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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.

In the razor-thin fluid disk dispersion relation, the cold limit is ${\omega }^2={\kappa }^2-2\pi G\Sigma |k|$. Rotation therefore suppresses wavelengths above ${\lambda }_T=2\pi /k_T=4{\pi }^{2}G\Sigma /{\kappa }^2$. With this convention, $k_T$ is the cold-disk boundary of instability, not the fastest-growing wave number when finite pressure is present. At marginal fluid stability $Q=1$, the minimum occurs at $|k|=2k_T$.