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.     

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.     

Voyager disk resolved photometry of the Saturnian satellites

Photometric analysis of Voyager images of the medium-sized icy satellites of Saturn shows that their surfaces exhibit a wide range of scattering properties. At low phase angles, Rhea and Dione closely follow lunar behavior with almost no limb darkening. Mimas, Tethys, and especially Enceladus shiw significant limb darkening at low phase angles, which suggests multiple scattering is important for their surfaces. A simple photometric function of the form I/F = f(α)Aμ₀/(μ + μ₀) + (1 ? A)μ₀ has been fit to the observations. For normal reflectances <0.6, we find lunar-like scattering properties (A = 1). No satellite's surface can be described by Lambert's Law (A = 0). Dione exhibits the widest albedo variations (about 50%). A longitudinal dark stripe which represents a 15% decrease in albedo is situated near the center of the trailing side of Tethys. A correlation is found between the albedo and color of the satellites: the darker objects are redder. Similarly, darker areas of each satellite are redder. Spectral reflectances of Mimas and Enceladus can be derived for the first time. After the proper calibrations to the Voyager color images are made, it is found that both satellites have remarkably flat spectra into the ultraviolet.     

The phase dependence of brightness and color of the lunar surface: a study based on integral photometric data

A procedure of an a posteriori correction of the available data on the integral photometry of the Moon is described. This procedure reduces the regular errors of the integral phase curves caused by variations of the libration parameters; the effect due to libration can reach 4%. A method allowing the integral measurements of the Moon to be compared correctly with the photometric measurements of the lunar areas or laboratory samples imitating the lunar soil has been developed. To approximate the phase curves of integral albedo in the phase-angle range from 6° to 120°, we proposed a simple empirical formula A _eq(α) = m _l e ^?ρα + m ₂ e ^?0.7α, where α is the phase angle, ρ is the factor of effective roughness, and m ₁ + m ₂ is the surface albedo at a zero phase angle. An empirical phase dependence of the slope of the lunar spectrum in the 360–1060 nm range has been obtained. The results may be used to test various theoretical models of the light scattering by the lunar surface and to calibrate the data of ground-based and space-borne spectrophotometric observations.     

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.     

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.     

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.     

Rational approximation formula for Chandrasekhar’s <Emphasis Type="Italic">H</Emphasis>-function for isotropic scattering

In this work, we first establish a simple procedure to obtain with 11-figure accuracy the values of Chandrasekhar’s H-function for isotropic scattering using a closed-form integral representation and the Gauss-Legendre quadrature. Based on the numerical values of the function produced by this method for various combinations of ? ₀, the single scattering albedo, and μ, the cosine of the zenith angle θ of the direction of radiation emergent from or incident upon a semi-infinite scattering-absorbing medium, we propose a rational approximation formula with μ ^1/4 and \(\sqrt{1-\varpi_{0}}\) as the independent variables. This allows us to reproduce the correct values of H(? ₀,μ) within a relative error of 2.1×10^?5 without recourse to any iterative procedure or root-finding process.     

The Umov effect for single irregularly shaped particles with sizes comparable with wavelength

The Umov effect manifests itself as an inverse correlation between the linear polarization maximum of an object’s scattered light P_max and its geometric albedo A. This effect is observed for the Moon, Mercury and Mars, and there are data suggesting this effect is valid for asteroids. The Umov effect is due to the contribution of interparticle multiple scattering that increases albedo and decreases polarization. We here study if the Umov effect can be extended to the case of single irregularly shaped particles with sizes comparable with the wavelength. This, in particular, is important for cometary dust polarimetry. We show the Umov effect being valid for weakly absorbing irregular particles (Im(m) ? 0.02) almost through the entire range of size parameters x considered. Highly absorbing particles (Im(m) > 0.02) follow the Umov effect only if x exceeds 14. In the case of weakly absorbing particles, the inverse correlation is essentially non-linear, which is caused by the contribution of particles with small x. However, averaging over many different types of irregularly shaped particles could make it significantly more linear. The size averaging does not change qualitatively the diagram log(P_max)-log(A) for weakly absorbing particles. For single irregular particles whose sizes are comparable with wavelength, there is no reliable correlation between the slope of the polarization curve h near the inversion phase angle and geometric albedo A. Using the extended Umov Law, we estimate the geometric albedo of dust particles forming cometary circumnuclear haloes A = 0.1 − 0.2, which is a few times larger than the average geometric albedo over the entire comae. Note that, using the obtained values for A of cometary particles, one can derive their number density in circumnuclear haloes from photometric observations.     

