Characteristics of the radiation emerging from the top of a rayleigh atmosphere—I Intensity and polarization

S. Chandrasekhar's solution of the equation of radiative transfer m a medium scattering according to Rayleigh's law has been used for an extensive series of computations of the scattered and diffusely reflected radiation directed outward from the top of a plane stratified model of the atmosphere. The results are discussed in two parts, the first of which deals with the intensity, degree of polarization, plane of polarization and the neutral points of the scattered and diffusely reflected radiation. Part II (see following article) will be a discussion of the upward flux, integrated over the hemisphere, as a function of wavelength of the incident radiation, and of the albedo of the atmospheric model for an energy distribution of the incident light similar to that of the extra-terrestrial solar spectrum. The results are compared with similar quantities for the diffusely scattered radiation from the sunlit sky.     

Emergent radiation from the atmosphere bounded by the two half Lambert surfaces composed of different albedo,II

For the evaluation of the effect of the non-uniform surface albedo on the emergent radiation from the atmosphere, the emergent radiation from the atmosphere bounded by the two half Lambert surfaces composed of different albedo is computed. This paper is the improved version of the previous paper (Takashima and Masuda, 1991). The atmosphere is assumed to be homogeneous, which is composed of aerosol, molecules, and absorbent gases. Their optical thicknesses are (1) 0.25, 0.23, and 0.02, and (2) 0.75, 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 calculated approximately by the contributions due to the multiple scattering in the atmosphere, and due to the diffusely or directly transmitted radiation through the atmosphere which is reflected by the surfaces once (4 interactive radiative modes between atmosphere and surface). Furthermore, to perform the hemispherical integration processing the radiative interaction, the transmission function based on the single scattering in the atmosphere is introduced and then the transmission function is averaged over the hemisphere with weighting function. The numerical simulation exhibits the extraordinary effect near the two half surface boundary of different albedoes. The effect decreases exponentially with the distance from the boundary. The effect depends on the atmospheric aerosol type, optical thickness, and surface albedo. The present version enables us to quantitatively discuss the radiative transfer trend near the boundary of two half surfaces. The upward radiance would simply be evaluated using the present scattering approximation method if the surface albedo is less than 0.3. The present method is thought of as a first step extending the one-dimensional radiative transfer model to two-dimensional using the doubling-adding method.     

Reflection and transmission by a vertically inhomogeneous planetary atmosphere

By appealing to the reciprocity principle simple expressions are derived for the plane albedo and the transmissivity of a vertically inhomogeneous, plane parallel atmosphere. The plane albedo is shown to equal the angular distribution of the reflected intensity for isotropie Illumination of unit intensity incident at the top of the atmosphere, while the transmissivity equals the angular distribution of the transmitted intensity for isotropie illumination of unit Intensity incident at the bottom of the atmosphere. Chandrasekhar's solution of the planetary problem (including a Lambert reflecting lower boundary) in terms of the solution to the standard problem (no reflecting ground) is extended to apply to an inhomogeneous atmosphere resting on a surface that reflects radiation anisotropically but with no dependence on the direction of incidence (anisotropic Lambert reflector). The computational aspects are discussed and a procedure for computing the planetary albedo and transmissivity Is outlined for a vertically inhomogeneous, anisotropically scattering atmosphere overlying a partially reflecting surface. Numerical verification and illustration are also provided and it is shown that the assumed vertical variation of the single scattering albedo strongly affects the plane albedo but only weakly the transmissivity.     

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.     

Accurate and approximate calculations of Raman scattering in the atmosphere of Neptune

