Interpreting photometry of regolith-like surfaces with different topographies: shadowing and multiple scattering

The effects of various types of topography on the shadow-hiding effect and multiple scattering in particulate surfaces are studied. Two bounding cases were examined: (1) the characteristic scale of the topography is much larger than the surface particle size, and (2) the characteristic scale of the topography is comparable to the surface particle size. A Monte Carlo ray-tracing method (i.e., geometric optics approximation) was used to simulate light scattering. The computer modeling shows that rocky topographies generated by randomly distributed stones over a flat surface reveal much steeper phase curves than surface with random topography generated from Gaussian statistics of heights and slopes. This is because rocks may have surface slopes greater than 90°. Consideration of rocky topography is important for interpreting rover observations. We show the roughness parameter in the Hapke model to be slightly underestimated for bright planetary surfaces, as the model neglects multiple scattering on large-scale topographies. The multiple scattering effect also explains the weak spectral dependences of the roughness parameter in Hapke's model found by some authors. Multiple scattering between different parts of a rough surface suppresses the effect of shadowing, thus the effects produced by increases in albedo on the photometric behavior of a surface can be compensated for with the proper decreases in surface roughness. This defines an effective (photometric) roughness for a surface. The interchangeability of albedo and roughness is shown to be possible with fairly high accuracy for large-scale random topography. For planetary surfaces that have a hierarchically arranged large-scale random topography, predictions made with the Hapke model can significantly differ from real values of roughness. Particulate media with surface borders complicated by Gaussian or clumpy random topographies with characteristic scale comparable to the particle size reveal different photometric behaviors in comparison with particulate surfaces that are flat or the scale of their topographies is much larger than the particle size.     

The Mars opposition effect at 20°N. latitude and 20°W. longitude

Low phase angle observations in the Chryse-Acidalia region have been obtained the Viking Orbiter 1 spacecraft under clearer atmospheric conditions than reported earlier. A variety of surface features were recorded, e.g., crater streaks, dark and bright patches. Several findings for this scene include: an abrupt brightness increase (10%) was found at phase angles less than 3°, an effect dependent on surface albedo and possibly particle distribution; a slight weakening of reflectance surge with decreasing wavelengths; a larger opposition effect for features of high albedo was recorded; and a greater reddening with increase phase angle took place for low albedo regions. Both reflectance and contrast values are provided at three wavelengths as a function of phase angle from 0.15 to 20°.     

Polarimetry of Pluto

The polarization of Pluto has been measured for a range of solar phase angles from 0.8 to 1.8°. A mean linear polarization of 0.29 ± 0.01% (error of the mean) was found. No dependence of both the amount of polarization and position angles with rotational phase or solar phase angle could be detected. The positional angles of polarization agree with calculated position angles of the defect of illumination and are therefore parallel to the scattering plane. The observed polarization cannot be explained as resulting purely from a surface material which is similar to asteroidal surfaces. A hypothesis of polarization from a thin atmosphere, in addition to the surface polarization, is advanced.     

Viking orbiter observations of the Mars opposition effect

The Viking Extended Mission has experienced two major dust storms that have changed the global photometric properties of Mars. Large quantities of atmospheric dust arising from the June 5, 1977, storm have been observed at very low phase angles to measure the opposition effect. These particles yield only a small increase in brightness at 0° phase angle with the least enhancement seen in violet light. The phase function is well modeled by nonspherical particles with a spectrally dependent single scattering albedo. It is doubtful, therefore, that atmospheric dust plays a significant role in the reported blue light brightness surge. Such particles as surface structure combined with a lunar photometric function could, however, produce the wavelength-dependent backscattering observed during the 1967 and 1969 oppositions under clearer conditions.     

