It is tempting to observe binary galaxies to obtain galactic masses. Typical periods are, however, so large that orbits cannot be observed. On the other hand, typical distances between the two galaxies are much larger than visible galactic sizes, and therefore we could, in principle, obtain total masses.

The observations permit the obtention of "projected
distances", *r*, and of "differences in the velocity
component along the line-of-sight", *v*. From both series of
data we must infer a mean *M*/*L* ratio. The analysis must be
statistical as no parameter of the individual orbits is known.

The first problem to resolve, and a very serious one, is the selection
of a suitable
sample. Chance superpositions must be avoided: including in the sample
two unbounded galaxies, for which the velocity difference is due to
Hubble's flow, could give a very high *v* and hence an
overestimation of the mean mass. If the pair is not isolated, the
influence of a third galaxy could produce a misinterpretation of the
results; galaxies are often in small or large clusters and are
rarely found in truly isolated pairs.

Usually, only pairs with a projected separation of less than a given value,
*R*, are accepted in the sample.
Binney and Tremaine (1987)
have warned of this danger. Suppose that we take
*R* as smaller than the mean
true pair distance. Then, the velocity should be mainly perpendicular
to the line-of-sight, and therefore much greater than the projected
velocity along the line of sight; then *v* would be
underestimated, as would the galactic masses.

Limit values of *r* and of *v* become necessary, but
then, we find the results that we expect. Indeed,
Sharp (1990)
after comparing the large discrepancies between different workers, even with
similar samples, was very pessimistic about the ability of these
statistical approaches to derive galactic masses.

To interpret statistical distributions of *r* and *v*
in order to obtain *M*, the mass of the two galaxies, it is necessary
to adopt a law for the distribution of the true separation *r*; for
example,
(*r*) *r*^{- }, deduced with the two-point
correlation function of galaxies
(Peebles, 1974).
Observations indicate that
1.8. This distribution might not be valid
for close binary systems. It is also necessary to adopt a hypothesis
about the orbits, and more precisely about the value of the parameter
, defined as

(63) |

where *v*_{} and *v*_{r} are the
azimuthal and radial components of
the velocity. If the orbits are radial,
< *v*_{}^{2} > = 0 and we
should take = 1. If the orbits are perfectly circular, then
< *v*_{r}^{2} > = 0 and
= - . An interesting intermediate
assumption is the condition of isotropy
< *v*_{r}^{2} > = < *v*_{}^{2} >,
hence = 0.

A natural way to study binary samples is the adoption of galaxies as
mass points. In a classical analysis by
White et al. (1983),
however, it was demonstrated that the mass point model does not
fit the data. This model predicts a correlation between *v* and
*r* (clearly
(*v*)^{2} should correlate with
(*r*)^{-1}), which is not found. This
negative result is highly
interesting, as it can be interpreted as being due to the existence of
greatly extended halos. If
the force, instead of - *GM*/*r*^{2}, were of the type
*r*^{-1}, the
correlation
[(*v*)^{2} (*r*)^{-1}] would not
exist.

This classical paper also claimed other evidence favouring the
existence of massive extended halos.
They found dark-to-luminous mass ratios higher than those found with
rotation curves. This is really to be expected, because mass
determinations from rotation curves are made at a maximum radius lower
than the rim of the halo, while a companion could be far away, at a
distance greater than the sum of both halos. Indeed their results are
compatible with extrapolations of observed rotation curves. These
authors found a relation of the type
*M*/*L* *L*^{-3/4}. Therefore, low luminosity
galaxies should contain large amounts of dark matter.
Lake and Schommer (1984)
confirmed very high
*M*/*L* values in a sample of dwarf irregular pairs.

However, Karachentsev (1983, 1985) found no evidence of dark matter in binary systems, using very large samples, even containing some galaxies with well observed flat rotation curves.

Honma (1999) found M/L for spiral pairs in the range 12-16, lower than M/L for ellipticals, confirming previous results by Schweizer (1987). These values are clearly lower than those previously reported.

Among the large list of workers who have attempted to obtain proof of dark matter with this method, noticeable are the studies by van Moorsel (1987), Charlton and Salpeter (1991) and others, favouring the scenario of a large common dark matter envelope, as we have seen in the pair formed by M31 and the Milky Way. From the cosmological point of view, whether dark matter lies in individual or in common halos is unimportant, but from the point of view of galactic structure and evolution, the two models are completely different.

As stated by
Binney and Tremaine (1987)
"the mass-to-light ratio of
binary galaxies is probably large, but not so large as the ratio of
the mass of papers on this subject to the light they have shed on
it". Even the concept of binary systems is controversial: with
typical velocities of 100 *kms*^{-1} and separations of 100
kpc, a
typical value for the orbital period is of the order of Hubble's
time. In most cases, such as in the M31-Milky Way pair, a simple orbit
has not been completed. The Universe is expanding with a typical time of
the orbital period. Nevertheless, despite the large variety of
results, it should be emphasized that the most widely
accepted point of view is that binary galaxies possess large amounts of
dark matter, either in individual halos or in common super halos.