City University of New York (CUNY) CUNY Academic Works Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects by Program 9-2020 An Accurate Solution of the Self-Similar Orbit-Averaged Fokker- Planck Equation for Core-Collapsing Isotropic Globular Clusters: Properties and Application Yuta Ito The Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/3969 Discover additional works at: https://academicworks.cuny.edu 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 by YUTA ITO A dissertation submitted to the Graduate Faculty in Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York 2020 © 2020 YUTA ITO All Rights Reserved ii An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-Planck Equation for Core-Collapsing Isotropic Globular Clusters: Properties and Application by Yuta Ito This manuscript has been read and accepted by the Graduate Faculty in Physics in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy. Date Carlo Lancellotti Chair of Examining Committee Date Alexios P. Polychronakos Executive Officer Supervisory Committee: Andrew C. Poje Rolf J. Ryham Tobias Schafer¨ Mark D. Shattuck THE CITY UNIVERSITY OF NEW YORK iii ABSTRACT An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-Planck Equation for Core-Collapsing Isotropic Globular Clusters: Properties and Application by Yuta Ito Adviser: Carlo Lancellotti Hundreds of dense star clusters exist in almost all galaxies. Each cluster is composed of approximately ten thousand through ten million stars. The stars orbit in the clusters due to the clusters’ self-gravity. Standard stellar dynamics expects that the clusters behave like collisionless self-gravitating systems on short time scales (∼ million years) and the stars travel in smooth contin- uous orbits. Such clusters temporally settle to dynamically stable states or quasi-stationary states (QSS). Two fundamental QSS models are the isothermal- and polytropic- spheres since they have similar structures to the actual core (central part) and halo (outskirt) of the clusters. The two QSS models are mathematically modeled by the Lane-Emden equations. On long time scales (∼ bil- lion years), the clusters experience a relaxation effect (Fokker-Planck process). This is due to the finiteness of total star number in the clusters that causes stars to deviate from their smooth orbits. This relaxation process forms a highly-dense relaxed core and sparse-collisionless halo in a self- similar fashion. The corresponding mathematical model is called the self-similar Orbit-Averaged Fokker-Planck (ss-OAFP) equation. However, any existing numerical works have never satisfac- torily solved the ss-OAFP equation last decades after it was proposed. This is since the works rely on finite difference (FD) methods and their accuracies were not enough to cover the large gap in the density of the ss-OAFP model. To overcome this numerical problem, we employ a Chebyshev pseudo-spectral method. Spectral methods are known to be accurate and efficient scheme com- iv pared with FD methods. The present work proposes a new method by combining the Chebyshev spectral method with an inverse mapping of variables. Our new method provides accurate numerical solutions of the Lane-Emden equations with large density gaps on MATLAB software. The maximum density ratio of the core to halo can reach the possible numerical (graphical) limit of MATLAB. The same method provides four sig- nificant figures of a spectral solution to the ss-OAFP equation. This spectral solution infers that existing solutions have at most one significant figure. Also, our numerical results provide three new findings. (i) We report new kinds of the end-point singularities for the Chebyshev expansion of the Lane-Emden- and ss-OAFP equations. (ii) Based on the spectral solution, we discuss the thermodynamic aspects of the ss-OAFP model and detail the cause of the negative heat capac- ity of the system. We suggest that to hold a ’negative’ heat capacity over relaxation time scales stars need to be not only in a deep potential well but also in a non-equilibrium state with the flow of heat and stars. (iii) We propose an energy-truncated ss-OAFP model that can fit the observed structural profiles of at least half of Milky Way globular clusters. The model can apply to not only normal clusters but also post collapsed-core clusters with resolved (observable) cores; those clusters can