Dynamical Evolution of Star Clusters Around a Rotating Black Hole With an Accretion Disk

The dynamical evolution of relativistic star clusters around a massive Kerr black hole with an accretion disk is examined, in the regime where the black hole dominates the potential and star-disk interactions dominate the evolution. A set of diagrams exhibiting the time development of the energy-dependent distribution function, f(E), and the distributions of semi-major axes, a, eccentricities, e, and inclinations, i, of the model system are presented; plots of the latter three quantities over time for a few illustrative orbits are also given. A simple approximation for the final radius of an orbit brought into the disk under star-disk interactions, namely a_f = a₀(1-e₀²) cos⁴(i₀/2) (or, in terms of angular momentum, L_f = (L₀+L_z,0)/2), is derived. It is found that the main effect of star-disk interactions on an isotropic cluster, besides the circularization and alignment of orbits, is to steepen an initial density profile ρ_* ∝ r^-n to the approximate `asymptotic' profile ρ_* ∝ r^-2.5 when n < 2.5, to leave the profile unchanged when n > 2.5, and in both cases to increase the central density (by several hundred very close to the black hole); initially anisotropic clusters are found to exhibit similar patterns. The numerical results can be explained in terms of a simple analytic model. Relativistic effects are found to affect the cluster properties significantly only at very small radii (< 10GM/c²); in particular, the location of the last stable orbit limits the cluster's inner extent. By significantly increasing the central stellar density, star-disk interactions could self-limit themselves by causing stellar collisions to become important; the future evolution of the cluster in this case will depend on the relative balance of the collisional, alignment, and stellar evolutionary timescales.

Keywords: accretion disks --- black hole physics --- galaxies: nuclei --- stellar dynamics