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

This is the first paper of a series our works on self-similar orbit-averaged Fokker-Planck (OAFP) equation to model distribution function for stars in isotropic dense spherical star clusters. At late stage relaxation evolution clusters, standard stellar dynamics predicts that clusters evolve fashion forming collapsing cores. However, corresponding mathematical model, OAFP equation, has never been solved whole energy domain (−1<E<0). Existing based kinds finite difference methods provide solutions only truncated −1<E⪅−0.2. To broaden range domain, present work resorts (highly accurate and efficient) Gauss-Chebyshev pseudo-spectral method. We spectral solution four significant figures domain. The can reduce semi-analytical form whose degree polynomials eighteen holding three figures. also new eigenvalues c1=9.0925×10−4, c2=1.1118×10−4, c3=7.1975×10−2 , c4=3.303×10−2, core-collapse rate ξ=3.64×10−3, scaled escape ,χesc=13.881 power-law exponent α=2.2305. Since unstable against Chebyshev we domains (−1<E<Emax, where −0.35≤Emax≤−0.03) explain how handle instability. By reformulating several ways, improve accuracies reproduce an existing solution, which infers are up one figure.