Single Particle Phase Function Research Articles

New Horizons Photometry of Pluto's Moon Charon

Abstract The New Horizons spacecraft extended the range in solar phase angle coverage for Pluto’s moon Charon from 1.°8—the maximum observable from Earth—to 170°. This extraordinary expansion in range has enabled photometric modeling and a robust determination of Charon’s phase integral and Bond albedo at visible wavelengths. Photometric modeling shows that Charon is similar in its photometric properties to other icy moons, except that its single particle phase function is more isotropic, suggesting the Kuiper Belt may represent a new regime for surface alteration processes. Charon’s phase integral is 0.70 ± 0.04 and its Bond albedo is 0.29 ± 0.05.

Cassini-VIMS observations of Saturn’s main rings: II. A spectrophotometric study by means of Monte Carlo ray-tracing and Hapke’s theory

This work is the second in a series of manuscripts devoted to the investigation of the spectrophotometric properties of Saturn’s rings from Cassini-VIMS (Visible and Infrared Mapping Spectrometer) observations. The dataset used for this analysis is represented by ten radial spectrograms of the rings which have been derived in Filacchione et al. (2014) by radial mosaics produced by VIMS. Spectrograms report the measured radiance factor I/F of the main rings of Saturn as a function of both radial distance (from 73500 to 141375 km) and wavelength (0.35–5.1 µm) for different observation geometries (phase angle ranging in the 2°–132° interval). We take advantage of a Monte Carlo ray-tracing routine (Ciarniello et al., 2014) to characterize the photometric behavior of the rings at each wavelength and derive the spectral Bond albedo of ring particles. This quantity is used to infer the composition of the regolith covering ring particles by applying Hapke’s theory. Four different regions, characterized by different optical depths, and respectively located in the C ring, inner B ring, mid B ring and A ring, have been investigated. Results from photometric modeling indicate that, in the VIS-NIR spectral range, B ring particles are intrinsically brighter than A and C ring particles, with the latter having the lowest albedo, while the single particle phase function of the ring’s particles is compatible with an Europa-like or Callisto-like formulation, depending on the investigated region. Spectral modeling of the inferred Bond albedo indicates that ring spectrum can be reproduced by water ice grains with inclusion of organic materials (tholin) as a UV absorber intimately mixed with variable amounts of other compounds in pure form (carbon, silicates) or embedded in water ice grains (nanophase hydrated iron oxides, carbon, silicates, crystalline hematite, metallic iron, troilite). The abundance of tholin decreases with radial distance from C ring (0.2–0.6%) to A ring (0.06%) for the selected regions. Its distribution is compatible with an intrinsic origin and is possibly related to the different plasma environment of the different ring regions. The identification of the other absorber(s) and its absolute volumetric abundance is uncertain, depending on the adopted grain size and mixing modality (intraparticle or intimate). However, assuming a common composition of the other absorber in the ring plane, we find that its abundance anti-correlates with the optical depth of the investigated regions, being maximum in the thinnest C ring and minimum in the thickest mid B ring. In the case of the C ring, an additional population of low-albedo grains is required to match the positive spectral slope of the continuum in the 0.55–2.2 µm interval, represented by an intraparticle mixture of water ice and a spectrum similar to troilite or metallic iron. The distribution of the darkening compounds is interpreted as the result of a contamination by exogenous material, which is more effective in the less dense regions of the rings because of their lower surface mass density of pure water ice.

Vis-NIR disk-integrated photometry of asteroid 25143 Itokawa around opposition by AMICA/Hayabusa

We present photometry of the S-type near-Earth asteroid 25143 Itokawa based on both ground-based observations in the UBVRI bands and measurements from the AMICA/Hayabusa spacecraft observations with ul-, b-, v-, w-, x-, and p-filters. Hayabusa observed Itokawa around opposition during the rendezvous, thus providing a unique set of observations of this asteroid. We fit the phase curve measurements with both the Classic Hapke Model (Hapke, 1981, 1984, 1986) and Modern Hapke Model (Hapke, 2002, 2008, 2012a) and thereby extract the physical properties of Itokawa's surface regolith. The single-scattering albedo (0.57 ± 0.05) is larger than that derived for Eros (0.43 ± 0.02), another S-type near-Earth asteroid visited by a spacecraft. Both models indicate a regolith that is forward-scattering in nature. From the hockey stick relationship derived for the single-particle phase function (Hapke, 2012b), both modeling results suggest a regolith comprised of rough surfaced particles with a low density of internal scatterers. Application of the Modern Hapke model derives porosity parameter values from 1 to 1.1, for BVR bands, which corresponds to porosity values between 77–79%. This suggests the surface of Itokawa is very fluffy and the large boulders may be bonded with smaller size particles, typical of the particle sizes observed in Muses Sea. Both models also provide similar geometric albedo values (0.27 ± 0.02) at the V-band wavelength, which are equivalent to Eros’ geometric albedo.

