Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense star 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 to model distribution function for stars in isotropic dense spherical star clusters. At the late stage of the 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). 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 accurate to four significant figures on the whole domain. 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 c1=9.0925×10−4,c2=1.1118×10−4,c3=7.1975×10−2 , and c4=3.303×10−2, corresponding to the core-collapse rate ξ=3.64×10−3, scaled escape energy ,χesc=13.881 and power-law exponent α=2.2305. Since the solution is unstable against the degree of Chebyshev polynomials on the whole domain, we also provide spectral solutions on truncated domains (−1<E<Emax, where −0.35≤Emax≤−0.03) to explain how to handle the instability. By reformulating the self-similar OAFP equation in several ways, we improve the accuracies of the spectral solutions and reproduce an existing self-similar solution, which infers that the existing solutions are accurate up to only one significant figure.