Stellar Dynamics around a Massive Black Hole III: Resonant Relaxation of Axisymmetric Discs

TL;DR

This paper develops a kinetic theory for resonant relaxation in axisymmetric stellar discs around black holes, deriving a Fokker-Planck equation that describes the evolution of stellar distributions considering relativistic effects and external potentials.

Contribution

It presents a simplified kinetic model for resonant relaxation in axisymmetric discs, including relativistic effects, and analyzes equilibrium states and star loss rates to the black hole.

Findings

Boltzmann entropy is non-decreasing during RR in lossless discs

Secular thermal equilibria are maximum entropy states with Boltzmann form DFs

Derived expressions for star feeding rates to the black hole

Abstract

We study the Resonant Relaxation (RR) of an axisymmetric low mass (or Keplerian) stellar disc orbiting a more massive black hole (MBH). Our recent work on the general kinetic theory of RR is simplified in the standard manner by ignoring the effects of `gravitational polarization', and applied to a zero-thickness, flat, axisymmetric disc. The wake of a stellar orbit is expressed in terms of the angular momenta exchanged with other orbits, and used to derive a kinetic equation for RR under the combined actions of self-gravity, 1 PN and 1.5 PN relativistic effects of the MBH and an arbitrary external axisymmetric potential. This is a Fokker-Planck equation for the stellar distribution function (DF), wherein the diffusion coefficients are given self-consistently in terms of contributions from apsidal resonances between pairs of stellar orbits. The physical kinetics is studied for the two main cases of interest. (1) `Lossless' discs in which the MBH is not a sink of stars, and disc mass, angular momentum and energy are conserved: we prove that general H-functions can increase or decrease during RR, but the Boltzmann entropy is (essentially) unique in being a non-decreasing function of time. Therefore secular thermal equilibria are maximum entropy states, with DFs of the Boltzmann form; the two-Ring correlation function at equilibrium is computed. (2) Discs that lose stars to the MBH through an `empty loss-cone': we derive expressions for the MBH feeding rates of mass, angular momentum and energy in terms of the diffusive flux at the loss-cone boundary.