thermal pressure of the gas, $$\frac{{c_s}^2}{\rho} \, \frac{d \rho }{dz} = -\frac{d \Phi}{dz} \; ,$$ with solution $$\rho(R,z) = \rho(R,0) \exp \left\{c_s^{-2}\left[\Phi(R,0) - \Phi(R,z) \right] \right\} \; .$$ Here $\Phi$ denotes the gravitational potential, $\rho$ denotes the density of the gas, and $R$ is the cylindrical radius. If we assume that the gravitational potential is dominated by the mass of the central black hole $M_\mathrm{BH}$ at these scales, then the density distribution is $$\rho(R,z) = \rho(R,0) \exp \left\{ \frac{G M_\mathrm{BH} }{R \, c_s^2}\left[\frac{1}{\sqrt{1 + z^2/R^2}} - 1 \right] \right \}\;.$$ In the limit of small $z/R$, this yields a Gaussian vertical structure. In this limit, the ratio of the squared sound speed to the squared Keplerian velocity $V_c$ functions as the ratio of the disk scale height to the cylindrical radius, and for convenience we choose to define a parameter that makes this identification universal: $$\frac{h_s}{R} \equiv \frac{c_s}{V_c} = c_s \left( \frac{G\, M_\mathrm{BH}}{R} \right )^{-1/2} \;.$$

Moderately large values of $h/R$ ($\gtrsim 0.2$) are suggested by the observed fraction of obscured versus unobscured quasars, although generally this fraction correlates strongly with luminosity [@Maiolino2007]. Meanwhile, typical values of $h_s/R$ found in @Hopkins2012-1 range from 0.1 to 0.5. In this study we 