Rebuttal to comment on “Modeling of opposition effects with ensembles of clusters: Interplay of various scattering mechanisms” by Elena V. Petrova, Victor P. Tishkovets, Klaus Jockers, 2007 [Icarus 188, 233-245]

Shkuratov and Zubko [Shkuratov, Yu.G., Zubko, E., 2008. Icarus 194, 850-852] criticize our paper [Petrova, E.V., Tishkovets, V.P., Jockers, K., 2007. Icarus 188, 233-245]. With this comment we reply to this criticism. We show that the experimental data and the modeling calculations presented by these authors cannot disprove the near-field effect as an important contributor to the scattering mechanisms considered in our paper.     

Comment on “Modeling of opposition effects with ensembles of clusters: Interplay of various scattering mechanisms” by Elena V. Petrova, Victor P. Tishkovets, Klaus Jockers, 2007 [Icarus 188, 233-245]

We show that the mechanism called “near-field effect” [e.g., Petrova, E.V., Tishkovets, V.P., Jockers, K., 2007. Icarus 188, 233-245], which is used to explain wide-phase-angle negative polarization branch observed for planetary regoliths and cometary comas, is not realistic as it contradicts laboratory experiments and results of modeling with discrete dipole approximation calculations.     

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.     

Scattering properties of planetary regolith analogs

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

Re-analysis of previous laboratory phase curves: 1. Variations of the opposition effect morphology with the textural properties,and an application to planetary surfaces

Typical variations in the opposition effect morphology of laboratory samples at optical wavelengths are investigated to probe the role of the textural properties of the surface (roughness, porosity and grain size). A previously published dataset of 34 laboratory phase curves is re-analyzed and fit with several morphological models. The retrieved morphological parameters that characterize the opposition surge, amplitude, width and slope (A, HWHM and S respectively) are correlated to the single scattering albedo, the roughness, the porosity and the grain size of the samples. To test the universality of the laboratory samples’ trends, we use previously published phase curves of planetary surfaces, including the Moon, satellites and rings of the giant planets. The morphological parameters of the surge (A and HWHM) for planetary surfaces are found to have a non-monotonic variation with the single scattering albedo, similar to that observed in asteroids (Belskaya, I.N., Shevchenko, V.G. [2000]. Icarus 147, 94–105), which is unexplained so far. The morphological parameters of the surge (A and HWHM) for laboratory samples seem to exhibit the same non-monotonic variation with single scattering albedo. While the non-monotonic variation with albedo was already observed by Nelson et al. (Nelson, R.M., Hapke, B.W., Smythe, W.D., Hale, A.S., Piatek, J.L. [2004]. Planetary regolith microstructure: An unexpected opposition effect result. In: Mackwell, S., Stansbery, E. (Eds.), Proc. Lunar Sci. Conf. 35, p. 1089), we report here the same variation for the angular width.     

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.     

A quantitative test of the ability of models based on the equation of radiative transfer to predict the bidirectional reflectance of a well-characterized medium

Predictions of two widely-used regolith reflectance models, a numerically exact computer code and an approximate analytic equation, based on the equation of radiative transfer were tested against the measured reflectance of a medium of close-packed spheres, whose properties supposedly can be well-characterized. Surprisingly, the approximate analytic model was a better match to the experimental data than the numerically exact computer solution. Other approximate regolith models were tested briefly with similar results. Discrepancies between the two models and between models and experiment can be explained if the phase functions and albedos of the spheres are not the same as when the particles are isolated. Differences include the absence of the Fraunhoffer diffraction peak, which is an intrinsic assumption of the approximate analytical model but not the exact numerical model, and increased scattering in the mid-range of phase angles, which the approximate analytic model fortuitously describes more accurately than the exact numerical model. These changes may be caused by the close proximity of surrounding particles. If they are taken into account, models based on the radiative transfer equation appear able to quantitatively predict the reflectances of regoliths and other particulate media. Interparticle perturbations are also predicted to cause a coherent backscatter opposition effect in the backward direction that was observed, but its angular width was found to be much larger than predicted by theories for sparsely-packed media.     

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.     

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.     

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.     

Albedo and size determination of potentially hazardous asteroids: (99942) Apophis

The near-Earth object (99942) Apophis will make an extremely close approach to the Earth in 2029, and currently has approximately a one-in-45,000 chance of impacting our planet in 2036 (JPL Sentry, November 2006). Computation of the orbital evolution of this object is limited by insufficient knowledge of physical properties required to determine the role played by non-gravitational effects. Using polarimetric observations, we have obtained the first reliable determination of the albedo of Apophis, obtaining 0.33±0.08. We also derive an updated estimate of the asteroid's absolute magnitude: H=19.7±0.4. Using this albedo and H, we find that Apophis has a diameter of 270±60 m, slightly smaller than preliminary estimates based upon an assumed albedo. Our observations demonstrate the feasibility of polarimetric observations aimed at obtaining albedos and sizes of small, potentially hazardous asteroids.     

