An Accurate Solution of the Self-Similar Orbit-Averaged Fokker- Planck Equation for Core-Collapsing Isotropic Globular Clusters: Properties and Application

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Yuta Ito The Graduate Center, City University of New York

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This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] AN ACCURATE SOLUTION OF THE SELF-SIMILAR ORBIT-AVERAGED FOKKER-PLANCK EQUATION FOR CORE-COLLAPSING ISOTROPIC GLOBULAR CLUSTERS: PROPERTIES AND APPLICATION

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

ii An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-Planck Equation for Core-Collapsing Isotropic Globular Clusters: Properties and Application

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

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 continuous 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 (∼ billion 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 satisfactorily 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 significant 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 capacity 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 numerical 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 Feje´r’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 coefficients ...... 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 functions ...... 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 structural 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 isothermal 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 distribution function 219

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

PN B.1.1 Truncated integral over the terms 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

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

(1) 8.2 3.1 Chebyshev coeﬃcients {an} for test function h (x) = (0.5 + 0.5x) 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 underlined. 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-

−4 4 eters (13 ≤ N ≤ 560, 10 ≤ FBC ≤ 10 , 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

0 {Qn} 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

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

−2 c4 = 3.52 × 10 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 statistical mechanics...... 132

xvi LIST OF TABLES

9.2 First turning points of caloric curves at small radii. The data for IS are the corresponding 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 compilation works based on the King- and Wilson- models...... 174

2 10.2 Values of χν 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, h i (2010 edition)). The total masses log M are from (Mandushev et al., 1991) where Mo

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, R 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-

∗ −1 ∗ −1 values are compared with c1ex ≡ c1o and c4ex ≡ 3.03155223×10 (= c4o +1×10 ) −12 obtained for (c, d) = (1, 10) and the value of 3I(x = 1) is 7.3 × 10 . The combination (c, d) = (∞, N/A) means the extrapolated function is constant...... 255

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

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 P |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

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

(1) extrapolate the function hN (x) at the points. The dashed horizontal lines show the

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

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

2 The graph compares φ¯N to A ≡ ln (¯r /2) at cutoﬀ radius rcut. The rcut was introduced to avoid numerical divergences atr ¯ → ∞ 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

h 2 i 1/2 6.5 (a) Asymptotic oscillation of LE2 spectral solutions, φ¯N − ln r¯ /2 r¯ , compared to existing semi-analytical solutions for N = 225, L = 36. (b) Relative

deviation of our semi-analytical asymptotic singular solution, φ¯a, from the LE2

spectral solution φ¯N. The φ¯SA is the semi-analytical asymptotic LE2 solution of ¯ r¯2 | − ¯ ¯ | (Soliman and Al-Zeghayer, 2015) and φs = ln 2 . The plot of 1 φN/φa ﬂattens around at r = 1013 because the calculation reaches machine precision...... 101

7.1 LE1 function φ¯N, 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 = φ¯ s for polytropes. The cutoﬀ radii rcut were chosen to avoid divergences atr ¯ → ∞ when MATLAB reaches machine precision...... 104

7.4 Comparison between the spectral solutions, φ¯N, and Horedt’s solutions, φ¯H. . . . . 105

7.5 Relative error of the spectral solutions, φ¯N, from Horedt’s solutions, φ¯H 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

m−3 (b) its magniﬁcation near ( m−1 , ω¯ ). The solution starts from (uE, vE) = (3, 0) and m−3 asymptotically approaches the singular point (us.p, vs.p) = m−1 , ω¯ ...... 108

xxii LIST OF FIGURES

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

α ω¯ m = 6. For larger ¯,r ¯ r¯ φ¯N/Am − 1 ≈ C1 cos (ω lnr ¯ + 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 c4o, and the absolute value of 3I(x = 1), (b) Condition number of the Jacobian matrix for the 4ODEs and Q-integral. The con-

old new −13 dition number was computed when {aq} − {aq} reached the order of 10 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 c4 from their refer-

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

new old Condition number of the Jacobian matrix calculated when {aq} − {aq} reached the order of 10−13 in the Newton iteration...... 119

∗ 8.6 (a) Relative errors of c1 and c4 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 c4 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 c4 from the reference values and characteristics of | 3I(x =

βo ∗ 1) | obtained at each Nbest. The guideline c1o(−Emax) /c4o 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

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

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

2 9.1 (a) Dimensionless m.f. potential Φ and velocity dispersion V(dis) and (b) dimensionless heat ﬂux Fh and stellar ﬂux Fp. The latter also depicts the inverse of

2 3/2 regularized local relaxation time T˜R = D(φ, tc)/[3 (r, tc)] for the sake of comparison...... 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 q P 2 9.5 Escape speeds of stars. The stellar ﬂux F and velocity dispersion V(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 increases 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 potentialϕ ¯(R) of the energy-truncated ss-OAFP model. The corresponding 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) concentrationc ¯ obtained from the energy-truncated ss-OAFP model (m = 3.9) applied to the PPC- and KM- clusters and (b) concentration c based on the King model reported in (Harris, 1996, (2010 edition)). . . . . 182 10.10Completion rate of core collapse against (a) concentrationc ¯ 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

3.9 10.13(a)Dimensionless density profiles Do, exp[−13.88(1 + E)] and δB(3.4, 1.5)(−E) 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 approximated 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 concentration 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) L mapped by x = sin πy , x = tanh √Lmy , and x = 1 − 2 1−y . The parameter values 2 1−y2 2

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

ηi D.3 Asymptotic absolute values of the Chebyshev coefficients for ϕ cos (ωi ln ϕ). 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 − c4/c4o | 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 c4/FBC from the reference eigenvalues c1o and c4o 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 | {a}new −

old −10 ∗ ∗ −3 −6 {a} |≈ 7 × 10 (resulting in | 1 − c4/c4o |≈ 6.0 × 10 | and 3I = 4.4 × 10 ). 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

(tes) F.4 Chebyshev Coeﬃcients for the Q-integral with a discontinuous test function 3R . Recall E = −(0.5 + 0.5x)L, here L = 1...... 257 q √ (m) F.5 Chebyshev Coeﬃcients for 3R and exp(3R) 0.5 + 0.5x for a discontinuous test (tes) function 3D . The dashed guidelines are added for the measure of slow decays. . . 258

(m) (m) G.1 Absolute values of Chebyshev coeﬃcients {Fn } for 3F . In the modiﬁed ss-OAFP

system, 3S , 3Q, 3G, 3I, 3J are not modiﬁed. The maximum values Emax of the truncated 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 I.2 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density 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 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 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 I.4 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density 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

I.5 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density profiles 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. 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 I.6 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density 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 I.7 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density profiles of NGC 6569, NGC 6638, NGC 6715, NGC 6864, NGC 6934, and NGC 7006 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). . . . 275 I.8 Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density 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 brightness 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 magnitude 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 approximation 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 density 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 normal 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 proﬁles of NGC 3201, NGC 6144, NGC 6273, NGC 6352, NGC 6388, and NGC 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

SBo − SB from the model...... 290

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

SBo − SB from the model...... 291 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 brightness 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 various 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 considers 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 particles0 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 CHAPTER 1. INTRODUCTION

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 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 CHAPTER 1. INTRODUCTION

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 CHAPTER 1. INTRODUCTION

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 with improvements of telescopes such as Virgo and Gaia 2. Stellar dynamicists have included in standard theories some realistic factors, such as the mass (distribution) function for stars, stellar evolution, tidal effect, close encounters, and binary formation/interaction. They discussed how the factors affect the relaxation evolution of star clusters. The new observations, however, showed unexpected clusters’ structures. Although dark matter itself has yet to be found in globular clusters, some observation implicated that the formation of globular clusters was possibly subject to mini dark-matter halos and that the halos might have caused the flattening of speed dispersion of stars in the outer skirt of the 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 be exactly correct. 5The dynamical time of a cluster is a few crossing time. The crossing time is time duration for a star to travel across the cluster. 6The m.f. approximation corresponds with the assumption that the total number of stars is large N >> 1 and the cluster is dense in phase space so that one can adequately define the distribution function for stars.

5 CHAPTER 1. INTRODUCTION 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; Binney and Tremaine, 2011) states that if the potential is Keplerian or Harmonic the orbit of particle is closed. On the one hand, 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 numerical estimation of the logarithm, (e.g., Binney and Tremaine, 2011, Ch.7) for the fundamental discussion and Appendix C.2 for the derivation of the logarithm based on statistical first principle.

6 CHAPTER 1. INTRODUCTION

Figure 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 P |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 astronomical 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 CHAPTER 1. INTRODUCTION

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 statistical 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 studies (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 of extensive quantities (entropy, energy. . . ). The examples of ”short-range” potentials are hard-sphere potentials and the Casimir-Polder (Lennard-Jones) potential for neutral gaseous systems, and the Yukawa (Debye-Huckel)¨ potential for classical collisional plasmas.

8 CHAPTER 1. INTRODUCTION

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 dynamics. 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 encounters. 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 statistical 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-wise Newtonian forces.

9 CHAPTER 1. INTRODUCTION

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) r232 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, limited at present. Hence, Appendices A, B, and C explain the fundamental kinetics employed in the present dissertation.

10 CHAPTER 1. INTRODUCTION

the DF in energy-space volume element d. The energy per unit mass is deﬁned as follows

32 = φ(r, t) + , (1.3.2) 2

The binding energy 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 and + d13

! ZZ 32 ∂q(, t) N(, t) d = δ − φ(r, t) + f (r, v, t) d3r d3v d = f (, t) d, (1.3.3) 2 ∂

where the function q(, t) is the phase space volume whose energy is bounded at from above. The q-integral (the integral associated with the radial action14) reads

1 Z rmax(,t) q(, t) = 2 − 2φ(r0, t)3/2 r02 dr0, (1.3.6) 3 0

−1 where rmax(, t) = φ ().

13The DF f (, t) does not represent an energy probability density DF but still phase-space density DF. The function f (, t) means that the function depends only on the energy at time t while N(, t) is exactly an energy density DF at time t. 14The q-integral is related to the radial action of star’s orbits

Z ra(,L) Ir(, L) = 2m 3r dr, (1.3.4) rp(,L) where ra and rp are the apocenter and pericenter of the orbits (or the maximum- and minimum- radii that can be found L2 solving = φ(r) + 2r2 for ﬁxed and L.). An anisotropic system can be described by the DF f (, L, t) of energy and modulus of angular momentum L. This is a complete system as a spherical self-gravitating system (e.g., Binney and Tremaine, 2011). Hence, to make the system isotropic, one must reduce the arguments of f (, L, t) to only . One typically employs ’isotropizing’, i.e., averaging over the modulus of the angular momentum. For example, the isotropized q-integral reads

2 Z Lc () 1 2 q(, t) = Ir(, L) dL , (1.3.5) 4 0 where Lc() is the modulus of the angular momentum of a star moving in a circular orbit. The isotropizing process corresponds with giving up the correct description that the radial velocity dispersion is greater than tangential one in the outer halo.

11 CHAPTER 1. INTRODUCTION

1.3.2 Collisionless relaxation evolution on dynamical time scales

The evolution of DF f (, 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 of f (, 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 interact 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

15 −11 −13 For galaxies and clusters of galaxies tdyn/trel ≈ 10 ∼ 10 . This corresponds that their relaxation times are longer than their ages.

12 CHAPTER 1. INTRODUCTION 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 primary 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 mixing 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 (), (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 () ∼ exp[−/ < 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 time dependent potential. Then, the system can reach a QSS. The phase mixing is based on an integrable potential. The mixing obeys the Liouville’s volume-element invariant theorem. It extends the near-by elements of the DF linearly with 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 an isothermal state in the beginning of evolution. This is since currently-observed globular clusters are considered to have experienced two-body relaxations. Hence, globular clusters might have experienced collisionless relaxation in a diﬀerent way from galaxies.

13 CHAPTER 1. INTRODUCTION

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 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.

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 relaxation- time 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 CHAPTER 1. INTRODUCTION

this sense, the explicit time-dependence of DF may be retrieved, i.e., f (r, v, t) ≈ f ((t), t).18 The corresponding m.f. potential is to be determined by the time-dependent Poisson’s equation

∂2φ 2 ∂φ Z 0 p ρ φ r, t ≡ π2Gm f , t 0 − φ r, t 0. 2 + = ( ) 16 ( ) 2 2 ( ) d (1.3.8) ∂r r ∂r φ(r,t)

The effect of many-body interaction has been conventionally modeled, in the first-order approximation 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 example, 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

! ∂ ∂ + v · ∇ − ∇φ(r, t) · f (r, v, t) = ∂t ∂v ←→ ! ∂ Z | v − v0 |2 I − (v − v0)(v − v0) ∂ ∂ − B · · − f (r, v, t) f (r, v0, t) d3v0, (1.3.9) ∂v 33 ∂v ∂v0 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 ((t), t) includes the effect of two-stage evolution. The time scales of the arguments (t) and t are different. The time dependence of (t) is determined through m.f. potential, meaning (t) changes ‘rapidly’ on dynamical times scales. On the one hand, the argument t is the order of relaxation time scales, hence the explicit time change in DF 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-exact treatment of 1/N-expansion of N-body Liouville equation or Klimontovich-Dupree equation which includes the effects of inhomogeneity in encounter, gravitational polarization, statistical acceleration and/or strong encounter.

15 CHAPTER 1. INTRODUCTION

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) ∂ f (, t) ∂ f (, t) ∂φ(r, t) = + , (1.3.10) ∂t ∂t ∂ ∂t

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 between 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

R 2 − 3 − · 3 3 cluster corresponds to operating the integral δ( 2 φ(r, t)) d rd v (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 applying the orbit-averaging to equation (1.3.9), one obtains the orbit-averaged Fokker-Planck (OAFP) equation

∂ f (, t) ∂q(, t) ∂q(, t) ∂ f (, t) + ∂t ∂ ∂t ∂ ( ) ∂ ∂ f (, t) = Γ f (, t) f (, t)q(, t) − j(, t) + i(, t) + q(, t)g(, t) , ∂ ∂ (1.3.11a)

Γ ≡ (4πGm)2 ln N (1.3.11b)

16 CHAPTER 1. INTRODUCTION

where m is the stellar mass and G is the gravitational constant. The integrals associated with dynamical friction and energy diﬀusion read

Z i(, t) ≡ f 0, t q 0, t d0, (1.3.12a) φ(0,t) Z ∂ f (0, t) ≡ 0 0 j(, t) 0 q , t d , (1.3.12b) φ(0,t) ∂ Z 0 g(, t) ≡ f 0, t d0. (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 dynamical 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 CHAPTER 1. INTRODUCTION

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 confirm 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 (, t) ∝ (−)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 → 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 (, t)/ f (φ(0, t), t) in the stellar DF reaches the order of machine precision at ≈ 0.01.21 One can correctly understand how the finer structure in f (, 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 DF reaches the precision at = 0.012.

18 CHAPTER 1. INTRODUCTION

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 considered ‘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 coefficients 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 explicit 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 degree 19 CHAPTER 1. INTRODUCTION

a few to tens if the solution is regular and analytic on a finite domain (See Chapter 3). The binding energy 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, introduces unused polynomials, and combines the conventional methods to other approaches (wavelet analysis, multi-domain, Adomian decomposition method, ...) (ii) One has employed the conventional 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 functions 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 CHAPTER 1. INTRODUCTION 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 extending 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 importance. 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 contribute 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 CHAPTER 1. INTRODUCTION

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 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 CHAPTER 1. INTRODUCTION

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 chapters. 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 CHAPTER 1. INTRODUCTION 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 equations 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 between 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 unbounded 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 and Infrared 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 () 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 collisionless 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 mathematical convenience). Hence, a mathematical program to establish a QSS model is to choose a predetermined time-independent DF f () first, and then find the m.f. potential due to the DF by solving time-independent Poisson’s equation

d2φ 2 dφ Z max q πGρ φ r ≡ π f − φ r 0, 2 + = 4 ( ) 4 ( ) 2 ( ) d (2.1.1) dr r dr φ(r)

where the domain is defined by ∈ φ(r), max). The max 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 CHAPTER 2. MATHEMATICAL MODELS

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)

ZZ Z ∂q(, t) S (t) = f (r, 3, t) ln f (r, 3, t) d33d3r = (4π2) f (, t) ln f (, t) d, (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)

!2 dS Γ ZZ max ∂ f (, t) ∂ f (0, t) 0 − 0 0 = f (, t) f , t 0 min q(, t), q , t dd , dt 2 φ(0,t) ∂ ∂ ≥ 0, (2.1.3) where min q(, t), q(0, t) is the minimum of q(, t) and q(0, t). Hence, the possible ﬁnal stationary state4 of the isotropic OAFP model is an isotropic Maxwell-Boltzmann DF (or Maxwellian DF)

ρc 2 f () = exp −/σc , (2.1.4) 23/2 2πσc 3The H-theorem is correct even for more rigorous models based on statistical first principles, including the effect of 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 isotropic Maxwellian 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 correct since 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 mathematical program 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 DF in collisionless dynamics (due to violent relaxation and mixing) will be found in (e.g., Nakamura, 2000; Bindoni and Secco, 2008).

27 CHAPTER 2. MATHEMATICAL MODELS

2 where ρc is the central density and σc the normalized temperature. The temperature satisﬁes the p R equipartition relation with the velocity dispersion 32 as follows < 32 > /2 = 4π f ()32d3 =

2 3σc/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

2 RcE/GM 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 review 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 isothermal 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 results (Taruya and Sakagami, 2003, 2005) predict that the globular cluster may reach the isothermal sphere subject to a certain initial DF.

28 CHAPTER 2. MATHEMATICAL MODELS

The radial density proﬁle of the isothermal sphere reads

" # φ(r) − φ(0) ρ r ρ − . ( ) = c exp 2 (2.1.5) σc

For numerical calculation, the potential φ(r), radius r, and density ρ(r) reduce to the following dimensionless forms

φ(r) − φ(0) φ¯ r − , ( ) = 2 (2.1.6a) σc s 4πGρ r r c , ¯ = 2 (2.1.6b) σc ρ(r) ρ¯(r) = . (2.1.6c) ρc

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¯ 2 dφ¯ + − e−φ¯ = 0, (2.1.7) d¯r2 r¯ d¯r with the following boundary conditions

dφ¯ φ¯(¯r = 0) = 0, (¯r = 0) = 0. (2.1.8) d¯r 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 analysis in applied mathematics (Davis, 1962). It describes a self-gravitating ideal gas of particles interacting through pairwise Newtonian potential and direct particle collisions. The gas is in a state of hydrostatic equilibrium at a constant temperature. 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 CHAPTER 2. MATHEMATICAL MODELS

One also can ﬁnd the LE2 equation for dimensionless densityρ ¯ using equation (2.1.5)

!2 ! d2ρ¯ d¯ρ 2 d¯ρ ρ¯ − + ρ¯ + ρ¯3 = 0, (2.1.9) d¯r2 d¯r r¯ d¯r with the following boundary conditions

d¯ρ ρ¯(¯r = 0) = 1, (¯r = 0) = 0. (2.1.10) d¯r

Using equations (2.1.8) and (2.1.10), one obtains the Taylor expansions aroundr ¯ = 0

r¯2 1 φ¯(¯r) = − r¯4 + ..., (2.1.11a) 6 120 r¯2 1 ρ¯(¯r) = 1 − + r¯4 − .... (2.1.11b) 6 45

The asymptotic approximations asr ¯ → ∞ are

 √  C3  7  φ¯(¯r) ≈ φ¯ s(¯r) + cos  lnr ¯ + C4 , (2.1.12a) r¯1/2  2    √   C3  7  ρ¯(¯r) ≈ ρ¯ s(¯r) 1 + cos  lnr ¯ + C4 , (2.1.12b)  r¯1/2  2 

where C3 and C4 are constants of integration. The isothermal sphere modeled by

2 φ¯ (¯r) = ln , (2.1.13a) s r¯2 2 ρ¯ (¯r) = , (2.1.13b) s r¯2 is called the singular isothermal sphere.

30 CHAPTER 2. MATHEMATICAL MODELS

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

1 ρ(r) ∝ , (2.1.14) rα

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

  2.20, Self-similar moment model (Louis, 1990),    2.208, Self-similar gaseous model (Lynden-Bell and Eggleton, 1980), α =  (2.1.15)  2.23, Time-dependent OAFP model (Cohn, 1980),    2.23, Self-similar OAFP model (Heggie and Stevenson, 1988).

31 CHAPTER 2. MATHEMATICAL MODELS

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

  2.36, for N = 3.2 × 104 (Makino, 1996),    3 2.33, for N = 4 and 8 × 10 (Baumgardt et al., 2002a)    4 2.26, for N = 6.4 × 10 (Baumgardt et al., 2003), α =  (2.1.16)  2.2, for N = 4 × 103 ∼ 6.5 × 104 (Gieles et al., 2013)    2.32 ± 0.07, for N = 1.0 × 104 (Pavl´ık and Subr,ˇ 2018)    4 ˇ 2.27 ± 0.03. for N = 5.0 × 10 (Pavl´ık and Subr, 2018)

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- and particle- flow occurs due to the spatial change in gas temperature through the collisions. Those models are numerically more 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 artificially cutoff the higher order moments for the moment models and heuristically determine the mathematical expression of thermal conductivity for the gaseous conductive models. 7The anisotropic models are more exact than isotropic ones since they include the effect of anisotropic velocity dispersion or the modulus of angular momentum. 8The N-body simulations include the effect of binaries, strong encounter and ejection while the continuum models do not.

32 CHAPTER 2. MATHEMATICAL MODELS

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

  2.2, NGC 7078 (or M15) (Gebhardt et al., 1997),   α = 3.5, MGC1 in NGC 224 (or M31) (Mackey et al., 2010), (2.1.17)    3.42 and 3.48. JM81GC-1 and JM81GC-2 (Jang et al., 2012),

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. Projected surface 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 employed for basic N-body direct simulations for isolated systems. The clusters exist far away from the host galaxies and the rest of the clusters.

33 CHAPTER 2. MATHEMATICAL MODELS

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

 !m−3/2  ρc m!  − , ( < 0)  23/2 (m − 3/2)! σ2 f () =  2πσc c (2.1.18)   0, ( > 0)

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

m! equation (2.1.18), the factor (m−3/2)! is inserted so that the corresponding density proﬁle has a ﬁnite central density ρc at r = 0

!m φ(r) ρ r ρ − . ( ) = c 2 (2.1.19) σc 11The original polytropic sphere for stellar structure in (Chandrasekhar, 1939) is at a state of local thermodynamic state. 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 CHAPTER 2. MATHEMATICAL MODELS

For numerical calculations, the potential φ(r), radius r, and density ρ(r) may reduce to dimensionless forms

φ(r) φ¯ r − , ( ) = 2 (2.1.20a) σc s 4πGρ r r c , ¯ = 2 (2.1.20b) σc ρ(r) ρ¯(r) = . (2.1.20c) ρc

Then, the time-independent Poisson’s equation (2.1.1) reduces to the Lane-Emden equation of the ﬁrst-kind (LE1 equation)

d2φ¯ 2 dφ¯ + + φ¯m = 0. (2.1.21) d¯r2 r¯ d¯r

The function φ¯ is called the LE1 function when it satisﬁes the LE1 equation with the boundary conditions

dφ¯ φ¯(¯r = 0) = 1, (¯r = 0) = 0. (2.1.22) d¯r

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¯

2 is ﬁxed to unity atr ¯ = 0 (in place of −φ(0)/σc) 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 engineering, 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 and isothermal 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 and Zweibel, 1995). As the most fundamental example, stars (like Sun) on the main-sequence may be modeled by a polytrope of m = 3 for the inner region (to include radiation pressure) and polytrope of m = 1.5 for the outer region (that is fully convective).

