Kinematics and Dynamics of Galactic Stellar Populations [1 ed.] 9781527514805, 9781527505629

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Kinematics and Dynamics of Galactic Stellar Populations

Kinematics and Dynamics of Galactic Stellar Populations By

Rafael Cubarsi

Kinematics and Dynamics of Galactic Stellar Populations By Rafael Cubarsi This book first published 2018 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright ©2018 by Rafael Cubarsi All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-0562-6 ISBN (13): 978-1-5275-0562-9

Contents Preface

ix

1

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Analytical dynamics 1.1 Introduction . . . . . . 1.2 Jeans’ problems . . . . 1.3 Isolating integrals . . . 1.4 Self-consistent models 1.5 Stellar statistics . . . . 1.6 Velocity moments . . .

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1 1 2 4 8 9 12

Stellar hydrodynamic equations 2.1 Comoving-frame equations . 2.2 Conservation of pressures . . 2.3 Conservation of moments . . 2.4 Closure example . . . . . . 2.5 Absolute reference frame . . 2.6 Remarks . . . . . . . . . . .

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15 15 17 21 24 27 28

Ellipsoidal distributions 3.1 Quadratic integral . . . . . . . . . . . . 3.2 Schwarzschild distribution . . . . . . . 3.3 Closure for Schwarzschild distributions 3.3.1 Even-order equations, n ≥ 2 . . 3.3.2 Odd-order equations, n ≥ 3 . . . 3.4 Chandrasekhar’s approach . . . . . . . 3.5 Generalised Schwarzschild distribution . 3.6 Remarks . . . . . . . . . . . . . . . . .

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31 31 33 35 36 37 38 40 42

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The closure problem 4.1 The Boltzmann and moment equations 4.2 Maximum entropy function . . . . . . 4.3 Fundamental system of equations . . . 4.4 Closure of moment equations . . . . . 4.4.1 Notation . . . . . . . . . . . 4.5 Arbitrary polynomial function . . . . 4.5.1 Moment equations . . . . . . 4.5.2 Equivalence . . . . . . . . . . 4.6 Remarks . . . . . . . . . . . . . . . .

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43 43 44 46 48 50 50 51 52 53

Maximum entropy approach 5.1 The problem of moments . . . 5.2 Maximum entropy distribution 5.2.1 Boundary conditions . 5.2.2 Properties . . . . . . . 5.2.3 Information entropy . 5.3 Moments problem . . . . . . . 5.4 Gramian system . . . . . . . . 5.4.1 Polynomial coefficients 5.5 Remarks . . . . . . . . . . . .

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57 57 58 60 62 63 66 69 72 75

Local velocity distribution 6.1 Stellar samples . . . . . . 6.2 Large-scale structure . . . 6.3 Truncated distributions . . 6.4 Small-scale structure . . . 6.5 Orbital eccentricity . . . . 6.6 Radial velocity distribution 6.7 Remarks . . . . . . . . . .

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77 77 79 81 85 92 93 97

Superposition of stellar populations 7.1 Mixture approach . . . . . . . . . . . . . . 7.2 Two-component mixture . . . . . . . . . . 7.3 Moment constraints . . . . . . . . . . . . . 7.4 Local velocity ellipsoids . . . . . . . . . . 7.5 Second moments of a n-population mixture 7.6 Remarks . . . . . . . . . . . . . . . . . . .

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101 101 102 104 109 112 116

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vi

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Axisymmetric system 8.1 Model hypotheses . . . . . . . . . . . 8.2 Dynamical model . . . . . . . . . . . 8.3 Chandrasekhar’s axial system . . . . . 8.4 Conditions of consistency for mixtures 8.4.1 Axisymmetric general case . . 8.4.2 Flat velocity distribution . . . 8.5 The solar neighbourhood . . . . . . . 8.5.1 Thin disc . . . . . . . . . . . 8.5.2 Thick disc . . . . . . . . . . . 8.5.3 Halo . . . . . . . . . . . . . . 8.6 Local values of the potential . . . . . 8.6.1 Separable cylindrical potential 8.6.2 Spherical potential . . . . . . 8.6.3 General case . . . . . . . . . 8.7 Remarks . . . . . . . . . . . . . . . .

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119 119 121 123 127 128 130 140 140 143 144 145 145 147 148 149

Point-axial symmetric system 9.1 Point-axial symmetry . . . . . . 9.2 Single point-axial system . . . . 9.3 The potential is axisymmetric . . 9.4 The potential is spherical . . . . 9.4.1 Separable potential . . . 9.4.2 Non-separable potential 9.5 Conditions of consistency . . . . 9.6 Remarks . . . . . . . . . . . . .

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153 153 154 156 157 159 160 161 162

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Appendices

167

Appendix A Chandrasekhar equations A.1 Equation of order n = 3 . . . . . A.2 Property . . . . . . . . . . . . . A.3 Equation of order n = 2 . . . . A.4 Property . . . . . . . . . . . . . A.5 Equation of order n = 1 . . . . A.6 Equation of order n = 0 . . . .

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169 169 170 171 172 173 173

Appendix B Power series

175

Appendix C Moment recurrence

179 vii

Appendix D Parameter estimation

181

Appendix E K-statistics

187

Appendix F Mixture equations 189 F.1 U-cumulants . . . . . . . . . . . . . . . . . . . . . . . . . 189 F.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Appendix G Axisymmetric stellar system 191 G.1 Components of A2 and v . . . . . . . . . . . . . . . . . . . 191 G.2 Second central moments . . . . . . . . . . . . . . . . . . . 192 G.3 Moment gradients . . . . . . . . . . . . . . . . . . . . . . . 194 Appendix H Epicycle model

195

Appendix I

199

Point-axial symmetric system

Bibliography

201

Index

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viii

Preface Stellar dynamics is an interdisciplinary field where mathematics, physics, and astronomy overlap. It describes systems of stars considered as many point mass particles whose mutual gravitational interactions determine their orbits. Theses interactions may arise from the smoothed-out stellar distribution of matter, which are then given through a gravitational potential, and from the effect of the stellar encounters. The collisional relaxation time is used to measure how long will it take before the cumulative effect of stellar encounters prevents us from considering the stars as independent, conservative dynamical systems. In large stellar systems, like a galaxy, the relaxation time is long and they may be assumed to be in statistical equilibrium according to specific phase space density and potential functions. This approach is generally done from the analytical dynamics viewpoint. That is, the stellar system is described as a conservative dynamical system from the canonical equations through a Hamiltonian function, and the hydrodynamical flow in the phase space is obtained according to Liouville’s theorem. In this monograph we shall focus our attention on analytical stellar dynamics, by considering the stellar system as a fluid. In contrast, in small globular clusters, and in processes of violent relaxation producing rapid fluctuations of the gravitational field, the collisions cannot be omitted, and statistical mechanics is generally used to describe the dynamics of the interacting particles through a many particle distribution function. Analytical stellar dynamics has its origins in the early 20th century, when the kinetic theory of gases was adapted to astronomical problems by J.H. Jeans. He showed that, under some regularity conditions, the fundamental equation of stellar dynamics is equivalent to the collisionless Boltzmann equation, so that the Liouville theorem is satisfied. Then, velocity moments of the collisionless Boltzmann equation yield the stellar hydrodynamic equations. These equations, written in a comoving reference frame, are comparable to the equations of motion of a compressible viscous fluid. ix

However, because short-range atomic interactions dominate fluids, it is a much better approximation to truncate the moment equations at low order for fluids (i.e. continuity and Navier-Stokes equations) than it is for stellar systems. In addition, the general stellar hydrodynamical equations are anisotropic in their spatial and velocity coordinates. Therefore, higher-order hydrodynamic equations are non-negligible for stellar dynamics. In the forties, S. Chandrasekhar gave an alternative formulation to explain the dynamics of collisionless systems and, in addition, he introduced a new statistical approach for collisional systems through a dynamical friction mechanism. Chandrasekhar’s alternative approach for collisionless systems is known as Jeans’ inverse problem and it is a functional approach for the phase space density function based in the assumption that the residual velocity distribution of any stellar population in statistical equilibrium satisfies a generalised ellipsoidal law. During the second half of last century, J.J. de Orús produced a rigorous mathematical formulation of Chandrasekhar’s theory which was collected in his Notes on Galactic Dynamics for the Astronomy Department in the University of Barcelona. He and his disciples thoroughly studied the direct and inverse Jeans’ problems. It was proven that, if the Chandrasekhar equations are fulfilled, the continuity equation and the Navier-Stokes equation are also satisfied and, even more, that the Chandrasekhar equations could be derived from the first four hydrodynamic equations. Solutions to the Chandrasekhar equations were given under hypotheses of axial (rotational) and point-axial symmetry, and for stellar population mixtures. The aim of the current monograph is to review, update, and make these topics available to a broader audience. It is a fascinating area that addresses issues on dynamical systems, information theory, numerical analysis, partial differential equations, probability, statistics, tensor algebra, and vector calculus, among other topics, in addition to astronomy and physics subjects. These nine chapters altogether provide the reader with a quite complete review of what are the main problems in this area at a level of postgraduate course. The first two chapters are devoted to the Jeans’ direct problem, where the full mathematical expression of an arbitrary n-order stellar hydrodynamic equation, either depending on the pressures or on the comoving moments, is derived. In this way, the stellar hydrodynamic equations can be compared to the equations of fluid dynamics, and general closure conditions can be studied in order to build up a dynamical model from a finite number of equations and variables, generally known as closure problem. The third and fourth chapters deal with the Jeans’ inverse problem in relation to the long-standing closure problem, which is one of the classic, x

unsolved problems in fluid dynamics, discovered even earlier, in the nineteenth century, in ordinary hydrodynamics by O. Reynolds. The equivalence of the Chandrasekhar equations and the stellar hydrodynamic equations is discussed by proving that, for a generalised ellipsoidal velocity distribution, some moment recurrence relationships act as closure conditions making the infinite hierarchy of the hydrodynamic equations equivalent to the collisionless Boltzmann equation. This result is generalised to maximum entropy velocity distributions and to any velocity distribution function depending on a polynomial function in the velocity variables. Chapters five and six focus on the distribution function and the moments problem. The maximum entropy approach for the solution of inverse problems, first introduced by E.T. Jaynes, illustrates how the velocity distribution function is connected to the eventual asymmetries collected through its population moments. The density function maximising Shannon’s information entropy provides the simplest and smoothest approach to the distribution function that fulfils a provided set of moment constraints, and gives a very good estimation of the density function and of its velocity derivatives involved in the collisionless Boltzmann equation. In particular, if an extended set of moments is available, the parameter estimation of the distribution function may be simply done by solving a linear system of equations. Several numerical applications of this functional approach, either to complete or truncated distributions, are presented to show how the above mathematical methods are able to describe the main kinematical features of the neighbourhood stars. As an alternative approach, in the seventh chapter, instead of using a higher-degree polynomial along with a maximum entropy function, the mixture model of Schwarzschild (Gaussian) density functions is studied in connection with the moments problem. This approach is useful to describe large-scale kinematic structures of the Galactic disc associated with kinematic stellar populations, that has been a very active research field during the last decades. Finally, in the last two chapters, Chandrasekhar’s dynamical models for axisymmetric and point-axial symmetric systems are studied, with a particular application to the superposition problem, which is the appropriate approach to the actual case of a Galaxy composed of several stellar populations. Under a common potential, a finite mixture of ellipsoidal velocity distributions satisfying the collisionless Boltzmann equation provides a set of integrability conditions that may constrain the population kinematics. These conditions determine which potentials are connected with a more flexible superposition of stellar populations. xi

The author wishes to record his gratitude to the late Prof. Juan J. de Orús. His earliest Notes on Galactic Dynamics and his always stimulating comments were, some years ago, the origin of the current notes.

xii

Chapter 1

Analytical dynamics 1.1 Introduction The aim of the first two chapters is to provide the complete mathematical expression for an arbitrary n-order hydrodynamic equation depending on the pressures, or alternatively on the comoving moments, without any additional hypotheses. The stellar hydrodynamic equations have been used in a number of works on galactic dynamics to study the stellar mass and velocity distributions, either from an analytical viewpoint (e.g., Vandervoort 1975, Hunter 1979, Evans & Lynden-Bell 1989, Evans et al. 2000, van de Ven et al. 2003, Evans et al. 2015, An & Evans 2016) or as a model for numerical simulations to investigate the shape of the velocity distribution, or to reproduce the spiral structure of galactic discs as an alternative way to the N-body approach (e.g., Korchagin et al. 2000, Orlova et al. 2002, Vorobyov & Theis 2006). However, only equations of mass, momentum and, in few cases, energy transfer are generally handled, and, in most cases, axial symmetry, steady-state stellar system, and other hypotheses are assumed. There are few works that, in a mathematical aspect, have gone beyond such a basic assumptions. Sala et al. (1985) proposed a general expression for the n-order equation, without steadiness and axisymmetry, although it was written depending on the absolute, non-comoving moments of the stellar velocity distribution, where, by substitution of the moments as a series of the pressures, they obtained a general but non-explicit expression of the equations. The explicit equations were, in the end, specifically written for 1

2

CHAPTER 1. ANALYTICAL DYNAMICS

orders n = 0, 1, 2, 3. However, it is well-known that stellar hydrodynamic equations are physically meaningful when they can be compared with the ordinary hydrodynamic equations of a compressible, viscous fluid, and this is only possible when they are written in terms of the tensors of comoving moments or of pressures, in the reference frame associated with the local centroid. Often, these expansions or computational procedures are provided instead of their explicit expression, and they are later used to simplify and to close the system of equations, for example to study a cool, pure rotating disc (Aoki 1985, Amendt & Cuddeford 1991). On the other hand, the work by Cuddeford & Amendt (1991) had also a general and more interesting mathematical scope, although it was restricted to steady-state systems, amid other hypotheses. They studied higher-order stellar hydrodynamic equations, by using central velocity moments up to eighth-order, and they investigated some quite general conditions over the velocity distribution in order to close the infinite hierarchy of the moment equations. The general expression for such an arbitrary order hydrodynamic equation in the comoving frame was first derived by Cubarsi (2007, 2013). It should be taken as a starting point in forthcoming works either to use improved observational data or to carry out more exhaustive numerical simulations. In addition, the exact n-order equation is also essential to study more general closure conditions or, under unrestrictive assumptions, for building up more accurate dynamical models from a finite number of equations and variables. Actual kinematic data (ESA 1997, Nordtröm et al. 2004) do not support any more the hypotheses of axisymmetry, steadiness, or pure galactic rotation (Cubarsi & Alcobé 2006). In addition, newer missions such as the RAdial Velocity Experiment (RAVE) survey (Siebert et al. 2011, Zwitter et al. 2008, Steinmetz et al. 2006) represent a major improvement, since the three velocity components are available for the largest number of stars ever collected, where an unbiased radial velocity component will provide essential information to kinematic and dynamic studies of the Galaxy.

1.2 Jeans’ problems From a macroscopic approach, a stellar system is described by giving its distribution in the phase space, which consists in couples of three-dimensional vectors r and V representing star position and velocity, measured from an inertial reference system. The stellar distribution is then given through the

1.2. JEANS’ PROBLEMS

3

phase space density function f (t, r, V ), which is assumed as continuous and differentiable in nearly every point, providing, at time t, the number of stars with position within the range r and r + dr, and velocity between V and V + dV. It is generally assumed that the Galaxy is at present in a state in which each star can be idealised as a conservative dynamical system to a very high degree of accuracy. In general, the forces acting in the system can be associated with a gravitational potential function per unit mass U(t, r), possibly non-stationary, so that the motion of a star is described in a Cartesian coordinates system by the Hamiltonian system of equations r˙ = V ,

V˙ = −U(t, r).

(1.1)

For the whole stellar system, the collisionless Boltzmann equation is satisfied, so that the phase space density function f (t, r, V ), with (t, r, V ) ∈ 1 R × Γr × ΓV , by using the Stokes operator D(·) Dt , fulfils Df ∂f ≡ + V · ∇r f − ∇r U · ∇V f = 0. Dt ∂t

(1.2)

The above equation is sometimes referred as Vlasov equation, Liouville equation, Boltzmann equation, or Jeans equation; however, Hénon (1982) clarifies the appropriate terminology. The collisionless Boltzmann equation is a consequence of the Hamiltonian flow, which preserves volume, i.e. satisfies the Liouville theorem: the density of any element of phase space remains constant during its motion. Jeans showed that the fundamental equation of stellar dynamics was a particular case of the Boltzmann equation from the kinetic theory of gases, ∂f + r˙ · ∇r f + V˙ · ∇V f = C( f, f ), ∂t

(1.3)

where the collision term of the right-hand side may be assumed to be null in two cases. First, if the effect of the irregular forces, such as star encounters, is negligible. Second, if the phase density is invariant with respect to the irregular forces, that is, when the number of points leaving any space volume as a result of encounters is balanced by those which enter the volume for the same reason. In both cases, the Liouville theorem is satisfied. Hilbert (1912) gave an equivalent mathematical condition to neglect the collision term C( f, f ), when it is orthogonal to 1, V , and |V |. 1 The Stokes operator D(·) is generally used to simplify the notation of the Lagrangian Dt ˙ derivative ∂(·) ∂t + V · ∇r (·) + V · ∇V (·).

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The collisional relaxation time is long in large stellar systems. The time of relaxation for stellar encounters in the solar neighbourhood is greater than 1013 years (Binney & Tremaine 2008), while the galactic rotation period is about 108 years. Hence, the encounters are entirely unimportant. The collisions cannot be omitted in a globular cluster which contain 105 stars, but for a galaxy of 1011 stars, the relaxation time turns out to be much larger than the age of the universe, and the encounters can be neglected. The collisionless Boltzmann equation may be regarded from two different viewpoints. It is either a linear and homogeneous partial differential equation for f , for a given potential U, which is known as Jeans’ direct problem, or a linear non-homogeneous partial differential equation for U, where the density function f is already known, which is called Jeans’ inverse problem. Both approaches have been largely studied since Eddington (1921) and Oort (1928), and among other works, those of Vandervoort (1979), de Zeeuw & Lynden-Bell (1985), Bienaymé (1999) and Famaey et al. (2002) may be pointed out. Obviously, neither the phase space density function nor the potential are observable quantities, while we do have enough large data sets of the full space motions in the solar neighbourhood for different types of stars to compute the kinematic statistics of the distribution. Then, in order to isolate information about the spatial properties of the stellar system, the collisionless Boltzmann equation may be integrated over the velocity space, or in a more general way, it may be multiplied through by any powers of the velocities before integrating, and each choice of powers leads to a different equation which involves the kinematic statistics describing the stellar system for fixed time and position, which are the mean velocity and the moments of the velocity distribution. Such a strategy, which is usually referred as moment or fluid approach, provides us with an infinite hierarchy of stellar hydrodynamic equations, which could be used as a dynamical model to study the stellar system, on condition that some closure relationships were available in order to work with a finite number of equations and unknowns.

1.3 Isolating integrals According to Jeans’ direct problem, Eq. 1.2 is a linear and homogeneous partial differential equation for f , for a given potential U, whose subsidiary

1.3. ISOLATING INTEGRALS

5

Lagrange system of equations is dr1 dr2 dr3 dV1 dV2 dV3 = = = ∂U = ∂U = ∂U = dt V1 V2 V3 ∂r ∂r ∂r 1

2

(1.4)

3

An immediate consequence of the Liouville theorem is that if I1 , I2 , ..., I6 are any six functional independent integrals of the stellar motion for a given potential satisfying the equations in Eq. 1.4, then the phase space density function must be of the form f (t, r, V ) = f (I1 , I2 , ..., I6 ), where the quantity on the right-hand side stands for an arbitrary function of the specified arguments, on the condition that the mass of the system be finite and that the density in the phase space be non-negative. The phase space density function is itself an integral of motion. The integrals of motion univocally determine the orbit of any star in the phase space. However, the phase density, by its physical significance, must be a onevalued function of the six phase coordinates. Therefore, only the integrals satisfying the condition of being one-valued in phase space, which are called isolating integrals, can appear as an argument of the phase density, although they may take several values in the space of integrals of motion. In 1953 G. Kuzmin was the first in suggesting this fact, and Lynden-Bell (1961) provided a rigorous demonstration that a continuous phase space density function must be independent of any non-isolating integral almost everywhere. More precisely, if in a bounded region of the phase space the equation Ik (r, V ) = Ck can be solved with respect to every variable and gives a finite number of solutions, then the integral is called isolating (e.g., Contopoulus 1963). On the other hand, if there exist at least one accumulation point at a finite distance, the integral is called non-isolating. Isolating integrals are important because they constrain the shapes of orbits by one dimension in the phase space. Analytic integrals in a simply connected region including the phase space are isolating, but non-classic integrals, which are implicit in a numerical integration of an orbit, are usually non-isolating. Thus, if the stellar system is time-independent, the phase space remains decomposed in a set of disjoint hypersurfaces corresponding to different integral values. Notice that if the system is time-dependent, the orbits may intersect for different times, although, for a fixed time t0 , the integrals must define a family of level curves of the phase space. Let us briefly review some typical examples of isolating integrals. In general, up to three isolating integrals are found for all orbits under steadystate and axisymmetric potentials. Thus, by expressing the velocity components (V1 , V2 , V3 ) in a Cartesian heliocentric coordinate system, with V1

6

CHAPTER 1. ANALYTICAL DYNAMICS

toward the Galactic centre, V2 in the rotational direction, and V3 perpendicular to Galactic plane, positive in the direction of the North Galactic pole, for a stationary potential U the energy integral can be written as I1 = V12 + V22 + V32 + 2U(r).

(1.5)

The integral for the axial component of the angular momentum under an axisymmetric potential is expressed, in cylindrical coordinates r = (r, θ, z), as (1.6) I2 = rV2 . Also, under a separable potential U = U1 (r) + U2 (z), which is valid near the Galactic plane, it is obtained a third integral, sometimes called Oort’s integral, I3 = V32 + 2U2 (z). (1.7) Of course, any combination of above integrals is also conserved. Let us point out two very simple cases. For a fixed position in the Galaxy, the quantity (1.8) I4 = V12 + V32 is also an isolating integral. Similarly, a quadratic function I5 = V12 + α(V2 − V0 )2 + βV32

(1.9)

for any V0 , α and β constants or depending on the position, is also conserved. The later may be generalised, under appropriate hypotheses, to arbitrary quadratic functions of the peculiar velocity components, which justifies the generalised use of Gaussian type velocity distributions. When some kinematic knowledge about the stellar system is available, such as that concerning the integrals of motion, if the density function f is already known, Eq. 1.2 may be interpreted according to the Jeans’ inverse problem as a linear, non-homogeneous partial differential equation for U. For example, the velocity distribution of some stellar groups can be assumed, after a transient period, of Maxwell type, Schwarzschild type, or ellipsoidal shaped (e.g., de Zeeuw & Lynden-Bell 1985). This viewpoint is a functional approach, which generally focuses on the study of a single stellar population (or may be used to define a statistical population) where the gravitational potential and the total stellar density N are related through the Newton-Poisson equation ∇2r U = 4πGN.

1.3. ISOLATING INTEGRALS

7

However, self-consistent models that use the above equation are very limited, since unknown stellar populations –including gas and dark matter– do contribute also to the gravitational field. In addition, it may be combined with a mixture model to get a more complete portrait of the velocity distribution (e.g., Cubarsi 1990, 1992). With this viewpoint, there is no need of the collisions term. On the contrary, it is assumed that there are sufficient collisions to keep the system in statistical equilibrium, according to the specific phase space density function or the particular integrals of motion. In other words, it is assumed that the phase space density function is invariant under the collisional operator C( f, f ). The idea comes from the original work on statistical dynamics (Chandrasekhar 1943), where the collisional term, accounting for diffusion and frictional processes, is exactly what is needed to conserve the energy of the whole system and leave the Maxwellian distribution invariant. Therefore, it is not surprising that Lynden-Bell (1967), in studying the equilibrium distribution achieved after a violent relaxation process, induced by rapid fluctuations of the gravitational field, obtains a similar smooth distribution function for a rotating elliptical system, which is quadratic as in Chandrasekhar (1942). Notice, however, that the former uses the statistical dynamics approach, while the latter in this case faces the problem from analytical dynamics. Other examples are described in Ogorodnikov (1965) when deriving the most probable phase distribution after an efficient relaxation mechanism. If P(t, r, V ) is an isolating integral of motion, continuous and differentiable in its arguments, for any fixed time t0 , the equation P(t0 , r, V ) = C must define a one-parameter family of five-dimensional surfaces filling all the six dimensional phase space Γr × ΓV , for all the possible values of the constant C ∈ IP . If we assume that the phase density depends only on P, that is f (t, r, V ) = f (P), then f (P) is also an isolating integral of motion, which must define, for the same fixed time t0 , another uniparametric family of curves f (P) = K ∈ I f (P) , associated with the same set of hypersurfaces filling Γr × ΓV . To each level curve of the former family corresponds one, and only one, level curve of the later family, so that K = f (C). Thus, we can assume that f is a diffeomorphism in the interior of its domain. Hence, the following inequality must be fulfilled, d f (P)  0. dP

(1.10)

In other words, f (P) is a strictly increasing or decreasing, smooth function of the argument in any open set within the interval IP . This is a basic prop-

8

CHAPTER 1. ANALYTICAL DYNAMICS

erty used in Chapter 4 for the general solution of the closure problem. A typical example of this situation is the generalised Schwarzschild distribution, with P = Q + σ, where Q = uT · A2 · u is a quadratic, positive definite form depending on the peculiar velocity u, where the second-rank symmetric tensor A2 and the scalar function σ depend only on time and position. Then, owing to Eq. 1.10, we can express the collisionless Boltzmann equation in either of the following forms DP D f (P) d f (P) DP = = 0 ⇐⇒ = 0. Dt dP Dt Dt

(1.11)

For the generalised Schwarzschild distribution, Chandrasekhar (1942) obtained a system of twenty partial differential equations for A2 , σ, the mean velocity, and the potential, which is equivalent to the collisionless Boltzmann equation.

1.4 Self-consistent models For isolated stellar systems the symmetries of the potential and the stellar density can be investigated from the self-consistency hypothesis according to a variant of the Jeans inverse problem. When the density function or the isolating integrals of the star’s motion are known, the collisionless Boltzmann equation allows to determine the potential or some properties of the potential such as symmetry properties. The variant consists in taking into account the Poisson equation by relating the potential to the stellar density and assuming that the gravitational potential generated by the stellar system satisfying the stationary collisionless Boltzmann equation is the unique origin of the stellar system force field (e.g., An et al. 2017, and references therein). Theoretically, from the isolating integrals the stellar density can be determined by integrating the distribution function in terms of either the velocities or the integrals themselves, arising the dependence of the distribution function on the potential and the space coordinates. Once established this functional dependence, several theorems about the symmetry of the solutions of elliptical partial differential equations (in particular for the Poisson equation) lead to particular symmetries for the potential and the stellar density, such as symmetry plane and axisymmetry, without the need of solving any differential equation. Nevertheless, the existence of such a joint solution is not guaranteed. Formally, the Poisson equation acts as a mathematical shortcut to deduce such symmetries instead of deducing them from the collisionless Boltzmann

1.5. STELLAR STATISTICS

9

equation by assuming that there is no external force that favours any direction of the symmetry axis and, therefore, the average behaviour of the stellar fluid is symmetric. In this sense, it is worth mentioning the work by Camm (1941). Camm considers the distribution function depending on a linear combination of the three integrals I1 , I2 , and I3 (i.e., he assumes the ellipsoidal hypothesis). He solves the stationary collisionless Boltzmann equation and obtains a plane of symmetry for the velocity ellipsoid and a potential symmetric with respect to this plane. This is obtained without using the hypothesis of self-consistency. He proves that the potential, in addition to be axisymmetric, is either: (a) spherically symmetric, i.e., the solution below Eq. (19), in which he is not interested; (b) separable in addition, viz., the expression below Eq. (22), not consistent with a finite system; (c) Eq. (23), depending on the latitude (actually on its absolute value); (d) symmetric about an arbitrary plane z=0, i.e., the general solution below Eq. (23), in the new coordinates, which are the roots of the quadratic equation. Without using the hypothesis of self-consistency, the solutions that Camm thinks of physical significance (a,c,d) satisfy both separability for the potential and symmetry plane. But what is more interesting is that when he adds the Poisson equation he finds that there is no mathematical solution satisfying such a joint solution. It could also occur that a possible solution was so simple that it was totally unrealistic. Therefore, one must be cautious in using the self-consistency model. Nevertheless, Camm forgets that, just as the ellipsoidal hypothesis is not valid for the whole system, neither the potential consistent with this model will be valid for the whole system. That is to say, in a certain region of the stellar system the potential, or its dominant term, can behave, for example, as the one associated to a quasi-elastic field of force (Ogorodnikov 1965), and therefore be separable in addition. The harmonic potential, according to Poisson equation, is the one created within a homogeneous spheroid. It would not be logical to pretend that such a model is extensible to an infinite galactic system but, in the central part of the system, the potential due to the galactic halo can certainly be modelled that way.

1.5 Stellar statistics For fixed values of time t and position r, the macroscopic properties of a stellar system can be described from the moments of the distribution, which provide indirect information on the phase-space density function f (t, r, V ).

CHAPTER 1. ANALYTICAL DYNAMICS

10

It is well-known that the first moments, accounting for the mean, give the more elementary property of the distribution; the second central moments describe how much the distribution is spread around the mean; the third moments describe distribution asymmetries like the skewness; the fourth moments are used to quantify how peaked the distribution is; and so forth (e.g., Stuart & Ord 1987). In particular, the stellar density is given by  f (t, r, V ) dV (1.12) N(t, r) = ΓV

and the stellar mean velocity, or velocity of the centroid, is  1 v(t, r) = V f (t, r, V ) dV . N(t, r) ΓV

(1.13)

The symmetric tensor of the n-order, non-centred trivariate moments is obtained from the expected value  1 n mn (t, r) = (V ) ≡ (V )n f (t, r, V ) dV , n ≥ 0 (1.14) N(t, r) ΓV   where (·)n stands for the n-tensor power. The tensor mn then has n+2 2 different elements according to the expression mi1 i2 ...in = Vi1 Vi2 . . . Vin ,

(1.15)

so that the indices belong to the set {1, 2, 3}, depending on the velocity component. Sometimes, instead of the component notation, namely Latin indices, it is also used a notation to make the velocity powers explicit, namely Greek indices, according to β

γ

mαβγ = V1α V2 V3 .

(1.16)

Obviously, m0 = 1 and m1 = v(t, r), which is the mean velocity, or velocity of the centroid. In a similar way, the symmetric tensor of the n-order centred moments is obtained by working from the peculiar velocity u = V − v(t, r), as the expected value μn (t, r) =

1 N(t, r)

(1.17)

 ΓV

(V − v(t, r))n f (t, r, V ) dV ,

n ≥ 0,

(1.18)

1.5. STELLAR STATISTICS

11

with elements μi1 i2 ...in = ui1 ui2 . . . uin .

(1.19)

In this case, μ0 = 1 and μ1 = 0. The second central moment μ2 is also known as covariance matrix. The second moment tensors, either centred or non-centred, are symmetric and positive-definite matrices, hence are diagonalizable with positive eigenvalues. When all the eigenvalues are equal, we say the tensor is isotropic. If an eigenvalue does not depend on r, we say it is isothermal in the direction of the corresponding eigenvector. The tensor of the central moments is related to the tensor of temperatures from the kinetic theory of gases, while the tensor of pressures is given by Pn = N μn .

(1.20)

Hereafter, when studying the velocity dependence of the distribution function from a statistical viewpoint, the variables of time and position are omitted, although they might be used in the framework of a dynamical model for the whole phase-space distribution function. Ellipsoidal distributions, such as the Schwarzschild distribution, can be described in terms of their central second moments μi j , which sometimes are written with Latin indices, such as σ2i j = Vi V j − Vi V j (Binney & Tremaine 2008). However, in other standard astronomy reference books, the Greek index notation is used (Gilmore, King & van der Kruit 1989), in particular when the velocity variables are expressed in the (U, V, W) coordinate system (without subindices), where the n-th moments mαβγ satisfy α + β + γ = n. The second central moments account for the shape and orientation of the velocity ellipsoid and for the variance σ2l of the velocity distribution function in an arbitrary direction l of the peculiar velocity space. According to the coordinate system, if c1 , c2 , and c3 are the corresponding direction cosines, we have  ci c j μi j ; i, j ∈ {1, 2, 3}. (1.21) σ2l = (c1 u1 + c2 u2 + c3 u3 )2 = i, j

The symmetric tensor μ−1 2 (inverse of the second central moments μ2 ) is then associated with the peculiar velocity ellipsoid uT · μ−1 2 · u = 1,

(1.22)

so that the velocity dispersions σ1 , σ2 , and σ3 are the semiaxes of the ellipsoid that refers to the same coordinate axes.

12

CHAPTER 1. ANALYTICAL DYNAMICS

Usually the direction u1 is taken as the radial direction, having the greater velocity dispersion. Then, the major semiaxis of the velocity ellipsoid has a direction close to that of the location vector of the centroid. The angle of such a deviation is referred to as vertex deviation. The tilt of the velocity ellipsoid is also significant. If one of the principal semiaxes points to the Galactic centre, then there is no tilt, if not, the tilt is the angle of such a deviation. A precise definition of vertex deviation and tilt of the velocity ellipsoid is given in Appendix G.2). For these distributions, higher order central moments can be computed depending on the second ones; in other words, they cannot take arbitrary values, as shown in Appendix C. However, for arbitrary distributions, the variances and the velocity ellipsoid are meaningless, unless they could be used as a Gaussian approximation. Similarly, the skewness and the kurtosis are also meaningless for multivariate distributions far from Gaussian, which have to be qualitatively described from moments of order higher than two, up to a sufficient degree of approximation of the basic distribution trends.

1.6 Velocity moments In the very beginning of the book, Chandrasekhar (1942) outlines the appropriate conditions to define a unique local standard of rest (LSR) for describing the motions in a given relatively small volume of the Galaxy. The conditions are related to a continuous estimation of the centroid2 velocity within this volume and to a slow varying distribution function, which could be referred to as regularity conditions. He concludes that the stellar systems can be divided into those for which the notion of LSR (and, by extension, higher velocity moments) is of significance and those for which it is not. Among the latter we can mention the systems dominated by a phase mixing process (e.g., Binney & Tremaine 2008), for which a macroscopic, coarse-grained distribution function may be defined in contrast with the true, fine-grained distribution function, although the coarse-grained distribution function of a mixing system would not satisfy the collisionless Boltzmann equation. We shall focus on the first class of stellar systems, for which the mean velocity and similar statistics are meaningful. 2 The centroid of motion corresponds to the mean velocity of the stars in a small volume of the Galaxy and is used as synonymous of LSR. It is not exactly the same as the centre of mass of this volume, since we know little about the masses of individual stars. However, for the whole Galaxy, the galactic standard of rest is identified with the centre of gravity and not with the centroid of motion of all stars.

1.6. VELOCITY MOMENTS

13

Let us remember that for the Galaxy and, in general, for stellar systems larger than globular clusters, the forces acting on a star can exclusively be associated with a mean gravitational field, by neglecting the random forces due to stellar encounters. In the solar neighbourhood, assuming that the Galaxy has reached an equilibrium configuration, the potential is usually taken as explicitly time-independent. Then, the Hamiltonian flow possesses the energy integral, is always nonergodic and, therefore, nonmixing (e.g., Arnold & Avez 1968). Thus, in order to introduce the kinematic statistics into the collisionless Boltzmann equation, Eq. 1.2 is multiplied by the n-tensor power of the star velocity and then integrated over the whole velocity space,  Df dV = (0)n , n ≥ 0 (V )n (1.23) Dt ΓV where in the integration process the following boundary conditions are assumed because there are no stars with velocity beyond ΓV V → ∂ΓV =⇒ (V )n f (t, r, V ) → (0)n , n ≥ 0.

(1.24)

It is always assumed that the foregoing integrals do exist, as those of the velocity moments. For each value of n, the tensor equation Eq. 1.23 leads to the n-order stellar hydrodynamic equation, which provides us with a conservation or transfer law along the centroid trajectory. The most basic cases are the continuity equation, for n = 0, which stands for mass conservation, and the momentum conservation equation, for n = 1. However, the methodology of most books on galactic dynamics which devote a chapter to obtain or discuss the stellar hydrodynamic equations (e.g., Chandrasekhar 1942, Kurth 1957, Ogorodnikov 1965, Mihalas 1968, Binney & Tremaine 2008) is to integrate Eq. 1.23 –for n = 0 and n = 1– over the absolute, non-peculiar velocities, leading to equations involving the absolute moments of the velocity distribution, and afterwards, in order to give a physical interpretation of each equation, the total moments are explicitly written in function of the central moments. Such a procedure is appropriate for the lowest order equations, but it is not adequate for an arbitrary n-order equation.

Chapter 2

Stellar hydrodynamic equations 2.1 Comoving-frame equations Let us write the collisionless Boltzmann equation, Eq. 1.2, in terms of the stellar mean velocity, Eq. 1.13, and of the peculiar velocities, Eq. 1.17, by expressing the phase space density function f in the form φ(t, r, u) = f (t, r, u + v(t, r))

(2.1)

where t, r and u are independent variables. Hence, the derivatives with respect to these variables are ∂φ ∂u ∂f ∂φ ∂v = + · ∇u φ = − · ∇u φ, ∂t ∂t ∂t ∂t ∂t ∇r f = ∇r φ + ∇r u · ∇u φ = ∇r φ − ∇r v · ∇u φ,

(2.2)

∇V f = ∇V u · ∇u φ = ∇u φ. Then Eq. 1.2 becomes ∂φ ∂v − · ∇u φ + (u + v) · (∇r φ − ∇r v · ∇u φ) − ∇r U · ∇u φ = 0 ∂t ∂t 15

(2.3)

CHAPTER 2. STELLAR HYDRODYNAMIC EQUATIONS

16

so that, by reorganising terms, it yields   ∂v ∂φ +v ·∇r φ− + v · ∇r v ·∇u φ+u·(∇r φ−∇r v ·∇u φ)−∇r U ·∇u φ = 0. ∂t ∂t (2.4) To simplify the notation, we use the material derivative (also called substantial derivative) associated with the motion of the centroid,   ∂ d (·) = + v · ∇r (·) . (2.5) dt ∂t Since r and u are independent variables, we take into account the identity u · ∇r φ = ∇r · (u φ)

(2.6)

and we also consider the following equality1 u · ∇r v · ∇u φ = ∇r v : (u ⊗ ∇u φ)

(2.7)

where each dot represents an inner product, and ⊗ a tensor product2. Notice that the colon stands for the dot products ∇r with u, and v with ∇u , respectively. Hence, Eq. 2.4 may be written as follows   dv dφ − + ∇r U · ∇u φ + ∇r · (uφ) − ∇r v : (u ⊗ ∇u φ) = 0. (2.8) dt dt We take now the tensor product of the foregoing equation with the n-tensor power of the peculiar velocity (u)n ,   dv dφ − (u)n ⊗ + ∇r U · ∇u φ + (u)n dt dt (2.9)



 n+1 n+1 n +∇r · (u) φ − ∇r v : (u) ⊗ ∇u φ = (0) 1 In

component notation the equality can be written as ui

  ∂v j ∂v j ∂φ ∂φ = ui , where ∂ri ∂u j ∂ri ∂u j

Einstein’s summation criterion for repeated indices is applied. 2 The notation used for nabla operators is the usual one. If x is a vector variable and Fn (x) a n-rank symmetric tensor field, then for n ≥ 1 the divergence ∇x · Fn is, in components, ∂x∂i Fi1 ...in , while for n ≥ 0, ∇x Fn is used instead of ∇x ⊗ Fn to represent the gradient ∂ ∂xi1

1

Fi2 ...in+1 .

2.2. CONSERVATION OF PRESSURES

17

and the resulting equation is then integrated over the peculiar velocities space Γu , where the factors depending only on r and t are left out of the integrals. Thus, we obtain d dt



 Γu

(u)n φ du −

  dv + ∇r U · (u)n ⊗ ∇u φ du+ dt Γu

 +∇r ·

Γu

 (u)n+1 φ du − ∇r v :

Γu

(2.10)

(u)n+1 ⊗ ∇u φ du = (0)n .

The first and third terms of the above relationship are directly expressed in function of the pressures, according to Eq. 1.14 and Eq. 1.20. Instead, for the other terms an auxiliary tensor may be defined as follows  (u)n ⊗ ∇u φ du, n ≥ 0 (2.11) Qn+1 = − Γu

so that Eq. 2.10 may be rewritten in a more compact notation,   dv dPn + + ∇r U · Qn+1 + ∇r · Pn+1 + ∇r v : Qn+2 = (0)n . dt dt

(2.12)

However, the tensors Qn are not directly computable in their current form.

2.2 Conservation of pressures The next step is to write the general hydrodynamic equation Eq. 2.12 explicitly depending on the pressures. Hence we shall find out how the tensors Qn can be expressed in terms of the pressures Pn . Let us note a particular case of Eq. 2.11. For n = 0, bearing in mind the boundary condition Eq. 1.24, we get  ∇u φ du = φ|u = 0. (2.13) Q1 = − Γu

For n = 1, the tensor product (u)n ⊗ ∇u φ within the integral of Eq. 2.11 verifies, in components, ui

∂(ui φ) ∂φ = − δi j φ, ∂u j ∂u j

(2.14)

18

CHAPTER 2. STELLAR HYDRODYNAMIC EQUATIONS

being δi j the Kronecker delta, and for n ≥ 2, ∂(ui1 . . . uin φ) ∂φ = − δi1 in+1 ui2 . . . uin + . . . + ∂uin+1 ∂uin+1

ui1 . . . uin



(2.15)

+δi j in+1 ui1 . . . u i j . . . uin + . . . + δin in+1 ui1 . . . uin−1 φ where the hat remarks the omitted factors. Then, the tensor Qn+1 can be evaluated by integrating Eq. 2.14 and Eq. 2.15. The conditions of Eq. 1.24 are once more applied over the integration boundary, so that the first term on the right-hand side of Eq. 2.15, when integrating over uin+1 , yields  uin+1

∂(ui1 . . . uin φ) duin+1 = ui1 . . . uin φ|uin+1 = 0. ∂uin+1

(2.16)

Hence, the tensor Qn+1 is obtained by integrating only the remaining terms, and by taking into account Eq. 1.19 and Eq. 1.20. Thus, for n = 1 we are led to (Q2 )i j = δi j P0

(2.17)

and for n ≥ 2, we get the following expression depending on the pressures, (Qn+1 )i1 ...in+1 = δi1 in+1 Pi2 ...in + . . .+ δi j in+1 Pi1 ... i j ...in + . . .+ δin in+1 Pi1 ...in−1 . (2.18) The foregoing relationships will be used to write both terms in Eq. 2.12, which involve the tensor Qn+1 . One of the terms contains a single dot product of this tensor with a vector, namely a · Qn+1 . Hence, by applying Eq. 2.18, we get (a · Qn+1 )i1 ...in =   = ain+1 δi1 in+1 Pi2 ...in + . . . + δi j in+1 Pi1 ... i j ...in + . . . + δin in+1 Pi1 ...in−1 =

(2.19)

= ai1 Pi2 ...in + . . . + ai j Pi1 ... i j ...in + . . . + ain Pi1 ...in−1 where Einstein’s summation convention is used. The result is the symmetrised tensor product in regard to permutations of indices, S(a ⊗ Pn−1 ),

2.2. CONSERVATION OF PRESSURES

19

which will be represented according to the following notation3 (Cubarsi 1992), S(a ⊗ Pn−1 ) = n a  Pn−1 , n ≥ 1, (2.21) so that the number of summation terms, which are needed in order to symmetrise the tensor product, is explicitly written. Although the star product is a non-standard notation, which is defined in the footnote, it worthy simplifies the forthcoming formulas and shall be used along the following chapters. Hence, Eq. 2.19 now stands for a · Qn+1 = n a  Pn−1 , n ≥ 1. and, therefore,     dv dv + ∇r U · Qn+1 = n + ∇r U  Pn−1 , n ≥ 1. dt dt

(2.22)

(2.23)

For the particular case n = 0, the relation 

 dv + ∇r U · Q1 = 0 dt

is fulfilled, which means, from an algebraic viewpoint, that Eq. 2.23 is also valid for n ≥ 0, since the factor n appearing in Eq. 2.23 would make null the equality, even though P−1 is not defined. 3 In general, if A and B are two m- and n-rank symmetric tensors, we can define the tenm n sor Am  Bn as the obtained by symmetrising the tensor product Am ⊗ Bn , and by normalising then with respect to the number of summation terms, T . The result is a (m + n)-rank symmetric tensor, whose components are

(Am  Bn )i1 i2 ...im+n = =

1 T

1 S (Am ⊗ Bn )i1 i2 ...im+n = T



Aαi1 ...αim Bαim+1 ...αim+n

(2.20)

αi1 < . . . < αim αim+1 < . . . < αim+n

where α belongs to the symmetric group S(m + n). If both tensors are different ones, then T = (m+n)! n!m! . (2n)! , and Notice that, in particular, if Am = Bn the number of summation terms is T = 2!n!n! k (kn)! An ) the number of terms is T = . for the symmetric tensor product S( k!(n!)k

CHAPTER 2. STELLAR HYDRODYNAMIC EQUATIONS

20

In a similar way, Eq. 2.12 will be calculated for the double product ∇r v : Qn+2 . Indeed, for n = 0 Eq. 2.17 simply leads to ∇r v : Q2 = (∇r · v) P0

(2.24)

while for n ≥ 1, according to Eq. 2.18, it can be written in components as follows (∇r v : Qn+2 )i1 ...in = =

∂vin+2  δi i Pi ...i + . . . + δi j in+2 Pi1 ... i j ...in+1 + . . . ∂rin+1 1 n+2 2 n+1

 +δin in+2 Pi1 ... in in+1 + δin+1 in+2 Pi1 ...in = =

∂vi j ∂vi1 Pi2 ...in+1 + . . . + P +... ∂rin+1 ∂rin+1 i1 ...i j ...in+1

+

∂vi ∂vin Pi1 ... in in+1 + n+1 Pi1 ...in . ∂rin+1 ∂rin+1

(2.25)

The last term of the expression above is equivalent to (∇r · v) Pn , while the first n terms are the components of the symmetrised tensor product n (Pn · ∇r )  v. Thus, for n ≥ 1, the foregoing equation can be written as ∇r v : Qn+2 = n (Pn · ∇r )  v + (∇r · v) Pn .

