Kinematic deprojection and mass inversion of spherical systems of known
  velocity anisotropy
  Traditionally, the mass / velocity anisotropy degeneracy (MAD) inherent in
the spherical, stationary, non-streaming Jeans equation has been handled by
assuming a mass profile and fitting models to the observed kinematical data.
Here, the opposite approach is considered: the equation of anisotropic
kinematic projection is inverted for known arbitrary anisotropy to yield the
space radial velocity dispersion profile in terms of an integral involving the
radial profiles of anisotropy and isotropic dynamical pressure. Then, through
the Jeans equation, the mass profile is derived in terms of double integrals of
observable quantities. Single integral formulas for both deprojection and mass
inversion are provided for several simple anisotropy models (isotropic, radial,
circular, general constant, Osipkov-Merritt, Mamon-Lokas and
Diemand-Moore-Stadel). Tests of the mass inversion on NFW models with these
anisotropy models yield accurate results in the case of perfect observational
data, and typically better than 70% (in 4 cases out of 5) accurate mass
profiles for the sampling errors expected from current observational data on
clusters of galaxies. For the NFW model with mildly increasing radial
anisotropy, the mass is found to be insensitive to the adopted anisotropy
profile at 7 scale radii and to the adopted anisotropy radius at 3 scale radii.
This anisotropic mass inversion method is a useful complementary tool to
analyze the mass and anisotropy profiles of spherical systems. It provides the
practical means to lift the MAD in quasi-spherical systems such as globular
clusters, round dwarf spheroidal and elliptical galaxies, as well as groups and
clusters of galaxies, when the anisotropy of the tracer is expected to be
linearly related to the slope of its density.
