Radiative Transfer: Asymptotic Solutions of the Kinetic Equation of Radiation Propagation, $n$th Order Asymptotic Approximation and Improved Boundary Conditions

In the article, new asymptotic approximation of the $n$th order is obtained and proposed to be used in calculations of radiation propagation without scattering in optically thick media; the asymptotic approximation is much simpler and more precise than the known diffusion approximation. The rigorous derivation of the diffusion approximation equation and the equation of the radiation heat conduction approximation is obtained from the constructed asymptotic solution of the kinetic equation of radiation propagation in optically thick media. It is shown, that for optically thick media the asymptotic solution of the kinetic equation of radiation propagation without scattering is asymptotic expansion of the exact integral solution of that kinetic equation. Improved boundary conditions, which are essential for practical application in calculations of radiation propagation, are derived (for inner boundaries and outer boundaries with vacuum).