 Finite-size secular gravitational criterion

For a dust layer of extent $L$, its smallest admissible wavenumber is of order $1/L$. The secular gravitational instability band ends at $k_{\mathrm{crit}}=2\pi G{\sigma }_0/c^2=2/(QH)$, with Toomre parameter $Q=c\Omega /(\pi G{\sigma }_0)$ and $H=c/\Omega$. Requiring an admissible mode in the band gives $Q\lesssim L/H$. Boundary conditions fix the order-unity prefactor; a finite disk lifetime also limits practical growth.