Asteroid photometric and polarimetric phase curves: Joint linear‐exponential modeling

Abstract— We present Markov‐Chain Monte‐Carlo methods (MCMC) for the derivation of empirical model parameters for photometric and polarimetric phase curves of asteroids. Here we model the two phase curves jointly at phase angles ≤25° using a linear‐exponential model, accounting for the opposition effect in disk‐integrated brightness and the negative branch in the degree of linear polarization. We apply the MCMC methods to V‐band phase curves of asteroids 419 Aurelia (taxonomic class F), 24 Themis (C), 1 Ceres (G), 20 Massalia (S), 55 Pandora (M), and 64 Angelina (E). We show that the photometric and polarimetric phase curves can be described using a common nonlinear parameter for the angular widths of the opposition effect and negative‐polarization branch, thus supporting the hypothesis of common physical mechanisms being responsible for the phenomena. Furthermore, incorporating polarimetric observations removes the indeterminacy of the opposition effect for 1 Ceres. We unveil a trend in the interrelation between the enhancement factor of the opposition effect and the angular width: the enhancement factor decreases with decreasing angular width. The minimum polarization and the polarimetric slope at the inversion angle show systematic trends when plotted against the angular width and the normalized photometric slope parameter. Our new approach allows improved analyses of possible similarities and differences among asteroidal surfaces.     

A Critical review of theoretical models of negatively polarized light scattered by atmosphereless solar system bodies

About a dozen physical mechanisms and models aspire to explain the negative polarization of light scattered by atmosphereless celestial bodies. This is too large a number for the reliable interpretation of observational data. Through a comparative analysis of the models, our main goal is to answer the question: Does any one model have an advantage over the others? Our analysis is based on new laboratory polarimetric and photometric data as well as on theoretical results. We show that the widely used models due to Hopfield and Wolff cannot realistically explain the phase-angle dependence of the degree of polarization observed at small phase angles. The so-called interference or coherent backscattering mechanism is the most promising model. Models based on that mechanism use well-defined physical parameters to explain both negative polarization and the opposition effect. They are supported by laboratory experiments, particularly those showing enhancement of negative polarization with decreasing particle size down to the wavelength of light. According to the interference mechanism, pronounced negative branches of polarization, like those of C-class asteroids, may indicate a high degree of optical inhomogeneity of light-scattering surfaces at small scales. The mechanism also seems appropriate for treating the negative polarization and opposition effects of cometary dust comae, planetary rings, and the zodiacal light.     

Polarization and brightness opposition effects for the E-type Asteroid 44 Nysa

We present new polarimetric and photometric observations of high-albedo E-type Asteroid 44 Nysa in the BVRI wavebands at phase angles ranging from 0.41° to 7.49° during the 2005 opposition. A bimodal phase-angle dependence of polarization was found for Nysa in the V band. The polarization opposition effect was revealed in the form of a secondary minimum of negative polarization with amplitude ∼0.3% centered at a phase angle ∼0.8°. It is superimposed on the regular negative polarization branch with minimal polarization −0.30% at a phase angle 5.8°. We analyzed all available polarimetric data for E-type Asteroids 44 Nysa, 64 Angelina, and 214 Ashera and confirmed the presence of the polarization opposition effect for high-albedo asteroids at phase angle ∼1° with an amplitude ∼0.35%. The magnitude-phase curves reveal the presence of spike-like opposition effect of brightness for 44 Nysa in the BVRI spectral bands. 44 Nysa is the second high-albedo asteroid after 64 Angelina for which both the polarization opposition effect and the brightness opposition effect are detected. The differences between the parameters of the opposition effects for silicate surfaces (44 Nysa, 64 Angelina, Io) and icy surfaces (Europa, Ganymede, Iapetus, Saturn's rings) are discussed. The specific morphological parameters of opposition effects, in particular the angular width of the polarization opposition effect is comparable to that of the brightness opposition effect, provide almost unequivocal evidence that they are caused by coherent backscattering. One of unexpected results of our investigation is that 44 Nysa becomes bluer with increasing phase angle, while 64 Angelina shows phase reddening.     

