One immediate result of eqs. (1)(2) is that at a
large radius around a mass *M*, the orbital speed on a circular
orbit becomes independent of radius. This indeed was a guiding
principle in the construction of MOND, which took asymptotic
flatness of galaxy rotation curves as an axiom (even though at the
time it was not clear how definite, and how universal, this is).
Second, this asymptotic rotational speed depends only on the total
mass *M* via *V*^{4} = *MGa*_{0}. This,
according to MOND, is the fact
underlying the observed Tully-Fisher-type relations, by which the
typical (mean) rotational velocity, *V*, in a disc galaxy is
strongly correlated with the total luminosity of the galaxy, *L*,
in a relation of the form
*L*
*V*^{}.
The power
is around 3-4,
and depends on the wavelength band at which *L* is measured.
The close agreement between this TF relation and the prediction
of MOND is encouraging; but,
to test MOND more precisely on this count, one would have
to bridge properly the mass-asymptotic-velocity MOND relation
with the commonly presented luminosity-bulk-velocity TF relation.
One should use the luminosity in a band
where it is a good representative of the stellar mass, take into
account not only the stellar mass, as represented by the
luminosity, but also the contribution of gas to the mass, and use
the asymptotic velocity, as opposed to other measures of the
rotational velocity. It has emerged recently (see
[Verheijen 2001] and
reference therein) that if one does all this one indeed obtains a
tight and accurate relation of the form predicted by MOND.

But, by far, the most clear-cut test of MOND is provided by
disc-galaxy rotation curves, simply because the astronomical
observations, and their interpretation, are the most complete and
best understood, if still not perfect. What we typically need to
know of a galaxy in order to apply this test has been discussed by
the various authors who conducted the test; for example,
[Begeman & al
(1991)],
[Sanders &
Verheijen (1998)],
[Sanders 1996],
[de Blok & McGaugh
(1998)].
On the whole, these tests speak cogently for MOND. These test involves
fitting the observed rotation curve of a galaxy by that predicted
by MOND. Such fits involve one free parameter per galaxy, which is
the assumed conversion factor from luminosity to mass in stars,
the so-called mass-to-light ratio. In fact, however, this
parameter is not totally free. It is constrained to an extent by
what theoretical understanding of galaxy composition tell us.
[Sanders &
Verheijen (1998)]
who have conducted a MOND rotation-curve analysis of
a sample of disc galaxies in the Ursa Major cluster, have compared
their deduced MOND best-fit *M*/*L* values with theoretical
results from stellar-population synthesis. They found a very good
agreement. This shows that, to some extent, the MOND rotation
curves might be looked at as definite prediction of MOND, which
use theoretical *M*/*L* values, and not as fits involving one free
parameter.

Regarding galactic systems other than galaxies, the comparison of
the systematics of the observed mass discrepancy with the
expectations from MOND are shown in Figure 2 in
[Milgrom (1998)]
based on analyzes referenced there. The agreement is uniform, with
one exception: The cores of rich x-ray clusters of galaxies show a
considerable mass discrepancy, while, according to MOND there
shouldn't be any, because the accelerations there are only of the
order of *a*_{0},
and not much smaller. (Application of MOND to the
clusters at large, say within a few megaparsecs of the center,
does predict correctly the mass discrepancy.) The resolution, by
MOND, will have to be that these cores harbor large quantities of
still undetected baryonic matter, perhaps in the form of dim
stars, perhaps as warm gas. The environment, and history, of these
cores is so unlike others that this would not be surprising.

In order to appreciate the message that the phenomenological
success of MOND carries, we should note the following. According
to MOND, the acceleration constant
*a*_{0} appears in
many independent
roles in the phenomenology of the mass discrepancy. For example,
in galaxies that have high central accelerations, the mass
discrepancy appears only beyond a certain radius; according to
MOND , the acceleration at this radius should always be
*a*_{0}.
*a*_{0}
also appears as the boundary acceleration between so called
high-surface-brightness galaxies (= high acceleration galaxies)
which do not show a mass discrepancy near the center, and
low-surface-brightness galaxies, where the discrepancy prevails
everywhere.
*a*_{0} appears in
the relation between the asymptotic
rotational velocity of a galaxy and its total mass, and in the
mass-velocity relation for all
sub-*a*_{0}
systems, etc., etc..

These roles that
*a*_{0} plays are
independent in the sense that in
the framework of the dark matter paradigm, it is easy to envisage
baryon-plus-dark-matter galactic systems that evince any of these
appearances of 0 without showing the others. In other words, in
the dark matter paradigm one role of
*a*_{0} in the
phenomenology does not follow from the others.

This is similar, for example, to the appearances of the Planck
constant in different quantum phenomena: in the black-body
spectrum, in the photoelectric effect, in the hydrogen spectrum,
in superconductivity, etc.. Phenomenologically, these roles of the
same constant seem totally unrelated. The only unifying frame is a
theory: non-relativistic quantum mechanics, in this case. MOND is,
likewise, a theory that in one fell swoop unifies all the above
appearances of
*a*_{0} in the
phenomenology of galaxy dynamics.

And finally, let me point out a possibly very significant
coincidence: The value of the acceleration constant
*a*_{0} that fits
all the data discussed above is about
10^{-8}*cm* *s*^{-2}. This
value of *a*_{0} is
of the order of some acceleration constants of
cosmological significance. It is of the same order as
*a*_{ex}
*cH*_{0}, where *H*_{0} is the Hubble
constant; and, it is also of the order of
*a*_{cc}
*c*( /
3)^{1/2},
where is the
emerging value of the cosmological
constant (or "dark energy"). So, for example, a body accelerating
at *a*_{0} from
rest will approach the speed of light in the life time of the universe.

Because the cosmological state of the universe changes, such a
connection, if it is a lasting one, may imply that galaxy
evolution does not occur in isolation, affected only by nearby
objects, but is, in fact, responding constantly to changes in the
state of the universe at large. For example, if the connection of
*a*_{0} with the Hubble constant always holds, the changing
of the
Hubble constant would imply that *a*_{0} must change over
cosmic
times, and with it the appearance of galactic systems, whose
dynamics *a*_{0} controls. If, on the other hand,
*a*_{0} is a
reflection of a true cosmological constant, then is might be a
veritable constant.