Tidal disruptions: A continuum theory for solid bodies

Although the theory of Roche 1847 for the tidal disruption limits of orbiting satellites assumes a fluid body, a length to diameter of exactly 2.07:1, and a particular body orientation, the theory is commonly applied to the satellites of the Solar System and to small asteroids and comets passing nearby a planet. Clearly these bodies are neither fluid nor generally are that elongated, so a more appropriate theory is needed. Here we present exact analytical results for the distortion and disruption limits of solid spinning ellipsoidal bodies subjected to tidal forces, using the Drucker-Prager strength model with zero cohesion. It is the appropriate model for dry granular materials such as sands and rocks, for rubble-pile asteroids and comets, and for larger satellites, asteroids and comets where the cohesion can be ignored. This study uses the same approach as the studies of spin limits for solid ellipsoidal bodies given in [Holsapple, K.A., 2001. Icarus 154, 432-448; Holsapple, K.A., 2004. Icarus 172, 272-303]. It is a static theory that predicts conditions for breakup and predicts the nature of the deformations at the limit state, but does not track the dynamics of the body as it comes apart. The strength is characterized by a single material parameter associated with an angle of friction, which can range from zero to 90°. The case with zero friction angle has no shear strength whatsoever, so it is then the model of a fluid or gas. The case of 90° represents a material that cannot fail in shear, but still has zero tensile strength. Typical dry soils have angles of friction of 30°-40°. Since the static fluid case is included in the theory as a special case, the classical results of Roche [Roche, E.A., 1847. Acad. Sci. Lett. Montpelier. Mem. Section Sci. 1, 243-262] and Jeans [Jeans, J.H., 1917. Mem. R. Astron. Soc. London 62, 1-48] are included and re-derived in their entirety; but the general solid case has much more variety and applicability. We consider both the spin-locked case, appropriate for most satellites of the Solar System; and the zero spin case, a possible case for a passing stray body. Detailed plots of many special cases are presented, in terms of shape, orientation and mass densities. A very typical result gives a closest approach d=1.5(ρ/ρP)^1/3R in terms of the planet radius R, and the satellite and planet mass densities ρ and ρP. We also use the theory to distinguish between conditions allowing global shape changes leading to new equilibrium states, or those leading to complete disruption. We apply the theory to the potentially hazardous Asteroid 99942 Apophis due to pass very near the Earth in 2029, and conclude it is extremely unlikely to experience any tidal readjustments during its passage. The states of many of the satellites of the Solar System are compared to the theory, and we find that all are well within their tidal disruption limits for expected values of the internal friction.     

Catastrophic disruption of asteroids and family formation: a review of numerical simulations including both fragmentation and gravitational reaccumulations

In the last few years, thanks to the development of sophisticated numerical codes, a major breakthrough has been achieved in our understanding of the processes involved in small body collisions. In this review, we summarize the most recent results provided by numerical simulations, accounting for both the fragmentation of an asteroid and the gravitational interactions of the generated fragments. These studies have greatly improved our knowledge of the mechanisms that are at the origin of some observed features in the asteroid belt. In particular, the simulations have demonstrated that, for bodies larger than several kilometers, the collisional process not only involves the fragmentation of the asteroid but also the gravitational interactions between the ejected fragments. This latter mechanism can lead to the formation of large aggregates by gravitational reaccumulation of smaller fragments, and helps explain the presence of large members within asteroid families. Numerical simulations of the complete process have thus reproduced successfully for the first time the main properties of asteroid families, each formed by the disruption of a large parent body, and provided information on the possible internal structure of the parent bodies. A large amount of work remains necessary, however, to understand in deeper detail the physical process as a function of material properties and internal structures that are relevant to asteroids, and to determine in a more quantitative way the outcome properties such as fragment shapes and rotational states.     