Polarization and brightness opposition effects for the E-type Asteroid 64 Angelina

We present new polarimetric and photometric observations of the high-albedo Asteroid 64 Angelina in the UBVRI wavebands at phase angles ranging from 0.43° to 13.02° during oppositions in 1995, 1999, and 2000/2001. The polarization opposition effect has been observed in the form of a sharp peak of negative polarization with amplitude of about −0.4% centered at αmin≈1.8°, which is superimposed on the regular negative polarization branch. The amplitude of the polarization opposition effect appears to be apparition-dependent. Our photometric data confirm the early detected by Harris et al. [1989. Phase relations of high-albedo asteroids: The unusual opposition brightening of 44 Nysa and 64 Angelina. Icarus 81, 365-374] of a very strong and unusually narrow opposition spike, i.e., brightness opposition effect, for Angelina. Thus, 64 Angelina is the first asteroid for which both the polarization opposition effect and the brightness opposition effect have been detected. We observed that the polarization opposition effect as well as the regular negative polarization branch depends on the wavelength of scattered light, but in different manners. In addition, the colors B-V and V-R show little phase-angle dependence, while the color U-B increases with increasing phase angle, thus indicating that the amplitude of the brightness opposition effect is larger in the U band and almost the same in the B, V, and R bands. It appears that all colors indices begin to increase with decreasing phase angle to zero. The composite lightcurve computed with a period of 8.752 h has amplitude of 0.13 magnitude.     

Reflectance model for densely packed media: Estimates of the surface properties of the high-albedo satellites of Saturn

Interpretation of photometric and polarimetric observations of atmosphereless celestial bodies faces the problems connected with both the insufficient accuracy and level of details in groundbased observations and the current state of the theory of the multiple scattering of light. In application to sparse media, where the electromagnetic waves, propagating between the scatterers, can be considered as spherical (the socalled far-field approximation), this theory is rather well developed for both the diffuse and coherent components of the scattered radiation. In this paper, we show that this theory can be also successfully applied to the measurements of polarization of light scattered by densely packed, though nonabsorbing or weakly absorbing, media. For this purpose, we calculated the models for a semi-infinite layer of the medium composed of randomly oriented clusters of spherical particles and compared them with the data of laboratory and astronomical measurements. The potential of the present approach is illustrated by an example of the interpretation of the polarization measurements of the ice satellites of Saturn—Rhea and Enceladus—which allowed some properties of the surface of these celestial bodies to be estimated. In particular, the ratio of the surface area that makes no contribution to the negative polarization of light reflected at small phase angles to the area producing the negative polarization branch was found. Under the assumption of the same albedo of these areas, this ratio turned out to be 3.31–3.66 and 1.7–3.8 for Rhea and Enceladus, respectively. For Enceladus, it is difficult to obtain a sufficiently narrow range of the estimated parameters, since the number of measurement points in the phase dependence of polarization of this satellite is small. For the surface of Rhea, the estimated packing density of particles, participating in the opposition effects, is approximately 15%, while their smallest size is of the order of the wavelength of visible light.     

Opposition polarimetry and photometry of S- and E-type asteroids

The first results of the observational program devoted to simultaneous investigation of asteroid polarimetric and photometric opposition phenomena are presented. UBVRI polarimetric and V-band photometric observations of the S-type Asteroid 20 Massalia and the E-type Asteroids 214 Aschera and 620 Drakonia were carried out in 1996-1999 down to phase angles of 0.08°, 0.7°, and 1.2°, correspondingly. The S-type Asteroid 20 Massalia is characterized by the pronounced brightness opposition surge with an amplitude larger than that observed for the E-type asteroids. A sharp peak of negative polarization at small phase angles was not observed for this asteroid. The value of polarization degree at phase angle α<1° is less than 0.5% for both S and E types. The negative polarization branches of S and especially E-asteroids have an asymmetrical shape. The phase angle at which the polarization minimum occurs is close to the angle at which non-linear increase begins in the asteroid magnitude phase curves. A relation of the observed effects to the mechanism of coherent backscattering is discussed.     

