Radiative transfer in spherical shell atmospheres: II. Asymmetric phase functions

In this paper we investigate the effects of sphericity on the radiation reflected from a planet with a homogeneous, conservative scattering atmosphere of optical thicknesses of 0.25 and 1.0. We considered a Henyey-Greenstein phase function with asymmetry factors of 0.5 and 0.7. Significant differences were found when these results were compared with the plane-parallel calculations. Also, large violations of the reciprocity theorem, which is only true for plane-parallel calculations, were noted. Results are presented for the radiance versus height distributions as a function of planetary phase angle. These results will be useful to researchers in the field of remote sensing and planetary spectroscopy.     

Circular polarization of sunlight reflected by planetary atmospheres

Multiple scattering calculations are performed in order to investigate the nature of the circular polarization of sunlight reflected by planetary atmospheres. Contour diagrams as a function of size parameter and phase angle are made for the integrated light from a spherical but locally plane-parallel atmosphere of spherical particles. To investigate the origin of the circular polarization, results are also computed for second-order scattering and for a simpler semiquantitative model of scattering by two particles. Observations of the circular polarization of the planets are presently too meager for accurate deduction of cloud particle properties. However, certain very broad constraints can be placed on the properties of the dominant cloud particles on Jupiter and Saturn. The cloud particle size and refractive index deduced for the Jupiter clouds by Loskutov, Morozhenko, and Yanovitskii from analyses of the linear polarization are not consistent with the circular polarization. The few available circular polarization observations of Venus are also examined.     

Rayleigh scattering in planetary atmospheres

The vector equation of radiative transfer is solved both for conservative and non-conservative planetary atmospheres using the method of discrete ordinates. The atmosphere, bounded by a Lambert bottom, is considered plane-parallel and homogeneous. The scattering in the atmosphere obeys the Rayleigh or Rayleigh-Cabannes law. The compiled package of FORTRAN codes allows us to find the Stokes parameters for such an atmosphere at arbitrary optical depth.     

Polarization effect on radiative transfer in planetary composite atmospheres with interacting interface

A procedure of computing the radiance and the polarization parameters of radiation diffusely reflected and transmitted by an inhomogeneous, plane-parallel terrestrial atmosphere bounded by a ruffled ocean surface is discussed with the aid of the adding method. If the atmosphere and the ocean are simulated by a number of homogeneous sublayers, the matrices of radiation reflected and transmitted diffusely by the atmosphere-ocean system can be expressed in terms of these matrices of sublayers by using only a couple of iterative equations in which the polarity effect of radiation is included. Furthermore, the upwelling radiance and the polarization degree of radiation at the top of the atmosphere can be calculated by using a single iterative equation without requiring the equation for the diffuse transmission matrix of radiation. The ruffled ocean surface can be treated as an interacting interface, where the transmitted radiation from beneath the ocean surface into the atmosphere is also taken into account into the derivation of equations. Finally, sample computations of the upwelling radiance and the polarization degree of radiation from the top of the atmosphere are carried out at the wavelength of 0.60 micron.     

A technique to deduce atmospheric temperature and constituent profiles from a planet's limb radiance profile

The concept is described of deducing the temperature and constituent profile of a planetary atmosphere from orbiter measurements of the planet's ir limb radiance profile. Expressions are derived for the weighting functions associated with the limb radiance profile for a Goody random band model. Analysis of the weighting functions for the Martian atmosphere indicates that a limb radiance profile in the 15 μm CO₂ band can be used to determine the Martian atmospheric temperature profile from 20 to 60 km. Simulation of the Martian limb radiance profile in the rotational water vapor band indicates that Martian water vapor mixing ratios can be inferred from limb radiance observations in a water vapor band.     

Exact solution of the problem of diffuse reflection and transmission in Rayleigh scattering by use of the Laplace transform and the theory of linear singular operators

We have considered six scalar equations which are obtained from the vector transport equation for radiative transfer to the problem of diffuse reflection and transmission in finite plane-parallel Rayleigh scattering atmosphere. By use of the Laplace transform and the theory of linear singular operators these equations have been solved exactly to get the angular distribution of the intensity diffusely reflected from the surface and diffusely transmitted below the surface.     

