On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a nonzero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet, it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term “quasi-satellite” more and more present in the literature. However, some authors rather use the term “retrograde satellite” when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper, we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.     

On the co-orbital motion of two planets in quasi-circular orbits

We develop an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. The Hamiltonian, averaged over one of the planetary mean longitudes, is expanded in power series of eccentricities and inclinations. The model, which is valid in the entire co-orbital region, possesses an integrable approximation modeling the planar and quasi-circular motions. First, focusing on the fixed points of this approximation, we highlight relations linking the eigenvectors of the associated linearized differential system and the existence of certain remarkable orbits like the elliptic Eulerian Lagrangian configurations, the anti-Lagrange (Giuppone et al. in MNRAS 407:390–398, 2010) orbits and some second sort orbits discovered by Poincaré. Then, the variational equation is studied in the vicinity of any quasi-circular periodic solution. The fundamental frequencies of the trajectory are deduced and possible occurrence of low order resonances are discussed. Finally, with the help of the construction of a Birkhoff normal form, we prove that the elliptic Lagrangian equilateral configurations and the anti-Lagrange orbits bifurcate from the same fixed point $L_4$ L 4 .     

Evolution of the Axial Geometry of the Southwest Indian Ocean Ridge between the Melville Fracture Zone and the Indian Ocean Triple Junction – Implications for Segmentation on Very Slow-Spreading Ridges

Geophysical data from 900 km of the Southwest Indian Ridge are used todescribe the pattern of evolution of the plate boundary between 61° Eand 70° E over the past 20 million years. The SWIR is anobliquely-opening, ultra slow-spreading axis, and east of61° E comprises a series of ridge sections, each about 100–120 kmin length. The orientation of these sections varies fromsub-orthogonal to oblique to the approximately N–S spreadingdirection. In general, the suborthogonal sections are shallower, commonlysubdivided into an array of discrete axial segments, and carry recognisablecentral magnetic anomalies. The majority of the oblique sections are single,continuous rifts without continuous axial magnetic signatures.Morphotectonics of the Southwest Indian Ridge crust have not previously beenwell constrained off-axis, and we here present sidescan sonar andswath bathymetric data up to 100 km from the ridge to demonstrate the complexities of its spatial and temporal evolution.A model is proposed that the segmentation style correlates with analong-axis variation between: (a) relatively thick crustal sections which overlie mantle sections with higher magmatic supply created in orthogonally-spreading segments and (b) those oblique sections associated with cooler, magmatically-starved mantle and thinner crust. These latter sections are formed at broad offset zones in theplate boundary, more precisely defined on faster-spreading ridges asnontransform discontinuities. The nonsystematic pattern of crustalconstruction, extensional basin formation and the absence of extension-parallel traces of discontinuities off-axis suggest that the oblique spreading sections are not fixed in space or time.     

Propagation of the Southwest Indian Ridge at the Rodrigues Triple Junction

The analysis of multibeam bathymetric data of the Southwest Indian Ridge(SWIR) domain between the triple junction traces from 68° E to theRodrigues Triple Junction (RTJ; 70° E) reveals the evolution of thisridge since magnetic anomaly 4 (8 Ma). Image processing has been used toshow that the horizontal component of strain due to a network of normal stepfaults increases dramatically between 69°30 E and the RTJ. Thisarea close to the RTJ is characterized by a deep graben at the foot of thetriple junction trace on the African plate and by a narrow fault-boundedridge that joins an offset of the trace on the Antarctic plate. In thatarea, spreading is primarily amagmatic and dominated by tectonic extensionprocesses. To the west of 69°30 E, some lobate bathymetricfeatures atop of a large topographic high suggest volcanic constructions.Between 68°10 E and 69°25 E the southern flank of theSWIR domain is wider than the northern one and is characterized by a series of 7 en echelon bathymetric highs similar in size,shape and orientation to the one centred at 69°30E near the present-day triple junction. Their en echelon organization along the triple junction trace on the Antarctic plate and the typical lack of conjugated parts on the northern flank show that these bathymetric highs have been shifted to the south by successive northward relocalisations of the SWIR rifting zone. This evolution results in the asymmetric spreading of the SWIR in the survey area. The off-axis bathymetric highs connect to the offsets of the triple junction trace on the Antarctic plate when the Southeast Indian Ridges lightly lengthenstoward the northwest and the triple junction is relocated to the north. We propose that the SWIR lengthens toward the northeast with two propagation modes: 1) a continuous and progressive propagation with distributed deformation in preexisting crust of the Central Indian Ridge, 2) a discontinuous propagation with focusing of the deformation in a rift zone when the triple junction migrates rapidly to the north. The modes of propagation of the SWIR are related to different localisation and distribution of strain which are in turn controlled by changes of the triple junction configurations due to propagation, recession or a symmetric spreading on the Central and Southeast Indian Ridges.     