Testing the Hapke photometric model: Improved inversion and the porosity correction

In conjunction with a companion paper (Shepard, M.K., Helfenstein, P. [2011]. Icarus, submitted for publication), we derive, test, and apply a detailed approach for visualizing the phase angle dependence of light scattering in particulate soils from both whole-disk and disk-resolved observations. To reduce the number of model parameters and provide stronger constraints on model fits, we combine Hapke’s (Hapke, B. [2008]. Icarus 195, 918-926) recent correction for effects of porosity with his (Hapke, B. [1986]. Icarus 67, 264-280) model of the shadow hiding opposition effect. We further develop our method as a tool for least-squares fitting of Hapke’s model to photometric data. Finally, we present an improved method for estimating uncertainties in retrieved values of Hapke model parameters. We perform a preliminary test of the model on spectrogoniometric measurements from three selected laboratory samples from Shepard and Helfenstein (Shepard, M.K., Helfenstein, P. [2007]. J. Geophys. Res. 112 (E03001), 17). Our preliminary suite of test samples is too small and selective to permit the drawing of general conclusions. However, our results suggest that Hapke’s porosity correction improves the fidelity of fits to samples composed of low- and moderate-albedo particles and may allow for more reliable retrieval of porosity estimates in these materials. However, we find preliminary evidence that in high-albedo surfaces, the effects of porosity may be difficult to detect.     

Photometric anomalies of the lunar surface studied with SMART-1 AMIE data

We present new results from the mapping of lunar photometric function parameters using images acquired by the spacecraft SMART-1 (European Space Agency). The source data for selected lunar areas imaged by the AMIE camera of SMART-1 and the data processing are described. We interpret the behavior of photometric function in terms of lunar regolith properties. Our study reveals photometric anomalies on both small (sub-kilometer) and large (tens of kilometers) scales. We found the regolith mesoscale roughness of lunar swirls to be similar in Mare Marginis, Mare Ingenii, and the surrounding terrains. Unique photometric properties related to peculiarities of the millimeter-scale regolith structure for the Reiner Gamma swirl are confirmed. We identified several impact craters of subkilometer sizes as the source of photometric anomalies created by an increase in mesoscale roughness within the proximal crater ejecta zones. The extended ray systems reveal differences in the photometric properties between proximal and distant ejecta blankets. Basaltic lava flows within Mare Imbrium and Oceanus Procellarum indicate higher regolith porosity for the redder soils due to differences in the chemical composition of lavas.     

Optical properties of aerosols in Titan's atmosphere

In the frame of fractal modeling of tholin aggregates we made a systematic analysis of their optical properties. Ballistic particle-cluster aggregation (BPCA) and diffusion-limited aggregation (DLA) of spherical primary particles (monomers) identical in material composition were considered. Aggregates composed of identical particles (monodisperse cluster), as well as of size-distributed particles (polydisperse cluster), were simulated. To calculate the light-scattering models, the code based on the superposition T-matrix method is used. Orientationally averaged properties of light scattering by model particles were extracted, and the normalized phase function and the degree of linear polarization were calculated as functions of scattering angle. We concluded that: (a) aggregation mechanism as well as specific internal structure of the clusters play only a minor role, and for the future it is not necessary to investigate aggregates of different types; (b) the intensity is very sensitive both to the size parameter of forming particles x and to the size parameter of the aggregates X; (c) characterization of the aggregates by the number of monomers is insufficient to retrieve physical properties of aggregates from optical measurement; and (d) it is very desirable to include into the analysis polarization data calculated for the different clusters.     

Polarimetry and BVRI photometry of the potentially hazardous near-Earth Asteroid (23187) 2000 PN9

The results of V-band polarimetric observations of the potentially hazardous near-Earth Asteroid (23187) 2000 PN₉ at large phase angles are presented as well as its photometric observations in BVRI bands. Observations were made in March-April 2006 during its close approach to the Earth using the 1.82-m Asiago telescope (Italy) and the 0.7-m telescope at the Chuguevskaya Observational Station (Ukraine). We obtained polarimetric measurements at the phase angle of 115°, the largest phase angle ever observed in asteroid polarimetry. Our data show that the maximum value of the polarization phase curve reached 7.7% and occurred in the phase angle range of 90-115°. The measured values of linear polarization degree, BVRI colors and magnitude-phase dependence correspond to the S-type composition of this asteroid. Based on our observations the following characteristics of the Asteroid (23187) 2000 PN₉ were obtained: a rotation period of 2.5325±0.0004 h, a lightcurve amplitude of 0.13 mag, an albedo of 0.24±0.06 and a diameter of 1.6±0.3 km.     

