Annu. Rev. Astron. Astrophys. 1979. 17:
135-87
Copyright © 1979 by . 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 10^{10} yr. Using convenient formulae given by
Gunn (1974), we
calculate the total mass of M31 plus the Milky Way to be 2.9 x 10^{12} *M*_{}
and 1.1 x 10^{12}
*M*_{} for local
circular velocities of 220 and 300 km s^{-1},
respectively. The luminosity of M31 is 2.7 x 10^{10} *L*_{} on our system, and
that of the Milky Way is 2.0 x 10^{10} *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.