Stellar Dynamics around a Massive Black Hole II: Resonant Relaxation

S. Sridhar, Jihad R. Touma

Published 2015-09-08Version 1

We present a first-principles theory of Resonant Relaxation (RR) of stellar systems orbiting within the sphere of influence of massive black holes in galactic nuclei. We extend the rigorous kinetic theory of Gilbert (1968) to include the Keplerian field of a black hole of mass $M_\bullet$, and specialize to a (Keplerian) stellar system of mass $M \ll M_\bullet$. Using the results of the secular collisionless theory of Paper I, we orbit-average the kinetic equation through perturbative development in the small parameter $\varepsilon = M/M_\bullet$. This is supplemented with contributions from general relativistic corrections up to 1.5 post-Newtonian order and external gravitational sources. The result is a kinetic equation for a secular distribution function (DF) in 5-dim (Gaussian Ring) space, with explicit forms for the fluctuation and dissipation components of the collision integral. For general DFs, both apsidal and nodal precessions contribute to RR; so the traditional, physically-motivated distinction between scalar-RR and vector-RR disappears. Irreversible 2-particle correlations, that build up through secular gravitational interactions, are the driving agents of RR. The correlation function can be written in terms of the wake function, which is the linear response of the system to the perturbation offered by any chosen stellar orbit. The relationship includes direct interactions between pairs of stars as well as collective effects (gravitational polarization). We discuss the interplay of secular dynamics and RR in the evolution toward secular thermodynamic equilibria. In Paper III we apply our RR theory to axisymmetric discs, and provide explicit formulae for the loss cone rates at which mass, energy and angular momentum are fed to the black hole.