35 CHAPTER 2. MATHEMATICAL MODELS

In stellar-dynamics studies, one may constrain the range of index m and power α for globular 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

2m α α = or m = , (2.1.23) m − 1 α − 2

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

  2  r¯ m 4 1 − + r¯ + ..., (¯r → 0) φ¯(¯r) =  6 120 (2.1.24)  C φ¯ (¯r) 1 + 1 cos (ω lnr ¯ + C ) ≡ φ¯ (¯r), (¯r → ∞)  s r¯η 2 a 13The Plummer model is not realistic since the radius of the model reaches inﬁnity. However, stellar dynamicists have used it to quantitatively discuss the relationships among the total mass, total energy, and velocity dispersion since their 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 CHAPTER 2. MATHEMATICAL MODELS

where C1 and C2 are the constants of integration, and ω and η read √ 7m2 − 22m − 1 ω = , (2.1.25a) 2(m − 1) m − 5 η = . (2.1.25b) 2(m − 1)

The φ¯ s(¯r) is called the singular solution of the LE1 equation. Its explicit expression is

A φ¯ (¯r) = m , (2.1.26a) s r¯ω¯ " #ω/¯ 2 2(m − 3) A = , (2.1.26b) m (m − 1)2 2 ω¯ = . (2.1.26c) m − 1

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 CHAPTER 2. MATHEMATICAL MODELS

(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 ≥ 0 and reduces to the lowered-Maxwellian DF

  ρ h i  c exp −/σ2 − 1 , ( < 0) (2πσ2)I(K) c f () =  c (2.1.27)  0, ( > 0)

where the constant K and function I(x) are

φ(0) K − , = 2 (2.1.28a) σc r " # √ 4x 2x I(x) = exp(x)erf x − 1 + , (2.1.28b) π 3

h i R x h i √ where erf(x) is the error function; erf(x) = 2 exp x2 exp −x2 dx/ π. We inserted I(x) in 0

equation (2.1.27) so that the density can have a ﬁnite density ρc at r = 0

2 I −φ(r)/σc ρ(r) = ρ . (2.1.29) c I(K)

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 potential φ¯ in terms of dimensionless radiusr ¯

d2φ¯ 2 dφ¯ I(φ¯) + − = 0. (2.1.30) d¯r2 r¯ d¯r I(K)

16The rest of the clusters are considered at a phase of post core collapse whose central core has a cusp (a steep slope in a power-law density proﬁle diverging at r = 0) unlike the King model and isothermal sphere whose cores are relatively ﬂattened (See, e.g., Lugger et al., 1995; Noyola and Gebhardt, 2006; de Boer et al., 2019, for the observed data of the cores of collapsed-core clusters). 38 CHAPTER 2. MATHEMATICAL MODELS

The King model satisﬁes the boundary conditions

dφ¯ φ¯(¯r = 0) = K, (¯r = 0) = 0. (2.1.31) d¯r

The local approximation near r = 0 is

√ q K 2 K K − r¯ 2 exp( )erf( ) π φ¯(¯r) = 1 − + r¯4 + ... (2.1.32) 6 120I(K)

The lowered-Maxwellian DF in the limit of → 0− behaves like the polytrope of index m = 2.5

ρ − f → − c . ( 0 ) = 2 2 (2.1.33) (2πσc)I(K) σc

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

5π I(φ¯ → 0) = − √ φ¯3/2. (2.1.34) 6 2

Correspondingly, the King model reduces to a kind of the LE1 equation of m = 2.5

" #2.5 d2φ¯ 2 dφ¯ φ¯(¯r) + + = 0. (φ¯ → 0) (2.1.35) d¯r2 r¯ d¯r K

The polytrope of m = 2.5 is known to be ﬁnite in size (Chandrasekhar, 1939). This implies that as the radiusr ¯ 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 CHAPTER 2. MATHEMATICAL MODELS

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 proﬁleρ ¯, 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

s 4πGρ r c . Kin = 2 (2.1.37) 9σc

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

" # r c = log tid . (2.1.38) rKin

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 clusters 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 (Pavl´ık

40 CHAPTER 2. MATHEMATICAL MODELS and Subr,ˇ 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 evolution 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 = /t(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 FP equation) was first done in (Henon,´ 1961, 1965). In the work, the system is discussed with the physical conditions that a star cluster is immersed in the host galaxy. This condition is different from the present work and (Cohn, 1980; Heggie and 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 conditions are strictly imposed with certain numerical values or in behavioral fashions. For example, Henon´ (1961) assumed that the system size is limited and the DF is zero at a certain tidal radius by the tidal effect due to the host-galaxy potential. Also, Henon´ (1965)’ model is a cluster with an infinite radius, but the potential of the system is assumed to follow 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 CHAPTER 2. MATHEMATICAL MODELS

and for dependent variables

F(E) = f (, t)/ ft(t), (2.2.2a)

Q(E) = q(, t)/qt(t), (2.2.2b)

Φ(R) = φ(r, t)/φt(t), (2.2.2c)

I(E) = i(, t)/it(t), (2.2.2d)

J(E) = j(, t)/ jt(t), (2.2.2e)

G(E) = g(, 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)

dF 2c − 3c [I(E) + G(E)Q(E)] = c Q(E)F(E) + 1 2 J(E) − F(E) [F(E)Q(E) − J(E)] , (2.2.3a) dE 1 4 dG = −F(E), (2.2.3b) dE dI = Q(E)F(E), (2.2.3c) dE dJ dF = Q(E) , (2.2.3d) dE dE

the Q-integral

Z Rmax(E) 1 03/2 03 0 −1 Q(E) = E − Φ R R dR , Rmax(E) = Φ (E) , (2.2.4) 3 0

and the Poisson’s equation

∂2Φ 2 ∂Φ Z 0 p D R ≡ F E0 E0 − R E0. 2 + = [Φ( )] Φ( )d (2.2.5) ∂R R ∂R Φ(R)

42 CHAPTER 2. MATHEMATICAL MODELS

The relationships among the time-dependent variables read

φt(t) = t(t), (2.2.6a) 1 r (t) = , (2.2.6b) t √ √ 1/2 2 16 2π mG ft(t) t(t) 3 2 2t(t)[rt(t)] q (t) = . (2.2.6c) t 2

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

1 d c1 = ln[ ft(t)], (2.2.7a) Γ ft(t) dt 1 d c2 = ln[Et(t)], (2.2.7b) Γ ft(t) dt

and the corresponding physical parameters are

2(3 + 2β) α = , (the exponent of power-law density proﬁle) (2.2.8a) 2β + 1 c + c ξ = 1 √2 , (core-collapse rate) (2.2.8b) 0.167 π

FBC(FBC − c1) χesc = , (scaled escape energy), (2.2.8c) c3

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 CHAPTER 2. MATHEMATICAL MODELS

The boundary conditions for the 4ODEs and the Q-integral are

β F(E → 0) = c4(β + 1)(−E) , F(E = −1) = FBC, (2.2.9a)

β+1 G(E → 0) = c4(−E) , G(E = −1) = c3, (2.2.9b) 4(β + 1) I(E → 0) = c (−E)β+1Q(E → 0), I(E = −1) = 0, (2.2.9c) 4 2β − 7 4β(β + 1) J(E → 0) = −c (−E)βQ(E → 0), J(E = −1) = 0, (2.2.9d) 4 2β − 3 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) = 0, Φ(R = 0) = −1. (2.2.10) dR

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 CHAPTER 2. MATHEMATICAL MODELS 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 Feje´r’s ﬁrst-rule Quadrature

The present chapter details the Gauss-Chebyshev pseudo-spectral method. Section 3.1 describes a series expansion of a given function by Chebyshev polynomials. For numerical calculations, 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 quadrature 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

h −1 i Tn(x) = cos n cos (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)

X∞ h(x) = anTn−1(x), (3.1.2) n=1

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 domain 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;

(N+1) h(x) = hN(x) + Res (x), (3.1.3)

47 CHAPTER 3. SPECTRAL METHOD

(N+1) where the truncated function hN(x) and the residue Res (x) read

XN hN(x) = anTn−1(x), (3.1.4a) n=1 ∞ (N+1) X Res (x) = anTn−1(x). (3.1.4b) n=N+1

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)

Z 1 Tn−1(x)Tm−1(x) π √ dx = δn,m, (m = 2, 3, ..., N) (3.1.5) −1 (1 + x)(1 − x) 2

one can ﬁnd the explicit form of Chebyshev coeﬃcients in terms of hN(x)

Z 1 2 hN(x)Tn−1(x) an = √ dx. (3.1.6) π −1 (1 + x)(1 − x)

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 problems (Boyd, 2001; Trefethen, 2000). One can resort to the Gauss-Radau nodes for an interval open at an end and closed 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- and Gauss-Radau- Chebyshev-collocation methods are applied to initial-value- and boundary-value- problems. 48 CHAPTER 3. SPECTRAL METHOD

method for the sake of brevity. Refer to (Ito et al., 2018) for the Gauss-Radau spectral method. Gauss-Chebyshev nodes are

! 2k − 1 x = cos(t ) ≡ cos π . (k = 1, 2, 3 ··· N) (3.1.7) k k 2N

The discrete Gauss-Chebyshev polynomials Tn x j of the ﬁrst kind satisfy the orthogonality condition (e.g., Mason and Handscomb (2002))

   0, (1 < n , m < N) N  X  di j ≡ Tn−1 x j Tm−1 x j = N, (n = m = 1) (3.1.8)  j=1   N ≤  2 . (1 < n = m N)

The discrete Gauss-Chebyshev polynomial expansion of h(x) and its derivative read

XN hN(x j) = anTn−1(x j), (3.1.9a) n=1 N −1 dhN x j X (n − 1) sin [n − 1] cos x j = an , (3.1.9b) dx −1 n=1 sin cos x j

Hence, the Chebyshev coeﬃcients in terms of the discrete function hN(x j) are

1 XN a = T (x )h (x ), (3.1.10a) 1 N 0 j N j j=1 2 XN a = T (x )h (x ). (2 ≤ n ≤ N) (3.1.10b) n N n−1 j N j j=1

49 CHAPTER 3. SPECTRAL METHOD

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))

X∞ (N+1) EN+1 = max Res (x) = max kh(x) − hN(x)k ≤ | ak |, (3.2.1) −1

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 CHAPTER 3. SPECTRAL METHOD

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)

!8.2 1 + x h(1)(x) = . (3.2.3) 2

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-

(1) tion hN (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

(1) (1) the degree-truncated function hN (x) from the original function h (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

2 coeﬃcient aN+1. This simple example well supports the idea of the heuristic relation in equation (3.2.2).

2 The error at N = 25 is the same order as the absolute value of coeﬃcient a26 since it reaches the order of machine precision.

51 CHAPTER 3. SPECTRAL METHOD

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

(1) 8.2 Table 3.1: Chebyshev coefficients {an} for test function h (x) = (0.5+0.5x) with degree N = 30. The coefficients {an} are divided by a1 so that the first coefficient is unity.

3 10 N = 5 N = 10 N = 15 N = 20 N = 25 100 | − ) 10 3 x ( (1) N −6 h 10 − )

x −9

( 10 (1) h

| 10−12

10−15

10−18 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

(1) (1) Figure 3.1: Absolute error between the test function h (x) and degree-truncated one hN (x). On the graph, the graphing points are taken at points on [−1, 1] with the step size 10−1. Although (1) the Gauss-Chebyshev nodes can not reach x = ±1, one still may extrapolate the function hN (x) at the points. The dashed horizontal lines show the absolute values of the Chebyshev coeﬃcients −2 {aN+1} that descends with degrees N + 1 = 6, 11, 16, 21, 26, corresponding to |aN+1/a1| =9.5 × 10 , 1.2 × 10−7, 1.5 × 10−11, 5.9 × 10−14,6.7 × 10−16.

52 CHAPTER 3. SPECTRAL METHOD

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

!8.0 1 + x h(2)(x) = , (3.2.4a) 2 " #! 1 − x h(3)(x) = exp − , (3.2.4b) 2 " #! 1 − x h(4)(x) = exp −13.85 . (3.2.4c) 2

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 (equation (2.1.4)) and the ss-OAFP model (Cohn, 1979; Heggie and Stevenson, 1988). Figure 3.2 depicts 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

h bi Geometric convergence The coeﬃcients {an} exponentially decay ∼ exp −an .

If b > 1, called Super-geometric convergence.

If b = 1, called Geometric convergence.

If b < 1, called Sub-geometric convergence. a Power-law convergence The coeﬃcients {a } decay in a power law ∼ n nb

where a and b are real numbers. To see a more rigorous deﬁnition for the geometric convergence, 53 CHAPTER 3. SPECTRAL METHOD

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 fractional exponent singularities on a complex plane (the effect of a fractional power on the convergence 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 reach machine 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 rapid convergences) besides Chebyshev polynomials.

54 CHAPTER 3. SPECTRAL METHOD

101

10−1

10−3

10−5

10−7

a.u. 10−9

10−11 h(1)(x) 10−13 h(2)(x) 10−15 h(3)(x) h(4)(x) 10−17 0 0.2 0.4 0.6 0.8 1 1+x 2

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

55 CHAPTER 3. SPECTRAL METHOD

107 h(1)(x) 107 h(3)(x) h(2)(x) h(4)(x) 4 4 10 n−2∗8.2−1 10 e−n1.4 −10−6n7.5 −0.065n1.9 101 e 101 e

10−2 10−2 | | n n

a 10−5 a 10−5 | |

10−8 10−8

10−11 10−11

10−14 10−14 (a) (b) 10−17 10−17 5 10 15 20 25 30 5 10 15 20 25 30 index n index n

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 determined ’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

1 Z h(z) h(x) = dz, (3.3.1) 2iπ C z − x

56 CHAPTER 3. SPECTRAL METHOD

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

b Z h(z) a = −1 dz, (3.3.2) n √ √ n−1 iπ C (z + 1)(z − 1) z ± (z + 1)(z − 1)

√ 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)

b h(z) = −1 , (3.3.3) z − zs

where b−1 is the residue. Hence, the Chebyshev coeﬃcients for h(z) read

2b | a |= −1 . (3.3.4) n √ √ n−1 | (zs + 1)(zs − 1) zs ± [zs + 1][zs − 1] |

For n → ∞, equation (3.3.4) is | an |≈ exp µ(zs)n , (3.3.5)

4 where the convergence rate µ(zs) is

p 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 ellipse passing through the point zs on the complex plane. Such a contour may be called the isoconvergence contour (Boyd, 2008b).

57 CHAPTER 3. SPECTRAL METHOD

(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 fractional 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 properties 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 properties 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-

2ψ+1 tion (shown in Section 3.2.2) is due to the effect of the end-point branch point an ∝ 1/n . 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

2 Z π an = h(cos t) cos nt dt. (3.3.7) π 0

The integrand is symmetric about the change of the sign with respect to t. Hence, the coeﬃcients reduce to 1 Z π an = exp (ln h[cos t]) cos nt dt. (3.3.8) π −π 58 CHAPTER 3. SPECTRAL METHOD

Function an(n → ∞) Ref. Branch point at end points h`(x) =| 1 − x |ψ ∝ 1/n2ψ+1 (Elliott, 1964) Logarithmic singularity at end points h`(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) Discontinuity h`(x) = sign(x) ∝ 1/n (Boyd, 2008b) dh` 2 dx = sign(x) ∝ 1/n (Boyd, 2008b)

Table 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.

An analytic continuation provides an ’unconventional’ form (Boyd, 2008b)

Z 1 (int+ln h[cos t]) an = < e dt, (3.3.9) π C 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 CHAPTER 3. SPECTRAL METHOD

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)

! x + 1 r r = rtid or x = 2 − 1. (3.4.1) 2 rtid

Equation (2.1.30) reduces to

d2φ 2 dφ r2 I(φ) + − tid = 0. (3.4.2) dx2 1 + x dx 4 I(K)

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

dφ φ (x) ≡ , (3.4.3a) p dx 2 dφp 2 r I(φ) + φ − tid = 0. (3.4.3b) dx 1 + x p 4 I(K) 60 CHAPTER 3. SPECTRAL METHOD

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 differentiation 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

(n) (n) φp (x) and eigenvalue rtid 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

(n) (n) (n) ’small’ correction functions δφ (x) and δφp (x) and correction eigenvalue δrtid

δφ(n)(x) = φ(x) − φ(n)(x), (3.4.4a)

(n) (n) δφp (x) = φp(x) − φp (x), (3.4.4b)

(n) (n) δrtid = rtid − rtid . (3.4.4c)

The smallness of the correction functions and eigenvalues can provide its ’quasi-linearized’ form

61 CHAPTER 3. SPECTRAL METHOD

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) − = φ(n)(x) − φ(n)(x), (3.4.5a) dx p dφ(n) δφ(n)(x) = −φ(n)(x) + , (3.4.5b) p p dx

and that of equation (3.4.3b)

r 2  (n)  (n) 2 (n) r δφp(n)  p 4φ  dφp 2 r I(φ ) − tid erf φ(n) exp (φ(n)) −  = − − φ(n) + tid , (3.4.6a) 4 I(K)  π  dx 1 + x p 4 I(K) (n) (n) 2 (n) dδφp 2 dφp 2 r I(φ ) + δφ(n) = − − φ(n) + tid . (3.4.6b) dx 1 + x p dx 1 + x p 4 I(K)

The linearized equation regarding the eigenvalue rtid is

h i (n) (n) (n) 2 (n) δr I φ dφp 2 r I(φ ) − tid = − − φ(n) + tid . (3.4.7) 2 I(K) dx 1 + x p 4 I(K)

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

h −1 i t Tˆ ≡ T jn = cos cos x (n − 1) , (3.4.8a)

t Tx=1 ≡ 1 , (3.4.8b)

t Tx=−1 ≡ cos − (n − 1) , (3.4.8c) ! ddT 1 h i = diag sin cos−1 x (n − 1)t diag [nˆ − 1ˆ]t , (3.4.8d) dx sin cos−1 x

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 CHAPTER 3. SPECTRAL METHOD

(n) and δφp (x) may be Chebyshev-expanded as follows

h it (n) ˆ (n) δφ (x) = T δcφ , (3.4.9a) h it δφ(n)(x) = Tˆ δc(n) . (3.4.9b) p φp

(n) (n) where δcφ and δcφp are the vector forms of the Chebyshev coeﬃcients for the correction func-

(n) (n) tions δφ (x) and δφp (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

Jˆ(n)δc(n) = O(n)(x), (3.4.10) where the vector δc represents the changes in Chebyshev coeﬃcients and eigenvalue

t δc = t t . (3.4.11) δcφ δcφp δrtid

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

 (n)   −φ(n)(x) + dφ x   p dx   (n) 2 (n)   dφp x (n) r I(φ (x))   − − 2 tid   dx diag 1+x φp (x) + 4 I(K)    (n)   O (x) =  φ(−1) − K  , (3.4.12)      φ (−1) − 0   p     φ(1) − 0  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 CHAPTER 3. SPECTRAL METHOD

rtid. The corresponding Jacobian Matrix Jˆ (a (2N + 1) × (2N + 1) matrix) reads

               −Tˆ Tˆ 0                        Jˆ(n) =   . (3.4.13)  " #!   2 q 3 (n) n   r 16(φ(n)(x)) δr I[φ( )(x)]   diag − tid 1 I φ(n)(x) + cdT + diag 2 − tid   4 I(K) 9π dx 1+xˆ 2 I(K)                   Tx=−1 0 0       0 Tx=−1 0       0 Tx=1 0 

One can solve equation (3.4.10) using a MATLAB built-in operator “mldvide” which is equivalent 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 polynomials less than 100 though, we constructed some matrices composed of large elements (400 ∼ 1000) for the Gaussian elimination 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-law growth factor due to partial pivoting. however, we simply accepted our numerical results based on large degrees of polynomials unless they showed an instability with increasing degree of polynomials.

64 CHAPTER 3. SPECTRAL METHOD where

t c = t t . (3.4.15) cφ cφp rtid

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 difficulties 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

−2mK −5 by (0.5 − 0.5x) 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 CHAPTER 3. SPECTRAL METHOD

3 3 The degree reaches the order of 10 for K = 15 at which the tidal radius rtid reaches the order of 10 (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 CHAPTER 3. SPECTRAL METHOD

101

| 10−1 ) r (¯ ¯ φ 10−3 |

K = 0.01 10−5 K = 1 K = 4 − potential 10 7 K = 8 K = 12 10−9 K = 15

10−3 10−2 10−1 100 101 102 103 104 dimensionless radiusr ¯ 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 CHAPTER 3. SPECTRAL METHOD

4 10 K = 1 104 K = 1 K = 0.01 K = 0.01 101 101 100n−2mK−5 exp(−n) 10−2 10−2 | 10−5 | −5

n 10 n a a | 10−8 | 10−8

10−11 10−11

10−14 −14 (a) 10 (b) 10−17 10−17 100 101 102 20 40 60 80 100 index n index n Figure 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 determined from equation (2.1.35).

68 CHAPTER 3. SPECTRAL METHOD

102 K = 4(N = 100) 10−2 K = 8(N = 300) K = 12(N = 600) K = 15(N = 1100) 10−6 |

n 10−10 a 0.1 exp(−0.033n) |

10−14 0.3 exp(−0.067n) 10−18 3 exp(−0.6n) 2 exp(−0.27n) 10−22 100 200 300 400 500 600 700 800 900 1,000 1,100 index n 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 references to present the geometrical convergences of the coefficients.

104

102 Kin r /

tid 0 r 10

10−2

10−3 10−2 10−1 100 101 K 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 CHAPTER 3. SPECTRAL METHOD

√ √ 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. 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).

only at approximatelyr ¯ = 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 radiusr ¯ = 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 CHAPTER 3. SPECTRAL METHOD

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 decomposition 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 Feje´r’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 compute 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 Feje´r’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 CHAPTER 3. SPECTRAL METHOD

over interval (−1, 1)

Z 1 Io ≡ h(x) dx. (3.5.1) −1

The discrete Chebyshev expansion (equation (3.1.4a)) can approximate h(x) and the integral Io reduces to

XN Io ≈ IN ≡ anKn, (3.5.2) n=1

where the factors {Kn} are

 1 − n2 Z π   , (if n is an even integer) K ≡ T (cos θ) sin θ dθ =  2 (3.5.3) n n−1  0  0. (if n is an odd integer)

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

XN−1 IN = h(x j)ω j, (3.5.4) k=1

where the weights {ω j} of the quadrature are

N−1 X KnTn− (x j) ω ≡ 1 , (3.5.5a) j d n=1 nn  [(N−1)/2]  ! 2  X cos(2nt j) 2 j − 1 = 1 − 2  , t = (3.5.5b) N  4n2 − 1  j N  n=1 

where, on the last line, the symbol [M] in the summation means the ﬂoor function of M.

72 CHAPTER 3. SPECTRAL METHOD

3.5.2 The convergence rate of the Feje´r’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)-

dk−1 dk th derivative dxk−1 h(x) and the k-th derivative dxk h(x) of bounded variation Vk, the convergence rate is

−2k P∞ −k−1 2Vk(1 + 2 ) p p | I − I |≤ =1 . (N ≥ max(8, k + 1)) (3.5.6) N o πN(N − 1) ··· (N − k)

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

−k | IN − Io |= O N . (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 −

−2 Io |= O N (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 CHAPTER 3. SPECTRAL METHOD 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 radiusr ¯. 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), max) 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¯ andr ¯ 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 densityρ ¯ in place of φ¯. The density also monotonically decreases with increasingr ¯ 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 equations. 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

 3   dA 1 d2A d2R  1  = , = −   . (4.1.2) dB dR dB2 dϕ2  dR    dϕ dϕ

where the ’old’ dependent and independent variables (A, B) are deﬁned as

   ¯ φ, (LE1)   A = ρ,¯ (LE2) (4.1.3a)    Φ, (ss-OAFP)   r¯, (LE1 and LE2)  B =  (4.1.3b)  R. (ss-OAFP)

76 CHAPTER 4. INVERSE FORMS

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 approximations.

4.2.1 Inverse mapping

The theorem on local inverse function (equations (4.1.2) and (4.1.3)) can reduce the LE2 equation (2.1.9) to !2 !3 d2R dR dR dR ϕ R + R − 2ϕ − ϕ3R = 0. (4.2.1) dϕ2 dϕ dϕ dϕ

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

" # p 2 283 R(ϕ ≈ 1) = 6(1 − ϕ) 1 + (1 − ϕ) + (1 − ϕ)2 + ... , (4.2.2) 5 1050

and as ϕ → 0 s   √  2  1/4  7  R(ϕ → 0) = 1 + C3i ϕ cos  ln ϕ + C4i , (4.2.3) ϕ   4 

where C3i and C4i are constants of integration.

