An asymptotic solution for the stellar case of the planar Three-Body Problem

A simplified model of the planar Three-Body Problem is considered in which two particles, forming a close binary, orbit a distant point. A small parameter , related to the distance separating the binary and the remaining mass is defined. The time is eliminated from the equations of motion and an angular variable is used instead. A two-variable expansion procedure is used to find an asymptotic solution. The solution obtained is known completely up to the order ten in , and it is valid for almost arbitrary initial conditions. Specification of the initial conditions leads to a solution which is periodic with respect to a modified time variable.

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Résumé On considère un modèle simplifé du Problème Plan des Trois Corps, dans lequel deux particules, formant une binaire proche, sont en orbite par rapport à un troisième point éloigné des deux autres. On définit un petit paramètre , lié à la distance séparant la binaire de la particule restante. On élimine le temps des équations du mouvement et on utilise une variable angulaire comme nouvelle variable indépendante. Une méthode de développement à deux échelles est utilisée permettant d'obtenir une solution asymptotique du problème. La solution obtenue est connue complètement jusqu'àl'ordre dix en , et elle est valable pour des conditions initiales très générales. Pour des conditions initiales bien déterminées, on obtient une solution périodique par rapport à une nouvelle variable temporelle.

     

Fracturation d'âge Aptien supérieur sur la bordure orientale de la plate-forme arabique (Haushi-Huqf,Oman)Aptian faulting (Shu'aiba Formation) on the eastern edge of the Arabian Platform (Haushi-Huqf,Oman)

In the Haushi-Huqf (Eastern Central Oman) as in other parts of the Arabian platform, a major sedimentary break is recorded between the Early Aptian carbonates (Shu'aiba Formation) and the Albian orbitolinid-rich marls (Nahr Umr Formation). The unconformity corresponds to a succession of events: (1) a brusque interruption of the regressive sequence of the Shu'aiba limestone (algae and small rudistid build-ups); (2) a stratigraphic gap related to the Late Aptian; (3) the development of a thick ferruginous crust (hardground) that covered the top surface of the Shu'aiba; the hardground is related to a forced flooding surface; (4) the Shu'aiba was rapidly drowned and buried under the Nahr Umr marls. Moreover, the Shu'aiba limestone was subject to faulting NW–SE-trending normal faults before lithification and formation of the ferruginous crust. The faulting episode is clearly dated: post-Early Aptian and pre-Albian. The signification of the faulting remains hypothetical. The syndiagenetic NW–SE normal faults may correspond to ‘en echelon’ faults, combined with transcurrent fault movements (for example the Haushi-Nafun Fault). The possible causes of these intra-platform transcurrent movements are discussed. To cite this article: C. Montenat, P. Barrier, C. R. Geoscience 334 (2002) 781–787.     

On the pair-instability supernovae and gamma-ray burst phenomenon

We present a model of a gamma-ray burst (GRB) related to “very massive” stars. In the framework of our model, the GRB phenomenon is a result of helium burning in degenerate conditions in a massive star (≳130 M _⊙), in which the thermal nuclear burning occurs in the deflagration regime and has a pulsating temporal pattern. The shock runs away from the burning (reaction) zone, which leads to the development of a coronal outflow (jet-like) structure. In our scenario the GRB observable prompt fast rise and decay part can be a result of photon propagation through the hot corona (Comptonization photosphere) of the star. On the other hand, the GRB afterglow is a cooling phase of the expanding and outflowing envelope. Presumably, the X-ray part of the GRB emergent spectrum is formed due to upscattering of soft photons of outer layers of the star off hot coronal electrons, and thus it should have a specific shape of the Comptonization spectrum.     

Wet Snow Avalanche Deposits in the French Alps: Structure and Sedimentology

We analyse the morphology and sedimentology of 25 dirty snow avalanche deposits in the French Alps. The deposits typically have either a snow-ball structure or a massive structure with sliding planes. The snow balls show a longitudinal and a vertical sorting that reflects a sieve effect, similar to that observed in other rapid inertial granular flows. The massive type results from snow compaction when the avalanche is channelled by a gully or when it reaches the distal part of the scree. Velocity decrease and compaction limit the deformation to a zone at the base of the snow mass and cause the formation of distinctive sliding planes. These appear as smooth recrystallised surfaces due to local melt from frictional heating. The flow can be assimilated to a frictional granular flow. No systematic variation of size and shape of the rock debris has been observed along the profiles in both types of deposit. The distribution of rock debris and its fabric suggest that the clasts are transported passively and do not undergo any sorting during displacement. Snow melt after avalanching causes a redistribution of rock debris particularly when the snow thickness is important. This redistribution does not generate new sedimentological characteristics such as enhanced sorting or fabric.     

Towards a 4D topographic view of the Norwegian sea margin

The present-day topography/bathymetry of the Norwegian mainland and passive margin is a product of complex interactions between large-scale tectonomagmatic and climatic processes that can be traced back in time to the Late Silurian Caledonian Orogeny. The isostatic balance of the crust and lithosphere was clearly influenced by orogenic thickening during the Caledonian Orogeny, but was soon affected by post-orogenic collapse including overprinting of the mountain root, and was subsequently affected by a number of discrete extensional events eventually leading to continental break-up in Early Eocene time. In the mid-Jurassic the land areas experienced deep erosion in the warm and humid climate, forming a regional paleic surface. Rift episodes in the Late Jurassic and Early Cretaceous, with differential uplift along major fault zones, led to more pronounced topographic contrasts during the Cretaceous, and thick sequences of clastic sediments accumulated in the subsiding basins on the shelf. Following renewed extension in the Late Cretaceous, a new paleic surface developed in the Paleocene. Following break-up the margin has largely subsided thermally, but several Cenozoic shortening events have generated positive contraction structures. On the western side of the on-shore drainage divide, deeper erosion took place along pre-existing weakness zones, creating the template of the present day valleys and fjords. In the Neogene the mainland and large portions of the Barents Sea were uplifted. It appears that this uplift permitted ice caps to nucleate and accumulate during the Late Pliocene northern hemisphere climatic deterioration. The Late Pliocene to Pleistocene glacial erosion caused huge sediment aprons to be shed on to the Norwegian Sea and Barents Sea margins. Upon removal of the ice load the landmass adjusted isostatically, and this still continues today.     

Subhaloes in scale-free cosmologies

We explore the dependence of the subhalo mass function on the spectral index n of the linear matter power spectrum using scale-free Einstein-de Sitter simulations with   n =−1  and −2.5. We carefully consider finite volume effects that may call into question previous simulations of   n < −2  power spectra. Subhaloes are found using a 6D friends-of-friends algorithm in all haloes originating from high-σ peaks. For   n =−1  , we find that the cumulative subhalo mass function is independent of the parameters used in the subhalo finding algorithm and is consistent with the subhalo mass function found in Λ cold dark matter (ΛCDM) simulations. In particular, the subhalo mass function is well fit by a power-law with an index of  α=−0.9  , that is the mass function has roughly equal mass in subhaloes per logarithmic interval in subhalo mass. Conversely, for   n =−2.5  , the algorithm parameters affect the subhalo mass function since subhaloes are more triaxial with less well-defined boundaries. We find that the index α is generally larger with  α≳−0.75  . We infer that although the subhalo mass function appears to be independent of n so long as   n ≳−2  , it begins to flatten as   n →−3  . Thus, the common practice of using  α≈−1.0  may greatly overestimate the number of subhaloes at the smallest scales in the CDM hierarchy.