna to ground-based radio antennas given that the orbiting antenna is in constant motion and therefore these distances are always changing. Now that some background information about radio astronomy and the techniques utilized in conducting astronomical observations at radio wavelengths has been presented, the question that was given at the start of this chapter—why make astronomical observations at radio frequencies?—may now be addressed. The question may be addressed by considering the following two salient points:

References

15

Radio observations can detect radiation from astronomical sources that may be hard to detect or which produce little to no radiation at other wavelengths. All astronomical observations of sources located at distances beyond the Solar System must contend with interstellar (and intergalactic, when considering extragalactic sources) absorption along the line of sight from the observer to the source. This phenomenon is known as interstellar extinction and it is associated with atoms and molecules of interstellar gas and dust grains along the line of sight that scatter or absorb electromagnetic radiation as it travels from the source to the observer. The magnitude of interstellar extinction is wavelength-dependent: generally (but not strictly!) lower energy photons are affected to a lesser extent than higher energy photons. For example, if a source located in the Milky Way Galaxy emits photons at both optical and radio wavelengths, the optical photons are far more likely than the radio photons to be scattered or absorbed by interstellar matter while in transit from the source to the observer. Therefore, an astronomical source that may be too faint to be readily detected by an optical telescope due to significant interstellar extinction along the line of sight may still be detected by a radio telescope. In this manner, radio telescopes can detect emission from sources that may be too weak to be detected by telescopes operating at other wavelengths. Radio observations complement observations made at other wavelengths and help to constrain models of emission from astronomical sources over large wavelength ranges. Earlier in this chapter, blackbody radiation and synchrotron radiation were described as the most commonly encountered emission mechanisms in radio observations of astronomical sources. For example, synchrotron radiation from astronomical sources may be produced over very broad ranges of the electromagnetic spectrum, from the radio domain and into shorter wavelengths like the infrared, optical, ultraviolet, and even the X-ray. By comparing observations made of the source at all of these wavelengths, the synchrotron spectrum of the source may be modeled in a rigorous manner to determine such properties as the curvature of the spectrum and the maximum energies of the population of synchrotron-emitting electrons.11

References 1. W.T. Sullivan, Cosmic Noise: A History of Early Radio Astronomy (Cambridge University, New York, 2009) 2. G.L. Verschuur, The Invisible Universe: The Story of Radio Astronomy, 2nd edn. (Springer, China, 2007) 3. P. Robertson, Beyond Southern Skies: Radio Astronomy and the Parkes Telescope (Cambridge Press, Melbourne, 1992) 4. F.J. Lockman, F.D. Ghigo, D.S. Balser, But It Was Fun: The First Forty Years of Radio Astronomy at Green Bank (National Radio Astronomy Observatory, Charlottesville, 2007)

11 Details

about the properties of synchrotron radiation are given in Sect. 3.3.

16

1 Introduction

5. B.K. Malphrus, The History of Radio Astronomy and the National Radio Astronomy Observatory: Evolution Toward Big Science (Krieger Publishing Company, Malabar, 1996) 6. K.G. Jansky, Proc. IRE 21, 1387–1398 (1933) 7. G. Reber, Astrophys. J. 91, 621–624 (1940) 8. G. Reber, Astrophys. J. 100, 279–287 (1944) 9. A. Beasley, A. Peck, From Planets to Dark Energy: The Modern Radio Universe, vol. 4 (2007), pp. 4–13 10. R.E. Hills, R.J. Kurz, A.B. Peck, in Proceedings of the SPIE 7733, Ground-Based and Airborne Telescopes III, vol. 773317 (2010). https://doi.org/10.1117/12.857017 11. H. Kobayashi, K. Wajima, H. Hirabayashi, et al., Adv. Space Res. 26, 597–602 (2000) 12. Yu.A. Alexandrov, V.V. Andreyanov, N.G. Babakin, et al. Sol. Syst. Res. 46, 466–475 (2012)

Chapter 2

Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

2.1 Angles 2.1.1 Trigonometry Recall that for a circle with a radius r, the relationship between an angle θ and the arc length s of the circumference of the circle that is subtended by that angle is s θ (radians) = , r

(2.1)

where θ is expressed in units of radians (see Fig. 2.1). Recall that—by definition— 2π radians = 360 degrees (denoted as ◦ ) and therefore 1 radian ≈ 57.2958◦ . Note that astronomers commonly measure angles using arcminutes (denoted as’) and arcseconds (denoted as ”). These units are collectively defined such that 1◦ = 60 , 1 = 60 and thus 1◦ = 3600 : therefore, 1 radian corresponds to ≈ 3437.748 or ≈ 206,265 . Recall as well that in the case of a right triangle, if one of the corners of the triangle (not the corner corresponding to the right angle) is considered and the angle for that corner is denoted as θ , the three legs of the triangle can then be labeled as “O” (for the side “opposite” to the angle of interest), “A” (for the side “adjacent” to the angle of interest) and “H ” (for the hypotenuse or the longest leg of the triangle, that is the leg that is opposite to the right angle). The well-known relations for sin θ , cos θ , and tan θ may be expressed as sin θ =

O , H

cos θ =

A H

and

tan θ =

O . A

(2.2)

The small angle approximation is now introduced: this approximation is commonly encountered and implemented in radio astronomy as well as all other © Springer International Publishing Switzerland 2020 T. G. Pannuti, The Physical Processes and Observing Techniques of Radio Astronomy, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-16982-8_2

17

18

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Fig. 2.1 A schematic diagram relating the radius r of a circle and an angle θ to the subtended arc length s of the circumference of the circle

branches of astronomy. Because the fields of view of many telescopes and the angular extents of sources are so small, this approximation is warranted and convenient. In the limit of small angles where the value of θ approaches zero radians (corresponding to zero degrees), the small angle approximation can be employed and these relations can be re-expressed as sin θ ≈ θ =

O , H

cos θ ≈ 1 and

tan θ ≈ θ =

O . A

(2.3)

This example can be extended to a well-known astronomical application. Imagine that the leg O corresponds to the Earth–Sun distance and that the leg A corresponds to the distance between the Sun and a star of interest. In this example, θ thus corresponds to the angle of parallax; that is, the observed angle of shift seen in the position of the star as a consequence of the Earth’s orbital motion around the Sun (see Fig. 2.2). In fact, O is the astronomical unit (often denoted as A.U.), the mean distance between the Earth and the Sun and corresponds to approximately 150 million kilometers. Thus, in the small angle approximation θ may be expressed (in terms of radians) as tan θ ≈ θ (radians) =

O 1 A.U. = . A A

(2.4)

Solving for A and converting into units of arcseconds yields A=

1 A.U. 206,265 × 1 A.U. 206,265 A.U. = = . θ (radians) θ (”) θ (”)

(2.5)

If θ = 1 arcsecond, then A is simply 206,265 A.U. This equation defines the parsec (abbreviated as pc) as the distance at which a star must lie from Earth such that it exhibits a parallax angle p of 1 arcsecond. One parsec corresponds

2.1 Angles

19

Fig. 2.2 A schematic diagram illustrating the concept of the parallax. As a consequence of the orbital motion of the Earth around the Sun, the observed location of the star appears to shift by an angle θ, which corresponds to the observed parallax angle p. Through knowledge of the Earth– Sun distance (defined as the Astronomical Unit and corresponding to “O” in the diagram) and measurement of p, the distance to the star (corresponding to “A” in the diagram) can be determined

to 206,265 AU or 3.086 × 1016 m. The parsec is the standard unit of distance used in modern astronomy: distances may also be expressed in terms of kiloparsecs (corresponding to 103 parsecs) or megaparsecs (corresponding to 106 parsecs). Equation (2.5) may be used to determine the distances to a star of interest by a measurement of its parallax shift. This equation is commonly expressed in the following well-known form: d (pc) =

1 . p (”)

(2.6)

Here, d is the distance to the star in units of parsecs and p is the observed parallax of the star measured in arcseconds. Notice that as the distance to the star increases, the parallax exhibited by the star decreases. A very useful relation in astronomy for calculating the linear extent x of a source based on its measured angular extent θ (that is, the source subtends an angle θ in the sky) and its distance d is

20

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

x (pc) =

θ (”) d (pc) , 206,265 (”/radians)

(2.7)

where θ is expressed in arcseconds and both x and d are expressed in parsecs. Notice that the constant of 206,265 in this equation originates from the relationship between a parsec and an astronomical unit, where once again one parsec corresponds to 206,265 astronomical units. Also notice how the units on both sides of Eq. (2.7) simplify out to parsecs and that the relation between the number of arcseconds per radian is explicitly expressed. Applications for Eq. (2.7) include calculating the linear extent of extended distant astronomical sources—such as extended Galactic radio sources like HII regions and supernova remnants located within the Milky Way (see Chap. 6) or even external galaxies—if the angular extent of the source is measured and its distance is known.

2.1.2 Solid Angle While the radian is the standard unit employed by astronomers for measuring one-dimensional angles, a steradian is the standard unit most frequently used by astronomers for measuring the angular extent of a source in two dimensions on the sky (such as the Moon or the Sun). A steradian (abbreviated as “sr”) is defined such that 1 sr = (180/π )2 square degrees. Specifically, the total area A covered by an object in the sky, its distance r and the solid angle Ω (Fig. 2.3) subtended by the object are all related through the relation

Fig. 2.3 A schematic diagram illustrating the concept of solid angle: the ratio of the subtended area A to the radius r is the solid angle Ω. Notice that this is a two-dimensional analog to the relationship between the ratio of the subtended arclength s to the radius r, which is the angle θ in radians

2.1 Angles

21

Ω (steradians) =

A . r2

(2.8)

It is instructive to also consider solid angles within the context of spherical coordinates, a natural coordinate system to consider for measuring angular extents of astronomical sources on the sky. Recall that in this coordinate system, r is the distance from the origin, θ is the polar angle measured from the z-axis in the Cartesian coordinate system and φ is the azimuthal angle measured around the x-axis in the Cartesian coordinate system. In other words, x = r sin θ cos φ

(2.9)

y = r sin θ sin φ

(2.10)

z = r cos θ .

(2.11)

Based on these relations, a differential solid angle dΩ may be expressed in spherical coordinates as dΩ = sin θ dθ dφ.

(2.12)

Take note of the factor of sin θ in Eq. (2.12): its origin may be described as Consider two different latitude locations on the Earth: note that the distance between two locations of the same longitude decreases as you consider latitudes that are farther from the equator and closer to a pole as follows. Therefore, this factor of sin θ accounts for the variation in projected horizontal distances of latitude as a function of θ itself. Note that integrating Eq. (2.12) over all solid angles (that is, for an entire sphere) to obtain the total amount Ω of steradians in a sphere and thus yields  Ω=





2π

dΩ=

π

dφ 0

0

sin θ dθ = (2π )[− cos θ ]π0 = (2π )(2) = 4π steradians.

(2.13) The integration is performed for φ between 0 and 2π to account for the entire azimuthal extent but only for θ between 0 and π because integrating through 2π would be redundant in that contributions along θ would be counted twice. Combining Eqs. (2.8), (2.12), and (2.13), the surface area A of a sphere of radius R may be computed as  A=

 R dΩ = R 2

2



2π

π

dφ 0

sin θ dθ = 4π R 2 .

(2.14)

0

To help further illustrate spherical coordinates, note that the differential length scales dlr , dlθ , and dlφ in the r, θ , and φ directions, respectively, may be expressed as dlr = dr

(2.15)

22

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

dlθ = r dθ

(2.16)

dlφ = r sin θ dφ,

(2.17)

where dr, dθ , and dφ are the infinitesimal units of length in the r, θ , and φ directions, respectively. Note that the expressions for dθ and dφ both depend on r— this factor of r ensures that these quantities do indeed have units of length. Based on these expressions for infinitesimal units of length, a differential volume element dV may be expressed in spherical coordinates as dV = dlr dlθ dlφ = r 2 sin θ dr dθ dφ.

(2.18)

Integrating Eq. (2.18) over the entire volume V of a sphere of radius R may be performed as 







4 R3 )(2)(2π ) = π R 3 , 3 3 r=0 θ=0 φ=0 (2.19) thus yielding the well-known expression for the volume of a sphere. V =

dV =

r=R

θ=π

φ=2π

r 2 sin θ dr dθ dφ = (

Example Problem 2.1 The distance between the Earth and the Moon is approximately r = 384,400 km and the diameter of the Earth is x = 12, 742 km. Based on this information, calculate the angular diameter (in radians) of the Earth as it appears in the sky as viewed from the Moon. Also calculate the solid angle (in steradians) subtended by the Earth in the sky as viewed from the Moon. Solution From Eq. (2.4) and adopting the small angle approximation, the angular diameter θ may be calculated as follows: θ (radians) =

x (km) 12,742 km = = 3.32 × 10−2 . d (km) 384,400 km

(2.20)

Recalling that 1 radian ≈ 57.2958◦ , θ can be expressed in degrees as θ (◦ ) = θ (radians)



57.2958◦ 1 radian



= 3.32 × 10−2



57.2958◦ 1 radian



= 1.90◦ .

(2.21) For comparison purposes, the apparent angular extents of the Moon and the Sun as viewed in the sky of the Earth are both approximately 0.5◦ (see Problem 2.3). This fortuitous coincidence facilities the phenomenon of total solar eclipses where the apparent disk of the Moon completely covers the apparent disk of the Sun. (continued)

2.1 Angles

23

Example Problem 2.1 (continued) The solid angle Ω subtended by the Earth in the sky of the Moon may be calculated as follows: Ω (steradians) =

π((12, 742 km)2 )/4 A = = 8.65 × 10−4 . r2 (384, 000 km)2

(2.22)

Notice in this calculation A = πr 2 = πd 2 /4. Alternatively, given that the Earth appears to be approximately circular in the sky, Ω could be calculated based on its measured diameter θ in radians: π(3.32 × 10−2 )2 π θ (radians)2 = = 8.66 × 10−4 . 4 4 (2.23) This yields approximately the same result. Ω (steradians) =

2.1.3 Sky Coordinate Systems There are two main coordinate systems employed commonly by modern astronomers: the altitude-azimuth coordinate system and the equatorial coordinate system. Both of these systems depend on a theoretical construct known as the celestial sphere, a sphere which completely encloses the Earth and upon which it may be imagined that all objects in the universe are affixed. The motion of objects across the sky may be attributed to the rotation of this sphere. While certainly the celestial sphere does not actually exist, it is quite useful for the purposes of measuring the positions of objects in the sky. Several crucial features of the celestial sphere are worthy of discussion (see Fig. 2.4). The first such feature is the celestial equator: this is a band on the celestial sphere that corresponds to a projection of the Earth’s equator on to the celestial sphere. The location of Orion’s belt in the sky roughly marks the location of the celestial equator. Two other crucial features of the celestial sphere are the celestial poles: if a line that connects the two poles of the Earth (and passes through the Earth’s center) is extended toward the celestial sphere, the celestial poles are the locations where the line intersects with the celestial sphere. Specifically, the North Celestial Pole is the point on the celestial sphere located in projection beyond the north pole of the Earth while similarly the South Celestial Pole is the point on the celestial sphere located in projection beyond the south pole of the Earth. Currently, the location of the North Celestial Pole is approximately marked by the bright star Polaris (also known as the North Star): there is no star of comparable brightness located near the South Celestial Pole. Note that observers located at the north and south poles of the Earth would see that the North and South celestial

24

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Fig. 2.4 A schematic diagram of the celestial sphere. Note that the celestial equator is an extension of the plane of the equator of the Earth while the North and South Celestial Pole correspond to an extension of a line connecting the north and south poles of the Earth and extended toward intersections on the celestial sphere

poles, respectively, are located directly overhead. Finally, the ecliptic is the apparent path taken by the Sun across the sky: it is also corresponds to the projection of the Earth’s orbital plane around the Sun into the sky as well. Broadly speaking, the Sun covers approximately one degree (that is, approximately two apparent solar diameters) every day as it moves along the ecliptic. The plane of the ecliptic is inclined with respect to the celestial equator by an angle of approximately 23.5◦ , which precisely corresponds to the inclination of the Earth’s axial tilt to its orbital plane. As a consequence of this tilt, as it moves along the ecliptic, the Sun reaches positions which are the most distant north and south of the ecliptic by 23.5◦ . When the Sun is located at these positions, the summer solstice and winter solstice occur (as observed from the Northern Hemisphere). Furthermore, at the two positions along the ecliptic which intersect with the celestial equator, the vernal equinox and the autumnal equinox occur.

2.1.3.1

Altitude-Azimuth Coordinate System

The basis of the altitude-azimuth coordinate system is the use of two coordinates— the altitude angle h and the azimuth angle a—to indicate the location of an object

2.1 Angles

25

Fig. 2.5 An illustration of the altitude-azimuth system to specify locations in the sky. The position (denoted with an “*”) of an object in the sky is indicated by two coordinates as measured by an observer located at the position “O.” The first coordinate (“A”) is the azimuth angle and the second coordinate (“h”) is the altitude angle. The latter coordinate is defined along with the zenith angle (“z”) such that the sum of the two angles is always 90◦

in the sky. The altitude angle is defined as the angle from the horizon to the location of the object along an arc called a “great circle”—that is, a curve originating from the intersection of the celestial sphere with a plane passing through the center of the celestial sphere. In turn, the great circle passes through the object and the point on the celestial sphere that is located directly overhead of the observer: this point is known as the zenith. A zenith angle z may also be defined as the angle from the zenith to the location of the object: note that the sum of h and z may be expressed as z + h = 90◦ .

(2.24)

The azimuth angle a is the angle along the horizon measured east of north to the great circle used to measure h: by this definition, due north corresponds to 0◦ azimuth. While the altitude-azimuth system is very simple to implement, its main drawback is that the location of an object in the sky clearly depends on the location of the observer on the Earth as well as the time that the measurement of the location is made. This situation motivates the choice of a coordinate system (as presented in the next section) that describes positions in the sky in a manner that is independent of the location of the observer on the Earth. Such a system is absolutely essential in effectively communicating the position of an object of interest on the sky to the larger international community of astronomers based at institutions and observatories that are located across the globe (Fig. 2.5).

26

2.1.3.2

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Equatorial Coordinate System

The most commonly-used coordinate system by modern astronomers is known as the equatorial coordinate system: this system is very analogous to the latitude and longitude system used to indicate positions on the surface of the Earth. This coordinate system is the system used essentially exclusively by modern astronomers in indicating the positions of sources on the sky and in generating catalogs of the positions of objects on the sky as well. Like the altitude-azimuth coordinate system, the coordinates of the equatorial coordinate system are measured on the celestial sphere, but unlike that coordinate system, the coordinates of the equatorial coordinate system do not depend on the location of the observer on the surface of the Earth. In this system, the Declination coordinate (denoted as δ and abbreviated as Dec) is measured in degrees north or south of the celestial equator: it ranges from −90◦ at the South Celestial Pole to 0◦ at the celestial equator to +90◦ at the North Celestial Pole and it is analogous to the latitude coordinate. The conjugate coordinate to Declination, Right Ascension (denoted as α and abbreviated as RA), is analogous to longitude: it is measured in degrees eastward along the celestial equator from the location along the ecliptic where the Sun is on the day of the vernal equinox as seen from the Northern Hemisphere (that is, approximately March 21st). Recall that the ecliptic and the celestial equator intersect along this position of the ecliptic: also, this position is commonly indicated by the symbol Υ . Right Ascension may then be imagined as the distance along the celestial equator from Υ to the hour circle of the source, where the hour circle is a circle of constant Right Ascension that extends across the whole perimeter of the celestial sphere and connects the position of the source with the North Celestial Pole. Such circles are also commonly known as great circles: an example of a great circle is the meridian, which is a great circle that passes through the zenith of an observer and connecting with the horizon at due north and at due south. The meridian does not correspond to fixed coordinates of Right Ascension and Declination but instead remains fixed to an observer as the celestial sphere rotates. Like Declination, Right Ascension may be measured in degrees (ranging from 0◦ through 360◦ ), but it is commonly measured in hours, minutes, and seconds,  where at the celestial equator (where δ = 0◦ ) 1 h = 15◦ , 1 min = 15 , and  1 s = 15 . Complementing this format, Declination may be expressed in units of degrees, arcminutes, and arcseconds (see Sect. 2.1.1). Expressing the Right Ascension coordinates of a source in terms of hours, minutes, and seconds and the Declination coordinates of the source in terms of degrees, arcminutes, and arcseconds is known as the sexagesimal coordinate system. A clear application of expressing equatorial coordinates in sexagesimal format is to relate these coordinates to the time frame of the observer. Sidereal time is defined as the time between two successive passings of a position on the celestial sphere (corresponding to the position of a star, for example) across the meridian of an observer. This period of time is 24 h and it provides the basis for expressing Right Ascension in terms of 24 h. More specifically, local sidereal time (LST) is defined as the amount of time since Υ has last traversed the meridian of an observer. The

2.1 Angles

27

latter is also equivalent to the hour angle H of the vernal equinox, which is defined as the angle between the celestial object and the observer’s meridian. A common reference time used for observers around the world is universal time (UT), which corresponds to the local time of Greenwich, England. Time zones are measured relative to this time frame: for example, for observers based in the eastern United States are located in the Eastern Daylight Time (EDT) time zone which runs 4 h behind UT (that is, UT—4:00 = EDT). Note that for circles of constant Declinations located away from the celestial equator (that is, δ = 0◦ ), the angles subtended Right Ascension change (recall the discussion of the factor of sin θ in the expression for the differential solid angle dΩ in Eq. (2.12)) such that—for example—1 min of Right Ascension at δ = 30◦ subtends a smaller angle than 1 min of Right Ascension at δ = 60◦ . In fact, the scaling factor for the size subtended by the same unit of Right Ascension as a function of Declination is cos δ. To illustrate the usage of the equatorial coordinate system, note that the equatorial coordinates of the radio luminous Galactic supernova remnant Cassiopeia A may be expressed in decimal degrees as (α, δ) = (350.8864◦ , +58.8118◦ ) or in sexagesimal coordinates as (α, δ) =   (23h 23m 28s , +58◦ 48 42 ). Notice that at the North and South Celestial Poles, Right Ascension coordinates are undefined (simply put, at the celestial poles, any value of Right Ascension coordinate corresponds to the position of the pole). Due to the gravitational interaction between the Earth, Sun, and Moon and due to the nonspherical shape of the Earth, the location of Υ (where the celestial equator and the ecliptic intersect) does not remain fixed but instead drifts along the celestial equator. The magnitude of this drift is approximately 50 arcseconds per year. This phenomenon is known as precession and it is commonly described as a “wobble” by the Earth that is similar to the spinning of a child’s toy top and the period of this wobble is approximately 25,770 years. This precession also manifests itself in the phenomenon of the North Star (also known as Polaris) which in the current era is coincident with the North Celestial Pole and thus appears to be fixed in the sky as the celestial sphere rotates. This coincidence is merely a fortuitous alignment and in time because of precession the direction of the North Celestial Pole will change and Polaris will no longer be coincident it. During the period of precession, the location of the North Celestial Pole will trace out a large circle in the sky: notice also that precession causes changes in the location of the South Celestial Pole in the sky as well. Because the equatorial coordinate system uses as its origin the intersection between the celestial equator and Υ and because this location changes due to precession, astronomers use an epoch when stating coordinates of Right Ascension and Declination. This epoch is usually a commonly accepted reference date: for example, in the current astronomical literature the epoch known as “(J2000.0)” is often reported when coordinates of sources are specified. In this case, the epoch here refers to the position of the source on 1 January 2000 at noon in Greenwich, England: in turn, the local time of Greenwich is the basis of the universal time or “UT” time system commonly used to report time in astronomy. Lastly, the “J” prefix in J2000.0 refers to the Julian calendar system, the calendar system introduced by Julius Ceasar in 46 BC that codified the standard length of the year of 365 days

28

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

and was used for the first time in 45 BC. Related to the Julian calendar is the Julian Date, another time system common in astronomy which counts the number of days since noon UT on 1 January 4713 BC. Prior to the year 2000, a commonly used epoch in the astronomical literature for reporting coordinates was the B1950.0 epoch, where the “B” prefix refers to an alternative definition of a year known as a “Bessellian” year (named after the German mathematician and astronomer Friedrich Bessel) that has become obsolete. To help illustrate applications of the equatorial system, consider the angular separation Δθ between two objects in the sky with positions (α1 , δ1 ) and (α2 , δ2 ), respectively, such that Δα = |α1 −α2 | and Δδ = |δ1 −δ2 | (see Figs. 2.6 and 2.7). The positions A and B are defined to correspond to (α1 , δ1 ) and (α2 , δ2 ), respectively, while the point P is defined to be coincident with the North Celestial Pole. Thus, it can be seen that AP , AB, and BP are each segments along great circles on the celestial sphere. The angle φ between AB and AP is known as the position angle and it is measured from the North Celestial Pole. If another segment NB is defined such that N is at the same Declination as B and that the angle  P NB = 90◦ , then the angle  AP B = Δα, AP = 90◦ −δ, and NP = BP = 90◦ −(δ +Δδ). Recalling the law of sines for spherical triangles, sin b sin c sin a = = , sin A sin B sin C

(2.25)

where A, B, and C are the angles opposite to the sides a, b, and c, respectively, and both the angles and the sides are expressed in degrees (equivalent to arc lengths in Fig. 2.6 A schematic diagram illustrating the concept of a spherical triangle. Notice that the triangle with legs a, b, and c (with opposite angles A, B, and C, respectively) is a segment of a great circle on the surface of a sphere. Note that all angles are less than 180◦ and that the legs are measured in angular units (that is, in degrees)

2.1 Angles

29

Fig. 2.7 A schematic diagram illustrating the determination of the distance between two positions in the sky (denoted as A and B) using the equatorial coordinate system. The two positions A and B are connected to a location at the North Celestial Pole (denoted as P and with the northern direction indicated by the letter N) to form a triangle with A, B, and P as its vertices. The offset in Right Ascension between the two positions is Δα and the offset in Declination between the two positions is Δδ

this case), the law of sines applied to this spherical triangle yields sin AB sin NP = sin Δα sin φ

or

sin Δθ sin(90◦ − (δ + Δδ)) = , sin Δα sin φ

(2.26)

which simplifies to sin Δα cos (δ + Δδ) = sin (Δθ ) sin φ

(2.27)

after applying the trigonometric identity sin *(90◦ − θ ) = cos θ . Applying the small angle approximation presented in Eq. (2.3) to α and δ yields Δα =

Δθ sin φ . cos δ

(2.28)

To obtain a complementary equation for Δθ , recall the law of cosines for the sides of spherical triangles, which is cos a = cos b cos c + sin b sin c cos A.

(2.29)

Applying this relation to the spherical triangle under consideration here yields cos (90◦ − (δ + Δδ)) = cos(90◦ − δ) cos Δδ + sin(90◦ − δ) sin Δθ cos φ

(2.30)

30

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

and once more utilizing small angle approximations (this time, the specific approximations are cos (90◦ − θ ) ≈ 0 and sin (90◦ − θ ) ≈ 1 for small values of θ , as well as the relation cos(90◦ − θ ) = sin θ ) for δ and θ produces Δδ = Δθ cos φ.

(2.31)

Combining Eqs. (2.28) and (2.31) through consideration of the Pythagorean Theorem produces (Δθ )2 = (Δα cos δ)2 + (Δδ)2 .

(2.32)

It is emphasized that throughout this derivation, it has been assumed that the small angle approximation has been valid at every step. This is usually a very valid assumption in astronomy, where angular positions (such as the distances between two sources) fit comfortably within the realm of small angles.

Example Problem 2.2 Mizar and Alcor are two stars associated with the constellation of Ursa Major that are only separated by a very narrow distance, often close to the limit of resolution by the human eye. The equatorial coordinates for Mizar are RA (J2000.0) 13h 23m 55s .5, Dec (J2000.0) +54◦   55 31 and the equatorial coordinates for Alcor are RA (J2000.0) 13h 25m   13s .5, Dec (J2000.0) +54◦ 59 17 . Based on the equatorial coordinates of these stars, calculate the angular separation between them in degrees. Solution The first step is to convert the given coordinates for both stars from the given sexagesimal coordinates into decimal degrees, thus making the application of Eq. (2.32) more straightforward. The conversions may be accomplished as follows: the coordinates in decimal degrees of Mizar may be computed as  αMizar =

  m    23 × 360◦ 55.5s × 360◦ 13h × 360◦ + + = 200.9813◦ 24h 1440m 86,400s (2.33)

and  ◦

δMizar = 54 +



55 × 1◦  60



 +



31 × 1◦  3600

 = 54.9253◦ .

(2.34)

In both of these calculations, note how the conversions are calculated. Similarly for Alcor, the coordinates in decimal degrees may be computed as (continued)

2.2 Gravity

31

Example Problem 2.2 (continued)   m     h 25 × 360◦ 13.5s × 360◦ 13 × 360◦ + + = 201.3063◦ αAlcor = 24h 1440m 86,400s (2.35) and       59 × 1◦ 17 × 1◦ ◦ δAlcor = 54 + (2.36) + = 54.9881◦ .   60 3600 Therefore Δα = |αMizar − αAlcor | = |200.9813◦ − 201.3063◦ | = 0.325◦

(2.37)

and Δδ = |δMizar − δAlcor | = |54.9253◦ − 54.9881◦ | = 0.0628◦ .

(2.38)

The mean value of the two declinations of the two stars is δ¯ =

δMizar − δAlcor 54.9253◦ + 54.9881◦ = = 54.9567◦ , 2 2

(2.39)

and applying Eq. (2.32) yields (Δθ )2 = (0.325◦ × cos (54.9567◦ ))2 + (0.0628◦ )2 = 0.039◦2 ,

(2.40)

and therefore Δθ may be expressed in degrees and in arcminutes as 

60  Δθ = 0.20 × ◦ ≈ 12 . 1 ◦

(2.41)

This angular distance corresponds to approximately one-third the apparent angular size of the Moon to Earth-based observers. Again, it is emphasized that these calculations are made based on assuming that the small angle approximation is valid and that the offset between the two positions is small enough that this approximation may be made.

2.2 Gravity Consider two masses (denoted as m1 and m2 ) separated by a distance r (see Fig. 2.8). Remember that the gravitational force Fgravitational between the two masses may be expressed as

32

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Fig. 2.8 A schematic diagram depicting the gravitational force between two masses M and m separated by a distance r

Fgravitational =

Gm1 m2 , r2

(2.42)

where G is the gravitational constant which corresponds to 6.67 × 10−11 N m2 kg−2 . Strictly speaking, r refers to the distances between the centers of the two masses, but in many astronomical applications where the sizes of objects are negligible compared to the distances that separate them, the physical extents of the masses may be safely ignored. Note that Fgravitational is always attractive and is directed along a line that connects the two masses: the gravitational force on m1 is directed toward m2 and vice versa. The potential energy V between the two masses is  −Gm1 m2 V = Fgravitational dr = . (2.43) r By convention, gravitational potential energy is negative and asymptotically approaches zero as the separation between the two masses approaches infinity. Adopting this convention helps to ensure that the conservation of energy is maintained; in other words, recalling that the kinetic energy K of a mass m may be expressed as K=

mv 2 , 2

(2.44)

then the total energy E of a system (the sum K and V ) remains a constant such that E = K + V = constant.

(2.45)

Recall the phenomenon of a pendulum, where potential energy V is converted into kinetic energy K and then back again into potential energy U as the pendulum swings through a full amplitude of motion. Recall also that for conservative forces that do not change appreciably over time (such as the force of gravity), the

2.2 Gravity

33

relationship between Fgravitational and U can be expressed as F = −∇U.

(2.46)

Several examples are presented below to help illustrate the terminology and concepts presented here. In the first example, consider the gravitational force Fgravitational between the Earth and an object located on its surface. Defining M⊕ , R⊕ , and m to be the mass of the Earth, the radius of the Earth and the mass of the object, respectively, Fgravitational may be expressed as Fgravitational =

GM⊕ m = mg, 2 R⊕

(2.47)

2 is also known as the surface gravity of Earth and where g = G M⊕ /R⊕ corresponds to approximately 9.8 m/s2 . In the second example, consider a mass m in orbital motion (namely a circular orbit with a radius r) around a larger mass M (that is, m is much less than M, or m M). Here, m may be a satellite in orbit around the Earth or a Moon in orbit around a planet. In this situation, two types of velocities are presented and discussed: the circular velocity vc (the velocity of m in a circular orbit around M) and the escape velocity ve (the velocity required by m to escape the gravitational pull of M). To determine vc , recall the expression for the centripetal force Fcentripetal on a mass m moving in a circular path with radius r and a velocity vc , that is,

Fcentripetal =

mvc2 . r

(2.48)

Setting the force of gravity Fg on m equal to Fc , that is (from Eqs. (2.42) and (2.48)), Fgravitational = Fcentripetal ⇒

GMm mvc2 , = r r2

(2.49)

and solving for vc yields  vc =

GM . r

(2.50)

To determine ve , the kinetic energy K of the mass m is set to be equal to the absolute magnitude of the gravitational potential energy U . Recalling Eq. (2.44) and setting K and the absolute value of U equal to each other using Eqs. (2.43) and (2.44) gives K = |U | ⇒

GMm mve2 =| |, 2 r

(2.51)

34

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

and finally solving for ve yields  ve =

√ 2GM = 2vc . r

(2.52)

Note that both vc and ve are proportional to r −1/2 .

Example Problem 2.3 Calculate the circular orbital velocity vc and the escape velocity ve for an object in orbit around the Earth. Solution Consider objects which are located close to the surface of the Earth (therefore r ∼ R⊕ ): the radius and mass of the Earth are R⊕ = 6.37 × 106 m and M⊕ = 5.97 × 1024 kg, respectively. From Eqs. (2.50) and (2.52), vc may therefore be calculated as   GM⊕ (6.67 × 10−11 N × m2 /kg2 )(5.97 × 1024 kg) = 7906 m/s. = vc = R⊕ (6.37 × 106 m) (2.53) and  √ √ 2GM⊕ ve = = 2vc = 2 × 7906 m/s = 11,181 m/s. (2.54) R⊕

2.3 Orbits and Orbital Motion A thorough analysis of the motion of astronomical sources demands a detailed understanding of orbital motion, such as member stars in a binary star system or a star in orbit around the center of the Milky Way Galaxy. Within the scope of this book, mastery of Kepler’s Laws of Planetary Motion (as summarized below) is crucial for the concepts that will be presented. These concepts are presented in an elementary manner: more thorough derivations and analyses are presented elsewhere by other authors.

2.3 Orbits and Orbital Motion

35

2.3.1 Kepler’s Laws of Planetary Motion 2.3.2 Kepler’s First Law A succinct way to state Kepler’s 1st law of planetary motion is “the orbits of planets are in the shape of ellipses with the Sun located at one focus.” In turn, an ellipse may be described as a conic section: such sections are the set of curves (such as a circle, a parabola, and a hyperbola as well as an ellipse) that are produced by slicing a cone at different angles with a plane. Mathematically, an ellipse is the locus of all points such that the sum of the distances from the two foci (plural of focus) to any point on the ellipse is a constant. Defining r and r  as the distances from a point on the ellipse to foci F and F  , respectively, then the equation of an ellipse may be expressed as r + r  = 2a = constant.

(2.55)

Here, the major axis is defined as a line that passes through F and F  and intersects the ellipse at the two vertices ψ and ψ  . By extension, a may be defined as the semi-major axis (that is, one-half of the length of the major axis). By the same token, the minor axis is the perpendicular bisector of the major axis and the semi-minor axis b may be defined as one-half of the length of the minor axis. The shapes of ellipses are characterized by the parameter known as the eccentricity e of the orbit of the planet: this parameter is defined such that the distance from each focus to the center of the ellipse is ae (Fig. 2.9). Note that r = r  at a point where the minor axis intersects the ellipse. Considering this location on the ellipse, a right triangle may be defined where the right angle is located at the intersection of the major and minor axes at the center of the ellipse, the legs have lengths b and ae and a hypotenuse with length r = r  = a (from Eq. (2.55)). Therefore, from the Pythagorean Theorem, the relationship between a, b, and e may be expressed as a 2 = b2 + a 2 e2 → b2 = a 2 − a 2 e2 = a 2 (1 − e2 ),

(2.56)

which yields at last the following equation: b = a 1 − e2 .

(2.57)

How do can the distance from one focus (say F , where the Sun is located) to an arbitrary position on the ellipse be determined? Recalling that just such a distance is r, a polar coordinate system may be applied which is centered on F and the angle θ is defined as the true anomaly. This angle is in turn defined such that the line connecting the foci and the vertices are θ =0 and angles are measured counterclockwise. Therefore, the angle made between r and the major axis is θ while the angle located opposite r  is π -θ (as measured in radians). Noting that the

36

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Fig. 2.9 A schematic diagram illustrating key elements of an orbital ellipse. The two foci of the ellipse are indicated as F and F  , respectively, and the distances from F and F  to the location of the planet are r and r  , respectively. The semi-major and semi-minor axes are indicated as a and b, respectively. The distances from each foci to the geometric center of the ellipse are ae, where e is the eccentricity of the ellipse. Assuming that the Sun is located at the focus F , the locations of perihelion and aphelion are indicated as ψ and ψ  , respectively. Lastly, the angle made between the locations of ψ, F , and the position of the planet (corresponding to the length r) is θ

distance from F to F  is 2ae, applying the law of cosines yields r 2 = r 2 + (2ae)2 − 2r (2ae) cos(π − θ ) = r 2 + (2ae)2 + 2r (2ae) cos θ

(2.58)

where the trigonometric identity cos (π - θ ) = −cos(θ ) has been applied. But from Eq. (2.55) it is also known that r  = 2a - r. Therefore, Eq. (2.58) becomes r=

a (1 − e2 ) . (1 + e cos θ )

(2.59)

This is the general equation for an ellipse with an eccentricity 0 ≤ e < 1. Keeping in mind that the Sun is located at one foci (again the focus F ), the terms perihelion and aphelion may be defined as the positions on the orbit which are closest to and farthest away from the Sun, respectively. If these positions correspond to the locations of the vertices ψ and ψ  , respectively (where the corresponding values of θ are θ = 0 and θ =π ), then the perihelion and aphelion distances rperihelion and raphelion from the Sun are (from Eq. (2.59)) rperihelion =

a(1 − e)(1 + e) a(1 − e2 ) = = a(1 − e) 1+e (1 + e)

(2.60)

2.3 Orbits and Orbital Motion

37

and raphelion =

a(1 − e)(1 + e) a(1 − e2 ) = = a(1 + e). 1−e (1 − e)

(2.61)

2.3.3 Kepler’s Second Law Kepler’s Second Law may be expressed qualitatively as “a line connecting the Sun with a planet in its orbit will sweep out an equal amount of area in an equal period of time .” Alternatively, for brevity this law may be stated as “equal areas swept out in equal times (see Fig. 2.10).” Fundamentally, the velocity of a body in an orbit around the other body changes as the distances between the two bodies change. If the motion of bodies in the Solar System is considered, it is readily apparent that the orbits of the planets have low eccentricities and do not (in general) differ greatly from circles. Therefore the orbital velocities of the planets do not change appreciably during the courses of their orbits (even when considering the perihelion and aphelion positions in the orbits). In contrast, the orbits of comets may have very high eccentricities and thus the orbital velocities change greatly in time (particularly at the perihelion and aphelion locations in the orbits). Imagine a planet in an elliptical orbit with an instantaneous velocity v at a distance r from the Sun: in a time interval Δt, the planet sweeps out an angle Δθ . For small angles, this angle can be expressed as Δθ ≈

vt Δt r

(2.62)

where vt is the instantaneous tangential component (perpendicular to r) of the velocity. During this time interval, the area ΔA swept out by r is ΔA ≈

rvt Δt , 2

(2.63)

where the well-known equation for the area of a triangle A = (1/2)bh has been applied. In this case, b and h are the lengths of the base and the height of the triangle, respectively. In the limit where t → 0, Eq. (2.63) becomes   ΔA rvt rvt r  r 2 Δθ Ψ = = = = , Δt 2 2 r 2 Δt 2

(2.64)

where Ψ —the angular momentum per unit mass—is a constant according to Kepler’s Second Law because r 2 (Δθ /Δt)—the rate of change of area with time—is itself a constant. Integrating Eq. (2.64) over the entire area A of the ellipse (where A=πab) and over the entire time period P of the orbit yields

38

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Fig. 2.10 A schematic diagram illustrating Kepler’s Second Law of Planetary Motion. The orbital ellipse of a planet in orbit around a central object (e.g., the Sun) located at one focus of the orbit is shown. The dots indicate locations of the planet on its orbit as separated by equal intervals of time. Notice how the density of the dots is greater when the planet is far from the central object (where the planet is traveling more slowly) and is lesser when the planet is close to the central object (where the planet is traveling more rapidly). The sectors of the orbit (alternating in white and shaded coloring) correspond to equal time intervals for the motion of the planet along its orbital path: note that an equal number of dots are contained in the perimeters of the marked sectors. Notice further that the areas in the sectors are equal, indicating indeed that “equal areas are swept out in equal times”

Ψ A = , P 2

(2.65)

and therefore v as a function of Ψ may be expressed as v=

Ψ 2A = . r Pr

(2.66)

At the locations of perihelion and aphelion in an orbit (where θ = 0 and θ = π , respectively), the velocity v of a planet at perihelion and at aphelion is completely tangential (that is, there is no radial component). Therefore, the velocities vperihelion

2.3 Orbits and Orbital Motion

39

and vaphelion —of course corresponding to the velocities at perihelion and aphelion, respectively—may be calculated from Eqs. (2.57), (2.60), (2.61), and (2.66), yielding

vperihelion

 √ 2π ab 2π a 1 − e2 2π a (1 + e) 2A = = = = Pr P a(1 − e) P (1 − e) P (1 − e)

(2.67)

and vaphelion

 √ 2π ab 2π a 1 − e2 2π a (1 − e) 2A = = = . = Pr P a(1 + e) P (1 + e) P (1 + e)

(2.68)

2.3.4 Kepler’s Third Law This law—also commonly known as the “Harmonic Law”—may be expressed as follows: “the square of the orbital period P of a planet is directly proportional to the cube of its semi-major axis a.” When considering motion in the Solar System, this law can be expressed in a particularly simply way: expressing P in units of years and a in units of Astronomical Units, Kepler’s Third Law becomes P 2 = a3.

(2.69)

Kepler’s Third Law may be derived as follows. Consider a planet with mass m in orbit around a star with mass M: the sum of the two masses is m + M = Mtotal . In this derivation, both m and M are treated as point masses—that is, their radii are negligible compared to the distance between them, therefore their physical extents may be safely ignored. For any two bodies in orbit around each other, the two bodies will orbit around a common position known as the center of mass: this location is crucial because—for mechanics calculations—the whole system may be treated such that all of its mass is concentrated at this location. Consider a line that connects these two masses and assume that the separation distance (denoted as r) between the two masses is a constant. The location of the center of mass (located along r) is defined to be rCM , while the distance from M to rCM is defined to be rM and finally the distance from m to rCM is defined to be rm (note that that r = rM + rm ). The location of rCM itself along r (say its distance from M) may be determined using the equation rCM =

1 MrM + mrm , mi ri = Mtotal M +m

(2.70)

i

where the index i here is for all the bodies in the system (which in this case is simply the two masses m and M). Both M and m will complete orbits with the same orbital period P around the center of mass: the orbital velocities for M and m will be V

40

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

and v, respectively. The centripetal force Fcentripetal M that M will experience in its orbit around the center of mass will be (from Eq. (2.48)) Fcentripetal M =

MV , 2 rM

(2.71)

but assuming that the orbit of M is circular, P and rM are related through the equation P =

2π rM , V

(2.72)

and therefore Fcentripetal M may be expressed as Fcentripetal M =

4π 2 MrM . P2

(2.73)

Similarly, considering m, Fcentripetal m and P may be expressed as Fcentripetal m =

mV , 2 rm

P =

2π rm , v

so therefore

Fcentripetal m =

4π 2 mrm . P2 (2.74)

But for the system to remain bound gravitationally, Fcentripetal M and Fcentripetal m must equal each other. Setting these two quantities equal to each other yields Fcentripetal M = Fcentripetal m →

4π 2 MrM 4π 2 mrm rM m . = → = 2 rm M P P2

(2.75)

Notice that this equation indicates that the larger mass (which will be located closer to the center of mass) will have a smaller radius to its orbit around the center of mass. Recalling that r = rM + rm , this equation can be re-expressed as rM = (r − rM )

m m m → rM 1 + =r M M M

(2.76)

which leads to rM =

rm . M +m

(2.77)

Recalling that the gravitational force that M exerts on m (and conversely the gravitational force exerted by m on M) is (see Eq. (2.42)) Fgravitational =

GMm . r2

(2.78)

2.3 Orbits and Orbital Motion

41

Setting Fgravitational equal to Fcentripetal M yields Fgravitational = Fcentripetal M →

GMm 4π 2 MrM = , r2 P2

(2.79)

and finally solving for P 2 yields P2 =

4π 2 r 2 rM . Gm

(2.80)

Inserting the result from Eq. (2.77) into this equation produces P2 =

4π 2 r 3 . G(M + m)

(2.81)

This equation is the general form of Kepler’s Third Law. In the scenario where m is much less than M, then the sum of the two masses may be approximated as m+M ≈ M. If a further approximation is made that r = a (that is, the semi-major axis of the orbit of m), then the following modified form of Kepler’s Third Law is obtained: P2 =

4π 2 a 3 . GM

(2.82)

While Eq. (2.81) is the most general form of Kepler’s Third Law, Eq. (2.82) is an approximate form that is frequently applied to systems of two bodies where the mass of one body is much greater than the mass of another. Examples of such systems include a Moon in orbit around a planet, a planet in orbit around a star or a cloud of gas in orbit around a supermassive black hole in the center of a galaxy. Therefore, despite its name implying an application solely to the motion of planets, Kepler’s Third Law may be applied to a wide range of systems: once P and a are measured for the orbiting mass, M—the mass of the larger body—may be readily computed using Eq. (2.82). The scope of this Law and its range of applications is indeed remarkably broad. This section concludes with one final comment about the total energy E of a particular orbiting body. Recalling the definition of E given in Eq. (2.45), E must be negative if the body is in a bound orbit, that is, E = K + U < 0.

(2.83)

The origin of the negative sign may be understood as follows: in the limit of an infinite separation between the orbiting body and the central body (that is, as r = ∞ between M and m), the gravitational potential energy U between the two masses is zero. A positive amount of energy must be inputted into the system to reduce the distance between the two masses and—to reflect the input of this energy into the system—the sign of U becomes negative.

42

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Example Problem 2.4 Titan—the largest Moon of Saturn—completes an orbit of Saturn with a period P = 15.95 days. The semi-major axis a of Titan’s orbit is 1.22 × 106 km. Based on this information and making the proper assumption that the mass of Titan is significantly less than the mass of Saturn, calculate the mass M of Saturn. Finally, if the mean orbital velocity of Titan around Saturn is v = 5570 m/s and the mass of Titan itself is m = 1.35 × 1023 kg, it shows that E for Titan is negative and that Titan’s orbit is indeed gravitationally bound. For this latter calculation, note that the eccentricity of Titan’s orbit around Saturn is only e = 0.03, so it is appropriate to assume that the orbit of Titan is effectively circular around Saturn. Solution The given values for a and P need to be converted as follows:  1000 m = 1.22 × 109 m and 1 km   86,400 s = 1.38 × 106 s. P = 15.95 days 1 day 

a = 1.22 × 106 km

(2.84)

Thus, from Eq. (2.82) M may be calculated as follows: 4π 2 a 3 4π 2 (1.22 × 109 m)3 = = 5.64 × 1026 kg. GP 2 (6.67 × 10−11 N m2 /kg2 )(1.38 × 106 s)2 (2.85) Finally, from Eq. (2.45) and applying the assumption that TItan’s orbit around Saturn is effectively circular, E for Titan may be expressed as M=

E=

GMm mv 2 − , 2 a

(2.86)

and computed as E=

(1.35 × 1023 kg)(5570 m/s)2 2 −

(2.87)

(6.67 × 10−11 N m2 /kg2 )(5.64 × 1026 kg)(1.35 × 1023 kg) 1.22 × 109 m

which yields at last E = −2.07 × 1030 J. This result verifies that Titan is indeed in a stable orbit around Saturn.

(2.88)

2.4 Light

43

2.4 Light 2.4.1 Wave-Like Properties of Light Consider a wave characterized by a wavelength λ (the distance from one peak of a wave to the next) and a frequency ν (the time required for the wave to complete one cycle) that is propagating with a velocity v through a medium. The relationship between v, λ, and ν may be expressed as v = λ × ν.

(2.89)

For light waves, where c is defined as the speed of light within a vacuum (c = 3×108 m/s), this equation becomes simply c = λ × ν.

(2.90)

The velocity of light through a medium depends on the index of refraction n(λ) for a particular material, which in turn is a function of ν of the light wave. The index of refraction may be quantified as v=

c , n(λ)

(2.91)

where it is emphasized that n is indeed a function of the wavelength of the light. By definition, n(λ) = 1 for all values of λ in the case of light propagating through a vacuum. The fact that the velocity of light through a medium depends on the wavelength is demonstrated in an aesthetically pleasing manner by the prisms: as white light (which can be considered to be a mixture of light of all wavelengths) passes through the prism. Because different wavelengths of light will travel with different velocities through the medium of the prism, by the time the light waves exit the prism, the shortest wavelength light (say the blue light) has been delayed the most and has its path bent the most from the normal angle to the edge of the prism upon its exit. In contrast, the longest wavelength light (say the red light) has been delayed the least and has had its path bent the least from that normal angle. Recall that in the case of a monochromatic light wave passing through the interface between two media (say a light wave passing from air into water), Snell’s Law may be used. This law is commonly expressed as n1 sin θ1 = n2 sin θ2 ,

(2.92)

to relate the indices of refraction n1 and n2 at the wavelength of the monochromatic light wave for the two media to the angles θ1 and θ2 at which the wave propagates through the media.

44

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Fig. 2.11 A schematic diagram depicting light as an electromagnetic wave. The electric wave component is depicted in red and oscillates in the yz-plane, while the magnetic wave component is depicted in blue and oscillates in the xy-plane. The wavelength of the wave λ can be defined as the distance from the peak of one electric wave to the next (or from the peak of one magnetic wave to the next). The electromagnetic wave as a whole is propagating in the +y direction: this direction also corresponds to the same direction as the Poynting vector S of the wave

Formally, light is an electromagnetic wave: it is composed of an oscillating electric wave component coupled with an oscillating magnetic wave component and these two components are perpendicular to each other. This phenomenon is illustrated in a schematic diagram shown in Fig. 2.11. Light is classified as a transverse wave in that the directions of the oscillations (for both the electric wave component and the magnetic field component) are themselves perpendicular (transverse) to the direction of propagation. The oscillating electric wave component E(x,t) (which has a dependence both on position and time) may be expressed as E(x, t) = E0 ei(kx−ωt) = E0 (cos(kx − ωt) − i sin(kx − ωt)),

(2.93)

and a similar equation may be created to express the complementary oscillating magnetic wave component. Equation (2.93) describes an electric wave propagating in the +x direction: the wave may be oscillating in either the plane of the y-axis or in the plane of the z-axis. Here a well-known identity (known as Euler’s formula)1 may be used. Euler’s Formula is commonly expressed as eiθ = cos θ + i sin θ.

1 This

(2.94)

equation contains the most commonly accepted form of Euler’s formula. The discrepancy between the form of this equation and the content of Eq. (2.93)—namely, where the sine term has a negative sign in Eq. (2.93) but a positive sign in Eq. (2.94)—can be understood simply as a convention, where the negative sign in Eq. (2.93) describes a wave propagating in the +x direction. Recall that the sine function is symmetric for negative and positive values of x, so in the end the choice of sign is arbitrary.

2.4 Light

45

While waves are often described simply in terms of the sine or cosine functions, Euler’s formula is very commonly used in both engineering and physics. The advantage of using this formula is that manipulating exponentials is significantly easier than manipulating trigonometric functions. In Eq. (2.93), we have used the term wavenumber k—a commonly encountered quantity when describing waverelated phenomena like light—such that k=

2π . λ

(2.95)

Similarly, the term angular frequency ω—another such commonly encountered wave-related quantity—is defined (in units of radians sec−1 ) such that ω = 2π ν.

(2.96)

Lastly, in Eq. (2.93) E0 is simply the amplitude of the oscillating electric wave component. Like all waves, electromagnetic waves must satisfy the wave equation ∇ 2 E(x, t) =

1 ∂ 2 E(x, t) . v2 ∂t 2

(2.97)

While Eq. (2.97) has been presented through consideration of the electric wave component of light, clearly the magnetic wave component must satisfy the wave equation as well. Note that if the oscillating magnetic wave component B(x,t) is expressed as B(x, t) = B0 ei(kx−ωt) ,

(2.98)

the relationship between E0 and B0 (and in turn the relationship between the magnitudes of the electric and magnetic fields of an electromagnetic wave) is E0 E(x, t) = c. = B0 B(x, t)

(2.99)

This result indicates that the ratio of the magnitude of the electric field to the magnitude of the magnetic field is equal to the speed of light at every instant. Another relevant quantity to the present discussion is the Poynting vector S. This vector is defined as the rate of transfer of energy by an electromagnetic wave and may be expressed as S=

1 E × B, μ0

(2.100)

where E and B are vector descriptions of the electric wave component and the magnetic wave component, respectively, of the light wave. Here, μ0 is the magnetic permeability of a vacuum: in general, the magnetic permeability of a material is a

46

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

measurement of the ability of a material to support the formation of a magnetic field, or how magnetized a material becomes when an external magnetic field is passed through the material. The magnetic permeability of a material is generally indicated with the symbol μ: it is measured in units of N×A−2 or H×m−1 and the exact value of μ0 is 4π ×10−7 N×A−2 = 4π × 10−7 H×m−1 . Complementing μ0 is another physical constant known as 0 , which is the permittivity of free space: furthermore, complementing the magnetic permeability of a medium is another quantifiable characteristic known as the permittivity, which describes the resistance encountered when forming an electric field in a medium. The permittivity of a medium is generally denoted with the symbol : it is measured in units of F/m2 and the exact value of 0 is 8.85 × 10−12 F/m2 . The physicist James Clerk Maxwell realized that c may be expressed in terms of 0 and μ0 as  c=

1 . 0 μ0

(2.101)

Finally the intensity I of a light wave may be defined as the power per unit area or the rate at which energy transported by the light wave transfers through a unit area perpendicular to the direction of travel of the wave. This quantity corresponds to the time-averaged value of the magnitude of S (that is, < S >) and may be expressed mathematically as I =< S >=

cB02 E02 E0 B0 = = . 2μ0 2μ0 c 2μ0

(2.102)

For completeness, Maxwell’s equations are presented here in differential and integral form: these equations form the underpinnings of all studies of electric and magnetic fields (as well as their related applications). In particular, these equations describe how electric fields E and magnetic fields B are generated and how these fields interact with each other. In differential form (specifically as vector divergences and vector curls), these equations may be written as follows: ∇ ·E =

ρ 0

∇ ·B = 0

Gauss’s Law Gauss’s Law for Magnetism

∂B Faraday’s Law of Induction ∂t   ∂E ∇ × B = μ0 J + 0 Ampere-Maxwell’s Law ∂t

∇ ×E = −

(2.103) (2.104) (2.105) (2.106)

Alternatively, in integral form (specifically as surface integrals and contour integrals) these equations may be written as follows:

2.4 Light

47

E · nˆ da =

S

qenc 0

B · nˆ da = 0 S

Gauss’s Law

(2.107)

Gauss’s Law for Magnetic Fields

(2.108)

 d B · nˆ da Faraday’s Law of Induction dt S C  

 d B · dl = μ0 Ienc +0 E · nˆ da Ampere-Maxwell’s Law dt S C E · dl = −

(2.109) (2.110)

In these equations, ρ is the total charge density, J is the current density, qenc is the amount of enclosed charge, nˆ is the normal vector to the differential area element da, dl is a differential line element, and Ienc is the enclosed current. The reader is referred to other references (such as [1]) for a more thorough treatment of Maxwell’s Equations.

2.4.2 Particle-Like Properties of Light In addition to the wave-like properties of light, light may also be treated as a particle. Taking into account that light has particle-like properties as well as wavelike properties allows a proper explanation of such phenomenon as the photoelectric effect. Regarding its particle properties, light may be treated as a bundle of energy known as a photon and the energy E of the photon may be expressed as (see Eq. (2.90)) E = hν =

hc . λ

(2.111)

Here, h is Planck’s constant = 6.63 × 10−34 J · s = 4.14 × 10−15 eV · s. From Eq. (2.111), the energy of a photon may be described is quantized: that is, the energies of photons only occur in discrete multiples (so-called quanta) of h. Furthermore, as a particle a momentum p may be assigned to a photon. This momentum may be expressed as (see Eqs. (2.90) and (2.111)) p=

hν h E = = . c c λ

(2.112)

This equation embodies the particle nature of photons in that a photon may be envisioned as a particle with momentum that may be imparted on a target. For example, momentum may be imparted on an electron by a photon in the phenomenon known as Compton scattering (see Sect. 7.3.6 for more discussion about this phenomenon).

48

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

2.4.3 Wave-Particle Duality for Light For the remainder of this book, photons will be envisioned as bundles of energy that are characterized by a particular wavelength or frequency. Light is thus best described through the wave-particle duality where it has wave-like properties and particle-like properties at the same time. In many physical phenomena, one or the other set of properties is more relevant but a complete description of all light-related phenomena requires treatment of light as a wave or a particle.

Example Problem 2.5 Astronomers frequently make observations at a wavelength of λ=21 cm=0.21 m: at this wavelength, a spin-flip transition of atomic hydrogen occurs, thus enabling astronomers to detect cold hydrogen gas (see Chap. 5 for a discussion of the importance of observing this transition in the study of cold hydrogen gas in the Milky Way Galaxy). Calculate the frequency ν and energy E of a photon with this wavelength. Solution From Eqs. (2.90) and (2.111), ν may be computed as c 3 × 108 m/s = = 1.4 × 109 Hz λ 0.21 m

(2.113)

(6.63 × 10−34 J s)(3 × 108 m/s) hc = = 9.47 × 10−25 J. λ 0.21 m

(2.114)

ν= and E=

2.4.4 Polarization of Light In Sect. 2.4.1, light was described as a combination of electric and magnetic waves orthogonal that are oscillating in planes that are perpendicular to the direction of propagation of the wave. The plane of polarization of a light wave is defined by the plane of the oscillating electric field for the propagating electromagnetic wave. The treatment here is patterned after the treatment presented in [2]. Consider a light wave propagating in the positive z-direction: the components of the electric field in the x, y, and z directions (that is, the components Ex , Ey , and Ez , respectively) may be expressed as Ex = E1 cos(kz − ωt + δ1 )

(2.115)

Ey = E2 cos(kz − ωt + δ2 )

(2.116)

Ez = 0,

(2.117)

2.4 Light

49

where E1 and E2 are the amplitudes of the Ex and Ey components, respectively. By extension, δ may be defined to be δ = δ1 − δ2 , which is the phase difference between Ex and Ey and finally E itself is the vector sum of the components Ex and Ey . In general, the observed polarization of waves is determined by the amplitudes and difference in phase between the components of the electric field. A commonly encountered type of light polarization is linear polarization: for example, if E1 = 0, then the wave is said to be linearly polarized in the y-direction. Also, if δ = 0 and E1 = E2 , the wave is linearly polarized in a plane that is offset 45◦ with respect to the x-axis. The case where E1 = E2 but instead δ = ±90◦ corresponds to the scenario known as circular polarization. When δ = +90◦ , the wave is said to be left-circularly polarized and when δ = −90◦ the wave is said to be right-circularly polarized. Finally, when the amplitudes E1 and E2 are out of phase with each other by an angle unequal to 0◦ or 90◦ , the wave is said to feature elliptical polarization. The angle of polarization measured in the vertical in the clockwise direction is also known as the position angle of the polarization. The convention within the physics community is that a rotation of the electric field vector that may be described as clockwise as viewed with the wave approaching is right-circular polarization (that is, the curl of the fingers on the right hand with the thumb pointing down the z-axis toward the direction of the origin of the wave). Note that this convention is opposite to the convention within the engineering community which would define this particular rotation of the electric field vector as left-circular polarization: the convention within the radio astronomy community has been to adopt the same convention as that followed by the engineering community (reflecting the close history between radio astronomy and electrical engineering). It is important to be aware of this difference in convention when considering descriptions of polarization as presented by authors from the two different communities. A common way to visualize the different types of polarization is to imagine the set of points traced out by the electric field vector of the wave as it propagates in the positive z-direction. This set of points generally forms an ellipse as shown in Fig. 2.12. In this figure, the amplitudes Ex and Ey of the components of the incoming electric field are envisioned to trace out a figure that may be circular or elliptical, depending on the inherent polarization of the wave itself. The polarization of light may be described in terms of the Stokes parameters I , Q, U , and V . For polarized light waves described by Eqs. (2.116)–(2.117), these parameters may be quantified as follows: I = E12 + E22

(2.118)

Q = E12 − E22

(2.119)

U = 2E1 E2 cos δ

(2.120)

V = 2E1 E2 sin δ.

(2.121)

Furthermore, for a fully polarized monochromatic light wave,

50

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Fig. 2.12 A schematic diagram of the concept of the polarization ellipse. In this diagram, an electric field component E of an electromagnetic wave with an arbitrary polarization angle is shown to project into components Ex and Ey onto the x- and y-axes, respectively. The direction of propagation is in the z-axis, which in this convention is directed toward the viewer. Image adapted from [2]

I 2 = Q2 + U 2 + V 2 ,

(2.122)

while for a completely unpolarized wave, Q = U = V = 0.

(2.123)

How can the Stokes parameters be physically interpreted? This discussion is patterned after the one presented by Heiles [3]. The first parameter, I , is proportional to the total energy flux associated with the light wave. In many instances in radio astronomy (as well as in other branches of astronomy), a goal may be to measure the total flux received from a source from light of all polarizations (and unpolarized light as well): in those cases, I is the relevant parameter. The second parameter Q describes the extent to which the light is linearly polarized along the x and y axes. δ = 0.5 tan−1



U Q

 (2.124)

and the fractional linear polarization pQU is  pQU =

Q U

2

 +

U I

2 1/2 .

