the black hole ($d_{\rm cl}/r$), the gas density in the clump ($n_{\rm cl}$), and the number of clumps in the simulation volume $N_{\rm cl}$. The choices of $N_{\rm cl}$ and $d_{\rm cl}$ set the average number of clumps per line of sight, which we compute by averaging over all ($\theta$, $\phi$) sightlines with a weighting to account for the solid angle subtended by each sightline.

We have chosen four combinations of clumping parameters to allow the average number of clumps per line of sight to take on values as low as 6 (in line with the results from @Mor2009) to as large as 105 in order to demonstrate a transition to the smooth density models. These parameter choices are listed in Table \[ClumpingParameters\].

[ccccc]{} 1 & .24 & $9.8 \times 10^4$ & 36864 & 105\
2 & .12 & $7.8 \times 10^5$ & 36864 & 25\
3 & .24 & $7.8 \times 10^5$ & 4608 & 13\
4 & .49 & $7.8 \times 10^5$ & 576 & 6.5\
\[ClumpingParameters\]

Ultimately we find that it is the volume filling fractions of the clumpy gas models that correlate most strongly with the integrated force exerted by the accretion radiation. If we let 