77 CHAPTER 4. INVERSE FORMS

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 !−L/2 " !# 1 − x 1 + x x + 1 R(ϕ) ≡ exp 3 , (4.2.4a) 2 2 r 2

r !− 3L " !# 1 dR 1 2 1 + x 2 x + 1 S(ϕ) ≡ ≡ − exp 3 , (4.2.4b) LϕL−1 dϕ L 1 − x 2 s 2

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

dvr 2(1 − x2) =L(1 − x) + (1 + x) − 4e3s−3r , (4.2.5a) dx dvs 4 2(1 − x2) =L(1 − x) − (1 + x) − 8e3s−3r + e23s , (4.2.5b) dx L

with the boundary conditions

r √ 3L 3 (x = 1) = ln 6L, 3 (x = 1) = ln . (4.2.6) r s 2

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

r √ L2 3 (x → −1) = ln 2, 3 (x → −1) = ln . (4.2.7) r s 2

78 CHAPTER 4. INVERSE FORMS

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 approximations.

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

!2 !3 d2R dR dR R − 2 − ϕmR = 0. (4.3.1) dϕ2 dϕ dϕ

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

" # p 3m 121m2 + 200m R(ϕ ≈ 1) = 6(1 − ϕ) 1 + (1 − ϕ) + (1 − ϕ)2 + ... . (4.3.2) 20 5600

and the asymptotic approximation of R as ϕ → 0 is

ηi R(ϕ → 0) = Rs(ϕ)[1 + C1i ϕ cos (ωi ln ϕ + 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 CHAPTER 4. INVERSE FORMS

In equation (4.3.3), Rs is the singular solution of the ILE1 equation

1/ω¯ Rs(ϕ) = Am/ϕ . (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 !−L/ω¯ " !# 1 − x 1 + x x + 1 R(ϕ) ≡ exp 3 , (4.3.6a) 2 2 r 2

r !−L( 1 +1) " !# 1 dR 1 2 1 + x ω¯ x + 1 S(ϕ) ≡ ≡ − exp 3 , (4.3.6b) LϕL−1 dϕ L 1 − x 2 s 2

x = 2ϕ1/L − 1. (4.3.6c)

The regularized functions 3r and 3s in equation (4.3.6) satisfy the ﬁrst-order ODEs

d3r 2(1 − x2) = (1 + x) − L(1 − m)(1 − x) − 4e3s−3r , (4.3.7a) dx d3s 4 2(1 − x2) = −(1 + x) + L(1 + m)(1 − x) − 8e3s−3r + e23s , (4.3.7b) dx L

with the boundary conditions

r √ 3L 3 (1) = ln 6L, 3 (1) = ln . (4.3.8) r s 2

80 CHAPTER 4. INVERSE FORMS

and the asymptotes as x → −1

s r 2(m − 3) m − 3 3 (x → −1) = ln , 3 (x → −1) = ln L2. (4.3.9) r (m − 1)2 s 2

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

!2 !3 d2R dR dR R(ϕ) − 2 + R(ϕ) D(ϕ) = 0. (4.4.1) dϕ2 dϕ dϕ

The local approximation of R near ϕ = −1 reads

s 6 (1 + ϕ) R(ϕ ≈ −1) = . (4.4.2) D(−1)

81 CHAPTER 4. INVERSE FORMS

Also, the asymptotic approximation of the dependent variable R as ϕ → 0 is

s ! ν + 1 2β + 1 ϕ → − −ϕ ν, ν − R( 0) = 2 ( ) = (4.4.3) ν 3D(ϕ → 0) 4

where

D(ϕ) 3D(ϕ → 0) ≡ lim . (4.4.4) ϕ→0 (−ϕ)ν

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 CHAPTER 4. INVERSE FORMS

(4.4.2), and (4.4.3)), one can regularize the dependent variables as follows

 !−1/2 !−Lν !  1 − y 1 + y  2β + 1 3R(y) ≡ ln R(y)  , ν = − , (4.4.6a)  2 2  4  !1/2 !1−Lν  1 − y 1 + y  3S (y) ≡ ln −S(y)  , (4.4.6b)  2 2   !−Lσ1/3 !  1 + x  6β − 3 3Q(x) ≡ Q(x)  , σ = − , (4.4.6c)  2  4  !−βL  1 + x  3F(x) ≡ ln F(x)  , (4.4.6d)  2  G(x) 3 (x) ≡ , (4.4.6e) G F(x) I(x) 3 (x) ≡ , (4.4.6f) I F(x)Q(x) (2β − 3)J(x) 3 (x) ≡ , (4.4.6g) J 4βF(x)Q(x) where the new dependent variable S is introduced for convenience

dR S(y) ≡ 2 . (4.4.7) dy

The boundary conditions for the regularized variables are

∗ 3F(x → −1) = ln c4, 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 c4 is deﬁned as

∗ c4 ≡ c4(β + 1). (4.4.9)

83 CHAPTER 4. INVERSE FORMS

∗ The 4ODEs (2.2.3a)-(2.2.3d) are ﬁrst order ODEs. Hence, c4 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

d3R 2(1 − y)(1 + y) + 2νL(1 − y) − (1 + y) + 4e3S −3R = 0, (4.4.10a) dy d3S 4 2(1 − y)(1 + y) + 2L(ν − 1)(1 − y) + (1 + y) + 8e3S −3R − e3S 3 = 0, (4.4.10b) dy L D

where

!−(β+3/2)L 1 + y 3 (y) ≡ D(y) D 2 !−L(β+3/2) Z y r 0 0 !L(β+1)−1 L y 0 x − x x 1 + 0 3F (x ) 1 + 0 = AL(y, x )e dx , (4.4.11) 2 2 −1 2 2

where

s !L !L !−1/2 1 + x 1 + x0 x − x0 A (x, x0) ≡ − . (4.4.12) L 2 2 2

The boundary conditions for the Poisson’s equations (4.4.10a)-(4.4.10b) are

s 6 3R(y = 1) = , (4.4.13a) 3D(y = 1) s 3 3S (y = 1) = . (4.4.13b) 23D(y = 1)

84 CHAPTER 4. INVERSE FORMS

The asymptotic approximations of 3R and 3S as ϕ → 0 or y → −1 are

s ! ν + 1 2β + 1 3 y → − − , ν − R( 1) = 2 = (4.4.14a) ν 3D(y → −1) 4 s ! ν + 1 2β + 1 3S (y → −1) = − . ν = − (4.4.14b) 3D(y → −1) 4

The explicit form of 3Q(x) is

!−σL Z 1 0 !3/2 0 !(3ν+1)L−1 L 1 + x 1 − y 0 1 + y 3 0 33R(y ) p 0 0 3Q(x) = √ AL(y , x)e y − x dy . (4.4.15) 6 2 2 x 2 2

Equations (4.4.6d)-(4.4.6g) reduce the 4ODEs (2.2.3) to;

!(β−1)L+1 2 1 + x d3 F [3 + 3 ] L 2 dx I G !(β−1)L  !L ! 1 + x  1 + x 4β3   J  −3F + β [3I + 3G] + − 1 + c1e [1 + 3J] = 0, (4.4.16a) 2  2 2β − 3  ! " # !L 1 + x d3 d3 −2β + 3 3(1 + x) d3 1 + x 3 I + 3 F + 3 3 + Q + 3 = 0, (4.4.16b) L Q dx I dx I 4 Q L dx 2 Q ! !L 1 + x d3 1 + x d3 x + 1 G + 3 F + β − = 0, (4.4.16c) L dx G L dx 2 ( " #) " # 1 + x d3 d3 2β − 3 −2β + 3 1 + x d3 2β − 3 3 J + F 3 − + 3 3 + 3 Q − 3 = 0. (4.4.16d) L Q dx dx J 4β J Q 4 L dx 4 Q

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 CHAPTER 4. INVERSE FORMS

(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)

1−L ! Z 1 " 0 # 0 0 !L(β+1)−1 L y 2 y (x ) 3 (y+1)(x +1) − √ x 1 + ( + 1) + 1 F 2 1 0 1 + 0 3D(y) = √ AL y, − 1 e 1 − x dx , 2 2 2 −1 2 2 (4.4.17a)

!3 !−σL Z 1 " 0 # 3 L 1 − x 1 + x (1 − x) (1 − y ) 3Q(x) = AL 1 − , x 24 2 2 −1 2 0 ν L− (1−x)(1−y ) " 0 #(3 +1) 1 33 1− p (1 − x)(1 − y ) × e R 2 1 + y0 1 − y03/2 1 − dy0. (4.4.17b) 4

(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

1 − x 1 − y z ≡ −2 + 1, w ≡ −2 + 1. (4.4.18) 1 − xmin 1 − xmin

We employed the following extrapolated DFs to extrapolate 3F(x) on (−1, xmin)

 ( c ) ∗ −c − 1+xmin  ∗ 2c4 d3F (w=−1) d(1+xmin ) 1 1+x ln c + e (x − xmin) , (smooth at x = xmin),  4 1−xmin dw 3(ex)(x) ≡  F   ∗ ln[c4], (non-smooth at x = xmin or c → ∞), (4.4.19)

(nonex) where c and d are numerical parameters. Hence, 3D(w) is composed of a total of integrals 3D (w) (ex) and 3D (w); the former is contribution to 3D(w) from the (non-extrapolated) DF 3F(x) and the latter

86 CHAPTER 4. INVERSE FORMS

(ex) is from an extrapolated DF 3F (x) as follows

(nonex) (ex) 3D(w) = 3D (w) + 3D (w). (4.4.20)

(ex) The explicit forms of 3F(x) and 3F (x) are

!3/2 " #−3/2 1 1 − x 1 2 + x (1 − w) 3(nonex)(w) = √ min + min D 2 2 2 2 2(1 + w) 0 β Z 1 (w+1)(z +1) ( 0 ) 3 −1 √ (1 − z )(1 + w)(1 − x ) × F 2 − 0 − min 0 e 1 z 1 0 dz , (4.4.21a) −1 2 [3 + xmin + z (1 − xmin)] 1 !β+3/2 00 r !β Z (ex) (1+xmin)(1+z ) 00 00 (ex) 1 1 + xmin 3F 2 −1 1 + y 1 + z 1 + z 00 3D (w) ≡ e − dz , (4.4.21b) 2 −1 1 + y 1 + xmin 2 2

1 − 3 where L = 1 is assumed for simplicity and y = 2 [1 + xmin + w(1 xmin)]. Lastly, Q(z) on the truncated domain is

!3 !−3/2 1 1 − x 1 + x 3 (z)3 = Q 24 2 2 0 ν Z 1 (1−z)(1−w ) ( 0 )3 33 1− √ (1 − z)(1 + w )(1 − x ) × e R 2 1 + w0 1 − w03/2 1 + min dw0, −1 4 (1 + x) (4.4.22)

1 − where L = 1 is assumed and x = 2 [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 regularized 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 arrangements 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

!0.1 !0.1 1 + x 1 − x v(0)(x) = v (x = 1) + v (x → −1) , (5.1.1a) r r 2 r 2 !0.1 !0.1 1 + x 1 − x v(0)(x) = v (x = 1) + v (x → −1) , (5.1.1b) s s 2 s 2

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 φ¯p (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-

(k) (k) (k) (k) tions 5.2.1 - 5.2.3, we assume we have already known the functions 3F (x), 3G (x), 3I (x), 3J (x), (k) and 3Q (x) at the k-th iteration.

89 CHAPTER 5. NUMERICAL ARRANGEMENTS

5.2.1 Step 1: Solving the regularized Poisson’s equation

(k) 1 At the k-th iteration, we plugged 3F (x) 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

new old −15 (k) old- coeﬃcients ({ap} − {ap} ) converged to the order of 10 . Then, we obtained 3R (y) and (k) 3S (y) for the k-th iteration.

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

(k) We plugged the 3R (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

1 The boundary conditions are known for 3F (x). Hence, we used the simple polynomials for zeroth iteration !n !m 1 + x 1 − x 3(0)(x) = F + c∗ , (5.2.1) F BC 2 4 2 where m and n are numerical parameters.

90 CHAPTER 5. NUMERICAL ARRANGEMENTS method2 as a 5N + 2 nonlinear equations

n o 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 c4 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)

(k+1) (k+1) (k+1) (k+1) (k+1) Then, we obtained the functions 3F (x), 3G (x), 3I (x), 3J (x), and 3Q (x) and eigenvalues (k+1) ∗(k+1) c2 and c4 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

new old −13 norm of the diﬀerence {aq} − {aq} reached the order of 10 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 iteration (k) (k) (k) (k) (k) for 3F (x), 3G (x), 3I (x), 3J (x), and 3Q (x). We employed simple polynomials like equation (5.2.1) only for the initial guess (k = 1) for the functions at zeroth iteration.

91 CHAPTER 5. NUMERICAL ARRANGEMENTS for the whole domain formulation explicitly read

N N X n−1 ∗ X Fn(−1) − ln[c4] = 0, Fn = 0, (5.2.6a) n=1 n=1 N X n−1 Gn(−1) = 0, (5.2.6b) n=1 N X n−1 In(−1) = 0, (5.2.6c) n=1 N N X n−1 X Jn(−1) + 1 = 0, Jn = 0, (5.2.6d) n=1 n=1 N X n−1 Qn(−1) − 3Q(x = −1) = 0, (5.2.6e) n=1

PN PN PN where the three conditions n=1 Gn − c3 = 0, n=1 Qn = 0, and 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 formulation 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 CHAPTER 5. NUMERICAL ARRANGEMENTS

we had a solution at the the ’old’ xmin with the ’old’ β. Hence, we next chose a new xmin that was

3 very close to the old xmin, and a new β that was very close to old β, . 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

N X linear 3D(y j) = Fn Dn(y j) (5.2.7) n=1

linear linear linear where {Fn } is the Chebyshev coeﬃcients for 3F (x)(≡ exp [3F(x)]). We obtained {Fn } 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

1−L ! 2 L 1 + y j Dn(y j) = √ 2 2 2 Z 1 " (y + 1)(x0 + 1) # " 0 # √ 0 !L(β+1)−1 j (y + 1)(x + 1) 0 1 + x 0 × AL y j, − 1 Tn − 1 1 − x dx . −1 2 2 2 (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).

3 For 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 reachesr ¯ ≈ 10100, and the solution is accurate to nine signiﬁcant ﬁgures. Section 6.2 discusses the asymptotic behavior of the spectral solution asr ¯ → ∞. 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 larger ¯ 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 variabler ¯ = ln[ex + e2x + 1], where x ∈ (−∞, ∞). Because of this transformation, the locations of Hermite-Gauss nodes condense nearr ¯ = 0. Also, the maximum radius of their solution is small (at mostr ¯ ≈ 20 even for N = 1000.) By contrast, our solution extends tor ¯ ≈ 10100 with the Chebyshev coefficients that converge at roughly the same rate as the Hermite coefficients.

103 102 101

N 0 ¯

φ 10 10−1 10−2 Isothermal Sphere 10−3 10−5 1011 1027 1043 1059 1075 1091 r¯ Figure 6.1: LE2 function φ¯ (N = 225 and L = 36.)

95 CHAPTER 6. RESULT: ISOTHERMAL SPHERE

10−1

10−6 | n r 10−11 |

10−16 ILE2 (Isothermal Sphere) 10−21 50 100 150 200 250 300 n Figure 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.

100

10−2 | −4 ) 10 cut

r −6 ( 10 N ¯ φ

/ 10−8 A

− 10−10 1 | 10−12

10−14 0 10 20 30 40 50 L Figure 6.3: Optimal value of the mapping parameter L for the isothermal sphere (N = 100). The 2 graph compares φ¯N to A ≡ ln (¯r /2) at cutoﬀ radius rcut. The rcut was introduced to avoid numerical divergences atr ¯ → ∞ when MATLAB reaches machine precision.

We consider that the signiﬁcant ﬁgures of our solution are nine digits in comparison with other

96 CHAPTER 6. RESULT: ISOTHERMAL SPHERE

Radiusr ¯ LE2 functions

Chebyshev expansion φ¯N RK (Horedt (1986)) φ¯H AD (Wazwaz (2001)) φ¯W 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’s solutions. Our solution’s first seven digits that match Horedt’s solutions are underlined. 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. 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 CHAPTER 6. RESULT: ISOTHERMAL SPHERE

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 tangent 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 be useful to understand the stability of the clusters’ cores.

98 CHAPTER 6. RESULT: ISOTHERMAL SPHERE

2.5 (a)

1.5 v

0.5 isothermal sphere(original scale) 0 −0.5 0 0.5 1 1.5 2 2.5 3 u

(b) isothermal sphere (magniﬁed) 5 · 10−13 s.p

v 0 − v

−5 · 10−13 0 13 13 13 13 13 13 − − − − − − 10 10 10 10 10 10 · · · · · · 8 6 4 2 2 4 − − − − u − us.p

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 reaches the 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 solution. Figure 6.5 compares the spectral solution to previous semi-analytical solutions (Liu, 1996;

99 CHAPTER 6. RESULT: ISOTHERMAL SPHERE

Hunter, 2001; Soliman and Al-Zeghayer, 2015) in which the existing solutions lose their accuracies signiﬁcantly at larger ¯. 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 φ¯N(¯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 using 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 nearr ¯ = 1010 gave the best results. We found the asymptotic LE2 function asr ¯ → ∞;

 √  2 ± × −7 r¯ 1.1779636 1 10  7 −7 φ¯a(¯r) = ln + cos  lnr ¯ + 1.0623271 ± 1 × 10  . (6.2.1) 2 r¯1/2  2 

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 atr ¯ = 105. Their values agree with the present values up to four-digit accuracy. Equation (6.2.1) is ﬁfteen-digit accurate at radii greater thanr ¯ ≈ 1013, where the oscillations have an ampli-

−7 13 tude ∼ 10 . At radiir ¯ less than 10 , the absolute deviations from φ¯N 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 onr ¯ as done in (Raga et al., 2013).

100 CHAPTER 6. RESULT: ISOTHERMAL SPHERE

2.5 Chebyshev expansion: φ¯N Soliman and Al-Zeghayer (2015) 2 Hunter (2001) Liu (1996) 2 /

1 1.5 Horedt (1986) ¯ r i

2 1 / 2 ¯ r h 0.5 ln

− 0 ¯ φ −0.5 (a) −1 101 102 103 r¯

100 | 1 − φ¯N/φ¯a | | 1 − φ¯N/φ¯SA | 10−4 | | 1 − φ¯N/φ¯ s | ¯ φ / 10−8 N ¯ φ

− 10−12 1 | 10−16 (b) 10−20 101 105 109 1013 1017 1021 1025 1029 r¯

h 2 i 1/2 Figure 6.5: (a) Asymptotic oscillation of LE2 spectral solutions, φ¯N − ln r¯ /2 r¯ , compared to existing semi-analytical solutions for N = 225, L = 36. (b) Relative deviation of our semi- analytical asymptotic singular solution, φ¯a, from the LE2 spectral solution φ¯N. The φ¯SA is the ¯ r¯2 semi-analytical asymptotic LE2 solution of (Soliman and Al-Zeghayer, 2015) and φs = ln 2 . The 13 plot of | 1 − φ¯N/φ¯a | ﬂattens around at r = 10 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 asr ¯ → ∞ 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 fromr ¯ ≈ 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

ω¯ the absolute error ofr ¯ φ¯(¯r) from the asymptotic value Am asr ¯ → ∞ 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.

100

10−18

10−36 N ¯ φ 10−54 m = 6 and L = 18 − 10 72 m = 10 and L = 5 m = 20 and L = 2 10−90 101 1031 1061 1091 10121 10151 10181 r¯

Figure 7.1: LE1 function φ¯N, for m = 6, 10 and 20 (N = 300.)

103 CHAPTER 7. RESULT: POLYTROPIC SPHERE

103 m = 6 and L = 18 m = 10 and L = 5 10−1 m = 20 and L = 2

| 10−5 n r | 10−9

10−13

10−17 50 100 150 200 250 300 n

Figure 7.2: Absolute values of Chebyshev coefficients for the regularized spectral ILE1 solution 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.

101 m = 6 (N = 200) m = 10 (N = 200) 10−1 m = 20 (N = 200) |

) 10−3 cut

r −5 ( 10 N ¯ φ −7

/ 10 A −9

− 10

1 −11 | 10

10−13

10−15 0 10 20 30 40 50 L Figure 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 = φ¯ s for polytropes. The cutoﬀ radii rcut were chosen to avoid divergences atr ¯ → ∞ when MATLAB reaches machine precision. 104 CHAPTER 7. RESULT: POLYTROPIC SPHERE

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.

100

10−1 ¯ φ

φ¯N (m = 6 and L = 18) φ¯N (m = 10 and L = 5) φ¯N (m = 20 and L = 2) −2 10 φ¯H (m = 6) φ¯H (m = 10) φ¯H (m = 20)

10−2 10−1 100 101 102 103 104 r¯

Figure 7.4: Comparison between the spectral solutions, φ¯N, and Horedt’s solutions, φ¯H.

105 CHAPTER 7. RESULT: POLYTROPIC SPHERE

(a) (b) 3 N = 60 N = 100 3 N = 30 N = 60 10 N = 200 N = 300 10 N = 100 N = 200 | N = 400 N = 500 | N = 300 H 0 H 0 ¯ ¯ φ 10 φ 10 / / N N ¯ ¯ φ 10−3 φ 10−3 − − 1 1 | 10−6 | 10−6

10−9 10−9 10−1 100 101 102 103 10−1 100 101 102 103 r¯ r¯ 0 10 N = 30 N = 60 (c) N = 100 N = 200 | 10−2 H ¯ φ / − N 10 4 ¯ φ − −6

1 10 |

10−8

10−1 100 101 102 103 r¯

Figure 7.5: Relative error of the spectral solutions, φ¯N, from Horedt’s solutions, φ¯H 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

m−3 singular point m−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 CHAPTER 7. RESULT: POLYTROPIC SPHERE

m−3 niﬁcation of the trajectory around the singular point m−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 tor ¯ ≈ 1033 for m = 20 andr ¯ ≈ 1070 for m = 6.

107 CHAPTER 7. RESULT: POLYTROPIC SPHERE

0.15 (a)

0.1 v

0.05

m = 20 (original scale) 0 0 0.5 1 1.5 2 2.5 3 u

6 · 10−14 (b)

4 · 10−14

2 · 10−14 s.p v

− 0 v −2 · 10−14 m = 20 −4 · 10−14 (magniﬁed) 0 13 13 12 12 12 − − − − − 10 10 10 10 10 · · · · · 5 5 1 2 5 . − 1 u − us.p

Figure 7.6: (a) Trajectory in the Milne (u, v) plane for m = 20 and L = 2 with N = 300 and m−3 (b) its magniﬁcation near ( m−1 , ω¯ ). The solution starts from (uE, vE) = (3, 0) and asymptotically m−3 approaches the singular point (us.p, vs.p) = m−1 , ω¯ .

108 CHAPTER 7. RESULT: POLYTROPIC SPHERE

0.1 (a) 1 0.05 − m A

/ 0 N ¯ φ ¯ ω ¯ r

−0.05 α ¯ r m = 20 and L = 2 −0.1 10−2 105 1012 1019 1026 1033 r¯

1 (b) 1 0.5 − m A

/ 0 N ¯ φ ¯ ω ¯ r

−0.5 α ¯ r m = 6 and L = 18 −1 10−2 1014 1030 1046 1062 1078 r¯ Figure 7.7: Asymptotic oscillations of the spectral solutions (N = 300) for (a) m = 20 and (b) α ω¯ m = 6. For larger ¯,r ¯ r¯ φ¯N/Am − 1 ≈ C1 cos (ω lnr ¯ + 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 solution. 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)

−12 We chose the value of βo so that 3I(x = 1) reached its minimum value (∼ 10 ). 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 CHAPTER 8. RESULT: THE SS-OAFP MODEL

F(E) F(E) 100 (a) (b) 10−1 10−1 10−6 10−2 10−11 10−3 −16 10 Spectral solution −4 Spectral solution HS’s solution 10 HS’s solution 10−21 −1 −0.8−0.6−0.4−0.2 0 −1 −0.8 −0.6 −0.4 E E Figure 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.)

