Fokker-Planck Models of Star Clusters with Anisotropic Velocity Distributions. I. Pre-Collapse Evolution

The evolution of a spherical single-mass star cluster is followed in detail up to core collapse by numerically solving the orbit-averaged two-dimensional FokkerPlanck equation in energy-angular momentum space. Velocity anisotropy is allowed in the two-dimensional Fokker-Planck model. Using improved numerical codes, the evolution has been followed until the central density increased by a factor of 10 with high numerical accuracy. The numerical results clearly show self-similar evolution of the core during the late stages of the core collapse. In the self-similar region between the isothermal core and the outer halo, the density profile is characterized by a power law ρ(r) ∝ r−2.23 and the ratio of the one-dimensional tangential velocity dispersion to the radial one is σ t /σ 2 r = 0.92. As the core collapse proceeds, the collapse rate ξ ≡ tr(0)d ln ρ(0)/dt tends to the limiting value of ξ = 2.9× 10 , which is 19% smaller than the value for isotropic clusters. When Plummer’s model is chosen as the initial condition, the collapse time is about 17.6 times the initial half-mass relaxation time. As the result of strong relaxation in the core, the halo becomes to be dominated by radial orbits. The degree of anisotropy monotonically increases as the radius increases. In the outer halo, the profiles of the density are approximated by ρ ∝ r−3.5. This work confirms that the generation of velocity anisotropy is an important process in collisional stellar systems.