Advertisement

Lunar Perturbation of the Metric Associated to the Averaged Orbital Transfer

  • Bernard BonnardEmail author
  • Helen Henninger
  • Jérémy Rouot
Part of the Springer INdAM Series book series (SINDAMS, volume 11)

Abstract

In a series of previous article (Bonnard and Caillau, Ann Inst H Poincaré Anal Non Linéaire 24(3):395–411, 2007; Forum Math 21(5):797–814, 2009), we introduced a Riemannian metric associated to the energy minimizing orbital transfer with low propulsion. The aim of this article is to study the deformation of this metric due to a standard perturbation in space mechanics, the lunar attraction. Using Hamiltonian formalism, we describe the effects of the perturbation on the orbital transfers and the deformation of the conjugate and cut loci of the original metric.

Keywords

Orbital Plane Orbital Transfer Jacobi Field Adjoint Vector Energy Minimization Problem 
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.

Notes

Acknowledgements

Work supported in part by the French Space Agency CNES, R&T action R-S13/BS-005-012 and by the region Provence-Alpes-Côte d’Azur.

References

  1. 1.
    Bonnard, B., Caillau, J.-B.: Riemannian metric of the averaged energy minimization problem in orbital transfer with low thrust. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(3), 395–411 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bonnard, B., Caillau, J.-B.: Geodesic flow of the averaged controlled Kepler equation. Forum Math. 21(5), 797–814 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bryson Jr., A.E., Ho, Y.C.: Applied Optimal Control. Hemisphere Publishing Corp. Washington, DC (1975)Google Scholar
  4. 4.
    Carathéodory, C.: Calculus of Variations and Partial Differential Equations of the First Order. Part I: Partial Differential Equations of the First Order. Chelsea Publishing Company, New York (1982)Google Scholar
  5. 5.
    Celletti, A., Chierchia, L.: KAM stability for a three-body problem of the solar system. Z. Angew. Math. Phys. 57(1), 33–41 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cots, O.: Contrôle optimal géométrique: méthodes homotopiques et applications. PhD thesis, Université de Bourgogne (2012)Google Scholar
  7. 7.
    Domingos, R.C., Vilhena de Moraes, R., Bertachini De Almeida Prado, A.F.: Third-body perturbation in the case of elliptic orbits for the disturbing body. Math. Probl. Eng. Art. ID 763654, 2008(14) (2008)Google Scholar
  8. 8.
    Edelbaum, T.N.: Optimum low-thrust rendezvous and station keeping. AIAA J. 2, 1196–1201 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Edelbaum, T.N.: Optimum power-limited orbit transfer in strong gravity fields. AIAA J. 3, 921–925 (1965)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Geffroy, S.: Généralisation des techniques de moyennation en contrôle optimal, application aux problèmes de rendez-vous orbitaux à poussée faible. PhD thesis, INPT (1997)Google Scholar
  11. 11.
    Geffroy, S., Epenoy, R.: Optimal low-thrust transfers with constraints—generalization of averaging techniques. Acta Astronaut. 41(3), 133–149 (1997)CrossRefGoogle Scholar
  12. 12.
    Nemytskii, V.V., Stepanov, V.V.: Qualitative Theory of Differential Equations. Princeton University Press, Princeton (1960)zbMATHGoogle Scholar
  13. 13.
    Pascoli, G.: Astronomie Fondamentale. Dunod, Paris (2000)Google Scholar
  14. 14.
    Poincaré, H.: Œuvres. Tome VII. Éditions Jacques Gabay, Sceaux (1996)Google Scholar
  15. 15.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience Publishers/Wiley, New York/London (1962)zbMATHGoogle Scholar
  16. 16.
    Vinti, J.P., Der, G.J., Bonavito, N.L.: Orbital and Celestial Mechanics. American Institute of Aeronautics and Astronautics, Reston (1998)zbMATHGoogle Scholar
  17. 17.
    Zarrouati, O.: Trajectoires Spatiales. CEPADUES-EDITIONS, Toulouse (1987)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Bernard Bonnard
    • 1
    Email author
  • Helen Henninger
    • 2
  • Jérémy Rouot
    • 2
  1. 1.Institut de Mathématiques de BourgogneDijonFrance
  2. 2.INRIASophia AntipolisFrance

Personalised recommendations