for all calculations in this study. We improve our statistics by mapping the energy and momentum deposited by the packets into a two-dimensional array of zones – a photon that scatters at spherical coordinates ($r$,$\phi$,$\theta$) is mapped to position ($r$,$\theta^\prime$) where $\theta^\prime = \theta$ if $0 \le \theta \le \pi/2$ and $\theta^\prime = \pi - \theta$ if $\pi/2 < \theta \le \pi$.

In Monte Carlo radiative transfer, the specific intensity of the radiation $I(\boldsymbol{r},\hat{\boldsymbol{n}},\lambda)$ is constructed by counting the number of photon packets with wavelength $\lambda$ that enter into each grid zone at position $\boldsymbol{r}$ and with direction vector $\boldsymbol{n}$ in a given interval of time. Specifically, the radiation force per volume $f_\mathrm{rad}$ at a given position is defined as $$\boldsymbol{f}_\mathrm{rad} \equiv \frac{1}{c}\int_{\omega,\lambda} {\rho \, \kappa_\lambda \, I \, \hat{\boldsymbol{n}} \, d\omega \, d\lambda} \; .
\label{RadiativeAcceleration}$$

To compute $f_\mathrm{rad}$ in a given zone of our computational domain with volume $\Delta V$ over a time interval $\Delta t$, we perform a sum a sum over all photon packets entering the zone. Each photon packet carries with it an energy $E_p$, a direction of travel $\hat{\boldsymbol{n}}_p$, and a wavelength $\lambda_p$. Associated with that wavelength is an opacity $\kappa(\lambda_p)$, measured per gram 