Reflection effect in close binaries. IV. Limb darkening of the reflected radiation incident from an extended surface of the secondary

The law of limb darkening has been calculated when the atmosphere of the primary component is illuminated by the extended surface of the secondary component in a binary system. The specific intensities calculated at infinity show marked changes when the plane-parallel approximation is replaced by the assumption of spherical symmetry. The middle portions of the illuminated surface reflect maximum radiation while the innermost and outermost layers show lesser amount of reflected radiation.     

Reflection effect in close binaries. II. Distribution of emergent radiation from the irradiated component along the line of sight

We have calculated the effects of irradiation from a point source observed at infinity. Plane-parallel approximation and spherically-symmetric approximations are employed in calculating the self-radiation field for the sake of comparison. It is found that there are considerable changes in the radiation received at infinity between the approximation of plane-parallel stratification and spherical symmetry.     

Emergent radiation from the atmosphere bounded by the two halves of the Lambert surface with different albedos

For the evaluation of the effect of the nonuniform surface albedo to the emergent radiation from the atmosphere, the emergent radiation from the atmosphere bounded by the two-halves of the Lambert surface with different albedos is computed. The principal plane is assumed to be perpendicular to the boundary of surfaces. The atmosphere is assumed to be homogeneous, which is composed of aerosol, molecules, and absorbent gases. Their optical thicknesses are 0.25, 0.23, and 0.02, respectively. The model aerosol is of the oceanic and water soluble types.In the computational procedure, the emergent radiation is approximated by the contributions due to the multiple scattering in the atmosphere, directly attenuated radiation, and radiation due to single scattering in the atmosphere which is reflected by the Lambert surface (up to 4 interactive radiative modes between atmosphere and surface). For quantitative analysis, results are compared with those of the atmosphere-uniform surface model, where the multiple scattering is considered. The numerical simulation exhibits the extraordinary effect near the surface boundary of different albedos. The effect decreases exponentially with the distance from the boundary. It is a function of the observational position, difference of surface albedos, optical thickness and aerosol type.The upward radiance would simply be evaluated using the present scattering approximation method if the atmosphere is in clear condition. Whereas in hazy condition, the effect of multiple scattering in the atmosphere should be considered more precisely, since the upward radiance exhibit a strong dependence on observational nadir angles due to multiple scattering in the atmosphere. Furthermore, it depends on the optical characteristics of aerosols.     

A new approach of the adding method for the computations of emergent radiation of an inhomogeneous plane-parallel planetary atmosphere

The procedure of computing the intensity and the polarization parameters of radiation diffusely reflected and transmitted by an inhomogeneous, plane-parallel planetary atmosphere is discussed with the aid of the adding method. If the atmosphere is simulated by a number of homogeneous sublayers (aerosols and ozone may be included), the matrices of radiation diffusely reflected and transmitted by the atmosphere can be expressed in terms of these matrices of sublayers by using only a couple of iterative equations with the polarity effect of radiation. This procedure is to be extended to the model atmosphere bounded by the surface reflector with a quite arbitrary phase matrix.     

The influence of forward-scattered light in transmission measurements of (exo)planetary atmospheres

The transmission of light through a planetary atmosphere can be studied as a function of altitude and wavelength using stellar or solar occultations, giving often unique constraints on the atmospheric composition. For exoplanets, a transit yields a limb-integrated, wavelength-dependent transmission spectrum of an atmosphere. When scattering haze and/or cloud particles are present in the planetary atmosphere, the amount of transmitted flux not only depends on the total optical thickness of the slant light path that is probed, but also on the amount of forward-scattering by the scattering particles. Here, we present results of calculations with a three-dimensional Monte Carlo code that simulates the transmitted flux during occultations or transits. For isotropically scattering particles, like gas molecules, the transmitted flux appears to be well-described by the total atmospheric optical thickness. Strongly forward-scattering particles, however, such as commonly found in atmospheres of Solar System planets, can increase the transmitted flux significantly. For exoplanets, such added flux can decrease the apparent radius of the planet by several scale heights, which is comparable to predicted and measured features in exoplanet transit spectra. We performed detailed calculations for Titan’s atmosphere between 2.0 and 2.8 μm and show that haze and gas abundances will be underestimated by about 8% if forward-scattering is ignored in the retrievals. At shorter wavelengths, errors in the gas and haze abundances and in the spectral slope of the haze particles can be several tens of percent, also for other Solar System planetary atmospheres. We also find that the contribution of forward-scattering can be fairly well described by modelling the atmosphere as a plane-parallel slab. This potentially reduces the need for a full three-dimensional Monte Carlo code for calculating transmission spectra of atmospheres that contain forward-scattering particles.     