(2.26)

Once more such a relation can formally be used also for n = 0, since, even though the dot product (P0 · ∇r ) which would appear in the first term of the right-hand side is not defined, it would become null being multiplied by n. Finally, by substitution of Eq. 2.23 and Eq. 2.26 into Eq. 2.12, and taking into account the definition of the material derivative, the general expression for an arbitrary n-order hydrodynamic equation becomes   ∂v ∂Pn + v · ∇r Pn + n + v · ∇r v + ∇r U  Pn−1 + ∂t ∂t (2.27) +∇r · Pn+1 + n (Pn · ∇r )  v + (∇r · v) Pn = (0)n . Therefore, for each n, the foregoing equation, which is written in terms of the generalised tensor of pressures Pn , is explicitly providing its conservation or transfer law.

2.3. CONSERVATION OF MOMENTS

21

2.3 Conservation of moments The lowest order hydrodynamic equations are generally used, together with some additional hypotheses like axisymmetry, steadiness, incompressible flow, etc., and together with some closure assumptions related to diffusion (e.g., by neglecting off-diagonal second moments), conductivity (e.g., by neglecting third moments and higher odd-order moments), etc., in order to estimate either kinematic parameters of the local stellar populations, or the local stellar density, similarly to the earliest works by Jeans (1922) and Oort (1932), or like more recent works by Bahcall (1984a,b), Jarvis & Freeman (1985), van der Marel (1991), Famaey & Dejonghe (2003), most of them by using also the Poisson equation for self-gravitating systems or Stäckel models in order to close the system of equations. For the lowest orders, it is easy to give a physical interpretation of the stellar hydrodynamic equations by comparing them with the ones of fluid dynamics. Thus, for n = 0, bearing in mind that P0 = N and P1 = 0, the continuity equation can be written in its transfer form, and by using the material derivative Eq. 2.5, as d ln N = −∇r · v. dt

(2.28)

−1

Hence, since d lndt N = − d lndtN , the divergence of the mean velocity yields the fractional time rate of change of the density N, as well as of the specific volume N −1 . For n = 1 the equation of momentum transfer, which is usually referred as Jeans equation, is 1 dv = −∇r U − ∇r · P2 dt N

(2.29)

and it is equivalent to the Navier-Stokes equation of fluid dynamics. Thus, the acceleration of the centroid is partially due to the force coming from the potential (per unit mass), and partially due to the force coming from relative pressure variations, that is, from the surface forces applied on a volume element. Let us remember that the second-order pressure tensor, which is also known as the comoving stress tensor, gives account, in its diagonal elements, of the normal stresses, and, in its non-diagonal elements, of the tangential stresses, which are associated with viscosity and diffusion effects. For n = 2, Eq. 2.27 yields dP2 = −(∇r · v) P2 − 2 (P2 · ∇r )  v − ∇r · P3 . dt

(2.30)

22

CHAPTER 2. STELLAR HYDRODYNAMIC EQUATIONS

The first and second terms in the right-hand side of Eq. 2.30 are related to the rate of strain. The first one contains the divergence of the mean velocity, or volumetric rate of strain, while the second term contains the symmetric tensor ∇r  v, which is the shear rate of strain. Although the trace of the above tensor equation provides the law for energy transfer, the six scalar equations involved in it give account of the work balance along the different transfer directions. Thus, according to the usual interpretation from fluid dynamics, the variation of internal energy in each direction is partially due to the work coming from the specific volume variation (first right-hand term), to the viscous dissipation through the surface of a volume element (second right-hand term), and from the heat added through conduction (third righthand term). In a general way, the transfer of the n-order pressure, Eq. 2.12, can be interpreted by taking into account Eq. 2.29, and by writing it in the following form, ∇r · P2 dPn = −Qn+1 · − Qn+2 : ∇r v − ∇r · Pn+1 . dt N

(2.31)

Thus, the changes in pressure Pn are explained from a first term linearly depending on the rate of stress variation per unit mass, through the tensor Qn+1 , from a second term linearly depending on the velocity gradient, through the tensor Qn+2 , and from a third term giving account of the nearest higher-order pressure variation Pn+1 . However, the general hydrodynamic equation Eq. 2.27 is not explicitly written in terms of data actually available from stellar velocity catalogues, since the pressures obviously depends on the stellar density, according to Eq. 1.20, while the central velocity moments can be directly computable from large stellar samples. By working from the velocity moments, together with the hydrodynamic equations and some appropriate closure conditions, it is possible to estimate or to model either the stellar density, the velocity distribution, or the potential function (e.g., Cuddeford & Amendt 1991). Similarly, numerical approaches and simulations by using the moment equations have not to be restricted either to orders n < 2, or to the assumption of vanishing odd-order moments (e.g., Vorobyov & Theis 2006). For that reason, the general n-order equation will be expressed in terms of the central moments μn . The continuity equation, Eq. 2.28, does not need to be rewritten, while for n ≥ 1, Eq. 2.28 can be used together with Eq. 1.20

2.3. CONSERVATION OF MOMENTS

23

to re-write the general relation Eq. 2.27 in the following form   ∂v ∂μn + v · ∇r μn + n + v · ∇r v + ∇r U  μn−1 + ∂t ∂t

(2.32)

+(∇r ln N + ∇r ) · μn+1 + n (μn · ∇r )  v = (0) . n

Hence, for n = 1, the momentum equation can be expressed as follows ∂v + v · ∇r v + ∇r U = −(∇r ln N + ∇r ) · μ2 ∂t

(2.33)

and for n = 2, since μ1 = 0, we have ∂μ2 + v · ∇r μ2 + (∇r ln N + ∇r ) · μ3 + 2 (μ2 · ∇r )  v = (0)2 . (2.34) ∂t In general, Eq. 2.33 may be introduced into the higher-order equations to replace the terms depending on the potential function, so that they remain written in terms of the comoving moments. Then, for n ≥ 2 we have ∂μn + v · ∇r μn − n [(∇r ln N + ∇r ) · μ2 ]  μn−1 + ∂t

(2.35)

+(∇r ln N + ∇r ) · μn+1 + n (μn · ∇r )  v = (0) , n

which is the general expression4 giving the contributing terms to the conservation of the n-order moment. Nevertheless, let us remember the typical situation we are led when working with hydrodynamic equations. The equations Eq. 2.28 and Eq. 2.33, for n = 0 and n = 1, contain four different scalar equations, which involve a set of eleven unknown scalar functions, namely N, U, v and the symmetric tensor μ2 . It is well-known that, even in the case of taking into component i1 . . . in for n ≥ 2 of the equation stands for   ∂μi ...i ∂μαi2 ∂μαi1 ∂μi1 ...in ∂μαin + vα 1 n − μi2 ...in + μi i ...in + . . . + μi1 ...in−1 + ∂t ∂rα ∂rα ∂rα 1 2 ∂rα

4 The

+

 ∂ ln N  μαi1 ...in − μαi1 μi2 ...in − μαi2 μi i ...in − . . . − μαin μi1 ...in−1 + 1 2 ∂rα

+

∂vi1 ∂vi2 ∂μαi1 ...in ∂vin + μαi2 ...in + μ + ... + μαi1 ...in−1 = 0. ∂rα ∂rα ∂rα αi1 i 2 ...in ∂rα

(2.36)

24

CHAPTER 2. STELLAR HYDRODYNAMIC EQUATIONS

account higher-order equations the system remains   always open, since by picking up the mth equation, which contains m+2 scalar equations, we are 2 m+3 also introducing as many as 2 new unknowns, which are the different components of the tensor μm+1 . In most cases, the lowest order equations are used under some particular assumptions to reduce the number of unknowns. For example, by assuming the epicyclic approach, like in Oort (1965), by assuming axial symmetry (e.g., Vandervoort 1975), or by taking a velocity distribution function depending on specific isolation integrals of the star motion (e.g., Jarvis & Freeman 1985). In such a way, some constraint relationships for the central moments may be reached (e.g., van der Marel 1991). Now, for specific velocity distributions and working from the general moment equation, the closure conditions could be studied in a more general way.

2.4 Closure example We shall see a simple example of how to use the n-order general expression, Eq. 2.35, to find out the closure conditions in terms of the velocity distribution statistics. Let us assume the simplest case of an isothermal velocity distribution of Maxwell type in the residual velocities, according to the Maxwell-Boltzmann law, which represents a system with the more basic thermal equilibrium. A more general case for Schwarzschild and generalised Schwarzschild distributions will be studied in detail in Chapter 3. Thus, for Eq. 2.1 we have φ(t, r, u) = e− 2 Q , 1

Q = μ−1 |u|2

(2.37)

where Q is a quadratic, positive definite form, and μ(t, r) is a continuous and differentiable function in both arguments, which gives account of the variance of the distribution. This is a well-known case of a spherical model, which can be solved by substitution of Eq. 2.37 in the collisionless Boltzmann equation Eq. 1.2, or directly in his form Eq. 2.4. By this way, the dynamical model is determined from a finite set of equations, but we may wonder about how is it related to the infinite hierarchy of hydrodynamic equations. Although we know that all the hydrodynamic equations are fulfilled, we may guess that there is a finite subset of hydrodynamic equations which are strictly equivalent to the collisionless Boltzmann equation. Then, which are the orders of these equations? Why and which are the redundant equations? Can we explicitly

2.4. CLOSURE EXAMPLE

25

write the conditions that make them redundant? In other words, which are the closure conditions? In general, the answers to the foregoing questions vary depending on the form of the velocity distribution function. This case example only claims to be the seed for more complete and general closure conditions. The Maxwellian distribution in the residual velocities, since it is a totally isotropic distribution, has all the odd-order central moments null. Its symmetric even-order moments are given through isotropic tensors accordingly to μ2n = S(

n

n

 I2 μ) = Cn μ I2  · · ·  I2 , n

Cn =

(2n)! 2n n!

(2.38)

where I2 is representing the second-rank identity tensor. Since Cn+1 = (2n + 1) Cn , it is easy to prove the relation μ2n+2 = (2n + 1) μ2n  I2 μ.

(2.39)

For the even-order equations, n = 2k and k ≥ 1, bearing in mind that the odd-moments are null, Eq. 2.35 becomes ∂μ2k + v · ∇r μ2k + 2k (μ2k · ∇r )  v = (0)2k , ∂t

(2.40)

which, by substitution of moment expressions Eq. 2.38 and Eq. 2.39, easily reduces to  ∂(I2 μ) + v · ∇r (I2 μ) + 2 (I2 μ · ∇r )  v = (0)2k . μ2k−2  (2.41) ∂t Thus, since μ2k−2 never vanishes, we conclude that all the even-order equations, n ≥ 2, are reduced to the second-order equation   ∂μ + v · ∇r μ + 2 μ∇r  v = (0)2 . I2 (2.42) ∂t Therefore, in virtue of Eq. 2.39, the foregoing relationship provides a closure condition in terms of the mean velocity, and of the velocity variance μ. In a similar way, for the odd-order equations, n = 2k+1, k ≥ 1, provided that the odd-order moments are null, from Eq. 2.35 we can write −(2k + 1) [(∇r ln N + ∇r ) · μ2 )]  μ2k + (∇r ln N + ∇r ) · μ2k+2 = (0)2k+1 . (2.43)

CHAPTER 2. STELLAR HYDRODYNAMIC EQUATIONS

26

Then, by substitution of moment expressions Eq. 2.38 and Eq. 2.39, we have −(2k + 1) μ∇r ln N  μ2k − (2k + 1) ∇r μ  μ2k + +(2k + 1) μ∇r ln N · (I2  μ2k ) + (2k + 1) ∇r · (μI2  μ2k ) = (0)2k+1 . (2.44) In addition, from Eq. 2.38 we can obtain the following identity, ∇r · (μI2  μ2k ) = ∇r μ  μ2k + k μ∇r  μ2k ,

(2.45)

which allow us to simplify Eq. 2.44, by leading to ∇r  μ2k = (0)2k+1 .

(2.46)

This expression stands for all the odd-order equations, n = 2k + 1 ≥ 3. Nevertheless, by taking into account Eq. 2.39, since μ2k−2 is always nonnull for k ≥ 1, then Eq. 2.46 can be reduced to the third-order equation I2  ∇r μ = (0)3 .

(2.47)

Hence, such a relation provides another closure condition in terms of the velocity variance μ. In other words, for a velocity distribution of Maxwell type, if the conditions given by Eq. 2.39, Eq. 2.42 and Eq. 2.47 are satisfied, then all the moment equations are reduced to the four equations of orders n = 0, 1, 2, 3, which are a set of twenty scalar equations, and they are then equivalent to the collisionless Boltzmann equation. Thus, such a simple example shows how to find out dependences of higher-order hydrodynamic equations on the lower-order ones, for a specific velocity distribution. Although this model has no interesting physical implications, it can be easily solved from the foregoing equations. Thus, from Eq. 2.47, we can find that μ is not a function of r. Also, from Eq. 2.42, the mean velocity components can be obtained5. Henceforth, from the equations corresponding to n = 0 and n = 1, and depending on the specific time-dependence or symmetry assumptions for the stellar system in general, or for the potential function in particular, the remaining unknowns can be determined in every situation. 5

∂μ ∂μ ∂μ In components, Eq. 2.47 becomes δi j ∂r + δik ∂r + δ jk ∂r = 0, which implies k j i

∂vi + all i. Then Eq. 2.42 reduces to ∂r j ∂v j ∂v3 ∂v1 ∂v2 ∂vi = = = κ ˙ and + ∂r1 ∂r2 ∂r3 ∂r j ∂ri =

∂v j ∂ri

=

μ − ∂ ln ∂t δi j

∂μ ∂ri

= 0 for

≡ 2˙κδi j . Thus, the mean velocity satisfies

0 for i  j. Hence, the shear strain rate is null, and there is some volumetric rate of strain only for a non-stationary system. By solving these equations, we easily obtain a mean velocity in the form v = κ˙ r +ω ∧r +τ , which means that the centroid moves accordingly to a radial expansion or contraction, in the case of a non-stationary system, and rotates like a rigid body, with angular velocity ω, in addition to an arbitrary translation τ .

2.5. ABSOLUTE REFERENCE FRAME

27

2.5 Absolute reference frame The hydrodynamic equations derived in the reference the frame of the absolute velocities in terms of the non-centred velocity moments, as appear in most text books, can be obtained in a more direct and easier way than in the comoving-frame. With the current notation, the general expression involves the non-centred moments of Eq. 1.14 and the tensor of temperatures from the kinetic theory of gases, defined as Tn = N m n .

(2.48)

Therefore, T0 = N and T1 = Nv. For the absolute velocity, a similar relationship to Eq. 2.11 can be defined. By taking into account Eq. 2.2, we write ∇V f = ∇u φ. Then, we define  Q n+1 = − (V )n ⊗ ∇V f dV , n ≥ 0. (2.49) ΓV

  The above tensor, similarly to Qn+1 , satisfies Q 1 = 0, Q 2 = δi j T0 , and ij for n ≥ 2,   = δi1 in+1 T i2 ...in + . . . + δi j in+1 T i1 ... i j ...in + . . . + δin in+1 T i1 ...in−1 . (2.50) Q n+1 i1 ...in+1

Instead of Eq. 2.22, we now have a · Q n+1 = n a  Tn−1 , n ≥ 1.

(2.51)

With these preliminaries the collisionless Boltzmann equation, Eq. 1.2, can be straightforward integrated over the velocity space. For n ≥ 1 we get ∂Tn + ∇r · Tn+1 + n ∇r U  Tn−1 = (0)n . ∂t

(2.52)

while n = 0 simply yields the continuity equation. Notice that the time derivative involves a temperature tensor of different parity than the other terms. This also applies to the velocity moments. In a simplified, stationary model, the remaining equation ∇r · Tn+1 + n ∇r U  Tn−1 = (0)n

(2.53)

would involve terms of the same parity. On the other hand, the integrals  Tn (t, r) = (V )n f (t, r, V ) dV , n ≥ 0 ΓV

28

CHAPTER 2. STELLAR HYDRODYNAMIC EQUATIONS

depend exclusively on the even-part f+ = f (V )+2f (−V ) of f if n is even, or on its odd-part f− = f (V )−2f (−V ) if n is odd. In other words, f− provides null even-order moments, while f+ does not contribute to the odd-order moments. Therefore, the families of even- and odd-order moments can be treated separately, as well as the functions f+ and f− , which are independent isolating integrals of motion and satisfy the collisionless Boltzmann equation and the whole set of hydrodynamic equations. Nevertheless, f− is not a density function, while f+ is. Therefore, f+ is a density function providing the same even-order moments than f .

2.6 Remarks The full expression for the stellar hydrodynamic equations of arbitrary order in a comoving reference frame has been deduced from the collisionless Boltzmann equation. It is written without any restrictive assumptions, like those of the steady-state system, axial symmetry, galactic plane of symmetry, pure rotating system, vanishing odd-order moments, etc., so that it can be used, for example under some of these hypotheses, to test analytical dependences of the phase space density function on the integrals of motion, or in its complete form to carry out numerical simulations about either the velocity distribution or stellar density variations. It is worth noting that most of the aforementioned hypotheses are not already valid in the solar neighbourhood. In Cubarsi & Alcobé (2004, 2006), Alcobé & Cubarsi (2005), and Cubarsi et al. (2010) it is discussed how the local thin disc is clearly non-axisymmetric, with a non-vanishing vertex deviation, whereas increasing nested subsamples of thick disc stars showed a trend to axisymmetry. Similarly, it is found that the local thin disc is not in steady-state, which is related to its net radial velocity towards the galactic centre, whereas thick disc stars show a trend to steady-state. Therefore, the moment equations have to be actually used in their complete form, at least for Galactic disc analysis. On the other hand, the general expression of moment equations may be also useful to study closure conditions, which are associated with specific velocity distributions, in order to reduce the infinite hierarchy of equations and unknowns to a finite number of them, so that a feasible dynamical model can be available. Obviously, when working with the Jeans equation alone, or whatever finite set of hydrodynamic equations, the collisionless Boltzmann equation is not generally fulfilled. It is an interesting mathematical problem to study how the system of equations can be closed, and which are

2.6. REMARKS

29

then the conditions over the velocity distribution function in order to exactly fulfil the Boltzmann equation. Thus, by using a similar procedure as in the case example, some more general and interesting cases are studied in the following chapters.

Chapter 3

Ellipsoidal distributions 3.1 Quadratic integral As it was suggested in the previous closure example, the exact n-order equation derived in Chapter 2 may be taken as the starting point to study the closure conditions, which make a finite set of moment equations equivalent to the collisionless Boltzmann equation. Two general cases of Gaussian and ellipsoidal trivariate velocity distributions, according to the general form of Chandrasekhar’s generalised Schwarzschild functions, are used to investigate the closure problem. The analysis shows that the set of hydrodynamic equations of orders n = 0, 1, 2, 3 and the equations obtained by Chandrasekhar (1942) for such a type of distributions generate the same dynamical model, since the higher-order equations are found to be redundant. This is first proven for Schwarzschild distributions, which are then taken as a basis to expand a generalised ellipsoidal function as a power series of them, so that the model is extended in a natural way to the family of quadratic functions in the peculiar velocities. In the next chapter, the above results will be generalised, as derived in Cubarsi (2010b, 2016), to any velocity distribution function depending on a polynomial function in the velocity variables. For an appropriate framework of the problem let us remember that a consequence of the collisionless Boltzmann equation is that if I1 , I2 , ..., I6 are any six independent isolating integrals of the equation of motion of a star for a given potential function per unit mass U(t, r), then the phase space density function must be of the form f (t, r, V ) = f (I1 , I2 , ..., I6). However, 31

32

CHAPTER 3. ELLIPSOIDAL DISTRIBUTIONS

isolating integrals like the energy integral, the integral for the axial component of the angular momentum, and sometimes a third integral, are only found for all orbits under steady-state, axisymmetric potentials, or other particular potentials. In order to avoid these limitations, a functional approach may be adopted, which takes advantage of some kinematic knowledge about the stellar system. Thus, after a transient period, the velocity distribution of some stellar groups tends to be of Maxwell type, Schwarzschild type, or, in a more general way, it is ellipsoidal shaped. Therefore, it is assumed the existence of a quadratic isolating integral resulting from an average of the classical integrals of motion. Such an approach was firstly explored by Eddington (1921) and Oort (1928), and was formulated in a more general way by Chandrasekhar (1942). For example, if the phase space density function is taken of Schwarzschild type, as Gilmore et al. (1989) point out, even though it is a quite simple case, the distribution then leads to a solution that predicts many details of the Galactic structure and kinematics, and it is possible that a realistic model could be build up as a superposition of such solutions, and it also leads in a natural way to Stäckel potentials and the quadratic third integral that goes with them. In order to allow some more degrees of freedom to the velocity distribution, as well as to the dynamical model, an arbitrary ellipsoidal function in the peculiar velocities was investigated by Chandrasekhar. Under such an approach the dynamical model can be derived from a finite set of equations by substitution of the phase space density function into the collisionless Boltzmann equation. But we may wonder about how is it related to the moment approach and, in particular, to the infinite hierarchy of hydrodynamic equations. As it was said in Chapter 2, although we know that all the hydrodynamic equations must be formally fulfilled, we may guess that there is a finite subset of hydrodynamic equations which are strictly equivalent to the collisionless Boltzmann equation. Then, which are the orders of these equations? Why and which are the redundant equations? Can we explicitly write the conditions that make them redundant? All of these questions, which are specific of the velocity distribution or the integral model that have been assumed, make up the closure problem. An interesting example of closure problem was studied by Cuddeford & Amendt (1991). Since a Schwarzschild velocity distribution may satisfy the collisionless Boltzmann equation, they adopted some closure assumptions involving the moments of the velocity distribution, which did match some known constraints between the moments of the Schwarzschild distribution, like those related to the skewness and the kurtosis of the distri-

3.2. SCHWARZSCHILD DISTRIBUTION

33

bution in specific directions. By this way they made the stellar hydrodynamic equations equivalent to the collisionless Boltzmann equation, as well as they obtained a phase space density function which was more general than of Schwarzschild type. However, the closure conditions they found, working even up to eighth-order moments, were only valid in a steady-state, cool and axisymmetric stellar system, with vanishing radial mean velocity. Nevertheless, most of those assumptions are not already valid in the solar neighbourhood according to recent catalogues. Therefore, in regard to actual data, the exact general expression of moment equations must be used to establish more general closure assumptions. In the current chapter the closure problem for Gaussian and ellipsoidal velocity distributions is analysed from a completely general approach, without any additional hypotheses, so that it is the basis to studies on some more general distribution functions of Chapter 4. Instead of studying at first hand the generalised Schwarzschild distribution, it is better to take advantage of the algebraic simplicity of the Gaussian distribution, and in a further step, to generalise the derived results to an arbitrary ellipsoidal velocity distribution.

3.2 Schwarzschild distribution The phase space density function f corresponding to a Schwarzschild distribution can be written in the form φ(t, r, u) = f (t, r, u + v(t, r)) according to 1 (3.1) φ(t, r, u) = e− 2 (Q+σ) , Q = uT · A2 · u where Q is a quadratic, positive definite form, with A2 (t, r) a second-rank symmetric tensor and σ(t, r) a scalar function, which are continuous and differentiable in both arguments. Hence, the distribution is of Gaussian type in the peculiar velocities, although it can be multiplied by an arbitrary function of time and position. In such a way the quadratic form Q can give account of the three aforesaid isolating integrals of star motions, so that, in general, it is allowing some friction phenomena which are quantified by the off-diagonal second central moments of the distribution. As it is known, the second moments of the Schwarzschild distribution satisfy (3.2) μ2 = A−1 2 and all the odd-order central moments are obviously null. Let us remark that, for the Schwarzschild distribution, the velocity moments do not depend on

CHAPTER 3. ELLIPSOIDAL DISTRIBUTIONS

34

the function σ appearing in Eq. 3.1 (e.g., de Orús 1977). Only the stellar density, which is obtained from Eq. 1.12, depends on it according to N(t, r) = (2π) 2 |A|− 2 e− 2 σ , 3

1

1

|A| = det A2 .

(3.3)

The more general way to characterise such a trivariate distribution is from its cumulants, which, in addition and opposite to the central moments, have unbiased sample estimators. In general, the relationship between moments with arbitrary mean μn and cumulants κn (Stuart & Ord 1987, §13.11-15) is given by μ1 = κ1 , μ2 = κ2 + S(κ1 ⊗ κ1 ), μ3 = κ3 + S(κ2 ⊗ κ1 ) + S(κ1 ⊗ κ1 ⊗ κ1 ),

(3.4)

μ4 = κ4 + S(κ3 ⊗ κ1 ) + S(κ2 ⊗ κ2 ) + S(κ2 ⊗ κ1 ⊗ κ1 )+ +S(κ1 ⊗ κ1 ⊗ κ1 ⊗ κ1 ), .. . where the notation for symmetrised tensors introduced in Eq. 2.21 will be applied. If centred variables are used, then the odd-order cumulants vanish and the Gaussian distribution remains characterised only from its second cumulants κ2 = μ2 . In other words, the symmetry properties of the Gaussian distribution do also provide vanishing even-order cumulants κn = 0 for n ≥ 4 (Stuart & Ord 1987, §15.3). Then, under those premises, the relationships of Eq. 3.4 are reduced to the following ones, which are written by using the star product notation, μ4 = S(μ2 ⊗ μ2 ) = 3 μ2  μ2 , μ6 = S(μ2 ⊗ μ2 ⊗ μ2 ) = 15 μ2  μ2  μ2 , .. . μ2n = S( .. .

n

n

 μ2 ) = C n μ2  · · ·  μ2 ,

(3.5)

(2n)! Cn = n , 2 n!

Therefore, we can easily obtain the relationship between two consecutive even-order moments. The previous coefficient Cn satisfies the recurrence

3.3. CLOSURE FOR SCHWARZSCHILD DISTRIBUTIONS relation Cn+1 =

(2n + 2)(2n + 1)!(2n)! = (2n + 1) Cn 2n 2(n + 1)n!

35

(3.6)

which allow us to write Eq. 3.5 as Cn+1  S( μ2 )  μ2 = (2n + 1) μ2n  μ2 . Cn n

μ2n+2 =

(3.7)

The relation Eq. 3.7 of moments recurrence will be used to simplify and to reduce higher-order moment equations to lower-order ones, so that such a relationship will provide the key to the closure problem.

3.3 Closure for Schwarzschild distributions In Chapter 2 the general expression for an arbitrary n-order hydrodynamic equation, was written in terms of the generalised tensor of pressures Pn as Eq. 2.27. Such a conservation law for pressures is written in terms of the comoving moments as follows. For n = 0 the continuity equation yields ∂N + v · ∇r N + N∇r · v = 0. ∂t

(3.8)

For n = 1 the momentum equation, also referred as Jeans equation, is expressed as ∂v + v · ∇r v + ∇r U = −(∇r ln N + ∇r ) · μ2 ∂t

(3.9)

and, in general, Eq. 3.9 may be introduced into the higher-order equations to replace the terms depending on the potential function, so that they remain written in terms of the comoving moments, for n ≥ 2, in the form ∂μn + v · ∇r μn − n [(∇r ln N + ∇r ) · μ2 ]  μn−1 + ∂t

(3.10)

+(∇r ln N + ∇r ) · μn+1 + n (μn · ∇r )  v = (0) . n

Let us remember that the equations Eq. 3.8 and Eq. 3.9, for n = 0 and n = 1, contain four different scalar equations, which involve a set of eleven unknown scalar functions (N, U, v and the symmetric tensor μ2 ) and, even in the case of taking into account higher-order equations, the system remains

CHAPTER 3. ELLIPSOIDAL DISTRIBUTIONS

36

  in general open, since by considering the mth equation, with m+2 scalar 2 m+3 equations, we are also introducing as many as 2 new unknowns, which are the different components of the tensor μm+1 . The current approach to the closure problem will consist in to investigate how the velocity distribution function, Eq. 3.1, and the recurrent moment relations it provides, Eq. 3.7, allow to reduce the infinity of equations involved in Eq. 3.10 to a finite subset.

3.3.1 Even-order equations, n ≥ 2 For even-order equations, n = 2k and k ≥ 1, bearing in mind that the oddmoments are null, Eq. 2.35 becomes ∂μ2k + v · ∇r μ2k + 2k (μ2k · ∇r )  v = (0)2k , (3.11) ∂t which, by substitution of the moment expression Eq. 3.5, is transformed into ⎡  ⎤ k k n  ⎢⎢⎢ ⎥⎥ ∂  S( μ2 ) + v · ∇r S( μ2 ) + 2k ⎣⎢S( μ2 ) · ∇r ⎥⎦⎥  v = (0)2k . ∂t (3.12) After some algebra, we have   Ck ∂μ2 Ck + μ2 )  μ2 )  (v · ∇r μ2 )+ k S( k S( Ck−1 ∂t Ck−1 k−1

k−1

 Ck μ2 )  (μ2 · ∇r )  v = (0)2k . + 2k S( Ck−1

(3.13)

k−1

And, by taking into account Eq. 3.7 we can write  ∂μ2 Ck + v · ∇r μ2 + 2 (μ2 · ∇r )  v = (0)2k . μ2k−2  Ck−1 ∂t

(3.14)

Since μ2k−2 never vanishes, all the even-order equations, n ≥ 2, are then reduced to the moment equation of second-order, ∂μ2 + v · ∇r μ2 + 2 (μ2 · ∇r )  v = (0)2 . (3.15) ∂t Therefore, such a relationship, along with the moments recurrence given by Eq. 3.7, provides a closure condition for the even-order hydrodynamic equations.

3.3. CLOSURE FOR SCHWARZSCHILD DISTRIBUTIONS

37

3.3.2 Odd-order equations, n ≥ 3 In a similar way, for odd-order equations, n = 2k + 1, k ≥ 1, and provided that the odd-order moments are null, from Eq. 2.35 we can write −(2k + 1) [(∇r ln N + ∇r ) · μ2 )]  μ2k + (∇r ln N + ∇r ) · μ2k+2 = (0)2k+1 . (3.16) Then, by substitution of the recurrence law for the moments, Eq. 3.7, into the foregoing equation we have −(2k + 1) (∇r ln N · μ2 )  μ2k − (2k + 1) (∇r · μ2 )  μ2k + +(2k + 1) ∇r ln N · (μ2  μ2k ) + (2k + 1) ∇r · (μ2  μ2k ) = (0)2k+1 . (3.17) In order to simplify the previous equation, by regarding the dependence of μ2k in terms of μ2 given by Eq. 3.5, and since ∇r ln N is a vector, we use the equivalence1 (∇r ln N · μ2 )  μ2k = ∇r ln N · (μ2  μ2k ),

(3.18)

so that Eq. 3.17 yields −(2k + 1) (∇r · μ2 )  μ2k + (2k + 1) ∇r · (μ2  μ2k ) = (0)2k+1 .

(3.19)

And now, to further simplify Eq. 3.19, and once more by taking into account Eq. 3.5, we use the following identity2 ∇r · (μ2  μ2k ) = (∇r · μ2 )  μ2k + k (μ2 · ∇r )  μ2k ,

(3.20)

1 In general, for a vector a and for any symmetric tensors A and B , the following equivm n alence is satisfied,

a · S(Am ⊗ Bn ) = S((a · Am ) ⊗ Bn ) + S(Am ⊗ (a · Bn )). 2 In

general, for any symmetric tensors Am and Bn , the following equality is satisfied,

∇ · S(Am ⊗ Bn ) = S((∇ · Am ) ⊗ Bn ) + S((Am · ∇) ⊗ Bn ) + S((∇ ⊗ Am ) · Bn ) + S(Am ⊗ (∇ · Bn )). In particular, if Am = Bn , we have ∇ · S(Am ⊗ Am ) = S((∇ · Am ) ⊗ Am ) + S((Am · ∇) ⊗ Am ) k

 and, if Bn = Am  · · ·  Am , the equality yields ∇ · S(Am ⊗ Bn ) = S((∇ · Am ) ⊗ Bn ) + k S((Am · ∇) ⊗ Bn ).

38

CHAPTER 3. ELLIPSOIDAL DISTRIBUTIONS

so that Eq. 3.19 takes the form (μ2 · ∇r )  μ2k = (0)2k+1 .

(3.21)

Finally, since μ2k = (2k − 1) μ2  μ2k−2 , and μ2k−2 is always nonnull for k ≥ 1, Eq. 3.21 is reduced to the third-order equation (μ2 · ∇r )  μ2 = (0)3 .

(3.22)

Thus, the foregoing expression stands for all the odd-order equations, n = 2k + 1 ≥ 3, and, in virtue of the moments recurrence Eq. 3.7, such a relation provides a closure condition for odd-order hydrodynamic equations in terms of the second central moments μ2 . In conclusion, for a velocity distribution of Schwarzschild type, if the closure conditions given by Eq. 3.7, Eq. 3.15 and Eq. 3.22 are satisfied, then all the moment equations are reduced to the four equations of orders n = 0, 1, 2, 3, which are a set of twenty scalar equations.

3.4 Chandrasekhar’s approach In the previous sections we have been left only with four independent hydrodynamic equations involving the statistics N, v and μ2 . In general, the equations for conservation of mass and momentum, n = 0, 1, do not provide the same model as the collisionless Boltzmann equation, but, for Schwarzschild distributions, since the higher even-order moments can be expressed in the recurrence form of Eq. 3.7, and, in virtue of the equations of orders n = 2 and n = 3 that are acting as closure conditions, then the four hydrodynamic equations are completely equivalent to the Boltzmann equation. Hereafter two implications of such a result are proved. Firstly, the referred set of equations is totally equivalent to the system of equations obtained by Chandrasekhar (1942) for generalised Schwarzschild distributions, which is the Jeans’ inverse problem approach. Secondly, our result for Gaussian velocity distributions is also valid for generalised ellipsoidal velocity distributions. The reason of proceeding in two steps is for mathematical simplicity. It is thus formally proved not only that Chandrasekhar equations are equivalent to a subset of hydrodynamic equations (de Orús 1952, Juan-Zornoza 1995) but also that, because of the closure conditions, they are equivalent to the infinite hierarchy of hydrodynamic equations. Indeed, de Orús (1952) had proved that if Chandrasekhar equations were fulfilled, then the continuity equation and the Jean’s equation were also satisfied. On

3.4. CHANDRASEKHAR’S APPROACH

39

the other hand, working from velocity moments up to fourth-order, JuanZornoza (1995) showed that Chandrasekhar equations could be derived from the first four hydrodynamic equations. Now, from a new and more general approach, the whole set of hydrodynamic equations and velocity moments will be taken into account. Therefore, since Chandrasekhar equations provide the functional dependence of A2 and σ, let us transform the moment equations in terms of those quantities. In Appendix A.1 we find the algebraic details showing that the Eq. 3.22, which corresponds to the equation of order n = 3, and only involves the tensor of moments μ2 , is equivalent to the following condition on the tensor A2 , 3 ∇r  A2 = (0)3 .

(3.23)

From Eq. 3.22, in Appendix A.2 it is also derived an auxiliary property which will be used later. The Eq. 3.23 represents a set of 10 scalar, firstorder linear equations in partial derivatives for the elements of the symmetric tensor A2 . Working from the hydrodynamic equation of order n = 2, Eq. 3.15, along with Eq. 3.23, we obtain in Appendix A.3 the following relationship, ∂A2 − 2 ∇r  (A2 · v) = (0)2 , ∂t

(3.24)

which stands for a set of 6 scalar first-order linear partial differential equations for A2 and v. Also, as an auxiliary property, the divergence of the mean velocity is determined in Appendix A.4. By using both aforesaid auxiliary properties, in Appendix A.5 it is obtained the relationship which is equivalent to the hydrodynamic equation of order n = 1, Eq. 3.9, ∂v 1 + v · ∇r v + ∇r U = A−1 · ∇r σ ∂t 2 2

(3.25)

whose components are 3 scalar equations, the only ones which involve the potential function. Notice that if the phase space density function is strictly of Gaussian type in the velocities with σ(t, r) = 0, then the motion of the centroid does not change neither due to pressure nor to viscosity. In other words, there are no transport phenomena and the centroid moves like a particle under the gravitational potential.

CHAPTER 3. ELLIPSOIDAL DISTRIBUTIONS

40

Finally, in Appendix A.6, the continuity equation Eq. 3.8, for order n = 0, is proved equivalent to the following condition ∂σ + v · ∇r σ = 0 (3.26) ∂t which is one scalar linear differential equation for σ, giving account of the conservation of such a quantity along the centroid path.

3.5 Generalised Schwarzschild distribution In this section it is shown that if the moment equations are fulfilled for a 1 Schwarzschild distribution e− 2 (Q+σ) , then they are also satisfied for a generalised ellipsoidal distribution in the form f (t, r, V ) = ψ(Q + σ)

(3.27)

where ψ is an arbitrary function of the specified argument as defined in Eq. 3.1. For such a distribution the even-order cumulants are not null, as they were in the Gaussian case, although all the even-order central moments can also be computed in terms of the second ones. Thus, the odd-order moments obviously vanish, and the even moments can be expressed from symmetrised tensor products of the second moments, similarly to the Gaussian case, but with a new factor that depends on σ (de Orús 1977) through the integral3  ∞

φn (σ) =

n

1

Q 2 ψ(Q + σ)Q 2 dQ

(3.28)

0

for n even or null. Then, Eq. 1.12 provides a stellar density in the form N(t, r) = 2π |A|− 2 φ0 (σ) 1

(3.29)

and, by using the notation which was introduced in Eq. 3.5 and Eq. 3.6, the even-order moments can be written as n

μ2n =

Cn+1 =

3 The

ϕ(s) =

 φ2n (σ) −1 Cn A−1 2  · · ·  A2 = φ0 (σ)

1

1 Cn+1

(3.30)

φ2n (σ)  −1 S( A2 ). φ0 (σ) n

 ∞integral is related to the Mellin transform, which, for a function f (x), is defined as xs−1 f (x)dx (e.g., Ditkin & Prudnikov 1965). 0

3.5. GENERALISED SCHWARZSCHILD DISTRIBUTION

41

Now, in order to investigate the closure conditions, a similar procedure as for Gaussian distributions could be applied, but in the current case it is quite more long and tedious, since the moments depend on the functions φk (σ) defined in Eq. 3.28. Hence we shall use an indirect and shorter approach. In Appendix B it is shown that the family of Schwarzschild functions 

 1 e− 2 (Q+σ) k k ,

k ∈ N − {0}

(3.31)

constitutes a non-orthogonal basis of the space of square-integrable functions over the interval (0, +∞) for the variable Q, so that the integrals of Eq. 3.28, which arise when computing the moments, are convergent. Then the velocity distribution, which we assume as to be continuous and differentiable, can be formally expressed as a linear combination of them, according to ψ(Q + σ) =

∞ 

γk−1 e− 2 (Q+σ) k . 1

(3.32)

k=1

Therefore, an arbitrary integrable, quadratic distribution ψ(Q+σ) can be written as a uniformly convergent series of Gaussian functions in the pecu1 liar velocities, e− 2 (Q+σ) k , for k ≥ 1, all of them having zero mean. Hence, all the centroids of the partial distributions –for each term of the series– have the same mean velocity v. Under those premises, not only the collisionless Boltzmann equation is linear for the phase space density function, but also all the hydrodynamic equations, Eq. 2.27, are linear in the pressures, since the total nth pressures is simply the sum of the partial nth pressures. Notice that if the mean velocities were different for each partial distribution, as the linearity for the pressures would not then hold, the hydrodynamic equations might be considered separately for each distribution component. Then, the tensor involved in the exponent of the kth term in Eq. 3.32 is k A2 , and the accompanying function of t and r is k σ. In addition, we can see that for each Schwarzschild component the moment equations in terms of A2 and σ, equations Eq. 3.23 to Eq. 3.26, remain invariant whether A2 and σ are respectively exchanged by k A2 and k σ. Therefore, due to the linear condition of the problem, if each Gaussian summation term in Eq. 3.31 satisfies the moment equations, then an arbitrary generalised ellipsoidal distribution ψ(Q + σ), according to Eq. 3.32, do satisfy them too.

42

CHAPTER 3. ELLIPSOIDAL DISTRIBUTIONS

3.6 Remarks The general expression for the moment equations that was derived in Chapter 2 has proved useful to study the closure problem for trivariate ellipsoidal velocity distributions, without any additional hypotheses. By this way, the infinite hierarchy of moment equations and unknowns are reduced to a finite number of them, which are equivalent to the collisionless Boltzmann equation, so that a feasible dynamical model is available. In general, under the functional approach, the statistical properties of the velocity distribution function may be derived and used to reduce the whole set of moment equations to a finite subset. For trivariate Schwarzschild distributions a recurrence relation between even central moments has been found, which allow to reduce the moment equations only to four different orders, being the even-order equations for n ≥ 2 equivalent to the one of order n = 2, and the odd-order equations for n ≥ 3 equivalent to the one of order n = 3. Therefore, the equations for mass and momentum transfer do not generate the same model as the collisionless Boltzmann equation, whereas if the moment equations of orders n = 2 and n = 3 are considered, along with the relation of moments recurrence, which are acting as closure conditions, the model they provide is the same as the one Chandrasekhar had derived by working from the Boltzmann equation. For generalised Schwarzschild distributions, a similar recurrence law for central moments would also provide some closure conditions, but, for the sake of mathematical simplicity, an alternative method has been preferred. The result derived for Schwarzschild distributions has been extended in a natural way, owing to the linear nature of the problem, to generalised Schwarzschild distributions, which can be expanded as a power series of Schwarzschild functions with the same mean velocity.

Chapter 4

The closure problem 4.1 The Boltzmann and moment equations It has been said that the infinite hierarchy of the stellar hydrodynamic equations cannot be used as a dynamical model to study a stellar system unless they are reduced to a finite number of equations and unknowns. In most applications, only the lowest order equations are generally used, in addition to some particular assumptions to restrict the number of unknowns, like the epicyclic approach, axial symmetry, or by taking a velocity distribution function depending on specific integrals of the star motion. When working with the Jeans equation alone, or with a finite set of hydrodynamic equations, the collisionless Boltzmann equation is not generally fulfilled and then arises the closure problem of how the infinite hierarchy of moment equations and unknowns are reduced to a finite subset, which is still equivalent to the collisionless Boltzmann equation. A first approach to the closure problem has been discussed and solved for ellipsoidal velocity distributions in the previous chapter. It has been proven for a Schwarzschild distribution that the whole set of hydrodynamic equations is reduced to the moment equations of the orders n = 0, 1, 2, 3, which were equivalent to the collisionless Boltzmann equation. In a more general case, for an arbitrary ellipsoidal trivariate velocity distribution, some analogous closure conditions have been obtained. In both cases, the closure conditions are related to some recurrent relationships of the velocity moments, so that moments of fourth and higher orders may be determined in terms of the lower-order moments. Therefore, it is suggested that the closure 43

44

CHAPTER 4. THE CLOSURE PROBLEM

problem is related to the macroscopic properties of the velocity distribution, although different velocity distributions may have analogous closure conditions and a similar set of independent moment equations. In the current chapter, the closure problem is studied in a more general way, by describing the whole family of phase space density functions, for which the collisionless Boltzmann equation is strictly equivalent to a finite subset of hydrodynamic equations. In addition, it is proven that the redundancy of the higher-order moment equations and the recurrence of the velocity moments are of similar nature. The method is based on the use of maximum entropy distributions, which allow an easy algebraic treatment. The closure problem is solved in two steps. First, a family of maximum entropy distributions is considered, which is a generalisation of an exponential quadratic function to an exponential polynomial function of degree n in the velocity variables. Second, the maximum entropy functions are taken as a basis of square-integrable functions, to expand any arbitrary, nonexponential function (assumed to be continuous and differentiable) in terms of a n-degree polynomial, as a uniformly convergent power series. For such a general family of functions, i.e., the phase space density functions depending on any isolating integral of motion expressed as polynomial function in the velocities, the equivalence between the first set of moment equations, up to order n + 1, and the collisionless Boltzmann equation is proven.

4.2 Maximum entropy function Although the closure problem will be studied for a wider family of density functions, a functional approach with maximum entropy distributions is firstly considered. Basically, they are generalised exponential distributions, including the exponential, normal, lognormal, gamma, and beta as special cases. The main properties of these density functions are reviewed in Chapter 5. From a thermodynamical viewpoint, a maximum entropy density is a function (a) that depends on a linear combination of the collisional invariants, i.e. mass, momentum, and energy; (b) for which the collision term of the Boltzmann equation is exactly zero, i.e., it is a solution of the collisionless Boltzmann equation; and (c) that minimises Boltzmann’s entropy functional. This solution represents a local equilibrium state, in the sense that other solutions to the Boltzmann equation become asymptotically close to it. But thermodynamical entropy is a particular case of information entropy, firstly introduced by Shannon (1948). If the conserved macroscopic quantities involve moments accounting for mass and energy, Shannon’s in-

4.2. MAXIMUM ENTROPY FUNCTION

45

formation entropy takes its maximum value in the form of Maxwellian distributions. If the conservation extends to all the second moments, the information entropy takes the extreme value for Gaussian distributions. In general, the number of constraints may involve higher-order moments, by reflecting a more complex situation in which the stars interact with the potential and with themselves. Indeed, the moment constraints are a direct consequence of the isolating integrals of the stellar motion, or more precisely, they reflect particular combinations of the integrals of motion which are conserved. Therefore, we can adopt the viewpoint of the Jeans’ inverse problem. By following Chandrasekhar’s functional approach for the phase space density function, to begin we may assume that the velocity distribution satisfies a polynomial exponential law. Indeed, this is just the form of a maximum entropy function in terms of the velocity variables. In the multivariate case, it is written as (e.g., Kouskoulas et al. 2004), f (t, r, V ) = eP(V ) ,

(4.1)

where P(V ) represents a power series in the velocity components with coefficients depending on time and position. It contains as many terms as the number of available moment constraints, so that each coefficient is related to a single moment constraint. A maximum entropy distribution has a very simple functional form, allowing us to show straightforwardly the equivalence of the collisionless Boltzmann equation and a finite set of moment equations. Afterwards, it can be easily generalised to non-maximum entropy distributions, i.e. to a non-exponential law f (P). We write the phase space density function as f = ePn ,

Pn =

 α+β+γ≤n

λαβγ (t, r) V1α V2β V3γ ,

(4.2)

where the subindex n does not represent the number of terms, but the polynomial degree. If the velocity domain ΓV is infinite, the polynomial Pn must be upper bounded to satisfy the integrability conditions given by Eq. 1.24. Otherwise, a truncated bell-shaped velocity distribution might be considered.

