  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.