Raman scattering by H₂ in Neptune's atmosphere has significant effects on its reflectivity for λ<0.5 μm, producing baseline decreases of ∼20% in a clear atmosphere and ∼10% in a hazy atmosphere. However, few accurate Raman calculations are carried out because of their complexity and computational costs. Here we present the first radiation transfer algorithm that includes both polarization and Raman scattering and facilitates computation of spatially resolved spectra. New calculations show that Cochran and Trafton's (1978, Astrophys. J. 219, 756-762) suggestion that light reflected in the deep CH₄ bands is mainly Raman scattered is not valid for current estimates of the CH₄ vertical distribution, which implies only a 4% Raman contribution. Comparisons with IUE, HST, and groundbased observations confirm that high altitude haze absorption is reducing Neptune's geometric albedo by ∼6% in the 0.22-0.26 μm range and by ∼13% in the 0.35-0.45 μm range. A sample haze model with 0.2 optical depths of 0.2-μm radius particles between 0.1 and 0.8 bars fits reasonably well, but is not a unique solution. We used accurate calculations to evaluate several approximations of Raman scattering. The Karkoschka (1994, Icarus 111, 174-192) method of applying Raman corrections to calculated spectra and removing Raman effects from observed spectra is shown to have limited applicability and to undercorrect the depths of weak CH₄ absorption bands. The relatively large Q-branch contribution observed by Karkoschka is shown to be consistent with current estimates of Raman cross-sections. The Wallace (1972, Astrophys. J. 176, 249-257) approximation, produces geometric albedo ∼5% low as originally proposed, but can be made much more accurate by including a scattering contribution from the vibrational transition. The original Pollack et al. (1986, Icarus 65, 442-466) approximation is inaccurate and unstable, but can be greatly improved by several simple modifications. A new approximation based on spectral tuning of the effective molecular single scattering albedo provides low errors for zenith angles below 70° in a clear atmosphere, although intermediate clouds present problems at longer wavelengths.     

Ultraviolet reflectivity and geometrical albedo of Titan

Intermediate resolution (6Å) photoelectric spectral scans of Titan, Saturn, Saturn's Rings and the Moon appear in two forms: ratio spectra of Titan vs the Rings and of Saturn vs the Rings, and relative reflectivities, which are compared to previously published results. Titan's geometrical albedo of 0.094 ± 0.012 was measured at 4255Å with a 50Å bandpass. From this and the spectral measurements, we derived the geometrical albedo as a function of wavelength. We find that the wavelength dependences of Titan's uv spectrum and the spectrum of Saturn's Rings are remarkably similar. No trace of any absorption bands is apparent. These results imply that uv gaseous absorption and Rayleigh scattering play a strongly subdued role in Titan's atmosphere. Any homogeneous atmospheric model implies that the absorber responsible for Titan's uv spectral albedo varies strongly with wavelength. On the other hand, we find that the uv observations can be satisfied by an absorber having a relatively weak dependence upon wavelength if an inhomogeneous atmospheric model is employed. In particular, a fine dust, which absorbs as 1/λ, can explain the uv observations provided that it is preferentially distributed high up in Titan's atmosphere where the optical depth from Rayleigh scattering is low. The likely presence of such a dust in Jupiter's atmosphere and the difficulty in explaining the nature of a continuous uv absorber which varies rapidly with wavelength suggest that the gas and aerosol in Titan's atmosphere are inhomogeneously distributed.     

Effects of Rayleigh-scattering polarization on reflected intensity: a fast and accurate approximation method for atmospheres with aerosols

Solar radiation reflected by the atmospheres of Neptune and Uranus is dominated by Rayleigh scattering at visible wavelengths, and thus subject to the effects of polarization. Ignoring these effects can lead to errors in reflected intensity of more than 9% in a clear atmosphere. But solving the full vector equation of transfer is computationally very costly, forcing approximations with limitations that are not well understood and not generally applicable to spatially resolved observations and complex atmospheric structures. Using accurate vector radiation transfer calculations, it is here shown that differences between vector and scalar results near zero phase angle have systematic dependencies on optical depth, single scattering albedo, and angle, that provide a basis for accurate approximation of the reflected intensities. With little computational cost, it is possible to calculate corrected spatially resolved scalar intensities that closely match vector intensities, with individual errors rarely exceeding 1%, and mean and RMS errors generally within a few tenths of 1%. The correction method accounts for the attenuating effects of clouds and molecular absorption.     

