5.2 Galactic Kepler-like laws of motion
- 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 , 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
- 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  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 . 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  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  with the more complex empirical parametrization of observed rotation curves
that has been proposed in . 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
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  in 1983 as a
unique scaling relation between the gravitational field generated by observed baryons and the total observed
gravitational force in galaxies.
Figure 14: The mass discrepancy (as in Figure 10) as a function of radius in observed spiral galaxies.
The curves for individual galaxies (lines) are color-coded by their characteristic baryonic surface
density (as in Figure 5). In order to be completely empirical and fully independent of any assumption
such as maximum disk, stellar masses have been estimated with population synthesis models .
The amplitude of the mass discrepancy is initially small in high–surface-density galaxies, and grows
only slowly at large radii. As the baryonic surface densities of galaxies decline, the mass discrepancy
becomes more severe and appears at smaller radii. This trend confirms one of the predictions of
Milgrom’s law .
Figure 15: The shapes of observed rotation curves depend on baryonic surface density (color coding
as per Figure 14). High–surface-density galaxies have rotation curves that rise steeply then become
flat, or even fall somewhat to the asymptotic flat velocity. Low–surface-density galaxies have rotation
curves that rise slowly to the asymptotic flat velocity. This trend confirms one of the predictions of
Milgrom’s law .
Figure 16: Centripetal acceleration as a function of radius and surface density (color coding as
per Figure 14). The critical acceleration is denoted by the dotted line. Milgrom’s formula
predicts that acceleration should decline with baryonic surface density, as observed. Moreover,
high–surface-density galaxies transition from the Newtonian regime at small radii to the weak-field
regime at large radii, whereas low–surface-density galaxies fall entirely in the regime of low
acceleration , as anticipated by Milgrom .