Planets in Mean-Motion Resonances and the System Around HD45364

In some planetary systems, the orbital periods of two of its members present a commensurability, usually known by mean-motion resonance. These resonances greatly enhance the mutual gravitational influence of the planets. As a consequence, these systems present uncommon behaviors, and their motions need to be studied with specific methods. Some features are unique and allow us a better understanding and characterization of these systems. Moreover, mean-motion resonances are a result of an early migration of the orbits in an accretion disk, so it is possible to derive constraints on their formation. Here we review the dynamics of a pair of resonant planets and explain how their orbits evolve in time. We apply our results to the HD 45365 planetary system.Comment: invited review, 17 pages, 6 figure

Imaging the symmetry breaking of molecular orbitals in carbon nanotubes

Carbon nanotubes have attracted considerable interest for their unique electronic properties. They are fascinating candidates for fundamental studies of one dimensional materials as well as for future molecular electronics applications. The molecular orbitals of nanotubes are of particular importance as they govern the transport properties and the chemical reactivity of the system. Here we show for the first time a complete experimental investigation of molecular orbitals of single wall carbon nanotubes using atomically resolved scanning tunneling spectroscopy. Local conductance measurements show spectacular carbon-carbon bond asymmetry at the Van Hove singularities for both semiconducting and metallic tubes, demonstrating the symmetry breaking of molecular orbitals in nanotubes. Whatever the tube, only two types of complementary orbitals are alternatively observed. An analytical tight-binding model describing the interference patterns of ? orbitals confirmed by ab initio calculations, perfectly reproduces the experimental results

Evolutionary Dynamics While Trapped in Resonance: A Keplerian Binary System Perturbed by Gravitational Radiation

The method of averaging is used to investigate the phenomenon of capture into resonance for a model that describes a Keplerian binary system influenced by radiation damping and external normally incident periodic gravitational radiation. The dynamical evolution of the binary orbit while trapped in resonance is elucidated using the second order partially averaged system. This method provides a theoretical framework that can be used to explain the main evolutionary dynamics of a physical system that has been trapped in resonance.Comment: REVTEX Style, Submitte

Dissipation in planar resonant planetary systems

Close-in planetary systems detected by the Kepler mission present an excess of periods ratio that are just slightly larger than some low order resonant values. This feature occurs naturally when resonant couples undergo dissipation that damps the eccentricities. However, the resonant angles appear to librate at the end of the migration process, which is often believed to be an evidence that the systems remain in resonance. Here we provide an analytical model for the dissipation in resonant planetary systems valid for low eccentricities. We confirm that dissipation accounts for an excess of pairs that lie just aside from the nominal periods ratios, as observed by the Kepler mission. In addition, by a global analysis of the phase space of the problem, we demonstrate that these final pairs are non-resonant. Indeed, the separatrices that exist in the resonant systems disappear with the dissipation, and remains only a circulation of the orbits around a single elliptical fixed point. Furthermore, the apparent libration of the resonant angles can be explained using the classical secular averaging method. We show that this artifact is only due to the severe damping of the amplitudes of the eigenmodes in the secular motion.Comment: 18 pages, 20 figures, accepted to A&

Stable manifolds and homoclinic points near resonances in the restricted three-body problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$ that circle each other with period equal to $2\pi$. For small $\mu$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2\pi p/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $\mu$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $\mu$ small enough in the unaveraged restricted three-body problem

Low frequency Raman studies of multi-wall carbon nanotubes: experiments and theory

In this paper, we investigate the low frequency Raman spectra of multi-wall carbon nanotubes (MWNT) prepared by the electric arc method. Low frequency Raman modes are unambiguously identified on purified samples thanks to the small internal diameter of the MWNT. We propose a model to describe these modes. They originate from the radial breathing vibrations of the individual walls coupled through the Van der Waals interaction between adjacent concentric walls. The intensity of the modes is described in the framework of bond polarization theory. Using this model and the structural characteristics of the nanotubes obtained from transmission electron microscopy allows to simulate the experimental low frequency Raman spectra with an excellent agreement. It suggests that Raman spectroscopy can be as useful regarding the characterization of MWNT as it is in the case of single-wall nanotubes.Comment: 4 pages, 2 eps fig., 2 jpeg fig., RevTex, submitted to Phys. Rev.

Multiple plasmon resonances in naturally-occurring multiwall nanotubes: infrared spectra of chrysotile asbestos

Chrysotile asbestos is formed by densely packed bundles of multiwall hollow nanotubes. Each wall in the nanotubes is a cylindrically wrapped layer of $Mg_3 Si_2 O_5 (OH)_4$. We show by experiment and theory that the infrared spectrum of chrysotile presents multiple plasmon resonances in the Si-O stretching bands. These collective charge excitations are universal features of the nanotubes that are obtained by cylindrically wrapping an anisotropic material. The multiple plasmons can be observed if the width of the resonances is sufficiently small as in chrysotile.Comment: 4 pages, 5 figures. Revtex4 compuscript. Misprint in Eq.(6) correcte

ORIGIN OF THE CHAOTIC MOTION OF THE SATURNIAN SATELLITE ATLAS

“The final publication is available at http://iopscience.iop.org/article/10.3847/0004-6256/151/5/122/meta

van der Waals interaction in nanotube bundles : consequences on vibrational modes

We have developed a pair-potential approach for the evaluation of van der Waals interaction between carbon nanotubes in bundles. Starting from a continuum model, we show that the intertube modes range from $5 cm^{-1}$ to $60 cm^{-1}$. Using a non-orthogonal tight-binding approximation for describing the covalent intra-tube bonding in addition, we confirme a slight chiral dependance of the breathing mode frequency and we found that this breathing mode frequency increase by $\sim$ 10 % if the nanotube lie inside a bundle as compared to the isolated tube.Comment: 5 pages, 2 figure

Secular dynamics of planetesimals in tight binary systems: Application to Gamma-Cephei

The secular dynamics of small planetesimals in tight binary systems play a fundamental role in establishing the possibility of accretional collisions in such extreme cases. The most important secular parameters are the forced eccentricity and secular frequency, which depend on the initial conditions of the particles, as well as on the mass and orbital parameters of the secondary star. We construct a second-order theory (with respect to the masses) for the planar secular motion of small planetasimals and deduce new expressions for the forced eccentricity and secular frequency. We also reanalyze the radial velocity data available for Gamma-Cephei and present a series of orbital solutions leading to residuals compatible with the best fits. Finally, we discuss how different orbital configurations for Gamma-Cephei may affect the dynamics of small bodies in circunmstellar motion. For Gamma-Cephei, we find that the classical first-order expressions for the secular frequency and forced eccentricity lead to large inaccuracies around 50 % for semimajor axes larger than one tenth the orbital separation between the stellar components. Low eccentricities and/or masses reduce the importance of the second-order terms. The dynamics of small planetesimals only show a weak dependence with the orbital fits of the stellar components, and the same result is found including the effects of a nonlinear gas drag. Thus, the possibility of planetary formation in this binary system largely appears insensitive to the orbital fits adopted for the stellar components, and any future alterations in the system parameters (due to new observations) should not change this picture. Finally, we show that planetesimals migrating because of gas drag may be trapped in mean-motion resonances with the binary, even though the migration is divergent.Comment: 11 pages, 9 figure