The Quantizer problem is a tessellation optimisation problem where point configurations are identified such that the Voronoi cells minimise the second moment of the volume distribution. While the ground state (optimal state) in 3D is almost certainly
the body-centered cubic lattice, disordered and effectively hyperuniform states with energies very close to the ground state exist that result as stable states in an evolution through the geometric Lloyds algorithm [Klatt et al. Nat. Commun., 10, 811 (2019)]. When considered as a statistical mechanics problem at finite temperature, the same system has been termed the Voronoi Liquid by [Ruscher et al. EPL 112, 66003 (2015)]. Here we investigate the cooling behaviour of the Voronoi liquid with a particular view to the stability of the effectively hyperuniform disordered state. As a confirmation of the results by Ruscher et al., we observe, by both molecular dynamics and Monte Carlo simulations, that upon slow quasi-static equilibrium cooling, the Voronoi liquid crystallises from a disordered configuration into the body-centered cubic configuration. By contrast, upon sufficiently fast non-equilibrium cooling (and not just in the limit of a maximally fast quench) the Voronoi liquid adopts similar states as the effectively hyperuniform inherent structures identified by Klatt et al. and prevents the ordering transition into a BCC ordered structure. This result is in line with the geometric intuition that the geometric Lloyds algorithm corresponds to a type of fast quench.
Dawns framing camera observed boulders on the surface of Vesta when the spacecraft was in its lowest orbit (LAMO). We identified, measured, and mapped boulders in LAMO images, which have a scale of 20 m per pixel. We estimate that our sample is virtu
ally complete down to a boulder size of 4 pixels (80 m). The largest boulder is a 400 m-sized block on the Marcia crater floor. Relatively few boulders reside in a large area of relatively low albedo, surmised to be the carbon-rich ejecta of the Veneneia basin, either because boulders form less easily here or live shorter. By comparing the density of boulders around craters with a known age, we find that the maximum boulder lifetime is about 300 Ma. The boulder size-frequency distribution (SFD) is generally assumed to follow a power law. We fit power laws to the Vesta SFD by means of the maximum likelihood method, but they do not fit well. Our analysis of power law exponents for boulders on other small Solar System bodies suggests that the derived exponent is primarily a function of boulder size range. The Weibull distribution mimics this behavior and fits the Vesta boulder SFD well. The Weibull distribution is often encountered in rock grinding experiments, and may result from the fractal nature of cracks propagating in the rock interior. We propose that, in general, the SFD of particles (including boulders) on the surface of small bodies follows a Weibull distribution rather than a power law.
We report a comprehensive analysis of the global spectrophotometric properties of Ceres using Dawn Framing Camera images collected from April to June 2015 during the RC3 and Survey mission phases. The single-scattering albedo of Ceres at 555 nm is 0.
14$pm$0.04, the geometric albedo is 0.096$pm$0.006, and the Bond albedo is 0.037$pm$0.002. The asymmetry factors calculated from the best-fit two-term Henyey-Greenstein (HG) single-particle phase function (SPPF) show a wavelength dependence, suggesting that the phase reddening of Ceres is dominated by single-particle scattering rather than multiple scattering or small-scale surface roughness. The Hapke roughness parameter of Ceres is derived to be 20$^circpm$6$^circ$ with no wavelength dependence. The phase function of Ceres shows appreciably strong scattering around 90$^circ$ phase angle that cannot be fitted with a single-term HG SPPF, suggesting possible stronger forward scattering than other asteroids previously analyzed with spacecraft data. We speculate that such a scattering characteristic of Ceres might be related to its unique surface composition. We grouped the reflectance data into a 1$^circ$ latitude-longitude grid and fitted each grid independently to study the spatial variations of photometric properties. The albedo and color maps are consistent with previous studies. The SPPF over the surface of Ceres shows stronger backscattering associated with lower albedo and vice versa, consistent with the general trend among asteroids. The Hapke roughness parameter does not vary much across the surface of Ceres, except for the ancient Vendimia Planitia region that has a slightly higher roughness. Based on the wavelength dependence of the SPPF of Ceres, we hypothesize that its regolith grains either contain a considerable fraction of $lessapproxmu$m-sized particles, or are strongly affected by internal scatterers of this size.
