Modeling mass independent of anisotropy

By manipulating the spherical Jeans equation, Wolf et al. (2010) show that the mass enclosed within the 3D deprojected half-light radius r_1/2 can be determined with only mild assumptions about the spatial variation of the stellar velocity dispersion anisotropy as long as the projected velocity dispersion profile is fairly flat near the half-light radius, as is typically observed. They find M_1/2 = 3 G^-1 <σ²_los> r_1/2 ~= 4 G^-1 <σ²_los> R_e, where <σ²_los> is the luminosity-weighted square of the line-of-sight velocity dispersion and R_e is the 2D projected half-light radius. This finding can be used to show that all of the Milky Way dwarf spheroidal galaxies (MW dSphs) are consistent with having formed within a halo of mass approximately 3 × 10⁹ M_solar, assuming a ΛCDM cosmology. In addition, the dynamical I-band mass-to-light ratio ϒ^I_1/2 vs. M_1/2 relation for dispersion-supported galaxies follows a U-shape, with a broad minimum near ϒ^I_1/2 ~= 3 that spans dwarf elliptical galaxies to normal ellipticals, a steep rise to ϒ^I_1/2 ~= 3,200 for ultra-faint dSphs, and a more shallow rise to ϒ^I_1/2 ~= 800 for galaxy cluster spheroids.