Skip to main content
SpringerLink
Log in

Families of Orbits in the Vicinity of the Collinear Libration Points

  • Original Article
  • Published:
The Journal of the Astronautical Sciences Aims and scope Submit manuscript

Abstract

In recent years, three-dimensional periodic and quasi-periodic orbits near the collinear libration points in the Sun-Earth/Moon three-body problem have been the focus of great interest for space mission design. Thus, more effort in the astrodynamics community has been directed toward analysis and computation of families of such orbits. But families of periodic orbits in the three-body problem have been the subject of much study for many decades. The first such orbit to be employed for a spacecraft trajectory (ISEE-3) was a member of a particular type of simply symmetric three-dimensional family, i.e., a halo orbit. Thus, attention shifted to further analysis of these periodic halo families. Some of the significant work in the development of periodic solutions in the three-body problem has been reviewed and a number of the highlights from the analysis and eventual numerical computation of halo families is presented here. The halo families of periodic orbits extend from each of the libration points to the nearest primary; they appear to exist for all values of the mass ratio. Thus, this further understanding may serve to support future spacecraft mission planning as well.

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

Similar content being viewed by others

References

  1. HOWELL, K. C. and FARQUHAR, R. W. “John Breakwell, the Restricted Problem, and Halo Orbits,” Acta Astronautica, Vol. 29, No. 6, 1993, pp. 485–488.

    Article  Google Scholar 

  2. FARQUHAR, R. W. “Station-Keeping in the Vicinity of Collinear Libration Points with an Application to a Lunar Communications Problem,” Space Flight Mechanics, Vol. 11, Science and Technology Series, American Astronautical Society, New York, 1967, pp. 519–535.

  3. FARQUHAR, R. W. and KAMEL, A. A. “Quasi-Periodic Orbits About the Translunar Libration Point,” Celestial Mechanics, Vol. 7, 1973, pp. 458–473.

    Article  Google Scholar 

  4. FARQUHAR, R. W. “The Control and Use of Libration Point Satellites,” NASA TR T-346, 1970.

  5. POINCARÉ, H. Les Méthodes Nouvelles de la Mécanique Céleste, Vol. I, 1892; New Methods of Celestial Mechanics (English Translation), Vol. 13, History of Modern Physics and Astronomy, American Institute of Physics, 1993.

  6. MOULTON, F. R. Periodic Orbits, Carnegie Institution of Washington Publications, No. 161, Washington, 1920.

  7. SZEBEHELY, V. Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, New York, 1967.

    MATH  Google Scholar 

  8. GOUDAS, C. L. “Three-Dimensional Periodic Orbits and Their Stability,” Icarus, Vol. 2, 1963, pp. 1–18.

    Article  Google Scholar 

  9. BRAY, T. A. and GOUDAS, C. L. “Three-Dimensional Periodic Oscillations about L1, L2, and L3,” Advances in Astronomy and Astrophysics (Z. Kopal, Editor), Vol. 5, Academic Press, New York, 1967, pp. 71–130.

    Google Scholar 

  10. HÉNON, M. “Vertical Stability of Periodic Orbits in the Restricted Problem, I. Equal Masses,” Astronomy and Astrophysics, Vol. 28, 1973, pp. 415–426.

    MATH  Google Scholar 

  11. KAZANTZIS, P. G. “The Structure of Periodic Solutions in the Restricted Problem of Three Bodies: Sun-Jupiter Case,” Astrophysics and Space Science, Vol. 59, 1978, pp. 355–371.

    Article  Google Scholar 

  12. ROBIN, I. A. and MARKELLOS, V. V. “Numerical Determination of Three-Dimensional Periodic Orbits Generated from Vertical Self-Resonant Satellite Orbits,” Celestial Mechanics, Vol. 21, 1980, pp. 395–434.

    Article  MathSciNet  Google Scholar 

  13. MICHALODIMITRAKIS, M. “A New Type of Connection Between Families of Periodic Orbits of the Restricted Problem,” Astronomy and Astrophysics, Vol. 64, 1978, pp. 83–86.

    MathSciNet  MATH  Google Scholar 

  14. ICHTIAROGLOU, S. S. and MICHALODIMITRAKIS, M. “Three-Body Problem: The Existence of Families of Three-Dimensional Periodic Orbits which Bifurcate from Planar Periodic Orbits,” Astronomy and Astrophysics, Vol. 81, 1980, pp. 30–32.

    MATH  Google Scholar 

  15. ZAGOURAS, C. G. and KAZANTZIS, P. G. “Three-Dimensional Periodic Oscillations Generating from Plane Periodic Ones Around the Collinear Lagrangian Points,” Astrophysics and Space Science, Vol. 61, 1979, pp. 389–409.

