ON STATIONARY SELF-SIMILAR DISTRIBUTIONS OF A COLLISIONLESS SELF-GRAVITATING GAS

We study in a systematic way stationary solutions to the coupled Vlasov and Poisson equations that have 'self-similar' or scaling symmetry in phase space. In particular, we find analytically all spherically symmetric distribution functions where the mass density and gravitational potential are strict power laws in r, the distance from the symmetry point. We treat as special cases systems built from purely radial orbits and systems that are isotropic in velocity space. We then discuss systems with arbitrary velocity space anisotropy and recover a general class of distribution functions. These distributions are mostly known and indeed have already proved to be useful in modelling galaxies. The cited references, however, use various ad hoc techniques to obtain the solutions, whereas we find them in a unified and self-contained manner. Distribution functions of the same type in cylindrical and planar geometries are also discussed briefly. Finally, we study as part of the same unified scheme the spatially spheroidal systems that exhibit strict power-law behaviour for the density and potential. The results are in agreement with the solutions published recently.