Interpreting photometry of regolith-like surfaces with different topographies: shadowing and multiple scattering

The effects of various types of topography on the shadow-hiding effect and multiple scattering in particulate surfaces are studied. Two bounding cases were examined: (1) the characteristic scale of the topography is much larger than the surface particle size, and (2) the characteristic scale of the topography is comparable to the surface particle size. A Monte Carlo ray-tracing method (i.e., geometric optics approximation) was used to simulate light scattering. The computer modeling shows that rocky topographies generated by randomly distributed stones over a flat surface reveal much steeper phase curves than surface with random topography generated from Gaussian statistics of heights and slopes. This is because rocks may have surface slopes greater than 90°. Consideration of rocky topography is important for interpreting rover observations. We show the roughness parameter in the Hapke model to be slightly underestimated for bright planetary surfaces, as the model neglects multiple scattering on large-scale topographies. The multiple scattering effect also explains the weak spectral dependences of the roughness parameter in Hapke's model found by some authors. Multiple scattering between different parts of a rough surface suppresses the effect of shadowing, thus the effects produced by increases in albedo on the photometric behavior of a surface can be compensated for with the proper decreases in surface roughness. This defines an effective (photometric) roughness for a surface. The interchangeability of albedo and roughness is shown to be possible with fairly high accuracy for large-scale random topography. For planetary surfaces that have a hierarchically arranged large-scale random topography, predictions made with the Hapke model can significantly differ from real values of roughness. Particulate media with surface borders complicated by Gaussian or clumpy random topographies with characteristic scale comparable to the particle size reveal different photometric behaviors in comparison with particulate surfaces that are flat or the scale of their topographies is much larger than the particle size.     

On the automatic parameter calibration of a hypoplastic soil model

Acta Geotechnica - This paper presents an approach for the automatic parameter calibration (AC) of a hypoplastic constitutive soil model. The calibration software developed in this work simplifies...     

TM/ETM+/LDCM Images for Studying Land Surface Temperature (LST) Interplay with Impervious Surfaces Changes over Time Within the Douala Metropolis,Cameroon

Douala, the most important metropolis of Cameroon, is a sub-Saharan wet coastal environment of which the anarchic urbanization is a socio-economic and environmental problem, significantly influencing the local climate. In this study, three Landsat images from 1986 (TM), 2007 (ETM+) and 2016 (LDCM), were utilized to investigate the effect of this urbanization on the increasing land surface temperature (LST) between these dates. Thus, the urban indices (UI), determined from the Landsat Visible and NIR channels were used to identify impervious areas (Urban Fabric and bare soil) of urban area. It has been shown from the UI images that, impervious areas have been increased from 1986 to 2016. The LST images derived have a continual expansion of zones and points of heat throughout these dates. The correlation analysis of LST and UI, at the pixel-scale, indicated the positive relationship between these parameters, which could show a real impact of urbanization on the increasing temperature in the area. These correlations are fairly low in 1986 (maximum R-square value is about 0.35) and in 2007 (maximum R-square value is about 0.44. In 2016, a high positive correlation (maximum R-square value is about 0.77) confirm that, the impervious areas strengthen the temperature and the Urban Heat Island effect in Douala urban zone. Overall, the earth observation images and the geographic information system techniques were effective approaches for aiming at environment monitoring and analyzing urban growth patterns and evaluating their impacts on urban climates.     

Lithium enrichment and isotopic variation in minerals from peridotite xenoliths from northwestern Ethiopian plateau

We report Lithium (Li) concentrations and isotopic compositions for co-existing olivine, orthopyroxene (opx), and clinopyroxene (cpx) mineral separates from depleted and metasomatised peridotite xenoliths hosted by basaltic lavas from northwestern Ethiopian plateau (Gundeweyn area). The peridotites contain five lherzolites and one harzburgite and are variably depleted and enriched in LREE relative to HREE. In both depleted and enriched lherzolites, Li is preferentially incorporated into olivine (2.4-3.3 ppm) compared to opx (1.4-2.1 ppm) and cpx (1.4-2.0 ppm) whereas the Li contents of olivines (5.4 ppm) from an enriched harzburgiteare higher than those of lherzolites. Olivines from the samples show higher Li abundances than normal mantle olivines (1.6-1.9 ppm) indicating the occurrence of Li enrichments through melt-preroditite interaction. The average δ⁷ Li values range from +2.2 to +6.0‰ in olivine, from -0.1 to +2.0‰ in opx and from -4.4 to -0.9‰ in cpx from the lherzolites. The Li isotopic composition (3.5‰) of olivines from harzburgite fall within the range of olivine from lherzolites but the opxs show low in δ⁷Li (-2.0‰). Overall Li isotopic compositions of olivines from the peridotites fall within the range of normal mantle olivine, δ⁷Li values of ~+4±2‰ within uncertainty, reflecting metasomatism (enrichment) of the peridotites by isotopically heavy Li-rich asthenospheric melt. Li isotope zonation is also observed in most peridotite minerals. Majority of olivine grains display isotopically heavy cores and light rims and the reverse case is observed for some olivine grains. Orthopyroxene and clinopyroxene grains show irregular distribution in δ⁷Li. These features of Li isotopic compositions within and between grains in the samples reflect the effect of diffusion-driven isotopic fractionation during meltperidotite interaction and cooling processes.     

