2010MNRAS.406.1220W

We derive an accurate mass estimator for dispersion-supported stellar systems and demonstrate its validity by analysing resolved line-of-sight velocity data for globular clusters, dwarf galaxies and elliptical galaxies. Specifically, by manipulating the spherical Jeans equation we show that the mass enclosed within the 3D deprojected half-light radius r_1/2can 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. We 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. While deceptively familiar in form, this formula is not the virial theorem, which cannot be used to determine accurate masses unless the radial profile of the total mass is known a priori. We utilize this finding to show that all of the Milky Way dwarf spheroidal galaxies (MW dSphs) are consistent with having formed within a halo of a mass of approximately 3x10⁹ M_☉, assuming a Λ cold dark matter cosmology. The faintest MW dSphs seem to have formed in dark matter haloes that are at least as massive as those of the brightest MW dSphs, despite the almost five orders of magnitude spread in luminosity between them. We expand our analysis to the full range of observed dispersion-supported stellar systems and examine their dynamical I-band mass-to-light ratios Υ^I_1/2. The Υ^I_1/2 versus 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≃ 3200 for ultra-faint dSphs and a more shallow rise to Υ^I_1/2 ≃ 800 for galaxy cluster spheroids.