Long-term monitoring of the long-period variable Y Cassiopeiae in the 1.35-cm water-vapor line

Observations of the circumstellar maser emission from the long-period variable star Y Cas in the 1.35-cm water-vapor line are presented. The observations were performed with the RT-22 radio telescope at the Pushchino Radio Astronomy Observatory (Astrospace Center, Lebedev Physical Institute, Russian Academy of Sciences) in the period 1982–2005. The variations in the integrated flux F_int in the H₂O line correlate with the visual light curve of the star. The phase delay Δ? between the F_int variations and the light curve is 0.2–0.4P (P is the period of the star). The H₂O maser Y Cas belongs to transient sources: peaks of high maser activity alternate with intervals of a low emission level when the H₂O-line flux does not exceed (0.1–0.5) × 10^?20 W m^?2. A “superperiod” of ～5.7 yr was found in the occurrence of activity peaks. A particularly strong maximum of maser radio emission took place at the end of 1997, when the flux F_int reached 15.6 × 10^?20 W m^?2. A model for the H₂O maser variability in Y Cas is discussed. The variability is caused by a periodic action of shock waves driven by stellar pulsations. The H₂O maser flares may be associated with short-lived episodes of enhanced mass loss by the star or with the propagation of a particularly strong shock wave when a planet orbiting the star passes through its periastron.     

Scattering in the Atmosphere of Venus. I. Line profiles and curves of growth for isotropic scattering

We have computed line profiles and curves of growth for both reflected and transmitted radiation for typical lines in CO₂ bands (in the photographic infrared) which occur in the spectrum of Venus. In our model the pressure variation with altitude was considered and the base of the cloud deck was set at the 2 bar level. The temperature was held constant at 250K and a Voigt profile was used for the lineshape. We also assumed that the scale height of the cloud particles was equal to the scale height of the gas. The calculations were made for four values of the scattering optical thickness (τ_c = 0.1, 1.0, 10, and 100) using a continuum single scattering albedo $\text{ω}{}_{\mathrm{<mtext>c</mtext>}}\text{= 0.9975}$ (which gives a Bond albedo of 0.896 for τ_c = 100, the value observed for Venus at these wavelengths). Curves of growth are also presented for reflected radiation which has been averaged over the visible disk for three values of the Venus phase angle (0, 86, and 166°).     

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.     

Upgraded technique to analyze the vertical structure of the aerosol component of the atmospheres of giant planets

To analyze the behavior of the optical thickness of aerosols or the ratio of the optical thicknesses of the aerosol and gas components in the spectral absorption bands of atmospheric gases with depth, we developed a software package. The package structure includes the units for the following operations: (1) to calculate the Legendre expansion coefficients x _i of the phase function and the volume scattering coefficient σ₀ of the polydisperse medium with the specified refractive index and the size distribution function N(r) with the use of the code developed by M.I. Mishchenko; (2) to generate the array containing the pairs of the single scattering albedo ω and the geometric albedo A _g for the models of a semi-infinite homogeneous medium with the parameters determined in the previous step; (3) to determine the single scattering albedo values from the comparison of the calculated and measured values of the geometric albedo for each of the measured points in the examined absorption band of the atmospheric gas (accounting for the change of the phase function due to Rayleigh scattering); (4) to calculate the spectral values of the effective optical depths τ_eff of the levels, where the intensity field of light diffusely reflected by the investigated atmosphere is formed; (5) to derive the scattering and absorption components of the effective optical depth (τ _eff ^s and τ _eff ^v ) from the values of ω and τ_eff; (6) from the values of τ _eff ^v to determine the amount of the absorbing gas NL (in km-amg) along the line of sight and, from these values, the atmospheric pressure p(NL) and the gas component of the scattering portion of the optical depth τ _g (λ₀) at the wavelength λ₀ = 887.2 nm; (7) from the values of τ _eff ^s (λ) and τ _g (λ, NL), to find the aerosol component τ _a (λ, NL); (8) to build the plots of τ _a (λ) or the ratio τ _a (λ)/τ _g (λ) reduced to λ₀ = 887.2 nm versus the pressure. The software package was validated in the analysis of the spectrophotometric data obtained in the measurements of the whole disk of Jupiter in the profiles of the strong absorption bands of methane centered at λλ 841.6, 864, and 887.2 nm under the assumption on two versions of the size distribution of particles (the modified gamma distribution and the log-normal one). It was found that the model with the gamma distribution is analyzed several times more quickly than the log-normal one that yields the close results in computations performed for the same medium.     

