The Analytical Distribution Function of Anisotropic Two‐Component Hernquist Models

The analytical phase-space distribution function (DF) of sphericalself--consistent galaxy (or cluster) models, embedded in a dark matter halo,where both density distributions follow the Hernquist profile, with differenttotal masses and core radii (hereafter called HH models), is presented. Theconcentration and the amount of the stellar and dark matter distributions aredescribed by four parameters: the mass and core radius of the {\it reference}component, and two dimensionless parameters describing the mass and core radiusof the {\it halo} component. A variable amount of orbital anisotropy is allowedin both components, following the widely used parameterization ofOsipkov-Merritt. An important case is obtained for a null core radius of thehalo, corresponding to the presence of a central black hole (BH). It is provedthat globally isotropic HH models are consistent for any mass ratio and coreradii ratio, even in the case of a central BH. In this last case the analyticalexpression for a lower limit of the anisotropy radius of the host system as afunction of the BH mass is given. In the particular case of global isotropy thestability of HH models is proved, and the stability of radially anisotropic HHmodels is briefly discussed. The expression derived for the DF is useful forunderstanding the relations between anisotropy, density shape and externalpotential well in a consistent stellar system, and to produce initialconditions for N-body simulations of two-component galaxies or galaxy clusters.Comment: 26 pages, Latex, plus 6 .ps figures and aaspp4.sty, uuencoded, gzipp'ed tar file -- accepted on Ap