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

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.

The isotope composition of selenium in chondrites constrains the depletion mechanism of volatile elements in solar system materials

Solar nebula processes led to a depletion of volatile elements in different chondrite groups when compared to the bulk chemical composition of the solar system deduced from the Sun's photosphere. For moderately-volatile elements, this depletion primarily correlates with the element condensation temperature and is possibly caused by incomplete condensation from a hot solar nebula, evaporative loss from the precursor dust, and/or inherited from the interstellar medium. Element concentrations and interelement ratios of volatile elements do not provide a clear picture about responsible mechanisms. Here, the abundance and stable isotope composition of the moderately- to highly-volatile element Se are investigated in carbonaceous, ordinary, and enstatite chondrites to constrain the mechanism responsible for the depletion of volatile elements in planetary bodies of the inner solar system and to define a δ(82/78)Se value for the bulk solar system. The δ(82/78)Se of the studied chondrite falls are identical within their measurement uncertainties with a mean of −0.20±0.26‰ (2 s.d., n=14n=14, relative to NIST SRM 3149) despite Se abundance depletions of up to a factor of 2.5 with respect to the CI group. The absence of resolvable Se isotope fractionation rules out a kinetic Rayleigh-type incomplete condensation of Se from the hot solar nebula or partial kinetic evaporative loss on the precursor material and/or the parent bodies. The Se depletion, if acquired during partial condensation or evaporative loss, therefore must have occurred under near equilibrium conditions to prevent measurable isotope fractionation. Alternatively, the depletion and cooling of the nebula could have occurred simultaneously due to the continuous removal of gas and fine particles by the solar wind accompanied by the quantitative condensation of elements from the pre-depleted gas. In this scenario the condensation of elements does not require equilibrium conditions to avoid isotope fractionation. The results further suggest that the processes causing the high variability of Se concentrations and depletions in ordinary and enstatite chondrites did not involve any measurable isotope fractionation. Different degrees of element depletions and isotope fractionations of the moderately-volatile elements Zn, S, and Se in ordinary and enstatite chondrites indicate that their volatility is controlled by the thermal stabilities of their host phases and not by the condensation temperature under canonical nebular conditions.