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 eigenvalues’ to compare them with eigenvalues obtained from other formulas and solutions

−4 c1o = 9.09254120455 × 10 , (8.1.2a)

∗ −1 c4o = 3.03155222 × 10 . (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 or basis functions (e.g., Legendre polynomials, Gegenbauer polynomials, and Hermite functions.) if the solution has no non-analytical- and non-regular properties. Hence, we call such a solution the semi-analytical solution.

112 CHAPTER 8. RESULT: THE SS-OAFP MODEL

Eigenvalues Spectral method HS T % relative error [%]

−4 −4 −4 c1 9.0925 × 10 9.10 × 10 9.1 × 10 0.1 −4 −4 c2 1.1118 × 10 1.12 × 10 − 0.9 −2 −2 c3 7.1975 × 10 7.21 × 10 − 0.1 −2 −2 c4 3.303 × 10 3.52 × 10 − 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’s eigenvalues and the present ones are also shown. The present eigenvalues are based on the results −4 4 for various combinations of numerical parameters (13 ≤ N ≤ 560, 10 ≤ FBC ≤ 10 , 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

0 0 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 minimum 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 CHAPTER 8. RESULT: THE SS-OAFP MODEL

index Coeﬃcients

0 n Fn Qn Rn 1 −0.9793 3.0496 2.0588 2 0.4515 3.1373 0.7337 3 0.3949 1.1575 0.1589 Function Semi-analytical expression 4 0.1751 0.2169 −0.0066 hP18 i β 5 −0.0171 −0.0123 −0.0182 F(E) exp n=1 FnTn−1 (−2E − 1) (−E) 3 hP12 0 i σ 3 Q(E) Q T − (−2E − 1) (−E) (1 + E) 6 −0.0381 −0.0028 0.0013 n=1 n n 1 √ hP12 i ν 7 0.0076 0.0054 0.0038 R(Φ) exp n=1 RnTn−1 (−2Φ − 1) (−Φ) 1 + Φ 8 0.0103 −0.0011 −0.0007 Notation − − − 9 0.0046 0.0010 0.0009 h −1 i E ∈ (−1, 0) Tn−1(−2E − 1) = cos (n − 1) cos (−2E − 1) 10 −0.0023 0.0006 0.0003 Φ ∈ (−1, 0) β = 8.1784 11 0.0023 0.0001 0.0002 3 σ = − (2β − 1) = −11.5176 12 0.0001 −0.0002 −0.0001 4 1 ν = − (2β + 1) = −4.3392 13 −0.0009 4 14 0.0002 15 0.0002 16 −0.0002 17 0.0000 18 0.0001

Table 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 −4 0 whole-domain solution is the order of 10 . The coeﬃcients {Qn} 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 CHAPTER 8. RESULT: THE SS-OAFP MODEL 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)

| Fn/F1 | | In/I1 | | Gn/G1 | 100 100 100 10−2 10−2 10−2 10−4 10−4 10−4 10−6 10−6 10−6 10−8 10−8 10−8 10−10 10−10 10−10 10−12 10−12 10−12 10−14 10−14 10−14 20 40 60 20 40 60 20 40 60 n n n

| Jn/J1 | | Rn/R1 | | Qn/Q1 | 100 100 100 10−2 10−2 10−2 10−4 10−4 10−4 10−6 10−6 10−6 10−8 10−8 10−8 10−10 10−10 10−10 10−12 10−12 10−12 10−14 10−14 10−14 20 40 60 20 40 60 20 40 60 n n n Figure 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.

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 that 115 CHAPTER 8. RESULT: THE SS-OAFP MODEL

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).

0 Spectral solution ) 0.4 E

) HS’s solution −

−0.5 ( E / (

) .

F 0 2 3 −1 E ( Spectral solution I

3 HS’s solution −1.5 0 10−2 10−1 100 10−2 10−1 100 −E −E

) 0 Spectral solution E HS’s solution ) −

( 0.1 E / − .

( 0 5 ) J E 3

( Spectral solution

G 0.08 −1 3 HS’s solution 10−2 10−1 100 10−2 10−1 100 −E −E

3 Spectral solution 1 ) ) 2.5 HS’s solution E Φ ( ( 0.5 R Q 3 2 3 Spectral solution 1.5 0 HS’s solution 10−2 10−1 100 10−2 10−1 100 −Φ −E Figure 8.3: Regularized spectral solution compared with the corresponding HS’s solution. (N = 70, FBC = 1, and L = 1.)

116 CHAPTER 8. RESULT: THE SS-OAFP MODEL

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 c4 can preserve ∗ seven- and five- significant figures compared with c1o and c4o. 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/c4o | 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. Condition number 100 1010 | 3I(x = 1) | −2 ∗ ∗ 9 10 | 1 − c4/c4o | 10 8 10−4 10 7 10−6 10 106 10−8 105 10−10 104 10−12 (a) (b) 103 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 degree N degree N

∗ Figure 8.4: (a) Relative error between c4 and c4o, and the absolute value of 3I(x = 1), (b) Condition number of the Jacobian matrix for the 4ODEs and Q-integral. The condition number was computed old new −13 when {aq} − {aq} reached the order of 10 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-domain 117 CHAPTER 8. RESULT: THE SS-OAFP MODEL 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-

(ex) OAFP system. Based on the result of Section 8.1, we extrapolated 3F(x) using 3F (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

2 maximum energy Emax. We found the following classes

(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.) 118 CHAPTER 8. RESULT: THE SS-OAFP MODEL

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 − c4/c4o |, 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’.

104 1012 | 3I(x = 1) | Condition number 103 | 1 − c1/c1o | 11 | 1 − c∗/c∗ | 10 102 4 4o 101 1010 100 109 10−1 10−2 108 10−3 107 − 10 4 (a) (b) 10−5 106 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −Emax −Emax

∗ Figure 8.5: (a) Characteristics of | 3I(x = 1) | and relative errors of c1 and c4 from their reference eigenvalues for diﬀerent Emax.(N = 40, β = 8.1783, FBC = 1, and L = 1.) (b) Condition number new old −13 of the Jacobian matrix calculated when {aq} − {aq} reached the order of 10 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−c4/c4o |, 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 CHAPTER 8. RESULT: THE SS-OAFP MODEL

The coefficients do not decay rapidly with high indexes n. We believe that the rapid decay was hindered 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.

100 1 10 (b) (a) 0 10−1 10 10−1 10−2 10−2 −3 10 10−3 10−4 n −4

F 10 −5 10−5 10 −6 − 10 10 6 | 3 (x = 1) | I 10−7 −7 | 1 − c1/c1o | N = 50 10 ∗ ∗ 10−8 | 1 − c4/c4o | N = 360 10−8 10−9 0 50 100 150 200 250 300 350 100 101 102 Degree N index n

∗ Figure 8.6: (a) Relative errors of c1 and c4 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. )

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,

3 we sought the truncated-domain solutions with ﬁxed β = βo 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.

3 The 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.

120 CHAPTER 8. RESULT: THE SS-OAFP MODEL

(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 c4 from the reference values. For Emax > −0.07, | 3I(x = 1) |, ∗ ∗ −9 −13 | 1 − c1/c1o |, and | 1 − c4/c4o | decrease with increasing N, and reach small values (≈ 10 ∼ 10 ) 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 − c4/c4o | stall with N while their minimum values still can be found at relatively-low degrees (N = 27 ∼ 35).

121 CHAPTER 8. RESULT: THE SS-OAFP MODEL

10−2 10−2

10−7 10−7

10−12 | 3I(x = 1) | 10−12 | 3I(x = 1) | | 1 − c1/c1o | | 1 − c1/c1o | | − ∗ ∗ | | − ∗ ∗ | 1 c4/c4o Emax = −0.10 1 c4/c4o Emax = −0.09 10−17 10−17 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 N N

10−2 10−2

10−7 10−7

10−12 | 3I(x = 1) | 10−12 | 3I(x = 1) | | 1 − c1/c1o | | 1 − c1/c1o | | − ∗ ∗ | | − ∗ ∗ | 1 c4/c4o Emax = −0.08 1 c4/c4o Emax = −0.07 10−17 10−17 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 N N

10−2 10−2

10−7 10−7

10−12 | 3I(x = 1) | 10−12 | 3I(x = 1) | | 1 − c1/c1o | | 1 − c1/c1o | | − ∗ ∗ | | − ∗ ∗ | 1 c4/c4o Emax = −0.06 1 c4/c4o Emax = −0.05 10−17 10−17 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 N N

10−2 10−2

10−7 10−7

10−12 | 3I(x = 1) | 10−12 | 3I(x = 1) | | 1 − c1/c1o | | 1 − c1/c1o | | − ∗ ∗ | | − ∗ ∗ | 1 c4/c4o Emax = −0.04 1 c4/c4o Emax = −0.03 10−17 10−17 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 N N

∗ Figure 8.7: Relative errors of c1 and c4 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 CHAPTER 8. RESULT: THE SS-OAFP MODEL

(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 − c4/c4o | obtained at each Nbest. The value of | 1 − c4/c4o | decreases in a power-law-like

βo ∗ 4 fashion with increasing Emax. We introduced a power-law proﬁle c1o(−Emax) /c4o. In Figure 8.8,

βo ∗ ∗ ∗ c1o(−Emax) /c4o decreases in a similar way to | 1 − c4/c4o |. 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

βo ∗ double precision. In addition to the result that c1o(−Emax) /c4o can characterize the accuracy of ∗ (old) (new) c4o (correspondingly the solution), we found the inﬁnity norm of | {Fn} − {Fn} | for the

−13 βo ∗ Newton method reaches the order of 10 at best. These findings mean that c1o(−Emax) /c4o −13 reaches the order of 10 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 to ∗ the accuracy in c4 (See Ito, 2021, for detailed discussions).

123 CHAPTER 8. RESULT: THE SS-OAFP MODEL

method in (Ito, 2021).

102 | 3I(x = 1) | −1 | 1 − c1/c1o | 10 ∗ ∗ | 1 − c4/c4o | βo ∗ c1o(−Emax) /c4o 10−4

10−7

10−10

10−13

10−16 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 −Emax

∗ Figure 8.8: Relative errors of c1 and c4 from the reference values and characteristics of | 3I(x = 1) | βo ∗ obtained at each Nbest. The guideline c1o(−Emax) /c4o 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-

∗ ∗ −9 tion. For Emax = −0.03 in Figure 8.7, | 1 − c4/c4o | reaches the order of 10 . This is the same ∗ ∗ order as the minimum value of | 1 − c4/c4o | computed against different β in Appendix F.1.1. Also, −12 3I(x = 1) = 7.3 × 10 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 solution 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 reference- solutions. Our optimal truncated-domain solution validates the reference solution.

124 CHAPTER 8. RESULT: THE SS-OAFP MODEL

The largest relative error between them is the order of 10−9.

| 1 − F(E)(N=65)/F(E)(N) | | 1 − F(E)(N=65)/F (E) | 102 N = 25 N = 30 o (a) 10−8 N = 35 N = 40 (b) 100 N = 45 N = 50 N = 55 N = 60 −9 10−2 N = 70 10 10−4 10−10 10−6 −11 10−8 10 10−10 10−12 0.2 0.4 0.6 0.8 1 10−1 100 E −E

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

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 CHAPTER 8. RESULT: THE SS-OAFP MODEL

HS’s and reference- solutions 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 reproducing 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

! 1 − x 3(m)(x) = [3 (x)]2 . (8.3.1) R R 2

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-

(m) ues. We reformulated the ss-OAFP system based on 3R 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

−4 is approximately min(|{Fn}|) ∼ 3I(x = 1) ∼ 10 . 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 CHAPTER 8. RESULT: THE SS-OAFP MODEL

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

−4 Table 8.3: Eigenvalues of the reproduced HS’s solution with eigenvalues c1 = 9.10 × 10 and −2 c4 = 3.52 × 10 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.

100 | 0.005/ ln(FHS) | Emax = −0.24 E = −0.225 E = −0.215 −1 max max | 10 ] HS F 10−2 ln[ / ] F −3

ln[ 10 − 1 | 10−4

10−5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −E

Figure 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 CHAPTER 8. RESULT: THE SS-OAFP MODEL

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

−4 −5 of the coeﬃcients {In} for 3I(x) reach the order of 10 ∼ 10 for Emax ≈ −0.25. One also can ﬁnd

−6 −7 the reference solution if the coefficients reach the order of 10 ∼ 10 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 formulations 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 reference- solutions, the

(m) −5 3R -formulation of Section 8.3.1 provided the smallest relative error ∼ 4 × 10 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

5 The eigenvalues c1, c2, and c3 were more stable than c4 for any formulations in our work.

128 CHAPTER 8. RESULT: THE SS-OAFP MODEL

10−5

10−6

−7 10 (m) | 1 − 3Ro/3R | | 1 − 3Fo/3F | 10−8 10−1 100 −E

(m) Figure 8.11: Relative errors of 3R 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 equilibrium 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 ridge of a cluster, two dimensional (anisotropic) effect, and escapers and escaping stars with high eccentricities under the influence 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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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 statistical mechanics.

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 assumptions. 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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

time t of the ss-OAFP model is ﬁxed to a certain time tc. This means the time-dependent variables

(e.g., ft, gt, t, ...) 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 concerned 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 existing 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 kinetic and 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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL the negative heat capacity. First, we refer to a thermodynamic-equilibrium small-N system of homogeneous subsystems, e.g., nanoclusters (Michaelian and Santamar´ıa-Holek, 2007; Lynden-Bell and Lynden-Bell, 2008; Michaelian and Santamar´ıa-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 process 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 inhomogeneous 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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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

2 h ss-OAFP equation), the velocity dispersion 3 (r, tc), pressure p(r, tc), heat ﬂux f (r, tc), and stellar

p ﬂux f (r, tc) may be deﬁned in terms of the moments of the DF

Z 0 2 1 3/2 3 (r, tc) ≡ 2 − 2φ(r, tc) f (, tc) d, (9.2.1a) ρ(r, tc) φ(r,tc) 32(r, t ) ρ(r, t ) p(r, t ) ≡ c c , (9.2.1b) c 3 ∂ 32(r, t ) fh(r, t ) ≡ −k c , (9.2.1c) c ∂ r ∂ρ(r, t ) fp(r, t ) ≡ −D c , (9.2.1d) c ∂ r

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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

√ 3 and 2CD ln[N]/3 for the diﬀusivity , where Ck and CD are dimensionless constants of unity.). Employing the self-similar variables of Chapter 4 and introducing new dimensionless variables

2 2 2 2 3 (r, tc) = V(dis)(R) 3t (t) ≡ 2V(dis)(R) t(t), (9.2.3a)

p(r, tc) ≡ P(R) pt(t), (9.2.3b) h f (r, tc) h h ≡ F (R) ft (tc), (9.2.3c) 3 G m Ck ln[N] p f (r, tc) p p ≡ F (R) ft (tc), (9.2.3d) 2 CD ln[N]/3 the local quantities in dimensionless form read

2 U3/2(Φ) V(dis)(R) = 2 , (9.2.4a) U1/2(Φ) 2 P(R) = U (Φ), (9.2.4b) 3 3/2 ! U (Φ) U (Φ) U− (Φ) h R − 1/2 3/2 1/2 − , F ( ) = q 2 3 (9.2.4c) 2 [U1/2(Φ)] S (Φ) V(dis)(Φ)

p 1 U−1/2(Φ) F (R) = − q , (9.2.4d) 2 S (Φ) V(dis)(Φ)

where S (Φ) = −2S (x) due to the deﬁnition and Un(Φ) is

Z 0 n Un(Φ) = F(E)(E − Φ) dE, (9.2.5) Φ

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

1 √ 1 C 1 32(r, t ) 8 π G m C ρ(φ, t ) ln[N] 2 C ln[N] D = l 32 ≈ D = c D c = D , (9.2.2) 2 32 3/2 p 2 3 3kJ TR 3 4πGρ(φ(r, tc)) [ (r, tc)] 3 3 (r, tc) 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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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

2 2 V(dis) and m.f. potential profile Φ. The constancy in V(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

4 −10 The value of m holds the relative error 6.3 × 10 % from the expected value βo + 3/2.

137 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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

Γ of 1/χesc given that the polytropic constant K (in p(r, tc) = Kρ(r, tc) ) is considered the constant temperature of the polytrope. The followings are the equations of local states in the two limits of R

1.0 P = D, (R ≈ 0) (9.2.6a) χesc 0.5 P = D1.1. (R → ∞) (9.2.6b) χesc

where the inverse temperature of the ss-OAFP model is the scaled escape energy χesc.

101 100 (a) (b)

10−4

10−9 10−1 10−14 Fh p | Φ | 10−19 F 2 ˜ V(dis) 1/TR 10−2 10−24 10−1 100 101 102 103 104 105 10−1 100 101 102 103 104 105 106 107 108 R R

2 Figure 9.1: (a) Dimensionless m.f. potential Φ and velocity dispersion V(dis) and (b) dimensionless heat ﬂux Fh and stellar ﬂux Fp. The latter also depicts the inverse of regularized local relaxation 2 3/2 time T˜R = D(φ, tc)/[3 (r, tc)] for the sake of comparison.

138 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

1.2 (a)

1.1 Γ

100 102 104 106 108 1010 1012 1014 1016 R (b) 150

100 m

0 100 102 104 106 108 1010 1012 1014 1016 R Figure 9.2: (a) Adiabatic index Γ and (b) polytropic index m of the ss-OAFP model.

1.2

Γ 0.8 D /

esc 0.6 χ

P 0.4

0.2

100 102 104 106 108 1010 1012 1014 1016 1018 R

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

139 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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

3 3 M(RM, tc) = m f (, tc) d 3 d r, (9.2.7a) 32 KE(R , t ) =" m f (, t ) d33 d3r, (9.2.7b) M c 2 c " 3 3 PE(RM, tc) = m φ(r, tc) f (, tc) d 3 d r, (9.2.7c)

" Etot(RM, tc) = KE + PE. (9.2.7d)

The corresponding dimensionless forms are

Z ΦM M(r, tc) 2 M(ΦM) ≡ = − DR S dΦ, (9.2.8a) Mt(tc) −1 Z ΦM KE(r, tc) 2 KE(ΦM) ≡ = − U3/2R S dΦ, (9.2.8b) PEt(tc) −1 Z ΦM PE(r, tc) 1 2 PE(ΦM) ≡ = ΦDR S dΦ, (9.2.8c) PEt(tc) 2 −1 Etot(r, tc) Etot(ΦM) ≡ , (9.2.8d) PEt(tc)

140 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

where ΦM is the potential at the wall radius RM. The corresponding time-dependent variables at

time tc read

√ 2 3/2 3 Mt(tc) = (4π) 2 m ft(tc)[Et(tc)] [rt(tc)] , (9.2.9a) KE (t ) 1 PE (t ) = t c = M (t) E (t ). (9.2.9b) t c 2 2 t t c

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)

2KE T (kin)(Φ ) ≡ , (9.2.10a) M 3M V2 T (loc)(R ) ≡ (dis) . (9.2.10b) M 3

Figure 9.4 depicts the local- and kinetic- temperatures. At the center of the system, the temperatures 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

χ β(con) ≡ esc , (9.2.11) t(tc)

This provides an approximate relationship at the center of the system

1 T (loc) R r t ≈ T (kin) R r t ≈ , M t( c) M t( c) (con) (9.2.12a) kBβ

(loc) (kin) T (ΦM) ≈ T (ΦM) ≈ χesc, (9.2.12b)

5Exactly speaking, the temperatures gradually decrease with the wall radius. The maximum temperatures are lowered 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 isolated system enclosed by an adiabatic wall.

141 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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

√ (PE) 3esc = −4PE, (9.2.13a) r (con) M 3esc = 4 . (9.2.13b) RM

(con) We introduced 3esc 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, q 2 velocity dispersion V(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, q 2 the decreasing density causes an increasing stellar flux. Beyond R = 2.2 ∼ 2.3 at which V(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.

100 T (loc) T (kin) 10−1

10−2

10−3

10−4 10−1 102 105 108 1011 1014 RM

Figure 9.4: Local and kinetic temperatures.

142 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

101

100

10−1

10−2 (PE) 3esc (con) 3esc 10−3 q 2 V(dis) P 10−4 F 10−1 100 101 102 103 104 105 106 RM q P 2 Figure 9.5: Escape speeds of stars. The stellar ﬂux F and velocity dispersion V(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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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 polytropes based on the Tsallis entropy (Taruya and Sakagami, 2002; Chavanis, 2002b). The normalized energy reads

rE (r, t ) R E (Φ ) tot c M tot M . Λ = 2 = 2 (9.3.1) GM (r, tc) [M(ΦM)]

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 homogeneous systems 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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

(9.3.2) determines only the relations among the magnitudes of quantities. The important point is

9 that there exists some reasonable dimensionless parameter as RM → ∞. Similarly, one can convert the reciprocal of the kinetic, local, and thermodynamic temperatures into their dimensionless forms

GM(R r (t ), t ) M η(kin) M t c c , = (kin) = (kin) (9.3.3a) kBT (RMrt(tc)) T RM GM(R r (t ), t ) M η(loc) M t c c , = (loc) = (loc) (9.3.3b) kBT (RMrt(tc)) T RM β(con)R r (t ) S (Φ ) η(con) M t c − M . = 2 = (9.3.3c) GM(RMrt(tc)) RM

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-

(con) 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 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 itself 4 3 is the expansion parameter. Rather, the corresponding (dimensionless) occupation number 3 na , where a is the size of particles, is the actual parameter.

145 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

0 0.4 ss-OAFP model Isothermal sphere −2 0.2

−4 Λ Λ 0 −6

−0.2 −8

−10 −0.4 10−1 104 109 1014 1019 10−1 104 109 1014 1019 RM RM

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. The curve for the isothermal sphere was obtained using the numerical code in (Ito et al., 2018).

146 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

ss-OAFP model 102 (kin) /η

1 100

10−1 102 105 108 1011 1014 1017 1020 ss-OAFP model 102 (loc) /η

1 100

10−1 102 105 108 1011 1014 1017 1020 104

102 (con) /η 1 100 ss-OAFP model 10−1 102 105 108 1011 1014 1017 1020 Isothermal sphere 102 /η 1 100

10−1 102 105 108 1011 1014 1017 1020 RM

Figure 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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

9.3.2 Caloric curve

A caloric curve provides the characteristics of the heat capacity of the ss-OAFP model. The dimensionless 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

h i2  1   1  (kin)  ∂   ∂  CV T  Λ  =  Λ  = . (9.3.4) ∂η(kin)  ∂η(kin)  ME2 R,M V,Ntot tot

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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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, the closeness among the singularities in radius may occur if one assumes the wall does not affect the structure of potential well. 11For an isolated self-gravitating system, the Virial theorem states the heat capacity is always negative. Hence, the heat capacity is CV ∝ Etot/KE < 0. As Thirring (1970) originally pointed out, the energy range of a system with negative heat capacity in a micro-canonical ensemble corresponds to the system undergoing a phase transition in a canonical ensemble. Simply, the canonical ensemble applies to the isothermal sphere at radii smaller than RM = 8.9 while the micro-canonical ensemble at radii smaller than RM = 34.9 (See Table 9.2 for the value of radii.). The heat capacity of the isothermal sphere is positive over 0 < RM < 8.9 and negative over 8.9 < RM < 34.9.

149 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

0 0 RM = 0 (a) RM = 0 (b) −5 · 10−2 −5 · 10−2

−0.1 −0.1 Λ

/ −0.15 −0.15 1

−0.2 R → ∞ −0.2 M RM → ∞

−0.25 −0.25

−0.3 −0.3 0 1 2 3 0 1 2 3 η(kin) η(con)

0 −0.22 RM = 0 (c) RM → ∞ (d) − −5 · 10−2 0.23 −0.24 −0.1 −0.25 Λ

/ −0.15

1 −0.26 −0.2 RM → ∞ −0.27

−0.25 −0.28

−0.3 −0.29 0 1 2 3 2.4 2.5 2.6 2.7 2.8 η(loc) 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 tangents to the curves represent the sign of CV.

