Modelling 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, where 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 = 0 model; on the contrary, 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. Subject headings: galaxies: elliptical – stellar dynamics – dark matter – black holes