Spiral density wave

A collective perturbation of surface density and gravitational potential in a galactic disk. Material can cross its spiral-shaped crests; the pattern angular speed need not equal the local material angular speed. A tightly wound fluid approximation has $(\omega -m\Omega {)}^2={\kappa }^2+c_s^2{k}^2-2\pi G\Sigma |k|$, while a collisionless stellar disk requires its own response function.

Two radii with different orbital frequencies develop a relative angular displacement $\Delta \varphi =\Delta \Omega t$. A spiral made of a fixed set of stars therefore winds increasingly tightly. A density pattern need not follow that material winding because matter can pass through it.

A self-gravitating disk can transiently amplify a leading disturbance as differential rotation shears it into a trailing disturbance. Its strength depends on disk self-gravity, random motions and disturbance wavelength. Amplification does not by itself establish one indefinitely steady spiral.

For a perturbation phase $m\varphi +\int kdr-\omega t$, a constant-phase pattern rotates at ${\Omega }_p=\omega /m$. At corotation the local orbit angular speed equals this value. One observed galaxy can contain several patterns with distinct angular speeds and lifetimes.