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 V4 = MGa0. 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 a0, 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 a0 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 a0. a0 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. a0 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-a0 systems, etc., etc..
These roles that a0 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 a0 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 a0 in the phenomenology of galaxy dynamics.
And finally, let me point out a possibly very significant coincidence: The value of the acceleration constant a0 that fits all the data discussed above is about 10-8cm s-2. This value of a0 is of the order of some acceleration constants of cosmological significance. It is of the same order as aex cH0, where H0 is the Hubble constant; and, it is also of the order of acc c( / 3)1/2, where is the emerging value of the cosmological constant (or "dark energy"). So, for example, a body accelerating at a0 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 a0 with the Hubble constant always holds, the changing of the Hubble constant would imply that a0 must change over cosmic times, and with it the appearance of galactic systems, whose dynamics a0 controls. If, on the other hand, a0 is a reflection of a true cosmological constant, then is might be a veritable constant.