Elastic collisions in ion irradiation experiments: A mechanism for space weathering of silicates

Ion irradiation experiments have been performed on silicates (bulk samples) rich of olivine, pyroxene, and serpentine to simulate the effects of space weathering induced on asteroids by solar wind ions. We have used different ions (H⁺, He⁺, Ar⁺, Ar²⁺) having different energies (from 60 to 400 keV) to weather the samples, probed by Raman spectroscopy and UV-vis-NIR reflectance spectroscopy. All the irradiated materials have shown reddening and darkening of reflectance spectra in the 0.25-2.7 μm spectral range. We have found that the increase of the spectral slope of the continuum across the 1-μm band is strongly related with the number of displacements caused by colliding ions because of elastic collisions with the target nuclei. The spectral slopes have been compared, at increasing ion fluence, with those from irradiated Epinal meteorite. We show that formation of nuclear displacements by solar wind ion irradiation is a physical mechanism that reddens the asteroidal surfaces on a time-scale lower than 10⁶ years.     

Spectral alteration of the Meteorite Epinal (H5) induced by heavy ion irradiation: a simulation of space weathering effects on near-Earth asteroids

We present results obtained for Epinal (H5), an ordinary chondrite meteorite, irradiated with 60 keV Ar⁺⁺ ions, simulating solar wind heavy particle irradiation. Bidirectional reflectance spectra (0.3-2.67 μm) measured after irradiating Epinal samples with different ion fluences exhibit a progressive reddening that is similar to the spread of spectra observed for S-type near-Earth asteroids. The timescales for inducing the same effects in space as those obtained in laboratory are estimated to be ¹⁰⁴-¹⁰⁶ yr. These results suggest irradiation by heavy ions may be a very efficient weathering process in near-Earth space.     

Spectral study of the Eunomia asteroid family : I. Eunomia

We present color ratio curves of the S-Asteroid 15 Eunomia, which have been extracted from high-precision photometric lightcurves obtained in three different VNIR wavelength bands at the Bochum Telescope, La Silla. The measured color ratio curves and near infrared spectra were used to derive a detailed surface composition model whose shape has been computed by V-lightcurve inversions. According to this analysis, the asteroid shows on one hemisphere a higher concentration of pyroxene, which causes an increased 440/700 nm and a reduced 940/700 nm reflectance ratio as well as a pronounced 2-μm absorption band. The remaining surface shows a higher concentration of olivine, leading to a reduced 440/700 nm and slightly increased 940/700 nm color ratio. In addition, we found that the maximum of the 440/700 nm color ratio curve coincide with the minimum of the 940/700 nm color ratio curve and vice versa. We demonstrate on the basis of USGS laboratory spectra that this anti-cyclical behavior can be explained by choosing Fe-rich olivine and a pyroxene with moderate Fe content as varying mineral phases. Furthermore, our observations confirm that 15 Eunomia is an irregular elongated and at least partially differentiated body. Previous spectral investigations of several smaller fragments of the Eunomia asteroid family revealed that the amount of fragments showing an increased pyroxene content exceeds the amount of pyroxene-poor fragments (Nathues, 2000, DLR Forschungsbericht, ISSN 1434-8454). This finding together with the observation that the major fraction of Eunomia's surface is enriched in olivine let us claim that a large fraction of the original pyroxene-enriched crust layer has been lost due to a major collision that created the Eunomia asteroid family. Significant spectral evidences, consistent with high concentrations of metals have been found neither in the rotational resolved spectra of 15 Eunomia nor in its fragments. This led to the conclusion that either no core consisting mainly of metals exists or that an eventual one has not been unearthed by the impact.     

Iapetus dark and bright material:: giving compositional interpretation some latitude

Narrowband reflectance spectra (0.53-1.0 μm) of Iapetus' leading and trailing sides were obtained in 2000 to test the presence of an absorption feature located near 0.67 μm seen in reflectance spectra of Iapetus' dark material and Hyperion's surface material. No feature was observed. The difference in reflectance across the UV/VIS/NIR spectral region, and the dependence of the presence or absence of this absorption feature on angular separation from the apex of Iapetus in its orbit, phase angle, and heliocentric distance (affecting temperature), were examined. A trend of increased reddening, and the presence of the absorption feature, correlate with an angular separation from the apex of ? approximately 10°. Spectral information is lost when the contribution of the bright water ice signal to the reflectance spectrum increases sufficiently. In order to optimize compositional studies of Iapetus, we encourage future ground-based and space-based spectral observations to maximize the concentration of dark material in the instrumental field of view.     

