Dust gravitational dispersion relation with gas drag ($s[(s+ϵ\Omega {)}^2+{\Omega }^2]+(c^2{k}^2-2\pi G{\sigma }_0|k|)(s+ϵ\Omega )=0$)

The local axisymmetric density/velocity system of a pressured dust layer with linear gas drag and a fixed Keplerian shearing sheet gas flow has this cubic dispersion relation. The $\Omega /2$ azimuthal velocity coefficient includes advection of the Keplerian shear. The razor-thin disk Poisson kernel supplies the $|k|$ self-gravity term. Keeping the full amplitude determinant retains the zero-frequency branch and avoids dividing out the secular mode.

For any nonzero positive drag, the cubic dust gravitational dispersion relation with gas drag is negative at $s=0$ if ${\omega }^2<{\Omega }^2$ and positive at sufficiently large positive $s$. The intermediate value theorem therefore guarantees a growing real root. This existence argument is independent of the weak-drag expansion and includes the dynamically unstable ${\omega }^2<0$ regime.

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.

For fixed ${\omega }^2={\Omega }^2-2\pi G{\sigma }_0|k|+c^2{k}^2>0$, the two oscillatory roots of the dust gravitational dispersion relation with gas drag acquire negative real parts at first order in $ϵ$. Their frequencies have no first-order shift. The expansion is not uniform near ${\omega }^2=0$, where the oscillation and damping scales can become comparable.