Spectropolarimetry of comets Austin and Churyumov-Gerasimenko

Spectropolarimetric observations from 5000 to 8000 Å have been obtained for comets P/Austin (1982g) and P/Churyumov-Gerasimenko (1982f). The observations were spaced over phase angles of 50–125° for comet Austin and 10–40° for comet Churyumov-Gerasimenko. The use of spectropolarimetry allowed an evaluation of continuum polarization without molecular line contamination. Especially for comet Churyumov-Gerasimenko, the curve of polarization versus phase angle resembles curves for asteroids, where the polarization is negative (electric vector maximum parallel to the scattering plane) for phase angles less than 20° and the most negative polarization is from ?1 to ?2%. The negative polarization at backscattering angles may be due to multiple scattering in agglomerated grains, as assumed for asteroids, or to Mie scattering by small dielectric particles. If multiple scattering is important in comet dust, polarization measurements may imply a low albedo, less than 0.08. The polarization of comet Austin remained steady during a large change in the dust production rate. Both comets increased continuum flux by a factor of 2 near perihelion. The continuum of comet Churyumov-Gerasimenko had the shape of the solar spectrum with derivations less than 5%. The equivalent width of spectral features of C₂, NH₂, and O varied as r^?2.     

A Monte Carlo approach to scattering off surfaces of triaxial ellipsoids

We developed a Monte Carlo procedure to calculate the scattering properties of ellipsoidal bodies. Although originally intended for application to Saturn's ring A, the technique can also be applied to surfaces of asteroids modeled as triaxial ellipsoids. A simple coordinate transformation allows the Monte Carlo approach to be quick and efficient. By modifying the well-known acceptance-rejection technique to reflect the dependence of surface albedo on the angle of incidence, we can apply this technique to any surface scattering function. The backscattered brightness and the phase curve at small phase angles can be calculated to a high degree of accuracy in a small amount of computing time. The technique is shown to be accurate for problems that have analytic solutions. Light curve amplitudes of biaxial ellipsoids as a function of the axial ratio are plotted for several scattering laws. The application of this technique to a direct integration approach is also considered.     

Martian phase function: Modeling the visible to near-infrared surface photometric function using HST-WFPC2 data

Images of Mars in the visible to near-infrared acquired from 1996 to 2005 using the Hubble Space Telescope WFPC2 have been used to model the martian surface photometric function at 502, 673, 953, and 1042 nm. These data range in spatial resolution from 12 to 70 km/pixel at the sub-Earth point, and in phase angle coverage from 0.34° to 40.5°. The WFPC2 images have been calibrated to radiance factor or I/F and projected to a cylindrical map for coregistration and comparison to similarly mapped spacecraft data sets of albedo, topography, thermal inertia, composition, and geology. We modeled the observed I/F as a function of phase angle using Minnaert, Lambert, lunar-Lambert, and Hapke photometric functions for numerous regions of interest binned into albedo units defined by Viking and TES albedo maps, and thermal-inertia units defined by TES thermal-inertia maps. Visibly opaque water-ice clouds and data acquired under high dust opacity conditions were excluded from the analysis. Our modeling suggests that under average to low atmospheric dust opacity conditions and over this range of phase angles, the photometric properties of the martian surface at 502, 673, 953, and 1042 nm are best modeled by lunar-Lambert functions with parameters derived for three surface units defined by low, moderate, and high TES bolometric albedos.     

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.     

Are the circular, dark features on Comet Borrelly's surface albedo variations or pits?

The highest resolution images of Comet 19P/Borrelly show many dark features which, upon casual inspection, appear to be low albedo markings, but which may also be shadows or other photometric variations caused by a depression in the local topography. In order to distinguish between these two possible interpretations we conducted a photometric analysis of three of the most prominent of these features using six of the highest quality images from the September 22, 2001 Deep Space 1 (DS1) flyby. We find that: 1. The radiance in the darkest parts of each feature increases as phase angle decreases, similarly to the radiance behavior of the higher albedo surrounding terrain. The dark features could be either fully illuminated low albedo spots or, alternatively, they could be depressions. No part of any of the three regions was in full shadow. 2. One of the regions has a radiance profile consistent with a rimmed depression, the second, with a simple depression with no rim, and the third with a low albedo spot. 3. The regolith particles are backscattering and carbon black is one of the few candidate regolith materials that might explain this low albedo. We conclude that Borrelly's surface is geologically complex to the limit of resolution of the images with a combination complex topography, pits, troughs, peaks and ridges, and some very dark albedo markings, perhaps a factor of two to three darker than the average 3-4% albedo of the surrounding terrains. Our technique utilizing measured radiance profiles through the dark regions is able to discriminate between rimmed depressions, rimless depressions and simple albedo changes not associated with topography.     

