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An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-

Planck Equation for Core-Collapsing Isotropic Globular Clusters: Planck Equation for Core-Collapsing Isotropic Globular Clusters:

Properties and Application Properties and Application

Yuta Ito The Graduate Center, City University of New York

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AN ACCURATE SOLUTION OF THE SELF-SIMILAR ORBIT-AVERAGED

FOKKER-PLANCK EQUATION FOR CORE-COLLAPSING ISOTROPIC GLOBULAR

CLUSTERS: PROPERTIES AND APPLICATION

by

YUTA ITO

A dissertation submitted to the Graduate Faculty in Physics in partial fulfillment of the

requirements for the degree of Doctor of Philosophy, The City University of New York

2020

© 2020

YUTA ITO

All Rights Reserved

ii

An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-Planck Equation for

Core-Collapsing Isotropic Globular Clusters: Properties and Application

by

Yuta Ito

This manuscript has been read and accepted by the Graduate Faculty in Physics in satisfaction of

the dissertation requirement for the degree of Doctor of Philosophy.

Date Carlo Lancellotti

Chair of Examining Committee

Date Alexios P. Polychronakos

Executive Officer

Supervisory Committee:

Andrew C. Poje

Rolf J. Ryham

Tobias Schafer

Mark D. Shattuck

THE CITY UNIVERSITY OF NEW YORK

iii

ABSTRACT

An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-Planck Equation for

Core-Collapsing Isotropic Globular Clusters: Properties and Application

by

Yuta Ito

Adviser: Carlo Lancellotti

Hundreds of dense star clusters exist in almost all galaxies. Each cluster is composed of

approximately ten thousand through ten million stars. The stars orbit in the clusters due to the

clusters’ self-gravity. Standard stellar dynamics expects that the clusters behave like collisionless

self-gravitating systems on short time scales (∼million years) and the stars travel in smooth contin-

uous orbits. Such clusters temporally settle to dynamically stable states or quasi-stationary states

(QSS). Two fundamental QSS models are the isothermal- and polytropic- spheres since they have

similar structures to the actual core (central part) and halo (outskirt) of the clusters. The two QSS

models are mathematically modeled by the Lane-Emden equations. On long time scales (∼ bil-

lion years), the clusters experience a relaxation effect (Fokker-Planck process). This is due to the

finiteness of total star number in the clusters that causes stars to deviate from their smooth orbits.

This relaxation process forms a highly-dense relaxed core and sparse-collisionless halo in a self-

similar fashion. The corresponding mathematical model is called the self-similar Orbit-Averaged

Fokker-Planck (ss-OAFP) equation. However, any existing numerical works have never satisfac-

torily solved the ss-OAFP equation last decades after it was proposed. This is since the works rely

on finite difference (FD) methods and their accuracies were not enough to cover the large gap in

the density of the ss-OAFP model. To overcome this numerical problem, we employ a Chebyshev

pseudo-spectral method. Spectral methods are known to be accurate and efficient scheme com-iv

pared with FD methods. The present work proposes a new method by combining the Chebyshev

spectral method with an inverse mapping of variables.

Our new method provides accurate numerical solutions of the Lane-Emden equations with

large density gaps on MATLAB software. The maximum density ratio of the core to halo can

reach the possible numerical (graphical) limit of MATLAB. The same method provides four sig-

nificant figures of a spectral solution to the ss-OAFP equation. This spectral solution infers that

existing solutions have at most one significant figure. Also, our numerical results provide three

new findings. (i) We report new kinds of the end-point singularities for the Chebyshev expansion

of the Lane-Emden- and ss-OAFP equations. (ii) Based on the spectral solution, we discuss the

thermodynamic aspects of the ss-OAFP model and detail the cause of the negative heat capac-

ity of the system. We suggest that to hold a ’negative’ heat capacity over relaxation time scales

stars need to be not only in a deep potential well but also in a non-equilibrium state with the flow

of heat and stars. (iii) We propose an energy-truncated ss-OAFP model that can fit the observed

structural profiles of at least half of Milky Way globular clusters. The model can apply to not

only normal clusters but also post collapsed-core clusters with resolved (observable) cores; those

clusters can not generally be fitted by a single model. The new model is phenomenological in the

sense that the energy-truncation is based on polytropic models while the truncation suggests that

low-concentration globular clusters are possibly polytropic clusters.

v

ACKNOWLEDGMENTS

Part of the formulations in Chapter 4, the numerical arrangements in Chapter 5, and the nu-

merical results in Chapters 6 and 7 are based on the following published paper. (*This work was

supported by ONR DRI Grant No. 0604946 and NSF DMS Award No. 1107307.)

Ito, Y., Poje, A., and Lancellotti, C. (2018). ”Very-large-scale spectral solutions for spherical

polytropes of index m > 5 and the isothermal sphere.” New Astronomy, 58: 15-28.

DOI: https://doi.org/10.1016%2Fj.newast.2017.07.003

Part of the formulations in Chapter 4, the numerical arrangements in Chapter 5, and the numerical

results in Chapter 8 are based on the following accepted paper.

Ito, Y. (2021). ”Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical

dense star clusters (i) accurate pre-collapse solution.” New Astronomy, 83:101474.

DOI: https://doi.org/10.1016/j.newast.2020.101474

vi

Contents

List of Tables xv

List of Figures xix

1 Introduction 1

1.1 A quantitative understanding of dense star clusters . . . . . . . . . . . . . . . . . 2

1.1.1 The physical features of globular clusters . . . . . . . . . . . . . . . . . . 2

1.1.2 Dynamical time and mean field Newtonian potential . . . . . . . . . . . . 5

1.1.3 Relaxation time and two-body relaxation (’collision’) process . . . . . . . 6

1.2 The interdisciplinary importance of stellar-dynamics studies . . . . . . . . . . . . 7

1.2.1 Astronomical importance . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Statistical-physics importance . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Computational-physics importance . . . . . . . . . . . . . . . . . . . . . . 9

1.3 A basic mathematical model: Fokker-Planck (FP) kinetics of isotropic spherical

star clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Distribution function for stars . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2 Collisionless relaxation evolution on dynamical time scales . . . . . . . . . 12

1.3.3 Collisional relaxation evolution and Orbit-Averaged Fokker-Planck (OAFP)

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 The motivation to apply spectral methods to the OAFP equation . . . . . . . . . . 17

1.4.1 Numerical difficulty in solving OAFP equation . . . . . . . . . . . . . . . 18

vii

CONTENTS

1.4.2 Spectral method and (semi-)infinite domain problem . . . . . . . . . . . . 19

1.5 The state-of-the-art and organization of the present dissertation . . . . . . . . . . . 21

2 Mathematical models for the evolution of globular clusters 25

2.1 Quasi-stationary-state star cluster models . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Isothermal sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.2 Polytropic sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1.3 King model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Self-similar Orbit-Averaged Fokker-Planck (ss-OAFP) model . . . . . . . . . . . . 40

2.2.1 Self-similar analysis on the OAFP equation . . . . . . . . . . . . . . . . . 41

2.2.2 Difficulties in the numerical integration of the ss-OAFP equation . . . . . . 44

3 Gauss-Chebyshev pseudo-spectral method and Fejer’s first-rule Quadrature 46

3.1 Gauss-Chebyshev polynomials expansion . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Continuous Chebyshev polynomial expansion (degree truncation) . . . . . 47

3.1.2 Gauss-Chebyshev polynomial expansion (domain discretization) . . . . . . 48

3.2 Truncation-error bound and the geometric convergence of Chebyshev-polynomial

expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 An example of degree-truncation error . . . . . . . . . . . . . . . . . . . . 51

3.2.2 The effects of the mathematical structures of functions on Chebyshev co-

efficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 The convergence rates of Chebyshev coefficients . . . . . . . . . . . . . . . . . . . 56

3.4 An example of Gauss-Chebyshev pseudo-spectral method applied to the King model 59

3.4.1 The mathematical reformulation of the King model . . . . . . . . . . . . . 60

3.4.2 Matrix-based Newton-Kantrovich iteration method (Quasi-Linearization) . 61

3.4.3 Spectral solutions for the King model and unbounded-domain problems . . 65

3.5 Fejer’s first-rule quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

viii

CONTENTS

3.5.1 Fejer’s first-rule Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5.2 The convergence rate of the Fejer’s first-rule quadrature . . . . . . . . . . . 73

4 The inverse forms of the Mathematical models 75

4.1 The inverse mapping of dimensionless potential and density . . . . . . . . . . . . . 75

4.2 Isothermal sphere (LE2 function) . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.1 Inverse mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.2 Regularized ILE2 equation . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Polytropic sphere (LE1 function) . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.1 Inverse mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.2 Regularized ILE1 equations . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 The ss-OAFP system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4.1 The inverse form of the Poisson’s equation . . . . . . . . . . . . . . . . . . 81

4.4.2 Regularized ss-OAFP system . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.3 Integral formulas on the whole- and truncated- domains . . . . . . . . . . . 85

5 Numerical arrangements and optimization on the spectral methods 88

5.1 The numerical treatments of the ILE1- and ILE2- equations . . . . . . . . . . . . . 88

5.2 The numerical arrangements of the regularized ss-OAFP system . . . . . . . . . . 89

5.2.1 Step 1: Solving the regularized Poisson’s equation . . . . . . . . . . . . . 90

5.2.2 Step 2: Solving the regularized 4ODEs and Q-integral . . . . . . . . . . . 90

5.2.3 Step 3: Repeating steps 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.4 Numerical arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Preliminary Result 1: A spectral solution for the isothermal-sphere model 94

6.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 The topological property and asymptotic approximation of the LE2 function . . . . 98

ix

CONTENTS

6.2.1 Milne variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2.2 Asymptotic semi-analytical singular solution . . . . . . . . . . . . . . . . 99

7 Preliminary result 2: Spectral solutions for the polytropic-sphere model 102

7.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2 The topological property and asymptotic approximation of the LE1 function . . . . 106

8 Main result: A spectral solution for the ss-OAFP model 110

8.1 A self-similar solution on the whole domain . . . . . . . . . . . . . . . . . . . . . 111

8.1.1 Numerical results (the main results of the present chapter) . . . . . . . . . 111

8.1.2 The Chebyshev coefficients for the reference solution and regularized func-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.1.3 The instability of the whole-domain solution . . . . . . . . . . . . . . . . . 117

8.2 Self-similar solutions on truncated domain . . . . . . . . . . . . . . . . . . . . . . 117

8.2.1 Stable solutions on truncated domains with −0.35 ≤ Emax ≤ −0.05 . . . . . 118

8.2.2 Optimal semi-stable solutions on truncated domains with −0.1 ≤ Emax ≤

−0.03 for fixed β = βo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.3 Discussion: Reproducing the HS’s solution . . . . . . . . . . . . . . . . . . . . . 125

8.3.1 Reproducing the HS’s solution and eigenvalues with limited degrees . . . . 126

8.3.2 Successfully reproducing the HS’s solution and accuracy of the reference

solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

9 Property: The thermodynamic and dynamic aspects of the ss-OAFP model 130

9.1 Statistical mechanics of self-gravitating systems . . . . . . . . . . . . . . . . . . . 131

9.1.1 Isotropic self-gravitating models employed for statistical mechanics studies

and assumption made for the ss-OAFP model . . . . . . . . . . . . . . . . 131

9.1.2 The cause of negative heat capacity reported in the previous works . . . . . 133

x

CONTENTS

9.2 The thermodynamic quantities of the ss-OAFP model . . . . . . . . . . . . . . . . 135

9.2.1 The local properties of the ss-OAFP model . . . . . . . . . . . . . . . . . 135

9.2.2 The global properties of the ss-OAFP model . . . . . . . . . . . . . . . . . 140

9.3 Normalized thermodynamic quantities and negative heat capacity . . . . . . . . . . 143

9.3.1 Normalized thermodynamic quantities . . . . . . . . . . . . . . . . . . . . 143

9.3.2 Caloric curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.4 The cause of negative heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 151

9.4.1 Virial and total energy to discuss heat capacity at the center of the core . . . 152

9.4.2 The cause of negative heat capacity compared with the previous works . . . 155

9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10 Application: Fitting the energy-truncated ss-OAFP model to the projected struc-

tural profiles of Galactic globular clusters 160

10.1 Energy-truncated ss-OAFP model . . . . . . . . . . . . . . . . . . . . . . . . . . 162

10.1.1 The relationship of the ss-OAFP model with the PCC clusters and isother-

mal sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

10.1.2 Energy-truncated ss-OAFP model . . . . . . . . . . . . . . . . . . . . . . 165

10.1.3 The regularization of the concentration and King radius . . . . . . . . . . . 170

10.1.4 Determining an optimal value of index m . . . . . . . . . . . . . . . . . . 172

10.2 Fitting the ss-OAFP model to Galactic PCC clusters . . . . . . . . . . . . . . . . . 177

10.3 (Main result) The relaxation time and completion rate of core collapse against the

concentrations of the clusters with resolved cores . . . . . . . . . . . . . . . . . . 179

10.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

10.4.1 Fitting the energy-truncated OAFP model to‘whole’ projected density pro-

files with higher indices m . . . . . . . . . . . . . . . . . . . . . . . . . . 185

xi

CONTENTS

10.4.2 Application limit and an approximated form of the energy-truncated ss-

OAFP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

10.4.3 Are low-concentration globular clusters like spherical polytropes? . . . . . 191

10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

11 Conclusion 201

Appendix A The basic kinetics and scaling of physical quantities for stellar dynamics 206

A.1 The N-body Liouville equation for stellar dynamics . . . . . . . . . . . . . . . . . 206

A.2 The s-tuple distribution function and correlation function of stars . . . . . . . . . . 208

A.3 The scaling of the orders of the magnitudes of physical quantities to characterize

stars’ encounter and star cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

A.4 The trajectories of a test star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Appendix B The derivation of the BBGKY hierarchy for a weakly-coupled distribu-

tion function 219

B.1 BBGKY hierarchy for the truncated DF . . . . . . . . . . . . . . . . . . . . . . . 219

B.1.1 Truncated integral over the terms∑N

i=1 vi · ∇iFN(1, · · · ,N, t) . . . . . . . . . 222

B.2 BBGKY hierarchy for the weakly-coupled DF . . . . . . . . . . . . . . . . . . . . 227

B.2.1 Weakly-coupled DF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

B.2.2 Truncation Condition for the weakly-coupled DF . . . . . . . . . . . . . . 229

B.2.3 The approximated form of the weakly-coupled DF . . . . . . . . . . . . . 231

Appendix C The derivations of the Coulomb logarithm and an FP equation 232

C.1 The derivation of the basic kinetic equation . . . . . . . . . . . . . . . . . . . . . 232

C.2 Coulomb Logarithm and Landau (FP) kinetic equation . . . . . . . . . . . . . . . 237

Appendix D Singularity analysis on the Lane-Emden functions 240

xii

CONTENTS

D.1 The effect of the parameter L on the LE functions . . . . . . . . . . . . . . . . . . 240

D.2 The asymptotics of Chebyshev coefficients in the presence of logarithmic endpoint

singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

D.3 Asymptotic coefficients for a decaying logarithmic oscillatory function . . . . . . . 246

Appendix E The Milne forms of the Lane-Emden equations 248

E.1 The Milne form of the LE1 equation . . . . . . . . . . . . . . . . . . . . . . . . . 248

E.2 The Milne form of the LE2 equation . . . . . . . . . . . . . . . . . . . . . . . . . 249

Appendix F Numerical Stability in the integration of the ss-OAFP system 251

F.1 The stability analyses of the whole-domain solution . . . . . . . . . . . . . . . . . 251

F.1.1 The stability of the whole-domain solution against β . . . . . . . . . . . . 251

F.1.2 The stability of the whole-domain solution against FBC . . . . . . . . . . . 252

F.1.3 The stability of the whole-domain solution against L . . . . . . . . . . . . 253

F.2 The stability of the truncated-domain solution against extrapolated DF . . . . . . . 255

F.3 Solving part of the ss-OAFP system with a fixed independent variable . . . . . . . 256

F.3.1 Q-integral with fixed discontinuous 3R . . . . . . . . . . . . . . . . . . . . 256

F.3.2 Poisson’s equation with fixed discontinuous 3D . . . . . . . . . . . . . . . 257

Appendix G Relationship between the reference- and Heggie-Stevenson’s- solutions of

the ss-OAFP system 259

G.1 Modifying the regularization of variable 3F . . . . . . . . . . . . . . . . . . . . . 259

G.2 Modified variables (3(m)R , 3(m)

F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Appendix H The normalization of total energy 263

Appendix I The fitting of the finite ss-OAFP model to the King-model clusters 265

I.1 KM cluster (Miocchi et al., 2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

xiii

CONTENTS

I.2 KM clusters (Kron et al., 1984) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

I.3 KM clusters (Noyola and Gebhardt, 2006) . . . . . . . . . . . . . . . . . . . . . . 276

Appendix J The fitting of the polytropic sphere model to low-concentration clusters 280

J.1 Polytropic clusters (Miocchi et al., 2013) and (Kron et al., 1984) . . . . . . . . . . 280

J.2 Polytropic clusters (Trager et al., 1995) . . . . . . . . . . . . . . . . . . . . . . . . 282

Appendix K The fitting of the finite ss-OAFP model to the PCC clusters 287

K.1 PCC? clusters (Kron et al., 1984) . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

K.2 PCC clusters (Lugger et al., 1995) and (Djorgovski and King, 1986) . . . . . . . . 288

K.3 PCC clusters (Noyola and Gebhardt, 2006) . . . . . . . . . . . . . . . . . . . . . . 292

Appendix L MATLAB code 295

Bibliography 322

xiv

List of Tables

1.1 Some recent spectral-method studies on (integro-) ordinary differential equations

on (semi-)infinite domain. The spectral methods are collocations methods unless

they are specified with other approaches (e.g., tau. Galerkin, operational matrix,...).

Due to enormous publications for the Lane-Emden equations, the list only includes

part of publications between 2013-2019. . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 Chebyshev coefficients an for test function h(1)(x) = (0.5 + 0.5x)8.2 with degree

N = 30. The coefficients an are divided by a1 so that the first coefficient is unity. . 52

3.2 Asymptotic Chebyshev coefficients an (n → ∞) for functions h(x) that have non-

regular and/or non-analytic properties. The ψ is a nonintegral real number and k is

an integer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Ratio of the tidal radius to the King radius for different K between 0.00001 and

15. The ratio is rescaled by multiplying the ratio by 3/√

K to compare the ratio to

the polytrope model. As K decreases, the rescaled ratio 3rtid/rKin/√

K gradually

approaches 5.35528 that is the tidal radius of the spherical polytrope of index m =

2.5 (Boyd, 2011). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xv

LIST OF TABLES

6.1 Comparison of the spectral LE2 solution with the singular, Horedt’s, and Wazwaz’s

solutions. Our solution’s first seven digits that match Horedt’s solutions are under-

lined. The comparison of our solution with the other two is also shown using

underlines. The spectral solution was obtained by choosing combinations of L and

N such that the smallest absolute Chebyshev coefficient reaches machine precision. 97

8.1 Comparison of the present eigenvalues and physical parameters with the results of

’HS’ (Heggie and Stevenson, 1988) and ’T’ (Takahashi, 1993). The relative error

between the HS’s eigenvalues and the present ones are also shown. The present

eigenvalues are based on the results for various combinations of numerical param-

eters (13 ≤ N ≤ 560, 10−4 ≤ FBC ≤ 104, and L = 1/2, 3/4, and 1), different

formulations (Sections 8.2 and 8.3 and Appendix F.1.3) and stability analyses (Ap-

pendix F.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.2 Semi-analytical forms of F(E), R(Φ), and Q(E) obtained from the whole-domain

spectral solution (N = 70, L = 1, and FBC = 1). The relative error of the semi-

analytic form from the whole-domain solution is the order of 10−4. The coefficients

Q′

n are the Chebyshev coefficients for 3Q/(0.5 − 0.5x), and were introduced to

make the accuracy of Q(E) comparable to those of F(E) and R(Φ). . . . . . . . . . 114

8.3 Eigenvalues of the reproduced HS’s solution with eigenvalues c1 = 9.10×10−4 and

c4 = 3.52 × 10−2 for N = 15. Heggie and Stevenson (1988) reported the numerical

values of their solution on −1 / E ≤ −0.317. They mentioned that the Newton

iteration method worked up to Emax ≈ −0.223 and it could work beyond -0.223. . . 127

9.1 Typical isotropic self-gravitating systems discussed for study on equilibrium sta-

tistical mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xvi

LIST OF TABLES

9.2 First turning points of caloric curves at small radii. The data for IS are the cor-

responding values for the isothermal sphere (e.g., Antonov, 1962; D. Lynden-Bell

and Royal, 1968; Padmanabhan, 1989; Chavanis, 2002a). . . . . . . . . . . . . . . 151

10.1 Concentration and core- and tidal- radii obtained from the energy-truncated ss-

OAFP model. The structural parameters are compared with the previous compila-

tion works based on the King- and Wilson- models. . . . . . . . . . . . . . . . . . 174

10.2 Values of χ2ν and number of points discarded from the calculation. The data used

for the fitting to the KM clusters are from (Miocchi et al., 2013), NGC 6397 from

(Drukier et al., 1993), Terzan 2 from (Djorgovski and King, 1986), NGC 6752 from

(Ferraro et al., 2003), and the rest of the PCC clusters from (Lugger et al., 1995).

Nb is the number of data points at large radii excluded from calculation. . . . . . . 180

10.3 Core relaxation times and ages of polytropic- and non-polytropic- clusters. The

current relaxation time tc.r. and concentration c are values reported in (Harris, 1996,

(2010 edition)). The total masses log[

MMo

]are from (Mandushev et al., 1991) where

Mo is solar mass. We adapted the ages of clusters from (Forbes and Bridges, 2010)

and we resorted to other sources when we found more recent data or could not find

the cluster age in (Forbes and Bridges, 2010). . . . . . . . . . . . . . . . . . . . . 197

A.1 A scaling of the order of magnitudes of physical quantities associated with the

evolution of a star cluster that has not gone through a core-collapse. The scaling

can be useful for systems whose density contrast is much less than the order of

N. Hence, we expect the cluster to be normal rather PCC. (Recall gravothermal

instability occurs with density contrast of 709 for the isothermal sphere (Antonov,

1962).). The OoMs are scaled by N and r12 except for the correlation time tcor,

which needs the change in velocity, δva

(=

∫a12 dtcor

), due to Newtonian two-body

interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

xvii

LIST OF TABLES

A.2 A scaling of physical quantities according to the effective interaction range of New-

tonian interaction accelerations and close encounter. . . . . . . . . . . . . . . . . . 214

F.1 Numerical results for the contracted-domain formulation with with L = 1/2 and

L = 3/4 for FBC = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

F.2 Numerical results for different extrapolated DF (L = 1 and FBC = 1). The eigen-

values are compared with c1ex ≡ c1o and c∗4ex ≡ 3.03155223×10−1(= c∗4o +1×10−1)

obtained for (c, d) = (1, 10) and the value of 3I(x = 1) is 7.3 × 10−12. The combi-

nation (c, d) = (∞,N/A) means the extrapolated function is constant. . . . . . . . . 255

G.1 Eigenvalues and 3I(x = 1) for combinations of 3(m)R and 3(m)

F . The upper three rows

are the data that reproduced the HS’s eigenvalues while the lower three rows are

the data that reproduced three significant figures of the reference eigenvalues. In

the modified ss-OAFP systems, the variables 3S , 3Q, 3G, 3I , and 3J are not modified . 261

I.1 Core- and tidal radii of the finite ss-OAFP model applied to the projected density

profiles reported in (Miocchi et al., 2013). . . . . . . . . . . . . . . . . . . . . . . 270

xviii

List of Figures

1.1 (Top) Schematic image of globular clusters in Milky way and (bottom) picture

”False-color image of the near-infrared sky as seen by the DIRBE” (credit:(NASA,

Goddard)). The d in the top panel is the diameter of object concerned. The actual

number of observed globular clusters is approximately 160 in Milky Way (For

example, the number is reported as 157 in (Harris, 1996, (2010 edition))’s catalog).

The Galactic center and disc are composed of a plentiful of gases, clouds of dust

and stars. Our Sun lies on the Galactic disc. Many globular clusters exist in the

Galactic halo in which stars and gaseous components are less found. . . . . . . . . 3

1.2 Globular cluster NGC362 (credit: NASA, Goddard) as an example of globular clus-

ters. The mass of NGC362 is 3.8 × 105 solar masses, the core radius 0.42 pc and

tidal radius 37 pc (Boyer et al., 2009). Refer to Chapter 5 for the core- and tidal

radii of a globular cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

xix

LIST OF FIGURES

1.3 Concept of the encounter and two-body relaxation. On time scales much larger

than dynamical time, ’test’ star may approach a ’field’ star closer than the average

distance 1/n1/3. One can characterize this encounter by the impact parameter b

and initial velocity (dispersion speed) 3 in analogy with gaseous kinetic theory.

During a single encounter, the test star gains small perpendicular velocity δ3 since

the potential due to the field star deflects test star from its smooth initial orbit. The

encounter may successively occur between the test star and many field stars. On

a long time scale, accumulation of velocity change∑|δ3| reaches the order of the

initial velocity 3. At the time, the trajectory of test star largely deviates from the

original orbit and test star ’forgets’ its initial orbit. This process is the encounter of

stars and sometimes termed the velocity relaxation. . . . . . . . . . . . . . . . . . 7

1.4 Schematic graph for the Virial ratio against dimensionless time for an isolated self-

gravitating system. Time t is normalized by the dynamical time tdyn. The ratio

-2KE/PE is the Virial ratio. If the system is dynamically stationary, the Virial (=-

2KE+PE) is zero according to the Virial theorem. The graph shows the system

is not dynamically stationary at t = 0. The Virial ratio reaches unity after a few

dynamical times due to the violent relaxation and mixing. . . . . . . . . . . . . . . 14

3.1 Absolute error between the test function h(1)(x) and degree-truncated one h(1)N

(x).

On the graph, the graphing points are taken at points on [−1, 1] with the step size

10−1. Although the Gauss-Chebyshev nodes can not reach x = ±1, one still may

extrapolate the function h(1)N

(x) at the points. The dashed horizontal lines show the

absolute values of the Chebyshev coefficients aN+1 that descends with degrees

N + 1 = 6, 11, 16, 21, 26, corresponding to |aN+1/a1| =9.5 × 10−2, 1.2 × 10−7, 1.5 ×

10−11, 5.9 × 10−14,6.7 × 10−16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Functions h(1)(x), h(2)(x), h(3)(x), and h(4)(x) on Log-Linear scale. . . . . . . . . . . 55

xx

LIST OF FIGURES

3.3 Chebyshev coefficients an for test functions with N = 30. The circle- and square-

marks corresponds to the Chebyshev coefficients for (a) functions h(1)(x) and h(2)(x)

and for (b) functions h(3)(x) and h(4)(x), respectively. The solid- and dashed- lines

(that were artificially determined ’by eyes’.) represent the convergence rates of

Chebyshev coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Spectral solution (dimensionless m.f. potential φ(r)) of the King model for K =

0.01, 1, 4, 8, 12, and 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Absolute values of the Chebyshev coefficients on (a) log-log scale and (b) log-

linear scale for the regularized spectral King model φ(r) of K = 1 and K = 0.01

(N = 100). The figures (a) and (b) include guidelines to depict power-law- and

geometrical- convergences. The mk = 2.5 in the power-law guideline can be deter-

mined from equation (2.1.35). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.6 Absolute values of Chebyshev coefficients on log-linear scale for φ(r) with K =

4, 8, 12, 15. The dashed lines and exponential expressions in terms of n on the

figure are references to present the geometrical convergences of the coefficients. . . 69

3.7 The ratio of the tidal radius to the King radius. . . . . . . . . . . . . . . . . . . . . 69

6.1 LE2 function φ (N = 225 and L = 36.) . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Absolute values of the Chebyshev coefficients for the regularized ILE2 spectral

solution 3r (N = 225, L = 36). Round-off error appears for n & 200. . . . . . . . . 96

6.3 Optimal value of the mapping parameter L for the isothermal sphere (N = 100).

The graph compares φN to A ≡ ln (r2/2) at cutoff radius rcut. The rcut was intro-

duced to avoid numerical divergences at r → ∞ when MATLAB reaches machine

precision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

xxi

LIST OF FIGURES

6.4 (a) Milne phase space (u, v) for the isothermal sphere (L = 36 and N = 225) and

(b) its magnification around (1, 2). The curve begins with (uE, vE) = (3, 0) and

asymptotically reaches the singular point (us.p, vs.p) = (1, 2). . . . . . . . . . . . . . 99

6.5 (a) Asymptotic oscillation of LE2 spectral solutions,(φN − ln

[r2/2

])r1/2, com-

pared to existing semi-analytical solutions for N = 225, L = 36. (b) Relative

deviation of our semi-analytical asymptotic singular solution, φa, from the LE2

spectral solution φN. The φSA is the semi-analytical asymptotic LE2 solution of

(Soliman and Al-Zeghayer, 2015) and φs = ln r2

2 . The plot of | 1 − φN/φa | flattens

around at r = 1013 because the calculation reaches machine precision. . . . . . . . 101

7.1 LE1 function φN, for m = 6, 10 and 20 (N = 300.) . . . . . . . . . . . . . . . . . . 103

7.2 Absolute values of Chebyshev coefficients for the regularized spectral ILE1 so-

lution 3r for m = 6, 10 and 20 (N = 300). For (m, L) = (20, 2) the data plots

flatten out at large n due to the round-off error; for (m, L) = (10, 5) the decay rate

decreases around N = 250 due to the end-point singularity. . . . . . . . . . . . . . 104

7.3 Optimal values of the mapping parameter L for m = 6, 10 and 20 (N = 200). The

calculated value of φ at the cutoff Chebyshev-Radau point is compared with the

corresponding A = φs for polytropes. The cutoff radii rcut were chosen to avoid

divergences at r → ∞ when MATLAB reaches machine precision. . . . . . . . . . 104

7.4 Comparison between the spectral solutions, φN, and Horedt’s solutions, φH. . . . . 105

7.5 Relative error of the spectral solutions, φN, from Horedt’s solutions, φH for different

N. The degree N was increased until the spectral solutions ensured seven-digit

accuracy for (a) m= 6, (b) m = 10 and (c) m = 20. . . . . . . . . . . . . . . . . . . 106

7.6 (a) Trajectory in the Milne (u, v) plane for m = 20 and L = 2 with N = 300 and

(b) its magnification near (m−3m−1 , ω). The solution starts from (uE, vE) = (3, 0) and

asymptotically approaches the singular point (us.p, vs.p) =(

m−3m−1 , ω

). . . . . . . . . . 108

xxii

LIST OF FIGURES

7.7 Asymptotic oscillations of the spectral solutions (N = 300) for (a) m = 20 and (b)

m = 6. For large r, rα(rωφN/Am − 1

)≈ C1 cos (ω ln r + C2). . . . . . . . . . . . . . 109

8.1 (a) Distribution function F(E) of stars on the whole domain and (b) its magnified

graph on −1 ≤ E < −0.25. (N = 70, FBC = 1, and L = 1.) . . . . . . . . . . . . . . 112

8.2 Absolute values of the normalized Chebyshev spectral coefficients for the reference

solutions. (N = 70, FBC = 1, and L = 1.) The coefficients are divided by their first

coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.3 Regularized spectral solution compared with the corresponding HS’s solution. (N =

70, FBC = 1, and L = 1.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.4 (a) Relative error between c4 and c∗4o, and the absolute value of 3I(x = 1), (b)

Condition number of the Jacobian matrix for the 4ODEs and Q-integral. The con-

dition number was computed when aqold − aq

new reached the order of 10−13 in

the Newton iteration process. (L = 1 and FBC = 1.) . . . . . . . . . . . . . . . . . 117

8.5 (a) Characteristics of | 3I(x = 1) | and relative errors of c1 and c∗4 from their refer-

ence eigenvalues for different Emax. (N = 40, β = 8.1783, FBC = 1, and L = 1.) (b)

Condition number of the Jacobian matrix calculated when aqnew − aq

old reached

the order of 10−13 in the Newton iteration. . . . . . . . . . . . . . . . . . . . . . . 119

8.6 (a) Relative errors of c1 and c∗4 from the reference eigenvalues, and the absolute

value of 3I(x = 1) for different N. (b) Chebyshev coefficients for 3F with N = 50

and N = 360. (Emax = −0.225 and β = 8.17837. ) . . . . . . . . . . . . . . . . . . 120

8.7 Relative errors of c1 and c∗4 from the reference eigenvalues and characteristics of

3I(x = 1) against N for the truncated-domain solutions with β = βo. (−0.1 ≤

Emax ≤ −0.03, L = 1, and FBC = 1.) . . . . . . . . . . . . . . . . . . . . . . . . . 122

xxiii

LIST OF FIGURES

8.8 Relative errors of c1 and c∗4 from the reference values and characteristics of | 3I(x =

1) | obtained at each Nbest. The guideline c1o(−Emax)βo/c∗4o is shown for the sake of

comparison. (L = 1, FBC = 1, and β = βo.) . . . . . . . . . . . . . . . . . . . . . . 124

8.9 (a) Relative errors of DF F(N)(E) with different N from the DF F(N=65)(E) with

Emax = −0.03. (L = 1 and FBC = 1) (b) Relative error between the reference DF

Fo(E) and the optimal truncated-domain DF. (L = 1, FBC = 1, and β = βo.) . . . . . 125

8.10 Relative error between DFs obtained from the HS’s solution ln[FHS] and spectral

solution ln[F] with N = 15 for Emax. . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.11 Relative errors of 3(m)R and 3F from 3Ro and 3Fo respectively. (Emax = −0.05, N =

540, L = 1, and FBC = 1.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

9.1 (a) Dimensionless m.f. potential Φ and velocity dispersion V2(dis) and (b) dimen-

sionless heat flux Fh and stellar flux Fp. The latter also depicts the inverse of

regularized local relaxation time TR

(= D(φ, tc)/[32(r, tc)]3/2

)for the sake of com-

parison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.2 (a) Adiabatic index Γ and (b) polytropic index m of the ss-OAFP model. . . . . . . 139

9.3 Local equation of state with the constant thermodynamic temperature 1/χesc. . . . . 139

9.4 Local and kinetic temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.5 Escape speeds of stars. The stellar flux FP and velocity dispersion√

V2(dis) are also

depicted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.6 Dimensionless normalized total energies for the ss-OAFP model and isothermal

sphere. As RM → 0, the energy monotonically decreases for the former and in-

creases for the latter. The curve for the isothermal sphere was obtained using the

numerical code in (Ito et al., 2018). . . . . . . . . . . . . . . . . . . . . . . . . . . 146

xxiv

LIST OF FIGURES

9.7 Dimensionless temperatures of the ss-OAFP model and temperature 1/η of the

isothermal sphere. The curve for the latter was obtained using the numerical code

in (Ito et al., 2018) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9.8 Caloric curves for the dimensionless- total energy Λ and temperatures (a) η(kin),

(b) η(con), and (c) η(loc). (d) Magnification of (c) around the turning point. All the

curves start at (0, 0) that corresponds to RM = 0. For graphing, the ordinates are a

reciprocal of Λ. Hence, the tangents to the curves represent the sign of CV. . . . . . 150

9.9 Ratio of -PE/KE to χesc/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.10 Dimensionless m.f. potential Φ and (a) total number of the stars in the ss-OAFP

model enclosed by an adiabatic wall at RM and (b) normalized mean energy per

unit mass of the ss-OAFP model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10.1 Dimensionless densities D(R) of the isothermal sphere and ss-OAFP model. . . . . 165

10.2 Dimensionless density D(R) of the energy-truncated ss-OAFP model for different

δ. The corresponding profile of the ss-OAFP model is also depicted. . . . . . . . . 168

10.3 Dimensionless potential ϕ(R) of the energy-truncated ss-OAFP model. The corre-

sponding potential of the ss-OAFP model is also depicted. . . . . . . . . . . . . . . 168

10.4 Projected density Σ of the energy-truncated ss-OAFP model for different δ. The

power-law R−1.23 corresponds to the asymptotic approximation of the ss-OAFP

model as R→ ∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

10.5 Concentration of the energy-truncated ss-OAFP model. The horizontally dashed

line represents the concentration for polytropic sphere of m = 3.9. . . . . . . . . . 172

xxv

LIST OF FIGURES

10.6 Fitting of the energy-truncated ss-OAFP model to the projected density profile for

NGC 6752 (Ferraro et al., 2003) for different m. The unit of Σ is the number per

square of arcminutes. Σ is normalized so that the density is unity at the smallest

radius of data points. In the left top panel, double-power law profile is depicted as

done in (Ferraro et al., 2003). In the two bottom panels, ∆ log[Σ] for m = 3.9 and

m = 4.2 depicts the corresponding deviation of Σ from our model. . . . . . . . . . 176

10.7 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected surface

density of NGC 6397 reported in (Drukier et al., 1993). The unit of Σ is the number

per square of arcminutes. Σ is normalized so that the density is unity at the smallest

radius for the observed data. In the legends, (c) means PCC cluster as judged so

in (Djorgovski and King, 1986). ∆ log[Σ] is the corresponding deviation of Σ from

the model on a log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

10.8 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the surface brightness

of Terzan 2 and NGC 6388 reported in (Djorgovski and King, 1986). The unit of

the surface brightness (SB) is B magnitude per square of arcseconds. The bright-

ness is normalized by the magnitude SBo obtained at the smallest radius point. In

the legends, (n) means normal or KM cluster and (c) means PCC cluster as judged

so in (Djorgovski and King, 1986). ∆(SBo − SB) is the corresponding deviation of

SBo − SB from the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

10.9 Core relaxation time against (a) concentration c obtained from the energy-truncated

ss-OAFP model (m = 3.9) applied to the PPC- and KM- clusters and (b) concen-

tration c based on the King model reported in (Harris, 1996, (2010 edition)). . . . . 182

10.10Completion rate of core collapse against (a) concentration c based on the energy-

truncated ss-OAFP model with m = 3.9 and (b) concentration c based on the King

model reported in (Harris, 1996, (2010 edition)). . . . . . . . . . . . . . . . . . . . 184

xxvi

LIST OF FIGURES

10.11Fitting of the energy-truncated ss-OAFP model to the surface brightness profiles

of NGC 6284 and NGC 6341 reported in (Noyola and Gebhardt, 2006). ‘Cheb.’

means the Chebyshev approximation of the surface brightness reported in (Trager

et al., 1995). The unit of the surface brightness (SB) is the V magnitude per square

of arcseconds. The brightness is normalized by the magnitude SBo observed at the

smallest radius point. In the legends, (n) means a normal or KM cluster and (c)

means a PCC cluster as judged so in (Djorgovski and King, 1986). . . . . . . . . . 186

10.12Fitting of the energy-truncated ss-OAFP model and its approximated form to the

surface brightness profile of NGC 6715 reported in (Noyola and Gebhardt, 2006).

‘Cheb.’ means the Chebyshev approximation of the surface brightness reported in

(Trager et al., 1995). The core radius rc and δ in the figure were acquired from

the approximated form. The unit of the surface brightness (SB) is V magnitude per

square of arcseconds. The brightness is normalized by the magnitude SBo obtained

at the smallest radius point. In the legends, (n) means normal or KM cluster as

judged so in (Djorgovski and King, 1986). . . . . . . . . . . . . . . . . . . . . . . 188

10.13(a)Dimensionless density profiles Do, exp[−13.88(1 + E)] and δB(3.4, 1.5)(−E)3.9

for δ = 0.01.(b) Projected density profiles for the energy-truncated ss-OAFP model,

and its approximation for δ = 0.1 and δ = 0.001. For the both, m is fixed to 3.9. . . 190

10.14Density profiles Do of the energy-truncated ss-OAFP model, Da of the approxi-

mated model and exp[−13.88(1 + E)]. . . . . . . . . . . . . . . . . . . . . . . . . 190

10.15Fitting of the polytropic sphere of index m to the projected density Σ of NGC 288

and NGC 6254 reported in (Miocchi et al., 2013). The unit of Σ is the number per

square of arcminutes and Σ is normalized so that the density is unity at smallest

radius for data. In the legends, (n) means a ‘normal’ or KM cluster as judged so

in (Djorgovski and King, 1986). ∆ log[Σ] is the corresponding deviation of Σ from

the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

xxvii

LIST OF FIGURES

10.16Fitting of the polytropic sphere of index m to the surface brightness of NGC 5139

from (Meylan, 1987). The unit of the surface brightness (SB) is the V magnitude

per square of arcminutes. The brightness is normalized by the magnitude SBo

observed at the smallest radius point. In the legends, (n) means a ’normal’ or

KM cluster as judged so in (Djorgovski and King, 1986). ∆(SBo − SB) is the

corresponding deviation of SBo − SB from the model. . . . . . . . . . . . . . . . . 193

10.17Secular relaxation parameter ηM against concentration c. The values of concentra-

tion are adapted from (Harris, 1996, (2010 edition)). . . . . . . . . . . . . . . . . . 196

A.1 Schematic description of the Landau sphere. For the truncated DF in (Grad, 1958),

if any two stars approach closer than the conventional Landau radius, the stochastic

dynamics turns into the deterministic one. For the weakly-coupled DF (Appendix

B.2), any stars can not approach each other closer than the Landau radius. . . . . . 216

D.1 Absolute values of the Chebyshev coefficients for Ra/Rs with various m and L

values. For (m, L) = (20, 1) or (6, 30) the plots flatten out at high n due to the

round-off error; for other choices of (m, L) the coefficients decay slowly due to the

end-point singularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

D.2 Absolute values of the Chebyshev coefficients for the function (1 − x)k ln (1 − x)

mapped by x = sin πy2 , x = tanh Lmy√

1−y2, and x = 1 − 2

(1−y

2

)L. The parameter values

are k = 2, L = 8 and Lm = 3 (top) and k = 0.5, L = 30, and Lm = 6 (bottom). For

the power mapping the plot flattens out due to round off error when n ≈ 25, while

for the tangent hyperbolic mapping machine precision is reached when n ≈ 110. . . 245

D.3 Asymptotic absolute values of the Chebyshev coefficients for ϕηi cos (ωi lnϕ). The

solid curves show the differences between the values of the coefficients based on

the steepest decent (dashed-curves) and the interpolatory quadrature (filled circles).

The parameters are L = 1 and m = 6 (top), m = 20 (middle) and m = 45 (bottom). . 247

xxviii

LIST OF FIGURES

F.1 Values of 3I(x = 1), | 1− c1/c1o |, and | 1− c∗4/c∗4o | against change ∆β in eigenvalue

β around the reference value βo (∆β ≡ β − βo). Numerical parameters are N = 70,

L = 1, and FBC = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

F.2 (Left panel) Condition number of the Jacobian matrix for the Q-integral and 4ODEs

regularized through F(E)/FBC. (Right panel) Relative deviations of the regularized

eigenvalues c1/FBC and c∗4/FBC from the reference eigenvalues c1o and c∗4o for the

4ODEs regularized through F(E)/FBC. (FBC = 1 , L = 1, and N = 70.) . . . . . . . 253

F.3 (Left Panel) Chebyshev coefficients for 3F(x) with L = 0.75 in the following cases

(a) N = 60 and β = 8.178371160 and (ii) N = 65 and β = 8.178371275. The

iteration method for the latter did not work satisfactorily stalling around | anew −

aold |≈ 7 × 10−10 (resulting in | 1 − c∗4/c∗4o |≈ 6.0 × 10−3 | and 3I = 4.4 × 10−6).

However, it is shown here for the sake of comparison. (Right panel) Chebyshev

coefficients for 3F(x) for L = 0.5 in the following cases (a) N = 35 and β =

8.178371160, and (ii) N = 50 and β = 8.1783712. . . . . . . . . . . . . . . . . . . 254

F.4 Chebyshev Coefficients for the Q-integral with a discontinuous test function 3(tes)R .

Recall E = −(0.5 + 0.5x)L, here L = 1. . . . . . . . . . . . . . . . . . . . . . . . . 257

F.5 Chebyshev Coefficients for√3

(m)R and exp(3R)

√0.5 + 0.5x for a discontinuous test

function 3(tes)D . The dashed guidelines are added for the measure of slow decays. . . 258

G.1 Absolute values of Chebyshev coefficients F(m)n for 3(m)

F . In the modified ss-OAFP

system, 3S , 3Q, 3G, 3I , 3J are not modified. The maximum values Emax of the trun-

cated domain are also depicted in the figure. . . . . . . . . . . . . . . . . . . . . . 261

G.2 Absolute values of Chebyshev coefficients In for 3I . In the modified ss-OAFP sys-

tem, 3S , 3Q, 3G, 3I , 3J are not modified. The maximum values Emax of the truncated

domain are also depicted in the figure. . . . . . . . . . . . . . . . . . . . . . . . . 262

xxix

LIST OF FIGURES

I.1 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density

Σ of NGC 288, NGC 1851, NGC 5466, and NGC 6121 reported in from (Miocchi

et al., 2013). The unit of Σ is the number per square of arcminutes. Σ is normalized

so that the density is unity at the smallest radius for data. In the legends, (n) means

a ‘normal’ cluster that can be fitted by the King model as judged so in (Djorgovski

and King, 1986). ∆ log[Σ] is the corresponding deviation of Σ from the model on

log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267


Σ of NGC 6254, NGC 6626, Palomar 3 and Palomar 4 reported in (Miocchi et al.,

2013). The unit of Σ is number per square of arcminutes and Σ is normalized so

that the density is unity at the smallest radius for data. In the legends, (n) means

‘normal’ cluster that can be fitted by the King model as judged so in (Djorgovski

and King, 1986). ∆ log[Σ] is the corresponding deviation of Σ from the model on

log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268


Σ of Palomar 14, Terzan 5, and NGC 6809 reported in (Miocchi et al., 2013). The

unit of Σ is the number per square of arcminutes. Σ is normalized so that the density

is unity at the smallest radius for data. In the legends, (n) means a ‘normal’ cluster

that can be fitted by the King model as judged so in (Djorgovski and King, 1986).

∆ log[Σ] is the corresponding deviation of Σ from the model on log scale. . . . . . . 269


profiles NGC 2419, NGC 4590, NGC 5272, NGC 5634, NGC 5694, and NGC

5824 reported in (Kron et al., 1984). The unit of the projected density Σ is the

number per square of arcminutes. In the legends, (n) means a normal cluster that

can be fitted by the King model as judged so in (Djorgovski and King, 1986). . . . 272

xxx

LIST OF FIGURES


profiles of NGC 6093, NGC 6205, NGC 5229, NGC 6273, NGC 6304, and NGC


number per square of arcminutes. In the legends, (n) means a ‘normal’ cluster that

can be fitted by the King model and (n?) a ‘probable normal’ cluster as judged so

in (Djorgovski and King, 1986). . . . . . . . . . . . . . . . . . . . . . . . . . . . 273


profiles NGC 6341, NGC 6356, NGC 6401, NGC 6440, NGC 6517, and NGC

6553 reported in (Kron et al., 1984). The unit of the projected density Σ is number

per square of arcminutes. In the legends, (n) means a normal cluster that can be

fitted by the King model as judged so in (Djorgovski and King, 1986). . . . . . . . 274




number per square of arcminutes. In the legends, (n) means a normal cluster that

can be fitted by the King model as judged so in (Djorgovski and King, 1986). . . . 275


profiles NGC 5053 and NGC 5897 reported in (Kron et al., 1984). The unit of

the projected density Σ is the number per square of arcminutes. In the legends,

(n) means normal cluster that can be fitted by the King model as judged so in

(Djorgovski and King, 1986). Following (Kron et al., 1984), data at small radii are

ignored due to the depletion of projected density profile. . . . . . . . . . . . . . . . 276

xxxi

LIST OF FIGURES

I.9 Fitting of the energy-truncated ss-OAFP model to the surface brightness of NGC

104, NGC 1904, NGC 2808, NGC 6205, NGC 6441, and NGC 6712 reported in

(Noyola and Gebhardt, 2006). The unit of the surface brightness (SB) is the V

magnitude per square of arcseconds. The brightness is normalized by the mag-

nitude SBo obtained at the smallest radius point. In the legends, ‘Cheb.’ means

the Chebyshev approximation of the surface brightness reported in (Trager et al.,

1995) and ‘(n)’ means a KM cluster as judged so in (Djorgovski and King, 1986). . 277

I.10 Failure of the fitting of the energy-truncated ss-OAFP model to the surface bright-

ness profiles of NGC 5286, NGC 6093, NGC 6535, and NGC 6541 reported in

(Noyola and Gebhardt, 2006). The unit of the surface brightness (SB) is the V

magnitude per square of arcseconds. The brightness is normalized by the mag-

nitude SBo obtained at the smallest radius point. In the legends, ‘Cheb.’ means

the Chebyshev approximation of the surface brightness reported in (Trager et al.,

1995) and ‘(n)’ means a KM cluster as judged so in (Djorgovski and King, 1986). . 278

I.11 Fitting of the energy-truncated ss-OAFP model and its approximated form to the

surface brightness profiles of NGC 1851, NGC 5694, and NGC 5824 reported in

(Noyola and Gebhardt, 2006). The unit of the surface brightness (SB) is the V mag-

nitude per square of arcseconds. The brightness is normalized by the magnitude

SBo obtained at the smallest radius point. The values of rc and δ were obtained

from the approximated form. In the legends, ‘Cheb.’ means the Chebyshev ap-

proximation of the surface brightness reported in (Trager et al., 1995) and ‘(n)’ a

means KM cluster as judged so in (Djorgovski and King, 1986). . . . . . . . . . . 279

xxxii

LIST OF FIGURES

J.1 Fitting of the polytropic sphere of index m to the projected density Σ of NGC 5466,

NGC 6809, Palomar 3, and Palomar 4 reported in (Miocchi et al., 2013). The unit

of Σ is number per square of arcminutes. Σ is normalized so that the density is unity

at smallest radius for the data. In the legends, (n) means a normal or KM cluster as

judged so in (Djorgovski and King, 1986). ∆ log[Σ] is the corresponding deviation

of Σ from the model on a log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . 281

J.2 Fitting of the polytropic sphere of index m to the projected density of Palomar 14

reported in (Miocchi et al., 2013). The unit of the projected density Σ is the number

per square of arcminutes. In the legends, (n) means a normal or KM cluster as

judged so in (Djorgovski and King, 1986). . . . . . . . . . . . . . . . . . . . . . . 282

J.3 Partial success of fitting of the polytropic sphere of index m to the projected den-

sity profile of NGC 4590 reported in (Kron et al., 1984). The unit of the projected

density Σ is the number per square of arcminutes. In the legends, (n) means a nor-

mal or KM cluster as judged so in (Djorgovski and King, 1986). Following (Kron

et al., 1984), the data at small radii are ignored due to the depletion of projected

density profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

J.4 Fitting of the polytropic sphere of index m to the surface brightness profiles of NGC

1261, NGC 5053, NGC 5897, NGC 5986, NGC 6101, and NGC 6205 reported in

Trager et al. (1995). The unit of the surface brightness (SB) is the V magnitude per

square of arcseconds. The brightness is normalized by the magnitude SBo obtained

at the smallest radius point. In the legends, ‘(n)’ means a KM cluster as judged so

in (Djorgovski and King, 1986). . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

xxxiii

LIST OF FIGURES

J.5 Fitting of the polytropic sphere of index m to the surface brightness profiles of

NGC 6402, NGC 6496, NGC 6712, NGC 6723, and NGC 6981 reported in (Trager

et al., 1995). The unit of the surface brightness (SB) is the V magnitude per square

of arcseconds. The brightness is normalized by the magnitude SBo obtained at

the smallest radius point. In the legends, ‘(n)’ means KM cluster as judged so in

(Djorgovski and King, 1986). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

J.6 Failure of the fitting of the polytropic sphere of index m to the surface brightness


6656 reported in (Trager et al., 1995). The unit of the surface brightness (SB) is

the V magnitude per square of arcseconds. The brightness is normalized by the

magnitude SBo obtained at the smallest radius point. In the legends, ‘(n)’ means a

KM cluster as judged so in (Djorgovski and King, 1986). . . . . . . . . . . . . . . 286

K.1 Fitting of the energy-truncated ss-OAFP model to the projected densities of NGC

1904, NGC 4147, NGC 6544, and NGC 6652 reported in (Kron et al., 1984). The

unit of the projected density Σ is the number per square of arcminutes. In the

legends, (c) means a PCC cluster, (c?) a probable/possible PCC, and (n?c?) weak

indications of PCC as judged so in (Djorgovski and King, 1986). . . . . . . . . . . 288

K.2 Fitting of the energy-truncated ss-OAFP model to the surface brightness profiles

of NGC 5946, NGC 6342, NGC 6624, and NGC 6453 reported in (Lugger et al.,

1995). The unit of the surface brightness (SB) is the U magnitude per square

of arcseconds. The brightness is normalized by the magnitude SBo obtained at

the smallest radius point. In the legend, ‘(c)’ means a PCC cluster as judged so

in (Djorgovski and King, 1986). ∆(SBo − SB) is the corresponding deviation of


xxxiv

LIST OF FIGURES

K.3 Fitting of the energy-truncated ss-OAFP model to the surface brightness profiles of

NGC 6522, NGC 6558, and NGC 7099 reported in (Lugger et al., 1995) and Terzan

1 reported in (Djorgovski and King, 1986). The unit of the surface brightness (SB)

is the U magnitude per square of arcseconds except for Terzan 1 for which the

B band is used. The brightness is normalized by the magnitude SBo obtained at

the smallest radius point. In the legend, ‘(c)’ means a PCC cluster as judged so

in (Djorgovski and King, 1986). ∆(SBo − SB) is the corresponding deviation of


K.4 Fitting of the energy-truncated ss-OAFP model to the surface brightness profiles

of NGC 6266, NGC 6293, NGC 6652, NGC 6681, NGC 7078, and NGC 7099

reported in (Noyola and Gebhardt, 2006). The unit of the surface brightness (SB)

is the V magnitude per square of arcseconds. The brightness is normalized by the

magnitude SBo obtained at the smallest radius point. In the legend, ‘Cheb.’ means

the Chebyshev approximation of the surface brightness profiles reported in (Trager

et al., 1995). The symbol ‘(c)’ means a PCC cluster as judged so in (Djorgovski

and King, 1986). ∆(SBo − SB) is the corresponding deviation of SBo − SB from

the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

K.5 Failure of the fitting of the energy-truncated ss-OAFP model to the surface bright-

ness profiles of NGC 6541 and NGC 6624 reported in (Noyola and Gebhardt,

2006). The unit of the surface brightness (SB) is the V magnitude per square

of arcseconds. The brightness is normalized by the magnitude SBo obtained at the

smallest radius point. In the legend, ‘Cheb.’ means the Chebyshev approximation

of the surface brightness profiles reported in (Trager et al., 1995). ‘(c)’ means a

PCC cluster and ‘(c?)’ a possibly PCC cluster as judged so in (Djorgovski and

King, 1986). ∆ (SBo − SB ) is the corresponding deviation of SBo − SB from the

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

xxxv

Chapter 1

Introduction

The present dissertation is concerned with the relaxation evolution of dense star clusters (such

as globular clusters). Stellar dynamicists have conventionally discussed the relaxation evolution

based on kinetic theory (Section 1.1). The relaxation evolution has attracted one’s interest in vari-

ous fields such as astronomy, statistical dynamics, and computational physics (Section 1.2). How-

ever, existing computational works have prevailed that the clusters undergo a large-scale change in

both the spatial structure and time evolution. Such large scale-gap has hindered the accurate- and

efficient- numerical integrations of some mathematical models for stellar dynamics. The present

dissertation focuses on the orbit-averaged Fokker-Planck (OAFP) kinetic equation to model the

relaxation evolution of isotropic spherical clusters (Section 1.3). While the OAFP equation is the

most simplified mathematical model among existing cluster models based on the first principle

of statistical dynamics, none of the previous works has satisfactorily solved the equation. This

is since the existing works have resorted to finite difference (FD) methods that are not accurate

enough to solve it. In the present work, we resort to an accurate and efficient numerical scheme,

a Gauss-Chebyshev pseudo-spectral method, to find an accurate solution of the equation (Section

1.4). Based on the solution, we aim at detailing the thermodynamic property of the model and

applying the model to observed structural profiles of globular clusters in Milky Way.

1

CHAPTER 1. INTRODUCTION

1.1 A quantitative understanding of dense star clusters

Physical systems of concern in the present dissertation are globular clusters. Their observed

physical features are relatively simple (Section 1.1.1). Standard statistical stellar dynamics consid-

ers the dynamical evolution of globular clusters as a two-stage evolution: collisionless evolution

on short time scales (Section 1.1.2) and collisional evolution on long time scales (Section 1.1.3).

The present section quantitatively explains the evolution based on observed data.

1.1.1 The physical features of globular clusters

For simplicity, we focus only on globular clusters that do not contain any exotic astronomical

objects such as blue stragglers, white dwarfs, massive black holes, and x-ray sources. Typical

galaxies include hundreds of globular clusters. The majority of the clusters are likely observed in

the Galactic halo (Figure 1.1). The globular clusters have simple physical and dynamical features

(e.g., Spitzer, 1988; Meylan and Heggie, 1997; Heggie and Hut, 2003; Binney and Tremaine,

2011). The size of globular clusters is typically about 0.1 ∼ 20 parsecs (pc)1. The globular clusters

are composed of 104 − 107 stars only (Figure 1.2). Observations revealed typical globular clusters

include little dark matter, gasses, and clouds of dust. Also, Hertzsprung–Russell color diagram 2

prevailed that the clusters have less young stars (that were born during the dynamical evolution of

clusters). This result implies that most stars were formed in globular clusters when or before the

stars formed the clusters. The ages of globular clusters in Milky Way are approximately 10 billion

years ago. Hence, the simplest assumption for stars is that they did not evolve by themselves

during the evolution of globular clusters and that the physical states did not change. Although the

actual stars are different in size, if one assumes the radii of stars in a star cluster are that of the Sun

(∼ 2.3 × 10−8 pc), one may consider the stars as ‘point particles′ interacting through Newtonian

11 parsec ≈ 3.09 × 1016 meters.2The Hertzsprung–Russell color diagram can identify the age of stars based on how bright the stars are at present

(Lightman and Shapiro, 1978)

2


Figure 1.1: (Top) Schematic image of globular clusters in Milky way and (bottom) picture ”False-color image of the near-infrared sky as seen by the DIRBE” (credit:(NASA, Goddard)). The d inthe top panel is the diameter of object concerned. The actual number of observed globular clustersis approximately 160 in Milky Way (For example, the number is reported as 157 in (Harris, 1996,(2010 edition))’s catalog). The Galactic center and disc are composed of a plentiful of gases,clouds of dust and stars. Our Sun lies on the Galactic disc. Many globular clusters exist in theGalactic halo in which stars and gaseous components are less found.

3


Figure 1.2: Globular cluster NGC362 (credit: NASA, Goddard) as an example of globular clusters.The mass of NGC362 is 3.8 × 105 solar masses, the core radius 0.42 pc and tidal radius 37 pc(Boyer et al., 2009). Refer to Chapter 5 for the core- and tidal radii of a globular cluster.

4


pair-wise potential. Some other simple physical features are that the stars form spherical clusters

whose ellipticities3 are ∼ 0.9 and the globular clusters are little rotating. Due to the ’ideal’ observed

features, conventional stellar dynamics4 has been concerned with how stars travel and populate in

the spherical system due to the Newtonian pair-wise interaction between stars. Astrophysicists are

basically concerned with how the clusters have evolved to reach the current structures and will

change in the future. Since the total number N of stars is large (N >> 1), one may resort to

statistical theories to understand the evolution of the clusters. Typically, the evolution is described

by a two-stage dynamical evolution on dynamical time scales (Section 1.1.2) and relaxation time

scales (Section 1.1.3).

1.1.2 Dynamical time and mean field Newtonian potential

The dynamical time5 is important to discuss the dynamics on short time scales during the

relaxation evolution. Considering that the observed speed dispersion of stars in globular clusters

is a few ∼ tens km/s, one can find the dynamical time is 0.1 ∼ 1 Million years. On the dynamical

time scales, standard stellar dynamics expects that stars do not approach each other closer than

the average distance of stars and travel in smooth orbits determined by the static self-consistent

mean-field (m.f.) potential φ(r) at stellar position r (Jeans, 1915)6. The m.f. approximation is the

3The ellipticity is defined as the ratio of major to minor axes of a spheroid.4Although the present work focuses on idealized modeling rather than realistic modeling of globular clusters,

we point out the gap between the idealization and more recent observation data that have been reported along withimprovements of telescopes such as Virgo and Gaia 2. Stellar dynamicists have included in standard theories somerealistic factors, such as the mass (distribution) function for stars, stellar evolution, tidal effect, close encounters, andbinary formation/interaction. They discussed how the factors affect the relaxation evolution of star clusters. The newobservations, however, showed unexpected clusters’ structures. Although dark matter itself has yet to be found inglobular clusters, some observation implicated that the formation of globular clusters was possibly subject to minidark-matter halos and that the halos might have caused the flattening of speed dispersion of stars in the outer skirt ofthe clusters (Claydon et al., 2019; de Boer et al., 2019). Many globular clusters are subject to some internal rotation(Fabricius et al., 2014; Bianchini et al., 2018; Sollima et al., 2019), hence the spherical approximation may not beexactly correct.

5The dynamical time of a cluster is a few crossing time. The crossing time is time duration for a star to travelacross the cluster.

6The m.f. approximation corresponds with the assumption that the total number of stars is large N >> 1 and thecluster is dense in phase space so that one can adequately define the distribution function for stars.

5


zeroth-order approximation (N → ∞) to determine the motion of stars and the physical state of

cluster. Under this assumption, the orbit of a star is unclosed7. Also, it is ‘smooth’ in the sense the

star can keep moving continuously with time.

1.1.3 Relaxation time and two-body relaxation (’collision’) process

On time scales much larger than the dynamical time scales, stars in a cluster can not keep

staying on the smooth orbits determined by the m.f. potential; as a result, they undergo a two-body

relaxation process. The actual potential the stars feel may be always fluctuating about the m.f.

potential since the total number of stars is finite. Also, a star may approach another star closer

than the average distance of stars and they may pass by each other with a relatively large deflection

angle of passage. The latter is conventionally called an encounter between stars (Charlier, 1917).

The encounter can occur between a single star and the other stars many times on a long time

scale, one can consider the accumulation of encounters can result in a random process (See Figure

1.3). The process is called the two-body relaxation and the time scale of the relaxation is termed

the relaxation time. The concept of two-body relaxation is heavily based on Chandrasekhar and

coworker’s studies. They developed stochastic theory by extending Holtsmark distribution to a

phase space (Chandrasekhar and von Neumann, 1942, 1943) and by using the Fokker-Planck model

(Chandrasekhar, 1943,a,b). Ambartsumian (1938) estimated the relaxation time as N/ ln[N] times

longer than the dynamical time, meaning 1 ∼ 10 billion years8. Considering the Hubble time

(≈13.7 billion years) and age of cluster (∼ 10 billion years), one can expect the effects of two-

body encounter and m.f. potential drive the dynamical evolution of globular clusters. This kind of

clusters may be termed the collisional star clusters and is the main system concerned in the present

7Standard mechanics for the motion of a particle under a fixed spherical potential (Goldstein et al., 2002; Binneyand Tremaine, 2011) states that if the potential is Keplerian or Harmonic the orbit of particle is closed. On the onehand, if the potential is one of the other attractive power-law profiles or general spherical potential the orbit is unclosed.

8The logarithmic factor ln[N] is called the Coulomb logarithm. See (e.g., e Silva et al., 2017) for numericalestimation of the logarithm, (e.g., Binney and Tremaine, 2011, Ch.7) for the fundamental discussion and AppendixC.2 for the derivation of the logarithm based on statistical first principle.

6


Figure 1.3: Concept of the encounter and two-body relaxation. On time scales much larger thandynamical time, ’test’ star may approach a ’field’ star closer than the average distance 1/n1/3. Onecan characterize this encounter by the impact parameter b and initial velocity (dispersion speed)3 in analogy with gaseous kinetic theory. During a single encounter, the test star gains smallperpendicular velocity δ3 since the potential due to the field star deflects test star from its smoothinitial orbit. The encounter may successively occur between the test star and many field stars. Ona long time scale, accumulation of velocity change

∑|δ3| reaches the order of the initial velocity 3.

At the time, the trajectory of test star largely deviates from the original orbit and test star ’forgets’its initial orbit. This process is the encounter of stars and sometimes termed the velocity relaxation.

dissertation.9

1.2 The interdisciplinary importance of stellar-dynamics

studies

Stellar-dynamics studies inherently include interdisciplinary significance (e.g., Heggie and

Hut, 2003). They have intellectually contributed to not only understandings of observed astronom-

ical objects (Section 1.2.1) but also statistical mechanics of particles interacting with long-range

potential (Section 1.2.2) and new kinds of direct N-body simulation (Section 1.2.3 ).

9See (Renaud, 2018) for strict definitions of star clusters and collisional clusters.

7


1.2.1 Astronomical importance

The importance of studying stellar dynamics is not only in its own right. Astronomers regard

the centers of globular clusters as the ‘factories’ of black holes (gravitational waves) and exotic

astrophysical objects (e.g., Blue stragglers, X-ray sources). The dynamics of globular clusters

provides information about how and when the special objects were formed. Also, astronomers

consider the globular clusters were formed at the same epoch as the host galaxy was formed.

