Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 1 / iii

The thin-sheet field used above requires the gas to sample heights beyond most of the stellar mass, while the stellar galactic disk still dominates vertical gravity and is approximately locally plane-parallel. In inner regions a cold gas layer can lie inside the thicker stellar layer, where reflection symmetry gives $K_z\simeq {\nu }_z^{2}z$ rather than a constant field. The vertical Jeans equation, or gas hydrostatic equilibrium, then givesHere $H_g$ is the Gaussian width and ${\nu }_z$ is the vertical epicyclic frequency; in a locally flat galaxy rotation curve, ${\nu }_z^2\simeq 4\pi G{\rho }_{\mathrm{total}}(R,0)$. A stellar layer of approximately fixed thickness then gives $H_g\propto {\Sigma }_{\ast }^{-1/2}$, rather than the inverse surface density law.

At large radii, the dark matter halo or the gas's own gravity can exceed the dwindling stellar restoring force, so replacing the total force by $2\pi G{\Sigma }_{\ast }$ fails. A nearly spherical dark matter halo instead gives ${\nu }_z\simeq v_c/R$ near its plane, and $H_g\simeq {\sigma }_{g}R/v_c$ for a flat galaxy rotation curve, much slower than exponential galactic disk flaring. Warps, radial gradients across a thick gas layer, non-equilibrium disturbances, and height-dependent velocity dispersion also invalidate the locally plane-parallel steady isothermal approximation. The inverse stellar surface-density law applies only while its force and geometric assumptions apply.