(2.125)

The final parameter V indicates whether the wave is circularly polarized or not. This parameter (along with the parameter I ) can be used to compute the fractional circular polarization pV , which is defined as pV =

V . I

(2.126)

2.4 Light

51

Lastly the degree of total polarization pd —the ratio of polarized power to total power—is pd =

Q2 + U 2 + V 2 . I

(2.127)

The Stokes parameters for a wave are often expressed in the form of a onedimensional vector P: if this is formatted such that ⎡ ⎤ I ⎢Q⎥ ⎥ P=⎢ ⎣U ⎦ ,

(2.128)

V then the following forms of P are commonly encountered: ⎡ ⎤ 1 ⎢0⎥ ⎥ P=⎢ ⎣0⎦ for an unpolarized wave,

(2.129)

0 ⎡

⎤ 1 ⎢−1⎥ ⎥ P=⎢ ⎣ 0 ⎦ for a linearly polarized wave in the vertical direction, 0

(2.130)

⎡ ⎤ 1 ⎢0⎥ ◦ ⎥ P=⎢ ⎣1⎦ for a linearly polarized wave at a 45 angle, 0

(2.131)

⎡

⎤ 1 ⎢0⎥ ◦ ⎥ P=⎢ ⎣−1⎦ for a linearly polarized wave at a − 45 angle, 0

(2.132)

⎡

⎤ 1 ⎢0⎥ ⎥ P=⎢ ⎣ 0 ⎦ for a right circularly polarized wave, and −1

(2.133)

52

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

⎡ ⎤ 1 ⎢0⎥ ⎥ P=⎢ ⎣0⎦ for a left circularly polarized wave. 1

(2.134)

Example Problem 2.6 A light wave propagating in the positive z-direction has the following components: Ex = (5V /m) cos(kz − ωt + 135◦ ), Ey = (5 V /m) cos(kz − ωt + 45◦ ) and Ez = 0. Calculate the Stokes Parameters I , Q, U , and V for this wave and the degree of total polarization pd for this wave. Is this wave linearly polarized, circularly polarized, or elliptically polarized? Solution From inspection of the expressions for Ex and Ey , δ = 135◦ − 45◦ = 90◦ . Therefore, from Eqs. (2.119)–(2.121), the Stokes Parameters may be calculated as follows: I = E12 + E22 = (5 V/m)2 + (5 V/m)2 = 50 V2 /m2 .

(2.135)

Q = E12 − E22 = (5 V/m)2 − (5 V/m)2 = 0.

(2.136)

U = 2E1 E2 cos δ = 2 (5 V/m) (5 V/m) cos 90◦ = 0.

(2.137)

V = 2E1 E2 sin δ = 2 (5 V/m) (5 V/m) sin 90◦ = 50 V2 /m2 .

(2.138)

Similarly, from Eq. (2.127), pd is pd =

02 + 02 + (50 V2 /m2 )2 50 V2 /m2

= 1.0.

(2.139)

Therefore, the wave is completely polarized. Based on the value of δ, the wave is left-hand circularly polarized.

Note that the polarization of a light wave may change as it passes through a medium: this is particularly crucial in astronomical observations, where the light from a source passes through the interstellar medium before it reaches a telescope. Indeed, the interstellar medium impinges its presence on virtually all astronomical observations. Whenever a polarized light wave passes through a medium with a magnetic field, the plane of polarization of the wave will be altered. This phenomenon is known as the Faraday effect and the rotation of the plane of polarization of the wave is known as Faraday rotation. The concepts of the Faraday effect and Faraday rotation will be addressed in Sect. 6.4.4.

2.4 Light

53

2.4.5 Doppler Effect Consider a source moving with a radial velocity vr from an observer. Suppose that a spectral line—normally observed at rest to occur at a wavelength λ0 —is instead observed to occur at a wavelength λ. The change in wavelength Δλ may be defined as Δλ = λ − λ0 . Therefore, the relationship between z, Δλ, vr , and c may be expressed as z=

Δλ vr λ − λ0 = = . λ0 λ0 c

(2.140)

In a similar vein, if a spectral line is observed to occur at a rest frequency ν0 and instead is observed at a frequency ν, then—defining the change in frequency Δν to be Δν = ν - ν0 —the relationship between z, Δν, vr , and c may be expressed as z=

Δν vr ν − ν0 = = . ν0 ν0 c

(2.141)

The efficacy of Eqs. (2.140) and 2.141 is clearly seen when determining vr by measuring the amount of shift of an observed spectral line in the spectrum of a source of interest. Note that this equation only determines the radial component of the velocity of a source of interest and it cannot provide any information about the tangential component of the velocity. By convention, a positive recessional velocity indicates that the source is receding from the observer: in this scenario, the light waves emitted by the source are stretched to longer wavelengths and astronomers say that a redshift is seen in the spectrum of the source. Conversely, a negative recessional velocity indicates that a source is approaching the observer: in such a case, the light waves emitted by the source are compressed to shorter wavelengths and astronomers say that a blueshift is seen in the spectrum of the source. Other useful and wholly equivalent forms of Eq. (2.140) that are commonly encountered are 1+z=

λ ν0 = , λ0 ν

(2.142)

where ν and ν0 are the observed and rest frequencies of the spectral line, respectively. When considering how motion may impart shifts in the spectral lines of source, it may also certainly be true that the observer is in motion relative to a source as well. The detection of a redshift or a blueshift in the spectrum of a source (or even the spectrum of the observer as observed by the source) is dictated by the net motion between the observer and the source. Redshifts are seen in the spectra of all galaxies located several Megaparsecs from our home galaxy, the Milky Way. These redshifts are amongst the strongest evidence known for an expanding universe that began

54

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

with a Big Bang. For this reason, z is often referred to as the redshift parameter in the context of galaxy spectra. Note that Eq. (2.140) is applicable only when vr c (that is, motion that is termed as non-relativistic motion). While this equation is perfectly suitable for objects moving with low velocities (such as planets in orbits around stars), it is not suitable for objects moving at velocities that approach the speed of light (such as particles accelerated to such high velocities in the jets of such objects as blazars). Relativistic motion is discussed in more detail in Sect. 2.6: in such cases of relativistic motion, z may be expressed as  z=

1 + vr /c − 1. 1 − vr /c

(2.143)

Realize that Eq. (2.143) reduces to Eq. (2.140) in the limit where vr c: therefore, Eq. (2.140) is used very widely when the motion under consideration is non-relativistic. It can be shown (see Problem 2.4) that Eq. (2.143) may be expressed as (z + 1)2 − 1 vr = . c (z + 1)2 + 1

(2.144)

2.5 The Bohr Model of the Atom In 1913, Niels Bohr presented the first model of an atom that helped to explain how electrons remain in stable orbits around atomic nuclei. Classical theory predicts that when an electron undergoes an acceleration (such as an electron continuously accelerating as it orbits an electron) it must emit a photon: according to classical physics, then, an electron was expected to radiate away all of its energy and spiral into the nucleus of the atom. To explain how an electron may exist in an orbit but not lose energy, Bohr postulated that the orbits of electrons are quantized such that these orbits may exist only at certain fixed distances from the nucleus rather than in a continuum of possible orbits (see Fig. 2.13). While in one of these orbits, the electron does not lose energy. A discussion of the quantization of electron orbits in atoms is presented here (also see [4] for additional details). Consider an atom where a single electron (with charge e and mass me ) orbits a nucleus with atomic number Z (that is, the nucleus contains Z protons and thus has a charge Ze). From Eq. (2.48) the centripetal force that maintains the orbit may be expressed as Fcentripetal =

me v 2 , r

(2.145)

2.5 The Bohr Model of the Atom

55

Fig. 2.13 A schematic diagram depicting the Bohr model of the atom. Electrons may only orbit the nucleus (shown in red) at certain discrete distances that correspond to energy levels. The orbits that are located farther from the nucleus are associated with greater potential energy than the orbits located closer to the nucleus. When an electron completes a transition from one orbit to another, the energy of the electron changes. Therefore, to ascend to a higher energy level, an amount of energy must be provided and absorbed by the electron. By the same token, to descend to a lower energy level, an amount of energy must be emitted into surrounding space by the electron

and this force originates from the electrostatic force (also known as the Coulomb force FCoulomb ). Recall that for two charges q1 and q2 separated by a distance r, FCoulomb may be expressed as FCoulomb =

1 q1 q2 , 4π 0 r 2

(2.146)

where 0 is once again the permittivity of free space (see Sect. 2.4.1). Note that for computation purposes, 1/4π 0 = 9 ×109 N×m2 /C2 . In the case of the Coulomb force between the nucleus and the electron, FCoulomb may be expressed as FCoulomb =

1 (Ze)e . 4π 0 r 2

(2.147)

Therefore, setting Eqs. (2.145) and (2.147) equal to each other yields Fcentripetal = FCoulomb ⇒

(Ze)e me v 2 = . r 4π 0 r 2

(2.148)

However, according to Bohr’s model of the atom, the electron is only allowed to orbit the nucleus in discrete orbits known as energy levels that have quantized values of angular momentum. These orbits are indexed using the principal quantum number n which is defined as

56

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

me vr =

nh , 2π

n = 1, 2, 3 . . .

(2.149)

Here, n may only attain integer values. Contrast the quantization of electron orbits with the orbits of bodies in the solar system (such as planets) around the Sun, for which such quantization of distance is not seen. From Eqs. (2.148) and (2.149) the following expressions for r are obtained: r=

Ze2 4π 0 me v 2

and

r=

nh . 2π me v

(2.150)

From these expressions for r it can be shown (see Problem 2.12) that r=

n2 h2 0 ∝ n2 , π me Ze2

(2.151)

where it should be noted that the radii of the permitted discrete orbits increases as the square of the principal quantum number. Also note that the smallest orbit occurs when n = 1: this orbit is known as the ground state and it is the lowest stable orbit that may be occupied by an electron. Our next consideration is to determine the total energy of each orbit. Recall from Eq. (2.45) that the total energy of an electron in each orbit is the sum of its kinetic energy and its potential energy. In this situation, the potential energy is the electrostatic potential between the electron and the nucleus, that is V =

−(Ze)e 4π 0 r

(2.152)

and therefore E =K +V =

me v 2 Ze2 − . 2 4π 0 r

(2.153)

Combining Eqs. (2.149), (2.151), and (2.153) yields the following expression for E as a function of n (see Problem 2.13): E=

−me e4 Z 2 1 ∝ 2. 2 2 2 n 80 n h

(2.154)

Here the negative sign indicates that the orbits are bound, with the orbits located closest to the nucleus (that is, the orbits for which n is approximately unity) are the most tightly bound and with the orbits located farthest from the nucleus are the most loosely bound. A crucial facet of the Bohr model of the atom is that to move from one orbit to another (where the principal quantum numbers of the atoms dictate the orbits),

2.5 The Bohr Model of the Atom

57

the energy of an electron must change. Such changes of orbit (and corresponding changes in energy) are termed transitions. This situation may be compared to the arrangement of the rungs of a ladder: the gravitational potential energy of a person is quantized as dictated by the height of a rung above the ground and changes in location of the person (either by ascending to a higher rung or descending to a lower rung) correspond to quantized changes in the gravitational potential energy. Consider two energy levels corresponding to two different electron orbits: one energy level corresponds to a lower energy orbit nlower and the other energy level corresponds to a higher energy orbit nhigher . The energies of these two levels are if difference in energies between the two orbits is Ediff , then for an electron to ascend from the lower orbit to the higher orbit, a photon with energy Ediff = hνdiff (see Eq. (2.111)) must be absorbed to account for the differences in the two energy levels, that is, E(nlower ) + hνdiff = E(nlower ) + Ediff = E(nhigher ).

(2.155)

Conversely, for an electron to descend from the higher orbit to the lower orbit, a photon must be emitted, again to account for the energy difference (that is, Ediff ) between the two levels. E(nhigher ) = E(nlower ) + hνdiff = E(nlower ) + Ediff .

(2.156)

Notice the role played by photons in ensuring the conservation of energy here as the energy of the electron changes as a result of the transition. From Eq. (2.154), the quantities νdiff , nlower , nhigher , E(nhigher ), E(nlower ), and Ediff may all be related together as   1 −me e4 Z 2 1 Ediff = hνdiff = E(nhigher ) − E(nlower ) = − 2 . 80 h2 n2lower nhigher (2.157) From Eq. (2.157) and again recalling Eqs. (2.90) and (2.111), λdiff —the wavelength of a photon with an energy corresponding to Ediff —may be expressed as 1

hνdiff Ediff νdiff = = = = c hc hc



1 hc



−me e4 Z 2 80 h2



1

−

1



. n2higher (2.158) All of the constants in Eq. (2.158) may be gathered to define a new constant R known as the Rydberg constant such that λdiff

R=

me e 4 = 1.10 × 10−7 m−1 . 80 h3 c

n2lower

(2.159)

In the case of the hydrogen atom (Z = 1), Eqs. (2.158) and (2.159) can be combined to yield

58

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity



1 λdiff

=R

1 n2lower

−



1



n2higher

−1

= 1.10 × 10 m 7

1 n2lower

−



1 n2higher

. (2.160)

Example Problem 2.7 Optical astronomers often make observations at a wavelength known as “Hα”: this wavelength is associated with a transition between the nhigher = 3 and nlower = 2 levels of the hydrogen atom. Calculate the wavelength of the photon produced when an electron makes a transition associated with “Hα” and the energy of this photon as well. Solution From Eqs. (2.111) and (2.158), λH α may be computed as 1 λH α

 =R

1 n2lower

−

1 n2higher

 = 1.10×107 m−1



1 1 − 2 22 3



= 1.53×106 m−1 (2.161)

or λH α = 6.54 × 10−9 m = 654 nm

(2.162)

and E=

(6.63 × 10−34 J s)(3 × 108 m/s) hc = 3.04×10−17 J. = λH α 6.54 × 10−9 m

(2.163)

The wavelength of the Hα photon has a prominent red color: many diffuse astronomical sources have been imaged at this wavelength because it is widely encountered in many physical processes associated with this class of source. For example, HII regions—regions where massive stars are forming— contain hydrogen that is mostly in ionized form (see Sect. 6.3.4). When free electrons recombine with protons to create bound atoms of hydrogen, the electrons descend to the ground state. As they descend downward, one possible transition they may undergo is precisely the transition associated with the Hα photon. The name of this transition originates from its membership in the Balmer series of transitions associated with the hydrogen atom: transitions in this series either originate or terminate with the n = 2 level. In standard notation, the Hα transition corresponds to an electron either ascending from the n = 2 level to the n = 3 level or descending from the n = 3 level to the n = 2 level. Likewise, the Hβ transition corresponds to an electron either ascending from the n = 2 level to the n = 4 level or descending from the n = 4 level to the n = 3 level. This notation continues onward through transitions labeled as Hγ , Hδ, and so on.

2.6 Relativity

59

2.6 Relativity The scope of radio astronomy encompasses the study of emission from particles moving at velocities that approach the speed of light: such velocities are termed relativistic. When considering the motions of such particles, it is important to note that simple transformations of velocities (such as the addition of velocities) between different reference frames are no longer valid. For example, consider two observers (denoted as Observers #1 and #2) in two different reference frames where Observer #2 is moving with a velocity v with respect to Observer #1 (see Fig. 2.14). Suppose that each observer has a meter stick and a clock and that each observer measures the lengths of the meter sticks and the passage of time in the other observer’s reference frame. Suppose also that Observer #2 tosses a projectile with a velocity u in direction of motion of its reference frame. If the velocities of both u and v are non-relativistic (such that u c and v c), each observer would measure the length of the meter stick and the passage of a length of time in the other frame to be identical to the same quantities in its own frame. Moreover, Observer #1 would measure the velocity of the projectile to be simply u + v. However, when considering the motion of particles that are traveling with relativistic velocities, simple transformations between reference frames are no longer valid. For example, if both u and v are relativistic, Observer #1 would measure the velocity of the projectile to be not simply u +v. In addition, if Observer

Fig. 2.14 A schematic diagram illustrating the phenomenon of length contraction and time dilation for two observers (denoted as Observer 1 and Observer 2) traveling in space in rocket ships. Observer 1 sees Observer 2 moving toward the right at a velocity v: conversely, Observer 2 would see Observer 1 moving toward the left at the same velocity v. Each observer may measure lengths and time intervals in their own reference frames and compare those measurements with those reported by the other observer. In this figure, Observer 1 would report a length scale in the reference frame of Observer 2 to have contracted from a rest length L0 to a contracted length L as well as a dilation of timescale from Δt0 to Δt in that reference frame. Observer 2 would report the converse of the findings of Observer 1

60

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

#2 shines a flashlight that emits a beam of photons that move with a velocity c in the direction of v, Observer #1 would measure the velocity of the beam to be c instead of v + c. In both scenarios, simple velocity addition is not applicable in the cases of relativistic velocities: in fact, the speed of light is constant and invariant in all reference frames. Albert Einstein (1879–1955) is credited for developing the mathematical formalism known as general relativity that is used to describe the kinematics of objects with relativistic velocities. Consider an object moving with a velocity v. The Lorentz factor γ as a function of v is defined such that 1 1 γ = = , 2 2 1 − v /c 1 − β2

(2.164)

where the definition β = v/c (which is utilized frequently in studies of relativity) has been applied. The Lorentz factor surfaces in descriptions of two crucial phenomena that manifest themselves when considering objects moving with relativistic velocities: these phenomena are known as length contraction and time dilation. Returning to Fig. 2.14, suppose Observer #2 is moving with a relativistic velocity v as measured by Observer #1. If Observers #1 and #2 were to compare the lengths of each other’s metersticks, each observer would claim that the stick of the other observer has contracted in length. Specifically, if the length of a meterstick in the reference frame of one observer is L0 , the measured length L of the other meterstick will be L=

L0 . γ

(2.165)

Note that as v increases, the amount of contraction becomes more and more pronounced and L decreases as indicated by Eq. (2.165). Similarly, if each observer compared how time elapsed on their own watch with the elapsing of time on the watch of the other observer, each would conclude that time was passing more slowly (i.e., it was dilated) in the reference frame of the other observer. Namely, if the length of time in the reference frame of one observer is Δt0 , then the observer measures in the reference frame of the other observer a corresponding time period Δt such that Δt = γ Δt0 .

(2.166)

Note that as v increases, the amount of dilation becomes more and more pronounced and t increases as indicated by Eq. (2.166). Lastly, recalling that Observer #2 launches a projectile with a velocity u in their observing frame and that Observer #2 is moving with a velocity v with respect to Observer #1, then Observer #1 would measure a velocity v  of the projectile to be v =

u+v . 1 + (uv)/c2

(2.167)

2.6 Relativity

61

Notice three salient points in the study of relativistic velocities. Firstly, as v approaches zero (which corresponds to the low velocities typically encountered in day-to-day life), L approaches L0 and Δt approaches Δt0 . For this reason, these phenomena of time dilation and length contraction that are associated intimately with relativistic motion seem very foreign from typical experience. Secondly, note that there is no universal rest frame in which an observer can measure “absolute” time periods and length scales: such observations are always subjective and there is no preferred reference frame in the Universe for making such measurements. Thirdly, length contraction and time dilation complement each other in that length contraction in one frame corresponds to time dilation in another frame, and appealing to both of these phenomena can help explain the occurrence of observed events. Finally, note that the masses of objects increase as the objects are accelerated to velocities comparable to the velocity of light. Remember that no mass can ever be accelerated to the velocity of light and the amount of energy required to accelerate a mass increases dramatically as the velocity of the mass asymptotically approaches the speed of light. If the term the rest mass of a body—the mass the body has when it is at rest—is defined as m0 , then the mass m as a function of γ is m = γ m0 ,

(2.168)

recalling that γ = 0 for a body at rest and therefore in such a situation m = m0 . Similarly, the relativistic momentum p of a particle with a velocity v is p = γ mv,

(2.169)

and finally the energy equivalent E contained by a mass m as E = mc2 = γ m0 c2 .

(2.170)

Note that this equation reduces to the familiar equation E = mc2 when the object is at rest. For this reason, the rest masses of bodies may be expressed as a function of energy: for example, the rest mass m0 of a proton may be expressed as 1.5 × 10−10 J/c2 = 1.5 × 10−3 ergs/c2 = 938 MeV/c2 . Another useful expression for the energy E of a relativistic particle with momentum p and rest mass m0 is E 2 = p 2 c 2 + m2 c 4 .

(2.171)

In this equation, notice that as p for the particle increases, the p2 c2 term dominates over the m2 c4 term and thus in this limit E ≈ pc,

(2.172)

which is similar to the energy of a photon as a function of its momentum as given in Eq. (2.112). Also note that in the case of a particle with a relativistic velocity v (that

62

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

is, where v approaches c), the momentum of a particle given in Eq. (2.169) becomes p = γ mv ≈ γ mc ≈

γ mc2 E ≈ , c c

(2.173)

where Eq. (2.170) has been used. Notice that this result is consistent with Eq. (2.172). In the example below, the above concepts and equations are illustrated, with an emphasis on the equivalence of length contraction measured in one reference frame to time dilation measured in another reference frame.

Example Problem 2.8 A hockey player with a relativistic slap shot fires a puck with a velocity v = 0.75c toward a goal from the red line of an ice rink. The shot is taken from the red line of the rink, which lies 20 m from the goal line. When the player shoots the puck, there are only 10−6 s left before time expires. Will the puck cross the goal line before time expires? Answer this question by considering separately the phenomena of length contraction and time dilation. Finally, if the mass of the puck is 170 g, calculate the mass of the accelerated puck and the equivalent energy contained by the puck. Solution The relevant Lorentz factor for these calculations may be determined from Eq. (2.164), yielding 1 1 1 γ = = =√ = 1.51. 2 1 − 0.5625 1 − v 2 /c2 1 − (0.75c) c2

(2.174)

The length contraction of the distance between the puck and the goal as observed by the puck itself is considered first. Note that because of this length contraction, the distance that the puck needs to travel is reduced. From Eq. (2.165), L may be computed as L=

20 m L0 = = 13.25 m γ 1.51

(2.175)

and the time required for the puck to cover this distance is t=

d 13.25 m = = 6 × 10−8 s, v 0.75 × 3 × 108 m/s

(2.176)

so the puck will cross the goal line well before time expires. Next, consider the complementary time dilation effect: specifically, an observer watching the puck will measure a time dilation in the frame of the puck. Using Eq. (2.166), (continued)

2.7 Cosmic Rays

63

Example Problem 2.8 (continued) the dilated time interval Δt in seconds in the frame of the puck corresponding to the time interval Δt0 = 10−6 s in the time frame of the observer is Δt = γ Δt0 = (1.51)(10−6 s) = 1.51 × 10−6 s

(2.177)

and the distance traveled by the puck in this time period is d = v Δt = 0.75 × 3 × 108 m/s × 1.51 × 10−6 s = 340 m,

(2.178)

so again the puck will reach the goal by a comfortable margin. Once more, note the complementary nature of length contraction and time dilation: length contraction in one frame is the equivalent of time dilation in another frame. From either perspective (either the puck or the observer), the puck will cross the goal line before time expires. The mass m of the relativistic puck is simply (from Eq. (2.168)) m = γ m0 = 1.51 × 0.17 kg = 0.26 kg

(2.179)

and the energy of the puck is (from Eq. (2.170)) E = γ m0 c2 = 1.51 × 0.26 kg × (3 × 108 m/s)2 = 2.3 × 1016 J,

(2.180)

which—for comparison purposes—exceeds by an order of magnitude the energy released by a 1 megaton bomb, that is, 1015 J!

2.7 Cosmic Rays As discussed in future chapters of this text, the detected radio emission from a wide range of astronomical sources is produced by relativistic particles called cosmic rays. Cosmic rays were first detected by German physicist Victor Hess in 1912: using a balloon, Hess measured the intensity of radiation from space and noticed that the intensity increased with increased altitude of the balloon. Decades of subsequent study of cosmic rays have revealed that these objects comprise a steady “rain” of energetic subatomic charged particles from deep space. Examples of particles that are detected as cosmic rays include not just electrons and protons, but also more exotic subatomic particles such as positrons and muons along with the nuclei of elements like carbon, oxygen, and even iron [5].

64

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

Fig. 2.15 A schematic diagram depicting the number of detected cosmic-ray particles as a function of energy. Note that the overall spectrum can be described as three “broken” power laws over an extremely broad range of energies. The two discontinuous breaks in the spectrum are attributed to different classes of sources accelerating the cosmic-ray particles. The break near ≈1015 electron-Volts (the so-called knee energy) is thought to correspond to the maximum energies of cosmic-ray particles accelerated by supernova remnants. Similarly, the break near ≈1018 electron-Volts (the so-called ankle energy) is thought to correspond to the maximum energies of cosmic-ray particles accelerated by a currently unknown class of Galactic source. Cosmic-ray particles with energies above the “ankle” energy are thought to be accelerated by extragalactic sources such as active Galactic nuclei

A schematic diagram of the flux of cosmic-ray particles as a function of energy is presented in Fig. 2.15. Several remarkable features are readily apparent from an inspection of this figure: for example, cosmic-ray particles have been detected over a stunning range of energies (over ten orders of magnitude!) and the slope of the cosmic-ray spectrum appears to be constant over a very broad range of energies. The slope of the spectrum appears to change at two key energies: one energy (denoted as the “knee” energy) corresponds to E ≈ 3 × 1015 eV, while the second energy (denoted as the “ankle” energy) corresponds to E ≈ 1018 eV. Such changes in slope are believed to indicate boundaries on the energy ranges over which particular classes of astronomical sources may accelerate cosmic rays. For example, Galactic supernova remnants—the expanding shells of stellar ejecta and swept-up interstellar material produced by supernova explosions (see Sect. 6.4.3)—

Problems

65

are believed to be the primary accelerants of cosmic-ray particles to energies ranging up to the knee energy. In addition, the most energetic cosmic-ray particles— that is, those particles with energies equal to or exceeding the ankle energy—are believed to be accelerated by active Galactic nuclei, the superluminous “engines” associated with the supermassive blackholes found at the centers of galaxies (see Sect. 7.3). Cosmic-ray particles are believed to be accelerated to such high energies due to multiple transits of the particles across the shock associated with the supernova remnant or the active Galactic nucleus: this acceleration process is known as diffusive shock acceleration and it is discussed in detail in Sect. 6.4.3. The accelerated cosmic-ray particles manifest themselves by a characteristic type of radiation at radio wavelengths known as synchrotron radiation. This emission is produced when relativistic particles are accelerated by ambient magnetic fields and emit radiation: the emission mechanism is discussed in detail in Sect. 3.3.

Problems 2.1 Europa is one of the largest Moons of Jupiter: the semi-major axis of its orbit is approximately 670,900 km. If Jupiter’s diameter is 140,000 km and assuming the planet has a spherical shape, calculate the apparent diameter θ (in degrees) of Jupiter as observed from the surface of Europa. How does this angular extent compare to the apparent angular extent of the Moon as observed from the Earth (approximately 0.5 degrees)? 2.2 By considering the Earth–Sun distance and the Earth–Moon distance along with the respective diameters of the Moon and Sun show that the angular extents of the Sun and the Moon as observed from Earth are indeed comparable and thus the disk of the Moon can completely obscure the Sun during a total solar eclipse. 2.3 Sirius—the brightest star in the sky—is actually a binary star system composed of two stars, namely Sirius A (located at RA(J2000.0) 06h 45m 08.9s , Dec (J2000.0)   −16◦ 42 58.2 ) and Sirius B (located at RA(J2000.0) 06h 45m 09.0s , Dec (J2000.0)   −16◦ 43 06.0 ). Based on these positions of the two stars, calculate the angular separation between Sirius A and Sirius B. 2.4 The Apollo space program of the 1960s and 1970s was successful in both sending astronauts to the Moon and then returning them safely. Consider a module carrying Apollo astronauts of the surface of the Moon. How fast must the escape module of the Apollo astronauts be traveling (in kilometers per second) to escape the gravitational pull of the Moon? The mass and mean radius of the Moon are 7.35 × 1022 kg and 1.74 × 106 m, respectively. 2.5 Comets are known to have particularly elliptical orbits around the Sun. Such orbits have high eccentricities and dramatic differences in orbital velocities at the perihelion and aphelion positions in their orbits. Consider the orbit of Halley’s Comet, which is characterized by a semi-major axis a = 17.8 A.U. and an

66

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

eccentricity e = 0.97. Based on this information, calculate the following properties of the orbit: (a) (b) (c) (d) (e)

The orbital period P . The semi-minor axis b. The perihelion and aphelion distances rperihelion and raphelion , respectively. The perihelion and aphelion velocities vperihelion and vaphelion , respectively. The ratio of the gravitational force between the Sun and the comet when the comet is at rperihelion to the gravitational force between the Sun and the comet when the comet is at raphelion . (f) The total energy of the orbit as computed when the comet is at a distance rperihelion from the Sun, thus verifying that the comet is in a bound orbit. Assume a mass mcomet for Halley’s Comet of 5 × 1014 kg.

2.6 Radio astronomers often make observations at a wavelength of λ = 6 cm. Light at this wavelength falls within one of the “protected” wavelength ranges over which terrestrial communications are banned, allowing astronomical observations to be conducted with minimal interference. For a photon with this wavelength, compute the following: (a) (b) (c) (d) (c)

The frequency ν in Hertz, The wavenumber k in m−1 , The angular frequency ω in radians s−1 , The energy E in Joules, and The momentum p in kg m s−1 .

2.7 Consider an electric wave propagating in the positive z-direction with components Ex = Ey = 10−5 V/m cos (kz − ω t) and Ez = 0. Calculate the four Stokes Parameters I , Q, U , and V for this wave, its degree of total polarization pd and describe its polarization. 2.8 Calculate the observed shift in wavelength Δλ in nanometers of the Hα line (which has a rest wavelength λ0 = 656.3 nm) in the spectra of the following stars. (a) A star moving with a radial velocity v = 40 km/s. Is this a blueshift or a redshift? (b) A star moving with a radial velocity v = −70 km/s. Is this a blueshift or a redshift? 2.9 A blazar is a galaxy with a luminous nuclei: these objects belong to the class of galaxies known as active Galactic nuclei (AGNs), which feature nuclei of exceptional luminosity.2 Suppose a blazar is seen to be receding from an observer with an apparent recessional velocity v = 0.95c. As discussed in future chapters, such large apparent recessional velocities seen in the spectra of distant galaxies do not originate from intrinsic rapid motions of these objects but instead from the

2 Blazars

and active Galactic nuclei are discussed in more detail in Chap. 7.

References

67

expansion of the Universe, which is best observed by analyzing spectra of very distant galaxies. (a) What is the associated redshift z of this blazar? (b) Predict the wavelength λ in nanometers at which the Hα line (which has a rest wavelength λ0 = 656.3 nm) will be detected in the spectrum of the blazar. 2.10 Derive Eq. (2.144) from Eq. (2.143). 2.11 Derive Equation (2.151) from the relations given for r in Eq. (2.150). 2.12 Derive Equation (2.154) from Eqs. (2.149), (2.151) and (2.153). 2.13 In his spare time, farmer Old Man Johnson practices pole-vaulting at relativistic velocities. A chicken on his farm watches as Johnson—training for an upcoming meet—runs at a relativistic velocity v with his pole (with a length of 5 m) through a small shed that is 3 m in length. (a) What are the corresponding values of γ and v (in terms of c and m/s) such that the chicken sees the pole contract so that it fits inside the shed? (b) The chicken has its own watch and makes a comparison with the passage of time on Johnson’s watch (the chicken is an amateur astrophysicist in its spare time). If Δt0 = 1 second passes on Johnson’s watch, what is the corresponding time interval Δt observed on the chicken’s watch? (c) A horsefly is walking along the pole with a velocity u = 4c/5 as measured in Johnson’s reference frame. The direction of the motion of the horsefly is the same as the direction of motion of the pole. Calculate the velocity of the horsefly as observed by the chicken in units of c. 2.14 The “Oh-My-God” particle—detected in 1991 by the “Fly’s Eye” CosmicRay Detector in Utah—was the most energetic cosmic-ray particle detected up to that time. Its energy was estimated to be 3.2 × 1020 eV or 51 Joules. Recall that such extremely energetic cosmic rays are believed to be accelerated to such high energies by active Galactic nuclei, which are described in more detail in Sect. 7.3. Assuming that this particle was an extremely energetic proton, calculate (a) the corresponding Lorentz factor γ of this particle and (b) the velocity in meters per second that a tennis ball would need to have to possess the same kinetic energy as this particle. Assume a mass of 57 grams for the tennis ball and a rest mass of the proton of 9.38 × 108 eV/c2 or 1.67 × 10−27 kg.

References 1. D. Fleisch, A Student’s Guide to Maxwell’s Equations, 1st edn. (Cambridge University, New York, 2008) 2. J.D. Kraus, Radio Astronomy, 2nd edn. (Cygnus-Quasar Books, New York, 1988), pp. 4-1–4-21

68

2 Angles, Gravity, Light, the Bohr Model of the Atom and Relativity

3. C. Heiles, S. Stanimirovi´c, D.R. Altschuler, P.F. Goldsmith, C.J. Salter, Single-dish radio astronomy: techniques and applications, in Astronomical Society of the Pacific Conference Series, vol. 278, 1st edn. (Astronomical Society of the Pacific, California, 2001/2002), pp. 131–152 4. M. Zeilik, S.A. Gregory, Introductory Astronomy and Astrophysics, 4th edn. (Brooks/Cole, New York, 1998), pp. 160–161 5. B.W. Carroll, D.A., Ostlie, An Introduction to Modern Astrophysics, 2nd edn. (Pearson/AddisonWesley, New York, 2006), pp. 550–553

Chapter 3

Emission Mechanisms: Blackbody Radiation, An Introduction to Radiative Transfer, Synchrotron Radiation, Thermal Bremsstrahlung, and Molecular Rotational Transitions

3.1 Blackbody Radiation Any object with a temperature above zero Kelvin will emit thermal radiation: during the late nineteenth century, physicists grappled with modeling properly the spectra of thermal radiation produced by bodies as function of their temperatures. The particular model of the light-emitting body used by these researchers for modeling purposes is known as a blackbody. Blackbodies are idealized constructs of objects that absorb all radiation that is incident upon them and re-radiate away all of that radiation. Such objects would therefore neither produce their own energy nor store any of it: their names originate from the phenomenon where black materials would absorb all incident radiation upon them. The discussion presented here is patterned after the treatment given in [1]. Early attempts to model thermal radiation relied upon classical electromagnetic wave theory of light. For example, Lord Rayleigh considered a hot oven with an interior cavity at a certain temperature T that contained blackbody radiation: the distance between the walls of the oven was L. Rayleigh modeled the thermal radiation within the oven as standing waves of permitted wavelengths (namely L/2, L, 2L, and so on), similar to the permitted wavelengths of standing waves for a string of length L that is fixed at both ends. In this construct, wavelengths of ever-decreasing wavelength were permitted to exist within the oven. Classical physics predicted that the amount of energy possessed by each light wave would be kB T , where kB is Boltzmann’s constant (defined as kB = 1.38 × 10−23 J/K = 1.38 × 10−16 erg/K). From here, Rayleigh predicted that the spectrum of light emitted by the blackbody as a function of wavelength λ would correspond to a function known as the radiant energy Bλ (λ, T )), which may be expressed as Bλ (λ, T ) =

2ckB T . λ4

© Springer International Publishing Switzerland 2020 T. G. Pannuti, The Physical Processes and Observing Techniques of Radio Astronomy, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-16982-8_3

(3.1)

69

70

3 Emission Mechanisms

Fig. 3.1 The emission profiles (corresponding to Planck functions) for five blackbodies with temperatures of 100, 500, 1000, 3000, and 10,000 K. Two additional profiles for blackbodies with temperatures of 300 and 5777 K (approximating the blackbody emission from the Earth and the Sun, respectively) have also been included. Notice that as the temperature of the blackbodies increase, the wavelengths of peak emission decrease (in accordance with Wien’s Displacement Law) and the total amount of emission from the blackbody increases (according to the Stefan– Boltzmann Law)

Here, c is once again the speed of light. This equation is known as the Rayleigh– Jeans approximation: while this equation can model adequately the emission from objects at long wavelengths (such as those in the radio domain), it fails to properly predict the emission seen from blackbodies at short wavelengths. This result can be seen by inspection of Eq. (3.1), where the predicted amount of emitted thermal radiation would grow without bound as λ approaches zero. This is not what is in fact seen in these spectra, where a peak wavelength is seen where a maximum amount of emission is produced (this particular wavelength is discussed in more detail below) and Bλ decreases on either side of this wavelength. Examples of blackbody spectra produced by blackbodies at different temperatures are given in Fig. 3.1. A successful model of the emission from blackbodies was produced by Max Planck, who based his model on assuming that the energy contained by the wavelengths of light could not attain an arbitrary amount of energy but instead could only possess quantized energy values known as quanta that correspond to integral multiples of a minimum amount of energy. This concept of a quantum of energy was encountered previously in the discussion of light as a photon (that is, a

3.1 Blackbody Radiation

71

bundle of energy) where the energy of the photon depended on the frequency or the inverse of the wavelength of the photon (see Sect. 2.4.2). The function produced by Planck that successfully describes blackbody radiation is now known as the Planck function and thus Bλ dλ (λ, T ) (that is, the differential amount of energy emitted over the wavelength range from λ to λ+dλ) may be expressed as Bλ (λ, T ) dλ =

dλ 2hc2 , λ5 ehc/λkB T − 1

(3.2)

where h is again Planck’s constant. Alternatively, the Planck function may be expressed as Bν (ν, T ) when frequency ν is considered instead of λ. To transform from Bλ (λ, T ) dλ to Bν (ν, T ) dν (that is, the differential amount of energy emitted over the frequency range from ν to ν + dν), the relation dν = −

c dλ λ2

(3.3)

must be applied and therefore the expression for Bν (ν, T ) dν becomes Bν (ν, T ) dν =

dν 2hν 3 . c2 ehν/kB T − 1

(3.4)

In obtaining this expression, the absolute value of Eq. (3.3) has been taken.

Example Problem 3.1 Derive the Rayleigh–Jeans approximation form of Bλ (λ, T ) given in Eq. (3.1) using in the limit of long wavelengths. Considering the Planck function expressed instead in terms of Bν (ν, T ), derive an alternative form of the Rayleigh–Jeans approximation in the limit of low frequencies. Solution Recall that the Taylor expansion of the expression ex may be written as ex = 1 + x +

x3 x2 + + ... 2! 3!

(3.5)

and therefore the exponential term given in the denominator in Eq. (3.2) becomes ehc/λkB T ≈ 1 +

hc hc −→ ehc/λkB T − 1 ≈ . λkB T λkB T

(3.6)

This yields (continued)

72

3 Emission Mechanisms

Example Problem 3.1 (continued)



1 2hc2 2hc2 Bλ (λ, T ) = 5 hc/λk T ≈ 5 B −1 λ e λ

λkB T hc

 =

2ckB T . λ4

(3.7)

The Taylor approximation can be applied in a similar way to the expression for Bν (ν, T ). In this manner, the exponential term in Eq. (3.4) becomes ehν/kB T ≈ 1 +

hν hν −→ ehν/kB T − 1 ≈ , kB T kB T

(3.8)

therefore Bν (ν, T ) =

2hν 3 1 2hν 3 ≈ 2 hν/k T 2 B −1 c e c



kB T hν

 =

2kB T ν 2 . c2

(3.9)

Inspection of plots of the Planck function for blackbodies (such as the ones presented in Fig. 3.1) indicate that a peak wavelength λmax clearly exists: specifically, at this wavelength, the body emits the most light. What is the relationship between the temperature T of an object that emits as a blackbody and λmax ? We can determine this relationship by differentiating Eq. (3.2) with respect to λ and setting the resulting equation equal to zero (recall that differentiating a function, setting its derivative to zero and solving the resulting equation for the differentiated variable determines the maxima and minima of the function), we obtain ∂ 2hc2 2hc2 hc ehc/λkB T 1 1 10hc2 = . − hc/λk T hc/λk 5 5 2 hc/λk T 2 6 B BT − 1 B ∂λ λ e −1 λ λ kB T (e − 1) λ e (3.10) Setting this equation equal to zero yields 2hc2



λ6 (ehc/λkB T − 1)

 hcehc/λkB T − 5 = 0, λkB T (ehc/λkB T − 1)

(3.11)

and by making the substitution x=

hc , λkB T

(3.12)

Equation (3.11) may be represented as xex − 5 = 0, (ex − 1)

(3.13)

3.1 Blackbody Radiation

73

which can be solved numerically to yield x ≈ 4.97. Setting λ = λmax , Eq. (3.12) becomes λmax =

hc , xkB T

(3.14)

but h, c, x, and kB are all constants and this equation becomes λmax T = 2.90 × 10−3 m · K,

(3.15)

which is known as Wien’s Displacement Law or Wien’s Law. Notice that because the product of λmax and T is a constant, a reciprocal relationship between these quantities exists. For example, note that as the wavelength of λmax increases, the temperature T decreases. This result reflects a standard rule of astrophysics where observations of hotter objects are conducted at shorter wavelengths with the goal of detecting a maximum amount of emission from them. For example, hot sources (like remnants of supernova explosions) attain temperatures of ∼106 K and emit copious amounts of thermal radiation at X-ray wavelengths while cool interstellar dust grains attain temperatures of ∼102 K and instead emit large amounts of thermal radiation at infrared wavelengths.

Example Problem 3.2 Determine the corresponding value of λmax for a blackbody with a temperature T = 300 K. Solution From Eq. (3.15), λmax may be computed as λmax =

2.90 × 10−3 m · K 2.90 × 10−3 m · K = = 9.67 × 10−6 m. T 300 K (3.16)

This temperature corresponds roughly to the surface temperature of the Earth as well as the temperature of human bodies, while this wavelength corresponds to the infrared portion of the electromagnetic spectrum.

In the analysis of emission from blackbodies, a topic of interest is determining a quantity known as the flux F . This quantity corresponds to the total amount of power emitted per unit surface area per unit wavelength or frequency emitted at a wavelength λ or frequency ν by a blackbody with a temperature T . Flux may be determined by integrating Bλ (λ, T ) with respect to λ over all wavelengths or by integrating Bν (ν, T ) with respect to ν over all frequencies. In both cases, the integration is also performed over all solid angle dΩ as well. Therefore, in terms of integrating Bν (ν, T ) with respect to ν over all frequencies and over all solid angles, flux may be expressed as

74

3 Emission Mechanisms

 F =



 cos θ dΩ =

Bν (ν, T ) dν





2π

Bν (ν, T ) dν

π/2

sin θ cos θ dθ.

dφ 0

0

(3.17) In this equation, dΩ has been expressed in terms of θ and φ (see Eq. (2.12)). Also note that the cosine term in this equation originates from the intensity of the emission observed along the surface of the blackbody (which is assumed to be spherical) will be the actual intensity multiplied by the cosine of θ . Finally, note that the bounds on the integration for θ originate from assuming that the surface of the blackbody is emitting out into a half sphere. This result may be contrasted with the integration performed over all solid angles as given in Eq. (2.13), where the integration is performed over different bounds for θ and φ to correspond to the entire sphere (rather than half of a sphere, as is the case here). Performing just the integration given in Eq. (3.17) over φ and θ yields 

π/2

  1 = π, dφ cos θ sin θ dθ = = (2π ) 2 φ=0 θ=0 0 (3.18) and therefore the expression for F given in Eq. (3.17) simplifies to 

φ=2π



θ=π/2

(φ)2π 0

 F =π

sin2 θ 2

ν=∞

Bν (ν, T ) dν.

(3.19)

ν=0

Substituting the explicit form of Bnu (ν, T ) given in Eq. (3.4) into this equation and performing the integration yields (see Problem 3.2) F =

2hπ c2



ν=∞

ν=0

2π 5 kB4 T 4 ν 3 dν = = σSB T 4 . −1 15h3 c2

ehν/kB T

(3.20)

Equation (3.20) is known as the Stefan–Boltzmann Law, and σSB is known as the Stefan–Boltzmann constant, which is defined as σSB =

2π 5 kB4 = 5.67 × 10−8 W m−2 K−4 . 15h3 c2

(3.21)

A crucial property of a blackbody is its luminosity L, which corresponds to the total amount of radiant energy emitted by the blackbody integrated over the entire surface area of the blackbody and over all directions of emitted radiation. As discussed for the remainder of this text, the luminosity of an object and the total power emitted by the object correspond to the same quantity. A critical assumption made in determining L is that the blackbody emits radiation isotropically, that is uniformly in all directions. The luminosity of a spherical blackbody with radius R may be envisioned as the product of the amount of light emitted from the surface of the blackbody per unit surface area (a quantity known as flux F ) and the total surface area of the sphere (which is simply 4πR 2 , where R is the radius of the blackbody).

3.1 Blackbody Radiation

75

In other words, L may be expressed as L = (Total surface area) × (Energy emitted per unit surface area) = (Total surface area) × (Flux).

(3.22)

From Eq. (2.14), the first quantity in this product is known to be 4πR 2 . The second quantity—that is, the flux—may be determined by integrating Bλ (λ, T ) over all wavelengths or Bν (ν, T ) over all frequencies. Recalling Eqs. (3.2) and (3.4) and performing this integration yields  F =

λ=∞

λ=0

 Bλ (λ, T ) dλ =

λ=∞

λ=0

2hc2 dλ dλ = σSB T 4 , hc/λk 5 BT − 1 λ e

(3.23)

Equation (3.23) is the well-known Stefan–Boltzmann Law, which describes that total amount of energy emitted per unit area from the surface of a blackbody with a temperature T . From Eqs. (3.22) and (3.23), the luminosity L of a spherical blackbody with a radius R and surface temperature T may be expressed as L = 4π R 2 σSB T 4 .

(3.24)

A related quantity is the radiation energy density urad corresponding to a bath of photons produced by a source that emits radiation as a blackbody with a temperature T . This quantity is defined as urad =

4σSB T 4 . c

(3.25)

Finally, the radiation pressure P corresponds to the pressure exerted by a volume of photons on hypothetical enclosing walls around the volume. This pressure originates from the momentum associated with individual photons (see Eq. (2.112)) and may be expressed as P =

urad . 3

(3.26)

Notice that while the expressions for urad and P differ solely by a scalar, these quantities are typically expressed in different units (reflecting the contexts in which they are defined): the former is expressed in units of energy per volume (such as J/m3 ) while the latter is expressed in units of force per area (such as N/m2 ). Several crucial conclusions about blackbody radiation may be gleaned from inspection of the Stefan–Boltzmann equation. Firstly, notice that the total amount of energy emitted by the blackbody increases dramatically as its temperature increases. Imagining two bodies of equal areas but different temperatures (say the surface temperature of one body is twice the surface temperature of the other body), then the hotter body will emit 24 = 16 times more energy than the cooler body.

76

3 Emission Mechanisms

Secondly, notice that the total luminosity of the blackbody depends on both its surface area and its surface temperature. For this reason, evolved red supergiant stars with low surface temperatures coupled and prodigious surface areas can attain high luminosities while white dwarf stars with high surface temperatures and tiny surface areas are low-luminosity objects. In a similar vein, the radiation energy density and the radiation pressure associated with a volume of a blackbody emitting source both increase dramatically with increasing temperature through an identical T 4 dependence. Lastly, it is useful to provide a short description of the difference between flux F and flux density S as these terms are commonly encountered. In addition to the definition of flux given in Eq. (3.20) from the surface of a blackbody, flux (denoted again with F ) may also describe the total amount of light received in units of Watts per square meter by a detector located at a distance r from a source (possibly a blackbody) that is emitting radiation with a luminosity L. In this context, F may be expressed as F =

L . 4π r 2

(3.27)

The relationship between flux and luminosity is illustrated in Fig. 3.2. Notice the inverse-square dependence of F on L, which is similar to the inverse-square dependence seen for the gravitational force and the electrostatic force. Notice also the presence of the 4π term, which reflects the assumption that the radiation from the source is emitted isotropically (recall that there is a total of 4π steradians of solid angle for a sphere). In this context, the luminosity from the source may be for a particular wavelength or frequency of radiation (thus described as a monochromatic luminosity) and therefore the measured flux may be described as monochromatic as well. Alternatively, L may correspond to the total luminosity of the source as

Fig. 3.2 A schematic diagram illustrating the concept of flux density and luminosity. A source emits a total power (that is, a luminosity) isotropically in all directions (that is, over all 4π steradians) and an observer located a distance r from the source measures a flux density S from the sources

3.1 Blackbody Radiation

77

integrated over a certain range of wavelength or frequency or radiation: in such a situation, F corresponds to the detected flux over the same range of wavelength or frequency. In contrast, the term flux density takes into account in an explicit manner the frequency dependence of the detected radiation. Note that flux density is expressed in units of Watts per square meter per Hertz and a commonly encountered unit of flux density is the Jansky (Jy), where 1 Jy = 10−26 W m−2 Hz−1 . Specifically, if the detector described above is sensitive to radiation from the source over a specific bandwidth Δν, then the detected flux density may be expressed as S=

L . 4π r 2 Δν

(3.28)

It is often assumed that the flux of radiation detected from the source is independent of particular frequency within the bandwidth: such an assumption is usually quite valid for small bandwidth values.

Example Problem 3.3 The surface temperature of the Sun is T = 5778 K and its radius is R = 6.96 × 108 m. Based on this information, calculate the wavelength λmax at which the Sun emits the most radiation in meters and nanometers and the luminosity L 1 for the Sun. Also calculate the radiation energy density urad of Joules per m3 contained in a bath of photons produced by the Sun and the radiation pressure Prad exerted by the photons on the side of a cubic volume with a length of 1 m on each side. Finally, assuming a Sun– Earth distance of 1 A.U. = 1.49 × 1011 m, calculate the flux of energy arriving from the Sun per second per square meter on the surface of the Earth and the corresponding flux density of energy over a bandwidth of 1 MHz (assuming that the flux of energy is constant over the entire bandwidth).

Solution From Eq. (3.15), λmax may be computed as λmax =

2.90 × 10−3 m · K =5.01 × 10−7 m = 501 nm. 5778 K

(3.29)

Notice that a photon with a wavelength of 501 nm will fall in the green portion of the visible spectrum but the Sun is perceived by human eyes to be yellow in color. The explanation of this discrepancy lies in part with effects associated with the Earth’s atmosphere: molecules in the Earth’s atmosphere are more effective at scattering short wavelength light compared to long wavelength (continued) 1 Note that the subscript “” is commonly used to denote a property of the Sun, such as its luminosity.

78

3 Emission Mechanisms

light (the phenomenon known as Rayleigh scattering, where scattering of light is proportional to ∼λ−4 ). This phenomenon manifests itself in both the blue skies seen on Earth during the day and red skies seen toward the horizons during dawn and dusk, as well as the Moon attaining a red color during a total lunar eclipse. Thus, short wavelength light from the Sun is scattered more by the atmosphere more than long wavelength light and the Sun appears to be more yellow than green. Strictly speaking, however, if the Sun were to be observed from space, its color would not be purely green but instead whitish because the Sun emits light at all wavelengths and does not emit as an idealized blackbody. It is emphasized here that in astronomy, blackbodies are useful approximations but they do not always characterize properly the emission from sources. The luminosity L can be calculated from Eq. (3.24) as L = 4π(6.96×108 m)2 (5.67×10−8 W m−2 K−4 )(5778 K)4 = 3.84×1026 W, (3.30) while urad may be calculated from Eq. (3.25) as urad =

4(5.67 × 10−8 W m−2 K−4 )(5778 K)4 = 0.84 J/m3 , (3 × 108 m/s)

(3.31)

and from Eq. (3.26), Prad may be computed as Prad =

4.74 × 1015 J/m3 urad = = 1.58 × 1013 N/m2 . 3 3

(3.32)

The flux F may be calculated from Eq. (3.27) F as F =

3.84 × 1026 W L = = 1.38 × 103 W/m2 . 4π r 2 4π(1.49 × 1011 m)2

(3.33)

This value calculated in Eq. (3.33) is also known as the solar constant and it is clearly important to such applications as solar panel development (for both Earth-based panels as well as panels on orbiting spacecraft). It describes the total amount of energy received from the Sun per square meter on the Earth: a formal calculation that takes into account the attenuating effects of Earth’s atmosphere would reduce the computed quantity slightly. Lastly, the flux density S may be calculated from Eq. (3.28) as S=

L 3.84 × 1026 W = = 1.38 × 10−3 W/m2 Hz. 4π r 2 Δν 4π(1.49 × 1011 m)2 (106 Hz) (3.34)

3.2 An Introduction to Radiative Transfer

79

3.2 An Introduction to Radiative Transfer When light passes through a medium as it travels from a distant source to an observer, the medium may dramatically attenuate the amount of light that eventually reaches the observer. In the case of objects located within our own galaxy, the interstellar medium—the diffuse gas and dust grains that fill the space between the stars—may have a considerable effect on light traveling from the source toward observers located on Earth. In the case of light at optical wavelengths, the interstellar medium serves to preferentially scatter short wavelength light (that is, blue light) from sources more than red light (a phenomenon known as interstellar reddening) as well as diminish the overall intensity of light from a source (a phenomenon known as interstellar extinction). Therefore, a source may appear to be redder and fainter than it really is due to the presence of the intervening interstellar medium: thus, in observational astronomy the effect of the interstellar medium along the line of sight to a source of interest must always be taken into account if we wish to obtain the most accurate understanding possible of the true nature of the source. The study of the passage of light through a medium—and how this medium may affect this passage of light—is known as radiative transfer. Radiative transfer is an extremely broad topic to which whole volumes have been devoted: a particularly important application of this field includes the modeling of the transfer of radiation in stellar interiors and stellar atmospheres. Only a basic introduction to radiative transfer is given in the present work: the reader is referred to other references to obtain a more thorough treatment of the subject. A broad summary—based on content presented elsewhere [2]—is presented here. The specific intensity—or, for the remainder of this text, simply the intensity Iλ —is defined as the amount of energy Eλ transmitted by a beam of photons each with a wavelength between λ and λ + dλ in a time interval dt passing through a surface area dA at an angle θ into a cone of solid angle dΩ. In this case, θ measures the direction perpendicular to the surface, therefore dA cos θ is the projection of dA onto a plane perpendicular to the direction in which the beam is traveling. Thus, Iλ may be expressed as Iλ =

Eλ dλ , dλ dt dA cosθ dΩ

(3.35)

and the mean intensity < Iλ > can be obtained by integrating Iλ over all directions in which radiation is emitted. We note that the units of Iλ —like the units for the Planck function—are W m−3 sr−1 . Adopting a spherical coordinate system and dividing by 4π steradians—the solid angle subtended for an entire sphere (see Eq. (2.13))—yields 1 < Iλ >= 4π



1 Iλ dΩ = 4π



φ=2π



θ=2π

Iλ sin θ dθ dφ, φ=0

θ=0

(3.36)

80

3 Emission Mechanisms

and in the case where the radiation is emitted isotropically, this expression simplifies to (3.37)

< Iλ >= Iλ .

An example of a function of intensity is the Planck function (see Eq. (3.2)) for the case of emission from a blackbody, that is, (3.38)

< Iλ >= Bλ .

Note that the radiation emitted by a blackbody is assumed to be isotropic. Continuing the discussion about radiative transfer, consider the following scenario: imagine a monochromatic beam of light with a wavelength λ (such as the type of light beam produced by a coherent light source like a laser) with intensity Iλ,0 passing through a medium with a path length l. An observer located at the end of the path measures a resultant intensity Iλ (see Fig. 3.3). The relationship between Iλ,0 and Iλ may be expressed as Iλ = Iλ,0 e−τλ ,

(3.39)

or alternatively  τλ = −ln

Iλ Iλ,0

 ,

(3.40)

where τλ is defined to be the optical depth of the medium. Note that τλ is a dimensionless quantity and that in the case where τλ = 0, the medium is completely transparent to the beam of radiation. Such a scenario corresponds to the idealized situation of light passing through a vacuum on its path to an observer. Astronomers may often characterize a medium as optically thick if the medium is essentially opaque to light at that wavelength (τλ  1); alternatively, the medium may be described as optically thin if it is essentially transparent to light at that wavelength (τλ 1). The λ-subscripts emphasize that the optical depth of a particular medium

Fig. 3.3 A schematic diagram illustrating the phenomenon of radiative transfer. Incident radiation with intensity Iλ,0 passes through a medium symbolized by a cylinder: the intensity of the radiation as measured after passing through the medium is Iλ . The path length of the medium traversed by the radiation is s and a differential element of the path length is ds. The medium is characterized by an absorption coefficient or opacity κλ

3.2 An Introduction to Radiative Transfer

81

in fact is often a function of the wavelength of the incident radiation. Certainly, a medium can be essentially transparent to radiation at one wavelength but essentially opaque to radiation at another wavelength. For example, a thick fog is opaque to light at optical wavelengths but simultaneously transparent to light at radio wavelengths.

Example Problem 3.4 Two beams possessing photons of two different wavelengths (namely λ1 and λ2 ) are directed through a medium in an experiment to determine the optical depth of the medium at these two wavelengths. The initial intensities of both beams as they enter the medium are Iλ1 ,0 = Iλ2 ,0 = 100 W m−3 sr−1 and the measured intensities of the beams as they exit the medium are Iλ1 = 95 W m−3 sr−1 and Iλ2 = 0.01 W m−3 sr−1 . Based on this information, calculate the optical depths τλ1 and τλ2 and determine whether the medium is optically thin or optically thick at these wavelengths. Solution From Eq. (3.40), τλ1 and τλ2 may be calculated as  τλ1 = − ln

95 W m−3 sr−1



= 0.05 and 100 W m−3 sr−1   0.01 W m−3 sr−1 τλ2 = − ln = 9.21. 100 W m−3 sr−1

(3.41)

Thus, by the definitions given above, the medium is optically thin to the photons with wavelengths of λ1 and optically thick to the photons with wavelengths of λ2 .

The optical depth may be defined in either integral or differential form, that is, as  τλ =

s

κλ ρ ds

or

dτλ = −κλ ρ ds,

(3.42)

0

where κλ is the absorption coefficient or the opacity of the medium (which may also be a function of the wavelength λ of the radiation), ρ is the mass density and s is the total path length traveled by the radiation through the medium (ds is a differential element of that path length). The opacity of a material may be described as the cross-section area per unit mass of the medium for absorbing photons of wavelength λ: the units of opacity are therefore m2 kg−1 . For the purposes of the treatment presented here, the units of mass density and path length are the conventional kg m−3 and m, respectively. Frequently, it is assumed that both κλ and ρ are constant throughout the entire path length, thus making any needed integrations or differentiations simpler. From Eqs. (3.39) and (3.42), a differential

82

3 Emission Mechanisms

equation that relates Iλ , τλ , κλ , ρ, and s may be expressed as dIλ = −κλ ρIλ ds = −Iλ dτλ ,

(3.43)

where dIλ is the differential change in intensity along a differential element ds of the path length s. This equation is of course Eq. (3.39) in differential form. In the discussion presented so far, the only scenario considered has been the case where the intensity of radiation may be either diminished or unchanged as it passes through a medium as it heads toward an observer. In reality, the situation may be more complicated; that is, it is entirely possible that the medium itself is producing radiation and that this emitted radiation augments the beam of radiation headed toward the observer. Alternatively, just as a medium may scatter radiation out of the beam due to the natural scattering properties of the medium, radiation may also be scattered into the beam. In observational astronomy, this scattering phenomenon corresponds to the phenomena of interstellar reddening and extinction mentioned previously in this chapter. In analogy to Eq. (3.43), the differential change in intensity dIλ may be expressed as dIλ = jλ ρ ds,

(3.44)

where jλ is the emission coefficient of the medium. The units of the emission coefficient are m s−3 sr−1 . Note that in this equation, the sign of dIλ is positive (corresponding to an increase in intensity) while in Eq. (3.43) the sign of dIλ is negative (corresponding to a decrease in intensity). Clearly, a complete understanding of the change of intensity of radiation passing through a medium must take into account both increases and decreases in intensity. Therefore, combining Eqs. (3.43) and (3.44) yields dIλ = −κν ρIλ ds + jλ ρ ds.

(3.45)

In analyzing the propagation of radiation through a medium, the ratio of the rates of emission and absorption of radiation are crucially important. Equation (3.45) may be recast into a form that explicitly includes this ratio by simply dividing the whole equation by -κλ ρds, yielding −

1 dIλ jλ = Iλ − . κλ ρ ds κλ

(3.46)

Defining the source function Sλ as the ratio jλ /κλ , Eq. (3.46) becomes −

1 dIλ = Iλ − Sλ . κλ ρ ds

(3.47)

3.2 An Introduction to Radiative Transfer

83

This equation is best known as the equation of radiative transfer or the transfer equation. Note that the units of the source function are identical to the units of intensity: that is, the units of both quantities are W m−3 sr−1 . The significance of Sλ is that it represents the net change in the number of photons in the beam of radiation: photons may be lost out of the beam due to scattering or absorption while other photons may be scattered into the beam. Note that if Iλ is greater than Sλ (that is, dIλ /ds > 0), the intensity decreases with distance; similarly, if Iλ is less than Sλ (that is, dIλ /ds < 0), the intensity increases with distance. These results may be interpreted to mean that Iλ tends to approach the local value of Sλ , assuming that Sλ does not vary strongly with distance. The particular case of the source function as produced by a blackbody radiator can be examined by imagining a box of gas particles: the gas itself is optically thick and it is maintained at a constant temperature. The particles that make up the gas and the box form a closed system in that there is no net flow of energy through the box to the outside world or between the gas particles and the ambient radiation field (which is itself blackbody radiation). If the particles and the photons of the ambient radiation field are indeed in equilibrium, every process of photon absorption are balanced by an inverse process of emission. Such a condition is known as thermodynamic equilibrium, and such a condition is often assumed (as a convenient mathematical approximation) for astronomical sources that are not changing temperature rapidly. In these cases, the intensity of the radiation may be equated with the Planck function (that is, Iλ = Bλ ) and—because the intensity is constant throughout the box—dIλ /ds = 0, so Sλ = Iλ . It can therefore be concluded that in the case of thermodynamic equilibrium. The source function is equivalent to the Planck function (that is, Sλ = Bλ ). In the case of thermodynamic equilibrium, Kirchoff’s Law relates jλ , κλ , and Bλ along with Sλ as (from Eq. (3.47)) Sλ = Bλ =

jλ . κλ

(3.48)

The case of radiation emitted by a cloud capable of emitting and absorbing radiation is now discussed. This case is applicable to the scenario where we are considering thermal emission from a cloud itself and this emission is escaping out of the cloud, but the effects of attenuation within the cloud itself need to be considered. It is assumed that radiation from background sources passing through the cloud can be safely ignored. Assuming again that the cloud is in thermodynamic equilibrium and that the optical depth through the entire path length of the cloud is τ , then Eq. (3.48) becomes Bλ,observed =

jλ (1 − e−τ ) = Bλ (1 − e−τ ), κλ

(3.49)

where Bλ is the intrinsic source function of the cloud and Bλ,observed is the apparent or observed source function of the cloud. In terms of temperature, this equation may be expressed as

84

3 Emission Mechanisms

TB = T (1 − e−τ ),

(3.50)

where T is the intrinsic temperature of the cloud and TB is the observed temperature of the cloud (also known as the brightness temperature—the concept of a brightness temperature is discussed in more detail in Sect. 4.4).