Hence, they can constraint the age of the host galaxy from the history of the evolution of globular

clusters.

1.2.2 Statistical-physics importance

From a statistical-physics point of view, the dynamics of star clusters have shown unique sta-

tistical phenomena, such as negative heat capacity, inequalities among statistical ensembles (e.g.,

micro-, canonical- and grand-canonical- ensembles), quasi-stationary states (See, e.g., Heggie and

Hut, 2003; Binney and Tremaine, 2011). The uniqueness originates from the long-range nature

of Newtonian interaction potential. The long-range interaction hampers one to employ standard

statistical methods that apply to systems of particles interacting via short-range forces10. Cur-

rently, statistical mechanics of systems of long-range interacting particles are under intensive stud-

ies (Campa et al., 2014; Levin et al., 2014; Soto, 2016). The examples are hot collisionless plasmas,

cold atoms, geofluid vortex, and free-electron lasers. Especially, stellar dynamics suffices unique

self-organized physical features, such as naturally finite dimensions in size, strong inhomogeneity

in the density and statistically completely-open systems.

10The following concepts do not work for systems of long-range interacting particles; Thermodynamics limit(N/V → constant if N → ∞ and the system volume V → ∞), thermodynamic zeroth-law and the additivity ofextensive quantities (entropy, energy. . . ). The examples of ”short-range” potentials are hard-sphere potentials and theCasimir-Polder (Lennard-Jones) potential for neutral gaseous systems, and the Yukawa (Debye-Huckel) potential forclassical collisional plasmas.

8


1.2.3 Computational-physics importance

From a mathematical/computational point of view, stellar dynamicists have developed new

software, hardware, and numerical schemes to handle the large time- and spatial- scale gaps in

the evolution of globular clusters (See, e.g., Aarseth and Heggie, 1998; Heggie and Hut, 2003).

Especially, the community has preferably employed special direct N-body numerical methods11 as

’experiments’ to imitate the evolution of globular clusters. The methods need a reasonable span of

CPU time (∼ 3 months) to examine the relaxation evolution. However, the exponential-instability

growth problem always gets involved (Goodman et al., 1993) in a similar way to molecular dy-

namics. Especially, N-body simulations inherently include their special cutoff problem (Aarseth

and Heggie, 1998). The simulations can not ignore (cutoff) the near-field divergence of pair-wise

Newtonian potential since the divergence is important to discuss high-density cores and close en-

counters. Also, the simulations must adequately include the effect of escaping star in the outer

halo (outskirt) on the evolution because of the long-range nature of the pair-wise potential. Those

numerical problems limit the applicability of N-body simulation up to N ∼ 106 at present (e.g.,

Dehnen and Read, 2011; Rodriguez et al., 2016; Wang et al., 2016; Varri et al., 2018; Rodriguez

et al., 2018). Hence, the stellar dynamicists also have numerically solved the approximated models

of the N-Newton equations of motion. Typical continuum models are fluid- and gaseous- models

described in configuration spaces and kinetic models in phase space. Numerical approaches to the

models also have been fundamental methods because of their numerical efficiencies (at most a few

weeks CPU time). The continuum models have deepened the conceptual understanding of stellar

dynamics. We focus on the kinetic models in the present dissertation since they are based on sta-

tistical principle hence less approximated and more physically sensible compared to the fluid- and

gaseous- models (e.g., Heggie and Hut, 2003; Binney and Tremaine, 2011).

11The N-body direct simulation solves the N-Newton equations of motions for N-particles interacting via pair-wiseNewtonian forces.

9


1.3 A basic mathematical model: Fokker-Planck (FP) kinetics

of isotropic spherical star clusters

The present section extends the quantitative aspects of globular clusters (explained in Section

1.1) to the corresponding mathematical modeling based on a statistical kinetic theory12 for the

relaxation evolution of spherical globular clusters. The ideal model of globular clusters is a collec-

tion of N equal-mass stars that is isotropic in velocity space and spherical in configuration space.

The finiteness of the total number N requires one to cautiously treat mathematical modeling for

the definition of the distribution function (DF) for stars (Section 1.3.1) and the two-stage evolution

of the DF; collisionless relaxation evolution (Section 1.3.2) and collisional relaxation evolution

(Section 1.3.3).

1.3.1 Distribution function for stars

The physical feature of globular clusters complicates the definition of the DF for stars in phase

space. One may first define the number probability density DF for a star at phase space point (r, v)

at time t in the sense of distribution for an isotropic cluster

N =

"f (r, v, t) d3r d3v = (4π)2

"f (r, 3, t) r2

32 dr d3. (1.3.1)

where v is the stellar velocity, 3 the stellar speed, and r the stellar position or the radius from the

center of the cluster. However, the DF in phase space is not suitable for globular clusters. This is

since the star’s population is sparse in the phase space volume element d3v d3r due to the finiteness

of total number N (or obviously as you can see in Figure 1.2). As a remedy, one typically discusses

12Available concise statistical theories for stellar dynamics (e.g., Spitzer, 1988; Chavanis, 2013) are, however, lim-ited at present. Hence, Appendices A, B, and C explain the fundamental kinetics employed in the present dissertation.

10


the DF in energy-space volume element dε. The energy per unit mass is defined as follows

ε = φ(r, t) +32

2, (1.3.2)

The binding energy ε can be finite for the globular clusters that are finite in size. This means the

star’s population can be dense in energy space. Hence, one may redefine the DF for stars in an

energy band between ε and ε + dε13

N(ε, t) dε =

∫ ∫δ

(ε − φ(r, t) +

32

2

)f (r, v, t) d3r d3v dε = f (ε, t)

∂q(ε, t)∂ε

dε, (1.3.3)

where the function q(ε, t) is the phase space volume whose energy is bounded at ε from above. The

q-integral (the integral associated with the radial action14) reads

q(ε, t) =13

∫ rmax(ε,t)

0

[2ε − 2φ(r′, t)

]3/2 r′2 dr′, (1.3.6)

where rmax(ε, t) = φ−1(ε).13The DF f (ε, t) does not represent an energy probability density DF but still phase-space density DF. The function

f (ε, t) means that the function depends only on the energy ε at time t while N(ε, t) is exactly an energy density DF attime t.

14The q-integral is related to the radial action of star’s orbits

Ir(ε, L) = 2m∫ ra(ε,L)

rp(ε,L)3r dr, (1.3.4)

where ra and rp are the apocenter and pericenter of the orbits (or the maximum- and minimum- radii that can be foundsolving ε = φ(r) + L2

2r2 for fixed ε and L.). An anisotropic system can be described by the DF f (ε, L, t) of energy εand modulus of angular momentum L. This is a complete system as a spherical self-gravitating system (e.g., Binneyand Tremaine, 2011). Hence, to make the system isotropic, one must reduce the arguments of f (ε, L, t) to only ε.One typically employs ’isotropizing’, i.e., averaging over the modulus of the angular momentum. For example, theisotropized q-integral reads

q(ε, t) =14

∫ L2c (ε)

0Ir(ε, L) dL2, (1.3.5)

where Lc(ε) is the modulus of the angular momentum of a star moving in a circular orbit. The isotropizing processcorresponds with giving up the correct description that the radial velocity dispersion is greater than tangential one inthe outer halo.

11


1.3.2 Collisionless relaxation evolution on dynamical time scales

The evolution of DF f (ε, t) is a two-stage process due to the large gap (∼ N/ ln[N]) between the

relaxation time (trel) and dynamical time (tdyn). The dynamics on dynamical time scales is based

on the zeroth-order approximation (called the collisionless approximation). The approximation

mathematically takes the limit tdyn/trel → 0 as N → ∞. On the dynamical time scales, stars rapidly

move in smooth orbits, hence the m.f. potential also rapidly changes. In this case, the argument

ε of f (ε, t) depends on time t implicitly through equation (1.3.2). The collisionless approximation

applies to the dynamics of galaxies and rich galaxy clusters15. One would expect such collisionless

systems adiabatically evolve with time. However, observations have shown that the systems are

currently well-relaxed. This observation seems against standard stellar dynamics since they are

supposed to have not experienced the two-body relaxation process (Lightman and Shapiro, 1978;

Saslaw, 1985).

The collisionless relaxation originates from the nature of the long-range interacting stars. To

understand this, imagine stars are initially situated randomly in a globular cluster. The stars first

interact with near-by stars more efficiently due to the stronger (near-field) pair-wise interaction.

Then, the stars can form groups of stars in the system. The groups interact with each other through

their Newtonian potentials. Besides, the individual stars belonging to different groups can inter-

act with each other without closely approaching. This group interaction is often considered an

imitation of the Pauli’s exclusion principle (refer to D. Lynden-Bell and Royal, 1968, for seeing

the original idea). The groups rapidly and collectively oscillate since stars in each group rapidly

change their positions. This collective behavior can be well-characterized by the Virial theorem

or the Virial ratio (= −2KE/PE, where KE and PE are the total kinetic- and potential- energies of

the system). The amplitude of oscillation of the Virial ratio gradually decreases until the groups

of stars settle to their optimal (dynamically stable) state in velocity- and configuration- spaces as

15For galaxies and clusters of galaxies tdyn/trel ≈ 10−11 ∼ 10−13. This corresponds that their relaxation times arelonger than their ages.

12


shown in Figure 1.4. This corresponds to that the system reaches a quasi-stationary state (QSS) or

a state of Virial equilibrium. This collisionless relaxation process is called the violent relaxation

(See Bindoni and Secco, 2008, for review). The violent relaxation, however, is not only the pri-

mary cause of a QSS for actual collisionless systems. The actual collisionless relaxation is more

complex (e.g., Merritt, 1999). One needs to consider the violent relaxation as only part of mix-

ing processes. The efficiency of the violent relaxation depends on how broadly stars are initially

populated in phase space. The population of stars in phase space largely depends on the phase

mixing and chaotic mixing16. Stellar dynamicists apply the idea of the collisionless relaxation

even to globular clusters analogically17. The effect of collisionless relaxation brings a newly-born

star cluster in a QSS after a first few dynamical times. In other words, the DF does not explicitly

depend on time t

f (r, v, t) ∼ f (ε), (t << trel) (1.3.7)

where we assumed that the cluster is isotropic. If the collisionless relaxation drives the system into

a relaxed state, f (ε) ∼ exp[−ε/ < 3 >2] where < 3 > is the velocity dispersion.

16The elements of the DF in phase space can efficiently redistribute themselves in phase space through a timedependent potential. Then, the system can reach a QSS. The phase mixing is based on an integrable potential. Themixing obeys the Liouville’s volume-element invariant theorem. It extends the near-by elements of the DF linearlywith time in phase space. On the one hand, the chaotic mixing is based on the exponential instability of the stars’orbits. It extends the nearby DF elements in phase space exponentially with time.

17It is not clear whether globular clusters have actually experienced the collisionless relaxation and reached anisothermal state in the beginning of evolution. This is since currently-observed globular clusters are considered tohave experienced two-body relaxations. Hence, globular clusters might have experienced collisionless relaxation in adifferent way from galaxies.

13


Figure 1.4: Schematic graph for the Virial ratio against dimensionless time for an isolated self-gravitating system. Time t is normalized by the dynamical time tdyn. The ratio -2KE/PE is theVirial ratio. If the system is dynamically stationary, the Virial (=-2KE+PE) is zero according tothe Virial theorem. The graph shows the system is not dynamically stationary at t = 0. The Virialratio reaches unity after a few dynamical times due to the violent relaxation and mixing.

1.3.3 Collisional relaxation evolution and Orbit-Averaged Fokker-Planck

(OAFP) equation

The collisionless dynamical-evolution scenario breaks down on relaxation time scales because

of the two-body relaxation. Stellar dynamicists typically resort to a Fokker-Planck model averaged

over a certain time duration to handle the gap between the dynamical- and relaxationtime scales.

The finiteness of the total number N of stars affects the motion of a star in a globular cluster at a

QSS. It gradually deviates the star from its original smooth orbit. This deviation is directly caused

by stochastic irregular forces from the rest of the stars, corresponding to many-body Newtonian

interaction forces. Even after the stars largely deviate from their original orbits on a relaxation

time scale, the collisionless relaxation ’soon’ brings the system into another QSS on dynamical

times scales. Hence, the globular cluster can reach various QSSs on every relaxation time scale. In

14


this sense, the explicit time-dependence of DF may be retrieved, i.e., f (r, v, t) ≈ f (ε(t), t).18 The

corresponding m.f. potential is to be determined by the time-dependent Poisson’s equation

∂2φ

∂r2 +2r∂φ

∂r= ρ

[φ(r, t)

]≡ 16π2Gm

∫ 0

φ(r,t)f (ε, t)

√2ε′ − 2φ(r, t) dε′. (1.3.8)

The effect of many-body interaction has been conventionally modeled, in the first-order approxima-

tion of 1/N (Refer to Appendix B for one of 1/N expansions of the N-body Liouville equation19).

One can obtain the strict forms of kinetic equations including the effect of many-body interaction.

However, the numerical integration of those equations has not be done due to the complicated

mathematical structure. Hence, stellar dynamicists conventionally makes assumptions. For ex-

ample, the many-body and strong encounters are negligible, two-body encounters occur against a

homogeneously distributed field stars, and stars can not approach each other closer than the Landau

length (See Appendix A for the definition of the Landau length and Appendix B for the detailed

mathematical treatment of the assumptions.). The assumptions provide the following analytically

tractable FP kinetic equation in phase space

(∂

∂t+ v · ∇ − ∇φ(r, t) ·

∂

∂v

)f (r, v, t) =

− B∂

∂v·

∫| v − v′ |2

←→I − (v − v′)(v − v′)33

·

(∂

∂v−

∂

∂v′

)f (r, v, t) f (r, v′, t) d3v′, (1.3.9)

where typical dyadic expressions are employed, and B is a Coulomb logarithm (defined in equation

(C.2.5c)). The trajectory of test star is assumed rectilinear (Appendix A.4). See Appendix C.2 for

the derivation of equation (1.3.9). Following the discussion in Appendix A, in equation (1.3.9)

18The DF f (ε(t), t) includes the effect of two-stage evolution. The time scales of the arguments ε(t) and t are differ-ent. The time dependence of ε(t) is determined through m.f. potential, meaning ε(t) changes ‘rapidly’ on dynamicaltimes scales. On the one hand, the argument t is the order of relaxation time scales, hence the explicit time change inDF f is ‘slow’, which is caused by the two-body relaxation.

19Also, see (Gilbert, 1968, 1971; Kandrup, 1981; Chavanis, 2008, 2012; Ito, 2018a,b) for more statistically-exacttreatment of 1/N-expansion of N-body Liouville equation or Klimontovich-Dupree equation which includes the effectsof inhomogeneity in encounter, gravitational polarization, statistical acceleration and/or strong encounter.

15


the orders of the magnitudes of terms are as follows; the first term on the left-hand side and all

the (collision) terms on the right-hand side are ∼ ln[N]/N while the dynamical (second and third)

terms on the left-hand side are ∼ 1. One can simplify the equation to an equation with only the

order of ln[N]/N. First, one must rewrite equation (1.3.9) in spherical coordinates using equation

(1.3.2). However, encounters perturb both the DF and the m.f. potential. Hence, change in the DF

depends on time t not only explicitly but also implicitly through equation (1.3.2)

∂ f (r, v, t)∂t

=∂ f (ε, t)∂t

+∂ f (ε, t)∂ε

∂φ(r, t)∂t

, (1.3.10)

where the first term changes on relaxation time scale while the second changes on dynamical times

scales on the right-hand side. Hence, one needs to fill the large gap (ln[N]/N) in magnitude be-

tween the terms for practical numerical calculations. Henon (1961) resolved this issue and has

‘orbit-averaged’ FP equation (1.3.9). The orbit-averaging means an averaging over the radial pe-

riod between the apocenter and pericenter of the orbits. Mathematically, the averaging for isotropic

cluster corresponds to operating the integral∫δ(ε − 3

2

2 − φ(r, t)) · d3rd3v (See, e.g., Henon, 1961;

Spitzer, 1988; Heggie and Hut, 2003; Freitag, 2008; Binney and Tremaine, 2011). This process is

the same as the operation done for equations (1.3.3) and (1.3.5) in the present work. After apply-

ing the orbit-averaging to equation (1.3.9), one obtains the orbit-averaged Fokker-Planck (OAFP)

equation

∂ f (ε, t)∂t

∂q(ε, t)∂ε

+∂q(ε, t)∂t

∂ f (ε, t)∂ε

= Γ∂

∂ε

f (ε, t)

[f (ε, t)q(ε, t) − j(ε, t)

]+∂ f (ε, t)∂ε

[i(ε, t) + q(ε, t)g(ε, t)

],

(1.3.11a)

Γ ≡ (4πGm)2 ln N (1.3.11b)

16


where m is the stellar mass and G is the gravitational constant. The integrals associated with

dynamical friction and energy diffusion read

i(ε, t) ≡∫ ε

φ(0,t)f(ε′, t

)q(ε′, t

)dε′, (1.3.12a)

j(ε, t) ≡∫ ε

φ(0,t)

∂ f (ε′, t)∂ε′

q(ε′, t

)dε′, (1.3.12b)

g(ε, t) ≡∫ 0

ε

f(ε′, t

)dε′. (1.3.12c)

where φ(r = rmax, t) = 0 at the maximum radius rmax is assumed. The OAFP model can provide a

good qualitative understandings of relaxation evolution (Cohn, 1980; Takahashi, 1995).20

1.4 The motivation to apply spectral methods to the OAFP

equation

The numerical values of a solution to the OAFP equation significantly change over the energy

domain (Section 1.4.1). The large gap fully develops in the late stage of the relaxation evolution

of globular clusters in a self-similar fashion. The late stage of the evolution can be modeled by

the self-similar OAFP (ss-OAFP) equation (explained in detail in Section 2.2). The present work

speculates that the previous numerical works were not accurate enough to handle the large-scale

gap in solutions of the OAFP- and ss-OAFP- equations. This speculation leads us to apply a (highly

accurate) spectral method to the self-similar OAFP equation (Section 1.4.2).

20More realistic star clusters must be modeled anisotropic systems in velocity space based on statistical and dy-namical principles (Polyachenko and Shukhman, 1982; Luciani and Pellat, 1987) and numerical results (Cohn, 1979;Takahashi, 1995; Giersz and Spurzem, 1994; Baumgardt et al., 2002a).

17


1.4.1 Numerical difficulty in solving OAFP equation

Existing works have solved the OAFP equation based on Chang-Cooper finite difference

scheme (Chang and Cooper, 1970). The scheme can manage the time-scale gap between the FP-

and Poisson’s equations. Those works tested the conservation of the total energy of stars to con-

firm their numerical accuracies. On the one hand, the spatial gap itself has never been carefully

discussed. Especially, we doubt their numerical treatments for the spatial gap of the structure in

the late stage of the evolution. Standard stellar dynamics expects that, as time elapses, the stellar

DF (the solution of the OAFP equation) develops a power-law profile in the halo

f (ε, t) ∝ (−ε)β, (1.4.1)

where β ≈ 8.2 (Cohn, 1980). Generally, finding the form of a solution like equation (1.4.1) is

not a difficult task unless one needs the accurate numerical values of the power-law profile in the

limit ε → 0. However, the OAFP equation includes special structures in the density integral of

the Poisson’s equation (1.3.8) and the q-integral, equation (1.3.6). A local solution at a point on

the energy domain depends on the global (whole-domain) solution. Accordingly, one must know

the global solution before finding a local solution. This feature implies that one must correctly

count the small values of the DF, equation (1.4.1). The ’internal gap’ f (ε, t)/ f (φ(0, t), t) in the

stellar DF reaches the order of machine precision at ε ≈ 0.01.21 One can correctly understand

how the finer structure in f (ε, t) affects the whole-domain solution only after solving the OAFP

equation with adequate accuracy. The present work especially focuses on the ss-OAFP equation.

This equation can be used to describe the late stage of relaxation evolution. Also, it describes

an asymptotic solution of the OAFP equation as time t elapses. Accordingly, the solution of the

ss-OAFP equation represents the state that the ’internal gap’ in the DF has fully developed. Hence,

21The present work uses MATLAB whose machine precision is approximately 2.2 × 10−16. The internal gap in DFreaches the precision at ε = 0.012.

18


examining the ss-OAFP equation is ideal to test our speculation on the existing works. Also, the

solution of the ss-OAFP equation can be a reference for solving the OAFP equations.

Up to date, only a few works reported the solution of the ss-OAFP equation (Heggie and Steven-

son, 1988; Takahashi and Inagaki, 1992; Takahashi, 1993). Their works employed fourth-order

finite difference (FD) methods (e.g., deferred correction method). However, the FD methods are

not accurate enough to achieve the accuracy that the present work would like to reach. Their works

obtained solutions only on the truncated domain −0.22 / E < 0. Hence, our goal is to find a

whole-domain solution or at least a solution obtained on −0.01 / E < 0. To do so, we need a more

accurate numerical scheme compared to the fourth-order FD methods. The present work employs

a Chebyshev spectral method (Section 1.4.2).

1.4.2 Spectral method and (semi-)infinite domain problem

Spectral methods (e.g., Trefethen, 2000; Boyd, 2001; Shen and Tang, 2006; Shen et al., 2011)

are relatively new numerical methods known for their high efficiency and accuracy compared to

conventional methods (e.g., finite difference method and Runge-Kutta method). The conventional

methods are considered ‘local methods’. For the methods, one locally defines dependent variables

(unknowns) in equations on given domains. On the one hand, the spectral methods are consid-

ered ‘global methods’. They first series-expand dependent variables in given equations using base

functions (e,g, Fourier series, Chebyshev polynomials, Legendre functions, etc.). Then, the coeffi-

cients of the series-expansion become unknown ’spectrum’ for which one must solve the equation.

Especially, Chebyshev polynomials are convenient base functions. The polynomials have the ex-

plicit expressions of derivatives, integrals, and roots. The outstanding property of the Chebyshev-

polynomials expansion is the ‘spectral accuracy’. The spectral accuracy means that the coefficients

for a function (expanded by a basis function) decay faster than any decaying power-law function of

the degree of polynomials. This spectral accuracy provides a highly accurate solution. The relative

error in a spectral solution can reach the order of double precision only with polynomials of degree19


a few to tens if the solution is regular and analytic on a finite domain (See Chapter 3).

The binding energy ε in the ss-OAFP equation is finitely bounded while the Poisson’s equation

is defined on a semi-infinite domain r ∈ [0,∞). The latter has been intensively discussed as an

unbounded-domain problem in spectral-method studies. One may apply the Chebyshev spectral

methods to an infinite-domain problem only after properly regularizing the independent variables

in the Poisson’s equation. Application of spectral methods to differential equations on unbounded

domains has been important (e.g., Boyd, 2001; Shen and Wang, 2009; Shen et al., 2011). In fact,

the unbounded-domain problems are an active research field in computational physics and applied

mathematics (Table 1.1). The extensive list of the publications on the table details two recent

trends of spectral-method studies (i) One has extended conventional spectral methods (Rational

Chebyshev, Legendre, Hermite function...) to its fractional order and operational matrix, intro-

duces unused polynomials, and combines the conventional methods to other approaches (wavelet

analysis, multi-domain, Adomian decomposition method, ...) (ii) One has employed the conven-

tional methods after properly regularizing equations and variables based on the asymptotic or local

approximations of solution. Both the methods aim to improve the convergence rate of coefficients

to gain the spectral accuracy. The latter method especially focuses on avoiding non-analytic and

non-regular properties (singularity, discontinuity, branch point, diverging behavior, etc.) in func-

tions and equations concerned. To properly avoid the singularities, even finding new kinds of

singularities have been important works (e.g., Boyd, 1989, 1999, 2008b). Among the unbounded-

domain problems, the application of spectral methods to the Poisson’s equations (especially, the

Lane-Emden equations (Chandrasekhar, 1939)) is a challenging problem because the solution has

singularities at both the endpoints r = 0 and r → ∞. For astrophysical applications, many works

have actively contributed to spectral-method studies on the Lane-Emden equations and the variants

(Table 1.1). However, the existing works typically focus on regions near r = 0, such as r ≤ 1 (e.g.,

Caruntu and Bota, 2013) and r ≤ 10 (e.g., Parand et al., 2010). We first focus on making the

available numerical domain larger for the Lane-Emden equations. By employing an inverse map-

20


ping method (Section 4.1), we propose a method to obtain accurate solutions of the Lane-Emden

equations for the isothermal- and polytropic- spheres on a broad range of radii as our preliminary

work. The spheres are the imitations of the core and halo of globular clusters (Chapter 2). By ex-

tending the proposed method, the present work also aims to find a spectral solution of the ss-OAFP

equation as our main work. Since none of the existing works has accurately solved the ss-OAFP

equation, the present work also plans to reveal the thermodynamic and dynamic properties of the

ss-OAFP model and apply the solution to the observed structural profiles of globular clusters in

Milky Way.

.

1.5 The state-of-the-art and organization of the present

dissertation

As explained in Section 1.2, stellar-dynamics studies inherently have interdisciplinary impor-

tance. The present work aims to apply the Chebyshev spectral methods to the isothermal sphere,

polytropic sphere, and ss-OAFP model to find their accurate spectral solutions. These works con-

tribute to different fields with the following intellectual developments;

Computational physics & Applied mathematics (Chapters 6, 7, and 8)

−We report new kinds of singularities to the spectral-method community that are obtained

from spectral solutions for the isothermal- and polytropic- spheres and ss-OAFP model.

21


Equations Spectral method reference

Thomas-Fermi Rational Chebyshev (Parand and Shahini, 2009; Boyd, 2013; Zhang and Boyd, 2019)

Fractional-order Chebyshev (Parand and Delkhosh, 2017)

Volterra’s population Rational Chebyshev tau (Parand and Razzaghi, 2004)

fractional-order Rational Chebyshev (Parand and Delkhosh, 2016)

Gauss-Jacob (Maleki and Kajani, 2015)

sinc-Gauss-Jacobi (Saadatmandi et al., 2018)

multi-domain Legendre–Gauss (Maleki and Kajani, 2015)

Volterra integro-differential Jacobi (Huang, 2011; Shi et al., 2018)

Chebyshev-Legendre (Chen and Tang, 2009)

Dawson’s integral Rational Chebyshev (Boyd, 2008a)

exponential Chebyshev (Ramadan et al., 2017)

Burger’s Laggure-Galerkin (Guo and Shen, 2000)

Hermite (Ma and Zhao, 2007; Albuohimad and Adibi, 2017)

exponential Chebyshev (Albuohimad and Adibi, 2017)

Lane-Emden Chebyshev operational matrix (Doha et al., 2013a, 2015)

Jacob Rational (Doha et al., 2013b)

Ultraspherical wavelet (Youssri et al., 2015)

Radial basis function (Parand and Hemami, 2016; Parand and Hashemi, 2016)

Rational Chebyshev (Parand and Khaleqi, 2016)

Shifted Legendre (Guo and Huang, 2017)

Legendre-scaling Operational matrix (Singh, 2018)

Shifted Gegenbauer integral (Elgindy and Refat, 2018)

Shifted-Legendre Operational matrix (Tripathi, 2018)

orthonormal Bernoulli’s polynomials (Sahu and Mallick, 2019)

Chebyshev Operational matrix (Sharma et al., 2019)

Table 1.1: Some recent spectral-method studies on (integro-) ordinary differential equations on(semi-)infinite domain. The spectral methods are collocations methods unless they are specifiedwith other approaches (e.g., tau. Galerkin, operational matrix,...). Due to enormous publicationsfor the Lane-Emden equations, the list only includes part of publications between 2013-2019.

22


Astronomy - Stellar dynamics (Chapters 8 and 10)

−We document the usefulness of spectral methods to the stellar-dynamics community by

solving an unsolved fundamental kinetic equation (the ss-OAFP equation) in the field.

−We report the physical parameters of globular clusters based on the ss-OAFP model.

−We propose an energy-truncated ss-OAFP model that can fit the structural profiles of

globular clusters with observable cores in Milky Way while existing models can not.

−We show that low-concentration globular clusters (clusters at the early stage of

relaxation evolution) have structural profiles modeled by polytropic spheres.

Statistical dynamics (Chapters 9 and 10)

−We detail the negative heat capacity of the ss-OAFP model and explain a mechanism

to hold negative heat capacity in a thermalized system on relaxation time scales.

−We suggest a relation between the generalized statistical mechanics based on Tsallis

entropy and many-body relaxation in star clusters with long core-relaxation time.

The organization of the present dissertation is as follows. Chapter 2 details the Lane-Emden

equations and the ss-OAFP equation. In the chapter, we also introduce the King model (King,

1966) as one of the QSS models. The King model is finite in size; hence, we can highlight

the difference between finite-domain and unbounded-domain problems for the rest of the chap-

ters. Chapter 3 explains the Gauss-Chebyshev collocation spectral method and Fejer’s first-rule

quadrature employed for our numerical calculation. We show a straightforward application of the

spectral method to the King model and explain the basic idea of the numerical scheme employed

in the present dissertation. Chapter 4 shows the mathematical formulation and regularization of

the mathematical models. The formulations are further arranged for the application of the spec-

23


tral method. Chapter 5 explains how to deal with difficulties in the numerical integration of the

ss-OAFP model. Chapters 6, 7, and 8 report accurate spectral solutions of the Lane-Emden equa-

tions for isothermal- and polytropic- spheres as the preliminary result, and the ss-OAFP equation

as the main result. To discuss the physical properties of the ss-OAFP model, Chapter 9 explains

the thermodynamic- and dynamic- aspects of the model. As an application of the ss-OAFP model,

Chapter 10 proposes a new phenomenological model, i.e., an energy-truncated ss-OAFP model

that fits the structural profiles of at least half of globular clusters with observable cores in Milky

Way. Chapter 11 concludes the dissertation.

24

Chapter 2

Mathematical models for the evolution of

globular clusters

An isotropic spherical globular cluster evolves in two-stage fashion due to the large gap be-

tween the dynamical- and relaxation time scales, hence one can model the evolution by either of

a QSS model and relaxation-evolution model. Section 2.1 details mathematical formulations for

the QSS models. The isothermal and polytropic spheres are basic QSS models defined on an un-

bounded domain. Also, the same section introduces the King model that is also a QSS model but

defined on a finite domain. The King model can effectively highlight the difference between finite-

and unbounded- domain problems in spectral methods for the rest of the chapters. Section 2.2

explains the ss-OAFP equation as the main relaxation-evolution model.

2.1 Quasi-stationary-state star cluster models

Astronomers observe globular clusters using photometry methods such as space-based and

ground-based telescopes1. The photometry methods reveal the current structures of globular clus-

1For example, Hubble space telescope and Chandra X-ray observatory are space-based, and VISTA(Visible andInfrared Survey Telescope for Astronomy) and VLT(Very Large Telescope) are ground-based in Chile.

25

CHAPTER 2. MATHEMATICAL MODELS

ters. For example, the spatial profiles (e.g., the surface brightness and surface number-density

profiles) and line-of-sight velocity dispersion profile 2. The profiles are directly related to the m.f.

potential φ(r) and DF f (ε) of stars (See, e.g., Binney and Tremaine, 2011, Ch.4). Hence, one is

concerned with finding a realistic m.f. potential and DF to imitate the observed profiles. The colli-

sionless relaxation evolution infers that the current structures of globular clusters are ’frozen’ into

one of the infinitely possible QSSs in the relaxation-evolution process. However, solving time-

dependent equations for the relaxation evolution is not a mathematically-easy task. One typically

prescribes a DF to model a globular cluster based on physical principles (or at least for mathe-

matical convenience). Hence, a mathematical program to establish a QSS model is to choose a

predetermined time-independent DF f (ε) first, and then find the m.f. potential due to the DF by

solving time-independent Poisson’s equation

d2φ

dr2 +2r

dφdr

= 4πGρ[φ(r)

]≡ 4π

∫ εmax

φ(r)f (ε)

√2[ε − φ(r)

]dε′, (2.1.1)

where the domain is defined by ε ∈[φ(r), εmax). The εmax takes 0 or ∞ depending on if the model

includes escaping stars or not in the system. The present work focuses on the three QSS models; the

isothermal sphere, polytropic sphere, and King model. They are fundamental QSS models in stellar

dynamics (Spitzer, 1988; Heggie and Hut, 2003; Binney and Tremaine, 2011). The isothermal

and polytropic spheres can imitate the structures of the cores and inner halos of globular clusters

(Sections 2.1.1 and 2.1.2). The King model is the most fundamental model for stellar dynamicists

to directly fit its solution to projected structural profiles of observed globular clusters (Section

2.1.3).2Refer to (e.g., Meylan and Pryor, 1993; Binney and Tremaine, 2011) for discussion regarding the relationship

between observed data and spatial- and velocity-dispersion profiles.

26


2.1.1 Isothermal sphere

The isothermal sphere directly reflects the effect of two-body relaxation process that occurs the

most frequently in the cores of globular clusters. In the relaxation process, one is concerned with

if the Boltzmann-Grad entropy (i.e., H-function)

S (t) =

∫ ∫f (r, 3, t) ln

[f (r, 3, t)

]d33d3r = (4π2)

∫f (ε, t) ln

[f (ε, t)

] ∂q(ε, t)∂ε

dε, (2.1.2)

can monotonically increase and reach a certain (at least local) maximum. Assume the isotropic

OAFP equation (1.3.11a) properly models the relaxation evolution. Then, one can prove the H-

theorem3 after executing simple calculation employing the OAFP equation (1.3.11a)

dSdt

=Γ

2

∫ ∫ εmax

φ(0,t)f (ε, t) f

(ε′, t

) (∂ f (ε, t)∂ε

−∂ f (ε′, t)∂ε′

)2

min[q(ε, t), q

(ε′, t

)]dεdε′,

≥ 0, (2.1.3)

where min[q(ε, t), q(ε′, t)

]is the minimum of q(ε, t) and q(ε′, t). Hence, the possible final station-

ary state4 of the isotropic OAFP model is an isotropic Maxwell-Boltzmann DF (or Maxwellian

DF)

f (ε) =ρc(

2πσ2c)3/2 exp

(−ε/σ2

c

), (2.1.4)

3The H-theorem is correct even for more rigorous models based on statistical first principles, including the effectof anisotropy (Polyachenko and Shukhman, 1982), the gravitational polarization (Heyvaerts, 2010; Chavanis, 2012)and statistical acceleration (Ito, 2018c). The stationary solution of the corresponding kinetic equations is an isotropicMaxwellian DF that satisfies the condition that the extremum of entropy is zero (δS = 0).

4In the present work, Maxwellian DF is considered one of the QSS models. Strictly speaking, it may not be correctsince a QSS means a long-lived (metastable) dynamical state approaching the state of thermal equilibrium. However,a finite isothermal sphere can not exist (Antonov, 1962; D. Lynden-Bell and Royal, 1968). Also, the mathematicalprogram to find the m.f. potential for the Maxwellian DF is the same as that for polytrope and King model. Hence,the isothermal sphere is still an imitation of the core of a globular cluster. Some discussion regarding Maxwellian DFin collisionless dynamics (due to violent relaxation and mixing) will be found in (e.g., Nakamura, 2000; Bindoni andSecco, 2008).

27


where ρc is the central density and σ2c the normalized temperature. The temperature satisfies the

equipartition relation with the velocity dispersion√⟨32

⟩as follows < 32 > /2 = 4π

∫f (ε)32d3 =

3σ2c/2. When the system is modeled by the Maxwellian DF and equation (2.1.1), it is called the

isothermal sphere model.

The isothermal sphere has an infinite total energy E, mass M, and radius R. This property is

not suitable for modeling globular clusters since the actual globular clusters are finite in size and

energy. Due to the nature of self-gravitation and two-body relaxation, a globular cluster keeps de-

veloping a denser core and sparser halo on relaxation time scales, which does not allow the cluster

to be at stationary state modeled by a finite-size isothermal sphere. Assume that the core behaves

like the isothermal sphere since the relaxation time is shorter in the denser part of the cluster. Also,

assume that the core is enclosed by an adiabatic wall of radius Rc to avoid the infinite radius. Then,

once the density contrast ρ(0)/ρ(Rc) reaches a value greater than 709 and/or the normalized energy

RcE/GM2 is less than −0.335 (Antonov criteria (Antonov, 1962)), the system begins to undergo

the ‘gravothermal catastrophe/instability (D. Lynden-Bell and Royal, 1968). This instability infers

that the enclosed core can not hold the stable state of (local) thermodynamics equilibrium (See

Padmanabhan, 1989; Chavanis, 2003; Casetti and Nardini, 2012; Sormani and Bertin, 2013, for re-

view and references therein). The gravothermal instability brings the cluster’s core into a collapse

(i.e., its radius reaches extremely small and central cluster density becomes infinitely large.), which

is called the core collapse of the relaxation evolution. Possible direct applications of the isother-

mal sphere are very limited. For example, one may apply it to the globular clusters at the early

stage of relaxation evolution before the system undergoes a core-collapse phase. Numerical re-

sults (Taruya and Sakagami, 2003, 2005) predict that the globular cluster may reach the isothermal

sphere subject to a certain initial DF.

28


The radial density profile of the isothermal sphere reads

ρ(r) = ρc exp[−φ(r) − φ(0)

σ2c

]. (2.1.5)

For numerical calculation, the potential φ(r), radius r, and density ρ(r) reduce to the following

dimensionless forms

φ(r) = −φ(r) − φ(0)

σ2c

, (2.1.6a)

r = r

√4πGρc

σ2c

, (2.1.6b)

ρ(r) =ρ(r)ρc

. (2.1.6c)

The function φ(r) is called the LE2 function if it satisfies the Lane-Emden equation of the

second kind (LE2 equation)5 for dimensionless potential

d2φ

dr2 +2r

dφdr− e−φ = 0, (2.1.7)

with the following boundary conditions

φ(r = 0) = 0,dφdr

(r = 0) = 0. (2.1.8)

5The LE2 equation has been the basic model of the isothermal gas sphere (IGS) in classical stellar-structure theory(Eddington, 1926; Chandrasekhar, 1939), stellar dynamics (Chandrasekhar, 1942; Kurth, 1957) and nonlinear analysisin applied mathematics (Davis, 1962). It describes a self-gravitating ideal gas of particles interacting through pairwiseNewtonian potential and direct particle collisions. The gas is in a state of hydrostatic equilibrium at a constant tem-perature. The IGS provides the simplest model for spherical self-gravitating gases in which collisional (relaxation)time scale is much shorter than dynamical/chemical/nuclear time scales. Despite neglecting important physical effects,the IGS remains an approximation for star clusters: cores of globular cluster (Spitzer, 1988; Heggie and Hut, 2003;Binney and Tremaine, 2011), and nuclei of dense galaxies without black holes (Bertin, 2000; Merritt, 2013).

29


One also can find the LE2 equation for dimensionless density ρ using equation (2.1.5)

ρd2ρ

dr2 −

(dρdr

)2

+2rρ

(dρdr

)+ ρ3 = 0, (2.1.9)

with the following boundary conditions

ρ(r = 0) = 1,dρdr

(r = 0) = 0. (2.1.10)

Using equations (2.1.8) and (2.1.10), one obtains the Taylor expansions around r = 0

φ(r) =r2

6−

1120

r4 + . . . , (2.1.11a)

ρ(r) = 1 −r2

6+

145

r4 − . . . . (2.1.11b)

The asymptotic approximations as r → ∞ are

φ(r) ≈ φs(r) +C3

r1/2 cos √7

2ln r + C4

, (2.1.12a)

ρ(r) ≈ ρs(r)1 +

C3

r1/2 cos √7

2ln r + C4

, (2.1.12b)

where C3 and C4 are constants of integration. The isothermal sphere modeled by

φs(r) = ln2r2 , (2.1.13a)

ρs(r) =2r2 , (2.1.13b)

is called the singular isothermal sphere.

30


2.1.2 Polytropic sphere

The present section introduces a polytropic-sphere model that could model the power-law

structural profiles of globular clusters. The previous numerical and observational results have

revealed that the densities of inner halos behave like a power-law profile

ρ(r) ∝1rα, (2.1.14)

for collapsing-core globular clusters. The power-law index α is an essential physical parameter

(observable) to characterize the structure of globular clusters. To model the power-law profile,

the polytropic-sphere model is one’s first choice. This is since it models any power-law density

profiles with various power-indexes. The present section (i) explains the power-law density profiles

obtained from numerical- and observational- results first, and then (ii) introduces the polytropic-

sphere model.

(i) Power-law density profiles

Plentiful numerical approaches have predicted the value of α (see Baumgardt et al., 2003, for

review). By borrowing the results of (Baumgardt et al., 2003) and limiting our focus into isotropic

globular clusters, we can show the numerical results for the index

α =

2.20, Self-similar moment model (Louis, 1990),

2.208, Self-similar gaseous model (Lynden-Bell and Eggleton, 1980),

2.23, Time-dependent OAFP model (Cohn, 1980),

2.23, Self-similar OAFP model (Heggie and Stevenson, 1988).

(2.1.15)

31


The values of α based on the moment- and gaseous- models6 are slightly different from the index

for the isotropic OAFP models. However, the corresponding anisotropic- moment model (Louis,

1990), gaseous model (Louis and Spurzem, 1991), and time-dependent OAFP model (Takahashi,

1995) showed α ≈ 2.23.7 Hence, stellar dynamicists accept the theoretical value α ≈ 2.23 as the

results of continuum models.

On the one hand, the ‘experiments’ or N-body simulations8 showed the reasonable values of

the exponent compatible to the continuum models

α =

2.36, for N = 3.2 × 104 (Makino, 1996),

2.33, for N = 4 and 8 × 103 (Baumgardt et al., 2002a)

2.26, for N = 6.4 × 104 (Baumgardt et al., 2003),

2.2, for N = 4 × 103 ∼ 6.5 × 104 (Gieles et al., 2013)

2.32 ± 0.07, for N = 1.0 × 104 (Pavlık and Subr, 2018)

2.27 ± 0.03. for N = 5.0 × 104 (Pavlık and Subr, 2018)

(2.1.16)

The relatively new result of (Gieles et al., 2013) showed a lower value (α = 2.2) compared to the

other results. For this, Gieles et al. (2013) explained that the difference occurred since they used

data acquired at the early stage of the core collapse while the rest of the works used data at the late

stage.

6The mathematical expression of a moment model is composed of equations for the time-evolution of the moments(density, kinetic energy, velocity dispersion,..., ) of the FP equation. The gaseous models are an imitation of a self-gravitating gas that has thermal conductivity (through direct physical collision between particles). Hence energy- andparticle- flow occurs due to the spatial change in gas temperature through the collisions. Those models are numericallymore efficient than the OAFP model since they do not include integrals in their mathematical expressions. However,one necessarily compares the models to the OAFP model or N-body simulation. This is since one must artificiallycutoff the higher order moments for the moment models and heuristically determine the mathematical expression ofthermal conductivity for the gaseous conductive models.

7The anisotropic models are more exact than isotropic ones since they include the effect of anisotropic velocitydispersion or the modulus of angular momentum.

8The N-body simulations include the effect of binaries, strong encounter and ejection while the continuum modelsdo not.

32


In the applications of the power-law profile to the actual globular clusters, the exponent α =

2.23 has only a fundamental importance. One would be hard to find a power-law profile of a core-

collapsing globular cluster with α = 2.23 in nature9. In the actual globular clusters, binary stars

release a gravitational energy in the core-collapsing core and hold the core collapse process (e.g.,

Binney and Tremaine, 2011). Also, the core-collapse process depends on realistic effects (e.g.,

mass function, evaporation, tidal shock and internal rotation). The followings are the examples of

the observed exponents

α =

2.2, NGC 7078 (or M15) (Gebhardt et al., 1997),

3.5, MGC1 in NGC 224 (or M31) (Mackey et al., 2010),

3.42 and 3.48. JM81GC-1 and JM81GC-2 (Jang et al., 2012),

(2.1.17)

The exponent for globular cluster NGC 7078 (New General Catalogue 7078) well agrees with the

exponent α = 2.23. However, one should not jump to a conclusion. NGC 7078 is considered to

have experienced a core-collapse at least once. One needs to deepen the understanding of NGC

7078 by comparing realistic FP models (e.g., Dull et al., 1997; Murphy et al., 2011) and N-body

simulations (e.g., Baumgardt et al., 2002b) that were arranged for NGC 7078 to fit the simulated

data to the more current observed data. On the one hand, the isolated globular clusters in M31

and M81 (NGC 3031) groups have a physically simple feature. One may directly compare the

physical structure to the results of N-body simulations for isolated globular clusters10. The clusters

show high exponents 3.4 ∼ 3.5 compared to 2.23. Mackey et al. (2010) interpreted that the large

value of the exponent is consistent with the exponents for both the inner halo of the N-body direct

simulation (Baumgardt et al., 2002a) and the outer halo of anisotropic OAFP model (Takahashi,

9Astronomers can observe projected structural profiles rather than the actual three dimensional structure. Projectedsurface profiles have power-law index is α − 1 = 1.23 as discussed in Chapter 10.

10The isolated globular clusters in M31 and M81 groups have relatively-similar astrophysical conditions employedfor basic N-body direct simulations for isolated systems. The clusters exist far away from the host galaxies and therest of the clusters.

33


1995). According to (Baumgardt et al., 2002a), the exponent α ≈ 2.33 appears at limited radii

smaller than the radii at which the high exponent (∼ 3.5) appears.

(ii) Polytropic sphere model

To model the fractional power-law density profile in the inner halo of a globular cluster, one

can employ a polytropic sphere model. The DF for stars in the polytropic sphere of index m can

be written as a power-law function11 of energy

f (ε) =

ρc(

2πσ2c)3/2

m!(m − 3/2)!

(−ε

σ2c

)m−3/2

, (ε < 0)

0, (ε > 0)

(2.1.18)

When a system is modeled by the Poisson’s equation (2.1.1) for the DF (2.1.18), it is called the

stellar polytrope (Taruya and Sakagami, 2002; Chavanis, 2002b; Lima and de Souza, 2005). In

equation (2.1.18), the factor m!(m−3/2)! is inserted so that the corresponding density profile has a finite

central density ρc at r = 0

ρ(r) = ρc

(−φ(r)σ2

c

)m

. (2.1.19)

11The original polytropic sphere for stellar structure in (Chandrasekhar, 1939) is at a state of local thermodynamicstate. Hence the corresponding DF is a local Maxwellian DF rather than the power law (See (e.g., Chavanis, 2002b)for discussion of gaseous and stellar polytropes).

34


For numerical calculations, the potential φ(r), radius r, and density ρ(r) may reduce to dimension-

less forms

φ(r) = −φ(r)σ2

c, (2.1.20a)

r = r

√4πGρc

σ2c

, (2.1.20b)

ρ(r) =ρ(r)ρc

. (2.1.20c)

Then, the time-independent Poisson’s equation (2.1.1) reduces to the Lane-Emden equation of the

first-kind (LE1 equation)

d2φ

dr2 +2r

dφdr

+ φm = 0. (2.1.21)

The function φ is called the LE1 function when it satisfies the LE1 equation with the boundary

conditions

φ(r = 0) = 1,dφdr

(r = 0) = 0. (2.1.22)

The power m in equation (2.1.21) is called the polytropic index. Also, a self-gravitating system

modeled by equation (2.1.21) is called the polytropic sphere of index m.12 The reason why φ

is fixed to unity at r = 0 (in place of −φ(0)/σ2c) is that the LE1 function is invariant under the

homological transformation (Appendix E).

12The polytrope itself is a heuristic model. It typically applies to physical problems in astrophysics, thermal en-gineering, and atmospheric science. Originally Thomson (1862) introduced the polytrope. Chandrasekhar (1939)developed it for stellar structures. One may use the polytrope model to discuss processes other than the adiabatic andisothermal process of a thermodynamic system by controlling the value of the index m (Horedt, 1986). For example,the index m → ∞ corresponds to the isothermal sphere (Hunter, 2001) and m = 1.5 to the isentropic sphere (e.g.,McKee and Holliman II, 1999), and the negative value of m to spheres subject to external effects(e.g., McKee andZweibel, 1995). As the most fundamental example, stars (like Sun) on the main-sequence may be modeled by a poly-trope of m = 3 for the inner region (to include radiation pressure) and polytrope of m = 1.5 for the outer region (thatis fully convective).

35


In stellar-dynamics studies, one may constrain the range of index m and power α for globu-

lar clusters undergoing relaxation evolution. For example, the power can range 2.0 < α < 2.5

according to the self-similar analysis on an isotropic gaseous model (Lynden-Bell and Eggleton,

1980). In fact, the range of exponents shows a good relation with the numerical and observed data

2.2 ≤ α ≤ 2.5 (equations (2.1.15), (2.1.16) and (2.1.17)). Accordingly, the present work focuses

on the range 2.0 < α < 2.5. Employing the relationship between the power α and index m

α =2m

m − 1or m =

α

α − 2, (2.1.23)

the corresponding polytrope index is limited to 5 < m < ∞. The polytrope of m = ∞ is essentially

the isothermal sphere (Hunter, 2001) while the polytrope of m = 5 corresponds to the Plummer

model (Heggie and Hut, 2003; Binney and Tremaine, 2011).13 As a reference, the power α = 2.23

corresponds to the index m = 9.7.

In a similar way to the isothermal sphere, the applications of polytrope spheres of m > 5 are

limited. The spheres also have an infinite radius, mass, and energy. Only some N-body direct

simulations (e.g., Taruya and Sakagami, 2003, 2005) showed that the polytrope of m > 5 might

characterize a certain relaxation evolution of a globular cluster before the cluster undergoes a core

collapse.

The known asymptotic- and local- approximations for the polytropes14 are

φ(r) =

1 −

r2

6+

m120

r4 + . . . , (r → 0)

φs(r)[1 +

C1

rηcos (ω ln r + C2)

]≡ φa(r), (r → ∞)

(2.1.24)

13The Plummer model is not realistic since the radius of the model reaches infinity. However, stellar dynamicistshave used it to quantitatively discuss the relationships among the total mass, total energy, and velocity dispersion sincetheir explicit analytical forms are known.

14The limit m → ∞ of the oscillatory part of φ corresponds to that of equation (2.1.12a) for the isothermal sphere.See (Hunter, 2001) for the relation between polytropic and isothermal gases, and (Chandrasekhar, 1939; Horedt, 1987,2004; Adler, 2011) for the more detailed discussions of the asymptotic behaviors of the LE functions.

36


where C1 and C2 are the constants of integration, and ω and η read

ω =

√7m2 − 22m − 1

2(m − 1), (2.1.25a)

η =m − 5

2(m − 1). (2.1.25b)

The φs(r) is called the singular solution of the LE1 equation. Its explicit expression is

φs(r) =Am

rω, (2.1.26a)

Am =

[2(m − 3)(m − 1)2

]ω/2, (2.1.26b)

ω =2

m − 1. (2.1.26c)

2.1.3 King model

The present section introduces the King model that is the most fundamental fitting model in

stellar dynamics and fits the structural profiles of the majority of globular clusters in Milky Way.

As explained in Sections 2.1.1 and 2.1.2, the isothermal sphere and polytropic spheres of m > 5

can not preferably apply to the actual globular cluster due to their infinite radii, total masses,

and total energies. The size of the actual clusters is finite due to the tidal effect from the host

galaxies and gravitational Jeans instability (Jeans, 1902). Hence, one must introduce an artificial

maximum radius of globular-cluster models. The most standard finite-size model is known as the

Michie-King model (Michie, 1963; King, 1966). Since the Michie model (Michie, 1963) describes

anisotropic systems, the present work focuses only on the King’s isotropic model (King, 1966)15.

King and the coworkers examined the applicability of the King model to the globular clusters in

Milky Way (King, 1966; Peterson and King, 1975) and identified that the King model could well

fit 80 % of the observed projected structural profiles of Milky-Way (Galactic) globular clusters

15See (Meylan and Heggie, 1997) and references therein to find the application of Michie-King models.

37


(Djorgovski and King, 1986; Trager et al., 1995)16.

The King model’s fundamental idea is to imitate the actual finite-size globular clusters by

modifying the isothermal-sphere model so that the system does not include stars escaping from the

isothermal sphere. Hence, the DF for the isothermal sphere is zero at energy ε ≥ 0 and reduces to

the lowered-Maxwellian DF

f (ε) =

ρc

(2πσ2c)I(K)

[exp

(−ε/σ2

c

)− 1

], (ε < 0)

0, (ε > 0)

(2.1.27)

where the constant K and function I(x) are

K = −φ(0)σ2

c, (2.1.28a)

I(x) = exp(x)erf(√

x)−

√4xπ

[1 +

2x3

], (2.1.28b)

where erf(x) is the error function; erf(x) = 2 exp[x2

] ∫ x

0exp

[−x2

]dx/√π. We inserted I(x) in

equation (2.1.27) so that the density can have a finite density ρc at r = 0

ρ(r) = ρc

I(−φ(r)/σ2

c

)I(K)

. (2.1.29)

Using the same change of variables (equations (2.1.20a)-(2.1.20c)) as the LE1 equation, one can

obtain the ordinary differential equation (ODE) to describe the King model for dimensionless po-

tential φ in terms of dimensionless radius r

d2φ

dr2 +2r

dφdr−

I(φ)I(K)

= 0. (2.1.30)

16The rest of the clusters are considered at a phase of post core collapse whose central core has a cusp (a steepslope in a power-law density profile diverging at r = 0) unlike the King model and isothermal sphere whose cores arerelatively flattened (See, e.g., Lugger et al., 1995; Noyola and Gebhardt, 2006; de Boer et al., 2019, for the observeddata of the cores of collapsed-core clusters).

38


The King model satisfies the boundary conditions

φ(r = 0) = K,dφdr

(r = 0) = 0. (2.1.31)

The local approximation near r = 0 is

φ(r) = 1 −r2

6+

2 exp(K)erf(√

K) −√

Kπ

120I(K)r4 + . . . (2.1.32)

The lowered-Maxwellian DF in the limit of ε → 0− behaves like the polytrope of index m = 2.5

f (ε → 0−) =ρc

(2πσ2c)I(K)

−ε

σ2c. (2.1.33)

If the central potential is vanishingly low, then only the first order of φ is significant in the function

I(φ) in the limit of φ→ 0

I(φ→ 0) = −5π

6√

2φ3/2. (2.1.34)

Correspondingly, the King model reduces to a kind of the LE1 equation of m = 2.5

d2φ

dr2 +2r

dφdr

+

[φ(r)K

]2.5

= 0. (φ→ 0) (2.1.35)

The polytrope of m = 2.5 is known to be finite in size (Chandrasekhar, 1939). This implies that

as the radius r increases, the potential φ decreases. Hence, the potential finally reaches zero. King

(1966) showed this property for the King model numerically (except for K → ∞). The radius at

which the potential reaches zero is called the tidal radius rtid (sometimes limiting radius) of the

39


King model

r = rtid if φ(r) = 0. (2.1.36)

The King model does not have a homological property (Appendix E), unlike the polytropic and

isothermal spheres. Hence, the density profile ρ, potential φ, and tidal radius rtid of the King model

have a parametric dependence. The parameter is the normalized potential K. Also, one needs a

reference radius in comparison with the tidal radius. King (1966) defined the King radius

rKin =

√4πGρc

9σ2c. (2.1.37)

The King radius is considered the core radius of the King model. The logarithmic ratio of the tidal

radius rtid to the King radius is called the concentration c of the King model

c = log[

rtid

rKin

]. (2.1.38)

2.2 Self-similar Orbit-Averaged Fokker-Planck (ss-OAFP)

model

Astronomers have commonly found self-similarities (scale-free symmetries) in the relaxation-

and dynamical- evolutions of self-gravitating systems. The examples of the systems that may have

self-similarities are stars, molecular clouds, galaxies, the clusters of galaxies (e.g., Wesson, 1979).

As partly discussed in Section 2.1.2, stellar dynamicists have also considered that globular clus-

ters would experience a self-similar evolution since the early works (e.g., Henon, 1961; Larson,

1970; Lynden-Bell and Eggleton, 1980). This was confirmed at a theoretical level for the (time-

dependent) OAFP model (Cohn, 1980; Takahashi, 1995) and an N-body direct simulation (Pavlık

40


and Subr, 2018). Cohn (1980) showed that the DF for stars is similar to the lowered-Maxwellian

DF at the early stage of relaxation evolution, while the OAFP model undergoes a self-similar evolu-

tion in the late stage. Heggie and Stevenson (1988) applied a self-similarity analysis on the OAFP

system for a collapsing-core globular cluster17 and found the self-similar solution of the OAFP

system, which confirmed (Cohn, 1980)’s results. Section 2.2.1 details the mathematical derivation

of the ss-OAFP equation, and Section 2.2.2 highlights difficulties in the numerical integration of

the ss-OAFP system.

2.2.1 Self-similar analysis on the OAFP equation

The present section reviews the formulation established by (Heggie and Stevenson, 1988). To

reflect the self-similar evolution of a collapsing-core isotropic globular cluster, Heggie and Steven-

son (1988) employed the following self-similar variables in the OAFP equation (1.3.11a) and Pois-

son’s equation (1.3.8) for independent variables

E = ε/εt(t), (2.2.1a)

R = r/rt(t), (2.2.1b)

17Strictly speaking, the self-similar analysis on the OAFP equation itself (and even the orbit-averaging of the FPequation) was first done in (Henon, 1961, 1965). In the work, the system is discussed with the physical conditions thata star cluster is immersed in the host galaxy. This condition is different from the present work and (Cohn, 1980; Heggieand Stevenson, 1988). The present work focuses on the self-consistent conditions by which we mean (the exponents of)power-law profiles in physical quantities are to be determined self-consistently by the OAFP- and Poisson’s equations.On the one hand Henons’ models expect the fixed boundary conditions by which we mean the boundary conditionsare strictly imposed with certain numerical values or in behavioral fashions. For example, Henon (1961) assumedthat the system size is limited and the DF is zero at a certain tidal radius by the tidal effect due to the host-galaxypotential. Also, Henon (1965)’ model is a cluster with an infinite radius, but the potential of the system is assumed tofollow Keplerian ∼ 1/r due to the host galaxy. Hence, one does not find the conceptually-important exponent α ≈ 2.23(discussed in Section 2.1.2) from Henon’s models.

41


and for dependent variables

F(E) = f (ε, t)/ ft(t), (2.2.2a)

Q(E) = q(ε, t)/qt(t), (2.2.2b)

Φ(R) = φ(r, t)/φt(t), (2.2.2c)

I(E) = i(ε, t)/it(t), (2.2.2d)

J(E) = j(ε, t)/ jt(t), (2.2.2e)

G(E) = g(ε, t)/gt(t), (2.2.2f)

where the subscript t means that the variables depend only on time t. Using equations (2.2.1) and

(2.2.2), we can reduce the OAFP system into the ss-OAFP system. The latter is composed of four

ordinary differential equations (4ODEs)

[I(E) + G(E)Q(E)]dFdE

= c1Q(E)F(E) +2c1 − 3c2

4J(E) − F(E) [F(E)Q(E) − J(E)] , (2.2.3a)

dGdE

= −F(E), (2.2.3b)

dIdE

= Q(E)F(E), (2.2.3c)

dJdE

= Q(E)dFdE

, (2.2.3d)

the Q-integral

Q(E) =13

∫ Rmax(E)

0

[E − Φ

(R′

)]3/2 R′3dR′,(Rmax(E) = Φ−1(E)

), (2.2.4)

and the Poisson’s equation

∂2Φ

∂R2 +2R∂Φ

∂R= D [Φ(R)] ≡

∫ 0

Φ(R)F

(E′

) √E′ − Φ(R)dE′. (2.2.5)

42


The relationships among the time-dependent variables read

φt(t) = εt(t), (2.2.6a)

rt(t) =1(

16√

2π2mG ft(t)√εt(t)

)1/2 , (2.2.6b)

qt(t) =

(2εt(t)[rt(t)]2

)3

2. (2.2.6c)

where rt is the core radius at time t (Lynden-Bell and Eggleton, 1980). This core radius corresponds

to that all the physical quantities in the ss-OAFP system are normalized to the dimensionless forms

(equations (2.2.1) and (2.2.2)) by the time-dependent quantities associated with the core size rt.

For example, ρ(r, t) (≡ ρt(t)D(R)) is the density of the halo and ρt(t) is the density at radius rt (at

which the density of the halo is the maximum). The self-similar parameters are defined as

c1 =1

Γ ft(t)ddt

ln[ ft(t)], (2.2.7a)

c2 =1

Γ ft(t)ddt

ln[Et(t)], (2.2.7b)

and the corresponding physical parameters are

α =2(3 + 2β)

2β + 1, (the exponent of power-law density profile) (2.2.8a)

ξ =c1 + c2

0.167√π, (core-collapse rate) (2.2.8b)

χesc =FBC(FBC − c1)

c3, (scaled escape energy), (2.2.8c)

where c3(≡ G(E = −1)) is the third eigenvalue, and FBC is a boundary value to be assigned. We

define the new eigenvalue β (≡ c1/c2) that characterizes the power-law profiles of stars in the halo.

43


The boundary conditions for the 4ODEs and the Q-integral are

F(E → 0) = c4(β + 1)(−E)β, F(E = −1) = FBC, (2.2.9a)

G(E → 0) = c4(−E)β+1, G(E = −1) = c3, (2.2.9b)

I(E → 0) = c44(β + 1)2β − 7

(−E)β+1Q(E → 0), I(E = −1) = 0, (2.2.9c)

J(E → 0) = −c44β(β + 1)

2β − 3(−E)βQ(E → 0), J(E = −1) = 0, (2.2.9d)

Q(E → 0) ∝ (−E)σ, Q(E = −1) = 0, (2.2.9e)

where σ = −3(2β−1)/4 and c4 is the fourth eigenvalue. The boundary condition for the Poisson’s

equation is

dΦ(R = 0)dR

= 0, Φ(R = 0) = −1. (2.2.10)

Due to the nature of homologous evolution, the minimum of variables Φ (and E) is fixed to −1.

Solving the ss-OAFP system is a nonlinear eigenvalue problem in integro-differential equations.

One can solve the system of equations (2.2.3)-(2.2.5) for the set of dependent variables F, G, J, I,

Q, Φ and four eigenvalues c1(or β), c2, c3, c4 using the boundary conditions (2.2.9) - (2.2.10).

2.2.2 Difficulties in the numerical integration of the ss-OAFP equation

The ss-OAFP system is essentially a simplified form of the (time-dependent) OAFP system.

Hence, One may think at first numerically solving the ss-OAFP system is a simple task compared

to the OAFP model. However, only a few works (Heggie and Stevenson, 1988; Takahashi and

Inagaki, 1992; Takahashi, 1993) studied the ss-OAFP model. Although they found their self-

similar solutions, their works are not complete due to the following reasons. First, they truncated

the system’s domain in energy space and avoided the fine structures of power-law profiles using an

44


extrapolation. This means their solutions may depend on the extrapolation of power-law profiles.

They did not discuss how their extrapolation affected their solutions in detail. Second, the value

of χesc obtained by Heggie and Stevenson (1988) is smaller than that of (Cohn, 1980)’s work. The

former reported that a complete core collapse occurs χesc = 13.85 while the latter implied that

χesc ≈ 13.9. One has yet to discuss which result is more accurate. Third, Takahashi and Inagaki

(1992); Takahashi (1993) employed a variational principle, but they could not reproduce (Heggie

and Stevenson, 1988)’s result. All the existing works struggled with the Newton iteration method.

It did not work unless their initial guess for the solution is very close to the ‘true’ solution.

45

Chapter 3

Gauss-Chebyshev pseudo-spectral method

and Fejer’s first-rule Quadrature

The present chapter details the Gauss-Chebyshev pseudo-spectral method. Section 3.1 de-

scribes a series expansion of a given function by Chebyshev polynomials. For numerical calcula-

tions, the section also explains how to truncate the expansion at a certain degree of polynomials,

and discretize the polynomials’ domain using Gauss-Chebyshev nodes. Section 3.2 explains the

error bound for the degree truncation. Section 3.3 shows how (end-point) singularities affect the

convergence rate of Chebyshev coefficients. Section 3.4 explains the Gauss-Chebyshev pseudo-

spectral method based on Newton-Kantrovich iteration method. As an example, the section details

how to apply the method to the King model. Lastly, Section 3.5 explains Fejer’s first rule quadra-

ture for the numerical integration of integrals.

46

CHAPTER 3. SPECTRAL METHOD

3.1 Gauss-Chebyshev polynomials expansion

Chebyshev polynomials of the first kind (e.g., Boyd, 2001; Mason and Handscomb, 2002) are

Tn(x) = cos[n cos−1(x)

], (n = 0, 1, 2, · · · ,∞). (3.1.1)

where the domain of the polynomials is x ∈ [−1, 1]. The first step of the Chebyshev pseudo-spectral

method is to series-expand a function h(x) concerned in terms of the polynomials. The polynomials

are simple cosine functions. Like the Fourier-series expansion method, one must assume that the

function h(x) is smooth, continuous, bounded (|h(x)| ≤ L so that L ∈ <1 and L < ∞) with compact

support (|x| ≤ 1), and ideally infinitely-differentiable everywhere on the domain. These conditions

allow one to sensibly ’Chebyshev-expand’ h(x)

h(x) =

∞∑n=1

anTn−1(x), (3.1.2)

where an is the n-th Chebyshev coefficient for the function h(x). For numerical applications, one

must truncate the Chebyshev expansion at a certain degree (Section 3.1.1) and discretize the do-

main of the expansion at Gauss-Chebyshev nodes (Section 3.1.2).

