region of the disk, where $\sin \theta$ is small. To correct for this, we allow the density profile to drop as a power law in the spherical radius $r$ rather than in the cylindrical radius $R$. This amounts to dropping the factor of $(\sin{\theta})^{-\gamma}$, which is only significant far from the disk mid-plane. This leaves $$\rho(r,\theta) = \rho_0 \left(\frac{r }{r_0}\right)^{-\gamma} \exp \left[(h_s/R)^{-2}(\sin{\theta} - 1) \right] \;,$$ which is quite similar to the phenomenological models of @Granato1994 and @Efstathiou1995 that were used to explain the properties of spectral energy distributions observed in dusty AGN.

The results from @Hopkins2012-1 indicate that $h_s/R$ does not change by more than a factor of order unity for all $R$. For simplicity, we take $h_s/R$ to be a constant for all $R$ and allow it to vary as a free parameter for different disk models. For all calculations in this study we assume a black hole mass $M_\mathrm{BH}$ of $10^8$ $M_\odot$, and we parametrize the luminosity as a fraction of the electron-scattering Eddington luminosity for that mass. As mentioned above, we also vary $\rho_0$, which sets the sightline-averaged column density $\overline{N}_\mathrm{H}$. Unless stated otherwise, $\overline{N}_\mathrm{H}$ corresponds to the column density integrated to a distance of $r_0 