Annu. Rev. Astron. Astrophys. 1979. 17: 135-87 Copyright © 1979 by Annual Reviews. All rights reserved

6.4 The Local Group

The seminal paper on Local Group dynamics was written by Kahn & Woltjer (1959). Lynden-Bell & Lin (1977) and Yahil, Tammann & Sandage (1977) have recently given thorough rediscussions of the problem, while de Vaucouleurs et al. (1977) have given a new solution for the solar motion relative to Local Group galaxies.

The dynamical analysis of the Local Group is deeply entwined with the determination of the velocity of the sun's motion about the galactic center. The Milky Way and M31 together dominate the kinetic and potential energies of the Local Group to the extent that the problem becomes in essence an ordinary two-body interaction. The velocity of M31 relative to the galaxy is therefore essential to a knowledge of their mutual orbit. By sheer bad luck, the apparent radial velocity of M31 with respect to the sun is in large measure simply a reflection of the sun's motion about the galactic center. We may take comfort in the fact that the coincidence is lessening with time: in 40 million years or so the two problems will be geometrically independent. For the present, however, we must struggle to disentangle the two motions.

The magnitude of the sun's orbital velocity is a matter of dispute, but the values widely discussed range between 220 km s^-1 (Section 2.2) and 300 km s^-1 (Lynden-Bell & Lin 1977, Yahil et al. 1977). Using these values to correct the radial velocity of M31 to the galactocentric value, we obtain -125 km s^-1 and -60 km s^-1, both negative. This is the key point: no matter how large a rotational velocity for the sun is assumed, within reasonable limits, we cannot convert the apparent approaching motion of M31 into one of recession. Barring a theoretically implausible ``slingshot'' effect in which M31 or the Milky Way caromed off some third galaxy in the past, the velocity of approach of the two galaxies must arise from their mutual gravitational interaction. Hence, there must have been time for the orbital motion to ``turn around'' during the lifetime of the universe, and this requirement in turn makes the galaxies considerably more massive than mere boundedness of the orbit would imply. We review this argument in some detail because, even though it was outlined quite clearly in Kahn & Woltjer's original paper (see also Peebles 1971), one still sees it ignored today in favor of estimates based on energy considerations alone. This omission seems hardly reasonable given the lack of any other convincing theory for the motion of approach of the two galaxies.

We assume that the orbital motion is radial and that the orbital time is 2 x 10¹⁰ yr. Using convenient formulae given by Gunn (1974), we calculate the total mass of M31 plus the Milky Way to be 2.9 x 10¹² M and 1.1 x 10¹² M for local circular velocities of 220 and 300 km s^-1, respectively. The luminosity of M31 is 2.7 x 10¹⁰ L on our system, and that of the Milky Way is 2.0 x 10¹⁰ L (Section 2.2). We then obtain Mass-to-light ratios of 60 and 25 respectively, values quite typical of small groups and binary galaxies. These values are lower limits because the assumption of radial motion yields the minimum possible mass. These results are quite consistent with the value of M / L_B 70 for the Milky Way alone found in Section 2.2.

We have so far neglected a second consideration. The orbital solution described above must also be consistent with the observed motion of the sun with respect to the center of the mass of the Local Group, which is assumed to be the center of mass of M31 and the Milky Way. Using this additional constraint, one obtains best-fit solutions of the solar orbital velocity close to 300 km s^-1 (Lynden-Bell & Lin 1977, Yahil et al. 1977), although 220 km s^-1 is still within the 90% probability contour. A more accurate value for the solar motion relative to Local Group galaxies would help greatly to narrow these possibilities, but it probably will never be forthcoming-there are simply too few Local Group members to serve as referents.