The organic composition of Comet C/2001 A2 (LINEAR): I. Evidence for an unusual organic chemistry

We used the NIRSPEC instrument on the Keck-2 telescope atop Mauna Kea, HI to observe Comet C/2001 A2 (LINEAR) in a Target of Opportunity campaign on UT 2001 July 9.5, 10.5 August 4.4, 10.5. We measured seven organic parent volatiles (C₂H₆, C₂H₂, HCN, CH₄, CO, CH₃OH, H₂CO) simultaneously with H₂O. We obtained absolute production rates and relative abundances for parent volatiles, and also measured rotational temperatures for several of these species. The chemical composition of C/2001 A2 differs substantially from any comet we have observed to date. The abundances we measure (relative to H₂O) for C₂H₆, C₂H₂, HCN, and CH₃OH are enriched by a factor of ∼2 to 3 in C/2001 A2 compared with most comets in our database. Other molecular species were detected within the typical range of measured abundances. C/2001 A2 presented a unique opportunity to study the chemistry of a fragmenting comet where pristine areas are exposed to the Sun.     

The role of organic haze in Titan's atmospheric chemistry: I. Laboratory investigation on heterogeneous reaction of atomic hydrogen with Titan tholin

In Titan's atmosphere consisting of N₂ and CH₄, large amounts of atomic hydrogen are produced by photochemical reactions during the formation of complex organics. This atomic hydrogen may undergo heterogeneous reactions with organic aerosol in the stratosphere and mesosphere of Titan. In order to investigate both the mechanisms and kinetics of the heterogeneous reactions, atomic deuterium is irradiated onto Titan tholin formed from N₂ and CH₄ gas mixtures at various surface-temperatures of the tholin ranging from 160 to 310 K. The combined analyses of the gas species and the exposed tholin indicate that the interaction mechanisms of atomic deuterium with the tholin are composed of three reactions; (a) abstraction of hydrogen from tholin resulting in gaseous HD formation (HD recombination), (b) addition of D atom into tholin (hydrogenation), and (c) removal of carbon and/or nitrogen (chemical erosion). The reaction probabilities of HD recombination and hydrogenation are obtained as ηabst=1.9(±0.6)×¹⁰⁻³×exp(−300/T) and ηhydro=2.08(±0.64)×exp(−1000/T), respectively. The chemical erosion process is very inefficient under the conditions of temperature range of Titan's stratosphere and mesosphere. Under Titan conditions, the rates of hydrogenation > HD recombination ? chemical erosion. Our measured HD recombination rate is about 10 times (with an uncertainty of a factor of 3-5) the prediction of previous theoretical model. These results imply that organic aerosol can remove atomic hydrogen efficiently from Titan's atmosphere through the heterogeneous reactions and that the presence of aerosol may affect the subsequent organic chemistry.     

Magnetic aggregation: II. Laboratory and microgravity experiments

In a previous publication (Dominik and Nübold, 2002, Icarus 157, 173-186), we presented analytical expressions and theoretical considerations concerning preplanetary dust aggregation with magnetized grains in the solar nebula. The present work is dedicated to the experimental study of magnetic aggregation in a ground-based laboratory as well as under microgravity conditions on parabolic flights. We conducted aggregation experiments with dust analogues in order to study the temporal evolution and the structural outcome of grain growth processes dominated by or comprising exclusively magnetic grains. Within aggregation times ranging from a couple of seconds to a few minutes only, formation of huge chain-like and/or web-like dust aggregates was observed. After aggregate retrieval we were able to study the sizes and structures of the aggregates in great detail. We established the fractal dimension of the aggregates as D_fs=1.20±0.05 for single chains and D_fc=1.50±0.21 for inter-connected web-like structures. This is considerably lower than for non-magnetic grain growth. The dynamic exponent z for the mass increase with time according to t^z was found to be z=2.7 from in-situ video images of the microgravity aggregation runs. The results are compared with the theoretical considerations presented earlier as well as with previous experimental work on the same and on related topics, respectively.     