CHAPTER 4. THE CLOSURE PROBLEM

46

4.3 Fundamental system of equations The conservation of the integral of motion Pn is expressed by substituting Eq. 4.2 into Eq. 1.11, by considering Eq. 1.10. The equation DPn =0 Dt

(4.3)

represents a linear and homogeneous system of partial differential equations for the coefficients of Pn . Thus, taking the collisionless form of Eq. 1.3 into account, the above equation can be explicitly written as ∂Pn ∂Pn ∂U ∂Pn + Vk − = 0. ∂t ∂rk ∂rk ∂Vk

(4.4)

For the remaining part of this section, it is more convenient to write Eq. 4.2 by using Latin indices, along with Einstein’s summation criterion for repeated indices, Pn = λ0 + λi Vi + λi j Vi V j + · · · + λi1 ...in Vi1 · · · Vin .

(4.5)

to the kth power of the veThe coefficients λi1 ...ik in the term corresponding k+2 locities are symmetric, so that we have 2 different coefficients. Hence,    n+3 up to the nth power, it makes nk=0 k+2 = 3 different coefficients. The k correspondence between the Greek and Latin indices notation for the coefficients of Pn , depending on the use of either Eq. 4.2 or Eq. 4.5, is given by k! (4.6) λαβγ = λ1 . . . 1 2 . . . 2 3 . . . 3 ; k = α + β + γ. α!β!γ!    α

β

γ

Then, Eq. 4.4 becomes ∂λi j ∂λi1 ...in DPn ∂λ0 ∂λi = + Vi + Vi V j + · · · + Vi1 · · · Vin + Dt ∂t ∂t ∂t ∂t +

∂λi j ∂λi1 ...in ∂λ0 ∂λi Vk + Vi Vk + Vi V j Vk + · · · + Vi1 · · · Vin Vk − ∂rk ∂rk ∂rk ∂rk

∂U  − λi δik + λi j δik V j + λi j δ jk Vi + · · · + λi1 ...in δi1 k Vi2 · · · Vin + ∂rk  + · · · + λi1 ...in δin k Vi1 · · · Vin−1 = 0.

(4.7)

4.3. FUNDAMENTAL SYSTEM OF EQUATIONS

47

To simplify the notation, by permuting some indices and considering the symmetry of the coefficients, according to Eq. 2.20 for symmetrised tensors we define the following tensors Λ0 =

∂λ0 ∂U − λi , ∂t ∂ri

∂λi ∂λ0 ∂U + −2 λik , ∂t ∂ri ∂rk   ∂λi j ∂U ∂λi Λi j = −3 λi jk , +S ∂t ∂r j ∂rk

Λi =

···

(4.8) 

Λi1 ...in−1 = Λi1 ...in



∂λi1 i2 ...in−2 ∂λi1 i2 ...in−1 ∂U +S −n λi i ...i , ∂t ∂rn−1 ∂rin 1 2 n

  ∂λi1 i2 ...in−1 ∂λi1 i2 ...in +S = , ∂t ∂rin 

Λi1 ...in+1

 ∂λi1 i2 ...in =S , ∂rin+1

which are symmetric quantities1 . Hence, it suffices to consider only the indices satisfying 1 ≤ i1 ≤ i2 ≤ . . . ≤ 3. Therefore, from equations Eq. 4.7 and Eq. 4.8, the collisionless Boltzmann equation can be written in terms of the elements of tensors Λk as DPn = Λ0 + Λi Vi + Λi j Vi V j + · · · + Λi1 ...in+1 Vi1 · · · Vin+1 = 0, Dt

(4.9)

which must be fulfilled for all possible values of thevelocity. Then, since each symmetric tensor Λk , k = 0, . . . , n + 1, has k+2 2 different elements, a 1 Remember

the equivalence  S

 ∂λi1 i2 ...ik−1 = k(∇r  λk−1 )i1 i2 ...ik . ∂rik

CHAPTER 4. THE CLOSURE PROBLEM

48 number of

n+4 3

scalar equations are in total involved. They are Λ0 = 0, Λi = 0, Λi j = 0, ··· Λi1 ...in+1 = 0.

(4.10)

In this way, the collisionless Boltzmann equation is equivalent to the above system of partial differential equations, depending on the variables t and r.

4.4 Closure of moment equations We are ready to see that the foregoing system of equations is equivalent to a similar set of independent moment equations, and that the remaining hydrodynamic equations are obtained from linear combinations of them. Firstly, from Eq. 1.23 and 4.3, we write the general expression of the moment equations for an arbitrary order l ≥ 0,  β γ DPn Pn 3 e d V = 0; α + β + γ = l. V1α V2 V3 (4.11) Mαβγ = Dt ΓV n The foregoing relationship can be interpreted as the inner product of DP Dt and α β γ an element φαβγ = V1 V2 V3 , belonging to the basis of the space of squareintegrable functions in ΓV , with respect to the weight function ePn . Hence, we can express the first equality in a more compact form,

Mαβγ = φαβγ ,

DPn . Dt

(4.12)

We also write the left-hand side of Eq. 4.9 by making explicit the velocity powers, in the Greek indices notation, similar to Eq. 4.2,  DPn = Λαβγ (t, r) V1α V2β V3γ . (4.13) Dt α+β+γ≤n+1 A relationship similar to Eq. 4.6 between capital lambdas is hold. Finally, by substitution of Eq. 4.13 into Eq. 4.12, we obtain how the moment equations are related to the tensors Λk ,  Mαβγ = φαβγ , φιμν Λιμν ; α + β + γ = l, (4.14) ι+μ+ν≤n+1

4.4. CLOSURE OF MOMENT EQUATIONS

49

Notice that the above expression involves the moments m(α+ι) (β+μ) (γ+ν) = φαβγ , φιμν . If the foregoing system of equations is restricted to the first set of indices, 0 ≤ l ≤ n + 1, we get the same number of equations as coefficients Λιμν . The above system matrix may be written as a squared matrix G2 of inner products of the velocity components, G(αβγ, ιμν) = φαβγ , φιμν ; α + β + γ ≤ n + 1, ι + μ + ν ≤ n + 1.

(4.15)

Then, G2 is a Gram matrix, symmetric, positive definite and, among other well-known properties, is invertible. In this notation, we write this subsystem contained in Eq. 4.14 as  Mαβγ = G(αβγ, ιμν) Λιμν ; α + β + γ ≤ n + 1. (4.16) ι+μ+ν≤n+1

From the above relationship, the first elementary conclusion is that the finite system of equations, Λιμν = 0; ι + μ + ν ≤ n + 1,

(4.17)

corresponding to the collisionless Boltzmann equation, is fulfilled if, and only if, the finite set of hydrodynamic equations, Mαβγ = 0; α + β + γ ≤ n + 1

(4.18)

is also fulfilled. A second consequence is that Eq. 4.16 can be inverted, so that the components Λιμν can be explicitly expressed in terms of linear combinations of the hydrodynamic equations Mαβγ , up to order n + 1, according to the following relation,  Λαβγ = G−1 (αβγ, ιμν) Mιμν ; α + β + γ ≤ n + 1, (4.19) ι+μ+ν≤n+1

where G−1 (αβγ, ιμν) is the corresponding element of the inverse of matrix G2 . Thus, the finite set of elements Λαβγ , α + β + γ ≤ n + 1, may be introduced in Eq. 4.14 to express the infinite hierarchy of higher-order moment equations in terms of the lower-order hydrodynamic equations Mιμν , ι + μ + ν ≤ n + 1,   φαβγ , φα β γ G−1 (α β γ , ιμν) Mιμν (4.20) Mαβγ = α +β +γ ≤n+1

ι+μ+ν≤n+1

50

CHAPTER 4. THE CLOSURE PROBLEM

for α + β + γ ≥ n + 2. The foregoing relationship clearly shows the recurrence of the higherorder equations in terms of a finite set of the lower-order ones. Therefore, for maximum entropy density functions, the closure of the stellar hydrodynamic equations is proven.

4.4.1 Notation The equation 4.16 may be written by using Latin indices as well, although the usual algebraic notation is also possible, i.e., matrices depending on two indices and vectors on a single index. To establish this exact correspondence, the summation terms of Eq. 4.14 might be ordered in the following form: any quantity ϕαβγ of order k = α + β + γ, written in Greek indices, with each index taking values from 0 to k, can be sorted under a triple loop, 0 ≤ γ ≤ k, k − γ ≤ β ≤ k, and k − γ − β ≤ α ≤ k, along with increasing k. In Latin indices, it is equivalent to writing ϕi1 i2 i3 ··· with i1 i2 i3 · · · ∈ {1, 2, 3} and 1 ≤ i1 ≤ i2 ≤ i3 ≤ · · · ≤ 3. For example, for order k = 4, the Greek indices sequence {400, 310, 301, 220, 211, 202, 130, 121, 112, 103, 040, 031, 022, 013, 004} is equivalent to the Latin indices sequence { 1111, 1112, 1113, 1122, 1123, 1133, 1222, 1223, 1233, 1333, 2222, 2223, 2233, 2333, 3333 }. In general, for Greek indices, the total infinite sequence may be represented as σ, the ith -element as σ[i], and a finite sequence up to order k as σk . Similarly, in Latin indices, we could refer to them as s, s[i], and sk , respectively. By this way, each element ϕαβγ could be written as depending on a single index, ϕσ[i] , and Eq. 4.16 would become  Gσ[l]σ[k] Λσ[k] ; σ[l] ∈ σn+1 . Mσ[l] = σ[k] ∈ σn+1

Therefore, mn+1 and Λn+1 may be regarded as vectors, whose components are the ones of the respective tensors up to order n + 1 and they are written according to a given sequencing criterion.

4.5 Arbitrary polynomial function The closure of the moment equations is also valid for any continuous and differentiable density function f (Pn ), even if it is not a maximum entropy function. By following the same steps as in the Appendix B, it can be shown that any integrable phase density f (Pn ), that can be written as f (Pn ) = F(Pn )ePn ,

4.5. ARBITRARY POLYNOMIAL FUNCTION

51

being F(Pn ) a square-integrable function with respect to the weight ePn in ΓV , can be expressed as the following uniformly convergent power series f (Pn ) =

∞ 

γk−1 ek Pn ,

(4.21)

k=1

where the coefficients γk−1 are constant, and f depends on time, space, and velocity through Pn , as in Eq. 4.2. Thus, in the interior of ΓV , this generalised Fourier series can be integrated or differentiated term by term without losing its uniformly convergence property. Instead of repeating the full derivation of the above property, the velocity distribution function may be written in similar terms as in Appendix B, where it is shown that any arbitrary quadratic density function f (Q + σ) 1 can be expressed as a convergent series of the Gaussian functions e− 2 (Q+σ) k with k ≥ 1. Let us assume an infinite velocity domain ΓV . Since Pn is upper bounded, there exists a value ζ, which may depend on time and position, such that Pn < ζ for all velocity V ∈ ΓV . Thus, we can write Pn = − 21 (Qn + σ) with Qn = −2(Pn − ζ) a positive definite form, and σ = −2ζ. Hence, we are in the appropriate conditions to show that any function f (Qn + σ) can be expressed as a convergent power series in terms 1 of e− 2 (Qn +σ) . The case of a finite velocity domain is quite similar but with the corresponding changes concerning the domain of the variables. Thus, if ζ1 < Pn < ζ2 , then the variable τ = 12 (Qn + σ) belongs to the interval I = (−ζ2 , −ζ1 ), and the variable η = e−τ belongs to the interval J = (e−ζ2 , e−ζ1 ). On the other hand, due to the boundary conditions, Eq. 1.24, when the velocity V approaches the boundary domain ΓV , then Pn → −∞ and f (Pn ) → 0. According to Eq. 1.10, since f (Pn ) > 0, f must be an increasing function of Pn , so that d f (Pn ) > 0, (4.22) dPn in the interior of the domain Γr × ΓV .

4.5.1 Moment equations By taking derivatives in Eq. 4.21, d f (Pn )  = γk−1 k ek Pn , dPn k=1 ∞

(4.23)

CHAPTER 4. THE CLOSURE PROBLEM

52

and bearing in mind Eq. 1.11, we have D(k Pn ) D f (Pn )  γk−1 ek Pn = = 0. Dt Dt k=1 ∞

(4.24)

Then, for any k  0, the equations Eq. 4.5 and Eq. 4.9 are also fulfilled with ˜ i = k λi , and Λ  i = k Λi ; since Pn is linearly dependent on the  n = k Pn , λ P DPn tensor elements λi , and Dt and Λi are similarly related. Then, each term of the above series satisfies the collisionless Boltzmann equation, so that a linear relationship, similar to Eq. 4.16, holds. Therefore, for each term of the series Eq. 4.24, we get the integrals  D(k Pn ) k Pn 3 (k) e d V = 0; α + β + γ = l, φαβγ (4.25) Mαβγ = Dt ΓV and we sum up, according to the coefficients of the series, so that we obtain ∞ 

 γk−1 M(k) αβγ =

ΓV

k=1

φαβγ

∞ 

γk−1

k=1

D(k Pn ) k Pn 3 e d V. Dt

(4.26)

Hence, according to Eq. 4.24, we are led to the general expression of the moment equations, which generalises Eq. 4.11  D f (Pn ) 3 d V = 0. φαβγ (4.27) Mαβγ = Dt ΓV For any order l = α + β + γ, the l-order moment equation is obtained as linear combination of the moment equations associated with each term of the series Eq. 4.24, ∞  Mαβγ = γk−1 M(k) (4.28) αβγ . k=1

4.5.2 Equivalence The relationship between the hydrodynamic equations and the collisionless Boltzmann equation is now established by substitution of Eq. 4.13 in Eq. 4.26, ⎛ ⎞  ∞  ⎜⎜⎜  ⎟⎟⎟ ⎜ Mαβγ = φαβγ γk−1 ⎜⎝⎜ Λιμν φιμν ⎟⎟⎠⎟ k ek Pn d3 V . (4.29) ΓV

k=1

ι+μ+ν≤n+1

4.6. REMARKS

53

By reordering terms, Mαβγ =



Λιμν

ι+μ+ν ≤n+1

⎡ ⎢⎢⎢ ⎢⎢⎣

ΓV

φαβγ φιμν

⎛∞ ⎞ ⎤ ⎜⎜⎜ ⎟⎟⎟ 3 ⎥⎥⎥ k P n ⎜⎜⎝ γk−1 k e ⎟⎟⎠ d V ⎥⎥⎦ ,

(4.30)

k=1

and bearing in mind Eq. 4.23, we may also write   d f (Pn ) 3 d V Λιμν . φαβγ φιμν Mαβγ = dPn ΓV

(4.31)

ι+μ+ν ≤n+1

Therefore, if only orders α + β + γ ≤ n + 1 are considered, we are led to a similar relationship as Eq. 4.14, but with the inner product calculated with the weight function given by Eq. 4.22, which will be notated with a Gram 2 . The resulting integrals are some generalised velocity moments, matrix G f (Pn ) so that when f (Pn ) is a maximum entropy function then f (Pn ) = d dP n and the generalised moments become ordinary velocity moments. Thus we write   Mαβγ = G(αβγ, ιμν) Λιμν ; α + β + γ ≤ n + 1. (4.32) ι+μ+ν ≤ n+1

The relationship between a finite set of hydrodynamic equations and the collisionless Boltzmann equation, given by the equations Eq. 4.16, Eq. 4.18, 2 Λn+1 , so that and Eq. 4.19, is now expressed as Mn+1 = G Λn+1 = 0n+1 ⇐⇒ Mn+1 = 0n+1 .

(4.33)

Therefore, for any density function depending on an n-degree polynomial function Pn , there is a finite set of independent moment equations, for the orders i = 0, 1, . . . , n + 1, which is equivalent to the collisionless Boltzmann equation. Furthermore, a recurrence law for moment equations similar to Eq. 4.20, but with the weight function given by Eq. 4.22, is satisfied.

4.6 Remarks The description of how the Galaxy relaxes towards a steady state is still a matter of debate, but there are two processes that likely play an important role: phase mixing and violent relaxation. Lynden-Bell (1967), in a seminal work, gave a statistical description of how a rapid fluctuating gravitational field produces a relaxation mechanism under the collisionless Boltzmann

54

CHAPTER 4. THE CLOSURE PROBLEM

equation, which involves phase mixing, by changing the coarse-grained phase-space density near the phase point of each star, and violent relaxation, analogous to collisions in a gas, by changing the energy per unit mass of a star. Lynden-Bell’s approach leads, for a non-degenerate stellar system, to a Maxwell-Boltzmann macroscopic distribution. Improvements to the previous approach (e.g., Chavanis et al. 1996, Chavanis 1998) take into account, among other aspects, the self-confinement of the Galaxy, which is related to the incomplete relaxation problem due to the hypothesis of ergodicity; the maximum-entropy production principle to obtain a closure of the relaxation equation of diffusion type (non collisionless) for the coarse-grained distribution function; and the estimation of the diffusion current, which generalises the Chandrasekhar (1943) and Lynden-Bell (1967) equations. These and similar approaches, from a statistical viewpoint, and following the Jeans’ direct problem, lead to the most probable distribution function for an equilibrium configuration of the Galaxy, and provide information about the functional form of the distribution function, or about the conserved quantities along the stellar motion, by leading to a distribution function that may take the form f (Pn ). In this stage, once the system has achieved relaxation, and according to Jeans’ inverse problem, the situation can be reversed. It can be approached not from the statistical dynamics viewpoint but from analytical dynamics, by assuming the regularity conditions about the definition of the LSR, the continuity and differentiability of its velocity, and the existence of higherorder velocity moments. We then ask under which circumstances the collisionless Boltzmann equation admits a solution of the form f (Pn ). This is indeed the appropriate context to study the motion of the centroid and the admissible form of the potential function. Therefore, dissipative forces are not considered in the collisionless Boltzmann equation, but are indirectly connected with the functional form of the distribution function. This situation can be also tackled by using the stellar hydrodynamic equations, which explicitly involves the velocity moments. Then, the closure problem necessarily arises of how the infinite hierarchy of moment equations is related to the finite character of the collisionless Boltzmann equation. In this context, the equivalence of Boltzmann and moment equations was investigated in Chapter 3 for a polynomial degree n = 2. In this case, Eq. 4.10 corresponds to the Chandrasekhar’s (1942) system of equations, which allows us to obtain, under a time-dependent model and different symmetry hypotheses, a quite general solution for the collisionless Boltzmann equation, with the possibility of describing a stellar system with arbitrary mean velocity and orientation of velocity ellipsoid (Sanz-Subirana

4.6. REMARKS

55

& Català-Poch 1987, Sala 1990, Juan-Zornoza & Sanz-Subirana 1991, JuanZornoza 1995). In this case, de Orús (1952) proved that if the Chandrasekhar equations are fulfilled, the continuity equation and the Jean’s equation are also satisfied. In the same vein, working from velocity moments up to fourth-order, Juan-Zornoza (1995) showed that Chandrasekhar equations could be derived from the first four hydrodynamic equations. A more general result (Cubarsi 2007, 2013) for generalised Schwarzschild distributions was derived in Chapter 3, that is, the first four hydrodynamic equations, along with a moment recurrence relationship acting as closure condition, make the infinite hierarchy of hydrodynamic equations equivalent to the collisionless Boltzmann equation. In Chapter 4 the above results were generalised (Cubarsi 2010b, 2016) to any velocity distribution function depending on a polynomial function in the velocity variables. The degree of this polynomial function, which is an isolating integral of the stellar motion, may be used to quantify the complexity of the velocity distribution. In Chapter 6, under a maximum entropy approach, this complexity will be in some way measured in terms of the necessary set of velocity moments, for obtaining a good fit of the velocity distribution. Only in the most basic situations the stellar systems can be approximated by an ellipsoidal distribution, such as for the thin disc, thick disc, or halo, as independent Galactic components. Those systems are well described with moments up to second order and, therefore, according to our results, they can be modelled by using the moment equations up to third order, or the system of Chandrasekhar equations, equivalent to the collisionless Boltzmann equation. Otherwise, stellar systems having a significant deviation from the ellipsoidal hypothesis, such as the whole Galactic disc, must be modelled through some more complex distributions and with higher-order hydrodynamic equations. We summarise the main results. The following statements are equivalent: (a) The velocity distribution depends on an integral of motion which is a polynomial function of degree n. (b) There is an independent set of velocity moments, up to an order n, so that the higher-order moments can be expressed in terms of the independent moments. (c) The collisionless Boltzmann equation is given by a set of differential equations expressed from symmetric tensors of rank up to n + 1. (d) The independent moment equations are those of an order of up to n+1.

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CHAPTER 4. THE CLOSURE PROBLEM

(e) The hydrodynamic equations of an order higher than n + 1 are redundant. Therefore, any analytic isolating integral of the stellar motion, obtained, for instance, as a superposition of products of the basic integrals reviewed in Chapter 1, when it is expanded in any finite power series of the velocities, it provides as many independent velocity moments and hydrodynamic equations as the number of coefficients corresponding to the maximum degree in the series.

Chapter 5

Maximum entropy approach 5.1 The problem of moments The asymmetry of the local velocity distribution was first studied in 1905 by Kapteyn in his theory of two star streams and further developed by Kapteyn (1922), Strömberg (1925), and Charlier (1926), which considered up to fourth moments of the velocity distribution. However, those moments were not determined with a sufficient degree of accuracy up to Erickson (1975). During the past decade, higher order velocity moments with better precision could be obtained from large and representative stellar samples of the solar neighbourhood (ESA 1997, Nordtröm et al. 2004), accounting for velocity discontinuities and kinematic populations in the solar neighbourhood (Alcobé & Cubarsi 2005). Several approaches have been tried to describe the asymmetry of the velocity distribution. In the beginning, an anisotropic velocity distribution was obtained by superposition of isotropic phase-density functions with different means. Later, the Schwarzschild distribution, based on a single trivariate Gaussian distribution, could easily handle the basic anisotropic features, and more parameters could be controlled by assuming non-Gaussian ellipsoidal distributions. However, to account for nonnull, odd-order central moments, it was once again necessary to return to mixture models. In addition to the works describing the actual velocity distribution from a mixture of stellar populations (e.g., Soubiran & Girard 2005, Vallerani et 57

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al. 2006), there is a wide variety of approaches that generally do not make use of the velocity moments, such as the two- or three-integral models based in Fricke (1952) components (Evans et al. 1997, Famaey et al. 2002, Jiang & Ossipkov 2007) or even a combination of a Gaussian part of the density function with a perturbation factor expressed in a polynomial form in terms of the integrals of motion (van der Marel & Franx 1993, Gerhard 1993, Kormendy et al. 1998). The velocity distribution is sometimes numerically estimated (Dehnen 1998, Skuljan et al. 1999, Bovy et al. 2009), although it is also frequent the analytical modelling (Famaey et al. 2005, Veltz et al. 2008). However, in the latter case, according to today’s observational data, some intricate trivariate distribution functions (or with a very high number of components) may be obtained. In most of these works, there is the job of describing the detailed structure of the velocity distribution, or of associating specific moving groups with the density function components, although in most cases the small groups do not have a clear visual impact on the overall density function. There is also a desire for a simple, qualitative description of the distribution in terms of basic measures of spread or asymmetry like the skew or for a comparison to Gaussian distributions, like the kurtosis. For trivariate distributions with strong asymmetries, e.g., the structure that lies under the groups of young and early-type stars, the statistical moments are the natural tool for such a description of the basic geometric trends. To this purpose, the method of moments is revisited here. An alternative analytical model based on the maximum entropy approach is described to account for the eventual asymmetries of the velocity distribution, which are collected through its sample moments. Even though such an approach has been widely used to solve many univariate technical and scientific problems, there were no general application to stellar kinematics until Cubarsi (2010a). There are several numeric algorithms for estimating the maximum entropy density function, which are not computationally trivial for the trivariate case. However, if an extended set of moments is available, the method described in this work allows a parameter estimation by solving a linear system of equations. Its simplicity makes it worthwhile using it to construct any ad hoc velocity distribution function.

5.2 Maximum entropy distribution The maximum entropy approach to the solution of inverse problems was introduced long ago by Jaynes (1957), so that it provides a unique solution that is the best one for not having to deal with missing information. It agrees

5.2. MAXIMUM ENTROPY DISTRIBUTION

59

with what is known, but expresses maximum uncertainty with respect to all other matters. It is a flexible and powerful tool for density approximation, which collects a complete family of generalised exponential distributions, including the exponential, normal, lognormal, gamma, and beta as special cases. Other properties of maximum entropy distributions are outlined in the following section. An interesting application of the maximum entropy approach is the problem of moments (Mead & Papanicolaou 1984), which is described while introducing the notation accordingly to the astronomical formulation. Therefore, we chose a density function maximising Shannon’s information entropy and we study the necessary complexity of the velocity distribution for satisfying a set of moment constraints. The current approach will simplify both analytical dependence and parameter estimation of the distribution function. Hereafter, when studying the velocity dependence of the distribution function from a statistical viewpoint, the variables of time and position will be omitted, although they might be used in studying the dynamical model for the whole phase-space distribution function. In Chapter 3 we saw that the Schwarzschild distribution can be described in terms of their central second moments μi j , however, more general and anisotropic distributions have a wider set of independent moments, and, in a more general case, for a large family of probability density functions, the exact distribution may be univocally determined by the infinite hierarchy of independent moments. Provided an order for a set of moments (for example according to the Latin indices notation 0, 1, 2, 3, 11, 12, 13, 22, and so on) if the first m moments are known, it is possible to find an infinite variety of functions whose first m moments coincide with the above set. Various approximation procedures exist to find a sequence of functions fm , which fulfils the foregoing moment constraints and converges to the true distribution as m approaches infinity. Fortunately, between those sequences of functions, a uniquely maximum entropy sequence exists that maximises the entropy functional  fm (V ) ln fm (V ) dV . (5.1) W( fm ) = − ΓV

Then, the maxima f = fm is usually called the least biassed sequence of approximations, and, by using Lagrangian multipliers, it can be shown (e.g., Kagan et al. 1973, Kouskoulas et al. 2004) that it has the form f (V ) = eP(V ) ,

(5.2)

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CHAPTER 5. MAXIMUM ENTROPY APPROACH

where P(V ) is a power series of the velocity components containing m terms, as many terms as the number of moment constraints, so that each coefficient is related to a single moment constraint. Then, the sequence fm of maximum entropy functions, such as increasing m, is able to fit more complex and informative systems, so that the higher the number of moment constraints, the lower the entropy, the symmetry, and the algebraic simplicity of fm . The solution of the maximum entropy problem usually consists in solving a set of m nonlinear equations in the form  mk = (V )k eP(V ) dV . (5.3) ΓV

However, these solution techniques are typically not easy to generalise to the multidimensional problem. On the other hand, even for the unidimensional problem, an analytical solution generally does not exist for higher than second moments. Generally, the numerical techniques for solving the coefficients of the polynomial P are based on nonlinear optimisation, Legendre transformation, etc. (e.g., de Bruin et al. 1999, Kouskoulas et al. 2004), and either way, they are not easy to implement. However, if an extended set of moments is known, then the parameters can be estimated by solving a linear system of equations. In the case of trivariate distributions, for a polynomial of degree n in three variables, it is necessary to compute moments up to order 2(n − 1). The current purpose is to infer the trivariate velocity distribution from a finite set of moment constraints. To simplify estimation of the polynomial coefficients of P(V ), an alternative method has been developed, based on a unique assumption that the velocity distribution satisfies the boundary conditions associated with the stellar hydrodynamic equations, also known as moment equations.

5.2.1 Boundary conditions If the phase-space distribution function f satisfies the collisionless Boltzmann equation, DDtf = 0, then by multiplying it by the n-tensor power of the star velocity and by integrating over the whole velocity space, the family of stellar hydrodynamic equations Eq. 1.23 was obtained. In Chapter 2, the above equations were derived in terms of the central or comoving moments, in a completely analytical way, for any order n and without any additional hypotheses. Then, if the above integrals exist and since there are no stars

5.2. MAXIMUM ENTROPY DISTRIBUTION

61

with velocity beyond ΓV , the following boundary conditions were, as usual, assumed in the integration process: (V )n f (t, r, V ) |V ∈∂ΓV = (0)n , n ≥ 0.

(5.4)

These boundary conditions are satisfied by a wide family of distributions that are bell-shaped in any direction of the velocity domain. One of the integral properties that was derived in Chapter 2 will allow establishment of a Gramian system of equations for solving our estimation problem. From a purely statistical inference viewpoint, the requirement of estimating the distribution parameters is not that the phase density function is the solution to the collisionless Boltzmann equation, but it is enough that it satisfies, or approximately satisfies, the above boundary conditions. The entropy functional W( f ), as defined in Eq. 5.1, is far from containing all the information about the Boltzmann equation (with or without collisions) since W( f ) only depends on the velocity space, similar to the collision operator of the complete Boltzmann equation. In the following section, we discuss how such a maximum entropy density function may or may not be a solution to the collisionless Boltzmann equation. In review, two typical cases of maximum entropy distribution function are solutions to the whole set of moment equations. The simplest case is an isothermal velocity distribution of Maxwell type in the peculiar velocities, Eq. 2.37, which according to the Maxwell-Boltzmann law, represents a system with the more basic thermal equilibrium. The distribution is totally isotropic, so that has equal diagonal second central moments, vanishing offdiagonal second moments, and zero odd-order moments as well. Another well-known example is the Schwarzschild distribution, Eq. 3.1. The distribution may have some different diagonal second central moments and nonvanishing off-diagonal moments, although the odd-order moments still vanish. Therefore, it is a maximum entropy function constrained by the whole set of moments up to second order. The above examples, which are integrable functions in an infinite velocity domain, satisfy the boundary conditions, Eq. 5.4, and can be generalised according to an exponential function, Eq. 5.2, with as many polynomial terms as available moment constraints, under the necessary conditions over the polynomial coefficients to obtain an integrable distribution function. For higher-degree polynomials, the distribution function is integrable if the polynomial is upper bounded, and therefore the polynomial must be even. On the other hand, truncated distributions, which are associated with velocity-bounded stellar samples, |V − V0 | ≤ const., have a finite velocity

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domain. Then the boundary conditions are still a good approximation if the truncated distribution vanishes enough when approaching the contour of the velocity domain, so that the density function may be assumed null out of this boundary. Thus, for a domain that is either bounded or unbounded, we assume that the velocity distribution is continuous, differentiable, and positive in the interior of the velocity domain ΓV and that the boundary conditions are fulfilled in its contour ∂ΓV .

5.2.2 Properties The maximum entropy density function is again explicitly written as Pn

f =e ,

Pn =

n   k=0 α+β+γ=k

β

γ

λαβγ (t, r) V1α V2 V3 ,

(5.5)

where the subindex n does not represent the number of polynomial terms, but rather the maximum polynomial power. If the velocity domain ΓV is all the space R3 , the polynomial Pn must be upper bounded to satisfy the integrability conditions. As a result, the power series of the velocities reaches a natural value n, which must be even, and,  for the highest degree k = n, the n-adic form α+β+γ=n λαβγ (t, r)V1α V2β V3γ must be negative definite. Let us remember that Eq. 4.5 allowed us to write Pn in Eq. 5.5 with Latin indices, instead of Greek indices. The same properties derived from Eq. 4.2 in Chapter 4 are now valid, in particular Eq. 4.6 relating the λ coefficients between Eq. 5.5 and Eq. 4.5. Some practical aspects of the maximum entropy distribution function may still be pointed out. Under maximum entropy distributions, the sample moments are maximum likelihood estimators of the population moments. Equation 5.5, in addition to including equations Eq. 2.37 and Eq. 3.27 as particular cases, it also contains, in general, any desired type of two- or three-integral functions (e.g., Hénon 1973, Dejonghe 1983, White 1985). It represents a general functional approach, in a similar way to Fricke (1952), with the difference that, while the distribution function in the Fricke-based models is either a linear combination or product of the powers of integrals of motion, in Eq. 5.5 the linear combination of powers of integrals of motion appears as the argument in the exponential function. When n → ∞, Eq. 5.5 converges to the true distribution. Then, if the

5.2. MAXIMUM ENTROPY DISTRIBUTION

63

velocity distribution is expressed as a power series of the velocities, we have ⎛ ⎞ ∞   ⎜⎜⎜ ⎟⎟ β γ⎟ α ⎜ f (t, r, V ) = c0 ⎜⎜⎝1 + cαβγ (t, r)V1 V2 V3 ⎟⎟⎟⎠ = k=1 α+β+γ=k

⎞ ⎛∞ ⎟⎟⎟ ⎜⎜⎜  β γ λαβγ (t, r)V1α V2 V3 ⎟⎟⎟⎠ = exp ⎜⎜⎜⎝

(5.6)

k=0 α+β+γ=k

which is a similar relationship between generalised moments cαβγ and cumulants λαβγ (Stuart & Ord 1987, p. 437), where the coefficient c0 = exp(λ0 ) provides the normalisation of the distribution1 . A maximum entropy distribution function can exhibit several modes. In the trivariate case, if Eq. 5.5 is a polynomial of even degree n, the distribution can exhibit (n/2)3 modes, since an univariate exponential with a polynomial of degree n may have up to n/2 modes. In general, it is necessary to estimate less number of parameters for Eq. 5.5 than for a mixture of trivariate Gaussian distributions accounting for the same number of modes.

5.2.3 Information entropy Let us briefly explain how to interpret a maximum entropy density function, or better, what is the appropriate context for its use. Up to a change of sign, Shannon’s information entropy is defined as the Boltzmann H-functional, which first appeared in statistical mechanics in works by Boltzmann and Gibbs in the 19th century. However, it is not exactly the same concept. Boltzmann’s functional is used for non-equilibrium systems and it is related to the irreversibility of dynamical processes in a uniform gas. For elastic collisions involving short-range forces and in the absence of boundaries, mass, momentum, and energy are conserved in binary encounters (e.g., Cercignani 1988). They are usually referred to as collisional invariants. There is only one distribution function, the Maxwellian distribution, fulfilling all of the following properties: it depends on a linear combination of the collisional invariants, the collision term of the Boltzmann equation is exactly zero, and it minimises Boltzmann’s entropy. This solution represents a local 1 When a similar relation holds for the characteristic function Φ, which is the Fourier transform of density function f , then the coefficients cαβγ become proportional to the population moments mαβγ , and λαβγ become proportional to the cumulants of the distribution καβγ , by a 1 . factor α!β!γ!

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64

equilibrium state, in the sense that other solutions to the Boltzmann equation will become closer to it as the time goes by. Depending on the potential, boundary conditions, and dissipative or collision effects, maximum entropy solutions can be non-Maxwellian (e.g., Villani 2002). Shannon’s information entropy2 was introduced in communication theory to measure the redundancy of a language and the maximal compression rate, which is applicable to a message without any loss of information. It is defined for complex systems and is related to Boltzmann’s entropy as a measure of the number of microstates associated with a given macroscopic configuration. On the other hand, the Fisher information was introduced as ˘ ˘ ˙I as a measure of the uncerpart of his theory of âAIJefficient statisticsâA tainty. It is also related to Shannon’s entropy, so that the entropy quantifies the variation of information. If we maximise the entropy subject to some constraints (e.g., statistics describing macroscopic properties) we get distributions containing maximum uncertainty that is compatible with these constraints. For given mass and energy, the Fisher information takes its minimum value and Shannon’s entropy its maximum value in the form of Maxwellian distributions. For a given covariance matrix, they take extreme values for Gaussian distributions. The number of constraints involved in the Lagrange multipliers may reach higher order moments, by reflecting more complex situations in which the stars interact with the potential and with themselves, as well as having different masses. We quote Jaworsky (1987) to point out that these two typical viewpoints for interpreting the entropy as uncertainty. In mathematical statistics and information theory, the entropy functional is maximised by attending to some constraints that express any available information of a complex physical system, which depend on the actual experimental situation. In statistical mechanics the entropy is used to study the thermodynamic equilibrium or non-equilibrium of a physical system, generally a uniform gas, in terms of the mean values of some physical quantities, which describe the macroscopic state of a physical system as a whole, like energy or number of particles. Thus, statistical mechanics based on this principle can be interpreted 2 The

following quotation by Shannon, extracted from Martin & England (1981), is amusing: My greatest concern was how to call it. I thought of calling it ‘information’. But the word was overly used, so I decided to call it ‘uncertainty’. When I discussed it with John Von Neumann, he had a better idea. He told me: “You should call it entropy, for two reasons. In the first place your uncertainty has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one knows what entropy really is, so in a debate you will always have the advantage”.

5.2. MAXIMUM ENTROPY DISTRIBUTION

65

as a special type of statistical inference. The use of higher order statistical moments in addition to the mean values represents a generalisation of the thermodynamic concept of entropy, which is used to approximate the exact probability distributions for a few specified random variables when a finite number of their moments is known. The maximum entropy principle implies that the resulting distribution belongs to the exponential family. The actual moment constraints are a direct consequence of the isolating integrals of the stellar motion, or more precisely, they reflect particular combinations of the isolating integrals that are conserved. More complex distributions exist than the Maxwellian, which are maximum entropy distributions and are solution of the collisionless Boltzmann equation. These solutions are generally obtained by assuming that Liouville’s theorem is satisfied, so that the essential information about the density function is provided by the isolating integrals of the motion of the stars. Thus, if we assume that the polynomial form P of Eq. 5.1 depends on the integrals of motion and is itself an integral of motion, Liouville’s theorem is equivalent to the collisionless Boltzmann approximation. Then, the collisionless Boltzmann equation obviously takes the form d f (P) DP =0 dP Dt

(5.7)

df so that the factor dP accounts for the maximum entropy condition, and the DP factor Dt is, in fact, the collisionless Boltzmann condition. Thus, both conditions are independent and compatible. If the maximum entropy criterion is fulfilled, then the function f (P) = eP takes the smoothest possible form, while the dependence of P in terms of the powers of the velocity, as well as in terms of time and position through its polynomial coefficients, is, in this approach, independent of the maximum entropy condition. Thus, we may affirm that the maximum entropy procedure is non-essential to the solution of the collisionless Boltzmann equation. The physical mechanism providing such a maximum entropy function is irrelevant to the statistical approach. In contrast, what is important is the set of statistical moments accounting for the macroscopic state, which, of course, have a dynamical significance in terms of viscosity, conductivity, or diffusion effects. The present statistical approach adopts the opposite viewpoint of studying possible warming mechanisms that modify a Schwarzschild distribution, and to then test how the distribution fits the actual velocity moments (e.g., Dehnen 1999). In the current method, the available information is condensed within the polynomial P(V ). The maximum entropy approach then gives a very good mathematical estimation of

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66

the density function and of its velocity derivatives involved in DP Dt = 0, although it may or may not match any physical model. On the other hand, the maximum uncertainty in the light of the missing information is guaranteed by the function f (P) = eP .

5.3 Moments problem We now study a quite general case for fitting a defined set of velocity moments, up to order 2(n − 1), with a maximum entropy velocity distribution containing a polynomial of degree n, which allows a simple and linear estimation of the polynomial coefficients. By using Latin indices notation for Pn , according to Eq. 4.5, we assume that all the moments of Eq. 5.3 exist, which is equivalent to considering the distribution function to be a squareintegrable function in the velocity domain ΓV . The scalar λ0 is the normalisation factor, and, in general, all of the above coefficients are symmetric elements of k-rank tensors λk ; k = 0, . . . n involved in Eq. 4.5. The other coefficients than λ0 can be obtained by using the property 

 ∇V (V )m ePn d3 V = (0)n+1 , (5.8) ΓV

which is a direct consequence of Eq. 5.4 and, in particular, is fulfilled by any solution of the moment equations. The above integral is an (n + 1)-rank tensor, which is symmetric with respect to the indices of the tensor power (V )m . Thus, when integrating Eq. 5.8 by components and the conditions of Eq. 5.4 are applied over the domain of any variable Vim+1 , we get  ΓVi

m+1

∂(Vi1 . . . Vim ePn ) dVim+1 = Vi1 . . . Vim ePn |ΓVi = 0. m+1 ∂Vim+1

(5.9)

In the case of a finite velocity domain, if the density function is bell-shaped, the null value of the right-hand side might be substituted by a tolerance error, namely εim+1 = Vi1 . . . Vim ePn |ΓVi , (5.10) m+1

such that this value can be neglected on condition of being significantly small.3 3 For example, the velocity density function f of local stars is approximately Gaussian in the component W perpendicular to the galactic plane, with dispersion σW  19 km s−1 . A finite

5.3. MOMENTS PROBLEM

67

In particular, for m = 0, since ∂ePn ∂Pn = ePn ∂Vk ∂Vk 

we have

ΓV

∂Pn Pn 3 e d V = 0. ∂Vk

Similarly, for m = 1,    ∂ ∂Pn Pn 3 (Vi ePn )d3 V = δik ePn d3 V + Vi e d V =0 ∂Vk ΓV ∂Vk ΓV ΓV

(5.11)

(5.12)

(5.13)

where δik is the Kronecker delta. And, in general, for m ≥ 2, we get ∂(Vi1 . . . Vim ePn )  &i j . . . Vim + = δi1 im+1 Vi2 . . . Vim + . . . + δi j im+1 Vi1 . . . V ∂Vim+1  ∂ePn + . . . + δim im+1 Vi1 . . . Vim−1 ePn + Vi1 . . . Vim ∂Vim+1

(5.14) where the hat indicates the omitted factors. Once more, bearing Eq. 5.11 in mind, the identity Eq. 5.8 yields   &i j . . . Vim + . . . + δi1 im+1 Vi2 . . . Vim + . . . + δi j im+1 Vi1 . . . V ΓV





∂Pn Pn 3 e d V = 0. ∂V im+1 ΓV (5.15) Since the first integral is symmetric with respect to permutation of indices, and in general it is not null, then the second integral  qm+1 = (V )m ⊗ (∇V Pn ) ePn d3 V (5.16) +δim im+1 Vi1 . . . Vim−1 e

Pn

d V + 3

Vi1 . . . Vim

ΓV

velocity domain IW = [−Wmax , Wmax ] could be then assumed, with Wmax = 220 km s−1 , where  n f) −5 for the integral I ∂(W ∂W dW is exactly null for even values of n, and remains less than ∼ 10 W odd values n < 13. Obviously, the integral is still lower for wider intervals. Similarly, the local young-disc stars, with absolute heliocentric velocity up to 51 km s−1 , have a velocity dispersion σW  11 km s−1 . In the similar situation above, we may then assume a finite velocity domain for the truncated velocity distribution with Wmax = 120 km s−1 , where such an integral can be neglected up to powers n = 12.

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must be symmetric, too. Indeed, equations Eq. 5.15 and Eq. 5.16 are equivalent to those obtained in Chapter 2 as equations Eq. 2.11 and Eq. 2.18, which were derived for expressing the conservation of pressures. In this new context, the above identities will provide a linear method of fitting any desired set of moments. In contrast to the usual maximum entropy methods for the moments problem, which are nonlinear and not well conditioned enough, the present method allows all of the coefficients to be determined with accuracy. First we evaluate ∇V Pn starting from Eq. 4.5, ∂Pn = λk + 2λ j1 k V j1 + 3λ j1 j2 k V j1 V j2 + · · · + nλ j1 j2 ... jn−1 k V j1 · · · V jn−1 . (5.17) ∂Vk To obtain all of the elements of tensors λk ; k = 1, . . . , n, we compute the integrals of Eq. 5.8 for m from 0 to n − 1. For m = 0, by taking equations Eq. 5.12 and Eq. 5.17 into account, and by using the moments definition Eq. 5.3, since m0 = 1, we have λk + 2λ j1 k m j1 + 3λ j1 j2 k m j1 j2 + · · · + nλ j1 j2 ... jn−1 k m j1 ... jn−1 = 0,

(5.18)

which stands for a set of 3 scalar equations, k = 1, 2, 3. For m = 1, also by taking equations Eq. 5.13 and Eq. 5.17 into account, we get δik + λk mi + 2λ j1 k m j1 i + 3λ j1 j2 k m j1 j2 i + · · ·+ nλ j1 j2 ... jn−1 k m j1 j2 ... jn−1 i = 0. (5.19) Hence, this set of relations, for i, k = 1, 2, 3, thanks to the symmetry of Eq. 5.16, provides 6 independent scalar equations. And, in general, for m = n − 1, from Eq. 5.15 we likewise get δi1 in mi2 ...in−1 + . . . + δi j in mi1 ... i j ...in−1 + . . . + δin−1 in mi1 ...in−2 + +λin mi1 ...in−1 + 2λ j1 in m j1 i1 ...in−1 + 3λ j1 j2 in m j1 j2 i1 ...in−1 + · · · +

(5.20)

+nλ j1 j2 ... jn−1 in m j1 ··· jn−1 i1 ...in−1 = 0,   which consists in a set of n+2 independent scalar equations, i1 , . . . , in = 2 1, 2, 3, owing to the symmetry of Eq. 5.16. Therefore, we have as many independent linear equations as unknowns composing the elements of the symmetric tensors λk , for k = 1, . . . n, whose elements are the coefficients of Pn . Such a non-homogeneous system can be associated with a Gramian matrix, as we see in the next section.

