Gravitational instability of an astrophysical disk

Disk self-gravity can overcome pressure and rotational support, producing growing density disturbances, spiral structure, or fragmentation.

For a razor-thin isothermal gas disk,Axisymmetric disturbances are stable when $Q\ge 1$: pressure stabilizes short wavelengths, epicyclic motion stabilizes long wavelengths, and self-gravity drives intermediate wavelengths.

The shearing sheet is a local Cartesian approximation to a differentially rotating disk. It retains Coriolis force, linearized tidal gravity, and a background linear shear.

Thermal relaxation drives the local sound speed toward an equilibrium value on a cooling time $\tau$. Fast relaxation gives an isothermal response, while slow relaxation gives an adiabatic response.