A simple, heuristic derivation of the Balescu-Lenard kinetic equation for stellar systems

TL;DR

This paper presents a simplified and intuitive derivation of the Balescu-Lenard kinetic equation for stellar systems, incorporating collective effects and inhomogeneity, and linking it to classical two-body relaxation theory.

Contribution

It introduces a short, heuristic derivation of the Balescu-Lenard equation for stellar systems using Rostoker's superposition principle, simplifying previous complex derivations.

Findings

Derivation of the Balescu-Lenard equation using Rostoker's principle

Connection established between the BL equation and Chandrasekhar's two-body relaxation theory

Simplified approach makes the kinetic theory more accessible

Abstract

The unshielded nature of gravity means that stellar systems are inherently inhomogeneous. As a result, stars do not move in straight lines. This obvious fact severely complicates the kinetic theory of stellar systems because position and velocity turn out to be poor coordinates with which to describe stellar orbits - instead, one must use angle-action variables. Moreover, the slow relaxation of star clusters and galaxies can be enhanced or suppressed by collective interactions ('polarisation' effects) involving many stars simultaneously. These collective effects are also present in plasmas; in that case, they are accounted for by the Balescu-Lenard (BL) equation, which is a kinetic equation in velocity space. Recently several authors have shown how to account for both inhomogeneity and collective effects in the kinetic theory of stellar systems by deriving an angle-action generalisation of the BL equation. Unfortunately their derivations are long and complicated, involving multiple coordinate transforms, contour integrals in the complex plane, and so on. On the other hand, Rostoker's superposition principle allows one to pretend that a long-range interacting $N$-body system, such as a plasma or star cluster, consists merely of uncorrelated particles that are 'dressed' by polarisation clouds. In this paper we use Rostoker's principle to provide a simple, intuitive derivation of the BL equation for stellar systems which is much shorter than others in the literature. It also allows us to straightforwardly connect the BL picture of self-gravitating kinetics to the classical 'two-body relaxation' theory of uncorrelated flybys pioneered by Chandrasekhar.