THE FIRST HIGH-PHASE OBSERVATIONS OF A KBO: NEW HORIZONS IMAGING OF (15810) 1994 JR1 FROM THE KUIPER BELT

ABSTRACT NASA’s New Horizons spacecraft observed (15810) 1994 JR1, a 3:2 resonant Kuiper belt object (KBO), using the LOng Range Reconnaissance Imager (LORRI) on 2015 November 2 from a distance of 1.85 au and again on 2016 April 7 from a distance of 0.71 au. These were the first close observations of any KBO other than Pluto. Combining ground-based and Hubble Space Telescope (HST) observations at small phase angles and the LORRI observations at higher phase angles, we produced the first disk-integrated solar phase curve of a typical KBO from α = 0.°6–58°. Observations at these geometries, attainable only from a spacecraft in the outer solar system, constrain surface properties such as macroscopic roughness and the single particle phase function. 1994 JR1 has a rough surface with a 37° ± 5° mean topographic slope angle and has a relatively rapid rotation period of 5.47 ± 0.33 hr. 1994 JR1 is currently 2.7 au from Pluto; our astrometric points enable high-precision orbit determination and integrations that show that it comes this close to Pluto every 2.4 million years (104 heliocentric orbits), causing Pluto to perturb 1994 JR1. During the November spacecraft observation, the KBO was simultaneously observed using HST in two colors, confirming its very red spectral slope. These observations have laid the groundwork for numerous potential future distant KBO observations in the New Horizons–Kuiper belt extended mission.

Photometric properties of Ceres from telescopic observations using Dawn Framing Camera color filters

The dwarf planet Ceres is likely differentiated similar to the terrestrial planets but with a water/ice dominated mantle and an aqueously altered crust. Detailed modeling of Ceres’ phase function has never been performed to understand its surface properties. The Dawn spacecraft began orbital science operations at the dwarf planet in April 2015. We observed Ceres with flight spares of the seven Dawn Framing Camera color filters mounted on ground-based telescopes over the course of three years to model its phase function versus wavelength. Our analysis shows that the modeled geometric albedos derived from both the IAU HG model and the Hapke model are consistent with a flat and featureless spectrum of Ceres, although the values are ∼10% higher than previous measurements. Our models also suggest a wavelength dependence of Ceres’ phase function. The IAU G-parameter and the Hapke single-particle phase function parameter, g, are both consistent with decreasing (shallower) phase slope with increasing wavelength. Such a wavelength dependence of phase function is consistent with reddening of spectral slope with increasing phase angle, or phase-reddening. This phase reddening is consistent with previous spectra of Ceres obtained at various phase angles archived in the literature, and consistent with the fact that the modeled geometric albedo spectrum of Ceres is the bluest of all spectra because it represents the spectrum at 0° phase angle. Ground-based FC color filter lightcurve data are consistent with HST albedo maps confirming that Ceres’ lightcurve is dominated by albedo and not shape. We detected a positive correlation between 1.1-μm absorption band depth and geometric albedo suggesting brighter areas on Ceres have absorption bands that are deeper. We did not see the “extreme” slope values measured by Perna et al. (Perna, D., et al. [2015]. Astron. Astrophys. 575 (L1-6)), which they have attributed to “resurfacing episodes” on Ceres.

