The secular evolution of discrete quasi-Keplerian systems

I. Kinetic theory of stellar clusters near black holes

¹ Institut d’Astrophysique de Paris and UPMC, CNRS (UMR 7095), 98bis boulevard Arago, 75014 Paris, France
e-mail: fouvry@iap.fr
² Korea Institute of Advanced Studies (KIAS), 85 Hoegiro, Dongdaemun-gu, 02455 Seoul, Republic of Korea
³ Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK

Received: 17 June 2016
Accepted: 6 October 2016

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.