3.1.1 Continuous Chebyshev polynomial expansion (degree truncation)

For numerical calculations, one needs to truncate the degree of the Chebyshev expansion (3.1.2)

at a specific value, say, N;

h(x) = hN(x) + Res(N+1)(x), (3.1.3)

47


where the truncated function hN(x) and the residue Res(N+1)(x) read

hN(x) =

N∑n=1

anTn−1(x), (3.1.4a)

Res(N+1)(x) =

∞∑n=N+1

anTn−1(x). (3.1.4b)

The error bound for Res(N+1)(x) is discussed in Section 3.2.

Due to the following orthogonality of Tn−1(x) for n = 2, 3, ...,N (Mason and Handscomb, 2002)

∫ 1

−1

Tn−1(x)Tm−1(x)√

(1 + x)(1 − x)dx =

π

2δn,m, (m = 2, 3, ...,N) (3.1.5)

one can find the explicit form of Chebyshev coefficients in terms of hN(x)

an =2π

∫ 1

−1

hN(x)Tn−1(x)√

(1 + x)(1 − x)dx. (3.1.6)

3.1.2 Gauss-Chebyshev polynomial expansion (domain discretization)

One needs to discretize the domain of the truncated Chebyshev expansion (3.1.4a) and choose

optimal Chebyshev nodes for the ss-OAFP system (equations (2.2.3), (2.2.4), and (2.2.5)). As

explained in Chapter 4, the ss-OAFP system has (regular) singularities at the endpoints of their

domains. Hence, one should solve the system on an open interval −1 < x < 1. The present

work employs the Gauss-Chebyshev- and Gauss-Radau-Chebyshev- collocation methods, which

can avoid imposing boundary conditions at both or either of x = ±1 (e.g., Bhrawy and Alofi, 2012;

Boyd, 2013).1 Since the mathematical formulation for the Gauss-Radau nodes is very similar

to that for the Gauss-Chebyshev nodes, we only show the formulation for the Gauss-Chebyshev

1One needs to employ Gauss-Lobatto nodes for a closed interval [−1, 1] to handle two-point boundary-value prob-lems (Boyd, 2001; Trefethen, 2000). One can resort to the Gauss-Radau nodes for an interval open at an end andclosed at the opposite to tackle one-point boundary-value problems (Chen et al., 2000; yu Guo et al., 2001; Guo et al.,2012). See numerical examples in (Mohammad Maleki and Abbasbandy, 2012) in which Gauss-, Gauss-Lobatto- andGauss-Radau- Chebyshev-collocation methods are applied to initial-value- and boundary-value- problems.

48


method for the sake of brevity. Refer to (Ito et al., 2018) for the Gauss-Radau spectral method.

Gauss-Chebyshev nodes are

xk = cos(tk) ≡ cos(2k − 1

2Nπ

). (k = 1, 2, 3 · · ·N) (3.1.7)

The discrete Gauss-Chebyshev polynomials Tn

(x j

)of the first kind satisfy the orthogonality con-

dition (e.g., Mason and Handscomb (2002))

di j ≡

N∑j=1

Tn−1

(x j

)Tm−1

(x j

)=

0,

N,

N2 .

(1 < n , m < N)

(n = m = 1)

(1 < n = m ≤ N)

(3.1.8)

The discrete Gauss-Chebyshev polynomial expansion of h(x) and its derivative read

hN(x j) =

N∑n=1

anTn−1(x j), (3.1.9a)

dhN

(x j

)dx

=

N∑n=1

an

(n − 1) sin([n − 1] cos−1 x j

)sin

(cos−1 x j

) , (3.1.9b)

Hence, the Chebyshev coefficients in terms of the discrete function hN(x j) are

a1 =1N

N∑j=1

T0(x j)hN(x j), (3.1.10a)

an =2N

N∑j=1

Tn−1(x j)hN(x j). (2 ≤ n ≤ N) (3.1.10b)

49


3.2 Truncation-error bound and the geometric convergence of

Chebyshev-polynomial expansion

The present section explains the error bound for the degree truncation of the Chebyshev-

polynomial expansion (equation (3.1.3)), and then detail the relationship between the truncation

and the convergence rate of the polynomials. Since the function h(x) concerned is bounded with

a compact support, one can find the error bound from the residue Res(N+1)(x) of the Chebyshev

expansion (equation (3.1.4b))

EN+1 = max−1<x<1

∥∥∥Res(N+1)(x)∥∥∥ = max

−1<x<1‖h(x) − hN(x)‖ ≤

∞∑k=N+1

| ak |, (3.2.1)

where max−1<x<1 ‖b(x)‖ represents the maximum value of a function b(x) over the interval −1 <

x < 1. If the Chebyshev coefficients rapidly converge to zero with increasing index k, then the

absolute error in the degree truncation is practically the order of the absolute value of the (N + 1)-

th Chebyshev coefficient (Elliott and Szekeres, 1965)

EN+1 ≈ |aN+1|. (3.2.2)

This implies that the error bound for the degree truncation depends on the mathematical structure

of functions. One can consider the same situation analogically as Fourier-series expansion. For

the Fourier expansion, one generally expects to need more degrees of Fourier coefficients if the

given physical system or function has a smaller (finer) physical structure or smaller change in the

numerical values of functions. The rest of the present section shows an example of the error bound

for the Chebyshev expansion in Section 3.2.1, then explains how the mathematical structure of

functions affects Chebyshev coefficients in Section 3.2.2.

50


3.2.1 An example of degree-truncation error

To evaluate the relation between the degree truncation error EN+1 and the Chebyshev coeffi-

cients an, one may Chebyshev-expand the following test function using equations (3.1.4a) and

(3.1.10)

h(1)(x) =

(1 + x

2

)8.2

. (3.2.3)

The function h(1)(x) represents the asymptotic approximation of the DF for stars in the ss-OAFP

system (as explained in Section 2.1.2). Table 3.1 shows the coefficients an/a1 for the func-

tion h(1)N

(x) with degree N = 30 around which the absolute values of the coefficients reach the

order of MATLAB machine precision (∼ 2.1 × 10−16). Figure 3.1 shows the absolute error of

the degree-truncated function h(1)N

(x) from the original function h(1)(x) for different degrees (N =

5, 10, 15, 20, 25). As expected, the absolute error decreases with degree N. In the figure, the

horizontal dashed lines depict the absolute values of Chebyshev coefficients at degrees N + 1

(= 6, 11, 16, 21, 26). Each truncation error for N is well below the absolute value of the Chebyshev

coefficient aN+1. 2 This simple example well supports the idea of the heuristic relation in equation

(3.2.2).2The error at N = 25 is the same order as the absolute value of coefficient a26 since it reaches the order of machine

precision.

51


index n an/a1 index n an/a1 index n an/a1

1 1.00 × 1000 11 −1.17 × 10−7 21 −5.92 × 10−14

2 1.78 × 1000 12 1.09 × 10−8 22 2.40 × 10−14

3 1.26 × 1000 13 −1.52 × 10−9 23 −1.02 × 10−14

4 6.97 × 10−1 14 2.72 × 10−10 24 4.40 × 10−15

5 2.97 × 10−1 15 −5.88 × 10−11 25 −2.38 × 10−15

6 9.45 × 10−2 16 1.47 × 10−11 26 6.71 × 10−16

7 2.13 × 10−2 17 −4.13 × 10−12 27 −1.13 × 10−15

8 3.08 × 10−3 18 1.28 × 10−12 28 −1.16 × 10−16

9 2.28 × 10−4 19 −4.29 × 10−13 29 −4.82 × 10−16

10 2.65 × 10−6 20 1.55 × 10−13 30 −4.49 × 10−16

Table 3.1: Chebyshev coefficients an for test function h(1)(x) = (0.5+0.5x)8.2 with degree N = 30.The coefficients an are divided by a1 so that the first coefficient is unity.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−18

10−15

10−12

10−9

10−6

10−3

100

103

x

|h(1

) (x)−

h(1)

N(x

)|

N = 5 N = 10 N = 15N = 20 N = 25

Figure 3.1: Absolute error between the test function h(1)(x) and degree-truncated one h(1)N

(x). Onthe graph, the graphing points are taken at points on [−1, 1] with the step size 10−1. Althoughthe Gauss-Chebyshev nodes can not reach x = ±1, one still may extrapolate the function h(1)

N(x)

at the points. The dashed horizontal lines show the absolute values of the Chebyshev coefficientsaN+1 that descends with degrees N+ 1 = 6, 11, 16, 21, 26, corresponding to |aN+1/a1| =9.5×10−2,1.2 × 10−7, 1.5 × 10−11, 5.9 × 10−14,6.7 × 10−16.

52


3.2.2 The effects of the mathematical structures of functions on Chebyshev

coefficients

In order to show how the mathematical structures of functions affect the convergence rate of

the coefficients, we Chebyshev-expand the following three test functions

h(2)(x) =

(1 + x

2

)8.0

, (3.2.4a)

h(3)(x) = exp(−

[1 − x

2

]), (3.2.4b)

h(4)(x) = exp(−13.85

[1 − x

2

]). (3.2.4c)

The function h(2)(x) is similar to function h(1)(x), but the exponent of power is integral. The

functions h(3)(x), and h(4)(x) mock the Maxwellian DFs in the cores of the isothermal sphere (equa-

tion (2.1.4)) and the ss-OAFP model (Cohn, 1979; Heggie and Stevenson, 1988). Figure 3.2 de-

picts the four functions h(1)(x), h(2)(x), h(3)(x), and h(4)(x) on a log-linear scale. Figure 3.3 shows

the Chebyshev coefficients for the four functions. All the corresponding Chebyshev coefficients

converge quite rapidly and reach the machine precision within degrees N = 30.

For the further discussion of the Chebyshev coefficients for h(1)(x), h(2)(x), h(3)(x) and h(4)(x),

we classify the convergence rates of Chebyshev coefficients (e.g., Boyd, 2001) as follows

Geometric convergence The coefficients an exponentially decay ∼ exp[−anb

].

If b > 1, called Super-geometric convergence.

If b = 1, called Geometric convergence.

If b < 1, called Sub-geometric convergence.

Power-law convergence The coefficients an decay in a power law ∼anb

where a and b are real numbers. To see a more rigorous definition for the geometric convergence,

53


refer to Section 3.3. According to the definitions, the Chebyshev coefficients of h(1)(x) decay with a

power-law convergence while the rest of the coefficients decay with super-geometric convergences.

While the functions h(1)(x) and h(2)(x) themselves are almost visually identical in Figure 3.2,

one can find a distinctive difference between their Chebyshev coefficients in Figure 3.3 (a). The

slow convergence rate of the Chebyshev coefficients for h(1)(x) originates from the effect of frac-

tional exponent singularities on a complex plane (the effect of a fractional power on the conver-

gence rate is explained in Section 3.3). On the one hand, the Chebyshev coefficients of h(4)(x)

decay more slowly than those for h(3)(x) in Figure 3.3 (b). This is because the function h(4)(x)

has more small values than h(3)(x) on the interval −1 < x < 1 (the former has numerical values

between 10−7 and 1 while the latter 10−1 and 1). More small numerical values require h(4)(x) to

have more degrees in Chebyshev expansion. Hence, one can expect that a spectral solution of the

ss-OAFP system costs more degrees of the polynomials compared with the isothermal sphere and

some ideal cases3.3In spectral-method studies (Table 1.1), applied mathematicians typically propose a spectral solution with a few

to ten degrees of polynomials to show (super-)geometric convergences. Some of their accuracies can easily reachmachine precision like the example for h(1)(x). We do not intend to achieve such an ideal case in the present work.We avoid resorting to other base functions (suitable to the Lane-Emden and ss-OAFP equations to gain more rapidconvergences) besides Chebyshev polynomials.

54


0 0.2 0.4 0.6 0.8 110−17

10−15

10−13

10−11

10−9

10−7

10−5

10−3

10−1

101

1+x2

a.u.

h(1)(x)h(2)(x)h(3)(x)h(4)(x)

Figure 3.2: Functions h(1)(x), h(2)(x), h(3)(x), and h(4)(x) on Log-Linear scale.

55


5 10 15 20 25 3010−17

10−14

10−11

10−8

10−5

10−2

101

104

107

(a)

index n

|a n|

h(1)(x)h(2)(x)

n−2∗8.2−1

e−10−6n7.5

5 10 15 20 25 3010−17

10−14

10−11

10−8

10−5

10−2

101

104

107

(b)

index n

|a n|

h(3)(x)h(4)(x)e−n1.4

e−0.065n1.9

Figure 3.3: Chebyshev coefficients an for test functions with N = 30. The circle- and square-marks corresponds to the Chebyshev coefficients for (a) functions h(1)(x) and h(2)(x) and for (b)functions h(3)(x) and h(4)(x), respectively. The solid- and dashed- lines (that were artificially deter-mined ’by eyes’.) represent the convergence rates of Chebyshev coefficients.

3.3 The convergence rates of Chebyshev coefficients

The present section theorizes how the Chebyshev coefficients for functions depends on the

functions’ mathematical structures using Elliott (1964)’s method. Elliott (1964) employed the

Cauchy’s integral formula and rewrote a function h(x) by a contour integral on a complex z−plane

h(x) =1

2iπ

∫C

h(z)z − x

dz, (3.3.1)

56


where the contour C includes the interval −1 < x < 1. Equation (3.3.1) also rewrites the Chebyshev

coefficients (equation (3.1.6)) on the complex plane

an =b−1

iπ

∫C

h(z)√

(z + 1)(z − 1)(z ±√

(z + 1)(z − 1))n−1 dz, (3.3.2)

where the signs (±) are chosen so that the condition |z ±√

(z + 1)(z − 1)| > 1 is satisfied.

Equation (3.3.2) can provide the analytical expression of the geometric convergence for ana-

lytic functions. Assume the function h(z) to have only a single pole at z = zs (e.g., Elliott, 1964)

h(z) =b−1

z − zs, (3.3.3)

where b−1 is the residue. Hence, the Chebyshev coefficients for h(z) read

| an |=2b−1

|√

(zs + 1)(zs − 1)(zs ±

√[zs + 1][zs − 1]

)n−1|

. (3.3.4)

For n→ ∞, equation (3.3.4) is

| an |≈ exp[µ(zs)n

], (3.3.5)

where the convergence rate µ(zs)4 is

µ(zs) = ln | zs ±√

(zs + 1)(zs − 1) | . (3.3.6)

Equation (3.3.5) provides a clear understanding of the geometric convergence. The geometric

convergence describes the tendency that Chebyshev coefficients decay at a rate faster than any-

power law n−m (m ≥ 1). This statement may be generally termed the spectral accuracy. Following

4The mathematical expression of equation (3.3.6) infers that the convergence rate is the same along the ellipsepassing through the point zs on the complex plane. Such a contour may be called the isoconvergence contour (Boyd,2008b).

57


(Boyd, 2001), the present work employs the terms geometrical- and power-law convergences for

further discussions rather than the spectral accuracy.

Figure 3.3 of Section 3.2 shows a slow converge rate of the Chebyshev coefficients for the frac-

tional power function, which can be explained as the effect of an endpoint branch point. In general,

the cause of slow decay in Chebyshev coefficients could be explained on the complex plane. The

slow convergence rate occurs when a function h(z) has some non-regular or non-analytic proper-

ties such as singularities (divergences), discontinuities, and branch points. Especially, Darboux’s

principle (e.g., Boyd, 1999, 2008b) states that only one of the non-regular and non-analytic prop-

erties that is the closest to the domain of a function can most contribute to the convergence- radius

and rate. In this sense, the method of steepest ascent also can be useful. In fact, the method of

steepest ascent and (Elliott, 1964)’s method provided the explicit forms of the convergence rates

(See, e.g., Boyd, 1999, 2001, 2008b). Table 3.2 shows some examples of the convergence rate

relevant to the present dissertation. The slow convergence rate of the fractional power-law func-

tion (shown in Section 3.2.2) is due to the effect of the end-point branch point an ∝ 1/n2ψ+1. This

power-law decay is one of the most important causes of slow decay in the present dissertation. All

the isothermal- and polytropic- spheres and the ss-OAFP model have fractional power-law profiles

in their solutions.

Lastly, we explain the method of steepest ascent following (Boyd, 2008b). For the method,

one starts with the Chebyshev coefficients defined on a continuous domain (equation (3.1.6)). The

coefficients can be rewritten using the change of variable, x = cos t, as

an =2π

∫ π

0h(cos t) cos nt dt. (3.3.7)

The integrand is symmetric about the change of the sign with respect to t. Hence, the coefficients

reduce to

an =1π

∫ π

−π

exp (ln h[cos t]) cos nt dt. (3.3.8)

58


Function an(n→ ∞) Ref.

Branch point at end pointsh(x) =| 1 − x |ψ ∝ 1/n2ψ+1 (Elliott, 1964)

Logarithmic singularity at end pointsh(x) =| 1 − x |ψ ln(1 − x) ∝ ln(n)/n2ψ+1 (Boyd, 2008b)

h(x) =| 1 − x |k ln(1 − x) ∝ (n − k − 1)!/(n + k)! (Boyd, 1989)

Discontinuityh(x) = sign(x) ∝ 1/n (Boyd, 2008b)dhdx = sign(x) ∝ 1/n2 (Boyd, 2008b)

Table 3.2: Asymptotic Chebyshev coefficients an (n→ ∞) for functions h(x) that have non-regularand/or non-analytic properties. The ψ is a nonintegral real number and k is an integer.

An analytic continuation provides an ’unconventional’ form (Boyd, 2008b)

an =1π<

∫C

e(int+ln h[cos t]) dt, (3.3.9)

where C is a contour on the complex t-plane starting at t = −π and ending at t = π. After this, one

needs the explicit form of the function h(cos t) to compute an.

3.4 An example of Gauss-Chebyshev pseudo-spectral method

applied to the King model

The present section explains the Gauss-Chebyshev pseudo-spectral method based on Newton-

Kantrovich iteration method. As an example, we apply the method to the King model (equation

(2.1.30)). This example is instructive for the rest of chapters since the spectral method can apply

to the LE1- and LE2- equations and ss-OAFP system in a quite similar way (except for differences

in mathematical formulation). Section 3.4.1 mathematically reformulates the King model adjusted

for the spectral method. Section 3.4.2 explains the Newton-Kantrovich iteration method. Section

59


3.4.3 shows the numerical results for the King model and the effects of end-point singularities.

Also, the section implicates numerical difficulties of unbounded-domain problems (the LE1- and

LE2- equations and ss-OAFP system) based on the spectral solution of the King model (a finite-

domain problem).

3.4.1 The mathematical reformulation of the King model

The King model (equation (2.1.30)) is a second-order ODE. The two boundary conditions

(equation (2.1.31)) are to be imposed at r = 0. The conditions determine the tidal radius rtid of

the model. Hence, solving the King model is tackling an eigenvalue problem to find the tidal

radius. This circumstance is the same as solving the Lane-Emden equation of polytropic index

m < 5 that was discussed in (Boyd, 2011) using a spectral method. Following (Boyd, 2011), the

present section reformulates the mathematical expression of the Kind model for the use of the

Gauss-Chebyshev spectral method.

One can rewrite the radius r ∈ [0, rtid) of the King model in dimensionless form x ∈ (−1, 1)

r = rtid

(x + 1

2

)or x = 2

rrtid− 1. (3.4.1)

Equation (2.1.30) reduces to

d2φ

dx2 +2

1 + xdφdx−

r2tid

4I(φ)I(K)

= 0. (3.4.2)

Boyd (2011) solved the Lane-Emden equation as a single ODE, but we make equation (3.4.2) a

system of two ODEs by introducing a new function φp(x) as follows

φp(x) ≡dφdx, (3.4.3a)

dφp

dx+

21 + x

φp −r2

tid

4I(φ)I(K)

= 0. (3.4.3b)

60


Possible advantages of solving only first-order ODEs is that (i) we can employ Gauss-Radau-

Chebyshev spectral method without a conflict with the boundary conditions, and (ii) we may be

able to avoid a high condition number that originates from the higher orders of the numerical dif-

ferentiation in spectral method (e.g., Boyd, 2001) (of course, the condition number itself depends

on the mathematical structure of the Jacobian matrix for the algebraic equations though).

3.4.2 Matrix-based Newton-Kantrovich iteration method

(Quasi-Linearization)

The present section shows how to apply to equation (3.4.3b) the Newton Kantrovich iteration

method (i.e., a quasi-linearization method based on the Newton-Raphson method) (See, e.g., Man-

delzweig and Tabakin, 2001; Boyd, 2001, 2011). All the mathematical expressions in the method

are written in matrix- and vector- representations to use MATLAB.

The Newton-Kantrovich method works well when ’initially-guessed’ solution is close to the

actual solution of equations, which is quite similar to the root-finding process of the Newton-

Raphson method. Assume at n-th iterative process (n = 0, 1, 2...) that one has solutions φ(n)(x) and

φ(n)p (x) and eigenvalue r(n)

tid for the King model. They are very close to the ’true’ solution φ(x) and

φp(x) and ’true’ eigenvalue rtid. Then, the differences between the solutions and eigenvalues are

also very small in the sense of a certain proper norm (say, infinite norm). Hence one can introduce

’small’ correction functions δφ(n)(x) and δφ(n)p (x) and correction eigenvalue δr(n)

tid

δφ(n)(x) = φ(x) − φ(n)(x), (3.4.4a)

δφ(n)p (x) = φp(x) − φ(n)

p (x), (3.4.4b)

δr(n)tid = rtid − r(n)

tid . (3.4.4c)

The smallness of the correction functions and eigenvalues can provide its ’quasi-linearized’ form

61


of the equations with respect to φ(x), φp(x), and rtid. Based on this idea, one can ‘quasi-linearize’

the King model. The linearized form of equation (3.4.3a) reads

−dδφ(n)

dx= φ(n)(x) − φ(n)

p (x), (3.4.5a)

δφ(n)p (x) = −φ(n)

p (x) +dφ(n)

dx, (3.4.5b)

and that of equation (3.4.3b)

−r2

tid

4δφp(n)I(K)

erf( √

φ(n))

exp (φ(n)) −

√4φ(n)

π

= −dφ(n)

p

dx−

21 + x

φ(n)p +

r2tid

4I(φ(n))I(K)

, (3.4.6a)

dδφ(n)p

dx+

21 + x

δφ(n)p = −

dφ(n)p

dx−

21 + x

φ(n)p +

r2tid

4I(φ(n))I(K)

. (3.4.6b)

The linearized equation regarding the eigenvalue rtid is

−δr(n)

tid

2

I[φ(n)

]I(K)

= −dφ(n)

p

dx−

21 + x

φ(n)p +

r2tid

4I(φ(n))I(K)

. (3.4.7)

For quasi-linearized equations (3.4.5), (3.4.6), and (3.4.7), one can employ the Gauss-Chebyshev

discretization (equation (3.1.9b)). To do so, first one may prepare the matrix- and vector- forms of

the Gauss-Chebyshev polynomials and differentiation as follows

T ≡ T jn = cos([

cos−1 x]

(n − 1)t), (3.4.8a)

Tx=1 ≡ 1t, (3.4.8b)

Tx=−1 ≡ cos(− (n − 1)t

), (3.4.8c)

dTdx

= diag(

1sin

(cos−1 x

)) sin([

cos−1 x]

(n − 1)t)

diag([n − 1]t

), (3.4.8d)

where the hat mark (ˆ) means a (N − 1) × N matrix and bold letter means (N − 1) × 1 vector or

1 × N vector if the transposed symbol (t) is applied or not. Also, the correction functions δφ(n)(x)

62


and δφ(n)p (x) may be Chebyshev-expanded as follows

δφ(n)(x) = T[δc(n)

φ

]t, (3.4.9a)

δφ(n)p (x) = T

[δc(n)

φp

]t. (3.4.9b)

where δcφ(n) and δcφp(n) are the vector forms of the Chebyshev coefficients for the correction func-

tions δφ(n)(x) and δφ(n)p (x) at n-th iteration. As a result, one obtains the quasi-linearized equations

in the form of a linear algebraic equation for the King model

J(n)δc(n) = O(n)(x), (3.4.10)

where the vector δc represents the changes in Chebyshev coefficients and eigenvalue

δc =

[δcφt δcφp

t δrtid

]t

. (3.4.11)

In equation (3.4.10), O(n)(x) is the vector representation of equations (3.4.3a) and (3.4.3b) at the

n-th iteration defined as a (2N + 1) × 1 vector

O(n)(x) =

−φ(n)p (x) +

dφ(n)xdx

−dφ(n)

p xdx − diag

(2

1+x

)φ(n)

p (x) +r2

tid4

I(φ(n)(x))I(K)

φ(−1) − K

φp(−1) − 0

φ(1) − 0

, (3.4.12)

where, to impose the boundary conditions, the equations at (N − 1)-th and N-th nodes (rows) are

replaced by the boundary conditions (2.1.31) and the last row is the condition to find the eigenvalue

63


rtid. The corresponding Jacobian Matrix J (a (2N + 1) × (2N + 1) matrix) reads

J(n) =

−T T 0

diag(−

r2tid4

1I(K)

[I(φ(n)(x)

)+

√16(φ(n)(x))3

9π

])dTdx + diag

(2

1+x

)−δr(n)

tid2

I[φ(n)(x)]I(K)

Tx=−1 0 0

0 Tx=−1 0

0 Tx=1 0

. (3.4.13)

One can solve equation (3.4.10) using a MATLAB built-in operator “mldvide” which is equiva-

lent to a Gaussian elimination with partial pivoting5 (also, LU decomposition is built in the operator

to make the elimination process faster (See, e.g., Higham, 2011, for LU decomposition.)). Hence,

from the ’old’ Chebyshev coefficients c(n), one obtains ’new’ coefficients c(n+1) employing

c(n+1) = c(n) + δc(n), (3.4.14)

5In the present work, the majority of the main results are based on numerical results based on degrees of polyno-mials less than 100 though, we constructed some matrices composed of large elements (400 ∼ 1000) for the Gaussianelimination process. For the latter case, we followed the empirical rule (Trefethen and Schreiber, 1990). Conserva-tively, one should employ the complete pivoting or rook pivoting (Foster, 1997) to avoid an instability with a power-lawgrowth factor due to partial pivoting. however, we simply accepted our numerical results based on large degrees ofpolynomials unless they showed an instability with increasing degree of polynomials.

64


where

c =

[cφt cφp

t rtid

]t

. (3.4.15)

The iterative process of the Newton-Kantrovich method can provide a quadratic convergence rate

(Mandelzweig and Tabakin, 2001) to reach the ’true’ Chebyshev coefficients and eigenvalue if the

’initial guess’ for coefficients c(0) are close to ’true’ ones.

3.4.3 Spectral solutions for the King model and unbounded-domain

problems

The present section (i) shows the spectral solution for the King model and (ii) explains diffi-

culties in solving (integro-) differential equations on an infinite domain for the rest of the chapters.

(i) Spectral solution for the King model

We found spectral solutions for the King model of K ≤ 15 that have quite rapid convergences

of Chebyshev coefficients. Figure 3.4 depicts spectral solutions (the dimensionless m.f. potential

φ(r)) for the King model of different K between 0.01 and 15. Figure 3.5 depicts the absolute values

of the Chebyshev coefficients for φ(r) only with small K (K = 1 and K = 0.01). The coefficients

reach the order of the machine precision around at index n = 60, and beyond the index, they flatten

due to the rounding error. Figure 3.5 (a) depicts the coefficients on a log-linear scale and highlights

the geometrical convergence of the coefficients for small n. On one hand, the corresponding log-log

plot (Figure 3.5 (b)) emphasizes a power-law convergence for large n. The power-law convergence

originates from the effect of the end-point branch point at x = 1 (or r = rtid) that can be expressed

by (0.5 − 0.5x)−2mK−5 where mK = 2.5 is correct in the limit of K → 0. This effect can be easily

derived from equation (2.1.35) as done for the LE1 equation of m < 5 (See Boyd, 2011). On the

one hand, Figure 3.6 shows that as K increases, degree N increases to reach the machine precision.65


The degree reaches the order of 103 for K = 15 at which the tidal radius rtid reaches the order of 103

(Figure 3.7). The factor K greater than 15 resulted in unrealistically high degrees of polynomials

and long CPU times.

Although we obtained the spectral solutions with only the limited values of K(≤ 15), they are

good enough for practical applications to globular clusters. Generally, the King model reasonably

fits the observed structural profiles of globular clusters with the concentrations 0.75 < c < 1.75

which correspond to 3 / K / 8 (e.g., King, 1966; Peterson and King, 1975; Djorgovski and King,

1986; Trager et al., 1995). The Chebyshev collocation method can easily handle such cases. For

K / 8, the Chebyshev coefficients quite rapidly converge at geometrical convergence rates. The

slowest decay of the coefficients (for K = 8) can be described by ∼ 2 exp [−0.27n]. Correspond-

ingly, the coefficients reach the order of 10−5 at N = 50, and even the machine precision at degree

N = 200. Figure 3.6 shows another unique feature. The Chebyshev coefficients with large K(≥ 4)

do not show the power-law convergence, which implies the polytrope approximation can not apply

to the King model, unlike small K(≤ 1). This situation can be well explained by the K-dependence

of the eigenvalue rtid. Table 3.3 shows the regularized ratio of the tidal radius to the King radius.

The ratio approaches the tidal radius of the polytrope of m = 2.5 with decreasing K .

66


10−3 10−2 10−1 100 101 102 103 104

10−9

10−7

10−5

10−3

10−1

101

dimensionless radius r

pote

ntia

l|φ

(r)|

K = 0.01K = 1K = 4K = 8K = 12K = 15

Figure 3.4: Spectral solution (dimensionless m.f. potential φ(r)) of the King model for K =

0.01, 1, 4, 8, 12, and 15.

67


100 101 10210−17

10−14

10−11

10−8

10−5

10−2

101

104

(a)

index n

|a n|

K = 1K = 0.01

100n−2mK−5

20 40 60 80 10010−17

10−14

10−11

10−8

10−5

10−2

101

104

(b)

index n

|a n|

K = 1K = 0.01exp(−n)

Figure 3.5: Absolute values of the Chebyshev coefficients on (a) log-log scale and (b) log-linearscale for the regularized spectral King model φ(r) of K = 1 and K = 0.01 (N = 100). The figures(a) and (b) include guidelines to depict power-law- and geometrical- convergences. The mk = 2.5in the power-law guideline can be determined from equation (2.1.35).

68


100 200 300 400 500 600 700 800 900 1,000 1,10010−22

10−18

10−14

10−10

10−6

10−2

102

3 exp(−0.6n) 2 exp(−0.27n)

0.3 exp(−0.067n)

0.1 exp(−0.033n)

index n

|a n|

K = 4(N = 100)K = 8(N = 300)K = 12(N = 600)

K = 15(N = 1100)

Figure 3.6: Absolute values of Chebyshev coefficients on log-linear scale for φ(r) with K =

4, 8, 12, 15. The dashed lines and exponential expressions in terms of n on the figure are refer-ences to present the geometrical convergences of the coefficients.

10−3 10−2 10−1 100 101

10−2

100

102

104

K

r tid/

r Kin

Figure 3.7: The ratio of the tidal radius to the King radius.

(ii) What we can learn from the spectral solution for King model

From the example for the King model, one can find a numerical difficulty in infinite domain

problems. The degrees of Chebyshev polynomials for the King model easily reach the order of 103

69


K 3rtid/rKin/√

K K 3rtid/rKin/√

K

0.00001 5.355 × 1000 6 2.204 × 10+1

0.0001 5.355 × 1000 7 3.822 × 10+1

0.001 5.356 × 1000 8 7.228 × 10+1

0.01 5.359 × 1000 9 1.314 × 10+2

0.1 5.402 × 1000 10 2.122 × 10+2

0.5 5.611 × 1000 11 3.194 × 10+2

1 5.924 × 1000 12 4.748 × 10+2

2 6.785 × 1000 13 7.164 × 10+2

3 8.140 × 1000 14 1.109 × 10+3

4 1.038 × 10+1 15 1.760 × 10+3

5 1.435 × 10+1

Table 3.3: Ratio of the tidal radius to the King radius for different K between 0.00001 and 15. Theratio is rescaled by multiplying the ratio by 3/

√K to compare the ratio to the polytrope model. As

K decreases, the rescaled ratio 3rtid/rKin/√

K gradually approaches 5.35528 that is the tidal radiusof the spherical polytrope of index m = 2.5 (Boyd, 2011).

only at approximately r = 103. One may recall the discussion of Chapter 1 in which (Heggie and

Stevenson, 1988)’s solution for the Poisson’s equation reaches the maximum radius r = 3 × 103,

and the present work needs a large dimensionless radius, at least R(E ≈ 0.01) ≈ (0.01)ν ∼ 109.

Hence, one needs a method to extend the radius more effectively without costing the degrees of

polynomials.

One may consider several known methods to deal with unbounded-domain problems, as shown

in Table 1.1. (i) One may employ basis functions (e.g., the Hermite function, Rational Chebyshev

polynomials, etc.) whose domains are defined on semi-infinite spaces r ∈ (0,∞). However, the

nodes of the basis functions are condensed near r = 0. For example, Boyd (2013) the Thomas-

Fermi equation using the Rational Chebyshev polynomials. Even if the degree of the polynomials

reaches 500, the maximum radius is at most the order of 104. (ii) Another useful method is a

domain-decomposition method. This method is powerful as long as one has already known the

singular properties of equations concerned and the higher orders of asymptotic approximation of

the solution at an analytical level. For example, MacLeod (1992) solved the Thomas-Fermi equa-

70


tion using the domain-decomposition method. The method provided a large radius ∼ 1010 and

small Chebyshev coefficients ∼ 10−10 only with degree N = 20. However, the domain decomposi-

tion is not useful for the ss-OAFP system since the system includes the 4ODEs. It is not realistic

to ‘domain-decompose’ the 4ODEs and solve new eight equations with the Poisson’s equation and

Q-integral. Also, the higher orders of asymptotic approximations are not known for the ss-OAFP

system. Hence, the present work resorts to a different new method. The new method is based

on an inverse mapping and the regularization of dependent variables using the known first-order

asymptotic approximations for the ss-OAFP systems. The new method can provide a solution

with a large radius and rapid convergence rate of the Chebyshev coefficients without any domain

decomposition. The method is detailed in Chapter 4.

3.5 Fejer’s first-rule quadrature

The ss-OAFP system includes the two integrals in equations (2.2.4) and (2.2.5). We employ

the Feje’s first- and second- rule quadratures, and the Clenshaw-Curtis rule quadratures to com-

pute the integrals. Their mathematical formulations are, however, quite similar and well known

(e.g., Mason and Handscomb, 2002; Dahlquist and Bjorck, 2008). For brevity, the present section

explains only the interpolatory integral based on the Fejer’s first-rule quadrature (Section 3.5.1).

Section 3.5.2 explains the convergence rate of the quadratures.

3.5.1 Fejer’s first-rule Quadrature

The Fejer’s first-rule quadrature is suitable to avoid (at least weaken) the non-analytic and non-

regular property of integrand at endpoints of some certain integrals since the rule relies on the

Gauss-Chebyshev nodes. Consider the integral of a smooth and finitely-bounded function h(x)

71


over interval (−1, 1)

Io ≡

∫ 1

−1h(x) dx. (3.5.1)

The discrete Chebyshev expansion (equation (3.1.4a)) can approximate h(x) and the integral Io

reduces to

Io ≈ IN ≡N∑

n=1

anKn, (3.5.2)

where the factors Kn are

Kn ≡

∫ π

0Tn−1 (cos θ) sin θ dθ =

1 − n2

2, (if n is an even integer)

0. (if n is an odd integer)(3.5.3)

By rewriting the coefficients an in terms of discretized function h(x j) using equation (3.1.10), one

obtains the interpolatory Fejer’s first-rule quadrature

IN =

N−1∑k=1

h(x j)ω j, (3.5.4)

where the weights ω j of the quadrature are

ω j ≡

N−1∑n=1

KnTn−1(x j)dnn

, (3.5.5a)

=2N

1 − 2[(N−1)/2]∑

n=1

cos(2nt j)4n2 − 1

,(t j =

2 j − 1N

)(3.5.5b)

where, on the last line, the symbol [M] in the summation means the floor function of M.

72


3.5.2 The convergence rate of the Fejer’s first-rule quadrature

The Fejer’s first-rule quadrature is based on the Chebyshev expansion. Hence, its convergence

rate can be the order of that for the discrete Chebyshev-expansion given that the function h(x) is

smooth, infinitely differentiable and bounded with a compact support. For example, Xiang (2013)

has shown this in a more limited and practical situation. If h(x) has an absolutely continuous (k−1)-

th derivative dk−1

dxk−1 h(x) and the k-th derivative dk

dxk h(x) of bounded variation Vk, the convergence rate

is

| IN − Io |≤2Vk(1 + 2−2k)

∑∞p=1 p−k−1

πN(N − 1) · · · (N − k). (N ≥ max(8, k + 1)) (3.5.6)

Xiang et al. (2010); Xiang (2013) pointed out that the convergence rate of Fejer’s first-rule is ap-

proximately equal to those of the Fejer’s second rule and Gauss-Chebyshev quadrature for large N.

Since the convergence rate between the Clenshaw-Curtis rule and Gauss Quadrature does not show

a significant difference (e.g., Trefethen, 2008) for large degrees, the order of all the quadratures is

| IN − Io |= O(N−k

). (N >> 1) (3.5.7)

Hence, the convergence rate could achieve a geometrical convergence for infinitely differentiable

functions in the sense that the convergence rate is faster than any power of N.

On the one hand, one must caution against functions whose k-th derivative has a discontinuity.

For example, Xiang (2013) numerically confirmed that if h(x) = |x| then the decay rate is | IN −

Io |= O(N−2

)(N >> 1). Such a slow convergence rate is relevant to that of the asymptotic

approximation of the Chebyshev coefficients for a function whose first derivative is discontinuous,

as shown in Table 3.2.

Lastly, one may enhance the numerical efficiency of the Fejer’s first-rule quadrature by using

Fast Fourier Transform (FFT) (e.g., Waldvogel, 2006). However, the present work does not employ

73


the FFT method since the numerical integration of the integrals in the ss-OAFP system can be

implemented before the Newton iteration process begins (as explained in Section 5.2). Hence,

only the Fejer’s rule quadrature is efficient enough to handle the integrals without the FFT method.

74

Chapter 4

The inverse forms of the Mathematical

models

Section 4.1 introduces an inverse mapping method to find the inverse forms of the LE1- and

LE2- equations and ss-OAFP system. The inverse forms can provide spectral solutions on a broad

range of radii. We further regularize the inverse forms by the known asymptotic approximations

for the LE1 equation (Section 4.2), LE2 equation (Section 4.3), and ss-OAFP system (Section 4.4).

4.1 The inverse mapping of dimensionless potential and

density

The potential φ(r) in the Poisson’s equation (2.1.1) monotonically increases with the radius

r if the density ρ(r) is finitely bounded. Accordingly, the dimensionless potential φ(r) (equation

(2.1.20a)) decreases with the dimensionless radius r. Based on the theorem on global inverse

functions, one can define an inverse function R of φ;

R ≡ φ−1, (4.1.1)

75

CHAPTER 4. INVERSE FORMS

where the range is R ∈ (0,∞). Also, one can introduce a new independent variable ϕ ∈ (φ(0), εmax)

for the new dependent variable R. Equation (4.1.1) can directly apply to the LE1 equation (2.1.21)

while it applies to the Poisson equation’s (2.2.5) of the ss-OAFP system with the modification that

one must replace φ and r by Φ(R) and R. However, the LE2 function is not finitely bounded in

the LE2 equation (2.1.7). Hence, one may apply the inverse function theorem to the dimensionless

density ρ in place of φ. The density also monotonically decreases with increasing r because of the

definition (equation (2.1.20a)).

For brevity, the present work employs only a set of ‘new’ variables (ϕ,R) for all the LE1-

and LE2- equations and ss-OAFP system after we apply the inverse function theorem to the equa-

tions. For example, the local inverse function theorem applies to the differentiations of the ‘old’

dependent variable A with respect to the ‘old’ independent variable B as follows

dAdB

=1

dRdϕ

,d2A

dB2 = −d2R

dϕ2

1dRdϕ

3

. (4.1.2)

where the ’old’ dependent and independent variables (A,B) are defined as

A =

φ, (LE1)

ρ, (LE2)

Φ, (ss-OAFP)

(4.1.3a)

B =

r, (LE1 and LE2)

R. (ss-OAFP)(4.1.3b)

76


4.2 Isothermal sphere (LE2 function)

Section 4.2.1 applies the inverse mapping to the LE2 equation and shows the inverse form.

Section 4.2.2 regularizes the inverse form of the LE2 equation by their known asymptotic approx-

imations.

4.2.1 Inverse mapping

The theorem on local inverse function (equations (4.1.2) and (4.1.3)) can reduce the LE2 equa-

tion (2.1.9) to

ϕRd2R

dϕ2 + RdRdϕ− 2ϕ

(dRdϕ

)2

− ϕ3R

(dRdϕ

)3

= 0. (4.2.1)

We call equation (4.2.1) and R satisfying equation (4.2.1) the ’ILE2 equation’ and ’ILE2 function’,

respectively. The asymptotic approximations of the dimensionless density (equations (2.1.11b) and

(2.1.12b)) transform into the expansion for R near ϕ = 1

R(ϕ ≈ 1) =√

6(1 − ϕ)[1 +

25

(1 − ϕ) +2831050

(1 − ϕ)2 + . . .

], (4.2.2)

and as ϕ→ 0

R(ϕ→ 0) =

√2ϕ

1 + C3i ϕ1/4 cos

√74

lnϕ + C4i

, (4.2.3)

where C3i and C4i are constants of integration.

77


4.2.2 Regularized ILE2 equation

By the known local and asymptotic approximations (equations (4.2.2) and (4.2.3)), one can

regularize the ILE2 function and introduce regularized independent variables 3r and 3s as follows

R(ϕ) ≡

√1 − x

2

(1 + x

2

)−L/2

exp[3r

(x + 1

2

)], (4.2.4a)

S(ϕ) ≡1

LϕL−1

dRdϕ≡ −

1L

√2

1 − x

(1 + x

2

)− 3L2

exp[3s

(x + 1

2

)], (4.2.4b)

x ≡ 2ϕ1/L − 1, (4.2.4c)

where L is a numerical parameter introduced to handle logarithmic end-point singularity (See

Appendix D) and refer to (Ito et al., 2018) for the detail of the mathematical formulation. The

ILE2 equation (4.2.1) reduces to

2(1 − x2)dvr

dx=L(1 − x) + (1 + x) − 4e3s−3r , (4.2.5a)

2(1 − x2)dvs

dx=L(1 − x) − (1 + x) − 8e3s−3r +

4L

e23s , (4.2.5b)

with the boundary conditions

3r(x = 1) = ln√

6L, 3s(x = 1) = ln

√3L2. (4.2.6)

On can find these boundary conditions by solving equations (4.2.5) for 3r(x) and 3s(x) at x = 1.

The same procedure at x = −1 gives the asymptotes

3r(x→ −1) = ln√

2, 3s(x→ −1) = ln

√L2

2. (4.2.7)

78


4.3 Polytropic sphere (LE1 function)

Section 4.3.1 applies the inverse mapping to the LE1 equation and shows its inverse form.

Section 4.3.2 regularizes the inverse form of the LE1 equation by their known asymptotic approx-

imations.

4.3.1 Inverse mapping

Equations (4.1.2) and (4.1.3) convert the LE1 equation (2.1.21) into an equation in terms of

the inverse function R and its differentiations

Rd2R

dϕ2 − 2(

dRdϕ

)2

− ϕmR

(dRdϕ

)3

= 0. (4.3.1)

We call equation (4.3.1) and R satisfying equation (4.3.1) the ’ILE1 equation’ and ’ILE1 function’,

respectively. The local approximation of R near ϕ = 1 is

R(ϕ ≈ 1) =√

6(1 − ϕ)[1 +

3m20

(1 − ϕ) +121m2 + 200m

5600(1 − ϕ)2 + . . .

]. (4.3.2)

and the asymptotic approximation of R as ϕ→ 0 is

R(ϕ→ 0) = Rs(ϕ)[1 + C1i ϕηi cos (ωi lnϕ + C2i)] ≡ Ra(ϕ), (4.3.3)

where C1i and C2i are constants of integration and

ηi = η/ω, (4.3.4a)

ωi = ω/ω. (4.3.4b)

79


In equation (4.3.3),Rs is the singular solution of the ILE1 equation

Rs(ϕ) =(Am/ϕ

)1/ω. (4.3.5)

Thus, the singular solution diverges algebraically as ϕ→ 0, and ηi and ωi become arbitrarily large

as m increases, unlike η and ω.

4.3.2 Regularized ILE1 equations

One can regularize ILE1 equation (4.3.1) employing the local and asymptotic approximations

(equations (4.3.2) and (4.3.3))

R(ϕ) ≡

√1 − x

2

(1 + x

2

)−L/ω

exp[3r

(x + 1

2

)], (4.3.6a)

S(ϕ) ≡1

LϕL−1

dRdϕ≡ −

1L

√2

1 − x

(1 + x

2

)−L( 1ω+1)

exp[3s

(x + 1

2

)], (4.3.6b)

x = 2ϕ1/L − 1. (4.3.6c)

The regularized functions 3r and 3s in equation (4.3.6) satisfy the first-order ODEs

2(1 − x2)d3rdx

= (1 + x) − L(1 − m)(1 − x) − 4e3s−3r , (4.3.7a)

2(1 − x2)d3sdx

= −(1 + x) + L(1 + m)(1 − x) − 8e3s−3r +4L

e23s , (4.3.7b)

with the boundary conditions

3r(1) = ln√

6L, 3s(1) = ln

√3L2. (4.3.8)

80


and the asymptotes as x→ −1

3r(x→ −1) = ln

√2(m − 3)(m − 1)2 , 3s(x→ −1) = ln

√m − 3

2L2. (4.3.9)

4.4 The ss-OAFP system

The present section reformulates the ss-OAFP system taking the following three steps. First,

the domain of the 4ODEs is finite (E ∈ [−1, 0)) while that of the Poisson’s equation (2.2.5) is

semi-infinite (R ∈ [0,∞)). Hence, Section 4.4.1 finds the inverse form of the Poisson’s equation so

that the equation has the same domain as the ss-OAFP system. Second, all the dependent variables

F,G, I, J,R, S , and Q in the ss-OAFP system have power-law profiles forming large-scale gaps.

Hence, Section 4.4.2 regularizes the variables by a combination of the factors (−E)β, DF F(E), and

integral Q(E). Lastly, as done in (Heggie and Stevenson, 1988), a domain truncation is essential to

solve the ss-AOFP system. Section 4.4.3 gives the explicit expressions of the Q- and D- integrals

on the whole- and truncated- domains.

4.4.1 The inverse form of the Poisson’s equation

Equations (4.1.2) and (4.1.3) transform the Poisson’s equation (2.2.5) into its inverse form

R(ϕ)d2R

dϕ2 − 2(

dRdϕ

)2

+ R(ϕ)(

dRdϕ

)3

D(ϕ) = 0. (4.4.1)

The local approximation of R near ϕ = −1 reads

R(ϕ ≈ −1) =

√6 (1 + ϕ)D(−1)

. (4.4.2)

81


Also, the asymptotic approximation of the dependent variable R as ϕ→ 0 is

R(ϕ→ 0) =

√−

ν + 1ν23D(ϕ→ 0)

(−ϕ)ν,(ν = −

2β + 14

)(4.4.3)

where

3D(ϕ→ 0) ≡ limϕ→0

D(ϕ)(−ϕ)ν

. (4.4.4)

4.4.2 Regularized ss-OAFP system

The independent variables E and ϕ are defined on (−1, 0) in the ss-OAFP system. One may

introduce new independent variables x and y to employ the Chebyshev expansion

x ≡ 2(−E)1/L − 1, y ≡ 2(−ϕ)1/L − 1, (4.4.5)

where x, y ∈ (−1, 1), and L is the numerical parameter introduced in Sections 4.2 and 4.3. By using

the known local and asymptotic approximations of dependent variables (i.e., equations (2.2.9),

82


(4.4.2), and (4.4.3)), one can regularize the dependent variables as follows

3R(y) ≡ lnR(y)

(1 − y

2

)−1/2 (1 + y

2

)−Lν , (ν = −

2β + 14

), (4.4.6a)

3S (y) ≡ ln−S(y)

(1 − y

2

)1/2 (1 + y

2

)1−Lν , (4.4.6b)

3Q(x) ≡Q(x)

(1 + x

2

)−Lσ1/3

,

(σ = −

6β − 34

), (4.4.6c)

3F(x) ≡ lnF(x)

(1 + x

2

)−βL , (4.4.6d)

3G(x) ≡G(x)F(x)

, (4.4.6e)

3I(x) ≡I(x)

F(x)Q(x), (4.4.6f)

3J(x) ≡(2β − 3)J(x)4βF(x)Q(x)

, (4.4.6g)

where the new dependent variable S is introduced for convenience

S(y) ≡ 2dRdy

. (4.4.7)

The boundary conditions for the regularized variables are

3F(x→ −1) = ln c∗4, 3F(x = 1) = ln FBC, (4.4.8a)

3I(x→ −1) = 0, 3I(x = 1) = 0, (4.4.8b)

3G(x→ −1) = 0, 3G(x = 1) = c3, (4.4.8c)

3J(x→ −1) = −1, 3J(x = 1) = 0, (4.4.8d)

where c∗4 is defined as

c∗4 ≡ c4(β + 1). (4.4.9)

83


The 4ODEs (2.2.3a)-(2.2.3d) are first order ODEs. Hence, c∗4 and c3 can be determined by the

boundary conditions assigned at the opposite ends. On the one hand, the eigenvalues c1 and c2 can

be determined by two out of the four boundary conditions for 3I(x) and 3J(x).

The inverse form of the Poisson’s equation (4.4.1) reduces to

2(1 − y)(1 + y)d3Rdy

+ 2νL(1 − y) − (1 + y) + 4e3S−3R = 0, (4.4.10a)

2(1 − y)(1 + y)d3Sdy

+ 2L(ν − 1)(1 − y) + (1 + y) + 8e3S−3R −4L

e3S 3D = 0, (4.4.10b)

where

3D(y) ≡ D(y)(1 + y

2

)−(β+3/2)L

=L2

(1 + y

2

)−L(β+3/2) ∫ y

−1AL(y, x′)e3F (x′)

√x − x′

2

(1 + x′

2

)L(β+1)−1

dx′, (4.4.11)

where

AL(x, x′) ≡

√(1 + x

2

)L

−

(1 + x′

2

)L (x − x′

2

)−1/2

. (4.4.12)

The boundary conditions for the Poisson’s equations (4.4.10a)-(4.4.10b) are

3R(y = 1) =

√6

3D(y = 1), (4.4.13a)

3S (y = 1) =

√3

23D(y = 1). (4.4.13b)

84


The asymptotic approximations of 3R and 3S as ϕ→ 0 or y→ −1 are

3R(y→ −1) =

√−

ν + 1ν23D(y→ −1)

,

(ν = −

2β + 14

)(4.4.14a)

3S (y→ −1) =

√−

ν + 13D(y→ −1)

.

(ν = −

2β + 14

)(4.4.14b)

The explicit form of 3Q(x) is

[3Q(x)

]3=

L

6√

2

(1 + x

2

)−σL ∫ 1

x

(1 − y′

2

)3/2

AL(y′, x)e33R(y′)√

y′ − x(1 + y′

2

)(3ν+1)L−1

dy′. (4.4.15)

Equations (4.4.6d)-(4.4.6g) reduce the 4ODEs (2.2.3) to;

2L

(1 + x

2

)(β−1)L+1 d3Fdx

[3I + 3G]

+

(1 + x

2

)(β−1)Lβ [3I + 3G] +

(1 + x

2

)L (4β3J

2β − 3− 1

) + c1e−3F [1 + 3J] = 0, (4.4.16a)

1 + xL3Q

(d3Idx

+ 3Id3Fdx

)+ 3I

[−2β + 3

43Q +

3(1 + x)L

d3Qdx

]+

(1 + x

2

)L

3Q = 0, (4.4.16b)

1 + xL

d3Gdx

+ 3G

(1 + x

Ld3Fdx

+ β

)−

(x + 1

2

)L

= 0, (4.4.16c)

1 + xL3Q

d3Jdx

+d3Fdx

[3J −

2β − 34β

]+ 3J

[3Q−2β + 3

4+ 3

1 + xL

d3Qdx

]−

2β − 343Q = 0. (4.4.16d)

4.4.3 Integral formulas on the whole- and truncated- domains

The present section shows (i) the whole-domain formulation and (ii) the truncated-domain

formulation for the integrals 3D(y) (equation (4.4.11)) and 3Q(x) (equation (4.4.15)).

85


(i) Whole-domain formulation

Given that we numerically solved the regularized ss-OAFP system (equation (4.4.16)) on the

whole-domains x, y ∈ (−1, 1), we employed the following expressions for the integrals 3D(y) and

3Q(x)

3D(y) =L

2√

2

(1 + y

2

) 1−L2

∫ 1

−1AL

[y,

(y + 1) (x′ + 1)2

− 1]

e3F[

(y+1)(x′+1)2 −1

]√

1 − x′(1 + x′

2

)L(β+1)−1

dx′,

(4.4.17a)

[3Q(x)

]3=

L24

(1 − x

2

)3 (1 + x

2

)−σL ∫ 1

−1AL

[1 −

(1 − x) (1 − y′)2

, x]

× e33R[1− (1−x)(1−y′)

2

] √1 + y′

(1 − y′

)3/2[1 −

(1 − x)(1 − y′)4

](3ν+1)L−1

dy′. (4.4.17b)

(ii) Truncated-domain formulation

For numerical integrations of the regularized ss-OAFP system on truncated domains x, y ∈

(xmin, 1) where −1 < xmin < 1, we introduced the new independent variables

z ≡ −21 − x

1 − xmin+ 1, w ≡ −2

1 − y1 − xmin

+ 1. (4.4.18)

We employed the following extrapolated DFs to extrapolate 3F(x) on (−1, xmin)

3(ex)F (x) ≡

ln

c∗4 +

2c∗41−xmin

d3F (w=−1)dw ed(1+xmin )−c

[1−

(1+xmin

1+x

)c](x − xmin)

, (smooth at x = xmin),

ln[c∗4], (non-smooth at x = xmin or c→ ∞),

(4.4.19)

where c and d are numerical parameters. Hence, 3D(w) is composed of a total of integrals 3(nonex)D (w)

and 3(ex)D (w); the former is contribution to 3D(w) from the (non-extrapolated) DF 3F(x) and the latter

86


is from an extrapolated DF 3(ex)F (x) as follows

3D(w) = 3(nonex)D (w) + 3

(ex)D (w). (4.4.20)

The explicit forms of 3F(x) and 3(ex)F (x) are

3(nonex)D (w) =

1

2√

2

(1 − xmin

2

)3/2 [12

+2 + xmin(1 − w)

2(1 + w)

]−3/2

×

∫ 1

−1e3F

[(w+1)(z′+1)

2 −1]√

1 − z′

1 −(1 − z′)(1 + w)(1 − xmin)2 [3 + xmin + z′(1 − xmin)]

βdz′, (4.4.21a)

3(ex)D (w) ≡

12

∫ 1

−1

(1 + xmin

1 + y

)β+3/2

e3(ex)F

[(1+xmin)(1+z′′)

2 −1] √

1 + y1 + xmin

−1 + z′′

2

(1 + z′′

2

)βdz′′, (4.4.21b)

where L = 1 is assumed for simplicity and y = 12 [1 + xmin + w(1 − xmin)]. Lastly, 3Q(z) on the

truncated domain is

[3Q(z)

]3=

124

(1 − x

2

)3 (1 + x

2

)−3/2

×

∫ 1

−1e33R

[1− (1−z)(1−w′)

2

]√

1 + w′(1 − w′

)3/2

1 +(1 − z)(1 + w′)(1 − xmin)

4 (1 + x)

3ν

dw′,

(4.4.22)

where L = 1 is assumed and x = 12 [1 + xmin + z(1 − xmin)]. Given that we sought a spectral solution

on the truncated domain, we employed equations (4.4.21) and (4.4.22).

87

Chapter 5

Numerical arrangements and optimization

on the spectral methods

The present chapter explains the numerical arrangements we made in solving the regular-

ized ILE1- and ILE2- equations (4.3.7) and (4.2.5) and the regularized ss-OAFP system (4ODEs

(4.4.16a)-(4.4.16d) and the Poisson’s equation (4.4.10)). Section 5.1 explains the numerical ar-

rangements for the ILE1- and ILE2- equations. Section 5.2 details the numerical difficulty that the

present work faced in the integration of the regularized ss-OAFP system and how we managed it.

5.1 The numerical treatments of the ILE1- and ILE2-

equations

The present work employed the following ‘initial solutions’ at the zeroth iteration

v(0)r (x) = vr(x = 1)

(1 + x

2

)0.1

+ vr(x→ −1)(1 − x

2

)0.1

, (5.1.1a)

v(0)s (x) = vs(x = 1)

(1 + x

2

)0.1

+ vs(x→ −1)(1 − x

2

)0.1

, (5.1.1b)

88

CHAPTER 5. NUMERICAL ARRANGEMENTS

to solve the regularized ILE1- and ILE2- equations based on the Newton-Kantrovich method.

Following the pseudo-spectral method (Section 3.4.2), one can solve the equations in a quite similar

way to the King model. A technical confusion would be that one must replace the independent

variables φ and φp (employed for the King model) by vr and vs.

The Newton iteration method quite well worked for the isothermal sphere and polytropic sphere

of m > 5.5. The Newton iteration method was hard to work with 5 < m < 5.5 though, the present

dissertation avoids the discussion for the polytropes with m < 5.5. This is since the core of the

ss-OAFP model behaves like a polytrope of m = ∞ and the inner halo like m ≈ 9.6. Hence, the

values 5.5 < m < ∞ are enough for this dissertation. Refer to (Ito et al., 2018) for the detailed

analysis on polytropic spheres of 5 < m < 5.5.

5.2 The numerical arrangements of the regularized ss-OAFP

system

Following (Heggie and Stevenson, 1988), we solved the 4ODEs (4.4.16a)-(4.4.16d) and Pois-

son’s equation (4.4.10). We took the three steps; step 1 is to solve the Poisson’s equation with

the Newton iteration method (Section 5.2.1), step 2 is to solve the 4ODEs and Q-integral with the

Newton method (Section 5.2.2), and step 3 is to repeat steps 1 and 2 (Section 5.2.3). These steps

successfully worked after we arranged numerical protocol (Section 5.2.4). For discussions in Sec-

tions 5.2.1 - 5.2.3, we assume we have already known the functions 3(k)F (x), 3(k)

G (x), 3(k)I (x), 3(k)

J (x),

and 3(k)Q (x) at the k-th iteration.

89


5.2.1 Step 1: Solving the regularized Poisson’s equation

At the k-th iteration, we plugged 3(k)F (x)1 into the integral 3D(y) (equation (4.4.17a) or (4.4.20))

and integrated the integral using the Fejer’s first-rule quadrature (Section 3.5). Then, we Chebyshev-

expanded 3R(y) and 3S (y) in the Poisson’s equation (4.4.10) with the coefficients Rn and S n for

3R(y) and 3S (y). After assigning Chebyshev coefficients an to Rn and S n as follows

an = Rn, an+N = S n, (n = 1, 2, 3, · · ·N), (5.2.2)

we solved the Poisson’s equation for an as 2N nonlinear algebraic equations

Ol(ap) = 0, (l = 1, 2, · · · , 2N & p = 1, 2, · · · , 2N) (5.2.3)

using the Newton iteration method until the infinity norm of the difference between the new- and

old- coefficients (apnew − ap

old) converged to the order of 10−15. Then, we obtained 3(k)R (y) and

3(k)S (y) for the k-th iteration.

5.2.2 Step 2: Solving the regularized 4ODEs and Q-integral

We plugged the 3(k)R (y) in the integral 3Q(x) (equation (4.4.17b) or (4.4.22)) and integrated

the integral using the Fejer’s first-rule quadrature. After Chebyshev-expanding 3F(x), 3G(x), 3I(x),

3J(x), and 3Q(x) in the 4ODEs and Q-integral, we solved the equations only once using the Newton

1The boundary conditions are known for 3F(x). Hence, we used the simple polynomials for zeroth iteration

3(0)F (x) = FBC

(1 + x

2

)n

+ c∗4

(1 − x

2

)m

, (5.2.1)

where m and n are numerical parameters.

90


method2 as a 5N + 2 nonlinear equations

Om

(aq

)= 0, (m = 1, 2, · · · , 5N + 2 & q = 1, 2, · · · , 5N + 2) (5.2.4)

where the Chebyshev coefficients Fn, Gn, In, Jn, and Qn for 3F(x), 3G(x), 3I(x), 3J(x), and

3Q(x) and eigenvalues c2 and c∗4 are

an = Fn, an+N = Gn, an+2N = In, an+3N = Jn, an+4N = Qn, (n = 1, 2, 3, · · ·N)

a5N+1 = c2, a5N+2 = c4. (5.2.5)

Then, we obtained the functions 3(k+1)F (x), 3(k+1)

G (x), 3(k+1)I (x), 3(k+1)

J (x), and 3(k+1)Q (x) and eigenvalues

c(k+1)2 and c∗(k+1)

4 at the (k + 1)-iteration.

5.2.3 Step 3: Repeating steps 1 and 2

We repeated steps 1 and 2 until the Newton iteration method satisfactorily converged for the

integral 3Q and 4ODEs. After we made proper numerical arrangements (Section 5.2.4), the infinity

norm of the difference aqnew − aq

old reached the order of 10−13 at best. The boundary conditions

2As the initial guess for the functions at (k + 1)-th iteration, we employed the functions obtained at k-th iterationfor 3(k)

F (x), 3(k)G (x), 3(k)

I (x), 3(k)J (x), and 3(k)

Q (x). We employed simple polynomials like equation (5.2.1) only for the initialguess (k = 1) for the functions at zeroth iteration.

91


for the whole domain formulation explicitly read

N∑n=1

Fn(−1)n−1 − ln[c∗4] = 0,N∑

n=1

Fn = 0, (5.2.6a)

N∑n=1

Gn(−1)n−1 = 0, (5.2.6b)

N∑n=1

In(−1)n−1 = 0, (5.2.6c)

N∑n=1

Jn(−1)n−1 + 1 = 0,N∑

n=1

Jn = 0, (5.2.6d)

N∑n=1

Qn(−1)n−1 − 3Q(x = −1) = 0, (5.2.6e)

where the three conditions(∑N

n=1 Gn − c3 = 0,∑N

n=1 Qn = 0, and∑N

n=1 In = 0)

are not included.

This is since the first two conditions were not necessary to solve the 4ODEs, and the third condition

was employed as an estimator of accuracy in the present work, as explained in Section 5.2.4.

5.2.4 Numerical arrangements

As stated in (Heggie and Stevenson, 1988), the Newton iteration method did not work well for

both the truncated-domain and whole-domain formulations. Even the truncated-domain formu-

lation worked for only xmin ' −0.2. We made many numerical arrangements (See Ito, 2021,

for the detail), including the arrangement that we truncated the whole-domain formulation at

x = xmin ≈ −0.7, like the truncated-domain formulation. To overcome the difficulty in the Newton

method’s convergence, we fixed the eigenvalue β to a specific value during each iteration process.

For the fixed β-value, we found a solution at a specific xmin. Then, we chose a new β that was close

to the old β. Using the ’old’ solution as the initial guess, we found a new solution for the new β

using the Newton iteration method. We repeated this process until 3I(x = 1) reached its minimum.

Hence, the boundary condition for 3I(x = 1) was excluded in equation (5.2.6). At this moment,

92


we had a solution at the the ’old’ xmin with the ’old’ β. Hence, we next chose a new xmin that was

very close to the old xmin, and a new β that was very close to old β,3. Then, we repeated the whole

process above. As a result, we obtained a whole-domain solution, and truncated-domain solution

on xmin = −0.96 < x < 1.

Also, we made the numerical integration of the integrals 3Q(x) and 3D(x) more efficient. First,

we discretized 3D(y) as follows

3D(y j) =

N∑n=1

F linearn Dn(y j) (5.2.7)

where F linearn is the Chebyshev coefficients for 3linear

F (x)(≡ exp [3F(x)]). We obtained F linearn from

3F(x) by using equations (3.1.4a) and (3.1.10). The matrix Dn(y j) is a preset matrix. It explicitly

reads

Dn(y j) =L

2√

2

(1 + y j

2

) 1−L2

×

∫ 1

−1AL

[y j,

(y j + 1)(x′ + 1)2

− 1]

Tn

[(y + 1)(x′ + 1)

2− 1

]√

1 − x′(1 + x′

2

)L(β+1)−1

dx′.

(5.2.8)

We applied the Fejer’s first rule quadrature to the integral before the Newton-iteration (loop) pro-

cess began. We also prepared a similar preset matrix for 3Q(x).

3For example, to find the whole-domain solution, the change δβ in β was 0.03 from xmin = −0.94 to −0.96,δβ ≈ 0.001 from xmin = −0.9994 to −0.9996, and δβ ≈ 0.000001 from xmin = −0.999994 to −0.999995.

