Modelling galaxies with Formula: a black hole in M32

A technique for the construction of axisymmetric distribution functions for individual galaxies is presented. It starts from the observed surface brightness distribution, which is deprojected to obtain the axisymmetric luminosity density, from which follows the stars' gravitational potential. After adding dark mass components, such as a central black hole, the two-integral distribution function $$(2I-DF) f(E,L_z)$$, which depends only on the classical integrals of motion in an axisymmetric potential, is constructed using the Richardson-Lucy algorithm. This algorithm proves to be very efficient in finding $$f(E,L_z)$$ provided that the integral equation to be solved has been properly modified. Once the 2I-DF is constructed, its kinematic can be computed and compared with those observed. Many discrepancies may be remedied by altering the assumed inclination angle, mass-to-light ratio, dark components, and odd part of the 2I-DF. Remaining discrepancies may indicate that the distribution function depends on the non-classical third integral, or is non-axisymmetric. The method has been applied to the nearby elliptical galaxy M32. A 2I-DF with ∼55° inclination and a central black hole (or other compact dark mass inside ∼1 pc) of $$1.6-2\times10^6\text M_\odot$$ fits the high-spatial-resolution kinematic data of van der Marel et al. remarkably well. 2I-DFs with a significantly less or more massive central dark mass or with edge-on orientation can be ruled out for M32. Based on this result, it is argued that even axisymmetric three-integral models cannot mannage without any central dark mass. Predictions are made for observations with the HST: spectroscopy using its smallest square aperture of 0.09×0.09 arcsec² should yield a non-Gaussian central velocity profile with broad wings, and true and Gaussian-fit velocity dispersions of $$150-170 \text {km s}^{-1}$$ and $$120-130 \text {km s}^{-1}$$, respectively.