Universal MOND relation between the baryonic and `dynamical' central surface densities of disc galaxies

I derive a new MOND relation for pure-disc galaxies: The dynamical' central surface density, $\Sigma^0_D$, deduced from the measured velocities, is a universal function of only the true, baryonic’ central surface density, ${\mathrm{\Sigma }}_B^0$: ${\mathrm{\Sigma }}_D^0={\mathrm{\Sigma }}_M\mathcal{S}({\mathrm{\Sigma }}_B^0\mathrm{/}{\mathrm{\Sigma }}_M)$, where ${\mathrm{\Sigma }}_M\equiv a_0\mathrm{/}2\pi G$ is the MOND surface density constant. This surprising result is shown to hold in both existing, nonrelativistic MOND theories. $\mathcal{S}(y)$ is derived: $\mathcal{S}(y)={\int }_0^y\nu (y^{\mathrm{\prime }})d{y}^{\mathrm{\prime }}$, with $\nu (y)$ the interpolating function of the theory. The relation aymptotes to ${\mathrm{\Sigma }}_D^0={\mathrm{\Sigma }}_B^0$ for ${\mathrm{\Sigma }}_B^0\gg {\mathrm{\Sigma }}_M$, and to ${\mathrm{\Sigma }}_D^0=(4{\mathrm{\Sigma }}_M{\mathrm{\Sigma }}_B^0{)}^{1\mathrm{/}2}$ for ${\mathrm{\Sigma }}_B^0\ll {\mathrm{\Sigma }}_M$. This study was prompted by the recent finding of a correlation between related attributes of disc galaxies by Lelli et al. (2016). The MOND central-surface-densities relation agrees very well with these results.