Skip to main content
SpringerLink
Log in

Periodic orbits in the restricted four-body problem with two equal masses

  • Original Article
  • Published:
Astrophysics and Space Science Aims and scope Submit manuscript

Abstract

The restricted (equilateral) four-body problem consists of three bodies of masses m 1, m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.

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

Similar content being viewed by others

References

  • Baltagiannis, A.N., Papadakis, K.E.: Families of periodic orbits in the restricted four-body problem. Astrophys. Space Sci. 336, 357–367 (2011)

    Article  ADS  MATH  Google Scholar 

  • Baltagiannis, A.N., Papadakis, K.E.: Equilibrium points and their stability in the restricted four-body problem. Int. J. Bifurc. Chaos Appl. Sci. Eng. 21, 2179–2193 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Broucke, R.A.: Periodic orbits in the restricted three-body problem with earth-moon masses. Technical report, JPL (1968)

  • Buffoni, B.: Shooting methods and topological transversality. Proc. R. Soc. Edinb. A 129, 1137–1155 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Delgado, J., Álvarez–Ramirez, M.: Central configurations of the symmetric restricted four-body problem. Celest. Mech. Dyn. Astron. 87, 371–381 (2003)

    Article  ADS  MATH  Google Scholar 

  • Hénon, M.: Exploration numérique du probléme restreint I. Masses égales, orbites périodiques. Ann. Astrophys. 28, 499–511 (1965a)

    ADS  Google Scholar 

  • Hénon, M.: Exploration numérique du probléme restreint II. Masses égales, stabilité des orbites périodiques. Ann. Astrophys. 28, 992–1007 (1965b)

    ADS  Google Scholar 

  • Hénon, M.: Generating Families in the Restricted Three Body Problem. Springer, Berlin (1997)

    MATH  Google Scholar 

  • Henrard, J.: Proof of a conjeture of E. Strömgren. Celest. Mech. 7, 449–457 (1973)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Leandro, E.S.G.: On the central configurations of the planar restricted four-body problem. J. Differ. Equ. 226, 323–351 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Meyer, K.: Bifurcation of a central configuration. Celest. Mech. 40(3–4), 273–282 (1987)

    ADS  MATH  Google Scholar 

  • Meyer, K., McSwiggen, P.D.: The evolution of invariant manifolds in Hamiltonian-Hopf bifurcations. J. Differ. Equ. 189, 538–555 (2002)

    MathSciNet  Google Scholar 

  • Pedersen, P.: Librationspunkte im restringierten vierkoerperproblem. Dan. Mat. Fys. Medd. 1–80 (1944)

  • Simó, C.: Relative equilibrium solutions in the four body problem. Cel. Mech. 165–184 (1978)

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

    Google Scholar 

Download references

Acknowledgement

Author Burgos-García has been supported by a CONACYT fellowship of doctoral studies.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, C.P. 09340, México, D.F., México

    Jaime Burgos-García & Joaquín Delgado

Authors
  1. Jaime Burgos-García
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joaquín Delgado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joaquín Delgado.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burgos-García, J., Delgado, J. Periodic orbits in the restricted four-body problem with two equal masses. Astrophys Space Sci 345, 247–263 (2013). https://doi.org/10.1007/s10509-012-1118-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10509-012-1118-2

Keywords

Search

Navigation