93

Chapter 6

Preliminary Result 1: A spectral solution

for the isothermal-sphere model

Section 6.1 shows a spectral solution of the LE2 equation for the isothermal sphere. The

dimensionless radius of the spectral solution reaches r ≈ 10100, and the solution is accurate to

nine significant figures. Section 6.2 discusses the asymptotic behavior of the spectral solution

as r → ∞. It shows the consistency of the spectral solution compared to known analytical and

numerical asymptotic solutions.

6.1 Numerical results

We found a Chebyshev-Gauss-Radau spectral solution of the ILE2 equation, and an optimal

value for L. Figure 6.1 depicts the dimensionless potential obtained from the spectral solution.

The Chebyshev coefficients for the ILE2 function show a geometrical convergence (Figure 6.2).

The function needs degree N = 180 to reach MATLAB machine precision with L = 36. We

determined this optimal value of L for N = 100 by computing the absolute error of φN from the

singular solution ln (r2/2) at large r and choosing the value of L at which the minimum error occurs

94

CHAPTER 6. RESULT: ISOTHERMAL SPHERE

(Figure 6.3). The optimal value of L generally depends on N. Hence, one should construct a two-

dimensional table like the one in Boyd (2013). However, even the non-optimal values of L did not

significantly ruin the results as long as N was sufficiently high.

We compare the spectral solution to an existing spectral solution based on a Hermite-functions

collocation method found in (Parand et al., 2010). Parand et al. (2010) converted the functions

to a form similar to the asymptotic oscillation of the LE2 function (equation (2.1.12b)) through a

change of variable r = ln[ex +√

e2x + 1], where x ∈ (−∞,∞). Because of this transformation, the

locations of Hermite-Gauss nodes condense near r = 0. Also, the maximum radius of their solution

is small (at most r ≈ 20 even for N = 1000.) By contrast, our solution extends to r ≈ 10100 with

the Chebyshev coefficients that converge at roughly the same rate as the Hermite coefficients.

10−5 1011 1027 1043 1059 1075 109110−3

10−2

10−1

100

101

102

103

r

φN

Isothermal Sphere

Figure 6.1: LE2 function φ (N = 225 and L = 36.)

95


50 100 150 200 250 30010−21

10−16

10−11

10−6

10−1

n

|r n|

ILE2 (Isothermal Sphere)

Figure 6.2: Absolute values of the Chebyshev coefficients for the regularized ILE2 spectral solution3r (N = 225, L = 36). Round-off error appears for n & 200.

0 10 20 30 40 5010−14

10−12

10−10

10−8

10−6

10−4

10−2

100

L

|1−

A/φ

N(r

cut)|

Figure 6.3: Optimal value of the mapping parameter L for the isothermal sphere (N = 100). Thegraph compares φN to A ≡ ln (r2/2) at cutoff radius rcut. The rcut was introduced to avoid numericaldivergences at r → ∞ when MATLAB reaches machine precision.

We consider that the significant figures of our solution are nine digits in comparison with other

96


Radius r LE2 functionsChebyshev expansion φN RK (Horedt (1986)) φH AD (Wazwaz (2001)) φW

1.00 × 10−4 1.66666667 × 10−9 1.666666665833333 × 10−9

1.00 × 10−3 1.66666658 × 10−7 1.666666583333339 × 10−7

1.00 × 10−2 1.6666583334 × 10−5 1.666658333386243 × 10−5

1.00 × 10−1 1.66583386206 × 10−3 1.665834 × 10−3 1.665833862060535 × 10−3

1.00 × 100 1.58827677524 × 10−1 1.588277 × 10−1 1.588273536734185 × 10−1

1.00 × 101 3.73655998054 × 100 3.736560 × 100 1.947976769458251 × 10+4

1.00 × 102 8.59605987886 × 100 8.596060 × 100 2.271557889889379 × 1014

1.00 × 103 1.309605719528 × 10+1 1.309606 × 10+1

2.20 × 103 1.470565441230 × 10+1 1.470565 × 10+1

3.00 × 103 1.533288674939 × 10+1

5.00 × 103 1.635748844091 × 10+1

1.00 × 104 1.773671917546 × 10+1

5.00 × 104 2.094143569673 × 10+1

1.00 × 105 2.232959968900 × 10+1

1.00 × 106 2.693891409852 × 10+1 Singular solution ln(r2/2)

1.00 × 108 3.61483269191 × 10+1 3.614821430734479 × 10+1

1.00 × 1010 4.53585663919 × 10+1 4.535855467932097 × 10+1

1.00 × 1020 9.14102565393 × 10+1 9.141025653920188 × 10+1

1.00 × 1040 1.83513660259 × 10+2 1.835136602589637 × 10+2

1.00 × 1060 2.75617063979 × 10+2 2.756170639787255 × 10+2

1.00 × 1080 3.67720467698 × 10+2 3.677204676984874 × 10+2

1.00 × 10100 4.59823871418 × 10+2 4.598238714182492 × 10+2

1.00 × 10150 6.9008238072 × 10+2 6.900823807176538 × 10+2

Table 6.1: Comparison of the spectral LE2 solution with the singular, Horedt’s, and Wazwaz’ssolutions. Our solution’s first seven digits that match Horedt’s solutions are underlined. Thecomparison of our solution with the other two is also shown using underlines. The spectral solutionwas obtained by choosing combinations of L and N such that the smallest absolute Chebyshevcoefficient reaches machine precision.

numerical results. Table 6.1 compares our spectral solution to existing numerical solutions and

the exact singular solution (equation (2.1.13a)). We confirm that our solution has a seven-digit

accuracy compared to the Horedt (1986)’s seven-digit accurate solution. Furthermore, we verified

the accuracy of our solution up to nine digits by comparing the solution to a solution obtained

by applying Runge-Kutta-Fehlberg (RKF) method to the regularized LE2 equation (4.2.5) with a

domain truncation near the endpoints E = 1 and E = −1.

97


6.2 The topological property and asymptotic approximation

of the LE2 function

Section 6.2.1 shows a topological feature of the LE2 function using Milne variables u and v

(see Appendix E for the details about the Milne variables.) Section 6.2.2 provides an asymptotic

semi-analytical approximation of the LE2 function.

6.2.1 Milne variables

We show the trajectory of the Milne variables following (Chandrasekhar, 1939) to see whether

our spectral solution can reproduce a known topological property of the LE2 function. Figure

6.4 (a) shows the trajectory on the Milne phase plane. The curve spirals about the singular point

(us.p, vs.p) = (1, 2) corresponding to the singular solution, as discussed in (Chandrasekhar, 1939).

The spiral represents the characteristics of the semi-analytical approximation (2.1.12a). The tan-

gent to the curve represents the stability of a finite isothermal sphere (e.g., Padmanabhan, 1989;

Chavanis, 2002a)1. The magnification of the curve (Figure 6.4 (b)) shows that the spectral solution

holds the topological property even at the order of 10−13 on the plane.

1The finite isothermal sphere is an imitation of the cores of globular clusters. Hence, the Milne variables can beuseful to understand the stability of the clusters’ cores.

98


−0.5 0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5 (a)

u

v

isothermal sphere(original scale)−

8·10−

13

−6·10−

13

−4·10−

13

−2·10−

13 0

2·10−

13

4·10−

13

−5 · 10−13

0

5 · 10−13(b)

u − us.p

v−

v s.p

isothermal sphere (magnified)

Figure 6.4: (a) Milne phase space (u, v) for the isothermal sphere (L = 36 and N = 225) and (b)its magnification around (1, 2). The curve begins with (uE, vE) = (3, 0) and asymptotically reachesthe singular point (us.p, vs.p) = (1, 2).

6.2.2 Asymptotic semi-analytical singular solution

We found an accurate semi-analytical asymptotic singular solution based on our spectral solu-

tion. Figure 6.5 compares the spectral solution to previous semi-analytical solutions (Liu, 1996;

99


Hunter, 2001; Soliman and Al-Zeghayer, 2015) in which the existing solutions lose their accura-

cies significantly at large r. We can obtain a more accurate semi-analytical solution based on our

spectral solution that is accurate over a broader range of radii. To find such a solution, we sub-

stitute into equation (2.1.12a) the values of φN(r) at two adjacent Chebyshev-Gauss-Radau nodes

first, and then solved for C3 and C4 in equation (2.1.12a). The absolute error was determined us-

ing data whose relative error between the spectral and semi-analytical solutions reached machine

precision at a point and did not show a large deviation from machine precision at radii greater than

that point. Two points near r = 1010 gave the best results. We found the asymptotic LE2 function

as r → ∞;

φa(r) = lnr2

2+

1.1779636 ± 1 × 10−7

r1/2 cos √7

2ln r + 1.0623271 ± 1 × 10−7

. (6.2.1)

Equation (6.2.1) is consistent with the semi-analytical singular solution of (Soliman and Al-

Zeghayer, 2015). Soliman and Al-Zeghayer (2015) numerically calculated C3 and C4 using data

around at r = 105. Their values agree with the present values up to four-digit accuracy. Equation

(6.2.1) is fifteen-digit accurate at radii greater than r ≈ 1013, where the oscillations have an ampli-

tude ∼ 10−7. At radii r less than 1013, the absolute deviations from φN becomes significant. At radii

1 << r . 1013, one may be able to further improve the accuracy of our semi-analytical solution

by replacing C3 and C4 in equation (6.2.1) with functions that depend on r as done in (Raga et al.,

2013).

100


101 102 103

−1

−0.5

0

0.5

1

1.5

2

2.5

(a)

r

( φ−

ln[ r2 /

2]) r1/2

Chebyshev expansion: φNSoliman and Al-Zeghayer (2015)

Hunter (2001)Liu (1996)

Horedt (1986)

101 105 109 1013 1017 1021 1025 102910−20

10−16

10−12

10−8

10−4

100

(b)

r

|1−φN/φ|

| 1 − φN/φa |

| 1 − φN/φSA |

| 1 − φN/φs |

Figure 6.5: (a) Asymptotic oscillation of LE2 spectral solutions,(φN − ln

[r2/2

])r1/2, compared

to existing semi-analytical solutions for N = 225, L = 36. (b) Relative deviation of our semi-analytical asymptotic singular solution, φa, from the LE2 spectral solution φN. The φSA is thesemi-analytical asymptotic LE2 solution of (Soliman and Al-Zeghayer, 2015) and φs = ln r2

2 . Theplot of | 1 − φN/φa | flattens around at r = 1013 because the calculation reaches machine precision.

101

Chapter 7

Preliminary result 2: Spectral solutions for

the polytropic-sphere model

Section 7.1 shows spectral solutions of the LE1 equations of m > 5.5. The significant figures of

the solutions are at least seven digits. The Chebyshev coefficients have geometrical convergences,

and the radii of the spheres reach the order of 10300 at most, corresponding to Matlab’s graphing

limit. Section 7.2 details the Milne variables and asymptotic approximations of the LE1 functions

as r → ∞ to show the consistency of our spectral solutions.

7.1 Numerical results

We found spectral solutions of the LE1 equation of m > 5.5 over a wide range of radii holding

geometric convergences. Figure 7.1 shows the spectral solutions for spherical polytrope of m =

6, 10 and 20 at dimensionless radii from r ≈ 10−5 up to 10200. Also, for larger m, the radius

r extends further and even reaches the order of 10300 which is the graphical limit of MATLAB.

Figure 7.2 depicts the corresponding Chebyshev coefficients for the regularized ILE1 function 3r

and their geometrical convergences. As m gets closer to five, the convergence rate decreases, and

102

CHAPTER 7. RESULT: POLYTROPIC SPHERE

the optimal value of L becomes greater. We determined the optimal value of L by computing

the absolute error of rωφ(r) from the asymptotic value Am as r → ∞ for m = 6, 10 and 20 with

N = 200 (Figure 7.3). The smallest Chebyshev coefficients consistently reached the round-off

error for large N and many different combinations of m(< 104

)and L. The result for m = 10

shows that the spectral method is capable of apply to the ss-OAFP model since the model behaves

like a polytrope of m ≈ 9.7 as R→ ∞ and we need a spectral solution at R ∼ 109 at least.

101 1031 1061 1091 10121 10151 1018110−90

10−72

10−54

10−36

10−18

100

r

φN

m = 6 and L = 18m = 10 and L = 5m = 20 and L = 2

Figure 7.1: LE1 function φN, for m = 6, 10 and 20 (N = 300.)

103


50 100 150 200 250 30010−17

10−13

10−9

10−5

10−1

103

n

|r n|


Figure 7.2: Absolute values of Chebyshev coefficients for the regularized spectral ILE1 solution 3rfor m = 6, 10 and 20 (N = 300). For (m, L) = (20, 2) the data plots flatten out at large n due to theround-off error; for (m, L) = (10, 5) the decay rate decreases around N = 250 due to the end-pointsingularity.

0 10 20 30 40 5010−15

10−13

10−11

10−9

10−7

10−5

10−3

10−1

101

L

|1−

A/φ

N(r

cut)|

m = 6 (N = 200)m = 10 (N = 200)m = 20 (N = 200)

Figure 7.3: Optimal values of the mapping parameter L for m = 6, 10 and 20 (N = 200). Thecalculated value of φ at the cutoff Chebyshev-Radau point is compared with the correspondingA = φs for polytropes. The cutoff radii rcut were chosen to avoid divergences at r → ∞ whenMATLAB reaches machine precision.

104


In order to verify the accuracy of the spectral solutions, Figure 7.4 compares them with Horedt

(1986)’s seven digit-accurate solutions at radii 0 < r ≤ 1000. Figure 7.5 depicts the relative error

between the solutions at several radii. The spectral solutions need approximately N = 400 to reach

a seven-digit accuracy for m = 6, N = 150 for m = 10, and N = 90 for m = 20.

10−2 10−1 100 101 102 103 104

10−2

10−1

100

r

φ

φN (m = 6 and L = 18)φN (m = 10 and L = 5)φN (m = 20 and L = 2)

φH (m = 6)φH (m = 10)φH (m = 20)

Figure 7.4: Comparison between the spectral solutions, φN, and Horedt’s solutions, φH.

105


10−1 100 101 102 10310−9

10−6

10−3

100

103(a)

r

|1−φN/φ

H|

N = 60 N = 100N = 200 N = 300N = 400 N = 500

10−1 100 101 102 10310−9

10−6

10−3

100

103(b)

r

|1−φN/φ

H|

N = 30 N = 60N = 100 N = 200N = 300

10−1 100 101 102 103

10−8

10−6

10−4

10−2

100(c)

r

|1−φN/φ

H|

N = 30 N = 60N = 100 N = 200

Figure 7.5: Relative error of the spectral solutions, φN, from Horedt’s solutions, φH for different N.The degree N was increased until the spectral solutions ensured seven-digit accuracy for (a) m= 6,(b) m = 10 and (c) m = 20.

7.2 The topological property and asymptotic approximation

of the LE1 function

Figure 7.6 (a) depicts the trajectory on the Milne phase plane for m = 20 (see Appendix E for

details about the Milne variables [u, v].) Like the isothermal sphere, the curve spirals about the

singular point(

m−3m−1 , ω

)that corresponds with the singular solution of the LE1 equation. The Milne

variables can be used to discuss the stability of finite stellar polytrope based on the generalized

statistical mechanics (Taruya and Sakagami, 2002; Chavanis, 2002b). Figure 7.6 (b) is the mag-

106


nification of the trajectory around the singular point(

m−3m−1 , ω

), and it consistently approaches the

point that was analytically derived by (Chandrasekhar, 1939).

Figure 7.7 graphically reproduces the large-r asymptotic behaviors in equation (2.1.24) for

m = 6 and m = 20. The amplitude of the oscillation gets smaller as m increases. The oscillation

preserves up to r ≈ 1033 for m = 20 and r ≈ 1070 for m = 6.

107


0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15(a)

u

v

m = 20 (original scale)−

5·10−

13 0

5·10−

13

1·10−

12

1.5·10−

12

2·10−

12

−4 · 10−14

−2 · 10−14

0

2 · 10−14

4 · 10−14

6 · 10−14 (b)

u − us.p

v−

v s.p

m = 20(magnified)

Figure 7.6: (a) Trajectory in the Milne (u, v) plane for m = 20 and L = 2 with N = 300 and(b) its magnification near (m−3

m−1 , ω). The solution starts from (uE, vE) = (3, 0) and asymptoticallyapproaches the singular point (us.p, vs.p) =

(m−3m−1 , ω

).

108


10−2 105 1012 1019 1026 1033−0.1

−0.05

0

0.05

0.1(a)

r

rα( rω

φN/A

m−

1)

m = 20 and L = 2

10−2 1014 1030 1046 1062 1078−1

−0.5

0

0.5

1(b)

r

rα( rω

φN/A

m−

1)

m = 6 and L = 18

Figure 7.7: Asymptotic oscillations of the spectral solutions (N = 300) for (a) m = 20 and (b)m = 6. For large r, rα

(rωφN/Am − 1

)≈ C1 cos (ω ln r + C2).

109

Chapter 8

Main result: A spectral solution for the

ss-OAFP model

Section 8.1 provides a Chebyshev-Gauss spectral solution of the ss-OAFP system on the whole

domain with the eigenvalues and semi-analytical expression of the solution. The solution is stable

against various numerical parameters to five significant figures. However, we found the solution

is unstable against degree N. To discuss the cause of the numerical instability, Section 8.2 shows

truncated-domain solutions. It identifies that the instability occurs when the minimum value of

the Chebyshev coefficients reaches the order of 10−13, which corresponds to the limit of double

precision for the ss-OAFP system. Up to date, the Heggie-Stevenson (HS) solution of (Heggie and

Stevenson, 1988) is the most reliable solution. Hence, Section 8.3 reproduces the HS’s solution

using the spectral method and change of independent variables. We conclude that our spectral

solution is accurate to four significant figures. One the one hand, the HS’s solution is accurate

only to one significant figure. For brevity, we do not detail two topics (i) how the mathematical

formulation affects spectral solutions and (ii) why the Newton iteration method is hard to work for

the ss-OAFP system. Refer to (Ito, 2021) for the details of the two topics.

110

CHAPTER 8. RESULT: THE SS-OAFP MODEL

8.1 A self-similar solution on the whole domain

We provide the whole-domain solution, its semi-analytical form, and eigenvalues in Section

8.1.1. Section 8.1.2 details the characteristics of the Chebyshev coefficients and regularized solu-

tion. Section 8.1.3 discusses the numerical stability of the solution, and reports that the solution

is unstable against degree N. A MATLAB code to find the whole-domain solution is given in

Appendix L.

8.1.1 Numerical results (the main results of the present chapter)

Figure 8.1 depicts DF F(E) obtained from the whole-domain spectral solution. In the figure,

the HS’s solution is also depicted. The spectral and HS’s solutions are graphically almost identical.

The optimal values of numerical parameters are N = 70, FBC = 1, and L = 1. The optimal

eigenvalue of β is

βo ≡ 8.1783711596581. (8.1.1)

We chose the value of βo so that 3I(x = 1) reached its minimum value (∼ 10−12). In order to make

the Newton iteration method work, we needed at least eight significant figures of βo (Appendix

F.1.1). Also, degree N = 70 is the minimum value among 70 ≤ N ≤ 400 for which the Newton

iteration method worked (Section 8.1.3).

111


−1 −0.8−0.6−0.4−0.2 010−21

10−16

10−11

10−6

10−1(a)

E

F(E)

Spectral solutionHS’s solution

−1 −0.8 −0.6 −0.4

10−4

10−3

10−2

10−1

100(b)

E

F(E)


Figure 8.1: (a) Distribution function F(E) of stars on the whole domain and (b) its magnified graphon −1 ≤ E < −0.25. (N = 70, FBC = 1, and L = 1.)

The eigenvalues that we found agree with those of HS up to three significant figures (Table

8.1). Our value of χesc is greater than the HS’s value 13.85. This is consistent with the result of

(Cohn, 1980). According to Cohn (1980), a complete core-collapse occurs at χesc ≈ 13.9.

We report the ’semi-analytical’ solution1 of the ss-OAFP system for applications. Table 8.2

lists the semi-analytical forms of F(E), Q(E), and R(Φ). The degree of polynomials is at most

eighteen, and % error is 0.1% compared with the whole-domain solution with degree N = 70.

In the rest of the present work, we call the following eigenvalues and βo the ’reference eigen-

values’ to compare them with eigenvalues obtained from other formulas and solutions

c1o = 9.09254120455 × 10−4, (8.1.2a)

c∗4o = 3.03155222 × 10−1. (8.1.2b)

The reference eigenvalues were obtained from the whole-domain solution when β = βo, N = 70,

1Spectral methods can provide a solution can be expanded in terms of a few to tens of degrees of polynomials orbasis functions (e.g., Legendre polynomials, Gegenbauer polynomials, and Hermite functions.) if the solution has nonon-analytical- and non-regular properties. Hence, we call such a solution the semi-analytical solution.

112


Eigenvalues Spectral method HS T % relative error [%]

c1 9.0925 × 10−4 9.10 × 10−4 9.1 × 10−4 0.1

c2 1.1118 × 10−4 1.12 × 10−4 − 0.9

c3 7.1975 × 10−2 7.21 × 10−2 − 0.1

c4 3.303 × 10−2 3.52 × 10−2 − 6.7

α 2.2305 2.23 2.23 0

χesc 13.881 13.85 − 0.3

ξ 3.64 × 10−3 3.64 × 10−3 − 0

Table 8.1: Comparison of the present eigenvalues and physical parameters with the results of ’HS’(Heggie and Stevenson, 1988) and ’T’ (Takahashi, 1993). The relative error between the HS’seigenvalues and the present ones are also shown. The present eigenvalues are based on the resultsfor various combinations of numerical parameters (13 ≤ N ≤ 560, 10−4 ≤ FBC ≤ 104, and L = 1/2,3/4, and 1), different formulations (Sections 8.2 and 8.3 and Appendix F.1.3) and stability analyses(Appendix F.1).

L = 1, and FBC = 1. Also. we call the whole-domain solution with N = 70, L = 1, FBC = 1, and

β = βo the ’reference solution’. The reference solution is labeled with subscript symbol ′o′; such

as, functions Fo(E) and Φo(Ro) and regularized functions 3Fo(x) and 3Go(x).

8.1.2 The Chebyshev coefficients for the reference solution and regularized

functions

Figure 8.2 depicts the Chebyshev coefficients for the regularized reference solution. The min-

imum absolute values of all the normalized coefficients reach as low as ∼ 10−12 at around n = 70.

This means that a possible relative error in the spectral solution is ∼ 10−10% at best. The coeffi-

113


index Coefficients

n Fn Q′

n Rn

1 −0.9793 3.0496 2.0588

2 0.4515 3.1373 0.7337

3 0.3949 1.1575 0.1589

4 0.1751 0.2169 −0.0066

5 −0.0171 −0.0123 −0.0182

6 −0.0381 −0.0028 0.0013

7 0.0076 0.0054 0.0038

8 0.0103 −0.0011 −0.0007

9 −0.0046 −0.0010 −0.0009

10 −0.0023 0.0006 0.0003

11 0.0023 0.0001 0.0002

12 0.0001 −0.0002 −0.0001

13 −0.0009

14 0.0002

15 0.0002

16 −0.0002

17 0.0000

18 0.0001

Function Semi-analytical expression

F(E) exp[∑18

n=1 FnTn−1 (−2E − 1)]

(−E)β

Q(E)[∑12

n=1 Q′

nTn−1 (−2E − 1)]3

(−E)σ(1 + E)3

R(Φ) exp[∑12

n=1 RnTn−1 (−2Φ − 1)]

(−Φ)ν√

1 + Φ

Notation

E ∈ (−1, 0) Tn−1(−2E − 1) = cos[(n − 1) cos−1(−2E − 1)

]Φ ∈ (−1, 0) β = 8.1784

σ = −34

(2β − 1) = −11.5176

ν = −14

(2β + 1) = −4.3392

Table 8.2: Semi-analytical forms of F(E), R(Φ), and Q(E) obtained from the whole-domain spec-tral solution (N = 70, L = 1, and FBC = 1). The relative error of the semi-analytic form from thewhole-domain solution is the order of 10−4. The coefficients Q

′

n are the Chebyshev coefficientsfor 3Q/(0.5 − 0.5x), and were introduced to make the accuracy of Q(E) comparable to those ofF(E) and R(Φ).

114


cients show geometric convergences as follows

| Fn/F1 |∼| Gn/G1 |∼| In/I1 |∼| Jn/J1 |∼ exp(−0.3n), (8.1.3a)

| Rn/R1 |∼| Qn/Q1 |∼ exp(−0.4n). (8.1.3b)

20 40 6010−14

10−12

10−10

10−8

10−6

10−4

10−2

100

n

| Fn/F1 |

20 40 6010−14

10−12

10−10

10−8

10−6

10−4

10−2

100

n

| In/I1 |

20 40 6010−14

10−12

10−10

10−8

10−6

10−4

10−2

100

n

| Gn/G1 |

20 40 6010−14

10−12

10−10

10−8

10−6

10−4

10−2

100

n

| Jn/J1 |

20 40 6010−14

10−12

10−10

10−8

10−6

10−4

10−2

100

n

| Rn/R1 |

20 40 6010−14

10−12

10−10

10−8

10−6

10−4

10−2

100

n

| Qn/Q1 |

Figure 8.2: Absolute values of the normalized Chebyshev spectral coefficients for the referencesolutions. (N = 70, FBC = 1, and L = 1.) The coefficients are divided by their first coefficients.

Figure 8.3 compares the regularized reference solution on a linear scale. Functions 3F(E),

3R(Φ), and 3Q(E) prevail a discrepancy between the two solutions as E → 0. The figure shows that115


Heggie and Stevenson (1988) obtained their solution outside the domain on which our functions

asymptotically behave as constant functions. This implies that the actual significant figures of the

HS’s solution may not be more than one digit. This matter is discussed in detail in Section 8.3.1

(and Appendix G).

10−2 10−1 100−1.5

−1

−0.5

0

−E

3F(E

)


10−2 10−1 1000

0.2

0.4

−E3

I(E

)/(−

E)


10−2 10−1 100

0.08

0.1

−E

3 G(E

)/(−

E)


10−2 10−1 100

−1

−0.5

0

−E

3J(

E)


10−2 10−1 1001.5

2

2.5

3

−Φ

3 R(Φ

)


10−2 10−1 1000

0.5

1

−E

3Q

(E)


Figure 8.3: Regularized spectral solution compared with the corresponding HS’s solution. (N =

70, FBC = 1, and L = 1.)

116


8.1.3 The instability of the whole-domain solution

As explained in Appendix F.1, the whole-domain solution is stable against various numerical

parameters. For a broad range of the values of β, FBC, and L, the eigenvalues c1 and c∗4 can preserve

seven- and five- significant figures compared with c1o and c∗4o. On the one hand, the whole-domain

solution is unstable against degree N. The Newton iteration method worked only for 70 ≤ N . 400.

Figure 8.4 shows the numerical instability of the whole-domain solution. The values of | 3I(x = 1) |

and | 1 − c4/c∗4o | increase with N. Also, the condition number is unstable against N. The number

was calculated for the Jacobian matrix of the 4ODEs and Q-integral in the Newton method.

50 100 150 200 250 300 350 40010−12

10−10

10−8

10−6

10−4

10−2

100

(a)

degree N

| 3I(x = 1) || 1 − c∗4/c

∗4o |

50 100 150 200 250 300 350 400103

104

105

106

107

108

109

1010

(b)

degree N

Condition number

Figure 8.4: (a) Relative error between c4 and c∗4o, and the absolute value of 3I(x = 1), (b) Conditionnumber of the Jacobian matrix for the 4ODEs and Q-integral. The condition number was computedwhen aq

old − aqnew reached the order of 10−13 in the Newton iteration process. (L = 1 and

FBC = 1.)

8.2 Self-similar solutions on truncated domain

The present section provides spectral solutions on several truncated domains, and explains

the cause of the numerical instability against N. It also provides an optimal truncated-domain117


solution whose relative error is the order 10−9 from the reference solution. The present work

relies on a collocation method. We need to construct a spectral solution whose accuracy improves

with increasing N, unlike the whole-domain solution. Hence, we truncated the domain of the ss-

OAFP system. Based on the result of Section 8.1, we extrapolated 3F(x) using 3(ex)F (x) (equation

(4.4.19)) on −0.2 . E . 0. On the interval, the regularized reference functions behave like constant

functions of E. This means that one may expect to obtain several classes of solutions for different

maximum energy Emax. We found the following classes2

(i) Emax . −0.25 (Incorrect solution)

Solutions and eigenvalues significantly differ from the existing results.

(ii) − 0.25 . Emax . −0.05 (Stable solution)

Chebyshev coefficients are relatively stable against degree N.

(iii) − 0.05 . Emax . −0.005 (Semi-stable solution)

Chebyshev coefficients are stable up to a certain degree Nc.

(iv) − 0.005 . Emax (unstable solution)

Chebyshev coefficients are unstable against degree N.

Section 8.2.1 explains a condition to obtain a truncated-domain solution by examining classes (i)

and (ii). Section 8.2.2 discusses cases (ii) and (iii) to find an optimal truncated-domain solution for

fixed β = βo. We also discuss the cause of the numerical instability of the whole-domain solution.

8.2.1 Stable solutions on truncated domains with −0.35 ≤ Emax ≤ −0.05

We found that the spectral solutions on the truncated domains with −0.35 ≤ Emax ≤ −0.05 have

a transition point at around Esoln(= −0.225). The energy Esoln separates the solutions into incorrect

2This classification was made based on the absolute value of each term in equation (4.4.16a) (Refer to (Ito, 2021)for the details of the classification.)

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and stable solutions. The present section shows truncated-domain solutions obtained near Esoln for

L = 1, FBC = 1, and β = 8.1783. Figure 8.5 (a) shows the values of | 1− c1/c1o |, | 1− c∗4/c∗4o |, and

| 3I(x = 1) | for −0.35 ≤ Emax ≤ −0.1. Figure 8.5 (b) depicts the condition number of the Jacobian

matrix for the 4ODEs and Q-integral. All the values depict significant changes around at Esoln.

Heggie and Stevenson (1988) reported this transition point as a difficulty in the convergence of the

Newton method. The eigenvalues for Esoln > Emax significantly deviate from both the existing and

our eigenvalues. Hence, we call solutions obtained for Esoln > Emax the ’incorrect solutions’.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410−5

10−4

10−3

10−2

10−1

100

101

102

103

104

(a)

−Emax

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4106

107

108

109

1010

1011

1012

(b)

−Emax

Condition number

Figure 8.5: (a) Characteristics of | 3I(x = 1) | and relative errors of c1 and c∗4 from their referenceeigenvalues for different Emax. (N = 40, β = 8.1783, FBC = 1, and L = 1.) (b) Condition numberof the Jacobian matrix calculated when aq

new − aqold reached the order of 10−13 in the Newton

iteration.

Spectral solutions for −0.225 ≤ Emax ≤ −0.05 are relatively stable against degree N. Espe-

cially, the solution is the most stable at Emax = −0.225. For Emax = −0.225 and β = 8.17837,

Figure 8.6(a) shows the characteristics of | 1−c1/c1o |, | 1−c∗4/c∗4o |, and | 3I(x = 1) | against degree

N. The Newton method worked even for N = 360. Also, a higher N provides smaller absolute

values of Chebyshev coefficients. Figure 8.6 (b) shows the coefficients for N = 50 and N = 360.

119


The coefficients do not decay rapidly with high indexes n. We believe that the rapid decay was hin-

dered by the discontinuous behavior in the Q-integral (Appendix F.3.1) or the Poisson’s equation

(F.3.2). The discontinuity would be a result that one can not correctly specify the value of β with

a high accuracy. In fact, we could not specify the significant figures of β more than six digits for

−0.225 ≤ Emax ≤ −0.1.

0 50 100 150 200 250 300 35010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

(a)

Degree N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

100 101 10210−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101(b)

index n

Fn

N = 50N = 360

Figure 8.6: (a) Relative errors of c1 and c∗4 from the reference eigenvalues, and the absolute valueof 3I(x = 1) for different N. (b) Chebyshev coefficients for 3F with N = 50 and N = 360.(Emax = −0.225 and β = 8.17837. )

8.2.2 Optimal semi-stable solutions on truncated domains with

−0.1 ≤ Emax ≤ −0.03 for fixed β = βo

To examine the direct relationships between the reference- and truncated-domain- solutions,

we sought the truncated-domain solutions with fixed β = βo3 for −0.1 ≤ Emax ≤ −0.03. We report

(i) semi-stable solutions with β = βo, (ii) the relation of the solutions with the numerical instability

against degree N, and (iii) an optimal semi-stable solution compared to the reference solution.

3The optimal eigenvalue of β agreed with βo up to eight significant figures for −0.1 ≤ Emax ≤ −0.03 (Refer to Ito,2021). Hence, we fixed the value of β to βo.

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(i) Semi-stable truncated-domain solutions with β = βo

Semi-stable solutions with β = βo approach the reference solution with N, but they lose their

accuracies beyond certain degrees. Figure 8.7 shows the characteristics of 3I(x = 1) against N,

and the relative errors of c1 and c∗4 from the reference values. For Emax > −0.07, | 3I(x = 1) |,

| 1− c1/c1o |, and | 1− c∗4/c∗4o | decrease with increasing N, and reach small values (≈ 10−9 ∼ 10−13)

at certain degrees. However, beyond the degrees, the spectral solutions significantly lost their

accuracies, or the Newton iteration method did not work. On the one hand, for Emax < −0.07, the

values of | 3I(x = 1) |, | 1 − c1/c1o |, and | 1 − c∗4/c∗4o | stall with N while their minimum values still

can be found at relatively-low degrees (N = 27 ∼ 35).

121


0 10 20 30 40 50 60 7010−17

10−12

10−7

10−2

Emax = −0.10

N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

0 10 20 30 40 50 60 7010−17

10−12

10−7

10−2

Emax = −0.09

N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

0 10 20 30 40 50 60 7010−17

10−12

10−7

10−2

Emax = −0.08

N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

0 10 20 30 40 50 60 7010−17

10−12

10−7

10−2

Emax = −0.07

N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

0 10 20 30 40 50 60 7010−17

10−12

10−7

10−2

Emax = −0.06

N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

0 10 20 30 40 50 60 7010−17

10−12

10−7

10−2

Emax = −0.05

N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

0 10 20 30 40 50 60 7010−17

10−12

10−7

10−2

Emax = −0.04

N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

0 10 20 30 40 50 60 7010−17

10−12

10−7

10−2

Emax = −0.03

N

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

Figure 8.7: Relative errors of c1 and c∗4 from the reference eigenvalues and characteristics of 3I(x =

1) against N for the truncated-domain solutions with β = βo. (−0.1 ≤ Emax ≤ −0.03, L = 1, andFBC = 1.)

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(ii) Truncated domains and numerical instability against N

We consider the numerical instability against N originates from the property that the ss-OAFP

system may not have a solution when the terms in the system reach the order of machine precision

at the equation level and beyond the accuracy. In Figure 8.7, the minimum of | 3I(x = 1) | occurs

at degrees Nbest = 27, 25, 35, 35, 47, 57, 65, 65 for Emax = -0.10, -0.09, -0.08, -0.07, -0.06,

-0.05, -0.04, -0.03 . Figure 8.8 depicts the Emax-dependence of | 3I(x = 1) |, | 1 − c1/c1o |, and

| 1 − c∗4/c∗4o | obtained at each Nbest. The value of | 1 − c∗4/c

∗4o | decreases in a power-law-like

fashion with increasing Emax. We introduced a power-law profile c1o(−Emax)βo/c∗4o.4 In Figure 8.8,

c1o(−Emax)βo/c∗4o decreases in a similar way to | 1 − c∗4/c∗4o |. On the one hand, | 1 − c1/c1o |

and | 3I(x = 1) | do not decrease at Emax larger than −0.05. This would be due to a limit of

double precision. In addition to the result that c1o(−Emax)βo/c∗4o can characterize the accuracy of

c∗4o (correspondingly the solution), we found the infinity norm of | Fn(old) − Fn

(new) | for the

Newton method reaches the order of 10−13 at best. These findings mean that c1o(−Emax)βo/c∗4o

reaches the order of 10−13 at Emax ≈ −0.0523. This energy would be the maximum value of Emax

to preserve the numerical accuracy.

The discussion above implies that machine (double) precision is not precise enough to obtain

more accurate truncated-domain solutions at Emax . −0.05. Hence, the solutions can not improve

the accuracy with N (since a higher N means smaller values of the solution in spectral methods.)

On the one hand, the whole-domain solution is not truncated at high E. Hence, the power-law

boundary conditions in the ss-OAFP system can be satisfied given that the solution reaches the

order of machine precision. Unlike the semi-stable solution, the whole-domain solution may not

exist for a low N that can not provide the small values of the boundary condition. It only loses its

accuracy with increasing degree N when the solution satisfies the boundary conditions. One can

find more detail discussions about machine precision and the convergence of the Newton iteration

4The power-law profile represents the order of the last term in equation (4.4.16a). The term is directly related tothe accuracy in c∗4 (See Ito, 2021, for detailed discussions).

123


method in (Ito, 2021).

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.1110−16

10−13

10−10

10−7

10−4

10−1

102

−Emax

| 3I(x = 1) || 1 − c1/c1o |

| 1 − c∗4/c∗4o |

c1o(−Emax)βo/c∗4o

Figure 8.8: Relative errors of c1 and c∗4 from the reference values and characteristics of | 3I(x = 1) |obtained at each Nbest. The guideline c1o(−Emax)βo/c∗4o is shown for the sake of comparison. (L = 1,FBC = 1, and β = βo.)

(iii) An optimal semi-stable solution

We propose an optimal truncated-domain solution that is compatible with the reference solu-

tion. For Emax = −0.03 in Figure 8.7, | 1 − c∗4/c∗4o | reaches the order of 10−9. This is the same

order as the minimum value of | 1 − c∗4/c∗4o | computed against different β in Appendix F.1.1. Also,

3I(x = 1) = 7.3 × 10−12 for N = 65 is one of the smallest values among the 3I(x = 1)-values

obtained for semi-stable truncated solutions. Figure 8.9 (a) shows that, as N increases, the solu-

tion for Emax = −0.03 gradually converges to that with N = 65. Hence, the parameters for this

optimal truncated-domain solution are Emax = −0.03, N = 65, and β = βo in the present work.

Also, we found that the extrapolated DF’s mathematical expression does not affect the optimal

solution significantly (Appendix F.2). Figure 8.9 (b) shows the relative error between the optimal

and referencesolutions. Our optimal truncated-domain solution validates the reference solution.

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The largest relative error between them is the order of 10−9.

0.2 0.4 0.6 0.8 110−10

10−8

10−6

10−4

10−2

100

102 (a)

E

| 1 − F(E)(N=65)/F(E)(N) |

N = 25 N = 30N = 35 N = 40N = 45 N = 50N = 55 N = 60N = 70

10−1 10010−12

10−11

10−10

10−9

10−8

(b)

−E

| 1 − F(E)(N=65)/Fo(E) |

Figure 8.9: (a) Relative errors of DF F(N)(E) with different N from the DF F(N=65)(E) with Emax =

−0.03. (L = 1 and FBC = 1) (b) Relative error between the reference DF Fo(E) and the optimaltruncated-domain DF. (L = 1, FBC = 1, and β = βo.)

8.3 Discussion: Reproducing the HS’s solution

The present section discusses how to reproduce the solution of (Heggie and Stevenson, 1988)

using the spectral method first, and then estimates the accuracy of the reference solution. In Sec-

tion 8.2.1, the stable truncated-domain solution on −0.35 ≤ E ≤ −0.1 is closer to the reference

solution than the HS’s solution. The latter was obtained on almost the same domain. To explain the

discrepancy, we examined several classes of the ss-OAFP solutions by modifying the regularized

dependent variables. We found that only the modifications of 3J(x), 3R(x), and 3F(x) significantly

changed the ss-OAFP solution while that of the rest of the regularized functions did not (Refer to

Ito, 2021, for detailed discussions). Based on the modification of 3R(x), Section 8.3.1 reproduces

the HS’s solution with limited degrees. It also shows that the modification can provide both the

125


HS’s and referencesolutions only by controlling xmin (or Emax). Section 8.3.2 discusses how to

obtain the HS’s solution without any limitation. For brevity, a further detail discussion on repro-

ducing the HS’s solution is included in Appendix G. For the sake of comparison, the HS’s solution

is labeled hereafter by the subscript ‘HS’, such as FHS for the DF.

8.3.1 Reproducing the HS’s solution and eigenvalues with limited degrees

The present section reproduces the HS’s solution with only low degrees (N < 20) of polyno-

mials by modifying the regularization for 3R as follows

3(m)R (x) = [3R(x)]2

(1 − x

2

). (8.3.1)

Heggie and Stevenson (1988) reported that their solution is “thought to be accurate about three

significant figures.” On the one hand, they described the value of χesc as “∼ 13.85.” They reported

three significant figures of the eigenvalues c1, c2, c3, and c4 at best. Their inconsistent expressions

of accuracy led us to reproduce at least two significant figures of the HS’s solutions and eigenval-

ues. We reformulated the ss-OAFP system based on 3(m)R first, and then solved the system using the

numerical procedure of Section 5.2.

We found spectral solutions whose eigenvalues reproduced those of HS (c1 = 9.10 and

c4 = 3.52) with N = 15 near Emax = −0.225. For the solutions, Table 8.3 shows the values of β and

χesc. It also displays the measures of accuracy min(|Fn|) and 3I(x = 1). The order of the measures

is approximately min(|Fn|) ∼ 3I(x = 1) ∼ 10−4. Our χesc agrees with that of HS (= 13.85)

for Emax = −0.225. The numerical values of ln[F] obtained from the spectral solutions agree with

those of ln[FHS] to 2 ∼ 4 significant figures. The relative error between ln[F] and ln[FHS] is at most

the order of 1× 10−3 for E ≥ −0.9 (Figure 8.10). This result would infer that the spectral solutions

reproduced “about three significant figures” of the HS’s solution with the same eigenvalues.

126


Emax β χesc min(|Fn|) | 3I(x = 1) |

−0.240 8.17310 13.840 4.1 × 10−4 2.1 × 10−4

−0.225 8.17460 13.845 4.5 × 10−4 2.3 × 10−4

−0.215 8.17536 13.847 4.7 × 10−4 2.4 × 10−4

−0.200 8.17560 13.850 4.6 × 10−4 2.3 × 10−4

Table 8.3: Eigenvalues of the reproduced HS’s solution with eigenvalues c1 = 9.10 × 10−4 andc4 = 3.52 × 10−2 for N = 15. Heggie and Stevenson (1988) reported the numerical values of theirsolution on −1 / E ≤ −0.317. They mentioned that the Newton iteration method worked up toEmax ≈ −0.223 and it could work beyond -0.223.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−5

10−4

10−3

10−2

10−1

100

−E

|1−

ln[F

]/ln

[FH

S]|

| 0.005/ ln(FHS) | Emax = −0.24Emax = −0.225 Emax = −0.215

Figure 8.10: Relative error between DFs obtained from the HS’s solution ln[FHS] and spectralsolution ln[F] with N = 15 for Emax.

127


8.3.2 Successfully reproducing the HS’s solution and accuracy of the

reference solution

We briefly explain how to obtain both the reference- and HS’s solution with a high degree

of polynomials on truncated domains based on a single mathematical formulation of the ss-OAFP

model. For brevity, the detailed discussion is made in Appendix G, and we explain only the results.

The most crucial result in Appendix G is that one can find the HS’s solution if the absolute values

of the coefficients In for 3I(x) reach the order of 10−4 ∼ 10−5 for Emax ≈ −0.25. One also can find

the reference solution if the coefficients reach the order of 10−6 ∼ 10−7 for Emax ≈ −0.05 (Figure

G.2). We believe that the decay rate of the Chebyshev coefficients is too rapid in Section 8.3.1.

Hence, we obtained the HS’s solution only with low degrees. In fact, for the numerical calculation

in Appendix G, we intentionally included non-analytic and non-regular properties into dependent

variables by reformulating the variables 3R and 3F .

We believe that the numerical accuracy of the reference solution is at least four significant

figures. This accuracy is based on the detailed analyses that we carried out for the various for-

mulations employed in the present section, Sections 8.1, 8.2, and 8.3.1 and Appendices F and

G. Among the various formulations that reproduced both the HS’s and referencesolutions, the

3(m)R -formulation of Section 8.3.1 provided the smallest relative error ∼ 4 × 10−5 compared with

the reference solution (Figure 8.11). This error corresponds with the relative error of c4 from the

reference eigenvalue. Hence, Table 8.1 lists four significant figures of c4 and five of the rest of the

eigenvalues.5

5The eigenvalues c1, c2, and c3 were more stable than c4 for any formulations in our work.

128


10−1 10010−8

10−7

10−6

10−5

−E

| 1 − 3Ro/3(m)R |

| 1 − 3Fo/3F |

Figure 8.11: Relative errors of 3(m)R and 3F from 3Ro and 3Fo respectively. (Emax = −0.05, N = 540,

L = 1, and FBC = 1.)

129

Chapter 9

Property: The thermodynamic and

dynamic aspects of the ss-OAFP model

Fitting of the ss-OAFP model is not proper to the observed structural and kinematic profiles of

globular clusters due to its infinite total mass and radius, while analyzing the model may deepen

our understandings of the dynamical evolution of star clusters. The present chapter details the

fundamental thermodynamic aspects of the ss-OAFP model. Mainly, we discuss the relationship

between the cores of the ss-OAFP model and isothermal sphere. The latter is an imitation of the

core of a star cluster. The core of a finite isothermal sphere shows a positive heat capacity at a

normal state. Its physical state may adequately describe the early stage of the relaxation evolution.

On the one hand, the cluster core at the late stage is expected to show a negative heat capacity that

can be modeled by the ss-OAFP model (as we detail in the present chapter). Hence, we can clarify

the difference between the physical states of the cores of the two models.

The present chapter is organized as follows. We first explain three introductory topics for this

chapter in Section 9.1 (i) isotropic self-gravitating models commonly employed for equilibrium

statistical-mechanics study, (ii) assumptions that we made for the ss-OAFP model, and (iii) the

cause of the negative heat capacity discussed in existing works. Section 9.2 shows the local and

130

CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

global physical properties of the ss-OAFP model. In Section 9.3, we construct an analog of a

caloric curve and show a negative heat capacity at constant volume for the ss-OAFP model after

normalizing the thermodynamic quantities. In Section 9.4, we discuss the origin of the negative

heat capacity by using a simple analytical method and comparing the core of the ss-OAFP model

to existing models. Section 9.5 concludes the present chapter.

9.1 Statistical mechanics of self-gravitating systems

Section 9.1.1 introduces isotropic models typically discussed from the point of view of equilib-

rium statistical mechanics first, and then explains the assumption we made for the ss-OAFP model.

Section 9.1.2 details existing works associated with the negative heat capacity of thermodynamic

systems to review the known causes of the negativeness.

9.1.1 Isotropic self-gravitating models employed for statistical mechanics

studies and assumption made for the ss-OAFP model

One may list the isotropic models that could establish a certain statistical equilibrium of self-

gravitating (gaseous) systems based on what temperature is employed for the models (Table 9.1).

To consider whether they are realistic, one has analyzed their stability problems. However, such a

discussion is hard to apply to the ss-OAFP model that does not rely on a simple form of temperature

or entropy. The ss-OAFP model may have (at least) three different (non-)equilibrium states. In the

core, stars could behave like the isothermal sphere due to the frequent two-body relaxation events.

The inner halo would be a receiver of heat- and stellar (particle-) fluxes due to the escaping stars

from the core, resulting in a non-equilibrium state. In the outer halo, the self-similar analysis1

1The accurate arguments for the finite outer halo need to include realistic effects. Such as, the fluxes at the ridgeof a cluster, two dimensional (anisotropic) effect, and escapers and escaping stars with high eccentricities under theinfluence of tidal effects. The discussions for the outer halo can be found in (e.g., Michie, 1962; Spitzer and Shapiro,1972; Claydon et al., 2019).

131


Model Temperature entropy Ref

Isothermal sphere Thermodynamic Boltzmann (Antonov, 1962; D. Lynden-Bell and Royal, 1968)

Energy-truncated models Thermodynamic Phenomenological (Katz, 1980; Katz and Taff, 1983)

(e.g. Woolly-, King-, and

Wilson- models)

Stellar Polytrope Polytropic (constant) Tsallis’ (Taruya and Sakagami, 2002; Chavanis, 2002b)

ss-OAFP models ? ?

Table 9.1: Typical isotropic self-gravitating systems discussed for study on equilibrium statisticalmechanics.

extends the physical state of the inner halo to an infinite radius. The stellar and energy- fluxes are

little in the outer halo. Hence, the outer halo would behave like a collisionless system forming a

power-law profile. The profile may be described by a state of equilibrium based on Tsallis entropy

(Refer to Tsallis, 2009, for detail of Tsallis’ generalized statistical dynamics).

To discuss the thermodynamic aspects of the ss-OAFP model, one needs to make several as-

sumptions. For example, the system can reasonably exist only in a time-averaged sense due to star

clusters’ discreteness (the finiteness of the total number of stars). Also, a self-similar model can

not achieve an equilibrium state while it can be at stationary state only in the inner- and outer- ha-

los. Hence, to consider the system a thermodynamic system, it must be an ’exotic’ self-gravitating

gas at a stationary non-equilibrium state(s), composed of N(>> 106 >> 1) particles whose DF and

m.f. potential can be obtained from the solution of the ss-OAFP model at a certain fixed time of the

self-similar evolution. The meaning of ’exotic’ is that the gas has a ’negative’ heat capacity intrin-

sically as its normal state (at least in the core of the system), as explained in the following sections.

Besides, the ss-OAFP model is not isolated. The non-isolation of a self-gravitating system implies

that the system can not strictly achieve a QSS in the sense that the Virial ratio can not reach unity.

Hence, following the classical discussion (Antonov, 1962; D. Lynden-Bell and Royal, 1968), we

consider the ss-OAFP model is enclosed by an adiabatic spherical wall of radius R(Φ) that elasti-

cally reflects stars on the inner surface. The present chapter employs the stellar DF Fo(E) and m.f.

potential Φ(Ro) obtained from the reference solution of the ss-OAFP model (Chapter 8 ). Also, the

132


time t of the ss-OAFP model is fixed to a certain time tc. This means the time-dependent variables

(e.g., ft, gt, εt, ...) in equation (2.2.2) turn into the factors that make the self-similar variables (e.g.,

F,G,R, ...) in dimensionless forms.

9.1.2 The cause of negative heat capacity reported in the previous works

Negative heat capacity has been discussed for self-gravitating astrophysical objects, such as;

stars, star clusters, and black holes (Lynden-Bell, 1999). More recently, various fields are con-

cerned with the negative heat capacity. Such as stratified gasses (Ingel, 2000), proteins (Prabhu

and Sharp, 2006), granular gasses (Brilliantov et al., 2018), and the systems of small N particles.

The examples of the small-N systems are Lennard-Jones gases (Thirring et al., 2003), melting

metal clusters (Aguado and Jarrold, 2011), and hot nuclei (Borderie and Frankland, 2019). Al-

though many existing works are based on numerical simulations and analytical approaches, the

negative heat capacity itself is not just a theoretical concept. Laboratory experiments have shown

the sign of the negative heat capacity of nuclear fragmentation (D'Agostino et al., 1999, 2000;

Srivastava, 2001; Gobet et al., 2002) and melting metal clusters (Schmidt et al., 2001). The exist-

ing works commonly yield that the negative heat capacity occurs in an energy band on which the

system undergoes a phase transition, such as; ’gas’-’liquid’ and ’liquid’-’solid’. In the case of star

clusters, the transition corresponds to ’gas’-’collapsed-state’. The ’gas’ corresponds with the state

of a self-gravitating gas with a positive heat capacity. The ’collapsed-state’ means the state of a

core-halo structure with a negative heat capacity. Also, the self-gravitating system is isolated from

the environment. Hence, the negative heat capacity has been reported for a self-gravitating system

in a micro-canonical ensemble2.

We review the systems that do not undergo a negative heat capacity to explain why statistical

dynamicists typically employ an inhomogeneous system in a micro-canonical ensemble to discuss

2For an isolated self-gravitating system, the Virial theorem states 2KE + PE = 0 where KE and PE are the kineticand potential energies of the system. Hence, the total energy satisfies Etot = −KE (e.g., Heggie and Stevenson, 1988;Binney and Tremaine, 2011). The heat capacity may be described by CV ∝ Etot/KE < 0.

133


the negative heat capacity. First, we refer to a thermodynamic-equilibrium small-N system of ho-

mogeneous subsystems, e.g., nanoclusters (Michaelian and Santamarıa-Holek, 2007; Lynden-Bell

and Lynden-Bell, 2008; Michaelian and Santamarıa-Holek, 2015, 2017). In the small-N system,

a negative heat capacity does not occur regardless of what kind of statistical ensembles being

employed. The small N systems can not achieve an ergodicity because constituent particles are

generally trapped in only part of the whole phase space. This trapping means a less mixing pro-

cess occurs in the phase space. Hence, the equilibrium state (and heat capacity) depends on the

initial condition for the dynamical evolution of the system. On the one hand, in the case of inho-

mogeneous equilibrium systems, a negative heat capacity does not occur for a canonical ensemble

(Thirring, 1970; Lynden-Bell and Lynden-Bell, 1977) and grand-canonical ensemble (Josephson,

1967) since the heat capacities are defined by the square of fluctuation in thermodynamic quantities

in the same way as the homogeneous case. These discussions typically encourage one to discuss a

negative heat capacity only in an isolated inhomogeneous system in a micro-canonical ensemble.

Although a negative heat capacity has been found in (nearly) isolated astrophysical systems

of particles or stars interacting via the Newtonian potentials, the long-range nature of pair-wise

potential itself is not only the cause of the negativeness. For example, even non-interacting particles

can have a negative heat capacity under certain background potentials. Based on the Virial theorem

and toy models, Einarsson (2004) showed that collisionless particles could provide a negative heat

capacity in the background potential profile ∼ ra (in three dimensions) if a = −1 while they do

not if a , −1. Thirring et al. (2003); Carignano and Gladich (2010) showed that a negative heat

capacity could occur to collisionless particles under a sudden change in a potential well from a

deep narrow well at small system radii to a shallow wide well at large radii. On the one hand,

even without a background potential well, particles interacting through short-range pair potential

can reveal a negative heat capacity. The examples are Lennard-Jones potentials for small N(∼ 10)

(Thirring et al., 2003) and Gaussian potentials (Posch et al., 1990; Posch and Thirring, 2005) for

N ∼ 100.

134


9.2 The thermodynamic quantities of the ss-OAFP model

Since the previous works (Heggie and Stevenson, 1988; Takahashi, 1993) did not discuss the

detailed structure of the ss-OAFP model, unlike (Lynden-Bell and Eggleton, 1980)’s work for a

self-similar conductive gaseous model, we extend (Lynden-Bell and Eggleton, 1980)’s analysis to

the ss-OAFP model. We especially focus on the core of the model. Sections 9.2.1 and 9.2.2 detail

the local- and global- properties of the ss-OAFP model.

9.2.1 The local properties of the ss-OAFP model

The stellar DF of the ss-OAFP model may not be even a local Maxwellian DF. Hence, we

need to find the moments of the DF as the analogs of local thermodynamic- or hydrodynamic-

quantities to capture the physical features. In addition to the density ρ(r, tc) (calculated for the

ss-OAFP equation), the velocity dispersion 32(r, tc), pressure p(r, tc), heat flux fh(r, tc), and stellar

flux fp(r, tc) may be defined in terms of the moments of the DF

32(r, tc) ≡

1ρ(r, tc)

∫ 0

φ(r,tc)

[2ε − 2φ(r, tc)

]3/2 f (ε, tc) dε, (9.2.1a)

p(r, tc) ≡32(r, tc) ρ(r, tc)

3, (9.2.1b)

fh(r, tc) ≡ −k∂ 32(r, tc)∂ r

, (9.2.1c)

fp(r, tc) ≡ −D∂ρ(r, tc)∂ r

, (9.2.1d)

where k is the thermal conductivity and D the diffusivity. The value of k (and D) depends on

the definition of relaxation time concerned (Lynden-Bell and Eggleton, 1980; Louis and Spurzem,

1991). Hence, we regularize the heat- and stellar fluxes by the constants in the coefficients that

appears in the self-similar analysis (that is, by the constants 3GmCk ln[N] for the conductivity

135


and√

2CD ln[N]/3 for the diffusivity3, where Ck and CD are dimensionless constants of unity.).

Employing the self-similar variables of Chapter 4 and introducing new dimensionless variables

32(r, tc) = V2

(dis)(R) 32t (t) ≡ 2V2(dis)(R) εt(t), (9.2.3a)

p(r, tc) ≡ P(R) pt(t), (9.2.3b)

fh(r, tc)3 G m Ck ln[N]

≡ Fh(R) fht (tc), (9.2.3c)

fp(r, tc)2 CD ln[N]/3

≡ Fp(R) fpt (tc), (9.2.3d)

the local quantities in dimensionless form read

V2(dis)(R) = 2

U3/2(Φ)U1/2(Φ)

, (9.2.4a)

P(R) =23

U3/2(Φ), (9.2.4b)

Fh(R) = −U1/2(Φ)

S (Φ)√

V2(dis)(Φ)

(U3/2(Φ) U−1/2(Φ)

[U1/2(Φ)]2 − 3), (9.2.4c)

Fp(R) = −1√

V2(dis)(Φ)

U−1/2(Φ)S (Φ)

, (9.2.4d)

where S (Φ) = −2S (x) due to the definition and Un(Φ) is

Un(Φ) =

∫ 0

Φ

F(E)(E − Φ)n dE, (9.2.5)

where n is a real number; for example, if n = 1/2 then Un(Φ) = D(Φ).

3For the conductivity, the expression of the constant is the same as that of (Lynden-Bell and Eggleton, 1980).Similarly, the diffusivity can be calculated as follows

D =13

l√32 ≈

13k2

J

CD

TR=

13

32(r, tc)4πGρ(φ(r, tc))

8 πG m CD ρ(φ, tc) ln[N][32(r, tc)]3/2 =

2 CD ln[N]

3√32(r, tc)

, (9.2.2)

where l is the mean free path of stars, kJ is the inverse of the Jeans length, and TR is the relaxation time from (Spitzer,1988).

136


The ss-OAFP model has characteristics similar to those of the self-similar conductive gaseous

model studied in (Lynden-Bell and Eggleton, 1980). Figure 9.1(a) depicts the velocity dispersion

V2(dis) and m.f. potential profile Φ. The constancy in V2

(dis) further extends in radius compared

with that in Φ. Figure 9.1(b) shows the heat flux Fh and stellar flux Fp. The flux Fh reaches

its maximum at R = 4.31. This radius is relatively close to R = 8.99 at which the gravothermal

instability occurs for the isothermal sphere in a canonical ensemble (D. Lynden-Bell and Royal,

1968). The flux Fp reaches its maximum at R = 3.11. As the local relaxation time increases

with radius, the fluxes Fh and Fp rapidly decrease while Fp decays slowly compared with Fh.

The location of the maximum stellar flux can be explained by the escape speed of stars. It is

discussed in Section 9.2.2 since the speed is a global property of the cluster. The turning points for

fluxes seem different from the graphed maximum value in (Lynden-Bell and Eggleton, 1980) but

the qualitative behaviors of our velocity dispersion, m.f. potential and fluxes little differ from the

gaseous models.

The present work further details the physical features of the self-similar model based on the

adiabatic index and the equation of state. Those features have not been discussed in detail for

the self-similar pre-collapse model. Following (Cohn, 1980)’s analyses on the time-dependent

OAFP model, we calculated the polytropic index m ≡ β + 3/2 of the ss-OAFP model. Figure

9.2 depicts the adiabatic index Γ(≡ 1 + 1/m) and polytropic index m against dimensionless radius

R. We calculated Γ by taking the logarithmic derivative d ln[P(R)]/ d ln[D(R)]. The result shows

the distinctive structures of the core and inner and outer halos. In the outer halo, m asymptotically

reaches 9.67837115...(Γ ≈ 1.10) for large R.4 For small R or in the core and inner halo, one can find

two features. (i) Index m reaches its minimum value 8.52 (Γ = 1.117) at R = 157, and (ii) index

m increases with decreasing R and reaches 177 (Γ = 1.00564) at the center of the core. A higher

m (or Γ close to unity) implies the core behaves more like the isothermal sphere. Given Γ, one

can find the asymptotic form for the equation of state (Figure 9.3). As expected, the core behaves

4The value of m holds the relative error 6.3 × 10−10% from the expected value βo + 3/2.

137


approximately an ideal gas with the thermodynamic temperature 1/χesc. On the other hand, the

outer halo behaves like a polytrope of m = βo + 3/2. It has a temperature approximately a half

of 1/χesc given that the polytropic constant K (in p(r, tc) = Kρ(r, tc)Γ) is considered the constant

temperature of the polytrope. The followings are the equations of local states in the two limits of

R

P =1.0χesc

D, (R ≈ 0) (9.2.6a)

P =0.5χesc

D1.1. (R→ ∞) (9.2.6b)

where the inverse temperature of the ss-OAFP model is the scaled escape energy χesc.

10−1 100 101 102 103 104 10510−2

10−1

100 (a)

R

| Φ |

V2(dis)

10−1 100 101 102 103 104 105 106 107 10810−24

10−19

10−14

10−9

10−4

101

(b)

R

Fh

Fp

1/TR

Figure 9.1: (a) Dimensionless m.f. potential Φ and velocity dispersion V2(dis) and (b) dimensionless

heat flux Fh and stellar flux Fp. The latter also depicts the inverse of regularized local relaxationtime TR

(= D(φ, tc)/[32(r, tc)]3/2

)for the sake of comparison.

138


100 102 104 106 108 1010 1012 1014 1016

1

1.1

1.2 (a)

R

Γ

100 102 104 106 108 1010 1012 1014 10160

50

100

150(b)

R

m

Figure 9.2: (a) Adiabatic index Γ and (b) polytropic index m of the ss-OAFP model.

100 102 104 106 108 1010 1012 1014 1016 1018

0.2

0.4

0.6

0.8

1

1.2

R

Pχ

esc/

DΓ

Figure 9.3: Local equation of state with the constant thermodynamic temperature 1/χesc.

139


9.2.2 The global properties of the ss-OAFP model

The global properties of the ss-OAFP model can provide an understanding of the macroscopic

structure in comparison with the local properties (Section 9.2.1). However, it is in general hard

to conceptualize some thermodynamic quantities for non-equilibrium states. Hence, the present

work avoids exactly defining the entropy of the model and discussing the thermodynamic laws.

We consider that stars that are confined by an adiabatic wall at radius RM. The mass and kinetic

and potential energies of the system read

M(RM, tc) =

"m f (ε, tc) d3

3 d3r, (9.2.7a)

KE(RM, tc) =

"m32

2f (ε, tc) d3

3 d3r, (9.2.7b)

PE(RM, tc) =

"m φ(r, tc) f (ε, tc) d3

3 d3r, (9.2.7c)

Etot(RM, tc) = KE + PE. (9.2.7d)

The corresponding dimensionless forms are

M(ΦM) ≡M(r, tc)Mt(tc)

= −

∫ ΦM

−1DR2S dΦ, (9.2.8a)

KE(ΦM) ≡KE(r, tc)PEt(tc)

= −

∫ ΦM

−1U3/2R2S dΦ, (9.2.8b)

PE(ΦM) ≡PE(r, tc)PEt(tc)

=12

∫ ΦM

−1ΦDR2S dΦ, (9.2.8c)

Etot(ΦM) ≡Etot(r, tc)PEt(tc)

, (9.2.8d)

140


where ΦM is the potential at the wall radius RM. The corresponding time-dependent variables at

time tc read

Mt(tc) = (4π)2√

2 m ft(tc) [Et(tc)]3/2[rt(tc)]3, (9.2.9a)

PEt(tc) =KEt(tc)

2=

12

Mt(t) Et(tc). (9.2.9b)

The thermodynamic temperature χesc can not be defined properly as a global quantity for the

ss-OAFP model. Hence, we introduce the kinetic temperature T (kin) and local temperature T (loc)

T (kin)(ΦM) ≡2KE3M

, (9.2.10a)

T (loc)(RM) ≡V2

(dis)

3. (9.2.10b)

Figure 9.4 depicts the local- and kinetic- temperatures. At the center of the system, the temper-

atures hold approximately a constant profile.5 Hence, if focusing on the core behavior, one may

loosely define the inverse temperature of the Maxwellian DF in the standard context of statistical

mechanics6 as follows

β(con) ≡χesc

εt(tc), (9.2.11)

This provides an approximate relationship at the center of the system

T (loc) RM rt(tc) ≈ T (kin) RM rt(tc) ≈1

kBβ(con) , (9.2.12a)

T (loc)(ΦM) ≈ T (kin)(ΦM) ≈ χesc, (9.2.12b)

5Exactly speaking, the temperatures gradually decrease with the wall radius. The maximum temperatures arelowered by 1 % at R ≈ 2.1 for the local temperature and at R ≈ 2.7 for the kinetic one.

6We still do not discuss the relation of the temperature with entropy even if the system is considered an isolatedsystem enclosed by an adiabatic wall.

141


where kB is the Boltzmann constant.