Light scattering by media composed of semitransparent particles of different shapes in ray optics approximation: consequences for spectroscopy, photometry, and polarimetry of planetary regoliths

Computer simulations of light scattering by particulate surfaces and single particles forming these surfaces are presented. The ray optics approximation is used. Three types of particles are studied: spheres, cubes, and very irregular particles that are generated with an auxiliary random Gaussian field. The surfaces of the particles are represented as an arrangement of triangular facets. For the Monte Carlo ray tracing 10⁶−10⁷ rays were used. The ray tracing verifies Shkuratov et al.'s (Icarus 137 (1999) 235-246) spectral albedo model for powder-like media. We derive a useful relationship between the hemispheric albedo, A_int, and the bi-directional reflectance, R, at phase angle 30°: logR(30°)=1.088logA_int. This relationship provides a way to estimate bi-directional reflectance spectra from laboratory spectra measured with an integrating sphere for surfaces composed of particles of irregular shapes. We study also phase angle curves of the nonzero scattering matrix elements, F₁₁, −F₁₂/F₁₁, F₂₂/F₁₁, F₃₃/F₁₁, F₃₄/F₁₁, F₄₄/F₁₁, for single particles and media thereof. Randomly shaped particles show smoother phase angle behavior than particles with regular shapes that display distinct features. For media consisting of spheres the glory and primary rainbow both are prominent even in the case of conservative (nonabsorbing) scattering. On the other hand, such media clearly exhibit the depolarization effect, showing a significant role of multiple scattering between particles. For media composed of semitransparent cubes the retroreflector spike and a very deep negative polarization branch at small phase angles are observed. We demonstrate that, in the geometric optics approximation, neither a medium of spherical particles nor one of cubic particles is appropriate for modeling light scattering behavior of regolith-like surfaces.     

Voyager disk resolved photometry of the Saturnian satellites

Photometric analysis of Voyager images of the medium-sized icy satellites of Saturn shows that their surfaces exhibit a wide range of scattering properties. At low phase angles, Rhea and Dione closely follow lunar behavior with almost no limb darkening. Mimas, Tethys, and especially Enceladus shiw significant limb darkening at low phase angles, which suggests multiple scattering is important for their surfaces. A simple photometric function of the form I/F = f(α)Aμ₀/(μ + μ₀) + (1 ? A)μ₀ has been fit to the observations. For normal reflectances <0.6, we find lunar-like scattering properties (A = 1). No satellite's surface can be described by Lambert's Law (A = 0). Dione exhibits the widest albedo variations (about 50%). A longitudinal dark stripe which represents a 15% decrease in albedo is situated near the center of the trailing side of Tethys. A correlation is found between the albedo and color of the satellites: the darker objects are redder. Similarly, darker areas of each satellite are redder. Spectral reflectances of Mimas and Enceladus can be derived for the first time. After the proper calibrations to the Voyager color images are made, it is found that both satellites have remarkably flat spectra into the ultraviolet.     

Photometry and polarimetry of Titan: Pioneer 11 observations and their implications for aerosol properties

