Toomre Stability Criterion

Let us consider a blob of fluid in a thin Keplerian disc. If the fluid element is "squished" in the radial direction, it might collapse to a clump, or return to its unperturbed state. The time scale for returning to its original shape is the inverse of the epicyclic frequency. For simplicity, we will use the Keplerian angular velocity $\Omega$ instead. The time scale for collapse is just the Jeans' time $\frac{1}{\sqrt{G \rho}}$, where $G$ is the universal constant of gravitation, and $\rho$ is the density. The criterion for stability is therefore that the period is much shorter than the Jeans time

$\frac{1}{\Omega} < \frac{1}{\sqrt{G \rho}}$

The volume density is just the surface density divided by the height $\rho = \frac{\Sigma}{h}$. The height is determined by pressure balance with gravity

$\frac{h}{R} = \frac{c_s}{v_k} \Rightarrow h = \frac{c_s}{\Omega}$

where $R$ is the distance from the centre of the disc, $c_s$ is the local speed of sound, and $v_k$ is the Keplerian velocity. Putting it all together yields that a blob is stable if

$\frac{c_s \Omega}{G \Sigma} > 1$

The region of the disc most susceptible to the Toomre instability is its outer edge. At that point the surface density is roughly given by $\Sigma \approx m_d / R^2$, where $m_d$ is the mass of the disc. The Keplerian velocity is $v_k \approx \sqrt{G M/R}$ where $M$ is the mass of the star around which the disc rotates. Substituting these relations we obtain a simplified version of the Toomre stability criterion

$m_d < \frac{h}{R} M$