Central-surface-densities correlation in general MOND theories

It is shown that the foundational axioms of MOND alone predict a strong correlation between a bulk measure of the baryonic surface density, ${\mathrm{\Sigma }}_B$, and the corresponding dynamical one, ${\mathrm{\Sigma }}_D$, of an isolated object, such as a galaxy. The correlation is encapsulated by its high- and low-${\mathrm{\Sigma }}_B$ behaviors. For ${\mathrm{\Sigma }}_B\gg {\mathrm{\Sigma }}_M\equiv a_0\mathrm{/}2\pi G$ (${\mathrm{\Sigma }}_M$ is the critical MOND surface density) one has ${\mathrm{\Sigma }}_D\approx {\mathrm{\Sigma }}_B$. Their difference – which would be interpreted as the contribution of dark matter – is ${\mathrm{\Sigma }}_P={\mathrm{\Sigma }}_D-{\mathrm{\Sigma }}_B\sim {\mathrm{\Sigma }}_M\ll {\mathrm{\Sigma }}_B$. In the deep-MOND limit, ${\mathrm{\Sigma }}_B\ll {\mathrm{\Sigma }}_M$, one has ${\mathrm{\Sigma }}_D\sim ({\mathrm{\Sigma }}_M{\mathrm{\Sigma }}_B{)}^{1\mathrm{/}2}$. This is a primary prediction of MOND, shared by all theories that embody its basic tenets. Sharper correlations, even strict algebraic relations, ${\mathrm{\Sigma }}_D({\mathrm{\Sigma }}_B)$, are predicted in specific MOND theories, for specific classes of mass distribution – e.g., pure discs, or spherical systems – and for specific definitions of the surface densities. I proceed to discuss such tighter correlations for the central surface densities of axisymmetric galactic systems, ${\mathrm{\Sigma }}_B^0$ and ${\mathrm{\Sigma }}_D^0$. Past work has demonstrated such relations for pure discs in the AQUAL and QUMOND theories. Here I consider them in broader classes of MOND theories. For most observed systems, ${\mathrm{\Sigma }}_D^0$ can not be determined directly at present, but, in many cases, a good proxy for it is the acceleration integral $\mathcal{G}\equiv {\int }_0^{\mathrm{\infty }}{g}_{r}d\ln r$, where $g_r$ is the radial acceleration along a reflection-symmetry plane of a system, such as a disc galaxy. $\mathcal{G}$ can be determined directly from the rotation curve. I discuss the extent to which $\mathcal{G}$ is a good proxy for ${\mathrm{\Sigma }}_D^0$, and how the relation between them depends on system geometry, from pure discs, through disc-plus-bulge ones, to quasi-spherical systems.