Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (i) accurate pre-collapse solution

Abstract

This is the first paper of a series of our works on the self-similar orbit-averaged Fokker-Planck (OAFP) equation for distribution function of stars in dense isotropic star clusters. At the late stage of relaxation evolution of the clusters, standard stellar dynamics predicts that the clusters evolve in a self-similar fashion forming collapsing cores. However, the corresponding mathematical model, the self-similar OAFP equation, has never been solved on the whole energy domain $(-1< E < 0)$. The existing works based on kinds of finite difference methods provide solutions only on the truncated domain $-1< E<-0.2$. To broaden the range of the truncated domain, the present work resorts to a (highly accurate and efficient) Gauss-Chebyshev pseudo-spectral method. We provide a spectral solution, whose number of significant figures is four, on the whole domain. Also, the solution can reduce to a semi-analytical form whose degree of polynomials is only eighteen holding three significant figures. We also provide the new eigenvalues; $c_{1}=9.0925\times10^{-4}$, $c_{2}=1.1118\times10^{-4}$, $c_{3}=7.1975\times10^{-2}$ and $c_{4}=3.303\times10^{-2}$, corresponding to the core collapse rate $ξ=3.64\times10^{-3}$, scaled escape energy $χ_\text{esc}=13.881$ and power-law exponent $α=2.2305$. Since the solution on the whole domain is unstable against degree of Chebyshev polynomials, we also provide spectral solutions on truncated domains ( $-1< E<E_\text{max}$, where $-0.35<E_\text{max}<-0.03$) to explain how to handle the instability. By reformulating the OAFP equation in several ways, we improve the accuracy of the spectral solution and reproduce an existing self-similar solution, which infers that existing solutions have only one significant figure at best.