The preliminary measurements by Pioneer 11 of the limb darkening and polarization of Titan at red and blue wavelenghts (M. G. Tomasko, 1980,J. Geophys. Res., 85, 5937–5942) are refined and the measurements of the brightness of the integrated disk at phase angles from 22 to 96° are reduced. At 28° phase, Titan's reflectivity in blue light at southern latitudes is as much as 25% greater than that at northern latitudes, comparable to the values observed by Voyager 1 (L. A. Sromovsky et al., 1981,Nature (London), 292, 698–702). In red light the reflectivity is constant to within a few percent for latitudes between 40°S and 60°N. Titan's phase coefficient between 22 and 96° phase angle averages about 0.014 magnitudes/degree in both colors—a value considerably greater than that observed at smaller phase from the Earth. Comparisons of the data with vertically homogeneous multiple-scattering models indicate that the single-scattering phase functions of the aerosols in both colors are rather flat at scattering angles between 80 and 150° with a small peak at larger scattering (i.e., small phase) angles. The models indicate that the phase integral, q, for Titan in both red and blue light is about 1.66 ± 0.1. Together with Younkin's value for the bolometric geometric albedo scaled to a radius of 2825 km, this implies an effective temperature in equilibrium with sunlight of 84 ± 2°K, in agreement with recent thermal measurements. The single-scattering polarizations produced by the particles at 90° scattering angle are quite large, >85% in blue light and >95% in red. A vertically homogeneous model in which the particles are assumed to scatter as spheres cannot simultaneously match the polarization observations in both colors for any refractive index. However, the observed polarizations are most sensitive to the particle properties near optical depth $\text{1}\text{2}$ in each color, and so models based on single scattering by spheres can be successful over a range of refractive indices if the size of the particles increases with depth and if the cross section of the particles increases sufficiently rapidly with decreasing wavelenght. For example, with n_r = 1.70, the polarization (and the photometry) are reproduced reasonably well in both colors when the area-weighted average radous of the particles, α, is given by α = (0.117 μm)(τ_red/0.5)^0.217. While this model does not reproduce the large increase in brightness from 129 to 160° phase observed by Voyager 1, the observed increase is determined by the properties of the particles in the top few hundredths of an optical depth. Thus the addition of a very thin layer of forward-scattering aerosols on top of the above model offers one way of satisfying both the Pioneer 11 and Voyager 1 observations. Of course, other models, using bimodal size distributions or scattering by nonspherical particles, may also be capable of reproducing these data.     

Photometric study of the major satellites of Uranus

In this paper, we analyze the results of ground-based and space-born photometric observations of the major satellites of Uranus—Miranda, Ariel, Umbriel, Titania, and Oberon. All sets of photometric observations of the satellites available in the literature were examined for uniformity and systematic differences and summarized to a unified set by wavelength ranging from 0.25 to 2.4 μm. This set covers the interval of phase angles from 0.034° to 35°. The compound phase curves of brightness of the satellites in the spectral bands at 0.25, 0.41, 0.48, 0.56, 0.75, 0.91, 1.4, and 1.8 μm, which include a pronounced opposition surge and linear part, were constructed. For each satellite, the geometric albedo was found in different spectral bands taking into account the brightness opposition effect, and its spectral dependence was studied. It has been shown that the reflectance of the satellites linearly depends on the wavelength at different phase angles, but has different spectral gradients. The parameters of the phase functions of brightness, including the amplitude and the angular width of the brightness opposition surge, the phase coefficient, and the phase angle at which the nonlinear increase in brightness starts, were determined and their dependences on wavelength and geometric albedo were analyzed. Our investigations show that, in their optical properties, the satellites Miranda and Ariel, Titania and Oberon, and Umbriel present three types of surfaces. The observed parameters of the brightness opposition effect for the Uranian satellites, some ice satellites of Jupiter and Saturn, and the E-and S-type asteroids are analyzed and compared within the framework of the coherent backscattering and mutual shadowing mechanisms.     

Interpretation of the surface brightness of Phobos

Analysis of disk resolved images of Phobos obtained by the Phobos 2 spacecraft allows us to study the surface scattering law and albedo variations. From low phase angle images we find variations in local geometric albedo approximately 10%, with a correlation length approximately 1km. The scattering law is reasonably well matched by the recent proposed LPI (Lumme et al. 1990a) model, which allows us to deduce a small scale (approximately 1 mm) surface roughness (approximately 0.5), defined here as the rms. tangent of the local surface normal relative to the mean surface normal in the Duxbury (1991) model of Phobos. This value is very close to what has been found for Mercury and the Moon.     

