Scattering And Absorption of Light in Planetary Regoliths

For our experiment, an aggregate nominally consisting of densely-packed Ø=0.5 µm spherical SiO₂ particles was selected^29,30 and polished further, to approximate a spherical shape, after which it was characterized by weighing and measuring its dimensions (Figure 4). The nearly spherical aggregate had a diameter of 1.16 mm and a volume density of 0.47. Light scattering was measured according to step 1. The beam was filtered to 488±5 nm, with a Gaussian spectrum. The measurement was averaged from three sweeps and the empty levitator signal was subtracted from the result.

From the intensities of the four different polarization configurations, we calculated the phase function, the degree of linear polarization for unpolarized incident light –M₁₂/M₁₁, and the depolarization M₂₂/M₁₁, as a function of the phase angle (Figure 5, Figure 6, Figure 7). One known systematic error source of our measurement is the extinction ratio of the linear polarizers, which is 300:1. For this sample, it is, however, adequate so that the leaked polarized light is below the detection threshold.

The numerical modeling consists of multiple software interlinked by scripts which handle the information flow according to the parameters given by the user. The scripts and software are preconfigured to work on the CSC – IT Center for Science Ltd.’s Taito cluster, and the user needs to modify the scripts and Makefiles themselves to get the modeling tool to work on other platforms. The tool starts by running the STMM solver²⁰, which computes volume-element characteristics as described by Väisänen et al.¹⁸. After that, the scattering and absorption characteristics of the volume element are used as input for two different software. A Mie-scattering solver is used to find the effective refractive index by matching the coherent scattering cross section of the volume element to a Mie sphere of equal size²⁰. Then the aggregate is modeled by running the SIRIS4 software with the volume element as a diffuse scatterer and with the effective refractive index on the surface of the aggregate. The coherent backscattering component is added separately because there is no software that can treat effective refractive medium and coherent backscattering simultaneously. Currently, the RT-CB is incapable of accounting for the effective refractive medium, whereas the SIRIS4 is incapable of accounting for coherent backscattering. The coherent backscattering is, however, added to the SIRIS4^23,24 results approximately by running the volume-element scattering characteristics through the scattering phase matrix decomposition software PMDEC which derives pure Mueller and Jones matrices required for the RT-CB⁹. The coherent backscattering component is then extracted by subtracting the radiative transfer component from the results of the RT-CB. Then, the extracted coherent backscattering component is added to the results obtained from the SIRIS4.

We simulated numerically the properties of the mm-sized (radius 580 µm) SiO₂ aggregate by following step 2. We used two kinds of volume elements, one consisting of nominal equisized particles (0.25 µm) and the other consisting of normally distributed (mean 0.25 µm, standard deviation 0.1 µm) particles truncated to the range of 0.1-0.2525 µm. Introducing the latter distribution of particles is based on the fact that essentially all SiO₂ samples with a given nominal particle size also have a significant alien distribution of smaller particles³¹. In total, 128 volume elements of size kR₀=10 were drawn from 128 periodic boxes containing about 10,000 particles packed to the volume density v=47 % each. From the specifications of the material, we have n=1.463+i0 at the wavelength of 0.488 µm, which is the wavelength used in the measurements.

With SIRIS4, the scattering properties of 100,000 aggregates, with radius of 580 µm, standard deviation of 5.8 µm, and with the power-law index of the correlation function 2, were solved and averaged. These results are plotted (see Figure 5, Figure 6, Figure 7) with the experimental measurements, and an additional simulation without the effective medium. Both choices for the particle distribution produce a match to the measured phase function (see Figure 5), although they result in different polarization characteristics as is seen in Figure 6. These differences can be used to identify the underlying distribution of the particles in the sample. The best choice is to use the truncated normal distribution instead of the equisized particles (see Figure 6). If only normalized phase functions are used, the underlying distributions are indistinguishable (compare Figure 5, Figure 6, Figure 7). In Figure 7 for the depolarization, the numerical results have features similar to the measured curve, but the functions are shifted by 10° to the backscattering direction. The effective refractive index positively corrects the results as is seen from the simulations obtained with and without the effective medium (see Figure 5, Figure 6, Figure 7). The differences in the polarization (Figure 6) indicate that the sample has presumably a more complex structure (e.g., a separate mantle and core) than our homogeneous model. It is, however, beyond the existing microscopic methods for sample characterization to retrieve the true structure of the aggregate. The coherent backscattering was added separately to the results. The measurements lack visible intensity spike observed at the backscattering angles, but the degree of linear polarization is more negative between 0-30° which cannot be produced without coherent backscattering (compare “distribution” with “no cb”, see Figure 5, Figure 6, Figure 7).