Cassini UVIS observations of the Io plasma torus: II. Radial variations

On January 14, 2001, shortly after the Cassini spacecraft's closest approach to Jupiter, the Ultraviolet Imaging Spectrometer (UVIS) made a radial scan through the midnight sector of Io plasma torus. The Io torus has not been previously observed at this local time. The UVIS data consist of 2-D spectrally dispersed images of the Io plasma torus in the wavelength range of 561-1912 Å. We developed a spectral emissions model that incorporates the latest atomic physics data contained in the CHIANTI database in order to derive the composition of the torus plasma as a function of radial distance. Electron temperatures derived from the UVIS torus spectra are generally less than those observed during the Voyager era. We find the torus ion composition derived from the UVIS spectra to be significantly different from the composition during the Voyager era. Notably, the torus contains substantially less oxygen, with a total oxygen-to-sulfur ion ratio of 0.9. The average ion charge state has increased to 1.7. We detect S(V) in the Io torus at the 3σ level. S(V) has a mixing ratio of 0.5%. The spectral emission model used can approximate the effects of a nonthermal distribution of electrons. The ion composition derived using a kappa distribution of electrons is identical to that derived using a Maxwellian electron distribution; however, the kappa distribution model requires a higher electron column density to match the observed brightness of the spectra. The derived value of the kappa parameter decreases with radial distance and is consistent with the value of κ=2.4 at 8R_J derived by the Ulysses URAP instrument (Meyer-Vernet et al., 1995). The observed radial profile of electron column density is consistent with a flux tube content, NL², that is proportional to r⁻².     

Cassini UVIS observations of the Io plasma torus: I. Initial results

During the Cassini spacecraft's flyby of Jupiter (October, 2000-March, 2001), the Ultraviolet Imaging Spectrograph (UVIS) produced an extensive dataset consisting of 3349 spectrally dispersed images of the Io plasma torus. Here we present an example of the raw data and representative EUV spectra (561-1181 Å) of the torus, obtained on October 1, 2000 and November 14, 2000. For most of the flyby period, the entire Io torus fit within the UVIS field-of-view, enabling the measurement of the total power radiated from the torus in the extreme ultraviolet. A typical value for the total power radiated in the wavelength range of 580-1181 Å is 1.7×10¹² W, with observed variations of up to 25%. Several brightening events were observed. These events lasted for roughly 20 hours, during which time the emitted power increased rapidly by ∼20% before slowly returning to the pre-event level. Observed variations in the relative intensities of torus spectral features provide strong evidence for compositional changes in the torus plasma with time. Spatial profiles of the EUV emission show no evidence for a sharply peaked “ribbon” feature. The ratio of the brightness of the dusk ansa to the brightness of the dawn ansa is observed to be highly variable, with an average value of 1.30. Weak longitudinal variations in the brightness of the torus ansae were observed at the 2% level.     

The organic composition of C/2001 A2 (LINEAR): II. Search for heterogeneity within a comet nucleus

The nucleus of Comet C/2001 A2 (LINEAR) split several times during its recent apparition, presenting an unusual opportunity to search for chemical differences in freshly exposed material. We conducted this search using NIRSPEC at the W.M. Keck Observatory on four dates in 2001: 9.5 and 10.5 July and 4.4 and 10.5 August. We detected the R0 and R1 lines of the ν₃ vibrational band of CH₄ near 3.3 μm on all dates. The R2 line was detected on 4.4 and 10.5 August. When we compare production rates of CH₄ to H₂O, we find evidence of a significant enhancement in August relative to that found in July. H₂CO was securely detected via its ν₁ and ν₅ bands on 9.5 July. On 10.5 July, H₂CO emission was much weaker, and its mixing ratio had dropped by a factor of about four. The mixing ratios for other detected volatile species did not change significantly over the course of the observations. We discuss the implications of this evidence for chemical heterogeneity in the nucleus of Comet C/2001 A2.     

Binary asteroid population: 1. Angular momentum content

We compiled a list of estimated parameters of binary systems among asteroids from near-Earth to trojan orbits. In this paper, we describe the construction of the list, and we present results of our study of angular momentum content in binary asteroids. The most abundant binary population is that of close binary systems among near-Earth, Mars-crossing, and main belt asteroids that have a primary diameter of about 10 km or smaller. They have a total angular momentum very close to, but not generally exceeding, the critical limit for a single body in a gravity regime. This suggests that they formed from parent bodies spinning at the critical rate (at the gravity spin limit for asteroids in the size range) by some sort of fission or mass shedding. The Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect is a candidate to be the dominant source of spin-up to instability. Gravitational interactions during close approaches to the terrestrial planets cannot be a primary mechanism of formation of the binaries, but it may affect properties of the NEA part of the binary population.     

