 Disk aspect ratio

The aspect ratio of an astrophysical disk is its vertical thickness scale divided by cylindrical radius. A thin disk has $H/r\ll 1$. In gas-supported vertical hydrostatic equilibrium with harmonic external gravity $-{\Omega }_z^{2}z$, $H/r\sim c_s/(r{\Omega }_z)$, where ${\Omega }_z$ is the vertical epicyclic frequency and $c_s$ is the sound speed. In a centrally dominated Keplerian disk, ${\Omega }_z=\Omega$, so this compares sound speed with orbital speed. If disk self-gravity supplies the vertical restoring force, the relation $H\sim c_s/\Omega$ requires replacement. A polytropic free-surface height and an isothermal scale height differ by model-dependent constants even though their aspect ratios have the same basic role.