Light scattering by morphologically complex objects and opposition effects (a review)

Over the last decade, considerable progress has been achieved in the theory of light scattering by morphologically complex objects, which extends the potential of correct interpretation of photometric and polarimetric observations. This especially concerns the backscattering domain, where the opposition effects in brightness and polarization are observed. Although the equations of radiative transfer and weak localization (coherent backscattering) are rigorously valid only for sparse media, the results of exact computer solutions of the Maxwell equations for a macroscopic volume filled with randomly positioned particles show that their application area can be wider. In particular, the observations can be correctly interpreted if the packing density of particles in the medium reaches 20–30%. The recently suggested approximate solution of the coherent backscattering problem allowed interesting effects in the spectra of Saturn’s satellites to be explained. In the densely packed media, the effects that are impossible in the sparse media and caused by the near-field contribution can be observed. To calculate the characteristics of radiation reflected by such a medium, it is not sufficient to solve the radiative transfer and weak localization equations, even if they are written in a form without the far-zone limitations. Nowadays, the influence of the interaction of particles in the near field can be analyzed only for the restricted ensembles of particles. It shows that the substantial increase of the packing density essentially changes the phase functions of intensity and polarization in the backscattering domain. This allows the packing density of particles in the medium and their absorbing properties to be estimated from the shape of the phase curves measured. However, the task of quantitative interpretation of the measurements of radiation reflected by a densely packed medium, in terms of sizes of particles, their refractive index, and packing density, still remains unsolved.     

Interaction of particles in the near field and opposition effects in regolith-like surfaces

The explanation of the opposition effects observed in brightness and polarization in different celestial bodies and laboratory samples is still far from being complete. The shadow hiding and coherent backscattering mechanisms are mentioned most frequently in this connection. In the present work, we consider one more scattering mechanism—the interaction of particles in the near field—and its influence on the brightness and polarization of light scattered by ensembles of particles at small phase angles. First, we analyze two manifestations of this mechanism: the field inhomogeneity in the vicinity of the scatterers and the shielding of particles by each other at distances compared with their sizes. Then, we use the model regolith described as an ensemble of clusters as constituents and compare the contributions of the coherent backscattering and the near-field effect to the intensity and polarization of light when the porosity of the ensemble is varied. The modeling confirms that the phase dependences of the intensity and polarization of light scattered by complex structures in the backscattering domain is mainly caused by these two mechanisms. The coherent backscattering works more effectively in sparse media, while the near-field effect manifests itself in more compact ensembles of wavelength-sized particles. However, it is difficult to distinguish quantitatively their contributions, even in models of simple structures. A number of observations, especially of moderate- and low-albedo objects, can be explained only by invoking the near-field effect.     

Light Scattering by Aggregate Particles Comparable in Size to Wavelength: Application to Cometary Dust

Many naturally occurring particles (including, most likely, cometary dust) have an aggregate structure. We study the scattering properties of polydisperse independent aggregate particles (clusters) comparable in size to visible wavelengths. The sizes of the monomers constituting a cluster play a significant role in forming the angular dependences of intensity and linear polarization of the scattered light. Irregularly structured aggregates composed of a moderate number of spheres (<50) with size parameters 1.3–1.65 exhibit properties typical of cometary dust particles: a slight increase in backscattering intensity, a negative polarization at small phase angles, an inversion phase angle close to the observed one, an increase in brightness, and a linear polarization with increasing wavelength. In this case, the imaginary part of the refractive index for particles can increase with decreasing wavelength in the visible spectral range, which is typical of silicates with an admixture of iron or organic material. The spectral dependence of extinction efficiency for aggregates is less steep than that for equivalent spherical particles, and its maximum is shifted to larger size parameters. Therefore, when analyzing extinction measurements, the scatterer shape must be taken into account to avoid underestimation of the scattering-particle sizes.     

The Opposition Effect and Negative Polarization of Structural Analogs for Planetary Regoliths

To better understand the negative polarization and brightness opposition effects observed on airless celestial bodies, we carried out simultaneous photometric and polarimetric measurements of laboratory samples that simulate the structure of planetary regoliths. Computer modeling of shadow-hiding and coherent backscatter in regolith-like media are also presented. The laboratory investigations were carried out with a photometer/polarimeter at phase angles covering 0.2°-4° and wavelengths of 0.63 and 0.45 μm. We studied samples that characterize a variety of microscopic structures and albedos. A particle-size dependence of the negative branch of polarization for powdered dielectric surfaces was found. Colored samples such as a powder Fe₂O₃ exhibit a very prominent wavelength dependence of the photometric and polarimetric opposition phenomena. Metallic powders usually exhibit a wide branch of the negative polarization independent of the size of particles. For fine dielectric powders, both opposition phenomena become more prominent when the samples were compressed. Our computer modeling based on ray tracing in particulate media shows that shadow-hiding affects the negative polarization only in combination with the coherent backscatter enhancement. Modeling reveals that scattering orders higher than second contribute to negative polarization even in dark particulate surfaces. Our model qualitatively reproduces the effects of varying sample-compression that we observed in the laboratory. Our experimental and computer modeling studies mutually confirm that the degree of polarization for highly reflective dielectric surfaces depends not only on phase angle but also on surface tilt. Even at exactly zero phase the degree of polarization for tilted surfaces can be nonzero. A tilt of the surface normal to the scattering plane gives a parallel shift of the negative polarization branch to large values of |P|. The tilt in the perpendicular plane gives the same shift in the direction of positive polarization. At exactly zero phase angle, a celestial body of irregular shape can exhibit nonzero polarization even in integral polarimetric observations.     

