# On the coplanar eccentric non-restricted co-orbital dynamics

- 271 Downloads
- 1 Citations

## Abstract

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.

## Keywords

Trojans Co-orbitals Lagrange Planetary problem Three-body problem High eccentricity Mean-motion resonance## Notes

### Acknowledgements

The authors acknowledge financial support from the Observatoire de Paris Scientific Council, CIDMA strategic project UID/MAT/04106/2013, ENGAGE SKA POCI-01- 0145-FEDER-022217 (funded by COMPETE 2020 and FCT, Portugal), and the Marie Curie Actions of the European Commission (FP7-COFUND). Parts of this work have been carried out within the frame of the National Centre for Competence in Research PlanetS supported by the SNSF. This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (Reference ANR-10-EQPX-29-01) of the programme Investissements dAvenir supervised by the Agence Nationale pour la Recherche. The authors thank the referees for useful suggestions that greatly improved the description of the results.

## References

- Batygin, K., Morbidelli, A.: Analytical treatment of planetary resonances. Astron. Astrophys.
**556**, A28 (2013)ADSCrossRefGoogle Scholar - Beaugé, C., Roig, F.: A semianalytical model for the motion of the trojan asteroids: proper elements and families. Icarus
**153**, 391–415 (2001)ADSCrossRefGoogle Scholar - Charlier, C.V.L.: Über den Planeten 1906 TG. Astron. Nachr.
**171**, 213 (1906)ADSCrossRefGoogle Scholar - Delisle, J.-B., Laskar, J., Correia, A.C.M.: Resonance breaking due to dissipation in planar planetary systems. Astron. Astrophys.
**566**, A137 (2014)CrossRefGoogle Scholar - Delisle, J.-B., Laskar, J., Correia, A.C.M., Boué, G.: Dissipation in planar resonant planetary systems. Astron. Astrophys.
**546**, A71 (2012)CrossRefGoogle Scholar - Dermott, S.F., Murray, C.D.: The dynamics of tadpole and horseshoe orbits. I - Theory. Icarus
**48**, 1–11 (1981)ADSCrossRefGoogle Scholar - Érdi, B.: An asymptotic solution for the trojan case of the plane elliptic restricted problem of three bodies. Celest. Mech.
**15**, 367–383 (1977)ADSCrossRefzbMATHGoogle Scholar - Érdi, B., Nagy, I., Sándor, Z., Süli, Á., Fröhlich, G.: Secondary resonances of co-orbital motions. MNRAS
**381**, 33–40 (2007)ADSCrossRefGoogle Scholar - Ford, E.B., Gaudi, B.S.: Observational constraints on Trojans of transiting extrasolar planets. Astrophys. J. Lett.
**652**, 137–140 (2006)ADSCrossRefGoogle Scholar - Garfinkel, B.: Theory of the Trojan asteroids. I. Astron. J.
**82**, 368–379 (1977)ADSCrossRefGoogle Scholar - Gascheau, G.: Examen d’une classe d’équations différentielles et application à un cas particulier du problème des trois corps. C. R. Acad. Sci. Paris
**16**(7), 393–394 (1843)Google Scholar - Gastineau, M., Laskar, J.: Trip: a computer algebra system dedicated to celestial mechanics and perturbation series. ACM Commun. Comput. Algebra
**44**(3/4), 194–197 (2011)zbMATHGoogle Scholar - Giuppone, C.A., Beaugé, C., Michtchenko, T.A., Ferraz-Mello, S.: Dynamics of two planets in co-orbital motion. MNRAS
**407**, 390–398 (2010)ADSCrossRefGoogle Scholar - Giuppone, C.A., Benitez-Llambay, P., Beaugé, C.: Origin and detectability of co-orbital planets from radial velocity data. MNRAS (2012)Google Scholar
- Hadjidemetriou, J.D., Psychoyos, D., Voyatzis, G.: The 1/1 resonance in extrasolar planetary systems. Celest. Mech. Dyn. Astron.
**104**, 23–38 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar - Hadjidemetriou, J.D., Voyatzis, G.: The 1/1 resonance in extrasolar systems. Migration from planetary to satellite orbits. Celest. Mech. Dyn. Astron.
**111**, 179–199 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar - Henrard, J., Caranicolas, N.D.: Motion near the 3/1 resonance of the planar elliptic restricted three body problem. Celest. Mech. Dyn. Astron.
**47**, 99–121 (1989)ADSMathSciNetCrossRefGoogle Scholar - Laskar, J.: The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zone. Icarus
**88**, 266–291 (1990)ADSCrossRefGoogle Scholar - Laskar, J., Robutel, P.: Stability of the planetary three-body problem I: expansion of the planetary hamiltonian. Celest. Mech. Dyn. Astron.
**62**, 193–217 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar - Laskar, J., Robutel, P.: High order symplectic integrators for perturbed Hamiltonian systems. Celest. Mech. Dyn. Astron.
**80**, 39–62 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar - Leleu, A.: Dynamics of co-orbital exoplanets. PhD thesis (2016)Google Scholar
- Leleu, A., Robutel, P., Correia, A.C.M.: Detectability of quasi-circular co-orbital planets: application to the radial velocity technique. Astron. Astrophys.
