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.     

On photometric properties of Saturn's rings

The photometric observations of Saturn's rings made by Camichel (1958a), and Focas and Dollfus (1969) are studied. It has been found that, at large elevation angles multiple scattering between ring particles, and at small elevation angles deviation from Seeliger's principal photometric theory can explain the observations. The geometric albedo 0.82 and the Bond albedo 0.90 have been suggested for the ring particles. The optical thickness of ring B is found to be 1.25 and that of ring A 0.30.     

Saturn's rings and Pioneer 11

Quantitative predictions of the diffuse reflection and transmission properties of Saturn's rings, relevant to the September 1979 Pioneer 11 flyby, are presented. Predictions are based on an elementary anisotropic scattering model. Interparticle separations are considered to be sufficiently large that mutual shadowing is negligible. Likely ranges in both the single scattering albedo and perpendicular optical thickness of the ring are considered. Situations of pronounced back-scattering and of isotropic scattering are treated individually. Spacecraft measurement of the radiation suffering diffuse scattering by the ring can provide a useful test of the basic ring model.     

Anisotropic optical scattering within Saturn's rings

Visual photometric function data for Saturn's rings are analyzed in terms of elementary anisotropic scattering radiative transfer models which involve the Henyey-Greenstein function. Limits are placed on the combinations of single scattering albedo and backscattering directivity which are permitted by observation. Particles with both microscopic and macrscopic lunar-like scattering properties are excluded by the analysis. Results are consistent with the ring particles being nearly pure spherical conglomerates of H₂O frost.     

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.     

Polarization Models for Rayleigh Scattering Planetary Atmospheres

We present Monte Carlo simulations for the polarization of light reflected from planetary atmospheres. We investigate dependencies of intensity and polarization on three main parameters: single scattering albedo, optical depth of a scattering layer, and albedo of a Lambert surface underneath. The main scattering process considered is Rayleigh scattering, but isotropic scattering and enhanced forward scattering on haze particles are also investigated. We discuss disk integrated results for all phase angles and radial profiles of the limb polarization at opposition. These results are useful to interpret available limb polarization measurements of solar system planets and to predict the polarization of extra-solar planets as a preparation for VLT/SPHERE. Most favorable for a detection are planets with an optically thick Rayleigh-scattering layer. The limb polarization of Uranus and Neptune is especially sensitive to the vertically stratified methane mixing ratio. From limb polarization measurements constraints on the polarization at large phase angles can be set.     

Five-color photometry of Saturn and its rings

Analysis of 206 high-quality plates from three recent apparitions taken in five colors has yielded several photometric parameters for Saturn and its A and B rings. Phase curves and geometric albedos are derived for two regions of Saturn and for each ring. The phase coefficients of the rings are found to be independent of the ring-plane inclination angle. A comparison of the phase curves shows that the particles of ring A exhibit a larger phase coefficient than do those of ring B. When examined with a multiple-scattering model using Henyey-Greenstein phase functions, the observations of the ring tilt effect indicate that the particles of ring A may also have lower single-scattering and geometric albedos. The color dependence of the geometric albedo of the particles in ring B is shown to be very similar to that of Europa (J II). We find for ring A an optical thickness of 0.50 (0.45 ≤ τ_A ≤ 0.57) and for the Cassini division, 0.018 ± 0.004.     

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.     

