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, $Σ^0_D$, deduced from the measured velocities, is a universal function of only the true, `baryonic' central surface density, $Σ^0_B$: $Σ^0_D=Σ_M \mathcal{S}(Σ^0_B/Σ_M)$, where $Σ_M\equiv a_0/2πG$ is the MOND surface density constant. This surprising result is shown to hold in both existing, nonrelativistic MOND theories. $\mathcal{S}(y)$ is derived: $\mathcal{S}(y)=\int_0^yν(y')dy'$, with $ν(y)$ the interpolating function of the theory. The relation aymptotes to $Σ^0_D=Σ^0_B$ for $Σ^0_B\ggΣ_M$, and to $Σ^0_D=(4Σ_MΣ^0_B)^{1/2}$ for $Σ^0_B\llΣ_M$. 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.

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