Two–component galaxy models: the effect of density profile at large radii on the phase–space consistency

It is well known that the density and anisotropy profile in the inner regions of a stellar system with positive phase-space distribution function are not fully independent. Here we study the interplay between density profile and orbital anisotropy at large radii in physically admissible (consistent) stellar systems. The analysis is carried out by using two-component (n-γ,γ1) spherical self-consistent galaxy models, in which one density distribution follows a generalized γ profile with external logarithmic slope n, and the other a standard γ1 profile (with external slope 4). The two density components have different “core” radii, the orbital anisotropy is controlled with the Osipkov-Merritt recipe, and for simplicity we assume that the mass of the γ1 component dominates the total potential everywhere. The necessary and sufficient conditions for phase-space consistency are determined analytically, also in presence of a dominant massive central black hole, and the analytical phase-space distribution function of (n-γ,1) models, and of n-γ models with a central black hole, is derived for γ = 0, 1, 2. It is found that the density slope in the external regions of a stellar system can play an important role in determining the amount of admissible anisotropy: in particular, for fixed density slopes in the central regions, systems with a steeper external density profile can support more radial anisotropy than externally flatter models. This is quantified by an inequality formally identical to the “cusp slope-central anisotropy” theorem (An & Evans 2006), relating at all radii (and not just at the center) the density logarithmic slope and the anisotropy indicator in all Osipkov-Merritt systems.