150 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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 corresponding 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 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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))

Z RM Z 0 2 3/2 V ≡ −2KE − PE = − dR dE Ωmic(E, Φ) R (E)(E − Φ) , (9.4.1a) 0 Φ Z RM Z 0 2 3/2 Etot = dR dE Ωcan(E, Φ) R (E)(E − Φ) , (9.4.1b) 0 Φ

where

Φ dF Ω (E, Φ) = 2F(E) + , (9.4.2a) mic 3 dE Φ dF Ω (E, Φ) = F(E) + . (9.4.2b) can 3 dE

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

χesc Ω (E, Φ) = eχescE 2 + Φ , (9.4.3a) mic 3 χesc Ω (E, Φ) = eχescE 1 + Φ . (9.4.3b) con 3

The change in the sign of heat capacity depends on

6 Φ(mic) = − , (9.4.4a) χesc 3 Φ(can) = − . (9.4.4b) χesc

152 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

1 KE 3PE esc

χ 0.5 −

0 100 102 104 106 108 1010 1012 1014 1016 1018

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] contributes 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 a self-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-gravitating system does not exist in a canonical ensemble. Under a proper thermodynamic limit, the dimensionless total energy Λ 2 reaches an extremum (singular point) as RM increases. For a large RM, Etot is proportional to M G/RM. At radii larger or smaller than the singular point, the sign of CV possibly changes.

153 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

Hence, as RM → ∞, the potential well gets deeper, and the total energy decreases. On the one

2 hand, βφ(0) = −RM/6 for small RM(<< 1). Hence, even if RM = 1, Etot and V are necessarily 13 positive. The energy Etot increases monotonically with RM (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

2 approximation, Φ(RM ≈ 0) = −1 + RM/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

13 Controlling the values of Etot and KE by RM originates from the RM-dependence of the normalized energy and temperature. As shown in Section 9.3, the normalized quantities implicitly depend on radius RM on caloric curves even if the dependence does not appear explicitly in the Legendre transformation. In other words, we expect that controlling RM corresponds to changing the state of the system from a non-equilibrium state to another non-equilibrium state. This process is an analogy of tracing inﬁnitely possible equilibria for a self-gravitating system. 14 2 9 The factor M /RM behaves like RM/900 as RM → 0. This behavior changes the KE and Etot into decreasing 2 functions with small increasing RM. However, M /RM still changes monotonically and much more rapidly compared with Etot and KE at RM ∼ 0. Hence, it does not alter the sign of heat capacity.

154 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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

16 energy . 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.

15 As RM >> 1, the halo density ρ(r, t) is independent of the dynamics in the core. The self-similar analysis provides −α −α dρ(r, t)/dt = 0. Hence, D(R) ∝ R and ρt ∝ rt . 16One may recall the failure of the Bohr model of the early quantum theory. An electron releases an electromagnetic radiation due to the acceleration of elctron and deeply penetrates into the potential well around the ion. In a similar way, for the stellar encounter, stars can approach each other even on the scales of stellar size.

155 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

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

−3 latter, the total number NM(≡ M/m) of stars in the core is a small fraction, e.g., NM ≈ 5.6 × 10 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 physically expand the wall itself.

156 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL

3 10 (a) NM −Φ 10−1

10−5 100 102 104 106 108 1010 1012 1014 1016 RM

0 (b) −0.2 −0.4 −0.6 Φ −0.8 Etot/M −1 0 100 200 300 400 500 600 700 800 900 1,000 RM

Figure 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.

(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 speaking, 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’ core collapse like the ss-OAFP model, other than using continuum models. Not only the eﬀect of binary stars stops the core collapse, but also large N ≈ 105 costs unfeasible CPU time to achieve the complete collapse. Hence, one needs to ﬁnd a similarity of the ss-OAFP model to a self-gravitating system of fewer N stars.

157 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL to the core of the ss-OAFP model are as follows. The system must be self-gravitating and composed 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

158 CHAPTER 9. PROPERTY: THE SS-OAFP MODEL 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) clusters 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 radius 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- clusters 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-parametric model (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). We focus 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 model for the PCC clusters (Meylan and Heggie, 1997). The generalization done for the King model directly applies to the model we propose in this chapter. Hence, we do not discuss the particular relation of our model with some generalized models. 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 causes heat to flow inward toward the center while it cools down the central region. However, kinematic surveys generally provide much more significant uncertainties in velocity-dispersion data compared with structural-profile data (Meylan and Heggie, 1997). Hence, one can not easily determine if a well-relaxed (or high-concentration) cluster is currently in pre-collapse- or post-collapse- phase.

164 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL 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.

100

10−1

10−2 D 10−3

10−4 Isothermal sphere ss-OAFP model 10−5 10−1 100 101 102 103 R 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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

ρ F (E) + δ (−E)m−3/2 F˜ ≡ √ c o , (10.1.1) 3 − − 4 2πσc Do(ϕ = 1) + δ B(m 1/2, 3/2)

where δ and m are positive real numbers. The Do(ϕ) is the reference density of the ss-OAFP model R 1 and B(a, b) is the beta function deﬁned as B(a, b) = 2 t2a−1(1 − t2)b−1 dt with a > 1/2 and b > 1. 0

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-Maxwellian DF. 5The index m in our model is to be determined at ﬁrst place. Our model has the same parameter-dependence as the truncated γ exponential (fractional-power) model discussed in (Gomez-Leyton and Velazquez, 2014). The outer halo is controlled by a polytropic sphere.

166 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

can be analytically derived from the polytrope’s DF. The density of the new model reads

m Do(ϕ) + δ B(m − 1/2, 3/2) (−ϕ) D(ϕ) = ρc . (10.1.2) Do(ϕ = −1) + δ B(m − 1/2, 3/2)

The Poisson’s equation in dimensionless form is

d2ϕ¯ 2 dϕ ¯ D (ϕ ¯) + δ B(m − 1/2, 3/2) (−ϕ¯)m o , 2 + = (10.1.3) d¯r r¯ d¯r Do(ϕ ¯ = −1) + δ B(m − 1/2, 3/2)

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

dψ¯ ψ¯(¯r = 0) = 1, (¯r = 0) = 0. (10.1.4) d¯r

Sinceϕ ¯ is an independent variable for the ss-OAFP model, we solved the the inverse form of the

Poisson’s equation for R(ϕ ¯)

!2 !3 d2R dR dR D (ϕ ¯) + δ B(m − 1/2, 3/2) (−ϕ¯)m − o . R 2 2 = (10.1.5) dϕ ¯ dϕ ¯ dϕ ¯ Do(ϕ ¯ = −1) + δ B(m − 1/2, 3/2)

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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

100

10−1

10−2

10−3 D ss-OAFP 10−4 δ = 1000

−5 δ = 1 10 δ = 0.1 10−6 δ = 0.01 δ = 0.0001 10−7 10−2 10−1 100 101 102 103 R 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.

100

10−1

−2 ss-OAFP ¯

ϕ 10 δ = 1000 δ = 1 10−3 δ = 0.1 δ = 0.01 δ = 0.001 −4 10 δ = 0.00001

10−3 10−2 10−1 100 101 102 103 104 105 106 R Figure 10.3: Dimensionless potentialϕ ¯(R) of the energy-truncated ss-OAFP model. The corresponding 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL proﬁle D(ϕ ¯) to the projected density proﬁle

Z ∞ D(φ) 0 Σ(r) = 2 p dr . (10.1.6) 0 1 − (r/r0)2

The corresponding inverse form with dimensionless variables is

s " # Z ϕ¯ 1 − µ (ϕ, ¯ ϕ¯ 0) dD R(ϕ ¯ 0) 1 + 2µ (ϕ, ¯ ϕ¯ 0) − R − 0 0 R 0 Σ(ϕ ¯) = 2 0 2 0 0 + D(ϕ ¯ )S(ϕ ¯ ) 0 dϕ ¯ , (10.1.7) 0 1 + µR(ϕ, ¯ ϕ¯ ) dϕ ¯ S(ϕ ¯ ) 1 + µR(ϕ, ¯ ϕ¯ )

0 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

−2.23 (as expected from the asymptotic density profile of the ss-OAFP model since Do ∝ R ). 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

100

10−1

10−2

Σ R−1.23 10−3 δ = 10 δ = 1 10−4 δ = 0.01 δ = 0.001 δ = 0.00001 10−5 10−2 10−1 100 101 102 103 R Figure 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 → ∞.

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 potential. 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

d2φ¯ 2 dφ¯ 1 I(φ¯) + − = 0. (10.1.8) d¯r2 r¯ d¯r K I(1)

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 asr ¯ = Kr¯. 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 and 170 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

concentration6 as follows

rKin r¯Kin ≡ √ , (10.1.9a) K(m) " # r √ c¯ ≡ log tid K(m) , (10.1.9b) rKing √ (m) where K 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

(m) 7 rtid = 13.444 (based on our calculation). Hence, K = 4.82 for m = 3.9. Figure 10.5 depicts the concentrationc ¯ 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 < c4(= 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 isc ¯ > 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 ofc ¯ → 1.

6The tidal radius for the energy-truncated ss-OAFP model should not be regularized since it is still the radius at which the projected density reaches zero. The tidal radius can be found after the model is properly ﬁtted to the projected 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 the King model.

171 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

Finite ss-OAFP 3 Polytrope of m = 3.9

c 2

10−3 10−2 10−1 100 101 102 δ Figure 10.5: Concentration of the energy-truncated ss-OAFP model. The horizontally dashed line represents 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

NGC 1904 NGC 2419 NGC 6205

c rc rtid c rc rtid c rc rtid Finite ss-OAFP The 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

c rc rtid c rc rtid c rc rtid Finite ss-OAFP The 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-OAFP model. The structural parameters are compared with the previous compilation works based on the King- and Wilson- models.

174 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

0.0 0.0

−1.0 −1.0 ]

Σ −2.0 −2.0 log[

−3.0 ss-OAFP −3.0 (m = 3.0) ss-OAFP modiﬁed power-law (m = 3.5) −4.0 NGC 6752 −4.0 NGC 6752 −2 −1 0 1 −2 −1 0 1

0.0 0.0

−1.0 −1.0 ]

Σ −2.0 −2.0 log[ (r = 00.149, δ = 0.0044) (r = 00.188, δ = 0.0079) −3.0 c −3.0 c ss-OAFP ss-OAFP (m = 3.9, c¯ = 2.56) (m = 4.2, c¯ = 2.46) −4.0 NGC 6752 −4.0 NGC 6752 −2 −1 0 1 −2 −1 0 1

] 0.5 0.5 Σ 0 0 log[

∆ −0.5 m = 3.9 −0.5 m = 4.2 −2 −1 0 1 −2 −1 0 1 log[R(arcmin)] log[R(arcmin)]

Figure 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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).

2 Our model reasonably fits the density profile of NGC 6397 with χν = 1.52 (Figure 10.7), where the reduced chi-square is defined as follows

2 2 X χ χν = , (10.2.1) nd.f.

2 where χ 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 the latter 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). This is spatially constant for single-component models (McLaughlin, 2003). The exact evaluation of the effects is not important here since all the data in the present work normalize the structural profiles at the smallest radius data point. 177 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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

2 shows the values of χν 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

2 6397 and NGC 6752) at R /1 arcminute with χν / 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.

0.0

−1.0 ] Σ −2.0 log[ 0 (rc = 0 .085, δ = 0.0012) ] 0.5 −3.0 Σ ss-OAFP 0

(¯c = 2.19) log[ −4.0 NGC 6397 (c) ∆ −0.5 NGC 6397 (c) −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1 1.5 log[R(arcmin)] log[R(arcmin)]

Figure 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

0.0 0.0

−1.0 −1.0 SB

− −2.0 −2.0 o 0 0 (rc = 0 .25, δ = 0.047) (rc = 0 .078, δ = 0.0064) SB −3.0 ss-OAFP −3.0 ss-OAFP (¯c = 1.62) (¯c = 2.60) NGC 6388 (n) Trz 2 (c) −4.0 −4.0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

0.5 0.5 SB) −

o 0 0

(SB −0.5 NGC 6388(n) −0.5 Trz 2 (c) ∆ −2.5 −2 −1.5 −1 −0.5 0 0.5 1 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 log[R(arcmin)] log[R(arcmin)]

Figure 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 brightness 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.

10.3 (Main result) The relaxation time and completion rate of

core collapse against the concentrations of the clusters

with resolved cores

Concentrationc ¯ is a possible measure to characterize the states of globular clusters in the relaxation evolution, especially for the cores. Hence the present section comparesc ¯ with the core relaxation time and completion rate of core collapse. The energy-truncated ss-OAFP model can 179 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

2 KM cluster χν Nb

NGC 288 0.45 0 2 PCC cluster χν Nb NGC 1851 0.56 0 NGC 6342 1.73 3 NGC 5466 2.07 0 NGC 6397 1.52 0 NGC 6121 0.72 0 NGC 6453 1.89 5 PCC cluster χ2 N NGC 6205 1.05 0 ν b NGC 6522 2.52 5 NGC 6254 0.57 0 NGC 5946 6.75 5 NGC 6558 2.17 5 NGC 6626 0.47 0 NGC 6624 7.18 5 NGC 6752 2.00 6 NGC 6809 0.44 0 NGC 7099 2.12 2 Pal 3 0.06 0 Trz 1 2.41 5 Pal 4 0.34 0 Trz 2 1.94 0 Pal 14 0.31 0 Trz 5 2.23 0

2 Table 10.2: Values of χν 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL 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 characteristics of the core relaxation time tc.r. against the concentrationc ¯. Figure 10.9(b) portrays the corresponding 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 catalog. 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 concentrationc ¯ 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 profiles 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

10 10 PCC PCC PCC? PCC? 9 Normal 9 Normal

8 8 c.r. c.r. t 7 t 7

6 6

5 (a) 5 (b) 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 c¯ c Figure 10.9: Core relaxation time against (a) concentrationc ¯ obtained from the energy-truncated ss-OAFP model (m = 3.9) applied to the PPC- and KM- clusters and (b) concentration c based on the King model reported in (Harris, 1996, (2010 edition)).

To see the completion rate of core collapse, we used the formula employed in (Lightman, 1982)

p 2 to,age to,age −(1 + Aqo) + (1 + Aqo) + 4ABqo ηc ≡ ≡ (10.3.1) tc.r.o tc.r. 2Bqo

10 where A = 35, B = 4.8 and qo = to,age/tc.r.. The time to,age is the order of the age of clusters ∼ 10

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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 thanc ¯ = 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 withc ¯ ≈ 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

(a) NGC6544 (b) NGC6544 1 1

05 . NGC6517 1 NGC6517

0.8 0) . 0.8 2 NGC1851 − c NGC6626 75(¯ . 4 . 0.6 0 0 0.6

0) . 2 0.4 −

completion rate c 0.4 completion rate

75(¯ . 0

0.2 PCC 0.2 PCC PCC? PCC? Normal Normal 0 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 c¯ c Figure 10.10: Completion rate of core collapse against (a) concentrationc ¯ 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)).

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).

184 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

0.0 0.0 −2.0 − . −4.0 2 0 0 (rc = 0 .25, δ = 0.038)

SB −6.0 − −4.0 o 0 −8.0 (rc = 0 .0040, δ = 0.0055) SB

−10.0 ss-OAFP −6.0 ss-OAFP (¯c = 2.16, m = 4.2) (¯c = 3.28, m = 4.2) −12.0 NGC 6341(n) NGC 6284(c) Cheb. −8.0 Cheb. −14.0 −3 −2 −1 0 1 −3 −2 −1 0 log[R(arcmin)] log[R(arcmin)]

Figure 10.11: Fitting 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 arcminutes) 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 example, 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

0.0

0 −5.0 (rc = 0 .0648, δ = 0.080) SB −

o ss-OAFP (¯c = 2.53, m = 4.4) SB approx. −10.0 (¯c = 2.79, m = 4.9) NGC 6715(n) Cheb. −3 −2 −1 0 1

Figure 10.12: Fitting 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). 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

exp [−13.88(1 + E)] + δ (−E)m F˜(E) = , (10.4.1) exp [−13.88] + δ

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

11 considering χesc as a new parameter allowing the four degrees of freedom . On the one hand, the

11 We have tried to employ the four parameters model, though the useful values of χesc were χesc ' 9 holding the index m to 3.9. Less than those values, the morphology of the approximated model has perfectly changed from the exact model. Especially, the approximated model with low χesc(/ 9) could not fit the structural profiles of the PCC cluster at all, Perhaps, this reflects the result of (Cohn, 1980) in which the signs of the self-similarity and core collapse appear for χesc(' 9). 188 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL approximated model should not be applied to the PCC clusters in the first place since the approximation 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 dependence 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 function 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

100 0 (a) (b)

−1

10−1 −2 ] Σ

log[ −3

10−2

−4 OAFP (δ = 10−3) − Do/Do(−1) approx. (δ = 10 3) 3.9 δB(3.4, 1.5)(−E) /Do(−1) OAFP (δ = 0.1) exp[−13.88(1 + E)] −5 approx. (δ = 0.1) 10−3 0.6 0.7 0.8 0.9 1 −3 −2 −1 0 1 2 −E log[R]

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

10−1 exp[−13.88(1 + E)] Da D 10−3

10−5

10−7

10−9

10−5 10−4 10−3 10−2 10−1 100 −E

Figure 10.14: Density proﬁles Do of the energy-truncated ss-OAFP model, Da of the approximated model and exp[−13.88(1 + E)]. 190 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 concentrationsc ¯ 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 structural 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 polytropic 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 for NGC 288. We found the same result (χν = 1.7 with Wo = 5.8) for NGC 288 as their result. However, we use the result of our calculation and consider NGC 288 is a low concentration cluster. This is since we found the concentration for NGC 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

0.0 0.0

−1.0 −1.0 2 2 ] (χν = 0.48) (χν = 5.8) Σ −2.0 −2.0 log[ −3.0 −3.0 polytope polytrope (m = 4.50) (m = 4.90) −4.0 NGC 288 (n) −4.0 NGC 6254 (n) −1 0 1 −1 0 1 0.5 0.5 ] Σ 0 0 log[

∆ NGC 288(n) NGC6254(n) −0.5 −0.5 −1 0 1 −1 0 1 log[R(arcmin)] log[R(arcmin)]

Figure 10.15: Fitting 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

0.0

(χν = 2.9) SB −2.0

− 0.5 o SB) SB polytrope −

o 0 −4.0 (m = 4.77)

NGC 5139 (n) (SB NGC 5139 (n) ∆ −0.5 −1 0 1 2 −1 0 1 2 log[R(arcmin)] log[R(arcmin)]

Figure 10.16: Fitting 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.

(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 maximum 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL 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

1/3 order of 1/n ∼ 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 integrals 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 examples 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 discussed 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 estimated 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

tage ηM ≡ , (10.4.2) tsec

13 1/3 The corresponding scales of the apocenter and pericenter are rtid/N and 1/n . 194 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

where the secular evolution time is deﬁned as

" # M tsec ≡ ln 0.11 tc.r.o, (10.4.3) M

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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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.

4 polytropic 3.5 polytropic? non-polytropic 3

2.5

2 M η 1.5

0.5

0 0 1 2 c

Figure 10.17: Secular relaxation parameter ηM against concentration c. The values of concentration are 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

h M i polytropic c tc.r. tc.r.o log ηM tage Reference for tage Mo cluster (Gyr) (Gyr) (Gyr)

NGC 288 0.99 0.98 2.0 4.64 0.63 10.62 (Forbes and Bridges, 2010) NGC 1261 1.16 0.39 1.15 5.17 0.913 10.24 (Forbes and Bridges, 2010) NGC 5053 0.74 6.5 8.2 4.41 0.19 12.29 (Forbes and Bridges, 2010) NGC 5139 1.31 4.0 5.5 6.38 0.17 11.52 (Forbes and Bridges, 2010) NGC 5466 1.04 2.2 3.6 4.85 0.41 13.57 (Forbes and Bridges, 2010) NGC 5897 0.86 2.1 3.5 4.83 0.40 12.3 (Forbes and Bridges, 2010) NGC 5986 1.23 0.38 1.24 5.48 0.94 12.16 (Forbes and Bridges, 2010) NGC 6101 0.80 1.6 2.9 4.83 0.48 12.54 (Forbes and Bridges, 2010) NGC 6205 1.53 0.32 1.12 5.59 0.98 11.65 (Forbes and Bridges, 2010) NGC 6402 0.99 1.14 2.35 5.89 0.47 12.6 (Santos and Piatti, 2004) NGC 6496 0.70 0.87 2.0 4.29 0.82 12.42 (Forbes and Bridges, 2010) NGC 6712 1.05 0.40 1.2 4.98 0.95 10.4 (Forbes and Bridges, 2010) NGC 6723 1.11 0.62 1.7 5.15 0.81 13.06 (Forbes and Bridges, 2010) NGC 6809 0.93 0.72 1.8 5.03 0.77 13.0±0.3 (Wang et al., 2016) NGC 6981 1.21 0.52 1.4 4.80 0.89 10.88 (Forbes and Bridges, 2010) 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)

h M i polytropic c tc.r. tc.r.o log ηM tage Reference for tage Mo cluster? (Gyr) (Gyr) (Gyr)

NGC 3201 1.29 0.41 1.2 5.05 0.92 10.24 (Forbes and Bridges, 2010) NGC 4590 1.41 0.28 1.1 4.95 1.29 13.0 ±1.0 (Dotter et al., 2009)

h M i non-polytropic c tc.r. tc.r.o log ηM tage Reference for tage Mo 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 5634 2.07 0.047 0.53 5.18 2.30 11.8 (Forbes and Bridges, 2010) NGC 6121 1.65 0.079 0.61 4.83 2.11 11.5±0.4 (Wang et al., 2016) NGC 6144 1.55 0.60 1.7 4.76 0.94 13.82 (Forbes and Bridges, 2010) NGC 6254 1.38 0.16 0.81 5.06 1.51 11.39±1.1 (Forbes and Bridges, 2010) NGC 6273 1.53 0.33 1.1 6.03 1.47 11.90 (Forbes and Bridges, 2010) NGC 6352 1.10 0.29 1.1 4.57 1.37 12.67 (Forbes and Bridges, 2010) NGC 6388 1.75 0.052 0.553 6.16 1.82 12.03 (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 6656 1.38 0.34 1.2 5.53 1.00 12.67 (Forbes and Bridges, 2010) 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 current relaxation time tc.r. and concentration c are values reported in (Harris, 1996, (2010 edition)). The h M i total masses log are from (Mandushev et al., 1991) where Mo is solar mass. We adapted the Mo 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 observation 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 relation 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL

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 numerical 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 applicability 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 CHAPTER 10. APPLICATION: THE SS-OAFP MODEL 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 unrealistically 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 structural 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 physical 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 relaxation- time 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 polytropic 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 corresponding 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 convergence 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

−3 α is 2.2305, the collapse rate ξ = 3.64 × 10 , and the scaled escape energy χesc = 13.88. On the 202 CHAPTER 11. CONCLUSION 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 c4 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.

