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, $\Sigma^0_B$: $\Sigma^0_D=\Sigma_M \mathcal{S}(\Sigma^0_B/\Sigma_M)$, where $\Sigma_M\equiv a_0/2\pi G$ is the MOND surface density constant. This surprising result is shown to hold in both existing, nonrelativistic MOND theories. $\mathcal{S}(y)$ is derived: $\mathcal{S}(y)=\int_0^y\nu(y')dy'$, with $\nu(y)$ the interpolating function of the theory. The relation aymptotes to $\Sigma^0_D=\Sigma^0_B$ for $\Sigma^0_B\gg\Sigma_M$, and to $\Sigma^0_D=(4\Sigma_M\Sigma^0_B)^{1/2}$ for $\Sigma^0_B\ll\Sigma_M$. 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.