User:Astronome de Meudon/sandbox



The radial acceleration relation, or RAR, is a relation between the centripetal acceleration, $$V^2/R$$, of a star orbiting in a disk (or spiral) galaxy, and the gravitational acceleration due to the matter in the galaxy. The existence of such a relation is implied by Mordehai Milgrom's alternate theory of gravity, called "modified Newtonian dynamics," or MOND. Newton's theory of gravity also predicts a relation between the same two quantities, but real galaxies do not respect the Newtonian prediction, and the discrepancy is addressed under the standard cosmological model by postulating additional forces from dark matter. Milgrom's theory does not postulate the existence of dark matter.

Milgrom's prediction has been confirmed by observational studies. Confirmation of the prediction demonstrates that the internal kinematics of spiral galaxies are predictable based on the distribution of the visible mass alone, which adds support to the hypothesis that dark matter does not exist.

The prediction and its observational verification
Newton's theory of gravity predicts a simple relation between the centripetal acceleration, $$a$$, of a star or gas cloud orbiting in the disk plane of a spiral galaxy, and the gravitational acceleration, $$ g_N $$. The two quantities are predicted to be equal, i.e. $$ a = g_N $$, or

where $$ V(R) $$ is the orbital speed at distance $$ R $$ from the center and $$ \Phi $$ is the gravitational potential. Observed galaxies do not obey Equation (1): the measured $$ V $$ greatly exceeds the predicted value at large distances from the galaxy center. Under the standard cosmological model, the discrepancy is explained by postulating the presence of dark matter in and around the disk, in whatever amount and distribution required to augment the total $$ \Phi $$ and satisfy Equation (1).

Milgrom's theory does not postulate the existence of dark matter. Instead, Newton's relation between $$ a $$ and $$ g_N $$ is assumed to be modified in the low-acceleration regime; that is, in the regime $$ a \lesssim a_0 $$, where $$ a_0 $$ is Milgrom's constant, which has a value

The precise form of the relation between $$ a $$ and $$ g_N $$ is not specified in Milgrom's theory, except in two limiting cases: (i) large accelerations, where Newton's relation $$ a = g_N $$ is recovered; (ii) small accelerations, i.e. $$ a \ll a_0 $$, where $$ a \rightarrow (a_0 g_N)^{1/2} $$. However, the theory predicts that a "functional" relation exists between the two variables, i.e. that $$ a = f(g_N/a_0)g_N $$, with the function $$f$$ having the correct limiting forms in the regimes of low and high acceleration.

Milgrom's prediction is testable by measuring the two quantities $$ a $$ and $$ g_N $$ at different radii in a large sample of galaxies and plotting one variable against the other. The result is called the radial-accleration relation, or RAR. The observed relation is found to be consistent with zero intrinsic scatter, as predicted.

Given the observed RAR, the form of the mathematical relation between $$a$$ and $$g_N$$ can be inferred, by (for instance) fitting a smooth curve to the trend defined by the measured points. This gives the function $$ f(g_N/a_0) $$. A recent study finds:

This function is plotted in the figure.

Once the function $$ f(y) $$ has been determined in this way, Milgrom's theory can be quantitatively extended beyond the low-acceleration regime, to regimes of any acceleration. The theory can then be used to predict (for instance) the full rotation curve, $$ V(R) $$, of any galaxy given its measured matter distribution. The results are striking: in virtually every galaxy yet studied in this way, Milgrom's theory correctly predicts the observed rotation curve. No algorithm capable of doing this has yet been presented under the standard cosmological model. That model simply postulates whatever distribution of dark matter is required to yield the observed rotation curve under Newtonian gravity.

Significance
That the rotation speeds in spiral galaxies should be predictable from the observed mass distribution alone, without the need to postulate the existence of dark matter, is a successful, novel prediction of Milgrom's MOND theory. Standard-model cosmologists have attempted to accommodate the existence of the RAR in a post-hoc sense by adjusting their computer simulations of galaxy formation. These attempts have had limited success however.

Even if standard-model cosmologists should manage to accommodate the RAR via their simulations, the fact that Milgrom's theory correctly predicts the relation without any adjustment of parameters means that the existence of the RAR provides stronger support for MOND than for the standard cosmological model. For instance, philosopher of science John Worrall writes

"when one theory has accounted for a set of facts by parameter-adjustment, while a rival accounts for the same facts directly and without contrivance, then the rival does, but the first does not, derive support from those facts"

and Peter Lipton concurs:

"When data need to be accommodated, there is a motive to force a theory and auxiliaries to make the accommodation. The scientist knows the answer she must get, and she does whatever it takes to get it. ... In the case of prediction, by contrast, there is no motive for fudging, since the scientist does not know the right answer in advance. ... As a result, if the prediction turns out to have been correct, it provides stronger reason to believe the theory that generated it."