will consider $h_s/R$ in the range 0.1 to 0.35.

Mid-IR interferometric observations find the mid-plane density may be well-fit with a power-law $R^{-\gamma}$ where $\gamma$ lies within a range of approximately 0.4 to 1.4, with a tendency toward larger values for more luminous AGN [@Kishimoto2011]. This is also in agreement with the simulations presented in @Hopkins2012-1 in which $\gamma \approx 1.5$. We find that the results for varying $\gamma$ correlate strongly with the corresponding change in torus mass within the computational domain, and so we choose to capture variations in torus mass and column density by varying the normalization of the radial density profile (the parameter $\rho_0$ described below), while fixing $\gamma$ to $1.5$ for all calculations presented in this paper.

Converting to spherical polar coordinates $r$ and $\theta$, where $\theta$ is taken to be zero along the z-axis, we obtain $$\rho(r,\theta) = \rho_0 \left(\frac{r \sin \theta}{r_0}\right)^{-\gamma} \exp \left[(h_s/R)^{-2}(\sin{\theta} - 1) \right] \; .$$ Here $r_0$ represents some inner cut-off radius where the density is $\rho_0$, and to prevent the radial column density from diverging we take $\rho(r < r_0) = \rho_0$.

One undesirable aspect of this model is that it leads to an accumulation of mass in the polar 