3.3 Synchrotron Radiation Synchrotron radiation is one of the most commonly encountered emission processes encountered in modern astrophysics and is particularly prominent in radio astronomy. This type of radiation originates from accelerated particles that are gyrating in magnetic fields: according to classical physics, charged particles emit radiation as they are accelerated. As shown below, the energy of the emitted photon depends on both the energy of the emitting particle and the strength of the magnetic field. In turn, magnetic fields are ubiquitous entities produced by variations in electric fields (such as charged particles in motion): these fields are intertwined with virtually every type of astronomical source and phenomenon investigated in modern astrophysics. In addition, particularly energetic particles (such as those accelerated to relativistic energies) emit synchrotron radiation and therefore the study of this emission from these particles serves to probe how particles are accelerated to such high energies.2 As described in future sections, examples of discrete sources that emit synchrotron radiation at radio wavelengths include pulsars, supernova remnants, and active Galactic nuclei. Diffuse synchrotron radiation emission is detected from the disks of galaxies and the intergalactic medium within galaxy clusters. Consider a charged particle with a charge q moving with a velocity v through a magnetic field with magnitude B: the component of the velocity of the charge perpendicular to the direction of B is v⊥ (see Fig. 3.4). In this scenario, the velocity of the charge is non-relativistic. The force FB (known as the Lorentz Force) that acts on the charged particle as it passes through the magnetic field is FB = qv × B = qv⊥ B,

(3.51)

while the centripetal force Fcentripetal acting on the charge is (from Eq. (2.48)) Fcentripetal =

2 mv⊥ . r

(3.52)

Setting Eqs. (3.51) and (3.52) equal to each other and solving for r, we obtain

2 An

example of one process of particle acceleration (known as diffusive shock acceleration) is discussed in Chap. 6.

3.3 Synchrotron Radiation

85

Fig. 3.4 A schematic diagram depicting synchrotron radiation. An electron traces a helical path along magnetic field lines: as the electron gyrates, electromagnetic radiation in the form of photons is emitted. The total emitted power of the radiation is proportional to the energy of the electron itself while the lifetime of the synchrotron-emitting electron is inversely proportional to the energy of the electron

FB = Fcentripetal −→ r =

mv⊥ , qB

(3.53)

where r is commonly known as the gyroradius of the path of the electron. While moving through a magnetic field, an electron will gyrate as it completes a helical path around magnetic field lines and the radius of the helix traced out by the electron corresponds to the gyroradius. Recall that—because all charged particles emit photons when they are accelerated the electron emits photons as it traces out its helical path. Because the orbital period P of the electron is (see Eq. (2.72)) P =

2π r , v⊥

(3.54)

the frequency νcyc of the gyration may be expressed simply as simply νcyc =

v⊥ qBr qB 1 = = = . P 2π r 2π rm 2π m

(3.55)

This quantity is also known as the cyclotron frequency: the term originates from the study of gyrating charged particles as cyclotrons, a type of particle accelerator where charged particles are accelerated by changing electric fields and pass through magnetic fields of fixed strengths, thus tracing out helical paths. The angular cyclotron frequency ωcyc is given as ωcyc = 2π νcyc =

qB m

(3.56)

86

3 Emission Mechanisms

and—in terms of ωcyc —the acceleration a⊥ experienced by the charged particle in the direction perpendicular to the direction of B may be expressed as a⊥ =

2 v⊥ qB = v⊥ ωcyc . = v⊥ v⊥ r mv⊥

(3.57)

Note that both νcyc and ωcyc are proportional to the magnitude of the charge of the electron and the magnetic field and inversely proportional to the mass of the electron. Of course, these quantities are in turn proportional to the charge of any particle and inversely proportional to the mass of any particle. Note also that both νcyc and ωcyc are independent of the velocity—and, by extension, the kinetic energy—of the charged particle. A crucial property of the gyrating electron is the total amount of energy that it radiates per second. This quantity is formally defined as the power P and is defined as  P = S · dA, (3.58) where S is the Poynting vector as defined in Eq. (2.100) (note that the time-average value of this quantity is the intensity I as defined in Eq. (2.102)) and dA is the differential area element into which the radiation is emitted. Note that if the electron is assumed to be radiating isotropically into all 4π steradians of solid angle, then—at a location r from the electron at which the intensity of the radiation is measured— the integration of dA over all solid angle is 4πr 2 . One way to determine P is to compare the strengths of the radial and tangential components of the electric field of a charged particle, as shown in Fig. 3.5.3 Recall that the radial component Er of an electric field as measured at a distance r from a charge q is (for large distances from the charge itself) Er =

1 q . 4π 0 r 2

(3.59)

This expression assumes that the charged particle is at rest: it is for this reason that the electric field E created by the charge is purely radial and has no tangential component (that is, for a charged particle at rest, E = Er and E⊥ = 0). Suppose that the charged particle experiences an acceleration to a velocity Δv in a short time Δt and that the velocity is non-relativistic (that is, Δv c). How does the electric field change as measured at the position r at a later time t? Considering that the charged particle is offset by a distance Δvt from its original position, E⊥ is no longer zero.

3 Many

textbooks on electrodynamics explore the amount of energy emitted by a charged particle through a discussion of retarded potentials. Such a discussion involves a formalism which is beyond the scope of the present text. The discussion presented here follows a derivation presented by the physicist J. J. Thomson.

3.3 Synchrotron Radiation

87

Fig. 3.5 A schematic diagram illustrating the change in the radial and tangential components of the electric field of a charged particle that experiences an acceleration to a velocity Δv in a short time Δt. At the initial location of the charged particle (indicated by a lightly shaded circle), the electric field is purely radial. When the accelerated charged particle moves to a new location (indicated by the darkly shaded circle), the tangential components of the electric field are no longer zero. In this diagram, θ is defined to be the angle between the acceleration vector and a line drawn in the radial direction from the offset position of the charge

Defining θ to be the angle between the acceleration vector and a line drawn from the offset position of the charge and r, the ratio of E⊥ and Er can be expressed as E⊥ Δvt sin θ . = Er cΔt

(3.60)

Notice here that this quantity is simply the ratio of the displacements of the charge in the tangential and radial directions. Combining Eqs. (3.59) and (3.60) and solving for E⊥ yields  E⊥ = Er

E⊥ Er

 =

1 q Δvt sin θ , 4π 0 r 2 cΔt

(3.61)

which becomes (after substituting t = r/c and gathering Δv and Δt together) 1 q E⊥ = 4π 0 r 2



Δv Δt



r sin θ 1 qa sin θ = , 4π 0 rc2 c2

(3.62)

where a⊥ is again the acceleration of the charged particle in the direction perpendicular to the direction of B. To calculate the total power P radiated by the charge integrated over all directions. recall the Poynting vector S, which describes the rate of transfer of energy by an electromagnetic wave (see Eq. (2.102)). In the context of the current analysis, it also describes the power emitted per unit area by the electron. Adopting E⊥ from Eq. (3.62) and relating the magnitude of the complementary

88

3 Emission Mechanisms

magnetic field component B to E⊥ through Eqs. (2.99) and (2.102) thus becomes 1 1 1 (E⊥ )2 = S= E × B −→ S = μ0 μ0 c μ0 c



1 qa⊥ sin θ 4π 0 rc2

2 ,

(3.63)

which in turn becomes (see Eq. (2.101)) 1 S= μ0 c



2 sin2 θ q 2 a⊥ 1 r 2 c4 16π 2 02

 =

2 sin2 θ q 2 a⊥ . 16π 2 0 r 2 c3

(3.64)

Noting that the time-average value of sin2 θ is 2/3, P may be determined using Eq. (3.58), yielding    2 2 q 2 a⊥ q 2 a⊥ 2 2 P = S · dA = = , 4π r 3 16π 2 0 r 2 c3 6π 0 c3 (3.65) which is the well-known Larmor Formula for the power emitted by a nonrelativistic charge. Note that the power emitted is proportional to the square of the acceleration experienced by the charge. The radiation pattern emitted by a nonrelativistic charge is a dipole which in three dimensions is toroidal in shape: the axis of the toroid is parallel to the direction of the acceleration a and the power received is proportional to the component of a that is perpendicular to the line of sight. This radiation pattern is illustrated in Fig. 3.6: notice the symmetry in the sizes of the leading and trailing lobes of this pattern. It is crucial to explore the dependence of the power emitted by the gyrating particle on the mass of the particle itself. For the purposes of illustration, consider two particles with unequal masses but with the same kinetic energy K. From Eq. (2.44), the velocity of each particle as a function of its mass m would be 



2 sin2 θ q 2 a⊥ dA = 16π 2 0 r 2 c3

 v=

2K . m

(3.66)

Setting v equal to v⊥ and considering Eqs. (3.56), (3.57), and (3.65) can be rewritten as  P =

q2 6π 0 c3

 

2K qB m m

2 ∝

K . m3

(3.67)

Therefore, in the case of a proton and electron (where the charges have the same magnitude but opposite signs and where the proton is much more massive than the electron) with the same kinetic energy, according to Eq. (3.67) the electron will emit much more power due to its much lesser mass.

3.3 Synchrotron Radiation

89

Fig. 3.6 Side-by-side comparison of the radiation lobes produced by charged particles moving with non-relativistic and relativistic velocities. (left) The radiation lobes produced via cyclotron emission by a non-relativistic charged particle moving with a velocity v (where v c) and with a gyroradius r. (right) The radiation lobes produced via synchrotron emission by a relativistic charged particle moving with a velocity v (where v ∼ c) and with a gyroradius r. Notice that the radiation lobes are symmetric in the case of cyclotron radiation but asymmetric in the case of synchrotron radiation. The width of the forward lobe corresponds to 1/γ radians, where γ is the Lorentz factor of the relativistic particle

Example Problem 3.5 Calculate the power in Watts emitted by a nonrelativistic electron that is subject to an acceleration equivalent to the gravitational acceleration on the surface of the Earth (that is, 9.8 m/s2 ). Solution From Eq. (3.65), the power emitted by the accelerated nonrelativistic electron is P =

(1.6 × 10−19 C)2 (9.8 m/s2 )2 = 1.47 × 10−26 W. 6π(8.85 × 10−12 F/m)

(3.68)

In the domain of radio astronomy, the charged particles that are gyrating in magnetic fields and emitting radiation are moving at relativistic rather than nonrelativistic velocities, and therefore relativistic effects must be taken into account when properly modeling their radiation. Emission produced by gyrating relativistic electron is known as synchrotron radiation: we compare characteristics of synchrotron radiation in turn with each characteristic described above. Starting with the gyroradius r of synchrotron radiation, note that—unlike cyclotron radiation— the gyroradius of synchrotron radiation does depend on the energy of the accelerated particle. Recalling the discussion of relativity that was presented in Sect. 2.6 and the Lorentz factor γ , the gyrofrequency νsync , and the angular frequency ωsync of synchrotron radiation may be expressed as (from Eqs. (3.55) and (3.56)) νsync =

νcyc qB = 2π mγ γ

and

ωsync =

2π νcyc ωcyc qB = = . γ γ γm

(3.69)

These equations indicate that the gyrofrequency becomes smaller as the motion of the particle becomes more relativistic. Notice that νsync and ωsync are less than νcyc and ωcyc , respectively, by a factor of 1/γ due to the increasing mass of the

90

3 Emission Mechanisms

particle (recall how the mass of a relativistic particle increases with increasing Lorentz factor—see Eq. (2.168)). The acceleration a⊥ experienced by the charged particle may be expressed as a⊥ = ωsync v⊥ =

qBv⊥ γm

(3.70)

and the gyroradius of the relativistic charged particle becomes r=

γ mv⊥ . qB

(3.71)

This equation (and the corresponding discussion about emission from the particle) can be broadened to consider the case of a charged particle moving with a pitch angle with respect to the magnetic field. Defining this angle as α, Eq. (3.71) becomes r=

γ mv sin α . qB

(3.72)

Furthermore, Eq. (3.71) can be expressed in terms of the energy E of the particle as r=

E γ mc c  γ mc2 = = , qB c qcB qcB

(3.73)

where v⊥ ∼ c (in the case of relativistic motion). The radiation pattern for a relativistic electron emitting synchrotron radiation is shown in Fig. 3.6. Unlike the radiation pattern for a non-relativistic electron emitting cyclotron radiation, which is symmetric, the radiation patter of a relativistic electron emitting synchrotron radiation is clearly asymmetric. While the trailing lobe is both reduced in intensity and compressed in geometry, the leading lobe is enhanced in intensity and greatly extended in geometry. This amplification of the emission from the leading lobe is known as the beaming effect: note that the opening angle of the forward lobe of emission is 1/γ . Both the origin of this amplification and the size of the leading lobe are discussed in more detail later in this section. To determine the amount of power P emitted by a single relativistic charge emitting synchrotron radiation, a transformation must be made between the reference frame (called S) of an observer measuring the radiation emitted by the charged particle and another reference frame (called S  ) where the moving charged particle is at rest for an infinitesimal period of time dt. Realize that in the reference frame S  , the emitted power P  by the charged particle that is undergoing an acceleration  can be expressed simply by using Larmor’s Formula (see Eq. (3.65)); that is, a⊥ P =

2 q 2 a⊥ . 6π 0 c3

(3.74)

3.3 Synchrotron Radiation

91

The infinitesimal time interval dt may be expressed in terms of the corresponding time interval dt  as measured in the S  reference frame as a time dilation (see Eq. (2.166)), that is, dt  =

dt . γ

(3.75)

Defining E and E  as the total amount of energy emitted by the charged particle in reference frames S and S  , respectively, then the relationship between these two quantities is E =

E . γ

(3.76)

Therefore the expressions for P and P  in terms of E, E  , t, and t  are P =

dE dt

and

P =

dE  , dt 

(3.77)

 and a as measured in the and because the relation between the accelerations a⊥ ⊥ two reference frames is simply  = γ 2 a⊥ , a⊥

(3.78)

the expression for P  becomes P =

2 2 γ4 q 2 a⊥ q 2 a⊥ = = P. 3 6π 0 c 6π 0 c3

(3.79)

Note the crucial result that P = P  . Such a result makes physical sense since the total amount of power emitted by a particle should be the same in all reference frames. From Eqs. (2.170) and (3.70), P  may be expressed as (see Problem 3.6) 2 B 2γ 2 2 B2 q 4 v⊥ q 4 v⊥ P =P = = 6π 0 c3 m2 6π 0 c3 m2 



E mc2

2 .

(3.80)

In the case of a relativistic electron with charge e, mass me and velocity v⊥ ∼ c, and denoting P as Psync for the remainder of the text, this equation becomes (see Problem 3.7) Psync =

e4 B 2 γ 2 = 2σT cuB γ 2 . 6π 0 m2e c

(3.81)

92

3 Emission Mechanisms

In this equation, uB is the magnetic field energy density and is defined as uB =

B2 , 2μ0

(3.82)

while σT is the Thomson cross-section for an electron and is commonly expressed as σT =

8π 3



e2 4π 0 me c2

2 =

8π 2 r = 6.65 × 10−29 m2 . 3 e

(3.83)

Finally, re is the classical radius of the electron. The remainder of this section will concentrate exclusively on the synchrotron emission produced by an electron. It is important to note that the expressions for νsync , ωsync , r, and Psync all reduce to the analogous expressions for the same quantities in cyclotron radiation derived previously when the velocity of the charged particle is non-relativistic (that is, when γ ∼ 1). One of the most important characteristics of synchrotron-emitting particles is the time t1/2 required for the particle to emit away half of its energy: this time period is known as the synchrotron lifetime of the particle (see the discussion presented in [3]). To determine this quantity in the case of synchrotron-emitting electron, the initial Lorentz factor of the relativistic electron is defined as γ0 (corresponding to a time t = 0) while the Lorentz factor of the electron at a later time is simply γ . Recall that the total power emitted is the time derivative of the energy of the electron, that is, Psync = −

d dE = − γ me c 2 , dt dt

(3.84)

or, alternatively (from Eq. (3.81)), −Psync dγ 1 = =− dt me c 2 me c 2



e4 c2 B 2 γ 2 6π 0 c3 m2e



 =−

e4 B 2 γ 2 6π 0 c3 m3e

 .

(3.85)

Here, taking the time derivative of γ reflects the electron losing energy and becoming less relativistic. This differential equation can be integrated readily to solve for γ : defining Y such that Y = (e4 B 2 /6π 0 c3 m3e ) and performing the integration with respect to t yields −

1 = −Y t + constant, γ

(3.86)

and from the initial conditions described above, a boundary condition can be applied to this equation (when t = 0 and γ = γ0 ) to determine that the integration constant is simply -1/γ0 . Thus Eq. (3.86) becomes

3.3 Synchrotron Radiation

93

γ =

γ0 . (Y γ0 + 1)

(3.87)

When t = t1/2 , the corresponding value of γ is γ0 /2: inserting these values into Eq. (3.86) and combining with Eq. (3.81) yields 1 6π 0 c3 m3 6π 0 c5 m4 4.26 × 10−13 T2 J s . = 4 2 e = 4 2 e −→ t1/2 [s] = Y γ0 e B γ0 e B E0 B 2 [T] E0 [J] (3.88) Note here that the lifetime is inversely proportional to the initial Lorentz factor of the electron (and thus the initial energy of the electron) and the square of the magnetic field in which the electron is gyrating. Note then that in an ensemble of synchrotron-emitting electrons, the highest energy electrons will radiate away their energies before the lower energy electrons. A crucial distinguishing characteristic of synchrotron radiation is the fact that this type of emission is polarized (see Sect. 2.4.4). The electric and magnetic fields of the emitted synchrotron radiation from the gyrating charged particle are shown in Fig. 3.7. The reason why synchrotron emission may be expected to be polarized is described here and is patterned after a discussion presented elsewhere [4]. Schematic diagrams showing how the cyclotron radiation and synchrotron radiation from a gyrating charged particle would be perceived by different observers are presented in Fig. 3.8. In the case of cyclotron radiation, an observer located in the plane of gyration of the electron would observe the emitted radiation to be linearly polarized (because the electron appears to be oscillating in a linear plane) while an observer located above or below the plane of gyration would observe the emitted radiation to be circularly polarized (because the electron appears to be oscillating in a circular plane). Also note that the radiation pattern of the cyclotron-emitting electron is symmetric. This situation may be contrasted with a synchrotron-emitting electron moving with a relativistic velocity: as noted above, the radiation pattern for such an electron is not symmetric but instead is concentrated in a forward beam. An observer in the plane of the gyration of the electron will detect the component of the linearly polarized radiation (the components of circularly polarized emission will cancel each other out). The polarized emission will be most prominent in the presence of a strong magnetic field with a uniform direction: in most cases, however, diffuse astronomical sources that emit such radiation do not feature such an orderly magnetic field and therefore the fraction of polarized emission is reduced. What is the frequency distribution of synchrotron radiation from a single charged particle? Detailed treatments are presented elsewhere [5] and only a brief description is presented here. Consider the emission from a synchrotron-emitting electron as detected by observers in two different reference frames: one frame corresponds to an outside observer (S) and another frame corresponds to the frame of the synchrotronemitting electron (S  ). The opening angle of the forward lobe of emission from the synchrotron-emitting electron is 1/γ and the observer will see radiation from the t1/2 =

94

3 Emission Mechanisms

Fig. 3.7 A schematic diagram depicting the electromagnetic radiation emitted by a relativistic electron gyrating in a magnetic field and emitting synchrotron radiation. Recall that for all forms of electromagnetic radiation, the planes of the electric field and the magnetic field components are orthogonal to each other. Therefore, the emission produced by the gyrating electron would appear to be linearly polarized: the polarization of the electric field is orthogonal to the polarization of the magnetic field

lobe from the instant when the beam first sweeps towards the observer until the instant when it has swept past the observer. If the distance traveled by the electron along its circular path is s, the corresponding interior angle θ that corresponds to s is simply θ = 2/γ , which is just twice the opening angle. If the radius of the circular path of the electron is r and the velocity of the electron is v, then the time Δt required for the electron to cover the distance s (as observed in the reference frame S  of the electron itself) is Δt  =

s 2 rθ 2r θ = . = = = v v vγ ωsync γ ωsync

(3.89)

What is the corresponding time interval Δt as measured by the outside observer in the frame S? It is stressed that the observer will see pulses of emission from the electron as its highly focused forward radiation lobe sweeps by. The electron will emit a photon in this lobe when it starts to cover the distance s and will almost catch

3.3 Synchrotron Radiation

95

Fig. 3.8 Schematic diagram depicting how the observed polarization produced by a cyclotronemitting electron and a synchrotron-emitting electron is dependent on the location of the observer. (Left) Cyclotron emission from a gyrating charged particle produces circularly polarized light as detected by observers located above and below the plane of gyration and linearly polarized light as detected by an observer located in the plane of gyration

up with the photon when it has finished covering this distance. Because of this, Δt is shorter than Δt  and may be expressed explicitly as s s s Δt = Δt − = − = s c v c 



1 1 − v c



 =s

c−v vc

 =

θ ωsync

(1 − β) ,

(3.90)

where β = v/c (see Eq. (2.164)). As β approaches unity, it may be approximated as  β=

1−

1 1 ≈1− 2 γ 2γ 2

(3.91)

and therefore Δt may be expressed as (see Eqs. (3.69) and (3.89)) Δt =

θ ωsync

     1 2 1 1 1 1− 1− = = 3 = 2 , 2 2 γ ωsync 2γ 2γ γ ωsync γ ωcyc (3.92)

or alternatively, Δt =

1 1 = . 3 2π γ νsync 2π γ 2 νcyc

(3.93)

96

3 Emission Mechanisms

Analysi4 of the components of peak emission corresponding to a pulse of period Δt reveals that the typical frequencies contained in such a pulse are ν ∼ (Δt)−1 . With this in mind, the critical frequency νc at which the synchrotron-emitting electron will emit the most radiation may be expressed as νc = γ 2 νcyc = γ 3 νsync =

γ 2 qB . 2π me

(3.94)

In actuality, the electron will emit the most synchrotron radiation at a frequency ≈0.3 νc : therefore, for the purposes of accuracy and rigor, the relations between νc , νcyc and νsync that will be adopted for the remainder of this text will be νc ≈

γ 3 νsync γ 2 νcyc γ 2 qB γ 2 qB = = = . 0.3 0.3 2(0.3)π me 0.6π me

(3.95)

With these equation in mind, it is helpful to re-express Eq. (3.95) for electrons such that γ is a function νc in Hertz, B in Tesla and a multiplicative constant, that is,  γ ≈ 3.28 × 10−6 Hz−1/2 T1/2

νc [Hz] . B[T ]

(3.96)

Example Problem 3.6 A relativistic electron is emitting synchrotron radiation at a peak frequency νc = 109 Hz (a standard radio frequency) as it gyrates in a magnetic field with a field strength B = 10−9 T. This particular value of the magnetic field strength corresponds to the field strengths of astronomical sources known to emit copious amounts of synchrotron radiation like supernova remnants. Assuming that this electron has been recently accelerated to its current energy, calculate the corresponding Lorentz factor γ0 . Also determine the initial energy E0 of this electron, the total initial power P0 emitted by the electron and the time t1/2 required for the electron to radiate away half of its initial power P0 . Solution From Eq. (3.96). the Lorentz factor may be calculated as follows:  γ0 ≈ 3.28 × 10−6 T1/2 Hz−1/2

109 Hz = 3280. 10−9 T

(3.97) (continued)

4 The

type of analysis mentioned here is known as Fourier Analysis and it is discussed in detail in Sect. 4.1

3.3 Synchrotron Radiation

97

Example Problem 3.6 (continued) From Eq. (2.170), E0 may be computed as E0 = γ0 mc2 = (3280)(9.11 × 10−31 kg)(3 × 108 m/s)2 = 2.69 × 10−10 J. (3.98) While this quantity is small in magnitude, note that it corresponds to the amount of energy associated with a single highly relativistic electron. Before P0 can be calculated, uB must be determined. Using Eq. (3.82), uB =

(10−9 T)2 4π × 10−7 N/A2

= 7.96 × 10−13 J/m3 ,

(3.99)

and therefore from Eq. (3.81) P0 may be calculated as P0 =2(6.65×10−29 m2 )(3×108 m/s)(7.96×10−13 J/m3 )(3280)2 =3.42×10−25 W. (3.100) Finally, from Eq. (3.88), t1/2 may be computed as t1/2 [s] =

(10−9

4.26 × 10−13 = 1.58 × 1015 s, T)2 (2.69 × 10−10 J)

(3.101)

or, expressed in years, 

t1/2

1 yr = 1.58 × 10 s 3.15 × 107 s 15

 = 5.02 × 107 yr.

(3.102)

The spectrum of synchrotron emission from a single electron is shown in Fig. 3.9: notice the sharply peaked nature of this curve with peak emission at ≈0.3νc . Based on this spectrum of a single synchrotron-emitting electron, the integrated spectrum of an astronomical source emitting synchrotron radiation may be developed by extension. A model of this integrated spectrum is shown in Fig. 3.10: if the astronomical source is envisioned as an ensemble of synchrotron-emitting electrons with a range of energies, the integrated spectrum is simply the sum of the spectra of the individual electrons. This integrated spectrum can be described adequately for many applications as a simple power law with a spectral index α defined such that the observed flux as a function of energy is proportional to E −α .5 For astronomical sources that produce radio emission that is primarily synchrotron radiation (such as

5 The

reader is cautioned that in some contexts, the spectral index is expressed such that the dependence of observed flux on frequency is ν α (that is, with an opposite sign convention). Care must then be taken when interpreting published values of α.

98

3 Emission Mechanisms

supernova remnants, which are discussed in Chap. 6), the observed values of α are approximately 0.5 through 0.9. At lower frequencies (specifically where hνc kB T ), the emission detected from a synchrotron-emitting source is attenuated due to self-absorption: this phenomenon is known as synchrotron self-absorption and is depicted in Fig. 3.11. At these frequencies, the ensemble of synchrotron-emitting electrons becomes optically thick to its own radiation and therefore a photon that is emitted at one of these low frequencies by one of the electrons in the ensemble is absorbed by another electron in the ensemble. To understand how this absorption occurs, the ensemble may be envisioned such that all of the member electrons possess the same energy E = γme c2 . Furthermore, each electron may be assumed to be emitting photons at a frequency νc and therefore the energy of the emitted photons is hνc . In this scenario, the integrated emission from the ensemble of electrons may be modeled as a blackbody with a temperature T based on the relationship between T , γ , me , c and kB as kB T ≈ γme c2 . This relationship may be easily re-expressed as T ≈

γ me c 2 . kB

(3.103)

From Eq. (3.103) and recalling the Rayleigh–Jeans approximation at low frequencies for Bν (ν,T ) given in Eq. (3.9), the observed integrated emission from the ensemble of electrons at the frequency ν = νc may be expressed as  Bν (νc , T ) ≈

2kB T νc2 c2

Fig. 3.9 The spectrum of emission from a single synchrotron-emitting electron. Notice the sharply peaked nature of the emission with a peak located at the frequency ν ≈ 0.3νc



 ≈

2kB νc2 c2



γ me c 2 kB

 ≈ 2γ me νc2 .

(3.104)

3.3 Synchrotron Radiation

99

Fig. 3.10 The integrated spectrum from an astronomical source which is emitting synchrotron radiation. The source may be envisioned as an ensemble of synchrotron-emitting electrons with a range of energies. The observed spectrum is then described as the sum of all of the spectra from the individual sources and thus resembles a power law

Fig. 3.11 The integrated spectrum of a synchrotron-emitting source. Notice the effect of selfabsorption that results in a ν 5/2 dependence at low frequencies

Noting that γ ∼

√ νc /B (see Eq. (3.96)) and (3.104) may be rewritten as Bν (νc , T ) ≈ 2

ν 1/2 c

B

5/2

me νc2 ≈

2me νc , B 1/2

(3.105)

100

3 Emission Mechanisms

and therefore for low values for ν where self-absorption is significant, Bν (ν, T ) becomes Bν (ν, T ) ≈

2me ν 5/2 . B 1/2

(3.106)

Therefore, in the low-frequency domain, the effect of synchrotron self-absorption results in the observed spectrum exhibiting a frequency dependence of ν 5/2 . Notice that this frequency dependence is independent of other properties of the electron ensemble (such as the exact value of νc ) and is thus expected to be seen in the lowfrequency radio spectra of astronomical sources that produce synchrotron radiation. At higher frequencies (specifically where hνc  kT ), the observed slope of the spectrum does in fact depend on properties of the ensemble of synchrotron-emitting electrons. Specifically, it depends on the numerical distribution of the electrons as a function of energy: this may be quantified as follows. If N(E) dE is defined as the number of electrons per unit volume and per unit solid angle moving in the direction of the observer and with energies between E and E + dE, then a simple power law distribution of energies of the particles with an index φ may be expressed as N(E)dE ∝ kE −φ dE,

(3.107)

where k is a constant. What is the relationship between φ and α? This may be explored by first making the approximation that all electrons in the ensemble are emitting all of their power at a frequency ν corresponding to the critical frequency νc . From Eq. (3.95), ν may be expressed as ν ≈ νc ≈ γ 2 νcyc .

(3.108)

Next, the quantity ν dν is introduced and defined as the emissivity of the synchrotron-emitting population of relativistic electrons for photons emitted over the frequency range from ν to ν + dν. This emissivity may be expressed as the product of the power P emitted by each electron and N(E) dE, that is, ν dν = P N (E)dE,

(3.109)

where the relationship between P and dE/dt as stated in Eq. (3.84) has been applied. By combining Eqs. (2.170) and (3.108), E may be expressed in terms of ν as  E = γ me c ≈ 2

ν νcyc

1/2 me c 2 ,

(3.110)

and differentiating this equation with respect to ν yields dE ≈

me c2 −1/2 ν dν. 2νcyc 1/2

(3.111)

3.3 Synchrotron Radiation

101

Based on this relationship between dE and dν, integrating Eq. (3.109) with respect to ν to obtain the integrated emissivity  (along with including the expressions given for P , N(E) dE, and E in Eqs. (3.81), (3.107), and (3.110), respectively) produces  ν dν = P N (E)dE = 2σT cuB γ k

ν

2

−φ

1/2 me c

νcyc

2

ν −1/2 me c2 1/2

dν.

2νcyc

(3.112) Concentrating solely on the quantities uB , γ , ν, and νcyc and ignoring all the constants, this equation may be simplified to  ν dν ∝ uB γ

2

1/2 −φ

ν

1 (ννcyc )1/2

νcyc

dν.

(3.113)

Inspection of Eq. (3.82) reveals that uB is proportional to B 2 , while inspection of Eq. (3.55) reveals that νcyc is proportional to B. Based on these relations and recalling the relationship between γ , ν and νc expressed in Eq. (3.108) and (3.113) can be rewritten in terms of B and ν as ν dν ∝ B 2

ν  ν −φ/2 B

B

1 dν. (νB)1/2

(3.114)

Gathering terms yields at last the dependence of ν dν on B and ν, specifically ν dν ∝ B (1+φ)/2 ν (1−φ)/2 dν.

(3.115)

Combining Eqs. (3.107), (3.109), and (3.115) establishes a relation between ν dν and N (E) dE such that ν dν ∝ N(E)dE ∝ E −φ dE ∝ B (1+φ)/2 ν (1−φ)/2 dν.

(3.116)

Recalling again that the observed integrated spectrum of a synchrotron-emitting source is a power law described by a spectral index α, such that ν dν ∝ ν −α dν. This relation coupled with Eq. (3.116) establishes the following relation between α and φ as α=

φ−1 . 2

(3.117)

Therefore, by measuring α for an astronomical source that is emitting synchrotron radiation, a value for φ may readily be determined. Measurements of φ may in turn be used to constrain models that describe how the synchrotron-emitting electrons may have been accelerated to relativistic velocities in the first place by dictating the distribution of energies of these electrons as a function of the electron energy itself.

102

3 Emission Mechanisms

Finally, note that the maximum degree Π of linear polarization at a single frequency in the spectrum of a synchrotron-emitting source may be expressed in terms of α and φ as Π=

φ+1 α+1 = . φ + 7/3 α + 5/3

(3.118)

3.4 Thermal Bremsstrahlung “Bremsstrahlung” (from German—meaning “braking radiation”) is the term given to radiation emitted by an electron encountering a nucleus and experiencing an acceleration due to the electrostatic attraction between the electron and the nucleus. It is also known as “free-free” radiation because the photon-emitting electron is not bound to the nucleus at the start of the emission process, nor is it bound to the nucleus at the end of the emission process. The study of bremsstrahlung radiation is most applicable to the cases of sources which are ionized plasmas, such as HII regions (see Sect. 6.3.4). Consider an electron with charge e and mass me moving with a velocity v that takes it near a nucleus with charge Ze. At its closest approach to the nucleus, the separation between the nucleus and the electron is b, a quantity that is known as the impact parameter. To describe the path taken by the electron as it approaches the electron, the angle ψ is defined to correspond to the angle between the b and l, the instantaneous distance between the electron and the nucleus. The relationship between b, l, and ψ is given as b = l cos ψ,

(3.119)

the instantaneous Coulombic force between the electron and the nucleus is (recall Eq. (2.146)) FCoulomb =

1 (Ze)e 1 Ze2 = , 4π 0 l 2 4π 0 l 2

(3.120)

and the corresponding acceleration a experienced by the electron is me a =

Ze2 4π 0 l 2

or

a=

Ze2 Ze2 cos2 ψ = . 4π 0 me l 2 4π 0 me b2

(3.121)

The magnitude of the acceleration is greatest when the direction of the acceleration is perpendicular to the orbit, so for simplicity only the perpendicular component of the acceleration is considered here, that is,

3.4 Thermal Bremsstrahlung

103

a⊥ = a cos ψ =

Ze2 cos3 ψ . 4π 0 me b2

(3.122)

Therefore the total energy E emitted by the electron can be determined by integrating over the total amount of power P emitted by the electron during its entire acceleration, that is (recalling Eq. (3.79))  E=

+∞ −∞

e2 P (t)dt = 6π 0 c3



+∞

−∞

2 a⊥

2e2 dt = 6π 0 c3



+∞

0

2 a⊥ dt,

(3.123)

where the bounds of the integration have been changed such that t = 0 corresponds to the instant when the electron is at its closest distance to the nucleus (this is the origin of the extra factor of two in the numerator of the constants outside of the integration). Inserting Eq. (3.122) into Eq. (3.123) yields E=

2e2 6π 0 c3

+∞  Ze2

 0

cos3 ψ 4π 0 me b2

2

Z2q 6

dt =

48π 3 03 m2e b4 c3



+∞

cos6 ψ(t) dt.

0

(3.124) How can the integral over cos6 ψ(t) dt be evaluated? Recall the discussion of Kepler’s Second Law of Motion as presented in Sect. 2.3.3: from that discussion, in the time interval dt, the angle dψ and corresponding area dA swept out by the electron moving with a velocity v and at a distance l from the nucleus is (from Eqs. (2.62), (2.63), and (2.64), and again assuming small values of ψ) dA l 2 dψ = = constant. dt 2 dt

(3.125)

Remember that this derivative must be a constant as dictated by Kepler’s Second Law. At the time t = 0, the electron makes its closest approach to the nucleus (that is l = b) with a velocity v=b (dψ/dt) and so Eq. (3.125) becomes dA b2 = dt 2



dψ dt

 =

bv . 2

(3.126)

Solving for dψ/dt and applying Eq. (3.119) yields dψ bv b v cos2 ψ = 2 = −→ dt = dψ. dt b l v cos2 ψ

(3.127)

Substituting this result into Eq. (3.123) yields E=

Z 2 e6 48π 3 03 m2e b4 c3



+∞ 0

cos6 ψ(t) dt =

Z 2 e6 48π 3 03 m2e b3 c3 v



π/2

cos4 ψ dψ.

0

(3.128)

104

3 Emission Mechanisms

In this equation, the integration bounds for ψ have been changed to 0 and π /2 to correspond to the effective range of angles over which acceleration occurs for the electron. After applying the identity 

π/2

0

cos4 x dx =

3π , 16

(3.129)

the final expression for W becomes E=

3Z 2 e6 768π 2 03 m2e b3 c3 v

−→ E[J] = 4.28 × 10−49

b3

Z2 . [m] v [m/s]

(3.130)

Thus, E corresponds to the total energy radiated by the electron as it moves in the field of the ion with charge Ze. This expression is valid only for low energy collisions where the velocity of the electron is not relativistic (v c) where it is possible to approximate the path of the electron as a straight line. Notice that E is proportional to the square of the atomic number of the ion and inversely proportional to the velocity of the electron and the cube of the impact parameter. The broadband spectrum of a source emitting bremsstrahlung radiation is depicted in Fig. 3.12. Why does the integrated spectrum of a bremsstrahlung emitting source exhibit a turnover at low frequencies (corresponding to long wavelengths)? Similar to the case of synchrotron radiation, an astronomical source that emits bremsstrahlung radiation becomes optically thick to its own radiation in the low frequency domain. This behavior may be explained by reconsidering Kirchoff’s

Fig. 3.12 Broadband spectrum of a source emitting bremsstrahlung radiation. Notice three salient features in this plot: the exponential cutoff at high frequencies, the flat plateau of emission at medium frequencies and the rapid decline in emission (due to a dramatic increase in optical depth) at low frequencies

3.4 Thermal Bremsstrahlung

105

Law, as given in Eq. (3.48). In the Rayleigh–Jeans limit that is appropriate for this frequency domain and considering Kirchoff’s Law, the opacity of the source may expressed as (note that jν is approximately constant in this frequency domain) κν =

jν jν c 2 ≈ ∝ ν −2 . Bν 2kT ν 2

(3.131)

Therefore, at low frequencies, a thermal bremsstrahlung spectrum exhibits selfabsorption with a ν −2 dependence. Notice that—similar to the self-absorption seen at low frequencies in the spectrum of a synchrotron-emitting source—this frequency dependence is independent of other properties (such as temperature) of the source. A remarkable feature of the bremsstrahlung spectrum is that above a certain frequency (denoted here as νcritical ), no emission is produced. This phenomenon is exhibited in the spectrum shown in Fig. 3.12. What is the physical explanation for this phenomenon? The electron emits photons as it is accelerated and the kinetic energy of the electron is converted into the energy of the emitted photons. Therefore, the maximum energy of a photon that may be emitted by an electron as bremsstrahlung radiation corresponds to the entire kinetic energy of the electron (assuming that all of the kinetic energy is converted into photons). In other words, for an electron with a mass me and velocity v, the possible range of photon energies that may be emitted is hνcritical ≤

me v 2 , 2

(3.132)

where νcritical corresponds to the maximum energy of a photon that may be emitted by the electron.

Example Problem 3.7 Calculate the total energy E emitted by an electron emitting bremsstrahlung radiation as it passes a hydrogen nucleus (Z = 1) with an impact parameter b = 10−10 m (the nominal approximate diameter of an atom) and a velocity v = 100 m/s. Also calculate the frequency νcritical of the maximum energy of the photon emitted by the accelerated electron, assuming that all of the kinetic energy of the electron is converted into energy of the photon. Solution From Eq. (3.130), E may be computed as follows: E=

3(1)2 (1.69 × 10−19 C)6 768π 2 (8.85 × 10−12 F/m)3 (9.11 × 10−31 kg)2 (10−10 m)3 (3 × 108 m/s)3 (100 m/s) (3.133)

= 5.94 × 10−21 J.

From Eq. (3.132), νcritical may be computed as follows: (continued)

106

3 Emission Mechanisms

Example Problem 3.7 (continued) νcritical =

me v 2 (9.11 × 10−31 kg) (100 m/s)2 = 6.87 × 108 Hz. = 2h 2(6.63 × 10−34 J · s) (3.134)

It is important to note that unlike synchrotron radiation, bremsstrahlung radiation is not polarized: that is, there is no frame of reference nor particular characteristic of the emission process where radiation of a particular polarization will be produced more than at another polarization. Also note that the amount of emission from synchrotron radiation is “flat” (that is, independent of frequency) over a broad range of frequencies before reaching νcritical (see Eq. (3.130)). At this frequency and higher, the amount of observed bremsstrahlung radiation from a source is seen to drop off precipitously. Therefore, these two key characteristics of bremsstrahlung radiation—the flat behavior over broad ranges of frequencies and the lack of polarization to the emission—are essential for distinguishing this type of emission from synchrotron emission and are crucial in the proper classification of astronomical sources.

3.5 Molecular Transitions From the Stefan–Boltzmann Law as presented in Eq. (3.24), it is readily apparent that the total amount of thermal emission produced by an object increases dramatically as its temperature of the object increases. By the same token, very little thermal emission is expected from objects with low temperatures: in such cases, it is still possible to detect emission from such sources. In such cases, spectral line emission may be helpful in revealing the presence of cold sources. Spectral lines may be produced by transitions between rotational states of the molecule, as dictated by the laws of quantum mechanics. A diatomic molecule can be envisioned as two masses in rotation around a center of mass (Fig. 3.13): the amount of orbital angular momentum L is quantized—similar to the quantization of the orbital energy of electrons in the hydrogen atom as discussed in Sect. 2.5—and therefore only distinct quanta of photons are emitted or absorbed when the molecule changes its orbital angular momentum state. This phenomenon is encountered routinely in the use of microwave ovens: these ovens illuminate the food contained in the oven cavity with electromagnetic radiation at microwave wavelengths. Water molecules within the food absorb this radiation and as a consequence vibrate and rotate: these actions deposit thermal energy within their surroundings within the food and thus raise the temperature of the food. Emission from rotational transitions by interstellar carbon monoxide (CO) molecules was first detected by Wilson et al. [6] in an observation made of the Orion Nebula. Rotational transitions of molecules may be contrasted with classical mechanical descriptions of connected rotating masses, where a continuous range of angular

3.5 Molecular Transitions

107

Fig. 3.13 A schematic diagram of a diatomic molecule composed of a larger nucleus of mass M and a smaller mass m separated by a distance r. The distance from the center of M to the center of mass of the molecule is rM while the distance from the center of m to the center of mass of the molecule is rm . The two masses rotate as a rigid rotor about the axis which masses through the center of mass. The total amount of angular momentum J of the rotor is quantized to discrete values of corresponding energies

momentum values are possible. The energies of the photons emitted by the transitions may be calculated as follows. Recall that the L of a system (a single mass, a diatomic molecule or a system of many masses) is given as L = I ω,

(3.135)

where I is the moment of inertia of the system and ω is the angular velocity. Based on these definitions, the rotational energy Erot of a system—in the specific case of a system of a diatomic molecule, composed of nuclei with masses M and m—can thus be expressed as Erot =

μr 2 ω2 I ω2 = , 2 2

(3.136)

where r is the distance between the two nuclei of the molecule, μ is the reduced mass of the molecule, and I may be expressed as (for a diatomic molecule) I = μr 2 ,

(3.137)

Furthermore, in the case of a two-particle system like the diatomic molecule, μ may be expressed as

108

3 Emission Mechanisms

μ=

mM . m+M

(3.138)

Recall the discussion about the concept of a center of mass in Sect. 2.3.4. A reduced mass is a useful construct in that—for the diatomic molecule—we can model the rotation properties of the molecule by imagining that the two masses of the atoms are concentrated into the reduced mass that is at the location of the center of mass. This idealization greatly simplifies considerations about modeling the orbital angular momentum of the molecule. Because L is quantized, the relation between L and its rotational energy E may be expressed as  L = Iω =

h 2π

 (3.139)

J,

where J is the total angular momentum quantum number. The total rotational energy Erot of a diatomic molecule with a total angular momentum quantum number J is  Erot =

h 2π

2

J (J + 1) . 2I

(3.140)

Therefore, the difference in energy ΔErot between an upper level rotational state J  and a lower level rotational state J is  ΔErot =

h 2π

2

1   (J (J + 1) − J (J + 1)). 2I

J = 0, 1, 2 . . .

(3.141)

Note that this frequency νrot of a photon with the equivalent amount of energy to ΔErot is simply νrot

ΔErot = = h



h 2I (2π )2



(J  (J  + 1) − J (J + 1)).

(3.142)

Expressions for ΔErot and νrot can be recast in terms of μ, r (recalling that I = μr 2 ), and h, ¯ a commonly encountered term in quantum mechanics that corresponds to h/2π . Using these quantities, the above expressions for ΔErot and νrot become ΔErot =

h¯ 2 (J  (J  + 1) − J (J + 1)), 2μr 2

J = 0, 1, 2 . . . .

(3.143)

and  νrot =

h¯ 4π μr 2



(J  (J  + 1) − J (J + 1)).

(3.144)

3.5 Molecular Transitions

109

In all of these equations, it is emphasized that J may only attain values of 0 or positive integers.

Example Problem 3.8 Calculate the energy ΔErot and the corresponding wavelength λ and frequency ν of the photon emitted when a molecule of carbon monoxide makes a transition from the J  = 2 rotational state to the J = 1 rotational state of carbon monoxide. Assume that the nuclei of the carbon and oxygen correspond to the most common isotopes of each element, that is 12 C (six protons and six neutrons) and 16 O (eight protons and eight neutrons). For simplicity, assume that the carbon and oxygen nuclei are composed of 12 and 16 nucleons, respectively, and each nucleon has a mass of approximately 1.67 × 10−27 kg (note that in reality the mass of a neutron is slightly more than the mass of a proton). Also assume a distance r between the two nuclei of 113 picometers = 113 × 10−12 m. Solution From Eq. (3.138) the reduced mass μ for the CO molecule is μ=

(12 × 1.67 × 10−27 kg) (16 × 1.67 × 10−27 kg) (12 × 1.67 × 10−27 kg) + (16 × 1.67 × 10−27 kg)

= 1.15 × 10−26 kg.

(3.145)

Therefore from Eqs. (3.137) and (3.141), ΔErot is  ΔErot =

6.63 × 10−34 J · s 2π

2

(2(2 + 1) − 1(1 + 1)) 2 × 1.15 × 10−26 kg × (1.13 × 10−10 m)2 (3.146)

= 1.52 × 10−22 J. From Eqs. (2.90) and (2.111), λ and ν can be computed as λ=

(6.63 × 10−34 J · s)(3 × 108 m s−1 ) = 1.31 × 10−3 m = 1.31 mm 1.52 × 10−22 J (3.147)

and ν=

3 × 108 m s−1 = 2.29 × 1011 Hz. 1.31 × 10−3 m

The actual observed frequency of this transition is ν = 230,538 MHz.

(3.148)

110

3 Emission Mechanisms

Note here that in this treatment, the masses of the electrons in the molecules have been omitted in discussions about the center of mass of the molecule. Given that the mass of a nucleon is ∼1800 times the mass of an electron, this omission is safely justified. Notice also that the spectrum of rotational energy transitions associated with this molecule forms a “ladder” of energy levels where the distances between the emission lines are all regularly spaced. Finally, note that a crucial application of measuring the energies of different rotational transitions is that the distance between the two nuclei in the diatomic molecule can be determined. In fact, one of the most common methods for determining the distances between nuclei in diatomic molecules is by measuring the energies of photons emitted by different rotational energy states of the molecule. What is the minimum temperature Tmin required to excite a particular rotational transition with associated energy Erot ? This temperature—also known as the excitation temperature—corresponds to Tmin ∼ Erot /kB . From Eq. (3.143), Tmin may be expressed as Tmin =

h¯ 2 h¯ 2   (J  (J  +1)−J (J +1)). (J (J +1)−J (J +1)) = 2kB I 2kB μr 2

(3.149)

Observations of these molecular transitions are often made of molecules with different isotopes of constituent atoms such as—in the case of CO—13 C16 O and 12 C18 O in addition to the most common version of this molecule discussed above, 12 C16 O. The purposes of these observations of additional molecules are often twofold: firstly, the ratios of abundances of isotopes of elements (such as the ratio of the abundances of 16 O to 18 O) can help lend insights into the chemical evolution of the interstellar medium. Astronomers believe that atoms of heavy elements are synthesized in abundances in the cores of stars and that the end processes of these stars release copious amounts of these heavier atoms into the surrounding interstellar medium. Studies of the abundance ratios of these isotopes in turn help to shape an understanding of how atoms of heavy elements are synthesized in the cores of stars and released into the surrounding interstellar medium, along with the ratios of the isotopes of these atoms. Secondly, in some cases, a cloud may be optically thick to emission at a particular wavelength: for example, a dense molecular cloud may have a high internal optical depth at the wavelength of a common transition such as the transition from the J = 1 rotational state to the J = 0 rotational state, but a significantly lower optical depth to a different transition, such as the J = 2 to J = 1 rotational state. By comparing the ratios of the strengths of these transitions to the expected strengths in the case of a cloud of negligible internal depth, an estimate of the number density of the particles within the cloud may be obtained.

Problems

111

Problems 3.1 Derive Eq. (3.15) by inserting the proper values for the constants in Eq. (3.14). 3.2 (a) Derive Eq. (3.20) through integration of the Planck function Bν (ν, T ) over all frequencies (that is, from ν = 0 to ν = ∞). A useful integral identity for this proof is 

∞ 0

π4 x 3 dx = . ex − 1 15

(3.150)

(b) Show that Eq. (3.20) may also be obtained by explicit integration of the Planck function Bλ (λ, T ) over all wavelengths (that is, from λ = 0 to λ = ∞). 3.3 Emission from stars can be approximated using the blackbody model. Consider the star Canopus, which has the following observed properties: λmax = 385 nm, p = 0.01 arcseconds, and R = 70R . Here, R is the radius of the Sun and corresponds to 6.96 × 108 m. Compute the following properties of this star: (a) The surface temperature T of the star in K. Recall that 1 nm = 10−9 m. (b) The flux F in Watts per square meter that is emitted per square meter of the surface Canopus. (c) The total luminosity L in Watts of Canopus. (d) The flux F in Watts per square meter of light received at Earth from Canopus. (e) The flux density S in Watts per square meter per Hertz received from Canopus using an optical filter with a bandwidth of Δν = 3 × 1013 Hz. 3.4 Consider an experiment to determine the optical depth of a gas at multiple frequencies. In the experiment, emission from a background continuum source passes through the gas and the flux density of the emission received by a detector is measured. The predicted and observed flux densities for different frequencies are tabulated below. The unit used to measure flux density is a Jansky (Jy): this unit is described in more detail in Sect. 4.4.

Observed frequency (MHz) 5000 840 408 74

Incident flux (Jy) 500 550 600 650

Observed flux (Jy) 498 525 3 1

112

3 Emission Mechanisms

For each frequency (74, 408, 840, and 5000 MHz), calculate the corresponding optical depth τ . Also state if the gas is optically thick or optically thin at each frequency. 3.5 Derive Eq. (3.80) from Eqs. (2.170) and (3.70). 3.6 Derive the final form of Eq. (3.81) by making the appropriate substitutions into the initial form of the equation for P . 3.7 Consider an electron moving at a relativistic velocity v = 0.99c through a magnetic field with a strength B = 5 × 10−5 T. Note that such a magnetic field strength corresponds to the nominal strength of the Earth’s magnetic field at its surface. (a) Calculate the Lorentz factor of this electron. (b) Calculate the energy E in Joules of this electron. Recall that the rest mass me of an electron is 9.11 × 10−31 kg. (c) Based on the calculate value of E, compute the gyroradius r in meters of the helical path traced by the electron. (d) Use your calculated value for γ to calculate the total power P in Watts radiated by the electron. (e) Calculate the opening angle θ in radians of the forward lobe of the emission generated by the electron. (f) Calculate the “half-life” time t1/2 in seconds required for the electron to radiate away half of its energy. (g) Calculate the peak frequency νc in Hertz of the emission produced by the electron. 3.8 The radio spectrum of the Galactic supernova remnant Cassiopeia A is best described as synchrotron in origin with a spectral index α = 0.77. Compute the index φ of the power law distribution of the energies of the relativistic electrons that produce the observed radiation. Also compute the maximum degree Π of linear polarization at a single frequency in the radio spectrum of Cassiopeia A. 3.9 The ambient interstellar magnetic fields within spiral galaxies are amplified in strength from approximately B = 3 × 10−10 T to B = 10−9 T due to compression by expanding supernova remnants. Adopting the latter value for the magnetic field strength B, calculate the corresponding Lorentz factor value γ of relativistic electrons that emit synchrotron radiation at 408 MHz, 1 GHz, and 5 GHz. Note that these frequencies correspond to standard frequency bands at which radio astronomical observations are conducted. 3.10 Consider a cosmic-ray electron that has been accelerated to the “knee” energy of the cosmic-ray spectrum (that is, E = 3 × 1015 eV). (a) Calculate the Lorentz factor γ of this electron and its momentum in kg·m/s. Recall that the rest energy of the electron is me = E/c2 = 5.11 × 105 eV/c2 .

Problems

113

(b) How many electrons with this energy would be needed to have the same aggregate momentum as the momentum of a snowball with a mass of 20 g that is thrown with a velocity of 10 m/s? 3.11 Determine the lifetime t1/2 in years of a synchrotron-emitting relativistic cosmic-ray electron moving through the interstellar medium where the component B⊥ of the interstellar medium perpendicular to the path of the electron is 3 µG and the energy of the electron is 5 GeV = 5 × 109 eV. (Hint: See Eq. (2.170) and recall that the mass of an electron is me = E/c2 = 5.11 × 105 eV/c2 .) 3.12 Consider an HII region that is emitting exclusively bremsstrahlung radiation at radio wavelengths. The HII region is located at a distance of 1000 pc and the measured flux of the emission from the HII region at a particular radio wavelength is 5 × 10−26 W/m2 . If all of the bremsstrahlung is produced by interactions between free electrons and hydrogen nuclei as described in Example 3.7, calculate the number of such interactions that would need to occur each second to produce the observed amount of radio emission from the HII region. 3.13 Consider bremsstrahlung emission from an interstellar plasma that is assumed to be composed entirely of ionized hydrogen gas. The temperature of the plasma is T = 106 K. (a) If the amount of thermal energy per particle is 3kT /2, calculate the velocity in meters per second of an electron in the plasma. Assume that the thermal energy of the electron is equal to its kinetic energy. (b) Calculate the critical frequency νcritical corresponding to the highest frequency photon emitted by an electron in this plasma via bremsstrahlung radiation. 3.14 Recall that the moment of inertia I of a system of i particles with masses mi located at distances ri from an origin may be expressed as I = Σi mi ri2 .

(3.151)

Use this equation, derive the moment of inertia I for a diatomic molecule as expressed in Eq. (3.137). 3.15 Consider the J  = 1 to J = 0 rotational transition of carbon monoxide (CO). Assume that the nuclei of the carbon and oxygen correspond to the most common isotopes of each element: this would be 12 C (six protons and six neutrons) and 16 O (eight protons and eight neutrons). For simplicity, assume that the carbon and oxygen nuclei are composed of 12 and 16 nucleons, respectively, and each nucleon has a mass of approximately 1.67 × 10−27 kg (note that in reality the mass of a neutron is slightly more than the mass of a proton). Also assume a distance r between the two nuclei of 113 picometers = 113 × 10−12 m. (a) Calculate the energy E in Joules, the wavelength λ in meters and the frequency ν in Hz of the transition. (b) Calculate the minimum excitation temperature Tmin in K of the transition.

114

3 Emission Mechanisms

3.16 Consider the molecular species cyanogen (CN). Assume that the nuclei of carbon and nitrogen correspond to the most common isotopes of each element, this would be 12 C (six protons and six neutrons) and 14 N (seven protons and seven neutrons). For simplicity, assume that the carbon and nitrogen nuclei are composed of 12 and 14 nucleons, respectively, and each nucleon has a mass of approximately 1.67 × 10−27 kg (note that in reality the mass of a neutron is slightly more than the mass of a proton). Further assume a distance r between the two nuclei of 116 picometers = 116 × 10−12 m. (a) Compute the reduced mass μ in kilograms of cyanogen. (b) Compute the first three rotation frequencies of cyanogen in Hertz (that is, for the J  = 1 to J = 0, J  = 2 to J = 1, and J  = 3 to J = 2 transitions). The actual observed frequencies of these transitions are 112, 224, and 336 GHz, respectively [7].

References 1. B.W. Carroll, D.A. Ostlie, An Introduction to Modern Astrophysics, 2nd edn. (Pearson/Addison Wesley, New York, 2007), pp. 71–75 2. B.W. Carroll, D.A. Ostlie, An Introduction to Modern Astrophysics, 2nd edn. (Pearson/Addison Wesley, New York, 2007), pp. 231–283 3. G.B. Rybicki, A.P. Lightman, Radiative Processes in Astrophysics, 1st edn. (Wiley, New York, 1979) 4. F. Melia, High-Energy Astrophysics, 1st edn. (Princeton Series in Astrophysics, New York, 2009) 5. K. Rohlfs, T.L. Wilson, Tools of Radio Astronomy, 4th edn. (Springer, New York, 2004), pp. 227–261 6. R.W. Wilson, K.B. Jefferts, A.A. Penzias, Astrophys. J. Lett. 161, L43-L43 (1970) 7. C.A. Gottlieb, S. Brünken, M.C. McCarthy, P. Thaddeus, J. Chem. Phys. 126, 191101 (2007)

Chapter 4

Radio Observations: An Introduction to Fourier Transforms, Convolution, Observing Through Earth’s Atmosphere, Single Dish Telescopes, and Interferometers

4.1 An Introduction to Fourier Transforms The study of Fourier transforms permeates the study of many systems of interest in modern physics and engineering. At its core, Fourier analysis—which is conducted with Fourier transforms as its basis—is a form of component analysis: a commonly invoked example to describe Fourier analysis is the decomposition of a musical chord to determine its component notes. Fourier transforms are very frequently encountered in modern astronomy and astrophysics, manifesting themselves in both the analysis of emission mechanisms (as outlined in the next two sections of this chapter) and in the analysis of resolution (both spatial and spectral) and imaging by detectors (as outlined in the next chapter). The description of Fourier transforms presented here is based on the treatment given in [1]. Consider a simple periodic function F (t) of time that is merely a sinusoidal wave with a period P and corresponding frequency ν0 . Such a simple wave may be expressed as the sum of a large number of component waves, expressed as sines and cosines. Each component wave has its own amplitude (denoted here as either an for the cosine wave components or bn for the sine wave components) and its own frequency that is an integer multiple of ν0 . Such component waves are known as the harmonics of F (t): based on these definitions and adopting the conventions that the amplitude of F (t) itself is a0 and that b0 is set to null, the function F (t) may then be expressed as F (t) = a0 + a1 cos(2π ν0 t) + b1 sin(2π ν0 t) + a2 cos(4π ν0 t) + b2 sin(4π ν0 t) + . . . , (4.1) or in terms of a summation (where the range of index values for n spans from −∞ to ∞ for mathematical expediency),

© Springer International Publishing Switzerland 2020 T. G. Pannuti, The Physical Processes and Observing Techniques of Radio Astronomy, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-16982-8_4

115

116

4 Radio Observations

F (t) =

∞

(an cos (2π nν0 t) + bn sin (2π nν0 t)) .

(4.2)

n=−∞

Recalling the trigonometric identities sin (x) = −sin (−x) and cos x = cos(−x), Eq. (4.2) may be rewritten as ∞

F (t) =

A0 + (An cos (2π nν0 t) + Bn sin (2π nν0 t)) , 2

(4.3)

n=1

where An = a−n + an and Bn = bn —b−n and—because by these definitions, A0 = a0 + a0 = 2a0 , A0 is divided by two to avoid counting a0 twice. The primary goal of Fourier analysis is to determine the values of the coefficients An and Bn when F (t) is measured or determined. How is this accomplished? Recall that sines and cosines are orthogonal functions, which may be described as follows: consider a sine and a cosine where the functions have the same fundamental frequency ν0 (or some multiple of this frequency). Taking the product of these two functions and integrating over one period of ν0 returns a value of zero except in particular cases described below. The same result holds true if the product of two sines and two cosines is also considered, as long as the functions have the same fundamental frequency or multiples of that frequency and the integration is over a period. Defining the period P = 1/ν0 , the explicit results of the integrations are 

t=P

cos (2π nν0 t) cos (2π mν0 t) dt =

1 if m = ±n and 0 otherwise. 2ν0

(4.4)

sin (2π nν0 t) sin (2π mν0 t) dt =

1 if m = ±n and 0 otherwise. 2ν0

(4.5)

t=0



t=P

t=0



t=P

cos (2π nν0 t) sin (2π mν0 t) dt = 0 for all values of m and n.

(4.6)

t=0

To determine the coefficients Am , both sides of Eq. (4.3) are multiplied by cos (2π ν0 t), yielding 

t=P

 F (t) cos (2π ν0 t) dt =

t=0

t=P t=0

A0 cos (2π ν0 t) dt + C1 , 2

(4.7)

where  C1 =

t=P

t=0

∞

 (An cos (2π nν0 t) + Bn sin (2π nν0 t)) cos (2π ν0 t)

dt.

n=1

(4.8)

4.1 An Introduction to Fourier Transforms

117

However, due to the orthogonality of the sine and cosine functions, all of the terms in Eq. (4.7) vanish except 

t=P

 Am cos (2π nν0 t) dt = Am 2

t=0

t=P

cos2 (2π mν0 t) dt =

t=0

Am Am P = 2ν0 2 (4.9)

therefore Am =

2 P



t=P

F (t) cos (2π mν0 t) dt

(4.10)

t=0

and a similar equation can be derived for Bm , that is, 2 Bm = P



t=P

F (t) sin (2π mν0 t) dt.

(4.11)

t=0

Assuming that the function F (t) is known over the interval t = 0 to t = P , the coefficients Am and Bm can be determined by performing the integration. The significance of Fourier analysis, then, is that the coefficients—which describe the strengths of the harmonics that comprise F (t)—are obtained through these integrations. Note that these integrations do not necessarily need to begin at t = 0 but instead need to be performed only for one period. If the integrations are simple enough, they may be performed analytically, but in most cases the integrations are complex and numerical approximations are obtained, sometimes through intense computational processing. Once the values of the coefficients are obtained, F (t) may be reconstructed. Using Euler’s Formula (see Eq. (2.94)), F (t) may be expressed in exponentials rather than with trigonometric functions as F (t) =

∞

Cm e2π imν0 t .

