The diffusion of stars through phase space

Abstract

Deviations of the potentials of stellar systems from integrability cause stars to diffuse through three-dimensional orbit space. The Fokker-Planck equation that describes this diffusion takes a particularly simple form when actions are used as orbit-space coordinates. The rate of diffusion is governed by a vector Δ and a tensor Δ2, which according to the circumstances of a particular problem should be calculated either from kinetic theory or from Hamiltonian perturbation theory. In many astrophysically interesting circumstances Δ is related to the divergence of the more readily calculated tensor Δ2. In addition to being computationally handy, this relationship ensures that the orbital diffusion described by the Fokker-Planck equation causes the system's entropy, derived from any H-function, to increase whenever the system is interacting with a hotter system of scatterers. An investigation of the heating of stellar discs in the light of these general results is discussed in detail.