5.4. GRAMIAN SYSTEM

69

Finally, the scalar λ0 left to be evaluated may be obtained to satisfy   1 Pn 3 −λ0 eλi Vi +λi j Vi V j +···+λi1 ...in Vi1 ···Vin d3 V . (5.21) e d V = Ne = N ΓV ΓV

5.4 Gramian system The three scalar equations involved in Eq. 5.18, corresponding to m = 0, for k = 1, 2, 3, are homogeneous in the elements of tensors λk . In Eq. 5.19, for m = 1, we group the terms containing the elements of λk , by writing the other ones on the right-hand side, and likewise for the general equation with m = n − 1, Eq. 5.20. Thus we obtain the following linear system of equations for the elements of tensors λk , λk + 2λ j1 k m j1 + 3λ j1 j2 k m j1 j2 + · · · + nλ j1 j2 ... jn−1 k m j1 ... jn−1 = 0, λk mi + 2λ j1 k m j1 i + 3λ j1 j2 k m j1 j2 i + · · · + nλ j1 j2 ... jn−1 k m j1 j2 ... jn−1 i = −δik , .. . λin mi1 ...in−1 + 2λ j1 in m j1 i1 ...in−1 + 3λ j1 j2 in m j1 j2 i1 ...in−1 + + · · · + nλ j1 j2 ... jn−1 in m j1 ··· jn−1 i1 ...in−1 =   = − δi1 in mi2 ...in−1 + . . . + δi j in mi1 ... i j ...in−1 + +δin−1 in mi1 ...in−2 . (5.22) Such a system of equations can be grouped according to three different vectors on its right-hand side, for k = 1, 2, 3 in the first two equations and for in = 1, 2, 3 in the general expression. A similar procedure can be applied to the λk coefficients. The system matrix G2 is displayed in Table 5.1 and can be interpreted as a symmetric matrix of inner products with respect to the weight ePn of the velocity components V0 Vi V j ..., V0 V p Vq ... according to the Latin indices notation, with V0 ≡ 1 and the other indices sorted as 1 ≤ i ≤ j ≤ . . . ≤ 3 and 1 ≤ p ≤ q ≤ . . . ≤ 3. Therefore, G2 is a Gram matrix, symmetric, positivedefinite and, among other well-known properties, it is invertible, meaning the system has a unique solution. Thus, we may define, according to Table 5.2, the following three column matrices X = [a, b, c] and Y = [A, B, C] so that the following equality is satisfied:

CHAPTER 5. MAXIMUM ENTROPY APPROACH 70

0 0 1 1 2 3 11 12 13 22 23 33 111 112 . .. 333 . ..

1 m1 m11

2 m2 m12 m22

3 m3 m13 m23 m33

11 m11 m111 m112 m113 m1111

12 m12 m112 m122 m123 m1112 m1122

13 m13 m113 m123 m133 m1113 m1123 m1133

22 m22 m122 m222 m223 m1122 m1222 m1223 m2222

23 m23 m123 m223 m233 m1123 m1223 m1233 m2223 m2233

33 m33 m133 m233 m333 m1133 m1233 m1333 m2233 m2333 m3333

111 m111 m1111 m1112 m1113 m11111 m11112 m11113 m11122 m11123 m11133 m111111

112 m112 m1112 m1122 m1123 m11112 m11122 m11123 m11222 m11223 m11233 m111112 m111122

··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· . ..

333 m333 m1333 m2333 m3333 m11333 m12333 m13333 m22333 m23333 m33333 m111333 m112333 . .. m333333

··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· . .. ··· . .. (5.23)

Table 5.1: Matrix G2 is a symmetric matrix of inner products of the velocity components V0 Vi V j ..., V0 V p Vq ... , according to Latin indices, with V0 ≡ 1 and the other indices sorted as 1 ≤ i ≤ j ≤ . . . ≤ 3 and 1 ≤ p ≤ q ≤ . . . ≤ 3. Note: The external first row and first column refer to the velocity indices. Since the matrix is symmetric, only the diagonal and upper triangular part are written.

5.4. GRAMIAN SYSTEM

71

Y = G2 X.

(5.24)

This is the numerical form of the system of equations Eq. 5.22, ready to be solved. The coefficients to compute are elements of the symmetric tensors λk , for orders k = 1, . . . , n, since   the zero order coefficient is the normalisation factor. In total there are n+3 3 − 1 independent coefficients. Each column of the matrix X is composed of: -one element of the symmetric tensor λ1 , which multiplies the moments of orders 0, 1, . . . , n − 1 in the first column of matrix G2 ; -three elements of λ2 , which multiply the moments of orders 1, 2, . . . , n in the next three columns  of matrix G2 ; -and, in general, k+2 elements of the symmetric tensor λk+1 , which 2 multiply the moments of orders k, .. . , k + n − 1, up to the value k = n − 1. Therefore, the matrix G2 has n+2 rows and columns, where the mo3 ments up to order 2(n − 1) are involved. For example, for n = 2 we use the matrix G2 with the first row containing moments up to first order (1+3=4 columns in total) and the last row containing moments up to order 2, a 4 × 4 matrix. For n = 4 we use the matrix G2 with the first row containing moments up to order 3 (1+3+6+10=20 columns in total) and the last row containing moments up to order 6, as a 20 × 20 matrix. Similarly, for n = 6 we use the matrix G2 with the first row containing moments up to order 5 (1+3+6+10+15+21=56 columns in total) and the last row containing moments up to order 10, as a 56 × 56 matrix. On the other hand, since the matrices X and Y consist of three column vectors, we dispose of a number of 3 n+2 3 equations. This number, for n > 1, is always greater than the number of independent unknowns (leaving out the normalisation factor). For example, in the case n = 2, we have 12 equations and 9 independent unknowns, because the symmetric coefficients λ12 , λ13 , and λ23 are equivalent to λ21 , λ31 , and λ32 , respectively, and similarly for higher values of n. In general, if the true distribution is indeed a maximum entropy distribution, the actual moments will be consistent with the symmetry of the coefficients, but a significant deviation from the maximum entropy property will produce some non-symmetric coefficients4 . To avoid this situation, an equivalent overdeterminate system of equations may be built up, as explained in Appendix D, where the symmetric coefficients of tensors λn will not be repeated in the vector of unknowns. The system 4 This is true for n > 2 but, for n = 2, Appendix C shows that the coefficients λ are related ij to the second central moments μ−1 i j , which are necessarily symmetric.

CHAPTER 5. MAXIMUM ENTROPY APPROACH

72

may be solved by applying a least squares method, so that to get the minimum squared error of the fit, it is weighted in terms of the inverse sampling variances σ 2 of the moments up to order n − 2, in the right-hand side of Eq. 5.22. In addition, a predictor-corrector method may be applied to evaluate the variance matrix of the unknowns.

5.4.1 Polynomial coefficients Similar to Eq. 5.25, but for the general case of a non-maximum entropy function, it is also possible to prove the linear relationship between the coefficients of the polynomial function Pn and a finite set of extended velocity moments. For the general case of an arbitrary density function f (Pn ), we compute the coefficients λk , 1 ≤ k ≤ n, of Eq. 4.5, in terms of an extended set of moment constraints, by integrating  ΓV

( ' ∇V (V )m f (Pn ) d3 V = (0)n+1 ,

(5.25)

as a result of applying in the integration process the boundary conditions 2 is now a matrix given by Eq. 5.4. The resulting Gramian system matrix G α β γ of inner products associated with the basis φαβγ = V1 V2 V3 , with regard to f (Pn ) given by Eq. 4.22. the weight d dP n We write the integrand of Eq. 5.25 in Greek indices and we assume α + β + γ = m, 0 ≤ m ≤ n − 1. By taking the V1 -derivative, we have ∂(V1α V2β V3γ f (Pn )) = ∂V1 = α V1α−1 V2β V3γ f (Pn ) + V1α V2β V3γ = α V1α−1 V2β V3γ f (Pn ) + V1α V2β V3γ

∂Pn d f (Pn ) = ∂V1 dPn  ι+μ+ν≤n

λιμν ι V1ι−1 V2μ V3ν

d f (Pn ) . dPn

The last summation can be carried out from ι ≥ 1 instead of ι ≥ 0. Thus, by

5.4. GRAMIAN SYSTEM m=0 m=1 m=2

m=3

m=4

m=5

.. .

A 0 −1 0 0 −2 m1 −m2 −m3 0 0 0 −3 m11 −2 m12 −2 m13 −m22 −m23 −m33 0 0 0 0 −4 m111 −3 m112 −3 m113 −2 m122 −2 m123 −2 m133 −m222 −m223 −m233 −m333 0 0 0 0 0 −5 m1111 −4 m1112 −4 m1113 −3 m1122 −3 m1123 −3 m1133 −2 m1222 −2 m1223 −2 m1233 −2 m1333 −m2222 −m2223 −m2233 −m2333 −m3333 0 0 0 0 0 0 .. .

B 0 0 −1 0 0 −m1 0 −2 m2 −m3 0 0 −m11 0 −2 m12 −m13 0 −3 m22 −2 m23 −m33 0 0 −m111 0 −2 m112 −m113 0 −3 m122 −2 m123 −m133 0 −4 m222 −3 m223 −2 m233 −m333 0 0 −m1111 0 −2 m1112 −m1113 0 −3 m1122 −2 m1123 −m1133 0 −4 m1222 −3 m1223 −2 m1233 −m1333 0 −5 m2222 −4 m2223 −3 m2233 −2 m2333 −m3333 0 .. .

73 C 0 0 0 −1 0 0 −m1 0 −m2 −2 m3 0 0 −m11 0 −m12 −2 m13 0 −m22 −2 m23 −3 m33 0 0 −m111 0 −m112 −2 m113 0 −m122 −2 m123 −3 m133 0 −m222 −2 m223 −3 m233 −4 m333 0 0 −m1111 0 −m1112 −2 m1113 0 −m1122 −2 m1123 −3 m1133 0 −m1222 −2 m1223 −3 m1233 −4 m1333 0 −m2222 −2 m2223 −3 m2233 −4 m2333 −5 m3333 .. .

a λ1 2λ11 2λ12 2λ13 3 λ111 2 · 3 λ112 2 · 3 λ113 3 λ122 2 · 3 λ123 3 λ133 4 λ1111 3 · 4 λ1112 3 · 4 λ1113 3 · 4 λ1122 6 · 4 λ1123 3 · 4 λ1133 4 λ1222 3 · 4 λ1223 3 · 4 λ1233 4 λ1333 5 λ11111 4 · 5 λ11112 4 · 5 λ11113 6 · 5 λ11122 12 · 5 λ11123 6 · 5 λ11133 4 · 5 λ11222 12 · 5 λ11223 12 · 5 λ11233 4 · 5 λ11333 5 λ12222 4 · 5 λ12223 6 · 5 λ12233 4 · 5 λ12333 5 λ13333 6 λ111111 5 · 6 λ111112 5 · 6 λ111113 10 · 6 λ111122 20 · 6 λ111123 10 · 6 λ111133 10 · 6 λ111222 30 · 6 λ111223 30 · 6 λ111233 10 · 6 λ111333 5 · 6 λ112222 20 · 6 λ112223 30 · 6 λ112233 20 · 6 λ112333 5 · 6 λ113333 6 λ122222 5 · 6 λ122223 10 · 6 λ122233 10 · 6 λ122333 5 · 6 λ123333 6 λ133333 .. .

b λ2 2λ21 2λ22 2λ23 3 λ211 2 · 3 λ212 2 · 3 λ213 3 λ222 2 · 3 λ223 3 λ233 4 λ2111 3 · 4 λ2112 3 · 4 λ2113 3 · 4 λ2122 6 · 4 λ2123 3 · 4 λ2133 4 λ2222 3 · 4 λ2223 3 · 4 λ2233 4 λ2333 5 λ21111 4 · 5 λ21112 4 · 5 λ21113 6 · 5 λ21122 12 · 5 λ21123 6 · 5 λ21133 4 · 5 λ21222 12 · 5 λ21223 12 · 5 λ21233 4 · 5 λ21333 5 λ22222 4 · 5 λ22223 6 · 5 λ22233 4 · 5 λ22333 5 λ23333 6 λ211111 5 · 6 λ211112 5 · 6 λ211113 10 · 6 λ211122 20 · 6 λ211123 10 · 6 λ211133 10 · 6 λ211222 30 · 6 λ211223 30 · 6 λ211233 10 · 6 λ211333 5 · 6 λ212222 20 · 6 λ212223 30 · 6 λ212233 20 · 6 λ212333 5 · 6 λ213333 6 λ222222 5 · 6 λ222223 10 · 6 λ222233 10 · 6 λ222333 5 · 6 λ223333 6 λ233333 .. .

c λ3 2λ31 2λ32 2λ33 3 λ311 2 · 3 λ312 2 · 3 λ313 3 λ322 2 · 3 λ323 3 λ333 4 λ3111 3 · 4 λ3112 3 · 4 λ3113 3 · 4 λ3122 6 · 4 λ3123 3 · 4 λ3133 4 λ3222 3 · 4 λ3223 3 · 4 λ3233 4 λ3333 5 λ31111 4 · 5 λ31112 4 · 5 λ31113 6 · 5 λ31122 12 · 5 λ31123 6 · 5 λ31133 4 · 5 λ31222 12 · 5 λ31223 12 · 5 λ31233 4 · 5 λ31333 5 λ32222 4 · 5 λ32223 6 · 5 λ32233 4 · 5 λ32333 5 λ33333 6 λ311111 5 · 6 λ311112 5 · 6 λ311113 10 · 6 λ311122 20 · 6 λ311123 10 · 6 λ311133 10 · 6 λ311222 30 · 6 λ311223 30 · 6 λ311233 10 · 6 λ311333 5 · 6 λ312222 20 · 6 λ312223 30 · 6 λ312233 20 · 6 λ312333 5 · 6 λ313333 6 λ322222 5 · 6 λ322223 10 · 6 λ322233 10 · 6 λ322333 5 · 6 λ323333 6 λ333333

Table 5.2: System of Eq. 5.22 is grouped as a three column matrix.

CHAPTER 5. MAXIMUM ENTROPY APPROACH

74

noting ι − 1 as ι, we then have β

γ

∂(V1α V2 V3 f (Pn )) = α V1α−1 V2β V3γ f (Pn )+ ∂V1 (5.26)



+V1α V2β V3γ

λ(ι+1)μν (ι +

ι+μ+ν≤n−1

d f (Pn ) 1) V1ι V2μ V3ν . dPn

Similarly, the other derivatives are ∂(V1α V2β V3γ f (Pn )) = β V1α V2β−1 V3γ f (Pn )+ ∂V2 +V1α V2β V3γ

 ι+μ+ν≤n−1

∂(V1α V2β V3γ

β

γ

d f (Pn ) , dPn (5.27)

f (Pn ))

∂V3 +V1α V2 V3

λι(μ+1)ν (μ + 1) V1ι V2μ V3ν



= γ V1α V2β V3γ−1 f (Pn )+ μ

ι+μ+ν≤n−1

λιμ(ν+1) (ν + 1) V1ι V2 V3ν

d f (Pn ) . dPn

Then, if the above expressions are substituted into Eq. 5.8, by using the notation of Eq. 4.32, we get   −α m(α−1)βγ = G(αβγ, ιμν) (ι + 1) λ(ι+1)μν , ι+μ+ν≤n−1

−β mα(β−1)γ =



 G(αβγ, ιμν) (μ + 1) λι(μ+1)ν ,

ι+μ+ν≤n−1

−γ mαβ(γ−1) =



(5.28)  G(αβγ, ιμν) (ν + 1) λιμ(ν+1) .

ι+μ+ν≤n−1

According to this notation, all the moments with a negative index must be considered null. The elements of tensors λk , 1 ≤ k ≤ n, involved in Eq. 4.5, are explicitly obtained in terms of the generalised moments up to an order 2(n − 1), as well as of the ordinary velocity moments up to an order n − 2,

5.5. REMARKS

75

by inverting the above system of equations:  −1 −1 (αβγ, ιμν) ι m(ι−1)μν , G λ(α+1)βγ = α + 1 ι+μ+ν≤n−1 λα(β+1)γ =

λαβ(γ+1) =

−1 β+1 −1 γ+1



−1 (αβγ, ιμν) μ mι(μ−1)ν , G

ι+μ+ν≤n−1



(5.29) −1 (αβγ, ιμν) ν mιμ(ν−1) . G

ι+μ+ν≤n−1

−1 (αβγ, ιμν) The foregoing expressions are valid for α + β + γ ≤ n − 1, and G 2 . is the corresponding element of the inverse of the matrix G Once the polynomial coefficients are calculated, it is possible to express the higher-order generalised moments in terms of them, making use of Eq. 5.28, for α+β+γ ≥ n, by using the corresponding inner products φαβγ , φιμν  with the new weight, instead of G(αβγ, ιμν). Also, as commented in the previous section, for a given density function, the quantities λk , k ≤ n, are univocally related to the minimum set of moments mk , k ≤ n, so that all the higher-order moments could be computed from the former set. However, the proven linear relationship makes use of an extended set of moments up to an order 2(n − 1). Only for n = 2, the minimum and the extended set of moments match up. Thus, for n > 2, the existence of a general analytical relationship involving only the minimum set of moments should be further investigated.

5.5 Remarks The system of equations involved in Eq. 5.22 allows us to compute the elements of tensors λk ; 1 ≤ k ≤ n in terms of the velocity moments up to order 2(n − 1), which is the highest order involved in Eq. 5.20. For the case n = 2, it is easy to solve the Gramian system in an analytical way and to find how moments of order higher than two depend on the second ones, as shown in Appendix C. For higher values of n, however, it might be done by using the numerical procedure outlined in Appendix D, which provides an optimal estimation. Also, for n = 2, the integrability of the distribution function in an infinite velocity domain is easily derived from the

76

CHAPTER 5. MAXIMUM ENTROPY APPROACH .

tensor λ2 , since λi j = − 12 μ−1 i j and the tensor of second central moments μ2 is positive-definite. For n ≥ 4, it is impossible to guarantee the definiteness of the tensor λn in a general way. This is a problem related to Hilbert’s 17th problem. In one dimension, real polynomials of a single variable has the fundamental property that every non-negative polynomial is a sum of squares of polynomials. In multiple dimensions, however, it is possible for a polynomial to be non-negative without being a sum of squares. Hilbert in his 17th problem conjectured that every non-negative polynomial can always be written as a sum of squares of rational functions. This conjecture was later proved to be correct by E. Artin in 1926. An immediate extension is to consider characterisations of polynomials that are non-negative on closed domains given by polynomial inequalities. This is a field to investigate, which is beyond the scope of the present monograph. However, by using a finite velocity domain defined from the velocity ellipsoid, one as wide as needed depending on the working stellar sample, such a problem may be easily avoided for the truncated distributions discussed in the next chapter.

Chapter 6

Local velocity distribution 6.1 Stellar samples As an application of the past chapters, the maximum entropy approach is used to describe the main kinematical features of the neighbourhood stars by working from a recent, large, kinematically representative local stellar sample. In a first case, the large-scale distribution of the local disc is inferred from Sample A coming from the GCS catalogue (Nordström et al. 2004, Holmberg et al. 2007, Holmberg et al. 2009). It has new and more accurate radial velocity data than previous samples, and contains the total velocity space of F and G dwarf stars, which are considered the favourite tracer populations of the history of the disc. The largest sample providing stable velocity moments is used. This application provides and confirms some general and well-known trends in the background velocity distribution, such as the overall vertex deviation, the skewness, or the symmetry plane of the distribution. Sample A, which mainly contains thin and thick disc stars, can be sufficiently described from an exponential density function with a four-degree polynomial, although a six-degree polynomial provides a more accurate portrait of the local velocity distribution. According to the maximum entropy modelling, it is possible to interpret the phase space density function as a product of two exponential functions, the one giving a background ellipsoidal shape of the distribution and the other, which is even and at least quadratic in the rotation velocity alone, acting as a perturbation factor that breaks the distribution symmetry. An alternative approach to infer the large-scale structure is to apply the 77

78

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

superposition principle. Large stellar populations can be identified with galactic components such as thin disc, thick disc, and halo (e.g., Freeman 1987, Gilmore & Wyse 1987, Sandage 1987, Strömgren 1987, Norris 1987), each one according to one or more Schwarzschild velocity distributions. The Boltzmann collisionless equation represents a linear differential operator over the phase space density function where the superposition principle is satisfied, thus, a mixture of space density functions may be assumed. Here, we shall only consider Schwarzschild distributions, since, laying aside astronomical considerations, from a Bayesian criterion, the Gaussian distribution is the less informative one with known means and covariances (Koch 1990), and hence it is the usual and less restrictive approach for the components of a mixture model without any other prior information. On the other hand, the small-scale velocity distribution of the local disc can be deduced from some truncated distributions. According to Alcobé & Cubarsi (2005), a selection of stars with an absolute value of the total space motion |V | ≤ 51 km s−1 leaves the older disc stars aside. Such a subsample contains a complex mixture of early-type and young disc stars for which a Gaussian mixture approach is not feasible. Thus, Sample B is built as a subsample of Sample A with |V | ≤ 51 km s−1 . It is also possible to obtain a more detailed structure of the velocity distribution for specific subsamples (Cubarsi 2010a, Cubarsi et al. 2017), allowing the results of our approach to be compared with the small-scale structure sustained by well-known moving groups. Among metallicity, colour, and other star properties, the planar eccentricity of the star’s orbit is found to behave as a very good sampling parameter that allows distinguishing between different eccentricity layers within the thin disc, and allowing visualisation of the underlying structure of the distribution. In particular, for maximum eccentricity 0.3 and maximum distance to the Galactic plane 0.5 kpc, we get a representative thin disc sample. For these truncated distributions, the density function needs a six-degree polynomial to describe their strong asymmetries and their main kinematic features. They provide a higher resolution contour plot for the inner thin disc, which in addition to describing a velocity distribution far from the ellipsoidal hypothesis, explains a clear bimodal structure. Therefore, the maximum entropy modelling can be applied where mixture models do not provide enough complexity to explain the velocity distribution.

6.2. LARGE-SCALE STRUCTURE

79

6.2 Large-scale structure The maximum entropy approach is used to describe the main kinematical features of the solar neighbourhood. In a first application, the whole velocity distribution of the local disc is considered, which alternatively is usually fitted by a mixture of trivariate Gaussian distributions. The method is applied to Sample A that contains the total velocity space of 13,240 stars. According to the authors, the main essential features of the sample are the lack of kinematic selection bias and the radial velocity data, which allowed to reject stars that have not taken part in the evolution of the local disc. The velocity components (U, V, W) are expressed in a Cartesian heliocentric coordinate system, with U toward the Galactic centre, V in the rotational direction, and W perpendicular to Galactic plane, positive in the direction of the North Galactic pole. According to the analysis by Cubarsi et al. (2010), the moments are computationally stable for all the velocity components in the range 400 ≤ |V | ≤ 500 km s−1 . The limitation up to an absolute velocity space of 500 km s−1 excludes five stars. The halo component is present in the total sample in a fraction less than 0.5 %. Therefore, for practical purposes, this sample may be considered as unbounded. To choose the optimal fitting, Eq. 5.22 up to n = 6 is used, by taking up to tenth moments into account. According to Appendix D, by normalising to the number N of equations, the squared error of the fit is then given by the expression N ))2 1  1 )) ) ) . − g x (6.1) y χ2 = i i j j N i=1 ε2i The maximum entropy procedure with n = 2 tries to represent the whole distribution from a unique ellipsoidal distribution. Thus, odd-order moments and even-order moments higher than four are not fitted. The contour plots of the velocity distribution on each velocity plane are displayed in Fig. 6.1. The coordinate system is centred in the mean velocity of the sample (−9.97, −18.59, −7.16) km s−1 , referring to the Sun. The resulting fitting error χ2 = 84. is not acceptable. Let us point out, however, that more than the value χ2 itself, what is more significant is the increase or decrease in this quantity, since it may become distorted because over or underestimating observational errors, or undesired error distribution. The approach with n = 4 by using up to sixth moments gives a clearly improved result χ2 = 1.1. A symmetric distribution around the plane W = 0 is also quite admissible, with χ2 = 2.5, which is the same order as the previous error. The approach

80

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

with n = 6 is also computed by fitting up to the tenth moments, which gives a very accurate fit, χ2 = 0.02, even with symmetry around the plane W = 0, χ2 = 0.6. In Table 6.1, centred and non-centred velocity moments up to order four are listed for Sample A, along with their standard errors. A moment is usually taken as zero if its value is lower than twice its standard error, so we say that we are working within a 2σ level. Hence, according to the table, in addition to the diagonal moments, two of the non-diagonal moments do not vanish. The moments provide thus some measures of spread and asymmetry of the distribution in the desired variables, as the nonnull skewness in the 3 rotation velocity V, γV = μ030 μ020 − 2 = −3 ± 0.5 (in the Greek indices notation), which is zero in the other components. The moments also lead to a non-significant kurtosis in the vertical velocity W, cW = μ004 μ002 −2 − 3 = 34 ± 40, or the non-vanishing vertex deviation on the UV plane δ = 1 −1 ◦ 2 arctan[2μ110 (μ200 − μ020 ) ] = 11.3 ± 2.3 (see Table 7.1). The main features of the maximum entropy distribution for Sample A, which show a plausible deviation from an ellipsoidal distribution, may be easily deduced from Fig. 6.1, either for n = 4 or n = 6. The case n=4, by fitting up to sixth moments, leads to more realistic contour plots than the pure ellipsoidal distribution, n=2, although n=6 provides a slightly improvement, by fitting up to tenth moments. The qualitative features of the distribution are: - The velocity distribution is not symmetric around the mean, mainly in the rotation direction. - The whole distribution has a clear vertex deviation on the plane UV and no deviation on other planes. - There is some skewness in the variable V. As a consequence of both previous situations there is a wider distribution wing towards lower U and V velocities, which is likely caused by thick disc stars. - The kurtosis in the W variable vanishes and is zero or very small in the U velocity. - The plane W = 0 is basically a symmetry plane. The resulting density function, according to the most significant polynomial coefficients, can be expressed as a product of two exponential functions in the form (6.2) f = ϕ1 (Q) ϕ2 (I2 ) where Q is a quadratic negative-definite form, which gives the background ellipsoidal shape of the distribution, with axis ratios 1:0.7:0.5, sym-

6.3. TRUNCATED DISTRIBUTIONS

81

metry plane W = 0, as expected for disc stellar samples, and overall vertex deviation in the UV velocity components of about 12o . The function ϕ2 (I2 ) can be expressed in terms of the angular momentum integral I2 , Eq. 1.6, and may be interpreted as a perturbation factor. It is even and is at least quadratic in the V velocity alone, which accounts for the skewness and the shift in the ellipsoidal isocontours in terms of the rotation velocity. The bulk of the local velocity distribution does not show any substructure reflecting the existing moving groups, even by associating these moving groups with different proxy Gaussian components (Bovy et al. 2009). Then, it results in a smooth background distribution. However, by selecting specific subsamples by colour, or by using different analysis techniques where the resolution scale may vary, several substructures of the velocity distribution may arise.

6.3 Truncated distributions The next example is used for two new purposes: first, to test the ability of the maximum entropy method in reconstructing a truncated velocity distribution associated with a velocity bounded sample; second, to try a magnifying glass effect over the distribution and to focus on a specific velocity domain. The selection of local stars with an absolute value of the total space motion |V | ≤ 51 km s−1 leave the older disc stars aside, which are the originators of an important softening of the distribution. Such a selected group of stars contained a complex mixture of early-type and young disc stars for which a Gaussian mixture approach was unreliable because of the large fitting errors. This small-scale structure of the velocity distribution was strongly asymmetric in comparison to the background distribution. It was also observed in other analyses of the solar neighbourhood (e.g., Famaey et al. 2005, Soubiran & Girard 2005). Sample B is composed of 9,733 stars with absolute velocity lower than 51 km s−1 . The maximum entropy approach for n = 2 gives a fitting error χ2 = 1.1, according to Eq. 6.1. Although it could seem a very low value compared to previous sample, we might bear in mind that Sample A contains stars with higher velocity than Sample B, which increases the uncertainty of the computed moments. Because of this, the fitting errors for Sample B are expected to be much smaller. Once again, we must pay attention to the variation in χ2 . For n = 4, the approach is able to provide a more realistic, non-ellipsoidal map of the truncated distribution by fitting moments up to sixth order. In this

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

82

UV

n=2

UW

VW

100

V

60

60

50

40

W

W 40

20

20

K100

K50

0

50

U

100

K100

K50 K20 K40 K60

K50

50

U

100

K100

K50

0

50

K20 K40 K60

100

V

K100

n=4 60 60

60

40

K100

U

K50 V

0 20

K K40 K60 K80 K100

40

40

20

20

20 50

100

K100

U

K50

0

K20

50

100

W K40

K100 K80 K60 K40 K20 0 V K20

K60 K80

W

20

40

60

K40 K60 K80

n=6 60 60

60

40

W

20

K100

U

K50 K20 K40 V K60 K80

W

40

50

100

K100

U

K50

0 20

K K40 K60

40 20

20 50

100

K80 K60 K40 K20 0 V K20

20

40

60

K40 K60

Figure 6.1: Contour plots of the local velocity distribution in terms of the peculiar velocities for Sample A. The coordinate system is centred in the mean velocity of the sample (−9.97, −18.59, −7.16) km s−1 , referring to the Sun. The contours indicate levels ( 21 )k ; k = 0, ..., 10, and the black contour line corresponds to an approximate level 10−2 surrounding nearly the whole distribution.

6.3. TRUNCATED DISTRIBUTIONS

83

αβγ 100 010 001

μαβγ 0.00 0.00 0.00

Δμαβγ ± 0.30 0.22 0.16

mαβγ -9.97 -18.59 -7.16

Δmαβγ ± 0.30 0.22 0.16

200 110 020 101 011 002

1205.49 114.33 657.33 -28.26 14.09 332.93

± 30.08 20.02 28.67 11.81 10.97 17.33

1304.84 299.63 1002.95 43.09 147.17 384.17

± 30.94 22.96 34.28 12.32 12.65 17.82

300 210 120 030 201 111 021 102 012 003

-5410.36 -29820.51 -1883.35 -50726.61 -1529.39 1567.29 -1486.53 -2581.18 -9440.44 -5661.87

± 6929.41 4514.01 3918.62 6922.64 2007.04 1482.29 2027.51 2450.85 1724.77 5733.57

-42446.74 -56357.61 -16130.85 -93813.02 -10306.27 -192.55 -9189.67 -6005.75 -16784.25 -13178.21

± 7693.93 5178.37 4722.18 8810.45 2143.30 1670.70 2515.32 2637.40 2090.32 6024.02

400 310 220 130 040 301 211 121 031 202 112 022 103 013 004

13430320.65 1460621.61 5317165.76 322235.30 11308141.98 -1192280.95 62741.79 -315560.92 234834.16 1845208.39 315298.09 1593831.34 139487.68 492111.63 4084082.40

± 2125166.40 1270588.16 948783.69 930673.62 1962964.12 478081.09 322480.47 289850.40 517841.17 421200.10 262885.04 251847.69 845660.69 552721.46 2190306.74

14374454.53 3175498.30 7064506.56 1480869.49 16562919.85 -851132.23 454274.09 -248097.44 1003877.53 2010427.12 506475.65 2140083.38 321923.19 941999.67 4351176.92

± 2291035.07 1469075.82 1153512.90 1171917.48 2562819.68 485712.04 354935.98 344247.93 647019.02 459870.47 323652.27 327148.87 926678.48 688288.39 2325987.63

Table 6.1: Centred moments μαβγ and non-centred moments mαβγ with their standard errors up to fourth order for Sample A.

84

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

case the fitting error is χ2 = 0.3. However, for n = 6, with a fitting error χ2 = 0.002, more than 102 lower than the case n = 4, the maximum entropy approach gives a detailed portrait of actual asymmetries, in particular on the UV plane. The contour plots of the velocity distribution on each velocity plane are displayed in Fig. 6.2. The coordinate system is centred in the mean heliocentric velocity (−6.12, −11.23, −6.18) km s−1 . In Table 6.2, centred and non-centred velocity moments up to fourth order are listed as well with their standard errors. Notice that the standard errors are much smaller than those of Table 6.1. The skewness in the rotation velocity V is small, γV = 0.23 ± 0.03, but non-zero, being similar in the U direction. The kurtosis in the vertical velocity W, cW = 0.7 ± 0.3, is also very low. The vertex deviation on the UV plane is δ = 8.7◦ ± 0.6. The results are summarised in Table 7.1. Therefore, for n=6, the bounded sample provides a higher resolution contour plot of the velocity distribution, which in addition to describing a velocity distribution far from the ellipsoidal hypothesis, shows a clear bimodal structure, as displayed in Fig. 6.2. The results are consistent with the contour plots obtained by Dehnen (1998) when inferring the velocity distribution of his total sample (AL), in particular for the innermost dark contour. Also, the shape of the velocity distribution for early-type stars (Skuljan et al. 1999) is similar to ours, which is now derived only from velocity moments. By using the GCS catalogue, Famaey et al. (2007) describe a similar small-scale structure of local stars; however, the entropy approach provides the smoothest density function that is also consistent with the data. In Fig. 6.2 (n=6) two regions with higher probability densities are clearly identified, even using a large sample containing most of the thin disc with 9,733 stars. The highest peak is placed around the Hyades-Pleiades moving groups, and the lower peak around the Sirius-UMa stream. However, our method works in the opposite direction of methods based on an arbitrary number of mixture components, or on wavelet transforms on arbitrary smaller scales. As Bovy et al. (2009) point out, adding a new component could substantially increase the goodness of the fit over the model with less complexity, while still being far from the truth. Similarly, Dehnen (1998) points out that structures on scales of a few km s−1 are likely to be spurious. On the contrary, the maximum entropy approach is a technique for computing the simplest and smoothest approach to the distribution function that fulfils the provided set of moment constraints. For a good estimation, the only requirement is that the sample is bell-shaped enough and the moments have enough accuracy. The method tends to smoothing all the statistical fluctuations of the sample, since the

6.4. SMALL-SCALE STRUCTURE UV

85

UW

VW

n=4 30 20

30

V

20

20

10

10

K40 K30 K20 K10 0 K10 K20 K30

10

20

30

U

40

50

K40 K30 K20 K10 0 K10 W K20 K30

10

10

20

30

U

40

50

K30

K20

K10

0

10

K10 W

20

V

30

K20 K30

n=6 30

20

20

20

10

10

K40 K30 K20 K10 0 K10 V K20 K30

10

20

30

U

40

50

K40 K30 K20 K10 0 K10 W K20 K30

10

10

20

30

U

40

50

K30

K20 V

K10

0

10

20

30

K10 W

K20 K30

Figure 6.2: Contour plots of the velocity distribution for Sample B, with |V| ≤ 51 km s−1 . The peculiar velocities are centred on the heliocentric mean velocity (−6.12, −11.23, −6.18) km s−1 . The strong asymmetry of the velocity distribution, mainly on the UV plane, may only be described in the case n=6, by fitting moments up to tenth order. moments are obviously means. However, as shown in the above examples and in applications of the next section, if more complexity or resolution is desirable, some specific subsamples must be selected.

6.4 Small-scale structure It is however possible to obtain a more detailed shape for the velocity distribution for specific subsamples, allowing comparison of the results of our approach with the small-scale structure sustained by moving groups as described by other authors. By selecting samples with bounded peculiar velocity, such as |u| ≤ 7.5 km s−1 (256 stars), 10 km s−1 (498 stars), or 20 km s−1 (2,817 stars), a more complex structure is manifest on the UV plane, also in the vertical direction. The shape of the distribution becomes softer while increasing the size of the sample. Because of this, the substructure of thin

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

86

αβγ 100 010 001

μαβγ 0.00 0.00 0.00

Δμαβγ ± 0.22 0.14 0.13

mαβγ -6.12 -11.23 -6.18

Δmαβγ ± 0.22 0.14 0.13

200 110 020 101 011 002

462.82 41.10 199.40 -5.00 -1.72 153.41

± 5.41 2.75 2.87 2.53 1.90 2.55

500.27 109.85 325.63 32.82 67.71 191.61

5.56 3.82 3.95 2.81 2.35 2.92

300 210 120 030 201 111 021 102 012 003

2173.59 -145.28 530.94 638.37 330.59 131.81 96.46 337.15 308.20 142.89

± 157.78 70.79 53.23 79.37 20.62 34.76 40.55 46.93 37.83 65.05

-6551.81 -6268.73 -2385.18 -7500.56 -2700.19 -480.42 -1877.37 -773.54 -1823.19 -2937.69

± 276.03 109.93 97.37 158.75 82.60 47.48 64.93 62.06 53.07 110.67

400 310 220 130 040 301 211 121 031 202 112 022 103 013 004

499489.15 34551.28 75113.83 6595.67 119660.84 -1064.18 -299.90 -102.22 3803.90 62272.91 2098.65 35099.21 1025.32 -1390.25 86642.60

± 10136.14 3316.06 1635.32 1901.12 4596.79 2965.11 1022.52 838.49 1762.48 1472.05 788.56 1149.57 1640.64 1394.34 3447.49

551669.32 115444.19 153794.57 50161.86 257920.40 32799.70 32365.54 10219.59 46257.99 78158.49 8713.72 58305.32 12177.32 25702.11 119730.52

± 9932.67 4479.28 2498.65 3041.40 6223.25 3070.77 1356.19 1199.62 2182.27 1552.90 991.04 1340.04 1815.65 1649.39 4100.98

Table 6.2: Centred and non-centred moments with their standard errors up to fourth order for Sample B.

6.4. SMALL-SCALE STRUCTURE

87

disc subsamples with less stars become statistical fluctuations within larger subsamples, up to describing a sufficiently complete distribution of the thin disc. Thus the clue is to find a clean and representative thin disc sample. The cut |V | ≤ 51 km s−1 therefore seems to be a good value that includes most of thin disc stars and excludes thick disc stars, but it is still far from being a complete thin disc sample. Samples selected from small peculiar velocities have some limitations. On one hand, they contain few stars, so that their distribution may not be bell-shaped enough. Furthermore, their moments have greater uncertainties. On the other hand, the boundary of the distribution is fixed by the velocity limit of the sample, which may cut down some well-defined structures. Fortunately, there is a way to avoid this problem. In Alcobé & Cubarsi (2005) and Cubarsi et al. (2010) consecutive stellar populations were merged to nested subsamples in terms of several sampling parameters: maximum absolute velocity, peculiar velocity, vertical velocity, etc. Optimal values of these sampling parameters allowed the segregation of these populations. For the complete GCS sample, once the stars are classified according to the probability of belonging to any of the local Galactic components, a highly significative correlation is obtained between the expected population of a star and its absolute velocity |V |. The expected population is similarly highly correlated with the planar eccentricity, and also correlated with rotation velocity, |zmax |, and metallicity. The colour is few correlated with the expected population and the other preceding properties. Therefore, significant partial correlations between couples of the former star properties exist. However, when the sample is bounded to |V | ≤ 51 km s−1 , by leaving aside thick disc and halo stars, the only significant partial correlation that is maintained is the absolute velocity and the eccentricity, as well as the expected population with them. That means that the other properties are only relevant for segregating thick-disc or halo stars, but are not useful within the very thin disc. As discussed in Cubarsi et al. (2010), the sampling parameter is related to the isolating integrals of the star motion. Both the absolute velocity and the eccentricity satisfy this requirement. The former is less discriminant, but is a direct measure from the star. The latter is more discriminant, but requires computing the orbital parameters, with the need of additional hypothesis on the potential, symmetries, stationarity, mean motion, solar position, etc. Therefore, it is possible to use the eccentricity not for segregating populations, as e.g., Pauli et al. (2005) or Vidojevi´c & Ninkovi´c (2009), but as an improved sampling parameter to select subsamples.

88

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

ecc=0.01

ecc=0.02

Figure 6.3: Series of contour plots and distributions on the UV plane subsamples selected from eccentricities up to 0.01 and 0.02 in Sample B. The origin is at the Solar velocity.

6.4. SMALL-SCALE STRUCTURE

89

NGC 1901 Coma Berenices

Pleiades

ecc=0.03

Middle branch

NGC 1901 Coma Berenices

Pleiades

ecc=0.05

Figure 6.4: Series of contour plots and distributions on the UV plane for Sample B subsamples selected from eccentricities up to 0.03 and 0.05.

90

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

Siriu Hyades Pleiades ecc=0.10

Sirius Hyades+Pleiades ecc=0.15

Figure 6.5: Series of contour plots and distributions on the UV plane for Sample B subsamples selected from eccentricities up to 0.10 and 0.15.

6.4. SMALL-SCALE STRUCTURE

91

ecc=0.20

ecc=0.30

Figure 6.6: Series of contour plots and distributions on the UV plane for Sample B subsamples selected from eccentricities up to 0.20 and 0.30.

92

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

6.5 Orbital eccentricity For disc samples with maximum eccentricities 0.01 (220 stars), 0.02 (591 stars), 0.03 (1,058 stars), 0.05 (2,465 stars), 0.1 (7,095 stars), 0.15 (9,545 stars), 0.2 (10,903 stars), and 0.3 (11,826 stars), with the additional condition |zmax | < 0.5 kpc to avoid contamination from stars not belonging to the thin disc, the maximum entropy approach provides the series of plots in figures Fig. 6.3 to Fig. 6.6. Both previous limitations introduced by the velocity boundary have now disappeared. For example, the structure derived from |u| ≤ 10 km s−1 with 498 stars is now completely described from the plot with maximum eccentricity 0.05 with 2,465 stars. Similarly, the shape of the distribution is no longer forced by the sampling parameter. The eccentricity then behaves as a very good sampling parameter that allows us to distinguish between different eccentricity layers within the thin disc and enables us to visualise the structure below each layer. In the lower layers, with maximum eccentricities 0.01 and 0.02, the velocity distribution shows a hole around the LSR, taken as (−10., −5.23, 7.17) km s−1 (Dehnen & Binney 1998), which is the mean of the distribution. Those lowest eccentricity stars are moving around the LSR and have velocities distributed on a ring with some peaks around the LSR. The radial velocities are symmetrically grouped into two main bulks at each side of the LSR. This behaviour is maintained up to eccentricity e=0.03, where the LSR hole begins to be filled by the group of stars corresponding to the Coma Berenices moving group, nearly at the same LSR velocity. In addition, three stellar groups around the LSR conform the basic structure: NGC 1901, a group that can be part of the middle branch (Skuljan et al. 1999), and a part of the Pleiades group. The structure is the same as described by Bovy et al. (2009) and by previous works of Dehnen (1998), Skuljan et al. (1999), Famaey et al. (2005, 2007), with the greatest peak in NGC 1901. For e=0.05 the structure is maintained and enlarged. It incorporates a new group of stars also associated with the middle branch, which is not referred to as a moving group by Bovy et al. (2009), but is the centre of their Gaussian component with the largest weight. In the range of eccentricities from 0.05 to 0.1, the small previous structures are diluted in a background distribution, and only the Pleiades group remains. The main weight of the distribution is now in the stars around the Hyades group. A stellar group around the Sirius/UMa stream arises at positive radial velocities. For e=0.15, that is, approximately the higher eccentricity before appearing thick disc stars, the distribution is divided into about half: one bulk with negative radial and rotation mean velocities with respect to the LSR, which contain

6.6. RADIAL VELOCITY DISTRIBUTION

93

the main groups Hyades and Pleiades; and another one with positive values around Sirius and UMa stream. For higher eccentricities, the distribution becomes similar to the one corresponding to the thin disc. In particular, for e=0.3, with a fraction of 90% of the whole sample, we get a distribution similar to the thin disc of Cubarsi et al. (2010) (obtained by two different methods: MEMPHIS algorithm and the method of Galactic orbits), with dispersions and vertex deviation (σU , σV , σW ; δ)=(29.1 ± 0.2, 18.1 ± 0.1, 11.6 ± 0.1; 10◦ ± 1). Then, as a main result, we may affirm that for eccentricities below 0.15, there is a general trend in the radial direction: the main weight of the distribution is symmetrically placed around the LSR. Thus, the velocity distribution of the thin disc is supported by two major stellar groups with opposite radial velocities around the LSR, with a dearth of stars at the LSR. One bulk, with positive radial velocity, has a mean velocity similar to the Sun or slightly higher, with a lower peak but a wider distribution. The other one, with radial velocity ≈ −30 km s−1 , has a mean rotation ≈ −20 km s−1 and a higher peak. This behaviour is definitively broken for eccentricities e ≥ 0.2.

