The Analytical Distribution Function of Anisotropic Two‐Component Hernquist Models

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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 radius ("hereafter 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 reference component and two dimensionless parameters that describe the mass and core radius of the halo component. A variable amount of orbital anisotropy is allowed in both components, following the widely used Osipkov-Merritt parameterization. An important case is obtained for a null core radius of the halo, corresponding to the presence of a central black hole (BH). Before giving the explicit form for the DF, the necessary and sufficient conditions that the model parameters must satisfy in order to correspond to a consistent system (i.e., a system for which each physically distinct component has a positive DF) are analytically derived. In this context, 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. These results are then compared with those obtained by direct inspection of the DF. In the particular case of global isotropy, the stability of HH models is proved, and the explicit formula for the differential energy distribution is derived. Finally, 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.
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