(4.12)

m=−∞

As described previously, the chief advantage of using exponentials in this context is that these functions are easier to manipulate mathematically than trigonometric functions. Note here that the coefficients Cm are in general complex numbers and ∗ holds, where C ∗ is the complex conjugate of C (in that the relation Cm = Cm m m ∗ = a−ib). The other words, if Cm = a+ib, where a and b are real numbers, then Cm following expressions—known as inversion formulae—can be used to determine Am , Bm , and Cm :  Am = 2ν0

t=1/ν0

F (t) cos (2π mν0 t) dt t=0

(4.13)

118

4 Radio Observations

 Bm = 2ν0

t=1/ν0

F (t) sin (2π mν0 t) dt

(4.14)

t=0

 Cm = 2ν0

t=1/ν0

F (t) e−2π mν0 t dt

(4.15)

t=0

Note that F (t) may be described as a sum of sines and cosines regardless of whether or not it is a periodic function. If it is not periodic, a proper expression for F (t) requires a consideration of all frequencies ν. In such a case, F (t) may be treated as a limiting case of a periodic function with a period approaching infinity and a frequency approaching zero. For such a function, the spacing between the harmonics decreases and a limit is approached where there is a continuum of harmonics, each one with infinitesimal amplitudes a(ν)dν and b(ν)dν. In this case, F (t) may be expressed as  F (t) =

∞

−∞

 a(ν) cos (2π νt) dν +

∞

−∞

b(ν) sin (2π νt) dν,

(4.16)

(note that the summation sign in Eq. (4.12) has been replaced by an integration and note the correspondence between summations and integrations), or alternatively  F (t) =

∞ −∞

f (ν)e2π iνt dν.

(4.17)

Therefore, f (ν) may be considered to be the Fourier transform of F (t). Conversely, the inverse of this transform, that is  f (ν) =

∞ −∞

F (t)e−2π iνt dt,

(4.18)

indicates that F (t) is the Fourier transform of f (ν). The relationship between F (t) and f (ν) as a conjugate Fourier pair may be described as f (ν) ⇔ F (t).

(4.19)

In the present text, there is an emphasis on considering the conjugate Fourier pairs of functions of time and frequency, which may be viewed as a pair of conjugate variables. Other pairs of conjugate variables may form conjugate Fourier pairs of functions, as long as the products of their units are dimensionless (for example, in the case of conjugate Fourier pairs of functions of time and frequency, the products of the units of time and frequency, that is seconds × Hertz, are dimensionless). Examples of functions f (x) of a general variable x that are commonly encountered in Fourier analysis and their corresponding Fourier transforms are presented in Table 4.1.

4.1 An Introduction to Fourier Transforms

119

Table 4.1 Examples of commonly encountered Fourier transforms in radio astronomy Function f (x) Gaussian Function e−π

Fourier Transform F (s) e−π s

x

 0 1

Rectangle Function Π (x) =  Triangle Function Λ(x) =

: :

|x| > 1/2 |x| ≤ 1/2 sinc (π s)

0 : 1 − |x| : 

Dirac Delta function δ(x) =

|x| > 1 |x| ≤ 1 sinc2 (π s)

+∞ : 0 :

x=0 x = 0 1

Below are commonly encountered theorems related to Fourier transforms: in each case, f (x) and F (s) form a conjugate Fourier pair, as do g(x) and G(s). Proofs of these theorems are left as exercises (see Problem 3.4): • Addition Theorem: f (x) + g(x) −→ F (s) + G(s)

(4.20)

• Shift Theorem: (where a is a constant) f (x − a) −→ e−i2π as F (s)

(4.21)

• Similarity Theorem: (where a is a constant) f (ax) −→ |a|−1 F (s/a)

(4.22)

• Modulation Theorem: f (x) cos(ωt) −→

1

ω  1

ω  F s− + F s+ 2 2π 2 2π

(4.23)

f  (x) −→ i2π sF (s)

(4.24)

• Derivative Theorem:

120

4 Radio Observations

Example Problem 4.1 Show that the Fourier transform of f (x) = Π (x) (the function that is known as the rectangle function) is F (s) = (sin(πs))/πs. This latter function is also known as the function sinc (s). Solution Inserting Π (x) into Eq. (4.17) and using the variables x and s for t and ν, respectively, yields  F (s) =

+∞

−∞

 f (x)e2π ixs dx =

+∞ −∞

 Π (x)e2π ixs dx =

+1/2

−1/2

e2π ixs dx,

(4.26) where the bounds of integration have been changed to correspond to the range of values of x for which Π (x) is non-zero. Recalling Euler’s formula in Eq. (2.94), this integration becomes  F (s) =

+1/2 −1/2

(cos (2π xs) - i sin(2π xs)) dx.

(4.27)

This integration may be evaluated to produce   sin(−2π s/2) i cos(−2π s/2) sin(2π s/2) i cos(2π s/2) + − + , F (s) = 2π s 2π s 2π s 2π s (4.28) which in turn—based on the trigonometric identities sin (−x) = − sin (x) and cos (−x) = cos(x)—simplifies to F (s) =

sin(π s) = sinc(s). πs

(4.29)

In this final step, the normalized sinc function has been introduced.1 This function—which is commonly encountered in both physics and engineering—is defined as above for s = 0; when s = 0, this function is defined to be the limiting value (continued)

1 For

completeness, the sinc function itself is defined (for s = 0) sinc(s) =

sin(s) , s

(4.25)

where the limiting value corresponding to s = 0 is likewise defined to be 1. In contrast to the normalized sinc function, the definite integral of this function over all real values of s is π .

4.2 Convolution

121

Example Problem 4.1 (continued) sin(π s) lim = 1. s→0 πs

(4.30)

Note that the normalization factor of 1/π ensures that the definite integral of this function over all real values of s is unity.

4.2 Convolution Convolutions are commonly encountered in all branches in observational astronomy where a detector measures a signal (either an image or a spectrum) from a source of interest. The resulting image or spectrum is modulated by the response of the detector to the signal and the modulation depends on the resolving capabilities (imaging or spectroscopic) of the detector. For example, if the same astronomical source is imaged by two detectors and one detector has superior angular resolution capabilities than the other, the image produced by the former detector reveals more fine detail in the source than the image produced by the latter detector. The differences in the amount of detected detail is due to the different capabilities of the detector instead of intrinsic properties of the observed astronomical source. A detailed description of convolution is presented in [2] and only a brief summary is presented here. Consider two functions, F1 (t) and F2 (t), which have Fourier transforms of f1 (ν) and f2 (ν), respectively. The convolution C(t) of F1 (t) and F2 (t) is defined as  C(t) =

∞

−∞

F1 (t  )F2 (t − t  )dt  ,

(4.31)

which is typically written symbolically in the literature as C(t) = F1 (t)  F2 (t).

(4.32)

Convolutions are related to Fourier transforms in that the convolution C(t) of the two functions F1 (t) and F2 (t) is equal to the product of their Fourier transforms f1 (ν) and f2 (ν). This may be proven as follows: defining the Fourier transform of C(t) as D(s) and taking the Fourier transform of both sides of Eq. (4.32) yields  D(s) =

∞ −∞

 C(t)e2π ist dt =

∞



∞

−∞ −∞

F1 (t  )F2 (t − t  )e2π ist dtdt  .

(4.33)

This integration may be performed through a change of variables: by making the substitution that u = t − t  , the variable t  is held constant during the integration over t and dt = du. Therefore, Eq. (4.33) becomes

122

4 Radio Observations

 D(s) =

∞



∞

−∞ −∞



F1 (t  )F2 (u)e2π is(t +u) dt  du.

(4.34)

These two integrals may be separated as  D(s) =

∞ −∞



F1 (t  )e2π ist dt 



∞ −∞

F2 (u)e2π isu du = f1 (ν) · f2 (ν).

(4.35)

What is the relevance of the convolution theorem to astronomy in general and radio astronomy in particular? The final image generated by an observatory (as collected by either a mirror for an observation at optical wavelengths or by an antenna for an observation at radio wavelengths) is produced by the convolution of an input flux of electromagnetic radiation from a source of interest with the response characteristics of the observatory. More broadly speaking, any measurement made by a detector can be envisioned as the convolution of an input signal with the properties of that detector to that input signal. How can the convolution theorem be imagined? Consider two optical reflecting telescopes featuring mirrors with different diameters observing the Moon. As discussed in Sect. 4.3, the angular resolution of a reflecting telescope—that is, a measurement of its ability to resolve fine angular structure—is inversely proportional to the diameter of its mirror such that larger mirrors are capable of discerning structures of smaller angular sizes. Therefore, even though both telescopes observe the Moon and thus receive the same input source, the telescope with the larger mirror will be able to resolve craters on the Moon with smaller angular extents because the convolution of the input radiation with its response characteristics (commonly defined as the point spread function of the telescope) produces images that contain smaller angular-sized structures in comparison to the angular sizes of the structures revealed in the convolution of the input radiation from the Moon with the point spread function of the smaller mirror. Finally, discussions about convolutions and Fourier transforms also include the Rayleigh Theorem, which is also known as the Power Theorem or Parseval’s Theorem. If F (t) is a function and f (ν) is its Fourier transform, than the Rayleigh Theorem may be expressed as 

+∞ −∞

 |F (t)|2 dt =

+∞

−∞

|f (ν)|2 dν.

(4.36)

According to this theorem, the total energy contained in F (t) summed across all values of t is equal to the total energy contained in f (ν) summed across all values of ν. The proof of this theorem as expressed in Eq. (4.36) is left as an exercise (see Problem 4.3).

4.3 Angular Resolution and Observing Through Earth’s Atmosphere

123

4.3 Angular Resolution and Observing Through Earth’s Atmosphere Properties of single dish radio telescopes are discussed here first with the intent of establishing key concepts that will be revisited when arrays of radio telescopes are discussed later in the chapter. The first crucial characteristic of any telescope (radio or otherwise) is its angular resolution, that is, its ability to reveal fine structure in a source of interest. Angular resolution may be illustrated by imagining an observer during night time scrutinizing the headlights of a distant oncoming car. At some point as the car approaches, the observer will be able to resolve the combined beam of the two headlights of the car into the individual beams of each headlight: precisely when the observer can resolve the combined beams into individual ones is dictated by the angular resolution capabilities of the observer’s eyes. Another frequently cited illustration of angular resolution is the scenario of a binary star system with a tiny apparent angular separation between its two component stars that cannot be resolved by a telescope with a small aperture but can be resolved by a telescope with a large aperture. Often, astronomers want to resolve the smallest-sized structure possible in a source of interest. The angular resolution Θ in radians of a telescope with an aperture diameter d making an observation at a wavelength of λ is Θradians (FWHM) = 1.22

λ λ ∼ , d d

(4.37)

or, in terms of arcseconds (see the discussion about angular sizes in Sect. 2.1.1), Θarcseconds (FWHM) =

2.06 × 105 λ d

(4.38)

Here, “FWHM” is an acronym for “full-width at half-maximum”: in general it refers to the range of values between which the function of interest is equal or greater than half of its maximum value. In the present case, it refers to the range of angle in which the amplitude of the signal as focused by the antenna into a typically Gaussian shape with a full-width at half-maximum as described by Eqs. (4.37) and (4.38). In radio astronomy as well as in engineering applications of antenna theory, the unit of decibel—abbreviated as a dB—is also used to describe the focusing and transmitting capabilities of an antenna. A decibel is actually a logarithmic unit used to express the ratio between two values of a physical quantity such as power or intensity: one decibel is one-tenth of one bel. It is crucial to note that—as a ratio—a decibel corresponds to a linear difference in amplitude of a signal but to a square-root difference in the power of a signal. For example, a difference of 10 dB corresponds to a difference in a factor of 10 in the power detected from two sources but a factor of 3.162 (the square root of 10) in the amplitude of a signal (recall that the power detected from a source is proportional to the square of the amplitude of the wave). Therefore, the “FWHM” also corresponds to a range where the power

124

4 Radio Observations

contained by a telescope drops by a factor of 3 dB (corresponding to a ratio of two in power), thus the correspondence with the FWHM definition. Defining Q as the ratio between a measured power Pm and a reference power P0 , then Q may be expressed in decibels as  Q = 10 log10

Pmeasured Preference

 (4.39)

Example Problem 4.2 Verify that a power ratio Pmeasured /Preference of 1000 corresponds to Q = 30 dB. Solution From Eq. (4.39), Q may be computed as Q (dB) = 10 log10 (1000) = 10(3) = 30 dB.

(4.40)

Equations (4.37) and (4.38) define the diffraction-limited angular resolution that may be attained by a telescope. This equation also defines the primary beam of the radio telescope. For several instrumental reasons, the shape of the primary beam of single dish radio telescopes more closely resembles a Gaussian shape, and the solid angle ΩB subtended by the primary beam in steradians is [3] 2 (FWHM). ΩB = 1.133Θradians

(4.41)

In the following example problem, Eq. (4.37) is derived through consideration of diffraction theory for a circular aperture.

Example Problem 4.3 Through Fraunhofer diffraction theory for a circular aperture, derive Eq. (4.37). Solution This derivation is based on the discussion presented by Arfken [4]. We first define a polar coordinate system centered at the middle of the circular aperture, with θ as the azimuthal angle in the plane of the circular aperture and r as the radius measured from that middle point. The radius of the aperture itself is a and α is the angle between a normal through the center point on the aperture and a point on the screen below the aperture. Using Euler’s formula (see Eq. (2.94)) expressed in polar coordinates to describe the passage of the light wavefront through the aperture in a mathematically expeditious manner, we may express the amplitude Φ of the diffracted wave as approximately (continued)

4.3 Angular Resolution and Observing Through Earth’s Atmosphere

125

Example Problem 4.3 (continued) (where the integrations are performed over all radii of the aperture and all azimuthal angles) 



a

Φ∼ 0

2π

eibr

cosθ

dθ r dr,

(4.42)

0

where b=

2π sin α λ

(4.43)

and λ itself is the wavelength of the incident wave. In Eq. (4.42), the exponent ibr cos θ corresponds to the phase of the wave on the screen (assumed to lie at a large distance from the aperture) at the angle α relative to the phase of the wave incident on the aperture at the point (r, θ ). The imaginary exponential form of this integrand means that the integral is technically a Fourier transform (see Sect. 4.1). In general, the Fraunhofer diffraction pattern is given by the Fourier transform of the aperture. Into Eq. (4.42) a substitution is made such that 1 J0 (x) = 2π



2π

eix cosθ dθ,

(4.44)

0

where J0 (x) is a Bessel function of the first kind. For the sake of completeness, it is noted that J0 (x) may also be expressed as J0 (x) =

1 2π



2π

eix sinθ dθ,

(4.45)

0

depending on the context and particular situation under consideration. A more thorough description of Bessel functions is presented in Appendix 4.5.3: after making the substitution expressed in Eq. (4.44), the new form of Eq. (4.42) becomes  a J0 (br) r dr. (4.46) Φ ∼ 2π 0

To perform this integration the recursion relations for Bessel functions of the first kind are utilized (see Appendix 4.5.3 and Eq. (4.124)). As a special case of the recursion relations presented in Appendix 4.5.3, J0 (x), may be expressed as J0 (x) = −J1 (x),

(4.47) (continued)

126

4 Radio Observations

Example Problem 4.3 (continued) so performing the integration in Eq. (4.46) yields λa 2π ab J1 J1 (ab) ∼ Φ∼ sin α b2



2π a sinα λ

 (4.48)

,

where we have made use of Eq. (4.43). The intensity I (α) of the diffraction pattern is proportional to Φ 2 , which may be expressed as (omitting the constant factor of λ2 a 2 when Φ is squared)  I (α) ∼ Φ ∼ 2

J1 ((2π a sin α)/λ) sin α

2 .

(4.49)

For completeness, also consider the values of 2πa sin α for which Eq. (4.49) equals zero: the first minimum of this equation would completely enclose the first maximum in the diffraction pattern as a function of α. The first zeros of J1 (x) are known to occur at (ignoring the trivial zero when x = 0), J1 (x) = 0 when

x = 3.8317, 7.0156, 10.1735 . . . ,

(4.50)

The first minimum is found at 3.83λ 2π a sin α ≈ 3.83 → sin α ≈ , λ 2π a

(4.51)

and—after collecting constants and substituting in the diameter d for the aperture (d = 2a), this equation becomes α = 1.22

λ λ ∼ . d d

(4.52)

This equation is best known as the Airy disk equation and it describes the angular extent of the spherical diffraction pattern that is known as the Airy disk. This equation indicates the location of the first dark ring seen in the diffraction pattern seen through a circular aperture and is commonly encountered in diffraction theory.

Regarding Eqs. (4.37) and (4.38), note that these equations are relevant only for a telescope that is diffraction-limited. Specifically, for a given telescope, this equation describes the smallest structure that may be resolved by a telescope and assumes that there is no interfering medium between the telescope and the source of interest. Such a situation may be imagined for an optical telescope placed on the surface of the Moon: due to the lack of a substantial atmosphere, the telescope

4.3 Angular Resolution and Observing Through Earth’s Atmosphere

127

observes through the vacuum of space and therefore it attains its theoretical resolution given by Eqs. (4.37) and (4.38). If the telescope was instead placed on the surface of the Earth, light from sources must pass through the atmosphere on its way toward the telescope, and passage through the atmosphere introduces effects (such as turbulence or the “twinkling” of stars) that prevent the telescope from attaining this optimum resolution. Astronomers employ the term seeing to describe the stability of the atmosphere above a telescope (and therefore describe the quality of images obtained by that telescope). “Good” seeing refers to a site where the atmosphere is particularly stable: seeing itself may be quantified as the magnitude of the smallest angular size that may be resolved readily by the telescope at the site (for example, a seeing of one arcsecond indicates that the atmosphere limits the resolution to an angle of that size). An optical telescope may have a theoretical angular resolution (according to Eq. (4.38)) of 0.5 arcseconds but the seeing at the location of the telescope may limit the resolution capabilities of a telescope to 1 arcsecond. If this is the case, we say that the telescope is seeing-limited rather than diffraction-limited because the atmosphere (rather than the diameter of the aperture of the telescope and the wavelength at which observations are conducted) limit the attained angular resolution. In most cases in radio astronomy, radio telescopes are diffractionlimited rather than seeing-limited because the atmosphere is broadly transparent to electromagnetic radiation at radio wavelengths. The transparency of the Earth’s atmosphere to radiation at different wavelengths is illustrated in Fig. 4.1: note that the two domains of the electromagnetic spectrum where the Earth’s atmosphere is essentially transparent are the optical and the radio domain. In relevance to the present study, electromagnetic radiation with wavelengths in the centimeter range can be readily detected by radio telescopes placed on the Earth’s surface. For shorter wavelengths (in the millimeter range, for example), the atmosphere of the Earth is less transparent to such radiation due to such phenomenon as absorption and emission by molecules that compose the Earth’s atmosphere, such as water vapor. To address this problem, radio observatories that operate at such low wavelengths are placed in locations that feature a high elevation and a low moisture content, such as the Atacama desert, the site of the ALMA observatory. For observations made at long wavelengths (such as a meter or greater), the ionosphere of the Earth—a high altitude band of electrons located just beneath the Earth’s magnetosphere—blocks all radiation at that wavelength from reaching the Earth’s surface. This domain of the electromagnetic spectrum represents an unexplored frontier in astronomy. While the Earth’s atmosphere is transparent to light at radio wavelengths, radio astronomers must still contend with the ever worsening problem of artificial interference. One way to provide some shielding from this interference is to place telescopes in remote regions with low population density, or to take advantage of natural geographic features on Earth. For example, placing a radio telescope in a valley reduces the exposure to artificial radiation that would be encountered by a telescope situated at an exposed location such as the top of a hill. Also, a site surrounded by mountains—such as the Plains of San Agustin in New Mexico, where the VLA is located—also provides some measure of protection against

128

4 Radio Observations

Atmospheric Opacity

100%

50%

0% 0.1 nm 1 nm

10 nm 100 nm 1 µm

Gamma Rays, X-Rays and Ultraviolet Light blocked by the upper atmosphere (best observed from space)

10 µm 100 µm 1 mm Wavelength

Visible Light observable from Earth, with some atmospheric distortion

1 cm

10 cm

1m

10 m

100 m

1 km

Most of the Long-wavelength Infrared spectrum Radio Waves observable Radio Waves absorbed by from Earth. blocked. atmospheric gasses (best observed from space).

Fig. 4.1 (top) The opacity of the Earth’s atmosphere as a function of the wavelength of radiation: notice that the atmosphere is completely opaque to short wavelength radiation (0.1–100 nm, corresponding to the gamma-ray through ultraviolet domains) and only at best partially transparent to infrared wavelengths (1 µm to 1 cm). Atmospheric windows exist in the optical (400–700 nm) range and in the radio (10 cm to 10 m) where the atmosphere is either completely or nearly completely transparent. The ionosphere of the Earth blocks light with wavelengths of 10 m and greater. (bottom) Illustration of telescopes based on the transparency of the atmosphere: the high opacity to gamma-ray, X-ray, ultraviolet and infrared light motivates the placement of observatories operating in these wavelength domains in orbit, while the transparency of the Earth’s atmosphere at optical and radio wavelengths motivates the placement of telescopes on the Earth’s surface. From Wikipedia

stray radiation. By international treaty, particular bands of the electromagnetic spectrum have been allocated for radio astronomy and terrestrial transmission at these wavelengths is prohibited. A prominent example of such a “protected” wavelength is the 21-cm line, which corresponds to a transition associated with atomic hydrogen that astronomers use to detect cold hydrogen in the Universe. The physical origin of this line and its utility in modern radio astronomy is described in Sect. 6.2.2.

4.4 Single Dish Radio Astronomy The following discussion defines and describes some of the most commonly encountered terminology that is relevant to making observations with single dish radio telescopes: considerable overlap exists between the terminology used for

4.4 Single Dish Radio Astronomy

129

single dish radio telescopes and the terminology used for interferometers, as discussed later in this chapter. A quick description of the hardware used in radio astronomy is now given.2 It is emphasized that astronomical sources are quite weak in nature (as described below) and therefore much of the detection of these sources relies on the amplification of this weak signal by a considerable margin. The most commonly encountered type of receiver used by radio telescopes is known as the superheterodyne receiver. For such a receiver, a signal power is received from an antenna operating at a radio frequency νRF : the signal power is input into a radio-frequency (RF) amplifier that applies a gain G to the signal. In this context, gain may be defined as the increase in the strength of the signal as attained by the amplifying hardware. The amplified signal power is then input into a mixer, where the signal is mixed with a much stronger signal produced by a local oscillator (LO) operating at a frequency νLO : the output from the LO is at an intermediate frequency (IF) and the signal power at the IF frequency is directly proportional to the input signal power at the RF frequency. From here, the signal is input into a detector. Consider now an astronomical source with a brightness function Bν that appears to be extended in the sky (that is, the source does not appear to be point-like but instead spans over a range of θ and φ in the sky). The total flux density Sν of the source may be expressed as   Sν =

Bν (θ, φ) dΩ dν,

(4.53)

where the integrations over θ and φ are performed over the bounds that encompass completely the apparent angular extent of the source in the sky. A flux density is commonly expressed in units of 1 Jansky = 1 Jy = 10−26 W m−2 Hz−1 = 10−26 J m−2 s−1 Hz−1 . Notice here the significance of the presence of “Hz−1 ” in this expression: it refers to the bandwidth Δν implemented by the detector when observations are being made. If two identical telescopes operating with differentsized bandwidths observed a source, the telescope with the larger bandwidth will detect more emission from the source (assuming that the source is a continuous source of radiation and the amount of radiation emitted does not vary significantly over the implemented bandwidths). For that reason, the sizes of the bandwidths of a detector must be taken into account when the flux densities of an observed source are calculated. In radio astronomy (the domain of long wavelength and low-frequency astronomy), emission from bodies that produce primarily thermal radiation can be modeled using the Rayleigh–Jeans approximation (see Sect. 3.1 and Eq. (3.1)). The relation may be re-expressed in terms of frequency ν as

2 Since

the emphasis of this textbook is on the natural phenomena and physical processes encountered in radio astronomy, it is by intent that only a quick summary is presented here in the present volume.

130

4 Radio Observations

Bν (ν, T ) =

2ν 2 kB T . c2

(4.54)

This relation is applicable when the condition hν/kB T 1 is valid: a useful relation to specify the appropriate range of ν and T is ν (GHz) 20.84T ,

(4.55)

where an emphasis is placed on the fact that ν is in units of GHz in this equation.3 In implementing the Rayleigh–Jeans approximation, radio astronomers often utilize the concept of a brightness temperature TB : when conducting an observation of an extended source, TB corresponds to the temperature that would produce the observed brightness if it is inserted for T into the Rayleigh–Jeans approximation equation (keep in mind the two forms of this equation that are presented in Eqs. (3.1) and (4.54)). Therefore the appropriate expressions for TB are TB =

Bν c 2 Bλ λ 2 = . 2 2kB 2kB ν

(4.56)

If the source that is emitting the radiation may in fact be treated as a blackbody, then TB may indeed correspond to the temperature of the emitting source. Note that if the emission from the source is non-thermal (for example, it has a synchrotron origin), TB cannot be interpreted as the temperature of a source but it still may be considered to be a useful quantity. For example, in the case where the nature of a source is uncertain, if the measured value of TB is unphysically high (say TB ∼ 1012 K or greater), it may be concluded that the nature of the emission from the source is unlikely to be thermal in origin. The concept of a brightness temperature can also be related to the optical depth τ of the medium: for the case of an optically thick medium (τ  1), the expression for TB becomes TB = τT , while in the case of an optically thin medium (τ 1), the expression is TB = T . A schematic diagram depicting the radiation pattern of a single dish antenna is shown in Fig. 4.2. The key characteristics to note in this figure include the numerous lobes which indicate the directions at which the telescope is sensitive to radiation: the sizes of the lobes are directly proportional to the sensitivity of the antenna as well. Note that—as expected—the sensitivity of the antenna is greatest in the 3 For

the sake of completeness, it is noted that in the other range of the ratio of ν and T —that is, where e−hν/kB T  1 and hν/kB T  1—the approximation for (3.4) becomes Bν =

2hν 3 −hν/kB T e , c2

(4.57)

which is commonly known as Wien’s approximation. While this approximation is encountered frequently at short wavelengths such as optical, it is not applicable to radio astronomy.

4.4 Single Dish Radio Astronomy

131

Fig. 4.2 The lobes that comprise the radiation pattern of an antenna. Note the primary lobe— the lobe located along in the forward direction and along the main lobe axis—is the most prominent. This prominence indicates that the antenna is most sensitive in the direction that— by the reciprocity theorem—a signal emitted by the antenna will be directed most strongly in the forward direction of the antenna. Note also that lobes exist even in the “sideways” and backward directions as well, so that the antenna is susceptible to emission from sources located off-axis and even behind the antenna as well. For this reason, the antenna can transmit in sideways and backwards directions as well

“forward” direction: this particular lobe corresponds to the primary beam of the antenna and its width can be calculated from Eqs. (4.37) and (4.38). Also notice that secondary lobes of smaller amplitude (corresponding to lower sensitivity) are arrayed on either side of the primary beam. Inspection of the locations of the secondary lobes illustrates why observations of low elevation sources can be undesirable: in those situations, one of these lobes may “touch” the surface of the Earth and thus intercept stray thermal radiation from the Earth (note that the Earth emits radiation much like a blackbody with a surface temperature of approximately 300 K). This stray radiation will manifest itself as confusing emission that will corrupt the observation of the source of interest. Finally, rear-facing lobes are also seen in this figure: these lobes indicate that the antenna will also be sensitive to radiation even from positions that are located directly opposite from the direction at which the antenna is pointed. Therefore, an observer must take into account stray radiation from multiple directions (even directly behind an antenna) when addressing the effects of noise that may interfere with an observation and confuse the detected signal.

132

4 Radio Observations

One of the most important concepts to discuss in the context of discussing the detection of radiation from an antenna is the reciprocity theorem. A fuller description is provided elsewhere [5] and only a brief description is provided here, imagine two antennae immersed in a medium that may be described as reciprocal (meaning here that the propagation properties through the medium are independent of the direction of propagation). Suppose a source of known intensity “illuminates” one of the two antennae, exciting it to emit radiation and the other antenna measures the power emitted by the excited antenna. The measured power by the second antenna is equivalent to the power that would be received if the source emitted its power through the first antenna through the connection of a transmission line, for example. This result is independent of the orientations of the two antennae to each other: also, the antennae need not be identical to each other in characteristics such as geometry. This result holds even if the source is moved to a large distance away from the receiving antenna, to the limit where the source may be considered to be point-like (that is, its apparent angular extent is negligible). In essence, the angular response of any antenna when used to transmit a signal is the same as when the antenna is used to receive a signal. Therefore, to model the sensitivity of an antenna to incoming radiation, it is entirely equivalent (and often easier conceptually) to model how the antenna emits radiation in different directions. Consider a single dish radio telescope with a diameter D and a corresponding geometric area A = π(d/2)2 . Because of factors such as blockage by a feed placed at the prime focus of the telescope as well as the support structure of the feed, a single dish telescope is not able to conduct observations using its full geometric area. Instead, the aperture efficiency η is the fraction of the geometric area that is actually “brought to bear” on the source during the observation: typical values of η are 0.5 through 0.7 and specific values are functions of the wavelengths at which the observations are conducted. The functional area of the telescope for the observation—known as the effective area Aeff — of the telescope may be expressed as a function of η as Aeff = ηA.

(4.58)

Furthermore, the total power Prec received by the telescope with an effective area Aeff from a source that emits a flux density Sν may be expressed as Prec = Aeff Sν Δν,

(4.59)

where Δν is the frequency bandwidth over which the observation is made. In this expression, the electrical power received by the source can also be envisioned as the power emitted by a resistor through which a time-dependent voltage V (t) is applied and thus a current I (t) passes. The origin of this current is the random motion of electrons: even though the mean value of I will be zero because of the random motion of the electrons, the root-mean-square value of the current will not be zero.

4.4 Single Dish Radio Astronomy

133

Therefore, the time average of the square of the current is also non-zero: recall that the square of a current is proportional to the power emitted by a resistor, and assuming that the resistor is in thermal equilibrium with its surroundings, the power emitted depends on the temperature. The noise generated by the random motion of electrons in a resistor is known as Johnson noise . Furthermore, the mathematical description of the random motion of electrons—through the modeling of the motion of the electron as a random walk and the relationship between this motion and an equivalent temperature—is known as the Nyquist sampling theorem and is described in more detail later in this section. With this background information and comparison with the thermal noise emission of a resistor, the total system noise power Ptotal may be expressed as Ptotal = kB TN Δν.

(4.60)

Here, kB is Boltzmann’s constant and TN is the noise temperature. This latter quantity is defined as the equivalent temperature TN that a resistor would need to be to emit a total power Ptotal through thermal noise. In a similar vein, the antenna temperature TA may be defined as the equivalent temperature that a blackbody must possess to produce the observed signal power through purely blackbody radiation. In terms of Sν and Aeff , TA may be expressed as TA =

Sν Aeff . 2kB

(4.61)

For a telescope making an observation at a certain wavelength λ, the gain G (a dimensionless quantity!) of the telescope is a crucial parameter in describing its sensitivity. Gain may be defined as the ratio of the maximum radiation intensity received by an antenna to the maximum radiation intensity received by an antenna from a reference antenna with the same power input [6]. Gain may also be defined as the increase in the strength of the signal by the hardware of the telescope. In effect, the gain describes the efficiency of an antenna to receive a signal. The directivity Δ is the maximum gain of a telescope (this may be realized when a telescope is pointed directly at a source) and it may be quantified as Δ = Gmaximum =

4π Aeff . λ2

(4.62)

Example Problem 4.4 Consider an observation made by a telescope with a diameter d = 30 m at an observing wavelength λ = 6 cm: at this wavelength, the aperture efficiency of the telescope is η = 0.6. The telescope is observing a source that has a flux density Sν = 100 Jy at this wavelength: the frequency bandwidth Δν of the detector used during the observation (continued)

134

4 Radio Observations

Example Problem 4.4 (continued) is 1 MHz. Calculate the angular resolution Θarcseconds of the telescope at this wavelength in arcseconds, the aperture efficiency Aeff in m2 , and the directivity Δ of this telescope, as well as the total power Prec in Watts received by the telescope, the noise temperature TN in Kelvin and the corresponding antenna temperature TA in Kelvin when this observation is made. Solution From Eq. (4.38), Θarcseconds is  Θarcseconds = 2.06 × 105



0.06 m 30 m

= 412 arcseconds.

(4.63)

From Eq. (4.58), Aeff is  Aeff = (0.6)π

(30 m)2 4

 = 424.11 m2 .

(4.64)

From Eq. (4.62), Δ is Δ = Gmaximum =

4π Aeff 4π(424.11 m2 ) = = 1.48 × 106 . 2 λ (0.06 m)2

(4.65)

From Eq. (4.59), Prec received per second is 

Prec

10−26 W m−2 Hz−1 = (424.11 m )(100 Jy) 1 Jy



2

(106 Hz) = 4.24 × 10−16 W.

(4.66)

Notice the remarkably low value for Prec : this result underscores the weakness of radio emission from astronomical sources, even when observations are made of sources with moderate radio brightness using medium aperture telescopes and receivers with large bandwidths. Using this value of Prec and recalling that 1 W = 1 J/s, the value of TN that corresponds to the observation of this particular source may be computed from Eq. (4.60) as TN =

4.24 × 10−16 J/s Prec = = 30.72 K. kB Δν (1.38 × 10−23 J/K)(106 Hz)

(4.67)

Lastly, from Eq. (4.61), TA is Sν Aeff (100 Jy)(424.11 m2 ) TA = = 2kB 2(1.38 × 10−23 J/K)



10−26 J m−2 s−1 Hz−1 1 Jy

 = 15.37 K. (4.68)

4.4 Single Dish Radio Astronomy

135

Regarding the hardware of a radio telescope, every component of equipment will introduce some noise that will propagate through the amplification of the received signal. Each of these components can be characterized by an equivalent temperature for the power emitted, and the sum of these component temperatures determines the system temperature Tsys of all of the hardware. Clearly, it is preferable to have all hardware contribute as little noise as possible, and, therefore, keeping Tsys to the lowest value possible. The value of Tsys also has a bearing on the minimum temperature fluctuation ΔTmin that can be measured by a system. An expression for ΔTmin can be derived from consideration of the Nyquist sampling theorem: if the total bandwidth of a receiver is Δν and samples are taken at a rate of Δt, then ΔνΔt = 1 is the time resolution of the taking of independent samples and any two samples taken in a time interval less than Δt = 1/Δν are not independent. For a sampling time ts , there are N = ts /Δt = ts Δν independent samples: because √ Gaussian statistics dictates that the total error for N samples is 1/ N, if the error of a single sample is Tsys , then the minimum detectable change in temperature ΔTmin is Tsys ΔTmin = √ , Δνts

(4.69)

and in terms of an arbitrary value for N samples, the expression for ΔT becomes Tsys . ΔTmin = √ Δνts N

(4.70)

Finally, the corresponding minimum detectable flux density ΔSmin may be expressed as ΔSmin =

2kB Tsys . √ Aeff Δνts N

(4.71)

Returning to the concept of gain mentioned previously, a challenge in amplifying signals from astronomical sources is that the gain from amplifiers is inherently variable. The effects of gain variations can be reduced if the input into the receiver is switched between the antenna and a comparison “noise” source. The frequency of the switching between the antenna and the noise source is high enough that the gain does not have enough time to change appreciably during one cycle. This device that strives for a more stable gain by switching inputs between the antenna and the noise source is known as the Dicke switch: specifically, in a Dicke switch the receiver is connected to the antenna for only half of the time and thus the minimum detectable change in temperature ΔTmin by the detector is π Tsys , ΔTmin = √ 2Δνts N

(4.72)

136

4 Radio Observations

and the corresponding minimum detectable flux density ΔSmin is ΔSmin =

2π kB Tsys , √ Aeff 2Δνts N

(4.73)

where the value of ΔTmin has been reduced and √ the value of ΔSmin has been increased by the same scalar factor, that is, π / 2. This particular scalar factor corresponds to a “best case” scenario for the Dicke switch when a square wave input modulation and sine wave multiplication is applied to the signal. In another scenario involving a Dicke switch, a square wave input modulation may be applied along with a square wave multiplication: in this latter case, the scalar factor becomes instead 2. Frequently, equations for ΔTmin and ΔSmin are expressed such that this scalar is given as a system constant ksys where—depending on the hardware setup—values of this constant range from 0.5 through 2.83. The use of a Dicke switch has the chief advantage of providing a very stable reference voltage for an observation: the main disadvantage is that time spent in connection with the reference voltage is at the expense of the time spent observing the source, thus necessitating a longer exposure time to detect a source at a similar level of significance. From Eqs. (4.72) and (4.73), a ratio of ΔSmin to ΔTmin may be expressed in the following very useful relation (in two different forms): ΔSmin ΔTmin





J m2 K

=

2kB Aeff

or

ΔSmin ΔTmin



 Jy 2 × 1026 kB = . K Aeff

(4.74)

Example Problem 4.5 (a) Assume that the system temperature Tsys of the telescope described in Example Problem 4.4 is Tsys = 70 K. Use Eqs. (4.70) and (4.71) to compute ΔTmin and ΔSmin for the telescope using the given hardware parameters. (b) Assuming that a Dicke switch is in place when the observations are made, use Eqs. (4.72) and (4.73) to recompute ΔTmin and ΔSmin . Solution (a) From Eq. (4.70), ΔTmin may be computed as ΔTmin =

70 K (106 Hz)(1 s)(1)

= 0.07 K.

(4.75)

From Eq. (4.71), ΔSmin may be computed as ΔSmin =

2(1.38 × 10−23 J/K)(70 K) = 0.46 Jy. (424.11 m2 ) (106 Hz)(1 s)(1)

(4.76) (continued)

4.5 Interferometry

137

Example Problem 4.5 (continued) (b) From Eq. (4.72), ΔTmin may be computed as ΔTmin =

π(70 K) 2(106 Hz)(1 s)(1)

= 0.15 K.

(4.77)

From Eq. (4.73), ΔSmin may be computed as ΔSmin =

2π(1.38 × 10−23 J/K)(70 K) = 1.01 Jy. (424.11 m2 ) 2(106 Hz)(1 s)(1)

(4.78)

The values for ΔTmin and ΔSmin in the case of the Dicke switch are noticeably larger than the previous case. Again, Dicke switches have the advantage of providing more sampling time for the calibration source (and thus providing better calibration for the observation) at the expense of time spent observing the source of interest.

4.5 Interferometry To make meaningful comparisons between imaging observations made at radio wavelengths and observations made at other wavelengths, comparable angular resolution must be achieved. For example, consider an optical telescope with an aperture diameter of 1 m making an observation at a wavelength λ = 500 nm = 500 × 10−9 m: from Eq. (4.38), the angular resolution attained by the optical telescope (assuming it is diffraction-limited) is ≈0.1 arcseconds. For comparison, the 110Meter Green Bank telescope—the largest steerable single dish radio telescope on the planet—observing at a commonly encountered radio astronomy wavelength of 21 cm attains an angular resolution of 1.9 × 10−3 radians ≈394 arcseconds, a factor of more than 104 greater! Costs (both for initial construction and for maintenance) and engineering limitations make the development of steerable radio telescopes with apertures larger than approximately 300 m prohibitive and impractical.4

4 The

largest single dish radio telescope in the world is the 305-m in Arecibo, Puerto Rico: it is also known as the National Astronomy and Ionosphere Center (NAIC). Operated by SRI International, the Universities Space Research Association and the Universidad Metropolitana of San Juan, Puerto Rico, this telescope was built inside the depression of a karst sinkhole. Rather than physically moving to receive radiation from different parts of the sky, the NAIC operates as a transit telescope, where radiation is detected from sources as the rotation of the Earth carries them through the field of view of the telescope [7]. The 600-Meter RATAN telescope—located in the Greater Caucasus mountains of Russia and operated by the Special Astrophysical Observatory of the Russian Academy of Sciences—consists of a 895 rectangular radio reflectors with dimensions

138

4 Radio Observations

How then can radio astronomers attain high angular resolution that is comparable to the angular resolution routinely attained at shorter wavelengths? Through the phenomenon of interferometry, where multiple telescopes are linked together to form a system known as an interferometer. As the individual telescopes of an interferometer observe a source, signals from each telescope are interfered together to produce a final map. The angular resolution attained by an interferometer corresponds to the maximum distance between two of its member telescopes. In that manner, an interferometer that has a maximum separation between its telescopes of 1 km, for example, attains the angular resolution of a single dish radio telescope with an aperture diameter of 1 km. Note that when used in the context of an interferometer, the term primary beam has a slightly different definition: in that case, the primary beam is the field of view of the interferometer. Quantitatively, it corresponds to the ratio of the wavelength at which the observations are conducted and the diameter of one of the individual telescopes that make up the interferometer. Frequently, the element telescopes of an interferometer all have essentially the same diameter, so the choice of a particular telescope for computing the primary beam is arbitrary. The principles of interferometers are discussed below.

4.5.1 Two-Element Interferometer Consider two telescopes in an interferometer that are observing the same source: the two telescopes #1 and #2 are continuously measuring voltages—denoted as V1 (t) and V2 (t)—and the voltages are combined by a unit of hardware known as a correlator. As shown in Fig. 4.3, the separation between the two telescopes is b and the unit vector in the direction of the telescopes toward the source is s, so for one telescope (in this case, telescope #1) the light from the source must travel an additional distance b · s to reach the detector compared to the other telescope (telescope #2). Therefore, the geometric delay τg —that is, the corresponding time delay for a signal traveling at the speed of light c to reach one telescope compared to the other—may be expressed as τg =

b·s . c

(4.79)

Assuming that the telescopes are making observations at the same frequency ν and at the same bandwidth Δν, the correlator—which multiplies the input voltages and then time averages them—will produce an output voltage that is proportional to , that is, the time average of the product of the voltages. The received of 2 × 11.4 m arranged in a circle with a diameter of 576 m. Each reflector points toward a central conical receiver and the resolving power of the entire array of reflectors is equivalent to a single dish radio telescope with a diameter of 600 m. Like the Arecibo radio telescope, the 600-Meter RATAN operates primarily as a transit telescope and has limited steering capabilities [8].

4.5 Interferometry

139

Fig. 4.3 A simplified schematic diagram of a two-element interferometer. From [9]

signals from the two telescopes may be described as quasi-monochromatic Fourier components of frequency ν as, respectively, V1 (t) = v1 cos 2π ν(t − τg )

and

V2 (t) = v2 cos 2π νt,

(4.80)

where v1 and v2 are the amplitudes of the two received signals, respectively. Therefore the output of the correlator—denoted as R(τg )—is R(τg ) =< V1 (t)V2 (t) >= v1 v2 cos 2π ντg .

(4.81)

Note here that τg varies slowly with time as the Earth rotates and the oscillations seen as a result of the cosine term in Eq. (4.81) describe the motion of the source through the interferometer fringe pattern. Also note that the v1 v2 is proportional to the received power from the source.

140

4 Radio Observations

How can R(τg ) be expressed in terms of the radio brightness of the source integrated over the entire angular extent Ω of the source? First, the quantity A(s) is defined as the effective collective area of an individual telescope in the direction of s (assuming that all telescopes have the same such effective area): the units of A(s) are simply m2 . Likewise, the quantity I (s)—with units W m−2 Hz−1 sr−1 —is defined as the brightness of the radio emission of the source in the direction of s as observed at frequency ν. The individual response of each telescope is worth mentioning, in that each telescope is only sensitive to a component of the input radiation field determined by the telescope’s polarization. For this reason, the polarization of the telescopes may be changed to probe the nature of the radiation from the source (see Chap. 3). The differential signal power dP received over the bandwidth Δν from a source element dΩ as viewed in the direction of the unit vector s is dP = A(s)I (s)Δν dΩ,

(4.82)

and the resulting output from the correlator is proportional to the resulting power and the cosine fringe term (see Eq. (4.81)), that is, dR(τg ) = A(s)I (s)Δν cos 2π ν dΩ,

(4.83)

and integrating this equation over all solid angle and expressing in terms of b and s yields    2π ν(b · s) dΩ. (4.84) R = Δν A(s)I (s) cos c S Several assumptions that have been built into the derivation of Eq. (4.84) need to be explicitly stated. Firstly, it is assumed that the bandwidth Δν is so narrow that variations of A and I as a function of ν may be safely ignored. Secondly, strictly speaking the integral in Eq. (4.84) should be performed over the entire celestial sphere (that is, all 4π steradians—see Eq. (2.13)) but practically speaking the integrand drops to negligible values outside of a small angular field as a result of such effects as the finite angular dimensions of the radio source and the beamwidth of the antenna. Thirdly, it is also assumed that the far-field approximation for the observation of the source is applicable (that is, incoming wavefronts from the source may be assumed to be coplanar): this assumption is clearly valid for very distant sources (such as Galactic and extragalactic objects) but may not be completely sound in the cases of observations made of Solar System objects at high frequencies and with wide spacings between telescopes. Finally, it is assumed that responses from different positions in the angular extent of the source can be added independently, which requires that no correlation exists between signal components emitted from different positions in the angular extent of the source. When interferometric observations of astronomical sources are made, a position on which the synthesized field of view is to be centered is often specified. This position is commonly known as the phase tracking center or the phase reference position: it is denoted as s0 and it is drawn as a vector from the center of the

4.5 Interferometry

141

Fig. 4.4 Diagram of position vectors used in deriving the interferometric response to an extended source, indicated by the contours of radio emission (corresponding to the radio brightness I(s) on the sky). From [9]

baseline b connecting the two telescopes to the center of the angular extent of the astronomical source of interest. From this definition, s may be expressed as s = s0 + σ , where σ is a vector in the plane of the sky between the phase tracking center of the interferometric image and the location of the differential solid angle element dΩ. The position vectors and their definitions are illustrated in Fig. 4.4: based on this figure and the definitions of s, s0 , and σ , Eq. (4.84) becomes    2π ν(b · σ ) 2π ν(b · s0 ) dΩ A(σ )I (σ ) cos c c S     2π ν(b · s0 ) 2π ν(b · σ ) dΩ − Δν sin A(σ )I (σ ) sin c c S 

R = Δν cos

(4.85)

The concept of visibility is now introduced here: historically, in the studies of optical interferometers this term was used to express the relative amplitude of optical fringes. In the present context, it is a complex quantity with units of W m−2 Hz−1 . Visibility may be defined as the unnormalized measure of the coherence of the electric field of the radiation (though characteristics of the interferometer may modify this quantity somewhat) and expressed as

142

4 Radio Observations



˜ )I (σ )e A(σ

V = |V |eiφV =

−2π iν(b·s) c

dΩ

(4.86)

S

˜ ) = A(σ )/A0 is the normalized antenna reception pattern (where Here, A(σ A0 is the response at the beam center) and φV is the complex phase of the visibility. The real and imaginary components of V as expressed in Eq. (4.86) can be separated using Euler’s formula (recall Eq. (2.94)) to yield 



2π ν(b · σ ) A0 |V | cos φV = A(σ )I (σ ) cos c S

 dΩ

(4.87)

dΩ.

(4.88)

and 



2π ν(b · σ ) A0 |V | sin φV = − A(σ )I (σ ) sin c S



Substituting Eqs. (4.87) and (4.88) into Eq. (4.85) yields 

   2π ν(b · s0 ) 2π ν(b · s0 ) R = Δν cos A0 |V | cos φV − Δν sin A0 |V | sin φV c c (4.89) or       2π ν(b · s0 ) 2π ν(b · s0 ) cos φV − sin sin φV R = A0 Δν|V | cos c c (4.90) which may be simplified at last to be  R = A0 Δν|V | cos

2π ν(b · s0 ) + φV c

 (4.91)

where the trigonometric identity cos(θ1 + θ2 ) = cosθ1 cosθ2 − sinθ1 sinθ2

(4.92)

has been applied. How is Eq. (4.91) applied in practice to obtain the brightness distribution of a source? The typical procedure is to measure the amplitude and phase of the fringe pattern that is described by the cosine term in this equation: the amplitude and phase of V is then determined from proper calibration. By inverting the transformation in Eq. (4.86), the brightness distribution of the source is obtained from the visibility data. For this reason, V must be measured over a broad range of values for the quantity ν(b · σ )/c, that is, the component of the baseline between the two antennas normal to the direction of the source and measured in wavelengths, or alternatively, the baseline viewed from the direction of the source.

4.5 Interferometry

143

Fig. 4.5 The (u,v,w) coordinate system used to express interferometer baselines and the (l,m,n) coordinate system used to express the source brightness distribution. From [9]

It is crucial to establish and discuss the coordinate systems used in imaging when conducting interferometric observations. These coordinate systems are depicted in Fig. 4.5 and described here. The baseline vector that connects the telescopes can be envisioned as existing in a three-dimensional space known as the (u, v, w) coordinate system: in this system, the w coordinate points in the direction of the source (that is, towards the position of s0 , the center of the synthesized image), while the u and v coordinates point in the north and east directions, respectively. Therefore, the baseline separation between two telescopes can be imagined being projected into the (u, v) plane. Note also the units of u and v, which are in wavelengths of the center frequency where the radio observations are being made. In the (l, m, n) coordinate system—which complements the (u, v, w) coordinate system—the coordinates l and m are used to measure positions on the sky, while the coordinate n is in a direction normal to the (l, m) plane and in the same direction as w and the vector s0 . These coordinates are the direction cosines of the u and v coordinates and a synthesized image in the (l,m) plane can be treated as a projection of the celestial sphere onto a tangent plane that has the l and m coordinates as its

144

4 Radio Observations

origin. The relations between the (u, v, w) and the (l, m, n) coordinate systems and the parameters used to derive Eqs. (4.85) and (4.91) may be expressed as ν(b · s) = ul + vm + wn c ν(b · s0 ) =w c dl dm dl dm =√ dΩ = . n 1 − l 2 − m2

(4.93) (4.94) (4.95)

and therefore the expression for V given in Eq. (4.86) may be expressed in terms of the (u,v,w) and (l,m,n) coordinate systems as  V (u, v, w) =

+∞  +∞

−∞

−∞

√ ˜ m)I (l, m)e−2π i(ul+vm+w( 1−l 2 −m2 −1)) √ dl dm . A(l, 1 − l 2 − m2 (4.96)

The value of this integrand is taken to be 0 when l 2 + m2 ≥ 1. In this equation, note that V (u, v, w) is a function of the modified brightness distribution ˜ m)I (l, m). Also note that V is expressed as a function of (u, v, w) which A(l, represent the spacings of the telescopes with respect to s0 . How can this equation be simplified into a more tractable two-dimensional Fourier transform in (u, v) space (see Sect. 4.1)? Imagine that the baseline between the two telescopes is coplanar: the tip of the vector b that corresponds to this baseline will trace out a complete circle as a consequence of the Earth’s rotation (see Fig. 4.6). For an East–West baseline, b will point in a direction perpendicular to the rotation axis: in such a case, the components of the baseline vector parallel to the Earth’s axis vanish. If the w-axis is chosen to correspond to the direction of the celestial pole, such that w = 0, Eq. (4.96) simplifies to (Fig. 4.7)  V (u, v) =

+∞  +∞ −∞

−∞

˜ m)I (l, m)e−2π i(ul+vm) √ dl dm . A(l, 1 − l 2 − m2

(4.97)

This equation is simply a two-dimensional Fourier transform, and the inverse of this equation is ˜ m)I (l, m)  +∞  +∞ A(l, = V (u, v)e2π i(ul+vm) du dv. √ 2 2 1−l −m −∞ −∞

(4.98)

At last, then, the fundamental relation for interferometry is obtained here: the measured visibilities V (u, v) in the u, v plane by the interferometer are the Fourier transform of the brightness distribution I (l, m) of the source on the sky, that is, V (u, v) ⇔ I (l, m).

(4.99)

4.5 Interferometry

145

Fig. 4.6 Illustration of the effect of the Earth’s rotation on the projection of the baseline vector b as observed by the observed source. The rotation of the Earth causes b to trace out a circular locus in the plane normal to the direction of Declination δ = 90◦ . In the case where b corresponds to an East–West baseline on the surface of the Earth, then b itself is normal to the rotation axis of the Earth. From [9]

Fig. 4.7 The projection of a source located at (α0 , δ0 ) onto the celestial sphere and the tangent plane at the North Celestial Pole (δ = 90◦ —corresponding to the direction of the w-axis). The spacing-vector loci correspond to an array with East–West baselines and lie in a plane parallel to the equator of the Earth. From [9]

146

4 Radio Observations

4.5.2 Multi-Element Interferometers Inspection of Eq. (4.99) reveals that to obtain the most accurate measurement of I (l, m) possible, the number of measured viabilities V (u,v) needs to be maximized. One way to realize this goal is to use as many pairs of antennas as possible to help improve the coverage in the (u, v) plane: note if an interferometer has N individual telescopes, then the number P of pairs of interfering telescopes is P =

N(N − 1) . 2

(4.100)

To maximize the coverage of the (u, v) plane, the method known as aperture synthesis—as pioneered by Sir Martin Ryle, for which he received the Nobel Prize in Physics in 1973—takes advantage of the rotation of the Earth to realize this goal and is commonly employed. As the telescopes in the interferometer observe a source (treated as a position fixed on the celestial sphere), the rotation of the Earth causes the u and v components of the baseline between any pair of two telescopes to trace out a set of points that is elliptical in shape. This ellipse corresponds to the projection onto the (u, v) plane of the circular path traced out by the tip of the baseline vector b as the Earth rotates (see Fig. 4.6). Because I (l, m) is a real quantity, V (−u, −v) = V ∗ (u, v) so that at any instant the correlator—which receives the input voltages from the telescopes in the interferometer—can provide a measure of the visibility at two points in the u, v plane. An array of telescopes has an associated set of elliptical curves of points in the u, v plane: this set of curves is known as the transfer function or the sampling function, denoted as T (u, v). The transfer function describes the values of u and v at which the visibility function V (u, v) is sampled: the function itself depends on both the declination of the observed source and on the spacings between the individual telescopes. In Fig. 4.8, the coverage in the (u, v) plane is shown for a “snapshot” observation (that is, an observation with an extremely short duration): notice the symmetrical coverage of the (u, v) plane. The value of V (u, v) for a point source at the (l, m) origin is a constant in both u and v: therefore, the Fourier transform of T (u, v) corresponds to the response of the array to a point source, or—in other words—the point spread function or the synthesized beam of the array.5 When designing an array, the primary goal is to arrange the telescopes such that the transfer functions generated by the observations cover as much of the (u, v) plane as possible. Toward this end, arrays take advantage of the Earth’s rotation so that variable baselines between the antenna (as seen from the perspective of the source) change and curves are traced out in the plane. In addition, the spacings between the antennas are crucial as well: they are typically chosen in a manner that reduces redundancies in physical baselines between the arrays. For example, note that the three “arms” of the VLA are not equally offset from each other, but instead the axis of the north arm is offset by 5 degrees so that it is that much closer to one arm and that much further from

5 Be

sure not to confuse the synthesized beam with the primary beam as defined earlier.

4.5 Interferometry

147

Fig. 4.8 An example of the uv coverage attained by a “snapshot” observation made with the Very Large Array. Notice the very sparse coverage of the uv-plane and the zero-spacing gap at (u, v) = 0, 0. From http://archive.nrao.edu/nvas/read.shtml

another. The wisdom to this choice is that it avoids redundancies in baselines and allows a wider coverage of the (u, v) plane. In Fig. 4.9, a sample spacing of the VLA telescopes is presented that illustrates how the physical arrangement of the array deviates from perfect symmetry (i.e., the distances between each arm is not an evenly spaced 120◦ ). Also, transfer functions for observations of sources at a range of Declinations are also presented. What is the effect of finite bandwidth Δν coverage in interferometric observations? Recalling that interferometers “interfere” signals and thus produce fringes, the finite bandwidth acts as a modulating function to the fringes with a width that is inversely proportional to Δν. This modulation requires an additional instrumental delay (denoted as τi ) to be inserted into the paths between the telescopes and the correlator to compensate for the geometrical delay τg described previously. Including this delay is only effective for the center of the field of the observation, and variation of τg causes the image to be blurred in a radial manner.

148

4 Radio Observations

Fig. 4.9 A schematic diagram of the configuration of the 27 telescopes of the VLA (north is toward the top of the figure). Notice that the north arm is offset by 5◦ from true north: this offset is purposely designed to reduce redundancies in coverage of the (u, v) plane. From [12]

When conducting the observation, the visibility data are usually assumed to be collected at the center frequency ν0 and the observer is interested in the average brightness of the source over the full bandwidth Δν (see Fig. 4.10). In fact, the coordinates—known as spatial frequencies—in the (u, v) plane are calculated based on their relative position to the center of the bandwidth: note that u and v are projected telescope spacings measured in wavelengths (recall the inverse relationship between wavelength and frequency for electromagnetic radiation, hence the origin of the term “spatial frequencies”). There is a relation between the observing frequency and the coverage in the (u, v) plane: if (u0 , v0 ) corresponds to frequency ν0 and (u, v) corresponds to another frequency ν within the bandwidth, then

ν u ν v  0 0 , (4.101) (u0 , v0 ) = ν ν A visibility that corresponds to a small portion of Δν contributes a component to the brightness I (l, m) through the Fourier transform relationship given in Eq. (4.99). Therefore, because the processes of correlation of visibilities from telescopes and Fourier transforms are linear, the final image obtained of the source may be considered to be a sum of contributions from different regions of the frequency bandwidth. When the image is created, values of u0 and v0 are assigned to the visibilities which are the original values of u and v multiplied by ν0 /ν. From the similarity theorem of Fourier transforms (see Eq. (4.22)), the original Fourier transform between V (u, v) and I (l, m) becomes

4.5 Interferometry

149

Fig. 4.10 (a) The (idealized) rectangular response for a detector centered on an observing frequency ν0 and with a bandwidth Δν. (b) The radial smearing of a point source (l1 , m1 ) in the synthesized image. From [9]

V

  2 

ν u ν v  ν νl νm 0 0 . , ⇔ I , ν ν ν0 ν0 ν0

(4.102)

Note that the coordinates of I (l, m) are multiplied by the reciprocal of the factor ν0 /ν by which the coordinates of V (u, v) and a factor of ν/ν0 appears in the amplitude to conserve the total integrated brightness of the source. Therefore, the procedure where the image is synthesized may be viewed as the combining of all of the visibilities over the entire bandwidth Δν is equivalent to averaging a series of images of the same sky brightness distribution. Each image is aligned at the origin of (l, m) and each has a slightly different scale factor, and the range of the value of the scale factor is equivalent to the variation of ν/ν0 over the full range of Δν. This averaging process introduces a radial smearing into the brightness distribution (see Fig. 4.10): this phenomenon is known as √ bandwidth smearing and the angular extent of the smearing at a radial distance l 2 + m2 from the origin is √ approximately equal to (Δν/ν0 ) l 2 + m2 . The magnitude of the effect of bandwidth smearing becomes important at distances for which the smearing is comparable to the synthesized beamwidth. To help build continuity with the discussion presented earlier in this chapter about single dish radio telescopes, analogs to the equations previously discussed

150

4 Radio Observations

are presented. Consider an interferometer comprised of N telescopes each with a diameter D and with spacings ranging from a minimum baseline denoted as bshortest and a maximum baseline denoted as blongest . First, the maximum angular resolution Θradians (FWHM) of the interferometer may be determined in an analogous manner to the angular resolution of a single telescope. Recalling that the angular resolution of a single telescope is inversely proportional to the diameter of its aperture, Θmaximum (FWHM) of an interferometer in units of radians is inversely proportional to blongest and may, therefore, be expressed as (see Eq. (4.37)) Θmaximum (FWHM) = 1.22

λ blongest

.

(4.103)

Here, λ once again is the wavelength at which the interferometer is making observations. Next, the diameter ΘFOV (FWHM) of the field of view (FOV) of the interferometer corresponds to merely the angular resolution of a single element within the interferometer (see Eq. (4.37)) and may be expressed as ΘFOV (FWHM) = 1.22

λ . D

(4.104)

Finally, according to the Ekers and Rots theorem [10], the range of angular scales Θ that may be imaged by the interferometer is λ λ = − < Utotal > .

(6.76)

214

6 Galactic Radio Astronomy

To illustrate the virial theorem, consider a cloud of hydrogen atoms that is destined to collapse and form a new star. Assume that initially all of the particles are separated by large distances (that is, Utotal = 0) and that all of the particles are at rest (that is, Ktotal = 0, and therefore Etotal = Ktotal + Utotal = 0). As the cloud collapses due to gravity, the particles will be accelerated toward each other and therefore Ktotal increases while Utotal decreases. According to the virial theorem, after the system has come into equilibrium, Etotal may be expressed as Etotal = Ktotal − 2Ktotal .

(6.77)

This result may be interpreted as the system radiates away half of its potential energy Utotal , leaving the bound system with a total energy Etotal such that Etotal = −Ktotal .

