 Dust gravitational dispersion relation with gas drag

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.