Weak-drag secular gravitational growth rate ($s_{\sec }=ϵ\Omega ({\Omega }^2-{\omega }^2)/{\omega }^2+O({ϵ}^2)$)

The mode originating at the neutral root grows for $0<{\omega }^2<{\Omega }^2$. At small nonzero $|k|$ this gives $s_{\sec }\sim 2\pi ϵG{\sigma }_0|k|/\Omega$. Gas drag transfers angular momentum to the fixed gas flow and permits secular gravitational instability even for a dust Toomre parameter above unity. If ${\omega }^2<0$, the small-root continuation is damped, while a separate dynamical root grows. The formula is singular at ${\omega }^2=0$.

Turbulence can raise dust random speeds, increase layer thickness and mix density enhancements by eddy diffusion, delaying secular gravitational instability. On sufficiently long wavelengths, weak-drag growth is proportional to $|k|$, whereas simple diffusive damping is proportional to $k^2$. Thus diffusion alone need not remove all long-wave growth in an infinite idealization, although finite size and lifetime can make it ineffective. Coherent turbulent concentration can also increase local surface density, so the net effect requires a stochastic transport model.

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.