The spheroids with scale-free densities are discussed in detail. These provide useful approximations to the behaviour of more realistic models in the limit of small and large radii. The self-consistent case is treated, as well as the case in which there are additional contributions to the potential from a central black hole or dark halo. The two-integral DFs for scale-free densities in a Kepler potential are particularly simple. These can be used to model power-law density cusps near a central black hole, or to model the outer parts of finite-mass systems. The range of axial ratios and density profile slopes is determined for which spheroidal power-law cusps with a central black hole have a physical two-integral DF.
More generally, the two-integral DFs are discussed for a set of
spheroidal `(alpha,beta)-models', characterized by a power-law density
cusp with slope alpha at small radii, and a power-law density fall-off
with slope alpha + 2beta at large radii. As an application, the DF is
constructed for the (alpha,beta)-model with a 1.8 x 10^6 solar mass
black hole used by van der Marel et al. to
interpret their high spatial resolution spectroscopic data for
M32. The line-of-sight velocity profiles are calculated. The results
confirm that the model fits the data remarkably well. The model is
used to calculate the predicted kinematic signatures of a central
black hole in M32 in spectroscopic observations through small
apertures, such as are now possible with the Faint Object Spectrograph
on board the Hubble Space Telescope.