6.6 Radial velocity distribution The situation described in the preceding section, where the star velocities are distributed, for low eccentricities, along the U direction with a local minimum at the LSR velocity, may have a simple explanation: a mixture of stars with a small range of eccentricity variation. In the special case of a nearly planar orbit, where the planar eccentricity is low enough that the amplitude of the vertical motion becomes independent of the radial motion, the motion of a star can be studied as a case of epicycle orbit (Appendix H). Let us remember that, if a nearly planar orbit is projected onto the Galactic plane, its distance to the Galactic centre oscillates between two limiting values p and a . The planar eccentricity e is then defined as e=

a − p , a + p

(6.3)

which is a dimensionless measure of the deviation from the circular motion in the plane of symmetry. The orbits in the Galactic disc are nearly planar, and their planar eccentricities may be significantly different from zero (Vidojevi´c & Ninkovi´c 2009). Thus, the epicycle approximation (Appendix H) can be used for the current disc sample. It is commonly assumed that moving groups of

94

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

young stars are born in nearly circular orbits (e.g., Dehnen 1998) and, with age, they are transformed into more eccentric orbits, which oscillate locally around the LSR (assumed to be in circular motion, ULSR = 0, for steady state and axisymmetric systems). The epicycle frequency depends on the potential function and is the same for all the stars. Thus, the radial velocity of a star oscillating around the LSR with a period T may be written as U = a sin

2πt , T

(6.4)

where the amplitude is proportional to the planar eccentricity, a = Tπ (a + p ) e. If the time t is measured in oscillation periods, we may assume T = 1. For a sufficiently great t, e.g., taking several periods, we may also assume that t is uniformly distributed within the interval [0, 1] (after a number of complete periods), so that its probability density function is ft (t) = 1, t ∈ [0, 1], and zero otherwise. We may ask for the distribution of U around the LSR velocity, that is, for the probability of finding a star velocity at 1 arcsin( Ua ), and this is a any given value within [−a, a]. Since t(U) = 2π two-valued function, the probability density function fU (U; a), for a given amplitude a, is easily obtained as 2 ft (t) |t (U)|. Thus, fU (U; a) =

1 1 , √ π a2 − U 2

U ∈ [−a, a],

(6.5)

and zero out of this interval. As seen in Fig. 6.6 (a), for an arbitrary value a = 1, it is less probable to find the star with nearly zero velocity, which means near the extreme positions p or a , than around the mean position (p + a )/2 with maximum absolute velocity, either negative by going toward the Galactic centre or positive toward the anticentre. We may think of a mixture of stars with the same oscillation period and different eccentricities, from zero eccentricity and amplitude, a = 0, up to greater amplitudes, say a = A. Let us assume a normalised density function ρ(a) of stars in terms of the amplitude a. Then the cumulative density function hU (U; A) is obtained by integration over a as  A fU (U; a) ρ(a) da. (6.6) hU (U; A) = 0

Depending on how the stars are distributed in terms of the amplitude, we may get different symmetric distributions around the LSR. For a fixed period T , this depends on the distribution of eccentricity. The distribution of eccentricity is approximately lognormal, similar to the distribution of wealth

6.6. RADIAL VELOCITY DISTRIBUTION

95

2,000

1,500

N 1,000

500

0 0

10

20

x

30

40

50

Figure 6.7: Distribution of eccentricities for Sample A. The probability density function obtained from the histogram is approximately lognormal. In the x-axis, the interval [0, 1] of eccentricities is divided into 50 bins. The variable x = 50 e is lognormal with m = 1.75 and σ = 0.5 (dashed line). in a country, as shown in Fig. 6.7 from the histogram of star eccentricities coming from the whole Sample A, where the interval [0, 1] of eccentricities is divided into 50 bins on the x-axis (x = 50 e is approximately lognormal with m = 1.75 and σ = 0.5). To find out the shape of hU (U; A) several simulations are carried out for arbitrary values of the amplitude, by assuming ρ(a) lognormal. Bimodal distributions around the origin (LSR) are always obtained, like the plots (b), (c), and (d) of Fig. 6.8. The reason is that the lognormal distribution vanishes in a neighbourhood of zero. (It is tangent to zero at the origin.) However, the wider the distribution wing, the less significant the bimodality, since the behaviour for low amplitudes becomes a smaller structure when larger amplitudes are considered. The distribution tends to be Gaussian. In the case of a discrete mixture of stars with different eccentricity distribution, we should get a mixture of densities hU (U; A) with similar properties, which could explain the radial velocity shape obtained for the inner thin disc. However, for a continuous mixture of populations in terms of eccentricities, the bimodal structure may be irrelevant. For example, apart from statistical fluctuations, the density function ρ(a) may become more populated around zero, so that it is no longer tangent to zero at the origin of amplitudes. Then, a high-peaked hU (U; A) may be obtained at the origin, for low values of A. In general, if

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

96

(a) fU (U; 1)

(b) hU (U; 0.25)

2.0

0.004

1.5

0.003

1.0 0.002

0.5 0.001

K

2

K

1

0

1

U

2

K

K

0.4

(c) hU (U; 1.25)

0.2

0

0.2

U

0.4

(d) hU (U; 6)

0.20

0.3

0.15

0.2

0.10

0.1 0.05

K

2

K

1

0

1

U

2

K

6

K

4

K

2

0

2

U

4

6

Figure 6.8: Simulated distribution of radial velocities for arbitrary values of the amplitude, by assuming the epicycle approximation and ρ(a) lognormal (0.1,0.5). (a) Probability density function fU (U; a) for amplitude a = 1. (b) Cumulative density function hU (U; A) integrated up to A = 0.25. (c) Cumulative density function hU (U; A) integrated up to A = 1.25. (d) Cumulative density function hU (U; A) integrated up to A = 6.

6.7. REMARKS

97

in a neighbourhood of the origin, ρ(a) behaves as a p , with 0 < p ≤ 1 we then get smooth unimodal distributions centred at the origin. On the other hand, it is easy to prove that, if the radial velocity does not oscillate symmetrically around the LSR, by meaning a slight deviation from the epicycle approximation, then the peaks of hU (U; A) become non-symmetric around the LSR. Therefore, these simple simulations reproduce the actual situation for low eccentricities approximately.

6.7 Remarks Significant results arise from the application of the maximum entropy approximation. The large-scale distribution shows a nearly constant vertex deviation in the pseudo ellipsoidal level curves, as well as a nearly constant axis ratio. According to the most significant polynomial coefficients, the resulting density function can be expressed as the product of two exponential functions in the form of Eq. 6.2. The background ellipsoidal shape has axis ratios 1:0.7:0.5 and a symmetry plane W = 0. The overall vertex deviation in the UV velocity components is approximately 11o . Some characteristic parameters of the distribution are summarised in Table 7.1. In addition, there is a perturbation factor which is given through a function of the angular momentum integral of motion, that is even and at least quadratic in the V velocity alone. It accounts for the skewness and the shift in the velocity ellipsoids in terms of the rotation velocity. This is clearly visible on the UV and VW planes of Fig. 6.1, and may be interpreted with regard to the heating of disc stars, which is also correlated with a decreasing galactocentric rotation velocity, as expected from Strömberg’s law. The resulting overall distribution has zero kurtosis in the W velocity, and, within 2σ level, in the U component. The large-scale structure of the distribution can also be deduced from a mixture model with two Schwarzschild distributions, although a stellar sample containing almost two pure populations is necessary. To this purpose, it is possible to filter the sample through a sampling parameter and to apply the mixture model recursively. Nevertheless, for complex distributions, such as the small-scale structure induced by enough large moving groups, the maximum entropy approach presents great advantages, being able to provide an accurate portrait of the distribution in a very simple way. Thus, the entropy method offers an excellent estimation of the truncated velocity distributions of Sample B and the ones obtained for different eccen-

98

CHAPTER 6. LOCAL VELOCITY DISTRIBUTION

tricities. For these samples, a mixture approach is impossible. This method can therefore be used as an alternative way to study multimodal distributions. For star velocities |V | ≤ 51 km s−1 , a tentative mixture (with a very large chi-squared fitting error) suggested (Alcobé & Cubarsi 2005) a superposition of two enlarged pseudo-ellipsoidal distributions, mainly along the radial direction, with very overlapped wings, and a separation of 28 ± 9 km s−1 between means. In contrast, with the maximum entropy approach, the separation of the two peaks (the “U-anomaly”) along the radial direction may now be straightforwardly estimated in 30 − 35 km s−1 from the contour plots, which show a strongly asymmetric velocity dispersion on the plane UV, and nearly laminar isocontours of the W velocity component along the radial direction. There is a large radial velocity dispersion on the UV plane, in the direction of the gravitational gradient, and a very small dispersion in the direction perpendicular to the galactic plane. On the VW plane, the isotropy is slightly recovered. In addition, a look at the UW plane of Fig. 6.2 (n=6) shows a core distribution with negative radial peculiar velocity, while there is a clear unimodal behaviour with positive peculiar rotation velocity on the VW plane. However, those high-density regions of the distribution are not simultaneous on the UV plane, but are associated with different large stellar groups. For subsamples obtained from eccentricities e < 0.15, the maximum entropy method is able to plot the classical moving groups composing the small-scale structure of the velocity distribution, as described from other algorithms based on an arbitrary number of mixture components, wavelet transforms, or maximum likelihood, especially those providing a modest amount of complexity. For these subsamples with small eccentricities, there is a general trend: the star velocities are approximately symmetrically distributed around the LSR in the radial direction. In most cases the distribution is bimodal, with a dearth of stars at the LSR. At the end, for e = 0.15, the core distribution of the thin disc is supported by two major stellar groups with opposite radial velocities, referred to the LSR. One bulk, with zero or small positive heliocentric radial mean velocity, has a lower peak but a wider distribution around Sirius/UMa stream. The other one, with radial velocity about −30 km s−1 , has mean rotation ≈ −20 km s−1 and a higher peak, containing the main groups Hyades and Pleiades. For stars with a similar period of oscillation around the LSR in the radial direction (under the epicycle approximation), several simulations allow us to confirm that such a two-peaked distribution of radial velocities is due to a lognormal distribution of the eccentricities. For a mixture of stars with a lognormal distribution of the velocity amplitude of the stellar orbits, the

6.7. REMARKS

99

bimodal shape is maintained. However, if the number of stars with nearly vanishing amplitude increases, then the radial velocity distribution becomes unimodal, similar to the total thin disc sample with e = 0.3. In conclusion, the degree of the polynomial function Pn may be used to quantify the complexity of the velocity distribution. Under a maximum entropy approach, this complexity is in some way measured in terms of the necessary set of velocity moments, for obtaining a good fit of the velocity distribution. In the most basic situations, the stellar systems can be approximated by an ellipsoidal distribution, such as for the thin disc, thick disc, or halo, as independent Galactic components. Those systems are well described with moments up to second order. However, other stellar systems have a clear deviation from the ellipsoidal hypothesis, such as the whole Galactic disc. For them, a good fit of the velocity distribution is obtained for a fourth-degree polynomial function. On the other hand, to study large enough stellar groups conforming disc substructures, like those obtained from finite velocity domains, or such as the samples drawn from limited stellar eccentricity, in particular, for planar eccentricities lower than 0.15, it is necessary to approach the velocity distribution with a sixth-degree polynomial function.

Chapter 7

Superposition of stellar populations 7.1 Mixture approach As seen in Chapter 4 the collisionless Boltzmann equation, or the equivalent system of Eq. 4.10, and the moment equations, when are referred to an absolute reference frame are linear and homogeneous equations. Therefore, it is valid the principle of superposition. However, they lose their linearity when they are referred to the comoving reference frame, although the physical meaning of a solution made by superposition of particular solutions is maintained. From a statistical viewpoint, the superposition assumption is a powerful approach since a well-mixed stellar sample resulting from a single quadratic velocity distribution is actually never found in kinematic surveys. For describing the main trends of local stellar samples it suffices that the sample can be approximated by a finite number of quadratic distributions. In particular, Schwarzschild distributions can be used instead of other less meaningful quadratic distributions, perhaps at the expense of increasing the number of populations slightly. Then, the whole velocity distribution will show non-vanishing odd-order central moments on condition that the radial and vertical mean velocities of the population components are nonnull. Therefore, the introduction of the mixture hypothesis is necessary to gain degrees of freedom for the velocity distribution. The mixture hypothesis is in agreement with the actual description of the 101

102

CHAPTER 7. SUPERPOSITION OF STELLAR POPULATIONS

Galaxy through stellar populations conforming its structural components, such as the stellar disc, the stellar halo, the central bulge, or the dark matter halo (e.g., Freeman & Bland-Hawthorn 2002). The local Galactic components are generally fitted from a trivariate Gaussian distribution, although, as will be discussed below, an accurate description of the thin disc may need several Gaussian components. Recently, Pasetto et al. (2012a, 2012b) adapted the approach based on the cumulants method proposed by Cubarsi (1992) and Cubarsi & Alcobé (2004) to the newest radial velocity data from the RAdial Velocity Experiment (RAVE) survey (Siebert et al. 2011, Zwitter et al. 2008, Steinmetz et al. 2006) for obtaining the kinematics of the thin and thick discs in the solar neighbourhood. They provided the mean velocities and the whole set and trends of the second central moments. Similarly, Moni Bidin et al. (2012) and Casetti-Dinescu et al. (2011) provided moments and gradients for the thick disc, and Carollo et al. 2010 and Smith et al. (2009a, 2009b) discussed the halo kinematics. Also, using the HIPPARCOS (ESA 1997) and the Geneva-Copenhagen Survey (GCS) catalogues (Nordtröm et al. 2004, Holmberg et al. 2007, Holmberg et al. 2009), Cubarsi et al. (2010) and Alcobé & Cubarsi (2005) provided a kinematic classification of the populations that compose the solar neighbourhood. In addition, smaller Galactic structures produced by a large enough number of stars, such as those of early-type, younger, and older disc stars within the thin disc, were described through more detailed Gaussian multicomponent mixtures (e.g., Bovy et al. 2009, Famaey et al. 2007, Soubiran & Girard 2005).

7.2 Two-component mixture We now may consider the mixture of two stellar populations in order to describe, from a different viewpoint, a smooth enough large-scale velocity distribution. The overall density function is then obtained from the superposition of two normal density functions according to Eq. 3.1, each one associated with the corresponding stellar component, ( ) or (

) for the first or second population, f (V ) = n ψ(Q ) + n

ψ(Q

),

(7.1)

where n and n

represents the respective mixing proportions, obviously satisfying n + n

= 1. The statistics mean velocity v and central moments μn for the total velocity distribution, defined in Eq. 1.13 and Eq. 1.18, may easily be deduced starting from those of the partial ones. Hence the total

7.2. TWO-COMPONENT MIXTURE

103

mean velocity, expressed from the population means, satisfies v = n v + n

v

. Let us review a way of computing the mixture moments and cumulants (see Appendix E for the relation between sample and population cumulants) which properties were reviewed in section 3.2. The total central moments are written, by taking Eq. 7.1 into account, by using the differential velocity of the centroids, w = v − v

. (7.2) The tensor of the total n-order central moments, expressed from the partial ones, has then the following form (Cubarsi 1992): μn =

 n   n {n n

k μ n−k + (−n )k n

μ

n−k }  (w)k . k

(7.3)

k=0

We explicitly write the total central moments up to fourth-order. Obviously, for n = 0 we get μ0 = μ 0 = μ

0 = 1, and for n = 1, μ1 = μ 1 = μ

1 = 0. The symmetric tensor μ2 of the second-order moments, with six different elements, becomes μ2 = n μ 2 + n

μ

2 + n n

(w)2 .

(7.4)

The tensor μ3 of the third moments, with ten different elements, being μ 3 = μ

3 = 0, is written as μ3 = 3n n

(μ 2 − μ

2 )  w + n n

(n

− n )(w)3 .

(7.5)

And the fourth moments μ4 , with fifteen different elements, can be expressed, by using the relationship μ4 = 3 μ2  μ2 , obtained in Eq. 3.5 for a single Schwarzschild distribution, as follows μ4 = 3n μ 2  μ 2 + 3n

μ

2  μ

2 + +6n n

(n

μ 2 + n μ

2 )  (w)2 + n n

(1 − 3n n

)(w)4 .

(7.6)

However, if the population cumulants are used, it is possible to write the above relationships in a shorter form. This is done by introducing the following new variables, D=

√

n n

w;

q=

*

n /n

−

*

n

/n

(7.7)

104

CHAPTER 7. SUPERPOSITION OF STELLAR POPULATIONS

(we can appoint the populations so that n ≥ n

, then q is non-negative) and the following second-rank tensors, a2 = n κ 2 + n

κ

2 ;

C2 = *

1 q2

+4

(κ 2 − κ

2 ) − q(D)2

(7.8)

(all the above square roots are taken as positive values). With the foregoing definitions, Eq. 7.4, Eq. 7.5, and Eq. 7.6 can be rewritten explicitly depending on the overall mixture cumulants in a shorter form, as follows: κ2 = a2 + (D)2 , κ3 = 3C2  D + 2q(D)3,

(7.9)

κ4 = 3C2  C2 − 2(q2 + 1)(D)4. From the total cumulants of the mixture the parameters of the partial distributions have to be determined by inverting the above non-linear system of equations. These unknowns are the partial second moments (or cumulants) –six components for μ 2 and six for μ

2 –, the population fraction n , and the three components of the centroid differential velocity w. Sixteen unknowns in total. Since we have a set of thirty-one scalar equations, involved in Eq. 7.9, we must also find a set of fifteen constraint equations, which can be used as a test in order to verify whether a given sample is consistent with a two normal mixture. In the following section we describe how the total cumulants of the mixture are related. A set of equations is reached, that generalise the first relationship of Eq. 3.5 for a single Schwarzschild distribution, providing some characteristic mixture constants, such as a vector d in the direction along both subcentroids, and two constants, A and B, that can be linearly estimated from total cumulants, with useful information about the geometry of the mixture.

7.3 Moment constraints We study the general case where the difference between population means, w, and hence the vector D of Eq. 7.7, is not null. Let us assume the vector component D3  0 (in order to minimise the error propagation this component may be chosen to be maxi |Di |), and let us define a normalised vector d = D/D3 in the direction containing both subcentroids C1 and C2

7.3. MOMENT CONSTRAINTS

105


u





w


 







d


Figure 7.1: Directions W1 , W2 , W3 where the peculiar velocity u is projected (Fig. 7.1). Since every normal distribution is symmetric with respect to its centroid, then the total velocity distribution will be symmetric in whatever direction within a plane Φ orthogonal to the vector d, and in particular the one containing the global centroid Ct . Thus, in order to take profit of this symmetry, it is convenient to work with a transformed vector U instead of the peculiar velocity u, whose components are three non-orthogonal projections of the peculiar velocity u on the directions W1 = (1, 0, −d1)t and W2 = (0, 1, −d2)t , on the plane Φ, and another independent direction, for example W3 = (0, 0, 1)t. The transformed peculiar velocity U can be expressed from the following isomorphic transformation of the vector u, ⎛ ⎜⎜⎜ 1 ⎜ U = H2 · u; H2 = ⎜⎜⎜⎜ 0 ⎝ 0

⎞ 0 −d1 ⎟⎟⎟ ⎟ 1 −d2 ⎟⎟⎟⎟ . ⎠ 0 1

(7.10)

Note that det(H2 ) = 1. Also note that U , as well as u, have null mean. In fact U3 = u3 is an invariant component. Indeed, the above transformation means a change of basis given by the matrix ⎛ ⎞ ⎜⎜⎜ 1 0 d1 ⎟⎟⎟ ⎜ ⎟ H2−1 = ⎜⎜⎜⎜ 0 1 d2 ⎟⎟⎟⎟ . (7.11) ⎝ ⎠ 0 0 1

106

CHAPTER 7. SUPERPOSITION OF STELLAR POPULATIONS

so that its columns are the vectors of the new basis. Therefore, two of the old vector basis are maintained, and the new third vector of the basis is d, along the direction of both subcentroids. From the definition of H2 the following equality is deduced, 

Hi j D j = D3 δ3i ; i, j ∈ {1, 2, 3}

(7.12)

j

where δ is the Kronecker delta. If the third and fourth moments of U are calculated in function of the central velocity moments μi jk and μi jkl , the following equalities are obtained, Uα Uβ Uγ =

 i jk

Uα Uβ Uγ Uδ =

Hαi Hβ j Hγk μi jk ;

 i jk

α, β, γ, δ, i, j, k, l ∈ {1, 2, 3};

Hαi Hβ j Hγk Hδl μi jkl .

(7.13) And also the third and fourth cumulants of the transformed peculiar velocity U can be computed in function of the corresponding cumulants of u. With the indices α, β, γ, δ ∈ {1, 2} and i, j, k, l ∈ {1, 2, 3}, the ten components of the third U -cumulants are 

oαβγ ≡



pαβ ≡ sα ≡

i jk

ij

Hαi Hβ j Hγk κi jk ,

Hαi Hβ j κi j3 , (7.14)



i Hαi κi33 ,

1 2

κ333 , and the fifteen components of the fourth U -cumulants are, Xαβγδ ≡ Yαβγ ≡ Zαβ ≡ Tα ≡ κ3333 .

 i jkl

 i jk

 ij

 i

Hαi Hβ j Hγk Hδl κi jkl ,

Hαi Hβ j Hγk κi jk3 ,

Hαi Hβ j κi j33 ,

Hαi κi333 ,

(7.15)

7.3. MOMENT CONSTRAINTS

107

Thus the U -cumulants can be grouped according to the above twodimensional tensors o3 , p2 , s, X4 , Y3 , Z2 , and T , depending on the total cumulants of the distribution. These tensor components are explicitly written in Appendix F.1. Let us remark that all the above quantities are explicitly dependent on the velocity component that remains invariant under the transformation of Eq. 7.10. Hence they should have to be noted, for example, with a superindex (3) indicating that component, since the described procedure is also valid under permutation of indices of the velocity components. However, in order to simplify the notation, this super-index has been omitted, although the unit value d3 has been written. The main properties and the steps in order to obtain the cumulant constraints, the mixture constants, and the population parameters are hereafter summarised. By substitution in Eq. 7.14 of κ3 , from Eq. 7.9, and taking into account Eq. 7.12, four independent vanishing linear combinations of the third U moments are obtained: oαβγ = 0; α, β, γ ∈ {1, 2}

(7.16)

These are the third cumulants of the vector components U1 and U2 , that vanish as consequence of having defined the vectors W1 and W2 on the plane Φ, with symmetrical properties of the distribution. Similar equalities are satisfied for higher odd-order cumulants. From the above equations the values d1 and d2 can be computed. The solution of this system of nonlinear equations provides us with the first characteristic constant of the mixture, d, that is the orientation of both subcentroids. Now it is already possible to calculate the elements of the matrix H2 , in Eq. 7.10, and the tensor elements of Eq. 7.14 and Eq. 7.15. Let us remark that, while the method of moments for an univariate normal mixture requires to solve a fundamental nonic equation (Cohen 1967), originally derived by Pearson, for a trivariate mixture only three-degree polynomials have to be solved, and the moments method loses, or substantially reduces, the consideration of ill-conditioned problem. For the elements of tensors p2 and s in Eq. 7.14, by substitution of the third cumulants κ3 from Eq. 7.9, and taking into account Eq. 7.12, the following equivalences are obtained,  pαβ = D3 i j Hαi Hβ jCi j , (7.17)  sα = D3 i Hαi Ci3 .

108

CHAPTER 7. SUPERPOSITION OF STELLAR POPULATIONS

These relationships are used to compute the elements of C2 other than C33 . Similarly, for the elements of tensors X4 , Y3 , Z2 , and T in Eq. 7.15, by substitution of the fourth cumulants κ4 from Eq. 7.9, and taking Eq. 7.17 into account, we get the following set of constraint equations, X 4 = 3 A p2  p2 , Y3 = 3 A p2  s, (7.18) Z2 = 2 A s  s + B p2 , T = 3B s. −1 where A = D−2 3 and B = C33 D3 . These fourteen scalar relationships are explicitly written in Appendix F.2. The set of relationships in Eq. 7.18 represents an overdeterminate linear system, which can be solved by means of weighted least squares in order to find optimal values for the mixture constants A and B. This step provides the absolute values of C33 and D3 . The mixing proportions are then evaluated from the parameter q, so that the following two relationships are fulfilled,

κ333 = 3C33 D3 + 2qD33 , (7.19) 2 − 2(q2 + 1)D43 . κ3333 = 3C33

Hence, by elimination of q, a new constraint equation is hold. The remaining five unknowns of the tensor C2 may be evaluated from Eq. 7.17, and finally, from Eq. 7.7 and Eq. 7.8, the population parameters n , w, μ 2 , and μ

2 , can be determined. The mixture model can be applied to Sample A according to an optimal sampling parameter to determine the kinematic parameters and the stellar population mixture of the thin disc, thick disc, and halo (Cubarsi et al. 2010). The sampling parameter |(U, V, W)|, which is the absolute heliocentric velocity, allows us to build an optimal subsample containing both thin and thick disc stars, omitting most of the halo population. The sampling parameter |W|, which is absolute perpendicular velocity, allows us to create an optimal subsample of all disc and halo stars, although it does not allow an optimal differentiation of thin and thick discs. Some results are shown in Table 7.2.

7.4. LOCAL VELOCITY ELLIPSOIDS

109

7.4 Local velocity ellipsoids The bimodal structure obtained either for Sample B or for eccentricities up to 0.15 showed that the thin disc contains two major streams moving with opposite radial directions around the LSR, one with small positive radial mean velocity and rotation similar to the Sun, and the other with negative radial mean velocity and lower rotation. For each subsystem, we could think of assuming some less restrictive hypotheses, such as point-axial symmetry (opposite points through an axis, allowing, in particular, spiral structures) or a time-dependent model, in order to describe the non-vanishing radial velocity of their centroids and the vertex deviation of their approximate velocity ellipsoids. Thus, a general Chandrasekhar point-axial model (Sanz-Subirana & Català-Poch 1987, Juan-Zornoza et al. 1990, Juan-Zornoza 1995) should be the simplest approximation, where, despite the non-cylindrical symmetry of the system, we shall see that the solution of Chandrasekhar’s equations system yields an axisymmetric potential. However, only the simple case of a harmonic potential allows some vertex deviation of the velocity ellipsoid. In such a case, even if only as a local approximation, it is interesting to recall the relationship between the vertex deviation and the radial mean velocity, . According to Eq. G.9, the vertex deviation δ of a velocity ellipsoid depends on the second central moments in the form tan(2δ) =

2μUV . μUU − μVV

(7.20)

Thus, if μUU −μVV > 0, the angle δ and the moment μUV have the same sign. As explained in Cubarsi & Alcobé (2006), the radial mean velocity referred to an axisymmetric (cylindrical) system can be related to the increment of rotation mean velocity with respect to the same axisymmetric system and, in particular, to the vertex deviation of the velocity ellipsoid, through the expression μUV (cyl) Θ0 (7.21) U0 − U0 = μVV (cyl)

where U0 is the radial mean velocity, U0 the radial mean velocity of an ideal axisymmetric system (which is now associated with the LSR), and Θ0 the galactocentric rotation mean velocity. This is not a special situation, and recent studies of the shape of velocity ellipsoids in spiral galaxies (Vorobyov & Theis 2008) confirm that the magnitude of the vertex deviation is not correlated with the gravitational potential (even though it is assumed nonaxisymmetric), but is strongly correlated with the spatial gradients of the

110

CHAPTER 7. SUPERPOSITION OF STELLAR POPULATIONS

mean stellar velocities, in particular with the radial gradient of the mean radial velocity. Thus, according to Eq. 7.21, a velocity ellipsoid with a positive [negative] vertex deviation might be associated with a loss of axisymmetry and with a radial motion towards [against] the Galactic centre. In Fig. 7.2 (top) a simulation is displayed. The two major stellar groups with opposite radial velocities around the LSR, which support the largest structure within the thin disc, are associated with two velocity ellipsoids with similar total dispersions. The one moving toward the Galactic centre, with radial and rotation galactocentric mean velocities U0 = 15, Θ0 = 220, and the other one, toward the anticentre, with mean motion U0 = −15, Θ0 = 200. The LSR is placed in the middle of both ellipsoids in a similar situation to the sample with maximum eccentricity 0.15. By assuming the same mixture proportions n = n

, the partial diagonal central moments μ ii = μ

ii are obtained from totals and from their deferential mean velocity wi according to Eq. 7.4 as μii = n μ ii + n

μ

ii + n n

w2i ;

i = 1, 2, 3.

(7.22)

The vertex deviation of each ellipsoid is obtained from Eq. 7.21. The total moments are taken from the sample with maximum eccentricity e = 0.3. The graph shows the partial ellipsoids uT · μ−1 2 · u = 1, in dark gray, and the thin disc velocity ellipsoids uT · μ−1 2 · u = 2, 3, in light gray, which are projected in the UV plane. The ellipsoid with positive radial velocity has null or very small, positive vertex deviation, and the one with negative radial peculiar velocity has null or very small, negative vertex deviation. The shape of the thin disc, in particular its apparent positive vertex deviation, is generated from the inner structure. The dashed partial ellipsoids represent a situation with the opposite radial motions, so that, in such a case, the apparent total vertex deviation should be negative. On the bottom, the contour plots in the UV plane (in heliocentric velocities) for the samples with maximum eccentricity e = 0.15 (dark gray) and e = 0.3 are superposed. Simulated and actual plots are totally consistent.

7.4. LOCAL VELOCITY ELLIPSOIDS

111

240 230

Q

220 210 200 190 180 170

K40 K20

0

20

P

40

20 10 0

V

K10 K20 K30 K40 K50

K60 K40 K20

U

0

20

40

Figure 7.2: (Top) Velocity ellipsoids, in dark gray, depicted according to equations Eq. 7.21 and Eq. 7.22, from total moments corresponding to the sample with eccentricities e ≤ 0.3. They are centred in galactocentric velocities Π0 = 15, Θ0 = 220, and Π0 = −15, Θ0 = 200, with the LSR placed in the middle of them. Thin disc isocontours, in light gray, with positive vertex deviation, are generated from the inner structure. The dashed partial ellipsoids represent a situation with the opposite radial motions. (Bottom) Contour plots in the UV plane (heliocentric velocities) for the samples with maximum eccentricity e = 0.15 (dark gray) and e = 0.3 (light gray).

112

CHAPTER 7. SUPERPOSITION OF STELLAR POPULATIONS

The bimodal behaviour of the central disc associated with the previous major subsystems can be explained from two different phenomena. On one hand, it may be a perturbation similar to a pressure wave acting in part along the radial direction that induces an oscillation of the radial velocity around the LSR. Let us remember that the oscillation of each subsystem centroid along the U direction is also the expected motion of axisymmetric systems under steady state potentials, as we shall see in the next chapter. On the other hand, both kinematical major groups, which actually are placed at the solar position, are in opposite oscillation states. In addition, both groups have a difference of about 20 km s−1 in rotation mean velocity, so that one group of stars actually surpasses the other group. Therefore, the apparent vertex deviation of the thin disc may stem from the swinging of those major kinematic groups. A scenario of a continuously changing orientation of the disc pseudo ellipsoid is then possible.

7.5 Second moments of a n-population mixture In Cubarsi (1992), Cubarsi & Alcobé (2004), and Pasetto et al. (2012a, 2012b), the velocity moments and cumulants of a two-component mixture were evaluated in terms of the partial statistics and the mean velocity differences between populations. In this appendix, these relationships are generalised to a n-population mixture, by obtaining the total mixture of the second moments in terms of the one-to-one mean velocity differences. For the ith population, for fixed time and position, let fi represent the velocity distribution function. According to Eq. 1.12, its stellar density is  given by Ni = Γ fi dV . Each normalised velocity distribution function V  ψi = fi /Ni defines the respective mean velocity as vi = Γ V ψi dV . If V the total mixture is written without a subindex and the population fractions are defined as ni = Ni /N, we may obtain the following basic relationships for the normalised density functions, the population fractions, and the mean velocities: f =

n  i=1

fi =⇒ ψ =

n  i=1

ni ψi ,

n  i=1

ni = 1,

v=

n 

n i vi .

(7.23)

i=1

The tensor of the second central moments μ ≡ μ2 , defined in Eq. 1.18 with n = 2, can be expressed working from the population components in terms

29.1 ± 0.2

e ≤ 0.3

18.1 ± 0.1

σV 26.1 ±0.6 14.1 ±0.1 11.6 ± 0.1

σW 18.6 ±0.5 12.4 ±0.1 9.6 ±0.6

δ [◦ ] 10.7 ±2.4 8.7 ±0.6 0.2 ±0.2

cU 5.5 ±2.8 -0.7 ±0.1 0.4 ±0.2

cV 22.7 ±9.3 0.0 ±0.2 -0.4 ±0.1

cW 35.2 ±42.4 0.7 ±0.3

0.1 ±0.0

γU -0.1 ±0.2 0.2 ±0.0

-0.3 ±0.0

γV -3.1 ±0.5 0.2 ±0.0

0.0 ±0.0

γW -0.9 ±1.0 0.1 ±0.0

Table 7.1: Distribution parameters for Sample A and Sample B. The last row is for the subsample obtained from eccentricities e ≤ 0.3. The displayed parameters are dispersions σU , σV , σW , vertex deviation δ on the UV-plane, kurtosis cU , cV , cW , and skewness γU , γV , γW .

σU 35.2 ±0.4 21.5 ±0.1

Sample Sample A Sample B

7.5. SECOND MOMENTS OF A N -POPULATION MIXTURE 113

CHAPTER 7. SUPERPOSITION OF STELLAR POPULATIONS 114

|W| =170

P |(U, V, W)| =230 D 99% H 1%

Pop. t 96% T 4% 32.5±0.7 151.0±14.7

σU 30.6±0.4 68.8±2.7

17.8±5.7 107.9± 54.0

σV 17.5±1.4 30.0±15.0

16.5±0.3 70.3± 5.7

σW 15.2±0.3 40.0±2.0

-9.9±0.3 -12.2±2.1

U -10.0±0.3 -8.6±0.7

-16.9±0.3 -230.0±24.1

V -15.2±0.3 -81.8±4.3

-7.1± 0.2 -18.3± 3.3

W -6.9±0.2 -11.6±0.5

8±4 −3 ± 7

δ [◦ ] 9±1 6±2

Table 7.2: Mixture parameters for subsamples drawn from Sample A . The displayed quantities are: sampling parameter, segregated population (t=thin disc, T=thick disc, D=total disc, H=halo) and mixture proportion, velocity dispersions, mean velocities (both in km s−1 ), and vertex deviation.

7.5. SECOND MOMENTS OF A N -POPULATION MIXTURE

115

of the peculiar velocity referred to each population, ui = V − vi , as follows: 

μ= = =

ΓV

n i=1



(V − v)2 ψ dV = 

ni

Γu

i=1 ni

Γu

+(vi − v)2

(V − v)2

n i=1

ni ψi dV =

(ui + vi − v)2 ψi dui =



n

ΓV

(ui )2 ψi dui + 2(vi − v) ⊗

(7.24)

 Γu

ui ψi dui +



 Γu

ψi dui .

  Since Γ ui ψi dui = 0 and Γ ψi dui = 1, by substitution of the mean u u velocity given in Eq. 7.23, we get μ= =

n i=1

n i=1

n

n i μi +

n i μi +

i=1

n i=1

ni

ni (vi − 

n j=1

n j=1

n j v j )2 = 2

(7.25)

n j (vi − v j ) .

We prove the following equality

I≡

n 

⎞2 ⎛ n ⎟⎟⎟ ⎜⎜⎜ ni ⎜⎜⎝⎜ n j (vi − v j )⎟⎟⎠⎟ = j=1

i=1

n 

ni n j (vi − v j )2

(7.26)

i, j = 1 i< j

by writing I as the addition of the series I1 + I2 , so that I≡

 

I1 ≡ =

ni

i

i



I2 ≡

i ni

 i



ni  ni

ji



   n j (vi − v j ) ⊗ ki nk (vi − vk ) , 

ji

ki, k= j

n j nk (vi − v j ) ⊗ (vi − vk ) = (7.27)

2 2 ji n j (vi − v j ) ,



 ji

ki, k j

n j nk (vi − v j ) ⊗ (vi − vk ).

CHAPTER 7. SUPERPOSITION OF STELLAR POPULATIONS

116

The series I1 may be written as 1   I1 ≡ i j>i (ni n2j + n j n2i )(vi − v j )2 = = = =

  i

j>i

ni n j (n j + ni )(vi − v j )2 =

j>i

ni n j (1 −

j>i

ni n j (vi − v j )2 −

  i

  i

(7.28)



ki, k j nk )(vi

− v j )2 =

  i

 j>i

ki, k j

ni n j nk (vi − v j )2 .

Similarly, the series I2 may be written as    I2 ≡ i ji k j, ki ni n j nk (vi − v j ) ⊗ (vi − vk ) = =

  i

 j>i

k j, ki

ni n j nk [(vi − v j ) ⊗ (vi − vk )+

+(v j − vi ) ⊗ (v j − vk )] = = =

  i

 j>i

  i

k j, ki

 j>i

k j, ki

(7.29)

ni n j nk (vi ⊗ vi − 2vi ⊗ v j + v j ⊗ v j ) = ni n j nk (vi − v j )2 .

Finally, by addition of Eq. 7.28 and Eq. 7.29 we are led to Eq. 7.26. Then, Eq. 7.25 becomes μ=

n  i=1

n i μi +

n 

ni n j (vi − v j )2 .

(7.30)

i, j = 1 i< j

Notice, however, that the set of 21 n(n − 1) differences {(vi − v j )}i, j for i < j are linearly dependent. A reduced set of n − 1 independent quantities is, e.g., {(vk − v1 )}k for k > 1, since any difference vi − v j can be obtained as (vi − v1 ) − (v j − v1 ).

7.6 Remarks In most of these cases the techniques to disentangle the mixture distribution yielded a characterisation of the population components that was totally independent from dynamical assumptions. For example, Pasetto et al. (2012a,   1  Any series i ji ai j over the elements ai j of an arbitrary matrix is equivalent to (a + a ). ji i j>i i j

7.6. REMARKS

117

2012b) based their segregation algorithm on previous works involving developments of the moments method in three dimensional velocity space to obtain the best fit for the total distribution cumulants, which are better statistics than the moments. This approach takes advantage of the symmetry shown by the distribution cumulants about the axis along the centroids of a two-component quadratic mixture distribution (Cubarsi 1992). In the beginning, when this method was applied to HIPPARCOS’s samples (Cubarsi & Alcobé 2004), it was only capable of disentangle two populations. Further improvements, however, based on the construction of a series of nested subsamples depending on optimal properties of a sampling parameter associated with an isolating integral of motion, allowed for identification of several stellar populations contained in the total sample. These included early-type and young disc stars within the thin disc, as well as thin disc, thick disc, and halo populations (Alcobé & Cubarsi 2005). In particular, the segregation of populations improved when the GCS catalogue was used. These populations were associated with partitions of the total sample providing the best estimates for population kinematical parameters and mixture proportions, according to both criteria of minimum chi squared error and maximum partition entropy (Cubarsi et al. 2010). After segregation, the population kinematics, described through the population means and the second central moments (or covariance matrix), needs to be interpreted in the framework of a dynamical model to test the consistency of these observables with the hypotheses and variables of the model. Most recent results show, among other features, that the disc populations have non-vanishing vertex deviation, the thick disc has a radial mean motion differing from the thin disc, and the halo velocity ellipsoid is slightly tilted. For a dynamical model to explain such a features, Pasetto et al. (2012b) and Steinmetz (2012) suggest that the axisymmetry assumption should be relaxed, perhaps towards a model with rotational symmetry of 180◦. This will be studied in Chapter 9. Since these results provide enough material to review in detail the dynamical model sustaining such a mixture of stellar populations in the solar neighbourhood, in the next chapter we shall try to answer that question by testing the consistency of the axisymmetry assumption against the kinematic observables. Often, the validation of the axial symmetry hypothesis is made by identifying the stellar system with a single population, and it ignores the fact that a mixture of populations with arbitrary mean velocities gives a totally different shape to the velocity distribution through their moments and gradients.

Chapter 8

Axisymmetric system 8.1 Model hypotheses In Chapter 3 we saw that Chandrasekhar’s classical approach to describing the local kinematics and dynamics of the Galaxy consists of solving the collisionless Boltzmann equation by assuming a phase space density function that depends on a quadratic integral of motion whose coefficients are functions of time and position. This involves a number of linear combinations of the classical integrals of the galactic dynamics (Sala 1990). Here it is necessary to point out that the central point of Chandrasekhar’s approach is not the shape of the integral Q+σ in Eq. 3.27, but to assume Q+σ = uT ·A2 ·u +σ as depending on time through A2 (t, r) and σ(t, r), as well as the potential U(t, r). I believe that the criticism that has received this model (e.g., Evans 2011) is because this issue has not been considered. If the stellar system had been assumed to be stationary, i.e., does not depending on time explicitly, his results would not have meant any significant progress to the results of Eddington (1915). We know that in the stationary case, the Stäckel conditions are satisfied (Pars 1965, Makarov et al. 1967), by providing orthogonal coordinate systems where the Hamilton-Jacobi equation is completely separable, and leading to separable potential solutions, the so called Stäckel potentials (e.g., Goldstein 1980, p.453 and Appendix D). However, the stationary model leads to axisymmetric potentials and restricts the differential motion of the centroids to rotation. Instead, for an explicitly time dependent system, such a separation of variables is not always guaranteed, although a priori the above limitations do not exist. Nevertheless, for a time de119

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CHAPTER 8. AXISYMMETRIC SYSTEM

pendent, three-dimensional system the solution of Chandrasekhar equations does also provide separable potentials (de Orús 1977, Sala 1990), although the Hamilton-Jacobi method was not used by this authors. Depending on some parameter values, the potential is separable either in spherical coordinates, prolate spheroidal coordinates, or cylindrical coordinates. To simplify the model, a velocity distribution of Schwarzschild type may be also assumed, that is, a trivariate Gaussian function in peculiar velocities, which is the maximum entropy distribution with known means and covariances. In particular, this means that the relationship given by Eq. 3.30 for n = 1 is exactly Eq. 3.2, without any factor depending on σ, although this has no consequences when solving the collisionless Boltzmann equation. In this way, the coefficients of the integral of motion are directly related to the mean and the central moments of the velocity distribution. The kinematics of the stellar system is then studied from a statistical viewpoint: the individual orbits of the stars are replaced by the average orbit of their centroid with its associated distribution statistics. Some symmetries are usually introduced for the mass and velocity distributions to simplify the solution of the collisionless Boltzmann equation. Commonly, it is assumed that the Galaxy is axisymmetric, has a plane of symmetry, and is in a steady state. However, we have already pointed out that these hypotheses provide serious limitations to the stellar kinematics, so that it is impossible to describe the local velocity distribution in a realistic way. Following are examples of these well-known limitations. The steady state hypothesis, which yields an axisymmetric potential (Chandrasekhar 1942), supports only stellar systems with differential motion in rotation. A quadratic integral provides a symmetric velocity distribution that cannot account for odd-order central velocity moments (Cubarsi 1992). A quadratic axially symmetric velocity distribution is not able to show any vertex deviation of the velocity ellipsoid (Sala 1990). In this context, i.e., already avoiding the stationary hypothesis and still assuming a quadratic integral of motion as an equilibrium solution of the collisionless Boltzmann equation, there are basically two complementary alternatives for allow a more flexible description of the stellar kinematics. One is to relax the hypothesis of axial symmetry. The other alternative is to introduce a mixture model associated with several kinematic populations under the ellipsoidal hypothesis. This chapter will explore the latter option, by trying to find out whether it is possible to fit the actual kinematic observables of the Galaxy working from a finite mixture distribution and still maintain the axisymmetric assumption. The next chapter will explore what happens if axial symmetry is exchanged by rotational symmetry of 180◦,

8.2. DYNAMICAL MODEL

121

i.e., point-axial symmetry. From Kapteyn’s theory of the two star streams in 1905, which probably is the first mixture model in astronomy, there have been different approaches to construct velocity distributions that fit specific observables of the Galaxy. The first statistical and numerical approaches were developed by Kapteyn (1922), Strömberg (1925), and Charlier (1926), in order to fit up to the fourth moments of the velocity distribution. Most kinematical models construct the velocity distribution from a function depending on two or three integrals of motion, and by addition or products of these functions. Quadratic distributions and mixtures of them are a particular case of this. On the other hand, more recent dynamical models use parametrised velocity distributions that are analytic functions of the action integrals of motion (Binney 2010, Binney & McMillan 2011, Binney 2012), which also allow to constrain the parameters of the Galactic potential (Ting et al. 2013). This approach benefits from both kinematic and dynamic models. When the quadratic integral of motion is introduced into the collisionless Boltzmann equation, a direct analytical relationship between the potential and the velocity distribution function of each stellar population emerges. On the other hand, if each stellar population is associated with a Gaussian distribution, the segregation of the populations composing the mixture may be worked out from a number of standard statistical and numerical techniques that are independent from the dynamical model.

8.2 Dynamical model From a dynamical viewpoint, stellar mixtures can be introduced for the sake of the superposition principle, since the collisionless Boltzmann equation is linear and homogeneous in regard to the phase space density function, for a given potential. Thus, we may assume that the whole stellar system is composed of a finite number of stellar populations in statistical equilibrium, which have the most probable phase distribution of Schwarzschild type (e.g., Ogorodnikov 1965, Lynden-Bell 1967). As seen Chapter 1, the term statistical equilibrium is a notion coming from statistical dynamics that, in analytical dynamics, should be interpreted as associated with an invariant density function in the phase space under the collisionless Boltzmann equation. Dissipative forces, such as dynamical friction, which are essential to statistical dynamics, emerge as solutions of the collisionless Boltzmann equation via non-steady state phase density functions and potentials. Then, for a given potential, the Jeans’ direct prob-

122

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lem yields the most probable distribution function for an equilibrium configuration of the Galaxy, and provides us with information about the functional form of the distribution function and the conserved quantities of the stellar motion. Otherwise, when there is some kinematic knowledge about the stars’ integrals of motion, or the velocity distribution function is already known, the Jeans’ inverse problem leads to the most probable potential function. For a mixture model, the natural approach is the Jeans’ inverse problem, once the populations have been characterised from their velocity distributions. In wide regions of the Galaxy, this is usually done by associating each stellar component with a Schwarzschild distribution function. It is assumed that each population has a centroid or LSR that moves with a mean velocity that is a continuous and differentiable function of time and position. This is known as the differential motion of the stellar system and it is guaranteed by the existence of a helicoidal symmetry axis (Chandrasekhar 1939). For that reason, it is generally assumed that the stellar system has rotational symmetry, also referred as axisymmetry. This hypothesis substantially simplifies the dynamical model, although other symmetries, such as point-axial symmetry, could be more appropriate in describing ellipsoidal or spiral mass distributions. The velocity distribution function should be a time dependent function to react to changes of internal gravitational forces and to allow arbitrary mean velocities of the populations. That is, the mean motion of the populations should not be restricted to rotation alone as in case of stationary systems. For the potential, it is assumed that the whole set of populations produces a total self-gravitating system yielding a unique galactic potential, so that the self-gravitation of a single population is negligible. Similar to the velocity distribution function, the potential is expected to be nonstationary, although it is possible for a stationary potential to coexist with a non-stationary velocity distribution. Then, the collisionless Boltzmann equation relates the dynamics of each stellar population to the common potential shared by all of the population components. When solving the collisionless Boltzmann equation for each population, a set of integrability conditions arises to obtain an admissible potential consistent with all the populations. They are referred to as conditions of consistency for a multicomponent stellar system. These conditions may force the potential function to adopt a specific functional form, by allowing the velocity and mass1 distributions a number of degrees of freedom. 1

In the current approach the Poisson equation is not used together with the collisionless

8.3. CHANDRASEKHAR’S AXIAL SYSTEM

123

For axisymmetric systems, these conditions were studied in Cubarsi (1990). It was shown that the more general solution for the potential provides populations differing only in mean rotation. However, particular families of potentials, which are independent of specific population parameters, give rise to kinematically independent populations. This means that the populations may have arbitrary mean velocities and unconstrained velocity ellipsoids. In this case, the centroids of the stellar populations that occupy the same position in the Galaxy do not follow a circular orbit around the symmetry axis. Instead, they may visit other centroid orbits and mix with other populations of their neighbourhood, which is a more realistic situation. Some aspects of the former study can now be reformulated and improved. In general, there is a tug of war between the potential function and the population velocity ellipsoids in that when the potential function is more general, the stellar populations are more kinematically constrained. This conjures up the well-known Bob Dylan song, "You’re gonna have to serve somebody." We are going to study in more detail the conditions of consistency for mixtures of axisymmetric systems by determining what potentials are connected with more flexible superposition cases in regard to the population kinematics. It will be tested against actual values of moments and gradients for the thin disc, the thick disc, and the halo, to determine whether an axisymmetric dynamical model is still able to describe the main kinematical features of the solar neighbourhood.

