Advertisement

Celestial Mechanics and Dynamical Astronomy

, Volume 117, Issue 4, pp 405–421 | Cite as

Retrograde resonance in the planar three-body problem

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

Abstract

We continue the investigation of the dynamics of retrograde resonances initiated in Morais and Giuppone (Mon Notices R Astron Soc 424:52–64, doi: 10.1111/j.1365-2966.2012.21151.x, 2012). After deriving a procedure to deduce the retrograde resonance terms from the standard expansion of the three-dimensional disturbing function, we concentrate on the planar problem and construct surfaces of section that explore phase-space in the vicinity of the main retrograde resonances (2/\(-\)1, 1/\(-\)1 and 1/\(-\)2). In the case of the 1/\(-\)1 resonance for which the standard expansion is not adequate to describe the dynamics, we develop a semi-analytic model based on numerical averaging of the unexpanded disturbing function, and show that the predicted libration modes are in agreement with the behavior seen in the surfaces of section.

Keywords

Resonance Three-body problem Surface of section Co-orbital resonance 

Notes

Acknowledgments

We thank both reviewers for helpful suggestions that improved the article’s clarity. We acknowledge financial support from FCT-Portugal (PEst-C/CTM/LA0025/2011). The surfaces of section computations were performed on the Blafis cluster at the University of Aveiro.

Supplementary material

10569_2013_9519_MOESM1_ESM.pdf (1.7 mb)
Supplementary material 1 (pdf 1732 KB)
10569_2013_9519_MOESM2_ESM.pdf (8.4 mb)
Supplementary material 2 (pdf 8596 KB)

References

  1. Ellis, K.M., Murray, C.D.: The disturbing function in solar system dynamics. Icarus 147, 129–144 (2000). doi: 10.1006/icar.2000.6399 ADSCrossRefGoogle Scholar
  2. Gayon, J., Bois, E.: Are retrograde resonances possible in multi-planet systems? Astron. Astrophys. 482, 665–672 (2008). doi: 10.1051/0004-6361:20078460 ADSCrossRefzbMATHGoogle Scholar
  3. Gayon, J., Bois, E., Scholl, H.: Dynamics of planets in retrograde mean motion resonance. Celest. Mech. Dyn. Astron. 103, 267–279 (2009). doi: 10.1007/s10569-009-9191-8 MathSciNetADSCrossRefzbMATHGoogle Scholar
  4. Morais, M.H.M., Giuppone, C.A.: Stability of prograde and retrograde planets in circular binary systems. Mon. Notices R. Astron. Soc. 424, 52–64 (2012). doi: 10.1111/j.1365-2966.2012.21151.x ADSCrossRefGoogle Scholar
  5. Morais, M.H.M., Namouni, F.: Asteroids in retrograde resonance with Jupiter and Saturn. MNRAS Lett. (2013). doi: 10.1093/mnrasl/slt106
  6. Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  7. Namouni, F.: Secular interactions of coorbiting objects. Icarus 137, 293–314 (1999). doi: 10.1006/icar.1998.6032 ADSCrossRefGoogle Scholar
  8. 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 Google Scholar
  9. Saha, P., Tremaine, S.: The orbits of the retrograde Jovian satellites. Icarus 106, 549 (1993). doi: 10.1006/icar.1993.1192 ADSCrossRefGoogle Scholar
  10. Triaud, A.H.M.J., Collier Cameron, A., Queloz, D., Anderson, D.R., Gillon, M., Hebb, L. et al.: Spin-orbit angle measurements for six southern transiting planets. New insights into the dynamical origins of hot Jupiters. Astron. Astrophys. 524, A25, doi: 10.1051/0004-6361/201014525 (2010)
  11. Winter, O.C., Murray, C.D.: Resonance and chaos. I. First-order interior resonances. Astron. Astrophys. 319, 290–304 (1997a)ADSGoogle Scholar
  12. Winter, O.C., Murray, C.D.: Resonance and chaos. II. Exterior resonances and asymmetric libration. Astron. Astrophys. 328, 399–408 (1997b)ADSGoogle Scholar
  13. Yokoyama, T., Do Nascimento, C., Santos, M.T.: Inner satellites of Neptune: I the disturbing function. Adv. Space Res. 36, 569–577 (2005). doi: 10.1016/j.asr.2005.08.002 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Physics & I3NUniversity of AveiroAveiroPortugal
  2. 2.Université de Nice, CNRSNiceFrance

Personalised recommendations