    Article  Google Scholar 

  16. MARCHAL, C. The Three-Body Problem, Elsevier Science Publishing Company, New York, 1990.

    MATH  Google Scholar 

  17. BARDEN, B. T. and HOWELL, K. C. “Fundamental Motions Near Collinear Libration Points and Their Transitions,” Journal of the Astronautical Sciences, Vol. 46, No. 4, October– December 1998, pp. 361–378.

    MathSciNet  Google Scholar 

  18. BARDEN, B. T. and HOWELL, K. C. “Formation Flying in the Vicinity of Libration Point Orbits,” Paper No. AAS 98-169, AAS/AIAA Space Flight Mechanics Meeting, Monterey, California, February 1998.

  19. HOWELL, K. C. and PERNICKA, H. J. “Numerical Determination of Lissajous Trajectories in the Restricted Three-Body Problem,” Celestial Mechanics, Vol. 41, Nos. 1–4, 1988, pp. 107– 124.

    MATH  Google Scholar 

  20. GÓMEZ, G., JORBA, À, MASDEMONT, J., and SIMÓ, C. “Study Refinement of Semianalytical Halo Orbit Theory,” Final Report, ESOC Contract No. 8625/89/D/MD(SC), 1991.

  21. SIMÓ, C. “Effective Computations in Hamiltonian Dynamics,” in Cent ans après les Méthodes Nouvelles de H. Poincaré, Société Mathématique de France, 1996, pp. 1–23.

  22. GÓMEZ, G., MASDEMONT, J., and SIMÓ, C. “Lissajous Orbits Around Halo Orbits,” Paper No. 97-106, AAS/AIAA Spaceflight Mechanics Meeting, Huntsville, Alabama, February 1997.

  23. FARQUHAR, R. W., MUHONEN, D. P., and RICHARDSON, D. L. “Mission Design for a Halo Orbiter of the Earth,” Journal of Spacecraft and Rockets, Vol. 14, 1977, pp. 170–177.

    Article  Google Scholar 

  24. RICHARDSON, D. L. “Halo-Orbit Formulation for the ISEE-3 Mission,” Journal of Guidance and Control, Vol. 3, No. 6, November–December 1980, pp. 543–548.

    Article  Google Scholar 

  25. BREAKWELL, J.V. and BROWN, J. V. “The ‘Halo’ Family of 3-Dimensional Periodic Orbits in the Earth-Moon Restricted 3-Body Problem,” Celestial Mechanics, Vol. 20, 1979, pp. 389–404.

    Article  Google Scholar 

  26. HOWELL, K. C. “Three-Dimensional, Periodic, ‘Halo’ Orbits,” Celestial Mechanics, Vol. 32, 1984, pp. 53–71.

    Article  MathSciNet  Google Scholar 

  27. BREAKWELL, J.V. and HOWELL, K.C. “Almost Rectilinear Halo Orbits,” Celestial Mechanics, Vol. 32, 1984, pp. 29–52.

    Article  MathSciNet  Google Scholar 

  28. HOWELL, K. C. and CAMPBELL, E. T. “Families of Periodic Orbits that Bifurcate from Halo Families in the Circular Restricted Three-Body Problem,” Paper No. AAS 99-161, AAS/AIAA Spaceflight Mechanics Conference, Breckenridge, Colorado, February 1999.

  29. GÓMEZ, G., JORBA, À., MASDEMONT, J., and SIMÓ, C. “Study of the Transfer from the Earth to a Halo Orbit Around the Equilibrium Point L1,” Celestial Mechanics and Dynamical Astronomy, Vol. 56, No. 4, 1993, pp. 541–562.

    Article  Google Scholar 

  30. HOWELL, K. C., BARDEN, B. T., and LO, M. W. “Application of Dynamical Systems Theory to Trajectory Design for a Libration Point Mission,” Journal of the Astronautical Sciences, Vol. 45, No. 2, April–June 1997, pp. 161–178.

    MathSciNet  Google Scholar 

  31. GUZMÁN, J. J., COOLEY, D. S., HOWELL, K. C., and FOLTA, D. “Trajectory Design Strategies for Libration Point Missions that Incorporate Invariant Manifolds and SWINGBY,” Paper No. AAS 98-349, AAS/GSFC International Symposium on Space Flight Dynamics, Greenbelt, Maryland, May 1998.

Download references

Author information

Authors and Affiliations

  1. School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, 47907, USA

    K. C. Howell (Professor)

Authors
  1. K. C. Howell
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Howell, K.C. Families of Orbits in the Vicinity of the Collinear Libration Points. J of Astronaut Sci 49, 107–125 (2001). https://doi.org/10.1007/BF03546339

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03546339

Search

Navigation