Phase-space models of the dwarf spheroidals

This paper introduces new phase-space models of dwarf spheroidal galaxies (dSphs). The stellar component has an isotropic, lowered isothermal (or King) distribution function. A physical basis for the isotropization of stellar velocities is given by the theory of tidal stirring, whilst the isothermality of the distribution function guarantees the observed flatness of the velocity dispersion profile in the inner parts. For any analytic dark matter potential - whether of cusped or of cored form - the stellar density and velocity dispersion are analytic. The origin of the observational correlation between half-light radius R-h and line-of-sight central velocity dispersion Sigma(p, 0) is investigated. We prove that a power-law correlation R-h proportional to Sigma D-p,D-0 can exist if, and only if, the dark halo potential is a power law of the radius. Although a power law is a good approximation in the central parts (D = 2 for a Navarro-Frenk-White halo, D = 1 for cored haloes), the theoretical correlation curve between R-h and Sigma(p, 0) dramatically steepens at larger half-light radii. Using our phase-space models, we show that different dark halo profiles - whether cored or cusped - lead to very similar mass estimates within one particular radius, approximate to 1.7R(h). The formula for the enclosed mass M(< 1.7R(h)) is approximate to 5.8 Sigma 2(p,0)R(h)/G and extends out to larger radii than those in previous investigations. This is a tight result for models with a flattish projected velocity dispersion profile (out to several half-light radii). We show that deviations between mass measures due to different density profiles are substantially smaller than the uncertainties propagated by the observational errors on the half-light radius and central velocity dispersion. We produce a mass measure for each of the dSphs and find that the two most massive of the Milky Way dSphs are the most luminous, namely Sgr (M(< 1.7R(h)) similar to 2.8 x 108 M-circle dot) and Fornax (similar to 1.3 x 108 M-circle dot). The least massive of the Milky Way satellites are Willman 1 (similar to 4 x 105 M-circle dot) and Segue 1 (similar to 6 x 105 M-circle dot).