= Collisionless correction to the fluid spiral-wave dispersion relation
A stellar disk does not respond to a <density> wave as a collisional gas with a local <pressure> law. Its stars follow individual orbits, and their spread in radial motions partly averages away the gravitational response to short-wavelength perturbations. The appropriate calculation uses a perturbed <distribution function> rather than only a fluid <sound speed>.
A characteristic parameter is $\chi=(k\sigma_R/\kappa)^2$, comparing the radial epicycle scale with the perturbation <wavelength>. Orbital averaging reduces the response to perturbations whose <wavelength> is small relative to the stellar excursions. Merely replacing the fluid <sound speed> with stellar <velocity dispersion> misses that orbital averaging and can give the wrong stability threshold.<On the Gravitational Stability of a Disk of Stars by Alar Toomre>[Sections IV–V: random stellar motions and axisymmetric stability]{ref}
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