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 non zero 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.Comment: 30 pages, 13 figures, 1 tabl

The family of Quasi-satellite periodic orbits in the circular co-planar RTBP

In the circular case of the coplanar Restricted Three-body Problem, we studied how the family of quasi-satellite (QS) periodic orbits allows to define an associated libration center. Using the averaged problem, we highlighted a validity limit of this one: for QS orbits with low eccentricities, the averaged problem does not correspond to the real problem. We do the same procedure to L 3 , L 4 and L 5 emerging periodic orbits families and remarked that for very high eccentricities F L4 and F L5 merge with F L3 which bifurcates to a stable family

A Comparison of TSV Etch Metrology Techniques

International audienceWe use three metrology techniques, vertical scanning interferometry (VSI), confocal chromatic microscopy (CCM), and time domain optical coherence tomography (TD-OCT), for depth measurement of through-silicon vias (TSVs) of various cross sections and depths. The merits of these techniques are discussed and compared. Introduction While sales of semiconductor equipment broke a new record this year, many metrology needs should be addressed to support the development and production of electronic chips based on "More than Moore" scaling. Among these scaling approaches, 3D integration based on TSVs offers superior integration density and reduces interconnect length/latency. Measurements are needed to evaluate the depth uniformity of etched TSVs. Indeed, upon metal filling, geometrical variations of TSVs can affect Cu nails coplanarity and can warp the wafer, resulting in a low stacking yield. Measuring the depth of TSVs is an increasingly challenging task as the diameter of many TSVs has now shrunk to only a few microns

GABA Receptors and the Pharmacology of Sleep

Current GABAergic sleep-promoting medications were developed pragmatically, without making use of the immense diversity of GABAA receptors. Pharmacogenetic experiments are leading to an understanding of the circuit mechanisms in the hypothalamus by which zolpidem and similar compounds induce sleep at α2βγ2-type GABAA receptors. Drugs acting at more selective receptor types, for example, at receptors containing the α2 and/or α3 subunits expressed in hypothalamic and brain stem areas, could in principle be useful as hypnotics/anxiolytics. A highly promising sleep-promoting drug, gaboxadol, which activates αβδ-type receptors failed in clinical trials. Thus, for the time being, drugs such as zolpidem, which work as positive allosteric modulators at GABAA receptors, continue to be some of the most effective compounds to treat primary insomnia

Coorbital resonance dynamics : an introduction to Quasi-satellite motion

International audienc

Coorbital motion in the co-planar RTBP: family of Quasi-satellite periodic orbits

International audienceIn the framework of the Restricted Three-body Problem (RTBP), we consider a primary whose mass is equal to one, a secondary on circular or eccentric motion with a mass # and a massless third body. The three bodies are in coplanar motion and in co-orbital resonance. We actually know three classes of regular coorbital motions: in rotating frame with the secondary, the tadpole orbits (TP) librate around Lagrangian equilibria L4 or L5; the horseshoe orbits (HS) encompass the three equilibrium points L3, L4 and L5; the quasi-satellite orbits (QS) are remote retrograde satellite around the secondary, but outside of its Hill sphere. Contrarily to TP orbits which emerge from a fixed point in rotating frame, QS orbits emanate from a oneparameter family of periodic orbits, denoted family-f by Henon (1969). In the averaged problem, this family can be understood as a family of fixed points. However, the eccentricity of these orbits can reach high values. Consequently a development in eccentricity will not be efficient. Using the method developed by Nesvorny et al. (2002) which is valid for every values of eccentricity, we study the QS periodic orbits family with a numerical averaging. In the circular case, I will present the validity domain of the average approximation and a particular orbit. Then, I will highlight an unexpected result for very high eccentricity on families of periodic orbits that originate from L3, L4 and L5. Finally, I will sketch out an analytic method adapted to QS motion and exhibit associated results in the eccentric case

Coorbital motion in the co-planar RTBP: family of Quasi-satellite periodic orbits

International audienceIn the framework of the Restricted Three-body Problem (RTBP), we consider a primary whose mass is equal to one, a secondary on circular or eccentric motion with a mass # and a massless third body. The three bodies are in coplanar motion and in co-orbital resonance. We actually know three classes of regular coorbital motions: in rotating frame with the secondary, the tadpole orbits (TP) librate around Lagrangian equilibria L4 or L5; the horseshoe orbits (HS) encompass the three equilibrium points L3, L4 and L5; the quasi-satellite orbits (QS) are remote retrograde satellite around the secondary, but outside of its Hill sphere. Contrarily to TP orbits which emerge from a fixed point in rotating frame, QS orbits emanate from a oneparameter family of periodic orbits, denoted family-f by Henon (1969). In the averaged problem, this family can be understood as a family of fixed points. However, the eccentricity of these orbits can reach high values. Consequently a development in eccentricity will not be efficient. Using the method developed by Nesvorny et al. (2002) which is valid for every values of eccentricity, we study the QS periodic orbits family with a numerical averaging. In the circular case, I will present the validity domain of the average approximation and a particular orbit. Then, I will highlight an unexpected result for very high eccentricity on families of periodic orbits that originate from L3, L4 and L5. Finally, I will sketch out an analytic method adapted to QS motion and exhibit associated results in the eccentric case

Coorbital resonance dynamics : an introduction to Quasi-satellite motion

International audienc

Coorbital motion in the co-planar RTBP: family of Quasi-satellite periodic orbits

International audienceIn the framework of the Restricted Three-body Problem (RTBP), we consider a primary whose mass is equal to one, a secondary on circular or eccentric motion with a mass # and a massless third body. The three bodies are in coplanar motion and in co-orbital resonance. We actually know three classes of regular coorbital motions: in rotating frame with the secondary, the tadpole orbits (TP) librate around Lagrangian equilibria L4 or L5; the horseshoe orbits (HS) encompass the three equilibrium points L3, L4 and L5; the quasi-satellite orbits (QS) are remote retrograde satellite around the secondary, but outside of its Hill sphere. Contrarily to TP orbits which emerge from a fixed point in rotating frame, QS orbits emanate from a oneparameter family of periodic orbits, denoted family-f by Henon (1969). In the averaged problem, this family can be understood as a family of fixed points. However, the eccentricity of these orbits can reach high values. Consequently a development in eccentricity will not be efficient. Using the method developed by Nesvorny et al. (2002) which is valid for every values of eccentricity, we study the QS periodic orbits family with a numerical averaging. In the circular case, I will present the validity domain of the average approximation and a particular orbit. Then, I will highlight an unexpected result for very high eccentricity on families of periodic orbits that originate from L3, L4 and L5. Finally, I will sketch out an analytic method adapted to QS motion and exhibit associated results in the eccentric case