(6.78)

Example Problem 6.10 The free-fall time tff of a cloud of gas particles is defined as the time required for the cloud to collapse upon itself due to gravity. Show that tff of a collapsing cloud of gas particles with a uniform mass density ρ is proportional to ρ −1/2 . Solution Assuming that the total energy of the system is conserved throughout, any change in potential energy U is equivalent to any change in kinetic energy K, that is, ΔU = ΔK. Recalling the expressions for U and K (Eqs. (2.43) and (2.44), respectively) and associating velocity v with the change in radius r of the cloud with time (that is, v = dr/dt), the conservation of energy can be expressed as mgas ΔK = ΔU −→ 2



dr dt

2 =

Gmgas m0 Gmgas m0 − r r0

(6.79)

where m0 and r0 are the total mass of the shell of gas and the initial radius of the shell of gas, respectively. This expression becomes  2Gm0 2Gm0 dt − = −→ r r0 dr

dr = dt



2Gm0 2Gm0 − r r0

−1/2 (6.80)

From here, tff may be expressed as  tff =

r=0  dt r=r0

dr



 dr = −

r=0  2Gm r=1

r

0

−

2Gm0 r0

−1/2 dr

(6.81) (continued)

6.3 Radio Observations of Star Formation Sites in the Milky Way Galaxy

215

Example Problem 6.10 (continued) This integration may be performed via a change of variables, specifically x=

r r0

dx =

and

dr , r0

which leads to 

 1/2  −1/2 x=1  x 1/2 r03 −1 dx= dx x 2Gm0 1−x x=0 x=0 (6.82) This integration may be performed through the following (second) change of variables:

r03 tff = 2Gm0

1/2 

x=1  1

x = sin2 θ −→ dx = 2 sin θ cos θ dθ. Note that through this change of variables, the bounds of the integration transform from x = 0 through x = 1 to θ = 0 through θ = π /2. Therefore the expression for tff becomes  tff =

r03 2Gm0

1/2 

tff =

θ=π/2



1/2

sin2 θ

θ=0

1 − sin2 θ



1/2

r03 2Gm0



θ=π/2

2

2 sin θ cos θ dθ

(6.83)

sin2 θ dθ

(6.84)

θ=0

This integration can be performed using the following identity: 

θ=π/2

sinn θ dθ =

θ=0

(n − 1)! π . n! 2

(6.85)

Thus the expression for tff becomes:  tff =

r03 2Gm0

1/2



π = 2× 4



r03 2Gm0

1/2

π 2

(6.86)

Recalling that the density ρ of a spherical cloud (in terms of m, volume V , and radius r0 ) is m m = 4 3 ρ= (6.87) V 3 π r0 (continued)

216

6 Galactic Radio Astronomy

Example Problem 6.10 (continued) where V = 43 πr03 for a spherical body, Eq. (6.87) can be substituted into Eq. (6.86), thus yielding for tff  tff =

3 8π Gρ

1/2

π = 2



3π 32Gρ

1/2 .

(6.88)

Notice that tff depends solely on ρ!

6.3.2 Molecular Clouds In the highest density clouds (both in number and in mass) within the interstellar medium, the hydrogen atoms are found in molecules rather than as individual atoms. For this reason, such clouds are known as molecular clouds: typical number densities and typical temperatures of these clouds are 108 m−3 and 15K, respectively [7]. In fact, the collapse of these molecular clouds is driven by the cooling, which in turn occurs due to resident molecules of hydrogen emitting radiation via rotational transitions (see Sect. 3.5) [8]. One challenge in the study of molecular clouds is the fact that molecular hydrogen produces very few spectral lines (such as rotational transitions), making it difficult to detect emission from that particular molecular species. To address this situation, astronomers instead observe emission associated with rotational transitions produced by carbon monoxide molecules present in the clouds. In contrast to H2 (the most common molecule in the Universe), CO—the second most common molecule in the Universe—produces a copious amount of spectral lines at such frequencies as 230 GHz (see Sect. 3.5). Astronomers use CO observations as an effective proxy for detections of molecular hydrogen resident in molecular clouds by assuming a ratio of the number of CO molecules to hydrogen molecules in the cloud. Furthermore, by observing redshifts and blueshifts of the CO emission lines, astronomers can determine the orbital motions of molecular clouds in the Milky Way around the Galactic center.

6.3.3 Masers Observations of diffuse astronomical sources like molecular cloud have revealed the presence of discrete “spots” of emission from the hydroxyl radical OH or water (H2 O). These diffuse astronomical sources are known as masers. The name for these sources is an acronym for “microwave amplification by the stimulated emission of radiation,” which in turn originates from the well-known acronym “laser”

6.3 Radio Observations of Star Formation Sites in the Milky Way Galaxy

217

Fig. 6.7 Energy-level diagram for a hypothetical three-level maser: From [30]

which stands for “light amplification by stimulated emission of radiation" and was originally applied to terrestrial emitting sources that produce electromagnetic radiation at optical wavelengths. The stimulated emission of radiation process may be explained as follows (see [30]). In Fig. 6.7, a schematic diagram is presented of a molecule with three energy states, specifcally (in order of increasing energy) a ground state with energy E 1 , a metastable state with energy E 2 and an excited state with energy E 3 . A molecule may be excited to the excited state by colliding with another particle or absorbing radiation (a process which is known as pumping). Note that for the excitation to occur, the energy of the collision or the absorbed photon corresponds to the energy difference between the two levels. Assume that once the molecule is in the excited state, the probability of de-excitation into the metastable state is greater than the probability of de-excitation into the ground state. Over time, a large number of molecules may be pumped and subsequently de-excite into the metastable state. If a photon with an energy equal to the difference between the metastable state and the ground state encounters the excited molecule, the molecule will be stimulated to de-excite by emitting a photon with an energy corresponding to precisely the difference in energy between the two levels (hence the term stimulated radiation). The emitted photon will travel in the same direction as the incident photon and these photons may now stimulate two other excited molecules to produce two more photons, and so on. By this chain reaction, the original photon may be amplified dramatically into many photons propagating through the gas. For this reason, the light from masers is observed to be very intense, narrow in direction and singular in frequency. Stimulated emission by photons incident upon the excited molecules rather than solely by collision must play a crucial role in the production of the emission because the required temperature of the gas to produce the emission solely through thermal excitation by collisions is excessive (over 1013 K). A consensus has not yet emerged regarding the exact process that excites the molecules in the first place. High angular resolution radio observations of masers have revealed that regions of maser emission must be very compact (on the scale of tens of AUs in size) and must be associated

218

6 Galactic Radio Astronomy

Fig. 6.8 Energy level diagram for the maser transitions associated with OH radical. From [30]

with regions of particularly high density. Masers also appear to be associated with regions of star formation and sites of interaction between expanding supernova remnants and molecular clouds. A schematic diagram of the maser process associated with the OH radical is presented in Fig. 6.8. The OH is pumped to the highest energy level (denoted in this figure as level 5) via (possibly) the absorption of ambient infrared photons. The radical is then most likely to de-excite to level 3, from which it is then most likely to de-excite to level 1 and emit a photon with a frequency of 1665 MHz. The radical may also de-excite to level 2 and emit a photon with a frequency of 1612 MHz, but this transition is less likely. If the radical de-excites from level 5 to level 4 instead, it may further de-excite to level 2 or level 1 and emit photons with frequencies of 1667 MHz or 1720 MHz, respectively.

6.3.4 HII Regions An HII region is a region where the majority of the hydrogen atoms are found in an ionized state. Such regions are known to be stellar nurseries where the most massive stars (known as O and B type stars) form and first shine: the surface temperatures of these stars are quite high (104 K and greater) and by Wien’s Displacement Law (recall Eq. (3.15)) those stars—approximated as blackbodies—emit copious amounts of ultraviolet radiation. Such photons have sufficient energy to photoionize any hydrogen atoms seen in close proximity to these stars, hence the nomenclature of these regions. Recall the discussion in Sect. 2.5 where the red emission frequently observed from these. This terminology itself is an extension of the nomenclature described in the discussion of HI in Sect. 6.2.2 where “HI” refers to hydrogen in the neutral state while “HII” indicates singly ionized hydrogen. This nomenclature is

6.3 Radio Observations of Star Formation Sites in the Milky Way Galaxy

219

employed throughout astronomy: for example, in the infrared a commonly observed species is singly ionized iron, which is denoted as [FeII]. HII regions can be detected by radio observations through both line emission and continuum (bremsstrahlung) emission. In the former case, HII regions are filled with free electrons that have been separated from nuclei in hydrogen atoms by photoionizing radiation: these free electrons—through Coulombic attraction— recombine with the protons, thus creating neutral hydrogen atoms. The electron transitions downward toward the ground level—the minimum energy state—and as it descends it emits photons (see Sect. 2.5). The exact set of transitions that the electron will take as it descends cannot be predicted in advance exactly: instead, quantum mechanics and statistics can only express probabilities that certain transitions may be more likely than others. If the electrons transition from an upper energy level to a lower energy level where the two levels are widely separated in energy, the emitted photon will of course have more energy. For example, as described in Example Problem 1.5, the transition from the nhigher = 3 level to the nlower = 2 level produces an energetic photon in the optical domain with a red color. This characteristic transition helps to identify emission from HII regions in optical surveys. The electron transitions that produce photons that are detected in the radio domain tend to be at much higher levels and of course contain much less energy. For example, a commonly studied radio recombination line emitted by HII regions in the radio is the H109α line, which corresponds to the electron transition from the nhigher = 110 level to the nlower = 109 level.

Example Problem 6.10 What is the frequency νH109α of the photon emitted by the H109α transition? Solution From Eq. (2.160) the wavelength of the transition is 1 λH109α

−1

= 1.10 × 10 m 7



1 1 − 2 109 1102



= 16.8 m−1 .

(6.89)

Therefore (recalling Eq. (2.90)) 3 × 108 m s−1 = 5.03 × 109 Hz. 5.96 × 10−2 m (6.90) This observing wavelength and the corresponding observing frequency are clearly within the domain of radio astronomy. λH109α = 5.96 cm and

νH109α =

Crucial physical insights can be determined by computing the ratio of the fluxes detected from two different transitions: for example, computing the ratio of the H109α flux to the flux seen at another transitions allows the computation of the number density of particles that compose the HII region. In fact, determining

220

6 Galactic Radio Astronomy

densities by comparing the ratios of fluxes for two closely spaced transitions is a very common technique for determining densities in these HII regions.2 In addition to the line emission produced by such transitions as H109α, HII regions are also known to emit continuum radiation in the form of bremsstrahlung radiation (Sect. 3.4). The spectral properties of this emission process contrast with the complementary process of synchrotron radiation, in that over broad ranges of observing frequency ν, the spectrum of bremsstrahlung emission is flat (with flux density as a function of ν is proportional to roughly ν −0.1 ). This may be contrasted with the phenomenon of synchrotron radiation 3.3, where over wide ranges of ν the spectrum of synchrotron radiation is steeper (in that case, the flux density as a function of ν is proportional to roughly ν −0.5 ). The latter case is made manifest in the case of supernova remnants, which will be discussed in Sect. 6.4.3. The differences in these spectral characteristics in the radio spectra of these sources aid in the classification of such objects both in the Milky Way galaxy and in nearby galaxies, where the similar diffuse and extended nature of these objects make them challenging to classify. An example of a radio spectrum of an HII region is depicted in Fig. 6.9, which depicts the observed radio spectrum of the prominent HII region known as the Orion Nebula (also known as Messier 42 and Orion A). As described above, the radio spectrum of the Orion nebula remains flat over a remarkably broad range of radiofrequencies and such a flat slope is characteristic of bremsstrahlung radiation. A noticeable turnover in flux is seen in the spectrum of the Orion Nebula as shown in Fig. 6.9 at approximately ν = 1 GHz. The frequency at which this turnover in flux occurs is known as the turnover frequency and is denoted as νturnover . The significance of the turnover frequency in the thermal bremsstrahlung spectra of HII regions is two-fold. First, this frequency delineates between two domains in the radio spectrum of an HII region: one domain is the low-frequency domain, below which the HII region becomes opaque to its own thermal bremsstrahlung radiation and significant self-absorption occurs. The other domain is the high frequency domain, where the HII region is transparent to its own thermal bremsstrahlung emission. Second, as discussed below the turnover frequency itself may be used to determine the number density of particles that comprise the HII region. The detailed discussion that follows examines these salient features of the thermal bremsstrahlung spectra of HII regions as observed at radio-frequencies: this discussion is patterned after the presentation given in [2]. The monochromatic volume emission coefficient jν (see Sect. 3.2 for a discussion about emission coefficients) for thermal bremsstrahlung emission may be expressed as jν = ψν np ne , 2 For

(6.91)

example, comparing the measured fluxes of two optical lines associated with [SII]—namely lines located in the optical domain of the electromagnetic spectrum at the rest wavelengths of 671.7 nm and 673.1 nm—is a commonly employed diagnostic for estimating the electron number density of a source of interest, such as an HII region. These number density estimates are based on an assumed temperature of the source, such as T = 104 K for an HII region [9].

6.3 Radio Observations of Star Formation Sites in the Milky Way Galaxy

221

Fig. 6.9 The radio spectrum of the Orion Nebula (also known as Orion A). Notice that the main spectral characteristics of an astronomical source emitting thermal bremsstrahlung radiation are evident: self-absorption at low frequencies that can be modeled as thermal emission, a turnover frequency and a flat plateau of steady emission beyond the turnover frequency. From [10]

where np and ne correspond to the number densities of protons and electrons in the HII region and ψν is a temperature-dependent function which may be approximated at radio-frequencies as ψν ≈ (3 × 10−51 J m3 sr−1 Hz−1 ) T −1/2 ν −0.1 .

(6.92)

In the low-frequency domain where ν is less than νcutoff , the thermal blackbody emission from the HII region that is responsible for the self-absorption can be described using the Rayleigh–Jeans approximation for the brightness Bν (ν, T ) (see Eq. (3.9)). In turn, Bν (ν, T ) can be equated to the source function Sν via Kirchhoff’s Law (see Sect. 3.2). This yields Bν ≈ Sν ≈

2kB T ν 2 . c2

(6.93)

Applying Kirchhoff’s Law (see Eq. (3.48)) to determine the opacity κν produces κν =

jν ≈ (10−11 m5 Hz2.1 K1.5 )ν −2.1 T −1.5 np ne . Sν

(6.94)

222

6 Galactic Radio Astronomy

The optical depth τν is obtained from integrating this equation for κν over the distance L corresponding to the radial extent of the HII region along the line of sight. If L is not known, the volume of the HII may be assumed to be spherical: in this case, by symmetry L can be equated to the linear extent of the HII region in the sky (as determined from the angular extent of the HII region on the sky and the distance to the HII region itself). If the mass dependence of τν and κν is ignored, this integration becomes  τν =

κν (L)dL ≈ (10−11 m5 Hz2.1 K1.5 )



ν −2.1 T −1.5 np ne dL

(6.95)

Defining the emission measure EM as the integration of np and ne over dL, that is, as  (6.96) EM = np ne dL, Equation (6.95) thus may be rewritten as τν ≈ (10−11 m5 Hz2.1 K1.5 )ν −2.1 T −1.5 EM.

(6.97)

This result is based on the assumption that the HII region has a constant temperature throughout (and therefore T may be taken outside of the integral) and that there is no dependence of ν on L (and therefore ν may be taken outside of the integral as well). Other useful information about the HII region that may be determined from radio observations include direct estimates of T and ne (and, by extension of the latter, np as well). Note that the observed intensity Iν of emission at radio-frequencies from the HII region (assuming that the temperature T is constant throughout the HII region and that stray background emission is negligible) via the thermal bremsstrahlung process is Iν = Sν (1 − e−τν ) = Bν (1 − e−τν ).

(6.98)

This relation defines the significance of a turnover frequency νturnover where τν = 1: this frequency demarcates the Iν into different frequency domains for ν  νturnover and ν νturnover . In the former frequency domain (where also τν 1, indicating that the HII region is optically thin to emission in this frequency range), Iν may be approximated by performing a Taylor expansion for e−τν , which produces (see Eq. (3.5)) e−τν ≈ 1 − τν −

τν + ... 2!

(6.99)

Applying this expansion to the term 1–e−τν and ignoring higher order terms in the expansion yields

6.3 Radio Observations of Star Formation Sites in the Milky Way Galaxy

223

τν + . . .) ≈ τν . 2!

(6.100)

1 − e−τν ≈ 1 − (1 − τν −

Finally, combining this result with Eqs. (6.97) and (6.98) yields the following equation for Iν in the frequency domain corresponding to ν  νturnover : Iν ≈ Bν τν ∝ ν −0.1 T −0.3 EM.

(6.101)

Similarly, in the latter low-frequency domain (where τν  1, indicating that the HII region is optically thick to emission in this frequency range), Iν corresponds to the brightness Bν which may once again be approximated as the Rayleigh–Jeans approximation (see Eq. (3.9)) as Iν ≈ Bν ≈

2kB T ν 2 . c2

(6.102)

Note that Eq. (6.102) may be rewritten as T ≈

c 2 Bν c2 Iν ≈ 2kB ν 2 2kB ν 2

(6.103)

and therefore T may be determined for the HII region by the measurement of Iν at a particular frequency ν. To determine ne for the HII region, note that Eq. (6.97) is equal to unity at νturnover , that is −2.1 EM = 1. (10−11 m5 Hz2.1 K1.5 )T −1.5 νturnover

(6.104)

Solving for EM in this equation yields 2.1 , EM ≈ (1011 m−5 Hz−2.1 K−1.5 )T 1.5 νturnover

(6.105)

and therefore np and ne may be determined from the definition of EM given in Eq. (6.96). This yields  np ≈ ne ≈

EM L

1/2 .

(6.106)

Here, it is assumed that the HII region is composed solely of ionized hydrogen and therefore the number density of protons and electrons will be at least approximately the same. The presence of trace amounts of ionized atoms of other elements (such as helium) present in the HII region and the electrons liberated from these atoms by ionization events (such as the same photoionization effects that liberate electrons from hydrogen atoms) will slightly affect this ratio.

224

6 Galactic Radio Astronomy

Example Problem 6.11 An HII region located at a distance of 250 pc and spanning angular extent of 5 arcminutes is the subject of pointed radio observations made at multiple frequencies. Based on the analysis of the radio spectrum of this source, the measured turnover frequency in the spectrum of this region is νturnover = 1 GHz. At this frequency, the measured intensity of radiation is Iν = 3.50 × 10−18 J/m2 . Calculate (a) the linear extent L of the HII region in parsecs and meters, (b) the temperature T of this HII region in Kelvins, (c) its emission measure EM in m−5 , (d) its electron number density ne , and (e) its mass M, assuming that the HII region is spherical in volume and that ne and np are identical. Solution (a) From Eq. (2.7), L is computed as   5 (60 /1 ) 250 pc 3.086 × 1016 m L (pc) = = 1.12×1016 m. = 0.36 pc 206,265 1 pc (6.107) (b) Using the provided information about the measured value of Iν at νturnoff , T may be calculated using Eq. (6.103) as T ≈

(3 × 108 m/s)2 (3.5 × 10−18 J/m2 ) = 1.14 × 104 K. 2(1.38 × 10−23 J/K)(109 Hz)2

(6.108)

(c) From Eq. (6.105), EM may be computed as EM ≈ (1011 m−5 Hz−2.1 K−1.5 )(1.14 × 104 K)1.5 (109 Hz)2.1 = 9.67 × 1035 m−5 .

(6.109)

(d) From Eq. (6.106), ne may be calculated as  ne ≈

9.67 × 1035 m−5 1.12 × 1016 m

1/2

= 1.86 × 109 m−3 .

(6.110)

The typical electron number densities for HII regions range approximately from 106 to 1012 particles per cubic meter. (e) Assuming a spherical geometry for the HII region and that its radius r corresponds to L/2, its volume V is simply V =

4π(1.12 × 1016 m/2)3 4π(5.6 × 1015 )3 4π(L/2)3 = = 3 3 3 (continued)

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

225

Example Problem 6.11 (continued) = 7.36 × 1047 m3 .

(6.111)

Therefore the mass M of the cloud is M = np mp V = (1.86 × 109 m−3 )(1.67 × 10−27 kg)(7.36 × 1047 m3 ) = 2.26 × 1030 kg.

(6.112)

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way 6.4.1 Maser Emission from Evolved Stars As stars deplete their stores of hydrogen fuel in their cores, internal dynamics to maintain equilibrium can cause a star to bloat out to an evolved evolutionary state known as the red giant or supergiant stage (the name stems from the large size coupled with low surface temperatures that characterize these evolved stars). In these advanced evolutionary stages, stars may undergo significant mass loss which in turn sculpt the final evolutionary stages of these stars, but the details regarding the physics of the mass loss in these older stars remain unknown. High angular resolution radio observations of such stars helps provide insights into the phenomenon of mass loss of these stars through detailed studies of thermal motions within the outer layers of the star as well as the strength and orientation of the star’s magnetic field, which is believed to play a critical role in mass loss processes. Such insights are yielded by the studies of emission from silicon monoxide (SiO) masers located in a shell in the inner portions around the star’s atmosphere, where these molecules form. A radio map made of the evolved star TX Camelopardalis with the VLBA is presented in Fig. 6.10. The regions of maser emission from SiO molecules are shown as “spots” and form a shell around the star. Because the rest frequency of the masing transition associated with this molecule is well known (43 GHz), along with the proper motion of the star itself, any redshifts and blueshifts in the observed frequencies of the masing transitions can be interpreted as motions of the masing material in the atmosphere of the star. Furthermore, the observed polarization of the emission from these masers can also be used to determine the orientation of the star’s magnetic field [11]. In the case of TX Camelopardalis, the observations of the polarization of the emission from the masers indicate that the magnetic field of this star is remarkably well-ordered (see Fig. 6.11). It is believed that accelerated mass loss is taking place in those locations where there is a significant disruption of the

226

35 30 25 MILLIARC SEC

Fig. 6.10 VLBA image of the star TX Camelopardalis, showing regions (“spots”) of maser emission from silicon monoxide (SiO) molecules forming a shell around the star. Approximate size of the star is indicated by red circle. Image Credit: NRAO/AUI/NSF

6 Galactic Radio Astronomy

20 15 10 5 0 –5 –10 10

0

–10

–20

–30

MILLIARC SEC

Fig. 6.11 VLBA image of the star TX Camelopardalis with blue lines showing the relative strength and orientation of the magnetic field of the star. Image Credit: NRAO/AUI/NSF

magnetic field pattern. Finally, the presence of the magnetic field serves to split the energy levels of the masing transitions via the Zeeman effect (see Sect. 6.2.5). The magnitude of the magnetic field can be estimated from the observed Doppler Effect upon the observed masing transition and the Zeeman splitting: through this method, the magnitude of the magnetic field of TX Camelopardalis has been estimated to be approximately 5–10 G [11, 12]. This value is comparable to estimates of the magnitudes of the magnetic field strengths of similar stars.

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

227

6.4.2 Microquasars Microquasars comprise one of the most remarkable classes of discrete radio sources detected in our Galaxy. These sources are composed of a black hole with a mass on the order of stellar masses that has a binary stellar companion. The black hole siphons material from the stellar companion, which forms an accretion disk around the black hole. This material in the accretion disk is heated by frictional forces between constituent particles to temperatures of approximately 106 K or greater, thereby producing X-ray emission via blackbody radiation (see Sect. 3.1). At radio wavelengths, prominent jets that originate from the accretion disk are detected: these jets are composed of particles accelerated to relativistic velocities that are emitting synchrotron radiation (see Sect. 3.3). Microquasars were named based on their resemblance to scaled-down versions of quasars, which are the bright nuclei of galaxies that feature large jets of relativistic particles emitted from the accretion disk surrounding the central supermassive black hole of the galaxy. The phenomena of quasars, jets and accretion disks are all described in more detail in Sect. 7.3. The best known Galactic microquasar is the source SS 433 (see Fig. 6.12). Longterm radio observations of SS 433 conducted with high angular resolution have

Fig. 6.12 A radio map of the Galactic microquasar SS 433, a binary star system that includes a compact stellar remnant (either a neutron star or a black hole) along with a stellar companion. Material is siphoned from the star to the compact object and an accretion disk forms around this object: this material tightly circles the compact object prior to being pulled into it. Jets of relativistic particles (with estimated velocities of approximately 0.25c) are emitted in close proximity from the surface of the compact object at its poles: these jets are collimated by the accretion disk. The jets will also wobble as the relativistic particles are ejected due to precession of the disk with a period of 162 days: this wobbling creates a swirling tracer pattern that is visible in this radio map. See [14]. Image credit: B. Saxton from data provided by R. M. Hjellming, K. L. Johnston, NRAO/AUI/NSF

228

6 Galactic Radio Astronomy

detected jets producing copious amounts of synchrotron emission as produced by relativistic electrons accelerated to approximately 0.25c. The jets are seen to precess around an axis inclined along the line of sight between Earth and SS 433: the period of the precession is 162 days. As a result of this precession, the jets appear to trace out a “corkscrew” pattern on the sky as they move away from the central black hole.

Example Problem 6.12 A study of the acceleration of cosmic-ray particles by the jets of SS 433—based on observations made by the High Altitude Water Cherenkov (HAWC) observatory by Abeysekara et al. [13]—argues for the presence of particles accelerated to 1015 eV = 1.6 × 10−3 J and a magnetic field strength B = 16 µG = 1.6 × 10−9 T based on an assumption of equipartition between the relativistic particles and the magnetic field within the jets (see Sect. 6.4.3.1). Compute the Lorentz factor γ and the half-life t1/2 in years of an electron with this energy and propagating through a magnetic field with this strength. Solution From Eq. (2.170), γ may be computed as γ =

E 1.6 × 10−3 J = = 1.95 × 1010 . 2 m0 c (9.11 × 10−31 kg)(3 × 108 m/s)2

(6.113)

From Eq. (3.88), t1/2 may be computed as t1/2 =

  4.26 × 10−13 1 yr 8s = 1.04 × 10 = 3.30 yr. (1.6 × 10−9 T)2 (1.6 × 10−3 J) 3.15 × 107 s (6.114)

6.4.3 Supernova Remnants A supernova (plural supernovae) is the violent, explosive death of a star. In these explosions, vast amounts of energy (in such forms as kinetic energy and radiation) are released into the surrounding interstellar medium. Currently, there are believed to be two scenarios through which a supernova explosion may occur: in the first scenario, a white dwarf star—an evolved star with a mass no greater than approximately 1.4 solar masses, a limit known as the Chandrasekhar limit— in a binary star system perishes by one of two different mechanisms. In the first mechanism, the white dwarf siphons a sufficient amount of mass from a companion star and collapses upon itself, resulting in a supernova explosion. In the second mechanism, it is two white dwarfs in a binary system, and over time these stars spiral into each other and collide, annihilating each other. The type of supernova explosion associated with the death of a white dwarf star or stars is known as a Type

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

229

Ia supernova: the first mechanism is known as the single degenerate supernova while the second mechanism is known as the double degenerate supernova.3 The Type Ia scenario of supernovae involving white dwarf stars may be contrasted with the death of a star with a mass of approximately eight solar masses or greater. In such a case, when a star can no longer generate enough energy through nuclear fusion to support itself against its own gravity, the star collapses upon itself and in the ensuing collapse the supernova explosion occurs. Depending on how much mass is left over toward the center of the star after the explosion, either a neutron star or a black hole may be created: the properties of neutron stars and pulsars—a particular type of neutron star—are discussed in Sect. 6.4.4. Based on characteristics of the observed spectra, supernovae associated with the deaths of massive stars may be classified as either Type Ib, Type Ic or Type II supernovae.4 Supernovae were first identified by telescopic observations made of nearby galaxies in the late nineteenth and early twentieth century, though in time it was realized based on eyewitness accounts that supernova explosions have been observed in the Milky Way galaxy as well. Approximately ten historical supernovae are believed to have been observed over the past two millennia. After the initial supernova explosion, an expanding shock front from the explosion enters the surrounding interstellar medium, sweeping up material in its path. Initially, the swept-up mass is composed primarily of stellar ejecta from the explosion but in time the amount of mass swept up from the interstellar medium dominates the total amount of swept-up mass. This expanding shock front of sweptup material is known as a supernova remnant (SNR): within the Milky Way galaxy, 294 Galactic supernova remnants are now known to exist at the time of the writing of this text [15, 16]. SNRs are powerful sources of synchrotron emission at radio wavelengths: as these sources expand, they compress and amplify the ambient magnetic field of the interstellar medium and in these regions with an enhanced magnetic field cosmic-ray electrons gyrate and emit synchrotron radiation. In fact, the resemblance between the spectrum of cosmic-ray electrons and the spectrum of synchrotron emission suggested that astronomical sources that emit synchrotron radiation (such as SNRs) may in fact be the accelerators of cosmic-ray particles [17]. SNRs are the leading candidates for the origin of cosmic-ray particles to 3 The origin of the term “degenerate” in this nomenclature is traced to white dwarfs being described

as composed of degenerate matter: such matter is compressed so intensely by gravity that only the nature of electrons such that more than one electron cannot be placed in the same energy state prevents further gravitational collapse. 4 The nomenclature for supernova explosions is based on tradition: when spectra of supernova in other galaxies were obtained, the classification system categorized supernovae Type I supernovae as those which lacked hydrogen lines in their spectra while Type II supernovae featured prominent hydrogen lines in their spectra. Furthermore, Type I supernovae that featured silicon lines were denoted as Type Ia while other Type I supernovae that lacked silicon lines but exhibited helium lines were denoted as Type Ib. Finally, Type I supernovae that lacked silicon and helium lines in their spectra were denoted as Type Ic. It was only after additional analysis that it was determined that supernovae associated with massive stars could manifest themselves as either Type II or a variation of a Type I supernova explosion.

230

6 Galactic Radio Astronomy

approximately the “knee” energy of the cosmic-ray spectrum (that is, approximately 1015 eV). The most widely accepted theory regarding the acceleration of cosmicray particles by supernova remnants is known as diffusive shock acceleration: this mechanism—also known as first-order Fermi acceleration describes the acceleration of cosmic-ray particles by the expanding shock front of an SNR [18– 20]. Diffusive shock acceleration by SNRs may be qualitatively described as occurring when particles trapped by a magnetic field undergo multiple crossings over the shock front of the SNR. The multiple crossings of the shock front can be imagined as a ping-pong ball bouncing between a paddle and a table, with the paddle falling toward the table. Just as the ping-pong ball would become more energetic as it bounces between the paddle and the table (with the distance between the paddle and the table becoming ever smaller), a particle would gain energy with each successive crossing of the shock. Eventually, the particle would become relativistic, escape the confines of the shock, and drift through the interstellar medium of the galaxy (Fig. 6.13). Quantitatively, diffusive shock acceleration may be described as follows. It is expected that the energy spectrum of cosmic-ray particles to have a power law dependence and any effective mechanism for accelerating such particles must reproduce this dependence to be acceptable. Consider a cosmic-ray particle with an initial energy E0 that is accelerated by the expanding shock front of an SNR. The energy E after one collision is E = κE0 ,

(6.115)

where κ is the fraction of energy change for the particle. Now considering an ensemble of cosmic-ray particles with initially N0 members, if P is the probability that an individual particle remains in the accelerating region after one collision, then after n crossings of the shock, the number of accelerated particles N remaining in the shock will be N = N0 P n ,

(6.116)

and each of the N particles will have energies E = κ n E0 .

(6.117)

Eliminating n from Eqs. (6.116) and (6.117) yields ln (N/N0 ) ln P = ln (E/E0 ) ln κ and solving for N/N0 gives

(6.118)

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

231

Fig. 6.13 A schematic diagram illustrating the phenomenon of diffuse shock acceleration from multiple perspectives on the propagation of a strong shock wave (that is, a shock wave with a supersonic velocity) through a medium. (upper left) In the rest frame of an outside observer, the shock propagates with a supersonic velocity u through the medium. Shocked material lies in its wake while unshocked material lies in its path. (upper right) In the rest frame of the shock itself, shocked material appears to recede from the shock with a velocity u/4 while unshocked material appears to approach the shock with a velocity u. (lower left) In the rest frame of the unshocked material, the shocked material appears to be in approach with a velocity 3u/4. (lower right) In the rest frame of the shocked material, the unshocked material appears to be in approach with a velocity 3u/4. Figure adopted from [21]

N E ln P /ln κ = . N0 E0

(6.119)

This quantity describes the number of particles with energies of E or greater: a fraction of the particles (proportional to E −1 ) are accelerated to higher energies. Thus, the differential energy spectrum N(E)dE—that is, the number of particles of energy E for a differential energy element dE—may be expressed as N(E) dE ∝ E

−1+ ln

P

ln κ dE.

(6.120)

How can the quantity ln P /ln κ be evaluated? Following the development presented elsewhere [22], consider the case of a plane shock front (corresponding to the

232

6 Galactic Radio Astronomy

leading edge of an SNR) that is moving through interstellar gas. In the case of an SNR, the supersonic expansion of the sphere of hot gas associated with the SNR creates a sharp discontinuity between the expanding gas and the swept-up material: this sharp boundary between the expanding gas and the shocked gas right in front of it is called the contact discontinuity. In addition, by convention, the region located before the shock front is known as the downstream region while the region located behind the shock front is known as the upstream region.5 What is the increase in energy of a highly energetic particle that crosses from the upstream region to the downstream region? The shock velocity vs through the ambient interstellar medium is assumed to be highly supersonic: in the following treatment, all descriptions will be made from the reference frame where the shock front is at rest. In this reference frame, gas in the ambient interstellar medium flows into the shock front with a velocity vu = vs and leaves the shock with a downstream velocity vd . Defining ρu and ρd as the mass densities of the gas upstream and downstream of the shock, respectively, and recognizing that mass must be conserved through the shock, the mass densities and the velocities are related as ρu vu = ρd vd .

(6.121)

In the domain of strong shocks (that is, shocks with associated values of vs that greatly exceed the speed of sound in a medium), the ratio of the mass densities is ρd /ρu = 4 and thus the ratio of the velocities is vd /vu = 1/4. This means that for a particle closing from the upstream to the downstream sides of the shock, the gas on the downstream side approaches the particle with a velocity v = (3/4)vshock . If the initial energy of the particle is E and its momentum in the x-direction is px , then the Lorentz transformation E  of E as the particle passes into the downstream region is E  = γ (E + px v).

(6.122)

Even though the particle is relativistic, note that the shock velocity is nonrelativistic. Therefore, the energy of the particle E and recalling Eq. (2.172) for the energy of a relativistic particle as a function of its momentum, px can be expressed as   E cos θ, (6.123) px = c where θ is the angle of incidence of the path of the particle: it is measured from a line normal to the shock plane. The change in energy ΔE of the particle as it crosses the shock and the ratio ΔE/E as ΔE = pv cos θ, 5 These

(6.124)

regions are determined relative to the reference frame of the expanding shock front.

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

233

and v cos θ ΔE = . E c

(6.125)

The probability p(θ ) that the particle will cross the shock as a function of θ needs to be calculated: this probability is the product of two quantities. The first quantity is the number of particles located within a range of angles θ to θ + dθ which is itself proportional to sin θ dθ . The second quantity is the rate at which particles approach the shock front: this quantity is proportional to the component of their velocities in the x-direction, that is, c cos θ (again, it is emphasized that the particles are moving at relativistic velocities such that v ≈ c). Therefore, the probability that the particle will cross the shock is proportional to sin θ cos θ dθ . Adopting a range of values of θ = 0 to θ = π /2 for the values of θ and normalizing the probability such that the integral of the probability is equal to unity over this range of values of θ , p(θ ) can then be expressed as p(θ ) = 2 sin θ cos θ dθ.

(6.126)

Furthermore, the average energy < ΔE/E > gained by the particle with each crossing of the shock is 

θ=π/2

ΔE v >= E c

p(θ ) cos θ dθ =

κ=

2v c



θ=π/2

2v . 3c θ=0 θ=0 (6.127) In addition to this shock acceleration event, the particle will experience a second acceleration when it recrosses the shock after random scattering without energy loss in the downstream region. It will thus gain energy a second time and in one round trip across the shock and back again, the total fractional increase in the energy of the particle will be 2< ΔE/E >= 4v/3c. Finally, from Eq. (6.117) the expression for κ becomes
0) as these sources gradually convert their rotational energy into radiation and accelerated particles. The Crab and Vela pulsars are known to undergo sudden decreases—known as glitches—where their periods suddenly decrease by approximately one part in 106 before slowly resuming a monotonic increase in period again. While glitches have been most readily observed in the Crab and Vela pulsars, it is generally believed that all pulsars may exhibit this behavior as well. The exact cause of glitches is unknown: one theory proposes that the interiors of pulsars are not uniform but instead differentiated into a core and a surface crust. The core is believed to rotate more rapidly than the crust so the rotation rates of these two portions of the pulsar are usually decoupled. On occasion, the rotation rate of the core and the crust may become coupled, perhaps due to a break in the magnetic dipole moment of the pulsar, which would result in a torque applied to the crust and in turn the rotation rates of the core and the crust would become coupled. When this coupling happens, a glitch may occur. Note that the minimum magnetic field strength Bmin of the pulsar given in Eq. (6.192) and the characteristic age τc of the pulsar given in Eq. (6.198) may also be expressed in terms of ν and ν˙ as  Bmin =

3π 0 c3 I 2R 6

  ν˙ − 3 ν

(6.206)

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

253

and τc =

ν , 2|˙ν |

(6.207)

respectively.

Example Problem 6.16 Radio observations of the young pulsar PSR B0540−69 have yielded values for ν, ν˙ , and ν¨ of 19.8 Hz, −1.8×10−10 Hz s−1 , and 3.7×10−21 Hz s−2 , respectively [35]. Based on these values, calculate Bmin , τc and n for PSR B0540−69, assuming that the pulsar has a mass M of 1.4 M and a radius R of 10 km. Solution To calculate Bmin , the moment of rotational inertia I of the rotating pulsar is first computed as follows (from Eq. (6.180)): I=

   2 × 1030 kg 2 2MR 2 = 1.4 M 5 5 1 M

2 10 × 10 m = 1.12 × 1038 kg · m2 .

(6.208)

3

Therefore, from Eq. (6.206), Bmin may be computed as

Bmin

     3π(8.85 × 10−12 F/m)(3 × 108 m/s)3 (1.12 × 1038 kg · m2 ) −1.8 × 10−10 Hz s−1 = − 2(10 × 103 m)6 (19.8 Hz)3

= 5.41 × 107 T.

(6.209)

Next, τc may be computed from Eq. (6.207) as   19.8 Hz 1 yr 10 = 5.5 × 10 = 1744 yr. s 2| − 1.8 × 10−10 Hz| 3.15 × 107 s (6.210) Finally, n may be computed from Eq. (6.204) as τc =

n=

(19.8 Hz)(3.7 × 10−21 Hz s−2 ) (−1.8 × 10−10 Hz s−1 )2

= 2.26.

(6.211)

Note that this value for n is comparable to the canonical value of 3 for the braking index of neutron stars.

254

6 Galactic Radio Astronomy

Fig. 6.18 A schematic diagram depicting the Faraday effect. As an electromagnetic wave passes through a medium with a path length L and a magnetic field with a component B parallel to the line of sight, the plane of polarization of the electric field is rotated through an angle Θ

6.4.5 Rotation Measure and Dispersion Measure An important observable in the study of pulsars is known as the rotation measure (RM). To understand this observable, it is important to develop first an understanding of the Faraday effect. This effect describes how the plane of polarization of a light wave rotates as it passes through a magnetic field directed along the direction of propagation (see Fig. 6.18. The phenomenon of the rotation itself of this plane is known as Faraday rotation: the magnitude of the rotation depends on both the strength of the component of the magnetic field B that is parallel to the direction of propagation, the total path length  through which the light travels and the number electron density ne of the medium. The total angle ΔΘ through which the plane of polarization is rotated may be computed from 



ΔΘ (radians) = 8.1 × 10 λ (m) 5

2

0

B (G) ne (cm−3 ) d (pc).

(6.212)

From this equation, RM—in units of radians m−2 —is defined in turn as the product of the integral multiplied by the constant with the dependence on λ omitted, that is, RM (radians m−2 ) = 8.1 × 105



 0

B (G) ne (cm−3 ) d (pc).

(6.213)

Alternatively, RM can also be defined based on observations made at two different wavelengths (say λ1 and λ2 ). If the total angles of rotation observed at these

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

255

wavelengths are ΔΘ1 and ΔΘ2 , respectively, then RM may be expressed as RM (radians m−2 ) =

ΔΘ1 (radians) − ΔΘ2 (radians) . λ21 (m) − λ22 (m)

(6.214)

Note that the plane of polarization of emission from pulsars is constant over all frequencies. How can the intrinsic plane of polarization (say, corresponding to a certain angle Θ0 ) of the emission from the pulsar be determined? Consider measurements made of ΔΘ over large ranges of wavelength or frequency: over such a wide range of measured values, ΔΘ will asymptotically approach a value that will correspond to Θ0 (because B and ne are physical quantities that are independent of the wavelength or frequency at which observations are made). By convention, if RM is positive, then B points toward the observer. Note that a clear application of Eq. (6.213) is as a distance indicator to a pulsar: assuming typical values for B and ne (such as 3 µG and 1 cm−3 , for example), a distance  to the pulsar can be calculated once ΔΘ is measured. Another application of this equation occurs in the cases where the distance  to the pulsar is known (such as by parallax measurements, for example): again, if a typical value for B is assumed, measurement of ΔΘ leads to an estimate for ne . In fact, estimates of the number density of the ambient interstellar medium have been performed in this manner. Example Problem 6.17 Calculate RM in radians m−2 for a pulsar located at a distance of 1000 pc: assume typical values for B and ne of 3 µG and 0.1 cm−3 , respectively, and assume that ne is uniform along the line of sight to the pulsar. Also calculate ΔΘ1 , ΔΘ2 , and ΔΘ3 in radians for observations made at the wavelengths λ1 = 20 cm, λ2 = 6 cm, and λ3 = 3.6 cm, respectively. Solution From Eq. (6.213), RM = (8.1 × 105 )(3 × 10−6 G)(0.1 cm−3 )(1000 pc) = 243 rad m−2 . (6.215) Successive applications of Eq. (6.212) yields ΔΘ1 = (8.1 × 105 )(0.2 m)2 (3 × 10−6 G)(0.1 cm−3 )(1000 pc) = 9.72 rad, (6.216) ΔΘ2 = (8.1 × 105 )(0.06 m)2 (3 × 10−6 G)(0.1 cm−3 )(1000 pc) = 0.87 rad, (6.217) and ΔΘ3 = (8.1 × 105 )(0.036 m)2 (3 × 10−6 G)(0.1 cm−3 )(1000 pc) = 0.31 rad. (6.218)

256

6 Galactic Radio Astronomy

A second crucial observable in the study of pulsars is known as the dispersion measure (DM). Consider again the discussion of the concept of an index of refraction as presented in Sect. 2.4.1 and recall Eq. (2.91). As described in that section and illustrated in that equation, the velocity of light through a medium is a function of the wavelength (and in turn the frequency) of the light: when considering the motion of light at radio-frequencies through the interstellar medium, the lower the frequency of the light, the lower the velocity of the light through the medium. Therefore, if two photons with frequencies ν1 and ν2 (such that ν1 < ν2 ) are emitted simultaneously from a pulsar and head toward an observer, the photon with frequency ν2 will be delayed less and reach the observer before the photon with frequency ν1 . Measurement of this delay (denoted as ΔτD ) can then be used to provide an estimate of the distance  to the pulsar. Formally speaking, ΔτD may be expressed as e2 ΔτD = 2π cme



1 1 − 2 2 ν1 ν2





ne ()d

(6.219)

0

where ne () is the number density of electron along the line of sight toward the pulsar. As described above, for the sake of simplicity, it is often assumed that ne () is a constant and therefore the integration is simply over the entire distance  to the pulsar. In this equation, the integral is commonly referred to in the literature as DM and expressed in terms of cm−3 pc, such that DM (cm

−3

 pc) =



ne (cm−3 ) d (pc).

(6.220)

0

Example Problem 6.18 The measured value for the DM observed toward a pulsar is 100 cm−3 pc. Assuming a distance to the pulsar of 1 kpc, calculate the electron number density ne in cm−3 along the line of sight to the pulsar, assuming that ne is uniform along the line of sight. Solution From Eq. (6.220), ne (cm−3 ) =

100 cm−3 pc DM (cm−3 pc) = = 0.1 cm−3 .  (pc) 1000 pc

(6.221)

It is emphasized that it was assumed that ne is uniform along the entire line of sight toward the pulsar in this example. Certainly, wide variations in values of ne along any given line of sight in the Galaxy are likely: such variations may be indicated by—for example—HI observations, which may reveal the presence of different HI clouds along the line of sight through the Doppler Effect (see Sects. 2.4.5 and 6.2.3). In addition, measurements of fluxes from individual clouds can provide estimates for ne as well.

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

257

Convenient forms of Eq. (6.219) where ν1 and ν2 are expressed in terms of MHz and ΔτD is expressed in terms of μs are the following [36]: 

−9

ΔτD [μs] = 1.34 × 10

−2

DM [cm

1 1 − 2 ] 2 ν1 [MHz] ν2 [MHz]

 (6.222)

and 

ΔτD [μs] = 4.148 × 109 DM [cm−3

 1 1 − pc] 2 , ν1 [MHz] ν22 [MHz]

(6.223)

or finally 2.410 × 10−10 ΔτD (μs) . DM [cm−3 pc] =  1 1 − 2 ν12 [MHz] ν2 [MHz]

(6.224)

In fact, RM and DM can be combined together to help provide one method of measuring B along a line of sight toward a pulsar. It is the ratio of these two quantities that provides an estimate of B , that is B (G) = 1.23 × 10−6

RM . DM

(6.225)

Such observations have indicated that the ambient magnetic field strength of the interstellar medium ranges from 0.3 µG to 3 µG and provide the canonical estimate of a Galactic magnetic field strength of 3 µG that is used throughout this text.

6.4.6 Galactic Center As discussed in Chap. 1, the first extraterrestrial radio source discovered by Karl Jansky was the Galactic Center. While high absorption makes it difficult to observe this source at optical wavelengths, the intervening medium is more transparent at radio wavelengths, and therefore radio observations of the Galactic Center and its surrounding environment have yielded many crucial insights into both the central object (denoted as Sgr A∗ and considered to be a supermassive black hole—see Example Problem 6.4) and the dynamics of its surrounding environment. A map made of the Galactic Center at a wavelength λ = 90 cm is presented in Fig. 6.19. Inspection of this image reveals a wide diversity of radio astronomical sources, including several HII regions and supernova remnants (see Sects. 6.3.4 and 6.4.3, respectively, for more information about these sources) along with diffuse radio emission. The presence of these types of sources is indicative of active star

258

6 Galactic Radio Astronomy

formation taking place in the vicinity of the Galactic Center. In addition, prominent non-thermal filaments that extend several parsecs above and below the Galactic plane are clearly seen as well. Such filaments are thought to be emission structures produced by cosmic-ray electrons emitting synchrotron radiation (as indicated by their non-thermal spectra) as they gyrate along the lines of the magnetic field created by the central “dynamo” Sgr A∗ .

Example Problem 6.3 Stars are useful probes in the measurements of the masses of central black holes in galaxies: their physical extents are negligible compared to the scale sizes of their orbits, making them ideal “test particles” for applying Kepler’s Third Law and estimating the masses of these black homes. In a study of the orbits of stars located in close proximity to the supermassive black hole at the center of the Milky Way Galaxy [37], the orbital elements of the star S0-2 were modeled assuming multiple scenarios for the motion of the central supermassive black hole Sgr A∗ . In the simple case where the black hole is assumed to be fixed in position, the authors measured an eccentricity for the orbit of e = 0.8866 ± 0.0059, an orbital period of P = 15.78 ± 0.35 yr and an angular extent for the semi-major axis of the orbit of θ = 124+2.4 −3.3 milliarcseconds. Assuming a distance to the Galactic Center of d = 8.5 kpc, calculate the pericenter distance rpericenter and the apocenter distance rapocenter (that is, the points in the orbit of the star where it is closest to and farthest from Sgr A∗ , respectively) for S0-2 and apply Kepler’s Third Law to determine the mass M of Sgr A∗ . Note that in this application of Kepler’s Third Law, the assumption that the mass of the star is negligible compared to the mass of Sgr A∗ is fully justified. Solution At the assumed distance to Sgr A∗ , the linear distance for the semimajor axis a that corresponds to θ is (from Eq. (2.7)) a (pc) =

(124 × 10−3 ”) × 8500 pc = 5.10 × 10−3 pc, 206,265

(6.226)

or in terms of kilometers, −3

a (km) = 5.10 × 10

 pc ×

3.086 × 1013 km 1 pc

 = 1.58 × 1011 km.

(6.227) and from Eqs. (2.60) and (2.61), the distances rpericenter and rapocenter are rpericenter = 1.58 × 1011 km(1 − 0.8866) = 1.79 × 1010 km

(6.228)

and (continued)

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

259

Example Problem 6.3 (continued) rapocenter = 1.58 × 1011 km(1 + 0.8866) = 2.98 × 1011 km.

(6.229)

Note the order of magnitude difference between these two distances, as indicated by the high eccentricity of the orbit. After performing the simple conversion of the measured period from years to seconds,  P = 15.78 yr ×

3.15 × 107 s 1 yr

 = 4.98 × 108 s,

(6.230)

and finally from Eq. (2.82) the mass M of Sgr A∗ is M=

4π 2 (1.58 × 1014 m)3 6.67 × 10−11 N m2 kg−2 × (4.98 × 108 s)2

= 9.4 × 1036 kg.

In terms of solar masses, this mass is   1 M = 4.71 × 106 M . M = 9.4 × 1036 kg × 2 × 1030 kg

(6.231)

(6.232)

Example Problem 6.4 Inspection of Fig. 6.19 (which was made at an observing wavelength λ = 90 cm which corresponds to an observing frequency ν = 333 MHz) reveals the presence of filamentary structures arcing from the central diffuse radio emission. Analysis of these structures indicates that they are produced by accelerated cosmic-ray electrons that are gyrating in an amplified magnetic field with an estimated strength of B = 6 mG = 6 × 10−7 T [38]. Consider an accelerated cosmic-ray electron associated with the filamentary structure that is emitting synchrotron radiation at a critical frequency νc corresponding to the given observing frequency. Calculate the half-life t1/2 of this cosmic-ray electron, assuming that the electron is freshly accelerated and that the time between the acceleration of the electron to this energy and the current observation time may be ignored. Solution To apply Eq. (3.88), the Lorentz factor γ of the electron must be calculated first. Recalling the discussion about synchrotron radiation in Sect. 3.3 where it is assumed that all of the synchrotron radiation is emitted at the frequency νc (corresponding to 333 MHz in this example). The Lorentz factor γ can be calculated from Eq. (3.94), yielding (continued)

260

6 Galactic Radio Astronomy

Example Problem 6.4 (continued) 3γ 3 νsync 3γ 3 (qB) 3γ 2 qB νc = −→ γ = = = 2 2(2π me γ ) 4π me

 4π me νc , 3qB

(6.233)

and recalling Eq. (3.69). Inserting values into this equation produces  4π(9.11 × 10−31 kg)(3.33 × 108 Hz) ≈ 112, (6.234) γ = 3(1.69 × 10−19 C) (6 × 10−7 T) and therefore from Eq. (3.88) the value for t1/2 is 6π(8.85×10−12 F/m)(3×108 m s−1 )3 (9.11×10−31 kg)3 =1.29×1011 s, (1.60×10−19 C)4 (6×10−7 T)2 (112) (6.235) which corresponds to over 4000 years. How far can an electron move along these field lines before it radiates away half of its energy? This distance can be estimated by assuming that the electrons are diffusing along the magnetic field lines with a velocity known as the Alfvén velocity vA , which describes the propagation of a wave within a plasma in the direction of a magnetic field, with ions moving in a direction transverse to the direction of propagation of the wave.9 In MKS units, vA may be expressed as

t1/2 =

vA = √

B , μ0 ni mi

(6.236)

where ni is the number density of ions in the plasma and mi is the average mass per ion. It is appropriate to approximate the velocity of the electrons along the filament lines as va and to assume that the filament is a plasma structure of purely ionized hydrogen where ni = ne ≈ 106 –107 m−3 and mi = mH = 1.67×10−24 kg. Adopting a mean value for ni = 5×106 m−3 along with the value for B given above and inserting these values into Eq. (6.236) yields 6×10−7 T va =  =1.85×105 m s−1 . −2 −7 6 −3 −24 (4π ×10 N · A )(5×10 m )(1.67×10 kg) (6.237) (continued)

9 It is believed that the electrons are scattering back and forth off small local magnetic irregularities: even though the electrons are moving at relativistic speeds, theoreticians estimate that the diffusion velocity of the electrons is approximately equal to vA , though a complete understanding of the relationship with vA in this context has not yet been realized.

6.4 Radio Observations of the End Points of Stellar Evolution in the Milky Way

261

and therefore the distance covered by an electron in the time t1/2 is d = va × t1/2 = 1.29 × 1011 s × 1.85 × 105 m s1   1 pc 16 = 0.77 pc. = 2.39 × 10 m 3.086 × 1016 m

(6.238)

Because the observed radio filaments seen toward the Galactic Center are much larger than this distance, the synchrotron emitting electrons must be continuously re-accelerated rather than their acceleration occurring only once. The exact process responsible for re-accelerating the electrons is uncertain: one possibility is that a process analogous to diffusive shock acceleration (described in Sect. 6.4.3) may be occurring in the filaments.

Fig. 6.19 A radio map made with the VLA at a wavelength of 90 cm. The home of the central supermassive black hole Sgr A∗ is indicated. Also note the presence of the birthplaces of massive stars (HII regions) and the remains of stars that have perished (supernova remnants). Image Credit: NRAO/AUI/NSF and N. E. Kassim, Naval Research Laboratory

262

6 Galactic Radio Astronomy

Problems 6.1 (a) The Galactic supernova remnant SN 1006 (G327.6+14.6) lies at a distance d = 2.0 kpc. Calculate the distance z in parsecs at which SN 1006 lies above the Galactic plane. (b) Consider the location of the supernova remnant G359.1−0.5 as shown in Fig. 6.19. If this supernova remnant lies at approximately the same distance from the Sun as the Galactic Center (that is, at a distance of R0 = 8.5 kpc), calculate the linear distance x and z in parsecs that G359.1−0.5 is offset in Galactic longitude  and in Galactic latitude b, respectively, from the Galactic Center. 6.2 (a) Calculate the frequency ν in Hertz corresponding to the 21-cm transition. (b) A hydrogen atom in interstellar cloud emits a 21-cm photon in the direction of the Sun. The measured recessional velocity of the cloud with respect to the Sun is 200 km/s. Calculate the observed Doppler shift Δν in Hertz of the received photon. 6.3 The detection of radio emission from a spin-flip transition of deuterium (D— an isotope of hydrogen which has one neutron in addition to one proton in the nucleus)—analogous to the spin-flip transition of hydrogen—has been reported in the literature [39]. The transition is observed to occur at a frequency ν = 327.384 MHz and the rate for this transition is φ = 4.69 × 10−17 s−1 . (a) Calculate the wavelength λ in meters at which this transition is observed to occur. (b) Calculate the lifetime t of this transition in years. (c) Calculate the excitation temperature T in Kelvin of this transition. (d) The measured ratio of the abundance of deuterium to hydrogen—that is, [D/H]—is approximately 21 parts per million [39]. Consider the cloud of hydrogen gas described in Example 6.2. Assuming that this ratio of the [D/H] abundance holds for this cloud, predict the total number of photons of the spinflip transition are emitted per second by the cloud. Based on your result from Part (c), why is it reasonable to assume that all of the deuterium atoms in the cloud are in the excited state and are therefore capable of emitting a photon from the spin-flip transition? 6.4 An astronomer makes an HI observation within the disk of the Milky Way Galaxy along the Galactic longitude  = 35◦ . (a) Determine the radial velocity vr and tangential velocity vt (in km s−1 ) of a cloud of hydrogen gas detected along this direction and at a distance d = 1 kpc from the observer.

Problems

263

Fig. 6.20 The measured flux density for the 21 cm along the Galactic longitude  = 35◦ . The four peaks correspond to four clouds (in order from left to right) that are denoted as A, B, C, and D. See Problem 6.4

(b) Determine the minimum distance Rmin (in kpc) between a tangent along the line of sight and the Galactic Center. (c) Determine the radial velocity vmax (in km s−1 ) for a cloud of gas located at this minimum distance from the Galactic Center and identify which of the clouds shown in Fig. 6.20 (labeled A, B, C, and D) is located at that minimum distance Rmin , as indicated by its observed radial velocity. (d) Determine the proper motion μ (in arcseconds per year) of the cloud located at Rmin , assuming that its radial velocity vmax is equivalent to its tangential velocity vt = vθ . 6.5 Calculate the plasma frequency νp in Hz for a region of the interstellar medium with a mean electron number density ne = 0.1 cm−3 . Also compute the corresponding wavelength λp of a photon with this frequency. 6.6 A spherical cloud of interstellar gas collapses upon itself (assume that the “freefall” approximation is sufficient throughout the collapse). If the radius of the cloud is 0.1 pc and the mass of the cloud is 1 M , calculate the density ρ of the cloud (in kg/m3 ) and the timescale tff of the collapse (in years). 6.7 Calculate the frequency ν in Hertz of the J  = 1 to J = 0 rotational transition of molecular hydrogen. Assume that the mass of each nucleus corresponds to the mass of a proton and that the separation between the two nuclei is 75 pm (75 × 10−12 m). Calculate the excitation temperature Tmin of this transition and verify that the typical temperatures of molecular clouds are far too low to excite this transition. 6.8 Verify that the observed rest frequency νH56α of the H56α transition (that is, the transition from n = 57 to n = 56 for a hydrogen atom) occurs at approximately 36.46 GHz.

264

6 Galactic Radio Astronomy

6.9 NGC 6357 is an HII region located within the Milky Way Galaxy. Its distance, measured angular extent, electron temperature Te and turnover frequency νturnover are 1740 pc, 0.95◦ , 7800 K, and 800 MHz, respectively. Assume that NGC 6357 is spherical, its density is uniform, and all of the hydrogen atoms within the HII region are completely ionized. (a) Calculate the emission measure EM in m−5 of NGC 6357. (b) Calculate the brightness Bν in J/m2 of NGC 6357 at the turnover frequency. (c) Calculate the electron number density ne in m−3 . 6.10 Consider an HII region that lies at a distance of 500 parsecs and spans an angular extent of 60 arcminutes = 3600 arcseconds on the sky. Assume that the HII region is spherical and that it is composed entirely of ionized hydrogen atoms. Further assume that the number density of particles in the HII is uniform, that the electron temperature of the HII region is Te = 8000 K and the measured turnover frequency νturnover is 900 MHz. (a) Calculate the linear diameter L of the HII region. (b) Calculate the emission measure EM in m−5 of the HII region at the turnover frequency. (c) Assuming that the Rayleigh–Jeans approximation is valid for the spectrum of the HII region at radio wavelengths, calculate the brightness Bν of the HII region in J/m2 at the turnover frequency. (d) Calculate the electron number density ne of the HII region in m−3 . (e) Calculate the total mass M of the HII region in kilograms and in solar masses. 6.11 Verify that the J  = 1 to J = 0 rotational transition of silicon monoxide (SiO) that is detected from evolved stars does occur at a frequency ν = 43.12 GHz. Assume that the mass of the silicon and oxygen atoms correspond to 28 and 16 nucleons, respectively, and that each nucleon has a mass of 1.67 × 10−27 kg. Also assume that the bond length between the two nuclei is 151 pm = 151 × 10−12 m. 6.12 Consider the synchrotron radiation emitted by relativistic cosmic-ray electrons gyrating in the compressed ambient magnetic field in a region of the interstellar medium near a supernova remnant. Assume that the magnetic field strength B in this region is 5 × 10−9 T. (a) Calculate the energies E in Joules and the Lorentz factors γ for two cosmic-ray electrons accelerated by the supernova remnant: the first electron (Electron A) is emitting most of its radiation at ν = 5 GHz (a typical radio photon) and the other electron (Electron B) is emitting most of its radiation at ν = 5 × 1017 Hz (a typical X-ray photon). (b) Estimate the sizes (in radians) of the beamwidths of the radiation emitted in the forward direction by Electron A and Electron B. (c) Calculate the half-life time (in years) of both electrons and the ratio of the total power emitted PB /PA by the two electrons. Physically interpret your results by stating how frequently electrons of energies comparable to Electron A and

Problems

265

Electron B would need to be replenished during the nominal lifetime of a supernova remnant (typical lifetimes of 104 –105 years). 6.13 The Crab Nebula pulsar has a period P = 0.033 s and a slowdown rate dP /dt = 4.22 × 10−13 s/s. Assuming a mass M = 1M and a radius R = 10 km, calculate its characteristic age τ in seconds, its rotational kinetic energy Erot in Joules, and the rate dErot /dt at which the pulsar loses rotational kinetic energy. 6.14 A pulsar is discovered: the period of the pulsar is P = 0.05 s and the period derivative of the pulsar is measured to be dP /dt = 5 × 10−13 s s−1 . Assume that the mass of the pulsar is M = 1.4M and that the radius of the pulsar is R = 10 km. Compute the following properties of the pulsar: (a) (b) (c) (d) (e) (f)

The minimum density of the pulsar ρ in kg m−3 . The moment of inertia of the pulsar I in kg m2 . The rotational energy Erot in Joules. The time derivative of the rotational energy dErot /dt in Watts. The minimum magnetic field Bmin in Tesla of the pulsar. The characteristic age τc in years of the pulsar.

6.15 Derive Eq. (6.205) from Eqs. (6.202) and (6.204). 6.16 In a recent study of a Galactic supernova remnant known to lie at a distance of 2000 parsecs, two radio pulsars (denoted as Pulsar A and Pulsar B) were detected along the line of sight toward the remnant. Observations were made to determine if either of these pulsars may be associated with the remnant or instead are unassociated with the remnant and are simply viewed along the same line of sight. Assume a constant interstellar density ne = 0.1 cm−3 and a constant Galactic magnetic field strength B = 3 µG along the lines of sight to both pulsars. (a) For Pulsar A, observations made at the frequencies of 480 MHz and 1400 MHz reveal a time delay of 1.59 µs between the detection of pulses at these two frequencies. Calculate the distance to Pulsar A in parsecs. Could it be possibly physically associated with the remnant? (b) For Pulsar B, the measured rotation measure RM along the line of sight toward the pulsar is 486 radians m−2 . Calculate the distance to Pulsar B in parsecs. Could it be possibly physically associated with the remnant? 6.17 Massive B-type stars (assume a typical mass M = 5M and a typical radius r = 3R ) are thought to perish in supernova explosions that ultimately produce neutron stars or pulsars. The rotation velocities of these types of stars are typically v = 20 km/s and the magnetic fields on their surfaces are B = 5000 G. Assume a B-type star does in fact explode in a supernova and produce a central pulsar with a radius of rf = 10 km and a mass of M = 2M . (a) Through the conservation of magnetic flux, calculate the magnetic field strength B in Gauss on the surface of the pulsar. (b) Through the conservation of angular momentum, calculate the expected rotational velocity of the pulsar vf (in km/s). How does this velocity compare to

266

6 Galactic Radio Astronomy

the speed of light? Does this rotational velocity seem reasonable? Would you conclude that strict conservation of angular momentum is applicable in this case? If not, where might some of the angular momentum be lost? (c) The strongest magnetic field strengths measured for pulsars are on the order of B ≈ 1014 Gauss. Comment on how this value compares to the value calculated in Part (a). 6.18 The table below presents a summary of radio observations of a pulsar (both the time the observation was conducted and the measured period P ): Time of observation t (s) 0 100,000 200,000 300,000

Measured period P (s) 0.5 0.500000010 0.500000018 0.500000032

Assuming a mass of 2 M and a radius of 10 km (typical values for a pulsar) calculate the following properties of this pulsar: (a) The rate of the change of the period (dP /dt) in s/s. (b) The characteristic age of the pulsar (in years). (c) The rotational kinetic energy of the pulsar Erot (in Joules). Hint: Calculate ω for the pulsar first. (d) The time derivative dErot /dt of Erot in Watts or Joules/s. 6.19 The measured dispersion measure and rotation measure along the line of sight to the Crab Nebula is 56.5 cm−3 pc [40] and −25 rad m−2 [41]. Based on these values, calculate B in Gauss along this line of sight. 6.20 As described previously, one of the first explanations proposed for the phenomenon of glitches was that quakes occurring on the surface of the pulsar would cause a sudden decrease in the radius R of the pulsar with a corresponding sudden decrease in the period P . Show that the change in radius dR may be expressed in terms of the change in period dP as dR =

P GM dP . 6π 2 R 2

(6.239)

What is the change in radius dR of the Crab Nebula when a glitch occurs with a corresponding change in period dP ? Assume that the period P , radius R, and mass M of the Crab Nebula pulsar is 0.033 s, 10 km, and 1.4 solar masses, respectively. Also assume that the change in period is dP = 10−8 s.

