  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.
