Uniformly rotating gas-sheet dispersion relation (${\omega }^2=c^2{k}^2-2\pi G{\Sigma }_0|k|+4{\Omega }^2$)

For a homogeneous inviscid razor-thin disk approximation with solid-body rotation and barotropic sound speed $c$, the compressive wave branch obeys ${\omega }^2=c^2{k}^2-2\pi G{\Sigma }_0|k|+4{\Omega }^2$. The razor-thin disk Poisson kernel supplies the self-gravity term, and the radial epicyclic frequency is $2|\Omega |$. Growing modes have ${\omega }^2<0$; a separate zero-frequency balanced mode can also exist.

At fixed ${\Sigma }_0,\Omega \ne 0$, slowly decreasing $c$ first reaches a zero of the wave-frequency minimum at $c=\pi G{\Sigma }_0/(2|\Omega |)$. The corresponding wavelength is $\ell ={\pi }^{2}G{\Sigma }_0/(2{\Omega }^2)$. It estimates the spacing and scale of growing density concentrations just below marginality; linear theory does not determine the final nonlinear clump radius.

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.