The secular evolution of discrete quasi-Keplerian systems. I. Kinetic theory of stellar clusters near black holes

TL;DR

This paper derives a kinetic equation for the long-term evolution of particles orbiting a massive object, capturing resonant relaxation effects like mass segregation and the Schwarzschild barrier.

Contribution

It introduces a Balescu-Lenard kinetic equation tailored for quasi-Keplerian systems, advancing understanding of secular dynamics near black holes.

Findings

Captures phenomena of mass segregation

Models the relativistic Schwarzschild barrier

Describes resonant relaxation effects

Abstract

We derive the kinetic equation that describes the secular evolution of a large set of particles orbiting a dominant massive object, such as stars bound to a supermassive black hole or a proto-planetary debris disc encircling a star. Because the particles move in a quasi-Keplerian potential, their orbits can be approximated by ellipses whose orientations remain fixed over many dynamical times. The kinetic equation is obtained by simply averaging the BBGKY equations over the fast angle that describes motion along these ellipses. This so-called Balescu-Lenard equation describes self-consistently the long-term evolution of the distribution of quasi-Keplerian orbits around the central object: it models the diffusion and drift of their actions, induced through their mutual resonant interaction. Hence, it is the master equation that describes the secular effects of resonant relaxation. We show how it captures the phenonema of mass segregation and of the relativistic Schwarzschild barrier recently discovered in $N$-body simulations.