A two-component equilibrium-diffusion limit

The equilibrium-diffusion limit of the radiative transfer equations is characterized by a medium that is optically thick and diffusive for photons of all frequencies. In reality, this condition is almost never met because the transport medium tends to be optically thin for photons of sufficiently high frequency. Motivated by this fact, we derive a new asymptotic limit of the radiative transfer equations that is characterized by two photon components: one for which the medium is optically thick and diffusive, and the other for which the medium is extremely optically thin. In this limit, the leading-order material temperature satisfies a time-dependent diffusion equation, and the leading-order radiation intensity for the optically thick photons is given by the Planck function evaluated at the leading-order material temperature, but the radiation intensity for the optically thin photons is zero through first order. The O(2) radiation intensity for the optically thin photons satisfies a quasi steady-state transport equation with zero interaction terms and a Planck emission term that depends upon the leading-order material temperature. We also discuss alternative scalings associated with the two-component limit that are characterized by stronger coupling between the material and the optically thin component