- Asymptotic flatness of rotation curves. The rotation curves of galaxies are asymptotically flat, even though this flatness is not always attained at the last observed point (see point hereafter about the shapes of rotation curves as a function of baryonic surface density). What is more, Milgrom’s law can be thought of as including the total acceleration with respect to a preferred frame, which can lead to the prediction of asymptotically-falling rotation curves for a galaxy embedded in a large external gravitational field (see Section 6.3).
- defining the zero-point of the baryonic Tully–Fisher relation. The plateau of a rotation curve is . The true Tully–Fisher relation is predicted to be a relation between this asymptotic velocity and baryonic mass, not luminosity. Milgrom’s law yields immediately the slope (precisely 4) and zero-point of this baryonic Tully–Fisher law. The observational baryonic Tully–Fisher relation should thus be consistent with zero scatter around this prediction of Milgrom’s law (the dotted line of Figure 3). And indeed it is. All rotationally-supported systems in the weak acceleration limit should fall on this relation, irrespective of their formation mechanism and history, meaning that completely isolated galaxies or tidal dwarf galaxies formed in interaction events all behave as every other galaxy in this respect.
- defining the zero-point of the Faber–Jackson relation. For quasi-isothermal systems [296], such as elliptical galaxies, the bulk velocity dispersion depends only on the total baryonic mass via . Indeed, since the equation of hydrostatic equilibrium for an isotropic isothermal system in the weak field regime reads , one has where . This underlies the Faber–Jackson relation for elliptical galaxies (Figure 7), which is, however, not predicted by Milgrom’s law to be as tight and precise (because it relies, e.g., on isothermality and on the slope of the density distribution) as the BTFR.
- Mass discrepancy defined by the inverse of the acceleration in units of . Or alternatively, defined by the inverse of the square-root of the gravitational acceleration generated by the baryons in units of . The mass discrepancy is precisely equal to this in the very–low-acceleration regime, and leads to the baryonic Tully–Fisher relation. In the low-acceleration limit, , so in the CDM language, inside the virial radius of any system whose virial radius is in the weak acceleration regime (well below ), the baryon fraction is given by the acceleration in units of . If we adopt a rough relation , we get that the acceleration at , and thus the system baryon fraction predicted by Milgrom’s formula, is . Divided by the cosmological baryon fraction, this explains the trend for with potential () in Figure 2, thereby naturally explaining the halo-by-halo missing baryon challenge in galaxies. No baryons are actually missing; rather, we infer their existence because the natural scaling between mass and circular velocity in CDM differs by a factor of from the observed scaling .
- as the characteristic acceleration at the effective radius of isothermal spheres. As a corollary to the Faber–Jackson relation for isothermal spheres, let us note that the baryonic isothermal sphere would not require any dark matter up to the point where the internal gravity falls below , and would thus resemble a purely baryonic Newtonian isothermal sphere up to that point. But at larger distances, in the presence of the added force due to Milgrom’s law, the baryonic isothermal sphere would fall [296] as , thereby making the radius at which the gravitational acceleration is the effective baryonic radius of the system, thereby explaining why, at this radius in quasi-isothermal systems, the typical acceleration is almost always observed to be on the order of . Of course, this is valid for systems where such a transition radius does exist, but going to very-LSB systems, if the internal gravity is everywhere below , one can then have typical accelerations as low as one wishes.
- as a critical mean surface density for stability. Disks with mean surface density have added stability. Most of the disk is then in the weak-acceleration regime, where accelerations scale as , instead of . Thus, instead of , leading to a weaker response to small mass perturbations [299]. This explains the Freeman limit (Figure 8).
- as a transition acceleration. The mass discrepancy in galaxies always appears (transition from baryon dominance to dark matter dominance) when , yielding a clear mass-discrepancy acceleration relation (Figure 10). This, again, is the case for every single rotationally-supported system irrespective of its formation mechanism and history. For HSB galaxies, where there exist two distinct regions where in the inner parts and in the outer parts, locally measured mass-to-light ratios should show no indication of hidden mass in the inner parts, but rise beyond the radius where (Figure 14). Note that this is the only role of that the scenario of [218] was poorly trying to address (forgetting, e.g., about the existence of LSB galaxies).
- as a transition central surface density. The acceleration defines the transition from HSB galaxies to LSB galaxies: baryons dominate in the inner parts of galaxies whose central surface density is higher than some critical value on the order of , while in galaxies whose central surface density is much smaller (LSB galaxies), DM dominates everywhere, and the magnitude of the mass discrepancy is given by the inverse of the acceleration in units of ; see (5). Thus, the mass discrepancy appears at smaller radii and is more severe in galaxies of lower baryonic surface densities (Figure 14). The shapes of rotation curves are predicted to depend on surface density: HSB galaxies are predicted to have rotation curves that rise steeply, then become flat, or even fall somewhat to the not-yet-reached asymptotic flat velocity, while LSB galaxies are supposed to have rotation curves that rise slowly to the asymptotic flat velocity. This is precisely what is observed (Figure 15), and is in accordance [162] with the more complex empirical parametrization of observed rotation curves that has been proposed in [376]. Finally, the total (baryons+DM) acceleration is predicted to decline with the mean baryonic surface density of galaxies, exactly as observed (Figure 16), in the form (see also Figure 9).
- as the central surface density of dark halos. Provided they are mostly in the
Newtonian regime, galaxies are predicted to be embedded in dark halos (whether real or virtual,
i.e., “phantom” dark matter) with a central surface density on the order of as
observed
^{19}. LSBs should have a halo surface density scaling as the square-root of the baryonic surface density, in a much more compressed range than for the HSB ones, explaining the consistency of observed data with a constant central surface density of dark matter [167, 313]. - Features in the baryonic distribution imply features in the rotation curve. Because a small variation in will be directly translated into a similar one in , Renzo’s rule (Section 4.3.4) is explained naturally.

As a conclusion, all the apparently independent roles that the characteristic acceleration plays in the unpredicted observations of Section 4.3 (see end of Section 4.3.3 for a summary), as well as Renzo’s rule (Section 4.3.4), have been elegantly unified by the single law proposed by Milgrom [293] in 1983 as a unique scaling relation between the gravitational field generated by observed baryons and the total observed gravitational force in galaxies.

Living Rev. Relativity 15, (2012), 10
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