Models for anisotropic spherical stellar systems with a central point mass and Keplerian velocity dispersions

We add to the lore of spherical, stellar-system models a two-parameter family with an anisotropic velocity dispersion, and a central point mass (``black hole''). The ratio of the tangential to radial dispersions, is constant--and constitutes the first parameter--while each decreases with radius as r^{-1/2}. The second parameter is the ratio of the central point mass to the total mass. The Jeans equation is solved to give the density law in closed form: rho\propto (r/r0)^{-c}/[1+(r/r0)^{3-c}]^2, where r0 is an arbitrary scale factor. The two parameters enter the density law only through their combination c. At the suggestion of Tremaine, we also explore models with only the root-sum-square of the velocities having a Keplerian run, but with a variable anisotropy ratio. This gives rise to a more versatile class of models, with analytic expressions for the density law and the dispersion runs, which contain more than one radius-scale parameter.