Skip to main content
SpringerLink
Log in

Families of periodic solutions in three-dimensional restricted three-body problem

  • Published:
Cosmic Research Aims and scope Submit manuscript

Abstract

Planar orbits of three-dimensional restricted circular three-body problem are considered as a special case of three-dimensional orbits, and the second-order monodromy matrices M (in coordinate z and velocity v z ) are calculated for them. Semi-trace s of matrix M determines vertical stability of an orbit. If |s| ≤ 1, then transformation of the subspace (z, v z ) in the neighborhood of solution for the period is reduced to deformation and a rotation through angle φ, cosφ = s. If the angle ϕ can be rationally expressed through 2π,φ = 2π·p/q, where p and q are integer, then a planar orbit generates the families of three-dimensional periodic solutions that have a period larger by a factor of q (second kind Poincareé periodic solutions). Directions of continuation in the subspace (z, v z ) are determined by matrix M. If |s| < 1, we have two new families, while only one exists at resonances 1: 1 (s = 1) and 2: 1 (s = −1). In the course of motion along the family of three-dimensional periodic solutions, a transition is possible from one family of planar solutions to another one, sometimes previously unknown family of planar solutions.

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. Kardashev, N.S., Kreisman, B.B., and Ponomarev, Yu.N., New Orbit and New Capabilities of the RADIOASTRON Project, in Radioastronomicheskaya tekhnika i metody (Radio Astronomy Instrumentation and Technique), vol. 228 of Trudy FIAN, Moscow, 2000, pp. 3–12.

  2. Szebehely, V., Theory of Orbits: The Restricted Problem of Three Bodies, New York: Academic, 1967. Translated under the title Teoriya orbit. Ogranichennaya zadacha trekh tel, Moscow: Nauka, 1982.

    Google Scholar 

  3. Poincare, H., Les Methodes Nouvelles de la Mecanique Celeste, vol. 1–3, Paris: Gauthier-Villars, 1899.

    Google Scholar 

  4. Bruno, A.D., Ogranichennaya zadacha trekh tel (The Restricted Three-Body Problem), Moscow: Nauka, 1990.

    MATH  Google Scholar 

  5. Kreisman, B.B., Symmetrical Periodic Solutions to the Planar Restricted Three-Body Problem, Preprint of P.N. Lebedev Physical Inst., Russ. Acad. Sci., Moscow, 1997, no. 66.

  6. Kreisman, B.B., Gravitational Maneuver Using the Families of Super-Unstable Orbits around Libration Points, Kosm. Issled, 2003, vol. 41, no. 1, pp. 57–68.

    Google Scholar 

  7. Kreisman, B.B., Families of Periodic Solutions to a Hamiltonian System with Two Degrees of Freedom: Asymmetric Periodic Solutions for a Planar Restricted Three-Body Problem, Preprint of P.N. Lebedev Physical Inst., Russ. Acad. Sci., Moscow, 2003, no. 30.

  8. Kreisman, B.B., Families of Periodic Solutions to Hamiltonian Systems: Nonsymmetrical Periodic Solutions for a Planar Restricted Three-Body Problem, Kosm. Issled. (Cosmic Res.), 2005, vol. 43, no. 2, pp. 88–110.

    ADS  Google Scholar 

  9. Kreisman, B.B., Periodic Solutions for a Three-Dimensional Restricted Three-Body Problem, Trudy Gos. Astr. Inst. im. P.K. Shternberga, 2004, vol. LXXV, pp. 210–211.

    Google Scholar 

  10. Goudas, C.L., Three-Dimensional Periodic Orbits and Their Stability, Icarus, 1963, vol. 2, pp. 1–18.

    Article  ADS  Google Scholar 

  11. Tychina, P.A., Evolution of Some Periodic Solutions to Three-Dimensional Restricted Circular Three-Body Problem, Preprint of P.N. Lebedev Physical Inst., Russ. Acad. Sci., Moscow, 1996, no. 62.

  12. Henon, M., Vertical Stability of Periodic Orbits in the Restricted Problem, Astron. and Astrophys., 1973, vol. 28, no. 3, pp. 415–426.

    ADS  Google Scholar 

  13. Lidov, M.L., About One Family of Three-Dimensional Periodic Orbits around the Moon and Planets, Dokl. Akad. Nauk SSSR, 1977, vol. 233, no. 6, pp. 1068–1071.

