Introduction: The OSIRIS-REx mission has revealed a dark, boulder-rich, apparently dust-poor surface of the B-type asteroid (101955) Bennu , therefore a challenge for bi-directional reflectance (rF) modeling. With an estimated geometric albedo of 4.5% , Bennu is darker than many comets, and its reflectance distribution is dominated by single-scattering processes. The general approach to model a dark asteroid’s bi-directional reflectance distribution is to apply the standard Hapke IMSA model and its shadowing function . However, this can imprecisely describe the roughness slopes for rocky surfaces . Assuming that surfaces are fully diffuse can negate a specular forward-scattering contribution from crystalline components in the regolith . To achieve a more complete photometric modeling of Bennu’s scattering curve, we rely on the radiative transfer semi-numerical model of Van Ginneken et al. [8,9].
Observations: MapCam is an optical imager  on-board the OSIRIS-REx spacecraft, equipped with four broadband color filters (60-90 nm wide) centered at 473 (b'), 550 (v'), 698 (w') and 847 (x') nm. We analyzed images acquired during the Equatorial Station (EQ), a campaign of the Detailed Survey mission phase , over a full rotation of (101955) Bennu at a nadir spatial resolution of ~33 cm/px. EQ comprehended seven phase angle configurations α =[7.5° ,30° ,45° ,90° ,130°] .
We analyzed the pixel subtended by the mission’s candidate sample collection sites, for which highly precise, laser altimeter-based digital terrain models (DTMs) were available [12,13]. These four candidate sites were called Sandpiper (latitude=-47°; longitude=322°), Osprey (11°; 88°), Nightingale ( 56°; 43°), and Kingfisher (11°; 56°). The varied latitude and longitudes provided the range of observational conditions required for our analysis.
Methodology: The methodology consists of the following steps:
a) NAIF SPICE kernels  and the DTMs are ingested into a ray-tracing code for rendering shadowed images. This allows us to discount for the effects of macroscopic shadows, leaving only the sub-facet texture to be modeled. These ancillary images provide the incidence, emergence, phase & azimuth for every facet.
b) We apply Van Ginneken’s model to every facet. Occlusion and shadowing as well as the retro-reflection among the reliefs are taken into account. The model has two major components: the analytical expression for the specular reflection; and the numerically-integrated diffusive reflection. At total, the model has three free parameters: ρ (single-scattering albedo), σ (RMS roughness slope), and g (specular-to-diffuse ratio), plus the three more related to the scattering phase function (bi-lobe Henyey-Greenstein function: c, b1, and b2)
c) Inverse problem: We run the Monte Carlo Markov Chain to sample the multi-parametric space in order to reconstruct a posteriori probability distribution of solutions for every free parameter, i.e., (ρ, σ, g, b1, b2, c), from which the statistics for every solution are estimated.
Results: The MCMC technique reveals some interesting aspects of Bennu's surface (Fig. 1): while the RMS roughness slope of 27+1-5 is in line with what has been obtained for other asteroids using Hapke shadowing function, we are puzzled by the indication of a non-zero specular reflection ratio from the surface (2.6+1−0.8 %). The specular reflection hints at a possible mono-crystalline component.
As for the diffuse rough component, the analysis of the photometric correction of OCAMS images taken at varied phase angles (α) indicates a more complex scenario. Up to α = 90°, the photometric correction is vastly improved by mixing two different solutions for roughness (one with low RMS σ and another with global RMS σ), a bi-modality already perceived from the MCMC a posteriori distributions. We have shown that most of Bennu's brightness variation can be explained by tuning the roughness slope statistical distribution.
Finally, we report a back-scatter phase function for the phase angle range between 7.5°, and 130°, without any expressive spectral trend in the visible range. The MCMC inversion hints at a possible second forward-scatter lobe of at least ~0.2 width. This leads to two possible solutions for the asymmetric factor (ξ (1) = −0.360 ± 0.030 and ξ(2) = −0.444 ± 0.020). We also report a dark global approximate single-scattering albedo at 550 nm from the collective analysis of all site candidates of 4.64+0.08-0.09 % . The single-scattering MapCam four-band colors show a similar spectral trend to the global average OVIRS EQ3 spectrum. The four sites together provide a general description of Bennu's colors.
Fig. 1. Parametric solutions after the MCMC technique for all sample sites together, and the scattering phase function (bottom row) for each MapCam filters.
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