An Eccentricity-Mass Relation for Galaxies from Tidally Disrupting Satellites

We infer the past orbit of the Sagittarius (Sgr) dwarf galaxy in the Milky Way halo by integrating backwards from its observed position and proper motions, including the effects of dynamical friction. Given measured proper motions, we show that there is a relation between the eccentricity ($e$) of Sgr's orbit and the mass of the Milky Way ($M_{T}$) in the limit of no dynamical friction. That relation can be fit by a power-law of the form: $e \approx 0.49 \left(M_{T}/10^{12} M_{\odot}\right)^{-0.88}$. At a fixed Milky Way mass, the dynamical friction term increases the mean eccentricity of the orbit and lowers the spread in eccentricities in proportion to the mass of the Sgr dwarf. We explore the implications of various observational constraints on Sgr's apocenter on the $e-M$ relation; Sgr masses outside the range $10^{9} M_{\odot} \la M_{\rm Sgr} \la 5 \times 10^{10} M_{\odot}$ are precluded, for Milky Way masses $\sim 1 - 2.5 \times 10^{12} M_{\odot}$. If Belokurov et al.'s (2014) observations represent the farthest point of Sgr's stream, then Milky Way masses in excess of $2 \times 10^{12} M_{\odot}$ are excluded for $M_{\rm Sgr} \la 10^{10} M_{\odot}$. Deeper observations of Sgr's tidal debris, from upcoming surveys such as GAIA, will allow better measurement of the Milky Way mass and of the Sgr dwarf.