Radiative transfer equation

The static radiative transfer equation in 3-D may be written

$$\hat{\vec n} \cdot \nabla I(\nu,\vec x,\hat{\vec n}) = \eta(\nu,\vec x) - \chi(\nu,\vec x)I(\nu,\vec x,\hat{\vec n})$$ (1)

where $I(\nu,\vec x,\hat{\vec n})$ is the specific intensity at frequency $\nu$, position $\vec x$, in the direction $\hat {\vec n}$, $\eta(\nu,\vec x)$ is the emissivity at frequency $\nu$ and position $\vec x$, and $\chi(\nu,\vec x)$ is the total extinction at frequency $\nu$ and position $\vec x$. The source function $S = {\eta}/{\chi}$. Here, we will work in the steady-state so that ${\partial I}/{\partial t} = 0$, and in Cartesian coordinates so the $\nabla = \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z}$ and the direction $\hat {\vec n}$ is defined by two angles $(\theta,\phi)$ at the position $\vec x$.