βo ∗ One loses the capability of handling the ss-OAFP equation when the power-law c1(−E) /c4 in the −13 4ODEs at Emin = −0.0523 reaches the order of 10 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 = 0.5 ρ1.1. The analog of caloric curves and a simple χesc 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 CHAPTER 11. CONCLUSION 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 arcminutes). 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 CHAPTER 11. CONCLUSION

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 cluster’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

Gm φi j(ri j) = − , (1 ≤ i, j ≤ N with i , j), (A.1.1) ri j

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

  XN  p2 XN   i  H =  + m φi j(ri j) , (A.1.2) 2m  i=1 j>i

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

  dF  XN  N = ∂ + [v · ∇ + a · ∂ ] F (1, ··· , N, t) = 0, (A.1.3) dt  t i i i i  N i=1

∂ ∂ ∂ where the symbols for the operators are abbreviated by ∂t = , ∇i = , and ∂i = . The ∂t ∂ri ∂vi

acceleration ai of star i due to the pair-wise Newtonian forces from the rest of stars is deﬁned as

XN XN ai ≡ ai j ≡ − ∇iφi j(ri j). (A.1.4) j=1(,i) j=1(,i)

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

Z 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 permutation 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 APPENDIX A. BASIC KINETICS

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 Z Fs(1 ··· s, t) = FN(1, ··· , N, t) ds+1 ··· dN, (A.2.1)

or the form of s-tuple DF: N! f (1 ··· s, t) = F (1 ··· s, t). (A.2.2) s (N − s)! s

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

  Z Z XN  p2 XN   i  E(t) = ···  + m φi j(ri j) FN(1 ··· N, t) d1 ··· dN, (A.2.3a) 2m  i=1 j>i Z p2 N(N − 1) Z = N 1 F (1, t) d + m φ (r )F (1, 2, t) d d , (A.2.3b) 2m 1 1 2 12 12 2 1 2 Z p2 m Z = 1 f (1, t) d + φ (r ) f (1, 2, t) d d . (A.2.3c) 2m 1 1 2 12 12 2 1 2

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 APPENDIX A. BASIC KINETICS 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 interactions, 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) " # f (1, t) f (2, t) f (1, 2, t) ≡ f (1, 2, t) = f (1, t) f (2, t) + g(1, 2, t) − , (A.2.5b) 2 N

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) 1 1 − (2g(1, 2, t) + f (1, t) f (2, t)) f (3, t) − (2g(2, 3, t) + f (2, t) f (3, t)) f (1, t) N N 1 − (2g(3, 1, t) + f (3, t) f (1, t)) f (2, t), (A.2.5c) N 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 APPENDIX A. BASIC KINETICS

Under the weak-coupling approximation, one can retrieve the form of the DFs2

! f (1, t) f (2, t) f (1, 2, 3, t) = f (1, t) f (2, t) f (3, t) + g(1, 2, t) − f (3, t) 3 N ! ! f (2, t) f (3, t) f (3, t) f (1, t) + g(2, 3, t) − f (1, t) + g(3, 1, t) − f (2, t). (A.2.6) N N

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

Z Z 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

Z p2 E(t) = 1 f (1, t) d + U (t) + U (t), (A.2.8) 2m 1 1 m.f. cor where

m Z U (t) = φ(r , t) f (1, t) d , (A.2.9a) m.f. 2 1 1 m Z U (t) = φ (r )g(1, 2, t) d d . (A.2.9b) cor 2 12 12 1 2

2The expressions were ﬁrst shown in (Gilbert, 1968) for stellar dynamics based on a functional derivative method (Bogoliubov, 1962).

210 APPENDIX A. BASIC KINETICS

The self-consistent gravitational m.f. potential is deﬁned as

! 1 Z φ(r , t) = 1 − φ (r ) f (2, t) d , (A.2.10) 1 N 12 12 2

1 where the factor 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. acceleration of star 1 reads

! 1 Z A(r , t) = − 1 − ∇ φ (r ) f (2, t) d . (A.2.11) 1 N 1 12 12 2

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 parameter, 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 APPENDIX A. BASIC KINETICS

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-

o tance b90 (the impact parameter to deﬂect test star thorough an encounter by 90 from the original direction of motion);

b90(v12(−∞)) r90(v12(−∞)) = √ , (A.3.2a) 1 + 2 2Gm −∞ b90(v12( )) = 2 , (A.3.2b) v12(−∞)

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 and passage of stars.

212 APPENDIX A. BASIC KINETICS

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, 1/2 aBG ∼ 1/N

G, ro, Kn ∼ 1/N,

g(1, 2, t) ∼ N/r12, for ro < r12 < R 2 ∼ N , for r12 < ro 2 a12 ∼ 1/(r12N) R tcor, δvA = A1 dtcor ∼ r12 for ro < r12 < R 2 ∼ δv12r12N for r12 < ro Table 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 r except for the correlation time t , R 12 cor 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

1 v1 · ∇1 f (1, t) ' , (A.3.3a) tdyn f (1, t) ! 1 1 ∂ f (1, t) ' , (A.3.3b) tsec f (1, t) ∂t ! 1 1 ∂g(1, 2, t) ' . (A.3.3c) tcor g(1, 2, t) ∂t

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 distance has been discussed for Newtonian interaction (Retterer, 1979; Ipser and Semenzato, 1983;

213 APPENDIX A. BASIC KINETICS

star 1 star 2

strong 2-body weak 2-body weak m.f m.f.(many-body) (Boltzmann) (Landau) (g-Landau) (g-Landau)

r12 ro aBG d R ∼ ∼ ∼ A1 1 1 1 ∼ 1

g(1, 2, t) ∼ N2 ∼ N3/2 ∼ N4/3 ∼ N −1/3 a12 ∼ N ∼ 1 ∼ N ∼ 1/N −1/2 −2/3 δva ∼ 1 ∼ N ∼ N ∼ 1/N −1 −1/2 −1/3 δvA, tcor ∼ N ∼ N ∼ N ∼ 1

Table A.2: A scaling of physical quantities according to the eﬀective interaction range of Newto- nian interaction accelerations and close encounter.

214 APPENDIX A. BASIC KINETICS

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

2Gm b ' ≡ b , (A.3.4a) 90 < v >2 o bo ro ≡ √ . (A.3.4b) 1 + 2

Assume the dispersion speed may be determined by the Virial theorem for a ﬁnite spherical star cluster of the radius of R as follows

r GmN < v >≡ ζ , (A.3.5) R

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

r 2 1 o = √ . (A.3.6) R 1 + 2 c2N

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 APPENDIX A. BASIC KINETICS

4 star k 4 star 1

4 star 2

Figure 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.

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;

Z t 0 0 ri(t) = ri(t − τ) + vi t dt , (i , j = 1, 2) (A.4.1a) t−τ Z t h 0 0i 0 vi(t) = vi(t − τ) + ai j t + Ai t dt . (A.4.1b) t−τ

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 APPENDIX A. BASIC KINETICS

two-body problem

Z t 0 0 ri(t) = ri(t − τ) + vi t dt , (A.4.2a) t−τ Z t 0 0 vi(t) = vi(t − τ) + ai j t dt . (A.4.2b) t−τ

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 collision description if one includes the Markovian approximation (e.g., Cercignani, 1988; Klimon- tovich, 1982)

Z t 0 0 r12(t) = r12(t − τ) + v12 t dt , (A.4.3a) t−τ r + r R = 1 2 ≈ r , (A.4.3b) 2 1 Z t 0 0 v1 (t ) + v2 (t ) 0 r1 = r1(t − τ) + dt , (A.4.3c) t−τ 2 Z t 0 0 vi(t) = vi(t − τ) + ai j t dt . (A.4.3d) t−τ

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 APPENDIX A. BASIC KINETICS

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

Z t 0 0 ri(t) = ri(t − τ) + vi t dt , (A.4.5a) t−τ Z t 0 0 vi(t) = vi(t − τ) + Ai t dt . (A.4.5b) t−τ

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 beginning 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

  1, (if r ≥ 4)  i j Θ(ri j − 4) =  (B.1.1a)  0, (otherwise)

≡ θ(i, j), (B.1.1b) with the following mathematical identity

ri j ∇iθ(i, j) = δ(ri j − 4). (B.1.2) ri j

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

N! Z f 4(1, ··· , s, t) = F (1, ··· , N, t) d ··· d , (B.1.3) s − N s+1 N (N s)! Ωs+1,N

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 → N! O(1) as N → ∞), meaning the small number s in the factor (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 the present work is slightly diﬀerent from the Cercignanni’s work due to the deﬁnition of DF.

220 APPENDIX B. BBGKY HIERARCH

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

 N i−1   Y Y n o Ω = {r , p ··· r , p } | r − r |> 4  . (B.1.4) s+1,N  s+1 s+1 N N i j  i=s+1 j=1

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

Z Is ≡ ds+1 ··· dNS (1, ··· , N, t), (B.1.6) Ωs+1,N

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 APPENDIX B. BBGKY HIERARCH

(B.1.4)), one may explicitly express the term as follows

Z Is = ds+1 θ(s+1,1) ··· θ(s+1,s) Z × ds+2 θ(s+2,1) ··· θ(s+2,s) θ(s+2,s+1) ...... Z × dN−1 θ(N−1,1) ··· θ(N−1,s)θ(N−1,s+1) ··· θ(N−1,N−2) Z × 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)

PN B.1.1 Truncated integral over the terms i=1 vi · ∇iFN(1, ··· , N, t)

PN For the function S (1, ··· , N, t) = 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

4 The goal here is to reduce the term Is associated with the terms vi ·∇iFs (1, ··· , s, t) by repeating the integral-by-parts method. For the numbers 1 ≤ i ≤ s, deﬁne the following term

s Z (1:s) X Is ≡ ds+1 ··· dNvi · ∇iFN(1, ··· N, t). (B.1.8) i=1 Ωs+1,N

222 APPENDIX B. BBGKY HIERARCH

Employing equation (B.1.2), one obtains

s (1:s) X 4 Is = vi · ∇iFs (1, ··· , s, t) i=1 Xs XN Z Z Z Z Z − vi · dN dN−1 ··· d j ··· ds+2 ds+1 i=1 j=s+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) . . . ..

ri j × θ( j,1) ··· δ( j,i) ··· θ( j,s) ··· θ( j, j−1) ri j . . . ..

× 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

Z Z I ri j 3 d jθ( j,1) ··· δ( j,i) ··· θ( j, j−1) = d p j dσi j, (B.1.10) ri j

ri j where the dσi j is the surface element of a sphere of the radius 4 with a radial unit vector around ri j the position r j. Employing equation (B.1.10), one obtains

s  N Z I  X  X  I(1:s) = v · ∇ F4 − d3p v · dσ F4 (1, ··· , s + 1, t) . (B.1.11) s  i i s j i i j s+1  i=1 j=s+1

223 APPENDIX B. BBGKY HIERARCH

Case 2: s + 1 ≤ i ≤ N

Deﬁne the term Is associated with the numbers s + 1 ≤ i ≤ N;

N Z (s+1:N) X Is ≡ ds+1 ··· dNvi · ∇iFN(1, ··· N, t). (B.1.12) i=s+1 Ωs+1,N

(s+1:N) To reduce the term Is , one must modify equation (B.1.9) as follows

N s Z I (s+1:N) X X 3 4 Is = − d pi vi · dσi jFs+1(1, ··· , s + 1, t) i=s+1 j=1 N Z  N i−1  X  Y Y  + v · ∇  θ F (1, ··· N, t) d ··· d i i  ( j,i) N  s+1 N i=s+1 i=s+1 j=1 XN XN − vi · i=s+1 j=s+1 Z Z Z Z Z Z × 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) ......

ri j × θ( j,1) ··· θ( j,s) θ( j,s+1) ··· δ( j,i) ··· θ( j, j−1) ri j ......

× 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 APPENDIX B. BBGKY HIERARCH

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

s N Z I (s+1:N) X X 3 4 Is = d p j v j · dσi jFs+1(1, ··· , s + 1, t) i=1 j=s+1 N N Z Z I X X 3 4 + di d p j v j · dσi j Fs+2(1, ··· , s + 2, t), (B.1.14) i=s+1 j=s+1

where the following relation is employed

Z Z Z Z I ri j 3 di d j θ( j,1) ··· δ( j,i) ··· θ( j, j−1) = di d p j v j · dσi j. (B.1.15) ri j

Combining equation (B.1.14) with the result of case 1 (1 ≤ i ≤ s) and making use of the dummy integral variables, one obtains

s s "Z # X 4 X 3 4 Is = vi · ∇iFs + (N − s) d vs+1 Fs+1vi,s+1 · dσi,s+1 i=1 i=1 (N − s)(N − s − 1) Z Z + d3v d × F4 v · dσ , (B.1.16) 2 s+2 s+1 s+2 s+1,s+2 s+1,s+2

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 APPENDIX B. BBGKY HIERARCH the following equation in terms of s-tuple DFs

s  s  s " Z # X  X  X ∂ f + v · ∇ + a · ∂  f + ∂ · f a d t s  i i i j i s i s+1 i,s+1 s+1 i=1 j=1(,i) i=1 Ωs+1,s+1 s " # X Z 1 Z Z = d3v f v · dσ + d3v d f v · dσ , s+1 s+1 i,s+1 i,s+1 2 s+2 s+1 s+2 s+1,s+2 s+1,s+2 i=1 (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 dσ . Also, in equation (B.1.17), the following approximation was i j made according to (Ito, 2018a) to hold the conservation of total number and energy

4 2 f1 (1, t) = f1(1, t) + O(1/N ), (B.1.18a)

4 f2 (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 and stellar 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 APPENDIX B. BBGKY HIERARCH

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

   fN(1, ··· , N, t), (if ri j ≥ 4with i , j) N  fN (1, ··· , N, t) =  (B.2.1)  0, (otherwise)

Following the deﬁnition of single- and double- truncated DFs in (Grad, 1958), the ﬁrst two s-tuple hard-sphere DFs explicitly read

N f1 (1, t) = f1(1, t), (B.2.2a)    f2(1, 2, t), if r12 ≥ 4 N  f2 (1, 2, t) =  (B.2.2b)  0, otherwise

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 APPENDIX B. BBGKY HIERARCH

double DF at the radius r12 = 4 is deﬁned as the limit value

f (1, 2, t) = lim f N(1, 2, t). (B.2.3) 2 r12=4 2 r12→4+

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) ! 1 O O(1), N 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

f (1, 2, t) = 0. (B.2.5) 2 r12=4

228 APPENDIX B. BBGKY HIERARCH

The corresponding deﬁnition for the Heaviside function is uniquely determined3 as follows

  1, (ri j > 4) Θ(r − 4) ≡  (B.2.7) i j   0. (ri j ≤ 4)

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 to hold the Leibniz rule, one needs to employ the following deﬁnition  1, (ri j > 4)  − 4 ≡  1 4 Θ(ri j )  2 , (ri j = ) (B.2.6)  0. (ri j < 4)

This formulation is signiﬁcant only to hold the surface integral terms when one assumes that equation (B.2.4) is correct. Accordingly, the use of equation (B.2.7) prohibits us from taking a derivative of any product of identical step functions 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 APPENDIX B. BBGKY HIERARCH

s-tuple hard-sphere DF. The double DF may be explicitly deﬁned as

   1  1 − f (1, t) f (2, t) + g(1, 2, t), (if r12 ≥ 4) N  N f2 (1, 2, t) =  (B.2.8a)  0, (otherwise) ! 1 ≡ 1 − f (1, t) f (2, t) + g(1, 2, t)r ≥4, (B.2.8b) N r12≥4 12

where the DFs f (1, t) and f (2, t) are not exactly statistically uncorrelated. This is since we must

N consider the geometrical condition assigned on the interaction range, r12 > 4. Only the DFs f (1, t) and f N(2, t) are statistically independent of each other. Hence,

f (1, t) f (2, t) f N(1, t) f N(2, t). (B.2.9) r12≥4 ,

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

N f (1, 2, t) ≡ Θ(r12 − 4) f (1, 2, t), (B.2.10a)

N f (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

! Z 1 Z f N(1, 2, t) d2 = (N − 1) f (1, t) 1 − f (2, t) d2 . (B.2.11) r12>4 N r12<4

To hold the consistent relationship between DF f (1, t) and higher orders of the DF, one may consider 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 APPENDIX B. BBGKY HIERARCH

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

! Z 1 f N(1, 2, t) d2 = (N − 1) f (1, t) + O , (B..12) r >4 N Z 12 N Θ(r13 − 4)Θ(r23 − 4) f (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¯, wheren ¯ is the (initial) mean density of the system in the relaxation evolution. Also, one may employ a standard definition for R the total number of stars N = f (1, t) d1. . If one neglects the effect of strong encounter following the conventional ideas in stellar dynamics, 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

s  s  s " Z # X  X  X ∂ f N + v · ∇ + a · ∂  f N + ∂ · f N a d = 0. (B.0.14) t s  i i i j i s i s+1 i,s+1 s+1 i=1 j=1(,i) i=1 Ωs+1,s+1

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

Z (∂t + v1 · ∇1) f1(1, t) = −∂1 · f2(1, 2, t)a12 d2, (C.1.1a) Ω2,2 Z (∂t + v1 · ∇1 + v2 · ∇2 + a12 · ∂12) f2(1, 2, t) = − a1,3 · ∂1 + a2,3 · ∂2 f3(1, 2, 3, t) d3, (C.1.1b) Ω3,3 Z (∂t + v2 · ∇2) f1(2, t) = −∂2 · f2(2, 3, t)a23 d3, (C.1.1c) r23>4

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

Z (2,2) ∂t + v1 · ∇1 + A1 · ∂1 f (1, t) = −∂1 · g(1, 2, t)a12 d2, (C.1.2a) Ω2,2

(3,3) (3,3) N ∂t + v1 · ∇1 + v2 · ∇2 + A1 · ∂1 + A2 · ∂2 g (1, 2, t) ! 1 1 = −a · ∂ + A(3,3) · ∂ + A(3,3) · ∂ f (1, t) f (2, t)Θ(r − 4) 12 12 N 1 1 N 2 2 12 ! 1 h i − 1 − A(2,2) − A(3,3) · ∂ f (1, t) f (2, t)Θ(r − 4) N 1 1 1 12 ! 1 h i − 1 − A(2,2) − A(3,3) · ∂ f (1, t) f (2, t)Θ(r − 4) N 2 2 2 12 Z Z ! − ∂1 · a13g(1, 3, t) d3 − a13g(1, 3, t) d3 f (2, t) Ω3,3 Ω2,2 Z Z ! − ∂2 · a23g(2, 3, t) d3 − a23g(2, 3, t) d3 f (1, t), (C.1.2b) Ω3,3 Ω2,2 where the lowest OoMs of the terms are left with O(1).1 The truncated m.f. accelerations are deﬁned as

" # Z (2,2) 1 Ai (ri, t) = 1 − f (3, t)ai3 d3, (i, j = 1, 2) (C.1.3a) N ri3>4 " # Z (3,3) 1 Ai (ri, r j, t) = 1 − f (3, t)ai3 d3. (C.1.3b) N Ω3,3

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 acceleration A(2,2)/N to hold the self-consistency of the kinetic equation.

233 APPENDIX C. COULOMB LOGARITHM AND FP EQUATION

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

Z Z 2 · d3 ≈ · d3 + O 1/N . (C.1.5) Ω2,2 Ω3,3

Hence, equation (C.1.2b) simply reduces to

(2,2) (2,2) 4 4 ∂t + v1 · ∇1 + v2 · ∇2 + A1 · ∂1 + A2 · ∂2 g(1, 2, t) = − a˜12 · ∂1 + a˜21 · ∂2 f (1, t) f (2, t), (C.1.6)

where

1 a˜4 = a − A(2,2), (C.1.7a) 12 12 N 1 1 a˜4 = a − A(2,2). (C.1.7b) 21 21 N 2

Employing the method of characteristics, one can solve equation (C.1.6) for the correlation function

gN(1, 2, t) = gN(1(t − τ), 2(t − τ), t − τ) Z t 4 4 0 0 0 0 0 0 − a˜12 · ∂1 + a˜21 · ∂2 t=t0 f 1(t ), t f 2(t ), t Θ(r12(t ) − 4) dt . (C.1.8) t−τ

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 APPENDIX C. COULOMB LOGARITHM AND FP EQUATION 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)

(2,2) ∂t + v1 · ∇1 + A1 · ∂1 f (1, t) = Z Z τ 0 4 4 0 0 0 0 ∂1 · d2a12 dτ a˜12 · ∂1 + a˜21 · ∂2 t−τ0 f (1(t − τ ), t − τ ) f (2(t − τ ), t − τ ). (C.1.9) Ω2,2 0

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

t O(1/N2) < cor . O(1/N). (C.1.11) trel

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 the changes in the momentum and position of test star in an encounter is not negligible due to the eﬀect of m.f. acceleration.

235 APPENDIX C. COULOMB LOGARITHM AND FP EQUATION

0 τ < tcor

f (1(t − τ0), t − τ0) f (2(t − τ0), t − τ0) ≈ f (1(t − τ0), t) f (2(t − τ0), t), (C.1.12)

Taking the limit of τ → ∞, one obtains

(2,2) ∂t + v1 · ∇1 + A1 · ∂1 f (1, t) Z Z ∞ 0 4 4 0 0 = ∂1 · d2a12 dτ a˜12 · ∂1 + a˜21 · ∂2 t−τ0 f (1(t − τ ), t) f (2(t − τ ), t). (C.1.13) Ω2,2 0

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) Z Z ∞ 0 0 0 = ∂1 · d2a˜12 dτ [a˜12 · ∂1 + a˜21 · ∂2]t−τ0 f (1(t − τ ), t) f (2(t − τ ), t), 0 (C.1.14) where the statistical acceleration can be found in the forms

1 a˜ = a − A , (C.1.15a) 12 12 N 1 1 a˜ = a − A . (C.1.15b) 21 21 N 2

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 APPENDIX C. COULOMB LOGARITHM AND FP EQUATION

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

Z Z ∞ N 0 0 3 3 I =∂ · a a t − τ 0 dτ d r · ∂ f (1, t) f (r , p , t) d p , (C.2.1) L 1 12 12 r12(t−τ )>4 12 12 1 2 2 Ω2,2 0

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

3 transform of the acceleration of star 1 due to star 2 at distances r12 > 4 is as follows

Z r Gm sin[k4] F [a (r > 4)] = −Gm exp(−ik · r ) 12 d3r = − kˆ. (C.2.3a) 12 12 12 3 12 iπ2k24 Ω2,2 r12 2

0 The same transform must be employed for a12(t−τ ) in equation (C.2.1), but the time of r is ﬁxed to t−τ0. 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

3 Fourier 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 transform. The factor can remove singularities of (generalized) functions on complex planes and slow decays of potentials in three-dimensional spaces (e.g., Adkins, 2013). However, one does not need to employ the factor in the Fourier transform of the truncated acceleration, a12Θ(r12 > 4), and even in the corresponding inverse Fourier transform, −1 −1 F [F [a12Θ(r12 > 4)]]. Rendering the transform, F [F [a12Θ(r12 > 4)]], is a simple task; hence, it is left for readers. One will need the following identity to ﬁnd the step function

Z ∞ sin k π dk = . (C.2.2) 0 k 2

237 APPENDIX C. COULOMB LOGARITHM AND FP EQUATION

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)

Gm lim F [a12] = − kˆ −−−→ F φ12 kˆ. (C.2.4) 4→0 2 1 2iπ k × −ik

After a proper calculation following (Chavanis, 2012, Appendix C), the collisional term results in

Z N ←→ 3 IL = ∂1 · T (p12, p2) · ∂12 f (r1, p1, t) f (r1, p2, t) d p2, (C.2.5a)

2 ←→ ←→ p I − p12p12 T ≡ −B 12 , (p ≡ p − p ) (C.2.5b) p3 12 1 2 Z ∞ sin2[k4] B ≡ πGm2 k. 2 3 2 d (C.2.5c) 0 k 4

←→ 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

X∞ (−16π2)m B = ln[N] + 1.5 + + O(1/N2), (C.2.6) 2m(2m)! m=1 where the following indeﬁnite integral formula (e.g., Zeidler et al., 2004) is employed4

m 2 Z cos[αk] X∞ −[αk] dk = ln[αk] + . (C.2.7) k 2m(2m)! m=1

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 their work, the upper limit is also assigned to the domain of the integration. 238 APPENDIX C. COULOMB LOGARITHM AND FP EQUATION

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

m X∞ −[4π]2 B − ln[N] = 1.5 + ≈ −3.11435, (C.2.9) 2m(2m)! m=1

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 functions (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 approximation of the LE functions to increase the decay rate of the Chebyshev coeﬃcients for the functions. 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¯ ,

p (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 coefficients. In fact, by adapting the method of (Elliott, 1964) to [0, 1], one easily sees that the p Chebyshev coefficients for 1 − ϕ decay algebraically like n−2 for high n. One can regularize both p (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

ηi (iii) Logarithmic derivative at ϕ ≈ 0 R/Rs − 1 ≈ ϕ cos (ωi ln ϕ + 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

−2ηi−1 coeﬃcients {an} decay like n 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

1/L ϕo = ϕ (D.1.1)

where L is a mapping parameter. This transformation is inspired by the fact that power-law mappings 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 APPENDIX D. SINGULARITY IN LE FUNCTIONS

104 m = 6 and L = 1 m = 10 and L = 1 100 m = 20 and L = 1

| 10−4 n a | 10−8

10−12

10−16 0 200 400 600 800 1,000 n

102 m = 6 and L = 1 m = 6 and L = 5 −1 10 m = 6 and L = 30 10−4 | n −7 a 10 |

10−10

10−13

10−16 0 200 400 600 800 1,000 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 other choices of (m, L) the coefficients decay slowly due to the end-point singularity.

