Toomre's stability criterion

Remarks

For the fluid disc, the density wave dispersion relation with $m = 0$ reads $\omega^2 = \kappa^2 - 2 \pi G \Sigma \abs{k} + c_s^2 k^2$. The right side is smallest at $\abs{k} = \pi G \Sigma / c_s^2$, where it equals $\kappa^2 - (\pi G \Sigma / c_s)^2$. So $\omega^2 \ge 0$ for all $k$ if and only if $c_s \kappa \ge \pi G \Sigma$; otherwise some $\omega$ is imaginary and the mode grows exponentially. The stellar case uses the stellar dispersion relation, and the constant $3.36$ comes from minimising its reduction factor.

The criterion combines two stabilising effects against the Jeans instability of the disc: pressure, which acts on short wavelengths, and rotation through $\kappa$, which acts on long wavelengths. Without rotation, $\kappa = 0$, every disc with $\Sigma > 0$ is unstable at long wavelengths. The result is due to A. Toomre (1964) for stellar discs; the fluid version is often credited to V. S. Safronov (1960) as well. $Q \ge 1$ does not guarantee stability against non-axisymmetric perturbations.

Going deeper

For a student

In galaxy discs, star formation is often observed where the gas has $Q$ near or below one, so $Q$ is a standard diagnostic. The same criterion governs the fragmentation of massive protoplanetary and accretion discs. It is the rotating-disc version of the Jeans criterion, with the extra $\kappa$ term coming from the epicyclic motion.