Skip to main content
SpringerLink
Log in

Out-of-plane motion about libration points: Nonlinearity and eccentricity effects

  • Published:
Celestial mechanics Aims and scope Submit manuscript

Abstract

Out-of-plane motion about libration points is studied within the framework of the elliptic restricted three-body problem. Nonlinear motion in the circular restricted problem is given to third order in the out-of-plane amplitudeA z by Jacobi elliptic functions. Linear motion in the elliptic problem is studied using Mathieu's and Hill's equations. Additional terms needed for a complete third-order theory are found using Lindsted's method. This theory is constructed for the case of collinear libration points; for the case of triangular points, a third-order nonlinear solution is given separately in terms of Jacobi elliptic functions.

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

  • Buck, T.: 1920, ‘Oscillating Satellites Near the Lagrangian Equilateral-Triangular Points’, in F. R. Moulton (ed.),Periodic Orbits, Carnegie Institute of Washington, pp. 300–324.

  • Cole, J. D.: 1965,Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, Mass., pp. 79–102.

    Google Scholar 

  • Davis, H. T.: 1962,Introduction to Nonlinear Differential and Integral Equations, Dover, New York.

    Google Scholar 

  • Farquhar, R. W. and Kamel, A. A.: 1972, ‘Quasi-Periodic Orbits about the Translunar Libration Point’, AIAA Paper 72-935, AIAA/AAS Astrodynamics Conf., Sept. 1972. (Solutions previously listed in NASA TN D-6365, July 1971.)

  • Goldstein, H.: 1965,Classical Mechanics, Addison-Wesley, Reading, Mass.

    Google Scholar 

  • Heppenheimer, T. A.: 1970, ‘Optimal Controls for Out-of-Plane Motion about the Translunar Libration Point’,J. Spacecraft Rockets 7, 1088–1092.

    Google Scholar 

  • Mathews, J. and Walker, R. L.: 1964,Mathematical Methods of Physics, W. A. Benjamin, New York.

    Google Scholar 

  • McLachlan, N. W.: 1947,Theory and Applications of Mathieu Functions, Clarendon Press, Oxford.

    Google Scholar 

  • Moulton, F. R.: 1920,Periodic Orbits, Carnegie Institute of Washington, pp. 151–180.

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

    Google Scholar 

  • Whittaker, E. T. and Watson, G. N.: 1963,A Course of Modern Analysis, Cambridge University Press.

Download references

Author information

Author notes

  1. T. A. Heppenheimer (Work performed as a National Science Foundation Graduate Fellow)

    Present address: Department of Aerospace Engineering, University of Michigan, 48104, Ann Arbor, Michigan

Authors and Affiliations

  1. Science Applications, Inc., Schiller Park, Ill., USA

    T. A. Heppenheimer (Work performed as a National Science Foundation Graduate Fellow)

Authors
  1. T. A. Heppenheimer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heppenheimer, T.A. Out-of-plane motion about libration points: Nonlinearity and eccentricity effects. Celestial Mechanics 7, 177–194 (1973). https://doi.org/10.1007/BF01229946

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01229946

Keywords

Search

Navigation