Estimating the water content of hydrated minerals using reflectance spectroscopy: II. Effects of particle size

Visible-near infrared reflectance spectra for five particle size fractions of a Hawaiian palagonite (HWMK101) and a nontronite (ferruginous smectite, Clay Minerals Society source clay SWa-1) were measured under ambient, purged, and heated conditions to characterize the effects of surface and volume scattering on the relationship between absolute H₂O content and the strength of the 3 μm absorption feature. Both materials were ground and dry sieved to particle sizes of <25, 25-45, 45-75, 75-125, and 125-250 μm. Particles of the bulk palagonite have an approximate bimodal distribution consisting of small, amorphous particles <5 μm in diameter mixed with crystalline and glass particles <1 mm in diameter, whereas the nontronite particles are polycrystalline aggregates. We find that band parameters value relating the strength of the 3 μm hydration feature to water content increase with particle size for a given water content, regardless of whether reflectance or single scattering albedo spectra are used. Spectra generally increase in reflectance as particle size decreases, a result of the relative increase in volume to surface scattering. Spectra of large particles are commonly saturated in the 3 μm region due to an increase in optical path length, making an accurate estimate of water content indeterminate until the samples dehydrate to the volume-scattering regime. We find that the presence of fines in several of the size fractions of palagonite cause their spectra to be representative of the finest fraction rather than the mean particle size. The nontronite spectra appear to be representative of an effective particle size within the range of the sieved size fractions. Many planetary surfaces are expected to have a large number of small particles which can dominate their spectral signature. Our results for particles <45 μm provide a reasonable model for estimating the H₂O content of hydrated asteroids and regions of Mars.     

Cassini UVIS observations of the Io plasma torus: IV. Modeling temporal and azimuthal variability

In this fourth paper in a series, we present a model of the remarkable temporal and azimuthal variability of the Io plasma torus observed during the Cassini encounter with Jupiter. Over a period of three months, the Cassini Ultraviolet Imaging Spectrograph (UVIS) observed a dramatic variation in the average torus composition. Superimposed on this long-term variation, is a 10.07-h periodicity caused by an azimuthal variation in plasma composition subcorotating relative to System III longitude. Quite surprisingly, the amplitude of the azimuthal variation appears to be modulated at the beat frequency between the System III period and the observed 10.07-h period. Previously, we have successfully modeled the months-long compositional change by supposing a factor of three increase in the amount of material supplied to Io's extended neutral clouds. Here, we extend our torus chemistry model to include an azimuthal dimension. We postulate the existence of two azimuthal variations in the number of superthermal electrons in the torus: a primary variation that subcorotates with a period of 10.07 h and a secondary variation that remains fixed in System III longitude. Using these two hot electron variations, our model can reproduce the observed temporal and azimuthal variations observed by Cassini UVIS.     

The role of organic haze in Titan's atmospheric chemistry: II. Effect of heterogeneous reaction to the hydrogen budget and chemical composition of the atmosphere

One of the key components controlling the chemical composition and climatology of Titan's atmosphere is the removal of reactive atomic hydrogen from the atmosphere. A proposed process of the removal of atomic hydrogen is the heterogeneous reaction with organic aerosol. In this study, we investigate the effect of heterogeneous reactions in Titan's atmospheric chemistry using new measurements of the heterogeneous reaction rate [Sekine, Y., Imanaka, H., Matsui, T., Khare, B.N., Bakes, E.L.O., McKay, C.P., Sugita, S., 2008. Icarus 194, 186-200] in a one-dimensional photochemical model. Our results indicate that 60-75% of the atomic hydrogen in the stratosphere and mesosphere are consumed by the heterogeneous reactions. This result implies that the heterogeneous reactions on the aerosol surface may predominantly remove atomic hydrogen in Titan's stratosphere and mesosphere. The results of our calculation also indicate that a low concentration of atomic hydrogen enhances the concentrations of unsaturated complex organics, such as C₄H₂ and phenyl radical, by more than two orders in magnitude around 400 km in altitude. Such an increase in unsaturated species may induce efficient haze production in Titan's mesosphere and upper stratosphere. These results imply a positive feedback mechanism in haze production in Titan's atmosphere. The increase in haze production would affect the chemical composition of the atmosphere, which might induce further haze production. Such a positive feedback could tend to dampen the loss and supply cycles of CH₄ due to an episodic CH₄ release into Titan's atmosphere.     

