E.E. Qian, P.T. de Zeeuw, R.P. van der Marel, C. Hunter

The contour integral method of Hunter \& 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 axis 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 $1.8\times 10^6 M_{\odot}$ 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 kinematic signatures of a central black hole in observations such as are now possible with the Hubble Space Telescope. The predicted Gaussian velocity dispersion for the M32 centre is $127 kms^{-1}$ with the $0.09''\times 0.09''$ square aperture of the Faint Object Spectrograph, and $105 kms^{-1}$ with the $0.26''$ diameter circular aperture, while the central dispersion measured from ground-based data is only $86 kms^{-1}$.

keywords: stellar dynamics -- galaxies: kinematics and dynamics -- galaxies: structure -- galaxies: central black holes -- galaxies: individual: M32