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.

Liquid–Vapor Interface of Formic Acid Solutions in Salt Water: A Comparison of Macroscopic Surface Tension and Microscopic in Situ X-ray Photoelectron Spectroscopy Measurements

The liquid–vapor interface is difficult to access experimentally but is of interest from a theoretical and applied point of view and has particular importance in atmospheric aerosol chemistry. Here we examine the liquid–vapor interface for mixtures of water, sodium chloride, and formic acid, an abundant chemical in the atmosphere. We compare the results of surface tension and X-ray photoelectron spectroscopy (XPS) measurements over a wide range of formic acid concentrations. Surface tension measurements provide a macroscopic characterization of solutions ranging from 0 to 3 M sodium chloride and from 0 to over 0.5 mole fraction formic acid. Sodium chloride was found to be a weak salting out agent for formic acid with surface excess depending only slightly on salt concentration. In situ XPS provides a complementary molecular level description about the liquid–vapor interface. XPS measurements over an experimental probe depth of 51 Å gave the C 1s to O 1s ratio for both total oxygen and oxygen from water. XPS also provides detailed electronic structure information that is inaccessible by surface tension. Density functional theory calculations were performed to understand the observed shift in C 1s binding energies to lower values with increasing formic acid concentration. Part of the experimental −0.2 eV shift can be assigned to the solution composition changing from predominantly monomers of formic acid to a combination of monomers and dimers; however, the lack of an appropriate reference to calibrate the absolute BE scale at high formic acid mole fraction complicates the interpretation. Our data are consistent with surface tension measurements yielding a significantly more surface sensitive measurement than XPS due to the relatively weak propensity of formic acid for the interface. A simple model allowed us to replicate the XPS results under the assumption that the surface excess was contained in the top four angstroms of solution.