Polarization opposition effect for the Galilean satellites of Jupiter

We present results of polarimetric observations of the Galilean satellites Io, Europa, Ganymede, and Callisto at phase angles ranging from 0.19° to 2.22°. The observations in the UBVR filters were performed using a one-channel photoelectric polarimeter attached to 70-cm telescope of the Chuguev Observation Station (Ukraine) on November 19-December 7, 2000. We have observed the polarization opposition effect for Io, Europa, and Ganymede to be a sharp secondary spike of negative polarization with an amplitude of about −0.4% centered at phase angles of 0.2°-0.7° and superimposed on the regular negative polarization branch. Although these minima for Io, Europa, and Ganymede show many similarities, they also exhibit a number of distinctions. The polarization opposition effect appears to be wavelength-dependent, at least for Europa and Ganymede. No polarization opposition effect was found for Callisto. The results obtained are discussed within the framework of different mechanisms of light scattering.     

Coherent backscattering and opposition effects observed in some atmosphereless bodies of the solar system

The results of photometric and polarimetric observations carried out for some bright atmosphere-less bodies of the Solar system near the zero phase angle reveal the simultaneous existence of two spectacular optical phenomena, the so-called brightness and polarization opposition effects. In a number of studies, these phenomena were explained by the influence of coherent backscattering. However, in general, the interference concept of coherent backscattering can be used only in the case where the particles are in the far-field zones of each other, i.e., when the scattering medium is rather rarefied. Because of this, it is important to prove rigorously and to demonstrate that the coherent backscattering effect may also exist in densely packed scattering media like regolith surface layers of celestial bodies. From the results of the computer modeling performed with the use of numerically exact solutions of the macroscopic Maxwell equations for discrete random media with different packing densities of particles, we studied the origin and evolution of all the opposition phenomena predicted by the coherent backscattering theory for low-packing-density media. It has been shown that the predictions of this theory remain valid for rather high packing densities of particles that are typical, in particular, of regolith surfaces of the Solar system bodies. The results allow us to conclude that both opposition effects observed simultaneously in some high-albedo atmosphereless bodies of the Solar system are caused precisely by coherent backscattering of solar light in the regolith layers composed of microscopic particles.     

Polarization of Light Scattered by Fluffy Particles (PROGRA Experiment)

This work was carried out with the PROGRA² experiment developed to measure the angular dependence of the polarization of light scattered by dust particles. The dust samples are fluffy aggregates (size range 0.01-1 mm) with constituent grains of about 10 nm. Various setups were used: samples deposited on surfaces, the same samples lifted under the effect of a draft, and particles levitating in microgravity conditions on board the CNES dedicated aircraft. For deposited particles, the maximum value of polarization (P_max) follows the Umov law. For a cloud of particles (P_max) near 100° phase angle decreases when: (i) multiple scattering between the particles—or between the grains inside the particles—increases, or (ii) the real part of the refractive index of the materials increases, or (iii) the size parameter of the constituent grains increases between 0.05 and 0.5. A negative branch in the polarization phase curve is found for deposited samples. For levitating particles made of a single material and a single size distribution, a positive increase of polarization appears at phase angles smaller than 20°; for mixtures of these materials the polarization is negative at the same phase angles. These results are compared to modeling results as well as to polarimetric observations of comets.     

Polarization of Light Scattered by Surfaces with Complex Microstructure at Phase Angles 0.1°–3.5°

We present laboratory measurements of the phase dependences of linear polarization for surfaces with a complex microstructure in the range of phase angles 0.1°–3.5° A sample of freshly fallen snow (with particle sizes of about 50 × 500 m) exhibits a nearly zero polarization. Surfaces with submicron structure show a narrow branch of negative polarization at small phase angles, irrespective of whether the surface is powderlike or solid with microcrystalline structure. This polarization is similar to that exhibited by Jupiter's satellites. The negative polarization branch becomes deeper with decreasing porosity of light dielectric surfaces. At the phase angles between 0.5° and 3.0°, the polarization for quartz powder with 10-m particles is almost constant. The polarization for light dielectric surfaces depends on the geometry of illumination and observation. An inclination of the surface in the scattering plane produces a parallel shift of the negative polarization branch toward large values of the polarization modulus. The same inclination in a perpendicular direction produces the same shift toward positive degrees of polarization.     

