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

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.