    ADS  Google Scholar 

  14. Lidov, M.L. and Rabinovich, V.Yu., A Study of Families of Three-Dimensional Periodic Orbits for the Three-Body Problem, Kosm. Issled. (Cosmic Res.), 1979, vol. 17, no. 3, pp. 323–332.

    ADS  Google Scholar 

  15. Lidov, M.L., A Method of Constructing the Families of Three-Dimensional Periodic Orbits in the Hill Problem, Kosm. Issled., 1982, vol. 20, no. 6, pp. 787–803.

    ADS  Google Scholar 

  16. Lidov, M.L. and Lyakhova, V.A., Families of Three-Dimensional Periodic Orbits of the Hill Problem and Their Stability, Kosm. Issled., 1983, vol. 21, no. 1, pp. 3–11.

    ADS  Google Scholar 

  17. Lidov, M.L. and Lyakhova, V.A., A Variant of the Orbit for Near-Earth Radio Interferometer, Pisma Astron. Zh., 1988, vol. 14, no. 9, pp. 851–855.

    Google Scholar 

  18. Karimov, S.R. and Sokol’skii, A.G., Three-Dimensional Periodic Solutions for a Circular Restricted Three-Body Problem, Pisma Astron. Zh., 1989, vol. 15, no. 4, pp. 377–384.

    ADS  MathSciNet  Google Scholar 

  19. Zagouras, C.. and Markellos, V.V., Axisymmetric Periodic Orbits in the Restricted Problem in Three Dimensions, Astron. and Astrophys., 1977, vol. 59, no. 1, pp. 79–89.

    MATH  ADS  MathSciNet  Google Scholar 

  20. Markellos, V.V., Bifurcation of Plane with Three-Dimensional Asymmetric Periodic Orbits in the Restricted Three-Body Problem, Mon. Not. Royal Astr. Soc., 1977, no. 180, pp. 103–116.

  21. Markellos, V.V., Asymmetric Periodic Orbits in Three Dimensions, Mon. Not. Royal Astr. Soc., 1978, no. 184, pp. 273–281.

  22. Kazantzis, P.G., The Structure of Periodic Solutions in the Restricted Problem of Three Bodies, Astrophysics and Space Science, 1978, vol. 59, pp. 355–371.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Kazantzis, P.G., Numerical Determination of Families of Three-Dimensional Double-Symmetric Periodic Orbits in the Restricted Three-Body Problem. I, Astrophysics and Space Science, 1979, vol. 65, pp. 493–513.

    Article  MATH  ADS  Google Scholar 

  24. Kazantzis, P.G., Numerical Determination of Families of Three-Dimensional Double-Symmetric Periodic Orbits in the Restricted Three-Body Problem. II, Astrophysics and Space Science, 1979, vol. 69, pp. 353–368.

    Article  ADS  Google Scholar 

  25. Yakubovich, V.A. and Starzhinskii, V.M., Lineinye differentsial’nye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya (Linear Differential Equations with Periodic Coefficients and Their Applications), Moscow: Nauka, 1972.

    Google Scholar 

  26. Broucke, R., Periodic Orbits in the Restricted Three-Body Problem with Earth-Moon Masses, NASA Technical Report, Pasadena, 1968, no. 32-1168.

  27. Broucke, R., Stable Orbits of Planets of Binary Star System in the Three-Dimensional Restricted Problem, Celest. Mech. Dyn. Astr., 2001, vol. 81, pp. 321–341.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Bruno, A.D., Zero-Multiple and Inverse Periodic Solutions for the Restricted Three-Body Problem, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., Moscow, 1996, no. 93.

Download references

Author information

Authors and Affiliations

  1. Astrospace Center of Lebedev Physical Institute, Russian Academy of Sciences, Moscow, Russia

    B. B. Kreisman

Authors
  1. B. B. Kreisman
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © B.B. Kreisman, 2009, published in Kosmicheskie Issledovaniya, 2009, Vol. 47, No. 1, pp. 64–78.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kreisman, B.B. Families of periodic solutions in three-dimensional restricted three-body problem. Cosmic Res 47, 53–67 (2009). https://doi.org/10.1134/S0010952509010079

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0010952509010079

PACS

Search

Navigation