Dynamical evolution of star clusters around a rotating black hole with an accretion disc

Abstract

The dynamical evolution of relativistic star clusters around a massive Kerr black hole with an accretion disc is examined, in the regime where the black hole dominates the potential and star-disc interactions dominate the evolution. A set of diagrams exhibiting the time development of the energy-dependent distribution function, f (∊), and the distributions of semimajor axes, a, eccentricities, e, and inclinations, i, of the model system are presented; plots of the latter three quantitities over time for a few illustrative orbits are also given. A simple approximation for the final radius of an orbit brought into the disc under star-disc interactions, namely |$a_\text f\approx a_0(1-e^2_0) \enspace\text {cos}^4(i_0/2)$|⁠) (or, in terms of angular momentum, |$L_\text f\approx(L_0+L_{z,0})/2$|⁠), is derived. It is found that the main effect of star-disc 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 |$\rho_\ast\propto r^{-2.5}$| 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 (≲ 10 GM/c2); in particular, the location of the last stable orbit limits the cluster's inner extent. By significantly increasing the central stellar density, star-disc interactions could be self-limiting 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 time-scales.