Saturation of spiral instabilities in disk galaxies

TL;DR

This paper proposes a nonlinear trapping mechanism as a key factor in the saturation of spiral density waves in galactic disks, providing analytical predictions that align with simulations and explaining the typical amplitude of spiral arms.

Contribution

It introduces a new analytical model for spiral saturation based on nonlinear stellar trapping near corotation resonance, advancing understanding of spiral arm amplitudes.

Findings

Saturation occurs when libration frequency reaches a threshold related to pattern speed and growth rate.

Maximum relative spiral surface density is proportional to the square of the growth rate over pattern speed, modulated by pitch angle.

Typical spiral amplitudes are a few tens of percent, with larger arms likely caused by external perturbations.

Abstract

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.