Skip to main content
SpringerLink
Log in

Image classification of retrograde resonance in the planar circular restricted three-body problem

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

The study of resonances in celestial mechanics is crucial for understanding the dynamics of planetary or stellar systems. This study focuses on presenting a method for investigating the topology and resonant structures of a dynamical system. To illustrate the strength of the method, we have applied our method to retrograde resonances in the planar circular restricted three-body problem within binary star systems. Because of the high mass ratio systems, the techniques based on perturbation of the two-body orbit are not the ideal to analyze the system. Consequently, resonant angles could be meaningless, necessitating alternative methods for resonance identification. To address this challenge, an image classification-based machine learning model is implemented to identify resonances based on the shape of orbits in the rotating frame. Initially, the model is trained on empirical cases with low mass ratios using the resonant angle as a starting point for resonance identification. The model’s performance is validated against existing literature results. The model results demonstrate successful classification and identification of retrograde resonances in both empirical and non-empirical cases. The model accurately captures the resonance patterns and provides initial insights into the short-term stability of the corresponding resonances.

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 data are available in appendix. Any additional data not found in these sources can be obtained from the first author upon reasonable request.

References

  • Beaugé, C., Nesvornỳ, D.: Multiple-planet scattering and the origin of hot jupiters. Astrophys. J. 751(2), 119 (2012)

    Article  ADS  Google Scholar 

  • 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  ADS  Google Scholar 

  • 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  ADS  Google Scholar 

  • Broucke, R.: Stability of periodic orbits in the elliptic, restricted three-body problem. AIAA J. 7(6), 1003–1009 (1969)

    Article  ADS  Google Scholar 

  • 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. Not. R. Astron. Soc. 515(2), 2280–2292 (2022)

    Article  ADS  Google Scholar 

  • Caritá, G., Signor, A., Morais, M., Carvalho, R.E., Prado, A.: Retrograde resonances at high mass ratio in the circular restricted 3-body problem. Nonlinear Dyn. 111(18), 17021–17035 (2023)

    Article  Google Scholar 

  • Carruba, V., Aljbaae, S., Domingos, R.C., Barletta, W.: Artificial neural network classification of asteroids in the m1: 2 mean-motion resonance with mars. Mon. Not. R. Astron. Soc. 504(1), 692–700 (2021)

    Article  ADS  Google Scholar 

  • Carruba, V., Aljbaae, S., Domingos, R., Huaman, M., Barletta, W.: Machine learning applied to asteroid dynamics. Celest. Mech. Dyn. Astron. 134(4), 36 (2022a)

    Article  MathSciNet  ADS  Google Scholar 

  • Carruba, V., Aljbaae, S., Domingos, R.C., Huaman, M., Martins, B.: Identifying the population of stable \(\nu \)6 resonant asteroids using large data bases. Mon. Not. R. Astron. Soc. 514(4), 4803–4815 (2022b)

    Article  ADS  Google Scholar 

  • Carruba, V., Aljbaae, S., Caritá, G., Domingos, R.C., Martins, B.: Optimization of artificial neural networks models applied to the identification of images of asteroids’ resonant arguments. Celest. Mech. Dyn. Astron. 134(6), 59 (2022c)

    Article  MathSciNet  ADS  Google Scholar 

  • Carruba, V., Aljbaae, S., Caritá, G., Lourenço, M., Martins, B., Alves, A.: Imbalanced classification applied to asteroid resonant dynamics. Front. Astron. Space Sci. 10, 1196223 (2023)

    Article  ADS  Google Scholar 

  • Chambers, J.E.: A hybrid symplectic integrator that permits close encounters between massive bodies. Mon. Not. R. Astron. Soc. 304(4), 793–799 (1999)

    Article  ADS  Google Scholar 

  • Cincotta, P.M., Simó, C.: Simple tools to study global dynamics in non-axisymmetric galactic potentials—I. Astron. Astrophys. Suppl. Ser. 147(2), 205–228 (2000)

    Article  ADS  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

  • Gayon-Markt, J., Bois, E.: On fitting planetary systems in counter-revolving configurations. Mon. Not. R. Astron. Soc. Lett. 399(1), 137–140 (2009)

    Article  ADS  Google Scholar 

  • Goździewski, K.: Stability of the HD 12661 planetary system. Astron. Astrophys. 398(3), 1151–1161 (2003)

    Article  ADS  Google Scholar 

  • 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 (2020a)

    Article  Google Scholar 

  • 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 (2020b)

    MathSciNet  Google Scholar 

  • Malmberg, D., Davies, M.B., Heggie, D.C.: The effects of fly-bys on planetary systems. Mon. Not. R. Astron. Soc. 411(2), 859–877 (2011)

    Article  ADS  Google Scholar 

  • Morais, M., Giuppone, C.: Stability of prograde and retrograde planets in circular binary systems. Mon. Not. R. Astron. Soc. 424(1), 52–64 (2012)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  • Morais, M., Namouni, F.: Asteroids in retrograde resonance with jupiter and saturn. Mon. Not. R. Astron. Soc. Lett. 436(1), 30–34 (2013)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Morais, H., Namouni, F.: Reckless orbiting in the solar system. Nature 543(7647), 635–636 (2017)