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).     

Modeling of opposition effects with ensembles of clusters: Interplay of various scattering mechanisms

Although the opposition phenomena observed in brightness and polarization for various astronomical objects and laboratory samples have been under intense study for many years, their explanation is still far from being complete. The shadow hiding and coherent backscattering effects are mentioned most frequently in this connection. In the present work we first discuss how other coherent scattering mechanisms, in particular interference and interaction between scatterers in the near field, influence brightness and polarization of complex ensembles of particles at small phase angles. Then we demonstrate the contribution of the different mechanisms to the scattering process in a model regolith described as an ensemble of wavelength-sized clusters as constituents. While the clusters are always densely packed, the porosity of the ensemble itself, i.e., the average distances between the clusters within the ensemble, is allowed to vary. The modeling confirms that the phase dependence of intensity and polarization of light scattered by complex structures in the backscattering domain is mainly caused by the interplay of (1) the constructive interference of waves traveling through the particle ensemble along similar paths but in opposite directions and (2) the near-field effect caused by the inhomogeneity of waves in the immediate vicinity of constituent particles. The first mechanism works more effectively in sparse ensembles, while the second one manifests itself in more compact structures of wavelength-sized scatterers at distances comparable to the wavelength. It is difficult to distinguish quantitatively their contributions in models of simple structures and even more in measurements. A number of observations, especially of moderate and low albedo objects, can, however, be explained only by invoking the near-field effect.     

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.     

Polarization Models for Rayleigh Scattering Planetary Atmospheres

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

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.     

Spectropolarimetry of comets Austin and Churyumov-Gerasimenko

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

Positive branch of asteroid polarization: Observational data and computer modeling

Observations of near-Earth asteroids at large phase angles made it possible to obtain a more complete (for ground-based observations) phase dependence of the polarization of the E-type asteroids’ radiation including the maximum of the positive branch of the linear polarization degree. It is shown that the position of the polarization maximum of high-albedo asteroids is noticeably shifted to the decrease of phase angles compared with S-type asteroids. Model calculations of polarimetric properties of random Gaussian particles that simulate dust particles on the regolith surface are carried out. Model calculations show a qualitatively similar behavior pattern of parameters of the positive polarization branch. The influence of the refractive index of individual scattering particles on the size and position of the maximum of the positive branch of the linear polarization degree is investigated within the considered model.     

A Negative Branch of Polarization for Comets and Atmosphereless Celestial Bodies and the Light Scattering by Aggregate Particles

Optical observations of comets and atmosphereless celestial bodies show that a change of sign of the linear polarization of scattered light from negative to positive at phase angles less than 20° is typical of the cometary coma, as well as of the regolith of Mercury, the Moon, planetary satellites, and asteroids. To explain a negative branch of polarization, this research suggests a unified approach to the treatment of cometary-dust particles and regolith grains as aggregate forms. A composite structure of aggregate particles resulting in the interaction of composing structural elements (monomers) in the light-scattering process is responsible for the negative polarization at small phase angles, if the monomer sizes are comparable to the wavelength. The characteristics of single scattering of light calculated for aggregates of this kind turned out to be close to the properties observed for cometary dust. Unlike the cometary coma, the regolith is an optically semi-infinite medium, where the interaction between particles is significant. To find the reflectance characteristics of regolith, the radiative-transfer equation should be solved for a regolith layer. In this case, the interaction between scatterers can be modeled to a certain extent by representing the regolith grains as aggregate structures consisting of several or many elements. Although real regolith grains are much larger than the particles considered here, laboratory measurements have shown that it is precisely the surface irregularities comparable to the wavelength that cause a negative branch of polarization. The main observed features of the phase and spectral dependence of the linear polarization of light scattered from comets and atmosphereless celestial bodies, which are due to the difference of the elementary scatterers in composition, size, and structure, can be successfully explained using the aggregate model of particles.     