HST observations of azimuthal asymmetry in Saturn's rings

From 378 Hubble Space Telescope WFPC2 images obtained between 1996-2004, we have measured the detailed nature of azimuthal brightness variations in Saturn's rings. The extensive geometric coverage, high spatial resolution (), and photometric precision of the UBVRI images have enabled us to determine the dependence of the asymmetry amplitude and longitude of minimum brightness on orbital radius, ring elevation, wavelength, solar phase angle, and solar longitude. We explore a suite of dynamical models of self-gravity wakes for two particle size distributions: a single size and a power law distribution spanning a decade in particle radius. From these N-body simulations, we calculate the resultant wake-driven brightness asymmetry for any given illumination and viewing geometry. The models reproduce many of the observed properties of the asymmetry, including the shape and location of the brightness minimum and the trends with ring elevation and solar longitude. They also account for the “tilt effect” in the A and B rings: the change in mean ring brightness with effective ring opening angle, |Beff|. The predicted asymmetry depends sensitively on dynamical ring particle properties such as the coefficient of restitution and internal mass density, and relatively weakly on photometric parameters such as albedo and scattering phase function. The asymmetry is strongest in the A ring, reaching a maximum amplitude A∼25% near a=128,000 km. Here, the observations are well-matched by an internal particle density near 450 kg m⁻³ and a narrow particle size distribution. The B ring shows significant asymmetry (∼5%) in regions of relatively low optical depth (τ∼0.7). In the middle and outer B ring, where τ?1, the asymmetry is much weaker (∼1%), and in the C ring, A<0.5%. The asymmetry diminishes near opposition and at shorter wavelengths, where the albedo of the ring particles is lower and multiple-scattering effects are diminished. The asymmetry amplitude varies strongly with ring elevation angle, reaching a peak near |Beff|=10° in the A ring and at |Beff|=15-20° in the B ring. These trends provide an estimate of the thickness of the self-gravity wakes responsible for the asymmetry. Local radial variations in the amplitude of the asymmetry within both the A and B rings are probably caused by regional differences in the particle size distribution.     

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.     

The first three orders of scattering in vertically inhomogeneous scattering-absorbing media

In order to facilitate the computations of the intensities of radiation reflected and/or transmitted by plane-parallel, vertically inhomogenous, scattering-absorbing media, we carry out the optical thickness integrations of the Cauchy systems (normally referred to as Invariant Imbedding Equations) for reflection and transmission functions originating from the first three orders of scattering: the medium in question is represented by a stack of a certain number of homogeneous slabs, each of which is characterized by a constant single scattering albedo and a constant phase function together with the optical thickness.The results are a set of recurrence formulae involving only the angular intergrations, a convenient feature for numerical computations, and should prove useful particularly for finding approximate values of the high frequency Fourier coefficients of reflection and transmission functions of inhomogeneous media or efficiently initializing the solution for a thin layer to perform rigorous multiple scattering computations by means of other techniques such as the Doubling-Adding method.     

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.     

A laboratory study of the bidirectional reflectance from particulate samples

The Hapke (Hapke, B. [1981]. J. Geophys. Res. 86, 3039-3054) photometric model and its modifications are widely used to characterize telescopic, spacecraft, and laboratory observations of the bidirectional reflectance of particulate surfaces. Following work and methods laid out in a companion paper (Helfenstein, P., Shepard, M.K. [2011]. Icarus, in press), we deconstruct the Hapke model and, separating all empirical and ad hoc parameters (opposition surge, particle phase function, surface roughness), combine them into a single parameter called the surface phase function, F(α). We illustrate how to extract this function from scattering data sets acquired with the Bloomsburg University Goniometer (BUG). We show how this method can be used to rapidly and accurately characterize bidirectional reflectance data sets from laboratory and spacecraft measurements, often giving better fits to the data. We examine samples with strong color contrasts in different wavelengths. This allows us to examine the exact same surface, changing only the albedo to investigate how the amplitude and the detailed shape of the surface phase function might systematically depend on wavelength and albedo. We also examine the changes in scattering behavior that result when samples are compacted and find the surface phase function and single scattering albedo to be significantly changed. We suggest that these observations support the hypothesis that much of the scattering behavior attributed to the single particle phase function is instead cause by the surface micro-structure.     

