Density wave theory

Definition

Consider a thin, differentially rotating disc with surface density $\Sigma_0(R)$, angular velocity $\Omega(R)$ and sound speed $c_s(R)$. Let $\kappa(R)$ be its epicyclic frequency, and let gravity be given by Poisson's equation. A spiral density wave is a small perturbation of the surface density of the form $$\Sigma_1(R, \varphi, t) = \operatorname{Re}\left[ S(R) \, e^{i (m \varphi - \omega t + \Phi(R))} \right],$$ with $m \ge 1$ arms and pattern speed $\Omega_p = \omega / m$. Its radial wavenumber $k = \Phi'(R)$ is assumed to satisfy $\abs{k} R \gg 1$ (tight winding). In this limit the waves obey the dispersion relation of Lin and Shu for a fluid disc, $$(\omega - m \Omega)^2 = \kappa^2 - 2 \pi G \Sigma_0 \abs{k} + c_s^2 k^2.$$ Density wave theory is the hypothesis that the spiral arms of disc galaxies are such waves, quasi-stationary in a frame rotating with $\Omega_p$.