Light Scattering from Regolith: Intensity Versus Particle Size Behavior

Nature of the photometric phase curves of the regolith like surfaces (like those of the asteroids) are believed to be dependent on the single particle characteristics like particle size, shape, composition etc. and physical characteristics of the surface like porosity and roughness. Most of the phase curves have a rapid surge of intensity at small phase angles (typically below 5^°) known as opposition effect, followed by a linear less decreasing trend at larger phase angles. Average intensity of the linear region has been found to be mostly dependent on the average particle size and its composition, in many laboratory observations. Generally, it is difficult to explain the nature of light scattering by an ensemble of irregular shaped inhomogeneous particles with a theoretical model, just by studying the phase curves. In the present work, we have investigated whether the theoretically expected variation of the scattered light intensity (at a given phase angle) with the average particle size of the grains constituting regoliths, for a given material of the particle is in agreement with the experimental results or not? If yes, this can be a simpler but efficient way to study light scattering by regolith like surfaces. For theoretical analysis, Hapke formula has been used with Mie theory for single particle phase function, where we have neglected the influence of porosity and roughness presently. The data are also fitted with an empirical formula. It has been found that this empirical formula may also be used to estimate the unknown average particle size of a real regolith with known composition.     

Interpretation of the lunar microwave brightness temperature spectrum: Feasibility of orbital heat flow mapping

A detailed model of the lunar regolith is analyzed to examine the feasibility of an orbital mapping of heat flow using microwave radiometers. For regolith thermal and electrical properties which are representative of Apollo findings, brightness temperature observations in the bandλ = 5–30 cm would be required for heat flow analysis. Spectral variations shortward of 5 cm are controlled primarily by the temperature dependencies of the thermal conductivity and electrical absorption within the diurnal-varying layer. For wavelengths longer than 30 cm, unwanted emission from high impedance subregolith layers can be significant and size limitations on spacecraft radiometers is a factor. Over the 5- to 30-cm band, lunation-averaged brightness temperature increases of 2–10°K are predicted for heat flow values representative of the Apollo measurements. The magnitude of this increase depends directly on the value of regolith microwave absorption. For absorption values consistent with Apollo laboratory measurements, a spectral increase of 5°K is predicted. This value is considered marginally sufficient for an orbital heat flow measurement. However, important non-heat flow effects must be accounted for. Spectral variations can occur due to surface topography and subsurface scattering. For nadir viewing, surface roughness effects are not expected to be significant and topographic effects are nearly constant with wavelength for λ > 5cm. Volume scattering due to subsurface rock fragments can cause emission darkening of 1–6°K. However, spectral variations will not be large unless the distribution of scatterer sizes is sharply skewed. For the Moon, the most serious spurious effect appears to be emissivity variations due to the near-surface density gradient. A brightness temperature decrease of 10°K is predicted from centimeter to decameter wavelengths. If the transition from porous surface fines to compacted regolith soil occurs rapidly (within the upper 3–5 cm), most of the emissivity decrease will occur in the 5- to 30-cm wavelength band. It is recommended that complementary radar measurements be utilized to augment constraints on regolith emissivity and scattering properties.     

Icy craters on the Galilean satellites?

A model which possibly accounts for the unusual radar scattering behavior observed for Europa, Ganymede, and Callisto postulates a thick surface layer of ice saturated with nearly hemispherical craters. In the development of this model it is noted that a single reflection at normal incidence reverses the rotational sense of circularly polarized incident radiation, in conflict with the radar observations which show an echo predominantly not reversed. Furthermore, an ensemble of backscattering events, each the result of a large number of successive dielectric reflections, tends to produce a weak and unpolarized echo. However, two coupled reflections can produce the observed backscattering behavior, provided the angles of incidence lie between the Brewster angle and its complement. The effect is maximum when the angles equal 45°, and, for water ice, yields a ratio of 1.9 for components of the echo received in rotational senses the same as, and opposite to, the sense transmitted. Randomly oriented reflecting facets, either of ice on the surface or of rocks in the interior, cannot yield the observed behavior since too few of the total possible backscattering configurations meet the above requirement. Hemispherical surface craters, on the other hand, favor 45° dual reflection. A model consisting of such craters in ice is investigated and found capable of explaining the observed results, not only in respect to polarization, but in respect to albedo and angular scattering law as well.     

