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 r1 / 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 M1 / 2 = 3G −1 _σ2 lo s _ r1 / 2 _ 4G −1 _σ2 lo s _Re, where _σ2 lo s _ is the luminosityweighted square of the line-of-sight velocity dispersion and Re 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 3×109M_, assuming a ΛCDM cosmology. In addition, the dynamical I-band mass-to-light ratio ΥI 1 / 2 vs. M1 / 2 relation for dispersion-supported galaxies follows a U-shape, with a broad minimum near ΥI 1 / 2 _ 3 that spans dwarf elliptical galaxies to normal ellipticals, a steep rise to ΥI 1 / 2 _ 3,200 for ultra-faint dSphs, and a more shallow rise to ΥI 1 / 2 _ 800 for galaxy cluster spheroids.