Effects of scattering and dust grain size on the temperature structure of protoplanetary discs: a three-layer approach

The temperature in the optically thick interior of protoplanetary discs is essential for the interpretation of millimetre observations of the discs, for the vertical structure of the discs, for models of the disc evolution and the planet formation, and for the chemistry in the discs. Since large icy grains have a large albedo even in the infrared, the effect of scattering of the diffuse radiation in the discs on the interior temperature should be examined. We have performed a series of numerical radiation transfer simulations, including isotropic scattering by grains with various typical sizes for the diffuse radiation as well as for the incident stellar radiation. We also have developed an analytic model including isotropic scattering to understand the physics concealed in the numerical results. With the analytic model, we have shown that the standard two-layer approach is valid only for grey opacity (i.e. grain size ≳10 μm) even without scattering. A three-layer interpretation is required for grain size ≲10 μm. When the grain size is 0.1–10 μm, the numerical simulations show that the isotropic scattering reduces the temperature of the disc interior. This reduction is nicely explained by the analytic three-layer model as a result of the energy loss by scatterings of the incident stellar radiation and of the warm diffuse radiation in the disc atmosphere. For grain size ≳10 μm (i.e. grey scattering), the numerical simulations show that the isotropic scattering does not affect the interior temperature. This is nicely explained by the analytic two-layer model; the energy loss by scattering in the disc atmosphere is exactly offset by the 'green-house effect' due to the scattering of the cold diffuse radiation in the interior.     

Inverse radiation modeling of Titan's atmosphere to assimilate solar aureole imager data of the Huygens probe

During the descent of the Huygens probe through Titan's atmosphere in January 2005, the Descent Imager/Spectral Radiometer (DISR) will perform upward and downward looking measurements at various spectral ranges and spatial resolutions. This internal radiation density could be estimated by radiative transfer calculations for Titan's atmosphere. However, to do this, the optical properties—i.e. volume extinction coefficient, single scattering albedo and scattering phase function—have to be prescribed at every altitude, and these are apriori not known. Herein, an inverse approach is investigated, which retrieves the single scattering albedo and the phase function of the aerosols from DISR observations. The method uses data from a DISR subinstrument, the Solar Aureole imager (SA), to estimate the optical properties of the atmospheric layer between two successive observation altitudes. A unique solution for one layer can in principle be calculated directly from a linear system of equations, but due to the sparseness of the data and the unavoidable noise in the measurements, the inverse problem is ill-posed. The problem is stabilized by the regularization method requiring smoothness of the resultant solution. A consistent set of solutions for all layers is obtained by iterating several times downward and upward through the layers. The method is tested in a simulated radiation density scenario for Titan, which is based on a microphysical aerosol model for the haze layer. Within this scenario, the expected coverage of SA data allows a reconstruction of the angular dependence of the scattering phase function with an explained variance better than 90%.     

Line Formation in a Purely Scattering, Optically Thick Atmosphere

A model problem in the theory of line formation in an optically thick, purely scattering, stellar atmosphere is considered. The integral equation of radiation transfer at line frequencies is solved numerically for a two-level atom in the approximation of complete frequency redistribution in scattering. The numerical results are compared with those calculated from equations of the asymptotic theory. On the basis of the asymptotic theory, the positions of intensity maxima in a line are found for different absorption profiles.     

Theoretical interpretation of photometric properties of the Martian surface and atmosphere

Earth-based UBV photometry, high-quality photographs from the Lowell Observatory collection, and Mariner 9 data have been combined with a new radiative transfer theory to derive physical parameters for the Martian surface and atmosphere, both before and during the 1971 dust storm. We find that the dust particles of the storm had a single-scattering albedo of 0.84 ± 0.02 and an asymmetry factor of 0.35 ± 0.10 in green (V) light. The geometric albedo of Mars was 0.15 and the phase integral 1.83, which yield 0.27 for the Bond albedo. The mean optical thickness of the “clear” atmosphere averaged over the whole planet was 0.15 ± 0.05 and was not detectably dependent on wavelength. Geometric albedos for the surface are 0.25 (light areas) and 0.17 (dark areas) in V, 0.095 in B (both areas), and 0.060 in U (both areas). The soil particles are moderately backward scattering with an asymmetry factor of ?0.20, indicating them to be rather opaque. The mean surface roughness, on a scale larger than that of individual dust particles and therefore large compared with the wavelength, is 0.57. This represents the depth/radius ratio of an average hole and it is only one-half as large as values typical for the Moon and asteroids.     

