Padé approximants for radiative transfer in inhomogeneous medium

The Padé approximants is used to estimate the albedo for an inhomogeneous semi-infinite medium. The single-scattering albedo is assumed to fall off exponentially with optical depth.     

The Shadowing Effect for Regolith-like Surfaces

We describe a numerical model that allows the shadowing probability and photometric characteristics to be calculated in the general case of a disconnected multi-valued surface (regolith-like medium) in the single-scattering approximation when the phase function of a surface element is known. We investigate a statistically homogeneous medium and a medium with a density gradient (along the normal to the mean surface) with Gaussian and fractal statistical properties. A comparison of the phase functions of reflectance for a disconnected multi-valued surface with those for the previously developed model of a medium composed of opaque spherical particles shows qualitative agreement. The results of our calculations are also compared with the phase function of brightness for the Moon's surface. The model is shown to yield a function that matches the observed one for phase angles 10° at very low densities of the medium. Achieving better agreement requires invoking additional mechanisms for the formation of the phase function of brightness.     

Energy albedo for multiregion slabs

An asymptotic solution for the equation of radiative transfer in an inhomogeneous medium was obtained on the basis of the corresponding solutions for homogeneous sub-layers in the slowing down region. Function relations between the reflection and transmission coefficients for the whole slab and those of the sublayers are given. The invariant embedding concepts are used to get the reflection and transmission coefficients for the sub-layers. We assumed different models for the slowing-down kernels. Laplace transform was used to transform the Boltzmann equation to one velocity approximation with re-scaled mean-free path and single-scattering albedo. Numerical results are given for energy albedo as a function of the mass number of the host medium.     

Silicon vidicon imaging of jupiter: 4100- to 8300-Å absolute reflectivities and limb darkening of spatially resolved regions

Jupiter was observed in six continuum wavelength channels in the region 4100–8300 Å, using a silicon vidicon imaging photometer. Spectral reflectivities and high spatial resolution limb-darkening curves for several belts and zones have been extracted from the data. Simple model fits to the data yield information regarding spectral and spatial variations in single-scattering albedos and shape of particle single-scattering phase functions. Belts appear to be more backscattering than zones, particularly in the blue. The data are in moderate agreement with limb-darkening predicted by models derived from the center-to-limb variation in equivalent width of the H₂ 4-0 S(1) quadrupole line (Cochran, 1976) in the South Tropical Zone, but strongly disagree with the results of such models for the North Equatorial Belt.     

Intensity of radiation reflected by a vertically inhomogeneous medium through multiple scattering

To facilitate the computation of the radiative intensity reflected, upon multiple scattering, by a vertically inhomogeneous medium, an implicit formula for integrating the invariant imbedding equation for Fourier-decomposed reflection function is derived starting with the formal solution: the height variations of the single-scattering albedo and the phase function characterizing the degree of inhomogeneity are thereby approximated by piece-wise, but continuous, linear functions of optical height , while the reflection function is approximated by a piece-wise quadratic polynomial in over each integration step. Using these approximations, the integration involved in the formal solution is then carried out analytically, yielding a correctortype formula for finding the reflection function at each step of . It is expected that this formula is capable of handling general cases of inhomogeneous media where both single-scattering albedo and phase function are allowed to vary continuously with height.Similar, but explicit expressions are also derived for the single and the second-order scattering solutions, with which the higher-order Fourier terms of reflection function are to be approximated, thereby enabling us to avoid the iterative process.     

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.     

On the nature of the blue and ultraviolet absorption in the Jovian atmosphere

Spatially resolved reflectivities from 3000 to 6600 Å of three positions from the center to the limb of the Jovian Equator, North Equatorial Belt, and North Tropical Zone are analyzed to determine the vertical distribution and wavelength dependence of various sources of blue and uv absorption. Six different models of the distribution of absorbing dust particles are examined. In each model, the variation of dust optical depth and cloud single-scattering albedo are determined. Only those models having dust above the upper NH₃ cloud layer will fit the data. The high altitude dust distribution is approximately uniform over the three regions examined. The contrast in reflectivity of the belts and zones may be modeled by a different cloud single-scattering albedo in the different regions.     

