MOND fiducial specific angular momentum of disc galaxies

Abstract

It is pointed out that MOND defines a fiducial specific angular momentum (SAM) for a galaxy of total (baryonic) mass $\mathcal{M}$: $j_M(\mathcal{M})\equiv\mathcal{M}^{3/4}(G^3/a_0)^{1/4}\approx 383(\mathcal{M}/10^{10}M_\odot)^{3/4}{\rm kpc~km/s}$. It plays important roles in disc-galaxy dynamics and evolution: It underlies scaling relations in virialized galaxies that involve their angular-momentum. I show that the disc SAM should be $j_D\approx[\langle r\rangle/r_M(\mathcal{M})]j_M(\mathcal{M})=[\Sigma_M/\langle \Sigma\rangle]^{1/2}j_M(\mathcal{M})$, with $\langle r\rangle$ the mean radius of the disc, $\langle \Sigma\rangle=\mathcal{M}/2\pi\langle r\rangle^2$ some mean surface density of the galaxy, $r_M=(\mathcal{M} G/a_0)^{1/2}$ is the MOND radius of the galaxy, and $\Sigma_M=a_0/2\pi G$ is the (universal) MOND surface density. So, e.g., for a fixed $\langle \Sigma\rangle$, $j_D\propto \mathcal{M}^{3/4}$, while for a fixed $\langle r\rangle$, $j_D\propto \mathcal{M}^{1/4}$. Furthermore, $j_M(\mathcal{M})$ is a reference predictor of the type of galaxy a protogalaxy will settle into, if it evolves in isolation: A protogalaxy of mass $\mathcal{M}$, and SAM $j\gg j_M(\mathcal{M})$ should settle into a low-surface-density disc -- with mean acceleration $\langle a\rangle/a_0\approx j_M/j\ll 1$. While a protogalaxy with $j\lesssim j_M(\mathcal{M})$ should end up with a disc of mass $\mathcal{M}_D\approx j\mathcal{M}/j_M(\mathcal{M})$, having a SAM $j_D\approx j_M(\mathcal{M})$, which is tantamount to $\langle a\rangle\approx a_0$ (i.e., at the `Freeman limit'); it should also develop a low-SAM bulge, taking up the rest of the mass $\mathcal{M}_B\approx\mathcal{M}-\mathcal{M}_D$.