Constraints on the Proper Motion of the Andromeda Galaxy Based on the Survival of Its Satellite M33

A major uncertainty in the dynamical history of the Local Group of galaxies originates from the unknown transverse speed of the Andromeda galaxy (M31) relative to the Milky Way. We show that the recent VLBA measurement of the proper motion of Andromeda's satellite, M33, severely constrains the possible values of M31's proper motion. The condition that M33's stellar disk has not been tidally disrupted by either M31 or the Milky Way over the past 10 billion years favors a proper motion amplitude of 100+/-20 km s<SUP>-1</SUP> for M31 with the quadrant of a negative velocity component along right ascension and a positive component along declination strongly ruled out. This inference can be tested by future astrometric measurements with SIM, Gaia, or the SKA. Our results imply that the dark halos of Andromeda and the Milky Way will pass through each other within the next 5-10 billion years