A Fluid-Dynamical Method for Computing the Evolution of Star Clusters
One way that can be used to study the development of a stellar system is the fluid-dynamical approach, whereby the stars are considered as the constituent particles of a continuous fluid whose behavior is described by moment equations derived from the Boltzmann equation. This method is most useful for systems with very many stars, where it complements the n-body technique which is feasible only for small systems. The fluid-dynamical approach begins by defining suitable moments of the velocity distribution at each point in space. In studying the evolution of a star cluster it is necessary to consider moments of at least fourth order in the velocities in order to represent all the essential physical effects, including an outward ‘heat flow’ caused by the escape of the most energetic stars, and an excess or deficiency of high velocity stars relative to a Maxwellian distribution. Allowing for unequal radial and transverse velocity dispersions, we find that six moments in all are required, for which six fluid-dynamical equations may be derived by taking the corresponding moments of the Boltzmann equation. The fluid-dynamical equations contain relaxation terms which may be evaluated from the Fokker-Planck equation, assuming that deviations from a Maxwellian velocity distribution are small. In the absence of other effects, the relaxation terms have the effect of making the various deviations from a Maxwellian velocity distribution decay exponentially with decay times which are closely related to the classical relaxation time. The resulting fluid-dynamical equations can then be solved numerically to yield the values of all quantities as functions of position and time.