242 APPENDIX D. SINGULARITY IN LE FUNCTIONS

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

πy x = sin , (Quadratic Mapping) (D.2.2a) 2 Lmy x = tanh p , (Hyperbolic-Tangent mapping) (D.2.2b) 1 − y2

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

!LL 1 − y x = −1 + (1 + x ) − 1 < x < x (D.2.3a) o 2 o !LR 1 + y x = 1 + (−1 + x ) x < x < 1 (D.2.3b) o 2 o

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 APPENDIX D. SINGULARITY IN LE FUNCTIONS endpoint. Hence, we apply a one-sided change of coordinates

!L 1 − y x = 1 − 2 , (D.2.4) 2 where L is a free parameter.

244 APPENDIX D. SINGULARITY IN LE FUNCTIONS

106 sin mapping tanh mapping (Lm = 3) 101 power mapping L = 8

−4

| 10 n a | 10−9

10−14

10−19 100 101 102 n 106 sin mapping tanh mapping (Lm = 6) 101 power mapping L = 30

−4

| 10 n a | 10−9

10−14

10−19 100 101 102 n Figure D.2: Absolute values of the Chebyshev coefficients for the function (1 − x)k ln (1 − x) L mapped by x = sin πy , x = tanh √Lmy , and x = 1 − 2 1−y . The parameter values are k = 2, L = 8 2 1−y2 2 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.

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 APPENDIX D. SINGULARITY IN LE FUNCTIONS

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

ηi 3.3), ﬁnding the asymptotic values of the Chebyshev coeﬃcients for ϕ cos (ωi ln ϕ) is a tedious task. We show only the result. As n → ∞,

n Ai an ≈ (−1) cos Θn, (D.3.1) n2ηi+1

where

r " !# 1 η 4 2ηi+ i Ai = γ 2 exp −2ηi + 2ωi arctan , (D.3.2a) π ωi " # 4ηi + 1 ηi 1 γ Θn = − arctan − π + 2ωi ln − 1 , (D.3.2b) 2 ωi 4 n q 2 2 γ = ηi + ωi . (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 APPENDIX D. SINGULARITY IN LE FUNCTIONS formula applies.

102 steepest descent interpolatory quadrature − | 10 2 diﬀerence n a | 10−6

10−10 0 200 400 600 800 1,000 n 103 steepest descent interpolatory quadrature 10−4 | diﬀerence n a | 10−11

10−18 0 100 200 300 400 n 103 steepest descent interpolatory quadrature 10−4 | diﬀerence n a | 10−11

10−18 0 50 100 150 200 250 300 350 400 n

ηi Figure D.3: Asymptotic absolute values of the Chebyshev coefficients for ϕ cos (ωi ln ϕ). 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 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−ω¯ φ¯(Ar¯), (E.1.1)

2 where A is a constant andω ¯ = m−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

r¯ φ¯m u = − , (E.1.2a) φ¯0 r¯ φ¯0 v = − . (E.1.2b) φ¯

These variables reduce the LE1 equation to the Milne form

du v u + v − 1 = − . (E.1.3) dv u u + mv − 3

Chandrasekhar (1939) used a geometrical method in the (u, v) plane. For the boundary conditions

0 φ¯(¯r = 0) = 1 and φ¯ (¯r = 0) = 0, the solution of the Milne equation starts at (uE, vE) = (3, 0), and

m−3 reaches asymptotically the singular point (us.p, vs.p) = m−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

u du u − 1 = − , (E.2.3) v dv u + v − 3

249 APPENDIX E. MILNE VARIABLES FOR LE FUNCTIONS

where

r¯ e−φ¯ r¯ ρ¯2 u = = − , (E.2.4a) φ¯0 ρ¯0 r¯ ρ¯0 v = r¯ φ¯0 = − . (E.2.4b) ρ¯

In the (u, v) plane, the E-solutions start at (uE, vE) = (3, 0), and approach asymptotically the singular 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 − c4/c4o |. 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

3 (x = 1) | 1 − c∗/c∗ | | 1 − c /c | ∼ | 1 − β/β | ∼ I ∼ 4 4o . (F.1.1) 1 1o o 102 105

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 c4.

1 10 3I(x = 1) (∆β < 0) | 1 − c1/c1o | (∆β < 0) ∗ ∗ | 1 − c4/c4o | (∆β < 0) 3I(x = 1) (∆β > 0) −2 ∗ ∗ 10 | 1 − c1/c1o | (∆β > 0) | 1 − c4/c4o | (∆β > 0)

10−5

10−8

10−11

10−14 10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 | ∆β/βo |

∗ ∗ Figure F.1: Values of 3I(x = 1), | 1 − c1/c1o |, and | 1 − c4/c4o | against change ∆β in eigenvalue β around the reference value βo (∆β ≡ β − βo). Numerical parameters are N = 70, L = 1, and FBC = 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 c4 appeared proportional to FBC while

β and c3 are stable to fifteen significant figures. However, the condition number of the Jacobian

12 4 matrix for the 4ODEs and Q-integral reached ∼ 10 for high FBC(∼ 10 ). 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 and 252 APPENDIX F. NUMERICAL STABILITY OF THE SS-OAFP SYSTEM

c2 are proportional to FBC with high accuracies. Figure F.2 implies the following relationships

between FBC and the eigenvalues

−16 c1(FBC) ∝ c1o + O 10 FBC, (F.1.2a)

∗ ∗ −10 c4(FBC) ∝ c4o + O 10 FBC, (F.1.2b)

−16 c3 = c3o + O(10 ), (F.1.2c)

−16 β(FBC) = βo + O(10 ), (F.1.2d)

−4 4 where −10 < FBC < 10 .

11 10 −4 10 | 1 − c1/FBC/c1o | 10 | − ∗ ∗ | 10 1 c4/FBC/c4o 10−7 109 10−10 108 −13 107 10 condition number 106 10−16 10−4 10−2 100 102 104 10−810−610−410−2100 102 104 FBC FBC

Figure 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 c4/FBC from the reference eigenvalues c1o and c4o for the 4ODEs regularized through F(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 APPENDIX F. NUMERICAL STABILITY OF THE SS-OAFP SYSTEM

L = 0.75 ∗ ∗ L = 0.5 N Eigenvalue β | 1 − c1/c1o | | 1 − c4/c4o | 3I(x = 1) ∗ ∗ −10 −7 −9 β | − c /c | | − c /c | 3 x 60 8.178371160 2.1 × 10 6.6 × 10 9.4 × 10 N Eigenvalue 1 1 1o 1 4 4o I( = 1) 55 8.178371160 2.1 × 10−10 1.3 × 10−6 1.7 × 10−9 35 8.17837104 4.9 × 10−8 2.3 × 10−6 2.1 × 10−10 50 8.178371160 8.0 × 10−10 9.8 × 10−6 1.3 × 10−9

Table F.1: Numerical results for the contracted-domain formulation with with L = 1/2 and L = 3/4 for 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’ L 1 1+2 is like an ∝ n . 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.

101 101 L = 3/4 L = 1/2 10−1 10−1 10−3 10−3 10−5 10−5 10−7 N = 60 10−7 N = 35 N = 65 N = 75 10−9 guide line n−2L−1 10−9 guide line n−2L−1 100 100.5 101 101.5 100 100.5 101 101.5 FBC FBC

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 method for the latter did not work satisfactorily stalling around | {a}new − {a}old |≈ 7 × 10−10 (resulting ∗ ∗ −3 −6 in | 1 − c4/c4o |≈ 6.0 × 10 | and 3I = 4.4 × 10 ). 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 APPENDIX F. NUMERICAL STABILITY OF THE SS-OAFP SYSTEM

∗ ∗ ∗ ∗ c d | 1 − c1/c1ex | | 1 − c4/c4ex | 3I(x = 1) c d | 1 − c1/c1ex | | 1 − c4/c4ex | 3I(x = 1) ∞ N/A 1.2 × 10−13 5.3 × 10−10 1.0 × 10−11 10 10 9.4 × 10−14 5.3 × 10−10 1.0 × 10−11 10 1 1.2 × 10−13 5.3 × 10−10 1.0 × 10−11 2 10 2.1 × 10−14 3.1 × 10−11 7.5 × 10−12 5 1 7.7 × 10−14 5.3 × 10−10 1.0 × 10−11 1 5 4.0 × 10−14 2.6 × 10−10 8.8 × 10−12 1 1 1.0 × 10−13 4.5 × 10−10 9.8 × 10−12 1 0.1 1.1 × 10−13 8.7 × 10−10 1.1 × 10−11 0.1 1 1.6 × 10−13 9.6 × 10−10 1.2 × 10−11 1 0.01 1.2 × 10−13 9.6 × 10−10 1.1 × 10−11 0.01 1 1.4 × 10−13 9.1 × 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 eigenvalues ∗ −1 ∗ −1 are compared with c1ex ≡ c1o and c4ex ≡ 3.03155223 × 10 (= c4o + 1 × 10 ) obtained for (c, d) = −12 (1, 10) and the value of 3I(x = 1) is 7.3 × 10 . The combination (c, d) = (∞, N/A) means the extrapolated 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 eigenvalues of the optimal truncated-domain solution (Table F.2). The set of parameters (c, d) = (1, 10)

−12 provided the best accuracy in the sense that 3I(x) reached the minimum value (∼ 7.3×10 ) among

−13 the chosen parameters. The relative deviation of all the eigenvalues is the order of 10 in c1 and

−9 ∗ 10 in c4 at most compared with those for (c, d) = (1, 10). Also, all the sets of parameters hold −11 small values of 3I(x = 1) ≈ 1 × 10 . 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 APPENDIX F. NUMERICAL STABILITY OF THE SS-OAFP SYSTEM

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 coefficients 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

! 1 + x − x 3(tes) = 0.1 Θ trans + 1, (F.3.1) R 2

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 APPENDIX F. NUMERICAL STABILITY OF THE SS-OAFP SYSTEM

100 1.15 10−2 1.1 | | −4 ) 1.05 n

10 E ( Q R | 3

−6 | 1 10 xtrans = 0.1 1/n2 x = 0.1 −8 0.95 trans 10 xtrans = 0.001 xtrans = 0.001 0.9 100 101 102 0.92 0.94 0.96 0.98 1 index n −E

(tes) Figure F.4: Chebyshev Coeﬃcients for the Q-integral with a discontinuous test function 3R . Recall E = −(0.5 + 0.5x)L, here L = 1.

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

(m) The present appendix shows how the modiﬁcation of the function from 3R to 3R changes (m) the numerical result. We examined how the dependent variable 3D aﬀected 3R in the Poisson’s equation. We employed the following test function for 3D

" # ! 1 + x − 0.001 3(tes) = 3 (x) A Θ + 1 , (F.3.2) D Do trans 2

where Atrans is a small positive number, and 3Do(x) is the regularized density obtained from the

(tes) reference solution. We solved the Poisson’s equation with the fixed 3D for different Atrans. When q √ (m) the value of Atrans is very small such as 0.00001, the solutions 3R 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

(m) at the order of 0.1 but also the coeﬃcients for 3R shows a slower decay compared with those for

257 APPENDIX F. NUMERICAL STABILITY OF THE SS-OAFP SYSTEM

3R (Left Panel, Figure F.5).

100 100

10−1 10−1

10−2 10−2

−3 −3

| 10 | 10 n n R R | 10−4 | 10−4

10−5 10−5 −4 Atrans = 0.1, 3R Atrans = 10 , 3R −6 (m) −6 −4 (m) 10 Atrans = 0.1, 3R 10 Atrans = 10 , 3R 0.4n−1 5n−2 10−7 10−7 100 101 102 100 101 102 n n q √ (m) Figure F.5: Chebyshev Coeﬃcients for 3R and exp(3R) 0.5 + 0.5x for a discontinuous test (tes) function 3D . 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 modifying 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 reference- solutions even for a high (∼ 200) degree of polynomials.

G.1 Modifying the regularization of variable 3F

We employed the following modiﬁcation of 3F

 " #β (m)  1 + x  3 = ln exp[3F]  . (G.1.1) F  2 

259 APPENDIX G. REFERENCE- AND HS’S SOLUTIONS OF THE SS-OAFP SYSTEM

(m) We solved the ss-OAFP system for 3F and unmodiﬁed variables 3S , 3Q, 3G, 3I, 3J, 3R using the pro- (m) cedure of Chapter 5. Like the modiﬁcation 3R (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

(m) (See Table G.1 in which 3F , 3R is the corresponding formula). Due to the logarithmic endpoint (m) (m) (m) singularity of 3F , the Chebyshev coeﬃcients {Fn } for 3F show slow decays for both high and low Emax (Figure G.1). A more distinct slow decay appears in the Chebyshev coeﬃcients {In} for

−4 3I, especially when Emax is high (Figure G.2). The value of 3I(x = 1) is still the order of 10 for

(m) high Emax that is the same order as that for the modiﬁcation 3R 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

(m) ≈ −0.225. However, the 3F -formulation provides the HS’s solution only for small N.

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

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 reference- solutions 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

(m) (m) 3R , 3F . 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

−4 (3I(x = −1) = O 10 ) while the reference solution can be found for high Emax(≈ −0.05) with

−6 a high accuracy (at least 3J(x = −1) = O 10 ). Also, the Chebyshev coeﬃcients {In} showed a

distinctive diﬀerence between the two solutions. The absolute values of {In} stall approximately at

−4 −5 −6 −7 10 ∼ 10 for Emax ≈ −0.25 and at 10 ∼ 10 for Emax ≈ −0.05 (Figure G.2).

260 APPENDIX G. REFERENCE- AND HS’S SOLUTIONS OF THE SS-OAFP SYSTEM

function N Emax β c1 c4 3I(x = 1)

(m) −4 −1 −4 3F , 3R 15 −0.24 8.181 9.1014 × 10 3.516 × 10 3.3 × 10 (m) −4 −1 −4 3F , 3R 15 −0.24 8.1731 9.1023 × 10 3.524 × 10 2.1 × 10 (m) (m) −4 −1 −4 3F , 3R 200 −0.225 8.175860 9.1018 × 10 3.523 × 10 2.9 × 10

(m) −4 −1 −6 3F , 3R 70 −0.00525 8.1783712 9.0925 × 10 3.304 × 10 2.2 × 10 (m) −4 −1 −7 3F , 3R 200 −0.04 8.178371129 9.0925 × 10 3.301 × 10 4.9 × 10 (m) (m) −4 −1 −7 3F , 3R 80 −0.04 8.1783683 9.0926 × 10 3.301 × 10 8.9 × 10

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

(m) (m) | Fn | | Fn |

100 100 Emax = −0.00525 −2 −2 10 Emax = −0.225 10 10−4 10−4 E = −0.04 −6 −6 max 10 Emax = −0.24 10 10−8 10−8 3(m), 3(m) 3(m), 3(m) F R F R −10 (m) −10 (m) 10 3F , 3R 10 3F , 3R 100 101 102 100 101 102 n n

(m) (m) Figure G.1: Absolute values of Chebyshev coefficients {Fn } for 3F . In the modified ss-OAFP system, 3S , 3Q, 3G, 3I, 3J are not modified. The maximum values Emax of the truncated domain are also depicted in the figure.

261 APPENDIX G. REFERENCE- AND HS’S SOLUTIONS OF THE SS-OAFP SYSTEM

| In | | In |

E = −0.04 10−1 10−1 max

10−3 10−3

10−5 10−5 E = −0.24 −7 max −7 10 Emax = −0.225 10 Emax = −0.00525 10−9 10−9 3(m), 3(m) 3(m), 3(m) F R F R −11 (m) −11 (m) 10 3F , 3R 10 3F , 3R 100 101 102 100 101 102 n n

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 also depicted 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

dΦ R2 = M. (H.0.1) dR

Since the ss-OAFP model has the power-law boundary condition Φ ∝ R−α at R → ∞, one can dΦ − employ the identity dR = αΦ/R. Accordingly, the Poisson’s equation at large R reduces to

M Φ = − . (H.0.2) Rα

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,

M2 E ∝ . (H.0.3) tot R

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 characteristics 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

2 values of fitting parameters so that χν 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

0.0 0.0

] (r = 10.43, δ = 0.13) (r = 00.10, δ = 0.017) Σ c c −2.0 −2.0 log[ ss-OAFP ss-OAFP (¯c = 1.30) (¯c = 2.04) −4.0 NGC 288(n) −4.0 NGC 1851(n) −1.5 −1 −0.5 0 0.5 1 1.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

0.5 0.5 ] Σ 0 0 log[

∆ NGC 288(n) NGC 1851(n) −0.5 −0.5 −1.5 −1 −0.5 0 0.5 1 1.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 log[R(arcmin)] log[R(arcmin)]

0.0 0.0

] (r = 1 .26, δ = 0.087) c 0 Σ −2.0 (rc = 1 .13, δ = 0.029) −2.0 log[ ss-OAFP ss-OAFP −4.0 (¯c = 1.41) (¯c = 1.81) NGC 5466(n) NGC 6121(n) −4.0 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5

] 0.5 0.5 Σ 0 0 log[

∆ −0.5 NGC 5466(n) −0.5 NGC 6121(n) −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 log[R(arcmin)] log[R(arcmin)]

Figure 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

0.0 0.0

] (r = 0068, δ = 0.044) (r = 00.19, δ = 0.014) Σ c c −2.0 −2.0 log[ ss-OAFP ss-OAFP (¯c = 1.64) (¯c = 2.12) −4.0 NGC 6254(n) −4.0 NGC 6626(n) −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1

0.5 0.5 ] Σ 0 0 log[

∆ NGC 6254(n) NGC 6626(n) −0.5 −0.5 −1.5 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 log[R(arcmin)] log[R(arcmin)]

0.0 0.0

] 0 0 Σ −2.0 (rc = 0 .55, δ = 1.3) (rc = 0 .46, δ = 0.27) −2.0 log[ ss-OAFP ss-OAFP −4.0 (¯c = 1.03) (¯c = 1.16) Pal 3 (n) Pal 4 (n) −4.0 −1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 −1.5 −1 −0.5 0 0.5

] 0.5 0.5 Σ 0 0 log[

∆ −0.5 Pal 3 (n) −0.5 Pal 4 (n) −1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 −1.5 −1 −0.5 0 0.5 log[R(arcmin)] log[R(arcmin)]

Figure I.2: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

0.0 0.0

] 0 Σ −2.0 (rc = 0 .85, δ = 1.2) 0 −2.0 (rc = 0 .15, δ = 0.039) log[ ss-OAFP ss-OAFP −4.0 (¯c = 1.04) (¯c = 1.69) Pal 14 (n) Trz 5 (n) −4.0 −1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 −1.5 −1 −0.5 0 0.5

] 0.5 0.5 Σ 0 0 log[

∆ −0.5 Pal 14 (n) −0.5 Trz 5 (n) −1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 −1.5 −1 −0.5 0 0.5 log[R(arcmin)] log[R(arcmin)]

0.0

] 0 Σ −2.0 (rc = 1 .63, δ = 0.1) log[ ss-OAFP −4.0 (¯c = 1.37) 6809 (n) −1.5 −1 −0.5 0 0.5 1

] 0.5 Σ 0 log[

∆ −0.5 NGC 6809 (n) −1.5 −1 −0.5 0 0.5 1 log[R(arcmin)]

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 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

NGC 288 NGC 1851 NGC 5466

c rc rtid c rc rtid c rc rtid 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

c rc rtid c rc rtid c rc rtid 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 proﬁles reported in (Miocchi et al., 2013).

270 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

100 100

10−1 10−1 0 0 (rc = 0 .41) (rc = 0 .72) Σ 10−2 10−2 δ = 0.18 δ = 0.098 10−3 (c = 1.24) 10−3 (c = 1.39) NGC 2419 NGC 4590 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .51) (rc = 0 .23) Σ 10−2 10−2 δ = 0.068 δ = 0.052 10−3 (c = 1.50) 10−3 (c = 1.60) NGC 5272 NGC5634 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .074) (rc = 0 .064) Σ 10−2 10−2 δ = 0.022 δ = 0.023 10−3 (c = 1.95) 10−3 (c = 1.93) NGC 5694 NGC 5824 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101 R(arcmin) R(arcmin)

Figure I.4: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

100 100

10−1 10−1 0 0 (rc = 0 .20) (rc = 0 .78) Σ 10−2 10−2 δ = 0.036 δ = 0.061 10−3 (c = 1.74) 10−3 (c = 1.54) NGC 6093 NGC 6205 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .18) (rc = 0 .49) Σ 10−2 10−2 δ = 0.080 δ = 0.045 10−3 (c = 1.45) 10−3 (c = 1.65) NGC 6229 NGC 6273 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .29) (rc = 0 .37) Σ 10−2 10−2 δ = 0.044 δ = 0.050 10−3 (c = 1.66) 10−3 (c = 1.61) NGC 6304 NGC 6333 (n?) (n?) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101 R(arcmin) R(arcmin)

Figure I.5: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density proﬁles 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. 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

100 100

10−1 10−1 0 0 (rc = 0 .31) (rc = 0 .27) Σ 10−2 10−2 δ = 0.042 δ = 0.042 10−3 (c = 1.68) 10−3 (c = 1.68) NGC 6341 NGC 6356 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .27) (rc = 0 .16) Σ 10−2 10−2 δ = 0.040 δ = 0.038 10−3 (c = 1.70) 10−3 (c = 1.72) NGC 6401 NGC 6440 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .16) (rc = 0 .55) Σ 10−2 10−2 δ = 0.020 δ = 0.108 10−3 (c = 1.99) 10−3 (c = 1.36) NGC 6517 NGC 6553 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101 R(arcmin) R(arcmin)

Figure I.6: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

100 100

10−1 10−1 0 0 (rc = 0 .37) (rc = 0 .25) Σ 10−2 10−2 δ = 0.15 δ = 0.105 10−3 (c = 1.28) 10−3 (c = 1.37) NGC 6569 NGC 6638 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .11) (rc = 0 .12) Σ 10−2 10−2 δ = 0.028 δ = 0.029 10−3 (c = 1.84) 10−3 (c = 1.83) NGC 6715 NGC 6864 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .24) (rc = 0 .18) Σ 10−2 10−2 δ = 0.065 δ = 0.095 10−3 (c = 1.52) 10−3 (c = 1.40) NGC 6934 NGC 7006 (n) (n) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101 R (arcmin) R (arcmin)

Figure I.7: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density proﬁles of NGC 6569, NGC 6638, NGC 6715, NGC 6864, NGC 6934, and NGC 7006 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).

275 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

100 100

10−1 10−1 0 0 (rc = 2 .7) (rc = 1 .64) Σ 10−2 10−2 δ = 5.5 δ = 0.78 10−3 (c = 1.00) 10−3 (c = 1.06) NGC 5053 NGC 5897 (n) (n) 10−4 10−4 10−1 100 101 10−1 100 101 R (arcmin) R (arcmin)

Figure I.8: Fitting of the energy-truncated ss-OAFP model (m = 3.9) to the projected density 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.

