A new family of distribution functions for spherical galaxies

A new family of anisotropic distribution functions for spherical stellar systems is described. These are quasi-separable functions of energy and angular momentum, and are specified in terms of a circularity function h(x). This function fixes the distribution of orbits on the potential’s energy surfaces outside some anisotropy radius. Self-consistent models are then found by solving for the remaining energy function. Such models are isotropic in their central parts and reach constant anisotropy in their envelopes. Detailed results are presented for a particular set of radially anisotropic circularity functions $$h_\alpha(x)$$. In the scale-free logarithmic potential exact analytic solutions are shown to exist for all scale-free circularity functions. In general potentials, analytic expressions are given for the distribution function in the envelope, while the complete dynamical model is found by iteratively solving a one-dimensional integral equation. As an example, the algorithm is applied to the isochrone sphere, and the resulting energy distributions are given in terms of simple functions. Intrinsic and projected velocity dispersions are calculated and show the expected properties. The distribution function of velocities along the line-of-sight can for many circularity functions be reduced to an integral over two variables instead of three. For the scale-free $$h_\alpha$$-models it can be calculated analytically and turns out to be a sum of Gaussians times powers of squared velocity. Several applications of the quasi-separable distribution functions are briefly discussed. These include the effects of anisotropy or a dark halo on line broadening functions, the radial orbit instability in anisotropic spherical systems, and violent relaxation in spherical collapse.