Photometry of rough planetary surfaces: The role of multiple scattering

Topographic features affect the scattering properties of planetary surfaces by casting shadows and altering the local incidence and emission angles. Measurements of this phenomenon were obtained on the Cornell goniometer for both high and low albedo surfaces. For the low albedo surface, the decrease in reflected radiation due to topography increases sharply with increasing phase angle, whereas for the high albedo sample the effects are approximately constant between phase angles of 30 and 70°. The observations are in good agreement with a theoretical model in the case of the dark surface. However, for the high albedo surface the model overestimates the effects by about a factor of 2, since it does not include the partial illumination of shadows by multiple scattering. For both high and low albedo surfaces, the effects of topography do not become significant until a phase angle of 30–40°.     

Shadow mechanism and the opposition effect of brightness of atmosphereless celestial bodies

We consider the Irvine-Yanovistkii modification of the shadow model developed by Hapke for the opposition effect of brightness. The relation between the single scattering albedo ω and the transparency coefficient of particles κ is suggested to be used in the form κ = (1 ? ω) ⁿ , which allows the number of unknowns in the model to be reduced to two parameters (the packing density of particles g and ω) and the single-scattering phase function χ(α). The analysis of spectrophotometric measurements of the moon and Mars showed that the data on the observed opposition effect and the changes in the color index with the phase angle α well agree if the values of n = 0.25 and g = 0.4 (the moon) and 0.6 (Mars) are assumed in calculations. When being applied to asteroids of several types, this method also yielded a satisfactory agreement. For the E-type asteroids, the sets of parameters are [g = 0.6, ω = 0.6, A _g = 0.21, and q = 0.83] or [g = 0.3, ω = 0.4, A _g = 0.15, and q = 0.71] under the Martian single-scattering phase function; for the M-type asteroids, it is [g = 0.4, ω ≤ 0.1, A _g ≤ 0.075, and q ≤ 0.42] under the lunar single-scattering phase function; for the S-type asteroids, it is [g = 0.4, ω = 0.4, A _g = 0.28, and q = 0.49] under the lunar single-scattering phase function; and for the C-type asteroids, it is [g = 0.6, ω ≤ 0.1, A _g ≤ 0.075, and q = 0.43] under the modified lunar single-scattering phase function. The polarization measurements fulfilled by Gehrels et al. (1964) for the bright feature on the lunar surface, Copernicus (L = -20°08′, φ = +10°11′), at a phase angle α = 1.6° revealed the deviations in the position of the polarization plane from that typical for the negative branch. They were 22° and 12° in the G and I filters, respectively. At the same time, the deviation was within the error (±3°) in the U filter and for the dark feature Plato (L = -10°32′, φ = +51°25′), which can be caused by the coherent mechanism of the formation of the polarization peak.     

Calculation of effective optical depth of absorption line formation in homogeneous semi-infinite planetary atmosphere during anisotropic scattering

The algorithm for determining effective optical thickness of absorption line formation in a plane-parallel homogeneous planetary atmosphere is presented. The case of anisotropic scattering is considered. The results of numerical calculations of τ _e (μ₀) at the scattering angle γ = π for some values of the single scattering albedo λ and the parameter of the Heyney-Greenstein scattering indicatrix g are given. The refined equation for the function T ^m (−μ, μ₀) is presented.     

A new look at photometry of the Moon

We use ROLO photometry (Kieffer, H.H., Stone, T.C. [2005]. Astron. J. 129, 2887-2901) to characterize the before and after full Moon radiance variation for a typical highlands site and a typical mare site. Focusing on the phase angle range 45° < α < 50°, we test two different physical models, macroscopic roughness and multiple scattering between regolith particles, for their ability to quantitatively reproduce the measured radiance difference. Our method for estimating the rms slope angle is unique and model-independent in the sense that the measured radiance factor I/F at small incidence angles (high Sun) is used as an estimate of I/F for zero roughness regolith. The roughness is determined from the change in I/F at larger incidence angles. We determine the roughness for 23 wavelengths from 350 to 939 nm. There is no significant wavelength dependence. The average rms slope angle is 22.2° ± 1.3° for the mare site and 34.1° ± 2.6° for the highland site. These large slopes, which are similar to previous “photometric roughness” estimates, require that sub-mm scale “micro-topography” dominates roughness measurements based on photometry, consistent with the conclusions of Helfenstein and Shepard (Helfenstein, P., Shepard, M.K. [1999]. Icarus 141, 107-131). We then tested an alternative and very different model for the before and after full Moon I/F variation: multiple scattering within a flat layer of realistic regolith particles. This model consists of a log normal size distribution of spheres that match the measured distribution of particles in a typical mature lunar soil 72141,1 (McKay, D.S., Fruland, R.M., Heiken, G.H. [1974]. Proc. Lunar Sci. Conf. 5, Geochim. Cosmochim. Acta 1 (5), 887-906). The model particles have a complex index of refraction 1.65-0.003i, where 1.65 is typical of impact-generated lunar glasses. Of the four model parameters, three were fixed at values determined from Apollo lunar soils: the mean radius and width of the log normal size distribution and the real part of the refraction index. We used FORTRAN programs from Mishchenko et al. (Mishchenko, M.I., Dlugach, J.M., Yanovitskij, E.G., Zakharova, N.T. [1999]. J. Quant. Spectrosc. Radiat. Trans. 63, 409-432; Mishchenko, M.I., Travis, L.D., Lacis, A.A. [2002]. Scattering, Absorption and Emission of Light by Small Particles. Cambridge Univ. Press, New York. <http://www.giss.nasa.gov/staff/mmishchenko/books.html>) to calculate the scattering matrix and solve the radiative transfer equation for I/F. The mean single scattering albedo is ω = 0.808, the asymmetry parameter is 〈cos Θ〉 = 0.77 and the phase function is very strongly peaked in both the forward and backward scattering directions. The fit to the observations for the highland site is excellent and multiply scattered photons contribute ?80% of I/F. We conclude that either model, roughness or multiple scattering, can match the observations, but that the strongly anisotropic phase functions of realistic particles require rigorous calculation of many orders of scattering or spurious photometric roughness estimates are guaranteed. Our multiple scattering calculation is the first to combine: (1) a regolith model matched to the measured particle size distribution and index of refraction of the lunar soil, (2) a rigorous calculation of the particle phase function and solution of the radiative transfer equation, and (3) application to lunar photometry with absolute radiance calibration.     

