Axisymmetric dynamical models are constructed for the E3 galaxy M32 to interpret high spatial resolution stellar kinematical data obtained with the Hubble Space Telescope ( HST ) .
Models are studied with two-integral , f ( E,L _ { z } ) , phase-space distribution functions , and with fully general three-integral distribution functions .
The latter are built using an extension of Schwarzschild ’ s approach : individual orbits in the axisymmetric potential are calculated numerically , and populated using non-negative least-squares fitting so as to reproduce all available kinematical data , including line-of-sight velocity profile shapes .
The details of this method are described in companion papers by Rix et al .
and Cretton et al .

Models are constructed for inclinations i = 90 ^ { \circ } ( edge-on ) and i = 55 ^ { \circ } .
No model without a nuclear dark object can fit the combined ground-based and HST data , independent of the dynamical structure of M32 .
Models with a nuclear dark object of mass M _ { \bullet } = 3.4 \times 10 ^ { 6 } \ > { M _ { \odot } } ( with 1 \sigma and 3 \sigma error bars of 0.7 \times 10 ^ { 6 } \ > { M _ { \odot } } and 1.6 \times 10 ^ { 6 } \ > { M _ { \odot } } , respectively ) do provide an excellent fit .
The inclined models provide the best fit , but the inferred M _ { \bullet } does not depend sensitively on the assumed inclination .
The models that best fit the data are not two-integral models , but like two-integral models they are azimuthally anisotropic .
Two-integral models therefore provide useful low-order approximations to the dynamical structure of M32 .
We use them to show that an extended dark object can fit the data only if its half-mass radius is r _ { h } \lesssim 0.08 ^ { \prime \prime } ( = 0.26 \ > { pc } ) , implying a central dark matter density exceeding 1 \times 10 ^ { 8 } \ > { M _ { \odot } } \ > { pc } ^ { -3 } .

The inferred M _ { \bullet } is consistent with that suggested previously by ground-based kinematical data .
However , radially anisotropic axisymmetric constant mass-to-light ratio models are now ruled out for the first time , and the limit on the dark matter density implied by the HST data is now stringent enough to rule out most plausible alternatives to a massive black hole .
Thus , the evidence for a massive black hole in the quiescent galaxy M32 is now very compelling .

The dynamically inferred M _ { \bullet } is identical to that suggested by existing models for HST photometry of M32 that assume adiabatic growth ( over a time scale exceeding 10 ^ { 6 } \ > { yr } ) of a black hole into a pre-existing core .
The low activity of the nucleus of M32 implies either that only a very small fraction of the gas that is shed by evolving stars is accreted onto the black hole , or alternatively , that accretion proceeds at very low efficiency , e.g .
in an advection-dominated mode .