Dynamical analysis of the Gliese-876 Laplace resonance

The existence of multiple planetary systems involved in mean motion conmensurabilities has increased significantly since the Kepler mission. Although most correspond to 2-planet resonances, multiple resonances have also been found. The Laplace resonance is a particular case of a three-body resonance where the period ratio between consecutive pairs is n_1/n_2 near to n_2/n_3 near to 2/1. It is not clear how this triple resonance can act in order to stabilize (or not) the systems. The most reliable extrasolar system located in a Laplace resonance is GJ876 because it has two independent confirmations. However best-fit parameters were obtained without previous knowledge of resonance structure and no exploration of all the possible stable solutions for the system where done. In the present work we explored the different configurations allowed by the Laplace resonance in the GJ876 system by varying the planetary parameters of the third outer planet. We find that in this case the Laplace resonance is a stabilization mechanism in itself, defined by a tiny island of regular motion surrounded by (unstable) highly chaotic orbits. Low eccentric orbits and mutual inclinations from -20 to 20 degrees are compatible with the observations. A definite range of mass ratio must be assumed to maintain orbital stability. Finally we give constrains for argument of pericenters and mean anomalies in order to assure stability for this kind of systems.Comment: 7 pages, 7 figures, accepted in MNRA

A semi-empirical stability criterion for real planetary systems

We test a crossing orbit stability criterion for eccentric planetary systems, based on Wisdom's criterion of first order mean motion resonance overlap (Wisdom, 1980). We show that this criterion fits the stability regions in real exoplanet systems quite well. In addition, we show that elliptical orbits can remain stable even for regions where the apocenter distance of the inner orbit is larger than the pericenter distance of the outer orbit, as long as the initial orbits are aligned. The analytical expressions provided here can be used to put rapid constraints on the stability zones of multi-planetary systems. As a byproduct of this research, we further show that the amplitude variations of the eccentricity can be used as a fast-computing stability indicator.Comment: 11 pages, 11 figures. MNRAS accepte

Origin and Detectability of coorbital planets from radial velocity data

We analyze the possibilities of detection of hypothetical exoplanets in coorbital motion from synthetic radial velocity (RV) signals, taking into account different types of stable planar configurations, orbital eccentricities and mass ratios. For each nominal solution corresponding to small-amplitude oscillations around the periodic solution, we generate a series of synthetic RV curves mimicking the stellar motion around the barycenter of the system. We then fit the data sets obtained assuming three possible different orbital architectures: (a) two planets in coorbital motion, (b) two planets in a 2/1 mean-motion resonance, and (c) a single planet. We compare the resulting residuals and the estimated orbital parameters. For synthetic data sets covering only a few orbital periods, we find that the discrete radial velocity signal generated by a coorbital configuration could be easily confused with other configurations/systems, and in many cases the best orbital fit corresponds to either a single planet or two bodies in a 2/1 resonance. However, most of the incorrect identifications are associated to dynamically unstable solutions. We also compare the orbital parameters obtained with two different fitting strategies: a simultaneous fit of two planets and a nested multi-Keplerian model. We find that the nested models can yield incorrect orbital configurations (sometimes close to fictitious mean-motion resonances) that are nevertheless dynamically stable and with orbital eccentricities lower than the correct nominal solutions. Finally, we discuss plausible mechanisms for the formation of coorbital configurations, by the interaction between two giant planets and an inner cavity in the gas disk. For equal mass planets, both Lagrangian and anti-Lagrangian configurations can be obtained from same initial condition depending on final time of integration.Comment: 14 pages, 16 figures.2012. MNRAS, 421, 35

Dynamical analysis and constraints for the HD 196885 system

The HD\,196885 system is composed of a binary star and a planet orbiting the primary. The orbit of the binary is fully constrained by astrometry, but for the planet the inclination with respect to the plane of the sky and the longitude of the node are unknown. Here we perform a full analysis of the HD\,196885 system by exploring the two free parameters of the planet and choosing different sets of angular variables. We find that the most likely configurations for the planet is either nearly coplanar orbits (prograde and retrograde), or highly inclined orbits near the Lidov-Kozai equilibrium points, i = 44^{\circ} or i = 137^{\circ} . Among coplanar orbits, the retrograde ones appear to be less chaotic, while for the orbits near the Lidov-Kozai equilibria, those around \omega= 270^{\circ} are more reliable, where \omega_k is the argument of pericenter of the planet's orbit with respect to the binary's orbit. From the observer's point of view (plane of the sky) stable areas are restricted to (I1, \Omega_1) \sim (65^{\circ}, 80^{\circ}), (65^{\circ},260^{\circ}), (115^{\circ},80^{\circ}), and (115^{\circ},260^{\circ}), where I1 is the inclination of the planet and \Omega_1 is the longitude of ascending node.Comment: 10 pages, 7 figures. A&A Accepte