Average photon path-length of radiation emerging from finite atmospheres

The determination of the average path-length of photons emerging from a finite planeparallel atmosphere with molecular scattering is discussed. We examine the effects of polarisation on the average path-length of the emergent radiation by comparing the results with those obtained for the atmosphere where the scattering obeys the scalar Rayleigh function. Only the axial radiation field is considered for both cases.To solve this problem we have used the integro-differential equations of Chandrasekhar for the diffuse scattering and transmission functions (or matrices). By differentiation of these equations with respect to the albedo of single scattering we obtain new equations the solution of which gives us the derivatives of the intensities of the emergent radiation at the boundaries.As in the case of scalar transfer the principles of invariance by Chandrasekhar may be used to find an adding scheme to obtain both the scattering and transmission matrices and their derivatives with respect to the albedo of single scattering. These derivatives are crucial in determining the average path length.The numerical experiments have shown that the impact of the polarisation on the average pathlength of the emergent radiation is the largest in the atmospheres with optical thickness less than, or equal to, three, reaching 6.9% in the reflected radiation.     

Exact solution of the transport equation for radiative transfer with scattering albedo ωO < 1 using the Laplace transform and the Wiener-Hopf technique and an expression ofH-function

We have considered the transport equation for radiative transfer to a problem in semi-infinite non-conservative atmosphere with no incident radiation and scattering albedo ₀ < 1. Usint the Laplace transform and the Wiener-Hopf technique, we have determined the emergent intensity and the intensity at any optical depth. We have obtained theH-function of Dasgupta (1977) by equating the emergent intensity with the intensity at zero optical depth.     

Application of methane band-model parameters to the visible and near-infrared spectrum of Uranus

Laboratory band-model absorption coefficients of CH₄ are used to calculate the Uranus spectrum from 5400 to 10,400 Å. A good fit of both strong and weak bands for the Uranus spectrum over the entire wavelength interval is achieved for the first time. Three different atmospheric models are employed: a reflecting layer model, a homogeneous scattering layer model, and a clear atmosphere sandwiched between two scattering layers. The spectrum for the reflecting layer model exhibits serious discrepancies but shows that large amounts of CH₄ (5–10 km-am) are necessary to reproduce the Uranus spectrum. Both scattering models give reasonably good fits. The homogeneous model requires a particle scattering albedo $\text{(}\text{g}\text{?}\text{w}{}_{\mathrm{<mtext>p</mtext>}}\text{) ? 0.998}$ and an abundance per scattering mean free path $\text{(}\text{a}\text{?}\text{)}\text{of}\text{a}\text{?}\text{1}\text{km-am}$. The parameters derived from the sandwich layer model are: forsb the upper scattering layer a continuum single scattering albedo ($\text{g}\text{?}\text{w}{}_0$) of 0.995 and a scattering optical depth variable with wavelength consistent with Rayleigh scattering; for the clear layer they are a CH₄ abundance (a) of 2.2 km-am and an effective pressure (p) ? 0.1 atm; for the lower cloud deck a Lambert reflectivity (L) of 0.9 resulted. A severe depletion of CH₄ in the upper scattering layer is required. An enrichment of CH₄/H₂ over the solar ratio by a factor of 4–14 in the lower atmosphere is, however, indicated.     

The range of validity of the Eddington approximation

The Eddington approximation is often assumed to be useful only for optically thick media having a single-scattering albedo near unity. We present detailed evidence in this paper that, for homogeneous layers illuminated by a beam of radiation, the Eddington approximation predicts albedo and absorptivity reasonably well for all values of optical depth and single-scattering albedo, for several scattering phase functions (Rayleigh, Henyey-Greenstein, and Mie) having asymmetry factors less than or equal to $\text{1}\text{2}$. The worst errors are in the neighborhood of optical depth unity and single-scattering albedo 0.5. The Eddington approximation is further found to maintain good accuracy over almost the full range of incident beam directions and surface albedos. It is least accurate for the Mie phase function example, where one can obtain a dramatic improvement in accuracy by going over to the δ-Eddington approximation; this shows that the forward peak of the Mie phase function, and not its detailed shape, is the primary cause of diminished accuracy in the Eddington approximation.     

Scattering of radiation by a large particle with a random rough surface

Intensity and polarization of scattered light by an absorbing spherical particle with a random rough surface and with a radius larger than the wavelength of radiation are investigated. Multiple reflections of incident light on the rough surface are treated based on the multiple scattering theory.Within the limits of our approximation the model gives good agreement with typical scattering features by irregular shaped particles derived by microwave analogue experiments and laser measurements, namely a backward enhancement of the intensity, and a reduction of magnitude of polarization.Our treatment will be provide a powerful tool for studying scattering problems of interstellar and/or interplanetary grains as well as those of asteroids and the Moon.     

