 Unstable wavenumber band of a rotating gas sheet

Put $Q=2|\Omega |c/(\pi G{\Sigma }_0)$. The uniformly rotating gas-sheet dispersion relation has growing waves precisely for $Q<1$, at $q_{-}<|k|<q_{+}$ with $q_{\pm }=(\pi G{\Sigma }_0/c^2)(1\pm \sqrt{1-Q^2})$. Pressure stabilizes short waves and rotation stabilizes long waves. Equality $Q=1$ is marginal; $Q\ge 1$ is the gas Toomre stability criterion.