Time-dependent Radiative Transfer Equation Research Articles

Effect of Radiation upon the Post-Shock State

The effect of radiation upon the state of the gas behind a strong shock wave is studied. The equations of radiation gas dynamics are used, and it is shown that the two characteristic times, τ the time for radiative cooling, and λ/c, the mean time of photon flight, determine whether a strong shock wave is optically thin. The steady-state shock jump conditions for hydrogen are determined for plasmas which are optically thin, and they are compared with the Saha equilibrium solutions. The temperature decay, plasma velocity, and density behind strong shock waves are determined for the case of an optically thin bremsstrahlung radiating plasma. The time-dependent equations of the motion of a fluid interacting with radiation are examined and the radiative terms in these equations are presented as part of a general solution of the time-dependent radiative transfer equation.

Radiative transport of heat in lunar and planetary interiors

The aim of the present investigation has been to formulate and solve the problem of the radiative transfer of energy in solid globes of planetary size-both in their interiors as well as surface layer-with the radiative loss into space taken into account. Following a general statement of this problem in the introductory section, the time-dependent partial differential equation of radiative transfer has been formulated in Section II in terms of spherical polar coordinates, and a method developed which permits us to construct a solution of this equation, to an arbitrary order of accuracy, both for the emissivity or temperature as the dependent variable. In the latter case, the well-known equation of heat conduction is shown to obtain a as a limiting case if the mean free path of the energy-carrying photons tends to zero. In the third section, the problem of the radiative transfer of energy in planetary interiors, with radiative loss at the boundary, is solved analytically for the case of a homogeneous globe characterized by constant absorption coefficient. It is pointed out that, in such a case, a suitable substitution can render the problem linear if the emissivity—rather than temperature—is adopted as the dependent variable; the problem is then solvable in terms of Fourier series arising by the inversion of a certain finite Fourier sine transform of the emissivity. In Section IV, the same technique is applied to the solution of a plane-parallel problem of radiative energy transfer in surface layers subjected to a periodic insolation from above; and Section V contains the derivation of an explicit analytical solution of this problem in a closed form. The Appendix has been devoted to a formulation of the integral equation of the time-dependent and spherically symmetrical transfer problem, and the errors arising from the neglect of the time-dependence of the intensity (as distinct from that of the emissivity) of the radiation, committed by all earlier investigators of the problem, are explicitly evaluated.