The F-type asteroids with small inversion angles of polarization

UBVRI polarimetric observations carried out in 1997-2004 for the F-type Asteroids 302 Clarissa, 419 Aurelia, 704 Interamnia, and 762 Pulcova (V band only) are presented. Asteroid 419 Aurelia is characterized by a negative polarization branch which is unusual for low-albedo asteroids. Its depth is about 1%, while the inversion angle, close to 14°, reaches the smallest value ever observed for asteroids. This is the first definite example of a minor body exhibiting a considerable decrease of both the depth and width of the negative polarization branch in comparison with polarization properties of other low-albedo bodies, mimicking a behavior previously found in laboratory measurements of extremely dark surfaces. The F-type Asteroids 302 Clarissa and 704 Interamnia are also characterized by unusually small inversion angles compared to other asteroid types, while Asteroid 762 Pulcova seems to have an ordinary negative polarization branch. Laboratory measurements of low-albedo samples and computer simulations of light scattering by particles of irregular shapes were made to interpret observational data. We find that an optical homogeneity of regolith microstructure at scales of the order of visible light wavelengths may be responsible for relatively small values of the depth of the negative polarization branch and of the inversion angle. Peculiar features of the F-type asteroids compared to other taxonomic classes are discussed.     

A Study of Saturn's Ring Phase Curves from HST Observations

Solar phase curves between 0.3° and 6.0° and color ratios at wavelengths λ=0.336 μm and λ=0.555 μm for Saturn's rings are presented using recent Hubble Space Telescope observations. We test the hypothesis that the phase reddening of the rings is less due to collective properties of the ring particles than to the individual properties of the ring particles. We use a modified Drossart model, the Hapke model, and the Shkuratov model to model reddening by either intraparticle shadow-hiding on fractal and normal surfaces, multiple scattering, or some combination. The modified Drossart model (including only shadowing) failed to reproduce the data. The Hapke model gives fair fits, except for the color ratios. A detailed study of the opposition effect suggests that coherent backscattering is the principal cause of the opposition surge at very small phase angles. The shape of the phase curve and color ratios of each main ring regions are accurately represented by the Shkuratov model, which includes both a shadow-hiding effect and coherent backscatter enhancement. Our analysis demonstrates that in terms of particle roughness, the C ring particles are comparable to the Moon, but the Cassini division and especially the A and B ring particles are significantly rougher, suggesting lumpy particles such as often seen in models. Another conspicuous difference between ring regions is in the effective size d of regolith grains (d∼λ for the C ring particles, d∼1-10 μm for the other rings).     

Polarimetric mapping of the Moon at a phase angle near the polarization minimum

At small phase angles the light scattered by the Moon reveals a negative polarization branch whose average amplitude is 1%. We present results of polarimetric mappings of the Moon in Pmin at a phase angle near 11°. The observations were carried out with the Kharkov 50-cm telescope at the Maidanak Observatory (Middle Asia) using a polarizing filter. A thorough calibration of the camera array allows for the reliable detection of significant variations of |Pmin| over the lunar surface, from 0.2 to 1.6%, at a wavelength of 0.52 μm. The smallest |Pmin| are characteristic of young bright craters, while the |Pmin| are the highest for the lunar highland and bright mare areas. The horse-shoe shape of the correlation dependence Pmin (albedo) is treated with data of our laboratory measurements of powdered surfaces and computer modeling of light scattering by small particles with the DDA (discrete dipole approximation) technique.     

