A numerical investigation of coorbital stability and libration in three dimensions

  • M. H. M. Morais
  • F. Namouni
Original Article


Motivated by the dynamics of resonance capture, we study numerically the coorbital resonance for inclination \(0\le I\le 180^\circ \) in the circular restricted three-body problem. We examine the similarities and differences between planar and three dimensional coorbital resonance capture and seek their origin in the stability of coorbital motion at arbitrary inclination. After we present stability maps of the planar prograde and retrograde coorbital resonances, we characterize the new coorbital modes in three dimensions. We see that retrograde mode I (R1) and mode II (R2) persist as we change the relative inclination, while retrograde mode III (R3) seems to exist only in the planar problem. A new coorbital mode (R4) appears in 3D which is a retrograde analogue to an horseshoe-orbit. The Kozai–Lidov resonance is active for retrograde orbits as well as prograde orbits and plays a key role in coorbital resonance capture. Stable coorbital modes exist at all inclinations, including retrograde and polar obits. This result confirms the robustness the coorbital resonance at large inclination and encourages the search for retrograde coorbital companions of the solar system’s planets.


Co-orbital resonance Resonance Three-body problem Retrograde resonances Kozai-Lidov mechanism Resonance trapping 



We thank Nelson Callegari Jr. for assistance with computational resources.


  1. Cincotta, P.M., Simó, C.: Simple tools to study global dynamics in non-axisymmetric galactic potentials—I. A&AS 147, 205–228 (2000). doi: 10.1051/aas:2000108 ADSCrossRefGoogle Scholar
  2. Goździewski, K.: Stability of the HD 12661 planetary system. Astron. Astrophys. 398, 1151–1161 (2003). doi: 10.1051/0004-6361:20021713 ADSCrossRefGoogle Scholar
  3. Mikkola, S., Brasser, R., Wiegert, P., Innanen, K.: Asteroid 2002 VE68, a quasi-satellite of Venus. Mon. Not. R. Astron. Soc. 351, L63–L65 (2004). doi: 10.1111/j.1365-2966.2004.07994.x ADSCrossRefGoogle Scholar
  4. Morais, M.H.M., Giuppone, C.A.: Stability of prograde and retrograde planets in circular binary systems. Mon. Not. R. Astron. Soc. 424, 52–64 (2012). doi: 10.1111/j.1365-2966.2012.21151.x. arXiv:1204.4718
  5. Morais, M.H.M., Namouni, F.: Asteroids in retrograde resonance with Jupiter and Saturn. Mon. Not. R. Astron. Soc. 436, L30–L34 (2013a). doi: 10.1093/mnrasl/slt106. arXiv:1308.0216
  6. Morais, M.H.M., Namouni, F.: Retrograde resonance in the planar three-body problem. Celest. Mech. Dyn. Astron. 117, 405–421 (2013b). doi: 10.1007/s10569-013-9519-2. arXiv:1305.0016
  7. Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  8. Namouni, F.: Secular interactions of coorbiting objects. Icarus 137, 293–314 (1999). doi: 10.1006/icar.1998.6032 ADSCrossRefGoogle Scholar
  9. Namouni, F., Morais, M.H.M.: Resonance capture at arbitrary inclination. Mon. Not. R. Astron. Soc. 446, 1998–2009 (2015). doi: 10.1093/mnras/stu2199. arXiv:1410.5383
  10. Namouni, F., Christou, A.A., Murray, C.D.: Coorbital dynamics at large eccentricity and inclination. Phys. Rev. Lett. 83, 2506–2509 (1999). doi: 10.1103/PhysRevLett.83.2506 ADSCrossRefGoogle Scholar
  11. Nesvorný, D., Thomas, F., Ferraz-Mello, S., Morbidelli, A.: A perturbative treatment of the co-orbital motion. Celest. Mech. Dyn. Astron. 82, 323–361 (2002). doi: 10.1023/A:1015219113959 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. Yu, Q., Tremaine, S.: Resonant Capture by Inward-migrating Planets. Astron. J. 121, 1736–1740 (2001). doi: 10.1086/319401. arXiv:astro-ph/0009255

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Instituto de Geociências e Ciências ExatasUniversidade Estadual PaulistaRio ClaroBrazil
  2. 2.Université Côte d’Azur, CNRS, ObservatoireNiceFrance

Personalised recommendations