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 } \lesssim M _ { Sgr } \lesssim 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 _ { Sgr } \lesssim 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 .