 Dust gravitational dispersion relation with gas drag

Consider a pressured dust layer in a prescribed Keplerian shearing sheet gas flow, neglecting gas backreaction. Let ${\sigma }_0$ be its unperturbed surface density, $c$ its constant effective random speed with pressure perturbation $c^2\delta \sigma$, and $\Omega >0$ the local angular velocity. Linear gas drag damps dust velocity at rate $\gamma =t_s^{-1}=ϵ\Omega >0$, where $t_s$ is the aerodynamic stopping time.

For axisymmetric perturbations proportional to $e^{st+ikx}$ with $k\ne 0$, write $\delta =\delta \sigma /{\sigma }_0$ and radial and azimuthal velocity amplitudes $u,v$. The linear mass and momentum balances areThe razor-thin disk Poisson kernel supplies the self-gravity term. The azimuthal coefficient $\Omega /2$ includes advection of the Keplerian shear. Taking the full amplitude determinant givesAt zero drag this factors as $s(s^2+{\omega }^2)=0$, with ${\omega }^2={\Omega }^2+c^2{k}^2-2\pi G{\sigma }_0|k|$. The zero root is the branch from which secular gravitational instability can develop when drag is restored. Dividing by $s$ before evaluating the determinant would discard that physical mode.