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 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.     

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.     

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.     

Light scattering by nonspherical ice grains: An application to noctilucent cloud particles

The light scattered by noctilucent cloud particles is nearly fully polarized at scattering angles in the vicinity of 90 . This was one of the reasons to conclude that the upper limit of their sizes is not larger than about 0.12 m. Nevertheless, this estimate was made on the basis of the Mie scattering theory for spherical particles, whereas many investigators noted usefulness of highly aspherical shapes of noctilucent cloud particles. In this paper, we used rigorous light scattering theory for randomly oriented nonspherical particles to calculate the degree of linear polarization of the scattered light for ice grains of different shape. By comparing these calculations with rocket polarization measurements of noctilucent clouds, we show that, as for spherical particles, the upper limit of particle equal-volume radii for slightly flattened and elongated grains is of about 0.12 m, while for highly aspherical plate-like and needle-like particles this upper limit is substantially larger and is of about 0.18–0.20 m. We also report calculations of the volumetric scattering cross-section for particles of different shape and show that randomly oriented spheroids have (slightly) smaller scattering cross section per unit particle mass than equal-volume spherical grains. Nevertheless, if in noctilucent clouds plate-like and needle-like grains grow to much larger sizes than spherical particles, their scattering efficiency may be much greater.     

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.     

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.     

Scattering matrices and expansion coefficients of martian analogue palagonite particles

We present measurements of ratios of elements of the scattering matrix of martian analogue palagonite particles for scattering angles ranging from 3° to 174° and a wavelength of 632.8 nm. To facilitate the use of these measurements in radiative transfer calculations we have devised a method that enables us to obtain, from these measurements, a normalized synthetic scattering matrix covering the complete scattering angle range from 0° to 180°. Our method is based on employing the coefficients of the expansions of scattering matrix elements into generalized spherical functions. The synthetic scattering matrix elements and/or the expansion coefficients obtained in this way, can be used to include multiple scattering by these irregularly shaped particles in (polarized) radiative transfer calculations, such as calculations of sunlight that is scattered in the dusty martian atmosphere.     

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

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

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.     

A Monte Carlo Code to Compute Energy Fluxes in Cometary Nuclei

A Monte Carlo model designed to compute both the input and output radiation fields from spherical-shell cometary atmospheres has been developed. The code is an improved version of that by H. Salo (1988, Icarus76, 253-269); it includes the computation of the full Stokes vector and can compute both the input fluxes impinging on the nucleus surface and the output radiation. This will have specific applications for the near-nucleus photometry, polarimetry, and imaging data collection planned in the near future from space probes. After carrying out some validation tests of the code, we consider here the effects of including the full 4×4 scattering matrix in the calculations of the radiative flux impinging on cometary nuclei. As input to the code we used realistic phase matrices derived by fitting the observed behavior of the linear polarization as a function of phase angle. The observed single scattering linear polarization phase curves of comets are fairly well represented by a mixture of magnesium-rich olivine particles and small carbonaceous particles. The input matrix of the code is thus given by the phase matrix for olivine as obtained in the laboratory plus a variable scattering fraction phase matrix for absorbing carbonaceous particles. These fractions are 3.5% for Comet Halley and 6% for Comet Hale-Bopp, the comet with the highest percentage of all those observed.The errors in the total input flux impinging on the nucleus surface caused by neglecting polarization are found to be within 10% for the full range of solar zenith angles. Additional tests on the resulting linear polarization of the light emerging from cometary nuclei in near-nucleus observation conditions at a variety of coma optical thicknesses show that the polarization phase curves do not experience any significant changes for optical thicknesses τ?0.25 and Halley-like surface albedo, except near 90° phase angle.     

Possible application of circular polarization for remote sensing of cosmic bodies

Phase dependences of circular polarization were obtained with a precision Stokes polarimeter designed and constructed at the Main Astronomical Observatory of AS Ukraine. A study was made of dielectric and metallic powders with grains of diameter 10–100 m. Metallic powders were found to produce an essential circular polarization - up to 3%, just as dielectric powders did not show circular polarization values more than 0.05% Change of circular polarization with phase angle V is greatly depended on surface structure. Loose powders give phase curves with the same sign of circular polarization everywhere and with maximum at large phase angles V > 120 . Measurements of compacted powders show curves which change the sign repeatedly and have additional maxima, including a maximum at small phase angles V < 40 . A theory was created which considers a circular polarization as a result of multiple reflections of light from particulate surface. The theory provides reasonable good fit to the experimental data. It was concluded that measurements of circular polarization can be used to find metals in surface material of cosmic bodies (especially asteroids) and to determine characteristics of surface structure, in particular, to establish presence of regolith on metal-rich bodies.     

Circular polarization of sunlight reflected by planetary atmospheres

Multiple scattering calculations are performed in order to investigate the nature of the circular polarization of sunlight reflected by planetary atmospheres. Contour diagrams as a function of size parameter and phase angle are made for the integrated light from a spherical but locally plane-parallel atmosphere of spherical particles. To investigate the origin of the circular polarization, results are also computed for second-order scattering and for a simpler semiquantitative model of scattering by two particles. Observations of the circular polarization of the planets are presently too meager for accurate deduction of cloud particle properties. However, certain very broad constraints can be placed on the properties of the dominant cloud particles on Jupiter and Saturn. The cloud particle size and refractive index deduced for the Jupiter clouds by Loskutov, Morozhenko, and Yanovitskii from analyses of the linear polarization are not consistent with the circular polarization. The few available circular polarization observations of Venus are also examined.     