not generally be fitted by a single model. The new model is phenomenological in the sense that the energy-truncation is based on polytropic models while the truncation suggests that low-concentration globular clusters are possibly polytropic clusters. v ACKNOWLEDGMENTS Part of the formulations in Chapter 4, the numerical arrangements in Chapter 5, and the nu- merical results in Chapters 6 and 7 are based on the following published paper. (*This work was supported by ONR DRI Grant No. 0604946 and NSF DMS Award No. 1107307.) Ito, Y., Poje, A., and Lancellotti, C. (2018). ”Very-large-scale spectral solutions for spherical polytropes of index m > 5 and the isothermal sphere.” New Astronomy, 58: 15-28. DOI: https://doi.org/10.1016%2Fj.newast.2017.07.003 Part of the formulations in Chapter 4, the numerical arrangements in Chapter 5, and the numerical results in Chapter 8 are based on the following accepted paper. Ito, Y. (2021). ”Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense star clusters (i) accurate pre-collapse solution.” New Astronomy, 83:101474. DOI: https://doi.org/10.1016/j.newast.2020.101474 vi Contents List of Tables xv List of Figures xix 1 Introduction 1 1.1 A quantitative understanding of dense star clusters . 2 1.1.1 The physical features of globular clusters . 2 1.1.2 Dynamical time and mean field Newtonian potential . 5 1.1.3 Relaxation time and two-body relaxation (’collision’) process . 6 1.2 The interdisciplinary importance of stellar-dynamics studies . 7 1.2.1 Astronomical importance . 8 1.2.2 Statistical-physics importance . 8 1.2.3 Computational-physics importance . 9 1.3 A basic mathematical model: Fokker-Planck (FP) kinetics of isotropic spherical star clusters . 10 1.3.1 Distribution function for stars . 10 1.3.2 Collisionless relaxation evolution on dynamical time scales . 12 1.3.3 Collisional relaxation evolution and Orbit-Averaged Fokker-Planck (OAFP) equation . 14 1.4 The motivation to apply spectral methods to the OAFP equation . 17 1.4.1 Numerical difficulty in solving OAFP equation . 18 vii CONTENTS 1.4.2 Spectral method and (semi-)infinite domain problem . 19 1.5 The state-of-the-art and organization of the present dissertation . 21 2 Mathematical models for the evolution of globular clusters 25 2.1 Quasi-stationary-state star cluster models . 25 2.1.1 Isothermal sphere . 27 2.1.2 Polytropic sphere . 31 2.1.3 King model . 37 2.2 Self-similar Orbit-Averaged Fokker-Planck (ss-OAFP) model . 40 2.2.1 Self-similar analysis on the OAFP equation . 41 2.2.2 Difficulties in the numerical integration of the ss-OAFP equation . 44 3 Gauss-Chebyshev pseudo-spectral method and 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 co- efficients . 53 3.3 The convergence rates of Chebyshev coefficients . 56 3.4 An example of Gauss-Chebyshev pseudo-spectral method applied to the King model 59 3.4.1 The mathematical reformulation of the King model . 60 3.4.2 Matrix-based Newton-Kantrovich iteration method (Quasi-Linearization) . 61 3.4.3 Spectral solutions for the King model and unbounded-domain problems . 65 3.5 Fejer’s´ first-rule quadrature . 71 viii CONTENTS 3.5.1 Fejer’s´ first-rule Quadrature . 71 3.5.2 The convergence rate of the Fejer’s´ first-rule quadrature . 73 4 The inverse forms of the Mathematical models 75 4.1 The inverse mapping of dimensionless potential and density . 75 4.2 Isothermal sphere (LE2 function) . 77 4.2.1 Inverse mapping . 77 4.2.2 Regularized ILE2 equation . 78 4.3 Polytropic sphere (LE1 function) . 79 4.3.1 Inverse mapping . 79 4.3.2 Regularized ILE1 equations . 80 4.4 The ss-OAFP system . 81 4.4.1 The inverse form of the Poisson’s equation . 81 4.4.2 Regularized ss-OAFP system . 82 4.4.3 Integral formulas on the whole- and truncated- domains . 85 5 Numerical arrangements and optimization on the spectral methods 88 5.1 The numerical treatments of the ILE1- and ILE2- equations . 88 5.2 The numerical arrangements of the regularized ss-OAFP system . 89 5.2.1 Step 1: Solving the regularized Poisson’s equation . 90 5.2.2 Step 2: Solving the regularized 4ODEs and Q-integral . 90 5.2.3 Step 3: Repeating steps 1 and 2 . 91 5.2.4 Numerical arrangements . 92 6 Preliminary Result 1: A spectral solution for the isothermal-sphere model 94 6.1 Numerical results . 94 6.2 The topological property and asymptotic approximation of the LE2 function . 98 ix CONTENTS 6.2.1 Milne variables .