Properties of the Hermean regolith: IV. Photometric parameters of Mercury and the Moon contrasted with Hapke modelling

A comparison of the photometric properties of Mercury and the Moon is performed, based on their integral phase curves and disk-resolved image data of Mercury obtained with the Swedish Vacuum Solar Telescope. Proper absolute calibration of integral V-band magnitude observations reveals that the near-side of the Moon is 10-15% brighter than average Mercury, and 0-5% brighter for the “bolometric” wavelength range 400-1000 nm. As shown, this is supported by recent estimates of their geometric albedos. Hapke photometric parameters of their surfaces are derived from identical approaches, allowing a contrasting study between their surface properties to be performed. Compared to the average near-side Moon, Mercury has a slightly lower single-scattering albedo, an opposition surge with smaller width and of marginally smaller amplitude, and a somewhat smoother surface with similar porosity. The width of the lobes of the single-particle scattering function are smaller for Mercury, and the backward scattering anisotropy is stronger. In terms of the double Henyey-Greenstein b-c parameter plot, the scattering properties of an average particle on Mercury is closer to the properties of lunar maria than highlands, indicating a higher density of internal scatterers than that of lunar particles. The photometric roughness of Mercury is well constrained by the recent study of Mallama et al. (2002, Icarus 155, 253-264) to a value of about 8°, suggesting that the surfaces sampled by the highest phase angle observations (Borealis, Susei, and Sobkou Planitia) are lunar mare-like in their textural properties. However, Mariner 10 disk brightness profiles obtained at intermediate phase angles indicate a surface roughness of about twice this value. The photometric parameters of the Moon are more difficult to constrain due to limited phase angle coverage, but the best Hapke fits are provided by rather small surface roughnesses. Better-calibrated, multiple-wavelength observations of the integral and disk-resolved brightnesses of both bodies, and obtained at higher phase angle values in the case of the Moon, are urgently needed to arrive at a more consistent picture of the contrasting light scattering properties of their surfaces.     

Voyager photometry of Europa

Ninety voyager images ranging in phase angle from 3 to 143° and covering the spectral range from 0.34 to 0.58 μm were analyzed to derive the photometric properties of Europa. At small phase angles the disk-integrated phase curve is remarkable in that it shows little or no evidence of an opposition effect (in agreement with earlier Earth-based observations by Millis and Thompson, Icarus26, 408, 1975). The phase integral determined in the Voyager clear filter (centered near 0.47 μm) is 1.09 ± 0.11, in good agreement with previous estimates based on radiometry. The bolometric Bond albedo is 0.62 ± 0.14. The scattering properties of Europa in general, and of the two major terrain types (bright plains and darker mottled terrain) in particular, cannot be represented by a lunar-like photometric law. However, an equation which is a linear superposition of a lunar-like scattering law and a Lambert component provides an adequate simple representation of the scattering properties. The plains are photometrically more homogeneous than the darker mottled terrain. In the Voyager clear filter, the average normal reflectance is 0.71 for the plains on both the leading and trailing hemispheres; for the darker mottled terrain the values are 0.60 on the leading hemisphere, and 0.48 on the trailing one.     

Theoretical interpretation of the ground-based photometry of Saturn's B ring

New photographic photometry at small tilt angles during the 1979 and 1981 apparitions is combined with earlier data to yield several physical parameters for Saturn's B ring in red and blue colors. Phase curves are obtained for a mean tilt angle B ? 6°. The value of the volume density D is 0.020±0.004 with no indication of dependence on either the color or the tilt angle for 6°<B<26°. This conclusion is not altered significantly if the individual ring particles have a phase function similar to the phase curves of bright solar system objects. For the geometric albedo of a single particle we derive 0.61±0.04 (red) and 0.41±0.03 (blue), which are superior to earlier estimates because of the additional data now available. These values and the derived amount of multiple scattering as a function of tilt angle constrain the particle phase function in the red to be moderately backscattering. Inferred values of the particle single-scattering albedo are $\text{0.7≤}\text{ω}{}_0\text{(}\text{red}\text{) ≤0.92}$ and $\text{0.5≤}\text{ω}{}_0\text{(}\text{blue}\text{) ≤0.7}$, depending on the choice of phase function. No indication was found that the particle photometric properties might depend on the vertical distance from the central plane. Our results show that the ground-based photometry is entirely consistent with the classical, many-particle-thick ring model.