One also can calculate the escape speed of stars escaping from the cluster core. The character-

istics of the speed allows us to explain the maximum and rapid decay in the stellar flux FP shown

in Figure 9.1. First, we introduce the following escape speeds in dimensionless form

3(PE)esc =

√−4PE, (9.2.13a)

3(con)esc =

√4

MRM

. (9.2.13b)

We introduced 3(con)esc based on the Virial theorem considering the core is the isothermal sphere

isolated from the halo. Figure 9.5 compares the escape speeds with the stellar flux FP. In the core,

velocity dispersion√

V2(dis) is greater than the escape speeds. However, at the center of the core,

there is little stellar flux due to the flattening in the density. On the one hand, as RM increases,

the decreasing density causes an increasing stellar flux. Beyond R = 2.2 ∼ 2.3 at which√

V2(dis)

is the order of the escape speeds, stars can hardly escape due to little two-body relaxation and the

cluster’s self-gravity. Hence, the stellar flux decays rapidly with increasing RM.

10−1 102 105 108 1011 101410−4

10−3

10−2

10−1

100

RM

T (loc)

T (kin)

Figure 9.4: Local and kinetic temperatures.

142


10−1 100 101 102 103 104 105 106

10−4

10−3

10−2

10−1

100

101

RM

3(PE)esc

3(con)esc√V2

(dis)FP

Figure 9.5: Escape speeds of stars. The stellar flux FP and velocity dispersion√

V2(dis) are also

depicted.

9.3 Normalized thermodynamic quantities and negative heat

capacity

The present section focuses on the core of the ss-OAFP model to discuss the negative heat

capacity. Section 9.3.1 normalizes the total energy and temperatures of the model, and Section

9.3.2 depicts the caloric curve to characterize the negative heat capacity.

9.3.1 Normalized thermodynamic quantities

In the present section, we normalize the total energy and temperatures to include into our

discussion the effect of the finite size of the system enclosed by the wall. One would like to

consider that there exists a thermodynamic limit even for inhomogeneous thermodynamic systems

143


of log-range interacting stars, like the standard thermodynamics.7 Also, the mass, energy, and size

of the ss-OAFP model are infinite. Hence, one must resort to a proper thermodynamic limit to

normalize thermodynamic quantities at the wall radius RM.

The same normalization of the total energy has been employed for the isothermal sphere based

on the Boltzmann entropy (Antonov, 1962; D. Lynden-Bell and Royal, 1968) and the stellar poly-

tropes based on the Tsallis entropy (Taruya and Sakagami, 2002; Chavanis, 2002b). The normal-

ized energy reads

Λ =rEtot(r, tc)GM2(r, tc)

=RMEtot(ΦM)[M(ΦM)]2 . (9.3.1)

This normalization originates from the feature of the Poisson’s equation (Appendix H). Figure 9.6

shows the normalized total energy Λ of the ss-OAFP model, compared with that of the isothermal

sphere. The energy of the ss-OAFP model is negative at all radii, meaning the potential energy

dominantly determines the state of the model. Statistical dynamicists term this state (or turning

point) a collapsed phase that occurs to the isothermal sphere (and polytropes of m > 5) enclosed

by an adiabatic wall of a large radius. The energy Λ reaches the order of unity under the following

relation8 with the limit N → ∞

RM ∝ N, (9.3.2a)

M(ΦM) ∝ N, (9.3.2b)

Etot(ΦM) ∼ KE(ΦM) ∼ PE(ΦM) ∝ N, (9.3.2c)

1/β(con) ∝ 1, (9.3.2d)

Refer to Appendix A.3 for the corresponding scaling for the kinetics of globular clusters. Equation

7By the standard thermodynamic limit, we mean that N/V →constant if N → ∞ and V → ∞ for homogeneoussystems of particles undergoing short-range interaction.

8Sometimes the relation and limit are called the dilute limit (e.g., de Vega and Sanchez, 2002; Destri and de Vega,2007). it corresponds with one of the Boltzmann-Knudsen limits in the standard gaseous kinetic theory.

144


(9.3.2) determines only the relations among the magnitudes of quantities. The important point is

that there exists some reasonable dimensionless parameter as RM → ∞.9

Similarly, one can convert the reciprocal of the kinetic, local, and thermodynamic temperatures

into their dimensionless forms

η(kin) =GM(RMrt(tc), tc)kBT (kin)(RMrt(tc))

=M

T (kin)RM, (9.3.3a)

η(loc) =GM(RMrt(tc), tc)kBT (loc)(RMrt(tc))

=M

T (loc)RM, (9.3.3b)

η(con) =β(con)RMrt(tc)

GM(RMrt(tc))2 = −S (ΦM)

RM. (9.3.3c)

Figure 9.7 depicts the dimensionless inverse temperatures η(kin), η(loc), and η(con) of the ss-OAFP

model, compared with the corresponding temperature η of the isothermal sphere. All the tempera-

tures monotonically decrease as RM increases until they reach their minimum values. Only 1/η(con)

diverges at large RM, which implies that the inverse temperature β(con) is not correctly normalized.

One must normalize β(con) using the normalization made in (Taruya and Sakagami, 2002; Chavanis,

2002b) to hold a constant (polytropic) temperature at large RM. This is since the ss-OAFP model

behaves like a polytropic sphere of m = βo + 3/2 as RM → ∞.

9For example, the density (cluster) expansion in the standard thermodynamic limit does not mean a density n itselfis the expansion parameter. Rather, the corresponding (dimensionless) occupation number 4

3 na3, where a is the size ofparticles, is the actual parameter.

145


10−1 104 109 1014 1019−10

−8

−6

−4

−2

0

RM

Λ

ss-OAFP model

10−1 104 109 1014 1019−0.4

−0.2

0

0.2

0.4

RMΛ

Isothermal sphere

Figure 9.6: Dimensionless normalized total energies for the ss-OAFP model and isothermal sphere.As RM → 0, the energy monotonically decreases for the former and increases for the latter. Thecurve for the isothermal sphere was obtained using the numerical code in (Ito et al., 2018).

146


10−1 102 105 108 1011 1014 1017 1020

100

102

1/η

(kin

)

ss-OAFP model

10−1 102 105 108 1011 1014 1017 1020

100

102

1/η

(loc

)

ss-OAFP model

10−1 102 105 108 1011 1014 1017 1020

100

102

104

1/η

(con

)

ss-OAFP model

10−1 102 105 108 1011 1014 1017 1020

100

102

RM

1/η

Isothermal sphere

Figure 9.7: Dimensionless temperatures of the ss-OAFP model and temperature 1/η of the isother-mal sphere. The curve for the latter was obtained using the numerical code in (Ito et al., 2018)

147


9.3.2 Caloric curve

A caloric curve provides the characteristics of the heat capacity of the ss-OAFP model. The di-

mensionless energy (Equation (9.3.1)) and temperatures (Equation (9.3.3)) construct caloric curves

at constant volume (Figure 9.8). Based on the Legendre transformation, the graphs are related to

the heat capacity CV. For example, the kinetic temperature relates to CV as follows

∂ 1Λ

∂η(kin)

R,M

=

∂ 1Λ

∂η(kin)

V,Ntot

=CV

[T (kin)

]2

ME2tot

. (9.3.4)

Hence, the slope of the tangent to the caloric curve shows the sign of the heat capacity. The advan-

tage of the caloric-curve approach is to provide a simple topological understanding for the (linear)

stability of systems in statistical (e.g., micro-canonical- and canonical-) ensembles without solving

the corresponding eigenvalue problem (Katz, 1978, 1979, 1980). Since the ss-OAFP model does

not achieve an equilibrium system, the rest of the present section analogically discusses the heat

capacity of the ss-OAFP model and its singularity. We compare the results with the corresponding

caloric curve for the isothermal sphere to understand the thermodynamic aspects.

Singularities in the caloric curve of the ss-OAFP model

Singularities in the heat capacity of the ss-OAFP model occur at radii close to each other,

which differs from the case of the isothermal sphere. Figure 9.8 depicts the caloric curves for the

normalized energy Λ against the normalized inverse temperatures η(kin), η(con), and η(loc). All the

caloric curves show negative heat capacities at small radii while they have different characteristics

at larger radii after passing through their turning points with increasing RM. The turning points of

the caloric curves (Table 9.2) occur very close to each other in radius (RM ≈ 18 ∼ 24 for CV → ∞

and RM ≈ 25 for CV = 0 ). Perhaps, this is consistent with the nature of gravothermal instability

148


if10 the negative heat capacity of the core originates from the core being nearly self-gravitating

and independent of the halo (D. Lynden-Bell and Royal, 1968; Thirring, 1970; Lynden-Bell and

Lynden-Bell, 1977) 11. This implies that the wall less affects the dynamics of the core regardless

of kinds of walls (adiabatic or thermal) and ensembles (micro-canonical or canonical). Hence, the

heat capacity’s singular behaviors could occur very close to each other at radii beyond the radii

RM = 4.31 and RM = 3.31 at which the heat- and stellar fluxes reach their maximum values.

This property is the distinct difference from the isothermal sphere. In the case of the isothermal

sphere, the radii at which the singularities occur are relatively away from each other at small radii

(RM = 34.9 for CV → ∞ and RM = 8.9 for CV = 0 ).10As we discuss in Section 9.4, the cause of the negative specific heat capacity is the relation between heat- and

stellar flows under a collisionless limit and deep potential well. Especially, the latter is more essential. Hence, thecloseness among the singularities in radius may occur if one assumes the wall does not affect the structure of potentialwell.

11For an isolated self-gravitating system, the Virial theorem states the heat capacity is always negative. Hence, theheat capacity is CV ∝ Etot/KE < 0. As Thirring (1970) originally pointed out, the energy range of a system withnegative heat capacity in a micro-canonical ensemble corresponds to the system undergoing a phase transition in acanonical ensemble. Simply, the canonical ensemble applies to the isothermal sphere at radii smaller than RM = 8.9while the micro-canonical ensemble at radii smaller than RM = 34.9 (See Table 9.2 for the value of radii.). The heatcapacity of the isothermal sphere is positive over 0 < RM < 8.9 and negative over 8.9 < RM < 34.9.

149


0 1 2 3−0.3

−0.25

−0.2

−0.15

−0.1

−5 · 10−2

0RM = 0

RM → ∞

(a)

η(kin)

1/Λ

0 1 2 3−0.3

−0.25

−0.2

−0.15

−0.1

−5 · 10−2

0RM = 0

RM → ∞

(b)

η(con)

0 1 2 3−0.3

−0.25

−0.2

−0.15

−0.1

−5 · 10−2

0RM = 0

RM → ∞

(c)

η(loc)

1/Λ

2.4 2.5 2.6 2.7 2.8−0.29

−0.28

−0.27

−0.26

−0.25

−0.24

−0.23

−0.22RM → ∞ (d)

η(loc)

Figure 9.8: Caloric curves for the dimensionless- total energy Λ and temperatures (a) η(kin), (b)η(con), and (c) η(loc). (d) Magnification of (c) around the turning point. All the curves start at (0, 0)that corresponds to RM = 0. For graphing, the ordinates are a reciprocal of Λ. Hence, the tangentsto the curves represent the sign of CV.

150


CV → ∞ CV = 0

η Λ RM D(0)/D(RM) η Λ RM D(0)/D(RM)

Kinetic 2.580 -3.599 19.2 36.6 2.540 -3.548 25.4 74.85

Local 2.712 -3.549 24.1 65.8 2.711 -3.548 25.4 74.85

Thermodynamic 2.492 -3.633 17.9 30.3 2.418 -3.548 25.4 74.85

IS -0.335 34.4 709 2.52 8.99 32.1

Table 9.2: First turning points of caloric curves at small radii. The data for IS are the corre-sponding values for the isothermal sphere (e.g., Antonov, 1962; D. Lynden-Bell and Royal, 1968;Padmanabhan, 1989; Chavanis, 2002a).

9.4 The cause of negative heat capacity

The ss-OAFP model is at a non-equilibrium state(s). Hence, the model’s self-gravity may

not cause a negative heat capacity, unlike (nearly) isolated self-gravitating systems at the states

of thermal equilibrium. The present section details possible causes of the negativeness, focusing

on the core of the ss-OAFP model. Accordingly, we discuss only the kinetic heat capacity for

simplicity. First, Section 9.4.1 introduces a simple analytical method to discuss the heat capacity

based on the Virial of the model and total energy. This method clarifies that the negative heat

capacity of the ss-OAFP model originates from the deep potential well (or high scaled-escape

energy) at the center of the core. Section 9.4.2 shows that the high temperature- and collisionless

nature of the core are also important to keep causing the negative heat capacity. This nature can be

inferred by comparing the present work to existing models.

151


9.4.1 Virial and total energy to discuss heat capacity at the center of the

core

One can discuss the heat capacity at the center of the ss-OAFP model by the Virial and total

energy Etot (equation (9.2.8d))

V ≡ −2KE − PE = −

∫ RM

0dR

∫ 0

Φ

dE Ωmic(E,Φ) R2(E) (E − Φ)3/2, (9.4.1a)

Etot =

∫ RM

0dR

∫ 0

Φ

dE Ωcan(E,Φ) R2(E) (E − Φ)3/2, (9.4.1b)

where

Ωmic(E,Φ) = 2F(E) +Φ

3dFdE

, (9.4.2a)

Ωcan(E,Φ) = F(E) +Φ

3dFdE

. (9.4.2b)

Equations (9.4.2a) and (9.4.2b) present a simple analytical method to determine the sign of heat

capacity in the core. Since the present focus is the central part of the ss-OAFP model, one may

approximate the reference solution Fo to a Maxwellian DF with a scaled-escape energy χesc. The

dimensionless DF reads F(E) = exp[χescE]. Then, Ωmic and Ωcon reduce to

Ωmic(E,Φ) = eχescE(2 +

χesc

3Φ

), (9.4.3a)

Ωcon(E,Φ) = eχescE(1 +

χesc

3Φ

). (9.4.3b)

The change in the sign of heat capacity depends on

Φ(mic) = −6χesc

, (9.4.4a)

Φ(can) = −3χesc

. (9.4.4b)

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100 102 104 106 108 1010 1012 1014 1016 10180

0.5

1

−3P

Eχ

escK

E

Figure 9.9: Ratio of -PE/KE to χesc/3

These equations may also be related to the singularities in heat capacity.12

Since we rely on the approximation F = eχescE, we first validate the approximation. One still

can use equations (9.4.2a) and (9.4.2b) with χesc = 13.88 for the ss-OAFP model. This is since the

reference DF Fo(E) well fits the exponential of χescE on −1 < E / −0.5, and exp[−χesc0.5] con-

tributes to the integrals (equations (9.4.1a) and (9.4.1b)) only by a small fraction (∼ 1×10−3). Also,

Figure 9.9 shows the ratio −PE/KE rescaled by 3/χesc, which is a direct graphical representation

for equation (9.4.2a). The value reaches unity at RM ≈ 0. This implies that the approximation is

consistent at the center of the core. The rest of the present section applies equations (9.4.3a) and

(9.4.3b) to the kinetic heat capacities of (i) the isothermal sphere and (ii) the ss-OAFP model.

(i) Positive kinetic heat capacity at the center of the isothermal sphere

First, the isothermal sphere (χesc = 1) is discussed whose kinetic heat capacity is positive at the

center of the core due to the shallow potential well. The relation between the potential well and

RM is known (see, e.g., D. Lynden-Bell and Royal, 1968; Padmanabhan, 1989). As RM → ∞, the

potential well deepens like βφ(0) = −2 ln[R] − 2 due to the relationship βφ(0) = −Φ(RM) − M/RM.

12Equation (9.4.4a) corresponds to the Virial being zero; that is, the thermal pressure at the adiabatic wall is zero.This determines the upper limit of the radius. Beyond the radius, a micro-canonical ensemble can not apply to aself-gravitating system if it is at a state of equilibrium. Equation (9.4.4b) corresponds to Etot being zero; that is,the potential energy dominates Etot beyond the radius. This is the least condition that an equilibrium self-gravitatingsystem does not exist in a canonical ensemble. Under a proper thermodynamic limit, the dimensionless total energy Λ

reaches an extremum (singular point) as RM increases. For a large RM, Etot is proportional to M2G/RM. At radii largeror smaller than the singular point, the sign of CV possibly changes.

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Hence, as RM → ∞, the potential well gets deeper, and the total energy decreases. On the one

hand, βφ(0) = −R2M/6 for small RM(<< 1). Hence, even if RM = 1, Etot and V are necessarily

positive. The energy Etot increases monotonically with RM13 (due to the integrand being positive).

Also, the kinetic energy monotonically increases with RM at all radii (since the local kinetic energy

is always positive). The energies KE and Etot are zero at RM = 0 and are made in dimensionless

form by the same factor (Section 9.3)14. As a result, the kinetic heat capacity is positive in the core

for the isothermal sphere due to the shallow potential well. This is the case that we consider the

core of the isothermal sphere as an ‘ordinary’ ideal gas (since the m.f. potential less affects the

state of the core).

(ii) Negative heat capacity at the center of the ss-OAFP model

In the case of the ss-OAFP model, the negative kinetic heat capacity at the center of the core

is caused by the deep potential well. Thanks to the ss-OAFP model’s self-similarity, the potential

Φ at RM ≈ 0 is not related to the wall radius RM, unlike the isothermal sphere. Hence, one can

discuss the heat capacity based only on the local property of the ss-OAFP model. The known

approximation, Φ(RM ≈ 0) = −1 + R2M/6, infers that Etot and V are negative at RM < 2.17 and

RM < 1.85. Accordingly, Etot decreases with RM at small radii (due to the integrand being negative).

Also, the kinetic energy increases at all radii as RM increases. In the same way as the isothermal

sphere, the energies KE and Etot are zero at RM = 0 and are made in dimensionless form by the

same factor (Section 9.3). Hence, the ss-OAFP model must have a negative heat capacity at small

radii (RM / 2.2) in the core. One may recall Φ(RM = 0) is set to −1 for the numerical integration

13Controlling the values of Etot and KE by RM originates from the RM-dependence of the normalized energy andtemperature. As shown in Section 9.3, the normalized quantities implicitly depend on radius RM on caloric curves evenif the dependence does not appear explicitly in the Legendre transformation. In other words, we expect that controllingRM corresponds to changing the state of the system from a non-equilibrium state to another non-equilibrium state. Thisprocess is an analogy of tracing infinitely possible equilibria for a self-gravitating system.

14The factor M2/RM behaves like R9M/900 as RM → 0. This behavior changes the KE and Etot into decreasing

functions with small increasing RM. However, M2/RM still changes monotonically and much more rapidly comparedwith Etot and KE at RM ∼ 0. Hence, it does not alter the sign of heat capacity.

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of the ss-OAFP system. However, through equations (9.4.2a) and (9.4.2b), a high χesc-value is

equivalent to a deep potential well. Hence, the negative heat capacity is the consequence of the

deep potential well. This is why the present work considers the core of the ss-OAFP model as an

‘exotic’ ideal gas.

9.4.2 The cause of negative heat capacity compared with the previous

works

The present section compares the negative heat capacity of the core of the ss-OAFP model to

that of (i) a simple model and (ii) a more realistic model.

(i) A simple understanding of the negative heat capacity in the core of the ss-OAFP model

In the case of the ss-OAFP model, a small number of stars are trapped in the deep potential

well due to the high escape energy (Figure 9.5), while the spatial profile of stars in the halo is

time-independent.15 In the core, the stars more tightly bounds each other every time the core loses

stars (particles) and kinetic energy through the heat- and stellar fluxes due to the conservation of

energy16. This means if one encloses the stars in the core by an adiabatic wall of small RM they

may behave like particles interacting via short-range pair potential discussed in (Posch et al., 1990;

Posch and Thirring, 2005; Thirring et al., 2003), but this is not the case. For such systems, the

majority of energy bands (levels) are available to particles. Hence, the particles can be well mixed,

and the negative heat capacity occurs to an only limited energy band. On the one hand, the present

work deals with only the collapsed state. The energy band that shows a negative heat capacity is

broad. It covers at least the energies that are available to stars in the potential well of the core.

15As RM >> 1, the halo density ρ(r, t) is independent of the dynamics in the core. The self-similar analysis providesdρ(r, t)/dt = 0. Hence, D(R) ∝ R−α and ρt ∝ r−αt .

16One may recall the failure of the Bohr model of the early quantum theory. An electron releases an electromagneticradiation due to the acceleration of elctron and deeply penetrates into the potential well around the ion. In a similarway, for the stellar encounter, stars can approach each other even on the scales of stellar size.

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The situation for stars in the core of the ss-OAFP model is alike that for collisionless particles

in a potential well that reported a negative heat capacity (Thirring et al., 2003; Carignano and

Gladich, 2010). The core of the ss-OAFP model is well mixed. Hence, the initial conditions for

the dynamical evolution are not important. The two key points here are that the temperature at

the center is high (Figure 9.7) and that the probability of finding stars in the core is low. For the

latter, the total number NM(≡ M/m) of stars in the core is a small fraction, e.g., NM ≈ 5.6 × 10−3

at RM = 1 and NM ≈ 1 at the radius at which the flattening in Φ ceases (Figure 9.10 (a)). Hence,

a proper zeroth-order approximation is the collisionless limit. In this limit, the core behaves like

a collisionless ideal gas as typically assumed for the isothermal sphere (e.g., Katz, 1978). This

means the stars behave as if they were non-interacting particles traveling only under the effect

of m.f. potential Φ. Figure 9.10 (b) shows the potential well Φ(RM) and mean total energy per

unit mass Etot/M. The stars in the core can stay in the deep potential well to develop the core-

collapse. However, due to the high temperature (kinetic energy), some of the stars need to spill out

of the potential well. The corresponding thought experiment is to ’expand’ the adiabatic wall17.

Then from Figure 9.10 (b), the total kinetic energy per unit mass decreases while the total energy

increases with radius. Hence, the negative heat capacity is the outcome of the deep potential well

together with the high temperature and low total number of stars in the core. This graphical method

(Thirring et al., 2003; Carignano and Gladich, 2010) would be the simplest way to understand

the negative heat capacity in the present case. The present result adds to their discussions the

importance of heat- and particle- flows to hold negative heat capacity on long (relaxation) time

scales; in short, the significance of a non-equilibrium state.

17We still follow the basic idea of the infinitely possible equilibria used for caloric curves (Section 9.3). The’expand’ means that one moves on to another non-equilibrium state as the radius increases, rather than physicallyexpand the wall itself.

156


100 102 104 106 108 1010 1012 1014 101610−5

10−1

103(a)

RM

NM−Φ

0 100 200 300 400 500 600 700 800 900 1,000−1

−0.8

−0.6

−0.4

−0.2

0(b)

RM

ΦEtot/M

Figure 9.10: Dimensionless m.f. potential Φ and (a) total number of the stars in the ss-OAFPmodel enclosed by an adiabatic wall at RM and (b) normalized mean energy per unit mass of thess-OAFP model.

(ii) Comparison with an existing realistic model

We compare the physical state of the core of the ss-OAFP model to the N-body simulations

(Komatsu et al., 2010, 2012) executed under physical conditions similar to the core. Strictly speak-

ing, systems similar to the ss-OAFP model do not exist not only in nature but also as a result of

a numerical direct N-body simulation18 since the complete core collapse itself is a mathematical

concept. However, some features of the complete core-collapse should appear at the early stage of

core collapse, as shown by the time-dependent OAFP model (Cohn, 1980). The conditions similar

18The former is explained in Section 2.1.2, while the latter is because it is not easy to achieve the ’complete’ corecollapse like the ss-OAFP model, other than using continuum models. Not only the effect of binary stars stops the corecollapse, but also large N ≈ 105 costs unfeasible CPU time to achieve the complete collapse. Hence, one needs to finda similarity of the ss-OAFP model to a self-gravitating system of fewer N stars.

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to the core of the ss-OAFP model are as follows. The system must be self-gravitating and com-

posed of small N particles. The stars need to be enclosed by a wall undergoing a core-collapse.

Simultaneously, the system needs to lose the kinetic energy and stars toward the outside of the wall

due to heat- and stellar- fluxes. Also, the system size must be large enough to form a core-halo

structure to make the fluxes occur. Komatsu et al. (2010, 2012) embodied such conditions by using

an N-body simulation for N = 125 ∼ 250. To achieve the condition, they used a partially perme-

able wall through which they change the evaporation (escape) rate of particles from the system by

controlling the escape energy; that is, the degree of permeability.

While our results show many physical features of star clusters in common with those of (Ko-

matsu et al., 2010, 2012), the former can reach the energy domain that the latter could not achieve.

Komatsu et al. (2012) obtained a Maxwellian-like velocity distribution function, and showed not

only a negative heat capacity, but also a steep velocity gradient with increasing time, like the result

of (Cohn, 1980). Also, they showed the density correlates with the velocity dispersion, like Figure

9.1(a). On the one hand, Komatsu et al. (2010) employed stellar polytropes to characterize the

non-equilibrium state of their model. They found the polytrope of m ∼ 9 is initially close to their

model with a negative value of Λ. However, it begins to deviate from the model with decreasing

Λ as the collapse proceeds. They insisted the deviation occurred because the kinetic temperature

is not suitable to describe the non-equilibrium state. However, it would less matter. A smaller Λ

means that the core size is relatively large and close to the wall radius. This corresponds to, in

the present model, the radius RM approaches the center of the core where the kinetic temperature

can be reasonably defined due to the frequent relaxation events. The deviation occurred because

the polytrope itself is not proper to describe the system with a negative heat capacity. The caloric

curve for the polytropes of m > 5 spirals as Λ decreases, showing marginal instabilities (or succes-

sive instability with increasing radius) (Chavanis, 2002b, 2003). This situation corresponds with a

collapsed state with a large RM. Such a state does not exist due to the instability. On the one hand,

the caloric curve for the ss-OAFP model provides only a negative heat capacity at a small Λ (or

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monotonically increasing 1/η(kin) with decreasing Λ). This characteristics matches the qualitative

nature of the curve reported in (Figure 5 in Komatsu et al., 2010).

9.5 Conclusion

Concluding the present chapter, we first showered the local and global properties of the ss-

OAFP model. Then, we discussed the negative heat capacity at constant volume of the model by

constructing the normalized energy and temperatures. A unique physical feature of the ss-OAFP

model is that it is a non-equilibrium system, but the core is at a well-relaxed state. The core can

achieve a negative heat capacity since stars in the core can behave like collisionless particles due to

the high temperature and low total number of stars in the deep potential. This cause of negative heat

capacity is different from equilibrium self-gravitating systems such as the isothermal sphere. The

isothermal sphere’s negative heat capacity occurs as a result of a phase transition. It also occurs

when the system size is large enough to be isolated from the ambient stars/gas. The isolation results

in the Virial reaches zero. On the other hand, the Virial of the ss-OAFP model is not zero, somewhat

positive. The ss-OAFP model is not related to the cases in which equilibrium systems can show a

negative heat capacity because of the initial conditions. The present analysis reemphasizes that the

negative heat capacity could be peculiar to inhomogeneous and non-equilibrium self-gravitating

systems on relaxation time scales.

159

Chapter 10

Application: Fitting the energy-truncated

ss-OAFP model to the projected structural

profiles of Galactic globular clusters

The present chapter proposes a phenomenological model that reasonably fits the projected

structural profiles of Galactic globular clusters1. It applies not only to normal (King model) clus-

ters but also to collapsed-core (or collapsing-core) clusters with resolved cores (finite-size cores

that can be observed). The most fundamental fitting model for the structures of globular clusters

is the King model (King, 1965). The King-model fitting relies on the three numerical parameters;

the dimensionless central potential K(= ϕ(r = 0)/σc), central projected density Σc and core ra-

dius rc. Only with the three degrees of freedom, the King model well fits the surface brightness

or projected density for approximately 80% of globular clusters in Milky Way. The rest of the

clusters currently undergo the first core collapse or have collapsed at least once (Djorgovski and

King, 1986). In this sense, one sometimes calls the clusters that can be fitted by the King model the

’normal’ or ’King-model (KM)’ clusters while those that can not is ’post-collapsed-core’ or ‘post-

1Galactic globular clusters are globular clusters belonging to Milky Way.

160

CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

core-collapse’ (PCC) clusters. The two primary differences between the KM- and PCC- clusters

are as follows. (i) The projected density profile of a typical KM cluster flattens in the core while a

PCC cluster has a cusp with a power-law profile like r−1. (ii) The concentrations of PCC clusters

are high c ' 2.0 while those of KM clusters are low 0.7 / c / 1.8 (See, e.g., Meylan and Heggie,

1997). The density profiles of PCC clusters may be fitted by the time-dependent post-collapse

OAFP models with realistic effects. However, it is not a simple task to self-consistently solve a

time-dependent OAFP equation coupled with the Poisson’s equation. Typically, a time-dependent

OAFP model applies to a certain globular cluster as a case study to discuss its detailed structure,

such as; (Murphy et al., 2011) for NGC 7088, (Drukier et al., 1992) for NGC 6838, and (Drukier,

1995) for NGC 6397. On the one hand, the King model is based only on the Poisson’s equation

that is easily solved for a spherical cluster’s potential. It can be used in a homogeneous survey to

capture the cluster’s common properties, e.g., the characteristic sizes and dynamical states. Hence,

it needs to apply to as many globular clusters as possible by neglecting each cluster’s detailed

physical features. The concentration and core- and tidal radii obtained from the King model have

been the fundamental structural parameters in compilation works for globular-cluster studies (e.g.,

Peterson and King, 1975; Trager et al., 1995; Miocchi et al., 2013; Merafina, 2017) and in (Harris,

1996, (2010 edition))’s catalog. To the best of our knowledge, there does not exist a single-mass

isotropic model only based on the Poisson’s equation that applies to both the PCC- and KM- clus-

ters due to their different core structures.2

The present chapter proposes an energy-truncated ss-OAFP model that can fit the density pro-

2For PCC clusters, a modified power-law profile (e.g., Lugger et al., 1995; Ferraro et al., 2003) and non-parametricmodel (e.g., Noyola and Gebhardt, 2006) have been employed in place of the King model, its variants (e.g., Woolley,1961; Wilson, 1975) and generalized models (e.g., Gomez-Leyton and Velazquez, 2014; Gieles and Zocchi, 2015). Wefocus on a single-mass model for simplicity while a multi-mass King model is known to be able to fit some PCC cluster(King et al., 1995). There is no any strict argument that rules out such a multi-mass King model from a proper modelfor the PCC clusters (Meylan and Heggie, 1997). The generalization done for the King model directly applies to themodel we propose in this chapter. Hence, we do not discuss the particular relation of our model with some generalizedmodels. For example, we do not consider more recent modeling (i) the fν model based on collisionless relaxation(de Vita et al., 2016) and the effect of escapers to discuss the elongated outer halos of some clusters (Claydon et al.,2019).

161


files of KM clusters and PCC clusters with resolved cores reported in (Kron et al., 1984; Djorgovski

and King, 1986; Trager et al., 1995; Lugger et al., 1995; Drukier et al., 1993; Ferraro et al., 2003;

Noyola and Gebhardt, 2006; Miocchi et al., 2013). The new model reasonably fits at least half of

the projected density profiles or surface brightness of globular clusters in the Milky Way. Since

we did not have access to the numerical data of (Djorgovski and King, 1986; Lugger et al., 1995;

Miocchi et al., 2013), we employed ’WebPlotDigitizer’ (Rohatgi, 2019) to extract the data points

and uncertainties for the projected structural profiles depicted on the figures of their works.

The present chapter is organized as follows. Section 10.1 introduces the energy truncated ss-

OAFP model. Section 10.2 explains the result of fitting the new model to PCC clusters in Milky

Way. Section 10.3 shows the relationship between the completion rate of core collapse and the

concentration. It is based on the results of the fitting of our model to the KM- and PCC- clusters.

Section 10.4 discusses three topics (i) the application of our model to Galactic globular clusters in a

broad range of radii, (ii) the approximated form of the new model, and (iii) the relationship between

low-concentration globular clusters and stellar polytropes of index m. Section 10.5 concludes the

chapter. For the sake of brevity, we show the majority of the structural profiles of the clusters fitted

by the new model in Appendixes I, J, and K.

10.1 Energy-truncated ss-OAFP model

The present section introduces an energy-truncated ss-OAFP model. First, Section 10.1.1

reviews the relaxation evolution before and after a core collapse and the motivation for truncating

the energy of the ss-OAFP model. Section 10.1.2 details the new model. The new model does not

depend on the dimensionless central potential K, unlike the King model. Hence, Section 10.1.3

explains how to regularize the concentration and core radius of the new model to compare the

new model with the King model. Also, our new model is composed of a polytrope of m and the

ss-OAFP model. We must select the value of m for the new model. Hence, Section 10.1.4 explains

162


how we found an optimal value of m.

10.1.1 The relationship of the ss-OAFP model with the PCC clusters and

isothermal sphere

We expect that the ss-OAFP model can reasonably model the structures of globular clusters in

the center and inner halo in the collapsing-core phase, compared with the King model. We expect

the structure of the ss-OAFP model to be similar to those of PCC clusters and the King model

(isothermal sphere) under certain conditions. This similarity motivated us to apply the ss-OAFP

model to globular clusters.

The ss-OAFP model for pre-collapse clusters can fit even the projected structural profiles of

PCC clusters due to the structural similarities between pre-collapse- and post-collapse- clusters. In

principle, the ss-OAFP model in the pre-collapse phase is the model that applies only to globular

clusters at the instant of the complete core collapse. Also, it may approximately model collapsing-

core clusters at the late stage of relaxation evolution before the complete core collapse. The ss-

OAFP model itself is unrealistic since binaries halt the core collapse before the infinite density

develops. After the core collapse holds, the time-dependent- and self-similar- conducting gaseous

models predict that the cluster periodically repeats core expansion (due to the energy released from

the binaries) and core collapse (due to the two body relaxation with the self-gravity) (Sugimoto and

Bettwieser, 1983; Bettwieser and Sugimoto, 1984; Goodman, 1984, 1987). This process is called

the gravothermal oscillation in the post-core-collapse phase since it shows a nonlinear oscillation

of the core density with time. Time-dependent OAFP models (Cohn et al., 1989; Murphy et al.,

1990; Takahashi, 1996) and N-body simulations (Makino, 1996; Breen and Heggie, 2012) also

predict the oscillation. Sugimoto and Bettwieser (1983); Bettwieser and Sugimoto (1984) found

that the velocity dispersion profiles of their models seem to approach that of the singular isothermal

sphere in the outer halo. On the one hand, the core structure is similar to that of the non-singular

163


isothermal sphere except at the instant of the core collapse. This feature is against the result of

the self-similar gaseous model with a central cusp (Inagaki and Lynden-Bell, 1983). The latter

seems proper to describe PCC clusters reported in (e.g., Djorgovski and King, 1986). However,

the formation of a cusp in the core is a conditional result. The self-similar gaseous model showed

that the core radius gets smaller with increasing N. Goodman (1987) examined how different N

affects the core collapse using the same model as Goodman (1984), but with different functional

forms of energy sources and more efficient binary heating. A large-N (' 7 × 103) system could

be overstable or unstable. As a result, the cluster could undergo a gravothermal oscillation. The

model of Goodman (1984) results in forming a cusp in the core, while that of Goodman (1987)

has a core, like the non-singular isothermal sphere. In fact, to avoid unrealistically small and

large core, efficient binary heating with primordial binaries is expected to occur. Efficient enough

heating can still form a possibly resolved core (Goodman and Hut, 1989). Furthermore, the density

profiles with resolved cores are similar to each other in between the post-collapse and pre-collapse

phases of the gaseous model (Goodman, 1987) and OAFP model (Takahashi, 1996). There is no

way to differentiate a density profile from the other in the two phases only from observational data

(Meylan and Heggie, 1997) unless one acquires accurate kinematic data to see the temperature

inversion.3 Hence, the pre-collapse ss-OAFP model can model some of the PCC clusters with an

efficient binary heating. This is a motivation to apply the ss-OAFP model to the PCC clusters.

Our goal is to find the structural parameters of the PCC clusters with resolved cores, rather than

establishing strict modeling of them.

Also, the ss-OAFP model can model Galactic KM clusters since it has a flat core, like the

isothermal core. The former’s core structure is similar to that of the isothermal sphere model,

3A distinct difference appears in the radial profiles for velocity dispersion between the pre-collapse and post-collapse OAFP models (Sugimoto and Bettwieser, 1983). When the collapsed core expands, the temperature (i.e.,velocity dispersion) increases with an increasing radius near the center of the cluster. This temperature gradient causesheat to flow inward toward the center while it cools down the central region. However, kinematic surveys generallyprovide much more significant uncertainties in velocity-dispersion data compared with structural-profile data (Meylanand Heggie, 1997). Hence, one can not easily determine if a well-relaxed (or high-concentration) cluster is currentlyin pre-collapse- or post-collapse- phase.

164


as discussed in Section 9.2.1. Figure 10.1 shows the density profiles of the ss-OAFP and the

isothermal sphere. They have almost the same morphology at small dimensionless radii. In the

figure, the radius of the ss-OAFP model is rescaled by multiplying by 3.739. Then, the core sizes

of the two models are approximately the same. These core structures infer that one can obtain a

model similar to the King model at small radii by properly truncating the ss-OAFP model’s energy.

We explain how to truncate the energy in Section 10.1.2.

10−1 100 101 102 10310−5

10−4

10−3

10−2

10−1

100

R

D

Isothermal spheress-OAFP model

Figure 10.1: Dimensionless densities D(R) of the isothermal sphere and ss-OAFP model.

10.1.2 Energy-truncated ss-OAFP model

We energy-truncate the ss-OAFP model so that the outer halo of the energy-truncated model

behaves like a polytrope of m. In this sense, our new model is a phenomenological model, unlike

the King model (King, 1966). The energy truncation of the King model is based on simple physical

arguments using ‘test particle’ method. They assume that particles (stars) other than the test particle

(star) follow the Maxwellian DF. The examples are a star cluster described by a stationary FP

model (Spitzer and Harm, 1958; Michie, 1962; King, 1965) and the OAFP model (Spitzer and

Shapiro, 1972). The DF for the test star is the lowered-Maxwellian DF. This DF corresponds

with the situation that the isothermal sphere is enclosed in a square well. Also, the stellar is DF

165


proportional to −E at the edge of the cluster (corresponding to polytrope of m = 2.5) (Spitzer

and Shapiro, 1972)4. This DF is an asymptotic stationary solution of the OAFP model when a

constant stellar flux occurs at the fringe. Our new model incorporates the effect of the escaping

stars like (King, 1966) by controlling high binding energies. However, the mathematical operation

for combining DFs is opposite to King (1966)’s method. To obtain the King model (or lowered-

Maxwellian DF), one must subtract the DF for the polytrope of m = 2.5 from the Maxwellian DF.

On the one hand, our energy-truncated ss-OAFP model adds the DF for a polytrope of m to the

reference DF Fo(E) for the ss-OAFP model as follows

F ≡ρc

4√

2πσ3c

Fo(E) + δ (−E)m−3/2

Do(ϕ = −1) + δ B(m − 1/2, 3/2), (10.1.1)

where δ and m are positive real numbers. The Do(ϕ) is the reference density of the ss-OAFP model

and B(a, b) is the beta function defined as B(a, b) = 2∫ 1

0t2a−1(1 − t2)b−1 dt with a > 1/2 and b > 1.

The factor 1/(Do(ϕ = −1)+δ B(m−1/2, 3/2)) is inserted in the DF so that the density profile of the

new DF has a certain central density ρc as R → 0. This new DF behaves like the ss-OAFP model

beyond the order of δ, while it is approximately a polytropic sphere of index m below δ. However,

the value of m must be further fixed5 based on physical arguments (with numerical experiments)

and observational data. In the present work, we do only the latter process (Section 10.1.4). In this

sense, we consider our model a phenomenological model.

The rest of the present section shows the numerically calculated density profile, m.f. potential,

and projected density profile of the energy-truncated ss-OAFP model. The density of polytropes

4The stellar DF proportional −E is the expression obtained in the higher-energy limit of the lowered-MaxwellianDF.

5The index m in our model is to be determined at first place. Our model has the same parameter-dependence as thetruncated γ exponential (fractional-power) model discussed in (Gomez-Leyton and Velazquez, 2014). The outer halois controlled by a polytropic sphere.

166


can be analytically derived from the polytrope’s DF. The density of the new model reads

D(ϕ) = ρcDo(ϕ) + δ B(m − 1/2, 3/2) (−ϕ)m

Do(ϕ = −1) + δ B(m − 1/2, 3/2). (10.1.2)

The Poisson’s equation in dimensionless form is

d2ϕ

dr2 +2r

dϕdr

=Do(ϕ) + δ B(m − 1/2, 3/2) (−ϕ)m

Do(ϕ = −1) + δ B(m − 1/2, 3/2), (10.1.3)

where the potential, radius, and density are made in dimensionless form using equations (2.1.20a)

- (2.1.20c). The boundary condition for the Poisson’s equation (10.1.3) is

ψ(r = 0) = 1,dψdr

(r = 0) = 0. (10.1.4)

Since ϕ is an independent variable for the ss-OAFP model, we solved the the inverse form of the

Poisson’s equation for R(ϕ)

Rd2R

dϕ2 − 2(

dRdϕ

)2

=

(dRdϕ

)3 Do(ϕ) + δ B(m − 1/2, 3/2) (−ϕ)m

Do(ϕ = −1) + δ B(m − 1/2, 3/2). (10.1.5)

The numerical integration of the Poisson’s equation (10.1.5) provided the density profile (Figure

10.2) and m.f. potential profile (Figure 10.3) for m = 3.9. (m = 3.9 is the optimal value for our

model and the reason is explained in Section 10.1.4.) In the figures, the value of δ spans 10−5

through 103. For large δ > 1, the both profiles do not change their morphology almost at all. This

is since they behave like a polytrope of m = 3.9 in the limit δ → ∞. On the one hand, the profiles

approach the ss-OAFP model for small δ (/ 10−2).

167


10−2 10−1 100 101 102 10310−7

10−6

10−5

10−4

10−3

10−2

10−1

100

R

D ss-OAFPδ = 1000δ = 1δ = 0.1δ = 0.01δ = 0.0001

Figure 10.2: Dimensionless density D(R) of the energy-truncated ss-OAFP model for different δ.The corresponding profile of the ss-OAFP model is also depicted.

10−3 10−2 10−1 100 101 102 103 104 105 106

10−4

10−3

10−2

10−1

100

R

ϕ ss-OAFPδ = 1000δ = 1δ = 0.1δ = 0.01δ = 0.001δ = 0.00001

Figure 10.3: Dimensionless potential ϕ(R) of the energy-truncated ss-OAFP model. The corre-sponding potential of the ss-OAFP model is also depicted.

For applications of the density profile to globular clusters, one needs to convert the density

168


profile D(ϕ) to the projected density profile

Σ(r) = 2∫ ∞

0

D(φ)√1 − (r/r′)2

dr′. (10.1.6)

The corresponding inverse form with dimensionless variables is

Σ(ϕ) = −2∫ ϕ

0

√1 − µR(ϕ, ϕ′)1 + µR(ϕ, ϕ′)

[−2

dDdϕ′

R(ϕ′)S(ϕ′)

+ D(ϕ′)S(ϕ′)1 + 2µR(ϕ, ϕ′)1 + µR(ϕ, ϕ′)

]dϕ′, (10.1.7)

where µR(ϕ, ϕ′) ≡ R(ϕ)/R(ϕ) and S ≡ −dR/dϕ(< 0). Figure 10.4 depicts the projected density

profiles for different δ. As δ decreases, the slope of R−1.23 in the inner halo develops more clearly

(as expected from the asymptotic density profile of the ss-OAFP model since Do ∝ R−2.23). This

power-law profile occurs at radii between R ∼ 10 and R ∼ 100 for δ = 10−4. One can also find a

similar power-law profile for larger δ. For δ = 10−2 and 10−3, Σ shows power-law-like structures

R−1.0 ∼ R−1.1 at radii between R ∼ 1 and R ∼ 10. This slope is a desirable feature to fit our model

to the projected density profiles of the PPC clusters whose projected density has similar power-law

profile near the core (e.g., Djorgovski and King, 1986; Lugger et al., 1995).

169


10−2 10−1 100 101 102 103

10−5

10−4

10−3

10−2

10−1

100

R

Σ

R−1.23

δ = 10δ = 1

δ = 0.01δ = 0.001δ = 0.00001

Figure 10.4: Projected density Σ of the energy-truncated ss-OAFP model for different δ. Thepower-law R−1.23 corresponds to the asymptotic approximation of the ss-OAFP model as R→ ∞.

10.1.3 The regularization of the concentration and King radius

The energy-truncated ss-OAFP model is different from the King model. Their mathematical

expressions depend differently on the concentration, core radius, and dimensionless central poten-

tial. Hence, one must properly regularize the structural parameters for the sake of comparison.

The King model (equation (2.1.30)) may be written in the following form by regularizing the m.f.

potential as ¯φ ≡ φ/K

d2 ¯φdr2 +

2r

d ¯φdr−

1K

I( ¯φ)I(1)

= 0. (10.1.8)

Due to the K-dependence of the equation, as K → 0 the concentration is also c → 0. Of course,

the minimum radius of the King model can be the tidal radius of polytrope of m = 2.5, that is

5.355275459... (e.g., Boyd, 2011) if one regularizes the radius as r =√

K ¯r. On the one hand,

the ss-OAFP model does not depend on K. To find the same value of concentration or at least

the same order as that of the King model (if necessary), one must regularize the core radius and170


concentration6 as follows

rKin ≡rKin√

K(m), (10.1.9a)

c ≡ log[

rtid

rKing

√K(m)

], (10.1.9b)

where K(m) is the dimensionless central potential that can be given when rtid/√

K of the King

model is approximately the same as that of the polytrope of m.

Using equation (10.1.9), one can obtain the concentration of the energy-truncated ss-OAFP

model. For example, if m = 3.9 is chosen, then the tidal radius of the polytrope of m = 3.9

is 13.4731 (based on our calculation). In this case, K of the King model must be chosen so

that rtid/√

K of the King model is close to 13.4731. This can be achieved when K = 4.82 and

rtid = 13.444 (based on our calculation). Hence, K(m) = 4.82 for m = 3.9.7 Figure 10.5 depicts the

concentration c for m = 3.9. As δ increases, the concentration approaches a constant value corre-

sponding to the concentration of polytrope of m = 3.9. The present focus is δ < c∗4(= 0.3032) with

which the energy-truncated ss-OAFP model behaves like the ss-OAFP model and can differentiate

itself from the isothermal sphere and King model. The corresponding concentration is c > 1.45.

On the one hand, our model is expected to behave like the King model for 1 < c < 1.45. Also, it

would behave like the polytrope of m = 3.9 in the limit of c→ 1.

6The tidal radius for the energy-truncated ss-OAFP model should not be regularized since it is still the radiusat which the projected density reaches zero. The tidal radius can be found after the model is properly fitted to theprojected structural profiles of a globular cluster on a graph.

7We show in Section 10.1.4 that the concentrations calculated by this scaling are reasonably close to those of theKing model.

171


10−3 10−2 10−1 100 101 102

1

2

3

δ

c

Finite ss-OAFPPolytrope of m = 3.9

Figure 10.5: Concentration of the energy-truncated ss-OAFP model. The horizontally dashed linerepresents the concentration for polytropic sphere of m = 3.9.

10.1.4 Determining an optimal value of index m

We determined the index m to be 3.9 in the energy-truncated ss-OAFP model after preliminarily

applying it to the projected surface densities of six KM clusters and a PCC cluster that we chose.

Initially, we expected m = 2.5 could be an optimal choice for the energy-truncated ss-OAFP model

following (Spitzer and Shapiro, 1972), though it was not the case. The useful values were found

on 3.5 ≤ m ≤ 4.4 with which our model reasonably fits the projected surface densities of the

KM clusters reported in (Kron et al., 1984; Noyola and Gebhardt, 2006; Miocchi et al., 2013) (See

Appendix I). Among the values of m, we chose 3.9 as an optimal value for the present work. This is

since it provides the same order of structural parameters for the six chosen clusters as those of the

existing works based on the King model, as shown in Table 10.1. The table provides the results for

the fitting of our model with m = 3.9 to the six Milky Way globular clusters reported in (Kron et al.,

1984). Only the six clusters were reported in all the compilation works (Peterson and King, 1975;

Kron et al., 1984; Chernoff and Djorgovski, 1989; Trager et al., 1993; Miocchi et al., 2013) that we

chose to compare this time. Miocchi et al. (2013) did not show the numeric values of the projected

density profiles, but the six clusters in Table 10.1 have approximately the same maximum radius

points on graphs reported in (Kron et al., 1984). Many of the compilation works based on the King

172


model are inhomogeneous surveys that depend on different instruments, photometry methods, and

statistical analyses. The structural parameters obtained from our model are reasonably close to

the compilation works’ results. On the one hand, If m < 3.8 or m > 4.3 is chosen for the fitting,

the structural parameters are small or large by over a factor of ten compared with the compilation

works’ results. Interestingly, our structural parameters are close to those obtained from the King

model, rather than the Wilson model (Wilson, 1975). The Wilson model relies on the polytrope of

m = 3.5 in the limit K → 0. The model can provide greater values of the structural parameters

since the polytrope of m = 3.5 reaches further in radius compared with that of m = 2.5. The

latter is the case when the same limit is taken for the King model (e.g., Chandrasekhar, 1939). The

reason why our model does not overestimate the structural parameters with high m(= 3.9) would be

that the density of the ss-OAFP model more rapidly decays compared with the isothermal sphere

in the inner halo (Figure 10.1). In Appendix I, one can find that the energy-truncated ss-OAFP

model with m = 3.9 can be reasonably fitted to the projected structural profiles of the KM clusters

reported in (Kron et al., 1984; Noyola and Gebhardt, 2006; Miocchi et al., 2013).

Another reason why we chose m = 3.9 is that the energy-truncated ss-OAFP model with m =

3.9 agreeably fits the relatively new data for NGC 6752 reported in (Ferraro et al., 2003). The

cluster is considered a PCC cluster. Ferraro et al. (2003) provided data points and error bars of the

projected surface density for NGC 6752. The given numeric data are convenient to test our model

(since we artificially do not have to extract data from their graph.). In their work, the King model

does not well fit the central part of the projected density profile. This is since the cluster is one of

(possible) PCC clusters with a power-law-cusp core. Hence, following (Lugger et al., 1995), they

employed a modified power-law profile ∼ (1 + (r/3.1)2)−0.525 where r is measured in log [arcsec].

This profile well fits the central part, as shown in Figure 10.6 (Left top). On the one hand, our

model with m = 3.0 is not close to the cluster’s morphology at all on the figure. However, we can

more reasonably fit our model to the same data with greater m. Especially, the model with m = 4.2

well fits the profile except in the tail of the cluster. Even with m = 3.9, one can find a reasonable

173


NGC 1904 NGC 2419 NGC 6205

c rc rtid c rc rtid c rc rtid

Finite ss-OAFPThe present work based on 1.86 0.191 13.9 1.24 0.410 7.16 1.54 0.779 27.0

data of (Kron et al., 1984)

King model(Miocchi et al., 2013) 1.76 0.15 9.32 1.51 0.27 9 1.32 0.825 18.5

(Harris, 1996, (2010 edition)) 1.70 0.16 8.0 1.37 0.32 7.5 1.53 0.62 21.0

(Trager et al., 1993) 1.72 0.159 8.35 1.4 0.348 8.74 1.49 0.875 27.0

(Chernoff and Djorgovski, 1989) 1.90 0.132 10.5 1.6 0.373 14.8 1.35 0.745 16.7

(Kron et al., 1984) 1.75 0.178 10.0 1.00 0.398 3.98 1.25 0.83 14.8

(Peterson and King, 1975) 1.60 0.27 10.7 1.41 0.42 10.7 1.55 0.76 26.9

Wilson model(Miocchi et al., 2013) 2.14 0.18 28 1.73 0.32 20 1.77 0.841 57

NGC 6229 NGC 6341 NGC 6864


Finite ss-OAFPThe present work based on 1.45 0.178 5.00 1.68 0.314 15.0 1.83 0.116 7.85

data of (Kron et al., 1984)

King model(Miocchi et al., 2013) 1.65 0.13 6.12 1.74 0.243 13.9 1.79 0.082 5

(Harris, 1996, (2010 edition)) 1.50 0.12 3.79 1.68 0.26 12.4 1.80 0.09 6.19

(Trager et al., 1993) 1.61 0.13 5.39 1.81 0.235 15.2 1.88 0.096 5.68

(Chernoff and Djorgovski, 1989) 1.40 0.167 4.19 1.70 0.132 6.64 1.85 0.084 5.91

(Kron et al., 1984) 1.25 0.173 3.08 1.50 0.308 9.75 1.75 0.095 5.34

(Peterson and King, 1975) 1.41 0.22 5.62 1.78 0.275 16.6 N/A 0.12 > 3.2

Wolley model(Miocchi et al., 2013) 1.82 0.16 12.0 2.17 0.33 46 2.38 0.095 25

Table 10.1: Concentration and core- and tidal- radii obtained from the energy-truncated ss-OAFPmodel. The structural parameters are compared with the previous compilation works based on theKing- and Wilson- models.

174


fit to the data. Hence, the present work chose m = 3.9 to consistently accumulate the data for both

the KM- and PPC- clusters.

175


−2 −1 0 1

−4.0

−3.0

−2.0

−1.0

0.0lo

g[Σ

]

ss-OAFP(m = 3.0)

modified power-lawNGC 6752

−2 −1 0 1

−4.0

−3.0

−2.0

−1.0

0.0

ss-OAFP(m = 3.5)

NGC 6752

−2 −1 0 1

−4.0

−3.0

−2.0

−1.0

0.0

(rc = 0′.149, δ = 0.0044)

log[

Σ]

ss-OAFP(m = 3.9, c = 2.56)

NGC 6752

−2 −1 0 1

−4.0

−3.0

−2.0

−1.0

0.0

(rc = 0′.188, δ = 0.0079)

ss-OAFP(m = 4.2, c = 2.46)

NGC 6752

−2 −1 0 1

−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

m = 3.9

−2 −1 0 1

−0.5

0

0.5

log[R(arcmin)]

m = 4.2

Figure 10.6: Fitting of the energy-truncated ss-OAFP model to the projected density profile forNGC 6752 (Ferraro et al., 2003) for different m. The unit of Σ is the number per square of arcmin-utes. Σ is normalized so that the density is unity at the smallest radius of data points. In the lefttop panel, double-power law profile is depicted as done in (Ferraro et al., 2003). In the two bottompanels, ∆ log[Σ] for m = 3.9 and m = 4.2 depicts the corresponding deviation of Σ from our model.

176


10.2 Fitting the ss-OAFP model to Galactic PCC clusters

The energy-truncated ss-OAFP model with m = 3.9 reasonably fits the projected structural

profiles of PCC clusters with resolved cores at R / 1 arcminute. Other than NGC 6752, we also

have access to numerical data of the projected density profile for NGC 6397 from (Drukier, 1995).

Our model reasonably fits the density profile of NGC 6397 with χ2ν = 1.52 (Figure 10.7), where

the reduced chi-square is defined as follows

χ2ν =

∑ χ2

nd.f., (10.2.1)

where χ2 is the chi-square value between the observed data and our model, and nd.f. is the degree

of freedom. We chose nd.f. = 3 in the same way as the King model since index m of our model

is fixed to 3.90. The parameters are the central density Σc, core radius rc, and δ. Since we did

not have access to numerical values for the rest of the projected structural profiles of PCC clusters

reported in (Djorgovski and King, 1986; Lugger et al., 1995), our error analysis becomes less

trustful hereafter. However, it appears enough to capture the applicability of our model to the

PCC clusters. For example, Meylan and Heggie (1997) introduced NGC 6388 and Terzan 2 as

an example of a KM cluster and PCC cluster by citing the B-band8 surface brightness profiles9 of

the clusters from Djorgovski and King (1986)’s work. Our model reasonably fits both the density

profiles at radii R / 1 arcminute (Figure 10.8). (As discussed in Section 10.4, to fit our model to

structural profiles at 10 ∼ 100 arcminutes, the value of m must be m ' 4.2.) Like NGC 6752 and

NGC 6397, we applied our model to the PCC clusters with resolved cores and some clusters with

8The B band means the brightness profiles were measured at around a wavelength of 440 nm. In the present work,the surface brightness with U and B bands are shown. The former was measured at approximately 360 nm and thelatter at 550 nm.

9The surface brightness (SB) has a relation with the density profiles as SB= −2.5 log(Σ)+const., where the ’const.’is the effects from the unit conversion from arcseconds to arcminutes, the conversion from luminosity to brightness,and reference- (or solar) brightness and mass. The ratio of the total mass to surface luminosity (Mass-to-light ratio)for globular clusters is approximately two times the Mass-to-light ratio of Sun (Binney and Tremaine, 2011). Thisis spatially constant for single-component models (McLaughlin, 2003). The exact evaluation of the effects is notimportant here since all the data in the present work normalize the structural profiles at the smallest radius data point.

177


unresolved cores reported in (Djorgovski and King, 1986; Lugger et al., 1995) (See Appendix K

in which the ’possible’ PCC clusters reported in (Kron et al., 1984) are also fitted). Table 10.2

shows the values of χ2ν for both the KM- and PCC- clusters that we could obtain the uncertainties

in observed densities from the numerical values or graphs. The result is evident that our model

can fit the KM clusters at all the data points given. The model fits only the PCC clusters with

resolved cores reported in (Lugger et al., 1995). For example, it reasonably fits the PCC clusters

with partially-resolved cores (NGC 6453, NGC 6522 and NGC 7099) and resolved cores (NGC

6397 and NGC 6752) at R /1 arcminute with χ2ν / 2. It fits even a PCC cluster with an unresolved

core (NGC 6342) similarly, though the present work does not account for the ‘seeing-effect’ that

comes from the finiteness of the seeing-disk. The structural profiles of the rest of the PCC clusters

with unresolved-core (e.g., NGC 5946 and NGC 6624) were hopeless to be fitted even only for

the cores. Their central parts have steeper power-law profiles, compared with our model. This

unfitness was expected since the present model does not correctly include the effect of binaries

whose heating effect is possibly inefficient to provide resolved cores.

−1 −0.5 0 0.5 1 1.5

−4.0

−3.0

−2.0

−1.0

0.0

(rc = 0′.085, δ = 0.0012)

log[R(arcmin)]

log[

Σ]

ss-OAFP(c = 2.19)

NGC 6397 (c)

−1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

NGC 6397 (c)

Figure 10.7: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected surfacedensity of NGC 6397 reported in (Drukier et al., 1993). The unit of Σ is the number per square ofarcminutes. Σ is normalized so that the density is unity at the smallest radius for the observed data.In the legends, (c) means PCC cluster as judged so in (Djorgovski and King, 1986). ∆ log[Σ] is thecorresponding deviation of Σ from the model on a log scale.

178


−2.5 −2 −1.5 −1 −0.5 0 0.5 1−4.0

−3.0

−2.0

−1.0

0.0

(rc = 0′.25, δ = 0.047)

SBo−

SB

ss-OAFP(c = 1.62)

NGC 6388 (n)

−2.5 −2 −1.5 −1 −0.5 0 0.5 1−4.0

−3.0

−2.0

−1.0

0.0

(rc = 0′.078, δ = 0.0064)

ss-OAFP(c = 2.60)Trz 2 (c)

−2.5 −2 −1.5 −1 −0.5 0 0.5 1

−0.5

0

0.5

log[R(arcmin)]

∆(S

Bo−

SB)

NGC 6388(n)

−2.5 −2 −1.5 −1 −0.5 0 0.5 1

−0.5

0

0.5

log[R(arcmin)]

Trz 2 (c)

Figure 10.8: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the surface brightnessof Terzan 2 and NGC 6388 reported in (Djorgovski and King, 1986). The unit of the surfacebrightness (SB) is B magnitude per square of arcseconds. The brightness is normalized by themagnitude SBo obtained at the smallest radius point. In the legends, (n) means normal or KMcluster and (c) means PCC cluster as judged so in (Djorgovski and King, 1986). ∆(SBo − SB) isthe corresponding deviation of SBo − SB from the model.

10.3 (Main result) The relaxation time and completion rate of

core collapse against the concentrations of the clusters

with resolved cores

Concentration c is a possible measure to characterize the states of globular clusters in the re-

laxation evolution, especially for the cores. Hence the present section compares c with the core

relaxation time and completion rate of core collapse. The energy-truncated ss-OAFP model can

179


KM cluster χ2ν Nb

NGC 288 0.45 0

NGC 1851 0.56 0

NGC 5466 2.07 0

NGC 6121 0.72 0

NGC 6205 1.05 0

NGC 6254 0.57 0

NGC 6626 0.47 0

NGC 6809 0.44 0

Pal 3 0.06 0

Pal 4 0.34 0

Pal 14 0.31 0

Trz 5 2.23 0

PCC cluster χ2ν Nb

NGC 6342 1.73 3

NGC 6397 1.52 0

NGC 6453 1.89 5

NGC 6522 2.52 5

NGC 6558 2.17 5

NGC 6752 2.00 6

NGC 7099 2.12 2

Trz 1 2.41 5

Trz 2 1.94 0

PCC cluster χ2ν Nb

NGC 5946 6.75 5

NGC 6624 7.18 5

Table 10.2: Values of χ2ν and number of points discarded from the calculation. The data used for

the fitting to the KM clusters are from (Miocchi et al., 2013), NGC 6397 from (Drukier et al.,1993), Terzan 2 from (Djorgovski and King, 1986), NGC 6752 from (Ferraro et al., 2003), and therest of the PCC clusters from (Lugger et al., 1995). Nb is the number of data points at large radiiexcluded from calculation.

180


reasonably apply not only to KM clusters (Appendix I) but also to PCC clusters (Appendix K).

Hence, one may systematically discuss their relationships. Figure 10.9(a) depicts the characteris-

tics of the core relaxation time tc.r. against the concentration c. Figure 10.9(b) portrays the corre-

sponding characteristics based on the King model reported in (Harris, 1996, (2010 edition)). For

both the figures, we employed the relaxation times reported in (Harris, 1996, (2010 edition))’s cata-

log. In the catalog, some of the concentrations are depicted as ‘2.50c’ for clusters whose structural

profiles are not well fitted by the King model. Hence, we assumed that the concentration of such

clusters is 2.50 in Figure 10.9(b). In Figure 10.9(b), the relaxation time decreases with increasing

concentration c for KM clusters. It is not clear if the PCC clusters have the same tendency. On the

one hand, Figure 10.9(a) shows not only that the relaxation time decreases with concentration c for

the KM clusters, but also that it drops down almost vertically for the PCC clusters with increasing

relaxation time. This tendency well captures the nature of the PCC clusters whose projected pro-

files can be close to the ss-OAFP model at the complete-core-collapse state. It is still similar to the

King model (KM clusters) in the expansion phase after their cores collapse.

181


0.5 1 1.5 2 2.5

5

6

7

8

9

10

(a)

c

t c.r.

PCCPCC?

Normal

0.5 1 1.5 2 2.5

5

6

7

8

9

10

(b)

c

t c.r.

PCCPCC?

Normal

Figure 10.9: Core relaxation time against (a) concentration c obtained from the energy-truncatedss-OAFP model (m = 3.9) applied to the PPC- and KM- clusters and (b) concentration c based onthe King model reported in (Harris, 1996, (2010 edition)).

To see the completion rate of core collapse, we used the formula employed in (Lightman, 1982)

ηc ≡to,age

tc.r.o≡

to,age

tc.r.

−(1 + Aqo) +√

(1 + Aqo)2 + 4ABqo

2Bqo(10.3.1)

where A = 35, B = 4.8 and qo = to,age/tc.r.. The time to,age is the order of the age of clusters ∼ 1010

years. The time tc.r.o is the estimated initial relaxation time of the evolution for a model cluster

based on N-body simulation. Figure 10.10 (a) shows the completion rate against the concentration

c obtained from the energy-truncated ss-OAFP model. The majority of data plots are within the

region between the lines ηc = 0.75(c − 2.0) + 1.05 and ηc = 0.75(c − 2.0) + 0.40. The lines

are empirical lines of equations, not based on physical arguments. Figure 10.10 (b) shows the

corresponding characteristics of ηc against concentration c based on the King model. The same

182


two lines of equations reasonably include the majority of data plots between them. From Figure

10.10 (a), one can find several conclusions. (i) The criterion explained in (Meylan and Heggie,

1997) still works. The clusters with c > 2.0 are PCC clusters but the completion rate is above

0.8. (ii) The clusters with over a completion rate of 0.8 are PPC clusters except for a cluster NGC

6517. (iii) The KM clusters with high concentrations (c ≥ 2.0) in (Harris, 1996, (2010 edition))’s

catalog are reasonably close to the other KM stars in the figure. Their concentrations are lowered

to values smaller than c = 2.0. Our model suggests that the two KM clusters (NGC 1851 and

NGC 6626) with high concentrations (c ≥ 2.0) have a morphology close to the complete-core-

collapse state. (iv) The PCC cluster (NGC 6544) differentiates itself from the KM- and PCC-

clusters. NGC 6544 has a high completion rate (0.989) compared with the KM clusters and a low

concentration (c = 1.61) compared with the rest of the PCC clusters. Hence, the cluster may be

a good candidate for the search of a PPC cluster that may have one of the most expanded cores.

The cluster was judged only as a ‘possible PCC’ cluster in (Djorgovski and King, 1986). (v) Our

model with c ≈ 1 fits the projected structural profiles of low-concentration clusters. This result

infers that the clusters may have structures similar to the polytrope of m ≈ 3.9. The conclusion (i)

just confirmed an expected property of the core-collapse process. The conclusions (ii) through (iv)

require a detailed case study for each cluster. Such a study is out of our scope. Hence, we further

discuss only the conclusion (v) in detail in Section 10.4.

