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Philippe Robutel

Publications and source records attributed to Philippe Robutel.

At least 19 recordsLinked to original sources
arXiv2026

Marchal's family of periodic orbits. I: Stability of inclined co-orbital planetary systems

At the Lagrange relative equilibrium of the three-body problem, for all values of the masses, the elliptic eigenvalues associated with vertical eigenvectors give rise to spatial quasi-periodic orbits, which become periodic in a rotating frame. In 2009, by averaging out the fast frequencies, Christian Marchal showed that these orbits, which are fixed points in the restricted average problem, form a oneparameter family connecting L 4 to L 5 . Using perturbation methods, we show the persistence of this family in the average three-body problem for nonzero masses in the limit where one mass is dominant over the other two (known as the planetary problem). We also give an analytical approximation valid for mutual inclinations less than 60 $\bullet$ . Then, using purely numerical methods, we show that this family exists in the full three-body problem (neither restricted nor average) for a wide range of masses, beyond the planetary case. We also show that the stability of its orbits evolves along the family, with inclined systems remaining stable for masses exceeding the Gascheau's value (also known as Routh's critical value). Finally, we show the impact of this family's stability on the global dynamics of the co-orbital region as well as its high instability for mutual inclinations exceeding 60 $\bullet$ .

math.DS↗
arXiv2024

Trojan Exoplanets

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.

astro-ph.EP↗
arXiv2024

Long-term dust dynamics in Didymos and Dimorphos system: production, stability, and transport

Target of NASA's DART mission, the system of Didymos and Dimorphos will once again be visited by a space mission -- ESA's Hera mission, scheduled to be launch in 2024. Hera will arrive in the system approximately 4 years after the DART impact, a long period compared to Dimorphos' orbital period (about 12 hours). It is therefore imperative to understand the dynamics of material in this environment on a long timescale. Here, we explore the long-term dynamics of the binary system (65038) Didymos, in the context of the perturbed, planar, circular and restricted 3-body problem. We design an analytical description for a symmetrical top-shaped object, the shape assumed for the Didymos, while the Dimorphos is considered an ellipsoid. In the absence of external effects, we identify seven stable equatorial regions where particles persist for more than a decade. However, in the presence of the solar radiation effect, the lifetime of small particles (<mm) is in the order of days, being unlikely that Hera spacecraft will encounter clusters of millimetre and sub-millimetre particles in stable equatorial orbits. Nonetheless, large objects may reside in the region for some years, particularly in quasi-satellite orbits, the most stable orbits in the system. Additionally, interplanetary dust impacts onto Didymos populate the region, extending up to a distance of approximately 1500 meters from the primary center, with young dust. These impacts are responsible for a transfer of dust mainly from Didymos to Dimorphos. If the interplanetary dust impacts generate metric-sized boulders, they may persist in the system for years, in first sort orbits around Didymos.

astro-ph.EP↗
arXiv2022

Dynamics of co-orbital exoplanets in a first order resonance chain with tidal dissipation

Co-orbital planets (in a $1:1$ mean motion resonance) can be formed within a Laplace resonance chain. Here, we develop a secular model to 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, and 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 one 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 one comes from non-linear contributions of the vector field and it is due to eccentricity damping. These two stabilising mechanisms increase the chances of a still-to-come detection of exoplanets in the co-orbital configuration.

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arXiv2021

An analytical model for tidal evolution in co-orbital systems $\,$I. Application to exoplanets

Close-in co-orbital planets (in a 1:1 mean motion resonance) can experience strong tidal interactions with the central star. Here, we develop an analytical model adapted to the study of the tidal evolution of those systems. We use a Hamiltonian version of the constant time-lag tidal model, which extends the Hamiltonian formalism developed for the point-mass case. We show that co-orbital systems undergoing tidal dissipation either favour the Lagrange or the anti-Lagrange configurations, depending on the system parameters. However, for all range of parameters and initial conditions, both configurations become unstable, although the timescale for the destruction of the system can be larger than the lifetime of the star. We provide an easy-to-use criterion to determine if an already known close-in exoplanet may have an undetected co-orbital companion.

astro-ph.EP↗
arXiv2019

Co-orbital exoplanets from close period candidates: The TOI-178 case

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 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 then focus on the three planet candidates system TOI-178: the two outer candidates of that system have similar orbital period and had an angular separation near $\pi/3$ 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-orbitals. 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 in more detail the co-orbital scenario. We show that despite the influence of an inner planet just outside the 2:3 mean-motion resonance, this potential co-orbital system can be stable on the Giga-year 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 configuration with a period of several hundred days. We then show how the mass of each planet can be retrieved from these TTVs.

