Analytical Models of Exoplanetary Atmospheres. II. Radiative Transfer via the Two-stream Approximation

摘要: We present a comprehensive analytical study of radiative transfer using the method moments and include effects non-isotropic scattering in coherent limit. Within this unified formalism, we derive governing equations solutions describing two-stream (which approximates passage radiation as pair outgoing incoming fluxes), flux-limited diffusion describes deep interior) for temperature-pressure profiles. Generally, problem is mathematically under-determined unless set closures (Eddington coefficients) specified. demonstrate that hemispheric (or hemi-isotropic) closure naturally derives from equation if energy conservation obeyed, while Eddington produces spurious enhancements both reflected light thermal emission. concoct recipes implementing stand-alone numerical calculations general circulation models. use our to construct toy models runaway greenhouse effect. new solution profiles with non-constant optical opacity elucidate infrared. generalized expressions spherical Bond albedos photon deposition depth. value depth corresponding photosphere not always 2/3 (Milne's solution) depends on combination stellar irradiation, internal heat properties Finally, total, net, fluxes convective regime.