 Solution

Without gas drag, the dispersion relation factorizes as $s(s^2+{\omega }^2)=0$. There is a neutral branch and two density-wave branches. If ${\omega }^2>0$, the waves have $s=\pm i\omega$; if ${\omega }^2<0$, one root is $\boxed{s=\sqrt{-{\omega }^2}>0}$ and the dust layer is gravitationally unstable.

For $c>0$, complete the square in the wavenumber magnitude:The minimum occurs at $|k|=\pi G{\sigma }_0/c^2$. Thus the Toomre stability criterion isAt $Q=1$ the minimizing mode is marginal. For $Q>1$ there is no growing axisymmetric wave in this razor-thin, pressure-supported, drag-free model. The unstable band for $Q<1$ isIn a finite layer, this band must contain an admissible mode; the continuum statement assumes an adequately large domain.