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.
