Date of Degree


Document Type


Degree Name





Carlo Lancellotti

Committee Members

Andrew C. Poje

Rolf J. Ryham

Tobias Schafer

Mark D. Shattuck

Subject Categories

Astrophysics and Astronomy | Ordinary Differential Equations and Applied Dynamics | Statistical, Nonlinear, and Soft Matter Physics


Dense star cluster, Orbit-averaged Fokker-Planck model, Pseudo-spectral method, Negative heat capacity, Galactic globular cluster


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 compared 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.