Axisymmetric galaxy models with central black holes, with an application
to M32.<P>

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

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.<P>

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.<P>

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 $86 kms^{-1}$.<P>

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