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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

0.0 0.0 0 0 (rc = 0 .13, δ = 0.023) (rc = 0 .43, δ = 0.023) −

SB 5.0 −5.0 − o ss-OAFP ss-OAFP

SB (¯c = 2.98, m = 4.3) (¯c = 2.50, m = 4.2) −10.0 NGC 104(n) NGC 1904(n) −10.0 Cheb. Cheb. −3 −2 −1 0 1 −3 −2 −1 0 1

0.0 0.0 0 (rc = 0 .24, δ = 0.06) 0 (rc = 0 .77, δ = 0.085) − −

SB 5.0 5.0 − o ss-OAFP ss-OAFP

SB (¯c = 2.41, m = 4.4) (¯c = 1.74, m = 4.2) −10.0 −10.0 NGC 2808(n) NGC 6205(n) Cheb. Cheb. −3 −2 −1 0 1 −3 −2 −1 0 1

0.0 0.0 −2.0 (r = 00.127, δ = 0.034) c − SB 2.0 0

− −4.0 (rc = 0 .69, δ = 0.15) o ss-OAFP ss-OAFP SB −6.0 (¯c = 3.50, m = 4.4) −4.0 (¯c = 1.54, m = 4.2) NGC 6441(n) NGC 6712(n) −8.0 Cheb. Cheb. −6.0 −3 −2 −1 0 1 −3 −2 −1 0 1 log[R(arcmin)] log[R(arcmin)]

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. The brightness 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

0.0 0.0

− 0

SB 5.0 (rc = 0 .230, δ = 0.044) −5.0 0 − (rc = 0 .121, δ = 0.025) o ss-OAFP ss-OAFP

SB (¯c = 2.08, m = 4.20) (¯c = 2.44, m = 4.2) −10.0 NGC 5286(n) NGC 6093(n) −10.0 Cheb. Cheb. −3 −2 −1 0 1 −3 −2 −1 0 1

0.0 0.0

− 0

SB 5.0 −5.0 0 (rc = 0 .168, δ = 0.020)

− (rc = 0 .272, δ = 0.048) o ss-OAFP ss-OAFP

SB (¯c = 2.03, m = 4.2) (¯c = 2.59, m = 4.2) −10.0 NGC 6535(n) NGC 6541(n) −10.0 Cheb. Cheb. −3 −2 −1 0 1 −3 −2 −1 0 1 log[R(arcmin)] log[R(arcmin)]

Figure I.10: Failure of the ﬁtting of the energy-truncated ss-OAFP model to the surface brightness 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 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 APPENDIX I. FITTING TO KING-MODEL CLUSTERS

0.0 0.0

−2.0 −2.0

−4.0 −4.0 SB −6.0 SB −6.0 − 0 − 0 o (rc = 0 .0528, δ = 0.055) o (rc = 0 .0388, δ = 0.10) SB −8.0 SB −8.0 ss-OAFP ss-OAFP (¯c = 2.59, m = 4.2) (¯c = 2.30, m = 4.2) −10.0 approx. −10.0 approx. (¯c = 2.72, m = 4.8) (¯c = 2.84, m = 4.95) −12.0 NGC 1851(n) −12.0 NGC 5694(n) Cheb. Cheb. −14.0 −14.0 −3 −2 −1 0 1 −3 −2 −1 0 1 log[R(arcmin)] log[R(arcmin)]

0.0

−2.0

−4.0

SB −6.0

− 0

o (rc = 0 .0388, δ = 0.090)

SB −8.0 ss-OAFP (¯c = 2.44, m = 4.2) −10.0 approx. (¯c = 2.83, m = 4.95) −12.0 NGC 5824(n) Cheb. −14.0 −3 −2 −1 0 1 log[R(arcmin)]

Figure 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 magnitude 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 approximation 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 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

2 minimized χν 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.

0.0 0.0

2 ] χ2 . (χν = 1.17) Σ ( ν = 0 37) − −2.0 2.0 log[ polytrope polytrope (m = 4.97) −4.0 (m = 4.6) −4.0 NGC 5466(n) NGC 6809(n) −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1 1.5 0.5

] 0.5 Σ 0 0 log[

∆ NGC 5466(n) −0.5 NGC 6809 (n) −0.5 −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1 1.5 log[R(arcmin)] log[R(arcmin)]

0.0 0.0

] χ2 . Σ 2 ( ν = 0 32) (χν = 0.004) −2.0 −2.0 log[ polytrope polytrope (m = 4.2) (m = 4.80) Pal 3(n) −4.0 Pal 4 (n) −4.0 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1

0.5

] 0.5 Σ 0 0 log[

∆ −0.5 Pal 3(n) Pal 4(n) −0.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 log[R(arcmin)] log[R(arcmin)]

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 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 APPENDIX J. FITTING TO LOW-CONCENTRATION CLUSTERS

0.0

] χ2 . Σ ( ν = 0 15) −2.0 0.5 ] log[ Σ polytrope 0

(m = 4.6) log[ −4.0 Pal14(n) ∆ Pal 14 (n) −0.5 −1.5 −1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 log[R] log[R(arcmin)]

Figure 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).

100

10−1

Σ 10−2

10−3 polytrope(m = 4.99) NGC 4590(n) 10−4 10−1 100 101 R (arcmin) . Figure J.3: Partial success of fitting of the polytropic sphere of index m to the projected density 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 normal 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.

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 approximation 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 APPENDIX J. FITTING TO LOW-CONCENTRATION CLUSTERS

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 asc ¯ = 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 APPENDIX J. FITTING TO LOW-CONCENTRATION CLUSTERS

0 0

−5 SB 0 − (rc = 0 .27) (r = 20.04) o −5 c c¯ = 3.34 c¯ = 1.11 SB −10 (m = 4.99) (m = 4.1) NGC 1261 NGC 5053 (n) (n) −15 −10 −3 −2 −1 0 1 −3 −2 −1 0 1

0 0 SB

− (r = 10.70) (r = 00.401) o −5 c −5 c c¯ = 1.11 c¯ = 3.34 SB (m = 4.1) (m = 4.99) NGC 5897 NGC 5986 (n) (n) −10 −10 −3 −2 −1 0 1 −3 −2 −1 0 1

0 0

0 −5 (rc = 0 .62) SB

− (r = 10.25) o −5 c c¯ = 0.685 c¯ = 3.34 SB (m = 3.3) −10 (m = 4.99) NGC 6101 NGC 6205 (n) (n) −10 −15 −3 −2 −1 0 1 −3 −2 −1 0 1 log[R(arcmin)] log[R(arcmin)]

Figure 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 APPENDIX J. FITTING TO LOW-CONCENTRATION CLUSTERS

0 0

0 −5 (rc = 0 .73) SB

− (r = 00.73) o −5 c c¯ = 2.24 c¯ = 2.24 SB (m = 4.9) −10 (m = 3.3) NGC 6402 NGC 6496 (n) (n) −10 −15 −3 −2 −1 0 1 −3 −2 −1 0 1

0 0 SB

− (r = 00.58) (r = 00.634) o −5 c −5 c c¯ = 3.34 c¯ = 3.34 SB (m = 4.99) (m = 4.99) NGC 6712 NGC 6723 (n) (n) −10 −10 −3 −2 −1 0 1 −3 −2 −1 0 1 log[R(arcmin)] log[R(arcmin)]

0 SB

− (r = 00.36) o −5 c c¯ = 3.34 SB (m = 4.99) NGC 6981 (n) −10 −3 −2 −1 0 1 log[R(arcmin)]

Figure 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 APPENDIX J. FITTING TO LOW-CONCENTRATION CLUSTERS

0 0

0 −5 −5 (rc = 0 .57) SB 0

− (rc = 0 .997) o c . c¯ = 3.34 SB ¯ = 3 33 −10 (m = 4.99) −10 (m = 4.99) NGC 3201 NGC 6144 (n) (n) −15 −15 −3 −2 −1 0 1 −3 −2 −1 0 1

0 0

− (r = 00.26)

SB 5 c

− (r = 00.622) o −5 c c¯ = 3.34 c¯ = 3.34 SB (m = 4.99) −10 (m = 4.99) NGC 6273 NGC 6352 (n) (n) −10 −3 −2 −1 0 1 −3 −2 −1 0 1

0 0

0 0 −5 (rc = 0 .233) −5 (rc = 0. 71) SB − o c¯ = 3.34 c¯ = 3.34 SB −10 (m = 4.99) −10 (m = 4.99) NGC 6388 NGC 6656 (n) (n) −15 −15 −3 −2 −1 0 1 −2 −1 0 1 log[R(arcmin)] log[R(arcmin)]

Figure J.6: Failure of the ﬁtting of the polytropic sphere of index m to the surface brightness proﬁles of NGC 3201, NGC 6144, NGC 6273, NGC 6352, NGC 6388, and NGC 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 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

2 the χν-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

100 100

10−1 10−1 0 0 (rc = 0 .19) (rc = 0 .14) Σ 10−2 10−2 δ = 0.027 δ = 0.025 10−3 (c = 1.86) 10−3 (c = 1.89) NGC 1904 NGC 4147 (c?) (n?c?) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101

100 100

10−1 10−1 0 0 (rc = 0 .50) (rc = 0 .14) Σ 10−2 10−2 δ = 0.050 δ = 0.028 10−3 (c = 1.61) 10−3 (c = 1.84) NGC 6544 NGC 6652 (c?) (n?c?) 10−4 10−4 10−2 10−1 100 101 10−2 10−1 100 101 R(arcmin) R (arcmin)

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 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).

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 APPENDIX K. FITTING TO PCC CLUSTERS

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 APPENDIX K. FITTING TO PCC CLUSTERS

0.0 0.0

−1.0 −1.0 SB

− 0 0 o (rc = 0 .093, δ = 0.082) (rc = 0 .068, δ = 0.0073)

SB −2.0 ss-OAFP −2.0 ss-OAFP (¯c = 2.34) (¯c = 2.38) NGC 5946 (c) NGC 6342(c) −3.0 −3.0 −2.5 −2 −1.5 −1 −0.5 0 0.5 −2.5 −2 −1.5 −1 −0.5 0 0.5

0.5 0.5 SB) −

o 0 0

(SB −0.5 NGC 5946(c) −0.5 NGC 6342(c) ∆ −2.5 −2 −1.5 −1 −0.5 0 0.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 log[R(arcmin)] log[R(arcmin)]

0.0 0.0

−1.0 −1.0 SB

− 0 0 o (rc = 0 .055, δ = 0.0060) (rc = 0 .096, δ = 0.011)

SB −2.0 ss-OAFP −2.0 ss-OAFP (¯c = 2.45) (¯c = 2.22) NGC 6624(c) NGC 6453(c) −3.0 −3.0 −2.5 −2 −1.5 −1 −0.5 0 0.5 −2.5 −2 −1.5 −1 −0.5 0 0.5

0.5 0.5 SB) −

o 0 0

(SB −0.5 NGC 6624(c) −0.5 NGC 6453(c) ∆ −2.5 −2 −1.5 −1 −0.5 0 0.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 log[R(arcmin)] log[R(arcmin)]

Figure 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 SBo − SB from the model.

290 APPENDIX K. FITTING TO PCC CLUSTERS

0.0 0.0

−1.0 −1.0 SB

− 0 0 o (rc = 0 .096, δ = 0.0094) (rc = 0 .085, δ = 0.0052)

SB −2.0 ss-OAFP −2.0 ss-OAFP (¯c = 2.29) (¯c = 2.51) NGC 6522(c) NGC 6558(c) −3.0 −3.0 −2.5 −2 −1.5 −1 −0.5 0 0.5 −2.5 −2 −1.5 −1 −0.5 0 0.5

0.5 0.5 SB) −

o 0 0

(SB −0.5 NGC 6522(c) −0.5 NGC 6558(c) ∆ −2.5 −2 −1.5 −1 −0.5 0 0.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 log[R(arcmin)] log[R(arcmin)]

0.0 0.0

−1.0 −1.0 SB

− 0 0 o (rc = 0 .086, δ = 0.0076) (rc = 0 .149, δ = 0.0087)

SB −2.0 ss-OAFP −2.0 ss-OAFP (¯c = 2.37) (¯c = 2.15) NGC 7099(c) Trz 1(c) −3.0 −3.0 −2.5 −2 −1.5 −1 −0.5 0 0.5 −2.5 −2 −1.5 −1 −0.5 0 0.5

0.5 0.5 SB) −

o 0 0

(SB −0.5 NGC7099(c) −0.5 Trz 1 (c) ∆ −2.5 −2 −1.5 −1 −0.5 0 0.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 log[R(arcmin)] log[R(arcmin)]

Figure 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 SBo −SB from the model.

291 APPENDIX K. FITTING TO PCC CLUSTERS

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 APPENDIX K. FITTING TO PCC CLUSTERS

0.0 0.0

0 −2.0 (rc = 0 .21, δ = 0028) SB −5.0 0 − −4.0 (rc = 0 .0050, δ = 0.0065) o ss-OAFP ss-OAFP

SB (¯c = 2.36, m = 4.2) (¯c = 3.22, m = 4.2) −6.0 NGC 6266(c?) NGC 6293(c) −10.0 Cheb. Cheb. −8.0 −3 −2 −1 0 1 −3 −2 −1 0

0.0 0.0

SB −5.0 0 0

− (rc = 0 .023, δ = 0.007) −5.0 (rc = 0 .0031, δ = 0.0053) o ss-OAFP ss-OAFP

SB (¯c = 3.19, m = 4.2) (¯c = 3.28, m = 4.2) NGC 6652(c) NGC 6681(c) −10.0 Cheb. −10.0 Cheb. −3 −2 −1 0 1 −3 −2 −1 0 1

0.0 0.0

−5.0 −5.0

SB 0 0 (rc = 0 .049, δ = 0.0055) − (rc = 0 .0038, δ = 0.0045) o ss-OAFP ss-OAFP SB −10.0 (¯c = 3.33, m = 4.2) −10.0 (¯c = 3.27., m = 4.2) NGC 7078(c) NGC 7099(c) Cheb. Cheb. −15.0 −15.0 −3 −2 −1 0 1 −3 −2 −1 0 1 log[R(arcmin)] log[R(arcmin)]

Figure 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 APPENDIX K. FITTING TO PCC CLUSTERS

0.0

− 0

SB 5.0 (rc = 0 .168, δ = 0.02) − o ss-OAFP

SB (¯c = 2.59, m = 4.2) −10.0 NGC 6541(c?) Cheb. −3 −2 −1 0 1 log[R(arcmin)] 0.0

SB −5.0 0 − (rc = 0 .0461, δ = 0.0066) o ss-OAFP

SB (¯c = 3.21, m = 4.2) NGC 6624(c) −10.0 Cheb. −3 −2 −1 0 1 log[R(arcmin)]

Figure K.5: Failure of the fitting of the energy-truncated ss-OAFP model to the surface brightness 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 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 ona 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;

23 [Kd]=Integral D standard prep(N,a,xmind,Pfac,Lc,Do);%%%%pre-integral ... forD

24 [Kq]=Integral Q standard prep Gauss(N,a,xminq,Pfac,Lc);%%%% ... pre-integral forQ

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';

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

296 APPENDIX L. MATLAB CODE

47 %%%%%%%%%%%%% Load files %%%%%%%%%%%%

48 cFKGLQ=load(LoadCheb name);ac1c2c3c4=load(LoadEigen name);

49 c2=ac1c2c3c4(2);c3=ac1c2c3c4(3);c4=ac1c2c3c4(4);

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));

56 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

57 %%%%%%% Initial guess of4 functions ... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

58 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

61 ifN >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

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 APPENDIX L. MATLAB CODE

77 %TrialL=TrialL./(0.5+0.5*Ec).ˆ((1)*Lc);

79 cF=TLEX*TrialF;

80 cL=TLEX*TrialL;cG=TLEX*TrialG;cK=TLEX*TrialK;

81 cQ=TLEX*TrialQ;

83 change=1;it=0;

84 c cheb=[cF;cK;cG;cL;cQ];c eign=[c2;c3;c4];c=[c cheb;c eign];

86 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

87 %%%%%%%%% Newron Kantrovich Iteration Process %%%%%%%%%%%

88 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

90 while change>1e-25&&change<1e7&&it

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);

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 APPENDIX L. MATLAB CODE

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 %%%% RedfineQ %%%%%%%%%%%%%%%%%% LogQ %%%%%%%%%

113 %Iq=log(Iq);

114 %Iqm1=Iq(end);Iqp1=Iq(1);

115 %Iq=Iq(1:end-1);

116 %Iq=Iq(2:end);

117 %%%% RedfineQ %%%%%%%%% LinearQ %%%%%

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 operatorO ... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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 operatorO %%%

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 APPENDIX L. MATLAB CODE

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 APPENDIX L. MATLAB CODE

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 APPENDIX L. MATLAB CODE

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

4 function [Tc,Tpc,Tppc,Ec,Bcl,Bcr]=...

5 cheb exact deriv Gauss(coeff,ingrid)

6 n=0:1:coeff-1; 302 APPENDIX L. MATLAB CODE

7 i=1:ingrid;i=i';

8 Bcl=zeros(3,coeff);

9 Bcr=zeros(3,coeff);

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 atE=-1)/Lobatto/Cheb

18 Ec=(cos(t));%Radau(BC atE=1)

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 atE=-1) standard

33 %Bcl(2,:)=(n.*n.*cos(n*tend)/(cos(tend)));

34 %%%Radau(BC atE=-1)

35 Bcr(2,:)=n.*n;%standard

36 %Bcr(2,:)=(n.*n.*cos(n*t1)/(cos(t1)));

37 %%%Radau(BC atE=-1)

303 APPENDIX L. MATLAB CODE

38 end

2 %%%%% Inverse form of the Gauss-Chebyshev expansion

4 function[TLEXc,Ecc,ttc]=Initial guess Cheb Gauss(coeff)

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,:);

13 end

1 %%%%%%%%% preset matrix for integralD ###############

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)

7 coeff=N*Pfac;Ing=N*Pfac;Grid=20;%total Nb <650

8 %max=b*Ing+1;%Radau;

9 max=Grid*Ing;%others

11 l=1:Ing;l=l';

12 %t(l)=pi*(l-1)/(coeff-1);%Gauss-Lobatto

304 APPENDIX L. MATLAB CODE

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)

18 x=cos(t);

19 z=1+(x-1)*(1-xmind)/2;

21 i=1:max;i=i';

24 %%%%%%%%%%%%% Integral weight %%%%%%%%%%%%%%

25 %%%%% Clenshaw Curtis, Fejer or Radau schemes %%%%%%%%

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)

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

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 APPENDIX L. MATLAB CODE

44 %w=2*(1-2*wcc)/(max-1/2);

45 %Radau(-1 and +1)

48 %w(end)=w(end)/2;%Radau(-1)

49 %w(1)=w(1)/2;%Radau(+1)

52 %%%%%%%%%%%%%%%%%%%%practice%%%%%%%%%%%

53 X=cos(Tb);%interior points

54 Lm=1;%numeric parameter in tanh transformaiton;

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 forL=k.%%%%%%%%%%%%%%%%%%%%%%%%%%

64 A=(z+1)*(y(j)+1)/4;B=(1+z)/2;

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 APPENDIX L. MATLAB CODE

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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

94 %%%%%%%%%%% loose xmin%%%%%%%%%%%%%%%%%%%%%

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 APPENDIX L. MATLAB CODE

1 %%%%%%%%% preset matrix for integralQ #########

2 %%%%%%% the nodes are chosen from Gauss

3 %%%%% Chebyshev, Lobtatto and Radau %%%%%%%%

5 function [Kq]=...

6 %Integral Q standard prep(N,a,xminq,Pfac,Lc)

8 coeff=N*Pfac;Ing=N*Pfac;b=20;%total Nb <650

9 %max=b*Ing+1;%Radau;

10 max=b*Ing;%others

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)

18 x=cos(t);

19 z=1+(x-1)*(1-xminq)/2;

23 i=1:max;i=i';

25 %%%%%%%%%%%%% Integral weight %%%%%%%%%%%%%%

26 %%% Clenshaw Curtis, Fejer or Radau schemes %%%

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 APPENDIX L. MATLAB CODE

31 %Tb(i)=(2*i-2)*pi/(2*max-1);%Radau(+1)

33 m=1:max;%trigonometric(Gauss-Lobatto)

34 wcc=sin(Tb*m)*((1-cos(m'*pi))./m');

35 %SINE TRIGOMETRIC(Gauss-Lobatto)

36 %form=1:(max)/2%(Cheb-Gauss)

37 %form=1:(max-1)/2%(Radau)

38 % wcc(i)=wcc(i)+cos(2*m*Tb(i))/(4*mˆ(2)-1);

39 %(Fejer)Gauss-Cheby& Radau

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)

46 %w(end)=w(end)/2;%Radau(-1)

47 %w(1)=w(1)/2;%Radau(+1)

50 %%%%%%%%%%%%%%%%%%%%practice%%%%%%%%%%%

51 nu=-(a+0.5)/2;sig=-1.5*(a-0.5);

52 %I=intˆ{1+E} {0}(2-y)ˆ{2}dy andE=(-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 inEˆ(L) transformaiton;

309 APPENDIX L. MATLAB CODE

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 forL=k.%%%%%%%%%%%%%%%%%%%%%%%%%%

69 A=(3+z+(1-z)*y(j))/4;B=(1+z)/2;

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);

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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

84 %%%%%%%%%%% loose xmin%%%%%%%%%%%%%%%%%%%%%

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;

310 APPENDIX L. MATLAB CODE

95 end

97 end

1 %%%%%%%%% Poisson equation solver %%%%%%%%%%%%

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)

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 atE=1

22 vrm1=(-(1+nu)/nuˆ(2)/vd1m)ˆ(1/2);

23 vsm1=(-Lcˆ(2)*(1+nu)/vd1m)ˆ(1/2);

24 %Boundary condition atE=-1 311 APPENDIX L. MATLAB CODE

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 of2N 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 the4N 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 APPENDIX L. MATLAB CODE

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;

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;

74 %OL(2*ingrid+1,1:coeff)=Br(1,:);

75 %OL(2*ingrid+2,coeff+1:2*coeff)=Br(1,:);

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

81 change=norm(cnew-c,inf);

82 c=cnew;

313 APPENDIX L. MATLAB CODE

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 %%%%%%%%%%%%%%%%%%

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 APPENDIX L. MATLAB CODE

8 Bcl=zeros(3,coeff);

9 Bcr=zeros(3,coeff);

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 atE=-1)/Lobatto/Cheb

17 Ec=(cos(t));%Radau(BC atE=1)

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);

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 atE=-1)

29 %Bcl(2,:)=n.*n.*cos(n*pi)/(cos(0));

30 %Radau(BC atE=1)

31 Bcr(2,:)=n.*n;

32 end

2 %%%%%%%%%%%%%%%%%%%%%%% Vector for the4ODE

3 %%%%%andQ integral $$$$$$$$$$$$$$$$$$$

4 315 APPENDIX L. MATLAB CODE

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);

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;

13 O=zeros(5*coeff+2,1);

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);

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);

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;

27 O(2*ingrid+1:3*ingrid)=vGp.*(2*Ep)/Lc...

28 +(vFp.*(2*Ep)/Lc+a+1-1/Lc).*vG-Ep;

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;

316 APPENDIX L. MATLAB CODE

34 O(4*ingrid+1:5*ingrid)=vQ-Iq;

35 %%Boundary conditions%%

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 andQ integral $$$$$$$$$$$$$$$$$$$

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);

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;

13 OL=zeros(5*coeff+2);

14 317 APPENDIX L. MATLAB CODE

15 %%%%% 1:coeff ≤> cf, coeff+1:2*coeff ≤>ck,

16 %%%%%2 *coeff+1:3*coeff ≤>cg,3 *coeff+1:4*coeff ≤>cL

18 %%%% Equation1 %%%%%%

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);

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);

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);

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 APPENDIX L. MATLAB CODE

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;

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;

56 %OL(1:1*ingrid,5*coeff+1)=a*exp(-vF);

60 %%%% Equation2 %%%%%%

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;

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;

319 APPENDIX L. MATLAB CODE

78 %%%% Equation3 %%%%%%

79 %O(2*ingrid+1:3*ingrid)=vGp.*(2*Ep)/Lc...

80 %+(vFp.*(2*Ep)/Lc+a+1-1/Lc).*vG-Ep;

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;

88 %%%% Equation4 %%%%%%

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;

95 OL(1+3*ingrid:4*ingrid,3*coeff+1:4*coeff)=...

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;

98 OL(1+3*ingrid:4*ingrid,4*coeff+1:5*coeff)=...

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 %%%% Equation5 %%%%%%

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 APPENDIX L. MATLAB CODE

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,:);

131 OL(5*ingrid+6,3*coeff+1:4*coeff)=Bl(1,:);

132 %OL(5*ingrid+8,4*coeff+1:5*coeff)=Br(1,:);

133 OL(5*ingrid+7,4*coeff+1:5*coeff)=Bl(1,:);

134

135 end

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