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 an instability grows exponentially in amplitude at some fixed growth rate $\beta$ before saturating nonlinearly. However, the mechanisms behind saturation, and the resulting saturated spiral amplitude, have received little attention. Here we argue that one important saturation mechanism is the nonlinear trapping of stars near the spiral's corotation resonance. Under this mechanism, we show analytically that an $m$-armed spiral instability will saturate when the libration frequency of resonantly trapped orbits reaches $\omega_\mathrm{lib} \sim \mathrm{a\,\, few}\times m^{1/2} \beta$. 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 reasonable agreement with recent $N$-body simulations, and suggests that spirals driven by internally-generated instabilities reach relative amplitudes of at most a few tens of percent; higher amplitude spirals, like in M51 and NGC 1300, are likely caused by very strong bars and/or tidal perturbations.