Proper motion of Milky Way dwarf spheroidals from line-of-sight velocities

Proper motions for several Milky Way dwarf spheroidal (dSph) galaxies have been determined using both ground- and space-based imaging. These measurements require long baselines and repeat observations, and typical errors are of order 10 mas century(-1). In this Letter, we utilize the effect of "perspective rotation" to show that the systematic proper motion of some dSphs can be determined to a similar precision using only stellar line-of-sight velocities. We show that including the effects of small intrinsic rotation in dSphs increases the proper motion errors by about a factor of 2. We provide error projections for future data sets and show that proposed 30 m class telescopes will measure the proper motion of a few dSphs with mas century(-1) precision.