Polarimetry of Pluto

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

The effect of aerosol shape in retrieving optical properties of cloud particles in the planetary atmospheres from the photopolarimetric data. Jupiter

The influence of the assumed shape of aerosols on the estimates of the refractive index and size of particles (based on the data of ground-based spectropolarimetric measurements) is investigated for Jupiters cloud layer. In the present analysis, we supposed the atmospheric particles to be chaotically oriented spheroids and cylinders with a gamma size distribution. Their single-scattering characteristics were calculated with the T-matrix method, and the intensity and degree of linear polarization of the radiation reflected by the center of the planetary disk were found by solving the vector radiative-transfer equation with consideration for multiple scattering in a plane-parallel atmosphere. We considered a spectral interval from 0.423 to 0.798 µm and phase angles from 0° to 11°. It has been shown that, if we use the optical characteristics of aerosols found within the frames of a spherical model (Mishchenko, 1990a), the models with the nonspherical particles considered here cannot fit the observational data. The refractive index and the sizes of spheroidal and cylindrical particles were estimated from a comparison of the data of measurements and calculations, and the simplest models for the Jovian cloud layer structure have been considered. We have concluded that the optical parameters of cloud particles (specifically, the refractive index) cannot be reliably estimated only on the basis of measurements made in a narrow range of phase angles.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 2, 2005, pp. 117–127.Original Russian Text Copyright © 2005 by Dlugach, Mishchenko.     

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.     

Polarization of Light Scattered by Solar System Bodies and the Aggregate Model of Dust Particles

The present study considers the dependence of characteristics of light scattering by aggregate particles on the refractive index, size, and number of spherical particles composing the aggregate, as well as on the structure and porosity of the cluster. The parameters were varied in sufficiently wide ranges to let a coherent picture of the polarimetric properties of relatively small aggregate particles emerge (the size parameter of the aggregate is less than 10). It was shown that, in the framework of the aggregate model, the behavior of polarization phase curves observed for both comets and regolith surfaces can be explained. The modeling carried out confirms that the sizes of the cometary dust particles are larger than the wavelength. However, the grains forming the cometary dust particles or the regolith (or details of the particle surface) have a size less than 0.3–0.5 m. This agrees with estimates obtained by other methods. The determining role in the formation of the polarization phase curve is played by the structure of the external layer of the clusters. The appearance of the negative branch of polarization and its shape substantially depend on the effectiveness of the interference of multiply scattered waves and on the interaction in the near field at these phase angles. Interference and interaction in the near field in turn are determined by the sizes of elementary scatterers and the structure of the ensemble. If the number of constituent particles in the aggregate is larger than several tens, its role in the formation of the negative branch of polarization is minor, while the influence on the polarization maximum position is rather substantial. The polarimetric data alone cannot provide a unique estimate of the refractive index: the brightness measurements must be invoked as well. For a more complete quantitative interpretation of the observations, the scattering matrix of aggregates comparable in size to or larger than the wavelength must be calculated in the short- and long-wavelength ranges, which still encounters serious theoretical and technical difficulties. Moreover, in order to obtain unique results, it is obvious that the spectral range of observations must be extended and that other types of measurements, such as spectroscopic ones, must also be used.     

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.     

Viking orbiter observations of the Mars opposition effect

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

Multiple Scattering of Light in Regolith-like Media: Geometric Optics Approximation

We present a numerical model that allows us to calculate the contribution of a specified scattering order in the geometric optics approximation for media composed of large particles with an arbitrary phase function. It has been demonstrated that the correlated propagation of the incident and emergent rays, which is disregarded in the classical radiative-transfer theory, markedly affects the contributions of different orders of scattering, especially the first-order scattering. If the theory describing the photometric properties of regolith-like media ignores the shadowing effect, the errors of its application may reach several tens of percent even for bright surfaces. The packing density of a medium essentially influences the phase dependence of the first-order scattering, although its effect on the value and the phase curve of the higher scattering orders is relatively weak. The backscattering peaks calculated on the basis of the Hapke theory are narrower than those obtained from the numerical simulations, because the Hapke theory is an approximate approach.     

Regolith Surface Reflectance: A New Attempt to Model

The reflectance coefficient of the regolith layer of celestial bodies has been studied in relation to the physical properties of regolith particles (size, refractive index, and packing density) on the basis of an accurate numerical radiative-transfer algorithm for a semi-infinite flat layer. Using the geometric-optics approximation, we have found that a shape mixture of randomly oriented spheroids can successfully model the single-scattering phase function of independent soil grains. In order to take into account the effect of packing density in a regolith layer, the concept of the so-called static structure factor was used. The main effect of increasing packing density is to suppress the forward-scattering peak of the phase function and to increase the albedo of the reflecting surface. We also investigated the influence of fine dust on the reflected light. An addition of small particles not only increases the surface albedo, but also changes the brightness profile and enhances the backscattering. Although the problem of unique solution, which is inherent in the retrieval of the properties of a medium from the measurements of the intensity of light scattered by this media, cannot be removed in the proposed model, the procedure used here, in contrast to widely used approximations, allows us to fit observational data with a set of real characteristics of the regolith. Semiempirical approaches are able to fit the measurements well with a small number of free parameters, but they do not explicitly contain crucial physical characteristics of the regolith such as grain sizes or the refractive index. We compared the numerical solution of the radiative-transfer equation with the Hapke approximation, which is most often used by investigators. The errors introduced by the Hapke model are small only for near-isotropic scattering by isolated particles. However, independent regolith grains are known to scatter light mainly in the forward direction.