On an anomalous polarization of the corona

At present data exist showing that in some regions of the corona the polarization degree has been found to be higher than the maximum possible value determined by Thomson scattering. Besides this, there exist regions where the direction of the prevailing vibration of the E-vector does not coincide with the tangential one. This may be caused by the velocity of scattered electrons. The theory of polarization taking the velocity into account is given, and the above-mentioned data are discussed. The direction of polarization turns out to be the sensitive detector of fact electrons (for energy of 5 keV, the deviation angle 10°).Very important data about accelerated electrons on the Sun may be received from precise measurements of the corona polarization.     

Spectropolarimetry of the Deep Impact target Comet 9P/Tempel 1 with HiVIS

High resolution spectropolarimetry of the Deep Impact target, Comet 9P/Tempel 1, was performed during the impact event on July 4th, 2005 with the HiVIS spectropolarimeter and the AEOS 3.67-m telescope on Haleakala, Maui. We observed atypical polarization spectra that changed significantly in the few hours after the impact. The polarization of scattered light as a function of wavelength is very sensitive to the size and composition (complex refractive index) of the scattering particles as well as the scattering geometry. As opposed to most observations of cometary dust, which show an increase in the linear polarization with the wavelength (at least in the visible domain and for phase angles greater than about 30, a red polarization spectrum) observations of 9P/Tempel 1 at a phase angle of 41° beginning 8 min after impact and centered at 6:30 UT showed a polarization of 4% at 650 nm falling to 3% at 950 nm. The next observation, centered an hour later showed a polarization of 7% at 650 nm falling to 2% at 950 nm. This corresponds to a spectropolarimetric gradient, or slope, of −0.9% per 1000 Å 40 min after impact, decreasing to a slope of −2.3% per 1000 Å an hour and a half after impact. This is an atypical blue polarization slope, which became more blue 1 h after impact. The polarization values of 4 and 7% at 650 nm are typical for comets at this scattering angle, whereas the low polarization of 2 and 3% at 950 nm is not. We compare observations of Comet 9P/Tempel 1 to that of a typical comet, C/2004 Machholz, at a phase angle of 30° which showed a typical red slope, rising from 2% at 650 nm to 3% at 950 nm in two different observations (+1.0 and +0.9% per 1000 Å).     

Limits on the size of aerosols from measurements of linear polarization in Titan’s atmosphere

The Descent Imager/Spectral Radiometer (DISR) instrument on the Huygens probe into the atmosphere of Titan yielded information on the size, shape, optical properties, and vertical distribution of haze aerosols in the atmosphere of Titan [Tomasko, M.G., Doose, L., Engel, S., Dafoe, L.E., West, R., Lemmon, M., Karkoschka, E., 2008. Planet. Space Sci. 56, 669-707] from photometric and spectroscopic measurements of sunlight in Titan’s atmosphere. This instrument also made measurements of the degree of linear polarization of sunlight in two spectral bands centered at 491 and 934 nm. Here we present the calibration and reduction of the polarization measurements and compare the polarization observations to models using fractal aggregate particles which have different sizes for the small dimension (monomer size) of which the aggregates are composed. We find that the Titan aerosols produce very large polarizations perpendicular to the scattering plane for scattering near 90° scattering angle. The size of the monomers is tightly constrained by the measurements to a radius of 0.04 ± 0.01 μm at altitudes from 150 km to the surface. The decrease in polarization with decreasing altitude observed in red and blue light is as expected by increasing dilution due to multiple scattering at decreasing altitudes. There is no indication of particles that produce small amounts of linear polarization at low altitudes.     

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.

     

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.     

Investigation of some of the principal geometric effects on planetary polarization

Three major geometric factors which are likely to influence theoretical interpretation of planetary polarization measurements, viz., observer—planet distance, horizontal inhomogeneity of planetary disk, and deviation from a spherical body, are investigated.The distance effect is examined for regional as well as global polarizations. For convenience of analysis, the expressions for zenith and azimuth angles of incident and emergent light appropriate for a snap-shot observation are derived as explicit functions of distance between observer and planet. Sample computations for Venus indicate that regional polarization near the planetary limb is significantly affected by the observer's distance. This effect should be particularly noticeable when an observation is made at a phase angle around which the single scattering polarization of atmospheric scattering agents exhibits a steep variation. The global polarization at large phase angles (measured at disk-center) is gradually moved toward smaller phase angles, as the observer approaches the planet. Any narrow polarization features such as rainbow and glory at small phase angles are heavily smoothed out.The effects of horizontal inhomogeneity are investigated with a planetary disk having highly polarizing regions at high latitudes. Comparison of theoretical global polarization computed for such a disk with the Pioneer Venus OCPP measurements shows a possible change in cloud-haze stratification approximately at 50° latitude, consistent with other imaging observations. An approximate analytical representation of residual polarization at zero phase angle is then derived to compare to the numerical results for Venus. An attempt is also made to explain the relatively large magnitude of residual polarization observed on Jupiter.Finally, to study the effects of nonsphericity of planetary body, the global polarizations are computed for a spheroidal planet. The global polarization tends to increase as the planet's oblateness increases. However, for Jupiter and Saturn, such effect may be of secondary importance.