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

Dynamics of two planets in co-orbital motion

We study the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration. We consider different ratios of planetary masses and orbital eccentricities, also we assume that both planets share the same orbital plane. Initially we perform numerical simulations over a grid of osculating initial conditions to map the regions of stable/chaotic motion and identify equilibrium solutions. These results are later analyzed in more detail using a semi-analytical model. Apart from the well known quasi-satellite (QS) orbits and the classical equilibrium Lagrangian points L4 and L5, we also find a new regime of asymmetric periodic solutions. For low eccentricities these are located at $(\sigma,\Delta\omega) = (\pm 60\deg, \mp 120\deg)$, where \sigma is the difference in mean longitudes and \Delta\omega is the difference in longitudes of pericenter. The position of these Anti-Lagrangian solutions changes with the mass ratio and the orbital eccentricities, and are found for eccentricities as high as ~ 0.7. Finally, we also applied a slow mass variation to one of the planets, and analyzed its effect on an initially asymmetric periodic orbit. We found that the resonant solution is preserved as long as the mass variation is adiabatic, with practically no change in the equilibrium values of the angles.Comment: 9 pages, 11 figure

Mapping the ν⊙\nu_\odot Secular Resonance for Retrograde Irregular Satellites

Constructing dynamical maps from the filtered output of numerical integrations, we analyze the structure of the $\nu_\odot$ secular resonance for fictitious irregular satellites in retrograde orbits. This commensurability is associated to the secular angle $\theta = \varpi - \varpi_\odot$, where $\varpi$ is the longitude of pericenter of the satellite and $\varpi_\odot$ corresponds to the (fixed) planetocentric orbit of the Sun. Our study is performed in the restricted three-body problem, where the satellites are considered as massless particles around a massive planet and perturbed by the Sun. Depending on the initial conditions, the resonance presents a diversity of possible resonant modes, including librations of $\theta$ around zero (as found for Sinope and Pasiphae) or 180 degrees, as well as asymmetric librations (e.g. Narvi). Symmetric modes are present in all giant planets, although each regime appears restricted to certain values of the satellite inclination. Asymmetric solutions, on the other hand, seem absent around Neptune due to its almost circular heliocentric orbit. Simulating the effects of a smooth orbital migration on the satellite, we find that the resonance lock is preserved as long as the induced change in semimajor axis is much slower compared to the period of the resonant angle (adiabatic limit). However, the librational mode may vary during the process, switching between symmetric and asymmetric oscillations. Finally, we present a simple scaling transformation that allows to estimate the resonant structure around any giant planet from the results calculated around a single primary mass.Comment: 11 pages, 13 figure

The 1:1 resonance in Extrasolar Systems: Migration from planetary to satellite orbits

We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves around the star. The stability regions around periodic orbits along the family are studied. Next, we study the dynamical evolution in time of a planetary system with two planets which is initially trapped in a stable 1/1 resonant periodic motion, when a drag force is included in the system. We prove that if we start with a 1/1 resonant planetary system with large eccentricities, the system migrates, due to the drag force, {\it along the family of periodic orbits} and is finally trapped in a satellite orbit. This, in principle, provides a mechanism for the generation of a satellite system: we start with a planetary system and the final stage is a system where the two small bodies form a close binary whose center of mass revolves around the star.Comment: to appear in Cel.Mech.Dyn.Ast

A new analysis of the GJ581 extrasolar planetary system

We have done a new analysis of the available observations for the GJ581 exoplanetary system. Today this system is controversial due to choices that can be done in the orbital determination. The main ones are the ocurrence of aliases and the additional bodies - the planets f and g - announced in Vogt et al. 2010. Any dynamical study of exoplanets requires the good knowledge of the orbital elements and the investigations involving the planet g are particularly interesting, since this body would lie in the Habitable Zone (HZ) of the star GJ581. This region,for this system, is very attractive of the dynamical point of view due to several resonances of two and three bodies present there. In this work, we investigate the conditions under which the planet g may exist. We stress the fact that the planet g is intimately related with the orbital elements of the planet d; more precisely, we conclude that it is not possible to disconnect its existence from the determination of the eccentricity of the planet d. Concerning the planet f, we have found one solution with period $\approx 450$ days, but we are judicious about any affirmation concernig this body because its signal is in the threshold of detection and the high period is in a spectral region where the ocorruence of aliases is very common. Besides, we outline some dynamical features of the habitable zone with the dynamical map and point out the role played by some resonances laying there.Comment: 12 pages, 9 figure

Stability of prograde and retrograde planets in circular binary systems

We investigate the stability of prograde versus retrograde planets in circular binary systems using numerical simulations. We show that retrograde planets are stable up to distances closer to the perturber than prograde planets. We develop an analytical model to compute the prograde and retrograde mean motion resonances' locations and separatrices. We show that instability is due to single resonance forcing, or caused by nearby resonances' overlap. We validate our results regarding the role of single resonances and resonances' overlap on orbit stability, by computing surfaces of section of the CR3BP. We conclude that the observed enhanced stability of retrograde planets with respect to prograde planets is due to essential differences between the phase-space topology of retrograde versus prograde resonances (at p/q mean motion ratio, prograde resonance is of order p - q while retrograde resonance is of order p + q).Comment: 13 pages, 10 figures; to appear in MNRA