Realistic uncertainties on Hapke model parameters from photometric measurement

The single particle phase function describes the manner in which an average element of a granular material diffuses the light in the angular space usually with two parameters: the asymmetry parameter b describing the width of the scattering lobe and the backscattering fraction c describing the main direction of the scattering lobe. Hapke proposed a convenient and widely used analytical model to describe the spectro-photometry of granular materials. Using a compilation of the published data, Hapke (Hapke, B. [2012]. Icarus 221, 1079–1083) recently studied the relationship of b and c for natural examples and proposed the hockey stick relation (excluding b>0.5 and c>0.5). For the moment, there is no theoretical explanation for this relationship. One goal of this article is to study a possible bias due to the retrieval method.We expand here an innovative Bayesian inversion method in order to study into detail the uncertainties of retrieved parameters. On Emission Phase Function (EPF) data, we demonstrate that the uncertainties of the retrieved parameters follow the same hockey stick relation, suggesting that this relation is due to the fact that b and c are coupled parameters in the Hapke model instead of a natural phenomena. Nevertheless, the data used in the Hapke (Hapke, B. [2012]. Icarus 221, 1079–1083) compilation generally are full Bidirectional Reflectance Diffusion Function (BRDF) that are shown not to be subject to this artifact.Moreover, the Bayesian method is a good tool to test if the sampling geometry is sufficient to constrain the parameters (single scattering albedo, surface roughness, b,c, opposition effect). We performed sensitivity tests by mimicking various surface scattering properties and various single image-like/disk resolved image, EPF-like and BRDF-like geometric sampling conditions. The second goal of this article is to estimate the favorable geometric conditions for an accurate estimation of photometric parameters in order to provide new constraints for future observation campaigns and instrumentations.

Optimization of the optical particle properties for a high temperature solar particle receiver

A non-homogeneous slab of particle dispersion composed of two-layers at high temperature, submitted to a concentrated and collimated solar radiation flux with a reflective receiver back wall is considered as a model of a solar particle receiver. The Radiative Transfer Equation (RTE) is solved using a two-stream method and an appropriate hybrid modified Eddington-delta function approximation. The single particle optical properties are modeled using the Lorenz–Mie theory, the single particle phase function is approximated by the Henyey–Greenstein phase function. A Particle Swarm Optimization (PSO) algorithm is used to optimize the particle radius (0.1μm⩽r≤100μm), the volume fraction (1×10-7⩽fv⩽1×10-4) and the refractive index (2.0⩽n≤4.5 and 0.0001⩽k≤25) of an ideal theoretical material to use in a solar particle receiver. Single- and two-layer receivers with known temperature profiles were optimized to maximize the receiver efficiencies. Spectral selective behavior of the optimized refractive index is discussed with the influence of particle radii and volume fractions. The theoretical ideal optical properties found for the particles have given the maximum efficiency reachable by the studied receivers and have shown that an optimized single-layer receiver will perform as well as a two-layer receiver.

Photometric properties of the nucleus of Comet 103P/Hartley 2

We have studied the photometric properties of the nucleus of a hyperactive comet, 103P/Hartley 2, at visible wavelengths using the DIXI flyby images with both disk-integrated and disk-resolved analyses. The disk-integrated phase function of the nucleus has a linear slope of 0.046±0.002mag/deg and an absolute magnitude of 18.4±0.1 at V-band. The nucleus displays an overall linear, featureless spectrum between 400nm and 850nm. The linear spectral slope is 7.6±3.6% per 100nm, corresponding to broadband solar-illuminated color indices B–V of 0.75±0.05 and V–R of 0.43±0.04. Disk-resolved photometric analysis with a Hapke model returns a best-fit single-scattering albedo of 0.036±0.006, an asymmetry factor of the Henyey–Greenstein single-particle phase function of −0.46±0.06, and a photometric roughness of 15±10°. The model yields a geometric albedo of 0.045±0.009 and a Bond albedo of 0.012±0.002. The overall photometric variations of the nucleus are small, with an equivalent albedo variation of 15% FWHM, and a color variation of 12% FWHM. Some areas near the terminator visible in the inbound images show an albedo of more than twice the global average value, and a much bluer color than the average nucleus. The overall photometric properties and variations of the nucleus of Hartley 2 are similar to those of the nuclei of Comets Wild 2 and Tempel 1 as studied from previous spacecraft flyby missions at similar resolutions.

Bidirectional reflectance spectroscopy 7: The single particle phase function hockey stick relation

The measured volume-average single particle angular scattering functions of a large number of types of particle of interest for planetary regoliths in the visible-near-IR wavelength region can be represented to a reasonable approximation by two-parameter, double Henyey–Greenstein functions. When the two parameters of this function are plotted against one another they are found to be inversely correlated and lie within a restricted zone shaped like a hockey stick within the parameter space. The centroid of the zone is a curve that can be represented by a simple empirical equation. The wide variety of types of particles used to construct the plot implies that this equation may represent most of the particles found in regoliths. This means that when modeling the bidirectional reflectance of a regolith it may be possible to reduce the number of parameters necessary to specify the reflectance, and also to characterize the entire single particle phase function from observations at phase angles less than 90°. Even if the hockey stick relation has a finite width, rather than being a line, it restricts the parameter space that must be searched when fitting data. The curve should also be useful for forward modeling particle phase functions.

