Cusped Spherical Stellar Systems with Anisotropic Velocities

Spherical models with constant velocity anisotropy β have distribution functions depending on binding energy E and angular momentum L of the form f (E, L) = L−2β fE(E). An inversion for the unknown function fE (E) is straightforward to derive and reduces to Eddington’s famous algorithm for isotropic models (β = 0). Importantly, if the anisotropy parameter of the model is a half-integer constant (i.e., β = 1/2, −1/2, ...), then the distribution function can be found by differentiation alone, without the need for any integration! This gives us a very simple way of constructing radially or tangentially anisotropic distribution functions of spherical models. The method is applied to two new families of cusped spherical galaxies. The first is a generalization of the isochrone sphere, the second a generalization of the hypervirial models. There are representatives of each family that possess the important cosmological cusps, with the density ρ behaving like r−1 or r−1.5 at small radii. The superposition of two or more components of the form L−2β fE(E) may permit very general anisotropy behavior to be modelled. Subject headings: celestial mechanics, stellar dynamics — galaxies: structure — galaxies : kinematics and dynamics — galaxies: elliptical and lenticular