Modeling elliptical galaxies: Phase-space constraints on two-component (γ1, γ2) models

In the context of the study of the properties of the mutual mass distribution of the bright and dark matter in elliptical galaxies, I present a family of two-component, spherical, self-consistent galaxy models, in which one density distribution follows a γ1 profile and the other a γ2 profile [hereafter (γ1, γ2) models], with different total masses and "core" radii. A variable amount of (radial) orbital anisotropy is allowed in both components, following the Osipkov-Merritt parameterization. For these models, I derive analytically the necessary and sufficient conditions that the model parameters must satisfy in order to correspond to a physical system (the so-called model consistency). Moreover, the possibility of adding a black hole at the center of radially anisotropic γ models is discussed, determining analytically a lower limit of the anisotropy radius as a function of γ. The analytical phase-space distribution function for (1, 0) models is presented, together with the solution of the Jeans equations and the quantities entering the scalar virial theorem. It is proved that a globally isotropic γ = 1 component is consistent for any mass and core radius of the superimposed γ = 1 model ; on the other hand, only a maximum value of the core radius is allowed for the γ = 0 model when a γ = 1 density distribution is added. The combined effects of mass concentration and orbital anisotropy are investigated, and an interesting behavior of the distribution function of the anisotropic γ = 0 component is found: there exists a region in the parameter space where a sufficient amount of anisotropy results in a consistent model, while the structurally identical but isotropic model would be inconsistent.