Axisymmetric galaxy models wth central BHs, with an application to M32

The contour integral method of Hunter and Qian is applied to axisymmetric galaxy models in which the distribution function (DF) is of the form f = f(E, L(z)), where E and L(z) are the classical integrals of motion in an axisymmetric potential, A practical way to construct the unique even part f(e)(E, L(z)) of the two-integral DF for such systems is presented, It is applied to models, both oblate and prolate, in which the mass density is stratified on similar concentric spheroids. 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 + 2 beta at large radii. As an application; the DF is constructed for the (alpha,beta)-model with a 1.8 x 10(6) M. 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.