Radar observations of the icy Galilean satellites

We report 12.6-cm-wavelength radar observations of Europa, Ganymede, and Callisto made at the Arecibo Observatory in November 1977 and February 1979. When combined with previous observations, our results establish firmly the distinguishing radar properties of these satellites: (i) high geometric albedos, α; (ii) circular polarization ratios, μ_C, which anomalously exceed unity; (iii) linear polarization ratios, μ_L, which are approximately 0.5; and (iv) diffuse scattering which varies as cosⁿθ, where θ is angle of incidence and 1 ? n ? 2. We tabulate weighted-mean values of α, μ_C, μ_L, and n derived from observations between 1975 and 1979. The values of μ_C for Ganymede and Europa are nearly identical and significantly larger than that for Callisto. The values of n for Ganymede and Callisto are nearly identical and significantly smaller than that for Europa. Although significant albedo and/or polarization features are common in the radar spectra, the fractional rms fluctuation in disk-integrated properties is only ～10%. No time variation in the radar properties has been evident during 1976–1979.     

Photometric properties of zodiacal light particles

From published ground-base, spacecraft, and rocket photometry and polarimetry of the zodiacal light, a number of optical and physical parameters have been derived. It was assumed that the number density, mean particle size, and albedo vary with heliocentric distance, and shown that average individual interplanetary particles have a small but definite opposition effect, a mean single-scattering albedo in the V band at 1-AU heliocentric distance of 0.09 ± 0.01, and a zero-phase geometric albedo of 0.04. Modeled by a power law, both albedos decrease with increasing heliocentric distance as r^?0.54. The corresponding exponents for changes in mean particle size and number density are related in a simple way. The median orbital inclination of zodiacal light particles with respect to the ecliptic is 12°, close to the observed median value for faint asteroids and short-period comets. Furthermore, the color of dust particles and its variation with solar phase angle closely resemble those of C asteroids. These findings are, at least, consistent with the zodiacal cloud originating primarily from collisions among asteroids. Finally, a value of ?10¹⁸?_ErmE g was derived for the mass of the zodiacal cloud, where ?_E is the mean particle radius (in micrometers) at 1-AU-heliocentric distance. For extinction in the ecliptic, $\text{Δm = 10}{}^{\mathrm{?5}}\text{?}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{mag}$ was obtained, where ? is the solar elongation in degrees.     

Pole coordinates of the asteroids 9 metis, 22 kalliope,and 44 Nysa

By means of new photoelectric observations made in 1974 an attempt to determine the poles of asteroids 9 and 44 was made. Following a method based upon the magnitude-aspect and amplitude-aspect relations, the coordinates of the poles for 9 and 44 were found to be, respectively, λ₀ = 191° ± 5°, β₀ = 56° ± 6° and λ₀ = 100° ± 10°, β₀ = 50° ± 10°. The previously published pole for asteroid 22, λ₀ = 215° ± 10°, β₀ = 45° ± 15°, was confirmed. From its phase relation we determined the phase coefficient of 44 Nysa, a very high albedo object (p_v = 0.377). The very low phase coefficient obtained (β_v = 0.018 mag/deg) agrees very well with an inverse relation between geometrical albedo and phase coefficient. The results are summarized in a table.