Solutions périodiques pour la resonance 4:1:4 dans les systèmes hamiltoniens a trois degrés de liberte

Resume On étudie l'existence de solutions périodiques pour un couplage d'oscillateurs harmoniques pour la resonance 4:4:1. Les résultats théoriques appliqués a la forme particulière de l'Hamiltonien permettent d'affirmer qu'il existe au moins une trajectoire pérodique pour une valeur p < µ₀ caractérisant le couplage et au moins deux trajectories pérodiques distinctes pour 0 < p < µ₁ < µ₀.

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The existence of periodic solutions for coupled oscillators with resonance 4:1 : 4 are investigated. The theoretical results allow the prediction of a periodic solution for p < µ₀, where p is the parameter describing the coupling, and two distinct periodic trajectories if 0 <_ p < pr < µ₀. The evaluation of µ₁ and µ₀ depends on the energy h of the system.

     

Sur les solutions Lagrangiennes du problème des trois corps solides avec la loi de Weber

In this work we consider the problem of translational-rotational motion of three solid bodies, for which the elementary particles attract each other according to different Weber's laws for each pair of bodies. This problem represents a special case of the generalized problem of three solids considered in a previous work, (Dubochin, 1974) and it gives an example of the verification of the existence conditions for the Lagrangian solutions. In these solutions, the centers of mass always for m an equilateral triangle. Each body has axial symmetry with the plane of symmetry perpendicular to the axis of symmetry rotates uniformly around this axis, which at any instant stays perpendicular to the plane of the triangle formed by the centers of mass. According to Weber's law (Tisserand, 1896) the elementary particles of two bodiesT _i andT _j (i, j=0, 1, 2) are attracted by forces which are proportional to the function $$F_{ij} (W) = \frac{{f_{ij} }}{{\Delta _{ij^2 } }}\left[ {1 - a_{ij} \dot \Delta _{ij^2 } + 2a_{ij} \Delta _{ij} \ddot \Delta _{ij} } \right]$$ wheref _ij anda _ij (in generalf _ji ≠f _ij anda _ji ≠a _ij ) are functions of the timet, and where the real quantities Δ_ij are the mutual distances between the particles of the bodiesT _i andT _j , and where \(\dot \Delta _{ij} \) and \(\ddot \Delta _{ij} \) are their derivatives with respect to the time. The analysis of the general conditions for the Lagrangian solutions gives the following results for the case of Weber's laws.

1. Only the invariant Lagrangian solutions, (the traingle of the centres of mass does not change in time) are possible in this problem.
2. Besides the conditions (NL) obtained in the case of the Newton-Coulomb law, (all thea _ij are zero), the complementary conditions (WL) must be satisfied.

In particular, if all the bodies are spheres or homogeneous ellipsoids, they must necessarily have the same dimensions, but they can have different masses.     

Un modele bidimensionnel du comportement de l'ozone dans la stratosphere

In order to study the behaviour of stratospheric minor constituents related to aeronomic processes and atmospheric transport in the meridional plane, a numerical two-dimensional model is established. The stratospheric dynamics is parametrized by mean motions and large scale eddy diffusion.This model is applied to the study of ozone in a nitrogen-hydrogen-oxygen atmosphere. The analysis of the various results indicates that the distribution of O₃ is related to the existence of a countergradient flux which is responsible for the transfer of these molecules from the equatorial regions where they are produced to the polar regions where they accumulate during the winter season. The model also shows the combined action of nitrogen and hydrogen compounds on the ozone layer. The particular role of nitric acid on the stability of stratospheric ozone is discussed.     

Etude de la collisionN-uple du probleme desN corps soumis a un potentiel homogene de degre-k aveck>0 cas ouk=2

Rèsumé Le changement de variables simple et adapté au problème de la collision triple utilisé dans un article précédent [3] était lié étroitement à l'homogénéité de l'énergie cinétiqueT (degré 2) et du potentielU (degré-1). Nous allons généraliser ce changement de variables au cas deU homogène de degré —k (k>0) et en déduire l'étude de la collision pour ces cas, en mettant en évidence un cas exceptionnel,k=2.

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Study of theN-tuple collision on theN-body problem submitted to a homogeneous potential of degree-k in the case ofk>0 ork=2

In a preceding paper, we used a change of variables which is simple and well adapted to the problem of triple collision and which is closely linked the homogeneousness of the kinetic energyT (degree 2) and the potentialU (degree-1). We will now generalize this change of variables to the case whereU is homogeneous of degree —k(k>0), in order to carry out the study of collision in these cases, bringing out the exceptional casek=2.

     

1:1 Ground-track resonance in a uniformly rotating 4th degree and order gravitational field

Using a gravitational field truncated at the 4th degree and order, the 1:1 ground-track resonance is studied. To address the main properties of this resonance, a 1-degree of freedom (1-DOF) system is firstly studied. Equilibrium points (EPs), stability and resonance width are obtained. Different from previous studies, the inclusion of non-spherical terms higher than degree and order 2 introduces new phenomena. For a further study about this resonance, a 2-DOF model which includes a main resonance term (the 1-DOF system) and a perturbing resonance term is studied. With the aid of Poincaré sections, the generation of chaos in the phase space is studied in detail by addressing the overlap process of these two resonances with arbitrary combinations of eccentricity (e) and inclination (i). Retrograde orbits, near circular orbits and near polar orbits are found to have better stability against the perturbation of the second resonance. The situations of complete chaos are estimated in the \(e-i\) plane. By applying the maximum Lyapunov Characteristic Exponent (LCE), chaos is characterized quantitatively and similar conclusions can be achieved. This study is applied to three asteroids 1996 HW1, Vesta and Betulia, but the conclusions are not restricted to them.     