The Standard Problem of Nonlocal Thermodynamic Equilibrium Radiative Transfer in the Rovibrational Band of the Planetary Atmosphere

The standard problem of radiative transfer in a rovibrational band is formulated for an optically semi-infinite plane-parallel planetary atmosphere using a model of a linear molecule with two vibrational states. The solution of the problem describes the variation with height of the population of the excited vibrational state due to the existence of the upper boundary of the atmosphere. We seek this solution as a function of the specially introduced dimensionless parameters—the atmosphere depth and four similarity parameters—and study it for the parameter values that can be realized in the planetary atmospheres, including the atmospheres of extrasolar planets. It is shown that an increase in the optical density of the atmosphere can reduce the population of an excited vibrational state in the band at the upper boundary of the atmosphere by as much as several orders of magnitude as compared to the population corresponding to the optically thin band limit. The anomalous decrease in the opacity of the atmosphere, when only several lines of the band are involved in radiative transfer, is predicted. We also determined the accuracy of calculating the population in the approximation of the Doppler line profile. An approximate formula is obtained for the dimensionless height of the boundary of the layer in which the local thermodynamic equilibrium exists for vibrational states. We propose a model and the formula following from this model to roughly evaluate the decrease in this height due to the impact on the population of additional radiative transitions between the state being considered and the underlying state belonging to another vibrational mode of the molecule.     

Radiative transfer in spherical shell atmospheres. III. Application to Venus

We have introduced a method of partitioning the radiance emerging from a planetary atmosphere in proportion to the average number of scatterings in each atmospheric layer in order to gain a more fundamental understanding of the so-called level of line formation. A realistic model of the Venus atmosphere was used to compute the radiance for a range of phase angles and two planetary colatitudes, namely, 20 and 90°. We computed the core and continuum radiances for the P(16) line of 8689-Å CO₂ band and introduced two ways of computing an effective temperature. Both definitions yielded similar results. We found that these effective temperatures varied little with phase angles up to 120°, but fell rather rapidly beyond this point. Also colder effective temperatures were found as we went from equator to pole. The results obtained are all consistent with the spectroscopic temperature determination from CO₂ band studies. We have also defined an effective optical depth, τ_eff, which we feel gives a better understanding of the level of line formation than other definitions used to date.     

Polarization of radiation from stellar atmospheres. The grey case

An accurate numerical method is presented for the solution of the transfer equations in a plane-parallel atmosphere in which scattering occurs according to Rayleigh's law. Some results are given for the polarization and limb darkening of both integrated and monochromatic radiation emerging from grey atmospheres with various ratios of scattering to absorption. The method is equally applicable to non-grey atmospheres.     

The effect of aerosol shape in retrieving optical properties of cloud particles in the planetary atmospheres from the photopolarimetric data. Jupiter

The influence of the assumed shape of aerosols on the estimates of the refractive index and size of particles (based on the data of ground-based spectropolarimetric measurements) is investigated for Jupiters cloud layer. In the present analysis, we supposed the atmospheric particles to be chaotically oriented spheroids and cylinders with a gamma size distribution. Their single-scattering characteristics were calculated with the T-matrix method, and the intensity and degree of linear polarization of the radiation reflected by the center of the planetary disk were found by solving the vector radiative-transfer equation with consideration for multiple scattering in a plane-parallel atmosphere. We considered a spectral interval from 0.423 to 0.798 µm and phase angles from 0° to 11°. It has been shown that, if we use the optical characteristics of aerosols found within the frames of a spherical model (Mishchenko, 1990a), the models with the nonspherical particles considered here cannot fit the observational data. The refractive index and the sizes of spheroidal and cylindrical particles were estimated from a comparison of the data of measurements and calculations, and the simplest models for the Jovian cloud layer structure have been considered. We have concluded that the optical parameters of cloud particles (specifically, the refractive index) cannot be reliably estimated only on the basis of measurements made in a narrow range of phase angles.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 2, 2005, pp. 117–127.Original Russian Text Copyright © 2005 by Dlugach, Mishchenko.     