The surface reflectance of planetary regoliths may increase dramatically towards zero phase angle, a phenomenon known as the opposition effect (OE). Two physical processes that are thought to be the dominant contributors to the brightness surge are s
hadow hiding (SH) and coherent backscatter (CB). The occurrence of shadow hiding in planetary regoliths is self-evident, but it has proved difficult to unambiguously demonstrate CB from remote sensing observations. One prediction of CB theory is the wavelength dependence of the OE angular width. The Dawn spacecraft observed the OE on the surface of dwarf planet Ceres. We characterize the OE over the resolved surface, including the bright Cerealia Facula, and to find evidence for SH and/or CB. We analyze images of the Dawn framing camera by means of photometric modeling of the phase curve. We find that the OE of most of the investigated surface has very similar characteristics, with an enhancement factor of 1.4 and a FWHM of 3{deg} (broad OE). A notable exception are the fresh ejecta of the Azacca crater, which display a very narrow brightness enhancement that is restricted to phase angles $< 0.5${deg} (narrow OE); suggestively, this is in the range in which CB is thought to dominate. We do not find a wavelength dependence for the width of the broad OE, and lack the data to investigate the dependence for the narrow OE. The prediction of a wavelength-dependent CB width is rather ambiguous. The zero-phase observations allow us to determine Ceres visible geometric albedo as $p_V = 0.094 pm 0.005$. A comparison with other asteroids suggests that Ceres broad OE is typical for an asteroid of its spectral type, with characteristics that are primarily linked to surface albedo. Our analysis suggests that CB may occur on the dark surface of Ceres in a highly localized fashion.
We present a global spectrophotometric characterization of the Ceres surface using Dawn Framing Camera (FC) images. We identify the photometric model that yields the best results for photometrically correcting images. Corrected FC images acquired on
approach to Ceres were assembled into global maps of albedo and color. Generally, albedo and color variations on Ceres are muted. The albedo map is dominated by a large, circular feature in Vendimia Planitia, known from HST images (Li et al., 2006), and dotted by smaller bright features mostly associated with fresh-looking craters. The dominant color variation over the surface is represented by the presence of blue material in and around such craters, which has a negative spectral slope over the visible wavelength range when compared to average terrain. We also mapped variations of the phase curve by employing an exponential photometric model, a technique previously applied to asteroid Vesta (Schroder et al., 2013b). The surface of Ceres scatters light differently from Vesta in the sense that the ejecta of several fresh-looking craters may be physically smooth rather than rough. High albedo, blue color, and physical smoothness all appear to be indicators of youth. The blue color may result from the desiccation of ejected material that is similar to the phyllosilicates/water ice mixtures in the experiments of Poch et al. (2016). The physical smoothness of some blue terrains would be consistent with an initially liquid condition, perhaps as a consequence of impact melting of subsurface water ice. We find red terrain (positive spectral slope) near Ernutet crater, where De Sanctis et al. (2017) detected organic material. The spectrophotometric properties of the large Vendimia Planitia feature suggest it is a palimpsest, consistent with the Marchi et al. (2016) impact basin hypothesis. The central bright area in Occator crater, Cerealia...
A scheme within density functional theory is proposed that provides a practical way to generalize to unrestricted geometries the method applied with some success to layered geometries [H. Rydberg, et al., Phys. Rev. Lett. 91, 126402 (2003)]. It inclu
des van der Waals forces in a seamless fashion. By expansion to second order in a carefully chosen quantity contained in the long range part of the correlation functional, the nonlocal correlations are expressed in terms of a density-density interaction formula. It contains a relatively simple parametrized kernel, with parameters determined by the local density and its gradient. The proposed functional is applied to rare gas and benzene dimers, where it is shown to give a realistic description.
