Continuum Equations for Stellar Dynamics

The description of a stellar system as a continuous fluid represents a convenient first approximation to stellar dynamics, and its derivation from the kinetic theory is standard. The challenge lies in providing adequate closure approximations for the higher-order moments of the phase-space density function that appear in the fluid dynamical equations. Such closure approximations may be found using representations of the phase-space density as embodied in the kinetic theory. In the classic approach of Chapman and Enskog, one is led to the Navier-Stokes equations, which are known to be inaccurate when the mean free paths of particles are long, as they are in many stellar systems. To improve on the fluid description, we derive here a modified closure relation using a Fokker-Planck collision operator. To illustrate the nature of our approximation, we apply it to the study of gravitational instability. The instability proceeds in a qualitative manner as given by the Navier-Stokes equations but, in our description, the damped modes are considerably closer to marginality, especially at small scales.