Toomre unstable band and fastest growth ($|k{|}_{\ast }=\pi G\Sigma /a_0^2$)

For the razor-thin fluid disk dispersion relation, set $Q=\kappa a_0/(\pi G\Sigma )$ and $k_T={\kappa }^2/(2\pi G\Sigma )$. When $Q<1$, the unstable band isThe quadratic dispersion has its minimum at $|k{|}_{\ast }=\pi G\Sigma /a_0^2$, with fastest growth ${\gamma }_{\ast }=\kappa \sqrt{Q^{-2}-1}$. At $Q=1$ the two band edges coincide and the mode is marginal. These are local axisymmetric fluid results, not a global stability theorem.