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.

For tightly wound perturbations in a locally uniform, razor-thin gas disk, continuity equation and rotating-fluid momentum equations couple the surface-density perturbation to radial and azimuthal velocities. The sheet potential is $\delta \Phi =-2\pi G\delta \Sigma /|k|$. Eliminating these perturbations gives the displayed dispersion relation. radial epicyclic frequency supplies the rotation term, the sound-speed term resists short waves, and self-gravity supplies the negative term. For an axisymmetric mode $n=0$, use ${\omega }^2$ directly; the pattern speed $\omega /n$ is undefined.

A stellar disk does not respond to a density wave as a collisional gas with a local pressure law. Its stars follow individual orbits, and their spread in radial motions partly averages away the gravitational response to short-wavelength perturbations. The appropriate calculation uses a perturbed distribution function rather than only a fluid sound speed.

A characteristic parameter is $\chi =(k{\sigma }_R/\kappa {)}^2$, comparing the radial epicycle scale with the perturbation wavelength. Orbital averaging reduces the response to perturbations whose wavelength is small relative to the stellar excursions. Merely replacing the fluid sound speed with stellar velocity dispersion misses that orbital averaging and can give the wrong stability threshold.^[ref]

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.