Philippe Robutel

The resonant structure of Jupiter's Trojan asteroids - II. What happens for different configurations of the planetary system

In a previous paper, we have found that the resonance structure of the present Jupiter Trojan swarms could be split up into four different families of resonances. Here, in a first step, we generalize these families in order to describe the resonances occurring in Trojan swarms embedded in a generic planetary system. The location of these families changes under a modification of the fundamental frequencies of the planets and we show how the resonant structure would evolve during a planetary migration. We present a general method, based on the knowledge of the fundamental frequencies of the planets and on those that can be reached by the Trojans, which makes it possible to predict and localize the main events arising in the swarms during migration. In particular, we show how the size and stability of the Trojan swarms are affected by the modification of the frequencies of the planets. Finally, we use this method to study the global dynamics of the Jovian Trojan swarms when Saturn migrates outwards. Besides the two resonances found by Morbidelli et al. which could have led to the capture of the current population just after the crossing of the 2:1 orbital resonance, we also point out several sequences of chaotic events that can influence the Trojan population.

The Observed Trojans and the Global Dynamics Around the Lagrangian Points of the Sun-Jupiter System

In this paper, we make a systematic study of the global dynamical structure of the Sun-Jupiter L 4 tadpole region. The results are based on long-time simulations of the Trojans in the Sun, Jupiter, Saturn system and on the frequency analysis of these orbits. We give some initial results in the description of the resonant structure that guides the long-term dynamics of this region. Moreover, we are able to connect this global view of the phase space with the observed Trojans and identify resonances in which some of the real bodies are located.

Frozen orbits at high eccentricity and inclination: application to Mercury orbiter

We hereby study the stability of a massless probe orbiting around an oblate central body (planet or planetary satellite) perturbed by a third body, assumed to lay in the equatorial plane (Sun or Jupiter for example) using a Hamiltonian formalism. We are able to determine, in the parameters space, the location of the frozen orbits, namely orbits whose orbital elements remain constant on average, to characterize their stability/unstability and to compute the periods of the equilibria. The proposed theory is general enough, to be applied to a wide range of probes around planet or natural planetary satellites. The BepiColombo mission is used to motivate our analysis and to provide specific numerical data to check our analytical results. Finally, we also bring to the light that the coefficient J2 is able to protect against the increasing of the eccentricity due to the Kozai-Lidov effect and the coefficient J3 determines a shift of the equilibria.

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

Analytical description of physical librations of saturnian coorbital satellites Janus and Epimetheus

L_4

Detectability of quasi-circular co-orbital planets. Application to the radial velocity technique

L4.

Spin-orbit coupling and chaotic rotation for coorbital bodies in quasi-circular orbits

We present a numerical study of several two-planet systems based on the motions of Jupiter and Saturn, in which the two giant planets move in low eccentric orbits close to a mean motion resonance. It is more likely to find two planets with similar characteristics in a system than a clone of the Jupiter-Saturn pair of our solar system. Therefore, we vary the distance between the two planets and their mass ratio by changing Saturns semimajor axis from 8 to 11 AU and increasing its mass by factors of 2-40. The different two-planets were analyzed for the interacting perturbations due to the mean motion resonances of the giant planets. We select several mass ratios for the gas giants, for which we study their influence on test bodies (with negligible mass) moving in the habitable zone (HZ) of a Sun-like star. The orbits are calculated for 2 × 107 yr. In all cases the HZ is dominated by a significant curved band, indicating higher eccentricity, which corresponds to a secular resonance with Jupiter. Interesting results of this study are finding (1) an increase of Venuss eccentricity for the real Jupiter and Saturn masses and the actual semimajor axis of Saturn; (2) an increase of the eccentricity of a test planet at Earths position when Saturns mass was increased by a factor of 3 or more; and (3) if the two giant planets are in 2:1 resonance, we observe a strong influence on the outer region of the HZ.

Rigorous treatment of the averaging process for co-orbital motions in the planetary problem

Abstract Janus and Epimetheus are famously known for their distinctive horseshoe-shaped orbits resulting from a 1:1 orbital resonance. Every 4 years these two satellites swap their orbits by a few tens of kilometers as a result of their close encounter. Recently Tiscareno et al. (Tiscareno, M.S., Thomas, P.C., Burns, J.A. [2009]. Icarus 204, 254–261) have proposed a model of rotation based on images from the Cassini orbiter. These authors inferred the amplitude of rotational librational motion in longitude at the orbital period by fitting a shape model to Cassini ISS images. By a quasi-periodic approximation of the orbital motion, we describe how the orbital swap impacts the rotation of the satellites. To that purpose, we have developed a formalism based on quasi-periodic series with long- and short-period librations. In this framework, the amplitude of the libration at the orbital period is found proportional to a term accounting for the orbital swap. We checked the analytical quasi-periodic development by performing a numerical simulation and find both results in good agreement. To complete this study, the results obtained for the short-period librations are studied with the help of an adiabatic-like approach.