Boundary condition for a semitransparent particle under diffuse irradiation using the \(\mathrm{P_{1}}\) approximation

Here we go. I am going to include most of the piddly mathematical steps to make sure it is clear.

Consider a semitransparent particle which we will denote region 1 surrounded by a transparent medium denoted region 2. The derivation starts with a statement that says when radiation at a particular frequency passes from region 1 to region 2, energy is conserved. So at the interface, we can write,

\begin{equation} dQ_{1}^{+}=dQ_{2}^{+}\\ \end{equation}

where the positive sign indicates the radiation is moving in the positive direction which we define to mean going from the particle to the surrounding medium. Then using the definition of spectral intensity \(I^{+}_{1,\nu}\) and spectral, directional reflectivity, \(\rho^{\prime}_{1\rightarrow 2,\nu}\)

\begin{equation} \label{eq:plusbalance}[1-\rho^{\prime}_{1\rightarrow 2,\nu}(\theta_{1})]I^{+}_{1,\nu}(\hat{s}_{1}\cdot\hat{n})\,\mathrm{d}\Omega_{1}\,\mathrm{d}A_{1}\,\mathrm{d}\nu_{1}=I^{+}_{2,\nu}(\hat{s}_{2}\cdot\hat{n})\,\mathrm{d}\Omega_{2}\,\mathrm{d}A_{2}\,\mathrm{d}\nu_{2}\\ \end{equation}

where \(\nu\) is the frequency of radiation traveling in the direction \(\hat{s}\) impinging upon a surface with a unit normal vector \(\hat{n}\) and area \(\,\mathrm{d}A\). This is precisely Eq. (17.39) in (Howell 2011). I will expound on the exact definition of the reflectivity and arrow notation I have adopted later. We have chosen to define intensity in terms of frequency. Since frequency does not change with refractive index, we can say

\begin{equation} \label{eq:nu}\nu_{1}=\nu_{2}.\\ \