8.3 Chandrasekhar’s axial system In a seminal work2 , Chandrasekhar adopted the Jeans’ inverse problem approach for a generalised Schwarzschild velocity distribution. He assumed the phase space density function depending on an integral of motion quadratic in the peculiar velocities and left free the functional dependency on time and space. Such a quadratic integral is the simplest way of labelling one statistical population through the whole set of first and second moments. However, the symmetry of this velocity distribution does not allow nonnull odd-order Boltzmann equation, since some population components like bulge, dark matter, etc., might have unknown velocity distribution. Instead, the mass distribution of each population is derived from each partial phase space density function. 2 Chandrasekhar’s (1939, 1940) works on stellar dynamics were compiled in 1942 in the book Principles of Stellar Dynamics. In 1960, the book was republished and three new capital articles were added. Hereafter several results will be referred to this particular edition.

124

CHAPTER 8. AXISYMMETRIC SYSTEM

central moments, which means that a mixture of populations is needed to account for other statistics of the distribution. Under particular symmetry hypotheses, basically for an axisymmetric velocity distribution of a disc, Chandrasekhar showed how a common potential shared by two stellar populations introduces links between their kinematic parameters (Chandrasekhar 1960, p.126). This approach is not likely to be valid for the entire Galaxy, but it is a good approximation for the populations existing in solar neighbourhood. Although the population velocity ellipsoids and their rotation curves have a local meaning, we may use the mixture model as a collage representation of the Galaxy. Nevertheless, some properties and symmetries of the velocity distribution and, in particular, the potential should be valid for the entire Galaxy with the exception of singular points such as the origin and, in some case, the symmetry axis. For the whole three dimensional space, under the axial symmetry hypothesis, Sala (1990) determined the family of potential functions that was consistent with a quadratic integral of motion, and Cubarsi (1990) studied what constraints would apply to a mixture of stellar populations. However, in 1990 the lag of accuracy in the stellar catalogues, especially in the radial velocity data, left open a long list of suitable models to explain the local kinematics, which now is possible to discuss in more detail (Cubarsi 2014a). Let us recall that a single stellar population is associated with a quadratic velocity distribution function in the peculiar velocities (u1 , u2 , u3 ). The phase  space density function is written as f (Q+σ(t, r)) with Q = i, j Ai j (t, r) ui u j , where Ai j are the elements of a symmetric, positive definite matrix. Hence, Q + σ is an isolating integral of the star’s motion, which is a combination of some of the classical integrals. Generally, from statistical criteria, the ve1 locity distribution is assumed of Schwarzschild type, f = e− 2 (Q+σ) , that is, a trivariate Gaussian function in the peculiar velocities. Then, the elements of the covariance matrix are μi j = A−1 i j , and the equation Q = 1 defines the velocity ellipsoid. From a Bayesian criterion, this is the less informative distribution with known means and covariances. By substitution of the quadratic density function in the collisionless Boltzmann equation, Chandrasekhar obtained a system of partial differential equations for the potential function U, the scalar function σ, the mean velocity v, namely velocity of the centroid, and the tensor A2 . We have already seen that these equations of Section §3.4 are equivalent to the infinite hierarchy of the stellar hydrodynamic equations, which can be reduced to equations of orders n = 0, 1, 2, 3, for the sake of a set of closure conditions. Under axial symmetry, we also assume z = 0 as a symmetry plane for the

8.3. CHANDRASEKHAR’S AXIAL SYSTEM

125

mass and the velocity distributions. We note the star’s position r = (, θ, z) and the velocity V = (Π, Θ, Z). The collisionless Boltzmann equation may be solved in two blocks. The first one composed of equations 3.23 and 3.24, which yields (Sala 1990) the functional form for the elements of A2 and the population mean velocity v. They are explicitly written in Appendix G. The second block, which generalises Chandrasekhar equations (Chandrasekhar 1960, Eq. 3.703), consists of equations 3.25 and 3.26. They provide the solution for the potential U and the function σ. These equations are not exactly written as derived by Chandrasekhar. They become equivalent if the new variables Δ, X are used instead v, σ. They are defined as Δ = A2 · v;

X = −Δ · v − σ.

(8.1)

3

Then, equation 3.25 and 3.26 become

1 ∂Δ = − ∇X, (8.2) ∂t 2 1 ∂X Δ · ∇U = . (8.3) 2 ∂t With the elimination of X between equations 8.2 and 8.3, with the new variables τ = 12 2 and ζ = 12 z2 , which are appropriate to the symmetry plane of the system, the following set of three second-order partial differential equations for the potential are obtained:  2 ∂2 U ∂2 U ∂U ∂U ∂ U −ζ 2 −2 − (τ − ζ) 2k4 τ 2 + 2 + ∂τ ∂ζ ∂τ∂ζ ∂τ ∂ζ (8.4) ∂2 U = 0, +(k1 − k3 ) ∂τ∂ζ   ∂ ∂U ∂U − 2k4 ζ + ∂t ∂τ ∂ζ (8.5) 2 2 2 ... 1 U ∂ U ∂U U ∂ ∂ + 2k˙1 + k1 + k˙3 ζ = 0, +k˙1 τ 2 + k1 ∂t∂τ ∂τ 2 ∂τ∂ζ ∂τ   ∂ ∂U ∂U 2k4 τ − + ∂t ∂ζ ∂τ (8.6) 2 2 2 ... 1 U ∂ U ∂U U ∂ ∂ + 2k˙3 + k3 + k˙1 τ = 0. +k˙3 ζ 2 + k3 ∂t∂ζ ∂ζ 2 ∂ζ∂τ ∂ζ A2 · ∇U +

3

Hereinafter we write ∇ instead of ∇r , since there is no possibility of misunderstanding.

126

CHAPTER 8. AXISYMMETRIC SYSTEM

Table 8.1: Potentials that are consistent with a flat velocity distribution. k4 = 0 k ≡ k1 = k3 ∂2 U =0 k ≡ k1 = k3 ; ∂τ∂ζ ∂2 U k1  k3 ; ∂τ∂ζ =0 k4  0 k ≡ k1 = k3 ∂2 U =0 k1  k3 ; ∂τ∂ζ

  U = A(t) (τ + ζ) + 1k U1 τk , ζk     U = A(t) (τ + ζ) + 1k U1 τk + 1k U2 ζk     U = A1 (t) τ + A2 (t) ζ + k11 U1 kτ1 + k13 U2 kζ3   U = A(t) (τ + ζ) + 1k U1 τ+ζ + k U = A(t) (τ + ζ) + Bτ + Cζ

U2 (ζ/τ) τ+ζ

(I) (II) (III) (IV) (V)

The particular case k4 → 0 corresponds to Chandrasekhar equations for a flat velocity distribution of a rotating disc. This yields the potentials shown in Table 8.3, equations I, II, and III. Then, the elements of A2 and the tensor of the second central moments μ do not depend on z. In particular, the moments μ and μzz do not depend on  and z (which is obvious from Appendix G.2, by taking k4 = 0). Thus, the velocity distribution is isothermal in these directions 4 . In addition, if k1 = k3 , the distribution is isotropic in the  and z directions. Alternatively, for k4  0 the foregoing equations describe a non isothermal three-dimensional distribution and provide two families of compatible potentials. One family of potentials not dependent on the constant k4 are displayed in Table 8.3, equations IV and V, corresponding to Eq. (2.7) of Sala (1990). Another family of potentials are dependent on that constant, which are given by Eq. (2.9) in Sala (1990), are not displayed in Table 8.3. The latter is a family of Stäckel potentials, separable in prolate ellipsoidal ˘ Zs ´ (1915) and Lynden-Bell (1962) stacoordinates, including EddingtonâA tionary potentials. For k4  0, the velocity distribution is in general non-isothermal regardless of the solution for the potential. The  and z gradients of the moments μ and μzz are in general nonnull out of the Galactic plane. The velocity ellipsoid of an axially symmetric population has no vertex deviation5, which 4 A mixture of such isothermal distributions in the Galactic plane is also isothermal in the z direction, but is not in the  direction, provided they have different radial mean velocities. Similarly, a mixture of two isotropic distributions in the  and z directions, with different radial mean velocities, is non-isotropic. 5 As explained in Cubarsi 1990, Appendix A, this is indeed a consequence of assuming z = 0 as a plane of symmetry for the elements the tensor of covariances and for the mean velocity. If

8.4. CONDITIONS OF CONSISTENCY FOR MIXTURES

127

is associated with the central moment μθ .

8.4 Conditions of consistency for mixtures Chandrasekhar calls the circumstances under which we can regard a stellar system as consisting of two or more independent populations sharing the same potential conditions of consistency. Since the potential may depend on the population parameters involved in the velocity distribution function, the less the potential depends on them, the less constrained the populations are. In particular, we are interested in potentials allowing the populations to have different mean velocities and arbitrary orientations of the velocity ellipsoids. For a stellar mixture, say two populations with fractions n and n

, the total moment μθ depends on the radial and rotational mean velocity differences, according to μθ = n μ θ + n

μ

θ + n n

(Π 0 − Π

0 )(Θ 0 − Θ

0 ).

(8.7)

In addition, according to Appendix G.2, for axisymmetric distributions the partial moments μ θ and μ

θ vanish. Therefore, if the radial and rotational mean velocity differences between both populations are nonnull, the total moment μθ is nonnull too. Since steady state systems are only capable of rotational differential motion, it is necessary to assume a mixture of time dependent systems. Depending on the potential function, the axisymmetric mixture model will lead to the following main cases. First, the general case of a potential depending on the constant k4 , hereafter referred as axisymmetric general case. Second, the case of a potential that does not depend on k4 , hereafter referred to as consistent with a flat velocity distribution, which leads to three particular situations: (a) a potential non-separable in cylindrical coordinates6 with k ≡ k1 = k3 , whose time dependency is explicitly expressed in terms of the population parameter k(t); (b) a non-separable potential with k ≡ k1 = k3 , referred to as quasi-stationary potential, whose time dependency is carried through a unique function A(t), allowing different mean velocities of the stellar populations, with untilted velocity ellipsoids; and ∂2 U = 0, with k1  k3 , also depending (c) a separable potential satisfying ∂τ∂ζ axial symmetry had been the only assumption, the vertex deviation would be null in z = 0, but nonnull on either side of the plane. 6 As we are studying the axisymmetric model, hereafter a non-separable potential will mean that it is not separable in addition in cylindrical coordinates.

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CHAPTER 8. AXISYMMETRIC SYSTEM

on time through A(t), allowing arbitrary mean velocities of the populations and arbitrary tilt of the velocity ellipsoids. Both main cases and their subcases are analysed in the following subsections, by ending with Table 2 that summarises them.

8.4.1 Axisymmetric general case For a single population, the elements of A2 (Appendix G.1) depend on the functions of time k1 (t), k3 (t), and on the constants k2 , k4 , β. However, the constants β and k2 do not appear in equations 8.4, 8.5, and 8.6. The latter constants related to the centroid rotation are responsible for phase mixing. Notice that, according to the expression for the mean rotation velocity in Eq. G.4, β , (8.8) Θ0 = − k + k2 2 + k4 z2 a value k2 = 0 at the symmetry plane would correspond to a constant angular velocity −β/k. Then, for each stellar population, the differential non-rigid rotation acts as the main factor of phase mixing, which will be intensified if differential radial motion exists. For instance, at a position 0 on the β Galactic plane, the rotation mean velocity can be expressed as Θ0 = − 1+α 2

μ  1  with α = 2 μθθ − 1 > 0. The value α = 0 would correspond to the 0 limiting case of a stellar population with rigid rotation. Fig. 8.1 simulates, for arbitrary values α = 0.1, β = 1 and in an arbitrary time scale, how the disc stars placed on a horizontal line across the Galactic centre become mixed after several rotation periods in the positive angular direction. On the contrary, for the equations of the potential, the remaining kinematical parameters play a significant role. There is a trivial case where the same potential is valid for all populations. It happens when the potential depends on the population parameters k1 , k3 , and k4 , but these parameters are proportional among populations, so that we get a system of equations identically planned for each population. Then, for a two-component mixture, the constant ratio between parameters is transferred to the second central moments (Appendix G.2), so that they satisfy μ  μ z μ zz = = , μ

 μ

z μ

zz

μ z μ

z = . μ  − μ zz μ

 − μ

zz

(8.9)

Then, the velocity ellipsoids, in addition to having two proportional semiaxes, have the same orientation in any z plane, that is, the same tilt. The mean velocity components satisfy Π 0 = Π

0 and Z0 = Z0

. Hence, according

8.4. CONDITIONS OF CONSISTENCY FOR MIXTURES

t = 0.

K K K K 8

6

4

2

t = .15000

8

8

6

6

6

4

4

4

2

2

2

0

2

4

6

8

K K K K 8

6

4

2

2

2

4

6

8

K K K K 8

6

4

2

K K K K

0

8

6

6

4

4

4

2

2

K K K K

2

4

6

8

8

K K K K 8

6

4

2

K K K K

0

2

4

6

8

2

4

6

8

t = 3.0000

8

6

8

8

6

4

6

6

t = 1.5000

2

4

4

8

0

2

2

8

t = .90000

4

2

6

8

6

0

4

6

8

K K K K

2

4

K K K K

t = .45000

8

K K K K

129

2

2

4

6

8

K K K K 8

6

4

2

K K K K

0

2

4

6

8

Figure 8.1: Simulation of how the mean differential rotation originates the phase mixing process, according to a rotation law as Eq. 8.8 at the Galactic plane.

CHAPTER 8. AXISYMMETRIC SYSTEM

130

to Eq. 8.7, the radial differential motion of the centroids is null and it is impossible to get a non-vanishing total central moment μθ . Since the rotation mean velocity Θ0 and the second moment μθθ depend on the constants β and k2 , these two quantities are not constrained by the potential. They are independent among populations. Thus, the axisymmetric assumption allows for the potential not to constraint the velocity distribution in the rotation direction. Such kinematical behaviour is not far from the actual portrait of the Galaxy in the solar neighbourhood; although, as is well-known, small violations of this ideal situation have long since been observed, such as the vertex deviation, mainly for the thin disc, and the existence of nonnull central moments, the forbidden moments, involving odd powers of the radial velocity (Erickson 1975).

8.4.2 Flat velocity distribution The solution for a potential that does not depend on k4 (but depends on k1 , k3 ) is the same as the potential obtained with k4 = 0, although the populations are still k4 -dependent. This is a generalisation of the case studied in Chandrasekhar (1960, pp.116-121), associated with an ideal rotating disc. An isothermal velocity distribution in the direction perpendicular to the disc does not imply an isothermal mass distribution, but a relatively small scale height, since the dependence of the mass distribution on z comes through the potential as well 7 . The general solution for the potential will not depend on k4 when the terms being multiplied by this constant in the equations 8.4, 8.5, and 8.6 are zero8 . This provides two additional and fundamental integrability conditions τ

∂2 U ∂2 U ∂2 U ∂U ∂U − ζ − (τ − ζ) = 0, + 2 − 2 ∂τ ∂ζ ∂τ∂ζ ∂τ2 ∂ζ 2   ∂ ∂U ∂U − = 0. ∂t ∂τ ∂ζ

(8.10)

(8.11)

The condition of Eq. 8.10 leads to a potential function in the form U = 1 U2 (ζ/τ, t) (a deduction can be found in Cubarsi 1990, U1 (τ + ζ, t) + τ+ζ stellar density is given by Eq. 3.3, where σ depends on the potential through the equations 3.25 and 3.26. 8 The general solution of a differential equation is a continuous and differentiable function of its parameters. 7 The

8.4. CONDITIONS OF CONSISTENCY FOR MIXTURES

131

Appendix C). The condition of Eq. 8.11 forces U2 to be time independent. Then, Eq. 8.4 simply remains as (k1 − k3 )

∂2 U = 0, ∂τ∂ζ

(8.12)

and the equations 8.5 and 8.6 become ∂2 U ∂U 1 ... ˙ ∂2 U ∂2 U k˙1 τ 2 + k1 + 2k˙1 + k1 + k3 ζ = 0, ∂t∂τ ∂τ 2 ∂τ∂ζ ∂τ 2 2 2 ∂U ∂ U ∂U 1 ... ˙ ∂ U k˙3 ζ 2 + k3 + 2k˙3 + k3 + k1 τ = 0. ∂ζ ∂t∂ζ ∂ζ 2 ∂ζ∂τ

(8.13)

The conditions of consistency now have to be studied in relation to Eq. 8.12. ∂2 U We investigate the cases where ∂τ∂ζ is null or nonnull separately. These correspond to a non-separable or separable potential in cylindrical coordinates, respectively9. For the particular case of a steady state system, where k1 and k3 are constant, and ∂U ∂t = 0, both relationships in Eq. 8.13 vanish. We then obtain the stationary potential U = U1 (τ + ζ) +

U2 (ζ/τ) , τ+ζ

(8.14)

which is independent from the population parameters. Thus, a steady state system is free from conditions of consistency, but only allows differential rotation of populations since k˙1 = k˙3 = 0, as is obvious from Eq. G.3. The overall vertex deviation of the mixture distribution is null and the most significant odd-order central moments are forbidden. Non-separable potential ∂ U  0 then k ≡ k1 = k3 . The condition is According to Eq. 8.12, if ∂τ∂ζ directly related to the tilt of the velocity ellipsoid. In a three-dimensional velocity distribution with k4  0, according to Eq. G.8, if k1 = k3 , one of the principal axes of the velocity ellipsoid points towards the Galactic centre. Then, the population velocity ellipsoids are not tilted. 2

9 The above reasoning can also be made âA ˘ Nâ ˘ Nas ´ A ´ follows. If Eq. 8.4 is divided by k4 , the expression becomes separated into two terms. One of them, Eq. 8.10, is independent of the population parameters. Then, the other term, Eq. 8.12, must vanish in order to allow the potential not to depend on the population parameters.

CHAPTER 8. AXISYMMETRIC SYSTEM

132

In this case, the existence of a quadratic integral involves three independent integrals of motion (Sala 1990) and allows for a general solution for the potential in the following form: + τ + ζ , U (ζ/τ) 1 2 U = A(t) (τ + ζ) + U1 + k k τ+ζ

(8.15)

with the function A(t) satisfying A(t) = −

k¨ k˙ 2 c + 2+ 2; 2k 4k k

c ∈ R.

(8.16)

We may assume that the two possible terms of U1 , which are propork tional to τ+ζ k and τ+ζ are already accounted for in the first and third terms of Eq. 8.15, respectively. The above potential, if expressed in spherical coordinates (R, θ, φ), with R2 = r2 + z2 and tan φ = z/r, satisfies the property  ∂2  2 R U(R, φ) = 0, ∂R∂φ

(8.17)

that is, R2 U(R, φ) is separable in addition in spherical coordinates. The first term of Eq. 8.15 is a harmonic potential. For a stellar mixture, A(t) must be the same computed from either populations, and determines the function k(t) according to a linear differential equation. Thus, if we eliminate the constant c in Eq. 8.16, by multiplying by k2 and taking the time derivative, we find ... ˙ = 0. (8.18) k + 4Ak˙ + 2 Ak For a given A(t), this is a third order homogeneous equation in k(t). If ϕ1 (t), ϕ2 (t), ϕ3 (t) are three linearly independent solutions of Eq. 8.18, then its general solution is k(t) = αϕ1 (t) + βϕ2 (t) + γϕ3 (t);

α, β, γ ∈ R.

(8.19)

Therefore, in each population, k(t) must be a particular solution to the above general solution10 . 10 On the other hand, for a given function k(t), the Eq. 8.18 is a first-order, linear, nonhomogeneous differential equation for A(t), that has Eq. 8.16 as its general solution, with the c term 2 being the general solution of its homogeneous part. k (t)

8.4. CONDITIONS OF CONSISTENCY FOR MIXTURES

133

An attractive force is associated with values A > 0, which guarantees the existence of stable orbits. ... In particular, if A > 0 is constant, the above differential equation becomes k + 4Ak˙ = 0, so that √ √ k(t) = α + β cos(2 A t) + γ sin(2 At). (8.20) The term U1 in Eq. 8.15 is a spherical potential associated with a generic central force. In the general case U1  0, the potential also depends on time through the function k(t), which is a population parameter. The term with U2 may depend on the elevation angle about the Galactic plane and it is responsible for producing a flattened potential. Similar to Binney & McMillan (2011), we find that non-spherical potentials may coexist with velocity distributions with a vanishing tilt of the velocity ellipsoid. (ζ/τ) is of the same type as the perturbation of a point-mass poNotice that U2τ+ζ tential induced by a tidal force, although as we assumed a symmetry plane, this force is symmetric about this plane. Hence, this force can only have the direction perpendicular to the Galactic plane or a direction along the same plane, with the application point in the Galactic centre. The interest of a potential sufficiently general like Eq. 8.15 is that its functional dependence may support its own perturbations. According to Appendix G.1, the non-rotational mean motion of a population is associated with time variations of the stellar system. In the current case, these velocity components satisfy Π0 =

1 k˙ , 2k

Z0 =

1 k˙ z, 2k

Z0 z = , Π0 

(8.21)

meaning that each population centroid moves on a circular conic surface with the apex in the Galactic centre. For a mixture of two populations, the second central moments satisfy μ z μ

z z = = 2 .



μ − μzz μ − μzz  − z2

(8.22)

Then, the semiaxes of the velocity ellipsoids are not proportional as in the general case, but have vanishing tilt. Nevertheless, as remarked above, the spherical potential term 1k U1 must be common among the populations. Then, for a two population mixture, it is easily deduced that k and k

must be proportional. Therefore, according to Eq. 8.21, the mean velocity differences satisfy Π 0 − Π

0 = Z0 − Z0

= 0. More precisely, being k and k

solutions of the same linear and homogeneous differential equation Eq. 8.18, k is proportional to k

if, and only

CHAPTER 8. AXISYMMETRIC SYSTEM

134 ˙

˙

if, kk = kk

. Hence, Π 0 − Π

0 = Z0 − Z0

= 0 if, and only if, k and k

are linearly dependent. For that reason, for a potential to allow different nonrotational motion of the centroids, we have to search for a particular solution ˙ to Eq. 8.15 that is independent from the quantity k/k. Therefore, a potential containing a general spherical term does not allow independent radial and vertical mean motions of the populations. Quasi-stationary potential For k ≡ k1 = k3 , in regard to equations 8.10 and 8.11, we define 2F(t, τ, ζ) ≡ τ

∂U ∂2 U ∂U ∂2 U ∂2 U ∂2 U +2 =ζ 2 +τ +2 , (8.23) +ζ 2 ∂τ∂ζ ∂τ ∂τ∂ζ ∂ζ ∂τ ∂ζ

∂2 U ∂2 U = . (8.24) ∂t∂τ ∂t∂ζ Then, both expressions in Eq. 8.13 can be written as a unique equation ... 1k k˙ = 0. (8.25) 2 F +G+ k 2k ... ˙ ˙ Eq. 8.25 then becomes By Eq. 8.18, since 12 kk = −2 kk A − A, G(t, τ, ζ) ≡

k˙ 2 (F − A) + G − A˙ = 0. k

(8.26)

As the functions A(t), F(t, τ, ζ), and G(t, τ, ζ) depend on the potential, a po˙ tential not dependent on kk , which is population dependent, must satisfy F = A. Therefore, G = A˙ is also held. Hence, by Eq. 8.24, the poten˙ tial must satisfy ∂U ∂t = A(t)(τ + ζ). Then, by Eq. 8.15 we find U1 = 0.

(8.27)

By taking Eq. 8.24 into account, the above condition for a potential allowing for different radial and vertical population mean velocities can also be expressed as ∂2 ∂U = 0. (8.28) ∂τ∂ζ ∂t The time derivative of the potential has to be separable in cylindrical coordinates. We are then left with the potential U = A(t) (τ + ζ) +

U2 (ζ/τ) . τ+ζ

(8.29)

8.4. CONDITIONS OF CONSISTENCY FOR MIXTURES

135

This potential will be referred to as a quasi-stationary potential, since if we force the general potential Eq. 8.15 to be stationary, we get the same solution as Eq. 8.29 (with U1 = 0), but with the condition A = constant. Thus, the quasi-stationary potential depends on time through a unique function A(t), which is population independent. For U2 > 0, the second term of Eq. 8.29 can be associated with an additional attractive force, which is more relevant at low distances from the Galactic centre. It may be interpreted as a gravitational force due to the outer mass of a dark matter halo. Otherwise, a term U2 < 0 is similar to the perturbation of a point-mass potential induced by a tidal force, according to variational principles. It is interesting to note the difference between the potential of a steady state system and the stationary potential of a non-stationary system. In steady state systems, k is constant. Then, Chandrasekhar equations yield ˙ =0 a stationary potential in the form of Eq. 8.14. Indeed, if we assume k(t) in Eq. 8.15, we are left with Eq. 8.14. However, if we force the potential of Eq. 8.15 to be stationary in a stellar system where k is not constant, which means that A(t) must be constant, we obtain the quasi-stationary potential of Eq. 8.29, which is a very particular case of Eq. 8.14. Therefore, in non-steady state systems, only a particular family of stationary potentials is allowed. As the functional dependence of k is given by Eq. 8.19, k(t) has three degrees of freedom for each population11. Hence, this functional dependence allows the mean velocities of each population to be decoupled, Π 0  Π

0 ,

Z0  Z0

,

which provides the possibility of an apparent vertex deviation for the total velocity distribution. However, as Eq. 8.21 is satisfied, the velocity differences are not totally independent. They still maintain the proportion Z0 − Z0

z = Π 0 − Π

0 

(8.30)

so that population centroids still move on a conic surface with the apex in the Galactic centre. 11 This result corrects the Papers I and II where it stated that, for non-stationary potentials, k and k

were proportional.

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CHAPTER 8. AXISYMMETRIC SYSTEM

Separable potential When k1 and k3 are non-equal functions of time, both satisfying Eq. 8.18, according to Eq. 8.12, the potential must separable in addition in cylindrical ∂2 U = 0. Then, the potential becomes a particular case coordinates, that is, ∂τ∂ζ of the quasi-stationary potential Eq. 8.29, with U1 = 0, U = A(t) (τ + ζ) +

B C + τ ζ

(8.31)

with B and C constant. We may assume C = 0 by continuity conditions in the plane z = 0. This is a first order approximation of the general potential Eq. 8.15 near the Galactic plane. Both relationships in Eq. 8.13 allow for a decoupling of kinematics in the directions  and z. This is related to the existence of an extra isolating integral of motion involving the vertical velocity alone (Sala 1990). However, the decoupling is not absolute like in Chandrasekhar’s case with k4 = 0 (Chandrasekhar 1960, Eq. 3.743), since the potential must be consistent with the general solution found in Eq. 8.15. According to Appendix G.1, the case k1  k3 allows for a total independence of the population mean velocities and of the shape of the velocity ellipsoids. According to Eq. G.7, their tilt may have a non-vanishing tilt. In addition, the overall distribution may show an apparent vertex deviation. This is the only axisymmetric case allowing for a total independence of the population’s kinematics. However, notice that the necessary and sufficient condition for a tilted velocity ellipsoid is k1  k3 , while the separability of the potential is only a necessary condition. A potential may be separable and the ellipsoid may have no tilt if k1 = k3 . If A(t) is constant, that is, if the potential is stationary, the population centroids move, in general, on a toroidal surface with elliptical section (Cubarsi et al. 1990) or, in particular cases, over a cone, a hyperboloid, an ellipsoid, or a circular band restricted to the Galactic plane. The second term of Eq. 8.31, as a particular case of the quasi-stationary potential, is related to the angular momentum integral of the stars with stable orbits. For disc stars, if B > 0, this term is associated with an additional attractive force (see Section §8.6.1, Eq. 8.37). It is then possible to have stable orbits even for those stars with no net angular velocity. Otherwise, if B < 0, the potential term is associated with an additional repulsive force that only allows stable orbits for stars trespassing a threshold angular velocity. In summary, Table 2 shows how some observables are related to the different potential cases arising from the conditions of consistency. The

8.4. CONDITIONS OF CONSISTENCY FOR MIXTURES

Z

-10

-10

-10

-5

-5

-5

0

Z

5

0

Z

5

10 -5

0 R

5

10

0

5

10 -10

137

10 -10

-5

0 R

5

10

-10

-5

0 R

5

10

Figure 8.2: Stellar density of a single population for the harmonic potential (left), for the non-separable quasi-stationary potential (centre), and for the potential separable in addition in cylindrical coordinates (right).

observables are the ratio between semiaxes of the velocity ellipsoids, the difference between radial and vertical mean velocities of the populations, the angles for the vertex deviation ε and the tilt δ of the population velocity ellipsoids, and the sign of the moment μz in terms of z (which is discussed below). The quantities without accents refer to the total stellar distribution and those with accents refer to two populations, which are easily generalised to any finite mixture. Each stellar population is assumed of Schwarzschild type, so that the deviation angles with accents apply to real velocity ellipsoids. However, the angles without accents measure the apparent vertex deviation and tilt of the whole stellar distribution from their total second central moments. The main cases for the potential are: an axisymmetric general case, for a potential depending on the population parameter k4 , and a flat velocity distribution, for a potential not dependent on the population parameter k4 . This case splits into: the non-separable potential (Eq. 8.15), when the potential depends on the population parameter k; the quasistationary potential (Eq. 8.29), when the potential does not depend on the population parameter k; and the separable potential (Eq. 8.31 ), when the potential does not depend on the population parameters k1  k3 . Therefore, regarding to the potential, the cases range from general to particular. Fig. 8.2 displays the stellar density of a single stellar population obtained for the harmonic potential according to an ideal quasi-elastic potential (left); and, in contrast, the stellar density for the quasi-stationary, nonseparable potential (centre), which allows for stellar orbits all around the Galactic centre; and the stellar density for the separable potential (right),

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for which the stars orbit around the rotation axis of the stellar system. The quasi-elastic potential might be understood as how an external halo affects a disc population. A couple of aspects are worth remarking. One is that the stellar density is obtained according to the collisionless Boltzmann equation and the corresponding solution for the potential, without making use of Poisson equation. The other is that such a stellar population is the one consistent with a quadratic integral of motion or, better, the one in a region where such an integral is preserved.

ε = ε = ε

= 0 δ = δ = δ

sign(z)

− Π

0 = 0 − Z0

= 0

ε = ε = ε

= 0 δ = δ = δ

= 0 sign(z)

Π 0 Z0

unconstrained

Non-separable

=

z 

ε  0; ε = ε

= 0 δ = δ = δ

= 0 sign(z)

Z0 −Z0

Π 0 −Π

0

Π 0 − Π

0  0 Z0 − Z0

 0

unconstrained

unconstrained

unconstrained

ε  0; ε = ε

= 0

Π 0 − Π

0  0 Z0 − Z0

 0

unconstrained

Flat velocity distribution Quasi-stationary Separable, k1  k3

Table 8.2: Some kinematic observables of the Galaxy are analytically related to the potential cases arising from the conditions of consistency for mixtures.

Vertex deviation Tilt sign(μz )

Mean velocity differences

− Π

0 = 0 − Z0

= 0

μ μ  μ

zz = =

μzz μzz μzz

Velocity ellipsoid semiaxes Π 0 Z0

Axisymmetric general case

Observables

8.4. CONDITIONS OF CONSISTENCY FOR MIXTURES 139

140

CHAPTER 8. AXISYMMETRIC SYSTEM

8.5 The solar neighbourhood We shall check the kinematic features of the local thin disc, thick disc, and halo components, in connection to the cases studied above to examine whether the hypothesis of axisymmetry is valid yet. We shall pay attention to the velocity moments and gradients, and, in particular, to four key observables: the vertex deviation, the trend of the moment μz around the sun, the tilt of the velocity ellipsoid, and the existence of stars without net rotation.

8.5.1 Thin disc The results obtained by Pasetto et al. (2012b) show that the velocity moments μ and μzz for the thin disc are approximately constant in the solar neighbourhood (z ≈ 0), the latter slightly decreasing towards the Galactic anticentre, with an anisotropy coefficient μzz /μ  1, with the same decreasing trend. In addition, the moment μθθ also increases towards the Galactic centre. According to the expressions of the moment gradients in  , Eq. G.10 (Appendix G.3), by fixing z = 0, we get a constant value for ∂μ∂ ∂μθθ ∂μzz while ∂ and ∂ are decreasing functions of . Therefore, the thin disc, or the stellar populations composing the thin disc, have a velocity distribution which, in good approximation, are consistent with such a trend and with a small value of k4 , close to that of a flat velocity distribution, and a small value of k2 , although nonnull. The thin disc shows a net vertex deviation and is the Galactic component in the solar neighbourhood more distant from the steady state. According to our analysis, it is possible to account for an apparent vertex deviation, even under axisymmetry, for the sake of the difference of radial and rotation mean velocities. Such a feature is in agreement with the previous analysis of Section §6.5, where the stellar subsystems around whose the kinematics of the thin disc is articulated became undisclosed by working from increasingly eccentricity layers. There, a maximum entropy method was applied to fit the velocity moments up to the tenth order, without assuming any type of symmetry or any specific velocity distribution other than a single maximum entropy density function. Its velocity distribution was close to a mixture of two ellipsoidal components, each one without vertex deviation, but producing an apparent vertex deviation of the whole distribution, as displayed in Fig. 7.2. Therefore, the thin disc may be approximately fitted from a mixture of two populations with no vertex deviation, with a significant radial mean velocity difference, in addition to an overlapped old disc component, all of them consistent with a mixture of axial symmetric populations. This

8.5. THE SOLAR NEIGHBOURHOOD

141

kinematic interpretation of the vertex deviation is also in agreement with recent studies of the shape of velocity ellipsoids in spiral galaxies (Vorobyov & Theis 2008), confirming that the magnitude of the vertex deviation is not correlated with the gravitational potential, but is strongly correlated with the spatial gradients of the mean stellar velocities. In contrast to our finite mixture distribution approach, other models prefer to describe the thin disc from a continuous mixture distribution (e.g., Binney 2010, 2012), by integrating the distribution functions associated with different ages and velocity dispersions, as a consequence of a continuous star formation rate. However, it is no doubt concerning that discrete populations exist in the solar neighbourhood, which, in addition, account for most stars of the local samples. It is not a bad approach, then, to handle the continuous mixture of the remaining disc stars through another discrete Gaussian component to apply the current dynamic model for quadratic distributions. It is more difficult to explain the behaviour of the moment μz around the Galactic plane, as described by Pasetto et al. (2012b), than it is to explain the vertex deviation. The moment takes opposite signs at different distances from the Galactic centre for nonnull z values. For instance, by picking up only values with uncertainty below 2σ levels, we see in Table 6 (and also the third column of Fig. 4) of Pasetto et al. (2012b) that stars with height −0.3 < z ≤ −0.1 kpc have increasing moment values −83 ≤ μz ≤ 34 km2 s−2 , as the distance to the Galactic centre increases from 8 to 8.8 kpc. Similarly, for the same range of distances to the Galactic centre, stars with height 0.1 < z ≤ 0.3 kpc have decreasing moments 121 ≤ μz ≤ −98 km2 s−2 . In both cases, if these values were meaningful, the moment μz would take opposite values around the solar position for a fixed height, z  0. Therefore, according to Eq. G.7, the apparent tilt of the thin disc distribution oscillates around that of an ellipsoid pointing to the Galactic centre. This feature is boosted if the stars belong to the bins z ∈ (−0.5, −0.3] and z ∈ (0.3, 0.5], although the error bars are then much greater. The central moment μz , from Eq. G.6 in Appendix G.2, is proportional to z by a factor that is always positive. Therefore, we might expect moment values with constant sign on both sides of the Galactic plane, positive if z > 0, negative if z < 0, and null on the plane. Then, for a single population, the moment μz can never show such an actual trend. To explain this possible behaviour of that moment it is necessary once more to use the mixture approach. For a two-component mixture we obtain,

CHAPTER 8. AXISYMMETRIC SYSTEM

142 similar to Eq. 8.7,

μz = n μ z + n

μ

z + n n

(Π 0 − Π

0 )(Z0 − Z0

),

(8.32)

now with nonnull partial moments, which have the same sign as z. To allow a change of sign in the total moment near the solar position for low values of |z|  0, it is necessary (1) that the third term involving the mean velocity differences is nonnull, and (2) that their sign is opposite to that of the partial moments. This is only possible if (Π 0 − Π

0 )(Z0 − Z0

) has the sign of −z. Thus, in a first instance, the general axisymmetric case and the nonseparable potential case of a flat velocity distribution have to be excluded because they provide values Π 0 = Π

0 and Z0 = Z0

. On the other hand, the particular case of a quasi-stationary potential with k ≡ k1 = k3 does allow unconstrained mean velocities (k˙ /k  k˙

/k

) and provides values Π 0 =

1 k˙ , 2 k

Z0 =

1 k˙ z, 2 k

Π

0 =

1 k˙

, 2 k

Z0

=

1 k˙

z 2 k

that yield nonnull values for Π 0 − Π

0 and Z0 − Z0

. However, we get (Π 0 − Π

0 )(Z0 − Z0

) =

2  1 k˙ k˙

− z , 4 k k

which has the same sign as z. Hence, this case would also have to be excluded, at least for a two-population mixture. Then, we could still wonder if a mixture of more than two populations with unconstrained mean velocities would be able to provide a contribution of the same sign than −z in the generalisation of Eq. 8.32. The answer is no. In Section §7.5 a general mixture with an arbitrary number of components was studied. It is then found that the contribution of the population mean velocity differences to the second moments is of the same type as for a twocomponent mixture, as shown in Eq. 7.30. Hence, the same reasoning about the sign of that moment is proven to be valid for a general mixture. If actual data are to be trusted, to explain the trend of this moment in the solar neighbourhood only the case k1  k3 with a separable potential would be left, where the product (Π 0 − Π

0 )(Z0 − Z0

) may have an arbitrary sign. Therefore, we cannot discard the possibility that such a behaviour is due to a sampling problem, although it could also be due to small local deviations from the non-separable, quasi-stationary potential.

8.5. THE SOLAR NEIGHBOURHOOD

143

8.5.2 Thick disc For the thick disc, Casetti-Dinescu et al. (2011) obtain vanishing values for θθ  and ∂μ the moment gradients ∂μ∂ ∂ , a nearly zero value within the error ∂μzz zz bars of ∂z , and clearly non-vanishing values of ∂μ∂z , ∂μ∂zθθ , and ∂μ ∂ . It is easy to verify that these quantities are qualitatively in agreement with the assumption of a single population with k4  0, although small. Thus, by naming  ≡ k4 , from Appendix G.3 we write explicitly the moment gradients in terms of  and z, for fixed . We then get the following order estimations: ∂μ = O( 2 z2 ), ∂

∂μzz = O( 2 z), ∂z

∂μθθ = O(z), ∂z

∂μ = O(z), ∂z

∂μzz = O(). ∂

(8.33)

θθ In addition, the value ∂μ ∂ is approximately constant and nonnull, unless k2 = 0. Notice that the gradients of moments that are odd in the variable z may show diminished values, since the samples were selected by different values of |z|, by averaging stars from both sides of the symmetry plane. Therefore, the thick disc kinematics may be described from a single quadratic population with small, but nonnull values of k4 and k2 . Similarly, 0 these authors find a small value for ∂Θ ∂z = O(z), which is also consistent with the small values of k4 and z. On the other hand, they find a non-vanishing tilt of 8.6◦ ±1.8 for the thick disc velocity ellipsoid, which should be associated with a separable potential, while the vertex deviation is consistent with zero. The non-vanishing tilt was also found by Siebert et al. (2008) and Fuchs et al. (2009). The general kinematic features of the thick disc are also in agreement with the results obtained by Carollo et al. (2010), Pasetto et al. (2012a), and Moni Bidin et al. (2012), although the latter find a nonnull vertex deviation of thick disc stars, which increases with the distance to the Galactic centre. It is worthwhile to remark that an axial symmetric model with a mean velocity non-symmetric about the plane z = 0 provides a vertex deviation proportional to z (Cubarsi 1990, Appendix A). In addition, for the thick disc there is evidence of a net peculiar radial mean velocity towards the Galactic centre of 9.2±1.1 km s−1 , that is, U ≈ 19 km s−1 referred to the Sun. The result, which agrees with those of Girard et al. (2006), Bramich et al. (2008), and Smith et al. (2009b), was also found by Cubarsi & Alcobé (2006) by working from nested HIPPARCOS subsamples, despite Casetti-Dinescu et al. (2011) saying that HIPPARCOS-based

144

CHAPTER 8. AXISYMMETRIC SYSTEM

results are unlikely to detect this. These nested HIPPARCOS subsamples were built to contain an increasing number of thick disc stars. They showed a trend that was interpreted as a single point-axial symmetric population (Juan-Zornoza 1995) approaching an ideal, axial, steady state population, with no net galactocentric radial mean velocity. The zero mean velocity of this ideal population was then extrapolated by leading to a heliocentric velocity of ca. 20 km s−1 towards the Galactic centre. Hence, this analysis confirms that a mixture of stellar populations with independent radial differential motions is required to explain the disc kinematics, which is still possible in axially symmetrical systems.

8.5.3 Halo Halo stars, either from the inner or the outer halo, despite the uncertainty of their velocity statistics, could have velocity ellipsoids with non-vanishing tilt (Carollo et al. 2010, Smith et al. 2009a, Chiba & Beers 2000), and also a differentiated mean radial motion (Smith et al. 2009b), in addition to a retrograde rotation of the outer halo. The outer halo has close values of μ and μθθ , by providing a nearly zero value of k2 , which, according to Eq. G.3, would correspond to an almost rigid body rotation for fixed values of z. However, a more detailed analysis (Smith et al. 2009a, Evans et al. 2015) suggests that the halo, in particular the inner halo, would be approximately at rest, close to steadiness and spherical symmetry, aside from a slight masking produced by some disc and bulge stars. These authors, based on an analysis of integrals of motion and Poisson equation, claim that the potential associated with a triaxial velocity dispersion tensor of the halo should be spherical and the ellipsoid should not show any tilt. However, according to the authors, although the potential of the ellipsoidal dark halo is the one dominant over the whole Galaxy, which is associated with the harmonic potential term, the total potential produced by the mixture of the Galactic components, and in particular by the disc, breaks the overall sphericity. Our approach is totally different from the approach of Smith et al. (2009a). We consider a time dependent potential and specifically avoid the Poisson equation, which they used as the main reason to reject the term depending on the elevation angle in Eq. 8.15. Further, we need streaming motions (which they rejected) to account for asymmetries in the disc velocity distribution. However, somehow, we are led to a similar result: a general spherical potential is inconsistent with a tilted velocity ellipsoid. The harmonic potential is the only spherical potential allowing tilted ellipsoids. However, the term

8.6. LOCAL VALUES OF THE POTENTIAL

145

depending on the elevation angle in Eq. 8.29 is not always responsible for the tilt. It only occurs when k1  k3 , by forcing the potential to be separable in cylindrical coordinates. In order to get unconstrained population mean velocities, the spherical terms of the potential must be proportional to (τ + ζ) and (τ + ζ)−1 . That is, must of the form of Eq. 8.29 with U2 constant. A similar analysis, as in the previous section, of the mean velocity gra0 dients provides an order estimation ∂Θ ∂z = O(z), which agrees with nonnull, but small values of k4 . Thus, each halo component could be fitted through a single Schwarzschild distribution with no vertex deviation.

8.6 Local values of the potential In Appendix H it is shown how local values of potential derivatives are related to the local planar (radial and transversal) and vertical epicycle frequencies κ, ν, and to the angular velocity Ωc . Radial and transversal epicycle frequencies are the same and, like the vertical frequency, they are constant, not depending on the star. This is an important result obtained in the twenties by Oort and Lindblad. Each region in a galaxy provides stars with a local oscillation mode, so that, in the axisymmetric model, the epicycle frequencies would vary depending on the distance to the Galactic centre and to the plane. These values can be used to test the reliability of the potential function. In such a case, we must assume a stationary potential, a particular case of the quasi-stationary potential given by Eq. 8.29 with A(t) constant, now written as 1 (8.34) U = M (2 + z2 ) + 2 F(z2 /2 ) .  In addition, we shall study two particular cases. One corresponding to a potential separable in addition in cylindrical coordinates and another one corresponding to the spherical potential.

8.6.1 Separable cylindrical potential We assume the potential of Eq. 8.34 separable in addition in cylindrical coordinates, F (8.35) U = M (2 + z2 ) + 2 ,  where F is constant.

146

CHAPTER 8. AXISYMMETRIC SYSTEM

The following linear operator, which appears in the condition of orbital stability of Eq. H.11 acting over the potential U,  L [ · ] =

 ∂2 3 ∂ + [·] ∂2  ∂

(8.36)

satisfies L [c1 −2 + c2 ] = 0, either with c1 , c2 constants or functions of z. Furthermore, L [M2 ] = 8M . Then, by assuming the local position  ≈ c , these potentials provide constant squared epicycle frequencies κ2 = L [U(c , 0)] = 8M and, according to Eq. H.15, ν2 = 2M regardless the point in the Galaxy. Therefore, on one hand, the existence of bounded orbits requires the factor M to be positive. On the other hand, the constant M determines both epicycle frequencies. Then, the ratio of the epicycle frequencies is νκ = 2, which is not the actual case, since, according to the commonly accepted values Ωc ≈ 27 km s−1 kpc−1 , κ ≈ 37 km s−1 kpc−1 , and ν ≈ 70 km s−1 kpc−1 (e.g., Binney & Tremaine 2008, Table 1.2), the vertical frequency must be higher than the planar (the rotation period is about 220 Myr and the vertical period of oscillation is approximately 87 Myr). By Eq. H.10, in the Galactic plane, for a star in circular motion and angular momentum integral J, the radius c of the circular orbit verifies 4c =

2F + J 2 . 2M

(8.37)

Since M > 0, then J 2 > −2F. This is the condition for a stable orbit. Thus, a factor F > 0 would allow stable orbits for all the stars, even for those with no rotation. This excludes the existence of circular orbits within a radius 1 lower that min = [F/M] 4 , corresponding to the value J = 0. Otherwise, a factor F < 0 would allow bounded orbits at any distance from the GC but only for stars trespassing a threshold angular velocity with 2 = −2F. Other orbits become a minimum angular momentum integral Jmin unstable. The angular velocity satisfies Eq. H.13. Then, for the potential in Eq. 8.34, the squared angular velocity in terms of the radius of any star in a circular orbit on the Galactic plane is given by Ω2c () = 2M −

2F , 4

4 ≥ F/M .

