Celestial mechanics

, Volume 32, Issue 1, pp 29–52 | Cite as

Almost rectilinear halo orbits

  • K. C. Howell
  • J. V. Breakwell


Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional ‘halo’ periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L3-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L1-family and L2-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.


Angular Momentum Periodic Orbit Entire Range Libration Point Halo Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bettis, D. G. and Szebehely, V.: 1971, ‘Treatment of Close Approaches in the Numerical Integration of the Gravitational Problem of N Bodies’,Astrophy. Space Sci. 14, 133.Google Scholar
  2. Breakwell, J. V. and Brown, J. V.: 1979, ‘The “Halo” Family of 3-Dimensional Periodic Orbits in the Earth-Moon Restricted 3-Body Problem’,Celest. Mech. 20, 389.Google Scholar
  3. Broucke, R.: 1969, ‘Stability of Periodic Orbits in the Elliptic, Restricted Three-Body Problem’,AIAA Journal 7, 1003.Google Scholar
  4. Farquhar, R. W. and Kamel, A. A.: 1973, ‘Quasi-Periodic Orbits About the Translunar Libration Point’,Celest. Mech. 7, 458.Google Scholar
  5. Howell, K. C.: 1984, ‘Three-Dimensional, Periodic, “Halo” Orbits,’Celest. Mech. 32, 53 (this issue).Google Scholar
  6. Kustaanheimo, P. and Stiefel, E.: 1965, ‘Perturbation Theory of Kepler Motion Based on Spinor Regularization’,Journal für die reine und angewandte Mathematik 218, 204.Google Scholar

Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • K. C. Howell
    • 1
  • J. V. Breakwell
    • 1
  1. 1.Dept. of Aeronautics and AstronauticsStanford UniversityStanfordUSA

Personalised recommendations