We present a comprehensive analytical study of radiative transfer using the
method of moments and include the effects of non-isotropic scattering in the
coherent limit. Within this unified formalism, we derive the governing
equations and solutions describing two-stream radiative transfer (which
approximates the passage of radiation as a pair of outgoing and incoming
fluxes), flux-limited diffusion (which describes radiative transfer in the deep
interior) and solutions for the temperature-pressure profiles. Generally, the
problem is mathematically under-determined unless a set of closures (Eddington
coefficients) is specified. We demonstrate that the hemispheric (or
hemi-isotropic) closure naturally derives from the radiative transfer equation
if energy conservation is obeyed, while the Eddington closure produces spurious
enhancements of both reflected light and thermal emission. We concoct recipes
for implementing two-stream radiative transfer in stand-alone numerical
calculations and general circulation models. We use our two-stream solutions to
construct toy models of the runaway greenhouse effect. We present a new
solution for temperature-pressure profiles with a non-constant optical opacity
and elucidate the effects of non-isotropic scattering in the optical and
infrared. We derive generalized expressions for the spherical and Bond albedos
and the photon deposition depth. We demonstrate that the value of the optical
depth corresponding to the photosphere is not always 2/3 (Milne's solution) and
depends on a combination of stellar irradiation, internal heat and the
properties of scattering both in optical and infrared. Finally, we derive
generalized expressions for the total, net, outgoing and incoming fluxes in the
convective regime.Comment: Accepted by ApJS. 23 pages, 11 figures, 3 tables, 158 equations. No
change from previous version except for title (to match ApJS convention
