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Galactic disk flaring is an outward increase of vertical thickness. A constant vertical velocity dispersion in an approximately thin-sheet stellar Newtonian gravitational field gives $h\propto 1/{\Sigma }_{\ast }$. An exponential galactic disk then predicts $h\propto e^{R/R_d}$ for a gas tracer. Inside a thick stellar layer the potential is instead approximately harmonic, and the tracer width is ${\sigma }_z/{\nu }_z$ with vertical epicyclic frequency ${\nu }_z$.

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.

In the thin-sheet regime, the gas scale height is $h_g(R)={\sigma }_g^2(R)/(2\pi G{\Sigma }_{\ast }(R))$. Hence radius-independent gas velocity dispersion givesFor an exponential galactic disk, ${\Sigma }_{\ast }(R)={\Sigma }_{\ast ,0}{e}^{-R/R_d}$, soThe gas therefore exhibits galactic disk flaring as stellar surface density falls outward. Stellar galactic disks, by contrast, commonly have roughly constant thickness over their main optical region; applying the same scale height relation to the stars requires ${\sigma }_{z,\ast }^2(R)\propto {\Sigma }_{\ast }(R)$, or ${\sigma }_{z,\ast }\propto e^{-R/(2R_d)}$ for an exponential galactic disk. This comparison does not follow by imposing a radially constant stellar velocity dispersion: that extra assumption would predict stellar galactic disk flaring too. The constancy used in the vertical calculation is constancy with height locally, not a mandatory constancy with radius.