The stability of multi-component gravitational systems described by a Boltzmann equation

This paper uses existing techniques developed by Ipser, Thorne, and Kandrup to examine the stability of multi-component systems, with a distribution of masses, whose evolution is described by the collisionless or collisional Boltzmann equation.The principal conclusions are as follows: (1) All static, spherically-symmetric solutions to the collisionless equations, appropriate for a star cluster, are guaranteed to be stable with respect to spherically-symmetric disturbances, provided only that the population of stars is, for fixed mass, a decreasing function of the mean field energy. (2) If, furthermore, the static configuration has an isotropic distribution of momenta, it will also be stable towards nonradial perturbations. (3) The unique static solution to the collisional equations for a spherically-symmetric, spatially truncated configuration can be stable only if that configuration is a local entropy maximum. (4) A simple expression is obtained for the Jeans length for a system with an arbitrary isotropic distribution of momenta.