Rosseland approximation

Definition

Consider a participating medium with refractive index $n$, spectral extinction coefficient $\beta_\lambda > 0$, and temperature field $T(x)$, in local thermodynamic equilibrium. Let $E_{b\lambda}(T)$ be the black-body spectral emissive power in the medium and $\sigma$ the Stefan–Boltzmann constant. The Rosseland mean extinction coefficient $\beta_R$ is defined by $$\frac{1}{\beta_R} = \frac{\int_0^\infty \beta_\lambda^{-1} \, \partial_T E_{b\lambda}(T) \, d\lambda}{\int_0^\infty \partial_T E_{b\lambda}(T) \, d\lambda}.$$ The Rosseland approximation replaces the radiative transfer equation by the radiative heat flux $$q_r = -\frac{16 n^2 \sigma T^3}{3 \beta_R} \nabla T.$$ It is meant for optically thick media, in which the mean free path $1/\beta_\lambda$ is much smaller than the length over which $T$ changes.