A study of the properties of a local dust storm with Mars Express OMEGA and PFS data

We present observations of a local dust storm performed by the OMEGA and PFS instruments aboard Mars Express. OMEGA observations are used to retrieve the dust single-scattering albedo in the spectral range 0.4-4.0 μm. The single-scattering albedo shows fairly constant values between 0.6 and 2.6 μm, and a sharp decrease at wavelengths shorter than 0.6 μm, in agreement with previous studies. It presents a small absorption feature due to ferric oxide at 0.9 μm, and a strong absorption feature due to hydrated minerals between 2.7 and 3.6 μm. We use a statistical method, the Independent Component Analysis, to determine that the dust spectral signature is decoupled from the surface albedo, proving that the retrieval of the single-scattering albedo is reliable, and we map the dust optical thickness with a conventional radiative transfer model. The effect of the dust storm on the atmospheric thermal structure is measured using PFS observations. We also simulate the thermal impact of the dust storm using a one-dimensional atmospheric model. A comparison of the retrieved and modeled temperature structures suggests that the dust in the storm should be confined to the 1-2 lowest scale heights of the atmosphere. However, the observed OMEGA reflectance in the CO₂ absorption bands does not support this suggestion.     

Calculation of single-scattering albedos: Comparison of Mie results with Hapke approximations

The grain size of water ice can be determined from its near-infrared spectrum, which has numerous diagnostic absorption bands of different opacities. Models that have been used to determine water ice grain size from infrared spectra of icy outer Solar System objects have shown discrepancies in modeled grain size of a factor of two or more. Here the single-scattering albedo calculated using the commonly used Hapke model given by Roush [Roush, T.L., 1994. Icarus 108, 243-254] is compared with the exact calculation for spheres from a Mie series. An earlier approximation of single-scattering albedo called the Hapke “slab” model is also used in the comparison. All three models are implemented using the same optical constants for water ice at ∼110 K. Results are displayed for a large range of grain sizes from 1 μm to 1 mm. In general neither Hapke model can mimic the Rayleigh effects from particles sized near the wavelength of light that the Mie model predicts. For 10 μm particles, the slab model matches the Mie calculation quite well, but larger sizes are more discrepant. The Hapke/Roush model grain size needs to be ∼2.5 times larger to mimic the Mie results, and there are additional discrepancies in the continuum levels and band strengths. The Mie calculation for spheres is recommended for analysis of unknown remote sensing measurements, as it can mimic the spectra of oblate, prolate, and hollow particles given by equivalent sphere theories.     

