2024MNRAS.528.5286H

Spiral density waves can arise in galactic discs as linear instabilities of the underlying stellar distribution function. Such an instability grows exponentially in amplitude at some fixed growth rate β before saturating non-linearly. 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 non-linear 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 Ω_p is the spiral pattern speed and α 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 per cent; higher amplitude spirals, like in M51 and NGC 1300, are likely caused by very strong bars and/or tidal perturbations.