Two-component fokker-planck models for the evolution of isolated globular clusters

Two-component (normal and degenerate stars) models are the simplest realization of clusters with a mass spectrum because high-mass stars evolve quickly into degenerates, while low-mass stars remain on the main sequence for the age of the universe. Here we examine the evolution of isolated globular clusters by using two-component Fokker-Planck (FP) models that include heating by binaries formed in tidal capture and in three-body encounters. Three-body binary heating dominates, and the postcollapse expansion is self-similar, at least in models with total mass M < 3 × 10⁵ M⊙, initial half-mass radius r_h,i > 5 pc, component mass ratio m₂/m₁ > 2, and number ratio N₁/N₂ ≤ 300, when m₂ = 1.4 M⊙. We derive scaling laws for ρ_c, v_i, r_h, and r_h as functions of m₁/m₂, N, M, and time t from simple energy-balance arguments; these agree well with the FP simulations. We have studied the conditions under which gravothermal oscillations (GTOs) occur. If E_tot and E_c are the energies of the cluster and of the core, respectively, and t_th and f_c are their relaxation times, then ε ≡ (E_tot/t_rh)/(E_c/t_rc) is a good predictor of GTOs: all models with ε > 0.01 are stable, and all but one with ε < 0.01 oscillate. We derive a scaling law for ε against N and m₁/m₂ and compare them with our numerical results. Clusters with larger m₂/m₁ or smaller N are stabler.