Axisymmetric galaxy models with central black holes, with an application to M32

Abstract

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 strati ed 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 nite{mass systems. The range of axis ratios and density pro le 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`( ; ){ models', characterized by a power{law density cusp with slope at small radii, and a power{law density fall{o with slope + 2 at large radii. As an application, the DF is constructed for the ( ; ) model with a 1:8 10 6 M black hole used by van der Marel et al. to interpret their high spatial resolution spectroscopic data for M32. The line-of-sight velocity pro les are calculated. The results con rm that the model ts 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 km s 1 with the 0:09 00 0:09 00 square aperture of the Faint Object Spectrograph, and 105 km s 1 with the 0:26 00 diameter circular aperture, while the central dispersion measured from ground{based data is only 86 km s 1 .

The old nuclear star cluster in the Milky Way: dynamics, mass, statistical parallax, and black hole mass

“…Subsequently, relaxation would lead to shrinking of the core by a factor of ∼ 2 in 10 Gyr towards a value similar to that observed (Merritt 2010). In the simulations of Antonini et al (2012), the final relaxed model has an inner slope of γ = 0.45, not far from our models (note that in flattened semi-isotropic models the minimum allowed slope for the density is also 0.5 (Qian et al 1995)). Their cluster also evolved towards a more spherical shape, however, starting from a configuration with much less rotation and flattening than we inferred here for the present Milky Way NSC.…”

“…For similar reasons, we also fix the value of γ2 to that used in the spherical case. The minimum value of γ1 for a semi-isotropic axisymmetric model with a black hole cannot be smaller than 0.5 (Qian et al 1995), as in the spherical case. For our current modeling we treat γ1 as a free parameter.…”

“…For the odd part of the DF we choose the two-parameter function from Qian et al (1995). This is a modified version of Dejonghe (1986) which was based on maximum entropy arguments:…”

“…Assuming constant mass-to-light ratio for the star cluster, we determine its potential using the relation from Qian et al (1995), which is compatible with their contour integral method (i.e. it can be used for complex R 2 and z 2 ).…”

“…We use the contour integral method of Hunter & Qian (1993, HQ) and Qian et al (1995). A 2I DF is the logical, next-simplest generalization of isotropic spherical models.…”