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, with 0 ≤ α ≤ 2. (2) Axially symmetric solutions in which the density varies as f(θ)/r, where r is the spherical radius and θ is the co-latitude. The singular isothermal sphere is a special case of the latter class with f(θ) = 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,∞, and their relevance to real systems is unclear. For each hydrodynamic equilibrium, we determine the phase-space distribution of the collisionless analog. Subject headings: galaxies: structure — galaxies: kinematics and dynamics — stars: kinematics — stars: formation — ISM: kinematics and dynamics — hydrodynamics