Axisymmetric Self-Similar Equilibria of Self-Gravitating Isothermal Systems

All axisymmetric self-similar equilibria of self-gravitating, rotating, isothermal systems are identified by solving the nonlinear Poisson equation analytically. There are two families of equilibria: (1) Cylindrically symmetric solutions in which the density varies with cylindrical radius as R^(-alpha), with 0 <= alpha <= 2. (2) Axially symmetric solutions in which the density varies as f(theta)/r^2, where r' is the spherical radius and theta’ is the co-latitude. The singular isothermal sphere is a special case of the latter class with f(theta)=constant. The axially symmetric equilibrium configurations form a two-parameter family of solutions and include equilibria which are surprisingly asymmetric with respect to the equatorial plane. The asymmetric equilibria are, however, not force-free at the singular points r=0, infinity, and their relevance to real systems is unclear. For each hydrodynamic equilibrium, we determine the phase-space distribution of the collisionless analog.