Estimates of the size of the particles in the rings of saturn and their cosmogonic implications

Very low values of the radio brightness temperature of the rings of Saturn indicate that their high refar reflectivity is not simply due to a gain effect in the backscattering direction. These two sets of observations are consistent with the ring particles having a very high single scattering albedo at radio wavelenghts, with multiple scattering effects being important. Comparison of scattering calculations for ice and silicate particles with the radio and radar observations imply a mean particle radius of ～1 cm. The ice bands observed in the rings' near-infrared reflectivity spectra are formed by scattering within a microstructure on the surface of the ring particles, with the scattering centers being 25–125 μm in size. The Poynting-Robertson effect has caused a significant spiraling-in of the ring particles, probably resulting in a broadening of the rings. The inferred mean size is consistent with a model in which meteoroid impacts have caused a substantial reduction in the mean particle size from its initial value.     

Gusev photometric variability as seen from orbit by HRSC/Mars-express

Minnaert and two-term phase function Hapke models are used to describe the photometric properties of the martian surface using HRSC (High Resolution Stereo Camera) multi-angular observations acquired along the ongoing Mars-Express mission. These observations can be pieced together to derive integrated phase functions over a wide range of phase angles. The photometric diversity at 675 nm, as seen from orbit, of the martian surface properties across Gusev is depicted with seven units. Three photometric units widespread across the flanks of Apollinaris Patera flank and the floor of Gusev Crater are identified as having high single scattering albedo with rather forward scattering properties, low to intermediate macroscopic roughness and porous or not compacted powdered surface state as indicated by the opposition parameters. Another unit has the highest single scattering albedo, the smoothest surface in terms of macroscopic roughness, associated with an extremely forward scattering behavior. The opposition parameters are consistent with the presence of transparent particles in the surface powder layer. The distribution of this unit appears quite intermittent across the crater and does not seem to indicate any relationship with a given morphological structure. It may correspond to sparse areas where the structure of the surface dust layer is the most preserved. The most pronounced photometric changes are observed in three units associated with the low-albedo features corresponding to dark wind streaks. These units have a low single scattering albedo, are the most backscattering surfaces across Gusev, have a high surface roughness and present variable surface states as shown by the opposition parameters estimates, consistent with the occurrence of large grains organized in more or less packed layers. Clear differences are seen among these units in terms of opposition effect. While one exhibits typical characteristics for the opposition effect, another appears more unusual in terms of lobe width and the third suggests the occurrence of a packed/compressed/narrow size distribution powder surface. The opposition effect thus appears to play a significant role suggesting that the surface state optical properties across Gusev are strongly influenced by the porosity and packing characteristics or grain size distribution of the upper layer of the martian soil. The mapping aspect of the investigation is quite useful to get a better sense of the meaning of the observed photometric variations. Indeed, the Hapke modeling suggests that surface organization (surface roughness, packing state) is more important than the simple physical characterization of the intrinsic optical properties of the constitutive particles. Given the overall spatial patterns derived from the photometric analysis, the variations, at least for the western and central part of Gusev Crater, are likely partly driven by the prevailing wind regimes, considered to be oriented north-northwest/south-southeast and disturbing the very upper surface layer. The present photometric results agree with independent investigations based on thermal inertia, reflectance spectroscopy, in situ photometric and microscopic imaging and support the idea of a thin layer of fine-grained dust, being stripped off in the low albedo units to reveal a dark basaltic substrate comprising coarse-grained materials.     

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%.     