References

267

References 1. E.B. Giacani, G.M. Dubner, A.J. Green, W.M. Goss, B.M. Gaensler, Astron. J. 119, 281–291 (2000) 2. D. Ward-Thompson, A.P. Whitworth, An Introduction to Star Formation (Cambridge University, Cambridge, 2011) 3. M. Feast, P. Whitelock, Mon. Not. R. Astron. Soc. 291, 683–693 (1997) 4. E.E. Barnard, Astron. J. 29, 181–183 (1916) 5. G.F. Benedict, B. McArthur, D.W. Chappell, et al., Astron. J. 118, 1086–1100 (1999) 6. V.V. Bobylev, Astron. Lett. 36, 220–226 (2010) 7. B.W. Carroll, D.A. Ostlie, An Introduction to Modern Astrophysics, 2nd edn. (Pearson/Addison-Wesley, New York, 2006), pp.406–408 8. P.J.E. Peebles, R.H. Dicke, Astrophys. J. 154, 891–908 (1968) 9. D.E. Osterbrock, G.J. Ferland, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei, 2nd edn. (University Science Books, California, 2006), p. 123 10. C. Goudis, Astrophys. Space Sci. 36, 105–109 (1975) 11. A.J. Kemball, P.J. Diamond, Astrophys. J. 481, L111-L114 (1997) 12. M. Elitzur, Astrophys. J. 415, 457 (1996) 13. HAWC Collaboration, A.U. Abeysekara, et al., Nature 563, 82–85 (2018) 14. K.M. Blundell, K.M. Bowler, Astrophys. J. Lett. 616, L159–162 (2004) 15. D.A. Green, A Catalogue of Galactic Supernova Remnants (2014 May version) (Cavendish Laboratory, Cambridge, 2014). http://www.mrao.cam.ac.uk/surveys/snrs/ 16. D.A. Green, Bull. Astron. Soc. India 42, 47–58 (2014) 17. V.L. Ginzburg, S.I. Syrovatskii, Annual Rev. Astronomy Astrophys. 3, 297–350 (1965) 18. A.R. Bell, Mon. Not. R. Astron. Soc. 182, 147–156 (1978) 19. A.R. Bell, Mon. Not. R. Astron. Soc. 182, 443–455 (1978) 20. R.D. Blandford, J.P. Ostriker, Astrophys. J. L. 221, L29-L32 (1978) 21. S. Rosswog, M. Brüggen, Introduction to High-Energy Astrophysics (Cambridge University, Cambridge, 2007), p. 52 22. M.A. Longair, High Energy Astrophysics, vol. 2, 2nd edn. (Cambridge University, New York, 1994), pp. 354–355 23. D.M. Matonick, R.A. Fesen, Astrophys. J. Suppl. Series 112, 49–107 (1997) 24. K.W. Weiler, R.A. Sramek, Annu. Rev. Astron. Astrophys. 26, 295–341 (1988) 25. S. Höfner, in Dynamical Evolution of Supernova Remnants. Lecture Notes (2010). http://www. astro.uu.se/~hoefner/astro/teach/apd$_$files/apd$_$SNR$_$dyn.pdf 26. S. Bhatnagar, U. Rau, D.A. Green, M.P. Rupen, Astrophys. J. Lett. 739, Article ID. L20 (2011) 27. B. Uyaniker, W. Reich, A. Yar, E. Fürst, Astron. Astrophys. 426, 909–924 (2004) 28. K. Rohlfs, T.L. Wilson, Tools of Radio Astronomy, 4th edn. (Springer, New York, 2004), pp. 259–260 29. R.N. Manchester, G.N. Hobbs, A. Teoh, M. Hobbs, Astron. J. 129, 1993–2006 (2005) 30. M. Zeilik, S.A. Gregory, Introductory Astronomy and Astrophysics, 4th edn. (Brooks/Cole, New York, 1998), pp. 297–298 31. K. Torii, H. Tsunemi, T. Dotani, K. Mitsuda, N. Kawai, K. Kinugasa, Y. Saito, S. Shibata, Astrophys. J. 523, L69–L72 (1999) 32. A.G. Lyne, C.A. Jordan, F. Graham-Smith, C.M. Espinoza, B.W. Stappers, P. Weltevrede, Mon. Not. R. Astron. Soc. 446, 857–864 (2015) 33. F.C. Michel, W.H. Tucker, Nature 223, 277–279 (1969) 34. R.D. Blandford, R.W. Romani, Mon. Not. R. Astron. Soc. 234, 57–60 (1988) 35. R.D. Ferdman, R.F. Archibald, V.M. Kaspi, Astrophys. J. 812, Article 95 (2015) 36. K. Rohlfs, D.L. Wilson, Tools of Radio Astronomy, 4th edn. (Springer, Berlin, 2004)

268

6 Galactic Radio Astronomy

37. A.M. Ghez, S. Salim, N.N. Weinberg, et al., Astrophys. J. 689, 1044–1062 (2008) 38. J.A. Davidson, Polarimetry of the Interstellar Medium (1996), p. 504 39. A.E.E. Rogers, K.A. Dudevoir, T.M. Bania, Astrophys. J. 133, 1625–1632 (2007) 40. K. Davidson, Y. Terzian, Astron. J. 74, 849–854 (1969) 41. A.S. Wilson, Mon. Not. R. Astron. Soc. 157, 229–253 (1972) 42. https://www.cv.nrao.edy/course/astr534/SynchrotronSrcs.html.

Chapter 7

Extragalactic Radio Astronomy: Galaxy Classification, Active Galactic Nuclei, Superluminal Motion, Galaxy Clusters, and the Cosmic Microwave Background

7.1 Extragalactic Radio Astronomy and Galaxy Classification Toward the start of the twentieth century, astronomers such as Edwin Hubble conclusively proved that the Milky Way Galaxy is just one of the innumerable galaxies in the Universe. Once astronomers had established that galaxies are external systems to our own galaxy, a need to classify galaxies based on such observable properties as morphology became immediate. Over time, galaxies were classified into four broad categories based on their apparent morphologies: this classification system based on morphology is often presented as the so-called Hubble tuning-fork diagram as shown in Fig. 7.1. The first category of galaxies—spiral galaxies— includes the Milky Way Galaxy and feature prominent spiral arm structure (as their name implies) with flat disk-like morphologies. Spiral galaxies possess extensive amounts of dust and gas as well as an elevated rate of star formation, as indicated by the presence of many short-lived high-mass stars. Some spiral galaxies feature a circular structure in their nuclei but other spiral galaxies—known as the barredspiral galaxies—exhibit a prominent bar-like structure at their centers. Spiral galaxies obey the following classification system: spiral galaxies with the most tightly wound spiral arms and the brightest nuclei are classified as “Sa” while spiral galaxies with the most loosely wound spiral arms and the faintest nuclei are classified as “Sd.” Spiral galaxies with intermediate properties in tightness of spiral arms and brightness of nuclei are classified as “Sb” and “Sc”: finer classification types (such as “Sbc” and “Scd”) are also found in the literature. The second category of galaxies is the elliptical galaxies: in contrast to spiral galaxies, elliptical galaxies have a more spherical morphology and lack distinct arm structure. They also are essentially bereft of dust and gas, exhibit low star formation rates, and possess essentially no high-mass stars. A prominent example of an elliptical galaxy is M87, also known as Virgo A: this galaxy is particularly well© Springer International Publishing Switzerland 2020 T. G. Pannuti, The Physical Processes and Observing Techniques of Radio Astronomy, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-16982-8_7

269

270

7 Extragalactic Radio Astronomy

Fig. 7.1 The well-known Hubble “tuning-fork” diagram depicting different morphological types of galaxies. Much of this classification system is dependent on the apparent morphology of the galaxy. On the left are the elliptical galaxies, which range from those which have an apparently circular morphology (denoted as “E0” types) to those which have an apparently oblate or prolate morphology (denoted as “E7” types). Spiral galaxies (normal and barred) are classified based on the tightness of their arms and the brightness of their nuclei. Normal spirals with tightly wound arms and bright nuclei are classified as “Sa” while normal spirals with more wide open arms and fainter nuclei are classified as “Sc,” with “Sb” designating an intermediate case between the two extremes. Barred spirals are indicated by a “B" in their types (such as “SBa”) and they follow a similar naming scheme as the normal spirals. Lenticular galaxies (denoted as “S0” types) have properties of both spirals and ellipticals: like spirals, they have appreciable amounts of dust and gas, and like ellipticals, they have rounded morphologies rather than disk-like morphologies. Lastly, the irregular galaxies (denoted as “Irr”) have amorphous morphologies and lack any kind of regular structure. Image credit: NASA and ESA

studied at radio wavelengths. The third category of galaxies is known as lenticular galaxies and members of this category can be envisioned as hybrids of the previous two categories. Like spiral galaxies, lenticular galaxies have disks and copious amounts of dust but like elliptical galaxies they lack gas and spiral arms. Lenticular galaxies also have extended rounded morphologies like elliptical galaxies as well. Finally, the fourth category of galaxies is known as the irregular galaxies and unlike the previous three categories of galaxies, members of this category feature an amorphous morphology rather than any kind of regular structure. For some irregular galaxies, there is no star formation activity but for other irregular galaxies very active star formation is seen localized to portions of the physical extent of the galaxy. The Large and Small Magellanic Clouds—two satellite galaxies of the Milky Way— are examples of irregular galaxies. Radio observations reveal a wealth of information about galaxies of all categories. For example, as discussed in this chapter, radio continuum observations can identify synchrotron emission from cosmic-ray electrons accelerated by supernova

7.1 Extragalactic Radio Astronomy and Galaxy Classification

271

remnants located within the galaxy. In fact, the amount of diffuse radio emission is believed to be proportional to the star formation rate of the galaxy. Also, HI and CO observations of galaxies reveal both the presence and the amount of atomic and molecular gas, respectively, in a galaxy. Finally, for nearby galaxies, radio observations made with sufficient angular resolution may resolve individual discrete sources within a galaxy, such as HII regions and supernova remnants. If the nucleus of the galaxy is a powerful radio source, it may be resolved from other radio-emitting components of the galaxy as well. In fact, radio observations of the luminous nuclei of galaxies yield particularly insightful results on the dynamics of supermassive central black holes and their interactions with their surrounding environments. The sheer significance of the study of external galaxies at radio wavelengths is illustrated in striking fashion in Fig. 7.2. This figure depicts a map made at a frequency of 4.85 GHz of the sky using the 300-foot radio telescope that operated at Green Bank in West Virginia [1]. While the figure shows several Galactic supernova remnants (seen as numerous extended arc-like structures), the vast majority of point sources that are seen correspond to distant galaxies and not Galactic stars. This figure helps underscore the very crucial role that radio astronomy plays in advancing understandings about the behavior of the Universe on the largest spatial scales through studies of galaxies, both individually and as members of larger gravitationally bound systems known as clusters and superclusters of galaxies.

7.1.1 Cosmology and Hubble’s Law Spectroscopic optical observations made of nearby galaxies by Edwin Hubble and other astronomers revealed a remarkable result: redshifts were detected in the spectra of these galaxies and the amounts of the redshift seen in these spectra were directly proportional to the distances to the galaxies [2]. Recall from Sect. 2.4.5 and Eqs. (2.140) and (2.141) that if there is a net increase in the distance between a source and an observer (as driven by the motion of the source away from the observer, for example), the observer will see that the lines in the spectrum of the source will be shifted toward long wavelengths (that is, redshifted) due to the light waves being stretched out due to the motion of the source as viewed by the observer. The amount of the redshift is directly proportional to the rate of increase in the distance between the observer and the source: for example, if the velocity of the source away from the observer increased, then the observed redshift (and the corresponding recessional velocity) would also increase. Based on his observations, Hubble derived a simple relationship between the measured recessional velocity of a galaxy is vr and the distance d to the galaxy. This relationship is known as Hubble’s Law and it is commonly expressed as vr = H0 d,

(7.1)

272

7 Extragalactic Radio Astronomy

Fig. 7.2 The “radio” sky: a map made at a frequency of 4.85 GHz of a broad portion of the sky. While Galactic objects (such as supernova remnants, as distinguished by their extended shell-like morphologies) can be seen, the sky is dominated by point sources that correspond to galaxies. This may be contrasted with a night sky as viewed at optical wavelengths, which is dominated by the light from stars. Superimposed on the figure is an image of the radio telescopes at Green Bank, West Virginia. From right to left, these telescopes are the 140-foot antenna, the 300-foot antenna (no longer extant), and the three 85-foot telescopes that compose an interferometer. The radio sky map was created from data collected by observations made with the 300-foot. Image credit: Image courtesy of NRAO/AUI

where H0 is known as Hubble’s Constant. The remarkable implication of Hubble’s Law that is revealed through inspection of Eq. (7.1) is that a linear relationship exists between the recessional velocity of a galaxy and its distance. Therefore, by measuring vr of a galaxy through the measurement of the redshift of its spectrum, d may be calculated readily if H0 is known. In fact, since Hubble’s original discovery and through the present era, vast amounts of telescope time have been devoted to measuring H0 through the measurements of the redshifts of many galaxies and calibrating Eq. (7.1) through the use of independent estimates of the distances to these galaxies. Hubble initially estimated the value of this constant to be 500 km

7.1 Extragalactic Radio Astronomy and Galaxy Classification

273

s−1 Mpc−1 : for much of the twentieth century, measured values of H0 typically ranged from 50 km s−1 Mpc−1 to 100 km s−1 Mpc−1 . For the remainder of this text, a value for H0 of 71 km s−1 Mpc−1 has been adopted for illustrative purposes. It is important to note that the recessional velocity vr that is considered in Hubble’s Law originates not from a physical motion of the galaxy but instead through the expansion of the Universe that occurs while the photon is traveling from the galaxy to the observer. The expansion of space causes the wavelength of the photon to stretch as the photon is in motion, thus producing a redshift. The component of the measured redshift of the galaxy that is due to the expansion of the universe is known as the cosmological redshift. Furthermore, photons emitted by more distant galaxies travel a larger distance to reach an observer than photons emitted by closer galaxies: photons in the former case are subject to more stretching over their longer light travel paths and therefore a larger redshift is seen. Certainly, galaxies may have their own physical motions (called peculiar motions) due to their orbital motions within gravitationally bound clusters of galaxies (see Sect. 7.4 for a detailed discussion about clusters of galaxies) but any redshift or blueshift imparted on the spectrum of the galaxy due to its peculiar motion would be far outstripped by the redshift imparted on the spectrum due to the expansion of the Universe.1 Lastly, note through inspection that the units of H0 are inverse time; therefore, the units of the reciprocal of H0 are time. This reciprocal corresponds to the time since the Big Bang and is thus used to estimate the age t0 of the Universe. This estimate assumes that the Universe has been expanding at a constant rate (specifically the observed present rate) of expansion since the Big Bang: while this is not strictly true it is a useful approximation for many applications.

1 Hubble’s Law can be most reliably applied as a distance indicator for galaxies located at a medium

distance region. It cannot be used to find distances to galaxies located in the Local Group (the local gravitationally bound assembly of galaxies to which the Milky Way Galaxy belongs) because on these small distance scales, orbital motions by the galaxy within the Local Group itself dominate over any redshift that may originate due to the expansion of the Universe. In fact, the spectrum of the Andromeda Galaxy (M31) exhibits an overall blueshift in its spectrum, indicating that it is actually approaching the Milky Way! At the other extreme in regard to very distant galaxies, the effects of dark energy manifest themselves: dark energy serves to accelerate the expansion of the Universe. Thus, at large distances, galaxies appear to be nearer than they really are and Hubble’s Law no longer provides accurate distances to galaxies. The true nature of dark energy is not known: its presence has only been revealed by the observations of distant galaxies and a detailed treatment of the subject is beyond the scope of this book. It is thus stressed that Hubble’s Law provides a proper description of the relationship between recessional velocity and distance for galaxies located at moderate distances and thus is indispensable as a distance indicator in cosmology.

274

7 Extragalactic Radio Astronomy

Example Problem 7.1 Assuming a value of 71 km s−1 Mpc−1 for H0 , estimate the age t0 of the Universe in years. Solution Noting that 1 Mpc = 3.086 × 1019 km and 1 yr = 3.15 × 107 s, t0 may be computed based on the assumed value of H0 as t0 =

1 = H0





1 71 km s−1 Mpc−1

3.086 × 1019 km 1 Mpc

= 13.8 × 109 yr.



1 yr 3.15 × 107 s



(7.2)

To accommodate different published estimates of the value of H0 , distances to galaxies are often reported in the literature in terms of the parameter h, which is defined as h=

H0 100 km s−1 Mpc−1

.

(7.3)

Therefore, distances to galaxies are expressed as a function of h−1 . The advantage of this notation is such that if an author quotes a distance to a galaxy in terms of h−1 , the reader may rapidly calculate their own distance to the galaxy based on their own assumed value of H0 . This is preferable to the reader needing to extract their own distance to the galaxy if the author adopted a different value for H0 . A concept that is frequently encountered in cosmology is the scale factor: this quantity (denoted as R) describes how length scales vary as the universe expands. The scale factor is defined as R=

R0 , (1 + z)

(7.4)

where R0 is a length scale in the rest frame of an observer (for example, the current size of the Universe for an observer in the present). The chief application of Eq. (7.4) is to relate the current size R0 of the Universe to a time when photons were emitted by a galaxy currently observed to have a redshift z in its spectrum: at this time, the size of the Universe was R. In a similar vein, we may use z to relate the time t when the photon was emitted by the galaxy to the present time t0 . Recall that according to Eq. (6.87), the radius R of a spherical object that is contracting due to gravity is proportional to ρ −1/2 , where ρ is the mass density of the object. Assuming that the geometry of the Universe is spherical, that gravity is the dominant force regarding its expansion and that the expansion of spherical object may be derived in a similar manner, the dependence of a particular age of the Universe (corresponding to time t) on R may be expressed as

7.2 Normal Galaxies

t ∝ ρ −1/2 =

275

V 1/2 ((4/3)π R 3 )1/2 = ∝ R 3/2 , M 1/2 M 1/2

(7.5)

where V is the volume of the Universe (again assumed to be spherical with a radius R) and M is its mass. Alternatively, this equation may be expressed as R ∝ t 2/3 and thus the relationship between t, the current age of the Universe t0 and z may be expressed as t=

t0 . (1 + z)3/2

(7.6)

Example Problem 7.2 The redshift of a distant galaxy is measured to be z = 1. Compare the size of the Universe R and the age of the Universe t at the time that the photon was emitted to the present size and age of the Universe (R0 and t0 , respectively). Solution From Eqs. (7.4) and (7.6), the values of R and t that correspond to redshift z=1 are R=

R0 R0 = (1 + 1) 2

and

t=

t0 t0 . = 3/2 2.83 (1 + 1)

(7.7)

Thus, the Universe was half its current size and approximately one-third its present age when the photon was emitted.

7.2 Normal Galaxies 7.2.1 HI Line Observations Observations of neutral HI were first described in Sect. 6.2.2 in studies of the Milky Way galaxy as a technique to probe cold hydrogen gas in the galaxy. Clearly, the cold hydrogen content of other galaxies may be probed by making HI observations of these sources as well: such observations can not only detect the content of HI from these galaxies, but also warps in the disks of galaxies due to tidal interactions with nearby galaxies. Such warps are not always obvious in optical observations of external galaxies, and therefore neutral HI observations of these warps provide details about how galaxies interact with each other. In Fig. 7.3, a multi-wavelength image of the nearby face-on spiral NGC 3596 is presented, which depicts both the optical disk of the galaxy and the HI disk. Note how much farther the HI disk extends beyond the optical disk of the galaxy: similar results have been seen in studies of other nearby spiral galaxies as well.

276

7 Extragalactic Radio Astronomy

Fig. 7.3 A multi-wavelength image of the nearby face-on spiral galaxy NGC 3586. Hα emission, visible light, and neutral hydrogen 21 cm gas emission are shown in red, white, and blue, respectively. Notice how far the HI extends beyond the nominal optical extent of the galaxy. Image credit: T. Burchell and B. Saxton (NRAO/AUI/NSF) from data provided by A. C. Boley and L. van Zee, Indiana University, D. Schade and S. Côté, Herzberg Institute for Astrophysics

In Fig. 7.4, the HI profiles (the flux density SHI of detected HI emission as a function of recessional velocity v) of a sample of nearby galaxies are presented. The measured fluxes from the galaxies are shown as a function of recessional velocity, which corresponds to redshift of the 21-cm line. The features of these profiles may be interpreted as follows: the central velocity of each galaxy’s profile is the systemic velocity, which corresponds to the velocity of the galaxy as a whole away from the observer due to the general expansion of the Universe. This velocity corresponds to a redshift in the 21-cm line to a longer wavelength and the velocity itself that corresponds to this longer wavelength is determined using Eq. (2.140). Peaks of flux are seen at velocities higher and lower than the systemic velocity: because spiral

7.2 Normal Galaxies

Fig. 7.4 HI profiles of select nearby spiral galaxies [7]

277

278

7 Extragalactic Radio Astronomy

galaxies rotate, it can be imagined that at any given time, this rotation carries a portion of the galaxy is in motion toward an observer (therefore the HI emission from that portion of the galaxy is blueshifted) while a portion of the galaxy is carried away from the observer (therefore the HI emission from that portion of the galaxy is redshifted). Therefore, peaks of HI flux seen at higher and lower velocities than the systemic velocity is particularly pronounced. For galaxies which are seen with orientations that are closer to face-on, peaks in emission at velocities above and below the systemic velocity are not as prominent. The total HI mass MHI of a galaxy may be determined by integrating over its entire HI flux profile using the following relation [3]: MHI (M ) = (2.36 × 105 M Mpc−2 Jy−1 s) D 2

 Sν dν

(7.8)

where MHI , D, and Sν is expressed in solar masses, Megaparsecs, and Janskys, respectively. The integration is performed over frequency ν, specifically the frequency range Δν over which HI emission is detected from the galaxy of interest. The following approximation to the integration is commonly used: typically the frequency range in this integration is instead expressed in terms of a range in velocity Δv (that is, the integration is performed over Δv instead of Δν) and the relation between these variables of integration may be derived as follows. Differentiating both sides of Eq. (2.90) with respect to ν and λ yields dν = −

c dλ, λ2

(7.9)

and from Eq. (2.140) dλ may be expressed in terms of Δλ as dλ = Δλ =

λ0 v . c

(7.10)

Combining these two equations yields the following equation for the integration of dν in terms of a certain wavelength λ0 of interest: Δν =

Δv , λ0

(7.11)

where Δv is simply the velocity range over which HI emission is detected from the galaxy of interest. Finally, for the purposes of simplification, the integration is replaced with merely < Sν > ≈ (Δv)/λ0 , where < Sν > is a median value of the measured flux density of the HI profile of the galaxy.

7.2 Normal Galaxies

279

Example Problem 7.3 Based on its HI profile as shown in Fig. 7.4, calculate the HI mass MHI of NGC 2403 in M . Assume a distance to NGC 2403 of 3.2 Mpc.[8] Solution Inspection of the HI profile of this galaxy as shown in Fig. 7.4 reveals that the velocity range Δv over which HI emission is detected from this galaxy is approximately 0–280 km/s. Furthermore, inspection of the HI profile of NGC 2403 suggests a median value of the measured flux density over the HI profile of the galaxy to be < Smin > ≈ 3 Jy. Therefore, Δv is 280 km/s and combining Eqs. (7.8) and (7.11) yields (with λ0 = 2.1 × 10−4 km for the 21-cm line) MHI (M ) =

(2.36 × 105 M Mpc−2 Jy−1 s) (3.2 Mpc)2 (3 Jy)(280 km/s) 2.1 × 10−4 km

= 9.67 × 1012 M .

(7.12)

Typical HI masses of spiral galaxies like NGC 2403 are 1012 –1014 M .

7.2.2 Tully–Fisher Relation As mentioned previously, determining accurate distances to sources is one of the most commonly encountered challenges in modern astronomy. The method using Hubble’s Law as described in Sect. 7.1.1 is a direct and often-employed method for finding distances but independent methods for determining distances are needed to provide a separate calibration of distance estimates. One such method is known as the Tully–Fisher relation and it relates the observed maximum rotational velocity of a spiral galaxy to its luminosity [11]. Consider a spiral galaxy of mass M and radius R: an object (say a cloud of hydrogen gas) located at the edge of the galaxy has an orbital velocity around the center of the galaxy of Vmax . The name for this velocity stems from the object being located at the farthest radius of the galaxy. A flat rotation curve is assumed for this galaxy, consistent with the flat rotation curves seen in other galaxies like the Milky Way (see the discussion regarding rotation curves of galaxies in Sect. 6.2.3) and interpreted as evidence for the presence of dark matter. The flat portion of the rotation curve at large radius is equated with Vmax , so the relationship between M, R and Vmax may be expressed as M=

2 R Vmax G

or

R=

GM . 2 Vmax

(7.13)

280

7 Extragalactic Radio Astronomy

Now consider a quantity known as the mass-to-light ratio: this ratio corresponds to the total mass of a galaxy to the total amount of light emitted by the galaxy. Defining the luminosity of a galaxy to be L and assuming that the mass-to-light ratio of all spirals is a constant (that is, M/L = 1/kML ), L may be expressed as L=

2 R kML Vmax . G

(7.14)

Making the additional assumption that the surface brightnesses at the centers of spiral galaxies are a constant value Σ such that Σ = L/R 2 , L can be expressed as L=

2 V4 kML max 4 ∝ Vmax . ΣG2

(7.15)

Therefore, by simply measuring the maximum rotational velocity Vmax of a spiral galaxy and assuming a value for its mass-to-light ratio M/L, the luminosity L of the galaxy may be determined and thus the distance to the galaxy may be estimated. The Tully–Fisher relation is therefore a very powerful tool for determining the distances to spiral galaxies. Empirically derived relations between Vmax for different spiral galaxy types defined along the Hubble tuning-fork diagram (see Fig. 7.1) are as follows [12]: MB = −9.95 log10 Vmax + 3.15

(Sa galaxies)

(7.16)

MB = −10.2 log10 Vmax + 2.71

(Sb galaxies)

(7.17)

MB = −11.0 log10 Vmax + 3.31

(Sc galaxies).

(7.18)

Here, Vmax is expressed in units of kilometers per second. In these relations and MB is the absolute magnitude (a measurement of luminosity based on a logarithmic scale) of the galaxy integrated over its entire angular extent over the range of optical wavelengths known as the “B” (“blue”) band. By convention, the B-band has a central wavelength of 440 nm and a bandwidth of 98 nm. The integrated luminosity LB of the galaxy in the B-band is related to MB through the relation   LB (7.19) MB = M − 2.5 log10 L where M and L are the absolute magnitude and the luminosity, respectively, of the Sun integrated over all wavelengths. These quantities correspond to +4.74 and 3.84×1026 W, respectively. Inserting this quantities into Eq. (7.19) and solving for LB yields LB [W] = 10(71.20−MB )/2.5 .

(7.20)

7.2 Normal Galaxies

281

Example Problem 7.4 An HI observation of an Sa galaxy yields a maximum rotation velocity for the galaxy of Vmax = 350 km/s. Based on this information, calculate (a) MB for the galaxy and (b) LB for the galaxy in units of Watts and solar luminosities. Solution From Eq. (7.14), MB may be computed as MB = −9.95 log10 (350 km/s) + 3.15 = −22.16.

(7.21)

Furthermore, from Eq. (7.20), LB may be computed in Watts as LB = 10(71.20−(−22.16))/2.5) = 2.19 × 1037 W,

(7.22)

or in solar luminosities as  LB = 2.19 × 1037 W

1 L 3.84 × 1026 W

 = 5.7 × 1010 L .

(7.23)

7.2.3 Radio Continuum Observations In addition to the HI observations made of galaxies that was described in Sect. 7.2.1, radio continuum observations of nearby galaxies have also been conducted. At lower angular resolution (or with single dish observatories), these observations detect disks of cosmic-ray particles emitting synchrotron radiation (as indicated by spectral analysis of the radio emission). These particles have been accelerated to relativistic energies by supernova explosions that have occurred within the galaxy. Because high-mass stars have shorter lifetimes than low-mass stars and because these objects will ultimately perish in supernova explosions, the higher amount of diffuse synchrotron emission from the disks of galaxies is linked with higher star formation rates in galaxies. In Fig. 7.5, a radio continuum observation made of the nearby spiral galaxy NGC 6946 is shown: this image was made by combining data from observations made with a single dish telescope and an interferometer with narrow spacing between the telescopes. This galaxy possesses a radio luminous disk of emission: it is also know to feature a remarkably high rate of star formation and supernovae. In this figure, the spiral arm structure of the galaxy is evident, along with emission from the radio-bright central nucleus and brighter discrete radio sources seen in projection within the disk of the galaxy.

282

7 Extragalactic Radio Astronomy

Fig. 7.5 A map of the nearby face-on spiral galaxy NGC 6946 made at a wavelength λ = 6.2 cm (from [13]). The map was made by combining data from observations made with the Very Large Array Interferometer (with a narrow spacing between the telescopes leading to a lower attained angular resolution) and the Effelsberg 100-m single dish radio telescope to help improve u,v plane coverage. The total intensity is represented in a logarithmic greyscale and the resolution of the image is 15 . At this angular resolution, diffuse emission from the spiral arms of the galaxy are clearly visible

High angular resolution radio interferometric observations of spiral galaxies reveal the discrete source population of a galaxy more clearly. In Fig. 7.6, a high angular resolution map is presented for NGC 6946: in this case, the map is dominated by discrete sources that may include supernova remnants and HII regions that are native to the galaxy or background galaxies seen in projection through the disk of NGC 6946. In the case of this particular galaxy, a prominent central nucleus is seen as well. Classification of these sources may be made based on spectral index measurements: HII regions which emit thermal bremsstrahlung radiation will have flatter spectral indices while supernova remnants, neutron stars and background galaxies which emit synchrotron radiation will have steeper spectral indices. More refined classification can be made by invoking observations made at other wavelengths: for example, background galaxies may be distinguished from neutron stars and supernova remnants on the basis of an absence of a clear optical

7.2 Normal Galaxies

283

Fig. 7.6 A map of the nearby face-on spiral galaxy NGC 6946 made at a wavelength λ=20 cm with the Very Large Array interferometer (with a wide spacing between the telescopes leading to a higher attained angular resolution (approximately 1 —from [15]). At this high angular resolution, emission from discrete sources resident to the galaxy—such as supernova remnants and HII regions—is apparent. Contrast this map with the map of the galaxy presented in Fig. 7.5 that was made using the Effelsberg 100-m single dish radio telescope and the Very Large Array with a narrower spacing between the telescopes: in the former map, diffuse emission from the spiral arms is visible while emission from discrete sources is not

counterpart (such as a region of diffuse Hα emission) within the disk of the galaxy. In addition, supernova remnants and neutron stars can be distinguished from each other based on observations made at other wavelengths: for example, an X-ray counterpart to a supernova remnant would be “softer”2 than such a counterpart for a neutron star, in that the X-ray emission from a supernova remnant is chiefly thermal bremsstrahlung emission, which has different spectral characteristics than the X-ray emission from neutron stars (namely synchrotron emission).

2 In

X-ray astronomy, the terms “soft” and “hard” refer to comparatively lower and higher energy photons. A soft source produces more photons at lower energies, while a hard source produces more photons at higher energies.

284

7 Extragalactic Radio Astronomy

Example Problem 7.5 If the ambient magnetic field strength of the Milky Way galaxy is B = 3 × 10−10 T, calculate and compare the relativistic gyroradii re and rp of a cosmic-ray electron and a cosmic-ray proton, respectively, each with Lorentz factors γe = γp = 106 (note that these particles are both clearly relativistic!). The rest masses of the electron and the proton—in units of energy—are me = 0.511 MeV/c2 = 0.511 × 106 eV/c2 , and mp = 938 MeV/c2 = 938 × 106 eV/c2 , respectively. Solution From Eq. (2.170), the energy Ee of the electron is  Ee = γ me c2 = (106 )(0.511×106 eV/c2 )(c2 )

1.6 × 10−19 J 1 eV



= 8.18×10−8 J.

(7.24) Therefore, from Eq. (3.73), the relativistic gyroradius re of the electron is E 8.18 × 10−8 J = −19 qcB (1.6 × 10 C)(3 × 108 m s−1 )(3 × 10−10 T)   1 pc 12 = 1.84 × 10−4 pc. = 5.68 × 10 m 3.086 × 1016 m

re =

(7.25)

Proceeding in the same manner for the cosmic-ray proton, the energy Ep of the proton is 

1.6 × 10−19 J Ep = (10 )(938 × 10 eV/c )(c ) 1 eV 6

6

2

2



= 1.50 × 10−4 J, (7.26)

and the relativistic gyroradius rp of the proton is 1.50 × 10−4 J C)(3 × 108 m s−1 )(3 × 10−10 T)   1 pc 16 = 0.33 pc. = 1.04 × 10 m 3.086 × 1016 m

rp =

(1.6 × 10−19

(7.27)

The ratios of the two radii are simply rp 0.33 pc ≈ 1800. = re 1.84 × 10−4 pc

(7.28)

Clearly, the gyroradius of the cosmic-ray proton is larger than the gyroradius of the cosmic-ray electron. Note that for the same value of γ for a cosmic-ray proton and a cosmic-ray electron, the ratios of the gyroradii are comparable (continued)

7.3 Radio Galaxies and Active Galactic Nuclei

285

Example Problem 7.5 (continued) to the ratios of the masses of the two particles. Therefore, in images of the diffuse continuum radio emission of galaxies (such as the image presented in Fig. 7.5), the observed emission is dominated by emission from accelerated cosmic-ray electrons in the disk of the galaxy because cosmic-ray protons will have gyroradii that will carry them out of the disk of the galaxy while the cosmic-ray electrons will have gyroradii that are small enough to contain them within the disk of the galaxy.

7.3 Radio Galaxies and Active Galactic Nuclei While radio astronomy was still in its infancy in the middle of the twentieth century, astronomers had already noticed that optical images of select spiral galaxies indicated the presence of unusually bright nuclei within these galaxies. Pioneering work in this field was accomplished by Carl Seyfert [14], who analyzed the optical spectra of the nuclei of six galaxies and noticed the presence of emission lines that appeared to be broadened in width due to Doppler motion. Little attention was paid to the work of Seyfert until radio observations of galaxies revealed the presence of powerful radio sources associated with two galaxies in Seyfert’s sample, namely NGC 1068 and NGC 1275 (the latter is also known as Perseus A). Galaxies that featured Doppler-broadened lines in their optical spectra are known as Seyfert galaxies: these galaxies are classified based on the presence of both Doppler-broadened permitted lines and narrower forbidden lines in their spectra (Type I) or the presence of only narrow permitted and forbidden lines (Type II). As the sensitivity of radio astronomy instrumentation improved, it was realized that many galaxies with bright optical nuclei featured radio luminous nuclei as well. During the first sky surveys conducted with radio telescopes, prominent radio sources were detected that appeared to be associated with point-like optical sources that resembled stars in appearance. Such a result was puzzling because typical stars are expected to be very weak radio sources: we only detect radio emission from the Sun because of our proximity to our star. It was eventually realized that these objects must be located at very large distances based on such studies as optical spectroscopic observations that revealed high redshifts in the spectra of these sources (thus implying large distances as indicated by Hubble’s Law—see Sect. 7.1.1). Furthermore, because they are detected over large distances, they must be intrinsically very luminous. Because these objects appear to be like star-like in appearance (but their properties are much different than stars), they are known as quasars (from “quasi-stellar” objects, where “quasi-” means false or in appearance only). A prominent example of a quasar is the object 3C 48: while the optical

286

7 Extragalactic Radio Astronomy

counterpart of this object seemed to resemble a star, detailed optical and radio observations proved that it was actually extragalactic [16, 17]. High angular resolution optical observations of quasars have detected extended diffuse structure surrounding the bright central source, confirming that these objects are galaxies with remarkably luminous nuclei. It appears that the central nucleus of these galaxies is so luminous that surrounding diffuse emission is swamped out, therefore the object appears to be point-like to a distant observer. The general class of galaxies that exhibit very luminous nuclei is known as active Galactic nuclei or AGNs. Monitoring observations made of quasars and other types of AGNs have indicated that these sources can exhibit variability in their emission on time scales as short as minutes. This short variability is used to place vital constraints on the size of the volume from which the observed emission is detected: the time scale can be envisioned to correspond broadly to the time required for a signal of light to travel across the diameter of the emitting volume.

Example Problem 7.6 Suppose an AGN shows a rapid change in brightness on the order of t = 10 min. Calculate the corresponding diameter d of the emitting region assuming that the “signal” for the source to change brightness travels throughout the source with the velocity c and compare d with the mean radius of the orbit of the Earth around the Sun (that is, an Astronomical Unit). Solution From the given value of t and assuming a velocity c, d is simply −1

d = ct = (3 × 10 m s 8



60 s )(10 min) 1 min

 = 1.8 × 1011 m.

(7.29)

The ratio of d to the distance corresponding to an Astronomical Unit (A.U.) is 1.8 × 1011 m d = = 1.2, 1 A.U. 1.49 × 1011 m

(7.30)

that is, slightly more than 1 A.U. It is important to note that this estimate is only an upper limit on the volume of the emitting area associated with an AGN in that c is treated as the velocity of a signal through the medium that composes the volume: certainly the volume of the emitting region may be considerably smaller. It is also important to note that since it is known that there is a large concentration of mass seen at the centers of galaxies (see, for example, the discussion presented in Sect. 6.4.6 about the supermassive black hole found in the center of the Milky Way) and because time variability implies that the volumes occupied by these central masses must be very small, the theory that supermassive black holes are located at the centers of galaxies has gained wide acceptance.

7.3 Radio Galaxies and Active Galactic Nuclei

287

7.3.1 Blazars A particular intriguing class of AGNs is known as blazars: which are radio-loud AGN that are dominated by bright cores. This class of course derives its name from the archetypical source BL Lac: this object was initially believed to be a variable star located within the Milky Way, as evidenced by its name:3 observations made at multiple wavelengths revealed that this source was extragalactic rather than Galactic in nature and the name of this source was the basis for ultimately naming this class of objects. In fact, blazars are in a way like quasars in that their optical counterparts appear to be point-like and thus resemble Galactic stars. Blazars may be imagined as AGNs where the jet of emission from the central black hole sources are viewed such that the jet of emission from the central black hole points essentially directly toward the observer. This scenario may be contrasted with how the lobes of radio galaxies are seen, which appear to extend in the plane of the sky. In fact, if blazars were observed such that their jets pointed in the plane of the sky rather than toward an observer, these sources would appear to be AGNs with radio lobes. Note that radio lobes are very diffuse structures such that they are best seen when the axis of the associated jets corresponds to the plane of the sky: the lobes become far more transparent (and thus harder to detect) when the observer is aligned with the axis of the jet. Clearly, how the emission of an AGN is perceived and its corresponding classification is highly dependent on the alignment of the jets to the line of sight of the observer. Blazars are frequently detected as gamma-ray sources through the effect of synchrotron self-Compton emission described in Sect. 7.3.6. Blazars are known to be highly variable at all wavelengths (even on short time scales) and as the total amount of detected flux from the blazar changes, the spectral energy distributions of AGNs are also seen to change. In the particular, peaks seen in these spectral energy distributions are seen to change wavelength. Blazars have been divided into numerous subclasses based on their (variable) spectral properties: for example, optically violent variable (OVV) quasars—as their name implies—rapidly change brightness with time, with fluxes changing by as much as 50% within a day. They form a distinct class within blazars in that their optical spectra feature broad emission lines, which are absent from the spectra of other types of blazars. Another class of blazars is known as the flat spectrum radio-loud quasars (FSRQs): like OVVs, FSRQs have broad emission lines in their spectra but their radio spectra are remarkably flat with spectral indices α ≈ 0.0, unlike other galaxies where the spectral indices are approximately α ≈ −0.8. These flat radio spectra may be interpreted as the superposition of the spectra of synchrotron-emitting electrons over a particularly broad range of energies. FSRQs are also observed at higher redshifts (and thus lie at greater distances) than other blazars. Other classes of

3 Variable

stars are typically named by a letter or combination of letters and the constellation in which the star is found: BL Lac derives its name from its apparent location within the bounds of the constellation Lacerta [28]

288

7 Extragalactic Radio Astronomy

blazars include low- and high-frequency peaked blazars (known as LBLs and HBLs, respectively): the former sources feature spectral peaks in their spectral energy distributions at long wavelengths (associated with the synchrotron emission component of the spectra of AGNs) in the infrared while the latter sources feature such peaks in the ultraviolet or X-ray. To classify all of these different types of blazars, a proposed system [29] considers the bolometric luminosity Lbol of the sources: in this system, the equivalent widths4 of broad lines in the spectra of “classical” blazars is 5Å = 0.5 nm or less while the equivalent widths of such lines in the spectra of FSRQs are broader.

7.3.2 Classification System of Radio Galaxies A classification system for the morphologies of radio galaxies with external lobes and based on the structure of their lobes as well as the luminosities of the lobes relative to the core of the galaxy has been established by Fanaroff and Riley [20]. This system—denoted as the Fanaroff and Riley classification system—divides galaxies into Classes I and II (also known as FR-I and FR-II in the literature). The galaxies that are classified as FR-I have a lower luminosity, are brighter in the cores than in the lobes and exhibit decreasing surface brightnesses toward the edges of the lobes. FR-I galaxies also feature lobes connected to their cores by clearly defined jets and the radio spectra of these sources is steep. In contrast, FR-II galaxies are more radio luminous than FR-I class galaxies and have more luminous lobes. These lobes exhibit both hot spots and limb-brightening: for these galaxies, the jets connecting the core to the lobes are less obvious: the lobes in turn may have comparable luminosities to the cores and may extend on scales of Megaparsecs. The transition luminosity L1.4 GHz at an observing frequency ν = 1.4 GHz between the two types of galaxies has been defined as L1.4 GHz = 1032 ergs s−1 Hz−1 [21]. For illustrative purposes, radio maps of examples of each type of FR galaxy—namely the FR-I galaxy 3C 296 and the FR-II galaxy 3C 334—are presented in Figs. 7.7 and 7.8, respectively.

7.3.3 Radio Jets from Active Galactic Nuclei Jets are commonly encountered radio structures associated with AGNs: these are large extended structures that appear to extend over thousands of parsecs from the AGN and into surrounding intergalactic space. The jets are known to be composed of subatomic particles that have been accelerated to very relativistic velocities in

4 An

equivalent width of a spectral line is defined as the width of a box that reaches the continuum and has the same area as the spectral feature itself [30].

7.3 Radio Galaxies and Active Galactic Nuclei

289

Fig. 7.7 A VLA image of the FR-I radio galaxy 3C 296 [22]. Image courtesy of the NRAO/AUI

close proximity to the central black hole of the galaxy. The particles then travel out of the plane of the galaxy (the “path of least resistance”) and into the intergalactic space: the detected radio emission from the jets is known to be synchrotron radiation from electrons that are emitting radiation as they decelerate (as indicated by the detection of polarized emission from the jets themselves). Jets are believed to be closely associated with the accretion disks that are suspected to surround the central supermassive black holes in AGNs. Approximately 10% of all AGNs are described as radio loud: this definition is based on the ratio of the radio flux density F5 GHz of the galaxy at a frequency ν = 5 GHz (corresponding to a wavelength λ = 6 cm) to the flux F4400 of the galaxy at the optical wavelength λ = 4400Å = 440 nm. If this ratio F5 GHz /F4400 is 10 or greater, the AGN is classified as radio loud; if it is less than 10, the AGN is classified as radio faint [18]. The majority of radio-loud galaxies are elliptical in type. The details of how jets are formed and extend out of the central volumes containing the central black hole are not well-understood. The consensus among astronomers who study these sources is that the jets are produced in systems where accretion occurs (in the form of an accretion disk) and that the jets themselves are powered by rotational energy from the accretion disk, the central supermassive

290

7 Extragalactic Radio Astronomy

Fig. 7.8 A VLA image of the FR-II radio galaxy 3C334 [6]. Image courtesy of the NRAO

black hole or by a combination of these two sources. Furthermore, it is also widely believed that the energy required to power the jets is released by accretion and tapped by magnetic torques and that magnetic fields are required for both the acceleration of the particles that comprise the jets and for the collimation of the jets such that the jets appear to be tightly focused over large spatial distances (Fig. 7.9). The most commonly accepted model that describes how energy is channeled from the accretion disk to the accelerated particles that form the jet is known as the Blandford–Znajek mechanism. [19] Recall the definition of the Lorentz Force FB from Eq. (3.51), FB = qv × B,

(7.31)

where particles with charge q moving with velocity v through a magnetic field of strength B will experience a force FB . This force will separate positive and negative charges and result in the formation of an electron field. If the magnetosphere of the black hole is relativistic such that its latent energy density Emagnetosphere is much greater than the rest energies of an electron, energetic photons will collide with each other and the physical process of pair production will occur, where electrons and their antiparticles—the counterpart to a particle that is identical in every property except charge, which is opposite—which are known as positrons (identical to electrons but with positive charge instead of negative charge). The magnetic field will separate the charges and the electric field—created by the

7.3 Radio Galaxies and Active Galactic Nuclei

291

Fig. 7.9 An artist’s conception of the formation region of a jet associated with the nucleus of a galaxy. The accretion disk (shown in yellow) surrounds the black hole and the magnetic field lines associated with the disk are twisted tightly as the disk rotates. As a result of this rotation, the outpouring subatomic particles are channeled into a narrow jet. Image Credit: STScI

charges—will accelerate particles. In the Blandford–Znajek mechanism, energy and angular momentum are extracted from a rotating black by the magnetic field and this energy is converted into the Poynting flux. This flux may be defined by recalling the definition of the Poynting vector S (from Eq. (2.100)) S=

1 E × B, μ0

(7.32)

where E and B are the electric and magnetic field vectors, respectively. From this equation, the Poynting flux corresponds to a flux of energy in the direction of S. Once particles are within the direction of the Poynting flux, they are accelerated and the magnetic field of the rotating accretion disk collimates the jet. Within the magnetic field, the particles emit synchrotron radiation and thus the jets are created. The accretion disk—coplanar with the equator of the black hole—also plays a role

292

7 Extragalactic Radio Astronomy

Fig. 7.10 The accretion disk (with its intrinsic magnetic field indicated with lines and arrows) orbiting the central rotating supermassive black hole of a galaxy

in helping to collimate the jets by resisting any motion by charged particles through the equatorial plane of the galaxy. A schematic diagram illustrating the interaction between the accretion disk and its magnetic field in orbit around a central rotating black hole is depicted in Fig. 7.10. The luminosity LBZ associated with the Blandford–Znajek mechanism may be expressed as LBZ ≈

16π G2 M 2 B 2 , μ0 c3

(7.33)

where M is the mass of the central black hole and B is the magnetic field strength associated with the black hole.

Example Problem 7.7 Compute LBZ in Watts for a central black hole with a mass M = 108 M and a magnetic field strength B = 1 T. Solution From Eq. (7.33), LBZ may be computed as (recalling that 1 M = 2 × 1030 kg) LBZ ≈

16π(6.67 × 10−11 m3 kg−1 s2 )2 (108 × 2 × 1030 kg)2 (1 T)2 (4π × 10−7 N/A2 )(3 × 108 m/s)3 (continued)

7.3 Radio Galaxies and Active Galactic Nuclei

293

Example Problem 7.7 (continued) = 8.28 × 1038 W.

(7.34)

Note that the black hole and the accretion disk do not have to rotate in the same direction. For this reason, it has been speculated that opposing spins between the black hole and the accretion disk may explain why some galaxies are radio loud and the others are radio quiet. Defining the prograde case as the scenario where the black hole and accretion disk spins are parallel and the retrograde as the case where the spins are antiparallel, it is noted that in the latter case, the location of the innermost stable orbit within the accretion disk around the black hole will be farther out, resulting in lower efficiency in converting energy and angular momentum from the accretion disk into the accelerated particles. In such a case, it has been argued that in the prograde case, the source would be radio faint because the ergosphere of the central black hole would be smaller, therefore there would be fewer particles subject to acceleration and forming jets. In contrast, the ergosphere of the central black hole is larger in the retrograde case, therefore the reservoir of particles that may be accelerated and form jets is larger [23, 24]. But the opposite conclusion— that in the prograde case more magnetic flux is trapped and therefore the efficiency of accelerating particles is greater than in the retrograde case—has been reached when the back-reaction of magnetic flux on the disk is taken into account [25]. By what process within the jets are the particles accelerated and the structure of the jets maintained? It is believed that diffuse shock acceleration—first encountered in Sect. 6.4.3 in the discussion about supernova remnants—is the responsible process: similar to the situation with supernova remnants, cosmic-ray particles gain energy with successive crossings of the shock. Compared to supernova remnants, cosmic-ray particles within the jets are believed to be accelerated to higher energies, as evidenced by features seen in the cosmic-ray spectrum. While supernova remnants are thought to accelerate cosmic-ray particles to approximately 3000 TeV (the so-called knee energy of the cosmic-ray spectrum), the jets within active Galactic nuclei are thought to be responsible for accelerating to energies above this knee energy (see the discussion about cosmic-ray particles presented in Sect. 2.7).

7.3.4 Superluminal Motion Radio observations made with high angular resolution of jets of ejected from AGNs reveal that the components of the jets may appear to be moving at a velocity that exceeded the speed of light. This phenomenon—known as superluminal motion—

294

7 Extragalactic Radio Astronomy

Fig. 7.11 A schematic diagram illustrating the phenomenon of superluminal motion. The knot is moving at a velocity v at an angle φ along the line of sight to Earth-based observers. The knot emits photons at two times—t = 0 and t = te —and it travels a distance vte . To the Earth-based observer, the motion of the knot is projected onto the plane of the sky and the velocity of the knot may appear to exceed the speed of light

would seem to violate the theory of general relativity which states that no object in the material may travel at a velocity comparable or in excess of the velocity of light. In reality, this apparent superluminal motion is actually simply a projection effect: it is an artifact of material ejected at nearly the speed of light at an angle that approaches the line of sight toward an observer (see Fig. 7.11). The mathematical and physical background of this phenomenon are explained below. Consider a knot of material that is ejected with a velocity v at an angle φ with respect to the line of sight toward an observer. Imagine that two flashes of light are emitted by the knot, one at time t1 at the location R and the other at a later time t2 at the location R  . The time required to reach the observer when the first flash of light is emitted is simply t, but when the second flash of light is emitted, the knot has traveled a distance r = vt along its path and is therefore located at a distance vt cos φ closer to the observer. The difference in time between when the two flashes have been emitted may be expressed as

7.3 Radio Galaxies and Active Galactic Nuclei

t2 − t1 = t −

295

 v vt cos φ = t 1 − cos φ c c

(7.35)

and the apparent transverse velocity vt of the knot is therefore vt =

v sin φ r sin φ vt sin φ = . = t2 − t1 t (1 − (v/c) cos φ) 1 − (v/c) cos φ

(7.36)

Note that in this expression, a value of vt that exceeds the speed of light is possible. In such cases, the phenomenon of superluminal motion is observed for the knot. It is emphasized that the true velocity of the knot is indeed less than the speed of light and that such apparent superluminal motion is simply a projection effect rather than a genuine phenomenon. The true velocity v of the knot can be expressed in terms of vt and φ as v (vt /c) = . c sin φ + (vt /c) cos φ

(7.37)

Constraints may be applied on the value of φ: in the physically meaningful case where v/c is less than one, the value of φ is constrained to be vt2 /c2 − 1 < cos φ < 1. vt2 /c2 + 1

(7.38)

Likewise, the smallest possible value for v—that is, vmin —can be calculated from the relation  vmin vt2 /c2 = , (7.39) c 1 + vt2 /c2 the corresponding value of φ in this case—that is, φmin —is cot φmin =

vt c

(7.40)

and finally the corresponding value of the Lorentz factor γ (see Eq. (2.164))— denoted here as γmin —is γmin = 

1 2 /c2 ) 1 − (vmin

=



2 /c2 ) = 1 + (vmin

1 . sin φmin

(7.41)

Conversely, when does vt attain a maximum value? In that scenario, cos φ = v/c and Eq. (7.36) becomes

296

7 Extragalactic Radio Astronomy

v (1 − cos2 φ) v (1 − (v/c)2 vt = = γ v. = 1 − (v/c)(v/c) 1 − (v/c)2

(7.42)

Example Problem 7.7 The galaxy OJ 287 is a well-studied active galaxy with a measured redshift of z = 0.306. High angular resolution radio observations made of OJ 287 with the VLBA have revealed the presence of knots moving away from the nucleus of the galaxy. For example, one of these knots—denoted as R—is moving away from the nucleus at an angular velocity μ of 0.68 milliarcseconds per year [4]. Assuming that the motion of the knot is entirely in the plane of the sky, calculate the apparent transverse velocity vt of this knot and show that this velocity is superluminal (and therefore unphysical). Also calculate both the maximum angle θ of the direction of motion of the knot along the line of sight from OJ 287 as well as the corresponding minimum velocity vmin of the knot. Assume that the distance to OJ 287 may be determined by computing its recessional velocity vr based on its redshift using Eq. (2.140). Solution From Eq. (2.140) the apparent recessional velocity vr of OJ 287 is z=

vr = 0.306 c

vr = z×c = 0.306×3×105 km s−1 = 91,800 km s−1

−→

(7.43)

and from Eq. (7.1) the distance d to OJ 287 is d=

vr 91,800 km s−1 = = 1293 Mpc. H0 71 km s−1 Mpc−1

(7.44)

At this distance, vt may be computed based on μ and d as  1 radian 206,265 ”   19 3.086 × 10 km = 1.32 × 1014 km yr−1 × 1293 Mpc × 1 Mpc

vt = μ × d = 0.68 × 10−3 ” yr−1 ×



(7.45)

or, in terms of kilometers per second, vt = 1.32×10

14

km yr

−1



1 yr × 3.15 × 107 s



= 4.19×106 km s−1

(7.46)

which in terms of c is (continued)

7.3 Radio Galaxies and Active Galactic Nuclei

297

Example Problem 7.7 (continued)   4.37 × 106 km s−1 vt = = 14.0 c. 3 × 105 km s−1 c−1

(7.47)

Clearly this motion is superluminal! From Eq. (7.38) the value of φmin is φmin < cos

−1



(14.0c)2 /c2 − 1 (14.0c)2 /c2 + 1



< 8.17◦ ,

(7.48)

and from Eq. (7.39) the value of vmin is  vmin = c

(14.6c)2 /c2 −→ vmin = 0.997c. 1 + (14.6c)2 /c2

(7.49)

The most plausible mechanism for powering the observed radio luminosity of active Galactic nuclei is accretion, where mass falling into the central black hole releases gravitational potential energy which is converted into radiation. The relationship between the observed radio luminosity L of the accretion-powered active Galactic nucleus, the rate M˙ of mass falling into the central black hole, and the efficiency η of the conversion from gravitational potential energy into radiation may be expressed as ˙ 2. L = ηMc

(7.50)

Note that typical values for η are between 0.05 and 0.20. A convenient way to state Eq. (7.50) with M˙ is expressed in units of solar masses per year is (see Problem 7.9) L37 L M yr−1 , M˙ = 2 ≈ 1.8 × 10−3 η ηc

(7.51)

where L37 is the luminosity expressed in units of 1037 W.

Example Problem 7.8 A nearby AGN is observed to feature a disk of material in a circular orbit around it: the mass of the disk is negligible compared to the mass of the central black hole. The orbital velocity of the material is v = 520 km per second and the radius of the circular orbit is 10 Astronomical Units. The observed radio luminosity of the AGN is L = 1037 W. Calculate both the mass M of the central black hole in solar (continued)

298

7 Extragalactic Radio Astronomy

Example Problem 7.8 (continued) masses and the rate M˙ of mass accretion onto the black hole, assuming an efficiency η = 0.20 at which gravitational potential energy is converted into radiation. Solution From Eq. (2.50), the mass M of the central black hole may be computed as v 2 R (520 × 103 m/s)2 (10 A.U.×3.086×1016 m/A.U.) = =1.25×1039 kg, G 6.67×10−11 N · m2 kg−2 (7.52) which may be expressed in solar masses as

M=

 M = 1.25 × 10 kg 39

1 M 2 × 1030 kg

 = 6.26 × 108 M .

(7.53)

Similarly, from Eq. (7.50), M˙ may be computed as 1037 W L = 5.56 × 1020 kg/s, M˙ = 2 = ηc (0.2)(3 × 108 m/s)2

(7.54)

or, using Eq. (7.51), ˙  yr−1 ) ≈ 1.8 × 10−3 L37 = 1.8 × 10−3 1 = 9 × 10−3 . M(M η 0.2

(7.55)

7.3.5 Radio Lobes Over time, the relativistic particles that comprise jets associated with active Galactic nuclei may collect to form giant structures called radio lobes that extend for large distances on the scales of tens of kiloparsecs beyond the galaxy itself. Prominent examples of radio lobes associated with the nearby galaxy Fornax A are shown in depicted in Fig. 7.12. Particles located at the leading tip of an jet will be decelerated as they encounter material both within the active galaxy itself and in the intergalactic medium as well, causing a shock to form. Trailing particles in the jet will accumulate and be decelerated by the shock, causing the radio lobe to form. The amounts of energy contained within the radio lobes is estimated to be quite vast, with energies ranging up to approximately 1053 –1054 J. Radio observations of these lobes have measured spectral index values of these lobes of approximately

7.3 Radio Galaxies and Active Galactic Nuclei

299

α = −0.5 along with a high degree of linear polarization in the observed emission. These two observational results help confirm that the observed emission from radio lobes is indeed synchrotron radiation produced by accelerated cosmic-ray electrons [34]. These particles—despite being relativistic—will not be able to travel even a significant part of the large scales of radio lobes before radiating away most of their energy. Therefore, it is suspected that the particles must be accelerated while they are within the lobes themselves: diffusive shock acceleration (see Sect. 6.4.3) has been proposed as a mechanism responsible for such acceleration. Analysis of the energetics of radio lobes often invokes the concept of equipartition discussed in Sect. 6.4.3.1 that assumes that the energies contained in the relativistic particles within the lobes and within the magnetic field associated with the lobe are equal. Here, applying equipartition permits an estimate of the average magnetic field strength Blobe of a lobe. If Elobe and Vlobe are the total energy and volume, respectively, of a lobe, then the assumption of equipartition states that half of Elobe is contained in the magnetic field with a magnetic field energy density uB = Blobe /2μ0 , that is, B 2 Vlobe Elobe = uB Vlobe = lobe . 2 2μ0

(7.56)

Solving this equation for Blobe yields  Blobe =

μ0 Elobe . Vlobe

(7.57)

Another important property of radio lobes is their lifetime tlobe which can be approximated as simply tlobe =

Elobe , L

(7.58)

where L is the measured radio luminosity of the lobe.

Example Problem 7.9 (a) Calculate the volume Vlobe and the radio luminosity L over the frequency range ν1 = 107 Hz to ν2 =1011 Hz of the eastern lobe of the nearby (z = 0.0059) radio galaxy Fornax A (see Fig. 7.12). The measured flux density of the eastern lobe is Sν = 44 Jy at 1.4 GHz (assume that the flux density of the lobe is constant across the entire bandwidth of interest), the measured angular size of the lobe (which is assumed to be spherical in volume) is 15 , [5] and the total energy contained in the lobe is E = 1052 J. Finally, assume a spectral index of the lobe to be α = −0.5, which is characteristic of synchrotron radiation. (b) Use the equipartition (continued)

300

7 Extragalactic Radio Astronomy

Fig. 7.12 A combined optical and radio (VLA) (in white and orange, respectively) image of the radio galaxy Fornax A. The visible galaxy NGC 1316 is seen in white at the center of the image. This galaxy is merging with the smaller spiral seen just above it (known as NGC 1317) and as a result of this merger, gas and dust is being stripped from NGC 1317 and deposited onto the central black hole in NGC 1316. As a result, the black hole is being sped up: in fact, more mass is being deposited upon this black hole than can be “devoured” and ultimately powerful jets comprised of spun-up and escaped material are formed. The large radio lobes seen in orange in this image lie at the ends of these jets: notice how large these lobes are in comparison to the size of a typical galaxy like NGC 1316 itself. Credit: NRAO/AUI and J. M. Uson

Example Problem 7.9 (continued) assumption about the energetics of the lobe to compute its average magnetic field strength Blobe in Tesla. (c) Calculate the lifetime tlobe in years. Solution (a) The volume of the lobe may be determined from the measured angular extent of the lobe multiplied by the distance to Fornax A. This latter quantity can be determined by first computing the recessional velocity vr of the galaxy from the given redshift. Applying Eq. (2.140), vr may be computed is (continued)

7.3 Radio Galaxies and Active Galactic Nuclei

301

Example Problem 7.9 (continued) vr z= = 0.0059 −→ vr = z×c = 0.0059×3×105 km s−1 = 1770 km s−1 c (7.59) and from Hubble’s Law (Eq. (7.1)) the distance to Fornax A is d=

vr 1770 km s−1 = = 24.93 Mpc. H0 71 km s−1 Mpc−1

(7.60)

At this distance, the linear scale corresponding to one arcsecond is (from Eq. (2.7)) 1 × 24.93 × 106 pc = 121 pc, 206,265

x (pc) =

(7.61)

and thus—at the computed distance to Fornax A—an angular size of 15 arcminutes corresponds to 15 arcminutes×(60 arcseconds/arcminute)×(121 pc/arcsecond) = 1.09×105 pc. Therefore— assuming the lobe is spherical—Vlobe may be computed as 4π(1.09 × 105 pc/2)3 4π(d/2)3 = 3 3 3  3.086 × 1016 m 14 3 = 1.99 × 1064 m3 . = 6.78 × 10 pc × 1 pc (7.62)

Vlobe =

To determine the luminosity L of the lobe, the flux density Sν of the emission detected from the lobe as a function of observing frequency ν may be expressed as  Sν = 44 Jy

ν 1.4 × 109 Hz

−0.5 (7.63)

.

