Diffusion of stars in a harmonic potential

Abstract

In a simple approximation, the evolution of a stellar system can be described in terms of the solutions to a diffusion equation for motion in a harmonic potential. This paper presents a discussion and characterization of the normal modes for this equation. These solutions are of particular interest in that they provide a simple example of the interplay between dynamical and relaxation phenomena. For the case of a large system, in which the relaxation timet _r is much greater than the dynamical timet _d,there exists a well-defined sense in which the effects of relaxation may be viewed as a perturbation of motion in the fixed field: the dynamical effects give rise to a purely oscillatory behavior, whereas collisions among stars provide a dissipative mechanism that drives the system towards the unique isothermal equilibrium. Alternatively, the presence of the fixed potential serves to alter the ‘e-folding’ time for the various modes. In the limit thatt _r ≫t _d , all characteristic relaxation times are essentially doubled. This suggests a danger in the use of ‘velocity space’ equations to model the effects of evaporation.