astro-ph.EP↗
arXiv2018

La stabilit\'e des lunes de Saturne, Janus et \'Epim\'eth\'ee: de l'observation astronomique \`a la th\'eorie KAM

Popular science article associated with the work `On the co-orbital motion in the three-body problem: existence of quasi-periodic horseshoe-shaped orbits" (arXiv:1806.07262) from the same authors. Janus and Epimetheus are two moons of Saturn which exhibit a really peculiar dynamics. As they orbit on circular trajectories whose radii are only 50 km apart (less than their respective diameters), every four (terrestrial) years the bodies are getting closer and their mutual gravitational influence leads to a swapping of the orbits: the outer moon becoming the inner one and vice-versa. In this article, we describe how, from this specific astronomical problem to the KAM theory, we came to prove the existence of perpetually stable trajectories associated with the Janus and Epimetheus orbits.

math.HO↗
arXiv2018

On the co-orbital motion in the three-body problem: existence of quasi-periodic horseshoe-shaped orbits

Janus and Epimetheus are two moons of Saturn with very peculiar motions. As they orbit around Saturn on quasi-coplanar and quasi-circular trajectories whose radii are only 50 km apart (less than their respective diameters), every four (terrestrial) years the bodies approach each other and their mutual gravitational influence lead to a swapping of the orbits: the outer moon becomes the inner one and vice-versa. This behavior generates horseshoe-shaped trajectories depicted in an appropriate rotating frame. In spite of analytical theories and numerical investigations developed to describe their long-term dynamics, so far very few rigorous long-time stability results on the "horseshoe motion" have been obtained even in the restricted three-body problem. Adapting the idea of Arnol'd (1963) to a resonant case (the co-orbital motion is associated with trajectories in 1:1 mean motion resonance), we provide a rigorous proof of existence of 2-dimensional elliptic invariant tori on which the trajectories are similar to those followed by Janus and Epimetheus. For this purpose, we apply KAM theory to the planar three-body problem.

math.DS↗
arXiv2018

Towards completing Planetary Systems: The role of minor bodies on life growth and survival

The search for extrasolar planets in the past decades has shown that planets abound in the Solar neighborhood. While we are still missing an Earth twin, the forthcoming space missions and ground-based instrumentation are already driven to achieve this goal. But, in order to fully understand the conditions for life appearing in the Solar System, we still miss some pieces of the planetary system jigsaw puzzle, namely a deeper understanding of the minor bodies. Trojans, moons, and comets are tracers of the formation and evolution processes of planetary systems. These missing pieces are also critical to understand the emergence and evolution of life over millions of years. With the large crop of planetary systems discovered so far and yet to be detected with the forthcoming missions, the hunt for minor bodies in extrasolar systems is a natural continuation of our search for real Solar System- and, in particular, Earth- analogs. This white paper is focused on detection of these minor components and their relevance in the emergence, evolution and survival of life.

astro-ph.EP↗
arXiv2017

Secular Dynamics of Multiplanetary Circumbinary Systems

We present an analytical formalism to study the secular dynamics of a system consisting of N-2 planets orbiting a binary star in outer orbits. We introduce a canonical coordinate system and expand the disturbing function in terms of canonical elliptic elements, combining both Legendre polynomials and Laplace coefficients, to obtain a general formalism for the secular description of this type of configuration. With a quadractic approximation of the development, we present a simplified analytical solution for the planetary orbits for both the single planet and the two-planet cases. From the two-planet model, we show that the inner planet accelerates the precession rate of the binary pericenter, which, in turn, may enter in resonance with the secular frequency of the outer planet, characterizing a secular resonance. We calculate an analytical expression for the approximate location of this resonance and apply it to known circumbinary systems, where we show that it can occur at relatively close orbits, for example at 2.4 au for the Kepler-38 system. With a more refined model, we analyse the dynamics of this secular resonance and we show that a bifurcation of the corresponding fixed points can affect the long term evolution and stability of planetary systems. By comparing our results with complete integrations of the exact equations of motion, we verified the accuracy of our analytical model.

astro-ph.EP↗
arXiv2017

Detection of co-orbital planets by combining transit and radial-velocity measurements