Note that this expression is normalized to the measured flux density at 1.4 GHz. To obtain L, Eq. (3.28) must be applied and Sν must be integrated over the given frequency bounds. This yields  L = 4π d S = 4π d 2

 Sν dν = 4π d

2

2

ν=1011 Hz ν=107 Hz

ν=1011 Hz

ν=107 Hz



ν 44 Jy 1.4 × 109 Hz

−0.5 dν.

(7.64) (continued)

302

7 Extragalactic Radio Astronomy

Example Problem 7.9 (continued) Evaluating the integral in this equation yields  S=



ν=1011 Hz

44 Jy ν=107 Hz

dν =

ν 1.4 × 109 Hz

−0.5

(44 × 10−26 W m−2 Hz−1 ) ν 0.5 (0.5)(1.4 × 109 Hz)−0.5

ν=1011 Hz ν=107 Hz

= 1.03 × 10−14 W/m2 , (7.65)

and therefore the luminosity L of the lobe is 2

   3.086×1022 m L = 4π d 2 S=4π 24.93 Mpc 1.03 × 10−14 W/m2 1 Mpc = 7.66 × 1034 W.

(7.66)

(b) From Eq. (7.57), Blobe may be computed as  Blobe =

(4π × 10−7 N/A2 )(1052 J) = 7.95 × 10−10 T. 1.99 × 1064 m3

(7.67)

(c) From Eq. (7.58), tlobe may be computed as tlobe

  1052 J 1 yr 17 = 6.53 × 10 s = 4.14 × 109 yr. = 7.66 × 1034 W 3.15 × 107 s (7.68)

7.3.6 Spectral Energy Distributions of Radio Galaxies What does the spectral energy distribution (SED) of a radio galaxy look like? A generalized SED for a radio galaxy is presented in Fig. 7.13: this SED (plotted with νFν on the abscissa—see Sect. 5.3 for a description about SEDs) features a gradual increase when progressing from radio frequencies to higher frequencies. This slope of emission originates from synchrotron emission from accelerated relativistic electrons. Note that the radio-loud population of radio galaxies (which comprises 10% of all radio galaxies) have a flatter slope in this frequency domain than the radio-quiet population (which comprises the remaining 90% of all radio galaxies).

7.3 Radio Galaxies and Active Galactic Nuclei

303

Fig. 7.13 An example spectral energy distribution of a radio galaxy

In approximately the infrared portion of the electromagnetic spectrum, the SEDs of radio galaxies feature a “turnover” (that is, a change in slope) and a peak of emission known as the “infrared (IR)” peak. The slope of this emission is subject to self-absorption effects by the population of synchrotron-emitting electrons (see Sect. 3.3): while this peak is typically seen in the infrared, in the cases of certain galaxies (specifically blazars) this peak may be instead found in the X-ray. In addition to the IR peak which originates from synchrotron radiation, there is a second peak known as the “big blue bump” that is typically found in the X-ray (and in some cases in the gamma-ray). The big blue bump originates from the process known as inverse Compton scattering. Inverse Compton scattering is the process where a photon and an electron collide: in this particular scenario, the electron has more energy than the photon. As a consequence of the collision, energy from the electron is transferred to the photon, thereby “up-scattering” the photon to a higher energy at the expense of reducing the kinetic energy of the electron. This process can be contrasted with Compton scattering, where in the collision between a photon and an electron (where the photon has more energy), the electron gains kinetic energy as a result of the collision while the photon is down-scattered to a lower energy.

304

7 Extragalactic Radio Astronomy

Fig. 7.14 A schematic diagram depicting the phenomenon of Compton scattering. A photon is traveling along the x-axis and is incident upon an electron that is assumed to be initially at rest. The photon scatters off the electron at an angle θ with the respect to the x-axis while the electron moves in a direction at an angle φ with respect to the x-axis. Notice that a portion of the initial energy of the photon is converted into kinetic energy of the electron so that the scattered photon has a lower energy (corresponding to a longer wavelength or lower frequency) in comparison to its initial value

To describe the process of inverse Compton scattering, it is easier to first describe Compton scattering. In this phenomenon, a photon with initial frequency νi scatters off an electron and loses energy in the recoil, such that its frequency is decreased to a new final value νf . The electron gains energy such that—if it is initially at rest with energy Ei = me c2 —its energy after the collision is now Ef = γme c2 = mc2 . A schematic diagram depicting the collision between the photon and the electron is shown in Fig. 7.14. Energy must be conserved by this collision, that is, hνi + Ei = hνf + Ef −→ hνi + me c2 = hνf + me c2 = hνf +



pe2 c2 + m2e c4 , (7.69) where pe is the momentum of the electron and Eq. (2.171) has been used. Similarly, the momenta of the electron and the photon must be conserved as well: this can be expressed as (recalling Eq. (2.112) which describes the momentum of a photon as a function of its energy) hνi hνf hνi hνf + pi = + pf −→ + me vi = + me vf , c c c c

(7.70)

where pi and pf are defined as the initial and final momentum of the electron, respectively, and vi and vf are defined as the initial and final velocities of the electron, respectively.

7.3 Radio Galaxies and Active Galactic Nuclei

305

If the angle made with the x-axis by the velocity vector of the scattered electron is φ and similarly the angle made with the x-axis by the path of the photon is θ , then by conservation of momentum (where the total momentum before and after the scattering event must be equal), the sum of the momenta of the photon and electron in the x-direction are hνi hνf +0= cos θ + me v cos φ, c c

(7.71)

where v = vf is the velocity of the electron after the collision: the electron is assumed to be at rest initially, so vi = 0. Similarly, from the conservation of momentum, the sum of the momenta of the photon and electron in the y-direction are 0=

hνi sin θ − me v sin φ, c

(7.72)

where the motion of the incoming photon is assumed to be entirely in the x-direction (that is, with no component in the y-direction) and that the electron is initially at rest. Defining Δν = νi − νf to describe the change in frequency of the scattered photon, a conjugate variable Δλ = λf − λi may be defined as the corresponding change in wavelength of the scattered photon. Therefore, Eqs. (7.71) and (7.72) can be manipulated to obtain (see Problem 7.11) Δλ =

h (1 − cos θ ) = λc (1 − cos θ ), me c

(7.73)

where λc is the Compton wavelength, defined as λc =

h = 2.43 × 10−12 m. me c

(7.74)

Notice that a photon with the wavelength λ = λc has the same energy Ec as the rest energy of an electron: Ec =

hc (hc)(me c) = me c2 = 5.11 × 105 eV. = λc h

(7.75)

The inverse Compton effect is embodied by a similar concept of a scattering event between an electron and a photon, except in the inverse Compton effect it is the photon that gains energy at the expense of the kinetic energy of the electron. In this case, an ensemble of photons may be boosted from low energies to higher energies: this phenomenon is seen in media that have high temperatures and a high number density of electrons, and such conditions are encountered in the jet of an AGN. The final energy EIC of the “down-scattered” electron after the scattering event is simply the difference between the initial energy Ei of the electron and the final energy Ef

306

7 Extragalactic Radio Astronomy

of the “up-scattered” photon, that is, EIC = Ei − Ef .

(7.76)

Note that the final energy of the up-scattered photon is assumed to be much greater than the initial energy of the photon before the scattering event takes place. The power PIC radiated by the down-scattered electron (which is equivalent to the energy loss dEIC /dt of the electron) moving with a velocity β = v/c via the inverse Compton process may be expressed as PIC =

dEIC 4σT cβ 2 γ 2 urad = . dt 3

(7.77)

Here, urad is the energy density of the radiation field: also recall that for a relativistic electron, v ≈ c and therefore β ≈ 1. In comparing Eqs. (3.81) and (7.77), it is crucial to note that the ratio of the power emitted by an electron via the inverse Compton process to the power emitted by an electron via the synchrotron process is merely the ratios of the energy densities urad and uB , that is, PIC urad = . Psync uB

(7.78)

The lifetime tIC of the electron that has been down-scattered to an energy EIC via the inverse Compton scattering process and which is radiating away its energy with a power PIC is simply tIC =

EIC . PIC

(7.79)

Finally, it is important to note that a photon with an initial frequency νi will be upscattered via the inverse Compton scattering process to a final frequency νf (where νf > νi ) and that the magnitude of the increase in frequency of the photon is proportional to γ 2 . This result may be derived as follows. The final energy Ef of the scattered photons may be expressed as Ef =

PIC , ψIC

(7.80)

where ψIC is the rate of photon scattering or alternatively the number of photons scattered per second by a single electron. This quantity may be expressed as ψIC =

σT curad . hνi

Combining Eqs. (2.111), (7.77), (7.80), and (7.81) yields

(7.81)

7.3 Radio Galaxies and Active Galactic Nuclei

PIC Ef = hνf = = ψIC



4σT cβ 2 γ 2 urad 3

307



hνi σT curad

 =

4β 2 γ 2 hνi . 3

(7.82)

Recalling that β is approximately unity for relativistic particles, this equation may be solved for νf , yielding νf =

4γ 2 νi ∝ γ 2. 3

(7.83)

For this reason, optical photons and X-ray photons may be up-scattered readily into γ -ray photons by the inverse Compton scattering process. As mentioned above, a second feature is seen in the spectra of radio galaxies and this feature originates from a process known as synchrotron self-Compton emission. As the name suggests, the origin of this emission lies with synchrotronemitting electrons that are gyrating in the magnetic field of the jet. In the case of synchrotron self-Compton emission, the photons emitted by the electrons are “seed photons” off of which the electrons scatter again and thus increase the energies of the photons. Like inverse Compton scattering, the photons are up-scattered from an initial frequency νi by a factor of γ 2 to a final frequency νf . In this process, a chain reaction may occur where the photons that have been up-scattered to higher energies may themselves Compton scatter off the electrons, thereby raising their energies. These electrons may then scatter off the photons and thus raise their energies in a continuing chain reaction. A requirement for this process is that the medium of the jet is optically thick to the emitted seed photons so that the initial up-scattering of electrons to higher energies may occur (see the discussion about the phenomenon of self-absorption in synchrotron spectra in Sect. 3.3). It is noted that this process may not continue indefinitely, as the most energetic photons (which have reached the gamma-ray domain of the electromagnetic spectrum) will ultimately perceive the medium as optically thin and escape out of the jet. In this manner, the most energetic photons serve as a “cooling” mechanism where the system of the photons and electrons within the jet experiences a net loss of energy to the surrounding intergalactic medium. These models that describe the interactions between photons and electrons in explaining the spectral characteristics of the jet are known as leptonic models, where a lepton is a low-mass subatomic particle exemplified by an electron. Other models of emission from jets have invoked interactions between photons and more massive protons are known as hadronic models, where a hadron is a high-mass subatomic particle exemplified by a proton. Hadronic models were first developed to explain cosmic-ray acceleration by the jets and—in same cases—hadronic models have had more success in modeling the high-energy emission seen from jets of AGNs [27].

308

7 Extragalactic Radio Astronomy

7.3.7 Doppler Boosting In the cases of some double-lobed galaxies, a jet connecting the central galaxy with the lobe is not always apparent: this effect is demonstrated in the radio image of the double-lobed galaxy 3C 334 as shown in Fig. 7.8. The explanation for the absence of the one missing jet in these images is that the jet is so tightly focused and that the angle of its emission is so narrow (and away from the direction of the observer) that its radiation is not detected by the radio observation. The observed flux density from a source traveling with a relativistic velocity may appear to be inflated or reduced dramatically, depending on whether the source appears to be moving toward or away from the observer. This phenomenon is known as Doppler boosting and the amount of enhancement or reduction of the flux density depends on the velocity of the source either toward or away from the observer. The amount of Doppler boosting for an observed signal may be derived as follows (see [35]). Imagine a knot of material (in this case, a component of a jet ejected from a Galactic nucleus) located at a distance R0 as observed in the reference frame of the galaxy: this knot is moving radially away from the galaxy itself. If the intrinsic luminosity of an ejected knot of material is L0 (ν), then the flux density S(ν0 ) dν0 in the frame of the galaxy of the knot as observed over the frequency range ν0 and ν0 + dν0 as it crosses a shell of radius R0 centered on the host galaxy is (from Eq. (3.28)) S(ν0 )dν0 =

L0 (ν)dν . 4π R02

(7.84)

If the knot is moving with a velocity v with respect to the galaxy at an angle θ along the line of sight to the observer, then the observed Doppler shift in terms of z of the emission may be expressed as (see Eq. (2.142)) 1+z=

1 − (v/c) cos θ ν0 λ = = , λ0 ν 1 − (v 2 /c2 )

(7.85)

where λ0 and ν0 are the intrinsic wavelength and frequency of the emission in the frame of the knot while λ and ν are the wavelength and frequency of the emission detected by an observer. Therefore, the detected flux density S(ν) over the frequency range dν is S(ν)dν =

S(ν(1 + z))dν S(ν0 )dν0 . = (1 + z) (1 + z)2

(7.86)

Note that the distance R0 from the knot to the observer as measured in the frame of the knot corresponds to a distance R = R0 /(1 + z) as measured by the observer (see Eq. (7.4)). Therefore the flux density S(ν) may be expressed as

7.4 Galaxy Clusters and Associated Diffuse Radio Emission

S(ν) =

L0 (ν(1 + z)) . 4π R 2 (1 + z)3

309

(7.87)

If the specific dependence of L0 (ν) on ν is a power law with a spectral index α such that L0 (ν) ∝ ν −α , then L0 (ν(1 + z)) ∝ ν −α (1 + z)−α . Therefore Eq. (7.87) becomes S(ν) =

L0 (ν)(1 + z)−3−α . 4π R 2

(7.88)

So finally, the observed flux density S(ν) in terms of S0 may be expressed in terms of z and α as S(ν) = S0 (ν0 )(1 + z)−(3+α) .

(7.89)

Example Problem 7.10 Compute 1 + z and the ratio S(ν)/S0 (ν0 ) in the case of a radio-emitting knot emitted from a galaxy with a velocity v = 0.95c at an angle θ = 15◦ to the line of sight to the observer. Assume a value for the spectral index of α = 1. Solution From Eq. (7.85), 1 + z may be computed as 1 − (0.95c/c) cos 15◦ 1+z= = 0.26. 1 − ((0.95c)2 /c2 )

(7.90)

From Eq. (7.89), S(ν)/S0 (ν0 ) may be computed as S(ν) = (0.26)−3+1 = 15. S0 (ν0 )

(7.91)

Therefore, in the scenario presented here, the Doppler boosting phenomenon would lead to an increase in the observed brightness of the knot by a factor of 15.

7.4 Galaxy Clusters and Associated Diffuse Radio Emission Large-scale surveys of the distributions of galaxies have revealed that galaxies are typically not found in isolation but instead within gravitationally bound gatherings known as clusters. Clusters themselves are also gathered in gravitationally bound assemblies called superclusters which themselves comprise the largest gravitationally bound systems in the Universe. Currently, the most widely accepted

310

7 Extragalactic Radio Astronomy

models for the formation of galaxy clusters argue that these structures from through a hierarchical sequence of merging events where individual galaxies and tiny assemblies of galaxies (known as groups or small clusters of galaxies) merge together due to gravitational attraction between these entities. In turn, collisions between clusters of galaxies form the vast structures of superclusters through continuing hierarchical merging events. Cluster mergers are tremendously violent events where typically between 1056 and 1057 J of energy are released on timescales of approximately 109 yr. During these mergers, shocks are driven into the intracluster medium of the clusters: these shocks dissipate their energy into heating the gas of the intracluster gas and initiating the turbulent motions in the intracluster medium that follow. However, some of this energy drives the acceleration (and in some cases the subsequent reacceleration) of relativistic particles and the amplification of the magnetic field within the intracluster medium itself. Therefore, these shocks may drive synchrotron emission from the cluster on spatial scales that approach the size of the cluster itself. Radio observations of galaxy clusters have indeed revealed the presence of extended diffuse structures located in the space between the individual galaxies, and detailed spectral analyses of these structures (based on measured spectral indices and the detection in some cases of strongly polarized emission) strongly favor a synchrotron origin for the observed emission. There are two main types of diffuse radio structures associated with galaxy clusters. One type of diffuse structure is termed a radio halo: such structures are seen toward the centers of clusters. Halos usually have smooth morphologies and their spatial distributions generally match the distribution of thermal X-ray emission from hot intracluster gas. Sensitive measurements of the polarization of the emission from halos reveal no polarization down to the level of several percent. The second type of diffuse structure is termed a radio relic: in contrast to halos, relics exhibit more irregular morphologies, are located at the peripheries of clusters rather than at their cores, and exhibit a high level of polarization (from 10 to 60%). Both types of structures are similar in spatial scale (between approximately 0.5 and 2.0 Mpc in size) and have similarly steep spectral indices (α ≈ −1.0 and lower). It is currently uncertain whether halos and relics differ only due to line of sight and projection effects or if they are truly different phenomena [9, 26]. Radio halos and relics appear to be preferentially associated with disturbed systems, such as clusters that are in the process of merging or have recently merged [38]. The details regarding the physical origin of halos and relics remain uncertain: any proposed theory regarding a physical origin must account for both the comparatively short lifetimes (≈108 yr) of accelerated electrons with the large spatial size and lifetimes of 109 yr associated with halos and relics. It is most likely that the electrons are subject to multiple acceleration events within the structures of the halos and relics themselves. In the studies of galaxy clusters, determining the rotation measure (see Sect. 6.4.5) is very useful in estimating the magnetic field strength of the intracluster medium [39]. For example, a background galaxy seen through the intracluster medium of the cluster itself may be used as a probe for estimating the rotation

7.4 Galaxy Clusters and Associated Diffuse Radio Emission

311

measure at multiple wavelengths. X-ray observations of the intracluster medium (which emits thermal bremsstrahlung radiation which registers prominently in the X-ray) can be used to derive the electron number density of the intracluster medium. With an assumption about the spatial scale of the intracluster medium as observed toward the background galaxy (such as assuming that the intracluster medium is spherical in size and that its depth is equal to its width), Eq. (6.213) may be applied to derive B (that is, the component of the intracluster magnetic field that is oriented parallel to the line of sight—see Example Problem 7.11). A prominent example of a galaxy cluster that exhibits both radio relics and a radio halo is Abell 2744: a multi-wavelength (X-ray, optical, and radio) image of this source is presented in Fig. 7.15. Abell 2744 may be envisioned as an ensemble of colliding sub-clusters of galaxies, each containing hundreds of galaxies. Two separate collisions (one occurring in the north-south or top-bottom direction, and the other occurring in the East–West or left–right direction) are currently believed

Fig. 7.15 A multi-wavelength image of Abell 2744, a large cluster of galaxies that may be envisioned as an ensemble of colliding sub-clusters of galaxies. X-ray emission (shown in blue) and radio emission (shown in orange) are superimposed on an optical image of the cluster. Image Credit: Pearce et al.; Bill Saxton, NRAO/AUI/NSF; Chandra, Subaru

312

7 Extragalactic Radio Astronomy

to be occurring, and a third collision is suspected as well. Note the presence of both significant diffuse central radio emission (toward the center of the cluster and coincident with central X-ray emission) and a prominent relic in the upper left (toward the northeast) in the image [36].

Example Problem 7.11 Radio observations are made of a galaxy cluster: the measured redshift of the galaxy cluster is z = 0.05 and the apparent angular extent of the cluster on the sky is 9 arcminutes. The emission measure EM of the cluster is 20 cm−6 pc and the rotation measure RM is determined to be 6 × 106 radians m−2 (based on a pointed observation made of a background galaxy seen through the diffuse intracluster medium of the galaxy cluster itself). From this information, calculate the component of the magnetic field B associated with the intercluster medium of the cluster along the line of sight toward this background galaxy. Solution For the sake of simplicity, it is assumed that any rotation of the plane of the electric field by the medium between the galaxy cluster and the interstellar medium within the Milky Way galaxy may be ignored. First, from Eq. (2.141), the apparent recessional velocity of the cluster may be computed from z as vr = cz = (3 × 108 m/s)(0.05) = 15,000 km/s.

(7.92)

Based on this value for vr , using Hubble’s Law as given in Eq. (7.1) and adopting a value for Hubble’s constant of H0 = 71 km s−1 Mpc−1 , the distance d to the cluster may be computed as d=

vr 15,000 km/s = = 211.27 Mpc. H0 71 km s−1 Mpc−1

(7.93)

With the given angular extent and the known distance, the diameter x of the cluster may be calculated from Eq. (2.7) as follows: x (pc)=

9 arcminutes (60 arcseconds/arcminute) (211.27×106 pc) =5.53×105 pc. 206,265 (7.94)

Assuming that the cluster is spherical in shape, we can adopt this value for the entire path length  through which the radiation from the background galaxy travels. The electron density ne can be determined from this value for  and using Eq. (6.106) to yield (continued)

7.5 The Cosmic Microwave Background

313

Example Problem 7.11 (continued)   EM (cm−6 pc) 20 cm−6 pc ne = = 6.01 × 10−3 cm−3 . =  5.53 × 105 pc

(7.95)

Note here that a uniform value of ne throughout the intercluster medium has been assumed. Finally, B can be calculated by re-expressing Eq. (6.213) as B [G] =

RM (radians m−2 ) (8.1 × 105 ) ne [cm−3 ]  [pc]

,

(7.96)

and solving this equation for B yields B [G]=

6×106 radians m−2 =2.23×10−6 G=2.23μG. (8.1×105 )(6.01×10−3 cm−3 )(5.53×105 pc) (7.97)

In performing this calculation, it has been assumed that B is uniform throughout the intracluster medium of the galaxy cluster.

7.5 The Cosmic Microwave Background Astronomers hypothesize that the Universe formed from an initial event where a singularity—a infinitesimally small point—began to expand rapidly: as a result of this expansion, the phenomenon of spacetime was defined. This event is known as the Big Bang and is estimated to have occurred 13.7 billion years ago. This estimate is based on taking the reciprocal of H0 (realize that this reciprocal will have units of time) and assuming that the Universe has been expanding at a rate that is approximately constant and corresponds to the current value of H0 . Therefore, determining a precise value for H0 is crucial not only for proper application of Hubble’s Law (see Eq. (7.1)) for determining distances to galaxies, but also for obtaining an estimate for the age of the Universe itself. Most likely, the Universe was expanding at a faster rate in the past than it is currently: therefore an estimate of the age of the Universe based on the reciprocal of H0 is best viewed as an upper limit. Since the Big Bang, the Universe has steadily expanded in size while the total amount of mass (in the form of baryonic and dark matter) and radiation (in the form of energy, namely photons) has remained essentially constant. For this reason, the densities of matter and radiation have both decreased as the Universe has increased in size. Furthermore, the expansion of the Universe drives the wavelengths of photons traveling in space to increase as well, therefore causing the energies

314

7 Extragalactic Radio Astronomy

of these photons to decrease as a consequence. Initially, the Universe was filled with a “soup” of particles and very high-energy gamma-ray photons that interacted with each other in a very dramatic manner. Specifically, these high-energy photons would easily photodissociate any nuclei that formed—that is, the photons had so much energy that collisions between these photons and nuclei would result in the disintegration of the nuclei into component nucleons. Therefore, the early Universe was dominated by solely the nuclei of the lightest atoms, principally hydrogen and helium with trace amounts of lithium and heavier elements. As the Universe continued to expand, the energies of these ambient photons continued to decrease and they lost the capability of photodissociating nuclei. Still, these ambient photons possessed enough energy to photodissociate any bound atoms that would form when nuclei and free electrons encountered each other. In time, however, as the photons continue to lose energy, they will eventually be unable to photodissociate bound atoms. The moment in the history of the Universe when this event occurred (and corresponding to a redshift of z = 1100) is known as the recombination epoch, and it corresponded to the first time in the history of the Universe where bound atoms formed and remained together. After the recombination epoch, the ambient photons lacked the energy to interact in a significant manner with matter, and as the Universe continued to expand, the energies of these photons decreased as the wavelengths of these photons increased. In the current epoch, the continuing expansion of the Universe has stretched the wavelengths of these photons to the extent that they now comprise a diffuse background radiation that fills all of space and may be best fit with a blackbody model with a peak wavelength of λmax = 1.06 × 10−3 m. This value for λmax corresponds to the microwave region of the electromagnetic spectrum and thus gives the name to the Cosmic Microwave Background (CMB) radiation. With the location of the peak of this radiation in the microwave range, analysis of this radiation is fully within the scope of radio astronomy.

Example Problem 7.12 Calculate the age trecombination of the Universe in years when recombination occurred. Assume that the current age of the Universe is t0 = 13.8 billion years. Solution From Eq. (7.6) and the given value of z for the epoch of recombination, t may be computed as t=

t0 13.8 × 109 yr = = 3.77 × 105 yr. 3/2 (1 + z) (1 + 1100)3/2

(7.98)

An all-sky picture made of the Cosmic Microwave background made using data collected by the Wilkinson Microwave Anisotropy Probe (WMAP) is presented in Fig. 7.16. Note that the color differences in this map correspond to temperature fluctuations in the CMB: the scales of these variations range up to about 200 μK from the fitted blackbody temperature of 2.73 K for the radiation. Aspects of the

7.5 The Cosmic Microwave Background

315

Fig. 7.16 An all-sky map of the Cosmic Microwave Background that was created by data collected by the Wilkinson Microwave Anisotropy Probe (WMAP). This map may be envisioned as a picture of the infant universe: color differences shown on this map correspond to temperature fluctuations in the cosmic microwave background: such fluctuations eventually became galaxies. Image credit: NASA/WMAP Science Team

CMB that have attracted considerable research efforts in modern astronomy is the remarkable smoothness of the radiation and the putative link between these variations and structures that eventually became galaxies and large-scale structure in the Universe. The photons that comprise the CMB interact with cosmic-ray particles (see Sect. 2.7) in a remarkable manner in that these photons establish a maximum energy (known as the Greisen–Zatsepin–Kuzmin Limit or the GZK Limit) for a cosmic-ray proton that is traveling in space between galaxies [31, 32]. As extremely relativistic cosmic-ray protons with energies that exceed the GZK Limit (estimated to be 5 × 1019 eV or 8 J) travel through this space, they interact with CMB photons. To these extremely energetic particles, the CMB photons appear to be blueshifted dramatically to much larger energies. These very energetic protons would interact with these apparently heavily blueshifted photons to produce pions, and the production of these particles would bleed energy from the protons. The interactions would continue with more pions produced and a further depletion of energy from the proton until the energy of the proton would drop below the GZK limit. The mean path associated with this interaction dictates a length scale such that cosmic-ray protons that have an energy in excess of the GZK limit and have traveled a distance greater than 50 Mpc (a distance known as the GZK horizon) should not be observed on Earth. Note that some cosmic-ray particles with energies greater than the GZK limit have in fact been detected (such as the “Oh-My-God” particle described in Problem 2.14). Such particles are thought to be atomic nuclei rather than protons, and atomic nuclei are not believed to be constrained in energy by the GZK limit.

316

7 Extragalactic Radio Astronomy

7.6 The Sunyaev–Zeldovich Effect As described in Sect. 7.5, space is filled with a background sea of photons associated with the cosmic microwave background. In the spaces between galaxies with galaxy clusters, a hot X-ray-emitting ionized plasma is present. Free electrons within the gas may interact with the photons in the cosmic microwave background through inverse Compton scattering (see Sect. 7.3.4): as a result of the interaction between free electrons in the gas and the cosmic microwave background photons, the photons are up-scattered to higher energies (that is, shorter wavelengths), and background photons with wavelengths of 1.6 mm and longer appear to vanish. This phenomenon is known as the Sunyaev–Zeldovich effect: the fractional decrease ΔT /T in the background cosmic microwave background temperature (recall that the characteristic temperature of the cosmic microwave background is TCMB = 2.73 K) may be expressed as 2kB Te σT ne  ΔT = . T me c 2

(7.99)

In this equation, Te is the thermal temperature of the electrons as determined from X-ray observations of the hot intercluster plasma, ne is the number density of electrons that comprise the plasma (also determined from X-ray observations), and  is the scale length of the volume of the X-ray emitting gas within the cluster (often assumed to be spherical in geometry). The remaining quantities—kB and σT —are Boltzmann’s constant and the Thomson cross-section of the electron and are just physical constants. The Sunyaev-Zelovich effect may be applied to derive an independent estimate of the value of Hubble’s Constant H0 (see Sect. 7.1.1). The method of application may be described as follows. If ΔT /T is measured for a galaxy cluster along with ne and Te for the diffuse X-ray emission associated with the cluster as determined by X-ray observations,  may then be determined from Eq. (7.99) (note that the remaining quantities in this equation are all constants). The computed value of  combined with a measurement of the angular extent of the cluster leads to a direct estimate of the distance to the cluster using Eq. (2.1). Finally, this distance—combined with a measurement of the recessional velocity of the cluster as a whole due to the expansion of the Universe (as indicated by the cosmological redshift in the spectra of member galaxies of the cluster)—yields an estimate for H0 using Eq. (7.1).

Example Problem 7.13 Reconsider the galaxy cluster described in Example Problem 7.11 and calculate the corresponding value of ΔT /T . Adopt the value for ne computed in that Example Problem and assume a value of 107 K for (continued)

Problems

317

Example Problem 7.13 (continued) Te . Also assume that —the linear diameter of the cluster—also corresponds to the scale length of the cluster as well. Solution From Eq. (7.99), ΔT /T may be computed as ΔT T =

2(1.38×10−23 J/K)(107 K)(6.65×10−29 m2 )(1.5×103 m−3 )(9.01×104 pc)(3.086×1016 m/pc) ≈ 10−6 . (9.11×10−31 kg)(3×108 m/s)2 (7.100)

Problems 7.1 The observed redshift of the radio galaxy 3C 123 is z = 0.2177 [33]. Based on this measured value of z and adopting a value for H0 of 71 km s−1 Mpc−1 , compute the following: (a) (b) (c) (d)

The apparent recessional velocity vr of 3C 123 in km s−1 . The distance d to 3C 123 in Mpc. The size R of the Universe in units of R0 corresponding to this redshift. The age t of the Universe in units of t0 corresponding to this redshift.

7.2 From inspection of Fig. 7.4, calculate the HI mass of the galaxy NGC 628, assuming a distance to this galaxy of D = 8.6 Mpc [10]. 7.3 M81 (NGC 3031) and M101 (NGC 5457) are two prominent nearby spiral galaxies of Hubble type Sb and Sc, respectively. HI observations are conducted of these two galaxies and values for Vmax of 161 km/s and 173 km/s are derived for M81 and M101, respectively. (a) Based on the measured values of Vmax , apply the appropriate versions of the Tully–Fisher relation to calculate MB and LB for M81 and M101. (b) The apparent magnitude mB in the B-band of an astronomical source is a measurement of its apparent brightness in the sky. This quantity is related to MB through the equation mB − MB = 5 log10 (d) − 5,

(7.101)

where d is the distance to the astronomical source in parsecs. In this equation, the effects of interstellar extinction have been ignored. If the measured values of mB for M81 and M101 are 7.96 and 7.99, respectively, use the computed values of MB for these two galaxies to compute their respective distances in parsecs.

318

7 Extragalactic Radio Astronomy

Note that M81 and M101 are both located at high Galactic latitudes (41◦ and 60◦ , respectively), so the effects of foreground Galactic extinction toward these galaxies is minimal and may safely be ignored in this problem. 7.4 As described in Sect. 7.2.3, high angular resolution radio observations of nearby galaxies can reveal the presence of discrete radio-emitting sources, such as supernova remnants, HII regions, and pulsar wind nebulae. The nominal flux densities at a frequency of 1 GHz of the Galactic supernova remnant Cassiopeia A and the Crab Nebula are 2720 and 1040 Jy, respectively. In addition, the distances to Cassiopeia A and the Crab Nebula are 3.4 and 2 kpc, respectively. If the limiting root-mean-square sensitivity of a 12 h observation performed with the Jansky Very Large Array at 1 GHz is 10 µJy/beam, calculate the distances of galaxies hosting analogs to Cassiopeia A and the Crab Nebula if these sources are detected at the 3 σ level (that is, with flux densities corresponding to 30 µJy). 7.5 The elliptical galaxy NGC 3842 is believed to host one of the most massive central black holes known, with an estimated mass of 9.7 billion solar masses [37]. Compute the corresponding luminosity LBZ for this black hole, assuming a magnetic field strength B = 1 T. 7.6 What is the observed transverse velocity vt (in units of c and m/s) for a jet of material ejected by an AGN with a velocity v = 0.8c at an angle φ = 60◦ to the line of sight? 7.7 Consider a blob of material ejected from a blazar with a velocity v = 0.9 c. (a) What is the Lorentz factor γ of the ejected blob as observed in the reference frame of the blazar? (b) At what angle φ (expressed in degrees) to the line of sight would produce a maximum value of the apparent transverse velocity vt ? (c) Assuming this value for φ, calculate the apparent transverse velocity vt (in units of both c and m/s). (d) Calculate the transverse distance (in meters and parsecs) traveled by the blob in 1 year in the reference frame of the galaxy. 7.8 Calculate the number of Earth masses required to “power” an observed active Galactic nuclei with a radio luminosity L = 1038 W assuming that the gravitational potential energy released by the masses as they fall into the central black hole is converted into radiation with an efficiency η of 0.1. 7.9 Show that Eq. (7.50) can be expressed as Eq. (7.51). 7.10 The western lobe of the radio galaxy Fornax A has an angular extent of 17 arcminutes and a measured flux density at 1.4 GHz of 74 Jy. [5] Consider radio observations made of this lobe over the frequency range from ν1 = 107 Hz to ν2 = 1011 Hz and assume that the given flux density of the lobe is constant across this entire bandwidth. Further assume that the total energy contained in the lobe is 1052 J.

Problems

319

Adopting the same redshift to the galaxy and therefore the same computed distance to Fornax A as derived in Example Problem 7.9, compute the following: (a) The volume Vlobe in m3 of the lobe (assume that it is spherical). (b) The integrated radio flux density S in Jy of the lobe and its integrated luminosity L in Watts (assume that spectral index of the lobe is α = −0.5). (c) The average magnetic field strength Blobe in Tesla of the lobe. (d) The lifetime tlobe of the lobe in years. 7.11 Derive Eq. (7.73) from Eqs. (7.71) and (7.72). Assume that the electron is initially at rest and utilize the conservation of momentum and energy for the system before and after the scattering event. 7.12 To illustrate the phenomenon of inverse Compton scattering, consider scattering events that take place between cosmic-ray electrons and cosmic microwave background (CMB) photons. (a) Assume that the emission from the CMB can be modeled as a blackbody with an effective temperature TCMB = 2.73 K. Recalling Wien’s displacement law, verify that the wavelength λmax in meters at which the CMB produces the most flux is 1.06 × 10−3 m. (b) Calculate the radiation energy density urad in eV/m3 of the CMB. (c) Let Ei correspond to the initial energy of a CMB photon before any scattering events occur. Calculate Ei for a CMB photon in electron-Volts. (d) What is the number of photons per m−3 that comprise the CMB? For simplicity, assume that all of the photons that comprise the CMB have the same energy, that is, the energy Ei calculated in Part (c). (e) The CMB photons scatter off a bath of relativistic cosmic-ray electrons, each with an energy E = 109 eV. Compute the Lorentz factor γ of the individual cosmic-ray electrons. (f) Calculate the final energy Ef in eV of the up-scattered CMB photon. (g) Calculate the power PIC in eV/s radiated away by the electron via the Compton scattering process. (h) Calculate the cooling time tIC in years of the electron as it emits the power calculated in Part (g). 7.13 Studies of the CMB must take into account both the orbital motion of the Earth around the Sun and the orbital motion of the Sun around the Galactic Center. These motions may impart a redshift and a blueshift to the observed emission of the CMB: these shifts manifest themselves as variations ΔT in temperature of the CMB that are not intrinsic to the emission from the CMB itself. Assuming that the orbital velocity of the Earth around the Sun is approximately 30 km/s and that the orbital velocity of the Sun around the Galactic Center is 230 km/s (see Sect. 6.2.3), estimate the scale sizes of the temperature variations in Kelvin that these orbital motions would impress upon the emission of the CMB. Note that in this case it is possible to relate the redshift z to temperature variations ΔT and the temperature T of the CMB as (from Eq. (2.140))

320

7 Extragalactic Radio Astronomy

z=

Δλ ΔT = . λ T

(7.102)

Recall that the temperature of the CMB is approximately TCMB = 2.73 K (see Sect. 7.5). 7.14 Calculate the Lorentz factor of a proton with an energy corresponding to the GZK limit. 7.15 Consider the Sunyaev–Zeldovich effect as applied to provide an independent estimate of Hubble’s constant H0 . The method to obtain this estimate is illustrated below. (a) Calculate  in parsecs for a particular galaxy cluster for which the measured values of ΔT /T , ne and Te are 10−5 , 103 m−3 , and 108 K, respectively. (b) Calculate d to the cluster if the measured angular extent θ of the cluster is 5 arcminutes. (c) Calculate H0 if the measured value of the recessional velocity v corresponding to the cosmological redshift is 7000 km/s.

References 1. J.J. Condon, J.J. Broderick, G.A. Seielstad, K. Douglas, P.C. Gregory, Astron. J. 107, 1829– 1833 (1994) 2. E.P. Hubble, Proc. Nat. Acad. Sci. U.S.A. 15(3), 168–173 (1929) 3. S. Phillips, The Structure and Evolution of Galaxies, 1st edn. (Wiley, New York, 2005), p. 116 4. I. Agudo, A.P. Marscher, S.G. Jorstad, et al., Astrophys. J. 747, Article id 63 (2012) 5. R.D. Ekers, W.M. Goss, K.J. Wellington, et al., Astron. Astrophys. 127, 361–165 (1983) 6. A.H. Bridle, D.H. Hough, C.J. Lonsdale, et al., Astron. J. 108, 766–820 (1994) 7. F. Walter, E. Brinks, W.J.G. de Blok, et al., Astron. J. 136, 2563 (2008) 8. W.L. Freedman, B.F. Madore, Astrophys. J. 332, L63 (1988) 9. L. Feretti, G. Giovannini, F. Govoni, M. Murgia, Astron. Astrophys. Rev. 20, 54 (2012) 10. K.A. Herrmann, R. Ciardullo, J.J. Feldmeier, et al. Astrophys. J. 683, 630 (2008) 11. R.B. Tully, J.R. Fisher, Astron. Astrophys. 54, 661–673 (1977) 12. V.C. Rubin, D. Burstein, W.K. Ford, Jr., N. Thonnard, Astrophys. J. 289, 81–104 (1985) 13. R. Beck, Astron. Astrophys. 470, 539–556 (2007) 14. C.R. Seyfert, Astrophys. J. 97, 28–40 (1943) 15. C.K. Lacey, N. Duric, W.M. Goss, Astrophys. J. Suppl. Ser. 109, 417–460 (1997) 16. T.A. Matthews, A.R. Sandage, Astrophys. J. 138, 30–56 (1963) 17. J.L. Greenstein, T.A. Matthews, Nature 197, 1041–1042 (1963) 18. K.I. Kellermann, R. Sramek, M. Schmidt, et al., Astron. J. 98, 1195–1207 (1989) 19. R.D. Blandford, R.L. Znajek, Mon. Not. R. Astron. Soc. 179, 433–456 (1977) 20. B. Fanaroff, J. Riley, Mon. Not. R. Astron. Soc. 167, 31–36 (1974) 21. A.H. Bridle, R.A. Perley, Ann. Rev. Astron. Astrophys. 22, 319–358 (1984) 22. J.P. Leahy, R.A. Perley, Astron. J. 102, 537–561 (1991) 23. C.S. Reynolds, D. Garofalo, M.C. Begelman, Astrophys. J. 651, 1023–1030 (2006) 24. D. Garofalo, D.A. Evans, R.M. Sambruna, Mon. Not. R. Astron. Soc. 106, 975–986 (2010) 25. A. Tchekhovskoy, J.C. McKinney, Mon. Not. R. Astron. Soc. Lett. 423, L55–L59 (2012) 26. L. Feretti, R. Rusco-Femiano, G. Giovannini, et al., Astron. Astrophys. 373, 106–112 (2001)

References

321

27. M. Böttcher, A. Reimer, Astrophys. J. 609, 576–588 (2004) 28. B.W. Carroll, D.A. Ostlie, An Introduction to Modern Astrophysics, 2nd edn. (Pearson/Addison Wesley, New York, 2007), p. 393 29. G. Fossati, L. Maraschi, A. Celotti, et al., Mon. Not. R. Astron. Soc. 299, 433–448 (1998) 30. B.W. Carroll, D.A. Ostlie, An Introduction to Modern Astrophysics, 2nd edn. (Pearson/Addison Wesley, New York, 2007), p. 267 31. K. Greisen, Phys. Rev. Lett. 16, 748–750 (1966) 32. G.T. Zatsepin, V.A. Kuz’min, Sov. J. Exp. Theor. Phys. Lett. 4, 78 (1966) 33. F. Massaro, D.E. Harris, G.R. Tremblay, E. Liuzzo, A. Bonafede, A. Paggi, Astrophys. J. Suppl. Ser. 206, 7 (2013) 34. B.W. Carroll, D.A. Ostlie, An Introduction to Modern Astrophysics, 2nd edn. (Pearson/Addison Wesley, New York, 2007), p. 1122–1124 35. K. Rohlfs, D.L. Wilson, Tools of Radio Astronomy, 4th Edn. (Springer, Berlin, 2004), pp. 299– 300 36. C.J.J. Pearce, et al., Astrophys. J. 845, 81 (2017) 37. N.J. McConnell, C.-P. Ma, J.D. Murphy, et al., Astrophys. J. 756, 179 (2012) 38. R. Cassano, S. Ettori, S. Giacintucci, et al., Astrophys. J. Lett. 721, L82 (2010) 39. C.L. Carilli, G.B. Taylor, ARA&A 40, 319 (2002)

Glossary

Accretion A physical process where matter falls onto a central body and gravitational potential energy is released, in part in the form of electromagnetic radiation. Altitude A coordinate in the altitude-azimuth coordinate system measured from a point on the horizon indicated by the azimuth coordinate to the altitude of the object of interest in the sky. Altitude-Azimuth Coordinate System A coordinate system which uses the altitude coordinate h and the azimuth angle a of an object to indicate its position on the sky. Angular Resolution The ability of a telescope to resolve fine detail in the angular structure of a source. Aperture The diameter of a radio telescope. Apogee The point in the orbit of an object around the Earth where it is farthest from the Earth. Arc Length A distance along the perimeter of a circle. Arcseconds A unit of angular measurement defined such that 3600 arcseconds corresponds to one degree. Arcminutes A unit of angular measurement defined such that 60 arcminutes corresponds to one degree. Astronomical Unit The average distance between the Earth and the Sun, corresponding to 1.50 × 108 km. Atmospheric Windows Wavelength ranges of the electromagnetic spectrum over which the Earth’s atmosphere is more transparent to radiation.

© Springer International Publishing Switzerland 2020 T. G. Pannuti, The Physical Processes and Observing Techniques of Radio Astronomy, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-16982-8

323

324

Glossary

Azimuth A coordinate in the altitude-azimuth coordinate system measured along the horizon from due north (which – by convention – corresponds to an azimuth of zero degrees). Blackbody Radiation A model of thermal radiation from astronomical sources. Big Bang The currrently favored scenario for the origin of the Universe from the rapid expansion of a singularity. Celestial Equator An extension of the equatorial plane of the Earth onto the celestial sphere. Celestial Poles (North and South) The projected locations of the North and South Pole of the Earth onto the Celestial Sphere. Celestial Sphere A theoretical construct of a sphere that completely encloses the Earth and upon which it may be imagined that all objects in the universe are affixed. Centripetal Force The force necessary to keep an object moving in a circular path about a central object or axis. Circular Velocity The orbital velocity of an object in a circular orbit around a parent body. Cluster A gravitationally bound system of galaxies. Conservation of Energy The principle that states that the total amount of energy involved in a physical process must always be conserved, though it may be transformed from one form into another. Conservative Forces Forces for which the total work done in moving a particle from one point to another does not depend on the path taken. Declination A coordinate within the equatorial coordinate system that is analogous to the latitude coordinate in the latitude-longitude coordinate system used to measure positions on the surface of the Earth. Degrees An angular measurement defined such that a complete circle corresponds to 360 degrees. Ecliptic The apparent path of the Sun across the sky as viewed from the Earth. Equivalently, the projection of the plane of the Earth’s orbit onto the celestial sphere. Epoch A reference time used with the Equatorial Coordinate System. Equatorial Coordinate System A coordinate system that uses the coordinates of Right Ascension and Declination to indicate the position of an object of interest on the sky. Equinoxes (Autumnal and Vernal) Calendar dates where the time periods of day and night are approximately equal.

Glossary

325

Equivalent Widths A property of a spectral line defined as the width of a box that reaches the continuum and has the same area as the spectral feature itself. Escape Velocity The minimum velocity required to escape a gravitational orbit around an object. Fanaroff and Riley Classification System A classification system of radio galaxies based on the structure of their radio lobes and the luminosities of the lobes relative to the core of the galaxy. FR-I galaxies have a lower luminosity, are brighter in the cores than in the loves, exhi Frequency The amount of time required for a wave to complete a cycle. Flux The amount of light emitted by an object or alternatively the amount of light received from an object by a detector. Greisen-Zatespin-Kuzmin Limit A theoretical upper limit on the energy of cosmic-ray protons of approximately 5 × 1019 eV or approximately 8 J. This upper limit is due to an interaction between the cosmic-ray protons and cosmic microwave background photons that produces pions and drains the energy of the protons. Group A gravitationally bound system of galaxies with fewer member sources than clusters. Hadron High-mass subatomic particle exemplified by a proton. Hertz A unit of frequency corresponding to one cycle per second. HII Region A region of massive star formation where hydrogen is observed to be principally in an ionized state. Interferometry This principle of combining (“interfering") signals from multiple radio telescopes to attain superior angular resolution. Interferometers Radio telescopes that operate through the principle of interferometry to attain superior angular resolution. Interstellar Extinction The reduction of intensity of electromagnetic radiation as it propagates through the interstellar medium. Ionosphere The ionized part of the Earth’s upper atmosphere that comprises the inner portion of the Earth’s magnetosphere. Kiloparsec A unit of distance corresponding to one thousand parsecs. Length Contraction A phenomenon related to relativistic motion where length scales appear to contract to ever shorter scales with increasing velocity. Lepton Low-mass subatomic particle exemplified by an electron. Light Electromagnetic radiation comprised of an alternating electric field and magnetic field.

326

Glossary

Megaparsec A unit of distance corresponding to one million parsecs. Milky Way Galaxy The galaxy to which the Sun belongs. Neutron Star A compact rapidly rotating body composed almost exclusively of neutrons and produced by the deaths of massive stars in supernova explosions. Parallax The apparent shift of position of an object due to the changing perspective of the observer. Parallax Angle The angular shift exhbited by a source due to its parallax. Parsec A unit of astronomical distance corresponding to the distance at which an object must lie from the Earth to exhibit a parallax of one arcsecond. Perigee The point in the orbit of an object around the Earth where it is closest to the Earth. Prime Focus Position where light is brought to a focus by a radio telescope. Pulsar A neutron star observed from such an angle that emission jets from the neutron star appear to sweep past the observer and apparent pulses of emission from the source are seen. Radians An angular measurement defined such that a complete circle corresponds to 2π radians. Radio Halo Diffuse structure of radio emission seen toward the centers of galaxy clusters. Radio Relic Diffuse source of radio emission seen toward the edges of galaxy clusters. Right Ascension A coordinate within the equatorial coordinate system that is analogous to the longitude coordinate in the latitude-longitude coordinate system used to measure positions on the surface of the Earth. Sensitivity A description of the ability of the telescope to detect signals from astronomical sources to ever lower intensities. Sexagesimal Coordinate System Sidereal Day The period of time (corresponding to 23 h 56 min) required for the Earth to complete one rotation relative to the reference frame of distant stars. Singularity An infinitesimally small point from which the Universe is believed to have originated through the Big Bang. Small Angle Approximation The approximation that for angular values that approach zero, the cosine of the angle may be approximated as unity while the sine and the tangent of the angle may be approximated as the angle itself expressed in radians.

Glossary

327

Solar Day The amount of time (corresponding to 24 h) required for the sun to complete one whole path across the sky. Solid Angle The apparent two dimensional angular area subtended by an object on the sky. Solstices (Summer and Winter) Calendar dates corresponding to days when the length of night is minimized and the length of day is maximized (for summer solstices as observed from the Northern Hemisphere) or when the length of night is maximized and the length of day is minimized (for winter solstices as observed from the Northern Hemisphere). Spherical Coordinates A coordinate system in three-dimensional space where positions of objects are determined using a single radial coordinate r and two angular coordinates, namely the polar angle θ and the azimuthal angle φ. Steradian A unit of measurement of solid angle such that 4π steradians corresponds to an entire sphere. Sunyaev–Zeldovich Effect The scattering of cosmic microwave background photons through the inverse Compton Scattering process of high-energy electrons in galaxy clusters. Supernova Remnant The expanding shock front of stellar ejecta and swept-up interstellar material produced by a supernova explosion. Synchrotron radiation A form of radiation emitted by relativistic electrons gyrating in magnetic fields. Time Dilation A phenomenon related to relativistic motion where time scales appear to dilate to ever longer intervals with increasing velocity. Thermal Bremsstrahlung An emission mechanism detected at radio wavelengths that is produced by free electrons experiencing near collisions with ions and emitting photons as they are accelerated. Wavelength The distance in meters from the peak of one wave to the next. Zenith The point in the sky located directly overhead of the observer. Zenith Angle The angle measured from the zenith to the location of an object of interest in the sky such that the sum of the zenith angle and the altitude angle is always equal to 90 degrees.

328 Constant A AU B c e EC G h h¯ H0 kB me mp MG M pc R0 R t0 TCMB 0 λC μ0 σSB σT Θ0 Ω0

Glossary Value 14.82±0.84 km s−1 kpc−1 1.50×108 km −12.37±0.64 km s−1 kpc−1 3×108 m s−1 1.60×10−19 C 511 keV 6.67×10−11 N m2 kg−2 6.63×10−34 J s 1.06×10−34 J s 71 km s−1 Mpc−1 1.38×10−23 J K−1 9.11×10−31 kg 5.11×105 eV/c2 1.67×10−27 kg 9.38×108 eV/c2 1.5×1011 M 1.99×1030 kg 3.09×1016 m 8.5±0.5 kpc 6.96×108 m 13.7 × 109 yr 2.73 K 8.85 × 10−12 F/m 2.43 × 10−12 m 1.26 × 10−6 N/A2 5.67×10−8 W m−2 K−4 6.65×10−29 m2 220 km s−1 27.19±0.87 km s−1 kpc−1

Description Oort Constant A Astronomical unit Oort Constant B Velocity of light in a vacuum Charge of the electron Compton energy Gravitational constant Planck’s constant Planck’s constant divided by 2π Hubble’s constant Boltzmann’s constant Mass of the electron (in kg) Mass of the electron (in eV/c2 ) Mass of the proton (in kg) Mass of the proton (in eV/c2 ) Mass of the Milky Way Galaxy Solar Mass Parsec Distance from Sun to Milky Way Center Solar Radius Age of the Universe Temperature of the Cosmic Microwave Background Permittivity of free space Compton wavelength Permeability of free space Stefan–Boltzmann constant Thomson cross-section Orbital velocity of Sun around Galactic Center Angular orbital velocity of Sun around Galactic Center

Glossary Term a A ΔA Aeff AU b B B B B⊥ Bmin c d D DM e E E EC EIC Erot Erad EM F F and F  F5 GHz F4400 FB Fcentripetal Fgravitational G h h¯ H I J k kb kML K  L LBZ

329 Definition Azimuth angle or semi-major axis or azimuth angle Oort’s Constant A or Adjacent leg of a right triangle or Area Area interval Effective area Astronomical Unit Semi-minor axis or Impact parameter or base of triangle or Galactic latitude Magnetic field strength or Oort’s Constant B Magnetic field vector Parallel component of the magnetic field Perpendicular component of the magnetic field Magnetic field corresponding to Emin Speed of light Distance Distance or Deuterium Dispersion Measure Eccentricity or Charge of electron Energy or Total energy Electric field vector Energy corresponding to the Compton wavelength Energy generated through Inverse Compton scattering Rotational kinetic energy Radiated energy Emission Measure Force or Flux Foci Flux of a galaxy at 5 GHz Flux of a galaxy at 4400 Å Lorentz force Centripetal force Gravitational force Gravitational constant Planck’s constant or altitude angle or height of triangle or H0 /100 km s−1 Mpc−1 Planck’s constant divided by 2π Hypotenuse Rotational inertia Total angular momentum quantum number Wavenumber Boltzmann’s constant Inverse of mass-to-luminosity ratio Kinetic Energy Linear diameter or Galactic longitude or Path length Angular momentum or Luminosity Blandford–Znajek Luminosity

330 Term m or M me MHI Mtotal M/L n ne np n(λ) O p pe pi pf P PIC Psync P˙ P¨ q r rCM R R0 Rmin RM s S Sν S t t0 Δt T Te ΔT uB urad U v v vapp vaphelion

Glossary Definition Mass Mass of electron HI mass Total mass Mass-to-light ratio Braking index Electron number density Proton number density Index of refraction as a function of wavelength λ Opposite leg of a right triangle Parallax or momentum Electron momentum Initial momentum Final momentum Orbital period or Power or Pulsation period Power emitted by Inverse Compton Scattering Power emitted by synchrotron radiation Time derivative dP /dt Second time derivative d 2 P /dt 2 Charge Radius or Radial distance Location of center of mass Radius or Scale Factor Distance from Sun to center of Milky Way Galaxy or Current radius of the Universe Minimum distance Rotation Measure Arc length Flux density Flux density at a given frequency ν Poynting vector Time Current age of the Universe Time interval Temperature Electron temperature Change in temperature Magnetic energy density Radiation energy density Gravitational potential energy Velocity Velocity vector Apparent velocity Orbital velocity at aphelion

Glossary

331

Term vperihelion vc ve vmin vmax vr vt V x z α β γ γmin δ η θ θmin Δθ Θ Θ0 ΔΘ λ λ0 λC λmax Δλ μ ν ν˙ ν¨ ... ν νi νf ρ Σ τ ΔτD Υ φ

Definition Orbital velocity at perihelion Centripetal velocity or Circular velocity Escape velocity Minimum velocity Maximum velocity Radial velocity Tangential or Transverse velocity Volume or Potential Energy Linear extent Zenith angle or Redshift or Linear distance above Galactic Plane Right Ascension coordinate or Spectral index Equivalent to v/c in Lorentz factor Lorentz factor (equivalent to 1/ 1 − v 2 /c2 ) Lorentz factor corresponding to vmin Declination coordinate Aperture efficiency Angle or polar angle in spherical coordinates Minimum angle corresponding to vmin Angular separation or angular extent Angular resolution Initial polarization angle Rotation of plane polarization angle Wavelength Rest wavelength Compton wavelength Wavelength of maximum emission for a blackbody Change in wavelength Reduced mass or Proper Motion Frequency First frequency derivative with respect to time Second frequency derivative with respect to time Third frequency derivative with respect to time Initial Frequency Final Frequency Mass density Surface brightness Optical depth or Characteristic age Time delay Intersection point of Celestial Equator and Ecliptic Azimuthal angle or azimuthal angle in spherical coordinates or position angle or transition rate Vertices of elliptical orbit corresponding to perihelion and aphelion, respectively Angular momentum per unit mass Angular frequency or angular velocity Solid angle

ψ and ψ  Ψ ω Ω

Solutions

Problems for Chap. 2 2.1 From Eq. (2.1), the angular extent θJupiter in radians as observed from Europa is θJupiter

140,000 km = 0.21 rad = 670,900 km



57.2958◦ 1 rad



= 12.03◦ .

(2)

Compared to the apparent angular size of the Moon in the sky of the Earth, Jupiter in the sky over Europa appears to be ≈12.03◦ /0.5◦ ≈ 24 times larger. 2.2 From Eq. (2.1) and noting that the diameter of the Sun is x = 1.39 × 106 km and that the mean distance to the Sun is r = 1.49 × 108 km, the angular extent θSun of the Sun as observed from the Earth is θSun (radians) =

1.39 × 106 km x = = 9.39 × 103 radians. r 1.49 × 108 km

(3)

From the same equation and noting that the diameter of the Moon is x = 3474 km and that the mean distance to the Moon is r = 384,000 km, the angular extent θMoon of the Moon as observed from the Earth is θMoon (radians) =

3474 km x = = 9.05 × 103 radians. r 384,000 km

(4)

Therefore the angular extents of the Moon and the Sun as observed by the Earth are indeed comparable. 2.3 Following the given example in the chapter for Mizar, the given sexagesimal coordinates for Sirius A and Sirius B are converted into decimal degrees as follows: The first step is to convert the given coordinates for both stars from sexagesimal to © Springer International Publishing Switzerland 2020 T. G. Pannuti, The Physical Processes and Observing Techniques of Radio Astronomy, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-16982-8

333

334

Solutions

decimal degrees, to make the application of Eq. (2.32) more straightforward. The conversions may be accomplished as follows:  αSirius A =

6h × 360◦ 24h



 +

45m × 360◦ 1440m



 +

08.9s × 360◦ 86,400s



= 101.2871◦ (5)

and  ◦

δSirius A = −16 −



42 × 1◦  60



 −



58.2 × 1◦  3600

 = −16.7162◦ .

(6)

Note how the calculation of δSirius A was performed. Because the Declination of this source is negative, the calculation is done by subtracting instead of summing to reflect the fact that Declination is measured such that a more negative Declination indicates that the source is farther south. Performing the same calculation for Sirius B yields  αSirius B =

6h × 360◦ 24h



 +

45m × 360◦ 1440m



 +

09.0s × 360◦ 86,400s



= 101.2875◦ (7)

and  ◦

δSirius B = −16 −



43 × 1◦  60



 −



6 × 1◦  3600

 = −16.7183◦ .

(8)

Therefore Δα = |αSirius A − αSirius B | = |101.2871◦ − 101.2875◦ | = 4.0 × 10−4◦

(9)

and Δδ = |δSirius A − δSirius B | = |(−16.7162◦ ) − (−16.7183◦ )| = 2.1 × 10−3 ◦. (10) The mean value of the two declinations of the two stars is δ − δSirius B (−16.7162◦ ) + (−16.7183◦ ) δ¯ = Sirius A = = −16.7173◦ , 2 2 (11) and applying Eq. (2.32) yields (Δθ )2 = (4.0 × 10−4◦ × cos (−16.7173◦ ))2 + (2.1 × 10−3◦ )2 = 2.13 × 10−3◦ (12) or

Solutions

335

Δθ = 2.13 × 10−3◦ ×



3600  ≈7 . ◦ 1

(13)

2.4 From Eq. (2.52) and the given properties of the Moon, the escape velocity ve is  ve =

 2GM = r

2(6.67 × 10−11 N m2 /kg2 )(7.35 × 1022 kg) = 2374 m/s. 1.74 × 106 m (14)

2.5 (a) Recalling that 1 A.U. = 1.49 × 1011 m, a can be converted into meters as follows:   1.49 × 1011 m = 2.65 × 1012 m. a = 17.8 A.U. (15) 1 A.U. Therefore, from Eq. (2.82), P may be computed as  P =

4π 2 (2.65 × 1012 m)3

(6.67 × 10−11 N m2 /kg2 )(2 × 1030 kg)   1 yr = 1.32 × 109 s = 74.5 yr. 3.15 × 107 s

(16)

Notice once more the approximation made regarding M, namely M = M + MHalley’s Comet ≈ M . As noted previously, this approximation is commonly made when analyzing the orbital properties of Solar System objects. (b) From Eq. (2.57), b may be calculated as follows:   1.49 × 1011 m b = a 1 − e2 = (17.8 A.U.) 1 − (0.97)2 = (4.33 A.U.) 1 A.U. (17) = 6.45 × 1011 m. (c) From Eqs. (2.60) and (2.61), rperihelion and raphelion can be computed as follows. For rperihelion , rperihelion = 2.65 × 1012 m(1 − 0.97) = 7.95 × 1010 m,

(18)

and for raphelion , rapihelion = 2.65 × 1012 m(1 + 0.97) = 5.22 × 1012 m.

(19)

336

Solutions

(d) From Eqs. (2.67) and (2.68), vperihelion and vaphelion can be computed as follows. For vperihelion , 2π(2.65 × 1012 m) = 1.32 × 109 s

vperihelion



(1 + 0.97) = 1.02 × 105 m/s, (1 − 0.97)

(20)

(1 − 0.97) = 1.56 × 103 m/s. (1 + 0.97)

(21)

and for vaphelion ,

vaphelion

2π(2.65 × 1012 m) = 1.32 × 109 s



(e) Denoting the mass of Halley’s Comet as mcomet , the gravitational force between the Sun and the comet when the comet is at rperihelion and raphelion (denoted as Fperihelion and Faphelion may be expressed as Fperihelion =

GM mcomet 2 rperihelion

and

Faphelion =

GM mcomet , 2 raphelion

(22)

respectively. From inspection of these expressions, the ratio Fperihelion /Faphelion may be expressed simply as 2 raphelion Fperihelion (5.22 × 1012 m)2 = 2 = = 4311. Faphelion (7.95 × 1010 m)2 rperihelion

(23)

(f) From Eq. (2.45), E may be expressed as E=

2 mcomet vperihelion

2

−

GM mcomet . rperihelion

(24)

Therefore E may be computed as E=

(5 × 1014 kg)(1.02 × 105 m/s)2 2 −

(25)

(6.67 × 10−11 N m2 /kg2 )(2 × 1030 kg)(5 × 1014 kg) 7.95 × 1010 m

which yields at last E = −8.39 × 1023 J. Thus, the orbit of Halley’s Comet is clearly stable gravitationally.