Opposition Effects in Brightness,Color, and Polarization of Comet 1P/Halley: Comparison with Atmosphereless Solar System Bodies

Systematic and uniform sets of photometric and polarimetric observations of comet 1P/Halley have been analyzed. The phase dependence of brightness for comet Halley was obtained at phase angles α ranging from 1.4° ≤ α ≤ 65°. The following parameters were determined: the amplitude of the opposition effect Δm = 0.75^m ± 0.06^m; the half-width at a half-maximum of intensity HWHM = 6.4° ± 1.6°; the linear phase coefficient β = 0.0045 ± 0.0001 mag/deg for α from 30° ≤ α ≤ 65°; and the phase angle at which a nonlinear increase in brightness starts, α _opp ≈ 31°. For the first time, the phase-angle dependence was obtained for the color of the dust of comet Halley: the color index BC-RC systematically decreases with increasing phase angle. Such a phase behavior of the dust color can be caused by the decrease in the mean size of dust particles that occurs when the comet approaches the Sun. For comet Halley, the negative polarization branch is almost symmetric; the minimal value of polarization is P _min = −1.54% at a phase angle α_min = 10.5°, and the inversion angle is α_inv = 21.7°. A comparative analysis of the phase functions of brightness and polarization has been performed for the cometary dust and atmosphereless bodies. Among the latter are low-albedo asteroids of the P and C types (102 Miriam and 47 Aglaja, respectively), as well as Deimos; high-albedo objects, such as the E-type asteroid 64 Angelina and the icy satellite of Jupiter Europa; and the Moon with its intermediate albedo. The possibility of a weak depression in the negative polarization branch of comets Halley and 47P/Ashbrook-Jackson at phase angles smaller than 2° is discussed.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 4, 2005, pp. 353–363.Original Russian Text Copyright © 2005 by Rosenbush.     

Asteroid photometric and polarimetric phase curves: empirical interpretation

A method for interpretation of asteroid phase curves, based on empirical modeling and laboratory measurements, is outlined and preliminary results are presented. A linear-exponential function is used to describe the opposition peaks and negative polarization surges of various asteroids and laboratory samples and a statistical algorithm is used in parameter estimation. The linear-exponential function describes well the phase curves, but dense phase angle coverage, particularly at small phase angles must be obtained to improve the results. Major emphasis should also be put on laboratory study: with an extensive library of laboratory measurements, a stronger connection between the phase curve properties and surface characteristics is possible.     

Prospects for Estimating the Properties of a Loose Surface from the Phase Profiles of Polarization and Intensity of the Scattered Light

It has been shown that the model of a scattering medium composed of clusters located in the far zones of each other allows some properties of regolith-like surfaces to be quantitatively estimated from the phase dependences of intensity and polarization measured in the backscattering domain. From the polarization profiles, the sizes of particles, the structure and porosity of the medium, and a portion of the surface area covered with a disperse material can be determined. At the same time, the intensity profiles of the scattered light weakly depend on the sizes and structure of particles; they are mainly controlled by the concentration of scatterers in the medium and the shadow-hiding contribution at small phase angles. Since the latter effect is beyond the considered model, a good agreement between the model and the measured intensity cannot be achieved. Nevertheless, if a portion of the surface that participates in coherent backscattering has been found from the phase profile of polarization, the present model makes it possible to determine the relative contribution of the shadow-hiding effect to the brightness surge measured at zero phase angle. This, in turn, may allow the roughness of the scattering surface to be estimated. The model contains no free parameters, but there is currently no possibility to verify it comprehensively by the data obtained in laboratory measurements of the samples with thoroughly controlled characteristics, because such measurements are rare for a wide range of the properties of particles in a medium, their packing density, and phase angles.