Yarkovsky detection opportunities: II. Binary systems

We consider the possibility of detecting the Yarkovsky orbital perturbation acting on binary systems among the near-Earth asteroids. This task is significantly more difficult than for solitary asteroids because the Yarkovsky force affects both the heliocentric orbit of the system's center of mass and the relative orbit of the two components. Nevertheless, we argue these are sufficiently well decoupled so that the major Yarkovsky perturbation is in the simpler heliocentric motion and is observable with the current means of radar astrometry. Over the long term, the Yarkovsky perturbation in the relative motion of the two components is also detectable for the best observed systems. However, here we consider a simplified version of the problem by ignoring mutual non-spherical gravitational perturbations between the two asteroids. With the orbital plane constant in space and the components' rotation poles fixed (and assumed perpendicular to the orbital plane), we do not examine the coupling between Yarkovsky and gravitational effects. While radar observations remain an essential element of Yarkovsky detections, lightcurve observations, with their ability to track occultation and eclipse phenomena, are also very important in the case of binaries. The nearest possible future detection of the Yarkovsky effect for a binary system occurs for (66063) 1998 RO₁ in September 2006. Farther out, even more statistically significant detections are possible for several other systems including 2000 DP₁₀₇, (66391) 1999 KW₄ and 1996 FG₃.     

Cassini UVIS observations of the Io plasma torus: III. Observations of temporal and azimuthal variability

In this third paper in a series presenting observations by the Cassini Ultraviolet Imaging Spectrometer (UVIS) of the Io plasma torus, we show remarkable, though subtle, spatio-temporal variations in torus properties. The Io torus is found to exhibit significant, near-sinusoidal variations in ion composition as a function of azimuthal position. The azimuthal variation in composition is such that the mixing ratio of S II is strongly correlated with the mixing ratio of S III and the equatorial electron density and strongly anti-correlated with the mixing ratios of both S IV and O II and the equatorial electron temperature. Surprisingly, the azimuthal variation in ion composition is observed to have a period of 10.07 h—1.5% longer than the System III rotation period of Jupiter, yet 1.3% shorter than the System IV period defined by [Brown, M.E., 1995. J. Geophys. Res. 100, 21683-21696]. Although the amplitude of the azimuthal variation of S III and O II remained in the range of 2-5%, the amplitude of the S II and S IV compositional variation ranged between 5 and 25% during the UVIS observations. Furthermore, the amplitude of the azimuthal variations of S II and S IV appears to be modulated by its location in System III longitude, such that when the region of maximum S II mixing ratio (minimum S IV mixing ratio) is aligned with a System III longitude of ∼200°±15°, the amplitude is a factor of ∼4 greater than when the variation is anti-aligned. This behavior can explain numerous, often apparently contradictory, observations of variations in the properties of the Io plasma torus with the System III and System IV coordinate systems.     

Impact evolution of asteroid shapes: 1. random mass redistribution

We explore whether the cumulative effect of small-scale meteoroid bombardment can drive asteroids into nonaxisymmetric shapes comparable to those of known objects (elongated prolate forms, twin-lobed binaries, etc). We simulate impact cratering as an excavation followed by the launch, orbit, and reimpact of ejecta. Orbits are determined by the gravity and rotation of the evolving asteroid, whose shape and spin change as cratering occurs repeatedly. For simplicity we consider an end-member evolution where impactors are all much smaller than the asteroid and where all ejecta remain bound. Given those assumptions, we find that cumulative small impacts on rotating asteroids lead to oblate shapes, irrespective of the chosen value for angle of repose or for initial angular momentum. The more rapidly a body is spinning, the more flattened the outcome, but oblateness prevails. Most actual asteroids, by contrast, appear spherical to prolate. We also evaluate the timescale for reshaping by small impacts and compare it to the timescale for catastrophic disruption. For all but the steepest size distributions of impactors, reshaping from small impacts takes more than an order of magnitude longer than catastrophic disruption. We conclude that small-scale cratering is probably not dominant in shaping asteroids, unless our assumptions are naive. We believe we have ruled out the end-member scenario; future modeling shall include angular momentum evolution from impacts, mass loss in the strength regime, and craters with diameters up to the disruption threshold. The ultimate goal is to find out how asteroids get their shapes and spins and whether tidal encounters in fact play a dominant role.     