A model of Titan's aerosols based on measurements made inside the atmosphere

The descent imager/spectral radiometer (DISR) instrument aboard the Huygens probe into the atmosphere of Titan measured the brightness of sunlight using a complement of spectrometers, photometers, and cameras that covered the spectral range from 350 to 1600 nm, looked both upward and downward, and made measurements at altitudes from 150 km to the surface. Measurements from the upward-looking visible and infrared spectrometers are described in Tomasko et al. [2008a. Measurements of methane absorption by the descent imager/spectral radiometer (DISR) during its descent through Titan's atmosphere. Planet. Space Sci., this volume]. Here, we very briefly review the measurements by the violet photometers, the downward-looking visible and infrared spectrometers, and the upward-looking solar aureole (SA) camera. Taken together, the DISR measurements constrain the vertical distribution and wavelength dependence of opacity, single-scattering albedo, and phase function of the aerosols in Titan's atmosphere.Comparison of the inferred aerosol properties with computations of scattering from fractal aggregate particles indicates the size and shape of the aerosols. We find that the aggregates require monomers of radius 0.05 μm or smaller and that the number of monomers in the loose aggregates is roughly 3000 above 60 km. The single-scattering albedo of the aerosols above 140 km altitude is similar to that predicted for some tholins measured in laboratory experiments, although we find that the single-scattering albedo of the aerosols increases with depth into the atmosphere between 140 and 80 km altitude, possibly due to condensation of other gases on the haze particles. The number density of aerosols is about 5/cm³ at 80 km altitude, and decreases with a scale height of 65 km to higher altitudes. The aerosol opacity above 80 km varies as the wavelength to the −2.34 power between 350 and 1600 nm.Between 80 and 30 km the cumulative aerosol opacity increases linearly with increasing depth in the atmosphere. The total aerosol opacity in this altitude range varies as the wavelength to the −1.41 power. The single-scattering phase function of the aerosols in this region is also consistent with the fractal particles found above 60 km.In the lower 30 km of the atmosphere, the wavelength dependence of the aerosol opacity varies as the wavelength to the −0.97 power, much less than at higher altitudes. This suggests that the aerosols here grow to still larger sizes, possibly by incorporation of methane into the aerosols. Here the cumulative opacity also increases linearly with depth, but at some wavelengths the rate is slightly different than above 30 km altitude.For purely fractal particles in the lowest few km, the intensity looking upward opposite to the azimuth of the sun decreases with increasing zenith angle faster than the observations in red light if the single-scattering albedo is assumed constant with altitude at these low altitudes. This discrepancy can be decreased if the single-scattering albedo decreases with altitude in this region. A possible explanation is that the brightest aerosols near 30 km altitude contain significant amounts of methane, and that the decreasing albedo at lower altitudes may reflect the evaporation of some of the methane as the aerosols fall into dryer layers of the atmosphere. An alternative explanation is that there may be spherical particles in the bottom few kilometers of the atmosphere.     

The tilt effect for Saturn's rings

Multiple-scattering computations are carried out to explain the variation of the observed brightness of the A and B rings of Saturn with declination of the Earth and Sun. These computations are performed by a doubling scheme for a homogeneous plane-parallel scattering medium. We test a range of choices for the phase function, albedo for single scattering, and optical depth of both the rings. Isotropic scattering and several other simple phase functions are ruled out, and we find that the phase function must be moderately peaked in both the forward and backward directions. The tilt effect can be explained by multiple scattering in a homogeneous layer, but, for ring B, this requires a single-scattering albedo in excess of 0.8. The brightest part of ring B must have an optical depth greater than 0.9. We find that the tilt effect for ring A can be reproduced by particles having the same properties as those in ring B with the optical depth for the A ring in the range 0.4 to 0.6.     

X-ray resonance scattering in a spherically symmetric coronal model

In the solar corona the opacities of some of the prominent X-ray emission lines are on the order of 1 over typical coronal path lengths. We present and discuss a particular solution of the radiative transfer problem involving an extended, spherically symmetric coronal shell radiating isotropic, homogeneous emission in which single-scattering also takes place. Within the context of this simplified model we find that scattered radiation is an important contribution to the total emergent resonance line flux and that for the He-like family of resonance (r), intercombination (i), and forbidden (f) lines, the ratio G=(f + i)/r would decrease as a function of optical depth for disk-center emission in an extended spherically symmetric corona.     