(8.38)

8.6. LOCAL VALUES OF THE POTENTIAL

147

By expressing M in terms of the planar epicycle frequency, for  = c we determine F from the local angular velocity, i.e.   4c κ2 2 − Ωc (c ) . (8.39) F= 2 4 The local values for these constants allow estimates κ4 ≈ 340 and Ω2c (c ) ≈ 730 (both values in km2 s−2 kpc−2 ). Then, F must be negative. Therefore, such an additional repulsive force requires κ < 2Ωc . 2

8.6.2 Spherical potential The spherical potential can be written as a particular case of Eq. 8.34, by ex  2 −1 pressing F(z2 /2 ) = N(z2 /2 ) 1 + z 2 , with N constant. The potential then becomes N . (8.40) U = M (2 + z2 ) + 2  + z2 In the symmetry plane, the planar epicycle frequency is, like in the previous case, constant κ2 = 8M. On the other hand, the vertical epicycle frequency at z = 0 is ) ∂2 U )) 2N ) = 2M − 4 . ν2 = (8.41) ∂z2 )c c Then, in the Galactic plane both epicycle frequencies are constrained according to the relationship   4c κ2 2 −ν . N= (8.42) 2 4 Hence, N is now related to the local vertical epicycle frequency. Since the actual values in the solar neighbourhood satisfy ν > κ, the above equation would provide a negative value for the constant N. Once again, the potential term associated with a repulsive force requires the condition κ < 2ν. However, by comparing Eq. 8.38 and Eq. 8.41, since the spherical and separable cylindrical potentials satisfy F = N and κ2 = 8M, we get ν2 = Ω2c (c ). Therefore, for the spherical potential in the Galactic plane, the local vertical epicycle frequency and the absolute value of the local angular velocity match at c . There is no alternative parameter to fit the local angular velocity. This could neither be the actual case in the solar neighbourhood.

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148

8.6.3 General case For the general case of Eq. 8.34, according to the term having the arbitrary function F(s) with s = z2 /2 , we get L [U(c , 0)] = 8M +

 1  8sF (s) + 4s2 F

(s) 4 c

(8.43)

and 1  ∂2 U = 2M + 4 2F (s) + 4sF

(s) . ∂z2 c

(8.44)

Then, in the Galactic plane, s = 0, the epicycle frequencies satisfy κ2 = 8M,

ν2 −

κ2 2F (0) = . 4 4c

(8.45)

In addition, Eq. 8.38 becomes Ω2c (c ) −

2F(0) κ2 =− . 4 4c

(8.46)

For the second term of Eq. 8.34, a repulsive force depending on the variables  and |z| is associated with local values F(0) < 0 and F (0) > 0. Therefore, it implies12 2Ωc > κ and 2ν > κ. The resulting potential is then weaker than the harmonic potential. Therefore, we have three independent parameters related to the three local constants. They can be adjusted according to actual values and provide 12 In the epicycle model, the guiding centre C in circular orbit and the projection S onto 0 the Galactic plane of the LSR are assumed as equal. Then, there is no asymmetric drift Δθ = Θc (S 0 ) − Θ0 (S 0 ). For the radial and transversal components, the central second moments are related to the local constants as 4Ω2 μrr γc2 ≡ 2c = . μθθ κ

For stellar systems dominated by rotation, according to Eq. 8.46, a potential associated with a value F ≤ 0 implies γc ≥ 1 and μrr ≥ μθθ . However, in a higher-order approximation than the epicycle model, where the asymmetric drift is nonnull, then it is satisfied γc2 =

μrr ≥1, μθθ + Δ2θ

so that μrr > μθθ is always fulfilled (e.g., Cubarsi et al. 2017).

(8.47)

8.7. REMARKS

149

the local derivatives of the potential function ) ∂U )) ) = Ω2c c , ∂ )(c ,0) ) ∂2 U )) ) = κ2 − 3Ω2c , ∂2 )(c ,0) ) ∂2 U )) ) = ν2 . ∂z2 )(c ,0)

(8.48)

As described in Cubarsi et al. (2017), a simple stationary potential, neither spherical nor separable in cylindrical coordinates, according to U = M (2 + z2 ) +

2

N ; + Q z2

M > 0, N < 0, Q > 0,

with F(s) = N(1 + Qs)−1 in Eq. 8.29, is able to fit the three local Galactic constants related to the first derivatives of the potential at the solar position, i.e., the two epicycle frequencies and the local angular velocity, also accounting for the ratios of the local velocity dispersions and the asymmetric drift.

8.7 Remarks To simplify the solution of the collisionless Boltzmann equation in either of its forms it is necessary to introduce some symmetries for the mass and the velocity distributions, such as the assumptions of axisymmetry, steady state, or Galactic plane of symmetry. These hypotheses provide serious limitations for describing, in a realistic way, the kinematic observables of the Galaxy. Several kinematic analyses using the newest radial velocity data from the RAVE survey confirms that the thin disc had non-vanishing vertex deviation, the thick disc has a radial mean motion differing from that of the thin disc, and the halo velocity ellipsoid is likely to be tilted. It was suggested (Pasetto et al. 2012b, Steinmetz 2012) that the axisymmetry assumption was the cause not allowing to explain some of these features. In this chapter a mixture model has been adopted to check the axisymmetric hypothesis to find what axisymmetric potentials are connected with a more flexible superposition of stellar populations, so that they can describe, in a realistic way, the main kinematic features of the solar neighbourhood. The conditions of consistency are integrability conditions allowing a stellar system composed of several independent populations to share the

150

CHAPTER 8. AXISYMMETRIC SYSTEM

same potential function in the collisionless Boltzmann equation. Since the potential may depend on the population parameters involved in the velocity distribution, the less the potential depends on them, the less kinematically constrained the populations will be. Thus, in solving the collisionless Boltzmann equation, we looked for potentials permitting different mean velocities of the populations and arbitrary orientation of the velocity ellipsoids. We found that these potentials do not depend on the population parameter k4 . They were consistent with a flat velocity distribution and did not constrain the axes of the velocity ellipsoids. For k ≡ k1 = k3 , we obtained the family of potentials of Eq. 8.29, designated as quasi-stationary, which, in general, are non-separable in cylindrical coordinates. Their time dependency is carried through a unique function A(t), which is population independent. Then, the stellar populations have untilted velocity ellipsoids and their centroids move on conic surfaces with the apex in the Galactic centre. The other possible solution was a potential separable in cylindrical coordinates, for which the values k1 and k3 may differ. Their time dependency is also carried through a unique function A(t). If k1  k3 , in addition to unconstrained mean velocities, the populations show an arbitrary tilt of the velocity ellipsoids. Therefore, with the mixture model, we should be able to fit the general features of the actual velocity distribution in the solar neighbourhood without the requirement of relaxing the axisymmetrical hypothesis. Four key observables have been evaluated to prove it. First, the vertex deviation, which is clearly non-zero for the thin disc. There is a tendency to associate the vertex deviation of the velocity ellipsoid with the break of axisymmetry, which is only true for a single population. The vertex deviation is easily explained as a result of the superposition of two or more axisymmetric populations with different radial and rotational mean velocities. Such a situation requires the quasi-stationary potential Eq. 8.29. This interpretation of the vertex deviation was confirmed by the fact that the isocontours of the velocity distribution obtained for disc stars with eccentricities lower than 0.15 gave a clear picture of a bimodal distribution, each mode approximately ellipsoidal with no vertex deviation, which were overlaid with a significant difference in the radial mean velocities. Upon completion of the sample with higher eccentricity stars, the overall thin disc distribution arose as a mixture of three axisymmetric populations with an apparent vertex deviation. The second key observable is the behaviour of the radial gradient of the moment μz in the solar neighbourhood. If the results of Pasetto et al.

8.7. REMARKS

151

(2012b) are right, although the moment is nearly null at the Galactic plane, there is a change of sign of this moment, above and below the plane, near the solar radius, instead of maintaining a constant sign, as a single population would do. To account for this behaviour, it is not sufficient to consider a population mixture with different radial and vertical mean velocities. It is necessary that the differences (Π 0 − Π

0 ) and (Z0 − Z0

) are non-proportional. This is only possible for a quasi-stationary potential with k1  k3 , which yields the potential separable in cylindrical coordinates of Eq. 8.31. The possibility of explaining such a behaviour from a mixture of three or more populations with a non-separable potential was also dismissed. The second moments of a mixture with an arbitrary number of populations were obtained with this purpose in Appendix 7.5. They were related to the one-toone mean velocity differences to verify that the change of sign of μz is only possible under a separable potential, regardless of the number of population components. The third point to check is about the tilt of the population velocity ellipsoids. The tilt could be nonnull for thick disc and halo stars. The most recent analyses show a preference for accepting a slight tilt of both Galactic components. If these results are conclusive, they are consistent with a separable potential with k1  k3 . According to these results, the axisymmetric quasi-stationary potential Eq. 8.29 is consistent with the local Galactic observables. The non-separable potential, with k1 = k3 , even containing a term depending on the elevation angle, would be admissible for a mixture of populations with non-tilted ellipsoids. Then, the slight tilt of the halo and the uncertain tilt of the thick disc should be interpreted as local perturbations or statistical fluctuations, instead of intrinsic characteristics of the stellar system. Otherwise, the potential separable in cylindrical coordinates, with k1  k3 , supports populations with tilted velocity ellipsoids as a general feature. The analysis undertaken is obviously limited by its own hypotheses. By taking the validity of the collisionless Boltzmann equation for granted, the two basic assumptions are: first, that a stellar population is associated with a quadratic velocity distribution; and, second, that the whole stellar system is obtained as a finite mixture of populations. In regards to the first assumption we may recall that, although the hypothesis fails when a quadratic distribution is associated with a moving group with a few hundreds of stars, if kinematically unbiased samples of thousands of stars are considered, the velocity distribution generally shows a multimodal shape, close to a superposition of quadratic distributions. Therefore, the first hypothesis should be accepted for describing general trends of

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CHAPTER 8. AXISYMMETRIC SYSTEM

large stellar populations, but admitting that local statistical fluctuations may exist depending on the size and bias of the sample. On the other hand, the limitations concerning the second assumption will depend on the goodness of the fit when a real continuous mixture of stellar populations is replaced by a single quadratic population. In two cases, the error produced by such an approximation is negligible. One case occurs when a continuous mixture represents a small fraction of the total sample, which is composed mostly of discrete populations. Then, the approximation may produce small changes in the wings of the discrete distributions. As mentioned above, this is the case in the solar neighbourhood. The other case involves a continuous mixture of Gaussian distributions with nearly the same mean takes place, by producing a positive excess of kurtosis. In that case, the resulting distribution is also quadratic (Cubarsi 2007) and, therefore, is consistent with the current model. The current analysis started with an additional assumption to be tested: the axial symmetry. We have concluded that, even under this hypothesis, it is possible to describe the general kinematic features of the solar neighbourhood from a finite mixture of quadratic populations. Although from a statistical viewpoint both basic hypotheses do not introduce serious limitations to our problem, from a dynamical viewpoint it could seem that the solutions for the potential have been greatly reduced. In this regard, a further analysis will be done to find out whether more general potentials are admissible. In the next chapter we shall investigate the conditions of consistency for point-axial symmetry models (Juan-Zornoza 1995, Sanz-Subirana, 1987). That is, to relax the axisymmetric hypothesis to see how this affects the potential.

Chapter 9

Point-axial symmetric system 9.1 Point-axial symmetry By following Cubarsi (2014b) we are going to complete some aspects of the analysis of conditions of consistency for mixtures of axisymmetric stellar systems by studying the more general point-axial symmetry (or bisymmetry) case, i.e., rotational symmetry of 180◦ for the potential and the phase space density functions. In the past chapter we proved that an axisymmetric mixture model is able to describe the actual velocity distribution in the solar neighbourhood provided that the potential has the quasi-stationary form of Eq. 8.29 and the phase space density function is time-dependent. This family of potentials is consistent with populations having different mean velocities producing a nonnull vertex deviation of the disc distribution. In addition, if the potential is separable in cylindrical coordinates, the velocity ellipsoids may have an arbitrary tilt. Unlike in the axisymmetric model, in steady state point-axial systems, nonnull radial and vertical differential motions are also possible (Sanz-Subirana 1987, Juan-Zornoza 1995). Point-axial symmetry is indeed not a relaxation of the axial symmetry, but a more informative symmetry which may account for ellipsoidal, spiral, or bar structures, and includes axial symmetry as a degenerate case. In particular, along with a quadratic velocity distribution, a point-axial symmetry model provides triaxial mass distributions and veloc153

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CHAPTER 9. POINT-AXIAL SYMMETRIC SYSTEM

ity ellipsoids with non-vanishing vertex deviation. However, a quadratic point-axial velocity distribution is still symmetric in the peculiar velocities, so that it has null odd-order central moments. Therefore, either in axial or in point-axial symmetric systems, a mixture model is compulsory to fit the full set of local velocity moments. Nevertheless, if each population of the mixture had a velocity ellipsoid with an arbitrary orientation, as, in principle, in the point-axial model, a lower number of populations would likely be required to fit the overall velocity distribution.

9.2 Single point-axial system For the generalised Schwarzschild velocity distribution, the collisionless Boltzmann equation yields the Chandrasekhar equations, which are equivalent to the moment equations. As in the axisymmetric case, their solution provides the tensor A2 , the function σ, the mean velocity v, and the potential U. The two first Chandrasekhar equations are equations 3.23 and 3.24. which yield the elements of the second-rank tensor A2 and the vector Δ given in Appendix I. The remaining Chandrasekhar equations are equations 3.25 and 3.26, which provide the potential U and the function σ. Once more, by using the variable X = −Δ · v − σ they can be easily written as equations 8.2 and 8.3. By elimination of X between equations 8.2 and 8.3, with the new variables τ = 21 2 and ζ = 12 z2 , which are appropriate to the symmetry plane of the system, six second-order partial differential equations for the potential are obtained. In their vector notation they can be found in Chandrasekhar (1960, equations (3.448) and (3.450), p.100). After substitution of the elements of A2 and the components of Δ, Sanz-Subirana (1987) and Juan-Zornoza (1995) proved that continuity conditions on the function X force the potential to be axisymmetric. A similar result was obtained by Vandervoort (1979) for point-axial systems, which he called galactic bars, although the study was limited to a two-dimensional disc with a steady-state potential. The resulting equations for the potential are as follows. The first three partial differential equations for the potential, obtained by taking the curl in Eq. 8.2, are

      ∂U ∂ ∂U ∂U ∂ ∂U ∂U − − − + τ + ζ + 8K4 τ 2 ∂U ∂τ ∂ζ ∂τ  ∂τ ∂ζ ∂ζ  ∂τ ∂ζ ∂2 U ∂U

∂2 U + 2K4 ∂θ∂ τ ∂U + U + ζ = 0, + K +4τ(K1 − k3 ) ∂τ∂ζ 1 ∂θ∂ζ ∂τ ∂ζ

(9.1)

9.2. SINGLE POINT-AXIAL SYSTEM

155 1

where a common factor proportional to ζ 2 was simplified;   

 ∂U ∂ ∂U ∂U ∂2 U + 2K1 τ ∂τ∂ζ + 4K4 τ 2 ∂U ∂τ − ∂ζ + ζ ∂ζ ∂τ − ∂ζ 2 2 ∂U ∂ U

∂ U ∗ +4K4 τ ∂θ2 + 2(3K4 − K4 ) ∂θ + 4K4 τ ∂θ∂τ − 2 −[2(k3 − K1∗ ) + 4(K4 − k2 )τ − 4K4∗ ζ] ∂∂θζU = 0 ,

(9.2)

1

where a common factor proportional to ζ 2 was simplified; and

   ∂ ∂U ∂U

∂2 U

∂2 U 4τ2 2K4 ζ ∂τ ∂τ − ∂ζ + K1 ∂τ2 − (K1 + 2K4 ζ) ∂θ2 + ∂2 U ∂2 U +8K4 τζ ∂θ∂ζ + 4τ[2k2 τ − (K1 − K1∗ ) − 2(K4 − K4∗ )ζ] ∂θ∂τ + ∂U ∗ ∗ ˙ +2[4k2τ + (K1 − K1 ) + 2(K4 − K4 )ζ] ∂θ = 8βτ .

(9.3)

1

Those equations which were proportional to ζ 2 become null at the Galactic plane. The remaining three equations, which are obtained by taking the gradient in Eq. 8.3 and the time derivative in Eq. 8.2, are   ∂U − 2K4 ζ ∂t∂ ∂U ∂τ ∂ζ +... 2 2 U 1 ˙ ∂2 U + 2K˙1 ∂U +K˙1 τ ∂∂τU2 + K1 ∂∂t∂τ ∂τ + 2 K1 + k3 ζ ∂τ∂ζ + 

2 2 1 ∂ U U + (K1 + 2K4 ζ) ∂∂θ∂t + (K˙ 1 + 2K˙ 4 ζ) ∂U + 4τ (K˙ 1 − 4β)τ ∂θ∂τ ∂θ = 0 ;   ∂U 2K4 τ ∂t∂ ∂U + ∂ζ − ∂τ ... ∂2 U 1 ∂2 U ∂2 U ∂U ˙ ˙ + +k3 ζ ∂ζ 2 + k3 ∂t∂ζ + 2k3 ∂ζ + 2 k3 + K˙1 τ ∂ζ∂τ 2 ∂2 U U − 14 K4 ∂∂θ∂t =0; +(K˙ 1 − 4β) ∂θ∂ζ

(9.4)

(9.5)

1

in the last one, a common factor ζ 2 was also simplified; and

  ...  ∂U

∂2 U ˙ ∂U 1 τ 2K4 ζ ∂t∂ ∂U ∂τ − ∂ζ + K1 ∂t∂τ + 2 K1 ∂τ + 2 K1 + 2 ∂2 U ∂2 U + 14 (K˙ 1 − 4β) ∂∂θU2 + k˙3 ζ ∂θ∂ζ + +K˙ 1 τ ∂θ∂τ 2 U ¨ +(K1∗ + 2k2 τ + 2K4 ζ) ∂∂θ∂t + 14 (K˙ 1

+ 4K1∗ ) ∂U ∂θ = 2βτ .

(9.6)

These equations complete the first set of three equations given by SanzSubirana (1987). They may be simplified by assuming that the parameter β is constant, which was actually derived by this author in solving them for the separable potential case, and by Juan-Zornoza (1995) for the general case. This fact is not relevant for our purpose.

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156

9.3 The potential is axisymmetric The solution of the above system of partial differential equations is tedious and long, and, unfortunately, the above-mentioned thesis papers cannot be accessed easily. Since this is one of the key properties of the point-axial model, we shall see a shorter, alternative justification to this crucial property. We note that in the Galactic plane ζ = 0, the three equations 9.1, 9.2, and 9.3 are reduced to Eq. 9.3, by providing the basic dependence of the potential on the radius and the angle variables. Hence, we focus on this equation in its complete form. First, we consider the main case K1  0, hence q  0, since if K1 does not depend on θ, the velocity distribution has no vertex deviation in the Galactic plane1 , which was one of the most important observables that justified trying a non-cylindrical model. If K1 is nonnull, we write Eq. 9.3 as . 4k2 τ+(K1 −K1∗ )+2(K4 −K4∗ )ζ ∂U 2K4 ζ ∂2 U ∂2 U 2 ∂2 U − 2 = 4τ −



2 2 K1 +2K4 ζ ∂θ K1 +2K4 ζ ∂τ∂ζ + ∂θ ∂τ (9.7) 2k2 τ−(K1 −K1∗ )−2(K4 −K4∗ )ζ ∂2 U 8K4 τζ ∂2 U +4τ K +2K ζ ∂τ∂θ + K +2K ζ ∂ζ∂θ . 1

4

1

4

∂U ∂θ

and bear in mind that the continuity and We define the function V = differentiability of the potential, at least up to the second derivative, implies that V is also differentiable. In the Galactic plane, the foregoing equation becomes ∂V ∂θ

− 4τ2 ∂∂τU2 = 2 2

4k2 τ+(K1 −K1∗ ) V K1

+ 4τ

2k2 τ−(K1 −K1∗ ) ∂V K1 ∂τ

.

(9.8)

Since K1 is a π-periodic function of the angle, a simple recall to the mean value theorem provides us with a value θ0 ∈ [0, π) for which K1 (θ0 ) = 0. 2 ∂2 U Then, if V and ∂V ∂θ − 4τ ∂τ2 are nonnull functions, in order to avoid any singularity, the right-hand side member of the above equation must vanish, at least for θ = θ0 . We see that the potential does not satisfy ∂2 U ∂θ2

− 4τ2 ∂∂τU2 = 0 . 2

If so, the solution would be that of the wave equation in the new variable x = ln τ, hence the solution satisfies U = F1 (x + 2θ) + F2 (x − 2θ), but the potential is a one-valued function and a periodic function of θ with period2 pointed out in the past chapter, the second-order central velocity moments satisfy μ2 ∝ A2 . In the Galactic plane, from Eq. I.1, we get μθ ∝ K1 . 2 Although we assume that the velocity distribution is π-periodic, we cannot discard that part of the potential function could still admit a 2π-periodic solution. 1 As

−1

9.4. THE POTENTIAL IS SPHERICAL

157

2π; therefore, U(x + 2θ) = U(x + 2(θ + 2kπ)) = U((x + 4kπ) + 2θ) is fulfilled for all k ∈ Z. However, as the Galaxy is of finite extent, such a potential taking the same value at all points x + 4kπ is unrealistic. On the other hand, the right-hand side of Eq. 9.8 is nonnull. If so, it would be a linear and homogeneous differential equation in V, with solu1 3 K −K ∗ tion V ∝ τ 2 (K1 − K1∗ − 2k2 τ)− 2 , which is discontinuous at τ = 12k2 1 . In particular, when K1 (θ0 ) = 0, according to Eq. I.3, the singularity takes place at τ = kq2 . It is worth noticing that for k2 = 0 such a singularity does not exist, so that we might have non-cylindrical potentials in that degenerate case of a rigid rotating bar. Therefore, the only admissible, continuous, and differentiable solutions to Eq. 9.8 are axisymmetric potentials satisfying V = 0, otherwise, in the Galactic plane, the potential is not differentiable. This means that, the axial symmetry is the way that the differential equations for the potential avoid the singularity produced by any root of the function K1 . It is actually a situation similar to the axisymmetric model, where the equations for the potential, in the quasi-stationary case in the past chapter, did avoid the singularity produced by the zero of the time-dependent function k˙ 1 by providing a solution that does not depend on k˙ 1 /k1 . The case where K1 is null, consequently K1 − K1∗ also vanishes, requires that K4 be nonnull, otherwise the velocity distribution is axisymmetric. Similarly, as in the above case, there is also an angle θ1 for which K4 (θ1 ) = 0 that would produce a singularity in the solution of Eq. 9.7, for ζ  0, unless the potential is axisymmetric3.

9.4 The potential is spherical Therefore, in a rotating point-axial system, the potential consistent with a quadratic velocity distribution is still axisymmetric. The set of partial differential equations for the potential generalises the ones for the axisymmetric model in the past chapter. We write these equations once they are simplified by taking advantage of the potential satisfying ∂U ∂θ = 0. The first three 3 In

this case, we first prove that, if

∂U ∂θ

 0, the equation

∂V ∂θ

− 4τ2

+

∂2 U ∂τ2

−

∂2 U ∂τ∂ζ

,

= 0, with

the variables x = ln τ, y = ζ/τ, provides a solution proportional to an exponential function on the argument x + θ, which is not periodic and, hence, unacceptable. We then verify that the remaining terms of Eq. 9.7 do not vanish, otherwise, its solution, which takes the general form  (k −K )τ−(K4 −K ∗ )ζ  1 − 3 k2 4 V=F 2 4 τ 2 ζ 2 K4 , has discontinuities either at ζ = 0 or at points satisfying k2 /K4 (k2 −K4 )ζ

(k2 − K4 )τ − (K4 − K4∗ )ζ = 0.

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158

equations, derived from equations 9.1, 9.2, and 9.3, are

     ∂U ∂ ∂U ∂U ∂ ∂U 2K4 2 ∂U ∂τ − ∂ζ + τ ∂τ ∂τ − ∂ζ + ζ ∂ζ ∂τ − ∂2 U +(K1 − k3 ) ∂τ∂ζ =0,

 2K4 2 ∂U ∂τ −

∂U ∂ζ



+ ζ ∂ζ∂

∂ 2K4 ζ ∂τ



∂U ∂τ



∂U ∂τ

−

−

∂U ∂ζ



∂U ∂ζ



∂U ∂ζ



∂ U =0, + K1 ∂τ∂ζ 2

+ K1 ∂∂τU2 = 0 . 2

+

(9.9)

(9.10) (9.11)

The remaining three equations, obtained from equations 9.4, 9.5, and 9.6, are   ∂U 2K4 ζ ∂t∂ ∂U + ∂τ − ∂ζ ... (9.12) ∂2 U 1 ∂2 U ∂2 U ∂U ˙ ˙ +K1 τ ∂τ2 + K1 ∂t∂τ + 2K1 ∂τ + 2 K1 + k˙3 ζ ∂τ∂ζ =0, +k˙3 ζ ∂∂ζU2 2

2K4 ζ ∂t∂



  ∂U 2K4 τ ∂t∂ ∂U ∂ζ − ∂τ... + 2 U 1 ˙ ∂2 U + k3 ∂∂t∂ζ + 2k˙3 ∂U ∂ζ + 2 k3 + K1 τ ∂ζ∂τ = 0 , ∂U ∂τ

−

∂U ∂ζ



... 2 U 1 + 2K˙1 ∂U + K1 ∂∂t∂τ ∂τ + 2 K1 = 0 .

(9.13)

(9.14)

However, equations 9.10 and 9.11 can be simplified further. By taking the θ-derivative in Eq. 9.9 and subtracting from Eq. 9.10, we get   ∂U ∂ ∂U K4 ∂τ − (9.15) ∂τ ∂ζ = 0 . Also, by taking into account Eq. 9.11, we get K1 ∂∂τU2 = 0 . 2

(9.16)

Similarly, Eq. 9.14 can be expressed in a simpler form. By taking the θ2 derivative in Eq. 9.12 and subtracting from Eq. 9.14 we get K˙ 1 ∂∂τU2 = 0, which does not add any new condition to the previous equation. Therefore, the equations for the potential in the point-axial model are the set of equations 9.9, 9.12, and 9.13, which are similar to the ones of the axisymmetric case, equations 8.4, 8.5, and 8.6, with the additional integrability conditions given by equations 9.15 and 9.16. As in the axisymmetric case, these equations do not involve the parameters k2 and β. In addition, the equations for the potential do not depend on the parameters K1∗ and K4∗ . Under axial symmetry, the conditions depending on the θ-derivatives K1 (θ, t)

9.4. THE POTENTIAL IS SPHERICAL

159

and K4 (θ) are identically null. Thus, in a mixture model these equations will be similarly planned for each population component, and will depend on the respective population parameters K1 (θ, t), k3 (t), and K4 (θ). In the axisymmetric case, when applying the conditions of consistency for a flat velocity distribution, that is, for a potential independent from the population parameter K4 , the potential becomes dramatically simplified. In the point-axial case, we shall see that a similar reasoning and solution are inherent to the point-axial symmetry assumption, since the reasoning can be done in regard to the angle dependence as well as to the population dependence of the parameters. In other words, a point-axial system is consistent with a flat velocity distribution unless it degenerates towards an axisymmetric system. Thus, being at least one of the population parameters K1 , K4 functions of the angle θ (otherwise the system is axisymmetric), Eq. 9.9, once divided by K4 , becomes separated into two parts, one independent from θ and the other depending on θ, which must be null separately4 ,       ∂U ∂U ∂ ∂U ∂U ∂ ∂U ∂U − − − 2 +τ +ζ =0, ∂τ ∂ζ ∂τ ∂τ ∂ζ ∂ζ ∂τ ∂ζ (9.17) ∂2 U =0. (K1 − k3 ) ∂τ∂ζ These equations are equivalent to the conditions of a potential independent from K4 in the axisymmetric case, equations 8.10 and 8.11. The latter equation leads to the two typical cases of a potential additively separable in cylindrical coordinates, or a non-separable potential.

9.4.1 Separable potential The separable potential satisfies ∂2 U =0. ∂τ∂ζ In the point-axial model, at least one of the parameters K1 or K4 depends on the angle. In particular, if K4 = 0, owing to Eq. 9.10, the potential must be separable, otherwise K1 = 0 would be held, rendering the axisymmetric model. 4 Similarly, by reasoning in regard to a population mixture, the potential is independent from the population parameters only if both parts are null separately.

160

CHAPTER 9. POINT-AXIAL SYMMETRIC SYSTEM

For a separable potential, either with K4 null or nonnull, we are led to the same equations as for the axisymmetric case, Eq. 8.13, with the addition 2 of equations 9.15 and 9.16, which add the new condition ∂∂τU2 = 0, yielding a separable potential in their harmonic form U = A(t) (τ + ζ) ,

(9.18)

where continuity conditions in the Galactic plane have been applied in order to neglect the term proportional to 1ζ . Therefore, for a point-axial system the separable potential reduces to the simple case of the harmonic function corresponding to the naive quasi-elastic force field, and does not depend on the population kinematic parameters except for the unique function A(t) discussed in the past chapter. Hence, under a separable potential, the kinematics of a point-axial symmetric system is totally free from conditions of consistency in regard to a mixture of populations. The population’s mean velocities, the semiaxes of the velocity ellipsoids, and their orientations remain unconstrained.

9.4.2 Non-separable potential The non-separable potential satisfies ∂2 U 0 ∂τ∂ζ and k(t) ≡ K1 = k3 . Then, K1 = 0. According to equations 9.15 and 9.16, the point-axial symmetry assumption requires K4  0. Hence, equations 9.10 and 9.11 provide the conditions     ∂U ∂U ∂ ∂U ∂U − − 2 +ζ =0, ∂τ ∂ζ ∂ζ ∂τ ∂ζ (9.19)   ∂ ∂U ∂U − =0, ∂τ ∂τ ∂ζ which separate Eq. 9.17 into two identically null equations. Hence, we can consider only one of them. Similarly, the same reasoning of the preceding section (either in regard to the dependency on the angle or on the population)

9.5. CONDITIONS OF CONSISTENCY applied to equations 9.12, 9.13, and 9.14 yields the conditions   ∂ ∂U ∂U − =0, ∂t ∂τ ∂ζ

161

(9.20)

2 2 2 ... ˙ ∂ U =0, ˙ ∂ U + k ∂ U + 2k˙ ∂U + 1 k + kζ kτ ∂t∂τ ∂τ 2 ∂τ∂ζ ∂τ2

(9.21) 2 2 2 ... ˙ ∂ U =0. ˙ ∂ U + k ∂ U + 2k˙ ∂U + 1 k + kτ kζ 2 ∂t∂ζ ∂ζ 2 ∂ζ∂τ ∂ζ

Thus, we reach the same set of equations as for an axisymmetric model consistent with a flat velocity distribution (equations 8.11 and 8.13), by pro  1 viding the potential U = A(t) (τ + ζ) + 1k U1 τ+ζ + k τ+ζ U2 (ζ/τ), although in the point-axial model we still have to submit it to Eq. 9.19. Therefore, the resulting potential must adopt the separable form U = f1 (τ + ζ) + f2 (ζ), so that +τ + ζ , 1 U = A(t) (τ + ζ) + U1 , (9.22) k k where, by continuity conditions in the Galactic plane, an additional term proportional to 1ζ is neglected. So far we have studied the potential for a single stellar population under point-axial symmetry. We now add the superposition hypothesis.

9.5 Conditions of consistency For a separable potential, there are no conditions of consistency, similarly to the axisymmetric model, since the equations for the potential, in the separable case, do not depend on the kinematical parameters. For a non-separable potential, all the system dependency on θ is carried through K4 (θ). Therefore, according to Eq. I.1, and bearing in mind that K1 = 0, in the Galactic plane the tensor elements Aθ and Aθz are null, as in the axisymmetric model. Hence, the velocity ellipsoid has no vertex deviation in z = 0. More precisely, according to the equations G.5 and 1 k3 K4 z2 . In I.1, the moment accounting for the vertex deviation is μθ = − 2|A| addition, similar to the axisymmetric case, since K1 = k3 the ellipsoid has no tilt in a meridional Galactic plane (i.e., the intersection of the ellipsoid with a meridional Galactic plane has an axis pointing toward the Galactic centre), and the mean velocities Π0 and Z0 are the same as in the axisymmetric case.

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CHAPTER 9. POINT-AXIAL SYMMETRIC SYSTEM

In the Galactic plane, the only moment depending on θ is μzz , whilst Θ0 and the other second moments are also axisymmetric. In summary, in the Galactic plane the velocity distribution of such a stellar system is basically axisymmetric and does not provide the most important feature we expected a point-axial system should provide, that is, the vertex deviation. Similarly, as for the axisymmetric case, the potential of Eq. 9.22 constrains the mean velocity components Π0 and Z0 to satisfy ΠZ00 = z . For a two population mixture we get Π 0 − Π

0 = 0 and Z0 − Z0

= 0, unless, according to the axisymmetric case, the function k(t) is linearly independent ˙ among populations and the potential does not depend on k(t)/k(t). In that case, an apparent vertex deviation of the mixture distribution is possible. The potential allowing unconstrained population mean velocities must then satisfy the condition ∂2 ∂U =0, (9.23) ∂τ∂ζ ∂t obtained in the axisymmetric case, and the potential takes the particular quasi-stationary form U = A(t) (τ + ζ) +

B , τ+ζ

(9.24)

with B = const, a particular spherical case of in Eq. 8.29.

9.6 Remarks The conditions of consistency studied in Chapter 8 for an axisymmetric model proved that a finite mixture of stellar populations was able to describe the main features of the local velocity distribution. In the Galactic plane, single populations had velocity ellipsoids without vertex deviation, so that the apparent vertex deviation of the disc velocity distribution was the result of different radial and rotation mean motions of the populations. However, we expected that under the point-axial hypothesis single populations had velocity ellipsoids with nonnull vertex deviation and non-vanishing tilt, as well as a point-axial mass distribution. The first important fact is that the potential must be axisymmetric in order to support a quadratic integral of motion for each population, which usually represents a stellar system in statistical equilibrium. That is, we assume that the stellar system has achieved relaxation and satisfies regularity

9.6. REMARKS

163

conditions about the definition of the LSR, continuity, and differentiability of its velocity, and that higher-order velocity moments exist. Although dissipative forces related to third and odd-order moments does not appear in the moment equations planned for a single population, they are indirectly connected with the assumption of the mixture model. The second result we have obtained is that the point-axial symmetry is, in a natural way, consistent with the flat velocity distribution of a disc population, by providing potentials not depending on the population parameter K4 , which is responsible for non-isothermal velocity distributions. In axisymmetric systems, only a particular family of potentials is consistent with a flat velocity distribution, while in point-axial systems any potential always is. The third result is that the potential is spherical, either in the separable or the non-separable case. We have found two possible solutions depending on the separability of the potential: (a) The point-axial model admits a potential additively separable in cylindrical coordinates. It is the harmonic potential of a quasi-elastic force field. As in the axisymmetric model, for a separable potential there is no need of conditions of consistency in regard to a mixture distribution, since the potential only constrains the population parameters through the function A(t). For each population, the radial and vertical mean velocities can be different, and their velocity ellipsoids can have different orientations, including the both vertex deviation and tilt. (b) For a non-separable potential, the condition given by Eq. 9.23 provides nearly non-constrained population kinematics. Then, the radial and the vertical mean velocities can differ among populations, although they are coupled, and they may produce an apparent vertex deviation of the whole velocity distribution. However, single population velocity ellipsoids have no vertex deviation in the Galactic plane and no tilt in their intersection with a meridional Galactic plane, similarly to the axisymmetric case. In both cases, the potential for the point-axial model becomes a particular function of the quasi-stationary potential for the axisymmetric model. The non-separable potential loses the dependency on the elevation angle, and the separable potential loses the non-harmonic term that they showed in the axisymmetric case. As discussed in Section §8.6, neither the separable potential nor the spherical one are able to fit the local Galactic constants, i.e., the local epicycle frequencies and the local angular velocity. Nevertheless, the spherical potential, Eq. 9.24, provides a non-harmonic term, which may be associated

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CHAPTER 9. POINT-AXIAL SYMMETRIC SYSTEM

with an additional repulsive force (if B < 0) similar to the perturbation of a point-mass potential induced by a tidal force. Such a potential constrains the population velocity ellipsoids to point toward the Galactic centre, although, out of the Galactic plane, the ellipsoids may show some vertex deviation. Then, according to the point-axial model, how could we explain the possible non-vanishing tilt for the thick disc and the halo ellipsoids that CasettiDinescu 2011, Carollo et al. 2010, Fuchs et al. 2009, Smith et al. 2009a, and Siebert et al. 2008 suggest? Once accepted that the harmonic potential is not realistic, under the point-axial model we cannot explain it. Similarly, the point-axial model is unable to explain the trend of the moment μz for the thin disc population described by Pasetto et al. (2012b), which was only possible under a separable potential. Instead, if the thick disc and the halo ellipsoids actually have a non-vanishing tilt, the axisymmetric model is still capable of making a reasonable approach to the local features of the local velocity distribution. On the other hand, for the stellar density, point-axial symmetry matters. We may assume a Schwarzschild velocity distribution without loss of generality to discuss the shape of mass distribution. In that case, the stellar 1 density is N ∝ √1|A| e− 2 σ and depends on the angle through K4 (θ). Remark that the stellar density is given that way, in relation to the potential and the collisionless Boltzmann equation, providing a finite mass of the stellar system, and with no connection to the Poisson equation. Then, leaving aside the simple and unrealistic case of a separable and harmonic potential, for the non-separable potential with k ≡ K1 = k3 , Eq. 9.24 is a particular case c2 c1 . The function σ involved in the of Eq. 9.22 with U1 = (τ + ζ) + k τ+ζ stellar density satisfies c1 c2 k β2 τ 1 σ = (τ + ζ) + − + const . 2 k τ + ζ k + 2k2 τ + 2K4 ζ

(9.25)

For ζ = 0, σ does not depend on θ. However, for ζ = 0, we have |A| = k(k + 2k2 τ)(k + 2K4 τ) ,

(9.26)

so that, in the Galactic plane, the stellar density N depends on θ. This dependency of the mass distribution on the angle is balanced out by the velocity distribution, which also depends on θ, while the potential maintains the axisymmetry. This is the only basic feature that the point-axial model adds to the axisymmetric model. While, in the Galactic plane, for the velocity distribution, according to equations I.1 and 9.26, the tensor element Azz and |A|,

9.6. REMARKS

165

which depend on θ, lead to moments μ , μzz also depending on the angle, although each population component is unable to provide a non-vanishing moment μθ . Thus, similarly to the axisymmetric case, for ellipsoidal velocity distributions under a non-separable potential, the apparent vertex deviation of the velocity distribution is a consequence of the coexistence of two or more populations with different radial and rotation mean velocities. Therefore, a point-axisymmetric stellar system would, in principle, be able to show a triaxial or bar-like structure in one of the population’s mass distributions, in particular the component responsible of star formation. It would be a matter of time that, for the specific population, the slow rotation of a density wave, with differential rotation similar to that of the rotation curve of the Galaxy, could transform a bar-like into a spiral-like structure. In that situation, the point-axial model would have the ability to describe a point-axial mass distribution, while the axial model would not. However, we note that the degenerate case of a rigid rotating bar, with k2 = 0, which was the only remaining case that could admit, a priori, a non-cylindrical potential as a solution of Eq. 9.8, cannot coexist with three-dimensional ellipsoidal velocity distributions under a common differentiable point-axial potential. Hence, any phase mixing process involving a rigid rotating bar with a non-cylindrical potential must be considered as a state previous to statistical equilibrium, which is not associated with stellar populations having ellipsoidal velocity distributions, even with point-axial symmetry. Then, notice what has happened. Looking for more flexibility in the kinematical description of the stellar system, we exchanged the axial symmetry hypothesis by the point-axial, allowing at the same time a mixture of populations. This has considerably limited the general form of the potential, by losing the possibility of the general case of the quasi-stationary potential of Eq. 8.29, with the azimuthal dependence, the one allowing a reasonable independence for the stellar populations. In conclusion, the local potential is consistent with the stationary case of Eq. 8.29, neither spherical nor separable in cylindrical coordinates, allowing the apparent vertex deviation of the velocity ellipsoid, although with vanishing tilt. The velocity distribution in the solar neighbourhood reflects a basically axisymmetric Galaxy, although a non-stationary system is required to account for the vertex deviation of the velocity ellipsoid and for describing the radial differential movement of population centroids. Among future alternatives for describing asymmetries of the velocity and mass distributions one would have to think about avoiding the assumption of a symmetry plane. It would be convenient that the general solution for the potential could account for self and external perturbations, such as

166

CHAPTER 9. POINT-AXIAL SYMMETRIC SYSTEM

those produced by tidal forces, which may occur in any direction out of the symmetry plane, yielding then to potentials and velocity distributions that are non-even functions in the variable z. It seems that this can be a fruitful field of future investigation, since isolated and stationary systems in equilibrium hardly will present asymmetric global trends.

Appendices

167

Appendix A

Chandrasekhar equations The moment equations of orders n = 0, 1, 2, 3 can be transformed into Chandrasekhar equations to find out the functional dependence of A2 and σ.

A.1

Equation of order n = 3

In components, Eq. 3.2 is equivalent to μiα Aα j = δi j .

(A.1)

By taking gradient in both sides we get the identity  ∂ ∂  μiα Aα j = (δi j ) = 0, ∂rk rk

(A.2)

which is valid for any index k of the derivative variable, either with or without contraction of indices, and it is also valid if the time derivative is taken. The following relationship is then satisfied ∂Aα j ∂μiα Aα j = −μiα . ∂rk ∂rk

(A.3)

If the equation Eq. 3.22 corresponding to order n = 3 is contracted three times with the tensor A2 ⊗ A2 , which is always nonnull, we have   ∂μ jk ∂μi j ∂μik Ai j Akl μiα + μ jα + μkα = 0. (A.4) ∂rα ∂rα ∂rα 169

170

APPENDIX A. CHANDRASEKHAR EQUATIONS

By taking into account Eq. A.1, we may then write ∂μ jk ∂μi j ∂μ jk ∂μi j ∂μik ∂μik + δiα Akl + δlα Ai j = Akl + Akl + Ai j =0 ∂rα ∂rα ∂rα ∂r j ∂ri ∂rl (A.5) and by Eq. A.3, as well as by changing the sign, we get δ jα Akl

μ jk

∂Ai j ∂Akl ∂Akl + μik + μi j = 0. ∂r j ∂ri ∂rl

If some repeated indices are changed, we can write   ∂Aik ∂A jk ∂Ai j = 0. + + μi j ∂r j ∂ri ∂rk

(A.6)

(A.7)

Therefore, since the double contraction of indices is carried out with the nonnull symmetric tensor μ2 , inverse of A2 , which is associated with a positive definite quadratic form, we are led to the equation 3 ∇r  A2 = (0)3 .

(A.8)

Such a relation gives then account of the spatial dependence of the tensor A2 .

A.2

Property

We deduce a consequence of Eq. A.7, which will be useful in the following sections. By applying the relation Eq. A.3, since the tensors there involved are symmetric, the left-hand side of Eq. A.7 can be written as   ∂Ai j ∂Aik ∂A jk ∂Ai j ∂Aik μi j + + + μi j = = 2μi j ∂r j ∂ri ∂rk ∂r j ∂rk (A.9) ∂μi j ∂Ai j = −2Aik + μi j . ∂r j ∂rk Now, if A¯i j denotes the cofactor of the element Ai j , which is the same one as for its transposed element A ji , then the relation Eq. 3.2 obviously implies μi j =

A¯i j , |A|

|A| ≡ det A2 .

(A.10)

A.3. EQUATION OF ORDER N = 2

171

Hence Eq. A.9 can be converted into   ∂μi j A¯i j ∂Ai j ∂Aik ∂A jk ∂Ai j μi j + + + . = −2Aik ∂r j ∂ri ∂rk ∂r j |A| ∂rk

(A.11)

It is well-known that if the tensor A2 depends on a variable ξ, then the relation ∂|A| ∂Ai j ¯ = (A.12) Ai j ∂ξ ∂ξ is satisfied1 . Hence, by writing ξ = rk we have   ∂μi j ∂Aik ∂A jk ∂Ai j 1 ∂|A| μi j + + + . = −2Aik ∂r j ∂ri ∂rk ∂r j |A| ∂rk

(A.13)

Therefore, bearing in mind that μ2 = A−1 2 , it is fulfilled 3μ2 : (∇r  A2 ) = −2A2 · (∇r · μ2 ) + ∇r ln |A|.

(A.14)

Nevertheless, in virtue of Eq. A.7, the left-hand side of the above equation is zero. Hence, the following relationship is satisfied, 1

A2 · (∇r · μ2 ) = ∇r ln |A| 2 ,

(A.15)

which, by taking dot product with μ2 , can also be written as 1

∇r · μ2 = μ2 · ∇r ln |A| 2 .

