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; $c_1=9.0925\times 10^{-4}$, $c_2=1.1118\times 10^{-4}$, $c_3=7.1975\times 10^{-2}$ and $c_4=3.303\times 10^{-2}$, corresponding to the core collapse rate $\xi =3.64\times 10^{-3}$, scaled escape energy ${\chi }_{\text{esc}}=13.881$ and power-law exponent $\alpha =2.2305$. 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.

References in corpus (11)

• Internal Dynamics of Globular Clusters
• Long-term evolution of isolated N-body sytems
• Comparing Direct N-Body Integration with Anisotropic Gaseous Models of Star Clusters
• Parameters of core-collapse
• Core Collapse via Coarse Dynamic Renormalization
• The hunt for self-similar core collapse
• Fokker-Planck Models of Star Clusters with Anisotropic Velocity Distributions. I. Pre-Collapse Evolution
• The Internal Dynamics of Globular Clusters
• Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (i) accurate pre-collapse solution
• A convergent kinetic theory of collisional star clusters (i) a self-consistent 'truncated' mean-field acceleration of stars
• Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (ii) physical properties and negative heat capacity of pre-collapse core

Cited by in corpus (3)

• Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (i) accurate pre-collapse solution
• When self-similarity meets mass spectrum and anisotropy
• Self-similar orbit-averaged Fokker-Planck equation for isotropic spherical dense clusters (iii) application of pre-collapse solution to Galactic globular clusters