Accurate masses for dispersion-supported galaxies
  We derive an accurate mass estimator for dispersion-supported stellar systems
and demonstrate its validity by analyzing 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 dynamical 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. We find M_1/2 = 3 \sigma_los^2 r_1/2 / G ~ 4
\sigma_los^2 R_eff / G, where \sigma_los^2 is the luminosity-weighted square of
the line-of-sight velocity dispersion and R_eff 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 mass approximately 3 x 10^9 M_sun in Lambda
CDM cosmology. The faintest MW dSphs seem to have formed in dark matter halos
that are at least as massive as those of the brightest MW dSphs, despite the
almost five orders of magnitude spread in luminosity. We expand our analysis to
the full range of observed dispersion-supported stellar systems and examine
their I-band mass-to-light ratios (M/L). The M/L vs. M_1/2 relation for
dispersion-supported galaxies follows a U-shape, with a broad minimum near M/L
~ 3 that spans dwarf elliptical galaxies to normal ellipticals, a steep rise to
M/L ~ 3,200 for ultra-faint dSphs, and a more shallow rise to M/L ~ 800 for
galaxy cluster spheroids.
