= Solution
<differential rotation> winds a material arm because $d\Omega/dr\ne0$. A <spiral density wave> instead has a <pattern speed> different from the material speed: stars and gas can pass through its crests. Collective <self-gravity>, <orbital resonances> and shear can sustain or repeatedly amplify non-axisymmetric patterns; a visible spiral need not be a single permanent rigid material structure.
For a razor-thin gas disk, local tightly wound disturbances satisfy
$$
(\omega-m\Omega)^2=\kappa^2+c_s^2k^2-2\pi G\Sigma|k|,\qquad
\kappa^2=4\Omega^2+r\frac{d\Omega^2}{dr}.
$$
Rotation stabilizes long <wavelengths> and <pressure> stabilizes short ones. Minimizing the axisymmetric right-hand side over $|k|$ gives $|k|_*=\pi G\Sigma/c_s^2$ and
$$
\boxed{Q_g=\frac{c_s\kappa}{\pi G\Sigma}\geq1}
$$
for local axisymmetric stability. A collisionless stellar disk instead has $Q_*=\sigma_R\kappa/(3.36G\Sigma_*)$; the radial random <velocity dispersion> is not literally a gas <sound speed>.<On the Gravitational Stability of a Disk of Stars by Alar Toomre>[Abstract: collisionless axisymmetric threshold and coefficient 3.36]{ref}
\b[Local axisymmetric stability does not guarantee global disk stability.] Gas and stars gravitate together, thickness modifies the short-scale force, and a disk with $Q>1$ may still have spiral or bar responses. Dissipation in gas can generate <shock waves> and concentrate <star formation> near a spiral crest; stellar random motions tend to reduce the collective response.
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