Skip to main content
SpringerLink
Log in

Retrograde resonances at high mass ratio in the circular restricted 3-body problem

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Studies involving retrograde orbits have been an emerging field in recent years, particularly in the case where there are resonances between objects orbiting in opposite directions. The high amount of data from space exploration missions increases the possibility of observing binary stellar systems which may have additional bodies with retrograde orbits. Furthermore, such orbits are relevant to understanding the dynamics of spacecrafts around binary asteroids, being essential to planning exploratory missions. In this work, we survey retrograde orbits around binary systems with mass ratio between 0.01 (hierarchical) and 0.5 (equal masses) in the framework of the planar circular restricted three-body problem (PCR3BP). We build surfaces of section and identify retrograde resonances up to fifth order, namely the 2/−1, 3/−2, 1/−1, 2/−3, 1/−2, 1/−3 and 1/−4 resonances. We conclude that retrograde resonances occur in binary systems at high mass ratio, including the co-orbital (1/−1) resonance. Period doubling bifurcations occur for the 1/−1 resonance, and period doubling and period tripling bifurcations are observed for the 1/−2 resonance. Asymmetric retrograde resonances of the type 1/−n occur for almost equal masses of the binary system. This study may be used for identifying retrograde planets in extrasolar systems and may possibly have applications to astrodynamics mission planning.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data availability

The surfaces of section and animations of the bifurcations are available: http://hdl.handle.net/11449/244803

Notes

  1. https://perezhz.github.io/TaylorIntegration.jl/latest/.

References

  1. Namouni, F., Morais, M.H.M.: An interstellar origin for jupiter’s retrograde co-orbital asteroid. Mon Notices R Astron Soc Lett 477(1), 117–121 (2018)

    Article  Google Scholar 

  2. Morais, M., Giuppone, C.: Stability of prograde and retrograde planets in circular binary systems. Mon Notices R Astron Soc 424(1), 52–64 (2012)

    Article  Google Scholar 

  3. Morais, M., Namouni, F.: Retrograde resonance in the planar three-body problem. Celest Mech Dyn Astron 117(4), 405–421 (2013)

    Article  MathSciNet  Google Scholar 

  4. Morais, M., Namouni, F.: Asteroids in retrograde resonance with jupiter and saturn. Mon Notices R Astron Soc Lett 436(1), 30–34 (2013)

    Article  Google Scholar 

  5. Morais, M., Namouni, F.: On retrograde orbits, resonances and stability. Comput Appl Math 35(3), 881–891 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Morais, M.H.M., Namouni, F.: A numerical investigation of coorbital stability and libration in three dimensions. Celest Mech Dyn Astron 125, 91–106 (2016)

    Article  MATH  Google Scholar 

  7. Morais, M., Namouni, F.: Periodic orbits of the retrograde coorbital problem. Mon Notices R Astron Socety 490(3), 3799–3805 (2019)

    Article  Google Scholar 

  8. Kotoulas, T., Voyatzis, G.: Retrograde periodic orbits in 1/2, 2/3 and 3/4 mean motion resonances with neptune. Celest Mech Dyn Astron 132(6), 1–16 (2020)

    MathSciNet  MATH  Google Scholar 

  9. Kotoulas, T., Voyatzis, G.: Planar retrograde periodic orbits of the asteroids trapped in two-body mean motion resonances with Jupiter. Planet Space Sci 182, 104846 (2020)

    Article  Google Scholar 

  10. Morais, M., Namouni, F., Voyatzis, G., Kotoulas, T.: A study of the 1/2 retrograde resonance: periodic orbits and resonant capture. Celest Mech Dyn Astron 133(5), 21 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kotoulas, T., Voyatzis, G., Morais, M.H.M.: Three-dimensional retrograde periodic orbits of asteroids moving in mean motion resonances with Jupiter. Planet Space Sci 210, 105374 (2022)

    Article  Google Scholar 

  12. Caritá, G.A., Cefali Signor, A., Morais, M.H.M.: A numerical study of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in planetary systems. Mon Notices R Astron Socety 515(2), 2280–2292 (2022)

    Article  Google Scholar 

  13. Signor, A.C., Caritá, G.A., Morais, M.H.M.: A numerical study of fourth and fifth order retrograde mean motion resonances in planetary systems. Mon Notices R Astron Soc, 336 (2023)

  14. Gayon, J., Bois, E.: Are retrograde resonances possible in multi-planet systems? Astron Astrophys 482(2), 665–672 (2008)

    Article  MATH  Google Scholar 

  15. Gayon-Markt, J., Bois, E.: On fitting planetary systems in counter-revolving configurations. Mon Notices R Astron Soc Lett 399(1), 137–140 (2009)

    Article  Google Scholar 

  16. Malmberg, D., Davies, M.B., Heggie, D.C.: The effects of fly-bys on planetary systems. Mon Notices R Astron Soc 411(2), 859–877 (2011)

    Article  Google Scholar 

  17. Buratti, B.J., Peter C. Thomas: Chapter 19 - planetary satellites. In: McFadden, L.-A., Weissman, P.R., Johnson, T.V. (eds.) Encyclopedia of the Solar System (Second Edition), Second edition edn., pp. 365–382. Academic Press, San Diego (2007). 10.1016/B978-012088589-3/50023-2 . https://www.sciencedirect.com/science/article/pii/B9780120885893500232