Multispectral imaging observations of Neptune’s cloud structure with Gemini-North

Observations of Neptune were made in September 2009 with the Gemini-North Telescope in Hawaii, using the NIFS instrument in the H-band covering the wavelength range 1.477–1.803 μm. Observations were acquired in adaptive optics mode and have a spatial resolution of approximately 0.15–0.25″.The observations were analysed with a multiple-scattering retrieval algorithm to determine the opacity of clouds at different levels in Neptune’s atmosphere. We find that the observed spectra at all locations are very well fit with a model that has two thin cloud layers, one at a pressure level of ∼2 bar all over the planet and an upper cloud whose pressure level varies from 0.02 to 0.08 bar in the bright mid-latitude region at 20–40°S to as deep as 0.2 bar near the equator. The opacity of the upper cloud is found to vary greatly with position, but the opacity of the lower cloud deck appears remarkably uniform, except for localised bright spots near 60°S and a possible slight clearing near the equator.A limb-darkening analysis of the observations suggests that the single-scattering albedo of the upper cloud particles varies from ∼0.4 in regions of low overall albedo to close to 1.0 in bright regions, while the lower cloud is consistent with particles that have a single-scattering albedo of ∼0.75 at this wavelength, similar to the value determined for the main cloud deck in Uranus’ atmosphere. The Henyey-Greenstein scattering particle asymmetry of particles in the upper cloud deck are found to be in the range g ∼ 0.6–0.7 (i.e. reasonably strongly forward scattering).Numerous bright clouds are seen near Neptune’s south pole at a range of pressure levels and at latitudes between 60 and 70°S. Discrete clouds were seen at the pressure level of the main cloud deck (∼2 bar) at 60°S on three of the six nights observed. Assuming they are the same feature we estimate the rotation rate at this latitude and pressure to be 13.2 ± 0.1 h. However, the observations are not entirely consistent with a single non-evolving cloud feature, which suggests that the cloud opacity or albedo may vary very rapidly at this level at a rate not seen in any other giant-planet atmosphere.     

Electromagnetic radiation from line sources interacting with a moving uniaxial dielectric or plasma half-space

The problem of electromagnetic radiation from electric and magnetic line sources interacting with a moving uniaxially anisotropic dielectric or plasma half-space is treated. The anisotropy of the plasma is due to an infinitely strong magnetizing field impressedparallel to the motion of plasma. The line source is oriented normal to the direction of medium motion. TheE and theH modes are excited independently in the medium by the magnetic and the electric line sources, respectively. Invoking the saddle-point method of integration, the far-zone radiation field and the radiation pattern are obtained for both line sources. It is found that the radiation from an electric line source is not affected by the anisotropy of the moving medium and that a magnetic line source invacuum has finite components of radiation in directions which are parallel and antiparallel to the direction of motion, for the case of a uniaxial plasma in contrast to the results for a uniaxial dielectric. Numerical results for the far-zone radiation pattern, referring to a magnetic line source, are presented for several values of parameters characterizing the non-dimensional velocity, the anisotropy of the medium, the electron-plasma density and the location of the line source.     

The amplitude of the coherent backscattering intensity peak for discrete random media: Effect of packing density

The amplitude of the coherent backscattering intensity peak is computed for a medium composed of densely packed, randomly positioned particles. The cyclical component of the Stokes reflection matrix at exactly the backscattering direction is expressed in terms of the ladder component, and the ladder component is rigorously computed by numerically solving the vector radiative transfer equation. The effect of packing density is accounted for by multiplying the single-scattering Mueller matrix by the static structure factor computed in the Percus-Yevick approximation. It is shown that increasing packing density can substantially reduce the amplitude of the copolarized coherent backscattering peak, especially for smaller particles, and can make it significantly lower than 2. The effect of packing density on the amplitude of the cross-polarized peak is significantly weaker.     

Photometry and polarimetry of Jupiter at large phase angles: I. Analysis of imaging data of a prominent belt and a zone from pioneer 10

