Kinematic deprojection and mass inversion of spherical systems of known velocity anisotropy

Traditionally, the degeneracy between the unknown radial profiles of total mass and velocity anisotropy 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. However, mass profiles are still not well known: there are discrepancies in the inner slopes of the density profiles of haloes found in dissipationless cosmological N-body simulations, and the inclusion of gas alters significantly the inner slopes of both the total mass and the dark matter component. 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 (itself a single integral of observable quantities). Then, through the Jeans equation, the mass profile of a spherical system 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 Navarro, Frenk and White (NFW) models with the first four of these anisotropy models yield accurate results in the case of perfect observational data, and typically better than 70 per cent (in four cases out of five) 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 analyse the mass and anisotropy profiles of spherical systems. It provides the practical means to lift the mass-anisotropy degeneracy 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.