A low-dimensional subsurface model for saturated and unsaturated flow processes: ability to address heterogeneity

A low-dimensional model that describes both saturated and unsaturated flow processes in a single equation is presented. Subsurface flow processes in the groundwater, the vadose zone, and the capillary fringe are accounted for through the computation of aggregated hydrodynamic parameters that result from the integration of the governing flow equations from the bedrock to the land surface. The three-dimensional subsurface flow dynamics are thus described by a two-dimensional equation, allowing for a drastic reduction of model unknowns and simplification of the model parameterizations. This approach is compared with a full resolution of the Richards equation in different synthetic test cases. Because the model reduction stems from the vertical integration of the flow equations, the test cases all use different configurations of heterogeneity for vertical cross-sections of a soil-aquifer system. The low-dimensional flow model shows strong consistency with results from a complete resolution of the Richards equation for both the water table and fluxes. The proposed approach is therefore well suited to the accurate reproduction of complex subsurface flow processes.     

Characterization of a major refractory component of marine dissolved organic matter

Refractory carboxyl-rich alicyclic molecules (CRAM) are characterized in marine dissolved organic matter (DOM) using nuclear magnetic resonance spectroscopy and ultrahigh resolution mass spectrometry. CRAM are distributed throughout the water column and are the most abundant components of deep ocean DOM ever characterized. CRAM are comprised of a complex mixture of carboxylated and fused alicyclic structures with a carboxyl-C:aliphatic-C ratio of 1:2 to 1:7. CRAM are expected to constitute a strong ligand for metal binding, and multiple coordination across cations could promote aggregation and marine gel formation thereby affecting CRAM reactivity and the bioavailability of nutrients and trace metals. It appears CRAM are ultimately derived from biomolecules with structural similarities to sterols and hopanoids. The occurrence of CRAM in freshwater and terrestrial environments seems likely, considering the global distribution of biomolecules and the similarities of biogeochemical processes among environments.     

XMCD at Fe L<Subscript>2,3</Subscript> edges,Fe and S K edges on Fe<Subscript>7</Subscript>S<Subscript>8</Subscript>

Pyrrhotite (Fe₇S₈) is a natural iron sulphide that can participate in rock magnetisation. Its electronic structure is not yet surely described. X-ray magnetic circular dichroism (XMCD) at Fe L_2,3 edges on Fe₇S₈, coupled with multiplet calculations, shows that iron is present only as Fe²⁺ in this magnetic iron sulphide. It reveals a strong magnetic orbital moment. XMCD at Fe and S K edges shows the quite strong polarization of both Fe and S in Fe₇S₈.     

Etude des Déformations et Inclinaisons du Sous-sol: Développements Instrumentaux et Applications dans la Région Sismique d’Arette,France

We present a modified version of a tilt and strain meter conceived to have great sensibility and good stability. These instruments were deployed in a cave at the seismic Pyrenean region of Arette (France). We set in evidence several types of phenomena, of which deformations brought about by water recharge filling up fissures on the massif are of importance. Owing to the redundance in the data, the study of the co-seismic deformations allows one to separate instrumental displacements and local readjustments from regional motions.     

Eurocode 8-compatible synthetic time-series as input to dynamic analysis

Nonlinear dynamic analysis of existing or planned structures often requires the use of accelerograms that match a target design spectrum. Here, our main concern is to generate a set of motions with a good level of fit to the Eurocode 8 design spectra for France. Synthetic time series are generated by means of a non-stationary stochastic method. To calibrate the input parameters in the stochastic approach, we select a reference set of accelerograms for a Eurocode 8 type B site category from the PEER Ground-Motion Database, which are then adjusted to the target spectrum through wavelet addition. Then, we compute nonlinear seismic responses of a soil column, including pore pressure effects, and brittle and ductile structures to the stochastic time-series, the natural accelerograms and time-series generated using stationary stochastic approaches. The results of these calculations reveal considerable variability in response despite the similarities in terms of spectral acceleration.