Toomre stability of disk galaxies in quasi-linear MOND

We consider disk stability in the quasi-linear formulation of MOND (QUMOND), the basis for some $N$-body integrators. We generalize the Toomre criterion for the stability of disks to tightly wound, axisymmetric perturbations. We apply this to a family of thin exponential disks with different central surface densities. By numerically calculating their QUMOND rotation curves, we obtain the minimum radial velocity dispersion required for stability against local self-gravitating collapse. MOND correctly predicts much higher rotation speeds in low surface brightness galaxies (LSBs) than does Newtonian dynamics without dark matter. Newtonian models thus require putative very massive halos, whose inert nature implies they would strongly stabilize the disk. MOND also increases the stability of galactic disks, but in contradistinction to Newtonian gravity, this extra stability is limited to a factor of 2. MOND is thus rather more conducive to the formation of bars and spiral arms. Therefore, observation of such features in LSBs could be problematic for Newtonian galaxy models. This could constitute a crucial discriminating test. We quantitatively account for these facts in QUMOND. We also compare numerical QUMOND rotation curves of thin exponential disks to those predicted by two algebraic expressions commonly used to calculate MOND rotation curves. For the choice that best approximates QUMOND, we find the circular velocities agree to within 1.5% beyond $\approx 0.5$ disk scale lengths, regardless of the central surface density. The other expression can underestimate the rotational speed by up to 12.5% at one scale length, though rather less so at larger radii.