Toomre stability radius of a flat-rotation disk ($r_Q=\sqrt{2}{v}_0{a}_0/(\pi G\Sigma )$)

For constant circular speed $v_0$, angular velocity is $v_0/r$ and radial epicyclic frequency is $\sqrt{2}{v}_0/r$. Constant effective random speed and surface density then give $Q=r_Q/r$. The local axisymmetric fluid model is stable inside $r_Q$ and unstable outside it. This statement is conditional on the thin-disk local approximation, rather than a proof of global or collisionless stability.