Velocity profiles of galaxies with claimed black holes - II. f(E, Lz) models for M32

Abstract

The galaxy M32 has been claimed to contain a massive central black hole. A major uncertainty in the existing models for M32 is the absence of observational constraints on the dynamical structure and velocity dispersion anisotropy. Here we determine such constraints for the first time. We recently measured kinematical quantities and line-of-sight velocity profile shapes for M32 along five different slit positions (Paper I). We construct axisymmetric dynamical models with distribution functions of the form f (E, Lz) for these data. Such models have σR = σz, and are flattened by an excess of azimuthal motion. We explore two approaches, one based on a set of constant mass-to-light ratio 'power-law' models recently discussed by Evans, the other based on the moment equations of the collisionless Boltzmann equation. In the latter approach we take into account the central surface brightness cusp observed with the HST, and we include a central black hole. We compare the even and the odd parts of the observed and predicted velocity profiles separately, and derive independent information on the parts of the distribution function that are even and odd in Lz, respectively. Models with f(E, Lz) and no central black hole cannot fit the observed central peak in the rms line-of-sight velocity and the steep central rotation velocity gradient. A good fit to the data in the central arcsec is obtained when M32 is assumed to have a central black hole with mass MBH ≈ 1.8 × 106 MΘ. The major axis rotation velocity of M32 is ∼ 90 per cent of that of a maximally streaming f(E, Lz) model. Outside the central arcsec, most of the data are remarkably well fitted by the/(E, Lz) models, with two exceptions. First, the even parts of the observed major axis velocity profiles are slightly more flat-topped than is predicted by the models. Secondly, the models predict too much mean streaming on the intermediate axis (major ±45°), relative to the major axis. So, both the even and the odd parts of the distribution function of M32 must in fact depend on a third integral of motion. Our models indicate that M32 most likely has a velocity distribution with ν2φ > ν2θ ≳ ν2r outside the central arcsec. Our models are more realistic than most previous models in that they take proper account of flattening, rotation and velocity profile data. Yet the models still require the presence of a massive central black hole. To fit the M32 data without a black hole requires a radially anisotropic velocity distribution in the central region and a tangentially anisotropic velocity distribution in the outer region. The measured excess of azimuthal motion outside the central arcsec is not inconsistent with this picture. However, the required excess of radial motion in the central region may be implausible, given that the central two-body relaxation time in the absence of a central black hole is a factor ∼ 102 shorter than the Hubble time.