Philippe Robutel

Co-orbital exoplanets are a by-product of the models of formation of planetary systems. However, none have been detected in nature thus far. Although challenging, the observation of co-orbital exoplanets would provide valuable information on the formation of planetary systems as well as on the interactions between planets and their host star. After a brief review of the stability and formation issues of co-orbital systems, some observational methods dedicated to their detection are presented.

Co-orbital planets (in a 1: 1 mean motion resonance) can be formed within a Laplace resonance chain. We develop a secular model tc study the dynamics of the resonance chain p: p : p + 1, where the co-orbital pair is in a first-order mean motion resonance with the outermost third planet. Our model takes into account tidal dissipation through the use of a Hamiltonian version of the constant time-lag model, which extends the Hamiltonian formalism of the point-mass case. We show the existence of several families of equilibria, anc how these equilibria extend to the complete system. In one family, which we call the main branch, a secular resonance between the libration frequency of the co-orbitals and the precession frequency of the pericentres has unexpected dynamical consequences when tidal dissipation is added. We report the existence of two distinct mechanisms that make co-orbital planets much more stable within the p : p : p + 1 resonance chain rather than outside it. The first is due to negative real parts of the eigenvalues of the linearised system with tides, in the region of the secular resonance mentioned above. The second comes from non-linear contributions of the vector fielt and is due to eccentricity damping. These two stabilising mechanisms increase the chances of a future detection of exoplanets in the co-orbital configuration.

Despite the existence of co-orbital bodies in the solar system, and the prediction of the formation of co-orbital planets by planetary system formation models, no co-orbital exoplanets (also called trojans) have been detected thus far. Here we study the signature of co-orbital exoplanets in transit surveys when two planet candidates in the system orbit the star with similar periods. Such a pair of candidates could be discarded as false positives because they are not Hill-stable. However, horseshoe or long-libration-period tadpole co-orbital configurations can explain such period similarity. This degeneracy can be solved by considering the transit timing variations (TTVs) of each planet. We subsequently focus on the three-planet-candidate system TOI-178: the two outer candidates of that system have similar orbital periods and were found to have an angular separation close to π∕3 during the TESS observation of sector 2. Based on the announced orbits, the long-term stability of the system requires the two close-period planets to be co-orbital. Our independent detrending and transit search recover and slightly favour the three orbits close to a 3:2:2 resonant chain found by the TESS pipeline, although we cannot exclude an alias that would put the system close to a 4:3:2 configuration. We then analyse the co-orbital scenario in more detail, and show that despite the influence of an inner planet just outside the 2:3 MMR, this potential co-orbital system could be stable on a gigayear time-scale for a variety of planetary masses, either on a trojan or a horseshoe orbit. We predict that large TTVs should arise in such a configuration with a period of several hundred days. We then show how the mass of each planet can be retrieved from these TTVs.

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.

Context.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 systems.