Stellar Occultations by turbulent planetary atmospheres: A wave-optical theory including a finite scale height

A generalized wave-optical theory of stellar occultations by a turbulent planetary atmosphere is developed. The finite scale height of the atmosphere is retained for the first time. It is found that the finite scale height of the atmosphere affects the scintillations observed during the occultation in a number of ways which are most easily understood in terms of an effective Fresnel scale. We demonstrate the validity of a phase-changing screen approximation for occultation by a turbulent atmosphere in parameter ranges of general interest. Using this approximation various statistical properties of the fluctuating intensity are calculated explicitly. We present expressions for the total scintillation power, correlation function of the intensity, the cross-correlation at two frequencies, and its application to refractivity determinations. All of these expressions are given as a function of occultation depth and of parameters of the mean atmosphere and turbulence.     

A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere

A discrete spherical harmonics method is developed for the radiative transfer problem in inhomogeneous polarized planar atmosphere illuminated at the top by a collimated sunlight while the bottom reflects the radiation. The method expands both the Stokes vector and the phase matrix in a finite series of generalized spherical functions and the resulting vector radiative transfer equation is expressed in a set of polar directions. Hence, the polarized characteristics of the radiance within the atmosphere at any polar direction and azimuthal angle can be determined without linearization and/or interpolations. The spatial dependent of the problem is solved using the spectral Chebyshev method. The emergent and transmitted radiative intensity and the degree of polarization are predicted for both Rayleigh and Mie scattering. The discrete spherical harmonics method predictions for optical thin atmosphere using 36 streams are found in good agreement with benchmark literature results. The maximum deviation between the proposed method and literature results and for polar directions \(\vert \mu \vert \geq0.1 \) is less than 0.5% and 0.9% for the Rayleigh and Mie scattering, respectively. These deviations for directions close to zero are about 3% and 10% for Rayleigh and Mie scattering, respectively.     

A method of computing the emergent radiation by the atmosphere in the region ranging from ultraviolet to infrared

A method of computing the diffuse reflection and transmission radiation by an inhomogeneous, plane-parallel planetary atmosphere with internal emission source is discussed by use of the adding method. If the atmosphere is simulated by a number of homogeneous sub-layers, the radiation diffusely reflected or transmitted by the atmosphere can be expressed in terms of the reflection and transmission matrices of the radiation of sub-layers. The diffusely transmitted radiation due to the internal emission source can be also easily computed in the same manner. These equations for the emergent radiation are in a quite general form and are applicable to radiative transfer in the atmosphere in the region from ultraviolet to infrared radiation. With this method, the tiresome treatment due to the polarity effect of radiation is overcome.     

Calculation of effective optical depth of absorption line formation in homogeneous semi-infinite planetary atmosphere during anisotropic scattering

The algorithm for determining effective optical thickness of absorption line formation in a plane-parallel homogeneous planetary atmosphere is presented. The case of anisotropic scattering is considered. The results of numerical calculations of τ _e (μ₀) at the scattering angle γ = π for some values of the single scattering albedo λ and the parameter of the Heyney-Greenstein scattering indicatrix g are given. The refined equation for the function T ^m (−μ, μ₀) is presented.     

A spherical model for computing polarized radiation in Titan's atmosphere

The Huygens descent through Titan's atmosphere in January 2005 will provide invaluable information about Titan's atmospheric composition and aerosol properties. The Descent Imager/Spectral Radiometer (DISR) will perform upward and downward looking radiation observations at various spectral ranges and spatial resolutions. To prepare the DISR data interpretation we have developed a new model for radiation transfer in Titan's atmosphere. The model solves for the full three-dimensional polarized radiation field in spherical geometry. However, the atmosphere itself is assumed to be spherically symmetric. The model is initialized with a fast-to-compute plane–parallel solution based on the doubling and adding algorithm that incorporates a spherical correction for the incoming direct solar beam. The full three-dimensional problem is then solved using the characteristics method combined with the Picard iterative approximation as described in Rozanov et al. (J. Quant. Spectrosc. Radiat. Transfer 69 (2001) 491). Aerosol scattering properties are calculated with a new microphysical model. In this formulation, aerosols are assumed to be fractal aggregates and include methane gas absorption embedded into the extinction coefficient. The resulting radiance of the model atmosphere's internal field is presented for two prescribed DISR wavelengths.