We present several different statistical methods to determine the transverse velocity vector of M31 .
The underlying assumptions are that the M31 satellites on average follow the motion of M31 through space , and that the galaxies in the outer parts of the Local Group on average follow the motion of the Local Group barycenter through space .
We apply the methods to the line-of-sight velocities of 17 M31 satellites , to the proper motions of the 2 satellites M33 and IC 10 , and to the line-of-sight velocities of 5 galaxies near the Local Group turn-around radius , respectively .
This yields 4 independent but mutually consistent determinations of the heliocentric M31 transverse velocities in the West and North directions , with weighted averages \langle v _ { W } \rangle = -78 \pm 41 \ > { km } { s } ^ { -1 } and \langle v _ { N } \rangle = -38 \pm 34 \ > { km } { s } ^ { -1 } .
The uncertainties correspond to proper motions of \sim 10 \ > \mu { as } { yr } ^ { -1 } , which is unlikely to be within reach of direct observational verification within the next decade .
The Galactocentric tangential velocity of M31 is 42 \ > { km } { s } ^ { -1 } , with 1 \sigma confidence interval V _ { tan } \leq 56 \ > { km } { s } ^ { -1 } .
The implied M31–Milky Way orbit is bound if the total Local Group mass M exceeds 1.72 ^ { +0.26 } _ { -0.25 } \times 10 ^ { 12 } \ > { M _ { \odot } } .
If the orbit is indeed bound , then the timing argument combined with the known age of the Universe implies that M = 5.58 ^ { +0.85 } _ { -0.72 } \times 10 ^ { 12 } \ > { M _ { \odot } } .
This is on the high end of the allowed mass range suggested by cosmologically motivated models for the individual structure and dynamics of M31 and the Milky Way , respectively .
It is therefore possible that the timing mass is an overestimate of the true mass , especially if one takes into account recent results from the Millennium Simulation that show that there is also a theoretical uncertainty of 41 % ( Gaussian dispersion ) in timing mass estimates .
The M31 transverse velocity implies that M33 is in a tightly bound orbit around M31 .
This may have led to some tidal deformation of M33 .
It will be worthwhile to search for observational evidence of this .