**581**, A128 (2015)ADSCrossRefGoogle Scholar - Leleu, A., Robutel, P., Correia, A.C.M., Lillo-Box, J.: Detection of co-orbital planets by combining transit and radial-velocity measurements. Astron. Astrophys.
**599**, L7 (2017)ADSCrossRefGoogle Scholar - Liouville, J.: Sur un cas particulier du problème des trois corps. C. R. Acad. Sci. Paris
**14**, 503–506 (1842)Google Scholar - Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian Dynamical Systems and the n-Body Problem. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
- Michtchenko, T.A., Ferraz-Mello, S., Beaugé, C.: Modeling the 3-d secular planetary three-body problem. Icarus
**181**, 555–571 (2006)ADSCrossRefzbMATHGoogle Scholar - Mikkola, S., Innanen, K., Wiegert, P., Connors, M., Brasser, R.: Stability limits for the quasi-satellite orbit. Mon. Not. R. Astron. Soc
**369**, 15–24 (2006)ADSCrossRefGoogle Scholar - Morais, M.H.M.: A secular theory for Trojan-type motion. Astron. Astrophys.
**350**, 318–326 (1999)ADSGoogle Scholar - Morais, M.H.M.: Hamiltonian formulation of the secular theory for Trojan-type motion. Astron. Astrophys.
**369**, 677–689 (2001)ADSCrossRefzbMATHGoogle Scholar - Morais, M.H.M., Namouni, F.: Retrograde resonance in the planar three-body problem. Celest. Mech. Dyn. Astron.
**117**, 405–421 (2013)ADSMathSciNetCrossRefGoogle Scholar - Morbidelli, A.: Modern Celestial Mechanics : Aspects of Solar System Dynamics. Taylor & Francis, London (2002). ISBN 0415279399Google Scholar
- Namouni, F.: Secular interactions of coorbiting objects. Icarus
**137**, 293–314 (1999)ADSCrossRefGoogle Scholar - Nauenberg, M.: Stability and eccentricity for two planets in a 1:1 resonance, and their possible occurrence in extrasolar planetary systems. Astron. J.
**124**, 2332–2338 (2002)ADSCrossRefGoogle Scholar - Nesvorný, D., Thomas, F., Ferraz-Mello, S., Morbidelli, A.: A perturbative treatment of the co-orbital motion. Celest. Mech. Dyn. Astron.
**82**(4), 323–361 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar - Páez, R.I., Efthymiopoulos, C.: Trojan resonant dynamics, stability, and chaotic diffusion, for parameters relevant to exoplanetary systems. Celest. Mech. Dyn. Astron.
**121**, 139–170 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar - Pousse, A., Robutel, P., Vienne, A.: On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited. Celest. Mech. Dyn. Astron. (2017)Google Scholar
- Roberts, G.: Linear stability of the elliptic Lagrangian triangle solutions in the three-body problem. JDIFE
**182**, 191–218 (2002)MathSciNetzbMATHGoogle Scholar - Robutel, P., Gabern, F.: The resonant structure of Jupiter’s Trojan asteroids I: long-term stability and diffusion. MNRAS
**372**, 1463–1482 (2006)ADSCrossRefGoogle Scholar - Robutel, P., Laskar, J.: Frequency map and global dynamics in the solar system I: short period dynamics of massless particles. Icarus
**152**, 4–28 (2001)ADSCrossRefGoogle Scholar - Robutel, P., Niederman, L., Pousse, A.: Rigorous treatment of the averaging process for co-orbital motions in the planetary problem. Comput. Appl. Math.
**35**(3), 675–699 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - Robutel, P., Pousse, A.: On the co-orbital motion of two planets in quasi-circular orbits. Celest. Mech. Dyn. Astron.
**117**, 17–40 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar - Sidorenko, V., Artemiev, A., Neishtadt, A., Zelenyi, L.: Quasi-satellite orbits in general context of dynamics at 1:1 mean motion resonance: a perturbative treatment (2014)Google Scholar