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

We study in a systematic way stationary solutions to the coupled Vlaso v and Poisson equations that have 'self-similar' or scaling symmetry i n phase space. In particular, we find analytically all spherically sym metric 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 orbi ts and systems that are isotropic in velocity space. We then discuss s ystems 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 th e solutions, whereas we find them in a unified and self-contained mann er. Distribution functions of the same type in cylindrical and planar geometries are also discussed briefly. Finally, we study as part of th e same unified scheme the spatially spheroidal systems that exhibit st rict power-law behaviour for the density and potential. The results ar e in agreement with the solutions published recently.