Marginal fragmentation wavelength of a rotating sheet (${\ell }_{\mathrm{crit}}={\pi }^{2}G{\Sigma }_0/(2{\Omega }^2)$)

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.