A.3

(A.16)

Equation of order n = 2

For the second order hydrodynamic equation, if we take the colon product of A2 ⊗ A2 with Eq. 3.15, we have   ∂μi j ∂v j ∂μi j ∂vi Aik A jl + vα + μiα + μ jα = 0. (A.17) ∂t ∂rα ∂rα ∂rα 1 The determinant of A can be expressed as |A| =  2 i1 ...in A1i1 · · · Anin , where i1 ...in denotes the Levi-Civitta tensor. Then,   ∂A1i1 ∂A2i2 ∂Anin ∂|A| = i1 ...in A2i2 . . . Anin + A1i1 . . . Anin + · · · + A1i1 . . . = ∂ξ ∂ξ ∂ξ ∂ξ

=

∂A2i2 ∂Ai j ∂A1i1 ∂Anin A¯1i1 + A¯2i2 + · · · + A¯nin = A¯i j . ∂ξ ∂ξ ∂ξ ∂ξ

APPENDIX A. CHANDRASEKHAR EQUATIONS

172

By taking into account Eq. A.3 we can write −Aik μi j

∂A jl ∂A jl ∂v j ∂vi + Aik μiα A jl + Aik A jl μ jα =0 − Aik μi j vα ∂t ∂rα ∂rα ∂rα

(A.18)

and now, by Eq. A.1, we have −δk j

∂A jl ∂A jl ∂v j ∂vi − δ k j vα + δkα A jl + Aik δ jα = ∂t ∂rα ∂rα ∂rα

∂v j ∂Akl ∂vi ∂Akl − vα =− + A jl + Aik = 0. ∂t ∂rα ∂rk ∂r j

(A.19)

On the other hand, in Eq. A.8, if we take the inner product with vα , we obtain the identity −vα

∂A jl ∂A jα ∂Aαl = vα + vα , ∂rα ∂rl ∂r j

(A.20)

which, by substitution in Eq. A.19, yields −

∂A jα ∂v j ∂Aαl ∂vi ∂Akl + vα + vα + A jl + Aik = 0. ∂t ∂rl ∂r j ∂rk ∂r j

(A.21)

Hence, by reordering and changing some repeated indices, we have −

∂Akl ∂(Akα vα ) ∂(Aαl vα ) + + = 0, ∂t ∂rl ∂rk

which can be written in the form ∂A2 − 2 ∇r  (A2 · v) = (0)2 . ∂t

A.4

(A.22)

(A.23)

Property

If we take the colon product of μ2 with Eq. A.23, by Eq. A.1 and by changing some repeated indices, we can write μi j

∂Ai j ∂A jα ∂vα ∂Aiα ∂vα − μi j vα − μi j A jα − μi j vα − μi j Aiα = ∂t ∂ri ∂ri ∂r j ∂r j

= μi j

∂Ai j ∂Aiα ∂vα ∂Aiα ∂vα − μi j vα − δiα − μi j vα − δ jα = ∂t ∂r j ∂ri ∂r j ∂r j

= μi j

∂Ai j ∂vα ∂Aiα − 2μi j vα − 2 = 0. ∂t ∂r j ∂rα

(A.24)

A.5. EQUATION OF ORDER N = 1

173

Now we apply Eq. A.11 and Eq. A.12 with ξ = t to the first summation term of the last equation, and we also apply the property expressed in Eq. A.13 to the second summation term, so that we obtain ∂ ln |A| ∂ ln |A| ∂vα + vα − 2 = 0. ∂t ∂rα ∂rα

(A.25)

Thus, the foregoing relation gives account of the divergence of the centroid velocity, which may be written as 1

1 ∂ ln |A| 2 ∇r · v = + v · ∇r ln |A| 2 . ∂t

A.5

(A.26)

Equation of order n = 1

By substitution of the stellar density N, Eq. 3.3, into the equation corresponding to n = 1, Eq. 3.9, we write ∂v 1 1 + v · ∇r v + ∇r U = ∇r ln |A| · μ2 + ∇r σ · μ2 − ∇r · μ2 . ∂t 2 2

(A.27)

Then, by taking into account Eq. A.16, we have 1 ∇r · μ2 − ∇r ln |A| · μ2 = (0). 2

(A.28)

Therefore, Eq. A.27 reduces to ∂v 1 + v · ∇r v + ∇r U = A−1 · ∇r σ. ∂t 2 2

A.6

(A.29)

Equation of order n = 0

Let us write the continuity equation Eq. 3.8 in the following form ∂ ln N + ∇r · v + v · ∇r N = 0. ∂t

(A.30)

Then, by substitution of N, Eq. 3.3, and by reordering terms, we can write ⎞  ⎛  1 ⎜⎜⎜ ∂ ln |A| 2 1⎟ 1 ∂σ ⎟ + v · ∇r σ − ⎜⎝ + v · ∇r ln |A| 2 ⎟⎟⎠ + ∇r · v = 0. − 2 ∂t ∂t

(A.31)

174

APPENDIX A. CHANDRASEKHAR EQUATIONS

Nevertheless, the relationship we obtained for ∇r · v in Eq. A.26 makes also null the above first term, which is independent from the tensor A2 . Hence we have ∂σ + v · ∇r σ = 0. (A.32) ∂t Thus, σ is conserved along the centroid trajectory.

Appendix B

Power series Let ψ(Q + σ) be a square-integrable function over the interval I = (0, +∞) in regard to the variable Q, so that it is denoted as ψ ∈ L2 (I). Notice that ψ cannot be constant. Let us remember that Q depends on the velocities through the positive definite quadratic form defined in Eq. 3.1, σ depends only on time and position, and ψ is the velocity distribution. Then ψ may be written as1 +Q + σ, 1 ψ(Q + σ) = F (B.1) e− 2 (Q+σ) , 2 where F can be expressed as a series of Laguerre polynomials, which are an orthogonal basis of the vector space L2 (I) with respect to the weight 1 w(Q) = e− 2 (Q+σ) on I. Hence, by defining the variable 1 (Q + σ) 2

τ= we can write F(τ) =

∞ 

αk L0k (τ)

(B.2)

(B.3)

k=0 1 Let us remember that any square-integrable function f (x) ∈ L2 (I) admits an expansion as a series of the associated Laguerre functions, ∞  f (x) = Lαk (x)xα e−x k=0

with α > −1, where {Lαk (x)}k∈N is the family of the associated Laguerre polynomials, which is an orthogonal basis of the space L2 (I) with respect to the weight xα e−x (e.g., Abramowitz & Stegun 1965).

175

APPENDIX B. POWER SERIES

176

so that each Fourier coefficient αk , which multiplies the Laguerre polynomial Lαk with α = 0, can be evaluated from the inner product defined as  ∞ f (τ)g(τ)e−τ dτ. (B.4)  f (τ), g(τ) = 0

On the other hand, the foregoing conditions do also allow to compute the central velocity moments, which are given through the integral Eq. 3.28. Thus, bearing in mind Eq. B.1, the integral takes the form  ∞ +Q + σ, 1 n 1 Q2 F (B.5) φn (σ) = e− 2 (Q+σ) Q 2 dQ. 2 0 Notice that such an integral is convergent because, after the change of variable given in Eq. B.2, F can be expressed through the associated Laguerre polynomials, now with α = 12 . The above polynomial form of F can be formally useful in the case we want to estimate it from all the available central moments. Nevertheless, to our current purpose, we need to write F depending on the new variable η = e−τ = e− 2 (Q+σ) . 1

(B.6)

Then, for a fixed σ it is possible to establish an isomorphism between the 1 respective domains of Q and η, namely I = (0, +∞) and J = (0, e− 2 σ ), so that with the notation ˜ F(η) = F(τ(η)) (B.7) ˜ the function F(η) can be expressed as depending on the new variable from a basis of orthogonal polynomials {Pk (η)}k∈N over J, according to an inner product which is equivalent to the previous one defined on I from Eq. B.4. That is,   f (τ), g(τ) I ≡

∞

−τ



f (τ)g(τ)e dτ =

1

e− 2 σ

f (τ(η))g(τ(η))e 0

0

 =

)) ))−1 −τ(η) )) dη )) ) dτ )

dη =

1

e− 2 σ

f˜(η)g˜ (η)dη ≡  f˜(η), g˜ (η) J .

0

(B.8) ˜ Thus, we can write F(η) as a series of polynomials {Pk (η)}k∈N or, by reorganising terms, as power series of η in the following form ˜ F(η) =

∞  k=0

βk Pk (η) =

∞  k=0

γk ηk .

(B.9)

B. POWER SERIES

177

Hence, according to Eq. B.7, by substitution of Eq. B.9 into Eq. B.1, as well as by writing it in terms of the variable Q, we finally have ψ(Q + σ) = e− 2 (Q+σ) 1

∞  k=0

γk e− 2 (Q+σ) k = 1

∞ 

γk−1 e− 2 (Q+σ) k . 1

(B.10)

k=1

Thus, any arbitrary integrable, quadratic function ψ(Q + σ) can be ex1 pressed as a convergent series of the Gaussian functions e− 2 (Q+σ) k , with k ≥ 1, although they are not an orthogonal system. This convergence is with respect to the quadratic norm in the vector space L2 (I) in regard to the corresponding weight. Under conditions of continuity in nearly every point and bounded variation of the function ψ, the series is uniformly convergent.

Appendix C

Moment recurrence The Gramian system and the moment recurrence can be solved straightforwardly for the case n = 2, which corresponds to a Schwarzschild distribution. For the sake of simplicity, and without losing generality, we use the central moments μn , so that μ1 = 0. Then, the equations Eq. 5.18 and Eq. 5.19, for m = 0, 1, become λk = 0;

k = 1, 2, 3;

(C.1)

and

1 δik + 2λ jk μ ji = 0 =⇒ λik = − μ−1 ; i, k = 1, 2, 3; (C.2) 2 ik where μ−1 ik are the elements of the inverse of the covariance matrix. Therefore, the above relation shows that the tensor of elements λik is a definite negative form, and it leads to an integrable distribution function. Now we can apply the same procedure for m ≥ 2, to obtain higher order moments in terms of the second moments. Thus, for m = 2, according to Eq. 5.20, with m = n − 1, and bearing in mind Eq. C.1, we have λk μi j + 2λlk μi jl = 0 =⇒ μi jk = 0,

i, j, k = 1, 2, 3.

(C.3)

The result accounts for the obvious symmetry of the distribution, with vanishing odd-order central moments. Similarly, for m = 3, we get δi1 i4 μi2 i3 + δi2 i4 μi1 i3 + δi3 i4 μi1 i2 + λi4 μi1 i2 i3 + 2λ ji4 μ j i1 i2 i3 = 0, for

i1 , i2 , i3 , i4 = 1, 2, 3. 179

(C.4)

180

APPENDIX C. MOMENT RECURRENCE

Then, taking into account Eq. C.2, we multiply by μki4 . Since the third moments are null, by reordering indices we obtain the following moment recurrence relation: μi1 i2 i3 i4 = μi1 i4 μi2 i3 + μi2 i4 μi1 i3 + μi3 i4 μi1 i2 .

(C.5)

The above relationship is the well-known property of a Gaussian distribution, which characterises it from having vanishing fourth cumulants. And, in general, according to Eq. 5.20, for even m, we obtain a vanishing set of odd-order central moments, and, for odd m, we obtain the relation δi1 im+1 μi2 ...im + . . . + δi j im+1 μi1 ... i j ...im + . . . + δim im+1 μi1 ...im−1 + 2λ jim+1 μ ji1 ...im = 0. (C.6) Once again, by multiplying by μkim+1 and by reordering indices, we get μi1 ...im im+1 = μi1 im+1 μi2 ...im + . . . + μi j im+1 μi1 ... i j ...im + . . . + μim im+1 μi1 ...im−1 . (C.7) This is the general relationship of moment recurrence for trivariate normal distributions, which leads to a vanishing set higher-order cumulants.

Appendix D

Parameter estimation The maximum entropy method yields the following Gramian system, corresponding to Eq. 5.22, Y = G2 X;

Y = [A, B, C], X = [a, b, c]

(D.1)

Due to the symmetry properties of moments mn and polynomial coefficients λn , for n > 1 the number of equations is always greater than the number of independent unknowns. Then, an equivalent overdeterminate system of equations must be built up, where the symmetric coefficients of tensors λn will not be repeated in the vector of unknowns. This overdeterminate system of equations, which is equivalent to Eq. D.1, is written as (D.2) y = g2 x, where the only unknowns are the non-identical elements of the symmetric tensors λn . It takes the following form: The first column A (Table 5.2) of matrix Y and the first column a of matrix X are related by the coefficient submatrix of Table D.1. The second column B of matrix Y and the first column b of matrix X are related by the coefficient submatrix of Table D.2. The third column C of matrix Y and the third column c of matrix X are related by the coefficient submatrix of Table D.3. The resulting g2 matrix is obtained by stacking the three foregoing submatrices. Vector x now takes the form x = (λ1 , λ2 , λ3 , 2λ11 , 2λ12 , 2λ13 , 2λ22 , 2λ23 , 2λ33 , 3 λ111 , 3 λ112 , 3 λ113 , 3 λ122 , 3 λ123 , 3 λ133 , 3 λ222 , 3 λ223 , 3 λ233 , 3 λ333 , ...)T . (D.3) 181

APPENDIX D. PARAMETER ESTIMATION 182

⎡ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎣⎢⎢ 2 m12 2 m113

2 m13

m222

m122

m22

2 m223

2 m123

2 m23

m333

m233

m133

m33

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0 ···

0 ···

0 ···

0 ···

0 ···

0 ···

0 ···

2 m1223

2 m1133

m2233

m2223

m2222

m1223

2 m2333

2 m2233

2 m2223

2 m1233

m3333

m2333

m2233

m1333

0 0

0 0

0 0

0

0

0

0 ···

0 ···

0 ···

m11

2 m123

2 m233

m1133

0 0

2 m1123

2 m1233

0

2 m112

m223

2 m1123

m1233

2 m1222

2 m1333

0

2 m122 2 m133

m1122

2 m1223

2 m1223

0 0 m112 2 m123

2 m1113

m1222

m1122

2 m1233

0 0

0 m113 2 m1112

2 m1123

m1123

m3

0 m1111 2 m1122

0

m1133

m2

0 0

0 m1112

0

m1

0 0

0

0 0

0

0 ···

0 0

0 0

0

0 0

0 ···

0 0

m33

0 0

0 0

0

1

m113

0 0

m233

0

m111

m23

m123 m1113

m333

0 0

m13

m112

m133

m223

0 0

m23

m13

m122

m223

m233

m11133

m12

0 0 m111

m123

m123

2 m11123

m22

m2

0 0 m112

m222

m133

m11122

m11

m3

0 0 m113

0 0

2 m11113

m12

m11

0 0 m122

m22

0 0

2 m11112

0 0

m12

0 0

m23

0 m11111

m1

m13

m33

0

0 ···

0 0

0 ···

0 0

0

m1113

0

m1112

0 0

m1111

0 0

0 0

m12233

0 0

m12333

m111

2 m12223

0 ···

2 m12233

0 m12222

0 ···

0 0

m12223

0 ···

m11233

2 m11223

0

0 ···

m11333

2 m11233

0

0 ···

2 m11223

2 m11222

0 0

0

2 m11233

2 m11223

0 0

0

m11222

0 m11122

m13333

0 0

m11223

0 m11123

2 m12333

m22233

0 0

2 m11123

0 0

m12233

2 m22223

m22333

2 m11133

0 0

2 m11333

m22222

2 m22233

2 m11122

m1223

2 m11233

2 m12223

m22223

2 m11123

m1233

0 m11133

2 m12222

2 m12233

m11112

m1222

0

m11222

2 m12223

0 m11113

m1223

0 0

0 m11223

0 0

m1122

0 0

0

m1123

m1123

m1333

0 0

m1133

0 0

m1233

m2223

0 0

m1122

m113

0 0 m1133

m2222

m2233

0 ···

m1123

m122

0 0

m1222

m2223

0

m1112

m123

0 0

m1223

0 0

m1113

m133

0 0

m23333

0 0

m222

0 0

m33333

m112

m223

2 m22333

. ..

2 m23333

. ..

m22233

. ..

m22333

. ..

2 m12333

. ..

2 m13333

. ..

2 m12233

. ..

2 m12333

. ..

m11233

. ..

0 m11333 . ..

0 0

. ..

m2333

. ..

m3333

. ..

m2233

. ..

m2333 . ..

m1233 . ..

m1333 . ..

0 0 . ..

m233 m333 . ..

Table D.1: Submatrix relating the first column A of matrix Y and the first column a of matrix X.

⎤ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥⎥ ⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎦⎥⎥

⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.. .

m12

m13

m22

m23

m33

0

0

0

0

0

0 m111

0 m112

0 m113

0 m122

0 m123

0 m133

0 m222

0 m223

0 m233

0 m333

.. .

0

m133

.. .

.. .

0 m2333

.. .

0 m1333

.. .

0 m2233

0 m1233

0 m2223

0 m2222

0 m1223

0 m1233

0 m1222

0 m1223

0 m1123

0 m1133

0 m1222

0 m1123

0 m1122

0 m1122

0 m1113

0 m1112

m233

m223

m222

m123

m122

m112

m23

m22

m12

m2

0 m1112

0

0

m122

m123

0

m113

0

0

m13

m112

0

m12

0

0

m11

m111

0

m1

0 m1111

0

0

0

0

0

0

0

0

0

0

.. .

m3333

m2333

m2233

m2223

m1333

m1233

m1223

m1133

m1123

m1113

m333

m233

m223

m133

m123

m113

m33

m23

m13

m3

.. .

.. .

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

.. .

m11333

m11233

m11223

m11222

m11133

m11123

m11122

m11113

m11112

m11111

m1133

m1123

m1122

m1113

m1112

m1111

m113

m112

m111

m11

.. .

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.. .

2 m12333

2 m12233

2 m12223

2 m12222

2 m11233

2 m11223

2 m11222

2 m11123

2 m11122

2 m11112

2 m1233

2 m1223

2 m1222

2 m1123

2 m1122

2 m1112

2 m123

2 m122

2 m112

2 m12

.. .

2 m13333

2 m12333

2 m12233

2 m12223

2 m11333

2 m11233

2 m11223

2 m11133

2 m11123

2 m11113

2 m1333

2 m1233

2 m1223

2 m1133

2 m1123

2 m1113

2 m133

2 m123

2 m113

2 m13

.. .

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.. .

m22333

m22233

m22223

m22222

m12233

m12223

m12222

m11223

m11222

m11122

m2233

m2223

m2222

m1223

m1222

m1122

m223

m222

m122

m22

.. .

2 m23333

2 m22333

2 m22233

2 m22223

2 m12333

2 m12233

2 m12223

2 m11233

2 m11223

2 m11123

2 m2333

2 m2233

2 m2223

2 m1233

2 m1223

2 m1123

2 m233

2 m223

2 m123

2 m23

.. .

m33333

m23333

m22333

m22233

m13333

m12333

m12233

m11333

m11233

m11133

m3333

m2333

m2233

m1333

m1233

m1133

m333

m233

m133

m33

⎤ 0 · · · ⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ ⎥ 0 · · · ⎥⎥⎥⎥ ⎥⎥⎥ 0 · · · ⎥⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎦ .. .

Table D.2: Submatrix relating the second column B of matrix Y and the first column b of matrix X.

.. .

0

m3

m11

0

m2

0

0

0

m1

0

0

0

1

0

D. PARAMETER ESTIMATION 183

APPENDIX D. PARAMETER ESTIMATION 184

⎡ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢⎢ ⎣⎢⎢ 0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

m123

m122

m113

m112

m111

m33

m23

m22

m13

m12

m11

m3

m2

m1

1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

m1123

m1122

m1113

m1112

m1111

m133

m123

m122

m113

m112

m111

m13

m12

m11

m1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

m1223

m1222

m1123

m1122

m1112

m233

m223

m222

m123

m122

m112

m23

m22

m12

m2

m1233

m1223

m1133

m1123

m1113

m333

m233

m223

m133

m123

m113

m33

m23

m13

m3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

m11123

m11122

m11113

m11112

m11111

m1133

m1123

m1122

m1113

m1112

m1111

m113

m112

m111

m11

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2 m11223

2 m11222

2 m11123

2 m11122

2 m11112

2 m1233

2 m1223

2 m1222

2 m1123

2 m1122

2 m1112

2 m123

2 m122

2 m112

2 m12

2 m11233

2 m11223

2 m11133

2 m11123

2 m11113

2 m1333

2 m1233

2 m1223

2 m1133

2 m1123

2 m1113

2 m133

2 m123

2 m113

2 m13

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

m12223

m12222

m11223

m11222

m11122

m2233

m2223

m2222

m1223

m1222

m1122

m223

m222

m122

m22

2 m12233

2 m12223

2 m11233

2 m11223

2 m11123

2 m2333

2 m2233

2 m2223

2 m1233

2 m1223

2 m1123

2 m233

2 m223

2 m123

2 m23

m12333

m12233

m11333

m11233

m11133

m3333

m2333

m2233

m1333

m1233

m1133

m333

m233

m133

m33

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

0

0

m11133

0

0

2 m11233

2 m11333

··· 0

0

···

m1333

m13333

m1233

2 m12333 0

m12233 0

0

m1133

0

0 0

···

0 0

m22233

m133

2 m22223

0 0

m22222

···

0 0

2 m12223

m22333

2 m12222

2 m22233

m11222

m22223

m2223

0

0

m2222

2 m12233

m1222

2 m12223

m222

0

0

m11223

···

0

m23333

0

m33333

0

2 m22333

0

2 m23333

m2233

m22233

m2223

m22333

0

0

0

2 m12333

. ..

m1223

2 m13333

. ..

0 0

2 m12233

. ..

0 0

2 m12333

. ..

m223

0

. ..

0 0

m11333

. ..

0 0

0

. ..

m11233 0

. ..

m2333

. ..

m3333

. ..

m2233

. ..

m2333 . ..

0 . ..

m1233 . ..

m1333 . ..

0 0 . ..

m233 . ..

m333 . ..

0 0 . ..

Table D.3: Submatrix relating the third column C of matrix Y and the third column c of matrix X.

⎤ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥⎥ ⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥⎥ ⎦⎥⎥

D. PARAMETER ESTIMATION

185

The factors multiplying the elements of tensors λn in Table 5.2, other than those appearing in vector x, have been carried over the elements of matrix g2 . It is well-known that, if Vy is the variance matrix for vector y, which is taken as the diagonal matrix of its sampling variances σy2 , then the least squares system weighted by Vy−1 provides minimum variance estimates for x according to (e.g., Stuart & Ord 1987) x = (g2T Vy−1 g2 )−1 g2T Vy−1 y.

(D.4)

The minimum fitting error is then obtained from the weighted norm of the difference between the observed values and their theoretical predictions, χ2 = (y T − xT g2T )Vy−1 (y − g2 x).

(D.5)

The error on the results of the least squares fit, that is the variance σ 2x of vector x, is obtained from the diagonal matrix of V x = (g2T Vy−1 g2 )−1 . Details of the fitting procedure are found in Cubarsi (2010b).

(D.6)

Appendix E

K-statistics It is known that the sample central moments, namely Mn , are biased estimators of the population central moments μn . Conversely, under the assumption of homogeneous observational errors, the population cumulants κn have as unbiased estimators the so-called k-statistics or sample cumulants, that here are notated as Kn . Let us remark that whereas the sample moments Mn are the same function of the sample values as the population moments μn , the same relation does not hold for Kn and κn . Basically the k-statistics are sums of products of the sample moments, which –like the cumulants and the central moments– are invariant under change of the origin, except for the first order. The tensor forms for the k-statistics of a multivariate distribution were published by Kaplan (1952). The first k-statistic is equal to the mean, and up to fourth-order they may be obtained depending on the sample moments according to K2 =

N M2 , N−1

K3 =

N2 M3 , (N − 1)(N − 2)

 N 2 (N + 1) (N − 1) M4 − 3 M2  M2 , K4 = (N − 1)(N − 2)(N − 3) (N + 1) where N is the size of the sample. In Stuart & Ord (1987, §12.10) there is a complete discussion and references on the subject.

187

Appendix F

Mixture equations F.1 U-cumulants o111 o222 o112 o122

= −κ333 d13 + 3κ133 d12 d3 − 3κ113 d1 d32 + κ111 d33 = 0 = −κ333 d23 + 3κ233 d3 d22 − 3κ223 d32 d2 + κ222 d33 = 0 = −κ333 d12 d2 + κ233 d12 d3 + 2κ133 d1 d3 d2 − 2κ123 d1 d32 − κ113 d32 d2 + κ112 d33 = 0 = −κ333 d1 d22 + κ133 d3 d22 + 2κ233 d1 d3 d2 − 2κ123 d32 d2 − κ223 d1 d32 + κ122 d33 = 0

p11 = κ333 d12 − 2κ133 d1 d3 + κ113 d32 p12 = κ333 d1 d3 d2 − κ233 d1 d3 − κ133 d3 d2 + κ123 d32 p22 = κ333 d22 − 2κ233 d3 d2 + κ223 d32 s1 = 12 (−κ333 d1 + κ133 d3 ) s2 = 12 (−κ333 d2 + κ233 d3 ) X1111 = κ3333 d14 − 4κ1333 d13 d3 + 6κ1133 d12 d32 − 4κ1113 d1 d33 + κ1111 d34 X1112 = κ3333 d13 d2 − κ2333 d13 d3 − 3κ1333 d12 d3 d2 + 3κ1233 d12 d32 + 3κ1133 d1 d32 d2 − −3κ1123 d1 d33 − κ1113 d33 d2 + κ1112 d34 X1122 = κ3333 d12 d22 − 2κ2333 d12 d3 d2 − 2κ1333 d1 d3 d22 + κ2233 d12 d32 + 4κ1233 d1 d32 d2 + +κ1133 d32 d22 − 2κ1223 d1 d33 − 2κ1123 d33 d2 + κ1122 d34 X1222 = κ3333 d1 d23 − κ1333 d3 d23 − 3κ2333 d1 d3 d22 + 3κ1233 d32 d22 + 3κ2233 d1 d32 d2 − −3κ1223 d33 d2 − κ2223 d1 d33 + κ1222 d34 X2222 = κ3333 d24 − 4κ2333 d3 d23 + 6κ2233 d32 d22 − 4κ2223 d33 d2 + κ2222 d34 189

APPENDIX F. MIXTURE EQUATIONS

190

Y111 Y112 Y122 Y222

= −κ3333 d13 + 3κ1333 d12 d3 − 3κ1133 d1 d32 + κ1113 d33 = −κ3333 d12 d2 + 2κ1333 d1 d3 d2 + κ2333 d12 d3 − 2κ1233 d1 d32 − κ1133 d2 d32 + κ1123 d33 = −κ3333 d1 d22 + 2κ2333 d1 d3 d2 + κ1333 d3 d22 − κ2233 d1 d32 − 2κ1233 d32 d2 + κ1223 d33 = −κ3333 d23 + 3κ2333 d3 d22 − 3κ2233 d32 d2 + κ2223 d33

Z11 = κ3333 d12 − 2κ1333 d1 d3 + κ1133 d32 Z12 = κ3333 d1 d2 − κ2333 d1 d3 − κ1333 d3 d2 + κ1233 d32 Z22 = κ3333 d22 − 2κ2333 d3 d2 + κ2233 d32 T 1 = −κ3333 d1 + κ1333 d3 T 2 = −κ3333 d2 + κ2333 d3

F.2 Constraints D23 =

3[p11 ]2 3[p22 ]2 3p11 p12 3p12 p22 p11 p22 + 2[p12 ]2 = = = = = X1111 X2222 X1112 X1222 X1122 =

3p11 s1 3p22 s2 p11 s2 + 2p12 s1 p22 s1 + 2p12 s2 = = = Y111 Y222 Y112 Y122

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ C33 1 ⎜⎜⎜ 2[s1 ]2 ⎟⎟⎟ 2s1 s2 ⎟⎟⎟ 2[s2 ]2 ⎟⎟⎟ 1 ⎜⎜⎜ 1 ⎜⎜⎜ ⎜⎝Z11 − ⎟⎠ = ⎜⎝Z12 − ⎟⎠ = ⎜⎝Z22 − ⎟⎠ = = D3 p11 p12 p22 D23 D23 D23 =

T2 T1 = 3s1 3s2

Appendix G

Axisymmetric stellar system The importance of the axial symmetry hypothesis arises from two of Chandrasekhar’s results (Chandrasekhar 1960, pp. 104-105). One result is that a stellar system in steady state with differential motions is characterised by an axis of helical symmetry. The other result is that if a stellar system is in steady state, the potential is axisymmetric. For this reason, it is worth studying a slightly more general situation of a non-steady state, axisymmetric system with differential motions not only in rotation, with a potential that may or may not be explicitly time dependent. In cylindrical coordinates, we note the star position and velocity as r = (, θ, z), V = (Π, Θ, Z), and the mean velocity of the stellar system as v = (Π0 , Θ0 , Z0 ). The radial direction is positive towards the Galactic anticentre, the rotation is positive in the direction of the Galactic rotation, and the vertical direction is positive towards the North Galactic Pole.

G.1 Components of A2 and v Under the axisymmetric hypothesis, by also assuming a symmetry plane for the mass and velocity distributions, the elements of the tensor A2 , which are solution of equations 3.23 and 3.24, are (we use the set of subindices {, θ, z} as in Sala 1990): A = k1 + k4 z2 , Azz = k3 + k4 2 ,

Aθθ = k1 + k2 2 + k4 z2 , Aθ = 0 , Az = −k4 z , 191

Aθz = 0.

(G.1)

192

APPENDIX G. AXISYMMETRIC STELLAR SYSTEM

Its determinant is |A| ≡ det A2 = (k1 + k2 2 + k4 z2 )(k1 k3 + k1 k4 2 + k3 k4 z2 ), where k1 , k3 are time dependent, positive functions, and k2 , k4 non negative constants1 . Similarly, the mean velocity v is obtained from equations 8.2 and 8.3 after solving Δ ans χ. The vector Δ has components Δ = 12 k˙ 1  ;

Δθ = −β ; Δz = 12 k˙ 3 z ;

(G.2)

and, by Eq. 8.1, the components of the mean velocity are Π0 =

 k˙1 k3 + k˙1 k4 2 + k˙3 k4 z2 , 2 k1 k3 + k1 k4 2 + k3 k4 z2

Θ0 = − Z0 =

β , k1 + k2 2 + k4 z2

(G.3)

z k˙3 k1 + k˙1 k4 2 + k˙3 k4 z2 2 k1 k3 + k1 k4 2 + k3 k4 z2

with β constant. The dots mean time derivatives. Therefore, a steady state system is only capable of differential rotation. In the case k ≡ k1 = k3 , we get 1 k˙ , 2k β Θ0 = − , k + k2 2 + k4 z2 ˙ 1k z. Z0 = 2k Π0 =

(G.4)

G.2 Second central moments According to Section 3.2, for a Schwarzschild velocity distribution, equations 3.2 and 3.3 are satisfied. Therefore, in general, the central moments 1 The model also provides two parameters, k , k , which depend on time and are generally 5 6 neglected. The first would determine a plane of symmetry for the velocity distribution at z = −k5 /k4 , while k6 would allow a non-symmetric mean velocity on either side of the Galactic plane.

G.2. SECOND CENTRAL MOMENTS

193

are obtained according to the following relationships μ2 = A−1 2

⎛ ⎜ A A − A2θz 1 ⎜⎜⎜⎜ θθ zz ⎜⎜ Aθz Az − Aθ Azz = |A| ⎜⎝ Aθ Aθz − Aθθ Az

Aθz Az − Aθ Azz A Azz − A2z Aθ Az − A Aθz

Aθ Aθz − Aθθ Az Aθ Az − A Aθz A Aθθ − A2θ

|A| = A Aθθ Azz − A A2θz − Aθθ A2z − Azz A2θ + 2 Aθ Aθz Az .

(G.5) Then, by Eq. G.1, the second central moment elements have the following functional dependency, 1 k3 + k4 2 , μθθ = , 2 2 k1 k3 + k1 k4  + k3 k4 z k1 + k2 2 + k4 z2 2 k1 + k4 z k4 z μzz = , μz = , 2 2 k1 k3 + k1 k4  + k3 k4 z k1 k3 + k1 k4 2 + k3 k4 z2 μ =

μθ = 0 ,

(G.6)

μθz = 0.

The tilt ϕ of the velocity ellipsoid, ut · μ−1 2 · u = 1, is the angle between one of the principal axes of the velocity ellipsoid and the radial direction. It satisfies 2k4 z 2μz = tan 2ϕ = . (G.7) μ − μzz k3 − k1 + k4 (2 − z2 ) In the case k ≡ k1 = k3 , the above relationship becomes tan 2ϕ =

2 z

1 − ( z )2

.

(G.8)

Then, ϕ = arctan( z ) or ϕ = arctan( z ) + π2 , so that one of the principal axes of the velocity ellipsoid points towards the Galactic centre. On the other hand, the vertex deviation ϑ of the velocity ellipsoid satisfies 2μθ (G.9) tan 2ϑ = μ − μθθ It is only considered the angle between the major semiaxis and the radial direction, since for a stellar system with net rotation, μ > μθθ is satisfied (see Eq. 8.47). If μθ = 0, as in the current axisymmetric case, the vertex deviation is null.

⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎠ ;

APPENDIX G. AXISYMMETRIC STELLAR SYSTEM

194

G.3 Moment gradients The  and z gradients of the central moments are ∂μ ∂

=

2k3 k42 z2 , (k1 k3 +k1 k4 2 +k3 k4 z2 )2

∂μθθ ∂

=

−2k2  , (k1 +k2 2 +k4 z2 )2

∂μzz ∂

=

−2k1 k4 (k1 +k4 z2 ) , (k1 k3 +k1 k4 2 +k3 k4 z2 )2

∂μz ∂

=

∂μθθ ∂z

k4 z(k1 k3 −k1 k4 2 +k3 k4 z2 ) , (k1 k3 +k1 k4 2 +k3 k4 z2 )2

∂μ ∂z

=

=

−2k3 k4 z(k3 +k4 2 ) , (k1 k3 +k1 k4 2 +k3 k4 z2 )2

−2k4 z , (k1 +k2 2 +k4 z2 )2

∂μzz ∂z

=

∂μz ∂z

(G.10)

2k1 k42 2 z , (k1 k3 +k1 k4 2 +k3 k4 z2 )2

=

k4 (k1 k3 +k1 k4 2 −k3 k4 z2 ) . (k1 k3 +k1 k4 2 +k3 k4 z2 )2

∂μz θθ ∂μzz Notice that, for z = 0, the values ∂μ ∂ , ∂ , and ∂z are expected to be nonnull, in general. In the Galactic plane, the kinematic parameters can be evaluated from the following relationships,   1 1 , k2 = 12 μ1θθ − μ , k1 = μ (G.11)   1 1 ∂μzz zz = − , k3 = μ1zz 1 + 2 μ1zz ∂μ , k 4 2 ∂ 2 μ ∂ zz

which, on the other side, provide the conditions μ > μθθ , zz − 2 ∂μ ∂ ≤ μzz .

∂μzz ∂

≤ 0, and

Appendix H

Epicycle model The epicycle approximation is a particular case of integration of the equations of motion under a minimum set of hypotheses allowing to obtain solutions for nearly circular orbits in the three dimensional space (e.g., Chandrasekhar 1960). In absence of collisions, the motion of a star is derived from the gravitational potential U, that in the current approach is assumed ∂U to be stationary and axially symmetric, i.e., ∂U ∂t = ∂θ = 0. In a cylindrical coordinates system, if we mark the star position as (, θ, z) and the velocity ˙ z˙), the equations of motion are written as of the star as (Π, Θ, Z) = (, ˙ θ, d2  dt2

= θ˙2 −

d 2˙ dt ( θ) d2 z dt2

∂U ∂

,

= − ∂U ∂θ ,

(H.1)

= − ∂U ∂z .

Two isolating integrals of motion exist. One is the energy integral I = Π2 + Θ2 + Z2 + 2 U(, z)

(H.2)

and the other one is the axial component of the angular momentum J = Θ = 2 θ˙ .

(H.3)

For a fixed integral of motion J, the energy integral may be written as I = Π2 + Z2 + 2 V(, z);

V(, z) = 195

J2 + U(, z) 22

(H.4)

APPENDIX H. EPICYCLE MODEL

196

where V(, z) is the effective potential energy. In addition, by assuming that) there is a Galactic plane of symmetry, z = 0, the potential satisfies ∂U )) ∂z z=0 = 0, which is equivalent to saying that U is a function even of z. Let us assume a star moving in a stable orbit on the Galactic plane z = 0 with a vertical velocity component z˙ = 0. The third equation of Eq. H.1, tells us that there is no acceleration in the vertical direction. Therefore, the motion of this star is restricted to the Galactic plane. We now fix the integral value J. By taking into account the energy integral of Eq. H.4 we have I= ˙ 2 + 2 V(, 0) .

(H.5)

Therefore, for each value of the energy integral I, the orbits, and, in particular, the values of  are constrained by the condition V(, 0) ≤

1 I. 2

In general, the equation V(, 0) = 12 I provides the extreme values of  for which  ˙ = 0, by delimiting a annular region p ≤  ≤ a where the motion takes place (e.g., Arnold 1989, p.35). In particular, by diminishing the value of I we reach a minimum value I = Ic , for which the pericentre and the apocentre coincide, say c ≡ p = a . In this case the orbit becomes circular, and satisfies  = c ,  ˙ = 0 where c is a local minimum satisfying ) ∂V(, 0) )) ) =0, (H.6) ∂ )c under the condition κ2 ≡

) ∂2 V(, 0) )) ) >0. ∂2 )c

(H.7)

To integrate the equations of motion for a circular orbit at radius  = c with no radial acceleration, we make use of the angular momentum integral Eq. (H.3) in the first expression of Eq. (H.1), and take into account the relation ∂V J2 ∂U =− 3 + . (H.8) ∂ ∂  Then, by introducing the condition given by Eq. (H.6) we get ) ∂V(, 0) )) ) =0.  ¨ =− (H.9) ∂ )c

H. EPICYCLE MODEL

197

Therefore, according to Eq. (H.8), for a star in circular motion in the plane z = 0 with angular momentum integral J = Jc , the radius c is obtained from the equality ) J2 ∂U(, 0) )) )) = c3 . (H.10) ∂ c c The condition of minimum of Eq. (H.7) is now given, by taking into account Eq. (H.10), as   3 ∂U(, 0) ∂2 U(, 0) 2 κ = + >0. (H.11)  ∂ ∂2 c The angular and circular velocities are constant, such that Jc θ˙c ≡ Ωc = 2 ; c

Θc =

Jc . c

(H.12)

According to Eq. (H.10) and Eq. (H.12), the value of the angular velocity is related to local properties of the potential as follows ) 1 ∂U(, 0) )) ) . (H.13) Ω2c = c ∂ )c Thus, an orbit in the Galactic plane with circular motion and constant angular velocity corresponds to a local minimum value of the energy integral J2 Ic = c2 + 2 U(c , 0) and other orbits with the same angular momentum c integral Jc are non-circular and have I > Ic energy integral. The epicycle approximation consists in to refer the orbit of a star with position (, θ, z) near the Galactic plane to a reference frame with centre in the position (c , θc , 0) of a star in the Galactic plane in circular motion with the same angular momentum integral Jc . It is the guiding centre C for which, given the position and velocity of a star, we may obtain c and Ωc which satisfy equations (H.10) and (H.13). For the first two coordinates, we write (H.14)  = c + ε , θ = θc + δ . Then, the orbit of the star is obtained by expanding the equations of motion up to the first order around the guiding centre C (e.g., Cubarsi et al. 2017). In the radial and axial components, it behaves as an harmonic oscillator around the guiding centre C with frequency κ, ε = a sin(κt − p) ,

δ=

2Ωc a cos(κt − p) . c κ

APPENDIX H. EPICYCLE MODEL

198

The positive value of κ is the planar epicycle frequency, and a and p are integration constants. In a similar way, the vertical component z of the star referred to the guiding centre C in the Galactic plane is obtained as z = b sin(νt − q) , being ν2 =

) ∂2 U )) ) >0, ∂z2 )c

(H.15)

where ν is the vertical epicycle frequency, and b and q integration constants. According the epicycle model, the local constants κ, ν, Ωv describe the stars motion in a nearly circular orbit around the guiding centre C at the Galactic plane. These are key values linked through equations H.11, H.13, and H.15 to local values of potential derivatives.

Appendix I

Point-axial symmetric system In the point-axial model the elements of the second-rank tensor A2 and the vector Δ have the following functional form (Juan-Zornoza et al. 1990, Juan-Zornoza & Sanz-Subirana 1991) A = K1 + K4 z2 ; Aθ = 12 (K1 + K4 z2 ) ; Az = −K4 z ; Aθθ = K1∗ + k2 2 + K4∗ z2 ; Aθz = − 12 K4 z ; Azz = k3 + K4 2 ;

(I.1)

Δ = 12 K˙ 1  ; Δθ = 14 (K˙ 1 − 4β) ; Δz = 12 k˙ 3 z ;

(I.2)

and

with K1 = k1 + q sin(2θ + ϕ1 ) ; K4 = k4 + n sin(2θ + ϕ2 ) ;

K1∗ = k1 − q sin(2θ + ϕ1 ) ; K4∗ = k4 − n sin(2θ + ϕ2 ) ;

(I.3)

where k1 , k3 , q, ϕ1 are time dependent functions and k2 , k4 , n, ϕ2 , β constants. The condition that A2 is positive-definite implies that k1 , k3 are positive functions and k2 , k4 non-negative constants, with the requirements k1 > q ≥ 0, k4 > n ≥ 0. If K4 is null, the velocity distribution is independent from z and, in addition, if k2 is null, then it is independent from  too, which makes no sense in a three-dimensional and finite Galaxy. Thus, in general, these constants are assumed to be positive, with the exception of the limiting case 199

200

APPENDIX I. POINT-AXIAL SYMMETRIC SYSTEM

K4 = 0 of a two-dimensional disc distribution1. The particular case k2 = 0 would correspond to a particular stellar component with constant angular rotation at fixed height z, similarly to the axisymmetric model. The uppercase letter K is used for a function also depending on θ. The accents mean derivatives with respect to the angle and the dots with respect to the time. As expected, the functional form of A2 is similar to the axisymmetric case in Paper I, with the difference that some parameters, those written in capital letters, have an additional term depending on cos 2θ and sin 2θ, responsible for the rotational symmetry of order 2. As in the axisymmetric case, the model also provides a time dependent parameter k5 that determines a plane of symmetry for the tensor A2 at z = −k5 /k4 . Without loss of generality (Camm 1941) this symmetry plane may be assumed as the Galactic plane, being fixed by taking k5 = 0, resulting then in a symmetric velocity distribution about this plane. Therefore, the point-axisymmetric model, with the inherent symmetry plane, also possesses point-to-point central symmetry.

1 In Section §8.4.2, the asymptotic case K → 0 was called flat velocity distribution, which 4 applies to the velocity distribution of an ideal disc. Although a disc stellar population can be approximated by this model, the other populations have a velocity distribution that must depend on z. Therefore, in general we must assume that K4 is nonnull.

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Index angular momentum integral, 6 asymmetric drift, 148, 149

distribution isothermal, 11, 24, 126, 130 isotropic, 11, 25, 61 modes, 63

Bayesian criterion, 78, 124 Bob Dylan, 123 Boltzmann collisionless equation, 3, 4, 8, 13, 15, 24, 26, 27, 31, 33, 38, 41, 43–45, 47–49, 52, 53, 61, 65, 78, 121, 122, 124

eccentricity, 78, 92–94 energy integral, 6 entropy functional, 44, 59, 61, 64 epicycle approximation, 93, 97, 145–148, 195, 197 Gram matrix, 49, 69 Greek indices, 10

centroid of motion, see local standard of rest centroid orbit, 123, 133, 135, 136 closure conditions, 4, 8, 21, 22, 24–26, 31–33, 36, 38, 41, 43, 44, 50, 124 continuity equation, 13, 21, 22, 27, 35, 38, 40, 173 covariance matrix, 11, 64, 78, 120, 124, 179 cumulants, 34, 40, 103, 107, 108, 180, 187 of a mixture, 103, 104, 112

Hamiltonian system, 3, 13 helicoidal symmetry axis, 122 hydrodynamic equations, 2, 4, 21–24, 27, 28, 31, 33, 35, 38, 39, 41, 43–45, 48–50, 52, 53, 60, 61, 66, 101, 124, 154, 169 information entropy, 44, 59, 63 uncertainty, 59, 64, 66 isolating integral of motion, 5, 7, 8, 28, 31, 44, 45, 65, 87, 124, 136

differential motion, 120, 122, 127, 130, 144, 191

Jeans 209

INDEX

210 direct problem, 4, 122 equation, 3, 21, 23, 35, 43 inverse problem, 4, 6, 8, 38, 45, 122, 123 kurtosis, 12, 32, 80, 84, 113 Lagrange multipliers, 64 subsidiary equations, 5 Liouville equation, 3 theorem, 3, 5, 65 local standard of rest, 92–94, 97, 98, 109, 110, 112 local standard of rest (LSR), 10, 12, 13, 16, 21, 39, 41, 104, 105, 120, 122–124, 173, 174 maximum entropy distribution, 44, 45, 58, 60–62, 65, 71, 120 moment equations, see hydrodynamic equations moments centred, 10–13, 22–24, 33, 35, 57, 60, 79, 80, 84, 120, 187 comoving, see centred extended set, 60, 72, 75 forbidden, 130, 131 generalised, 53, 63, 74 non-centred, 10, 13, 27, 49, 60, 66, 80, 84 of a mixture, 101, 103, 110, 112 momentum equation, see Jeans equation

Oort’s integral, 6 peculiar velocity, 10, 85, 115 transformed, 105 phase density, 3–5, 7, 15, 31, 33, 39, 41, 45, 50, 61, 77, 119, 121, 123, 124, 153 phase mixing, 12, 53, 128, 165 Poisson equation, 6, 8, 9, 21, 122, 138, 144, 164 pressures, tensor of, 11, 17, 20, 22, 35, 41, 68 skewness, 10, 12, 32, 80, 84, 113 stellar population, 6, 78, 102, 108, 117, 121–123 symmetrised tensor, 18–20, 34, 40, 47 temperatures, tensor of, 11, 27 tidal force, 133, 135, 164, 166 U-anomaly, 98 velocity distribution ellipsoidal, 7, 11, 31–33, 38, 40, 41, 43, 44, 51, 57, 79, 80, 101, 120, 121, 123, 124, 140, 141, 151–153 flat, 126, 127, 130, 137, 140, 142, 150, 159, 161, 163, 200 generalised Schwarzschild, 8, 31, 123, 154 Maxwellian, 7, 24, 32, 45, 61, 63–65 quadratic, see ellipsoidal Schwarzschild, 11, 24, 31–33, 38, 40, 41, 43, 57, 59, 61, 65, 78, 101,

INDEX 103, 104, 120–122, 124, 137, 145 trivariate Gaussian, see Schwarzschild velocity ellipsoid, 11, 109, 110, 123, 124, 126–128, 133, 136, 137, 140,

211 151, 153, 160 tilt, 12, 131, 143, 144, 161, 193 vertex deviation, 12, 109, 110, 120, 126, 135, 140, 141, 161, 193 Vlasov equation, 3