Analytical Galactic Models with Mild Central Cusps

We present a new class of spherical galactic models with mild cusps. The mass density goes like ρ(r) ∼ r^α-2, 0< α ≤ 2 for small r and falls off like ρ(r) ∼ r^-2-α(γ+1), γ > 0, ?γα ≤ ? 1, for large r and hence covers a large range of observed density distributions. We consider isotropic and anisotropic (Osipkov-Merritt type) models. The distribution functions (DFs) and intrinsic velocity dispersions (IVs) can be represented analytically in a unified way in terms of hypergeometric functions for a large number of parameters. This allows an easy comparison of the DFs and IVs of models having varying degrees of central cuspiness or outer density falloff. For some values of the parameters, the surface brightness and projected velocity dispersion (PV) can be also reduced to analytic expressions. In particular, we study the models for the innermost regions of galaxies harbouring mild cuspy centers with or without supermassive black holes (BH). It is shown that the anisotropy affects the DFs only outside the central parts for which they do not fall off as steeply with decreasing Q=E-L²/(2r_a ²) (r_a the anisotropy radius) as for the isotropic models, whereas the increase for large E or Q is dominated by the cusp parameter ? for both models. Moreover, the IVs decrease more rapidly for less anisotropic models and the IVs and PV run flatter for decreasing cuspiness. For fixed ?, the presence of a central BH causes the velocity dispersions to rise steeply for small radii for increasing BH mass which also lead to higher velocity dispersions over a wider radial range. On the other hand, the DFs are lower for increasing BH mass with fixed α or increasing α with fixed BH mass respectively. So the models are able to reproduce important properties of galaxies which have been observed over the last years and can be used to model elliptical galaxies and the central regions of spiral galaxies.