Ozone abundance on Mars from infrared heterodyne spectra: II. Validating photochemical models

Ozone is an important observable tracer of martian photochemistry, including odd hydrogen (HO_x) species important to the chemistry and stability of the martian atmosphere. Infrared heterodyne spectroscopy with spectral resolution ?¹⁰⁶ provides the only ground-based direct access to ozone absorption features in the martian atmosphere. Ozone abundances were measured with the Goddard Infrared Heterodyne Spectrometer and the Heterodyne Instrument for Planetary Wind and Composition at the NASA Infrared Telescope Facility on Mauna Kea, Hawai'i. Retrieved total ozone column abundances from various latitudes and orbital positions (LS=40°, 74°, 102°, 115°, 202°, 208°, 291°) are compared to those predicted by the first three-dimensional gas phase photochemical model of the martian atmosphere [Lefèvre, F., Lebonnois, S., Montmessin, F., Forget, F., 2004. J. Geophys. Res. 109, doi:10.1029/2004JE002268. E07004]. Observed and modeled ozone abundances show good agreement at all latitudes at perihelion orbital positions (LS=202°, 208°, 291°). Observed low-latitude ozone abundances are significantly higher than those predicted by the model at aphelion orbital positions (LS=40°, 74°, 115°). Heterogeneous loss of odd hydrogen onto water ice cloud particles would explain the discrepancy, as clouds are observed at low latitudes around aphelion on Mars.     

Saturn's icy satellites investigated by Cassini-VIMS: I. Full-disk properties: 350-5100 nm reflectance spectra and phase curves

Saturn's icy satellites are among the main scientific objectives of the Cassini-VIMS (Visual and Infrared Mapping Spectrometer) experiment. This paper contains a first systematic and comparative analysis of the full-disk spectral properties of Dione, Enceladus, Epimetheus, Hyperion, Iapetus, Mimas, Phoebe, Rhea and Tethys as observed by VIMS from July 2004 to June 2005. The disk integrated properties (350-5100 nm reflectance spectra and phase curves at 550-2232 nm) and images of satellites are reported and discussed in detail together with the observed geometry. In general, the spectra in the visible spectral range are almost featureless and can be classified according to the spectral slopes: from the bluish Enceladus and Phoebe to the redder Iapetus, Hyperion and Epimetheus. In the 1000-1300 nm range the spectra of Enceladus, Tethys, Mimas and Rhea are characterized by a negative slope, consistent with a surface largely dominated by water ice, while the spectra of Iapetus, Hyperion and Phoebe show a considerable reddening pointing out the relevant role played by darkening materials present on the surface. In between these two classes are Dione and Epimetheus, which have a flat spectrum in this range. The main absorption bands identified in the infrared are the 1520, 2020, 3000 nm H₂O/OH bands (for all satellites), although Iapetus dark terrains show mostly a deep 3000 nm band while the 1520 and 2020 nm bands are very faint. In this spectral range, the Iapetus spectrum is characterized by a strong reddening. The CO₂ band at 4260 nm and the Fresnel ice peak around 3100 nm are evident only on Hyperion, Phoebe and Iapetus. The phase curves at 550 and at 2232 nm are reported for all the available observations in the 0°-144° range; Rhea shows an opposition surge at visible wavelengths in the 0.5°-1.17° interval. The improvement on the retrieval of the full-disk reflectance spectra can be appreciated by a direct comparison with ground-based telescopic data available from literature. Finally, data processing strategies and recent upgrades introduced in the VIMS-V calibration pipeline (flat-field and destriping-despiking algorithm) are discussed in appendices.     

Bidirectional reflectance spectroscopy: 6. Effects of porosity

It is well known that the bidirectional reflectance of a particulate medium such as a planetary regolith depends on the porosity, in contrast to predictions of models based on the equation of radiative transfer as usually formulated. It is shown that this failure to predict porosity dependence arises from an incorrect treatment of the light that passes between the particles. In this paper a more physically correct treatment that takes account of the necessity of preventing particles from interpenetrating is used together with the two-stream approximation to solve the radiative transfer equation and derive improved expressions for the bidirectional and directional-hemispherical reflectances. It is found that increasing the filling factor (decreasing the porosity) increases the reflectance of low and medium albedo powders, but decreases it for ones with very high albedos. The model agrees qualitatively with measured data.