Limb-darkening curves are derived from Pioneer 10 imaging data for Jupiter's STrZ (?18 to ?21° latitude) and SEBn (?5 to ?8° latitude) in red and blue light at phase angles of 12, 23, 34, 109, 120, 127, and 150°. Inhomogeneous scattering models are computed and compared with the data to constrain the vertical structure and the single-scattering phase functions of the belt and the zone in each color. The very high brightness observed at a 150° phase angle seems to require the presence of at lleast a thin layer of reasonably bright and strongly forward-scattering haze particles at pressure levelsof about 100 mbar or less above both belts and zones. Marginally successful models have been constructed in which a moderate optical thickness (τ ≥ 0.5) of haze particles was uniformly distributed in the upper 25 km-amagats of H₂. Excellent fits to the data were obtained with models having a thin (optical depths of a few tenths) haze conentraated above most of the gas. Following recent spectrospcopicanalyses, we have placed the main “cloud” layer or layers beneath about 25 km-amagats of H₂, although successful fits to our continuum data probably could be achieved also if the clouds were permitted to extend all the way up to the thin haze layer. Similarly, below the haze level our data cannot distinguish between models having two clouds separated by a clear space as suggested by R. E. Danielson and M. G. Tomasko and models with a single extensive diffuse cloud having an H₂ abundance of a few kilometer-amagats per scattering mean free path as described by W. D. Cochran. In either case, the relative brightness of the planet at each phase angle primarily serves to constrain the single-scattering phase functions of the Jovian clouds at the corresponding scattering angles. The clouds in these models are characterized by single-scattering phase functions having strong forward peaks and modest backward-scattering peaks, indicating cloud particles with dimensions larger than about 0.6 μm. In our models, a lower single-scattering albedo of the cloud particles in the belt relative to the zone accounts for the contrast between these regions. If an increased abundance of absorbing dust above uniformly bright clouds is used to explain the contrast between belts and zones at visible wavelengths, the limb darkening is steeper than that observed for the SEBn in blue light at small phase angles. The phase integral for the planet calculated for either the belt or the zone model in either color lies in the range 1.2 to 1.3. If a value of 1.25 is used with D.J. Taylor's bolometric geometric albedo of 0.28, the planet emits 2.25 or 1.7 times the energy it absorbs from the Sun if it effective temperature is 134 or 125°K, respectively—roughly as expected from current theories of the cooling of Jupiter's interior.     

Shadow mechanism and the opposition effect of brightness of atmosphereless celestial bodies

We consider the Irvine-Yanovistkii modification of the shadow model developed by Hapke for the opposition effect of brightness. The relation between the single scattering albedo ω and the transparency coefficient of particles κ is suggested to be used in the form κ = (1 ? ω) ⁿ , which allows the number of unknowns in the model to be reduced to two parameters (the packing density of particles g and ω) and the single-scattering phase function χ(α). The analysis of spectrophotometric measurements of the moon and Mars showed that the data on the observed opposition effect and the changes in the color index with the phase angle α well agree if the values of n = 0.25 and g = 0.4 (the moon) and 0.6 (Mars) are assumed in calculations. When being applied to asteroids of several types, this method also yielded a satisfactory agreement. For the E-type asteroids, the sets of parameters are [g = 0.6, ω = 0.6, A _g = 0.21, and q = 0.83] or [g = 0.3, ω = 0.4, A _g = 0.15, and q = 0.71] under the Martian single-scattering phase function; for the M-type asteroids, it is [g = 0.4, ω ≤ 0.1, A _g ≤ 0.075, and q ≤ 0.42] under the lunar single-scattering phase function; for the S-type asteroids, it is [g = 0.4, ω = 0.4, A _g = 0.28, and q = 0.49] under the lunar single-scattering phase function; and for the C-type asteroids, it is [g = 0.6, ω ≤ 0.1, A _g ≤ 0.075, and q = 0.43] under the modified lunar single-scattering phase function. The polarization measurements fulfilled by Gehrels et al. (1964) for the bright feature on the lunar surface, Copernicus (L = -20°08′, φ = +10°11′), at a phase angle α = 1.6° revealed the deviations in the position of the polarization plane from that typical for the negative branch. They were 22° and 12° in the G and I filters, respectively. At the same time, the deviation was within the error (±3°) in the U filter and for the dark feature Plato (L = -10°32′, φ = +51°25′), which can be caused by the coherent mechanism of the formation of the polarization peak.     