Analysis of disk‐resolved OMEGA and CRISM spectral observations of Phobos and Deimos

Disk‐resolved observations of Phobos acquired by OMEGA at a range of lighting and viewing geometries were fit with the Hapke photometric function to solve for the single particle phase function and single scattering albedos from 0.4 to 2.5 μm. Single scattering albedos were recovered from CRISM observations of Phobos using the OMEGA derived single particle phase function and are similar to those from OMEGA data. Both the ubiquitous red unit and the blue unit around the crater Stickney exhibit a smooth red‐sloped spectrum, with a steeper continuum in the redder unit. Single scattering albedos retrieved from CRISM measurements of Deimos are similar to those for the red unit on Phobos. Retrieval of single scattering albedos from OMEGA data at 2.8 to 5.0 μm has greater uncertainty, but results in this wavelength range are also consistent with a smooth, red‐sloped spectrum. Phobos' and Deimos' low reflectances, lack of mafic absorption features, and red spectral slopes are incompatible with even highly space weathered chondritic or basaltic compositions. These results, coupled with similarities to laboratory spectra of Tagish Lake (possible D‐type asteroid analog) and CM carbonaceous chondrite meteorites, show that Phobos and Deimos have primitive compositions. If the moons formed in situ rather than by capture of primitive bodies, primitive materials must have been added to the Martian system during accretion or a late stage impact.

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.

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.

Hapke modeling of Rhea surface properties through Cassini-VIMS spectra

The surface properties of the icy bodies in the saturnian system have been investigated by means of the Cassini-VIMS (Visual Infrared Mapping Spectrometer) hyperspectral imager which operates in the 0.35–5.1 μm wavelength range. In particular, we have analyzed 111 full disk hyperspectral images of Rhea ranging in solar phase between 0.08° and 109.8°. These data have been previously analyzed by Filacchione et al. (Filacchione, G. et al. [2007]. Icarus 186, 259–290; Filacchione, G. et al. [2010]. Icarus 206, 507–523) to study, adopting various “spectral indicators” (such as spectral slopes, band depth, and continuum level), the relations among various saturnian satellites. As a further step we proceed in this paper to a quantitative evaluation of the physical parameters determining the spectrophotometric properties of Rhea’s surface. To do this we have applied Hapke (Hapke, B. [1993]. Theory of Reflectance and Emittance Spectroscopy, Topics in Remote Sensing: 3. Springer, Berlin) IMSA model (Isotropic Multiple Scattering Approximation) which allow us to model the phase function at VIS–IR (visible–infrared) wavelengths as well as the spectra taking into account various types of mixtures of surface materials. Thanks to this method we have been able to constrain the size of water ice particles covering the surface, the amount of organic contaminants, the large scale surface roughness and the opposition effect surge. From our analysis it appears that wavelength dependent parameters, e.g. opposition surge width ( h) and single-particle phase function parameters ( b, v), are strongly correlated to the estimated single-scattering albedo of particles. For Rhea the best fit solution is obtained by assuming: (1) an intraparticle mixture of crystalline water ice and a small amount (0.4%) of Triton tholin; (2) a monodisperse grain size distribution having a particle diameter a m = 38 μm; and (3) a surface roughness parameter value of 33°. The study of phase function shows that both shadow hiding and coherent backscattering contribute to the opposition surge. This study represents the first attempt, in the case of Rhea, to join the spectral and the photometric analysis. The surface model we derived gives a good quantitative description of both spectrum and phase curve of the satellite. The same approach and model, with appropriate modifications, shall be applied to VIMS data of the other icy satellites of Saturn, in order to reveal similarities and differences in the surface characteristics to understand how these bodies interact with their environment.