Exsolution Lamellae in Olivine Grains of Dunite Units from Different Types of Mafic‐Ultramafic Complexes

Exsolution microstructures in olivine grains from dunite units in a few selected tectonic environments are reported here. They include lamellae of clinopyroxene and clinopyroxene-magnetite intergrowth in the Gaositai and Yellow Hill Alaskan-type complexes, clinopyroxene-magnetite intergrowth in the K?z?lda? ophiolite, and chromite lamellae in the Hongshishan mafic-ultramafic intrusive complex. These lamellae commonly occur as needle-or rod-like features and are oriented in olivine grains. The host olivine grains have Fo contents of 92.5–92.6 in the Gaositai complex, 86.5–90.1 in the Yellow Hill complex, 93.2–93.4 in the K?z?lda? ophiolite and 86.9–88.3 in the Hongshishan complex. Clinopyroxene in the rod-like intergrowth exsolved in olivine grains in the Gaositai and Yellow Hill is diopside with similar major element compositions of Ca O(23.6–24.3 wt%), SiO_2(52.2–54.0 wt%), Al_2O_3(0.67–2.15 wt%), Cr_2O_3(0.10–0.42 wt%) and Na_2O(0.14–0.26 wt%). It falls into the compositional field of hydrothermal clinopyroxene and its origin is thus probably related to reaction between dunite and fluids. The enrichment of the fluids in Ca~(2+), Fe~(3+), Cr~(3+) and Na+, resulted in elevated concentrations of these cations in olivine solid solutions via the reaction. With decreasing temperature, the olivine solid solutions altered to an intergrowth of magnetite and clinopyroxene. The Fe~(3+) and Cr~(3+) preferentially partitioned into magnetite, while Ca~(2+) and Na+ entered clinopyroxene. Since the studied Alaskan-type complexes and ophiolite formed in a subduction environment, the fluids were probably released from the subducted slab. In contrast, the exsolved chromite in olivine grains from the Hongshishan complex that formed in post-orogenic extension setting can be related to olivine equilibrated with Cr-bearing liquid. Similarly, these lamellae have all been observed in serpentine surrounding olivine grains, indicating genetic relations with serpentinization.     

A Thickness Gauge for the Lithosphere Based on Ce/Yb and Sm/Yb of Mantle–Derived Magmatic Rocks

A new method for determining?the partial melting depth of mantle-derived magma and lithospheric thickness in continental regions is derived from REE geochemistry. This effective technique uses variations in the Ce/Yb and Sm/Yb ratios found in mainly volcanic rocks in continental China. The ratios change with the depth of origin consistent with the correlation between lithospheric thickness and the Ce/Yb and Sm/Yb ratios found in oceanic basalt. These ratios increase exponentially with the depth of origin, the lithospheric thickness, of a wide variety of Cenozoic volcanic basalt and Paleozoic kimberlite in the North China Craton, northeastern China continent and vicinity. This functional relationship with depth is shown in a plot of the ratios that forms a concordia curve, which is closely expressed by formulas using 8–degree polynomials. These provide a more accurate gage in measuring the lithospheric thickness than the traditional geophysical methods. When applied to volcanic rock of different ages it also reveals how the thickness has changed over time and thus, greatly aids the understanding of the tectonic history. Relations between the CO2 content, mineral reactions and pressure in the upper asthenosphere beneath the base of the lithosphere appears to affect the proportions of REE in partial melts and brings about a close correlation between lithospheric thickness and the Ce/Yb and Sm/Yb ratios in mantle–derived magmatic rock. This thickness gauge, for both continental and oceanic lithosphere, provides a new approach in analyzing the lithospheric thickness in different tectonic settings and geologic times.