Skip to main content
SpringerLink
Log in

Lunar capture in the planar restricted three-body problem

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

Abstract

The capture dynamics is an important field in Astronomy and Astronautics. In this paper, the near-optimal lunar capture in the Earth–Moon transfer is investigated under the frame of the planar circular restricted three-body problem. We try to work out how to achieve the permanent lunar capture with the minimum maneuver consumption. This problem is decomposed into two parts: the pre-maneuver part and the post-maneuver part. In the pre-maneuver part, considering the criteria of the gravitational capture, we obtain the minimum pre-maneuver velocity via the numerical backward integration. In the post-maneuver part, using the Poincaré section and the KAM theory, we find the maximum post-maneuver velocity to achieve the permanent capture. Synthesized the results of the two parts, a new method is presented to find the near-optimal maneuver position and the minimum maneuver consumption. The method presented is simple and visible, and can provide abundant capture orbits for the design of low energy Earth–Moon transfers.

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
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

References

  • Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems III Series: Encyclopaedia of Mathematical Sciences, vol. 3, 3rd edn. Springer, Berlin (2006)

  • Astakhov, S., Burbanks, A., Wiggins, S., Farrelly, D.: Order and chaos in stellar and planetary systems. ASP Conf. Ser. 316, 80–85 (2004)

    ADS  Google Scholar 

  • Bate, R.R., Mueller, D.D., White, J.E.: Fundamentals of Astrodynamics, pp. 333–334. Dover Publications, New York (1971)

  • Belbruno, E.A.: Lunar capture orbits, a method of constructing Earth Moon trajectories and the Lunar Gas mission, AIAA-87-1054. In: 19th AIAA/DGLR/JSASS International Electric Propulsion Conference, Colorado Springs, CO, May (1987)

  • Belbruno, E.A.: Examples of the nonlinear dynamics of ballistic capture and escape in the Earth-Moon system, AIAA-90-2896. In: AIAA Astrodynamics Conference, Portland, OR, August (1990)

  • Belbruno, E.A.: Ballistic lunar capture transfer using the fuzzy boundary and solar perturbations: a survey. In: Proceedings for the International Congress of SETI Sail and Astrodynamics, Turin, Italy (1992)

  • Belbruno, E.A.: Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers, pp. 111–114, 145–150. Princeton University Press, Princeton (2004)

  • Belbruno, E.A., Miller, J.K.: A ballistic lunar capture trajectory for Japanese spacecraft Hiten, Jet Propulsion Laboratory, JPL IOM 312/90.4-1731, Internal Document, Pasadena, CA, June (1990)

  • Belbruno, E.A., Topputo, F., Gidea, M.: Resonance transitions associated to weak capture in the restricted three-body problem. Adv. Space Res. 42, 1330–135 (2008)

    Article  ADS  Google Scholar 

  • Ceccaroni, M., Biggs, J., Biasco, L.: Analytic estimates and topological properties of the weak stability boundary. Celest. Mech. Dyn. Astron. 114, 1–24 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Circi, C., Teofilatto, P.: Effect of planetary eccentricity on ballistic capture in the solar system. Celest. Mech. Dyn. Astron. 93, 69–86 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Fantino, E., Gomez, G., Masdemont, J.J., Ren, Y.: A note on libration point orbits, temporary capture and low-energy transfers. Acta Astronaut. 67, 1038–1052 (2010)

    Article  ADS  Google Scholar 

  • Garcia, F., Gomez, G.: A note on weak stability boundaries. Celest. Mech. Dyn. Astron. 97, 87–100 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Hamilton, D.P., Burns, J.A.: Orbital stability zones about asteroids. II—the destabilizing effects of eccentric orbits and of solar radiation. Icarus 96, 43–64 (1992)

    Article  ADS  Google Scholar 

  • Heppenheimer, T.A., Porco, C.: New contributions to the problem of capture. Icarus 30(2), 385–401 (1977)

    Article  ADS  Google Scholar 

  • Horedt, G.P.: Capture of planetary satellites. Astron. J. 81, 675–678 (1976)