Infrared (0.83–5.1 μm) photometry of Phoebe from the Cassini Visual Infrared Mapping Spectrometer

Three weeks prior to the commencement of Cassini's 4 year tour of the saturnian system, the spacecraft executed a close flyby of the outer satellite Phoebe. The infrared channel of the Visual Infrared Mapping Spectrometer (VIMS) obtained images of reflected light over the 0.83–5.1 μm spectral range with an average spectral resolution of 16.5 nm, spatial resolution up to 2 km, and over a range of solar phase angles not observed before. These images have been analyzed to derive fundamental photometric parameters including the phase curve and phase integral, spectral geometric albedo, bolometric Bond albedo, and the single scattering albedo. Physical properties of the surface, including macroscopic roughness and the single particle phase function, have also been characterized. Maps of normal reflectance show the existence of two major albedo regimes in the infrared, with gradations between the two regimes and much terrain with substantially higher albedos. The phase integral of Phoebe is 0.29 ± 0.03 , with no significant wavelength dependence. The bolometric Bond albedo is 0.023 ± 007 . We find that the surface of Phoebe is rough, with a mean slope angle of 33°. The satellite's surface has a substantial forward scattering component, suggesting that its surface is dusty, perhaps from a history of outgassing. The spectrum of Phoebe is best matched by a composition including water ice, amorphous carbon, iron-bearing minerals, carbon dioxide, and Triton tholin. The characteristics of Phoebe suggest that it originated outside the saturnian system, perhaps in the Kuiper Belt, and was captured on its journey inward, as suggested by Johnson and Lunine (2005).

Near-infrared spectrophotometry of Asteroid 25143 Itokawa from NIRS on the Hayabusa spacecraft

A photometric analysis of the S-type Asteroid 25143 Itokawa is performed over multiple wavelengths ranging from 0.85 to 2.10 μm based on disk-resolved reflectance spectra obtained with the Hayabusa near-infrared spectrometer (NIRS). We derive the global photometric properties of Itokawa in terms of Hapke's photometric model. We find that Itokawa has a single-scatter albedo that is 35–40% less than that of Asteroid 433 Eros. Itokawa also has a single-particle phase function that is more strongly back-scattering than that of Eros. Despite its hummocky surface strewn with large boulders, Itokawa exhibits an opposition effect. However, the total amplitude of the opposition surge for Itokawa was estimated to be less than unity while Eros and other S-type asteroids have been found to have model values exceeding unity. The wavelength dependence of the opposition surge width reveals that coherent backscatter contributes to the opposition effect on Itokawa's surface. The photometric roughness of Itokawa is well constrained to a value of 26° ± 1° which is similar to Eros, suggesting that photometric roughness models the smallest surface roughness scale for which shadows exist.

Deep Impact photometry of Comet 9P/Tempel 1

The photometric properties of the nucleus of Comet 9P/Tempel 1 are studied from the disk-resolved color images obtained by Deep Impact (DI). Comet Tempel 1 has typical photometric properties for comets and dark asteroids. The disk-integrated spectrum of the nucleus of Tempel 1 between 309 and 950 nm is linear without any features at the spectral resolution of the filtered images. At V-band, the red slope of the nucleus is 12.5 ± 1 % per 100 nm at 63° phase angle, translating to B – V = 0.84 ± 0.01 , V – R = 0.50 ± 0.01 , and R – I = 0.49 ± 0.02 . No phase reddening is confirmed. The phase function of the nucleus of Tempel 1 is constructed from DI images and earlier ground-based observations found from the literature. The phase coefficient is determined to be β = 0.046 ± 0.007 mag / deg between 4° and 117° phase angle. Hapke's theoretical scattering model was used to model the photometric properties of this comet. Assuming a single Henyey–Greenstein function for the single-particle phase function, the asymmetry factor of Tempel 1 was fitted to be g = − 0.49 ± 0.02 , and the corresponding single-scattering albedo (SSA) was modeled to be 0.039 ± 0.005 at 550 nm wavelength. The SSA spectrum shows a similar linear slope to that of the disk-integrated spectrum. The roughness parameter is found to be 16 ° ± 8 ° , and independent of wavelength. The Minnaert k parameter is modeled to be 0.680 ± 0.014 . The photometric variations on Tempel 1 are relatively small compared to other comets and asteroids, with a ∼ 20 % full width at half maximum of albedo variation histogram, and ∼ 3 % for color. Roughness variations are evident in one small area, with a roughness parameter about twice the average and appearing to correlate with the complex morphological texture seen in high-resolution images.

