A family of models for spherical stellar systems

We describe a one-parameter family of models of stable spherical stellar systems in which the phase-space distribution function depends only on energy. The models have similar density profiles in their outer parts (p ∝ r^-4) and central power-law density cusps, p ∝ r^3-η, 0<η≤3. The family contains the Jaffe [MNRAS, 202, 995 (1983)] and Hernquist [ApJ, 356, 359 (1990)] models as special cases. We evaluate the surface brightness profile, the line-of-sight velocity dispersion profile, and the distribution function, and discuss analogs of King's core-fitting formula for determining mass-to-light ratio. We also generalize the models to a two-parameter family, in which the galaxy contains a central black hole; the second parameter is the mass of the black hole. Our models can be used to estimate the detectability of central black holes and the velocity-dispersion profiles of galaxies that contain central cusps, with or without a central black hole.