(26)

Solutions

337

2.6 (a) From Eq. (2.90), ν may be calculated as follows: ν=

3 × 108 m/s c = = 5 × 109 Hz. λ 0.06 m

(27)

(b) From Eq. (2.95), k may be calculated as follows: k=

2π 2π = = 104.72 m−1 . λ 0.06 m

(28)

(c) From Eq. (2.96), ω may be calculated as follows: ω = 2π ν = 2π(5 × 109 Hz) = 3.14 × 1010 radians s−1 .

(29)

(d) From Eq. (2.111), E may be calculated as follows: E = hν = (6.63 × 10−34 J · s)(5 × 109 Hz) = 3.32 × 10−24 J.

(30)

(e) From Eq. (2.112), p may be calculated as follows: p=

3.32 × 10−24 J E = = 1.11 × 10−32 kg m s−1 . c 3 × 108 m/s

(31)

2.7 By inspection of the given expressions for Ex , Ey , and Ez , the value for δ is zero. From Eqs. (2.119) to (2.121), the Stokes Parameters may be calculated as follows: I = (10−5 V/m)2 + (10−5 V/m)2 = 2 × 10−10 V2 /m2 .

(32)

Q = (10−5 V/m)2 − (10−5 V/m)2 = 0 V2 /m2 . −5

U = 2(10

−5

V/m)(10

◦

(33) −10

V/m) cos 0 = 2 × 10

2

2

V /m .

V = 2(10−5 V/m)(10−5 V/m) sin 0◦ = 0 V2 /m2 .

(34) (35)

From Eq. (2.127), pd is  pd =

(0 V2 /m2 )2 + (2 × 10−10 V2 /m2 )2 + (0 V2 /m2 )2 2 × 10−10 V2 /m2

= 1.00.

(36)

Because the difference δ in phase between the Ex and Ey components is null, the wave is linearly polarized at a position angle of 45◦ . Because the value of pd is unity, the wave is fully linearly polarized.

338

Solutions

2.8 (a) From Eq. (2.140), Δλ = λ0

v  r

c



40 km/sec = 656.3 nm 3 × 105 km/sec

 = 0.09 nm.

(37)

Because the recessional velocity is positive, the line appears to be shifted toward long wavelengths. Hence, this is a redshift. (b) From Eq. (2.140), Δλ = λ0

v  r

c

 = 656.3 nm

−70 km/sec 3 × 105 km/sec

 = −0.15 nm.

(38)

Because the recessional velocity is negative, the line appears to be shifted toward short wavelengths. Hence, this is a blueshift. 2.9 (a) From Eq. (2.143), z may be calculated as follows:  z=

1 + vr /c −1= 1 − vr /c



1 + (0.95c)/c − 1 = 5.24. 1 − (0.95c)/c

(39)

(b) From Eq. (2.140), λ can be determined as follows: z=

λ − λ0 −→ λ = λ0 (z + 1) = 656.3 nm (5.24 + 1) = 4.095 μm. λ0

(40)

Keep in mind that this observed shift of the spectral line is a redshift of significant magnitude: a line with a rest wavelength in the optical appears to have been shifted well into the infrared! 2.10 Equation (2.143) can be rewritten as (z + 1)2 =

1 + vr /c −→ (z + 1)2 − (z + 1)2 vr /c = 1 + vr /c, 1 − vr /c

(41)

and collecting terms for vr yields (z + 1)2 − 1 = ((z + 1)2 − 1)vr /c −→

(z + 1)2 − 1 vr = . c (z + 1)2 + 1

(42)

2.11 Taking each expression for r given in Eq. (2.150) and solving each one for v yields two expressions for r, namely

Solutions

339

Ze2 r= −→ v = 4π 0 me v 2

 Ze2 4π 0 me r

(43)

and r=

nh nh −→ v = . 2π me v 2π me r

(44)

Setting these expressions equal to each other and solving for r yields 

Ze2 nh Ze2 n2 h2 = −→ = 4π 0 me r 2π me r 4π 0 me r 4π 2 m2e r 2

(45)

and therefore n2 h2 0 . π me Ze2

r=

(46)

2.12 From Eq. (2.149), v may be re-expressed as v=

nh , 2π me r

(47)

and inserting the expression for r given in Eq. (2.151), v becomes nh v= 2π me



π me Ze2 n2 h2 0

 =

Ze2 . 2nh0

(48)

Thus, the kinetic energy K of the electron may be expressed as K=

me me v 2 = 2 2



Ze2 2nh0

2 =

me e 4 Z 2 . 8n2 h2 02

(49)

In a similar vein, from the expression for r given in Eq. (2.151), the potential energy V may be expressed as Ze2 Ze2 =− V =− 4π 0 r 4π 0



π me Ze2 n2 h2 0

 =

me e 4 Z 2 . 4n2 h2 02

(50)

Therefore, the expression for E becomes E=K+V =

me e 4 Z 2 me e 4 Z 2 − 8n2 h2 02 4n2 h2 02

  2 me e4 Z 2 2me e4 Z 2 me e 4 Z 2 = − =− . 2 2 2 8n2 h2 0 8n2 h2 0 8n2 h2 02 (51)

340

Solutions

2.13 (a) Note that in this situation, the chicken sees the pole (with a rest length L0 = 5 meters) contracted to a length L = 3 m (that is, to the length of the shed). Therefore, the appropriate value for γ can be determined from Eq. (2.165) as follows: γ =

L0 5m = = 1.67. L 3m

(52)

The corresponding value of v can be found by rearranging Eq. (2.164) such that  1 1 γ = −→ 1 − v /c = 2 −→ v = c 1 − 2 , γ γ 1 − v 2 /c2 1

so therefore  v =c 1−

2

2

1 = 0.8c = 0.8(3 × 108 m/s) = 2.4 × 108 m/s. 1.672

(53)

(54)

(b) From Eq. (2.166) and the value of γ computed above, t may be calculated as follows: Δt = γ Δt0 = (1.67) (1 s) = 1.67 s.

(55)

(c) From Eq. (2.167), the observed velocity v  of the horsefly will be v=

(8c/5) 40c u+v (4c/5)+(4c/5) (8c/5) = = . = = 1+(uv)/c2 1+((4c/5)(4c/5))/c2 1 + (16/25) (41/25) 41 (56)

2.14 (a) From Eq. (2.170), the Lorentz factor γ of the proton is γ =

E 3.2 × 1020 eV = = 3.41 × 1011 . 2 m0 c (9.38 × 108 eV/c2 )(c2 )

(57)

(b) From Eq. (2.44), the velocity √ v of a particle with mass m and a kinetic energy K may be expressed as v = 2K/m. Substituting in the given values for K and m for the tennis ball yields  v=

 2K = m

2(51 J) = 42 m/s, 0.057 kg

which corresponds to approximately 94 miles per hour!

(58)

Solutions

341

Problems for Chap. 3 3.1 Inserting the appropriate values for the constants h, c, and k along with the approximate value for x of 4.97 yields λmax T =

hc (6.63 × 10−34 J · s)(3 × 108 m/s) = 2.9 × 10−3 m · K. = xk (4.97)(1.38 × 10−23 J/K)

(59)

3.2 (a) The explicit integration that needs to be evaluated (beginning from the expression given for Bnu (ν, T ) in Eq. (3.4)) is 

ν=∞

F =π

ν=0

2hν 3 dν = hν/k 2 BT − 1 c e



2π h c2



∞

ν3 ehν/kB T

0

−1

dν.

(60)

Note here that the factor of π originates from the integration over solid angle (see Eq. (3.18)). Adopting a substitution of variables such that x=

hν kB T

and

dx =

h dν, kB T







(61)

the integration becomes 



∞ k

3

BT x

 F =

x3 ex −1

2π kB4 T 4 dx= c 2 h3



∞

x3 dx, h ex −1 0 0 (62) and applying the integration identity given in Eq. (3.150) produces at last

F =

2π h c2

2π kB4 T 4 h3 c 2

kB T h



π4 15

 = σSB T 4 ,

(63)

as expected (refer to Eq. (3.21) for the complete expression for σSB in terms of fundamental constants). (b) Similar to the result from Part (a), the explicit integration that needs to be evaluated (beginning with Eq. (3.2) and once again recalling the factor of π from the integration over solid angle as derived in Eq. (3.18)) is  F =π

λ=∞

λ=0

2hc2 dλ = 2π hc2 hc/λk 5 BT − 1 λ e



λ=∞ λ=0

dλ (λ5 )(ehc/λkB T

− 1) (64)

Adopting a substitution of variables such that

342

Solutions

x=

hc λkB T

dx = −

and

hc dλ, λ 2 kB T

(65)

and recognizing that 1 = λ5



=−

xkB T hc

5 dλ = −

and

kB T λ 2 kB T dx = − hc hc



hc xkB T

2 dx

(66)

hc dx, kB T x 2

the integration becomes  F = 2π hc

  5  hc dx − hc kB T x 2 e x − 1

x=0  xk

2

x=∞

 = 2π hc2

kB T hc

4 

BT

x=∞

x=0

(67)

x3 dx. −1

ex

Note that the integration bounds became inverted because of the inverse relationship between x and λ, but that the integration bounds may be switched again due to the presence of the negative sign that originates from the substitution of dx for dλ. Applying the integration identity given in Eq. (3.150) once more yields  F =

2π kB4 T 4 h3 c 2



π4 15

 = σSB T 4 ,

(68)

as expected. 3.3 (a) From Wien’s Displacement Law given in Eq. (3.15), T may be computed as follows: T =

2.90 × 10−3 m · K = 7532 K. 385 × 10−9 m

(69)

(b) From the Stefan–Boltzmann Law given in Eq. (3.23), F may be calculated as follows: F = (5.67 × 10−8 W m−2 K−4 )(7532 K)4 = 1.83 × 108 W m−2 . (c) From Eq. (3.24), L may be calculated as follows:

(70)

Solutions

343

 2  6.96 × 108 m L = 4π 70 R (5.67 × 10−8 W m−2 K−4 )(7532 K)4 1 R (71) = 5.44 × 1030 W. (d) First, using Eq. (2.6), the distance d to Canopus is calculated based on the given parallax value as follows: d (pc) =

1 = 100 pc. 0.01 arcseconds

(72)

Therefore, from Eq. (3.28), F may be calculated as follows: F =

5.44 × 1030 W −8 2 ! ! ""2 = 4.55 × 10 W/m . 4π (100 pc) 3.086 × 1016 m/pc

(73)

(e) From Eq. (3.28), S may be computed as S=

5.44 × 1030 W = 1.52×10−21 W/m2 . ! ! ""2 4π (100 pc) 3.086 × 1016 m/pc (3 × 1013 Hz) (74)

3.4 Noting that Eq. (3.39) can be rewritten as τ = −ln

Iλ , Iλ,0

(75)

the optical depths at each frequency may be calculated as follows: 498 Jy = 0.004. τ5000 MHz = −ln 500 Jy

(76)

525 Jy τ840 MHz = −ln = 0.047. 550 Jy

(77)

3 Jy τ408 MHz = −ln = 5.298. 600 Jy

(78)

1 Jy = 6.477. τ74 MHz = −ln 650 Jy

(79)

By inspection of these values, the gas is optically thin at the frequencies of 840 and 5000 MHz and optically thick at the frequencies 74 and 408 MHz.

344

Solutions

2 may be expressed as 3.5 From Eq. (3.70), a⊥

 2 a⊥ =

qBv⊥ γm

2 =

2 q 2 B 2 v⊥ , γ 2 m2

(80)

and therefore the expression for P  becomes q 2γ 4 P = 6π 0 c3 



2 q 2 B 2 v⊥ γ 2 m2

 =

2 q 4 γ 2 B 2 v⊥ . 6π 0 c3 m2

(81)

Lastly, rewriting Eq. (2.170) as E , mc2

(82)

 2 B2  2 B2  q 4 v⊥ q 4 v⊥ E 2 2 γ = . 6π 0 c3 m2 6π 0 c3 m2 mc2

(83)

γ = the expression for P  becomes P =

3.6 Inserting q = e, m = me , v⊥ = c and recalling Eq. (2.170) which gives the relationship between the Lorentz factor γ of a particle and its energy, Eq. (3.80) may be rewritten as e4 c2 B 2 P = 6π 0 c3 m2e



E me c 2

2 =

e4 c2 B 2 γ 2 e4 B 2 γ 2 = . 6π 0 c3 m2e 6π 0 cm2e

(84)

The expression for σT given in Eq. (3.83) maybe expanded out into  σT =

8π 3



e4 e4 = , 16π 2 02 m2e c4 6π 02 m2e c4

(85)

and with this definition the expression for P may be rewritten as (in terms of σT ) P = σT c3 0 B 2 γ 2 .

(86)

From the given expression for uB , P may be rewritten yet once more as P = 2σT 0 μ0 uB γ 2 c3 = 2σT cuB γ 2 , where the relation between c, μ0 , and 0 given in Eq. (2.101) has been applied. 3.7 (a) From Eq. (2.164), the Lorentz factor γ of the electron is

(87)

Solutions

345

γ =

1 1 − (v 2 )(c2 )

1 1 = = 7.09. =√ 2 2 1 − 0.98 1 − ((0.99c) /c )

(88)

(b) From Eq. (2.170), the energy E of the electron is E = γ me c2 = (7.09)(9.11 × 10−31 kg)(3 × 108 m/s)2 = 5.81 × 10−13 J. (89) (c) From Eq. (3.71), the gyroradius r of the electron is r=

(5.81 × 10−13 J) E = = 242.17 m. qcB (1.6 × 10−19 C)(3 × 108 m/s)(5 × 10−5 T)

(90) (d) From Eq. (3.81) and recalling that uB = B 2 /2μ0 , the total power P radiated by the electron may be rewritten as P = 2σT cuB γ 2 =

σT cB 2 γ 2 , μ0

(91)

which may be computed as (6.65 × 10−29 m−2 )(3×108 m/s)(5×10−5 T)2 (7.09)2 = 2.00 × 10−21 W. (4π × 10−7 N/A) (92) (e) The opening angle θ of the forward lobe of emission from the relativistic electron is   1 1 57.2958◦ θ= = = 0.14 radians ≈ 8◦ . (93) γ 7.09 1 radians P=

(f) From Eq. (3.88), the half-life t1/2 of the synchrotron-emitting electron is 4.26 × 10−13 4.26 × 10−14 = 2.93 × 108 s. = B 2 E0 (5 × 10−5 T)2 (5.81 × 10−13 J) (94) (g) From Eq. (3.95), the emitting frequency νc is t1/2 [s] ≈

νc =

γ 2 qB (7.09)2 (1.6 × 10−19 C)(5 × 10−5 T) = 2.34 × 108 Hz. = 2(0.3)π me 2(0.3)π(9.11 × 10−31 kg) (95)

3.8 From Eq. (3.117), φ may be computed as α=

φ−1 −→ φ = 2α + 1 = 2(0.77) + 1 = 2.54. 2

(96)

346

Solutions

From Eq. (3.118), Π may be computed as (first using the computed value for φ and then using the given value for α) Π=

0.7 + 1 2.54 + 1 = = 0.73. 2.54 + 7/3 0.7 + 5/3

(97)

The same value for Π is obtained whether φ or α is used in the computations. Thus, the maximum degree of linear polarization at a single frequency in the radio spectrum of Cassiopeia A is 73%. 3.9 Noting that after inserting appropriate values for me , q, and B, Eq. (3.95) may be rewritten in terms of νc as 

 γ =

2(0.3)π me νc = qB

2(0.3)π(9.11 × 10−31 kg)νc 1/2 = 0.33 s1/2 νc . (1.6 × 10−19 C)(10−10 T)

(98)

Defining the three Lorentz factors of interest as γ408 MHz , γ1 GHz , and γ5 GHz as the values of γ for which the critical frequency of emission is νc = 408 MHz, νc = 1 GHz, and νc = 5 GHz, respectively, these quantities may be computed using Eq. (98) as follows: 408 × 106 Hz = 6667, γ1 GHz = 0.33 s1/2 109 Hz = 10,435, and γ5 GHz = 0.33 s1/2 5 × 109 Hz = 23,335.

γ408 MHz = 0.33 s1/2

3.10 (a) Recalling that the rest mass m0 of an electron is 5.11 × 105 eV/c2 , the Lorentz factor of the electron is (from Eq. (2.170)) γ =

3 × 1015 eV E = = 5.87 × 109 . 2 m0 c (5.11 × 105 eV/c2 )(c2 )

(99)

From Eq. (2.172), the momentum pelectron of each relativistic electron is 3 × 1015 eV)(1.6 × 10−19 J/eV) E = = 1.6 × 10−12 kg · m/s, c 3 × 108 m/s (100) (b) From classical physics, the momentum psnowball is pelectron =

psnowball = mv = (0.02 kg)(10 m/s) = 0.2 kg · m/s.

(101)

Therefore, the number nelectrons of 3 × 1015 eV electrons needed to have the same aggregate momentum as the momentum of the snowball is

Solutions

347

nelectrons =

psnowball 0.2 kg · m/s = 1.25 × 1011 . = pelectron 1.6 × 10−12 kg · m/s

(102)

3.11 From Eq. (2.170), γ may be calculated as follows: γ =

5 × 109 eV E = 9785. = mc2 (5.11 × 105 keV/c2 )(c2 )

(103)

Recalling that 1 T = 104 G, the given magnetic field strength may be re-expressed as B⊥ = 3 × 10−10 T. Thus, from Eq. (3.88), t1/2 is 6π(8.85 × 10−12 F/m)(3 × 108 m/s)3 (9.11 × 10−31 kg)3 = 5.9 × 1015 s, (1.6 × 10−19 C)4 (3 × 10−10 T)2 (9785) (104) or in terms of years, t1/2 =



t1/2

1 yr = 5.9 × 10 s 31,536,000 s



15

= 1.87 × 108 yr.

(105)

Notice that while the electron is extremely relativistic, because the magnetic field is so weak, the lifetime of the electron is quite lengthy. 3.12 The amount of energy emitted in the interaction described in Example 3.2 can be envisioned as a luminosity L corresponding to an energy emitted per second. Therefore, at the given distance to the HII region, the interaction described in the Example would have a corresponding flux density S of (from Eq. (3.28)) L 5.94 × 10−21 W = = 4.96 × 10−61 J/m2 . 4π r 2 4π(1000 pc × (3.086 × 1016 m/pc))2 (106) Therefore, to determine the number of interactions N that would be needed to produce the observed amount of emission would be the ratio of the detected flux density from the entire HII region (that is, S = 5 Jy = 5 × 10−26 J/m2 ) to the flux density to be expected from one transition, that is, S=

N=

5 × 10−26 J/m2 4.96 × 10−61 J/m2

≈ 1035 .

(107)

This result emphasizes the large number of interactions required to produce the study flux of radio emission that is detected from HII regions. 3.13 (a) By setting the thermal energy per electron equal to the kinetic energy per electron, the velocity v of the particle may be calculated as

348

Solutions

  mv 2 3kT 3kT 3(1.38 × 10−23 J/K)(106 K) = 4.77 × 106 m/s. = → v= = 2 2 2m 2(9.11 × 10−31 kg) (108) (b) From Eq. (3.132), νcritical may be calculated as me v 2 (9.11 × 10−31 kg)(4.77 × 106 m/s)2 = 1.56 × 1016 Hz. = 2h 2(6.63 × 10−34 J · s) (109) Note that this is the frequency of the radiation emitted by the electron if ALL of the kinetic energy of the electron is converted into electromagnetic radiation. Certainly it is much more typical for the electron to still retain some kinetic energy after the scattering event. ν=

3.14 First, a coordinate system is defined such that its origin corresponds to the center of mass location (denoted as CM) of the two atoms. If the masses of the two nuclei are defined as m1 and m2 and the distances from the two masses to the origin (expressed as vectors) are r1 and r2 , respectively, then the location of the center of mass CM may be expressed as CM =

m1 r1 + m2 r2 = 0. m1 + m2

(110)

Based on this relation, r1 and r2 may be expressed as r1 =

−m2 r and m1 + m2

r2 =

m1 r, m1 + m2

(111)

where |r| = |r1 | + |r2 |. Based on the definition given in Equation X, the reduced mass μ of the system is μ=

m1 m2 , m1 + m2

(112)

and therefore r1 and r2 may be expressed as r1 =

−μ r and m1

r2 =

μ r, m2

(113)

respectively. Recalling the expression for the total moment of inertia I of a system of i particles given in Eq. (3.151), I in the case of a diatomic molecule may be expressed as  I = m1

−μ r m1



2

which can be rearranged as

+ m2

 2  μ μ μ μr 2 , r = + m1 m1 m2

(114)

Solutions

349

 I=

m2 m2 + m1 + m2 m1 + m2

 μr 2 = μr 2 .

(115)

3.15 (a) From Eq. (3.138) the reduced mass μ for the CO molecule is μ=

m12 C m16 O (12 × 1.67 × 10−27 kg) (16 × 1.67 × 10−27 kg) = m12 C + m16 O (12 × 1.67 × 10−27 kg) + (16 × 1.67 × 10−27 kg) (116)

= 1.15 × 10−26 kg. Therefore from Eq. (3.141), E is  E=

6.63 × 10−34 J · s 2π



1(1 + 1) 2 × 1.15 × 10−26 kg × (1.13 × 10−10 m)2 (117)

= 7.58 × 10−23 J. From Eq. (2.111), λ and ν can be computed as λ=

(6.63 × 10−34 J · s)(3 × 108 m s−1 ) = 2.62 × 10−3 m = 2.62 mm 7.58 × 10−23 J (118)

and ν=

3 × 108 m s−1 = 1.15 × 1011 Hz. 2.62 × 10−3 m

(119)

The actual observed frequency of this transition is ν = 1,152,672 MHz. (b) From Eq. (3.149), Tmin may be computed as  (6.63 × 10−34 J s)2 1 (1+1) =5.49 K. Tmin = (2π )2 2(1.38×10−23 J/K)(1.15×10−26 kg)(1.13×10−10 m)2 (120) 

3.16 (a) From Eq. (3.138) the reduced mass μ of cyanogen is μ=

m12 C m14 N (12 × 1.67 × 10−27 kg) (14 × 1.67 × 10−27 kg) = m12 C + m14 N (12 × 1.67 × 10−27 kg) + (14 × 1.67 × 10−27 kg) (121)

= 1.08 × 10−26 kg.

350

Solutions

(b) From Eq. (3.144), the rotational frequency for the J  = 1 to J = 0 transition may be computed as 

 (6.63×10−34 J · s)/(2π ) νrot = (1(1 + 1)−0(0+1))=116 GHz. 4π(1.08 × 10−26 kg)(116×10−12 m)2 (122) Likewise, the rotational frequency for the J  = 2 to J = 1 transition may be computed as  (6.63×10−34 J · s)/(2π ) (2(2+1)−1(1+1))=231 GHz, νrot = 4π(1.08×10−26 kg)(116×10−12 m)2 (123) and the rotational frequency for the J  = 3 to J = 2 transition may be computed as   (6.63 × 10−34 J · s)/(2π ) (3(3+1)−2(2+1))=347 GHz. νrot = 4π(1.08 × 10−26 kg)(116×10−12 m)2 (124) 

Problems for Chap. 4 4.1 Proof of Addition Theorem:   +∞ [f (x) + g(x)]e−i2π xs dx = −∞

+∞

f (x)e−i2π xs dx +

−∞



+∞ −∞

g(x)e−i2π xs dx

= F (s) + G(s) Proof of Shift Theorem:   +∞ f (x − a)e−i2π xs dx = −∞

=e

+∞ −∞

f (x − a)e−i2π as e−i2π(x−a)s d(x − a)

−i2π as

F (s)

Proof of Similarity Theorem: 

+∞

−∞

f (ax)e−i2π xs dx =

1 |a|



+∞

−∞

f (ax)e−i2π ax(s/a) d(ax)

1 = F (s/a) |a| Proof of Modulation Theorem:

Solutions



351

+∞

−∞

f (x) cos(ωx) e−i2π xs dx

1 = 2 1 = 2

 

+∞ −∞

f (x) e f (x) e

−i2π x(s−(ω/2π ))

+∞ −∞

1 dx + 2

iωx −i2π xs

e



+∞

−∞

1 dx + 2 ω 

ω  1

1

F s− + F s+ 2 2π 2 2π

=



f (x) e−iωx e−i2π xs dx +∞

−∞

f (x) e−i2π x(s+(ω/2π )) dx

Proof of Derivative Theorem df (t) d = dt dt



+∞ −∞



= −2π i

F (ν)e−2π iνt dν =

+∞

−∞





+∞

−∞

F (ν)

d −2π iνt e dt

 dν

νF (ν)e−2π iνt dν = −2π iνF (ν).

4.2 (a) Note that the given function may be simplified such that f (t) = cos ωt for |t| ≤ 1/2 and f (t) = 0 for all other values of |t| > 1/2. Therefore, the Fourier Transform F (ν) of f (t) may be derived as follows: 

+∞

−∞

Π (t) cos (ωt) e−2π iνt dt

1 = 2 =

1 2

 

+∞ −∞ +∞ −∞

Π (t)e

iωt −2π iνt

e

1 dt + 2



+∞

−∞

Π (t)e−2π it (ν−(ω/2π )) dt +

1 2



Π (t)e−iωt e−2π iνt dt +∞ −∞

Π (t)e−2π it (ν+(ω/2π )) dt

1 1 = F (ν − (ω/2π )) + F (ν + (ω/2π )) 2 2 1 = (sinc(ν − (ω/2π )) + sinc(ν + (ω/2π ))). 2 Note here that the substitution (eiθ + e−iθ )/2 = cos θ was used in this derivation: this substitution comes from Euler’s Formula (see Eq. (2.94)). (b) The Fourier Transform of f (t) = Π (t) sin(2π t) is derived as follows: 

+∞

−∞

Π (t) sin (2π t) e−i2π iνt dt

352

Solutions

1 = 2i 1 = 2i



+∞

Π (t) e

−∞



+∞

−∞

 1 +∞ dt − Π (t) e−i2π νt ei2π t dt 2i −∞  1 +∞ dt − Π (t) e−i2π t (ν+1) dt 2i −∞

−i2π νt −i2π t

e

Π (t) e−i2π t (ν−1)

1 1 1 F (ν − 1) − F (ν + 1) = (sinc(ν − 1) − sinc(ν + 1)). 2i 2i 2i

=

Note here that the substitution (eiθ − e−iθ )/2 = i sin θ has been applied in this derivation, which again comes from Euler’s Formula. (c) The Fourier Transform of f (t) = Λ(t) sin(ωt) is derived as follows: 

+∞ −∞

=

Λ(t) sin (ωt) e−2π iνt dt

1 2i

1 = 2i =

 

+∞ −∞

Λ(t) e−i2π νt e−iωt dt −

+∞ −∞

Λ(t) e

−i2π t (ν−ω/2π )

1 2i



1 dt − 2i

+∞

Λ(t) e−i2π νt eiωt dt

−∞



+∞ −∞

Λ(t) e−i2π t (ν+ω/2π ) dt

1 1 1 F (ν−ω/2π )− F (ν+ω/2π )= (sinc2 (ν−ω/2π )−sinc2 (ν+ω/2π )). 2i 2i 2i

Note that the substitution (eiθ − e−iθ )/2 = i sin θ has been applied once more in this derivation. 4.3 From the convolution theorem, Eq. (4.36) becomes 

+∞ −∞

 |f (x)| dx = 2

−∞

 =

+∞

+∞

−∞

f (x)f ∗ (x) dx 

f (x)f ∗ (x)e−i2π xs dx (where s  = 0)

= F (s)  F ∗ (−s  ) =  =

+∞

−∞



+∞ −∞

F (s)F ∗ (s) ds =

F (s)F ∗ (s − s  ) ds



+∞

−∞

|F (s)|2 ds.

4.4 (a) Recalling the definition of a Jansky and that 1 J = 107 ergs and 1 W = 1 J s−1 , the units of a Jansky may be rewritten as

Solutions

353

1 Jy=10−26 W m−2 Hz−1 =10−26 J m−2 sec−1 Hz−1



107 ergs 1J



1m 100 cm

2

(125) or 1 Jy = 10−23 ergs sec−1 cm−2 Hz−1 .

(126)

(b) From Eq. (4.39), a power ratio of 2 may be expressed in terms of decibels as Q as Q = 10 log10 (2) ≈ 10(0.30) = 3 dB.

(127)

4.5 From Eq. (4.37), the angular resolution Θradians attained by the optical telescope is Θradians (FWHM) =

(1.22)(500 × 10−9 m) 1.22 λ = = 1.22 × 10−7 radians. d 5m

Therefore, the diameter d of the radio telescope would need to be d=

(1.22)(0.2 m) 1.22 λ = = 2 × 106 m = 2 × 103 km! Θradians (FWHM) 1.22 × 10−7 radians

4.6 (a) From Eq. (3.28), the flux density Sν is 5 × 104 W L = =5.53 × 10−11 W m−2 Hz−1 . 4π r 2 Δν 4π(60 × 103 m)2 (20 × 103 Hz) (128) (b) Recalling that 1 Jansky = 10−26 W m−2 Hz−1 , Sν expressed in Janskys is Sν =s

Sν = 5.53 × 10−11 W m−2 Hz−1



1 Jy

 = 5.53 × 1015 Jy.

10−26 W m−2 Hz−1

(129) Notice here just how more flux is received from a typical broadcast radio signal when compared to the flux received from even the brightest astronomical radio sources. This result illustrates the severity of the impact of artificial noise on astronomical observations at radio wavelengths. 4.7 (a) From the given information about the radius of the raindrop, its volume V is 4 4 V = π r 3= π 3 3



d3 8



 =

4 π 3



(10 mm)3 8



1m 103 mm

3

=5.24 × 10−7 m3 , (130)

354

Solutions

and thus its mass m is  6    10 cm3 1 kg 1g (5.24×10−7 m3 )=5.24×10−4 kg. m=ρ V = 1000 g cm3 1 m3 (131) Recalling that the potential energy of a particle with mass m raised to a height h above the surface of the Earth is mgh, where g is the surface gravity of the Earth (see Eq. (2.47)) and corresponds to 9.8 m/s2 , the potential energy in Joules released by the falling raindrop is U = mgh = (5.24 × 10−4 kg)(9.8 m/s2 )(2000 m) = 10.27 J.

(132)

(b) To answer this question, the time t in question can be envisioned as the ratio U of the energy released by the raindrop to the power Prec collected per second by the telescope, that is t=

U . Prec

(133)

While U has been calculated in Part (a), Prec can be calculated from Eqs. (4.58) and (4.59), yielding Prec = η A S Δν,

(134)

where the Δν factor is included to account for the bandwidth used in making the observations. Therefore, the power Prec received by the antenna (that is, the energy received per second) is   π(25 m)2 (40×10−26 W m−2 Hz−1 )(500×106 Hz)=9.82×10−14 W. 4 (135) Thus, from the results of Part (a) and Eq. (133), t is

Prec =(1.0)

 t=

10.27 J 9.82 × 10−14 W





1 yr = (1.05 × 10 s) 31,536,000 s 14

 = 3.32 × 106 yr. (136)

4.8 (a) From Eq. (4.37), ΘFWHM is ΘFWHM (radians) =

1.22 × (3.6 × 10−2 m) 1.22 λ = = 8.78 × 10−4 d 50 m (137)

Solutions

355



360◦ × radians 2π rad



= 0.05◦ .

(b) From Eq. (4.58), Aeff is Aeff = ηA =

ηπ d 2 0.7π(50 m)2 = = 1374 m2 . 4 4

(138)

(c) From Eq. (4.62), Δ = Gmaximum is Δ = Gmaximum =

4π Aeff 4π (1374 m2 ) = = 1.33 × 107 . λ2 (3.6 × 10−2 m)2

(139)

(d) From Eq. (4.60), P is P = kB Tsys Δν = (1.38 × 10−23 J/K)(50 K)(80 × 106 Hz) = 5.52 × 10−14 W. (140) (e) From Eq. (4.61), TA may be computed as follows: TA =

Sν Aeff (200 × 10−26 W m−2 Hz−1 ) (1374 m2 = 99.6 K. = 2kB 2(1.38 × 10−23 J/K)

(141)

(f) From Eq. (4.70). ΔTmin may be computed as follows: ΔTmin =

50 K (80 × 106 Hz)(1 s)(1)

= 5.59 × 10−3 K.

(142)

(g) From Eq. (4.71), ΔSmin may be computed as follows: 2(1.38×10−23 J/K)(50 K) =1.12×10−28 J/m2 =1.12 × 10−2 Jy. (1374 m2 ) (80×106 Hz)(1 s)(1) (143) (h) From Eq. (4.72), ΔTmin may be computed as follows: ΔSmin =

π(50 K) = 1.24 × 10−2 K. ΔTmin = 2(80 × 106 Hz)(1 s)(1)

(144)

From Eq. (4.73), ΔSmin may be computed as follows: 2π(1.38×10−23 J/K)(50 K) = 2.49×10−28 J/m2 =2.49×10−2 Jy. ΔSmin = (1374 m2 ) 2(80×106 Hz)(1 s)(1) (145)

356

Solutions

4.9 (a) From Eq. (4.73), ΔSmin may be calculated as 2π(1.38 × 10−23 J/K)(50 K) (146) (1374 m2 ) (2)(80 × 106 Hz)(1 s)(1)   1 Jy = 2.49 × 10−2 Jy. = 2.49 × 10−28 J/m2 10−26 J/m2

ΔSmin =

(b) The source cannot be detected with the antenna. To determine the proper value of Tsys , Eq. (4.73) can be rewritten as (assuming ΔSmin = 0.01 Jy = 10−28 J/m2 ) √ ΔSmin Aeff 2Δνts n (10−28 J/m2 )(1374 m2 ) 2(80×106 Hz)(1 s)(1) =20 K. = Tsys = 2π kB 2π(1.38×10−23 J/K) (147) 4.10 (a) Rewriting Eq. (4.38) as Θarcseconds (FWHM) =

(2.06 × 105 )c (2.06 × 105 )λ = , d νd

(148)

Θarcseconds (FWHM) may be computed as (2.06 × 105 )(3 × 108 m/s) = 14.72 arcseconds. (1.4 × 109 Hz)(3 × 103 m) (149) Therefore, the angular resolution of the VLA in this configuration and at this observing frequency is insufficient to resolve the structures of interest. (b) Similar to Part (a) Θarcseconds (FWHM) may be computed as Θarcseconds (FWHM) =

(2.06 × 105 )(3 × 108 m/s) (2.06 × 105 ) c = = 0.19 arcseconds. νd (1.4 × 109 Hz)(3 × 103 m) (150) Therefore, the angular resolution of the VLBA at this observing frequency is sufficient to resolve the structures of interest.

Θarcseconds =

4.11 (a) From Eq. (4.100), P may be computed as P =

(10)(10 − 1) = 45 pairs. 2

(151)

Solutions

357

(b) From Eq. (4.58), the effective area Aeff of an individual telescope within the interferometer may be computed as 

Aeff

π(20 m)2 = 0.6 4

 = 188.50 m2 .

(152)

(c) From Eq. (4.103), Θmaximum (FWHM) may be computed as  Θmaximum (FWHM) = 1.22  ×

0.06 m 1000 m



= 7.32 × 10−5 radians

206,265 arcseconds 1 radians

(153)

 = 15.10 arcseconds.

(d) From Eq. (4.104), ΘFOV (FWHM) may be computed as    0.06 m 360◦ −3 =3.66×10 radians =0.21◦ . ΘFOV (FWHM)=1.22 20 m 2π radians (154) (e) From Eq. (4.105), Θ may be computed as 

0.06 m 0.06 m ≈ 3.5 Jy over the above velocity range. Recalling that the rest wavelength of the 21-cm line is λ0 = 2.1 × 10−4 km and applying Eqs. (7.8) and (7.11), the HI mass of NGC 628 may be computed as MHI (M ) =

(2.36 × 105 M Mpc−2 Jy−1 s) (8.6 Mpc)2 (3.5 Jy)(140 km/s) 2.1 × 10−4 km (310)

= 4.07 × 1013 M .

380

Solutions

7.3 (a) For M81, MB may be calculated as MB = −10.2 log10 Vmax + 2.71 = −10.2 log10 (161 km/s) + 2.71 = −19.82, (311) and from Eq. (7.20), LB may be calculated as LB [W] = 10(71.20−MB )/2.5 = 10(71.20−(−19.82))/2.5 = 2.56×1036 W.

(312)

For M101, MB may be calculated as MB = −11.0 log10 Vmax + 3.31 = −11.0 log10 (172 km/s) + 3.31 = −21.28, (313) and from Eq. (7.20), LB may be calculated as LB [W] = 10(71.20−MB )/2.5 = 10(71.20−(−21.28))/2.5 = 9.82×1036 W.

(314)

(b) Rewriting Eq. (7.101) as d = 10(mB −MB +5)/5 ,

(315)

the distance to M81 may be computed as d = 10(7.96−(19.82)+5)/5 = 3.60 × 106 pc,

(316)

and similarly the distance to M101 may be computed as d = 10(7.96−(−21.28)+5)/5 = 7.05 × 106 pc

(317)

7.4 First, the distances that need to be determined are defined as rExtragalactic Cas A and rExtragalactic Crab , respectively, and the distances that are known are defined as rGalactic Cas A and rGalactic Crab , respectively. Recall Eq. (3.28) that states the relationship between the intrinsic luminosity L of the source, its distance r and its flux density S at that distance to be L = 4π r 2 S.

(318)

Because analogs of Cassiopeia A and the Crab Nebula are under consideration in this problem, the luminosities of these sources are treated as intrinsic properties of the sources themselves. Therefore, the relationship between the flux densities SExtragalactic Cas A and SGalactic Cas A for the extragalactic and Galactic analogs of Cassiopeia A, respectively, is

Solutions

381 2 L = 4π rExtragalactic Cas A SExtragalactic Cas A 2 = 4π rGalactic Cas A SGalactic Cas A .

(319)

Solving Eq. (319) for rExtragalactic Cas A yields 

SGalactic Cas A (320) rExtragalactic Cas A = rGalactic Cas A SExtragalactic Cas A  2720 × 10−26 Jy = 3.24 × 107 pc. = (3400 pc) 30 × 10−32 Jy In a similar vein, the relationship between the flux densities SExtragalactic Crab and SGalactic Crab for the extragalactic and Galactic analogs of the Crab Nebula, respectively, is 2 2 L=4π rExtragalactic Crab SExtragalactic Crab =4π rGalactic Cas A SGalactic Crab . (321) Solving Eq. (321) for rExtragalactic Crab yields



SGalactic Crab (322) rExtragalactic Crab = rGalactic Crab SExtragalactic Crab  1040 × 10−26 Jy = 1.18 × 107 pc. = (2000 pc) 30 × 10−32 Jy 7.5 From Eq. (7.33), LBZ may be computed as (recalling that 1 M = 2×1030 kg) LBZ ≈

16π(6.67 × 10−11 N m2 kg−2 )2 (9.7 × 109 × 2 × 1030 kg)2 (1 T)2 (4π × 10−7 N/A2 )(3 × 108 m/s)3 (323)

= 2.48 × 1042 W. 7.6 From Eq. (7.36), vt is 0.8c sin 60◦ v sin θ (0.87)(0.8)c = = 1.16c, = 1 − (v/c) cos θ 1 − (0.8c/c) cos 60◦ 1 − (0.8)(0.5) (324) or, in terms of m/s, vt =

vt = 1.16(3 × 108 m/s) = 3.48 × 108 m/s.

(325)

382

Solutions

7.7 (a) From Eq. (2.164), γ is 1 1 γ = = = 2.27 2 2 1 − v /c 1 − (0.9c)2 /c2

(326)

(b) Recalling that cos φ = v/c, φ may be expressed as cos φ =

0.9 c = 0.9 −→ φ = cos−1 0.9 = 25.8◦ . c

(327)

(c) From Eq. (7.36), vt is   (0.9c) sin 25.8◦ v sin φ 0.44 = = 2.08c, = 0.9c vt = 1 − (v/c) cos φ 1 − 0.9 cos 25.8◦ 0.19 (328) and in m/s, this is simply vobs = 2.08c = 2.08(3.0 × 108 m/s) = 6.24 × 108 m/s.

(329)

(d) The traveled distance in units of meters and parsecs by the blob is simply the product of velocity and time, that is, d = v t = 0.9 (3 × 108 m/s)(3.15 × 107 s)   1 pc = 0.28 pc. = 8.51 × 1015 m 3.086 × 1016 m

(330)

7.8 From Eq. (7.50), M˙ may be computed as 1038 W L = 1.11 × 1022 kg/s. M˙ = 2 = ηc (0.1)(3 × 108 m/s)2

(331)

Recalling that the mass M⊕ of the Earth is 5.97 × 1024 kg, this rate may be expressed as M˙ = 1.11 × 1022 kg/s



1 M⊕ 5.97 × 1024 kg



= 1.86 × 10−3 M⊕ /s.

(332)

This small mass (when compared to the typical masses of supermassive black holes) illustrates that only a comparatively small amount of infalling mass is required to power the observed luminosities of AGNs. 7.9 Recall that c = 3 × 108 m/s, 1 M = 2 × 1030 kg and the number of seconds in a year may be calculated as

Solutions

383



60 s 1 min



60 m 1h



24 h 1 day



365 days 1 year

 =

3.15 × 107 s . year

(333)

Inserting these quantities into Eq. (7.51) yields M˙ =

L (3.15 × 107 s/yr)(1037 W) L37 ≈ 1.8 × 10−3 M yr−1 . 8 2 30 η η(3 × 10 m/s) (2 × 10 kg/M )

(334)

7.10 (a) At the derived distance to Fornax A of 24.93 Mpc, one arcsecond corresponds to 121 pc (as shown in Example Problem 7.9). Therefore, the given angular extent of the lobe (that is, 17 arcminutes) corresponds to a linear extent of 17 arcminutes × (60 arcseconds/arcminute) × (121 pc/arcsecond) = 1.23 × 105 pc. Therefore—assuming the lobe is spherical—Vlobe may be computed as Vlobe =

4π(d/2)3 4π(1.23 × 105 pc/2)3 = = 9.84 × 1014 pc3 3 3 3  3.086 × 1016 m × = 2.89 × 1064 m3 . 1 pc

(335)

(b) To determine the luminosity L of the lobe, the flux density Sν of the emission detected from the lobe as a function of observing frequency ν may be expressed as  Sν = 74 Jy

ν 1.4 × 109 Hz

−0.5 (336)

.

Note that this expression is normalized to the measured flux density at 1.4 GHz. To obtain L, Eq. (3.28) must be applied and Sν must be integrated over the given frequency bounds. This yields  L = 4π d S = 4π d 2

 = 4π d 2

2

ν=1011 Hz ν=107 Hz



ν=1011 Hz

74 Jy ν=107 Hz

(337)

Sν dν

ν 1.4 × 109 Hz

−0.5 dν.

Evaluating the integral in this equation yields  S=

ν=1011 Hz ν=107 Hz



ν 74 Jy 1.4 × 109 Hz

−0.5 dν

(338)

384

Solutions

=

(74 × 10−26 W m−2 Hz−1 ) ν 0.5 (0.5)(1.4 × 109 Hz)−0.5

ν=1011 Hz ν=107 Hz

= 1.73 × 10−14 W/m2 ,

and therefore the luminosity L of the lobe is 2

   3.086 × 1022 m 1.73 × 10−14 W/m2 L = 4π d 2 S = 4π 24.93 Mpc 1 Mpc (339) = 1.29 × 1035 W. (c) From Eq. (7.57), Blobe may be computed as  Blobe =

(4π × 10−7 N/A2 )(1052 J) = 6.59 × 10−10 T. 2.89 × 1064 m3

(340)

(d) From Eq. (7.58), tlobe may be computed as tlobe =

  1052 J 1 yr 16 = 7.75 × 10 = 2.46 × 109 yr. s 1.29 × 1035 W 3.15 × 107 s (341)

7.11 Recall that Ei and Ef have been defined as the initial and final energies, respectively, of the electron. Again, Ei = me c2 (simply the rest mass of the electron) while Ef = me c2 + E = Ei + E, where E is the energy gained by the electron after the collision. In a similar vein, i and f are the energies of the photon before and after the collision, respectively. Finally, pi and pf are the momenta of the photon before and after the collision, respectively, while P is the momentum of the electron after the collision. Notice that the momentum of the electron before the collision is zero because the electron is assumed to be at rest. From conservation of momentum, the relationship between pi , pf , and P is pi − pf = P

(342)

from pi = P + pf , which describes the total momentum of the system before and after the collision. Squaring both sides of Eq. (342) yields pi2 − 2pi · pf + pf2 = P 2 ,

(343)

and multiplying through by c2 gives c2 pi2 − 2c2 pi · pf + c2 pf2 = c2 P 2 .

(344)

From Eq. (2.112), the corresponding energies i and f of the photons before and after the collision are cpi and cpf , respectively. Also, from Eq. (2.172), E may be

Solutions

385

expressed in terms of P as E 2 = P 2 c2 + m2e c4 ,

(345)

and therefore Eq. (344) can be rewritten as i2 − 2i f cos θ + f2 = E 2 − m2e c4 .

(346)

Now through conservation of energy, the relationship between i , f , Ei = me c2 and E may be expressed as i + Ei = f + E,

(347)

i − f = E − m e c 2 .

(348)

or alternatively as

Squaring both sides of Eq. (348) produces i2 − 2i f + f2 = E 2 − 2Eme c2 + m2e c4 .

(349)

Subtracting Eq. (349) from Eq. (346) yields 2i f (1 − cos θ ) = 2Eme c2 − 2m2e c4 = 2me c2 (E − me c2 ) = 2me c2 (i − f ), (350) and dividing Eq. (350) through by 2 i f produces 1 − cos θ =

me c 2 (i − f ) = me c2 i f



1 1 − f i

 .

(351)

Recalling the definitions of λi and λf as the wavelengths of the photon before and after the scattering event, respectively, along with the relationship between the energy of a photon and its wavelength as given in Eq. (2.111) and (351) may be rewritten as   λi me c 2 me c 2 λf − = (λf − λi ) = Δλ. (352) 1 − cos θ = me c hc hc hc h This last equation can be written at last as Δλ = which is equivalent to Eq. (7.73).

h (1 − cos θ ), me c

(353)

386

Solutions

7.12 (a) From Eq. (3.15), the wavelength λmax at which the CMB produces the most flux is λmax =

2.90 × 10−3 m · K = 1.06 × 10−3 m. 2.73 K

(354)

(b) From Eq. (3.25), urad may be calculated as 4(5.67 × 10−8 W m−2 K−4 )(2.73 K)4 (355) (3 × 108 m/s)   1 eV −14 3 = 4.20 × 10 = 2.49 × 105 eV/m3 . J/m 1.69 × 10−19 J

urad =

(c) From Eq. (2.111), the energy Ei of this photon is hc (4.14 × 10−15 eV · s)(3 × 108 m/s) = 1.17 × 10−3 eV. = λmax 1.06 × 10−3 m (356) (d) Using the energy density urad computed in Part (b) and the energy Ei per CMB photon computed in Part (c), the number of CMB photons per cubic meter Ei =

n=

urad 2.48 × 105 eV/m3 = 2.12 × 108 photons/m3 . = Ei 1.17 × 10−3 eV

(357)

(e) From Eq. (2.170) and recalling that the rest mass of the electron may be expressed as 5.11 × 105 eV/c2 , the Lorentz factor γ of the cosmic-ray electron is γ =

E 109 eV = = 1957. m0 c 2 (5.11 × 105 eV/c2 )c2

(358)

(f) From Eq. (7.83), the final energy Ef of the up-scattered CMB photon may be expressed as Ef = hνf =

4(1957)2 (1.17 × 10−3 eV) = 5975 eV. 3

(359)

(g) The final energy of the down-scattered electron after the scattering event may be computed using Eq. (7.76), yielding EIC = 109 eV − 5975 eV ≈ 109 eV.

(360)

Note that the final energy of the scattered electron is essentially the same as the initial energy of the electron. From Eq. (7.77), PIC may be calculated as

Solutions

387

PIC = =

dEIC dt

(361)

4(6.65 × 10−29 m2 )(3 × 108 m/s)(12 )(19572 )(2.49 × 105 eV/m3 ) 3

= 2.54 × 10−8 eV/s. For the sake of simplicity, the approximation β ≈ 1 has been made in this calculation. (h) From Eq. (7.79), the lifetime t of the inverse Compton-scattered electron is   1 yr 109 eV 16 = 3.94 × 10 s = 1.25 × 109 yr. t= 2.54 × 10−8 eV/s 3.15 × 107 s (362) 7.13 From Eq. (2.140), the sum v of the two orbital velocities under consideration may be expressed in terms of redshift as z=

v . c

(363)

Assuming the case where the two orbital velocities interfere constructively, the corresponding velocity v is simply 30 km/s + 230 km/s = 260 km/s. Therefore, z may be calculated as z=

260 km/s = 9 × 10−4 . 3 × 105 km/s

(364)

Finally, from Eq. (7.102), Δ T may be calculated as ΔT = z × T = (9 × 10−4 )(2.73 K) = 2.46 × 10−3 K.

(365)

Note that by symmetry, considering a velocity v = −260 km/s, the scale size of ΔT = −2.46 × 10−3 K: while the redshift would manifest itself as a positive temperature variation, a blueshift would manifest itself as a negative temperature variation. 7.14 From Eq. (2.170), the Lorentz factor of a proton with an energy corresponding to the GZK limit is γ =

E 8J = = 5.32 × 1010 . 2 −27 m0 c (1.67 × 10 kg)(3 × 108 m/s)2

7.15 (a) Equation (7.99) may be rewritten as

(366)

388

Solutions

=

(ΔT /T )me c2 . 2kB Te σT ne

(367)

Inserting values into this equation yields = =

(10−5 )(9.11 × 1031 kg)(3 × 108 m/s)2 2(1.38 × 10−23 J/K)(108 K)(6.65 × 10−29 m2 )(1000 m3 )

(368)

4.47 × 1021 m = 1.45 × 105 pc. 3.086 × 1016 m/pc

(b) From Eq. (2.1), the relationship between , the angular extent θ , and the distance r to the cluster may be expressed as 1.45 × 105 pc  = = 108 pc = 100 Mpc. θ (5 arcmin)((1 rad)/(3437.748 arcmin)) (369) (c) From Eq. (7.1), H0 may be calculated as follows: r=

H0 =

7000 km s1 v = = 70 km s−1 Mpc−1 . d 100 Mpc

(370)

Index

A Absorption coefficient, 81 Accretion, 297 Acronyms, list of, xvii Active Galactic nuclei (AGNs), 65, 66, 286 prograde rotation, 293 radio-loud, 289 retrograde rotation, 293 Airy disk, 126 Airy disk equation, 126 Albedo, 169 Alfvén velocity, 260 Altitude-azimuth coordinate system, 23 Andromeda Galaxy (M31), 273 Angular frequency, 45 Angular resolution, 7, 122, 123 Angular velocity, 107 Antenna temperature, 133 Antiparticles, 290 Aperture efficiency, 132 Aperture synthesis, 146 Aphelion, 36 Apocenter, 258 Apogee, 13 Apparent magnitude, 317 Arc length, 17 Arcminute, 17 Arcsecond, 17 Astronomical unit (AU), 18 Atacama Large Millimeter Array (ALMA), 9 Atmospheric windows, 10, 128 Atomic number, 54 Australia Telescope Compact Array (ATCA), 9 Autumnal equinox, 24

Azimuthal angle, 21 Azimuthal quantum number, 194

B Balmer series, 58 Bandwidth, 129 Bandwidth smearing, 149 Beaming effect, 90 Bessel functions, 125, 161 Big Bang, 313 Bistatic radar, 169 Blackbody, 69 Blackbody radiation, 6 Blandford–Znajek mechanism, 290 Blazars, 66, 287 Blueshift, 53 Boltzmann’s constant, 69 Braking index, 251 Bremsstrahlung, 102 Brightness temperature, 84, 130

C Cassiopeia A, 27 Celestial equator, 23 Celestial sphere, 23 Center of mass, 39 Centripetal force, 33 Chandrasekhar limit, 228 Chromosphere, 172 Circular velocity, 33 Classical radius of the electron, 92 Complex conjugate, 117

© Springer International Publishing Switzerland 2020 T. G. Pannuti, The Physical Processes and Observing Techniques of Radio Astronomy, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-16982-8

389

390 Compton scattering, 303 Compton wavelength, 305 Conic section, 35 Conjugate Fourier pair, 118 Conservation of energy, 32 Conservative force, 32 Contact discontinuity, 232, 235 Convolution, 121 Correlator, 138 Cosmic microwave background, 314 Cosmic rays, 63 Cosmological redshift, 273 Coulomb Force, 55 Critical frequency, 96 Cyclotron, 85 Cyclotron frequency, 85

D Dark energy, 273 Decibel, 123 Declination, 26 Degeneracy, 194 Degenerate energy levels, 194 Degenerate matter, 229 Degrees, 17 Deuterium, 262 Dicke switch, 135 Differential Galactic rotation, 199 Diffuse shock acceleration, 293 Diffusive shock acceleration, 65, 230, 261 Direction cosines, 143 Directivity, 133 Dispersion measure, 256 Dominion Radio Astrophysical Observatory (DRAO), 9 Doppler boosting, 308 Doppler effect, 53 Dynamical local standard of rest, 199 Dynamo effect, 179

E Eccentricity, 35 Effective area, 132 Electromagnetic wave, 44 Ellipse, 35 Elliptical polarization, 49 Emission coefficient, 82 Emissivity, 100 Energy levels, 55 Epoch, 27 Equation of radiative transfer, 83 Equatorial coordinate system, 23, 26

Index Equivalent width, 288 Escape velocity, 33 Euler’s formula, 44 Excitation temperature, 110

F Fanaroff and Riley classification system, 288 Faraday effect, 52 Faraday rotation, 52, 254 First-order Fermi acceleration, 230 Flat spectrum radio-loud quasars (FSRQs), 287 Flux, 6, 73, 76 Flux density, 76 Focus, 35 Fourier analysis, 115 Fourier transforms, 115 Addition theorem, 119 Derivative theorem, 119 Modularity theorem, 119 Parseval’s theorem, 122 Power theorem, 122 Rayleigh theorem, 122 Shift theorem, 119 Similarity theorem, 119 Fractional circular polarization, 50 Fractional linear polarization, 50 Fraunhofer diffraction theory, 124 Free-fall time, 214 Frequency, 2, 43 Full-width at half maximum, 123

G Gain, 129, 133 Galactic Center, 192 Galactic coordinate system, 191 Galactic equator, 192 Galactic latitude, 192 Galactic longitude, 192 Galactic plane, 192 Galactic poles, 192 Galaxies elliptical, 269 irregular, 270 lenticular, 270 spiral, 269 Galaxy clusters, 309 Galaxy superclusters, 309 General relativity, 60 Geometric delay, 138 Giant Metrewave Radio Telescope (GMRT), 9 Gigahertz (GHz), 2 Glitches, 252

Index Gravitational force, 31 Great circle, 26 Greisen–Zatespin–Kuzmin (GZK) Limit, 315 Ground state, 56 Groups, 310 Gyroradius, 85 GZK horizon, 315

H Hadron, 307 Harmonics, 115 Hercules A (3C 348), 156 Hertz, 2 High-frequency peaked blazars (HBLs), 288 HII regions, 58, 218 Hour angle, 27 Hour circle, 26 Hubble’s constant, 272 Hubble’s Law, 271 Hyperfine splitting, 194 Hyperfine transition, 195

I Impact parameter, 102 Index of refraction, 43, 210 Intensity, 46, 79 Interferometers, 8, 138 Interferometry, 8, 138 Intermediate frequency, 129 Interstellar extinction, 15, 79 Interstellar medium, 79 Interstellar reddening, 79 Inverse Compton scattering, 303, 316 Inversion formulae, 117 Ionosphere, 10, 127 Isotropic emission, 74

J Jansky, 111, 129 Jets, 288 hadronic models, 307 leptonic models, 307 Johnson noise, 133 Julian Date, 28

K Kepler’s laws of planetary motion, 34 Kiloparsec, 19 Kinetic energy, 32 Kirchoff’s Law, 83

391 L Larmor Formula, 88 Length contraction, 60 Leptons, 307 Light, 2 L,m,n coordinate system, 143 Local Group, 273 Local oscillator, 129 Local sidereal time, 26 Lorentz factor, 60 Lorentz Force (FB ), 84 Low-frequency peaked blazars (LBLs), 288 Luminosity, 74 pulsar, 248

M Magnetic field energy density, 92 Magnetic permeability, 45 Magnetic quantum number, 194 Mass-to-light ratio, 280 Maxwell-Boltzmann equation, 196 Maxwell-Boltzmann velocity distribution of particles, 173 Maxwell’s equations, 46 Mean intensity, 79 Megahertz (MHz), 2 Megaparsec, 19 Meridian, 26 Metallic hydrogen, 179 Milky Way Galaxy, 4 Minimum detectable change in temperature, 135 Minimum detectable flux density, 135 Minor axis, 35 Missing zero-space flux, 155 Mixer, 129 Molecular clouds, 216 Moment of inertia, 107 Moment of rotational inertia, 247 Momentum of photon, 47 Monostatic radar, 169

N Neutron star, 244 characteristic age, 249 minimum magnetic field, 249 Noise temperature, 133 Normalized antenna reception pattern, 142 Normalized sinc function, 120 Nyquist sampling theorem, 133, 135

392 O Oort constants, 203 Opacity, 81 Optical depth, 80 Optically thick medium, 80 Optically thin medium, 80 Optically violent variable (OVV) quasars, 287 Orbital angular momentum, 106, 107

P Pair production, 290 Parallax, 18 Parallax angle, 18 Parsec, 18, 19 Peak wavelength, 72 Peculiar motions, 273 Pericenter, 258 Perigee, 13 Perihelion, 36 Permittivity, 46 Permittivity of free space, 46, 55 Phase reference position, 140 Phase tracking center, 140 Photon, 47 Photosphere, 172 Pitch angle, 90 Planck function, 71 Planck’s constant, 47 Plasma frequency, 210 Point spread function, 122 Polar angle, 21 Polarization circular, 49 linear, 49 Polarization of light, 48 Position angle, 28, 49 Positrons, 290 Potential energy, 32 Poynting vector, 45, 291 Precession, 27 Primary beam interferometer, 138 single dish, 124 Primary lobe, 131 Prime focus, 5 Principal quantum number, 55 Propagation coefficient, 169 Pulsars, 245 rotation-powered, 247

Q Quanta, 47, 70

Index Quantization of energy, 47 Quasars, 285

R Radar, 169 Radial velocity, 53 Radian, 17 Radiant energy, 69 Radiation energy density, 75 Radiation pattern, 130 Radiation pressure, 75 Radiative transfer, 79 Radio frequency amplifier, 129 Radio galaxy, 156 Radio halo, 310 Radio lobes, 298 Radio relic, 310 Rayleigh scattering, 78 Rayleigh–Jeans approximation, 70, 129 Reciprocity theorem, 132 Recombination, 219 Recombination epoch, 314 Rectangle function, 118, 120 Redshift, 53 Redshift parameter, 54 Resolution diffraction-limited, 126 seeing-limited, 127 Rest mass, 61 Right Ascension (RA), 26 Rigid-body rotation, 199 Rotation curve, 201 Rotation measure, 254 Rotation period, 245 Rotational angular velocity, 247 Rotational kinetic energy, 247 Rydberg constant, 57

S Sampling function, 146 Scale factor, 274 Seeing, 127 Semi-major axis, 35 Sensitivity, 7 Sexagesimal coordinate system, 26 Seyfert galaxies, 285 Sgr A∗ , 257 Sidereal day, 4 Sidereal time, 26 Sinc function, 120 Singularity, 313 Small-angle approximation, 17

Index Snell’s Law, 43 Solar circle, 199 Solar constant, 78 Solar corona, 173 Solar day, 4 Solid angle, 20 Source function, 82 Spatial frequencies, 148 Specific intensity, 79 Spectral energy distribution, 174 Speed of light, 43 Spherical coordinate system, 21 Spin electron and proton, 194 Spin quantum number, 194 Spin-down time, 245 Spin-flip transition, 195 Statistical weights, 197 Stefan–Boltzmann constant, 74 Stefan–Boltzmann law, 74, 75 Steradian, 20 Stokes parameters, 49 Summer solstice, 24 Sunspot, 173 Sunyaev–Zeldovich effect, 316 Superluminal motion, 294 Supernova, 228 double degenerate, 229 single degenerate, 229 Type Ia, 229 Type Ib, 229 Type Ic, 229 Type II, 229 Supernova remnants, 64, 229 adiabatic stage, 236 dissipative stage, 237 free expansion stage, 235 radiative stage, 236 Sedov stage, 236 snowplow stage, 237 Surface brightness, 167 Surface gravity, 33 Symbols, list of, xvii Synchrotron lifetime, 92 Synchrotron radiation, 7, 65 Synchrotron self-absorption, 98 Synchrotron self-Compton emission, 307 Synthesized beam, 146 System constant, 136, 151 System temperature, 135 Systemic velocity, 276

393 T Thermodynamic equilibrium, 83 Thomson cross section, 92 Time dilation, 60 Total angular momentum quantum number, 108 Total flux density, 129 Total system noise power, 133 Transfer equation, 83 Transfer function, 146 Transitions, 57 Transverse wave, 44 Triangle Function, 118 True anomaly, 35 Tully–Fisher relation, 279 Tuning-fork diagram, 269 Turnover frequency, 220

U Universal time, 27 U,v,w coordinate system, 143

V Velocities relativistic, 59 Vernal equinox, 24 Very Large Array, 9 Very long baseline interferometry (VLBI), 12 Virial theorem, 212 Visibility, 141

W Wave equation, 45 Wavelength, 2, 43 Wavenumber, 45 Wave-particle duality, 48 Wien’s approximation, 130 Wien’s Law, 73 Wilkinson Microwave Anisotropy Probe (WMAP), 314 Winter solstice, 24

Z Zeeman effect, 208 Zeeman splitting, 208 Zenith, 25 Zenith angle, 25