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