Saturation of spiral instabilities in disk galaxies

Spiral density waves can arise in galactic disks as linear instabilities of the underlying stellar distribution function. Such instabilities grow exponentially in amplitude at some fixed growth rate $\beta$ before saturating nonlinearly. However, the mechanisms behind nonlinear saturation, and the resulting saturated spiral amplitude, have not received much attention. Here we argue that the most important nonlinear saturation mechanism is likely trapping of stars near the spiral's corotation resonance. Under these circumstances, we show analytically that an $m$-armed spiral will saturate when the libration frequency of resonantly trapped orbits reaches $\omega_\mathrm{lib} \sim 3\, m^{1/2} \beta$. For a galaxy with a flat rotation curve this implies a maximum relative spiral surface density $\vert \delta\Sigma/\Sigma_0\vert \sim \mathrm{a\,\,few} \times (\beta/\Omega_\mathrm{p})^2 \cot \alpha$, where $\Omega_\mathrm{p}$ is the spiral pattern speed and $\alpha$ is its pitch angle. This result is in good agreement with recent N-body simulations of isolated stellar disks, and is consistent with observed trends from spectroscopic surveys.