For solar system applications, we compared the observed Vesta spectra and the modeled spectrum obtained by following protocol 3. The results are shown in Figure 3 and Figure 8 and they suggest that howardite particles, with more than 75% of them having a particle size smaller than 25 µm, dominate Vesta’s regolith. Although the overall match is quite satisfactory, the modeled and observed spectra differ slightly: the absorption band centers of the model spectrum are shifted to longer wavelengths, and the spectral minima and maxima tend to be shallow as compared to the observed spectra. The differences in the minima and maxima could be explained by the fact that mutual shadowing effects among the regolith particles have not been accounted for: the shadowing effects are stronger for low reflectances and weaker for high reflectances and, in the relative sense, would decrease the spectral minima and increase the spectral maxima when accounted for in the modeling. Furthermore, the imaginary part of the complex refractive indices for howardite was derived without taking into account wavelength-scale surface-roughness, and thus the derived values can be too small to explain the spectral minima. When further using these values in our model by utilizing geometric optics, the band depths in the modeled spectrum can become too shallow. These wavelength-scale effects could also play a part at longer wavelengths together with a small contribution from the low-end tail of the thermal emission spectrum. The differences can also be caused by a compositional mismatch of our howardite sample and Vesta minerals and by a different particle size distribution needed for the model. Finally, the reflectance spectra of Vesta were observed at 180-200 K, and our howardite sample was measured in room temperature. Reddy et al.³² have shown that the absorption band centers shift to longer wavelengths with increasing temperature.

The photometric and polarimetric phase curve observations for asteroid (4) Vesta are from Gehrels³³ and the NASA Planetary Data System’s Small Bodies Node (http://pdssbn.astro. umd.edu/sbnhtml), respectively. Their modeling follows step 4 and starts from the particle refractive index and size distribution available from the spectrometric modeling at the wavelength of 0.45 µm. These particles have sizes larger than 5 µm, that is, much larger than the wavelength and are thus in the geometric optics regime, termed large-particle population. For the phase curve modeling, an additional small-particle population of densely-packed subwavelength-scale particles is also incorporated, with due attention paid to avoid conflicts with the spectrometric modeling above.

The complex refractive index has been set to 1.8+i0.000168. The effective particle sizes and single-scattering albedos in the large-particle and small-particle populations equal (9.385 µm, 0.791) and (0.716 µm, 0.8935), respectively. The mean free path lengths in the large-particle and small-particle media are 16.39 µm and 0.56 µm. The large-particle medium has a volume density of 0.4, whereas the small-particle medium has a volume density of 0.3. The fractions of large-particle and small-particle media in the Vesta regolith are assumed to be 99% and 1%, respectively, giving a total single-scattering albedo of 0.815 and a total mean free path length of 12.78 µm. Following step 4, the Vesta geometric albedo at 0.45 µm turns out to be 0.32 in fair agreement with the observations (cf. Figure 8 when extrapolated to zero phase angle).

Figure 9, Figure 10, Figure 11 depict the photometric and polarimetric phase curve modeling for Vesta. For the photometric phase curve (Figure 10, left), the model phase curve from RT-CB has been accompanied with a linear dependence on the magnitude scale (slope coefficient -0.0179 mag/°), mimicking the effect of shadowing in a densely-packed, high-albedo regolith. No alteration has been invoked for the degree of polarization (Figure 10, right; Figure 11). The model successfully explains the observed photometric and polarimetric phase curves and offers a realistic prediction for the maximum polarization near the phase angle of 100° as well as for the characteristics at small phase angles <3°.

It is striking how the minute fraction of the small-particle population is capable of completing the explanation of the phase curves (Figure 10, Figure 11). There are intriguing modeling aspects involved. First, as shown in Figure 9 (left), the single-scattering phase functions for the large-particle and small-particle populations are quite similar, whereas the linear polarization elements are significantly different. Second, in the RT-CB computations, both particle populations contribute to the coherent backscattering effects. Third, in order to obtain realistic polarization maxima, there has to be a significant large-particle population in the regolith (in agreement with the spectral modeling). With the current independent mixing of the small-particle and large-particle media, it remains possible to assign a part of the small-particle contribution to the large-particle surfaces. However, in order for coherent backscattering effects to take place and to explain the observations, it is obligatory to incorporate a small-particle population.

