A Balescu-Lenard-type kinetic equation for the collisional evolution of stable self-gravitating systems

Abstract

A kinetic equation for the collisional evolution of stable, bound, self-gravitating and slowly relaxing systems is established, which is valid when the number of constituents is very large. It accounts for the detailed dynamics and self-consistent dressing by collective gravitational interaction of the colliding particles, for the system's inhomogeneity and for different constituents' masses. It describes the coupled evolution of collisionally interacting populations, such as stars in a thick disc and the molecular clouds off which they scatter. The kinetic equation derives from the BBGKY hierarchy in the limit of weak, but non-vanishing, binary correlations, an approximation which is well justified for large stellar systems. The evolution of the 1-body distribution function is described in action–angle space. The collective response is calculated using a biorthogonal basis of pairs of density–potential functions. The collision operators are expressed in terms of the collective response function allowed by the existing distribution functions at any given time and involve particles in resonant motion. These equations are shown to satisfy an H theorem. Because of the inhomogeneous character of the system, the relaxation causes the potential as well as the orbits of the particles to secularly evolve. The changing orbits also cause the angle Fourier coefficients of the basis potentials to change with time. We derive the set of equations which describes this coupled evolution of distribution functions, potential and basis Fourier coefficients for spherically symmetric systems. In the homogeneous limit, which sacrifices the description of the evolution of the spatial structure of the system but retains the effect of collective gravitational dressing, the kinetic equation reduces to a form similar to the Balescu–Lenard equation of plasma physics.