    Article  CAS  PubMed  ADS  Google Scholar 

  • Morais, M., Namouni, F.: Periodic orbits of the retrograde coorbital problem. Mon. Not. R. Astron. Soc. 490(3), 3799–3805 (2019)

    Article  ADS  Google Scholar 

  • 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), 1–14 (2021)

    Article  MathSciNet  ADS  Google Scholar 

  • Namouni, F., Morais, M.H.M.: Resonance capture at arbitrary inclination. Mon. Not. R. Astron. Soc. 446(2), 1998–2009 (2015)

    Article  ADS  Google Scholar 

  • Namouni, F., Morais, H.: Coorbital capture at arbitrary inclination. Comput. Appl. Math. 37(1), 65–71 (2018)

    Article  MathSciNet  Google Scholar 

  • Rein, H., Liu, S., Spiegel, D.S., Fujii, A., Tamayo, D., Silburt, A., et al.: REBOUND. https://rebound.readthedocs.io/en/latest/ (2001)

  • 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. Not. R. Astron. Soc. 520(3), 4696–4714 (2023)

    Article  ADS  Google Scholar 

  • Smirnov, E.A., Markov, A.B.: Identification of asteroids trapped inside three-body mean motion resonances: a machine-learning approach. Mon. Not. R. Astron. Soc. 469(2), 2024–2031 (2017)

    Article  ADS  Google Scholar 

  • Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., Salakhutdinov, R.: Dropout: a simple way to prevent neural networks from overfitting. J. Mach. Learn. Res. 15(1), 1929–1958 (2014)

    MathSciNet  Google Scholar 

  • Wiegert, P., Connors, M., Veillet, C.: A retrograde co-orbital asteroid of jupiter. Nature 543(7647), 687–689 (2017)

    Article  CAS  PubMed  ADS  Google Scholar 

Download references

Acknowledgements

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 Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001/PRInt. Carruba V. acknowledges the support of the Brazilian National Research Council (CNPq, Grant 304168/2021-1). We acknowledge the support Grants 2022/08716-4, 2021/08716-4, and 2021/08274-9 from São Paulo Research Foundation (FAPESP).

Author information

Authors and Affiliations

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

    G. A. Caritá, S. Aljbaae & A. F. B. A. Prado

  2. Institute of Planetary Research, DLR, Rutherfordstraße 2, 12489, Berlin, Germany

    G. A. Caritá & H. Hussmann

  3. Make the Way, R. Elvira Ferraz 250 - FL Office 305/306, Vila Olimpia, São Paulo, São Paulo, 04545-015, Brazil

    S. Aljbaae

  4. Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista (UNESP), Av. 24-A, 1515, Rio Claro, São Paulo, 13506-900, Brazil

    M. H. M. Morais & A. C. Signor

  5. School of Natural Sciences and Engineering, São Paulo State University (UNESP), Guaratinguetá, São Paulo, 12516-410, Brazil

    V. Carruba

Authors
  1. G. A. Caritá
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Aljbaae
    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. A. C. Signor
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. V. Carruba
    View author publications

    You can also search for this author in PubMed Google Scholar

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

    You can also search for this author in PubMed Google Scholar

  7. H. Hussmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

GC, SA, and AP contributed equally to the study’s conception and design. Material preparation and data collection were performed by GC, AS, SA, and HM. Data interpretation was done by GC, HM, VC, AP, and HH. The first draft of the manuscript was written by GC, SA, and HM. The numerical simulations were done using the computational resources of AP. 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.

Appendices

Appendix A: Codes and repositories

The codes, sample of the training data, and the model are available in the: https://github.com/gacarita/image_classification_retrograde_resonances.

Appendix B: Additional data

See Figs. 13, 14, 15, 16, and 17.

Fig. 13
figure 13

Status map for each family presented in Fig. 8. The blue points represent the ones that have survived for 50000 periods of the binary. Gray and red colors illustrate collisions and escape events

Full size image
Fig. 14
figure 14

Classification of the resonances of the initial conditions presented on the grid of \(x_0\) versus C (Jacobi constant) for the binary mass ratio of \(\mu =0.1\). The classification includes the 1/− 1, 2/− 1, 1/− 2, and circular orbits

Full size image
Fig. 15
figure 15

Classification of the resonances of the initial conditions presented on the grid of \(x_0\) versus C (Jacobi constant) for the binary mass ratio of \(\mu =0.2\). The classification includes the 1/− 1, 2/− 1, 1/− 2, and circular orbits

Full size image
Fig. 16
figure 16

Classification of the resonances of the initial conditions presented on the grid of \(x_0\) versus C (Jacobi constant) for the binary mass ratio of \(\mu =0.3\). The classification includes the 1/− 1, 2/− 1, 1/− 2, and circular orbits

Full size image
Fig. 17
figure 17

Classification of the resonances of the initial conditions presented on the grid of \(x_0\) versus C (Jacobi constant) for the binary mass ratio of \(\mu =0.4\). The classification includes the 1/− 1, 2/− 1, 1/− 2, and circular orbits

Full size image

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., Aljbaae, S., Morais, M.H.M. et al. Image classification of retrograde resonance in the planar circular restricted three-body problem. Celest Mech Dyn Astron 136, 10 (2024). https://doi.org/10.1007/s10569-024-10181-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10569-024-10181-8

Keywords

Search

Navigation