Scattering properties of planetary regolith analogs

The physics of scattering of electromagnetic waves by media in which the particles are in contact, such as planetary regoliths, has been thought to be relatively well understood when the particles are larger than the wavelength. However, this is not true when the particles are comparable with or smaller than the wavelength. We have measured the scattering parameters of planetary regolith analogs consisting of suites of well-sorted abrasives whose particles ranged from larger to smaller than the wavelength. We measured the variation of reflectance as the phase angle varied from 0.05° to 140°. The following parameters of the media were then deduced: the single scattering albedo, single scattering phase function, transport mean free path, and scattering, absorption, and extinction coefficients. A scattering model based on the equation of radiative transfer was empirically able to describe quantitatively the variation of intensity with angle for each sample. Thus, such models can be used to characterize scattering from regoliths even when the particles are smaller than the wavelength. The scattering parameters were remarkably insensitive to particle size. These results are contrary to theoretical predictions, but are consistent with earlier measurements of alumina abrasives that were restricted to small phase angles. They imply that a basic assumption made by virtually all regolith scattering models, that the regolith particles are the fundamental scattering units of the medium, is incorrect. Our understanding of scattering by regoliths appears to be incomplete, even when the particles are larger than the wavelength.     

A Monte Carlo Code to Compute Energy Fluxes in Cometary Nuclei

A Monte Carlo model designed to compute both the input and output radiation fields from spherical-shell cometary atmospheres has been developed. The code is an improved version of that by H. Salo (1988, Icarus76, 253-269); it includes the computation of the full Stokes vector and can compute both the input fluxes impinging on the nucleus surface and the output radiation. This will have specific applications for the near-nucleus photometry, polarimetry, and imaging data collection planned in the near future from space probes. After carrying out some validation tests of the code, we consider here the effects of including the full 4×4 scattering matrix in the calculations of the radiative flux impinging on cometary nuclei. As input to the code we used realistic phase matrices derived by fitting the observed behavior of the linear polarization as a function of phase angle. The observed single scattering linear polarization phase curves of comets are fairly well represented by a mixture of magnesium-rich olivine particles and small carbonaceous particles. The input matrix of the code is thus given by the phase matrix for olivine as obtained in the laboratory plus a variable scattering fraction phase matrix for absorbing carbonaceous particles. These fractions are 3.5% for Comet Halley and 6% for Comet Hale-Bopp, the comet with the highest percentage of all those observed.The errors in the total input flux impinging on the nucleus surface caused by neglecting polarization are found to be within 10% for the full range of solar zenith angles. Additional tests on the resulting linear polarization of the light emerging from cometary nuclei in near-nucleus observation conditions at a variety of coma optical thicknesses show that the polarization phase curves do not experience any significant changes for optical thicknesses τ?0.25 and Halley-like surface albedo, except near 90° phase angle.     

On the azimuthal brightness variations of Saturn's rings

We present a simple, semiquantitative explanation that accounts both for the presence of the azimuthal brightness variations in Saturn's ring A and for their absence in ring B. Our explanation avoids any ad hoc reliance on albedo variations and/or synchronous rotation of ring particles. Instead, it requires only some degree of self-gravitation between nearby orbiting bodies. A bias in the particle distribution and corresponding photometric effects are thereby produced—the latter corresponding very closely to the variations observed in ring A. Their absence in ring B is primarily a consequence of the higher optical thickness and decreasing importance of self-gravitation in that ring.     

The clouds of Venus: An anisotropic scattering model

The phase function $\text{ω}\text{?}\text{(1 + a}\text{cos}\text{θ)}$ for anisotropic scattering is applied to a homogeneous atmospheric model to ascertain the effects of anisotropy on the near-infrared spectrum of Venus. L. D. G. Young's equivalent widths for the 820 Å CO₂ band are analyzed to derive allowed combinations of CO₂ specific abundance, continuum albedo, pressure, and degree of anisotropy. From these combinations, values are derived for the photon mean-free path and total optical thickness of the clouds. The Venera 10 measurement of the transmitted flux at 7200 Å is then used in conjunction with spherical albedo requirements to reduce the range of possible solutions to the equivalent-width analysis. Through the application of approximate similarity relations, the “true” (i.e., anisotropic) atmospheric parameters are derived from the isotropic values. For an assumed Mie scatterer with an average anisotropy factor of 〈cos θ〉 = 0.7, the results indicate a total optical thickness of about 55 with a single-scattering albedo of 0.9992 to 0.9995.