Hubble Space Telescope imaging polarimetry of Mars during the 2003 opposition

We report results of polarimetric imaging observations of Mars with the Hubble Space Telescope during the 2003 opposition. Through careful calibration, the observations with the ACS camera allow measurements of the polarization degree with an absolute accuracy better than 0.5% and detection of features with polarization degree contrast as small as 0.2%. The general distribution of linear polarization parameters over the Mars disk and their dependence on phase angle and wavelength are well explained qualitatively by a combination of scattering separately by the martian surface and atmosphere. We have discovered transient polarization phenomena interpreted as clouds that are best observed in ultraviolet light. These clouds are optically thin but strongly polarizing, and their origin may be related to atmospheric ice condensation processes.     

Polarimetry of Centaurs (2060) Chiron, (5145) Pholus and (10199) Chariklo

Results of the first polarimetric observations of Centaurs (5145) Pholus and (10199) Chariklo, and new observations of (2060) Chiron are presented together with the estimates of their absolute magnitudes. Observations were carried out at the 8 m ESO Very Large Telescope in 2007-2008. They revealed noticeable negative polarization in the phase-angle range 0.5-4.4° with a minimum varying from −1% to −2.1% in the R band. All three objects show diverse polarization phase-angle behaviour, each distinctly different from that of transneptunian objects. We found evidence of surface heterogeneity for Chariklo while Chiron and Pholus appear to have rather homogeneous surfaces. Polarization phase behaviours of Chiron and Pholus are significantly different from any other Solar System bodies studied so far. A shift of negative polarization minima toward small phase angles seems to be a characteristic feature of polarization properties of Centaurs. Presence of a small amount of water frost on a dark surface is considered as one of the possible ways to explain these properties.     

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.     

The opposition and tilt effects of Saturn’s rings from HST observations

The two major factors contributing to the opposition brightening of Saturn’s rings are (i) the intrinsic brightening of particles due to coherent backscattering and/or shadow hiding on their surfaces, and (ii) the reduced interparticle shadowing when the solar phase angle α → 0°. We utilize the extensive set of Hubble Space Telescope observations (Cuzzi, J.N., French, R.G., Dones, L. [2002]. Icarus 158, 199–223) for different elevation angles B and wavelengths λ to disentangle these contributions. We assume that the intrinsic contribution is independent of B, so that any B dependence of the phase curves is due to interparticle shadowing, which must also act similarly for all λ’s. Our study complements that of Poulet et al. (Poulet, F., Cuzzi, J.N., French, R.G., Dones, L. [2002]. Icarus 158, 224), who used a subset of data for a single B ∼ 10°, and the French et al. (French, R.G., Verbiscer, A., Salo, H., McGhee, C.A., Dones, L. [2007b] PASP 119, 623–642) study for the B ∼ 23° data set that included exact opposition. We construct a grid of dynamical/photometric simulation models, with the method of Salo and Karjalainen (Salo and Karjalainen [2003]. Icarus 164, 428–460), and use these simulations to fit the elevation-dependent part of opposition brightening. Eliminating the modeled interparticle component yields the intrinsic contribution to the opposition effect: for the B and A rings it is almost entirely due to coherent backscattering; for the C ring, an intraparticle shadow hiding contribution may also be present.Based on our simulations, the width of the interparticle shadowing effect is roughly proportional to B. This follows from the observation that as B decreases, the scattering is primarily from the rarefied low filling factor upper ring layers, whereas at larger B’s the dense inner parts are visible. Vertical segregation of particle sizes further enhances this effect. The elevation angle dependence of interparticle shadowing also explains most of the B ring tilt effect (the increase of brightness with elevation). From comparison of the magnitude of the tilt effect at different filters, we show that multiple scattering can account for at most a 10% brightness increase as B → 26°, whereas the remaining 20% brightening is due to a variable degree of interparticle shadowing. The negative tilt effect of the middle A ring is well explained by the the same self-gravity wake models that account for the observed A ring azimuthal brightness asymmetry (Salo, H., Karjalainen, R., French, R.G. [2004]. Icarus 170, 70–90; French, R.G., Salo, H., McGhee, C.A., Dones, L. [2007]. Icarus 189, 493–522).