183


0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

0.75

(c−

2.0)

+1.

05

0.75

(c−

2.0)

+0.

4

NGC6544

NGC6517

NGC1851

NGC6626

(a)

c

com

plet

ion

rate

PCCPCC?

Normal

0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1NGC6544

NGC6517

(b)

c

com

plet

ion

rate

PCCPCC?

Normal

Figure 10.10: Completion rate of core collapse against (a) concentration c based on the energy-truncated ss-OAFP model with m = 3.9 and (b) concentration c based on the King model reportedin (Harris, 1996, (2010 edition)).

10.4 Discussion

The present section discusses the three topics that we could not detail in the main results.

(i) We consider a possible application of the energy-truncated ss-OAFP model with high index

m (other than m = 3.9) to the structural profiles of globular clusters in a broad range of radii

10−2 ∼ 101 arcminutes (Section 10.4.1) (ii) We found an application limit of the model and propose

an approximated form of the model (Section 10.4.2). (iii) We discuss why the model can fit the

projected density profiles of low-concentration (c ≈ 1) globular clusters (Section 10.4.3).

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10.4.1 Fitting the energy-truncated OAFP model to‘whole’ projected

density profiles with higher indices m

In the present section, we show that the energy-truncated ss-OAFP model with m = 4.2 well

fits the projected structural profiles of some PCC clusters and KM clusters in a broad range of radii

(approximately between 0.01 and 10 arcminutes). In the main result (Section 10.2), we employed

m = 3.9 to consistently apply the model to various clusters, though as explained in Section 10.1.4,

index m higher than 3.9 (e.g., m = 4.2) may provide a better fitting. Also, more recent surveys such

as Gaia 2 provided structural profiles with elongated outer halos for some clusters that can not be

well fitted by the King model, and rather the Wilson model showed a better fitting (de Boer et al.,

2019). A similar situation was also reported in (Miocchi et al., 2013). Under the circumstances, we

applied our model to the V-band magnitudes of globular clusters reported in (Noyola and Gebhardt,

2006) combined with those of (Trager et al., 1995)10. Following (Noyola and Gebhardt, 2006), we

overlapped their magnitudes to those of (Trager et al., 1995). For example, Figure 10.11 shows

the fitting of our model with m = 4.2 to the surface brightness of NGC 6341 and NGC 6284.

The former is a KM cluster while the latter a PCC. Following the fitting done in (Noyola and

Gebhardt, 2006), we fitted our model to the surface brightness so that our model fits the data plots

(Chebyshev approximation) of (Trager et al., 1995) in the outer halo, rather than the corresponding

plots of (Noyola and Gebhardt, 2006). We would need more realistic effects (e.g., mass function)

to more reasonably fit our model to NGC 6284. However, our model appears capable of capturing

the structure (the core- and tidal radius) of the cluster. For the rest of the fitting of the model, see

Appendix I.3 for KM clusters and Appendix K.3 for PCC clusters. We needed our model with

m = 4.2 for the PCC clusters while indexes higher than m = 4.2 for the KM clusters. Also, in

the fitting, we did not include some clusters whose central cores have a negative slope reported in

(Noyola and Gebhardt, 2006).

10Unfortunately, we did not have access to data of (de Boer et al., 2019) and could not employ ’WebPlotDigitizer’to extract their data since many of their data plots and error bars are overlapped each other.

185


−3 −2 −1 0 1−14.0

−12.0

−10.0

−8.0

−6.0

−4.0

−2.0

0.0

(rc = 0′.25, δ = 0.038)

log[R(arcmin)]

SBo−

SB

ss-OAFP(c = 2.16,m = 4.2)

NGC 6341(n)Cheb.

−3 −2 −1 0

−8.0

−6.0

−4.0

−2.0

0.0

(rc = 0′.0040, δ = 0.0055)

log[R(arcmin)]

ss-OAFP(c = 3.28,m = 4.2)

NGC 6284(c)Cheb.

Figure 10.11: Fitting of the energy-truncated ss-OAFP model to the surface brightness profiles ofNGC 6284 and NGC 6341 reported in (Noyola and Gebhardt, 2006). ‘Cheb.’ means the Chebyshevapproximation of the surface brightness reported in (Trager et al., 1995). The unit of the surfacebrightness (SB) is the V magnitude per square of arcseconds. The brightness is normalized by themagnitude SBo observed at the smallest radius point. In the legends, (n) means a normal or KMcluster and (c) means a PCC cluster as judged so in (Djorgovski and King, 1986).

186


10.4.2 Application limit and an approximated form of the

energy-truncated ss-OAFP model

The energy-truncated ss-OAFP model suffers from unrealistically large tidal radii (∼ 105 ar-

cminutes) beyond m = 4.4 and can not approach m = 5. These large radii prevented us from fitting

the model to some globular clusters. Hence, the present section introduces an approximated form

of the model as a remedy. The results of Sections 10.1.4, 10.2, and 10.4.1 show the applicability

of the energy-truncated ss-OAFP model to many globular clusters for m = 3.9 and m = 4.2 ∼ 4.4.

However, it turns out that some globular clusters have more elongated structures in the outer halos

that our model can not reach. For example, Figure 10.12 shows the surface brightness profile of

NGC 6715 reported in (Noyola and Gebhardt, 2006) and fitted by our model with m = 4.4. The

observed density has a calmer slope in the halo compared with our model. Our model can reach

further large radii by increasing m; however, the tidal radius unrealistically diverges soon. For ex-

ample, the tidal radius is approximately 105 arcminutes for m = 4.4 and δ = 0.01. (Even the tidal

radius of polytrope of m = 4.5 is approximately only 31.54.). This large radius can be remedied

by artificially introducing the system’s anisotropy, like Michie (1963)’s model. Instead, we stick

to an isotropic model to avoid adding an extra fitting parameter to our model.

To resolve the unrealistically large tidal radius of the ss-OAFP model with high indexes m, we

introduce an approximated form of the energy-truncated ss-OAFP model that well fits the structural

profiles of Galactic globular clusters. First, one must understand that the values of δ that apply to

globular clusters are relatively high (δ > 0.004). Also, they would not show a distinctive sign

of the power-law profile of Σ ∝ r−1.23 in the inner halo (Recall Figure 10.3.) The power-law

profile r−2.23 in the density is recognizable when the value of δ is small ∼ 10−3. This feature infers

that we may approximate the model by excluding the contribution from the asymptotic power-law

profile (Σ ∝ r−1.23) in the halo. In fact, the density profile of the energy-truncated ss-OAFP model

can be well fitted by the exponential profile exp[−13.88(1 + E)] at low energies E < −0.7, as

187


−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.0648, δ = 0.080)SB

o−

SB

ss-OAFP(c = 2.53,m = 4.4)approx.(c = 2.79,m = 4.9)

NGC 6715(n)Cheb.

Figure 10.12: Fitting of the energy-truncated ss-OAFP model and its approximated form to thesurface brightness profile of NGC 6715 reported in (Noyola and Gebhardt, 2006). ‘Cheb.’ meansthe Chebyshev approximation of the surface brightness reported in (Trager et al., 1995). The coreradius rc and δ in the figure were acquired from the approximated form. The unit of the surfacebrightness (SB) is V magnitude per square of arcseconds. The brightness is normalized by themagnitude SBo obtained at the smallest radius point. In the legends, (n) means normal or KMcluster as judged so in (Djorgovski and King, 1986).

shown in Figure 10.13 (a). In the figure, the exact model switches from the exponential decay

(the isothermal sphere) to the power-law decay (the polytropic sphere) around E = −0.72 as E

increases. Accordingly, we may define an approximate form of the DF for our model as follows

F(E) =exp [−13.88(1 + E)] + δ (−E)m

exp [−13.88] + δ, (10.4.1)

where 13.88 is the value of the scaled escape energy χesc for the complete core collapse. One needs

to know the advantages and disadvantages of using the approximated model. This approximation

would be handy since one does not have to resort to the inverse form of the Poisson’s equation,

unlike our exact model. Also, one can employ the scaled escape energy other than the 13.88 if

considering χesc as a new parameter allowing the four degrees of freedom11. On the one hand, the

11We have tried to employ the four parameters model, though the useful values of χesc were χesc ' 9 holding theindex m to 3.9. Less than those values, the morphology of the approximated model has perfectly changed from theexact model. Especially, the approximated model with low χesc(/ 9) could not fit the structural profiles of the PCCcluster at all, Perhaps, this reflects the result of (Cohn, 1980) in which the signs of the self-similarity and core collapseappear for χesc(' 9).

188


approximated model should not be applied to the PCC clusters in the first place since the approx-

imation can be reasonable when δ >> 10−3, as shown in Figure 10.13 (b). The projected density

profiles of the exact and approximated models are almost identical for large δ = 0.1. However, they

deviate from each other for small δ. More importantly, the approximated model loses the physical

significance to examine how close the states of globular clusters are to the complete-core-collapse

state.

The approximated form well fits even for the elongated structure in the outer halo of NGC

6715 (Figure 10.12). The reason why the approximated model may take higher m(> 4.4) is that

the density profile follows the exponential decay exp[−χesc(1+ E)] at large E. This exponential de-

pendence could shorten the tidal radius, like the Woolley’s model does. This feature can be seen in

Figure 10.14. The density profile Da of the approximated model behaves like an exponential func-

tion at large E. See Appendices I.3 and K.3 to find the rest of the applications of the approximated

model to KM- and PCC- clusters in a broad range of radii.

189


0.6 0.7 0.8 0.9 110−3

10−2

10−1

100(a)

−E

Do/Do(−1)δB(3.4, 1.5)(−E)3.9/Do(−1)

exp[−13.88(1 + E)]

−3 −2 −1 0 1 2

−5

−4

−3

−2

−1

0(b)

log[R]

log[

Σ]

OAFP (δ = 10−3)approx. (δ = 10−3)

OAFP (δ = 0.1)approx. (δ = 0.1)

Figure 10.13: (a)Dimensionless density profiles Do, exp[−13.88(1 + E)] and δB(3.4, 1.5)(−E)3.9

for δ = 0.01.(b) Projected density profiles for the energy-truncated ss-OAFP model, and its ap-proximation for δ = 0.1 and δ = 0.001. For the both, m is fixed to 3.9.

10−5 10−4 10−3 10−2 10−1 100

10−9

10−7

10−5

10−3

10−1

−E

exp[−13.88(1 + E)]DaD

Figure 10.14: Density profiles Do of the energy-truncated ss-OAFP model, Da of the approximatedmodel and exp[−13.88(1 + E)].

190


10.4.3 Are low-concentration globular clusters like spherical polytropes?

The result of Section 10.3 indicates that the projected structural profiles of low-concentration

globular clusters are quite similar to those of polytropic spheres. Some clusters (such as Palomar 3

and Palomar 4) have concentrations c close to one. The low concentration means that the projected

structural profiles are alike those of the polytrope of m = 3.9 rather than the isothermal sphere.

However, fitting the energy-truncated ss-OAFP model to those clusters is overfitting due to the low

number of data points and the large size of error bars. We examined whether the projected struc-

tural profiles of the low-concentration globular clusters are polytropic. For brevity, the majority of

the results are included in Appendix J. The appendix shows the structural profile data (Kron et al.,

1984; Trager et al., 1995; Miocchi et al., 2013) fitted by the polytropic sphere model. We found that

the polytropic-sphere model can fit the structural profiles of eighteen polytropic globular clusters.

In the present section, we show the example of NGC 288 and NGC 6254. Their concentrations

based on the energy-truncated ss-OAFP model are 1.30 and 1.64 while those based on the King

model are 1.012 and 1.41. NGC 288 is a good example of a polytropic globular cluster while NGC

6254 for a non-polytropic cluster. Figure 10.16 also depicts another example of the polytropic-like

globular cluster NGC 5139. Its surface brightness profile was reported in (Meylan, 1987). The

central part of the cluster deviate from polytrope model due to the weak cusp, though the poly-

tropic sphere well fits the inner- and outer- halos. In the rest, we consider (i) possible physical

arguments that the low-concentration globular clusters may have structures like polytropic spheres

and (ii) its criticism.12We consider the concentration of NGC 288 is 1.0 based on our fitting of the King model. We confirmed the

concentrations c and values of χν reported in Table 2 of (Miocchi et al., 2013) based on our calculation, but not forNGC 288. We found the same result (χν = 1.7 with Wo = 5.8) for NGC 288 as their result. However, we use the resultof our calculation and consider NGC 288 is a low concentration cluster. This is since we found the concentration forNGC 288 is c = 1.0 for Wo = 5.0 that provides χν = 0.48 which is smaller than their value χν = 1.7 with Wo = 5.8.Also, our value is close to unity.

191


−1 0 1

−4.0

−3.0

−2.0

−1.0

0.0

(χ2ν = 0.48)

log[

Σ]

polytope(m = 4.50)

NGC 288 (n)

−1 0 1

−4.0

−3.0

−2.0

−1.0

0.0

(χ2ν = 5.8)

polytrope(m = 4.90)

NGC 6254 (n)

−1 0 1−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

NGC 288(n)

−1 0 1−0.5

0

0.5

log[R(arcmin)]

NGC6254(n)

Figure 10.15: Fitting of the polytropic sphere of index m to the projected density Σ of NGC 288 andNGC 6254 reported in (Miocchi et al., 2013). The unit of Σ is the number per square of arcminutesand Σ is normalized so that the density is unity at smallest radius for data. In the legends, (n)means a ‘normal’ or KM cluster as judged so in (Djorgovski and King, 1986). ∆ log[Σ] is thecorresponding deviation of Σ from the model.

192


−1 0 1 2

−4.0

−2.0

0.0

(χν = 2.9)

log[R(arcmin)]

SBo−

SB

polytrope(m = 4.77)

NGC 5139 (n)

−1 0 1 2−0.5

0

0.5

log[R(arcmin)]

∆(S

Bo−

SB)

NGC 5139 (n)

Figure 10.16: Fitting of the polytropic sphere of index m to the surface brightness of NGC 5139from (Meylan, 1987). The unit of the surface brightness (SB) is the V magnitude per square ofarcminutes. The brightness is normalized by the magnitude SBo observed at the smallest radiuspoint. In the legends, (n) means a ’normal’ or KM cluster as judged so in (Djorgovski and King,1986). ∆(SBo − SB) is the corresponding deviation of SBo − SB from the model.

(i) The discussion of low-concentration clusters being polytropic stellar polytrope

The low-concentration clusters may have cores that are in the states of non-equilibrium. It may

be modeled by polytropic spheres rather than a state of (local) thermodynamical equilibrium if the

mass loss from the clusters is no significant. Considering that the King- and energy-truncated-ss-

OAFP- models also fit the projected structural profiles of low-concentration globular clusters, the

cores of such clusters are well relaxed. However, it is not clear whether the DFs have reached a

local Maxwellian DF considering their long core relaxation times tc.r.. Even the extreme case of

the relaxation evolution, the self-similar solution to the OAFP equation, can not strictly provide

the Maxwellian DF, as shown in Chapter 9.

At the initial stage of the relaxation evolution of globular clusters, the relaxation process may

be dominated by its non-dominant effect on the order of secular-evolution time scale tsec ∼ Ntcross.

The scale of relaxation time tr/tcross ∼ N/ ln[N] is the case when stars can orbit at possible max-

imum apocenter that is the tidal radius rtid and minimum pericenter that is the order of rtid/N.

However, at the early stage of the evolution, the pericenter may be much larger on average. The

193


extreme cases were discussed and mathematically formulated by Kandrup (1981, 1988). Consider

that stars are far from each other, then many of them could ideally separate from the others at the

order of 1/n1/3 ∼ rtid (Refer to Appendix A.3 for the scaling of physical quantities). The dominant

effect discussed in (Chandrasekhar, 1943)13 may not be important. The (non-dominant) many-

body effect can be more effective. For homogeneous self-gravitating systems, we can directly use

the (local) relaxation time as the measure of relaxation, while we need to consider the many-body

relaxation effect for the case of inhomogenous systems. The many-body effect is effective on time

scales longer than tc.r by a factor of ∼ ln[0.11N]. The mathematical formulation, including the

many-body effect for the relaxation time, is no longer logarithmic. It mitigates in collision inte-

grals as the summation of Fourier series expansion under the orbit-averaging of kinetic equation

(Polyachenko and Shukhman, 1982; Ito, 2018c). The order of the many-body relaxation time is

∼ Ntcross rather than N/ ln[N]tcross at the early stage. Kandrup (1985) discussed some simple ex-

amples of this matter by neglecting the effect of evaporation and gravothermal instability. These

conditions are the cases when the two-body relaxation processes are not dominant yet. Kandrup

(1985) considered the self-gravitating system is confined in a box to avoid evaporation. He dis-

cussed a secular evolution by terming relaxation process the ’anomalous’ collision. The collision

may cause a deviation of the stellar DF from the Maxwellian DF on the secular evolution time

scale.

Table 10.3 shows the time-scales of globular-cluster dynamics. They are the current and esti-

mated initial relaxation times tc.r. and tc.r.o, and the age tage of the clusters with the total mass M. We

estimated the values of tc.r.o using the analysis of (Lightman, 1982). What we would like to know

is how many initial secular-evolution times have already passed so far. This may be measured by

the following ’secular-evolution’ parameter

ηM ≡tage

tsec, (10.4.2)

13The corresponding scales of the apocenter and pericenter are rtid/N and 1/n1/3.

194


where the secular evolution time is defined as

tsec ≡ ln[0.11

MM

]tc.r.o, (10.4.3)

where M is the solar mass and M the dynamical mass for each cluster reported in (Mandushev

et al., 1991). The natural log and factor 0.11 were assumed. Hence, the mathematical expression

for tc.r.o follows the core relaxation time of (Spitzer, 1988) and quantitatively the results of N-body

simulations (Aarseth and Heggie, 1998). In (Lightman, 1982), to calculate the completion rate,

the initial relaxation time tc.r.o was compared with the order of cluster age to,age. We would like to

compare tsec and tage to see whether the clusters could have reached a state described by a (local)

Maxwellian DF in their cores. If ηM ' 1, the core of a cluster may be a state described by the

Maxwellian DF. On the one hand, if ηM / 1, then the cluster may be in a non-equilibrium state at

present. The latter would provide some insight of a polytrope model being a possible model for

the low-concentration clusters. Table 10.3 shows that the globular clusters that were well fitted by

polytropic spheres. They have small ηM (0.20 < ηM / 1). On the one hand, the parameter ηM of

the clusters that could not be fitted by polytropes are 1 / ηM / 3.77. The maximum value was

achieved by NGC 7099 (one of the PCC clusters). NGC 3201 and NGC 4590 are classified into

the intermediate class. A polytrope model fitted the projected structure of them at part of cluster

radii. Figure 10.17 shows the secular evolution rate against concentration c. For the concentration,

we used the (Harris, 1996, (2010 edition))’s values. It appears that ηM = 1 is a good threshold to

separate polytropic- and non-polytropic clusters. Especially when c ≈ 1.5 and ηM ≈ 1, both the

polytropic- and non-polytropic clusters coexist.

The realization of polytropic clusters was discussed by (Taruya and Sakagami, 2003) based

on N-body simulations. They assumed that a self-gravitating system of equal-mass was enclosed

by an adiabatic container. They found that stellar polytropes can well approximate the simulated

distribution function even on time scales much longer than half-mass relaxation time. This feature

195


was also confirmed using an isotropic time-dependent Fokker-Planck model (Taruya and Sak-

agami, 2004). Taruya and Sakagami (2003) also tested the system without an adiabatic wall. Due

to the evaporation, the stellar DF largely deviates from the stellar polytrope, while the simulated

DF seems well fitted by the DF for polytropes at the early stage of evolution. In their work, m = 5.7

at T = 50 seems to provide a DF reasonably close to the DF for a polytrope. Also, the polytropes

can well fit the inner parts of the systems and stellar DF at low energy regardless of the effect of

escaping stars. Their results imply that the stellar DF and structural profile of self-gravitating can

be like a polytrope unless the effect of evaporation is dominant.

0 1 2

0

0.5

1

1.5

2

2.5

3

3.5

4

c

ηM

polytropicpolytropic?

non-polytropic

Figure 10.17: Secular relaxation parameter ηM against concentration c. The values of concentrationare adapted from (Harris, 1996, (2010 edition)).

(ii) Criticism on polytropic globular clusters

The above discussion for polytropic globular clusters is oversimplified in the sense that actual

globular clusters are subject to mass spectrum (segregation) with stellar evolution and tidal effects

(shock). The actual clusters are supposed to have lost a significant amount of stars while the re-

196


polytropic c tc.r. tc.r.o log[

MMo

]ηM tage Reference for tage

cluster (Gyr) (Gyr) (Gyr)

NGC 288 0.99 0.98 2.0 4.64 0.63 10.62 (Forbes and Bridges, 2010)









NGC 6402 0.99 1.14 2.35 5.89 0.47 12.6 (Santos and Piatti, 2004)




NGC 6809 0.93 0.72 1.8 5.03 0.77 13.0±0.3 (Wang et al., 2016)


Pal 3 0.99 4.5 5.8 4.36 0.21 9.7 (Forbes and Bridges, 2010)

Pal 4 0.93 5.2 6.5 4.21 0.19 9.5 (Forbes and Bridges, 2010)

Pal 14 0.80 7.1 8.6 3.83 0.22 13.2 ± 0.3 (Sollima et al., 2010)

polytropic c tc.r. tc.r.o log[

MMo


cluster? (Gyr) (Gyr) (Gyr)


NGC 4590 1.41 0.28 1.1 4.95 1.29 13.0 ±1.0 (Dotter et al., 2009)

non-polytropic c tc.r. tc.r.o log[

MMo


cluster (Gyr) (Gyr) (Gyr)

NGC 1851 1.86 0.027 0.38 5.42 2.36 9.2±1.1 (Salaris and Weiss, 2002)


NGC 6121 1.65 0.079 0.61 4.83 2.11 11.5±0.4 (Wang et al., 2016)


NGC 6254 1.38 0.16 0.81 5.06 1.51 11.39±1.1 (Forbes and Bridges, 2010)




NGC 6626 1.67 0.042 0.52 5.36 2.29 12.1±1.0 (Kerber et al., 2018)DSED method


NGC 7099(c) 2.50 0.0023 0.35 4.91 2.97 12.93 (Forbes and Bridges, 2010)

Table 10.3: Core relaxation times and ages of polytropic- and non-polytropic- clusters. The currentrelaxation time tc.r. and concentration c are values reported in (Harris, 1996, (2010 edition)). Thetotal masses log

[MMo

]are from (Mandushev et al., 1991) where Mo is solar mass. We adapted the

ages of clusters from (Forbes and Bridges, 2010) and we resorted to other sources when we foundmore recent data or could not find the cluster age in (Forbes and Bridges, 2010).

197


lation of mass spectrum and polytrope sphere has not been discussed. Hence, polytropic globular

clusters are a phenomenological concept. In the case of an isolated N-body system of equal stellar

masses, the cluster loses a small fraction (∼ 0.1%) of the total mass in the first five initial relaxation

time scale (Baumgardt et al., 2002a). However, mass segregation and tidal effect generally make

the mass-loss process faster. The mass loss leads the cluster to undergo a core collapse earlier than

the system of equal-mass systems (Spitzer, 1988; Binney and Tremaine, 2011).14 A relatively-new

observation showed an unexpected feature of low-concentration clusters. They have more depleted

mass functions of low-mass stars than high-concentration ones (Marchi et al., 2007). This obser-

vation means that the lower-concentration clusters might have lost more stars due to evaporation or

tidal stripping. The former can be caused by mass segregation through two-body relaxation while

the latter by the tidal effect from the Galaxy. However, the excessive loss of low-mass stars from

low-concentration clusters contradicts standard stellar dynamics. Generally, higher-concentration

clusters are supposed to have lost more low-mass stars due to more frequent two-body relaxation

process. Based on direct an N-body simulation, (Baumgardt et al., 2008) explained a possible

interpretation of this issue. Baumgardt et al. (2008) showed that the low-concentration clusters

had already undergone primordial mass segregation in the early stage of evolution due to stellar

evolutions. This idea was extended to a sophisticated case study for one of the low-concentration

clusters, Palomar 4 (Zonoozi et al., 2017). The total mass of the model cluster rapidly decreases

only in the first 0.1 Gyr. The mass of the cluster calmly keeps decreasing with time. The decrease

in mass depends on the orbit of Palomar 4, though the total stellar number decreases in 10 Gyr by

approximately 60 %. It appears that the reason why the low-concentration clusters have polytropic

structures is not directly because of little loss of stars. Hence, one needs to directly discuss the re-

lation between the DF for polytropes and globular clusters that have experienced mass segregation.

The relation has not been detailed in the previous works.

14The relation of mass-loss and relaxation time was originally discussed for both the multi-mass OAFP model (e.g.,Chernoff and Weinberg, 1990; Takahashi and Lee, 2000) and N-body simulation (e.g., Fukushige and Heggie, 1996;Portegies Zwart et al., 1998; Baumgardt and Makino, 2003) in tidal field.

198


Also, the present work does not discuss the projected line-of-sight velocity dispersion profiles

of the energy-truncated ss-OAFP model. Many of the polytropic clusters are low-concentration

clusters. This result implies that accurate observational data are hard to be obtained compared

with high-concentration clusters (e.g., Meylan and Heggie, 1997). Perhaps, one can differentiate

the ss-OAFP model from the other models, including the King model, using accurate kinematic

data. The data may be useful from Gaia 2 (e.g., Baumgardt et al., 2018), the ESO Multi-instrument

Kinematic Survey (MIKiS) (Ferraro et al., 2018), or more accurate surveys in the future.

In conclusion, we consider the polytropic globular clusters may ’phenomenologically’ apply to

the structure of low-concentration clusters. The discussion is not matured on the relation between

mass segregation (spectrum) and stellar DF for polytrope. Hence, we further need to examine this

topic in the future with more accurate observed structural- and kinematic- data and realistic nu-

merical simulations. We believe that, especially, the kinematic data can draw a line in applicability

between the King and our models.

10.5 Conclusion

The present chapter introduced a phenomenological model, i.e., the energy-truncated ss-OAFP

model that can fit the projected density profiles for at least half of Milky Way globular clusters,

including PCC clusters with resolved cores. We aimed at establishing a model with broad appli-

cability compared with the classical isotropic one-component King model. Our model is a linear

algebraic combination of the DFs for the ss-OAFP model and the polytropic sphere of m. The

latter was weighted by a factor of δ. The optimal value of m was identified as 3.9 by comparing the

concentrations and tidal (limiting) radii between the King- and our models. After this procedure,

this new model has only three degrees of freedom that are the same as those of the King model.

On the one hand, our model can fit the projected structural profiles of Galactic PCC clusters with

resolved cores in addition to those of KM clusters. The fitting results provided a completion rate of

199


core collapse against the concentration, including PCC clusters with resolved-core. It is consistent

with standard stellar dynamics. Also, our model is more useful than the King model to single out

KM clusters whose morphology is close to the collapsing-core cluster, or high concentration is

c ≥ 2.0.

Our model can also apply to globular clusters in a broad range of radii 0.01 ∼ 10 arcminutes

with higher m(≥ 4.2). However, the energy-truncation based on a polytrope provides an unrealis-

tically large tidal radius. Hence, we also proposed an approximated form of the energy-truncated

ss-OAFP model to avoid unrealistic tidal radii. Lastly, we discussed that the low-concentration

globular clusters may be polytropic since polytrope-sphere models well fit their projected struc-

tural profiles. However, the physical arguments and previous numerical results for the polytropic

globular clusters are not well established. Hence, we consider the polytropic clusters are a heuristic

idea for now. This situation intrigues us to work in the future on examining three topics. (i) We

will discuss the relationship between mass spectrum (segregation) of star clusters and stellar DFs

for polytropes. (ii) We will analyze the relationship of the low-concentration clusters with stellar

polytropes that obey the generalized statistical mechanics based on the Tsallis entropy. (iii) We will

examine the applicability of the King and our models to the kinematic profiles of low-concentration

clusters.

200

Chapter 11

Conclusion

The present dissertation aimed to apply Chebyshev pseudo-spectral methods to the ss-OAFP

system to model the relaxation evolution of dense star clusters. It also aimed at showing the physi-

cal properties of the model and an application to globular clusters in Milky Way. The star clusters’

dynamical properties have been under long-term studies over 100 years due to its simple observed

characteristics to test stellar dynamics that have a cross-field importance (Chapter 1). The large

gap in dynamical and relaxationtime scales in the evolution of the clusters implies that the stellar

DF is always at a QSS on dynamical times scales (Chapter 2). The isothermal sphere and poly-

tropic sphere are important models at QSSs since they are considered an imitation of the core and

halo of star clusters. Standard stellar dynamics predicts that the relaxation evolution reaches a

core-collapse phase in a self-similar fashion in the late stage of the evolution. However, the cor-

responding mathematical model, the ss-OAFP system, has never been satisfactorily solved due to

the complicated mathematical structures and difficulty in the convergence of the Newton iteration

method. The present work employed Chebyshev collocation-spectral methods to overcome the

numerical difficulty (Chapter 3). We applied the spectral methods to the Poisson’s equations that

model the isothermal- and polytropic- spheres as the preliminary works. We also applied them to

the ss-OAFP system as the main work. These applications were made after we rearranged the ex-

201

CHAPTER 11. CONCLUSION

pressions of the mathematical models to apply the spectral method (Chapter 4). Also, we arranged

the spectral methods themselves for the ss-OAFP model (Chapter 5). We showed the numerical

results in Chapters 6, 7, and 8. We employed the results to discuss the thermodynamic aspects of

the ss-OAFP model (Chapter 9) and applied it to Galactic globular clusters (Chapter 10).

In Chapter 6, we obtained a highly accurate solution for the isothermal-sphere model in a broad

range of logarithmic radius scales and reproduced the known fine structures (analytically discussed

in (Chandrasekhar, 1939)). By adjusting the mapping parameter L, we achieved a geometrical con-

vergence for the Chebyshev expansion, and the coefficients reach the order of machine precision

only with approximately one hundred of polynomials. The numerical result enabled us to produce

a new reference table of the numerical values for the LE2 function covering the logarithmic radius

range (r ≈ 10−5 through r ≈ 10150). We also found an accurate semi-analytical asymptotic form of

the LE2 function as r → ∞.

In Chapter 7, we found highly-accurate spectral solutions for the polytropic spheres with

5.5 < m < 104 in a broad range of logarithmic radius scales. Our solutions can achieve Horedt

(1986)’s seven-digit accurate solutions with at most a few hundreds of polynomials. The results

also reproduced the known asymptotic forms of the LE 1 functions. The results of Chapters 6 and

7 showed enough accuracy to apply the established spectral method to the ss-OAFP system. The

latter was expected to have a huge numerical gap in stellar the DF and Q-integral requiring high

accuracy.

Chapter 8 found an accurate spectral solution for the ss-OAFP model on the whole domain that

we called the reference solution. In Section 8.1, we showed that the degree of polynomials for

the reference solution is seventy, and the coefficients reach the order of 10−12. We obtained the

eigenvalues more mathematically satisfactorily compared with existing works. Also, we provided

a semi-analytical solution whose degree of polynomials is at most eighteen. The corresponding

physical parameters to characterize the self-similar evolution are as follows. The power-law index

α is 2.2305, the collapse rate ξ = 3.64 × 10−3, and the scaled escape energy χesc = 13.88. On the

202


one hand, we found that the whole-domain solution is unstable against degree N of polynomials.

In Section 8.2 we aimed at finding the cause of the numerical instability and an optimal solution

by seeking truncated-domain solutions that can improve the accuracy with increasing degree N.

To find an optimal truncated-domain solution that is close to the reference solution, we obtained

truncated-domain solutions with β = βo for −0.1 ≤ Emax ≤ −0.03. Those solutions are stable

against up to specific degrees of polynomials. For Emax = −0.03, the truncated-domain solution

has the same order of accuracy in c∗4 as the reference solution. Hence, we compared the reference-

and truncated-domain solution with N = 65 and Emax = −0.03. The relative error between the

solutions is approximately 10−9.

From the result of Section 8.2, we believe the reason why the whole-domain solution is unstable

against the degree of polynomials is that one can correctly solve the ss-OAFP equation with a

limited accuracy due to the limit of machine precision and a special structure of the OAFP model.

One loses the capability of handling the ss-OAFP equation when the power-law c1(−E)βo/c∗4 in the

4ODEs at Emin = −0.0523 reaches the order of 10−13 corresponding to the minimum of infinity

norms for Chebyshev coefficients in the Newton iteration process.

In Section 8.3, by modifying the regularized dependent variables 3R and 3F , we reproduced the

Heggie and Stevenson (1988)’s solution around at Emax = −0.225 while it still can approach the

reference solution around at Emax = −0.05. This result inferred that the significant figures of our

spectral solution is accurate to four digits at least, while the existing solutions are one digit at most.

The results for the ss-OAFP model provided us with an opportunity to discuss the model’s

detailed characteristics that the existing works could not show. In Chapter 9, we discussed some

thermodynamic aspects of the ss-OAFP model. The local thermodynamic quantities showed that

the core locally follows the isothermal sphere at a state of ideal gas while the outer halo behaves

like a polytrope with an equation of state p = 0.5χescρ1.1. The analog of caloric curves and a simple

analytical calculation systematically showed that the core has the expected negative heat capacity.

They also showed that the singularities in the heat capacity occur closely in radius. Especially

203


in the chapter, we focused on discussing the cause of the negative heat capacity of the core. We

showed that the negativity originates from the deep potential well based on a simple discussion

of the Virial and total energy of the model. We concluded that the negative heat capacity of the

ss-OAFP model is driven by its collisionless state and high temperature on a long time duration

(∼ relaxation time scales). The isolation of the core from the surroundings or self-gravity does not

cause the negativity. Rather, the negative heat capacity is a result of a rapid change in the mean-

field potential. The structural change is related to a non-equilibrium state through stellar- and heat-

flows from the core to the halo.

As an application of the ss-OAFP model, Chapter 10 proposed an energy-truncated ss-OAFP

model that can fit the projected structural profiles of KM- and PCC- clusters with resolved cores

in Milky Way. Our model can have broader applicability than the most standard model, the King

model. The latter applies to only KM clusters. Using the results of the fitting of the new model to

Galactic globular clusters, we provided the characteristics of the completion rate of core collapse.

We showed that the new model can apply to the clusters in a broad range of radii (0.01∼10 ar-

cminutes). We also provided the approximated form of the new model to avoid the unrealistically

large tidal radii of our model. Also, our model showed that the low-concentration clusters have

structures close to polytropic clusters. However, it is not clear if the concept of polytropic globular

clusters can be well explained based on currently-known physical arguments. Also, our new model

is heuristic in the sense that a polytrope of index m controls the outer halo of our model. Hence, we

consider that our model is phenomenological. It is useful to discuss how globular clusters’ states

are close to complete-core-collapse or polytropic based on their structural profiles.

Other than the topics discussed in Section 9.5, the possible direct extension works for the

present work are relatively limited. Only one reasonable extension work is to apply the spectral

method to the ss-OAFP model with a point source for the post-core collapse phase as done in (Heg-

gie and Stevenson, 1988; Takahashi, 1993). Adding other physical effects and astrophysical objects

to the collapsing-core model is prohibited since they could desperately change the mathematical

204


structure of the ss-OAFP model. They may also hinder the convergence of the Newton method.

Anisotropic models are currently standard in stellar-dynamics studies, even for the OAFP model.

The extension to the anisotropic model is unrealistic since the ss-OAFP system have too many

complicated integrals (twelve twofold integrals), compared with the current four single integrals

in the isotropic one. Since the practical CPU time for the numerical integration of the ss-OAFP

model is ∼ three months that are the same CPU time cost as an N-body simulation. Rather, we

need to move on to optimization processes for the ss-OAFP system by employing the Anderson

accelerator (Anderson, 1965; Walker and Ni, 2011). However, it is very unclear if one can resort

to a line search method, quasi-Newton method and trust-region method since the high condition

number (∼ 108) in the present work would prevent us from obtaining twelve-digit eigenvalues (for

the reference solution) since the high condition number is against the Wolfe condition (e.g., No-

cedal and Wright, 1999). Accordingly, it would be helpful to develop a spectral numerical scheme

for the time-dependent OAFP model. The model is more realistic than the ss-OAFP model and has

been solved for anisotropic cases. Essentially, it has nothing to do with the convergence of Newton

iteration method.

205

Appendix A

The basic kinetics and scaling of physical

quantities for stellar dynamics

Good review works (e.g., Chavanis, 2013, and references therein) specializing in the star clus-

ter’s kinetics based on the first principles are limited at present. Hence, the present appendix

explains the fundamental kinetics of star clusters. In Appendices A.1 and A.2, we explain the

fundamental concepts of kinetics arranged for star clusters, beginning with the N-body Liouville

equation. Appendix A.3 shows the scaling of the orders of the magnitudes (OoM) of physical

quantities to describe the structure and evolution of star clusters and encounters. Appendix A.4

explains the trajectory of test star in encounters.

A.1 The N-body Liouville equation for stellar dynamics

Consider a star cluster of N-’point’ stars of equal masses m interacting with each other purely

via Newtonian gravitational potential

φi j(ri j) = −Gmri j

, (1 ≤ i, j ≤ N with i , j), (A.1.1)

206

APPENDIX A. BASIC KINETICS

where ri j

(=| ri − r j |

)is the distance between star i at position ri and star j at r j. The Hamiltonian

for the motions of stars reads

H =

N∑i=1

p2i

2m+ m

N∑j>i

φi j(ri j)

, (A.1.2)

where pi(= mvi) is the momentum of star i moving at velocity vi. Assume the corresponding 6N

Hamiltonian equations can be alternatively written in the form of the N-body Liouville equation

dFN

dt=

∂t +

N∑i=1

[vi · ∇i + ai · ∂i]

FN(1, · · · ,N, t) = 0, (A.1.3)

where the symbols for the operators are abbreviated by ∂t = ∂∂t , ∇i = ∂

∂ri, and ∂i = ∂

∂vi. The

acceleration ai of star i due to the pair-wise Newtonian forces from the rest of stars is defined as

ai ≡

N∑j=1(,i)

ai j ≡ −

N∑j=1(,i)

∇iφi j(ri j). (A.1.4)

The arguments 1, · · · ,N of the N-body (joint-probability) DF FN are the Eulerian position coor-

dinates and momenta r1,p1, · · · , rN ,pN of stars in the system at time t. The N-body DF FN is

interpreted as the phase-space probability density of finding stars 1, 2, · · · ,N at phase-space points

(r1,p1), (r2,p2), · · · and (rN ,pN) respectively at time t. The DF FN is normalized as

∫FN(1, · · · ,N, t) d1 · · · dN = 1, (A.1.5)

where an abbreviated notation is employed for the phase-space volume elements, d1 · · · dN (=

dr1 dp1 · · · drN dpN). For simplicity, we assume that the function FN is symmetric about a per-

mutation between any two phase-space states of stars (Balescu, 1997; Liboff, 2003). Also, the

Hamiltonian equation (A.1.2) in phase space holds the same symmetry. Hence, we may consider

stars 1, · · · , N to be identical and indistinguishable, respectively, though the assumptions are not

207


proper for stellar dynamics.

A.2 The s-tuple distribution function and correlation function

of stars

One may introduce the reduced DF for stars in a star cluster in the form of s-body (joint-

probability) DF

Fs(1 · · · s, t) =

∫FN(1, · · · ,N, t) ds+1 · · · dN , (A.2.1)

or the form of s-tuple DF:

fs(1 · · · s, t) =N!

(N − s)!Fs(1 · · · s, t). (A.2.2)

The s-tuple DF describes the probable number (phase-space) density of finding stars 1, 2, · · · , s at

phase-space points 1, 2, · · · , s respectively. The s-tuple DF simplifies the relation of macroscopic

quantities with irreducible s-body dynamical functions. For example, the total energy of the system

at time t may read

E(t) =

∫· · ·

∫ N∑i=1

p2i

2m+ m

N∑j>i

φi j(ri j)

FN(1 · · ·N, t) d1 · · · dN , (A.2.3a)

= N∫ p2

1

2mF1(1, t) d1 + m

N(N − 1)2

∫φ12(r12)F2(1, 2, t) d1 d2, (A.2.3b)

=

∫ p21

2mf1(1, t) d1 +

m2

∫φ12(r12) f2(1, 2, t) d1 d2. (A.2.3c)

where we employed the symmetry of permutation between two phase-space points for both the

Hamiltonian and the s-body DF. The total energy E(t) can turn into a more physically meaningful

208


form by introducing s-ary DFs to understand the effect of the correlation between stars.

f (1, t) : (unary) DF

f (1, 2, t) : binary DF

g(1, 2, t) : (binary) correlation function

f (1, 2, 3, t) : ternary DF

T (1, 2, 3, t) : ternary correlation function

Ignoring the effect of ternary correlation function T (1, 2, 3, t) (i.e., the effect of three-body interac-

tions, e.g., triple encounters of stars), we may rewrite the single-, double-, and triple- DFs using

the Mayer cluster expansion (e.g., Mayer and MG, 1940; Green, 1956)1

f1(1, t) ≡ f (1, t), (A.2.5a)

f2(1, 2, t) ≡ f (1, 2, t) = f (1, t) f (2, t) +

[g(1, 2, t) −

f (1, t) f (2, t)N

], (A.2.5b)

f3(1, 2, 3, t) = f (1, t) f (2, t) f (3, t) + g(1, 2, t) f (3, t) + g(2, 3, t) f (1, t) + g(3, 1, t) f (2, t)

−1N

(2g(1, 2, t) + f (1, t) f (2, t)) f (3, t) −1N

(2g(2, 3, t) + f (2, t) f (3, t)) f (1, t)

−1N

(2g(3, 1, t) + f (3, t) f (1, t)) f (2, t), (A.2.5c)

1The DFs and correlation functions for stars may generally depend on the stellar number N as follows

f (1, t), f (2, t), f (3, t) ∝ N,

g(1, 2, t), g(2, 3, t), g(3, 1, t) ∝ N(N − 1), (A.2.4)

where the normalization condition for DFs and correlation functions follows (Liboff, 1966).

209


Under the weak-coupling approximation, one can retrieve the form of the DFs2

f3(1, 2, 3, t) = f (1, t) f (2, t) f (3, t) +

(g(1, 2, t) −

f (1, t) f (2, t)N

)f (3, t)

+

(g(2, 3, t) −

f (2, t) f (3, t)N

)f (1, t) +

(g(3, 1, t) −

f (3, t) f (1, t)N

)f (2, t). (A.2.6)

There exists an essential difference of star clusters from classical plasmas and ordinary neutral

gases. The difference can be characterized by the effects of smallness parameter, 1/N, in equations

(A.2.5b) and (A.2.6). The parameter is not ignorable for dense star clusters(104 . N . 107

). The

correlation function g(i, j, t) has the anti-normalization property for self-gravitating systems

∫g(i, j, t) di =

∫g(i, j, t) d j = 0. (i, j = 1, 2, or 3 with i , j). (A.2.7)

as proved under the weak-coupling approximation by Liboff (1965, 1966) and employed by Gilbert

(1968, 1971). Employing the correlation function g(1, 2, t), the total energy, equation (A.2.3c), may

be rewritten as

E(t) =

∫ p21

2mf1(1, t) d1 + Um.f.(t) + Ucor(t), (A.2.8)

where

Um.f.(t) =m2

∫φ(r1, t) f (1, t) d1, (A.2.9a)

Ucor(t) =m2

∫φ12(r12)g(1, 2, t) d1 d2. (A.2.9b)

2The expressions were first shown in (Gilbert, 1968) for stellar dynamics based on a functional derivative method(Bogoliubov, 1962).

210


The self-consistent gravitational m.f. potential is defined as

φ(r1, t) =

(1 −

1N

) ∫φ12(r12) f (2, t) d2, (A.2.10)

where the factor(1 − 1

N

)is the effect of discreteness. The m.f. potential on a star is due to (N − 1)-

field stars (e.g., Kandrup, 1986). Also, the corresponding self-consistent gravitational m.f. accel-

eration of star 1 reads

A(r1, t) = −

(1 −

1N

) ∫∇1φ12(r12) f (2, t) d2. (A.2.11)

A.3 The scaling of the orders of the magnitudes of physical

quantities to characterize stars’ encounter and star

cluster

One needs two scaling parameters for the kinetic theory of star clusters; the discreteness pa-

rameter, 1/N, and the distance r12 between two stars (say, star 1 is test star at r1 and star 2 is

one of the field stars at r2.). The fundamental scaling of physical quantities associated with the

discreteness parameter follows the scaling employed in (Chavanis, 2013, Appendix A) except for

the correlation function g(1, 2, t) (equation (A.2.4)). For r12, following the scaling of the order

of magnitudes (OoM) of physical quantities for classical electron-ion plasmas (Montgomery and

Tidman, 1964, pg 22), one may classify the effective distance of two-body Newtonian interaction

and m.f. acceleration into the following four ranges of distances between two stars depending on

the magnitudes of forces exerted on test star in a star cluster system;

1. m.f.(many-body) interaction d< r12 < R

2. weak m.f.(many-body) interaction aBG < r12 < d

211


3. weak two-body interaction ro < r12 < aBG

4. strong two-body interaction 0 < r12 < ro

where R is the characteristic size of a finite star cluster (e.g., the Jeans length and tidal radius), d the

average distance of stars in the system, ro the ’conventional’ Landau radius (explained in equation

A.3.4b), and aBG the Boltzmann-Grad (BG) radius. The BG radius separates distance r12 at the

distance below which two-body encounters are more dominant with decreasing radius compared

with the effect of m.f. acceleration (many-body encounters). The radius aBG corresponds with the

scaling of the Boltzmann-Grad limit (Grad, 1958)3. For the relaxation processes in plasmas (Mont-

gomery and Tidman, 1964), the BG radius aBG is not essential since the fundamental mathematical

formulation assumes homogeneous plasmas and the Thermodynamic limit,

n = N/V → O(1) (with V → ∞ and N → ∞), (A.3.1)

where V is the system volume of plasmas.

In the present work, the ’Landau radius’ r90 is newly defined as the closest spatial separation of

two stars. This separation occurs when the impact parameter of test star is equal to the Landau dis-

tance b90 (the impact parameter to deflect test star thorough an encounter by 90o from the original

direction of motion);

r90(v12(−∞)) =b90(v12(−∞))

1 +√

2, (A.3.2a)

b90(v12(−∞)) =2Gm

v212(−∞)

, (A.3.2b)

where v12(−∞) is the relative speed between star 1 and star 2 before encounter. Refer to Tables A.1

and A.2 for the scaling of basic physical quantities and (Ito, 2018a, Appendix) for more details.

3The BG radius is in essence the same as the ’encounter radius (Ogorodnikov, 1965)’ to separate the encounter andpassage of stars.

212


quantities order of magnitude

tr ∼ N/ ln[N],

f (1, t), tsec ∼ N,

A1,R,m, v1, v12, r1, tdyn ∼ 1,

d ∼ 1/N1/3,

aBG ∼ 1/N1/2

G, ro,Kn ∼ 1/N,

g(1, 2, t) ∼ N/r12, for ro < r12 < R

∼ N2, for r12 < ro

a12 ∼ 1/(r212N)

tcor, δvA

(=

∫A1 dtcor

)∼ r12 for ro < r12 < R

∼ δv12r212N for r12 < ro

Table A.1: A scaling of the order of magnitudes of physical quantities associated with the evolutionof a star cluster that has not gone through a core-collapse. The scaling can be useful for systemswhose density contrast is much less than the order of N. Hence, we expect the cluster to be normalrather PCC. (Recall gravothermal instability occurs with density contrast of 709 for the isothermalsphere (Antonov, 1962).). The OoMs are scaled by N and r12 except for the correlation time tcor,which needs the change in velocity, δva

(=

∫a12 dtcor

), due to Newtonian two-body interaction.

The characteristic time scales of the relevant evolution of DFs and correlation function are defined

as

1tdyn'

∣∣∣∣∣v1 · ∇1 f (1, t)f (1, t)

∣∣∣∣∣ , (A.3.3a)

1tsec'

∣∣∣∣∣∣ 1f (1, t)

(∂ f (1, t)∂t

)∣∣∣∣∣∣ , (A.3.3b)

1tcor'

∣∣∣∣∣∣ 1g(1, 2, t)

(∂g(1, 2, t)

∂t

)∣∣∣∣∣∣ . (A.3.3c)

A particular focus is the scaling of the Landau radius r90, equation (A.3.2a). It depends on

the relative speed between the two stars. A mathematically strict treatment on the Landau dis-

tance has been discussed for Newtonian interaction (Retterer, 1979; Ipser and Semenzato, 1983;

213


RdaBGror12

strong 2-body(Boltzmann)

weak 2-body(Landau)

weak m.f(g-Landau)

m.f.(many-body)(g-Landau)

∼ 1∼ 1∼ 1∼ 1A1

∼ N∼ N4/3∼ N3/2∼ N2g(1, 2, t)

∼ 1/N∼ N−1/3∼ 1∼ Na12

∼ 1/N∼ N−2/3∼ N−1/2∼ 1δva

∼ 1∼ N−1/3∼ N−1/2∼ N−1δvA, tcor

star 1 star 2

Table A.2: A scaling of physical quantities according to the effective interaction range of Newto-nian interaction accelerations and close encounter.

214


Shoub, 1992) and Coulombian one (Chang, 1992), until then one had simplified the Landau dis-

tance by approximating the relative speed v12 to the velocity dispersion < v > of the system. The

’conventional’ Landau- distance bo and radius ro are defined as

b90 '2Gm< v >2 ≡ bo, (A.3.4a)

ro ≡bo

1 +√

2. (A.3.4b)

Assume the dispersion speed may be determined by the Virial theorem for a finite spherical star

cluster of the radius of R as follows

< v >≡ ζ

√GmN

R, (A.3.5)

where ζ is a constant and the radius R may be the Jeans length or tidal radius to hold the finiteness

of the system size. Simple examples of the values of the constants ζ are ζ =√

3/5 if the system

is finite and spatially homogeneous, and ζ is the order of unity if the system follows the King

model (King, 1966). In the present appendix, the dispersion approximation is still employed since

it simplifies the scaling of the Landau distance without losing the essential property of strong

encounters. Employing equations (A.3.4a), (A.3.4b), and (A.3.5), one finds the relation between

the system radius and the conventional Landau distance as follows

ro

R=

2

1 +√

2

1c2N

. (A.3.6)

See (Ito, 2018a) for the detailed discussion of the relation among close encounters, encounters with

large-deflection angles, and large-speed changes. The present work defines a Landau sphere by a

sphere whose center is situated at the location of a test star with the conventional Landau radius

r90 (Figure A.1).

215


4

star 1

4

star 2

4

star k

Figure A.1: Schematic description of the Landau sphere. For the truncated DF in (Grad, 1958), ifany two stars approach closer than the conventional Landau radius, the stochastic dynamics turnsinto the deterministic one. For the weakly-coupled DF (Appendix B.2), any stars can not approacheach other closer than the Landau radius.

A.4 The trajectories of a test star

The complete (Lagrangian) trajectory of star i can be discussed by taking the sum of the m.f.

acceleration of star i due to smooth m.f. potential force and the Newtonian pair-wise acceleration

via interaction with star j;

ri(t) = ri(t − τ) +

∫ t

t−τvi

(t′)

dt′, (i , j = 1, 2) (A.4.1a)

vi(t) = vi(t − τ) +

∫ t

t−τ

[ai j

(t′)

+ Ai(t′)]

dt′. (A.4.1b)

One can approximate the complete trajectory to a simpler form in each range of distance between

stars i and j, following the scaling of Appendix A.3. At distances ri j < aBG, where two-body New-

tonian interaction dominates the other effects, the trajectory perfectly follows a pure Newtonian

216


two-body problem

ri(t) = ri(t − τ) +

∫ t

t−τvi

(t′)

dt′, (A.4.2a)

vi(t) = vi(t − τ) +

∫ t

t−τai j

(t′)

dt′. (A.4.2b)

At relatively short distances (ri j . ro), we may consider the trajectory due to a strong-close

encounter as the local Newtonian interaction between two stars, i.e., the Boltzmann two-body col-

lision description if one includes the Markovian approximation (e.g., Cercignani, 1988; Klimon-

tovich, 1982)

r12(t) = r12(t − τ) +

∫ t

t−τv12

(t′)

dt′, (A.4.3a)

R =r1 + r2

2≈ r1, (A.4.3b)

r1 = r1(t − τ) +

∫ t

t−τ

v1 (t′) + v2 (t′)2

dt′, (A.4.3c)

vi(t) = vi(t − τ) +

∫ t

t−τai j

(t′)

dt′. (A.4.3d)

Equation (A.4.3) is still applicable to stellar dynamics (Agekyan, 1959; Retterer, 1979; Ipser and

Kandrup, 1980; Ito, 2018a). At intermediate distances (ro << ri j . aBG), the trajectory due to two-

body weak-distant encounters may take a rectilinear motion local in space with weak-coupling

limit

r12(t) = r12(t − τ) + v12τ, (A.4.4a)

R ≈ r1, (A.4.4b)

vi(t) = vi(t − τ), (A.4.4c)

217


Equation (A.4.4) has been employed as the most fundamental approximation to derive Landau

(FP) collision term from the first principles (Severne and Haggerty, 1976; Kandrup, 1980; Cha-

vanis, 2013; Ito, 2018a,b) or see Appendix C.2. Equation (A.4.4) is the trajectory used for the

ss-OAFP model. Lastly, at large distances (aBG << ri j < R), the trajectory due to many-body

weak-distant encounter may purely follow the motion of star under the effect of m.f. acceleration

with the weak-coupling limit

ri(t) = ri(t − τ) +

∫ t

t−τvi

(t′)

dt′, (A.4.5a)

vi(t) = vi(t − τ) +

∫ t

t−τAi

(t′)

dt′. (A.4.5b)

Equation (A.4.5) has been employed to find kinds of the orbit-averaged kinetic equation to describe

the effect of many-body interaction relaxation by assuming test star can move in orbit due to the

(quasi-)stationary orbit (Polyachenko and Shukhman, 1982; Heyvaerts, 2010; Chavanis, 2012; Ito,

2018c). Equation (A.4.5) corresponds to an approximation to be employed for the trajectory for

low-concentration clusters if the many-body relaxation is important for the clusters, as discussed

in Chapter 10.

218

Appendix B

The derivation of the BBGKY hierarchy for

a weakly-coupled distribution function

Appendix B.1 shows a BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy begin-

ning from the N-body Liouville equation by extending (Grad, 1958)’s truncated DF. We assume

that stars can not approach each other closer than the conventional Landau radius ro. This is the

most basic assumption made last decades in stellar dynamics. In Appendix B.2, the BBGKY

hierarchy is arranged for stellar dynamics by introducing a new kind of DF, weakly-coupled DF.

B.1 BBGKY hierarchy for the truncated DF

According to Appendix A.3, the effect of strong encounter, which corresponds with removing

the effect from stars inside of the Landau sphere, is the order of 1/N. This implies that one needs

to keep the effect in a 1/N-expansion of the Liouville equation. The present Appendix resorts to

Cercignani (1988)’s work for neutral gases. In (Cercignani, 1988), the derivation of the BBGKY

hierarchy for the hard-sphere DFs was made in a mathematically strict manner, by employing the

Gauss’s lemma and integration-by-parts. However, counting the correct patterns of combinations

219

APPENDIX B. BBGKY HIERARCH

for the Gauss’s lemma is confusing, and the BBGKY hierarchy for the truncated DF is not shown

explicitly. In the present appendix, the latter hierarchy is derived by exploiting integration-by-parts

and a general Heaviside function

Θ(ri j − 4) =

1, (if ri j ≥ 4)

0, (otherwise)(B.1.1a)

≡ θ(i, j), (B.1.1b)

with the following mathematical identity

∇iθ(i, j) =ri j

ri jδ(ri j − 4). (B.1.2)

The use of the Heaviside function Θ(ri j − 4) may admit of violating a mathematical strictness

in distribution theory. The N-body distribution function FN(1, · · · , t)(Cercignani, 1988) and the

function Θ(ri j − 4) are both generalized functions. The product of two generalized functions may

not be well-defined in the sense of distribution (e.g., Griffel, 2002). On the one hand, one can

still find its convenience of exploiting the Heaviside function to derive the (Cercignani, 1972)’s

hierarchy below.

First, one needs to remove the volume inside the Landau sphere from the Liouville dynamics.

We define an s-tuple truncated DF for stars1

f 4s (1, · · · , s, t) =N!

(N − s)!

∫Ωs+1,N

FN(1, · · · ,N, t) ds+1 · · · dN , (B.1.3)

where 1 ≤ s ≤ N − 1. We consider the effective interaction range 4 as the conventional Landau

1Cercignani (1972, 1988) used the s-body (symmetric) joint-probability DF and the Boltzmann-Grad limit (N42 →

O(1) as N → ∞), meaning the small number s in the factor N!(N−s)! is not important. On the one hand, stellar dynamics

needs to include the small s to discuss the granularity in the formulation. Accordingly, the formulation shown in thepresent work is slightly different from the Cercignanni’s work due to the definition of DF.

220


radius ro throughout the present work. Equation (B.1.3) is essentially the same as the definition

for the truncated DF used in (Cercignani, 1972). It can be rewritten in a reduced form due to the

symmetry in the s-body DFs about the permutation between two phase-space states of stars. The

domain of the integration in equation (B.1.3) must be taken over the limited phase-space volumes

Ωs+1,N defined by

Ωs+1,N =

rs+1,ps+1 · · · rN ,pN

∣∣∣∣∣∣ N∏i=s+1

i−1∏j=1

| ri − r j |> 4

. (B.1.4)

For example,

Ω2,2 =

(r2,p2

∣∣∣∣| r1 − r2 |> 4), (B.1.5a)

Ω3,3 =

(r3,p3

∣∣∣∣| r1 − r3 |> 4 × | r2 − r3 |> 4). (B.1.5b)

The fundamental idea is to deprive the Landau spheres around stars from the Liouville equation

(A.1.3) and reduce the equation in the form of the s-tuple DF (equation (B.1.3)). Also, we define

the following term

Is ≡

∫Ωs+1,N

ds+1 · · · dNS (1, · · · ,N, t), (B.1.6)

where S (1, · · · ,N, t) is any function of arguments 1, · · · ,N, t and is to be replaced by the term(s)

in equation (B.1.3) later. Considering the domain of the integral for the truncated DF (equation

221


(B.1.4)), one may explicitly express the term as follows

Is =

∫ds+1 θ(s+1,1) · · · θ(s+1,s)

×

∫ds+2 θ(s+2,1) · · · θ(s+2,s) θ(s+2,s+1)

......

.... . .

×

∫dN−1 θ(N−1,1) · · · θ(N−1,s)θ(N−1,s+1) · · · θ(N−1,N−2)

×

∫dN θ(N,1) · · · θ(N,s) θ(N,s+1) · · · θ(N,N−2)θ(N,N−1)

× S (1, · · · ,N, t). (B.1.7)

B.1.1 Truncated integral over the terms∑N

i=1 vi · ∇iFN(1, · · · ,N, t)

For the function S (1, · · · ,N, t) =∑N

i=1 vi · ∇iFN(1, · · · ,N, t) (the dynamical terms in the Li-

ouville equation), the pattern of subscripts of the distance ri j in equation (B.1.7) is simple. The

number 1 ≤ i ≤ s appears only as the first letter in the subscript. Hence one may separate the

summation in the function S (1, · · · ,N, t) into Case 1 (1 ≤ i ≤ s) and Case 2 (s + 1 ≤ i ≤ N).

Case 1: 1 ≤ i ≤ s

The goal here is to reduce the term Is associated with the terms vi ·∇iF4s (1, · · · , s, t) by repeating

the integral-by-parts method. For the numbers 1 ≤ i ≤ s, define the following term

I(1:s)s ≡

s∑i=1

∫Ωs+1,N

ds+1 · · · dNvi · ∇iFN(1, · · ·N, t). (B.1.8)

222


Employing equation (B.1.2), one obtains

I(1:s)s =

s∑i=1

vi · ∇iF4s (1, · · · , s, t)

−

s∑i=1

vi ·

N∑j=s+1

∫dN

∫dN−1 · · ·

∫d j · · ·

∫ds+2

∫ds+1

× θ(s+1,1) · · · θ(s+1,i) · · · θ(s+1,s)

× θ(s+2,1) · · · θ(s+2,i) · · · θ(s+2,s) θ(s+2,s+1)

.... . .

× θ( j,1) · · ·ri j

ri jδ( j,i) · · · θ( j,s) · · · θ( j, j−1)

.... . .

× θ(N−1,1) · · · θ(N−1,i) · · · θ(N−1,s) · · · θ(N−1,N−2)

× θ(N,1) · · · θ(N,i) · · · θ(N,s) · · · θ(N,N−1)

× FN(1, · · · ,N, t), (B.1.9)

where δ( j,i) ≡ δ(ri j −4). Due to the delta function δ( j,i), one can convert the volume integral into the

surface integral

∫d jθ( j,1) · · ·

ri j

ri jδ( j,i) · · · θ( j, j−1) =

∫d3p j

∮dσi j, (B.1.10)

where the dσi j is the surface element of a sphere of the radius 4 with a radial unit vector ri j

ri jaround

the position r j. Employing equation (B.1.10), one obtains

I(1:s)s =

s∑i=1

vi · ∇iF4s −N∑

j=s+1

∫d3p j

∮vi · dσi jF4s+1(1, · · · , s + 1, t)

. (B.1.11)

223


Case 2: s + 1 ≤ i ≤ N

Define the term Is associated with the numbers s + 1 ≤ i ≤ N;

I(s+1:N)s ≡

N∑i=s+1

∫Ωs+1,N

ds+1 · · · dNvi · ∇iFN(1, · · ·N, t). (B.1.12)

To reduce the term I(s+1:N)s , one must modify equation (B.1.9) as follows

I(s+1:N)s = −

N∑i=s+1

s∑j=1

∫d3pi

∮vi · dσi jF4s+1(1, · · · , s + 1, t)

+

N∑i=s+1

∫vi · ∇i

N∏i=s+1

i−1∏j=1

θ( j,i)FN(1, · · ·N, t)

ds+1 · · · dN

−

N∑i=s+1

vi ·

N∑j=s+1

×

∫dN

∫dN−1 · · ·

∫di · · ·

∫d j · · ·

∫ds+2

∫ds+1

× θ(s+1,1) · · · θ(s+1,s)

× θ(s+2,1) · · · θ(s+2,s)θ(s+2,s+1)

......

.... . .

× θ(i+1,1) · · · θ(i+1,s)θ(i+1,s+1) · · · θ(i+1,i)

......

......

. . .

× θ( j,1) · · · θ( j,s) θ( j,s+1) · · ·ri j

ri jδ( j,i) · · · θ( j, j−1)

......

......

.... . .

× θ(N,1) · · · θ(N,s)θ(N,s+1) · · · θ(N,i) · · · θ(N, j−1) · · · θ(N,N−1)

× FN(1, · · · ,N, t), (B.1.13)

224


where the first summation of the terms is obtained in the same way as done for equation (B.1.11).

However, this time we differentiated the functions of the displacement vector r j (associated with

the latter subscript j in the distance ri j). The second summation of the terms on the right-hand-side

in equation (B.1.13) vanishes if one assumes the function Fi(1, · · · , i, t) approaches zero rapidly at

the surfaces of the integral domain. Since the delta function in the third summation of the terms

links the two volume integrals to a surface integral, one obtains

I(s+1:N)s =

s∑i=1

N∑j=s+1

∫d3p j

∮v j · dσi jF4s+1(1, · · · , s + 1, t)

+

N∑i=s+1

N∑j=s+1

∫di

∫d3p j

∮v j · dσi j F4s+2(1, · · · , s + 2, t), (B.1.14)

where the following relation is employed

∫di

∫d j θ( j,1) · · ·

ri j

ri jδ( j,i) · · · θ( j, j−1) =

∫di

∫d3p j

∮v j · dσi j. (B.1.15)

Combining equation (B.1.14) with the result of case 1 (1 ≤ i ≤ s) and making use of the dummy

integral variables, one obtains

Is =

s∑i=1

vi · ∇iF4s +

s∑i=1

(N − s)[∫

d3vs+1

F4s+1vi,s+1 · dσi,s+1

]+

(N − s)(N − s − 1)2

∫d3vs+2

∫ds+1 ×

F4s+2vs+1,s+2 · dσs+1,s+2, (B.1.16)

where vi j = vi − v j. Lastly, we must deprive only the configuration space inside the Landau sphere

for the truncated DF from the Liouville equation (A.1.3). Hence, the rest of the treatment for

the other terms in the Liouville equation is the same as that for standard BBGKY hierarchy (e.g.,

Lifshitz and Pitaevskii, 1981; Saslaw, 1985; McQuarrie, 2000). The Liouville equation results in

225


the following equation in terms of s-tuple DFs

∂t fs +

s∑i=1

vi · ∇i +

s∑j=1(,i)

ai j · ∂i

fs +

s∑i=1

[∂i ·

∫Ωs+1,s+1

fs+1ai,s+1 ds+1

]

=

s∑i=1

[∫d3vs+1

fs+1vi,s+1 · dσi,s+1

]+

12

∫d3vs+2

∫ds+1

fs+2vs+1,s+2 · dσs+1,s+2,

(B.1.17)

where σi j is the normal surface vector perpendicular to the surface of the Landau sphere spanned

by the radial vector 4(ri − r j)/ri j around from the position r j and the surface integral

is taken

over the surface components dσi j. Also, in equation (B.1.17), the following approximation was

made according to (Ito, 2018a) to hold the conservation of total number and energy

f 41 (1, t) = f1(1, t) + O(1/N2), (B.1.18a)

f 42 (1, 2, t) = f2(1, 2, t) + O(1/N). (B.1.18b)

Hence, the truncated s-tuple DFs of stars may be treated as the standard DFs, equations (A.2.5a)

and (A.2.5b). The left-hand side of equation (B.1.17) is identical to the standard BBGKY hierarchy

except for the truncated DF, while the two terms on the right-hand side describe the effects of the

stars entering or leaving the surface of the Landau sphere. Those two extra terms may turn into

collisional terms to describe inelastic collisions.2 Equation (B.1.17) is a generalization of the

BBGKY hierarchy derived in (Gilbert, 1968, 1971). The latter is the most rigorous derivation in

the statistical dynamics of star clusters up to date. Hence, due to the flexibility and rigorousness,

equation (B.1.17) would be one of the proper BBGKY hierarchies to model the evolution of star

clusters.2One may recognize the form of the second integral term that can be used to describe the effect of non-ideal

encounters and inelastic direct physical collision (the loss of kinetic energy due to tidal effects, coalescence andstellar mass evaporation) between two finite-size stars (Quinlan and Shapiro, 1987) modeled by the Smoluchowski-coagulation collision term if the system has a continuous mass distribution.

