$f(r)$ denote the ratio of the volume occupied by at least one clump to the total volume within a sphere of radius $r$ centered on the black hole, then we find that it can be well approximated via broken power laws. For model 1, $$\begin{aligned}
f(r) &\approx 0.1 \times \left(\frac{r}{0.27 {\rm \; pc}}\right)^{-0.33} \qquad & {\rm for } \; 0.1 {\rm \; pc} < r < 2 {\rm \; pc} \; , \nonumber \\
  &\approx 0.05 \times \left(\frac{r}{2.3 {\rm \; pc}}\right)^{-1.5} \qquad & {\rm for } \; r > 2 {\rm \; pc} \; .\end{aligned}$$ For models 2 through 4, $$\begin{aligned}
f(r) &\approx 0.1 \times \left(\frac{r}{0.19 {\rm \; pc}}\right)^{-0.75} \qquad & {\rm for } \; 0.1 {\rm \; pc} < r < 1 {\rm \; pc} \;, \nonumber \\
  &\approx 0.03 \times \left(\frac{r}{1 {\rm \; pc}}\right)^{-1.5} \qquad & {\rm for } \; r > 1 {\rm \; pc} \; .\end{aligned}$$

Figure \[ColumnDensityHistograms\] shows the distribution of column density along randomly sampled sightlines for both our smooth and clumpy models. All column density values quoted in this study assume a mean mass per particle of 1.5 times the proton mass.

Making the gas clumpy leads to a larger number of sightlines 