The European Space Agency’s (ESA) Rosetta mission to the comet 67P/Churyumov-Gerasimenko provided an opportunity to measure the photometric phase function of the coma and the nucleus over a wide phase-angle range within just a few hours³⁴. The measured coma phase functions show a strong variation with time and a local position of the spacecraft. The coma phase function has been successfully modeled²⁰ with a particle model composed of submicrometer-sized organic and silicate particles using the numerical methods (steps 5 and 2) as shown in Figure 12. The results suggest that the size distribution of dust varies in the coma due to the comet’s activity and the dynamical evolution of the dust. By modeling scattering by a 1-km-sized object whose surface is covered with the dust particles, we have shown that scattering by the nucleus of the comet is dominated with the same type of particles that also dominate the scattering in the coma (Figure 13).

Figure 1: Asteroid (4) Vesta (left) and Comet 67P/Churyumov-Gerasimenko (right) visited by the NASA Dawn mission and by the ESA Rosetta mission, respectively. Image credits: NASA/JPL/MPS/DLR/IDA/Björn Jónsson (left), ESA/Rosetta/NAVCAM (right). Please click here to view a larger version of this figure.

Figure 2: Light scattering measurement instrument. Photo (above) and top view schematic (below) showing: (1) fiber-coupled light source with collimator, (2) focusing lens (optional), (3) bandpass filter for wavelength selection, (4) adjustable aperture for beam shaping, (5) motorized linear polarizer, (6) high-speed camera, (7) high-magnification objective, (8) acoustic levitator for sample trapping, (9) measurement head, comprising an IR filter, motorized shutter, motorized linear polarizer, and a photomultiplier tube (PMT), (10) motorized rotation stage for adjusting measurement head angle, (11) optical flat for Fresnel reflection, (12) neutral density filter, and (13) reference PMT, for monitoring beam intensity. The system is divided into three enclosed compartments to eliminate stray light. Please click here to view a larger version of this figure.

Figure 3: The imaginary part of the refractive index for howardite as a function of wavelength. The imaginary part of the refractive Im(n) obtained for the howardite mineral by following protocol 3.1. The refractive index is utilized in modeling the scattering characteristics of asteroid (4) Vesta. Please click here to view a larger version of this figure.

Figure 4: The measurement sample composed of densely-packed spherical SiO₂ particles. The sample has been carefully polished in order to obtain a nearly spherical shape that allows for both efficient scattering experiments and numerical modeling. Please click here to view a larger version of this figure.

Figure 5: Phase function. The phase functions of the sample aggregate obtained by following the experimental protocols 1 and the numerical modeling step 2. The phase functions are normalized to give unity when integrated from 15.1° to 165.04°. Please click here to view a larger version of this figure.

Figure 6: Degree of linear polarization. As in Figure 5 for the degree of linear polarization for unpolarized incident light –M₁₂/M₁₁ (in %). Please click here to view a larger version of this figure.

Figure 7: Depolarization. As in Figure 5 for the depolarization M₂₂/M₁₁. Please click here to view a larger version of this figure.

Figure 8: Absolute reflectance spectra. Asteroid (4) Vesta’s modeled and observed absolute reflectance spectra at 17.4-degree phase angle. Please click here to view a larger version of this figure.

Figure 9: Scattering phase function P₁₁ and degree of linear polarization for unpolarized incident light -P₂₁/P₁₁ as a function of the scattering angle for volume elements of large particles (red) and small particles (blue) in the regolith of asteroid (4) Vesta. The dotted line indicates a hypothetical isotropic phase function (left) and a zero level of polarization (right). Please click here to view a larger version of this figure.

Figure 10: Observed (blue) and modeled (red) disk-integrated brightness in the magnitude scale as well as the degree of linear polarization for unpolarized incident light as a function of phase angle for asteroid (4) Vesta. The photometric and polarimetric observations are from Gehrels (1967) and the Small Bodies Node of the Planetary Data System (http://pdssbn.astro.umd.edu/sbnhtml), respectively. Please click here to view a larger version of this figure.

Figure 11: Degree of linear polarization. The degree of linear polarization for asteroid (4) Vesta predicted for large phase angles based on the numerical multiple-scattering modeling. Please click here to view a larger version of this figure.

Figure 12: Modeled and measured photometric phase functions in the coma of comet 67P/Churyumov-Gerasimenko. The variations in the measured phase functions in time can be explained by varying dust size distribution in the coma. Please click here to view a larger version of this figure.

Figure 13: Phase functions. Modeled and measured phase functions of the nucleus of comet 67P. Please click here to view a larger version of this figure.