226


B.2 BBGKY hierarchy for the weakly-coupled DF

We may consider the lower-limit of interaction radius in encounters as the conventional Landau

radius ro if one neglects the effect of the strong encounters. In the present appendix, we extend the

hard-sphere DF introduced by Cercignani (1972) to the ‘weakly-coupled’ DF of stars (Appendix

B.2.1 ). We also discuss the condition to employ the introduced DF and its approximated form for

the BBGKY hierarchy, equation (B.1.17), in Appendices B.2.2 and B.2.3.

B.2.1 Weakly-coupled DF

Originally Cercignani (1972) extended the Grad (1958)’s truncated DF into the hard-sphere DF to

derive the collisional Boltzmann equation for rarefied gases of hard-sphere particles. The hard-

sphere model does not allow any particles of radius 4 exist inside the other particles of radius 4 in

a rarefied gas. The N-body DF is defined as

f NN (1, · · · ,N, t) =

fN(1, · · · ,N, t), (if ri j ≥ 4with i , j)

0, (otherwise)(B.2.1)

Following the definition of single- and double- truncated DFs in (Grad, 1958), the first two s-tuple

hard-sphere DFs explicitly read

f N1 (1, t) = f1(1, t), (B.2.2a)

f N2 (1, 2, t) =

f2(1, 2, t), if r12 ≥ 4

0, otherwise(B.2.2b)

In equation (B.2.2b), the hard-sphere double DF is smooth and continuous, well-defined in the

limit of r12 → 4+, while it can be discontinuous in the limit of r12 → 4

−. Hence, the value of the

227


double DF at the radius r12 = 4 is defined as the limit value

[f2(1, 2, t)

]r12=4 = lim

r12→4+f N2 (1, 2, t). (B.2.3)

Assume a strong constraint on a star cluster that a star can not approach any other stars in space

closer than the Landau radius. Then, the weak-coupling approximation may be embodied, like

equation (A.3.6);

ro < r12 ≤ R, (B.2.4)

O

(1N

)O(1),

where the system size bounds the maximum separation between stars. This ideal mathematical

condition is interpreted as a special case of the hard-sphere DF (equation (B.2.1)) in the limit value

of zero for f (1, 2, t) at r12 = 4

[f2(1, 2, t)

]r12=4 = 0. (B.2.5)

228


The corresponding definition for the Heaviside function is uniquely determined3 as follows

Θ(ri j − 4) ≡

1, (ri j > 4)

0. (ri j ≤ 4)(B.2.7)

We call the hard-sphere DF with the condition (equation (B.2.5)) the ’weakly-coupled DF’ in the

present work to isolate itself from the hard-sphere DF. The weakly-coupled DF corresponds with

the ’Rough approximation (Takase, 1950)’ of the random factor for the Holtsmark distribution

of Newtonian force strength. This means that we may neglect the relative velocity dependence

between test- and a field- star when test star enters the Landau sphere.

B.2.2 Truncation Condition for the weakly-coupled DF

The definition for the weakly-coupled DF gives the thresh point (r12 = 4) a physical causality

in space. The latter means that a direct collision between two spheres occurs only from the outside

of each sphere. One must carefully deal with the explicit form of the double or higher order of

3Although the present work relies on the formulation based on a Heaviside function and its derivatives, at least tohold the Leibniz rule, one needs to employ the following definition

Θ(ri j − 4) ≡

1, (ri j > 4)12 , (ri j = 4)0. (ri j < 4)

(B.2.6)

This formulation is significant only to hold the surface integral terms when one assumes that equation (B.2.4) iscorrect. Accordingly, the use of equation (B.2.7) prohibits us from taking a derivative of any product of identical stepfunctions with respect to r1, r2, and r12. This may be possible at the BBGKY-hierarchy level since the s-tuple DFs,including correlation functions, are linearly independent.

229


s-tuple hard-sphere DF. The double DF may be explicitly defined as

f N2 (1, 2, t) =

(1 − 1

N

)f (1, t) f (2, t) + g(1, 2, t), (if r12 ≥ 4)

0, (otherwise)(B.2.8a)

≡

(1 −

1N

) [f (1, t) f (2, t)

]r12≥4

+ g(1, 2, t)r12≥4, (B.2.8b)

where the DFs f (1, t) and f (2, t) are not exactly statistically uncorrelated. This is since we must

consider the geometrical condition assigned on the interaction range, r12 > 4. Only the DFs f N(1, t)

and f N(2, t) are statistically independent of each other. Hence,

[f (1, t) f (2, t)

]r12≥4, f N(1, t) f N(2, t). (B.2.9)

Also, the hard-sphere DF does not have a domain itself in the Landau sphere. To specify the

explicit form of the DFs, one may employ the following forms

f N(1, 2, t) ≡ Θ(r12 − 4) f (1, 2, t), (B.2.10a)

f N(1, 2, 3, t) ≡ Θ(r12 − 4)Θ(r13 − 4)Θ(r23 − 4) f (1, 2, 3, t). (B.2.10b)

For the sake of a self-consistent relation, the total number is

∫r12>4

f N(1, 2, t) d2 = (N − 1) f (1, t)(1 −

1N

∫r12<4

f (2, t) d2). (B.2.11)

To hold the consistent relationship between DF f (1, t) and higher orders of the DF, one may con-

sider two cases (i) the approximated form of the DF and (ii) the exact form of the weakly-coupled

DF. The former is explained in Appendix B.2.3. For the latter, refer to (Ito, 2018a,b).

230


B.2.3 The approximated form of the weakly-coupled DF

To employ standard DF f (1, t), one may approximate the second term on the right-hand side of

equation (B.2.11) to

∫r12>4

f N(1, 2, t) d2 = (N − 1) f (1, t) + O

(1N

), (B..12)∫

Θ(r13 − 4)Θ(r23 − 4) f N(1, 2, 3, t) d3 = (N − 2) f (1, 2, t) + O(1), (B..13)

where the second terms on the right-hand sides of the equations are N3 times weaker than the first

term in the order of magnitudes. This implies that one may keep employing the weakly-coupled

DF until the density of the system concerned reaches up to N2n, where n is the (initial) mean

density of the system in the relaxation evolution. Also, one may employ a standard definition for

the total number of stars(N =

∫f (1, t) d1.

).

If one neglects the effect of strong encounter following the conventional ideas in stellar dy-

namics, the contributions from the surface integral vanish. The two terms on the right-hand side

of equation (B.1.17) vanish. This is since any star does no exist inside the Landau sphere, i.e.,

equations (B.2.4) and (B.2.5) are valid. Hence, the BBGKY hierarchy for the weakly-coupled DF

is

∂t f Ns +

s∑i=1

vi · ∇i +

s∑j=1(,i)

ai j · ∂i

f Ns +

s∑i=1

[∂i ·

∫Ωs+1,s+1

f Ns+1ai,s+1 ds+1

]= 0. (B.0.14)

The use of the weakly-coupled DF means that we must consider the caveats. The weakly-coupled

DF misses counting the effects from a few stars traveling inside the Landau sphere. It also under-

estimates the effect of strong encounters. On the one hand, the weakly-coupled DF is based on

the first principle and the strict realization of the accepted assumption made for stellar dynamics

(Chandrasekhar, 1943; Rosenbluth et al., 1957).

231

Appendix C

The derivations of the Coulomb logarithm

and an FP equation

Appendix C.1 derives a basic kinetic equation for modeling the evolution of star clusters in

which stars can not approach closer than the conventional Landau radius ro. The corresponding

Coulomb logarithm and FP equation are derived in Appendix C.2.

C.1 The derivation of the basic kinetic equation

Assume that the weakly-coupled DFs for stars can model a star cluster at the early stage of

relaxation evolution. The first two equations of the hierarchy (equation (B.0.14)) for DF f1(1, t)

and the first equation of the hierarchy for DF f1(2, t) respectively read

(∂t + v1 · ∇1) f1(1, t) = −∂1 ·

∫Ω2,2

f2(1, 2, t)a12 d2, (C.1.1a)

(∂t + v1 · ∇1 + v2 · ∇2 + a12 · ∂12) f2(1, 2, t) = −

∫Ω3,3

[a1,3 · ∂1 + a2,3 · ∂2

]f3(1, 2, 3, t) d3, (C.1.1b)

(∂t + v2 · ∇2) f1(2, t) = −∂2 ·

∫r23>4

f2(2, 3, t)a23 d3, (C.1.1c)

232

APPENDIX C. COULOMB LOGARITHM AND FP EQUATION

where ∂12 = ∂1 − ∂2 and the domains of DFs and the accelerations are defined only at distances

ri j > 4. To simplify equations (C.1.1a) and (C.1.1b), one may employ the correlation formulations

(equations (A.2.4), (A.2.5a), (A.2.5b), and (A.2.6)).

By assuming the system is not gravitationally-polarizable, one obtains

(∂t + v1 · ∇1 + A(2,2)

1 · ∂1

)f (1, t) = −∂1 ·

∫Ω2,2

g(1, 2, t)a12 d2, (C.1.2a)(∂t + v1 · ∇1 + v2 · ∇2 + A(3,3)

1 · ∂1 + A(3,3)2 · ∂2

)gN(1, 2, t)

=

(−a12 · ∂12 +

1N

A(3,3)1 · ∂1 +

1N

A(3,3)2 · ∂2

) [f (1, t) f (2, t)Θ(r12 − 4)

]−

(1 −

1N

) [A(2,2)

1 − A(3,3)1

]· ∂1

[f (1, t) f (2, t)Θ(r12 − 4)

]−

(1 −

1N

) [A(2,2)

2 − A(3,3)2

]· ∂2

[f (1, t) f (2, t)Θ(r12 − 4)

]− ∂1 ·

(∫Ω3,3

a13g(1, 3, t) d3 −

∫Ω2,2

a13g(1, 3, t) d3

)f (2, t)

− ∂2 ·

(∫Ω3,3

a23g(2, 3, t) d3 −

∫Ω2,2

a23g(2, 3, t) d3

)f (1, t), (C.1.2b)

where the lowest OoMs of the terms are left with O(1).1 The truncated m.f. accelerations are

defined as

A(2,2)i (ri, t) =

[1 −

1N

] ∫ri3>4

f (3, t)ai3 d3, (i, j = 1, 2) (C.1.3a)

A(3,3)i (ri, r j, t) =

[1 −

1N

] ∫Ω3,3

f (3, t)ai3 d3. (C.1.3b)

The last six terms on the right-hand side of equation (C.1.2b) may be simplified by neglecting the

1One may realize that the lowest order at the kinetic equation level is ∼ O(1/N2) due to the truncated accelerationA(2,2)/N to hold the self-consistency of the kinetic equation.

233


existence of the third star in the two-body encounter between two stars of concern;

Ω2,2 ≈ Ω3,3. (C.1.4)

This is possible since the truncated phase-space volume of the DF for the third star contributes to

equation (C.1.2b) only as a margin of error with the order of O(1/N2). This corresponds to

∫Ω2,2

· d3 ≈

∫Ω3,3

· d3 + O(1/N2

). (C.1.5)

Hence, equation (C.1.2b) simply reduces to

(∂t + v1 · ∇1 + v2 · ∇2 + A(2,2)

1 · ∂1 + A(2,2)2 · ∂2

)g(1, 2, t) = −

[a412 · ∂1 + a421 · ∂2

]f (1, t) f (2, t),

(C.1.6)

where

a412 = a12 −1N

A(2,2)1 , (C.1.7a)

a421 = a21 −1N

A(2,2)2 . (C.1.7b)

Employing the method of characteristics, one can solve equation (C.1.6) for the correlation func-

tion

gN(1, 2, t) = gN(1(t − τ), 2(t − τ), t − τ)

−

∫ t

t−τ

[a412 · ∂1 + a421 · ∂2

]t=t′ f

(1(t′), t′

)f(2(t′), t′

)Θ(r12(t′) − 4) dt′. (C.1.8)

In the scenario for the generalized-Landau (g-Landau) equation in (Kandrup, 1981), all the

stars in a star cluster are perfectly uncorrelated at the beginning of correlation time t − τ, implying

234


that the destructive term g(1(t − τ), 2(t − τ), t − τ) vanishes at the two-body DF level. To apply

the same simplification to the secular evolution of the system of concern, one must necessarily

consider the memory effect. The effect is essential if the time duration between encounters is

comparable to the correlation-time scale. However, the memory effect may be no significant in

stellar dynamics due to the violent relaxation, mixing, short-range two-body encounters, spatial

inhomogeneities, and anisotropy (e.g., Saslaw, 1985, pg. 34). Hence, the destructive term on the

right-hand side of equation (C.1.8) may vanish. Then, one obtains the g-Landau equation with the

effect of discreteness from equations (C.1.2a) and (C.1.8)

(∂t + v1 · ∇1 + A(2,2)

1 · ∂1

)f (1, t) =

∂1 ·

∫Ω2,2

d2a12

∫ τ

0dτ′

[a412 · ∂1 + a421 · ∂2

]t−τ′ f (1(t − τ′), t − τ′) f (2(t − τ′), t − τ′). (C.1.9)

One may discuss the effect of the retardation in the collision term of equation (C.1.9). Since the

trajectory of test star is characterized by equation (A.4.5), the correlation time would be the order

of the free-fall time of test star under the effect of the m.f. acceleration. On the one hand, the

shortest correlation time scale is longer than the time scale for test star to travel across a Landau

sphere to hold the weak-coupling approximation

O(1/N) < tcor . O(1), (C.1.10)

meaning the non-Markovian effect on the relaxation process is no significant

O(1/N2) <tcor

trel. O(1/N). (C.1.11)

Hence, one may assume the Markovian limit2 of the collision term for the correlation time 0 <

2For the Markovian limit, one should not change the other arguments of the DF in the collision term since thechanges in the momentum and position of test star in an encounter is not negligible due to the effect of m.f. acceleration.

235


τ′ < tcor

f (1(t − τ′), t − τ′) f (2(t − τ′), t − τ′) ≈ f (1(t − τ′), t) f (2(t − τ′), t), (C.1.12)

Taking the limit of τ→ ∞, one obtains

(∂t + v1 · ∇1 + A(2,2)

1 · ∂1

)f (1, t)

= ∂1 ·

∫Ω2,2

d2a12

∫ ∞

0dτ′

[a412 · ∂1 + a421 · ∂2

]t−τ′ f (1(t − τ′), t) f (2(t − τ′), t). (C.1.13)

Employing the anti-normalization condition (equation (A.2.7)), for the correlation function and

taking the limit of 4 → 0, one may retrieve the Kandrup (1981)’s g-Landau equation

(∂t + v1 · ∇1 + A1 · ∂1) f (1, t)

= ∂1 ·

∫d2a12

∫ ∞

0dτ′ [a12 · ∂1 + a21 · ∂2]t−τ′ f (1(t − τ′), t) f (2(t − τ′), t),

(C.1.14)

where the statistical acceleration can be found in the forms

a12 = a12 −1N

A1, (C.1.15a)

a21 = a21 −1N

A2. (C.1.15b)

The truncated g-Landau equation (C.1.9) is different from the g-Landau equation (C.1.14), not

only in the domain of interaction range but also in the form of physical quantities.

236


C.2 Coulomb Logarithm and Landau (FP) kinetic equation

To derive the Coulomb logarithm and FP kinetic equation, assume that test star follows the

rectilinear motion (equation (A.4.4)). Also, assume that the encounter is local for the truncated

g-Landau collision term in equation (C.1.13). This means that the truncated Landau collision term

reduces to

INL =∂1 ·

∫Ω2,2

a12

∫ ∞

0

[a12

(t − τ′

)]r12(t−τ′)>4 dτ′ d3r12 · ∂12 f (1, t) f (r1,p2, t) d3p2, (C.2.1)

where we neglected the effects of non-ideality, i.e., the retardation and spatial non-locality (See,

e.g., Klimontovich, 1982; Ito, 2018a) for the Landau collision term for simplicity. The Fourier

transform of the acceleration of star 1 due to star 2 at distances r12 > 4 is as follows3

F [a12(r12 > 4)] = −Gm∫

Ω2,2

exp(−ik · r12)r12

r312

d3r12 = −Gm sin[k4]

2iπ2k24k. (C.2.3a)

The same transform must be employed for a12(t−τ′) in equation (C.2.1), but the time of r is fixed to

t−τ′. The corresponding wavenumber must be exploited. Equation (C.2.3a) is essentially the same

as the Fourier transform of the truncated acceleration a12Θ(r12 − 4), meaning the corresponding

acceleration of star 1 is null within the volume of the Landau sphere. This is since the existence

of stars in the Landau sphere is not essential due to the spatial locality. Also, the effect of the

truncation on the DF must be controlled through the truncation of the acceleration. In the limit of

3Fourier transform of the potential φ12 typically done to find the explicit form of the Landau collision term ne-cessitates a ’convergent factor’,e−λr12 , where λ is a vanishing low number to be taken as zero after the Fourier trans-form. The factor can remove singularities of (generalized) functions on complex planes and slow decays of potentialsin three-dimensional spaces (e.g., Adkins, 2013). However, one does not need to employ the factor in the Fouriertransform of the truncated acceleration, a12Θ(r12 > 4), and even in the corresponding inverse Fourier transform,F−1 [F [a12Θ(r12 > 4)]]. Rendering the transform, F−1 [F [a12Θ(r12 > 4)]], is a simple task; hence, it is left forreaders. One will need the following identity to find the step function∫ ∞

0

sin kk

dk =π

2. (C.2.2)

237


4 → 0, equation (C.2.3a) results in a well-known Fourier transform of acceleration or pair-wise

Newtonian potential in star-cluster kinetic theory (See, e.g., Chavanis, 2012, Appendix C)

lim4→0

F [a12] = −Gm

2iπ2kk −−−→× 1−ik

F[φ12

]k. (C.2.4)

After a proper calculation following (Chavanis, 2012, Appendix C), the collisional term results in

INL = ∂1 ·

∫←→T (p12,p2) · ∂12 f (r1,p1, t) f (r1,p2, t) d3p2, (C.2.5a)

←→T ≡ −B

p212←→I − p12p12

p3 , (p12 ≡ p1 − p2) (C.2.5b)

B ≡ 2πGm2∫ ∞

0

sin2[k4]k342 dk. (C.2.5c)

where for expressions of the tensor←→T , typical dyadics are exploited. Following the works (Severne

and Haggerty, 1976; Kandrup, 1981; Chavanis, 2013) if one assumes the cut-offs k ∈ [2π/R, 2π/ro]

at each limit of the integral domain of the collision term, the factor B, equation (C.2.5c), explicitly

reads

B = ln[N] + 1.5 +

∞∑m=1

(−16π2)m

2m(2m)!+ O(1/N2), (C.2.6)

where the following indefinite integral formula (e.g., Zeidler et al., 2004) is employed4

∫cos[αk]

kdk = ln[αk] +

∞∑m=1

(−[αk]2

)m

2m(2m)!. (C.2.7)

The lower limit of the distance r12, the Landau radius, can not remove the logarithmic singularity in

the collision term, as shown in equation (C.2.6). This is since the application of the weak-coupling

approximation to the g-Landau collision term is inconsistent, especially at r12 → ro. To avoid

4A similar calculation for a weakly-nonideal self-gravitating system appears in (Bose and Janaki, 2012). In theirwork, the upper limit is also assigned to the domain of the integration.

238


the singularity associated with high wavenumbers, one needs all the higher orders of the weak-

coupling approximation as the correction to the rectilinear-motion approximation. The trajectory

of test star must follow a pure Newtonian two-body problem (equation (A.4.2)). Since the value

of the parameter ζ in equation (A.3.6) essentially depends on one’s choice, the following ideal

relation is assumed for simplicity

R = Nro. (C.2.8)

The result of the numerical integration of factor B (equation (C.2.5c)) is as follows

B − ln[N] = 1.5 +

∞∑m=1

(−[4π]2

)m

2m(2m)!≈ −3.11435, (C.2.9)

The term ln[N] is the factor included in the relaxation time as the ’Coulomb logarithm.’ Refer to

(Ito, 2018b) for the detailed discussion of the effect of discreteness on the relaxation time and m.f.

potential.

239

Appendix D

Singularity analysis on the Lane-Emden

functions

In the present appendix, we explain the effects of the numerical parameter L on the LE func-

tions (Appendix D.1) and logarithmic functions (Appendix D.2). We also show the Chebyshev

coefficients for the asymptotic approximation of the LE functions (Appendix D.3).

D.1 The effect of the parameter L on the LE functions

The present appendix explains the effectiveness of the parameter L on the asymptotic approx-

imation of the LE functions to increase the decay rate of the Chebyshev coefficients for the func-

tions. For brevity, we only discuss the ILE1 functions; the ILE2 situation is very similar. Based on

equations (4.3.2) and (4.3.3), we have two obvious singularities

(i) Vertical asymptote at ϕ ≈ 0 R ≈ ϕ−1/ω,

(ii) Branch point at ϕ ≈ 1 R ≈√

1 − ϕ.

Clearly, we must regularize the blow-up (i) before we approximate the solution by Chebyshev

240

APPENDIX D. SINGULARITY IN LE FUNCTIONS

polynomials. Also, the branch point (ii) disturbs the geometrical convergence of the spectral co-

efficients. In fact, by adapting the method of (Elliott, 1964) to [0, 1], one easily sees that the

Chebyshev coefficients for√

1 − ϕ decay algebraically like n−2 for high n. One can regularize both

(i) and (ii) by rescaling R by ϕ1/ω and 1/√

1 − ϕ. However, there is also a third, weaker singularity

affecting the derivatives of R

(iii) Logarithmic derivative at ϕ ≈ 0 R/Rs − 1 ≈ ϕηi cos (ωi lnϕ + C2i).

This singularity also leads to an algebraic decay of the Chebyshev coefficients, but computing the

rate of decay for (iii) is more complicated than (ii). Appendix D.3 presents the Chebyshev coef-

ficients for Ra/Rs − 1 based on the method of steepest ascent. It also shows a direct numerical

calculation of the corresponding Chebyshev coefficients an. Both support the conclusion that the

coefficients an decay like n−2ηi−1 at their leading order. To increase the decay rate of the Cheby-

shev coefficients for (iii), we employ the transformation from (0, 1] to itself

ϕo = ϕ1/L (D.1.1)

where L is a mapping parameter. This transformation is inspired by the fact that power-law map-

pings work well for end-point logarithmic singularities (Boyd, 1989) (See Appendix D.2 for a brief

discussion.) For the sake of illustration, we calculated the coefficients for Ra/Rs using a Cheby-

shev interpolatory quadrature with Chebyshev-Gauss nodes. Figure D.1 shows that greater m and

L lead to more rapid decays of an, and push the “cross-over” point (Boyd, 1989, 2013) toward

higher n. In this sense, the mapping parameter L is the analog of the ”map parameter” for Rational

Chebyshev expansions.

241


0 200 400 600 800 1,00010−16

10−12

10−8

10−4

100

104

n

|a n|


0 200 400 600 800 1,00010−16

10−13

10−10

10−7

10−4

10−1

102

n

|a n|


Figure D.1: Absolute values of the Chebyshev coefficients for Ra/Rs with various m and L values.For (m, L) = (20, 1) or (6, 30) the plots flatten out at high n due to the round-off error; for otherchoices of (m, L) the coefficients decay slowly due to the end-point singularity.

242


D.2 The asymptotics of Chebyshev coefficients in the presence

of logarithmic endpoint singularities

Boyd (1989) discussed the asymptotic Chebyshev coefficients for a class of function with

logarithmic endpoint singularities

a(1 + x)k ln(1 + x) ± b(1 − x)k ln(1 − x), x ∈ (−1, 1) (D.2.1)

where a and b are constants, and k is a real number. In particular, Boyd (1989) analyzed the effect

on the coefficients of two possible mappings

x = sinπy2, (Quadratic Mapping) (D.2.2a)

x = tanhLmy√1 − y2

, (Hyperbolic-Tangent mapping) (D.2.2b)

where x, y ∈ (−1, 1), and Lm is a map parameter. Here, we would like to improve those mapping

to increase the convergence rate of Chebyshev coefficients. The easiest approach is to extend the

quadratic mapping to a power-law mapping by the method of a domain decomposition

x = −1 + (1 + xo)(1 − y

2

)LL

− 1 < x < xo (D.2.3a)

x = 1 + (−1 + xo)(1 + y

2

)LR

xo < x < 1 (D.2.3b)

where xo is the patching point, and LL and LR are free parameters such that LL = LR = 2 gives

the quadratic mapping. This transformation was introduced by MacLeod (1992), who was able to

successfully handle the strong singularities of the Thomas-Fermi function at both endpoints of a

semi-infinite domain. However, the LE functions have a logarithmic-type singularity only at one

243


endpoint. Hence, we apply a one-sided change of coordinates

x = 1 − 2(1 − y

2

)L

, (D.2.4)

where L is a free parameter.

244


100 101 10210−19

10−14

10−9

10−4

101

106

n

|a n|

sin mappingtanh mapping (Lm = 3)power mapping L = 8

100 101 10210−19

10−14

10−9

10−4

101

106

n

|a n|

sin mappingtanh mapping (Lm = 6)power mapping L = 30

Figure D.2: Absolute values of the Chebyshev coefficients for the function (1 − x)k ln (1 − x)mapped by x = sin πy

2 , x = tanh Lmy√1−y2

, and x = 1 − 2(

1−y2

)L. The parameter values are k = 2, L = 8

and Lm = 3 (top) and k = 0.5, L = 30, and Lm = 6 (bottom). For the power mapping the plotflattens out due to round off error when n ≈ 25, while for the tangent hyperbolic mapping machineprecision is reached when n ≈ 110.

Figure D.2(Top panel) shows the results (for quadratic and hyperbolic-tangent mappings) of

(Boyd, 1989) compared with those based on the transformation in equation (D.2.4) with L =8.

245


The power-law mapping is superior for n ≈ 20. Figure D.2(Bottom panel) shows the results for

k < 1. As expected, the quadratic and hyperbolic-tangent mappings fail due to slow convergence

rates and the endpoint singularities. However, the power-law mapping shows a better convergence.

The round-off plateau occurs just after n ≈ 25. For the ILE function on (0, 1], equation (D.2.4)

becomes equation (D.1.1).

D.3 Asymptotic coefficients for a decaying logarithmic

oscillatory function

Following the general approach in (Boyd, 2008b), i.e., the method of steepest ascent (Section

3.3), finding the asymptotic values of the Chebyshev coefficients for ϕηi cos (ωi lnϕ) is a tedious

task. We show only the result. As n→ ∞,

an ≈ (−1)n Ai

n2ηi+1 cos Θn, (D.3.1)

where

Ai =

√4πγ2ηi+

12 exp

[−2ηi + 2ωi arctan

(ηi

ωi

)], (D.3.2a)

Θn = −4ηi + 1

2arctan

[ηi

ωi

]−

14π + 2ωi

[ln

(γ

n

)− 1

], (D.3.2b)

γ =

√η2

i + ω2i . (D.3.2c)

The values in equation (D.3.1) are plotted in Figure D.3. The figure also shows the absolute

deviation from the values obtained by Gauss-Chebyshev interpolatory quadrature. For 5 < m < 40,

the asymptotic coefficients match the numerical solution, but for m > 45 they do not because

the coefficients rapidly reach machine precision before reaching high n at which the asymptotic

246


formula applies.

0 200 400 600 800 1,00010−10

10−6

10−2

102

n

|a n|

steepest descentinterpolatory quadrature

difference

0 100 200 300 40010−18

10−11

10−4

103

n

|a n|


difference

0 50 100 150 200 250 300 350 40010−18

10−11

10−4

103

n

|a n|


difference

Figure D.3: Asymptotic absolute values of the Chebyshev coefficients for ϕηi cos (ωi lnϕ). Thesolid curves show the differences between the values of the coefficients based on the steepestdecent (dashed-curves) and the interpolatory quadrature (filled circles). The parameters are L = 1and m = 6 (top), m = 20 (middle) and m = 45 (bottom).

247

Appendix E

The Milne forms of the Lane-Emden

equations

The LE1- and LE2- equations are homologically invariant under certain transformations. Al-

though the transformations of functions and change of variables can be done in infinitely many

ways, we introduce the Milne variables (one of the most-often employed variables for the Lane-

Emden equations) for the LE1 function in Appendix E.1 and for the LE2 function in Appendix

E.2.

E.1 The Milne form of the LE1 equation

The LE1 equation for m > 5 is invariant under the transformation (Chandrasekhar, 1939)

φ(r)→ A−ωφ(Ar), (E.1.1)

where A is a constant and ω = 2m−1 . Therefore, the second-order differential equation (2.1.21)

reduces to a first-order equation by introducing homologically invariant variables. One transfor-

248

APPENDIX E. MILNE VARIABLES FOR LE FUNCTIONS

mation studied by Chandrasekhar (1939) uses the Milne variables

u = −r φm

φ′, (E.1.2a)

v = −r φ′

φ. (E.1.2b)

These variables reduce the LE1 equation to the Milne form

dudv

= −vu

u + v − 1u + mv − 3

. (E.1.3)

Chandrasekhar (1939) used a geometrical method in the (u, v) plane. For the boundary conditions

φ(r = 0) = 1 and φ′(r = 0) = 0, the solution of the Milne equation starts at (uE, vE) = (3, 0), and

reaches asymptotically the singular point (us.p, vs.p) =(

m−3m−1 , ω

)(See Figure 7.6). Such solutions are

called a homologous family, or the E-solutions.

E.2 The Milne form of the LE2 equation

The LE2 equation is invariant under the transformation for the dimensionless m.f. potential

and radius

φ(r)→ φ(Ar) + 2 ln A, (E.2.1)

where A is constant and the corresponding transformation for the density and radius is

ρ(r)→ A−2ρ(rA). (E.2.2)

Hence, the LE2 equation reduces to the Milne form

uv

= −dudv

u − 1u + v − 3

, (E.2.3)

249

APPENDIX E. MILNE VARIABLES FOR LE FUNCTIONS

where

u =r e−φ

φ′= −

r ρ2

ρ′, (E.2.4a)

v = r φ′ = −r ρ′

ρ. (E.2.4b)

In the (u, v) plane, the E-solutions start at (uE, vE) = (3, 0), and approach asymptotically the singu-

lar point (us.p, vs.p) = (1, 2) (See Figure 6.4).

250

Appendix F

Numerical Stability in the integration of the

ss-OAFP system

Appendixes F.1 and F.2 discuss the numerical stability of the whole-domain and truncated-domain

solutions. Appendix F.3 shows the numerical results for the ss-OAFP system with fixed variables.

F.1 The stability analyses of the whole-domain solution

The present appendix details how the whole-domain solution depends on the eigenvalue β

(Appendix F.1.1), the boundary condition for 3F(x) (Appendix F.1.2), and the numerical parameter

L (Appendix F.1.3).

F.1.1 The stability of the whole-domain solution against β

We solved the ss-OAFP system with L = 1, FBC = 1, and N = 70 for different β on the

whole domain. Figure F.1 depicts the β-dependence of the values of 3I(x = 1), | 1 − c1/c1o |, and

| 1 − c∗4/c∗4o |. The figure provides the following approximate relationship in the order of values on

251

APPENDIX F. NUMERICAL STABILITY OF THE SS-OAFP SYSTEM

a broad range of β-values

| 1 − c1/c1o | ∼ | 1 − β/βo | ∼3I(x = 1)

102 ∼| 1 − c∗4/c

∗4o |

105 . (F.1.1)

Equation (F.1.1) implies that when one finds 5 ∼ 6 significant figures of β and c1, one can find only

one significant figure of c∗4.

10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−710−14

10−11

10−8

10−5

10−2

101

| ∆β/βo |

3I(x = 1) (∆β < 0) | 1 − c1/c1o | (∆β < 0)| 1 − c∗4/c

∗4o | (∆β < 0) 3I(x = 1) (∆β > 0)

| 1 − c1/c1o | (∆β > 0) | 1 − c∗4/c∗4o | (∆β > 0)

Figure F.1: Values of 3I(x = 1), | 1 − c1/c1o |, and | 1 − c∗4/c∗4o | against change ∆β in eigenvalue

β around the reference value βo (∆β ≡ β − βo). Numerical parameters are N = 70, L = 1, andFBC = 1.

F.1.2 The stability of the whole-domain solution against FBC

We solved the ss-OAFP system on the whole-domain with L = 1, β = βo, and N = 70

for different FBC. We found that the eigenvalues c1 and c∗4 appeared proportional to FBC while

β and c3 are stable to fifteen significant figures. However, the condition number of the Jacobian

matrix for the 4ODEs and Q-integral reached ∼ 1012 for high FBC(∼ 104). To avoid the high

condition number, we regularized the ss-OAFP system by dividing the function F(E) by FBC. We

again solved the regularized ss-OAFP system, and Figure F.2 shows the results. The condition

number does not change significantly against FBC. Also, we confirmed the eigenvalues c1 and252


c2 are proportional to FBC with high accuracies. Figure F.2 implies the following relationships

between FBC and the eigenvalues

c1(FBC) ∝(c1o + O

(10−16

))FBC, (F.1.2a)

c∗4(FBC) ∝(c∗4o + O

(10−10

))FBC, (F.1.2b)

c3 = c3o + O(10−16), (F.1.2c)

β(FBC) = βo + O(10−16), (F.1.2d)

where −10−4 < FBC < 104.

10−4 10−2 100 102 104106

107

108

109

1010

1011

FBC

cond

ition

num

ber

10−810−610−410−2100 102 10410−16

10−13

10−10

10−7

10−4

FBC

| 1 − c1/FBC/c1o |

| 1 − c∗4/FBC/c∗4o |

Figure F.2: (Left panel) Condition number of the Jacobian matrix for the Q-integral and 4ODEsregularized through F(E)/FBC. (Right panel) Relative deviations of the regularized eigenvaluesc1/FBC and c∗4/FBC from the reference eigenvalues c1o and c∗4o for the 4ODEs regularized throughF(E)/FBC. (FBC = 1 , L = 1, and N = 70.)

F.1.3 The stability of the whole-domain solution against L

We solved the ss-OAFP system with FBC = 1, β = βo, and N = 70 for different L. We found

spectral solutions for L = 1/2 and L = 3/4 (Table F.1), while the Newton method for L > 1

was hard to work. Hence, we call a solution of the ss-OAFP system for L(< 1) the ’contracted-

253


L = 0.75

N Eigenvalue β | 1 − c1/c1o | | 1 − c∗4/c∗4o | 3I(x = 1)

60 8.178371160 2.1 × 10−10 6.6 × 10−7 9.4 × 10−9

55 8.178371160 2.1 × 10−10 1.3 × 10−6 1.7 × 10−9

50 8.178371160 8.0 × 10−10 9.8 × 10−6 1.3 × 10−9

L = 0.5

N Eigenvalue β | 1 − c1/c1o | | 1 − c∗4/c∗4o | 3I(x = 1)

35 8.17837104 4.9 × 10−8 2.3 × 10−6 2.1 × 10−10

Table F.1: Numerical results for the contracted-domain formulation with with L = 1/2 and L = 3/4for FBC = 1.

domain’ solution. The contracted-domain solutions are still whole-domain solutions while they

need lower degrees of polynomials. The convergence rate of Chebyshev coefficients for ‘large n’

is like an ∝(

1n

)1+2L. The Newton method converged only when the Chebyshev coefficients reach

as low as of the order of 10−9 for L = 3/4 and 10−6 for L = 1/2. Also, the Newton method worked

for L = 1 when the coefficients reach as low as 10−12 (Table 8.2). Hence, we consider the Newton

iteration method works when the Chebyshev coefficients reach as low as of the order of 10−12L.

100 100.5 101 101.5

10−9

10−7

10−5

10−3

10−1

101

L = 3/4

FBC

N = 60N = 65

guide line n−2L−1

100 100.5 101 101.5

10−9

10−7

10−5

10−3

10−1

101

L = 1/2

FBC

N = 35N = 75

guide line n−2L−1

Figure F.3: (Left Panel) Chebyshev coefficients for 3F(x) with L = 0.75 in the following cases(a) N = 60 and β = 8.178371160 and (ii) N = 65 and β = 8.178371275. The iteration methodfor the latter did not work satisfactorily stalling around | anew − aold |≈ 7 × 10−10 (resultingin | 1 − c∗4/c

∗4o |≈ 6.0 × 10−3 | and 3I = 4.4 × 10−6). However, it is shown here for the sake of

comparison. (Right panel) Chebyshev coefficients for 3F(x) for L = 0.5 in the following cases (a)N = 35 and β = 8.178371160, and (ii) N = 50 and β = 8.1783712.

254


c d | 1 − c1/c1ex | | 1 − c∗4/c∗4ex | 3I(x = 1)

∞ N/A 1.2 × 10−13 5.3 × 10−10 1.0 × 10−11

10 1 1.2 × 10−13 5.3 × 10−10 1.0 × 10−11

5 1 7.7 × 10−14 5.3 × 10−10 1.0 × 10−11

1 1 1.0 × 10−13 4.5 × 10−10 9.8 × 10−12

0.1 1 1.6 × 10−13 9.6 × 10−10 1.2 × 10−11

0.01 1 1.4 × 10−13 9.1 × 10−10 1.1 × 10−11

c d | 1 − c1/c1ex | | 1 − c∗4/c∗4ex | 3I(x = 1)

10 10 9.4 × 10−14 5.3 × 10−10 1.0 × 10−11

2 10 2.1 × 10−14 3.1 × 10−11 7.5 × 10−12

1 5 4.0 × 10−14 2.6 × 10−10 8.8 × 10−12

1 0.1 1.1 × 10−13 8.7 × 10−10 1.1 × 10−11

1 0.01 1.2 × 10−13 9.6 × 10−10 1.1 × 10−11

0.1 0.1 1.2 × 10−13 8.4 × 10−10 1.0 × 10−11

Table F.2: Numerical results for different extrapolated DF (L = 1 and FBC = 1). The eigenvaluesare compared with c1ex ≡ c1o and c∗4ex ≡ 3.03155223 × 10−1(= c∗4o + 1 × 10−1) obtained for (c, d) =

(1, 10) and the value of 3I(x = 1) is 7.3 × 10−12. The combination (c, d) = (∞,N/A) means theextrapolated function is constant.

F.2 The stability of the truncated-domain solution against

extrapolated DF

We examined how the change in the extrapolated DF (equation (4.4.19)) affects the eigenval-

ues of the optimal truncated-domain solution (Table F.2). The set of parameters (c, d) = (1, 10)

provided the best accuracy in the sense that 3I(x) reached the minimum value (∼ 7.3×10−12) among

the chosen parameters. The relative deviation of all the eigenvalues is the order of 10−13 in c1 and

10−9 in c∗4 at most compared with those for (c, d) = (1, 10). Also, all the sets of parameters hold

small values of 3I(x = 1) ≈ 1 × 10−11. Even the effect of discontinuity in the derivative of the

extrapolated DF at E = Emin (c → ∞) is not significant compared with that of large values of

c(= 5, 10). Hence, we conclude that the optimal truncated-domain solution is less sensitive to the

expression of the extrapolated DF.

255


F.3 Solving part of the ss-OAFP system with a fixed

independent variable

The present appendix shows the results of the numerical integration of part of the ss-OAFP

system that we solved with some fixed independent variables. Appendices F.3.1 and F.3.2 show

the effects of discontinuities in independent variables on the convergence rates of Chebyshev co-

efficients for the integration of the Poisson’s equation and Q-integral, respectively.

F.3.1 Q-integral with fixed discontinuous 3R

In the present work, the truncated-domain solutions for low Emax include a certain flattening

in Chebyshev coefficients as index n becomes high. We examined the relationship between the

flattening and discontinuity in dependent variables. First, we calculated the Chebyshev coefficients

for the Q-integral with the discontinuous test function for 3R

3(tes)R = 0.1 Θ

(1 + x − xtrans

2

)+ 1, (F.3.1)

where xtrans is a small positive number and Θ(·) the Heaviside function. When the location of

discontinuity is relatively close to the order of unity, say xtrans = 0.1, the Chebyshev coefficients

for the Q-integral slowly decay like ∼ 1/n2 for large n (Left panel, Figure F.4). This decay rate

is similar to that of the Chebyshev coefficients for discontinuous functions (See Section 3.5). On

the one hand, if the location of the discontinuity is closer to an endpoint of the domain, such as

xtrans = 0.001 (Right panel, Figure F.4), then the coefficients show a flattening with high n. The

majority of the domain for xtrans = 0.001 is covered by a constant function. Hence, one can find a

rapid decay in the coefficients for low n.

256


100 101 102

10−8

10−6

10−4

10−2

100

index n

|Q

n|

xtrans = 0.11/n2

xtrans = 0.001

0.92 0.94 0.96 0.98 10.9

0.95

1

1.05

1.1

1.15

−E|3 R

(E)|

xtrans = 0.1xtrans = 0.001

Figure F.4: Chebyshev Coefficients for the Q-integral with a discontinuous test function 3(tes)R .

Recall E = −(0.5 + 0.5x)L, here L = 1.

F.3.2 Poisson’s equation with fixed discontinuous 3D

The present appendix shows how the modification of the function from 3R to 3(m)R changes

the numerical result. We examined how the dependent variable 3D affected 3(m)R in the Poisson’s

equation. We employed the following test function for 3D

3(tes)D = 3Do(x)

(Atrans Θ

[1 + x − 0.001

2

]+ 1

), (F.3.2)

where Atrans is a small positive number, and 3Do(x) is the regularized density obtained from the

reference solution. We solved the Poisson’s equation with the fixed 3(tes)D for different Atrans. When

the value of Atrans is very small such as 0.00001, the solutions√3

(m)R and exp(3R)

√0.5 + 0.5x (that

are supposed to be the same if the Poisson’s equation is successfully integrated) are compared in

the right panel of Figure F.5. The difference appears only at the order of 10−4. On the one hand,

when Atrans is close to unity such as 0.1, not only the difference appears in the value of coefficients

at the order of 0.1 but also the coefficients for 3(m)R shows a slower decay compared with those for

257


3R (Left Panel, Figure F.5).

100 101 10210−7

10−6

10−5

10−4

10−3

10−2

10−1

100

n

|R

n|

Atrans = 0.1, 3RAtrans = 0.1, 3(m)

R0.4n−1

100 101 10210−7

10−6

10−5

10−4

10−3

10−2

10−1

100

n

|R

n|

Atrans = 10−4, 3RAtrans = 10−4, 3(m)

R5n−2

Figure F.5: Chebyshev Coefficients for√3

(m)R and exp(3R)

√0.5 + 0.5x for a discontinuous test

function 3(tes)D . The dashed guidelines are added for the measure of slow decays.

258

Appendix G

Relationship between the reference- and

Heggie-Stevenson’s- solutions of the

ss-OAFP system

Appendix G.1 shows the numerical results for the ss-OAFP system that we solved after modify-

ing the regularization of variable 3F . This modification provided the reference- and HS’s solutions

with a reasonable accuracy, but still available degrees were limited, like Section 8.3.1. This led us

to apply to the ss-OAFP system both the modified variables employed in Appendix G.1 and Sec-

tion 8.3.1 (Appendix G.2). This double modification reproduces the HS- and referencesolutions

even for a high (∼ 200) degree of polynomials.

G.1 Modifying the regularization of variable 3F

We employed the following modification of 3F

3(m)F = ln

exp[3F][1 + x

2

]β . (G.1.1)

259

APPENDIX G. REFERENCE- AND HS’S SOLUTIONS OF THE SS-OAFP SYSTEM

We solved the ss-OAFP system for 3(m)F and unmodified variables 3S , 3Q, 3G, 3I , 3J, 3R using the pro-

cedure of Chapter 5. Like the modification 3(m)R (Section 8.3.1), the spectral solution is close to

the HS’s solution for high Emax, while it also can be close to the reference solution for low Emax

(See Table G.1 in which(3

(m)F , 3R

)is the corresponding formula). Due to the logarithmic endpoint

singularity of 3(m)F , the Chebyshev coefficients F(m)

n for 3(m)F show slow decays for both high and

low Emax (Figure G.1). A more distinct slow decay appears in the Chebyshev coefficients In for

3I , especially when Emax is high (Figure G.2). The value of 3I(x = 1) is still the order of 10−4 for

high Emax that is the same order as that for the modification 3(m)R in Section 8.3.1. This infers that

the HS’s solution may be obtained when a numerical scheme has a low accuracy, and Emax is low

≈ −0.225. However, the 3(m)F -formulation provides the HS’s solution only for small N.

G.2 Modified variables (3(m)R , 3(m)

F )

The results of Appendix G.1 show that slowing the rapid decay in Chebyshev coefficients is

a key to finding both the HS’s and referencesolutions based on a single formulation. Hence,

we combined the two formulations of Appendix G.1 and Section 8.3.1. We found that the HS-

and reference solutions only by controlling the value of Emax based on the double modification(3

(m)R , 3(m)

F

). This modification provided the Chebyshev coefficients for the solutions that can reach

a high degree, such as N = 200 for Emax = −0.225. On the one hand, it also provided a spectral

solution close to the reference solution with high N(= 80) for Emax = −0.04 (Table G.1). One

may conclude that the HS’s solution can be found for low Emax(≈ −0.225) with a low accuracy

(3I(x = −1) = O(10−4

)) while the reference solution can be found for high Emax(≈ −0.05) with

a high accuracy (at least 3J(x = −1) = O(10−6

)). Also, the Chebyshev coefficients In showed a

distinctive difference between the two solutions. The absolute values of In stall approximately at

10−4 ∼ 10−5 for Emax ≈ −0.25 and at 10−6 ∼ 10−7 for Emax ≈ −0.05 (Figure G.2).

260


function N Emax β c1 c4 3I(x = 1)(3

(m)F , 3R

)15 −0.24 8.181 9.1014 × 10−4 3.516 × 10−1 3.3 × 10−4(

3F , 3(m)R

)15 −0.24 8.1731 9.1023 × 10−4 3.524 × 10−1 2.1 × 10−4(

3(m)F , 3(m)

R

)200 −0.225 8.175860 9.1018 × 10−4 3.523 × 10−1 2.9 × 10−4(

3(m)F , 3R

)70 −0.00525 8.1783712 9.0925 × 10−4 3.304 × 10−1 2.2 × 10−6(

3F , 3(m)R

)200 −0.04 8.178371129 9.0925 × 10−4 3.301 × 10−1 4.9 × 10−7(

3(m)F , 3(m)

R

)80 −0.04 8.1783683 9.0926 × 10−4 3.301 × 10−1 8.9 × 10−7

Table G.1: Eigenvalues and 3I(x = 1) for combinations of 3(m)R and 3(m)

F . The upper three rowsare the data that reproduced the HS’s eigenvalues while the lower three rows are the data thatreproduced three significant figures of the reference eigenvalues. In the modified ss-OAFP systems,the variables 3S , 3Q, 3G, 3I , and 3J are not modified

100 101 102

10−10

10−8

10−6

10−4

10−2

100

Emax = −0.225

Emax = −0.24

n

| F(m)n |

(3

(m)F , 3(m)

R

)(3

(m)F , 3R

)100 101 102

10−10

10−8

10−6

10−4

10−2

100

Emax = −0.04

Emax = −0.00525

n

| F(m)n |

(3

(m)F , 3(m)

R

)(3

(m)F , 3R

)

Figure G.1: Absolute values of Chebyshev coefficients F(m)n for 3(m)

F . In the modified ss-OAFPsystem, 3S , 3Q, 3G, 3I , 3J are not modified. The maximum values Emax of the truncated domain arealso depicted in the figure.

261


100 101 102

10−11

10−9

10−7

10−5

10−3

10−1

Emax = −0.225Emax = −0.24

n

| In |

(3

(m)F , 3(m)

R

)(3

(m)F , 3R

)100 101 102

10−11

10−9

10−7

10−5

10−3

10−1 Emax = −0.04

Emax = −0.00525

n

| In |

(3

(m)F , 3(m)

R

)(3

(m)F , 3R

)

Figure G.2: Absolute values of Chebyshev coefficients In for 3I . In the modified ss-OAFP system,3S , 3Q, 3G, 3I , 3J are not modified. The maximum values Emax of the truncated domain are alsodepicted in the figure.

262

Appendix H

The normalization of total energy

The present appendix explains the relationship between the normalization of the total energy

Etot and Poisson’s equation for the ss-OAFP model. A proper integration of the Poisson’s equation

(2.2.5) with respect to R reduces to

R2 dΦ

dR= M. (H.0.1)

Since the ss-OAFP model has the power-law boundary condition Φ ∝ R−α at R → ∞, one can

employ the identity dΦdR = −αΦ/R. Accordingly, the Poisson’s equation at large R reduces to

Φ = −MRα

. (H.0.2)

The total energy Etot is proportional to M as R → ∞. The domain of E is (Φ, 0). Hence, a proper

change of variable makes the factor (E − Φ) be proportional to Φ for the Etot/M (Recall the form

of equation (9.4.1b).). As a result,

Etot ∝M2

R. (H.0.3)

263

APPENDIX H. DIMENSIONLESS POISSON’S EQUATION

Although one may choose different quantities to regularize Etot, the present work interests in heat

capacity at a constant volume. This means the total number N and volume V are constant; that

is, the total mass M and radius R are constant (under the Legendre transformation). Hence, the

mathematical expression of the normalized energy Λ (equation (9.3.1)) originates from the char-

acteristics of the Poisson’s equation.

264

Appendix I

The fitting of the finite ss-OAFP model to

the King-model clusters

The present appendix shows the fitting of the energy-truncated ss-OAFP model to projected

structural profiles of Galactic KM clusters reported in (Kron et al., 1984; Noyola and Gebhardt,

2006; Miocchi et al., 2013). For the fitting to (Miocchi et al., 2013)’s data, we determined the

values of fitting parameters so that χ2ν was minimized. For the fitting to (Kron et al., 1984; Noyola

and Gebhardt, 2006)’s data, the deviation of the model from the data plots were minimized. Ap-

pendices I.1, I.2, and I.3 show the fitting of our model to the KM stars reported in (Miocchi et al.,

2013), (Kron et al., 1984), and (Noyola and Gebhardt, 2006).

I.1 KM cluster (Miocchi et al., 2013)

Figures I.1, I.2, and I.3 depict the energy-truncated ss-OAFP model fitted to the projected

density profiles of KM clusters reported in (Miocchi et al., 2013). Table I.1 compares the structural

parameters of our model, the King model, and the Wilson model. Many parameters our model

provided are greater than those of the King model while less than the Wilson model. The energy-

265

APPENDIX I. FITTING TO KING-MODEL CLUSTERS

truncated ss-OAFP model does not completely fit the structures of NGC 5466 and Terzan 5 while

the King- and Wilson- models do. This result implies that the clusters are close to neither of the

states of core collapse (or in the gravothermal instability phase) nor polytropic sphere (possibly in

a collisionless relaxation phase).

266


−1.5 −1 −0.5 0 0.5 1 1.5

−4.0

−2.0

0.0

(rc = 1′.43, δ = 0.13)

log[

Σ]

ss-OAFP(c = 1.30)

NGC 288(n)

−2.5 −2 −1.5 −1 −0.5 0 0.5 1

−4.0

−2.0

0.0

(rc = 0′.10, δ = 0.017)

ss-OAFP(c = 2.04)

NGC 1851(n)

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

NGC 288(n)

−2.5 −2 −1.5 −1 −0.5 0 0.5 1−0.5

0

0.5

log[R(arcmin)]

NGC 1851(n)

−1 −0.5 0 0.5 1 1.5

−4.0

−2.0

0.0

(rc = 1′.26, δ = 0.087)

log[

Σ]

ss-OAFP(c = 1.41)

NGC 5466(n)

−1.5 −1 −0.5 0 0.5 1 1.5−4.0

−2.0

0.0

(rc = 1′.13, δ = 0.029)

ss-OAFP(c = 1.81)

NGC 6121(n)

−1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

NGC 5466(n)

−1.5 −1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

log[R(arcmin)]

NGC 6121(n)

Figure I.1: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density Σ ofNGC 288, NGC 1851, NGC 5466, and NGC 6121 reported in from (Miocchi et al., 2013). Theunit of Σ is the number per square of arcminutes. Σ is normalized so that the density is unity at thesmallest radius for data. In the legends, (n) means a ‘normal’ cluster that can be fitted by the Kingmodel as judged so in (Djorgovski and King, 1986). ∆ log[Σ] is the corresponding deviation of Σ

from the model on log scale.

267


−1.5 −1 −0.5 0 0.5 1

−4.0

−2.0

0.0

(rc = 0′68, δ = 0.044)

log[

Σ]

ss-OAFP(c = 1.64)

NGC 6254(n)

−2 −1.5 −1 −0.5 0 0.5 1

−4.0

−2.0

0.0

(rc = 0′.19, δ = 0.014)

ss-OAFP(c = 2.12)

NGC 6626(n)

−1.5 −1 −0.5 0 0.5 1−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

NGC 6254(n)

−2 −1.5 −1 −0.5 0 0.5 1−0.5

0

0.5

log[R(arcmin)]

NGC 6626(n)

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6

−4.0

−2.0

0.0

(rc = 0′.55, δ = 1.3)

log[

Σ]

ss-OAFP(c = 1.03)Pal 3 (n)

−1.5 −1 −0.5 0 0.5−4.0

−2.0

0.0

(rc = 0′.46, δ = 0.27)


−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6

−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

Pal 3 (n)

−1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

Pal 4 (n)

Figure I.2: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density Σ ofNGC 6254, NGC 6626, Palomar 3 and Palomar 4 reported in (Miocchi et al., 2013). The unit of Σ

is number per square of arcminutes and Σ is normalized so that the density is unity at the smallestradius for data. In the legends, (n) means ‘normal’ cluster that can be fitted by the King model asjudged so in (Djorgovski and King, 1986). ∆ log[Σ] is the corresponding deviation of Σ from themodel on log scale.

268


−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6

−4.0

−2.0

0.0

(rc = 0′.85, δ = 1.2)

log[

Σ]


−1.5 −1 −0.5 0 0.5−4.0

−2.0

0.0

(rc = 0′.15, δ = 0.039)

ss-OAFP(c = 1.69)Trz 5 (n)

−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6

−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

Pal 14 (n)

−1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

Trz 5 (n)

−1.5 −1 −0.5 0 0.5 1

−4.0

−2.0

0.0

(rc = 1′.63, δ = 0.1)

log[

Σ]

ss-OAFP(c = 1.37)6809 (n)

−1.5 −1 −0.5 0 0.5 1

−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

NGC 6809 (n)

Figure I.3: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density Σ

of Palomar 14, Terzan 5, and NGC 6809 reported in (Miocchi et al., 2013). The unit of Σ is thenumber per square of arcminutes. Σ is normalized so that the density is unity at the smallest radiusfor data. In the legends, (n) means a ‘normal’ cluster that can be fitted by the King model as judgedso in (Djorgovski and King, 1986). ∆ log[Σ] is the corresponding deviation of Σ from the modelon log scale.

269


NGC 288 NGC 1851 NGC 5466


ss-OAFP model 1.30 1.43 28.9 2.04 0.10 11.1 1.41 1.26 32.4

King model(Miocchi et al., 2013) 1.20 1.17 21 1.95 0.09 8.3 1.31 1.20 26.3

Wilson model(Miocchi et al., 2013) 1.10 1.53 25.8 3.33 0.09 204 1.42 1.33 40

NGC 6121 NGC 6254 NGC 6626


ss-OAFP model 1.81 1.13 73.6 1.64 0.68 30.1 2.12 0.19 25.4

King model(Miocchi et al., 2013) 1.68 1.07 53 1.41 0.68 19.0 1.79 0.26 16

Wilson model(Miocchi et al., 2013) 2.08 1.08 1200 1.80 0.73 52 3.1 0.26 380

Pal 3 Pal 4 Pal 14 Trz 5

c rc rtid c rc rtid c rc rtid c rc rtid

ss-OAFP model 1.03 0.55 5.91 1.16 0.46 6.79 1.04 0.85 9.23 1.69 0.15 7.25

King model(Miocchi et al., 2013) 0.8 0.47 3.6 1.1 0.37 4.9 0.9 0.68 6.4 1.59 0.13 5.2

Wilson model(Miocchi et al., 2013) 0.81 0.49 5.33 1.3 0.38 9 1.0 0.70 10 2.4 0.14 39

Table I.1: Core- and tidal radii of the finite ss-OAFP model applied to the projected density profilesreported in (Miocchi et al., 2013).

270


I.2 KM clusters (Kron et al., 1984)

Figures I.4- I.7 show the projected density profiles reported in (Kron et al., 1984) fitted by the

energy-truncated ss-OAFP model. In (Kron et al., 1984), the first several points are not included

in the fitting of the King model due to uncertainty in data originating from too high brightness.

However, the present work included them since our model well fits almost all the data plots.

271


10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.41)Σ

δ = 0.18(c = 1.24)NGC 2419(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.72)

δ = 0.098(c = 1.39)NGC 4590(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.51)

Σ

δ = 0.068(c = 1.50)NGC 5272(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.23)

δ = 0.052(c = 1.60)NGC5634(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.074)

R(arcmin)

Σ

δ = 0.022(c = 1.95)NGC 5694(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.064)

R(arcmin)

δ = 0.023(c = 1.93)NGC 5824(n)

Figure I.4: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected densityprofiles NGC 2419, NGC 4590, NGC 5272, NGC 5634, NGC 5694, and NGC 5824 reported in(Kron et al., 1984). The unit of the projected density Σ is the number per square of arcminutes.In the legends, (n) means a normal cluster that can be fitted by the King model as judged so in(Djorgovski and King, 1986).

272


10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.20)Σ

δ = 0.036(c = 1.74)NGC 6093(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.78)

δ = 0.061(c = 1.54)NGC 6205(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.18)

Σ

δ = 0.080(c = 1.45)NGC 6229(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.49)

δ = 0.045(c = 1.65)NGC 6273(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.29)

R(arcmin)

Σ

δ = 0.044(c = 1.66)NGC 6304(n?)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.37)

R(arcmin)

δ = 0.050(c = 1.61)NGC 6333(n?)

Figure I.5: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected densityprofiles of NGC 6093, NGC 6205, NGC 5229, NGC 6273, NGC 6304, and NGC 6333 reported in(Kron et al., 1984). The unit of the projected density Σ is the number per square of arcminutes. Inthe legends, (n) means a ‘normal’ cluster that can be fitted by the King model and (n?) a ‘probablenormal’ cluster as judged so in (Djorgovski and King, 1986).

273


10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.31)Σ

δ = 0.042(c = 1.68)NGC 6341(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.27)

δ = 0.042(c = 1.68)NGC 6356(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.27)

Σ

δ = 0.040(c = 1.70)NGC 6401(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.16)

δ = 0.038(c = 1.72)NGC 6440(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.16)

R(arcmin)

Σ

δ = 0.020(c = 1.99)NGC 6517(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.55)

R(arcmin)

δ = 0.108(c = 1.36)NGC 6553(n)

Figure I.6: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected densityprofiles NGC 6341, NGC 6356, NGC 6401, NGC 6440, NGC 6517, and NGC 6553 reported in(Kron et al., 1984). The unit of the projected density Σ is number per square of arcminutes. In thelegends, (n) means a normal cluster that can be fitted by the King model as judged so in (Djorgovskiand King, 1986).

274


10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.37)Σ

δ = 0.15(c = 1.28)NGC 6569(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.25)

δ = 0.105(c = 1.37)NGC 6638(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.11)

Σ

δ = 0.028(c = 1.84)NGC 6715(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.12)

δ = 0.029(c = 1.83)NGC 6864(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.24)

R (arcmin)

Σ

δ = 0.065(c = 1.52)NGC 6934(n)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.18)

R (arcmin)

δ = 0.095(c = 1.40)NGC 7006(n)

Figure I.7: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected densityprofiles of NGC 6569, NGC 6638, NGC 6715, NGC 6864, NGC 6934, and NGC 7006 reportedin (Kron et al., 1984). The unit of the projected density Σ is the number per square of arcminutes.In the legends, (n) means a normal cluster that can be fitted by the King model as judged so in(Djorgovski and King, 1986).

275


10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 2′.7)

R (arcmin)

Σ

δ = 5.5(c = 1.00)NGC 5053(n)

10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 1′.64)

R (arcmin)

δ = 0.78(c = 1.06)NGC 5897(n)

Figure I.8: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected densityprofiles NGC 5053 and NGC 5897 reported in (Kron et al., 1984). The unit of the projecteddensity Σ is the number per square of arcminutes. In the legends, (n) means normal cluster that canbe fitted by the King model as judged so in (Djorgovski and King, 1986). Following (Kron et al.,1984), data at small radii are ignored due to the depletion of projected density profile.

I.3 KM clusters (Noyola and Gebhardt, 2006)

Figure I.9 shows the fitting of the energy-truncated ss-OAFP model to the surface brightness

(V-band magnitude per arcseconds squared) with flat cores reported in (Noyola and Gebhardt,

2006). The brightness is depicted with the Chebyshev approximation of the brightness reported

in (Trager et al., 1995). To fit our model to the profiles, we employed polytropic indices m = 4.2

through m = 4.4. Figure I.10 depicts the surface brightness profile of the globular clusters that our

model could not fit due to the cusps in the cores. Figure I.11 depicts the surface brightness profiles

fitted by the energy-truncated ss-OAFP model and its approximated model. For approximated

models, we needed to employ high polytropic indices m = 4.8 ∼ 4.95.

276


−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.43, δ = 0.023)

SBo−

SB

ss-OAFP(c = 2.98,m = 4.3)

NGC 104(n)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.13, δ = 0.023)

ss-OAFP(c = 2.50,m = 4.2)

NGC 1904(n)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.24, δ = 0.06)

SBo−

SB

ss-OAFP(c = 2.41,m = 4.4)

NGC 2808(n)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.77, δ = 0.085)

ss-OAFP(c = 1.74,m = 4.2)

NGC 6205(n)Cheb.

−3 −2 −1 0 1

−8.0

−6.0

−4.0

−2.0

0.0

(rc = 0′.127, δ = 0.034)

log[R(arcmin)]

SBo−

SB

ss-OAFP(c = 3.50,m = 4.4)

NGC 6441(n)Cheb.

−3 −2 −1 0 1−6.0

−4.0

−2.0

0.0

(rc = 0′.69, δ = 0.15)

log[R(arcmin)]

ss-OAFP(c = 1.54,m = 4.2)

NGC 6712(n)Cheb.

Figure I.9: Fitting of the energy-truncated ss-OAFP model to the surface brightness of NGC 104,NGC 1904, NGC 2808, NGC 6205, NGC 6441, and NGC 6712 reported in (Noyola and Gebhardt,2006). The unit of the surface brightness (SB) is the V magnitude per square of arcseconds. Thebrightness is normalized by the magnitude SBo obtained at the smallest radius point. In the legends,‘Cheb.’ means the Chebyshev approximation of the surface brightness reported in (Trager et al.,1995) and ‘(n)’ means a KM cluster as judged so in (Djorgovski and King, 1986).

277


−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.230, δ = 0.044)

SBo−

SB

ss-OAFP(c = 2.08,m = 4.20)

NGC 5286(n)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.121, δ = 0.025)ss-OAFP(c = 2.44,m = 4.2)

NGC 6093(n)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.272, δ = 0.048)

log[R(arcmin)]

SBo−

SB

ss-OAFP(c = 2.03,m = 4.2)

NGC 6535(n)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.168, δ = 0.020)

log[R(arcmin)]

ss-OAFP(c = 2.59,m = 4.2)

NGC 6541(n)Cheb.

Figure I.10: Failure of the fitting of the energy-truncated ss-OAFP model to the surface brightnessprofiles of NGC 5286, NGC 6093, NGC 6535, and NGC 6541 reported in (Noyola and Gebhardt,2006). The unit of the surface brightness (SB) is the V magnitude per square of arcseconds. Thebrightness is normalized by the magnitude SBo obtained at the smallest radius point. In the legends,‘Cheb.’ means the Chebyshev approximation of the surface brightness reported in (Trager et al.,1995) and ‘(n)’ means a KM cluster as judged so in (Djorgovski and King, 1986).

