Two-component Jaffe models with a central black hole. I: the spherical case

TL;DR

This paper develops analytical models of spherical galaxies with Jaffe profiles for stars and dark matter, including a central black hole, deriving conditions for physical consistency and key dynamical properties.

Contribution

It introduces a new class of analytical spherical galaxy models with dual Jaffe profiles and a central black hole, extending previous models and providing explicit formulas for dynamics.

Findings

Analytical phase-space distribution functions derived using Lambert-Euler W function.

Conditions for positive, decreasing dark matter density profiles established.

Analytical solutions for Jeans equations and velocity dispersions obtained.

Abstract

Dynamical properties of spherically symmetric galaxy models where both the stellar and total mass density distributions are described by the Jaffe (1983) profile (with different scale-lenghts and masses), are presented. The orbital structure of the stellar component is described by Osipkov--Merritt anisotropy, and a black hole (BH) is added at the center of the galaxy; the dark matter halo is isotropic. First, the conditions required to have a nowhere negative and monothonically decreasing dark matter halo density profile, are derived. We then show that the phase-space distribution function can be recovered by using the Lambert-Euler $W$ function, while in absence of the central BH only elementary functions appears in the integrand of the inversion formula. The minimum value of the anisotropy radius for consistency is derived in terms of the galaxy parameters. The Jeans equations for the stellar component are solved analytically, and the projected velocity dispersion at the center and at large radii are also obtained analytically for generic values of the anisotropy radius. Finally, the relevant global quantities entering the Virial Theorem are computed analytically, and the fiducial anisotropy limit required to prevent the onset of Radial Orbit Instability is determined as a function of the galaxy parameters. The presented models, even though highly idealized, represent a substantial generalization of the models presentd in Ciotti et al. (2009), and can be useful as starting point for more advanced modeling the dynamics and the mass distribution of elliptical galaxies.