On the stability of a galactic disk in modified gravity

We find the dispersion relation for tightly wound spiral density waves in the surface of rotating, self-gravitating disks in the framework of Modified Gravity (MOG). Also, the Toomre-like stability criterion for differentially rotating disks has been derived for both fluid and stellar disks. More specifically, the stability criterion can be expressed in terms of a matter density threshold over which the instability occurs. In other words the local stability criterion can be written as $\Sigma_0<\Sigma_{\text{crit}}(v_s,\kappa,\alpha,\mu_0)$, where $\Sigma_{\text{crit}}$ is a function of $v_s$ (sound speed), $\kappa$ (epicycle frequency) and $\alpha$ and $\mu_0$ are the free parameters of the theory. In the case of a stellar disk the radial velocity dispersion $\sigma_r$ appears in $\Sigma_{\text{crit}}$ instead of $v_s$. We find the exact form of the function $\Sigma_{\text{crit}}$ for both stellar and fluid self-gravitating disks. Also, we use a sub-sample of THINGS catalog of spiral galaxies in order to compare the local stability criteria. In this perspective, we have compared MOG with Newtonian gravity and investigated the possible and detectable differences between these theories.