The Analytical Distribution Function of Anisotropic Hernquist+Hernquist Models

Abstract

The analytical phase-space distribution function (DF) of spherical self--consistent galaxy (or cluster) models, embedded in a dark matter halo, where both density distributions follow the Hernquist profile, with different total masses and core radii (hereafter called HH models), is presented. The concentration and the amount of the stellar and dark matter distributions are described by four parameters: the mass and core radius of the {\it reference} component, and two dimensionless parameters describing the mass and core radius of the {\it halo} component. A variable amount of orbital anisotropy is allowed in both components, following the widely used parameterization of Osipkov-Merritt. An important case is obtained for a null core radius of the halo, corresponding to the presence of a central black hole (BH). It is proved that globally isotropic HH models are consistent for any mass ratio and core radii ratio, even in the case of a central BH. In this last case the analytical expression for a lower limit of the anisotropy radius of the host system as a function of the BH mass is given. In the particular case of global isotropy the stability of HH models is proved, and the stability of radially anisotropic HH models is briefly discussed. The expression derived for the DF is useful for understanding the relations between anisotropy, density shape and external potential well in a consistent stellar system, and to produce initial conditions for N-body simulations of two-component galaxies or galaxy clusters.