Two-component galaxy models with a central BH -- II. The ellipsoidal case

TL;DR

This paper extends spherical two-component galaxy models with a central black hole to ellipsoidal axisymmetric forms, deriving analytical conditions and solutions for density and velocity fields, useful for theoretical and simulation studies.

Contribution

It introduces ellipsoidal JJe and J3e galaxy models with analytical density and velocity solutions, including the effects of flattening and a central black hole.

Findings

Derived conditions for positive dark matter density

Analytical solutions for stellar velocity dispersions

Formulas for velocity fields near galaxy centers and at large radii

Abstract

Recently, two-component spherical galaxy models have been presented, where the stellar profile is described by a Jaffe law, and the total density by another Jaffe law, or by an $r^{-3}$ law at large radii. We extend these two families to their ellipsoidal axisymmetric counterparts: the JJe and J3e models. The total and stellar density distributions can have different flattenings and scale lengths, and the dark matter halo is defined by difference. First, the analytical conditions required to have a nowhere negative dark matter halo density are derived. The Jeans equations for the stellar component are then solved analytically, in the limit of small flattenings, also in presence of a central BH. The azimuthal velocity dispersion anisotropy is described by the Satoh $k$-decomposition. Finally, we present the analytical formulae for velocity fields near the center and at large radii, together with the various terms entering the Virial Theorem. The JJe and J3e models can be useful in a number of theoretical applications, e.g. to explore the role of the various parameters (flattening, relative scale lengths, mass ratios, rotational support) in determining the behavior of the stellar kinematical fields before performing more time-expensive integrations with specific galaxy models, to test codes of stellar dynamics, and in numerical simulations of gas flows in galaxies.