Deep Impact photometry of Comet 9P/Tempel 1

The photometric properties of the nucleus of Comet 9P/Tempel 1 are studied from the disk-resolved color images obtained by Deep Impact (DI). Comet Tempel 1 has typical photometric properties for comets and dark asteroids. The disk-integrated spectrum of the nucleus of Tempel 1 between 309 and 950 nm is linear without any features at the spectral resolution of the filtered images. At V-band, the red slope of the nucleus is 12.5 ± 1 % per 100 nm at 63° phase angle, translating to B – V = 0.84 ± 0.01 , V – R = 0.50 ± 0.01 , and R – I = 0.49 ± 0.02 . No phase reddening is confirmed. The phase function of the nucleus of Tempel 1 is constructed from DI images and earlier ground-based observations found from the literature. The phase coefficient is determined to be β = 0.046 ± 0.007 mag / deg between 4° and 117° phase angle. Hapke's theoretical scattering model was used to model the photometric properties of this comet. Assuming a single Henyey–Greenstein function for the single-particle phase function, the asymmetry factor of Tempel 1 was fitted to be g = − 0.49 ± 0.02 , and the corresponding single-scattering albedo (SSA) was modeled to be 0.039 ± 0.005 at 550 nm wavelength. The SSA spectrum shows a similar linear slope to that of the disk-integrated spectrum. The roughness parameter is found to be 16 ° ± 8 ° , and independent of wavelength. The Minnaert k parameter is modeled to be 0.680 ± 0.014 . The photometric variations on Tempel 1 are relatively small compared to other comets and asteroids, with a ∼ 20 % full width at half maximum of albedo variation histogram, and ∼ 3 % for color. Roughness variations are evident in one small area, with a roughness parameter about twice the average and appearing to correlate with the complex morphological texture seen in high-resolution images.

Diffuse reflectance of TiO2 pigmented paints: Spectral dependence of the average pathlength parameter and the forward scattering ratio

Diffuse reflectance spectra of paint coatings with different pigment concentrations, normally illuminated with unpolarized radiation, have been measured. A four-flux radiative transfer approach is used to model the diffuse reflectance of TiO2 (rutile) pigmented coatings through the solar spectral range. The spectral dependence of the average pathlength parameter and of the forward scattering ratio for diffuse radiation, are explicitly incorporated into this four-flux model from two novel approximations. The size distribution of the pigments has been taken into account to obtain the averages of the four-flux parameters: scattering and absorption cross sections, forward scattering ratios for collimated and isotropic diffuse radiation, and coefficients involved in the expansion of the single particle phase function in terms of Legendre polynomials.

Photometric analysis of Eros from NEAR data

A photometric model of (433) Eros at wavelengths from 450 to 1050 nm is constructed using the combination of the images from the multispectral imager (MSI) obtained during the one-year long orbital phase of the NEAR mission, ground-based lightcurves from earlier observations, and our theoretical forward modeling simulations coupled with the NEAR shape model. The single scattering albedo is found to be 0.33 ± 0.03 at 550 nm, which is smaller than past findings by 30%. The amplitude and width of the opposition effect are 1.4 ± 0.1 and 0.010 ± 0.004 from ground based lightcurves. It is confirmed that the asymmetry factor of the single-particle phase function and the surface roughness parameter do not depend on wavelength from 450 to 1050 nm, and their values are estimated to be − 0.25 ± 0.02 and 28 ° ± 3 ° , respectively, comparable with the earlier measurements from the NEAR NIS data. The geometric albedo and the Bond albedo at 550 nm are calculated to be 0.23 and 0.093, respectively, which make Eros less reflective than previous models, but still slightly more reflective than average S-type asteroids. The lower albedos of Eros are more consistent with our forward modeling simulations, as well as with its spectrum. Eros is a typical S-type asteroid like (951) Gaspra and (243) Ida, and has similar surface regolith properties. Combining the single-scattering albedo with the olivine composition of ordinary chondrites, taking into account space weathering darkening, we constrain the grain size of the regolith particles on Eros to a range of 50 to 100 μm.

Effective phase function for light scattered by disperse systems—the small-angle approximation

We focus on the quasi-ballistic regime of the scattering process induced in a diffusive medium by a collimated laser beam. The main goal is to find an approximate analytic solution for the radiative transport equation. Particles with characteristic dimensions larger than the light wavelength and relative refraction indices close to unity have a scattering phase function strongly peaked in the forward direction. This specific angular dependence was used to approximate, for any scattering order, the angular dependence of the scattered light fluxes and to obtain simple analytic expressions. Even more, in the small-angle approximation the scattered far field can be described by an effective phase function given by the same analytic formula as the single-particle phase function.