Average photon path-length in isotropically scattering finite atmospheres

The determination of the average path-length of photons in a finite isotropically scattering plane-parallel homogeneous atmosphere is discussed. To solve this problem we have used the kernel approximation method which easily allows us to find the derivatives of the intensity with respect to optical depth, optical thickness and albedo of single scattering.In order to check the results we have used another approach by exploiting the set of integrodifferential equations of Chandrasekhar for theX- andY-functions. This approach allows us to find the average path length only at the boundaries of the atmosphere but on the other hand it gives also the dispersion of the path-length distribution function, thus generating the input parameters for determining the approximate path-length distribution function. It occurred that the set so obtained is stable and the results are highly accurate.As a by-product we obtain the first two derivatives of theX- andY-functions with respect to the albedo of single scattering and optical thickness, and the mixed derivative.     

International conference on planetary satellites

Further analysis of visual (V) wavelength photometric function data for Saturn's ring is presented. Evidence indicating both that primary scattering dominates, and that mutual shadowing is an irrelevant concept, is reviewed. Simple anisotropic scattering radiative transfer models are used to define the probable ranges in the single scattering albedo, and in the general shape of the scattering phase function of the individual particles. Limitations on the mean perpendicular optical thickness of the ring are also obtained. Results indicate that the ring particles are highly efficient back-scatterers of visual radiation. Macroscopic particles account for the basic shape of the scattering phase function. Based on an infinite optical thickness for the ring, a minimum single scattering albedo ≈0.75 is found. Use of conservative scattering leads to a minimum optical thickness ≈0.7. The analysis is consistent with the ring particles being centimeter-size pieces of ice.     

Studies related to satellite thermal control: Measurements of earth-reflected sunlight and stability of thermal-control coatings

As part of a study of satellite thermal control, data were obtained on earth-reflected solar radiation and on the stability of thermal-control coatings. The intensity distribution of incoming solar radiation to earth was found to be shifted toward the ultraviolet region upon reflection by the atmosphere. As a result of the shift, the intensity of reflected radiation reaches a maximum in the near ultraviolet. At shorter wavelengths in the ultraviolet region, the intensity drops sharply to zero as a result of absorption of incoming radiation by ozone in the upper atmosphere. Comparisons of calculated intensity distributions with measured distributions for two different atmospheric conditions gave good agreement, except at the shorter ultraviolet wavelengths where the calculations did not adequately include the effect of ozone absorption. Measurements of stability of thermal-control coatings showed a lower degradation rate of white paints than was obtained on other flight experiments flown outside of the protective influence of earth's magnetosphere. The differences in degradation rate were much larger than expected, indicating that further study is required in the development of white coatings for spacecraft thermal control.     

Water distribution in Martian permafrost regions from joint analysis of HEND (Mars Odyssey) and MOLA (Mars Global Surveyor) data

We jointly analyze data from the High-Energy Neutron Detector (HEND) onboard the NASA Mars Odyssey spacecraft and data from the Mars Orbiter Laser Altimeter (MOLA) onboard the Mars Global Surveyor spacecraft. The former instrument measures the content of hydrogen (in the form of H₂O or OH) in the subsurface layer of soil and the latter instrument measures the surface albedo with respect to the flux of solar energy. We have checked the presence of a correlation between these two data sets in various Martian latitude bands. A significant correlation has been found between these data at latitudes poleward of 40° in the northern hemisphere and at latitudes 40°–60° in the southern hemisphere. This correlation is interpreted as evidence for the presence of stable water ice in these regions under a dry layer of soil whose thickness is determined by the condition for equilibrium between the condensation of water from the atmosphere and its sublimation when heated by solar radiation. For these regions, we have derived an empirical relation between the flux of absorbed solar radiation and the thickness of the top dry layer. It allows the burial depth of the water ice table to be predicted with a sub-kilometer resolution based on near-infrared albedo measurements. We have found no correlation in the southern hemisphere at latitudes >60°, although neutron data also suggest that water ice is present in this region under a layer of dry soil. We conclude that the thickness of the dry layer in this region does not correspond to the equilibrium condition between the water ice table and the atmosphere.