= Toomre unstable band and fastest growth
{c}
{title2=$|k|_*=\pi G\Sigma/a_0^2$}
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 is
$$
\frac{|k|}{k_T}\in\left(\frac2{Q^2}(1-\sqrt{1-Q^2}),
\frac2{Q^2}(1+\sqrt{1-Q^2})\right).
$$
The quadratic dispersion has its minimum at $|k|_*=\pi G\Sigma/a_0^2$, with fastest growth $\gamma_*=\kappa\sqrt{Q^{-2}-1}$. At $Q=1$ the two band edges coincide and the mode is marginal. These are local axisymmetric fluid results, not a global stability theorem.
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