     

Characteristics of Light Scattering at Small Phase Angles by Surfaces Consisting of Spherical Particles

We present the results of measurements of the phase dependences of brightness and of the polarization and depolarization characteristics for surfaces consisting of spherical glass particles in the phase-angle range from 0.1° to 5.0°. The measurements are performed using the laboratory photometer/polarimeter of Kharkov Astronomical Observatory and the photometer of Jet Propulsion Laboratory. An optically thick layer of transparent-glass spheres with mean size of about 57 m and refractive index of 1.44 exhibits a strong opposition effect due to single scattering. The contribution of interparticle scattering is nearly independent of the phase angle. At an angle of 0.4°, the spheres exhibit a glory ring that manifests itself in the phase behavior of all characteristics investigated. Small details are seen on the curves when a monochromatic radiation source is used for measurements. Their occurrence is confirmed by calculations based on the Mie theory. The unusual behavior of the phase dependences of reflectivity, degree of polarization, and color index for layers composed of spherical particles can be used to search for sites of possible deposits of spherical glass (or ice) particles in regoliths of atmosphereless celestial bodies.     

Polarimetry of main belt asteroids: Wavelength dependence

New UBVRI polarimetric observations of ten asteroids, including space mission targets 1 Ceres and 21 Lutetia, are presented. These observations were obtained with the 1.25-m telescope of the Crimean Astrophysical Observatory and have been used to study the wavelength dependence of polarization for a sample of asteroids belonging to the M and low albedo classes. A more general analysis including also a larger data set of UBVRI polarimetric observations available in the literature for more than 50 main belt asteroids belonging to different taxonomic classes shows that the variation of the polarization degree Pr as a function of wavelength is generally well described by a linear trend. It typically does not exceed 0.2% in the studied spectral range 0.37-0.83 microns and tends to increase for increasing phase angle. Asteroids belonging to the S and M classes are found to exhibit a deeper negative branch and smaller positive polarization for increasing wavelength (negative sign of the slope of ΔPr/Δλ). Since the objects belonging to these classes are known to exhibit reddish reflectance spectra, the observed wavelength behavior of negative polarization contradicts the well-known inverse correlation of Pmin and albedo. Low albedo asteroids show larger dispersion of spectral slopes, but the overall trend is characterized by a shallower negative branch and a larger positive polarization for increasing wavelength (positive sign of the slope of ΔPr/Δλ). A few exceptions from this general trend are discussed. The observed variety in the wavelength dependence of asteroid polarization seems to be mainly attributed to surface composition.     

NEAR Infrared Spectrometer Photometry of Asteroid 433 Eros

We present near-infrared spectrometer (NIS) observations (0.8 to 2.4 μm) of the S-type asteroid 433 Eros obtained by the NEAR Shoemaker spacecraft and report results of our Hapke photometric model analysis of data obtained at phase angles ranging from 1.2° to 111.0° and at spatial resolutions of 1.25×2.5 to 2.75×5.5 km/spectrum. Our Hapke model fits successfully to the NEAR spectroscopic data for systematic color variations that accompany changing viewing and illumination geometry. Model parameters imply a geometric albedo at 0.946 μm of 0.27±0.04, which corresponds to a geometric albedo at 0.550 μm of 0.25±0.05. We find that Eros exhibits phase reddening of up to 10% across the phase angle range of 0-100°. We observe a 10% increase in the 1-μm band depth at high phase angles. In contrast, we observe only a 5% increase in continuum slope from 1.486 to 2.363 μm and essentially no difference in the 2-μm band depth at higher phase angles. These contrasting phase effects imply that there are phase-dependent differences in the parametric measurements of 1- and 2-μm band areas, and in their ratio. The Hapke model fits suggest that Eros exhibits a weaker opposition surge than either 951 Gaspra or 243 Ida (the only other S-type asteroids for which we possess disk-resolved photometric observations). On average, we find that Eros at 0.946 μm has a higher geometric albedo and a higher single-scatter albedo than Gaspra or Ida at 0.56 μm; however, Eros's single-particle phase function asymmetry and average surface macroscopic roughness parameters are intermediate between Gaspra and Ida. Only two of the five Hapke model parameters exhibit a notable wavelength dependence: (1) The single-scatter albedo mimics the spectrum of Eros, and (2) there is a decrease in angular width of the opposition surge with increasing wavelength from 0.8 to 1.7 μm. Such opposition surge behavior is not adequately modeled with our shadow-hiding Hapke model, consistent with coherent backscattering phenomena near zero phase.     

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.