Galileo Photopolarimetry of Jupiter at 678.5 nm

Brightness and linear polarization measurements at 678.5 nm for four south-north strips of Jupiter are studied. These measurements were obtained in 1997 by the Galileo photopolarimeter/radiometer. The observed brightness exhibits latitudinal variations consistent with the belt/zone structure of Jupiter. The observed degree of linear polarization is small at low latitudes and increases steeply toward higher latitudes. No clear correlations were observed between the degree of linear polarization and the brightness. The observed direction of polarization changes from approximately parallel to the local scattering plane at low latitudes to perpendicular at higher latitudes. For our studies, we used atmospheric models that include a haze layer above a cloud layer. Parameterized scattering matrices were employed for the haze and cloud particles. On a pixel-wise basis, the haze optical thickness and the single-scattering albedo of the cloud particles were derived from the observed brightness and degree of linear polarization; results were accepted only if they were compatible with the observed direction of polarization. Using atmospheric parameter values obtained from Pioneer 10 and 11 photopolarimetry for the South Tropical Zone and the north component of the South Equatorial Belt, this analysis yielded acceptable results for very few pixels, particularly at small phase angles. However, for almost all pixels, acceptable results were found when the parameterized scattering matrix of the cloud particles was adjusted to produce more negative polarization for single scattering of unpolarized light, especially at large scattering angles, similar to some laboratory measurements of ammonia ice crystals. Using this adjusted model, it was found that the derived latitudinal variation of the single-scattering albedo of the cloud particles is consistent with the belt/zone structure, and that the haze optical thickness steeply increases toward higher latitudes.     

Electromagnetic radiation from line sources interacting with a moving magnetoplasma slab backed by a finitely conducting medium

The problem of electromagnetic radiation from electric and magnetic line sources interacting with a moving magnetoplasma slab backed by a finitely conducting medium is treated. The local magnetostatic field is aligned parallel with the line source and is perpendicular to the direction of slab motion. For the configuration, theE andH modes are excited independently by a magnetic and an electric line source respectively. Expressions for the far zone radiation fields and the radiation pattern have been obtained for both the line sources. It is found that the radiation due to an electric line source is not affected by the presence of a static magnetic field and the motion of the slab medium. Numerical results for the radiation pattern referring to both the line sources have been presented for a wide range of parameters characterizing the finite magnetostatic field, the conductivity of the medium backing the plasma, the thickness of the slab and the location of the line source.     

Local burst model of CMB temperature fluctuations: scattering in primordial hydrogen lines

The propagation of an instantaneous burst of isotropic radiation from the time of its onset at some redshift z ₀ to the time of its detection at the present epoch (at z = 0) is considered within the framework of a flat Universe. Thomson scattering by free electrons and scattering in primordial hydrogen lines (Hα, Hβ, Pα, and Pβ) are believed to be the sources of opacity, with the single-scattering albedo in the lines being calculated by taking into account the deexcitation of the upper levels of the transitions being considered under the action of background blackbody radiation. The profiles of these lines in the burst spectrum at the present epoch have been constructed for various z0 at various distances from the burst center. To a first approximation, these profiles do not depend on the burst radiation spectrum and intensity. It is shown that the lines are purely absorption ones at a sufficiently large distance, but an emission component can appear with decreasing distance, which strengthens as the distance decreases, while the absorption component weakens. The absorption depth in the combined profile can reach 2 ×10^?4 for the Hα and Hβ lines and 7 × 10^?6 for the Pα and Pβ lines. In this case, the relative amplitude of the temperature fluctuations lies within the range 10^?7?10^?9. The calculations have been performed for bursts with different characteristic initial sizes. At the same z ₀, the hydrogen line profiles essentially coincide for sizes smaller than some value, and the contrast of the lines decreases with increasing burst size for greater ones.