Periodic orbit families near the 4:1 jovian resonance

An enlarged averaged Hamiltonian is introduced to compute several families of periodic orbits of the planar elliptic 3-body problem, in the Sun–Jupiter–Asteroid system, near the 4:1 resonance. Four resonant critical point families are found and their stability is studied. The families of symmetric periodic orbits of the elliptic problem appear near the corresponding fixed points computed in this model. There is a good agreement for moderate eccentricity of the asteroid for three of these families, whereas the remaining family cannot be considered as a family of periodic orbits of the real model. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Villalbeto de la Peña meteorite fall: II. Determination of atmospheric trajectory and orbit

Abstract— The L6 ordinary chondrite Villalbeto de la Peña fall occurred on January 4, 2004, at 16: 46: 45 ± 2 s UTC. The related daylight fireball was witnessed by thousands of people from Spain, Portugal, and southern France, and was also photographed and videotaped from different locations of León and Palencia provinces in Spain. From accurate astrometric calibrations of these records, we have determined the atmospheric trajectory of the meteoroid. The initial fireball velocity, calculated from measurements of 86 video frames, was 16.9 ± 0.4 km/s. The slope of the trajectory was 29.0 ± 0.6° to the horizontal, the recorded velocity during the main fragmentation at a height of 27.9 ± 0.4 km was 14.2 ± 0.2 km/s, and the fireball terminal height was 22.2 ± 0.2 km. The heliocentric orbit of the meteoroid resided in the ecliptic plane (i = 0.0 ± 0.2°), having a perihelion distance of 0.860 ± 0.007 AU and a semimajor axis of 2.3 ± 0.2 AU. Therefore, the meteorite progenitor body came from the Main Belt, like all previous determined meteorite orbits. The Villalbeto de la Peña fireball analysis has provided the ninth known orbit of a meteorite in the solar system.     

The Villalbeto de la Peña meteorite fall: I. Fireball energy,meteorite recovery,strewn field,and petrography

Abstract— An impressive daylight fireball was observed from Spain, Portugal, and the south of France at 16h46m45s UTC on January 4, 2004. The meteoroid penetrated into the atmosphere, generating shock waves that reached the ground and produced audible booms. The associated airwave was recorded at a seismic station located 90 km north of the fireball trajectory in Spain, and at an infrasound station in France located 750 km north‐east of the fireball. The absolute magnitude of the bolide has been determined to be ?18 ± 1 from a casual video record. The energy released in the atmosphere determined from photometric, seismic, and infrasound data was about 0.02 kilotons (kt). A massive fragmentation occurred at a height of 28 ± 0.2 km, resulting in a meteorite strewn field of 20 × 6 km. The first meteorite specimen was found on January 11, 2004, near the village of Villalbeto de la Peña, in northern Palencia (Spain). To date, about 4.6 kg of meteorite mass have been recovered during several recovery campaigns. The meteorite is a moderately shocked (S4) L6 ordinary chondrite with a cosmic‐ray‐exposure age of 48 ± 5 Ma. Radioisotope analysis shows that the original body had a mass of 760 ± 150 kg, which is in agreement with the estimated mass obtained from photometric and seismic measurements.     

New observations on high‐pressure phases in a shock melt vein in the Villalbeto de la Peña meteorite: Insights into the shock behavior of diopside

The petrology and mineralogy of shock melt veins in the L6 ordinary chondrite host of Villalbeto de la Peña, a highly shocked, L chondrite polymict breccia, have been investigated in detail using scanning electron microscopy, transmission electron microscopy, Raman spectroscopy, and electron probe microanalysis. Entrained olivine, enstatite, diopside, and plagioclase are transformed into ringwoodite, low‐Ca majorite, high‐Ca majorite, and an assemblage of jadeite‐lingunite, respectively, in several shock melt veins and pockets. We have focused on the shock behavior of diopside in a particularly large shock melt vein (10 mm long and up to 4 mm wide) in order to provide additional insights into its high‐pressure polymorphic phase transformation mechanisms. We report the first evidence of diopside undergoing shock‐induced melting, and the occurrence of natural Ca‐majorite formed by solid‐state transformation from diopside. Magnesiowüstite has also been found as veins injected into diopside in the form of nanocrystalline grains that crystallized from a melt and also occurs interstitially between majorite‐pyrope grains in the melt‐vein matrix. In addition, we have observed compositional zoning in majorite‐pyrope grains in the matrix of the shock‐melt vein, which has not been described previously in any shocked meteorite. Collectively, all these different lines of evidence are suggestive of a major shock event with high cooling rates. The minimum peak shock conditions are difficult to constrain, because of the uncertainties in applying experimentally determined high‐pressure phase equilibria to complex natural systems. However, our results suggest that conditions between 16 and 28 GPa and 2000–2200 °C were reached.