278


−3 −2 −1 0 1−14.0

−12.0

−10.0

−8.0

−6.0

−4.0

−2.0

0.0

(rc = 0′.0528, δ = 0.055)

log[R(arcmin)]

SBo−

SB

ss-OAFP(c = 2.59,m = 4.2)approx.(c = 2.72,m = 4.8)

NGC 1851(n)Cheb.

−3 −2 −1 0 1−14.0

−12.0

−10.0

−8.0

−6.0

−4.0

−2.0

0.0

(rc = 0′.0388, δ = 0.10)

log[R(arcmin)]

SBo−

SB

ss-OAFP(c = 2.30,m = 4.2)

approx.(c = 2.84,m = 4.95)

NGC 5694(n)Cheb.

−3 −2 −1 0 1−14.0

−12.0

−10.0

−8.0

−6.0

−4.0

−2.0

0.0

(rc = 0′.0388, δ = 0.090)

log[R(arcmin)]

SBo−

SB

ss-OAFP(c = 2.44,m = 4.2)

approx.(c = 2.83,m = 4.95)

NGC 5824(n)Cheb.

Figure I.11: Fitting of the energy-truncated ss-OAFP model and its approximated form to thesurface brightness profiles of NGC 1851, NGC 5694, and NGC 5824 reported in (Noyola andGebhardt, 2006). The unit of the surface brightness (SB) is the V magnitude per square of arcsec-onds. The brightness is normalized by the magnitude SBo obtained at the smallest radius point.The values of rc and δ were obtained from the approximated form. In the legends, ‘Cheb.’ meansthe Chebyshev approximation of the surface brightness reported in (Trager et al., 1995) and ‘(n)’ ameans KM cluster as judged so in (Djorgovski and King, 1986).

279

Appendix J

The fitting of the polytropic sphere model to

low-concentration clusters

The present appendix shows the fitting of the polytropic sphere of index m to the projected

structural profiles of low-concentration globular clusters reported in (Miocchi et al., 2013; Trager

et al., 1995; Kron et al., 1984). In fitting the polytrope models to (Miocchi et al., 2013)’s data, we

minimized χ2ν while (Kron et al., 1984; Trager et al., 1995)’s data we minimized the infinite norm

of difference between the model and data.

J.1 Polytropic clusters (Miocchi et al., 2013) and (Kron et al.,

1984)

Figures J.1 and J.2 show successful applications of the polytrope model to the projected density

profiles of NGC 5466, NGC 6809, Palomar 3, Palomar 4, and Palomar 14 reported in (Miocchi

et al., 2013). The polytropes of 3.0 < m < 5 reasonably apply to the globular clusters whose

concentrations range 1 < c < 1.4. Figure J.3 shows the projected density of NGC 4590 fitted

with a polytrope. NGC 4590 is one of the examples that could fall in either of polytropic and

280

APPENDIX J. FITTING TO LOW-CONCENTRATION CLUSTERS

non-polytropic.

−1.5 −1 −0.5 0 0.5 1 1.5

−4.0

−2.0

0.0

(χ2ν = 0.37)

log[

Σ]

polytrope(m = 4.97)

NGC 5466(n)

−1 −0.5 0 0.5 1 1.5

−4.0

−2.0

0.0

(χ2ν = 1.17)

polytrope(m = 4.6)

NGC 6809(n)

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

NGC 5466(n)

−1 −0.5 0 0.5 1 1.5

−0.5

0

0.5

log[R(arcmin)]

NGC 6809 (n)

−1.5 −1 −0.5 0 0.5 1−4.0

−2.0

0.0

(χ2ν = 0.004)

log[

Σ]

polytrope(m = 4.2)Pal 3(n)

−1.5 −1 −0.5 0 0.5 1

−4.0

−2.0

0.0

(χ2ν = 0.32)

polytrope(m = 4.80)Pal 4 (n)

−1.5 −1 −0.5 0 0.5 1

−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

Pal 3(n)

−1.5 −1 −0.5 0 0.5 1−0.5

0

0.5

log[R(arcmin)]

Pal 4(n)

Figure J.1: Fitting of the polytropic sphere of index m to the projected density Σ of NGC 5466,NGC 6809, Palomar 3, and Palomar 4 reported in (Miocchi et al., 2013). The unit of Σ is numberper square of arcminutes. Σ is normalized so that the density is unity at smallest radius for the data.In the legends, (n) means a normal or KM cluster as judged so in (Djorgovski and King, 1986).∆ log[Σ] is the corresponding deviation of Σ from the model on a log scale.

281


−1.5 −1 −0.5 0 0.5 1

−4.0

−2.0

0.0

(χ2ν = 0.15)

log[R]

log[

Σ]

polytrope(m = 4.6)Pal14(n)

−1.5 −1 −0.5 0 0.5 1−0.5

0

0.5

log[R(arcmin)]

∆lo

g[Σ

]

Pal 14 (n)

Figure J.2: Fitting of the polytropic sphere of index m to the projected density of Palomar 14reported in (Miocchi et al., 2013). The unit of the projected density Σ is the number per squareof arcminutes. In the legends, (n) means a normal or KM cluster as judged so in (Djorgovski andKing, 1986).

.

10−1 100 10110−4

10−3

10−2

10−1

100

R (arcmin)

Σ

polytrope(m = 4.99)NGC 4590(n)

Figure J.3: Partial success of fitting of the polytropic sphere of index m to the projected densityprofile of NGC 4590 reported in (Kron et al., 1984). The unit of the projected density Σ is thenumber per square of arcminutes. In the legends, (n) means a normal or KM cluster as judged soin (Djorgovski and King, 1986). Following (Kron et al., 1984), the data at small radii are ignoreddue to the depletion of projected density profile.

J.2 Polytropic clusters (Trager et al., 1995)

Figures J.4 and J.5 depict the fitting of the polytropic-sphere model to the Chebyshev approx-

imation to the surface brightness profiles reported in (Trager et al., 1995). The polytropic indices

m = 3.3 ∼ 4.99 are useful to fit the polytrope model to the low-concentration clusters whose core

282


relaxation times are the order of 1 Gyr (from (Harris, 1996, (2010 edition))’s catalog). Since the

polytrope itself does not rapidly decay near its limiting radius, the corresponding concentrations of

the polytrope are relatively high, such as c = 3.34 for m = 4.99, compared with those of the King

model. On the one hand, Figure J.6 shows the clusters whose surface brightness profiles are not

close to the polytropes. Such clusters have shorter core relaxation times < 0.5 Gyr and relatively

high concentrations c > 1.5 based on the King model, as shown in Table 10.3.

283


−3 −2 −1 0 1−15

−10

−5

0

(rc = 0′.27)

SBo−

SB

c = 3.34(m = 4.99)NGC 1261(n)

−3 −2 −1 0 1−10

−5

0

(rc = 2′.04)c = 1.11(m = 4.1)

NGC 5053(n)

−3 −2 −1 0 1−10

−5

0

(rc = 1′.70)

SBo−

SB

c = 1.11(m = 4.1)

NGC 5897(n)

−3 −2 −1 0 1−10

−5

0

(rc = 0′.401)c = 3.34(m = 4.99)NGC 5986(n)

−3 −2 −1 0 1−10

−5

0

(rc = 1′.25)

log[R(arcmin)]

SBo−

SB

c = 0.685(m = 3.3)NGC 6101(n)

−3 −2 −1 0 1−15

−10

−5

0

(rc = 0′.62)

log[R(arcmin)]

c = 3.34(m = 4.99)NGC 6205(n)

Figure J.4: Fitting of the polytropic sphere of index m to the surface brightness profiles of NGC1261, NGC 5053, NGC 5897, NGC 5986, NGC 6101, and NGC 6205 reported in Trager et al.(1995). The unit of the surface brightness (SB) is the V magnitude per square of arcseconds. Thebrightness is normalized by the magnitude SBo obtained at the smallest radius point. In the legends,‘(n)’ means a KM cluster as judged so in (Djorgovski and King, 1986).

284


−3 −2 −1 0 1−10

−5

0

(rc = 0′.73)

SBo−

SB

c = 2.24(m = 4.9)

NGC 6402(n)

−3 −2 −1 0 1−15

−10

−5

0

(rc = 0′.73)

c = 2.24(m = 3.3)

NGC 6496(n)

−3 −2 −1 0 1−10

−5

0

(rc = 0′.58)

log[R(arcmin)]

SBo−

SB

c = 3.34(m = 4.99)NGC 6712(n)

−3 −2 −1 0 1−10

−5

0

(rc = 0′.634)

log[R(arcmin)]

c = 3.34(m = 4.99)NGC 6723(n)

−3 −2 −1 0 1−10

−5

0

(rc = 0′.36)

log[R(arcmin)]

SBo−

SB

c = 3.34(m = 4.99)NGC 6981(n)

Figure J.5: Fitting of the polytropic sphere of index m to the surface brightness profiles of NGC6402, NGC 6496, NGC 6712, NGC 6723, and NGC 6981 reported in (Trager et al., 1995). Theunit of the surface brightness (SB) is the V magnitude per square of arcseconds. The brightness isnormalized by the magnitude SBo obtained at the smallest radius point. In the legends, ‘(n)’ meansKM cluster as judged so in (Djorgovski and King, 1986).

285


−3 −2 −1 0 1−15

−10

−5

0

(rc = 0′.997)

SBo−

SB

c = 3.33(m = 4.99)NGC 3201(n)

−3 −2 −1 0 1−15

−10

−5

0

(rc = 0′.57)

c = 3.34(m = 4.99)NGC 6144(n)

−3 −2 −1 0 1−10

−5

0

(rc = 0′.622)

SBo−

SB

c = 3.34(m = 4.99)NGC 6273(n)

−3 −2 −1 0 1

−10

−5

0

(rc = 0′.26)

c = 3.34(m = 4.99)NGC 6352(n)

−3 −2 −1 0 1−15

−10

−5

0

(rc = 0′.233)

log[R(arcmin)]

SBo−

SB

c = 3.34(m = 4.99)NGC 6388(n)

−2 −1 0 1−15

−10

−5

0

(rc = 0.′71)

log[R(arcmin)]

c = 3.34(m = 4.99)NGC 6656(n)

Figure J.6: Failure of the fitting of the polytropic sphere of index m to the surface brightnessprofiles of NGC 3201, NGC 6144, NGC 6273, NGC 6352, NGC 6388, and NGC 6656 reportedin (Trager et al., 1995). The unit of the surface brightness (SB) is the V magnitude per square ofarcseconds. The brightness is normalized by the magnitude SBo obtained at the smallest radiuspoint. In the legends, ‘(n)’ means a KM cluster as judged so in (Djorgovski and King, 1986).

286

Appendix K

The fitting of the finite ss-OAFP model to

the PCC clusters

The present appendix shows the results of the applications of the energy-truncated ss-OAFP

model to the PCC clusters reported in (Kron et al., 1984; Lugger et al., 1995; Noyola and Gebhardt,

2006). For the fitting of the model to (Kron et al., 1984; Lugger et al., 1995)’s data, the infinite

norm of the deviation of the model from the data was minimized while (Lugger et al., 1995)’s data

the χ2ν-value was minimized. Appendices K.1, K.2, and K.3 show the fitting to the PCC clusters

reported in (Kron et al., 1984), (Lugger et al., 1995), and (Noyola and Gebhardt, 2006).

K.1 PCC? clusters (Kron et al., 1984)

Figure K.1 depicts the energy-truncated ss-OAFP model with m = 3.9 fitted to the projected

density profiles of the possible PCC clusters reported in (Kron et al., 1984). We consider NGC

1904, NGC 4147, NGC 6544, and NGC 6652 to PCC clusters, as described in (Djorgovski and

King, 1986) as ‘probable/possible’ PCC clusters or ‘the weak indications of PCC’ clusters.

287

APPENDIX K. FITTING TO PCC CLUSTERS

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.19)

Σ

δ = 0.027(c = 1.86)NGC 1904(c?)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.14)

δ = 0.025(c = 1.89)

NGC 4147(n?c?)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.50)

R(arcmin)

Σ

δ = 0.050(c = 1.61)NGC 6544(c?)

10−2 10−1 100 10110−4

10−3

10−2

10−1

100

(rc = 0′.14)

R (arcmin)

δ = 0.028(c = 1.84)NGC 6652(n?c?)

Figure K.1: Fitting of the energy-truncated ss-OAFP model to the projected densities of NGC 1904,NGC 4147, NGC 6544, and NGC 6652 reported in (Kron et al., 1984). The unit of the projecteddensity Σ is the number per square of arcminutes. In the legends, (c) means a PCC cluster, (c?)a probable/possible PCC, and (n?c?) weak indications of PCC as judged so in (Djorgovski andKing, 1986).

K.2 PCC clusters (Lugger et al., 1995) and (Djorgovski and

King, 1986)

Figures K.2 and K.3 show the results of the fitting of the energy-truncated ss-OAFP model with

m = 3.9 to the surface brightness profiles of the PCC clusters reported in (Lugger et al., 1995) and

288


Terzan 1 in (Djorgovski and King, 1986). For the clusters with resolved cores, our model well fits

the core and halo up to 1 arcminute. On the one hand, the fitting of our model to the clusters with

unresolved core (NGC 5946 and NGC 6624) is not satisfactory as expected. Interestingly, NGC

6342 is one of the PCC clusters with an unresolved core, but it appears reasonably fitted by our

model.

289


−2.5 −2 −1.5 −1 −0.5 0 0.5−3.0

−2.0

−1.0

0.0

(rc = 0′.093, δ = 0.082)

SBo−

SB

ss-OAFP(c = 2.34)

NGC 5946 (c)

−2.5 −2 −1.5 −1 −0.5 0 0.5−3.0

−2.0

−1.0

0.0

(rc = 0′.068, δ = 0.0073)

ss-OAFP(c = 2.38)

NGC 6342(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

∆(S

Bo−

SB)

NGC 5946(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

NGC 6342(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5−3.0

−2.0

−1.0

0.0

(rc = 0′.055, δ = 0.0060)

SBo−

SB

ss-OAFP(c = 2.45)

NGC 6624(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5−3.0

−2.0

−1.0

0.0

(rc = 0′.096, δ = 0.011)

ss-OAFP(c = 2.22)

NGC 6453(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

∆(S

Bo−

SB)

NGC 6624(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

NGC 6453(c)

Figure K.2: Fitting of the energy-truncated ss-OAFP model to the surface brightness profiles ofNGC 5946, NGC 6342, NGC 6624, and NGC 6453 reported in (Lugger et al., 1995). The unitof the surface brightness (SB) is the U magnitude per square of arcseconds. The brightness isnormalized by the magnitude SBo obtained at the smallest radius point. In the legend, ‘(c)’ meansa PCC cluster as judged so in (Djorgovski and King, 1986). ∆(SBo − SB) is the correspondingdeviation of SBo − SB from the model.

290


−2.5 −2 −1.5 −1 −0.5 0 0.5−3.0

−2.0

−1.0

0.0

(rc = 0′.096, δ = 0.0094)

SBo−

SB

ss-OAFP(c = 2.29)

NGC 6522(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5−3.0

−2.0

−1.0

0.0

(rc = 0′.085, δ = 0.0052)

ss-OAFP(c = 2.51)

NGC 6558(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

∆(S

Bo−

SB)

NGC 6522(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

NGC 6558(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5−3.0

−2.0

−1.0

0.0

(rc = 0′.086, δ = 0.0076)

SBo−

SB

ss-OAFP(c = 2.37)

NGC 7099(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5−3.0

−2.0

−1.0

0.0

(rc = 0′.149, δ = 0.0087)

ss-OAFP(c = 2.15)

Trz 1(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

∆(S

Bo−

SB)

NGC7099(c)

−2.5 −2 −1.5 −1 −0.5 0 0.5

−0.5

0

0.5

log[R(arcmin)]

Trz 1 (c)

Figure K.3: Fitting of the energy-truncated ss-OAFP model to the surface brightness profiles ofNGC 6522, NGC 6558, and NGC 7099 reported in (Lugger et al., 1995) and Terzan 1 reported in(Djorgovski and King, 1986). The unit of the surface brightness (SB) is the U magnitude per squareof arcseconds except for Terzan 1 for which the B band is used. The brightness is normalized bythe magnitude SBo obtained at the smallest radius point. In the legend, ‘(c)’ means a PCC cluster asjudged so in (Djorgovski and King, 1986). ∆(SBo−SB) is the corresponding deviation of SBo−SBfrom the model.

291


K.3 PCC clusters (Noyola and Gebhardt, 2006)

The energy-truncated ss-OAFP model with m = 4.2 well fits the surface brightness profiles

of the PCC clusters reported in (Noyola and Gebhardt, 2006), as shown in Figure K.4. However,

NGC 6624 is an example of our model’s failure, as shown in Figure K.5. This failure also occurred

to the same cluster but reported in (Lugger et al., 1995) (Appendix K.2). It appears that NGC 6624

needs more realistic effects, such the binary heating and mass function. Also, NGC 6541 has also

a cusp in the core that our model can not fit.

292


−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.21, δ = 0028)SB

o−

SB

ss-OAFP(c = 2.36,m = 4.2)

NGC 6266(c?)Cheb.

−3 −2 −1 0−8.0

−6.0

−4.0

−2.0

0.0

(rc = 0′.0050, δ = 0.0065)ss-OAFP(c = 3.22,m = 4.2)

NGC 6293(c)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.023, δ = 0.007)

SBo−

SB

ss-OAFP(c = 3.19,m = 4.2)

NGC 6652(c)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.0031, δ = 0.0053)ss-OAFP(c = 3.28,m = 4.2)

NGC 6681(c)Cheb.

−3 −2 −1 0 1−15.0

−10.0

−5.0

0.0

(rc = 0′.0038, δ = 0.0045)

log[R(arcmin)]

SBo−

SB

ss-OAFP(c = 3.33,m = 4.2)

NGC 7078(c)Cheb.

−3 −2 −1 0 1−15.0

−10.0

−5.0

0.0

(rc = 0′.049, δ = 0.0055)

log[R(arcmin)]

ss-OAFP(c = 3.27.,m = 4.2)

NGC 7099(c)Cheb.

Figure K.4: Fitting of the energy-truncated ss-OAFP model to the surface brightness profiles ofNGC 6266, NGC 6293, NGC 6652, NGC 6681, NGC 7078, and NGC 7099 reported in (Noyolaand Gebhardt, 2006). The unit of the surface brightness (SB) is the V magnitude per square ofarcseconds. The brightness is normalized by the magnitude SBo obtained at the smallest radiuspoint. In the legend, ‘Cheb.’ means the Chebyshev approximation of the surface brightness profilesreported in (Trager et al., 1995). The symbol ‘(c)’ means a PCC cluster as judged so in (Djorgovskiand King, 1986). ∆(SBo − SB) is the corresponding deviation of SBo − SB from the model

293


−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.168, δ = 0.02)

log[R(arcmin)]

SBo−

SBss-OAFP(c = 2.59,m = 4.2)

NGC 6541(c?)Cheb.

−3 −2 −1 0 1

−10.0

−5.0

0.0

(rc = 0′.0461, δ = 0.0066)

log[R(arcmin)]

SBo−

SB

ss-OAFP(c = 3.21,m = 4.2)

NGC 6624(c)Cheb.

Figure K.5: Failure of the fitting of the energy-truncated ss-OAFP model to the surface brightnessprofiles of NGC 6541 and NGC 6624 reported in (Noyola and Gebhardt, 2006). The unit of thesurface brightness (SB) is the V magnitude per square of arcseconds. The brightness is normalizedby the magnitude SBo obtained at the smallest radius point. In the legend, ‘Cheb.’ means theChebyshev approximation of the surface brightness profiles reported in (Trager et al., 1995). ‘(c)’means a PCC cluster and ‘(c?)’ a possibly PCC cluster as judged so in (Djorgovski and King,1986). ∆ (SBo − SB ) is the corresponding deviation of SBo − SB from the model.

294

Appendix L

MATLAB code

1 Repmax=1;

2 for Rep=1:Repmax

3 clear all;clf;tic;

4 format long e;

5 %%%%%%%%%%%%%%%%%%%%%%%%%30%%%%%%%%%%%%

6 %%%%%%% parameters and set-up %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

8 %%%% Iteration process appears on a graph if GraphOn=1, it does not ...

otherwise

9 GraphOn=11;

10 SaveOn=11;%%%% result is saved if SaveOn=1, it does not otherwise

11 %%%% parameters %%%%%%%%%%

12 itmax=1;xmin=-1;a=8.178371159658;Lc=1;Lcc=1;d=1e-8;

13 Pfac=1;xmind=-1;xminq=xmin;xminr=-1;Do=1e0;

14 No=200;%total number of Input solutions

15 N=200;coeff=N;ingrid=N-1;

16 p2=1;b=a-6;

17 %%%%%% numerical diff %%%%%%%%%%%%%%%%%%%%%%

295

APPENDIX L. MATLAB CODE

18 [T,Tp,Tpp,E,Bl,Br]=cheb exact deriv Gauss(coeff,ingrid);

19 [TLEX,Ec,tt]=Initial guess Cheb Gauss(coeff);

20 [TLEXc,Ecc,ttc]=Cheb poly guess Radau p(coeff);

21 kk=1:N;

22

23 [Kd]=Integral D standard prep(N,a,xmind,Pfac,Lc,Do);%%%%pre-integral ...

for D

24 [Kq]=Integral Q standard prep Gauss(N,a,xminq,Pfac,Lc);%%%% ...

pre-integral for Q

25

26 %%%%%%%%%%%%%%% File names %%%%%%%%%%%%

27 LoadCheb name=...

28 'cFKGLQ FPQ linQ fixed a No200 xm 1 a8178371159658 tes.txt';

29 LoadEigen name=...

30 'ac1c2c3c4 FPQ linQ fixed a No200 xm 1 a8178371159658 tes.txt';

31 SaveCheb name=...

32 'cFKGLQ FPQ linQ fixed a No200 xm 1 a8178371159658 tes.txt';

33 SaveEigen name=...

34 'ac1c2c3c4 FPQ linQ fixed a No200 xm 1 a8178371159658 tes.txt';

35

36 %%%%%%%%%% extra parameters %%%%%%%%%%%%

37 po=0.2;mu=100;%relaxation oparameter

38 fo=1e-0;

39 % multiplication factor of power index of functions

40 ∆=1;

41 lambda=1.3e-0;%Tikhonov parameter

42 beta=10;%damping parameter

43 power=40;fac=1;%halving paramters

44 s=0;%homotopy parameter

45 qlamb=1;%Q parameters

46

296


47 %%%%%%%%%%%%% Load files %%%%%%%%%%%%

48 cFKGLQ=load(LoadCheb name);ac1c2c3c4=load(LoadEigen name);

49 c2=ac1c2c3c4(2);c3=ac1c2c3c4(3);c4=ac1c2c3c4(4);

50

51

52 %%%%%%%%%%%% (x-xmin)ˆa forimuala %%%%%%%%%%%%%%%%

53 y=1+(E-1)*(1-xmin)/2;Ep=(1+E)/2; Em=(1-y)/2;Tp=Tp*(2/(1-xmin));

54

55

56 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

57 %%%%%%% Initial guess of 4 functions ...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

58 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

59

60

61 if N>No

62 cF=cFKGLQ(:,1);cF=[cF;zeros(N-No,1)];cK=cFKGLQ(:,2);cK=[cK;zeros(N-No,1)];

63 cG=cFKGLQ(:,3);cG=[cG;zeros(N-No,1)];cL=cFKGLQ(:,4);cL=[cL;zeros(N-No,1)];

64 cQ=cFKGLQ(:,5);cQ=[cQ;zeros(N-No,1)];

65 else

66 cF=cFKGLQ(1:N,1);cK=cFKGLQ(1:N,2);cG=cFKGLQ(1:N,3);

67 cL=cFKGLQ(1:N,4);cQ=cFKGLQ(1:N,5);

68 end

69

70 Tc=cos(acos(2*(0.5*Ec+0.5).ˆ(1)-1)*(kk-1));

71 TLc=cos(acos(2*(0.5*Ec+0.5).ˆ(Lcc)-1)*(kk-1));

72 %TrialG=Tc*cG./(0.5+0.5*Ec);TrialK=Tc*cK./(0.5+0.5*Ec);

73 TrialL=Tc*cL/((2*a-3)/4/a);

74 TrialF=TLc*cF;TrialG=TLc*cG;TrialK=TLc*cK;TrialL=TLc*cL;

75 TrialQ=TLc*cQ;

76 %TrialL=(1+(2*a-3)*TrialL/4/a)./(0.5+0.5*Ec).ˆ((a-b)*Lc);

297


77 %TrialL=TrialL./(0.5+0.5*Ec).ˆ((1)*Lc);

78

79 cF=TLEX*TrialF;

80 cL=TLEX*TrialL;cG=TLEX*TrialG;cK=TLEX*TrialK;

81 cQ=TLEX*TrialQ;

82

83 change=1;it=0;

84 c cheb=[cF;cK;cG;cL;cQ];c eign=[c2;c3;c4];c=[c cheb;c eign];

85

86 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

87 %%%%%%%%% Newron Kantrovich Iteration Process %%%%%%%%%%%

88 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

89

90 while change>1e-25&&change<1e7&&it<itmax

91

92 %%%%%%%%%%%% D integral %%%%%%%%%%%%%%%%%

93 vf1=cos(acos(Ec)*(kk-1))*cF(kk);vf1=exp(vf1);

94 cF2=TLEX*vf1;vd=Kd*cF2; %%(loose)

95 yc=1+(Ec-1)*(1-xmin)/2;

96 %vdex=Im*cF;vd=vd+(vdex+I1)*c4; %%(Strict)

97 %[Ed,vd,vdp,cd,diffd]=...

98 Integral D strict(N,a,c4,cF,xmind,Pfac,Lc,fo,Bl);

99

100 %%%%%%%%%%%% Poisson eqn %%%%%%%%%%%%%%%%%

101 [Er,vr,cr,ratio,diffr,change r]=...

102 Poisson eqn(N,a,vd,xminr,Pfac,Lc);

103 %[Er,vr,cr,cs,ratio,diffr,change r]=...

104 Poisson eqn BC(N,a,vd,xminr,Pfac,Lc);

105 %%%%%%%%%%%% Q integral %%%%%%%%%%%%%%%%%

106 vr1=cos(acos(Ec)*(kk-1))*cr(kk);vr1=exp(3*vr1);

107 cr2=TLEX*vr1;Iq=Kq*cr2;

298


108 %vr1=cos(acos(Ec)*(kk-1))*cr(kk);vr1=(vr1).ˆ(3/2);

109 cr2=TLEX*vr1;Iq=abs(Kq*cr2);%PoissBC2

110 %[Eq, Iq]=Integral Q strict(N,a,cr,xminq,Lc);

111 %[Eq, Iq]=Integral Q standard(N,a,cr,xminq,Lc);

112 %%%% Redfine Q %%%%%%%%%%%%%%%%%% Log Q %%%%%%%%%

113 %Iq=log(Iq);

114 %Iqm1=Iq(end);Iqp1=Iq(1);

115 %Iq=Iq(1:end-1);

116 %Iq=Iq(2:end);

117 %%%% Redfine Q %%%%%%%%% Linear Q %%%%%

118 Iq=(0.5-0.5*Ec).*(Iq).ˆ(1/3);

119 %Iq=(Iq).ˆ(1/3);%PoissonBC2

120 Iqm1=Iq(end);Iqp1=Iq(1);

121 Iq=Iq(1:end-1);

122 %Iq=Iq(2:end);

123

124

125

126

127

128

129 %%%%%%%% Nonlinear operator O ...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

130 [O]=Equations linQ fixed a BC Radau m(a, c2, c3, c4, cF, cL, cK, cG, ...

cQ, T, Tp, Tpp, Iq, Iqp1, Iqm1, ingrid, coeff, Lc, Ep, Em, y, Br, Bl);

131 %%%%%%%%%%%%%% Frechet derivative of the nonlinear operator O %%%

132 [OL]=Jacobian linQ fixed a BC Radau m(a, c2, c3, c4, cF, cL, cK, cG, ...

cQ, T, Tp, Tpp, Iq, Iqp1, Iqm1, ingrid, coeff, Lc, Ep, Em, y, Br, Bl);

133

134 %%%%Newton iteration %%%

135 OL=full(OL);Del=-OL\O;

299


136

137 c cheb=[cF;cK;cG;cL;cQ];c eign=[c2;c4];

138 c=[c cheb;c eign];

139

140 cnew=c+p2*Del;

141 cFnew=cnew(1:N);cKnew=cnew(N+1:2*N);cGnew=cnew(2*N+1:3*N);

142 cLnew=cnew(3*N+1:4*N);cQnew=cnew(4*N+1:5*N);

143 c2new=cnew(5*N+1);c4new=cnew(5*N+2);

144 changeF=norm(cnew(1:N)-c(1:N),inf);

145 changeK=norm(cnew(N+1:2*N)-c(N+1:2*N),inf);

146 changeG=norm(cnew(2*N+1:3*N)-c(2*N+1:3*N),inf);

147 changeL=norm(cnew(3*N+1:4*N)-c(3*N+1:4*N),inf);

148 changeQ=norm(cnew(4*N+1:5*N)-c(4*N+1:5*N),inf);

149 changec2=norm(cnew(1+5*N)-c(1+5*N),inf);

150 changec4=norm(cnew(2+5*N)-c(2+5*N),inf);

151

152 change=norm(cnew-c,inf);

153 change cheb=norm(cnew(1:5*coeff)-c(1:5*coeff),inf);

154 change eign=norm(cnew(5*coeff+1:5*coeff+2)-c(5*coeff+1:5*coeff+2),inf);

155 c=cnew;

156 cF=c(1:coeff);cK=c(coeff+1:2*coeff);cG=c(2*coeff+1:3*coeff);

157 cL=c(3*coeff+1:4*coeff);cQ=c(4*coeff+1:5*coeff);

158 c2=c(5*coeff+1);c4=c(5*coeff+2);

159 it=it+1;

160

161 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

162 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

163

164

165

166 if GraphOn==1

300


167 figure(84);semilogy(it,change,'*','Markersize',15);hold on;

168 end

169

170 end

171 vKp2=cos(acos(1)*(kk-1))*c(N+1:2*N)

172 Change min=change

173 cFKGLQ=[cF cK cG cL cQ];ac1c2c3c4=[ a ;c2;c3; c4];

174

175 if Change min<1e-0&&SaveOn==1

176

177 save(SaveCheb name,'cFKGLQ','-ascii','-double');

178 save(SaveEigen name,'ac1c2c3c4','-ascii','-double');

179 end

180

181

182

183 end

184

185 %%%%%%%%%%%% Graphing %%%%%%%%%%%%%%%%%%%%%%%%

186 y=-1:10ˆ-3:1;y=y';

187

188

189 F=zeros(length(y),1);

190 K=zeros(length(y),1);

191 G=zeros(length(y),1);

192 L=zeros(length(y),1);

193 R=zeros(length(y),1);

194 %S=zeros(length(y),1);

195 Q=zeros(length(y),1);

196

197 for m=1:coeff

301


198 F=F+c(m)*cos((m-1)*(acos(y)));

199 K=K+c(m+coeff)*cos((m-1)*(acos(y)));

200 G=G+c(m+2*coeff)*cos((m-1)*(acos(y)));

201 L=L+c(m+3*coeff)*cos((m-1)*(acos(y)));

202 R=R+cr(m)*cos((m-1)*(acos(y)));

203 %S=S+cs(m)*cos((m-1)*(acos(y)));

204 Q=Q+cQ(m)*cos((m-1)*(acos(y)));

205 end

206 y=1+(y-1)*(1-xmin)/2;

207

208

209 eigen a=a;

210 eigen c2=c2;

211 eigen c1=c2*a

212 eigen c3=G(end)

213 eigen c4=c4

214 alpha=(6+4*a)/(2*a+1)

215 gap=(0.5+0.5*xmin)ˆ(a*Lc)

216 cond num=cond(OL)

217 cFmin=cF(end)

218 vKp1=K(end)

219 vKm1=K(1)

220 toc

1 %%%%%%%%%%%%%%% Numerical differentiation $$$$$$$$$

2 %%%%% User choice: Gauss-Chebychev, Lobatto, Radau

3

4 function [Tc,Tpc,Tppc,Ec,Bcl,Bcr]=...

5 cheb exact deriv Gauss(coeff,ingrid)

6 n=0:1:coeff-1;

302


7 i=1:ingrid;i=i';

8 Bcl=zeros(3,coeff);

9 Bcr=zeros(3,coeff);

10

11

12 %t=(i)*pi/(coeff-1);%Gauss cheb Lobatto

13 %t=(pi*(2*i)/(2*coeff-1));%Gauss Cheb Radau p

14 %t=pi*(2*i-1)/(2*coeff-1);%Gauss-Radau m

15 t=(2*i-1)*pi/(2*coeff);%Gauss CHeb m

16 %t=(2*i+1)*pi/(2*coeff);%Gauss CHeb p

17 %E=cos(t);%Radau(BC at E=-1)/Lobatto/Cheb

18 Ec=(cos(t));%Radau(BC at E=1)

19

20

21 Tc=cos(t*n);

22 Tpc=diag(1./sin(t))*sin(t*n)*diag(n);

23 Tppc=diag(cos(t)./(sin(t)).ˆ3)*sin(t*n)*diag(n)...

24 -diag(1./(sin(t)).ˆ2)*cos(t*n)*diag(n.ˆ2);

25 tend=(2*coeff-1)*pi/(2*coeff);

26 t1=(1)*pi/(2*coeff);

27 Bcl(1,:)=(cos(n*pi));

28 %Bcl(1,:)=(cos(n*(tend)));

29 Bcr(1,:)=ones(size(n));

30 %Bcr(1,:)=(cos(n*t1));

31 Bcl(2,:)=(n.*n.*cos(n*pi)/(cos(pi)));

32 %%%Radau(BC at E=-1) standard

33 %Bcl(2,:)=(n.*n.*cos(n*tend)/(cos(tend)));

34 %%%Radau(BC at E=-1)

35 Bcr(2,:)=n.*n;%standard

36 %Bcr(2,:)=(n.*n.*cos(n*t1)/(cos(t1)));

37 %%%Radau(BC at E=-1)

303


38 end

1

2 %%%%% Inverse form of the Gauss-Chebyshev expansion

3

4 function[TLEXc,Ecc,ttc]=Initial guess Cheb Gauss(coeff)

5

6 points=1:coeff;

7 ttc=((2*points-1)*pi/(2*(coeff)));

8 Ecc=(cos(ttc));Ecc=Ecc';

9 p=1:coeff;nn=p-1;nn=nn';

10 TLEXc=(2*cos(nn*ttc)/(coeff));

11 TLEXc(1,:)=0.5*TLEXc(1,:);

12

13 end

1 %%%%%%%%% preset matrix for integral D ###############

2 % the nodes are chosen from Gauss Chebyshev,

3 %Lobtatto and Radau %%%%%%%%

4 function [Kd]=...

5 Integral D standard prep(N,a,xmind,Pfac,Lc,Do)

6

7 coeff=N*Pfac;Ing=N*Pfac;Grid=20; %total Nb<650

8 %max=b*Ing+1;%Radau;

9 max=Grid*Ing;%others

10

11 l=1:Ing;l=l';

12 %t(l)=pi*(l-1)/(coeff-1);%Gauss-Lobatto

304


13 t=pi*(2*l-1)/(2*coeff);%Gauss Chev m

14 %t=(2*l+1)*pi/(2*coeff);%Gauss CHeb p

15 %t=(2*l-1)*pi/(2*coeff-1);%Radau(-1)

16 %t=(2*l-2)*pi/(2*coeff-1);%Radau(+1)

17

18 x=cos(t);

19 z=1+(x-1)*(1-xmind)/2;

20

21 i=1:max;i=i';

22

23

24 %%%%%%%%%%%%% Integral weight %%%%%%%%%%%%%%

25 %%%%% Clenshaw Curtis, Fejer or Radau schemes %%%%%%%%

26

27 %Tb=pi*(i)/(b*Ing+1);%SINE TRIGOMETRIC(Clenshaw Curtis)

28 Tb=(2*i-1)*pi/(2*Grid*Ing);%(Fejer)Gauss-Cheby

29 %Tb=(2*i-1)*pi/(2*b*Ing-1);%Radau(-1)

30 %Tb=(2*i-2)*pi/(2*max-1);%Radau(+1)

31

32 %m=1:max;%trigonometric (Gauss-Lobatto)

33 %wcc=sin(Tb*m)*((1-cos(m'*pi))./m');

34 %SINE TRIGOMETRIC( Clenshaw Curtis)

35 m=1:(max)/2;%(Cheb-Gauss)

36 % m=1:(max-1)/2;%(Radau)

37 wcc=cos(2*Tb*m)*(1./(4*m'.ˆ(2)-1))

38 ;%(Fejer)Gauss-Cheby & Radau

39

40 %w=sin(Tb).*wcc*2/(max+1);

41 %SINE TRIGOMETRIC( Clenshaw Curtis)

42 w=2*(1-2*wcc)/(Grid*Ing);

43 %(Fejer)Gauss-Cheby

305


44 %w=2*(1-2*wcc)/(max-1/2);

45 %Radau(-1 and +1)

46

47

48 %w(end)=w(end)/2;%Radau(-1)

49 %w(1)=w(1)/2;%Radau(+1)

50

51

52 %%%%%%%%%%%%%%%%%%%%practice%%%%%%%%%%%

53 X=cos(Tb);%interior points

54 Lm=1;%numeric parameter in tanh transformaiton;

55

56

57

58 Kd=zeros(length(z),N);

59 kk=1:N;

60 y=zeros(1,length(Tb));

61 for j=1:length(Tb)

62 y(j)=tanh(Lm*X(j)/sqrt(1-X(j)ˆ(2)));

63 %%%%%%% Use ABk for L=k.%%%%%%%%%%%%%%%%%%%%%%%%%%

64 A=(z+1)*(y(j)+1)/4;B=(1+z)/2;

65

66 if Lc==1

67 AB=1;

68 elseif Lc==3/2

69 AB=sqrt(A.ˆ(2)+B.ˆ(2)+A.*B)./sqrt(A.ˆ(3/2)+B.ˆ(3/2));

70 elseif Lc==5/4

71 AB=sqrt((A.ˆ3+B.ˆ3).*(A+B)+(A.*B).ˆ(2))./sqrt(A.ˆ(5/2)...

72 +B.ˆ(5/2))./sqrt(A.ˆ(5/4)+B.ˆ(5/4));

73 elseif Lc==2

74 AB=sqrt(A+B);

306


75 elseif Lc==3

76 AB=sqrt(A.ˆ(2)+A.*B+B.ˆ(2));

77 elseif Lc==4

78 AB=sqrt(A+B).*sqrt(A.ˆ(2)+B.ˆ(2));

79 elseif Lc==5

80 AB=sqrt((A.ˆ3+B.ˆ3).*(A+B)+(A.*B).ˆ(2));

81 %AB6=AB3.*sqrt(A.ˆ3+B.ˆ3);

82 elseif Lc==3/4

83 AB=sqrt(A.ˆ(2)+A.*B+B.ˆ(2))./sqrt(A.ˆ(9/4)+...

84 A.ˆ(3/2).*B.ˆ(3/4)+B.ˆ(3/2).*A.ˆ(3/4)+B.ˆ(9/4));

85 elseif Lc==1/2

86 AB=1./sqrt(sqrt(A)+sqrt(B));

87 elseif Lc==1/3

88 AB=1./sqrt(A.ˆ(2/3)+A.ˆ(1/3).*B.ˆ(1/3)+B.ˆ(2/3));

89 elseif Lc==1/4

90 AB=(1./sqrt(sqrt(A)+sqrt(B)))./sqrt(A.ˆ(1/4)+B.ˆ(1/4));

91 end

92 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

93

94 %%%%%%%%%%% loose xmin%%%%%%%%%%%%%%%%%%%%%

95

96 Kd=Kd+diag(Do*(0.5+z/2).ˆ(0.5-0.5*Lc).*AB)*...

97 cos((acos(0.5*(z-1+(z+1)*y(j))))*(kk-1))...

98 *w(j)*Lc*Lm*sqrt(2)*sqrt(1-y(j))*...

99 ((1+y(j))/2)ˆ(Lc*(a+1)-1)...

100 .*(1-X(j)ˆ(2))ˆ(-3/2)*(cosh(Lm*X(j)...

101 /sqrt(1-X(j)ˆ(2))))ˆ(-2)/4;

102

103 end

104

105 end

307


1 %%%%%%%%% preset matrix for integralQ #########

2 %%%%%%% the nodes are chosen from Gauss

3 %%%%% Chebyshev, Lobtatto and Radau %%%%%%%%

4

5 function [Kq]=...

6 %Integral Q standard prep(N,a,xminq,Pfac,Lc)

7

8 coeff=N*Pfac;Ing=N*Pfac;b=20; %total Nb<650

9 %max=b*Ing+1;%Radau;

10 max=b*Ing;%others

11

12 l=1:Ing;l=l';

13 %t=pi*(l-1)/(coeff-1);%Gauss-Lobatto

14 t=pi*(2*l-1)/(2*coeff);%Gauss Chev

15 %t=(2*l-1)*pi/(2*coeff-1);%Radau(-1)

16 %t=(2*l-2)*pi/(2*coeff-1);%Radau(+1)

17

18 x=cos(t);

19 z=1+(x-1)*(1-xminq)/2;

20

21

22

23 i=1:max;i=i';

24

25 %%%%%%%%%%%%% Integral weight %%%%%%%%%%%%%%

26 %%% Clenshaw Curtis, Fejer or Radau schemes %%%

27

28 Tb=pi*(i)/(b*Ing+1);%SINE TRIGOMETRIC(Gauss-Lobatto)

29 %Tb(i)=(2*i-1)*pi/(2*b*Ing);%(Fejer)Gauss-Cheby

30 %Tb(i)=(2*i-1)*pi/(2*b*Ing-1);%Radau(-1)

308


31 %Tb(i)=(2*i-2)*pi/(2*max-1);%Radau(+1)

32

33 m=1:max;%trigonometric (Gauss-Lobatto)

34 wcc=sin(Tb*m)*((1-cos(m'*pi))./m');

35 %SINE TRIGOMETRIC(Gauss-Lobatto)

36 %for m=1:(max)/2%(Cheb-Gauss)

37 %for m=1:(max-1)/2%(Radau)

38 % wcc(i)=wcc(i)+cos(2*m*Tb(i))/(4*mˆ(2)-1);

39 %(Fejer)Gauss-Cheby & Radau

40

41 w=sin(Tb).*wcc*2/(max+1);%SINE TRIGOMETRIC(Gauss-Lobatto)

42 %w(i)=2*(1-2*wcc(i))/(b*Ing);%(Fejer)Gauss-Cheby

43 %w(i)=2*(1-2*wcc(i))/(max-1/2);%Radau(-1 and +1)

44

45

46 %w(end)=w(end)/2;%Radau(-1)

47 %w(1)=w(1)/2;%Radau(+1)

48

49

50 %%%%%%%%%%%%%%%%%%%%practice%%%%%%%%%%%

51 nu=-(a+0.5)/2;sig=-1.5*(a-0.5);

52 %I=intˆ1+E 0(2-y)ˆ2dy and E=(-1,1)

53 X=cos(Tb);%interior points of Chebyshev-lobatto nodes

54 %tanh transformaitons

55 %X(j)=>tanh(Lm*X(j)/sqrt(1-X(j)ˆ(2)))

56 %dX(j)=>Lm*(1-X(j)ˆ(2))ˆ(-3/2)*...

57 (cosh(Lm*X(j)/sqrt(1-X(j)ˆ(2))))ˆ(-2)

58 Lm=1;%numeric parameter in tanh transformaiton;

59 %Lc;numeric parameter in Eˆ(L) transformaiton;

60

61

309


62

63 Kq=zeros(length(z),N);

64 kk=1:N;

65 y=zeros(1,length(Tb));

66 for j=1:length(Tb)

67 y(j)=tanh(Lm*X(j)/sqrt(1-X(j)ˆ(2)));

68 %%%%%%% Use ABk for L=k.%%%%%%%%%%%%%%%%%%%%%%%%%%

69 A=(3+z+(1-z)*y(j))/4;B=(1+z)/2;

70

71 AB1=1;

72 AB2=sqrt(A+B);

73 AB3=sqrt(A.ˆ(2)+A.*B+B.ˆ(2));

74 AB4=AB2.*sqrt(A.ˆ(2)+B.ˆ(2));

75 AB5=sqrt((A.ˆ3+B.ˆ3).*(A+B)+(A.*B).ˆ(2));

76 AB6=AB3.*sqrt(A.ˆ3+B.ˆ3);

77

78 AB3 4=AB3./sqrt(A.ˆ(9/4)+A.ˆ(3/2).*B.ˆ(3/4)...

79 +B.ˆ(3/2).*A.ˆ(3/4)+B.ˆ(9/4));

80 AB 2=1./sqrt(sqrt(A)+sqrt(B));

81 AB 3=1./sqrt(A.ˆ(2/3)+A.ˆ(1/3).*B.ˆ(1/3)+B.ˆ(2/3));

82 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

83

84 %%%%%%%%%%% loose xmin%%%%%%%%%%%%%%%%%%%%%

85

86 Kq=Kq+diag((0.5+z/2).ˆ(-sig*Lc).*AB1.*...

87 ((3+z+(1-z)*y(j))/4).ˆ((3*nu+1)*Lc-1))...

88 *cos((acos(0.5*(1+y(j)+z*(1-y(j)))))*(kk-1))...

89 *w(j)*Lc*Lm*(1-y(j)).ˆ(1.5).*sqrt(1+y(j))...

90 .*(1-X(j)ˆ(2))ˆ(-3/2)*(cosh(Lm*X(j)...

91 /sqrt(1-X(j)ˆ(2))))ˆ(-2)/24;

92

310


93

94

95 end

96

97 end

1 %%%%%%%%% Poisson equation solver %%%%%%%%%%%%

2

3 function [E,r,cr,ratio,diff,change r]=...

4 Poisson eqn(N,a,vd,xminr,Pfac,Lc)

5 format long e;

6 %Lc;mapping paramter

7 nu=-(a+0.5)/2;

8 coeff=N*Pfac;%number of coefficients

9 ingrid=N*Pfac;% number of interpolated grid points

10 % (Gauss-Chebyshev-Radau)

11

12 %%%%%Read functions for Matrix of Chebyshev

13 %%%%% matrices and initial guess of the

14 %%%%%coefficients

15 [T,Tp,Tpp,E,Bl,Br]...

16 =cheb exact deriv Poiss(coeff,ingrid);

17 [TLEX,Ec,tt]=Initial guess Cheb Gauss(coeff);

18 y=1+(E-1)*(1-xminr)/2;

19 vd1=vd(1); vd1m=vd(end);

20 vr1=(6*Lc/vd1)ˆ(1/2);vs1=(3*Lc/vd1/2)ˆ(1/2);

21 %Boundary condition at E=1

22 vrm1=(-(1+nu)/nuˆ(2)/vd1m)ˆ(1/2);

23 vsm1=(-Lcˆ(2)*(1+nu)/vd1m)ˆ(1/2);

24 %Boundary condition at E=-1

311


25

26 %%%%%%%Initial guess of LE2 function %%%%%%

27 Ec=1+(Ec-1)*(1-xminr)/2;

28 Guess r=log(vr1)*(1/2+Ec/2).ˆ(0.1)...

29 +log(vrm1)*(1/2-Ec/2).ˆ(0.1);

30 Guess s=log(vs1)*(1/2+Ec/2).ˆ(0.1)...

31 +log(vsm1)*(1/2-Ec/2).ˆ(0.1);

32 cr=TLEX*Guess r;

33 cs=TLEX*Guess s;

34 change=1;%norm of the correction term

35 it=0;%iteration number

36 c=zeros(2*coeff,1);%define the coefficients of 2N vector

37 while change>1e-100&&change<1e15&&it<30

38 vr=T*cr;vrp=Tp*cr;

39 vs=T*cs;vsp=Tp*cs;

40 %Nonlinear operator for the 4N coefficients

41 O=zeros(2*coeff,1);

42 %O(1:ingrid)=2*vrp.*(1-E).*(1+E)+...

43 (2*nu*L*(1-E)-(1+E))+4*L*exp(vs-vr);

44 %O(ingrid+1:2*ingrid)=...

45 %2*vsp.*(1-E).*(1+E)+((1+E)+2*L*(nu-1)*(1-E))+...

46 4*L*(1+E).*exp(-vr+vs)-4*exp(2*vs).*vd;

47 %

48 O(1:ingrid)=2*(1-y).*(1+y).*vrp*(2/(1-xminr))+...

49 (-(1+y)+2*Lc*nu*(1-y))+4*exp(vs-vr);

50 O(ingrid+1:2*ingrid)=2*vsp.*(1-y).*(1+y)*(2/(1-xminr))...

51 +(1+y)+2*Lc*(nu-1)*(1-y)...

52 +8*exp(vs-vr)-4*exp(2*vs).*vd/Lc;

53 %O(2*ingrid+1)=Br(1,:)*cr-log(vr1);

54 %O(2*ingrid+2)=Br(1,:)*cs-log(vs1);

55 %N-by-N Jacobian

312


56 OL=zeros(2*coeff);

57 OL(1:ingrid,1:coeff)=diag(-4*exp(vs-vr))*T...

58 +diag(2*(1-y).*(1+y))*Tp*(2/(1-xminr));

59 OL(1:ingrid,coeff+1:2*coeff)=diag(4*exp(vs-vr))*T;

60 OL(1+ingrid:2*ingrid,1:coeff)=diag(-8*exp(-vr+vs))*T;

61 OL(1+ingrid:2*ingrid,coeff+1:2*coeff)=...

62 diag(2*(1-y).*(1+y))*Tp*(2/(1-xminr))+...

63 diag(8*exp(-vr+vs)-8*exp(2*vs).*vd/Lc)*T;

64

65 %OL(1:ingrid,1:coeff)=diag(-4*L*exp(vs-vr))*T...

66 +diag(2*(1-E).*(1+E))*Tp;

67 %OL(1:ingrid,coeff+1:2*coeff)=diag(4*L*exp(vs-vr))*T;

68 %OL(1+ingrid:2*ingrid,1:coeff)=...

69 diag(-4*L*(1+E).*exp(-vr+vs))*T;

70 %OL(1+ingrid:2*ingrid,coeff+1:2*coeff)=...

71 diag(2*(1-E).*(1+E))*Tp+...

72 % diag(4*L*(1+E).*exp(-vr+vs)-8*exp(2*vs).*vd)*T;

73

74 %OL(2*ingrid+1,1:coeff)=Br(1,:);

75 %OL(2*ingrid+2,coeff+1:2*coeff)=Br(1,:);

76

77 Del=-OL\O;%correction to the coeffcients an

78 c(1:coeff)=cr;c(coeff+1:2*coeff)=cs;

79 cnew=c+Del;%iterated coefficients

80

81 change=norm(cnew-c,inf);

82 c=cnew;

83

84

85

86

313


87 cr=c(1:coeff);cs=c(coeff+1:2*coeff);

88 it=it+1;

89 end

90 r=zeros(length(E),1);

91 s=zeros(length(E),1);

92 for lg=1:coeff

93 r=r+c(lg)*cos((lg-1)*(acos(E)));

94 s=s+c(lg+coeff)*cos((lg-1)*acos(E));

95 end

96 ratio=zeros(N-1,1);

97 for p=1:coeff

98 ratio(p)=abs(cr(p)/cr(1));

99 end

100 %r=exp(r);s=-exp(s);

101 cr1=TLEX*r;

102 vr1=T*cr1;

103

104 diff=abs(r-vr1);

105 change r=change;

106 end

107 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

108 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

1 %%%%%%%%%%%% numerical differentiation

2 %%%%% for Poisson equaiton %%%%%%%%%%%%%%%%%%

3

4 function [Tc,Tpc,Tppc,Ec,Bcl,Bcr]=...

5 cheb exact deriv Poiss(coeff,ingrid)

6 n=0:1:coeff-1;

7 i=1:ingrid;i=i';

314


8 Bcl=zeros(3,coeff);

9 Bcr=zeros(3,coeff);

10

11

12 %t=(i)*pi/(coeff-1);%Gauss cheb Lobatto

13 %t=(pi*(2*i)/(2*coeff-1));%Gauss Cheb Radau p

14 %t=pi*(2*i-1)/(2*coeff-1);%Gauss-Radau m

15 t=(2*i-1)*pi/(2*coeff);%Gauss CHeb

16 %E=cos(t);%Radau(BC at E=-1)/Lobatto/Cheb

17 Ec=(cos(t));%Radau(BC at E=1)

18

19

20 Tc=cos(t*n);

21 Tpc=diag(1./sin(t))*sin(t*n)*diag(n);

22 Tppc=diag(cos(t)./(sin(t)).ˆ3)*sin(t*n)*diag(n)...

23 -diag(1./(sin(t)).ˆ2)*cos(t*n)*diag(n.ˆ2);

24

25 Bcl(1,:)=(cos(n*pi));

26 Bcr(1,:)=ones(size(n));

27 Bcl(2,:)=(n.*n.*cos(n*pi)/(cos(pi)));

28 %Radau(BC at E=-1)

29 %Bcl(2,:)=n.*n.*cos(n*pi)/(cos(0));

30 %Radau(BC at E=1)

31 Bcr(2,:)=n.*n;

32 end

1

2 %%%%%%%%%%%%%%%%%%%%%%% Vector for the 4ODE

3 %%%%%and Q integral $$$$$$$$$$$$$$$$$$$

4

315


5 function [O]=Equations linQ fixed a BC Radau m(a, c2, c3, c4, cF, cL, ...

cK, cG, cQ, T, Tp, Tpp, Iq, Iqp1, Iqm1, ingrid, coeff, Lc, Ep, Em, ...

y, Br, Bl);

6

7 vL=T*cL;vLp=Tp*cL;

8 vG=T*cG;vGp=Tp*cG;

9 vF=T*cF;vFp=Tp*cF;

10 vK=T*cK;vKp=Tp*cK;

11 vQ=T*cQ;vQp=Tp*cQ;

12

13 O=zeros(5*coeff+2,1);

14

15 %% 2 ODES %%

16 O(1:1*ingrid)=(Ep).ˆ(a*Lc-1).*(2*Ep).*vFp.*(vK+vG)...

17 /Lc+(Ep).ˆ(a*Lc-1).*(a*(vK+vG)...

18 +Ep.*(4*a*vL/(2*a-3)-1))+a*c2*exp(-vF).*(1+vL);

19

20 %O(1:1*ingrid)=(Ep).ˆ(a*Lc-1).*(2*Ep).*vFp.*(vK+vG)...

21 ./(1+vL)/Lc+(Ep).ˆ(a*Lc-1).*(a*(vK+vG)...

22 % +Ep.*(4*a*vL/(2*a-3)-1))./(1+vL)+a*c2*exp(-vF);

23

24 O(ingrid+1:2*ingrid)=(2*Ep).*vQ.*(vKp+vK.*vFp)/Lc...

25 +vK.*((-0.5*a+7/4-1/Lc).*vQ+3*(2*Ep).*vQp/Lc)+(Ep).*vQ;

26

27 O(2*ingrid+1:3*ingrid)=vGp.*(2*Ep)/Lc...

28 +(vFp.*(2*Ep)/Lc+a+1-1/Lc).*vG-Ep;

29

30 O(3*ingrid+1:4*ingrid)=(2*Ep).*vQ.*...

31 (vLp+vFp.*(vL-(2*a-3)/4/a))/Lc...

32 +vL.*((-0.5*a+3/4).*vQ+3*(2*Ep).*vQp/Lc)-vQ*(2*a-3)/4;

33

316


34 O(4*ingrid+1:5*ingrid)=vQ-Iq;

35 %%Boundary conditions %%

36

37 O(5*ingrid+1)=Br(1,:)*cF-log(1e-0);

38 O(5*ingrid+2)=Bl(1,:)*cF-log(c4);

39 O(5*ingrid+3)=Bl(1,:)*cK-0;

40 %O(4*ingrid+4)=Br(1,:)*cG-c3;

41 O(5*ingrid+4)=Bl(1,:)*cG-0;

42 O(5*ingrid+5)=Br(1,:)*cL-0;

43 O(5*ingrid+6)=Bl(1,:)*cL+1;

44 %O(5*ingrid+8)=Br(1,:)*cQ-Iqp1;

45 O(5*ingrid+7)=Bl(1,:)*cQ-Iqm1;

46 end

1 %%%%%%%%%%%%%%%%%%%%%%% Jacobian for the

2 %%%%% 4ODE and Q integral $$$$$$$$$$$$$$$$$$$

3

4 function [OL]=Jacobian linQ fixed a BC Radau m(a, c2, c3, c4, cF, cL, ...

cK, cG, cQ, T, Tp, Tpp, Iq, Iqp1, Iqm1, ingrid, coeff, Lc, Ep, Em, ...

y, Br, Bl);

5

6

7 vL=T*cL;vLp=Tp*cL;

8 vG=T*cG;vGp=Tp*cG;

9 vF=T*cF;vFp=Tp*cF;

10 vK=T*cK;vKp=Tp*cK;

11 vQ=T*cQ;vQp=Tp*cQ;

12

13 OL=zeros(5*coeff+2);

14

317


15 %%%%% 1:coeff≤> cf, coeff+1:2*coeff ≤>ck,

16 %%%%% 2*coeff+1:3*coeff ≤>cg, 3*coeff+1:4*coeff ≤>cL

17

18 %%%% Equation 1 %%%%%%

19 %

20 %O(1:1*ingrid)=(Ep).ˆ(a*Lc-1).*(2*Ep).*vFp.*(vK+vG)/Lc...

21 % +(Ep).ˆ(a*Lc-1).*(a*(vK+vG)...

22 %+Ep.*(4*a*vL/(2*a-3)-1))+a*c2*exp(-vF).*(1+vL);

23

24 OL(1:1*ingrid,1:1*coeff)=...

25 diag(-a*c2*exp(-vF).*(1+vL))*T...

26 +diag((Ep).ˆ(a*Lc-1).*(2*Ep).*(vK+vG)/Lc)*Tp;

27 OL(1:1*ingrid,1+coeff:2*coeff)=...

28 diag((Ep).ˆ(a*Lc-1).*(2*Ep).*vFp/Lc...

29 +(Ep).ˆ(a*Lc-1).*(a))*T;

30 OL(1:1*ingrid,1+2*coeff:3*coeff)=diag((Ep).ˆ(a*Lc-1)...

31 .*(2*Ep).*vFp/Lc+(Ep).ˆ(a*Lc-1).*a)*T;

32 OL(1:1*ingrid,1+3*coeff:4*coeff)=diag((Ep).ˆ(a*Lc-1)...

33 .*Ep*4*a/(2*a-3)+a*c2*exp(-vF))*T;

34 OL(1:1*ingrid,5*coeff+1)=a*exp(-vF).*(1+vL);

35

36 %O(1:1*ingrid)=(Ep).ˆ(a*Lc-1).*(2*Ep).*vFp...

37 %.*(vK+vG)./(1+vL)/Lc+(Ep).ˆ(a*Lc-1).*(a*(vK+vG)...

38 % +Ep.*(4*a*vL/(2*a-3)-1))./(1+vL)+a*c2*exp(-vF);

39

40 %OL(1:1*ingrid,1:1*coeff)=diag(-a*c2*exp(-vF))*T...

41 % +diag((Ep).ˆ(a*Lc-1).*(2*Ep)...

42 %.*(vK+vG)./(1+vL)/Lc)*Tp;

43 %OL(1:1*ingrid,1+coeff:2*coeff)=...

44 %diag((Ep).ˆ(a*Lc-1).*(2*Ep).*vFp./(1+vL)/Lc...

45 % +(Ep).ˆ(a*Lc-1).*(a)./(1+vL))*T;

318


46 %OL(1:1*ingrid,1+2*coeff:3*coeff)=...

47 %diag((Ep).ˆ(a*Lc-1).*(2*Ep).*vFp./(1+vL)/Lc...

48 % +(Ep).ˆ(a*Lc-1)./(1+vL).*a)*T;

49

50 %OL(1:1*ingrid,1+3*coeff:4*coeff)=diag(-(Ep)...

51 %.ˆ(a*Lc-1).*(2*Ep).*vFp.*(vK+vG)./(1+vL).ˆ2/Lc...

52 % -(Ep).ˆ(a*Lc-1).*(a*(vK+vG)...

53 %+Ep.*(4*a*vL/(2*a-3)-1))./(1+vL).ˆ(2)+(Ep)...

54 %.ˆ(a*Lc-1).*(Ep.*(4*a/(2*a-3)))./(1+vL))*T;

55

56 %OL(1:1*ingrid,5*coeff+1)=a*exp(-vF);

57

58

59

60 %%%% Equation 2 %%%%%%

61 %

62 %O(ingrid+1:2*ingrid)=(2*Ep).*vQ.*(vKp+vK.*vFp)/Lc...

63 %+vK.*((-0.5*a+7/4-1/Lc).*vQ+3*(2*Ep).*vQp/Lc)+(Ep).*vQ;

64

65

66

67 OL(1+ingrid:2*ingrid,1:1*coeff)=...

68 diag(vQ.*(2*Ep).*(vK)/Lc)*Tp;

69 OL(1+ingrid:2*ingrid,coeff+1:2*coeff)=...

70 diag(vQ.*(2*Ep).*vFp/Lc+(-0.5*a+7/4-1/Lc)...

71 .*vQ+3*(2*Ep).*vQp/Lc)*T+diag(vQ.*(2*Ep)/Lc)*Tp;

72 OL(1+ingrid:2*ingrid,4*coeff+1:5*coeff)=...

73 diag(vK.*3.*(2*Ep)/Lc)*Tp+diag((2*Ep)...

74 .*(vKp+vK.*vFp)/Lc+vK.*(-0.5*a+7/4-1/Lc)+Ep)*T;

75

76

319


77

78 %%%% Equation 3 %%%%%%

79 %O(2*ingrid+1:3*ingrid)=vGp.*(2*Ep)/Lc...

80 %+(vFp.*(2*Ep)/Lc+a+1-1/Lc).*vG-Ep;

81

82 OL(1+2*ingrid:3*ingrid,1:1*coeff)=...

83 diag((2*Ep).*vG/Lc)*Tp;

84 OL(1+2*ingrid:3*ingrid,2*coeff+1:3*coeff)=...

85 diag((2*Ep)/Lc)*Tp+diag(vFp.*(2*Ep)/Lc+a+1-1/Lc)*T;

86

87

88 %%%% Equation 4 %%%%%%

89 %O(3*ingrid+1:4*ingrid)=(2*Ep).*vQ...

90 %.*(vLp+vFp.*(vL-(2*a-3)/4/a))/Lc...

91 % +vL.*((-0.5*a+3/4).*vQ+3*(2*Ep).*vQp/Lc)...

92 %-vQ*(2*a-3)/4;

93 OL(1+3*ingrid:4*ingrid,1:1*coeff)=...

94 diag(vQ.*(2*Ep).*(vL-(2*a-3)/4/a)/Lc)*Tp;


96 diag((2*Ep).*vQ.*vFp/Lc+((-0.5*a+3/4).*vQ...

97 +3*(2*Ep).*vQp/Lc))*T+diag((2*Ep).*vQ/Lc)*Tp;


99 diag(vL.*3.*(2*Ep)/Lc)*Tp+diag((2*Ep).*(vLp+vFp.*...

100 (vL-(2*a-3)/4/a))/Lc+vL.*(-0.5*a+3/4)-(2*a-3)/4)*T;

101

102

103 %%%% Equation 5 %%%%%%

104 %O(4*ingrid+1:5*ingrid)=vQ-Iq;

105

106 OL(4*ingrid+1:5*ingrid,4*coeff+1:5*coeff)=T;

107

320


108 %%%% BCs %%%%%%

109 %O(5*ingrid+1)=Br(1,:)*cF-log(1);

110 %O(5*ingrid+2)=Bl(1,:)*cF-log(c4);

111 %O(5*ingrid+3)=Bl(1,:)*cK-0;

112 %O(4*ingrid+4)=Br(1,:)*cG-c3;

113 %O(5*ingrid+4)=Bl(1,:)*cG-0;

114 %O(5*ingrid+5)=Br(1,:)*cL-0;

115 %O(5*ingrid+6)=Bl(1,:)*cL+1;

116 %O(5*ingrid+8)=Br(1,:)*cQ-Iqp1;

117 %O(5*ingrid+7)=Bl(1,:)*cQ-Iqm1;

118

119 %%%%%%%%%%%%%% Frechet derivative of the

120 %%%%%nonlinear operator

121 %%%%% 0th derivative %%%%%%

122 OL(5*ingrid+1,1:1*coeff)=Br(1,:);

123 OL(5*ingrid+2,1:1*coeff)=Bl(1,:);

124 %OL(5*ingrid+2,5*coeff+1)=-1/(a+1);

125 OL(5*ingrid+2,5*coeff+2)=-1/c4;

126 OL(5*ingrid+3,coeff+1:2*coeff)=Bl(1,:);

127 %OL(4*ingrid+5,2*coeff+1:3*coeff)=Br(1,:);

128 %OL(4*ingrid+5,4*coeff+3)=-1;

129 OL(5*ingrid+4,2*coeff+1:3*coeff)=Bl(1,:);

130 OL(5*ingrid+5,3*coeff+1:4*coeff)=Br(1,:);


132 %OL(5*ingrid+8,4*coeff+1:5*coeff)=Br(1,:);


134

135 end

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