High order symplectic integrators for perturbed Hamiltonian systems

We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+\epsilon B$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps with a remainder of order $O(\tau^p\epsilon +\tau^2\epsilon^2)$, where $\tau$ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes $O(\tau^p\epsilon +\tau^4\epsilon^2)$. The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.Comment: 24 pages, 6 figurre

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus' equilateral equilibrium points is investigated with respect to the inclination of the asteroid's orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L4 and L5, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 billion years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 < a < 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids' orbit was varied in the range 0 < i < 60 and the semimajor axes were fixed. It turned out that only four 'windows' of stable orbits survive: these are the orbits for the initial inclinations 0 < i < 7, 9 < i < 13, 31 < i < 36 and 38 < i < 50. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.Comment: 15 pages, 12 figures, submitted to CMD

Influence of the coorbital resonance on the rotation of the Trojan satellites of Saturn

The Cassini spacecraft collects high resolution images of the saturnian satellites and reveals the surface of these new worlds. The shape and rotation of the satellites can be determined from the Cassini Imaging Science Subsystem data, employing limb coordinates and stereogrammetric control points. This is the case for Epimetheus (Tiscareno et al. 2009) that opens elaboration of new rotational models (Tiscareno et al. 2009; Noyelles 2010; Robutel et al. 2011). Especially, Epimetheus is characterized by its horseshoe shape orbit and the presence of the swap is essential to introduce explicitly into rotational models. During its journey in the saturnian system, Cassini spacecraft accumulates the observational data of the other satellites and it will be possible to determine the rotational parameters of several of them. To prepare these future observations, we built rotational models of the coorbital (also called Trojan) satellites Telesto, Calypso, Helene, and Polydeuces, in addition to Janus and Epimetheus. Indeed, Telesto and Calypso orbit around the L_4 and L_5 Lagrange points of Saturn-Tethys while Helene and Polydeuces are coorbital of Dione. The goal of this study is to understand how the departure from the Keplerian motion induced by the perturbations of the coorbital body, influences the rotation of these satellites. To this aim, we introduce explicitly the perturbation in the rotational equations by using the formalism developed by Erdi (1977) to represent the coorbital motions, and so we describe the rotational motion of the coorbitals, Janus and Epimetheus included, in compact form

(1173) Anchises - Thermophysical and Dynamical Studies of a Dynamically Unstable Jovian Trojan

We have performed detailed thermophysical and dynamical modelling of Jovian Trojan (1173) Anchises. Our results reveal a most unusual object. By examining observational data taken by IRAS, Akari and WISE between 11.5 and 60 microns, along with variations in its optical lightcurve, we find Anchises is most likely an elongated body, with an axes-ratio of ~1.4. This yields calculated best-fit dimensions of 170x121x121km (an equivalent diameter of 136+18/-11km). We find the observations are best fit by Anchises having a retrograde sense of rotation, and an unusually high thermal inertia (25 to 100 Jm-2s-0.5K-1). The geometric albedo is found to be 0.027 (+0.006/-0.007). Anchises therefore has one of the highest published thermal inertias of any object larger than 100km in diameter, at such large heliocentric distances, and is one of the lowest albedo objects ever observed. More observations are needed to see if there is a link between the very shallow phase curve, with almost no opposition effect, and the derived thermal properties for this large Trojan asteroid. Our dynamical investigation of Anchises' orbit has revealed it to be dynamically unstable on timescales of hundreds of Myr, similar to the unstable Neptunian Trojans 2001 QR322 and 2008 LC18. Unlike those objects, we find that Anchises' dynamical stability is not a function of its initial orbital elements, the result of the exceptional precision with which its orbit is known. This is the first time that a Jovian Trojan has been shown to be dynamically unstable, and adds weight to the idea that planetary Trojans represent a significant ongoing contribution to the Centaur population, the parents of the short-period comets. The observed instability does not rule out a primordial origin for Anchises, but when taken in concert with the result of our thermophysical analysis, suggest that it would be a fascinating target for future study.Comment: 5 figures, 3 tables, accepted for publication in Monthly Notices of the Royal Astronomical Societ

The TROY project III. Exploring co-orbitals around low-mass stars

Co-orbital objects, also known as trojans, are frequently found in simulations of planetary system formation. In these configurations, a planet shares its orbit with other massive bodies. It is still unclear why there have not been any co-orbitals discovered thus far in exoplanetary systems or even pairs of planets found in such a 1:1 mean motion resonance. Reconciling observations and theory is an open subject in the field. The main objective of the TROY project is to conduct an exhaustive search for exotrojans using diverse observational techniques. In this work, we analyze the radial velocity time series informed by transits, focusing the search around low-mass stars. We employed the alpha-test method on confirmed planets searching for shifts between spectral and photometric mid-transit times. This technique is sensitive to mass imbalances within the planetary orbit, allowing us to identify non-negligible co-orbital masses. Among the 95 transiting planets examined, we find one robust exotrojan candidate with a significant 3-sigma detection. Additionally, 25 exoplanets show compatibility with the presence of exotrojan companions at a 1-sigma level, requiring further observations to better constrain their presence. For two of those weak candidates, we find dimmings in their light curves within the predicted Lagrangian region. We established upper limits on the co-orbital masses for either the candidates and null detections. Our analysis reveals that current high-resolution spectrographs effectively rule out co-orbitals more massive than Saturn around low-mass stars. This work points out to dozens of targets that have the potential to better constraint their exotrojan upper mass limit with dedicated radial velocity observations. We also explored the potential of observing the secondary eclipses of the confirmed exoplanets to enhance the exotrojan search.Comment: 41 pages, 9 figures, 6 tables. Accepted in Astronomy & Astrophysic

On the coplanar eccentric non-restricted co-orbital dynamics

We study the phase space of eccentric coplanar co-orbitals in the non-restricted case. Departing from the quasi-circular case, we describe the evolution of the phase space as the eccentricities increase. We find that over a given value of the eccentricity, around 0.5 for equal mass co-orbitals, important topological changes occur in the phase space. These changes lead to the emergence of new co-orbital configurations and open a continuous path between the previously distinct trojan domains near the L_4 and L_5 eccentric Lagrangian equilibria. These topological changes are shown to be linked with the reconnection of families of quasi-periodic orbits of non-maximal dimension.publishe

The TROY project: II. Multi-technique constraints on exotrojans in nine planetary systems

Co-orbital bodies are the byproduct of planet formation and evolution, as we know from the solar system. Although planet-size co-orbitals do not exists in our planetary system, dynamical studies show that they can remain stable for long periods of time in the gravitational well of massive planets. Should they exist, their detection is feasible with the current instrumentation. Aims. In this paper, we present new ground-based observations searching for these bodies co-orbiting with nine close-in (P < 5 days) planets, using various observing techniques. The combination of all of these techniques allows us to restrict the parameter space of any possible trojan in the system. Methods. We used multi-Technique observations, comprised of radial velocity, precision photometry, and transit timing variations, both newly acquired in the context of the TROY project and publicly available, to constrain the presence of planet-size trojans in the Lagrangian points of nine known exoplanets. Results. We find no clear evidence of trojans in these nine systems through any of the techniques used down to the precision of the observations. However, this allows us to constrain the presence of any potential trojan in the system, especially in the trojan mass or radius vs. libration amplitude plane. In particular, we can set upper mass limits in the super-Earth mass regime for six of the studied systemspublishe

A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems

We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.Comment: 31 pages, 4 figure