Co-orbital planets have not yet been discovered, although they constitute a frequent by-product of planetary formation and evolution models. This lack may be due to observational biases, since the main detection methods are unable to spot co-orbital companions when they are small or near the Lagrangian equilibrium points. However, for a system with one known transiting planet (with mass $m_1$), we can detect a co-orbital companion (with mass $m_2$) by combining the time of mid-transit with the radial-velocity data of the star. Here, we propose a simple method that allows the detection of co-orbital companions, valid for eccentric orbits, that relies on a single parameter $α$, which is proportional to the mass ratio $m_2/m_1$. Therefore, when $α$ is statistically different from zero, we have a strong candidate to harbour a co-orbital companion. We also discuss the relevance of false positives generated by different planetary configurations.

astro-ph.EP↗
arXiv2016

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

Several celestial bodies in co-orbital configurations exist in the solar system. However, co-orbital exoplanets have not yet been discovered. This lack may result from a degeneracy between the signal induced by co-orbital planets and other orbital configurations. Here we determine a criterion for the detectability of quasi-circular co-orbital planets and develop a demodulation method to bring out their signature from the observational data. We show that the precision required to identify a pair of co-orbital planets depends only on the libration amplitude and on the planet's mass ratio. We apply our method to synthetic radial velocity data, and show that for tadpole orbits we are able to determine the inclination of the system to the line of sight. Our method is also valid for planets detected through the transit and astrometry techniques.

astro-ph.EP↗
arXiv2016

On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

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.

astro-ph.EP↗
arXiv2015

Secular Dynamics of S-type Planetary Orbits in Binary Star Systems: Applicability Domains of First- and Second-Order Theories

We analyse the secular dynamics of planets on S-type coplanar orbits in tight binary systems, based on first- and second-order analytical models, and compare their predictions with full N-body simulations. The perturbation parameter adopted for the development of these models depends on the masses of the stars and on the semimajor axis ratio between the planet and the binary. We show that each model has both advantages and limitations. While the first-order analytical model is algebraically simple and easy to implement, it is only applicable in regions of the parameter space where the perturbations are sufficiently small. The second-order model, although more complex, has a larger range of validity and must be taken into account for dynamical studies of some real exoplanetary systems such as $γ$-Cephei and HD 41004A. However, in some extreme cases, neither of these analytical models yields quantitatively correct results, requiring either higher-order theories or direct numerical simulations. Finally, we determine the limits of applicability of each analytical model in the parameter space of the system, giving an important visual aid to decode which secular theory should be adopted for any given planetary system in a close binary.

astro-ph.EP↗
arXiv2015

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

We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamilto-nian, we estimate the size of the transformation that maps this Hamil-tonian to its first order averaged over one of the fast angles. After having derived an integrable approximation of the averaged problem, we bound the distance between this integrable approximation and the averaged Hamiltonian. This finally allows to prove rigorous theorems on the behavior of co-orbital motions over a finite but large timescale.

astro-ph.EP↗
arXiv2015

Spin-orbit coupling and chaotic rotation for circumbinary bodies. Application to the small satellites of the Pluto-Charon system

We investigate the resonant rotation of circumbinary bodies in planar quasi-circular orbits. Denoting $n_b$ and $n$ the orbital mean motion of the inner binary and of the circumbinary body, respectively, we show that spin-orbit resonances exist at the frequencies $n\pm kν/2$, where $ν= n_b - n$, and $k$ is an integer. Moreover, when the libration at natural frequency has the same magnitude as $ν$, the resonances overlap and the rotation becomes chaotic. We apply these results to the small satellites in the Pluto-Charon system, and conclude that their rotations are likely chaotic. However, the rotation can also be stable and not synchronous for small axial asymmetries.

astro-ph.EP↗
arXiv2014

The family of Quasi-satellite periodic orbits in the circular co-planar RTBP

In the circular case of the coplanar Restricted Three-body Problem, we studied how the family of quasi-satellite (QS) periodic orbits allows to define an associated libration center. Using the averaged problem, we highlighted a validity limit of this one: for QS orbits with low eccentricities, the averaged problem does not correspond to the real problem. We do the same procedure to L 3 , L 4 and L 5 emerging periodic orbits families and remarked that for very high eccentricities F L4 and F L5 merge with F L3 which bifurcates to a stable family.

astro-ph.EP↗