= Long-wave and short-wave branches at marginal Toomre stability
{title2=$|k|=[\kappa\pm n|\Omega_P-\Omega|]/a_0$}
At fluid <Toomre parameter> $Q=1$, the local spiral-wave <dispersion relation> becomes $n^2(\Omega_P-\Omega)^2=(a_0|k|-\kappa)^2$. The minus-sign, longer-wavelength root is nonnegative only between the two <Lindblad resonance> conditions. The plus-sign, shorter-wavelength root remains positive outside that interval. Thus this <dispersion relation> alone does not prove that every <density> wave is confined between <Lindblad resonances>; an additional physical restriction is needed.
Back to article page