  18. Dell’Elce, L., Baresi, N., Naidu, S.P., Benner, L.A.M., Scheeres, D.J.: Numerical investigation of the dynamical environment of 65803 Didymos. Adv Space Res 59, 1304–1320 (2017). https://doi.org/10.1016/j.asr.2016.12.018

    Article  Google Scholar 

  19. Ćuk, M., Nesvornỳ, D.: Orbital evolution of small binary asteroids. Icarus 207(2), 732–743 (2010)

    Article  Google Scholar 

  20. Damme, F., Hussmann, H., Oberst, J.: Spacecraft orbit lifetime within two binary near-earth asteroid systems. Planet Space Scince 146, 1–9 (2017)

    Article  Google Scholar 

  21. Aljbaae, S., Chanut, T.G., Carruba, V., Souchay, J., Prado, A.F., Amarante, A.: The dynamical environment of asteroid 21 lutetia according to different internal models. Mon Notices R Astron Soc 464(3), 3552–3560 (2017)

    Article  Google Scholar 

  22. Aljbaae, S., Prado, A.F., Sanchez, D.M., Hussmann, H.: Analysis of the orbital stability close to the binary asteroid (90) antiope. Mon Notices R Astron Soc 496(2), 1645–1654 (2020)

    Article  Google Scholar 

  23. Benest, D.: Effects of the mass ratio on the existence of retrograde satellites in the circular plane restricted problem. Astron Astrophys 32, 39 (1974)

    Google Scholar 

  24. Benest, D.: Effects of the mass ratio on the existence of retrograde satellites in the circular plane restricted problem. ii. Astron Astrophys 45, 353–363 (1975)

    Google Scholar 

  25. Benest, D.: Effects of the mass ratio on the existence of retrograde satellites in the circular plane restricted problem. iii. Astron Astrophys 53, 231–236 (1976)

    Google Scholar 

  26. Gabern, F., Koon, W.S., Marsden, J.E.: Spacecraft dynamics near a binary asteroid. In: Conference Publications, vol. 2005, p. 297 (2005). American Institute of Mathematical Sciences

  27. Hussmann, H., Oberst, J., Wickhusen, K., Shi, X., Damme, F., Lüdicke, F., Lupovka, V., Bauer, S.: Stability and evolution of orbits around the binary asteroid 175706 (1996 fg3): implications for the marcopolo-r mission. Planet Space Sci 70(1), 102–113 (2012)

    Article  Google Scholar 

  28. Bosanac, N., Howell, K.C., Fischbach, E.: Stability of orbits near large mass ratio binary systems. Celest Mech Dyn Astron 122(1), 27–52 (2015)

    Article  MathSciNet  Google Scholar 

  29. Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge university press, ??? (1999)

  30. Benet, L., Sanders, D.P.: Taylorseries. jl: Taylor expansions in one and several variables in julia. J. Open Source Softw 4(36), 1043 (2019)

    Article  Google Scholar 

  31. Broucke, R.: Stability of periodic orbits in the elliptic, restricted three-body problem. AIAA J. 7(6), 1003–1009 (1969). https://doi.org/10.2514/3.5267

    Article  MathSciNet  MATH  Google Scholar 

  32. Hadjedemetriou, J.D.: Periodic orbits in gravitational systems. In: Chaotic Worlds: from Order to Disorder in Gravitational N-Body Dynamical Systems, pp. 43–79 (2006). Springer

Download references

Acknowledgements

We thank George Voyatzis for directing us to the technical report on periodic orbits by Roger Broucke. The computational resources were supplied in part by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP).

Funding

This study was financed in part by the Coordenação de Aperfeicoamento de Pessoal de Nivel Superior—Brasil (CAPES)—Finance Code 001. The authors wish to express their appreciation for the support provided by Grant 309089/2021-2 from the National Council for Scientific and Technological Development (CNPq), Grants from São Paulo Research Foundation FAPESP/2021/11982-5, FAPESP/2022/08716-4, FAPESP/2019/07329-4, and FAPESP/2016/24561-0. This publication has been supported by the RUDN University Scientific Projects Grant System, Project No. 202235-2-000.

Author information

Authors and Affiliations

  1. Universidade Estadual Paulista (Unesp), Instituto de Geociências e Ciências Exatas, Rio Claro, São Paulo, 13506-900, Brazil

    G. A. Caritá, A. C. Signor, M. H. M. Morais & R. Egydio de Carvalho

  2. PPG em Engenharia e Tecnologia Espacial, CEPE, INPE, Av. dos Astronautas, 1.758 - Jardim da Granja, São José dos Campos, São Paulo, 12227-010, Brazil

    G. A. Caritá & A. F. B. A. Prado

  3. Academy of Engineering, RUDN University, Miklukho-Maklaya Street 6, Moscow, Moscow, Russia, 117198

    A. F. B. A. Prado

Authors
  1. G. A. Caritá
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. C. Signor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. H. M. Morais
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Egydio de Carvalho
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. F. B. A. Prado
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

HM and GC contributed equally to the study’s conception and design. Material preparation and data collection were performed by GC, AS, and HM. Data interpretation was done by GC, HM, RC, and AS. The first draft of the manuscript was written by GC, AS, HM. The numerical simulations were done using the computational resources of AP and HM. All authors have commented on and suggested previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to G. A. Caritá.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caritá, G.A., Signor, A.C., Morais, M.H.M. et al. Retrograde resonances at high mass ratio in the circular restricted 3-body problem. Nonlinear Dyn 111, 17021–17035 (2023). https://doi.org/10.1007/s11071-023-08779-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08779-y

Keywords

Search

Navigation