    Article  ADS  Google Scholar 

  • Hyeraci, N., Topputo, F.: The role of true anomaly in ballistic capture. Celest. Mech. Dyn. Astron. 116, 175–193 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  • Jehn, R., Campagnola, S., García, D., Kemble, S.: Low-thrust approach and gravitational capture at mercury, In: Proceedings of the 18th International Symposium on Space Flights Dynamics, vol. 584, p. 487. ESA, Noordwick, The Netherlands (2004)

  • Jewitt, D., Sheppard, S.: Irregular Satellites in the context of planet formation. Space Sci. Rev. 114, 407–421 (2004)

    Google Scholar 

  • Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical systems, the three-body problem and space mission design. Springer, Berlin (2006)

    Google Scholar 

  • Krish, V., Belbruno, E.A., Hollister, W.M.: An investigation into critical aspects of a new form of low energy lunar transfer, the Belbruno-Miller trajectories. In: Proceedings of the AIAA/AAS Astrodynamics Conference, AIAA, pp. 435–444. Washington, DC (1992)

  • Landis, R.R.: Comet P/Shoemaker-Levy’s collision with Jupiter: covering HST’s planned observations from your planetarium. In: Proceedings of the International Planetarium Society Conference held at the Astronaut Memorial Planetarium and Observatory, Cocoa, Florida (1994)

  • Machuy, A.L., Prado, A.F.B.A., Stuchi, T.J.: Numerical study of the time required for the gravitational capture in the bi-circular four-body problem. Adv. Space Res. 40, 118–124 (2007)

    Article  ADS  Google Scholar 

  • Prado, A.F.B.A.: Numerical study and analytic estimation of forces acting in ballistic gravitational capture. J. Guid. Control Dyn. 25(2), 368–375 (2002)

    Article  Google Scholar 

  • Prado, A.F.B.A.: Numerical and analytical study of the gravitational capture in the bicircular problem. Adv. Space Res. 36, 578–584 (2005)

    Article  ADS  Google Scholar 

  • Prado, A.F.B.A., Vieira Neto, E.: Study of the gravitational capture in the elliptical restricted three-body problem. J. Astronaut. Sci. 54(3 & 4), 567–582 (2006)

  • Qi, Y., Xu, S.J., Qi, R.: Celest. Mech. Dyn. Astron. 120, 1–17 (2014). doi:10.1007/s10569-014-9554-7

  • Roncoli, R.B., Fujii, K.K.: Mission design overview for the gravity recovery and interior laboratory (GRAIL) mission, AIAA/AAS Astrodynamics Specialist Conference, Toronto, August (2010)

  • Rosaev, A.: On a possibility of a comet’s capture to a satellite orbit by splitting in the gravitational sphere of Jupiter. In: Proceedings of the 1-st International Conference CAMMAC-99, Vinnytsia, pp. 78–80, September 26–October 1 (1999)

  • Schoenmaekers, J., Horas, D., Pulido, J.A.: SMART-1 with solar electric propulsion to the Moon. In: Proceeding of the 16th International Symposium on Space Flight Dynamics. Pasadena, CA (2001)

  • Sousa Silva, P.A., Terra, M.: Applicability and dynamical characterization of the associated sets of the algorithmic weak stability boundary in the lunar sphere of influence. Celest.Mech.Dyn.Astron. 113, 141–168 (2012)

  • Szebehely, V.: Theory of Orbits. Academic Press, New York (1967)

  • Vieira Neto, E., Prado, A.F.B.A.: Time-of-flight analyses for the gravitational capture maneuver. J. Guid. Control Dyn. 21(1), 122–126 (1998)

  • Yamakawa, H., Kawaguchi, J., Ishii, N., Matsuo, H.: A numerical study of gravitational capture orbit in Earth–Moon system, pp. 92–186. AAS/AIAA Spaceflight Mechanics Meeting, Colorado Springs, Colorado (1992)

  • Yegorov, V.A.: The capture problem in the three-body restricted orbital problem, Technical Translation, F-9, NASA (1960)

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 11432001.

Author information

Authors and Affiliations

  1. School of Astronautics, Beihang University, Beijing, 100191, China

    Yi Qi & Shijie Xu

Authors
  1. Yi Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shijie Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Qi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, Y., Xu, S. Lunar capture in the planar restricted three-body problem. Celest Mech Dyn Astr 120, 401–422 (2014). https://doi.org/10.1007/s10569-014-9582-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10569-014-9582-3

Keywords

Search

Navigation