Celestial Mechanics and Dynamical Astronomy

, Volume 94, Issue 1, pp 105–134 | Cite as

An Analytical Solution of the Lagrange Equations Valid also for Very Low Eccentricities: Influence of a Central Potential

  • Florent Deleflie
  • Gilles Métris
  • Pierre Exertier


This paper presents an analytic solution of the equations of motion of an artificial satellite, obtained using non singular elements for eccentricity. The satellite is under the influence of the gravity field of a central body, expanded in spherical harmonics up to an arbitrary degree and order. We discuss in details the solution we give for the components of the eccentricity vector. For each element, we have divided the Lagrange equations into two parts: the first part is integrated exactly, and the second part is integrated with a perturbation method. The complete solution is the sum of the so-called “main” solution and of the so-called “complementary” solution. To test the accuracy of our method, we compare it to numerical integration and to the method developed in Kaula (Theory of Satellite Geodesy, Blaisdell publ. Co., New York. 1966), expressed in classical orbital elements. For eccentricities which are not very small, the two analytical methods are almost equivalent. For low eccentricities, our method is much more accurate.


analytic integration artificial satellites eccentricity vector short and long period terms spherical harmonics zero eccentricities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Cook, G.E. 1966Perturbations of near-circular orbits by the Earth’s gravitational potentialPlanet. Space Sci.14433444ADSGoogle Scholar
  2. Deleflie F., Métris G., Exertier P. (2006). Long-period variations of the eccentricity vector valid also for near-circular orbits around a non-spherical body, Celest. Mech. Dynam. Astron., this issue.Google Scholar
  3. Demailly J.-P. (1996). Analyse numérique et équations différentielles, EDP Sciences.Google Scholar
  4. Exertier, P. 1990Precise determination of mean orbital elements from osculating elements, by semi-analytical filteringManuscripta Geodaetica15115123Google Scholar
  5. Giacaglia, G.E.O. 1977The equations of motion of an artificial satellite in non singular variablesCelest. Mech.15191215ADSMATHGoogle Scholar
  6. Kaula, W.M. 1966Theory of Satellite GeodesyBlaisdell publ. Co.New YorkGoogle Scholar
  7. Lemoine, F.G., Kenyon, S.C., Factor, J.K., Trimmer, R.G., Pavlis, N.K., Chinn, D.S., Cox, C.M., Klosko, S.M., Luthcke, S.B., Torrence, M.H., Wang, Y.M., Williamson, R.G., Pavlis, E.C., Rapp, R.H., Olson, T.R. 1998The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96Goddard Space Flight CenterGreenbelt MDNASA/TP-1998-206861Google Scholar
  8. Murray, C.D., Dermott, S.F. 1999Solar System DynamicsCambridge University PressCambridgeGoogle Scholar
  9. Nacozy, P.E., Dallas, S.S. 1977The geopotential in non singular orbital elementsCelest. Mech.15453466ADSMathSciNetGoogle Scholar
  10. Tisserand F. 1890, Traité de Mécanique Céleste, Gauthiers Villards, reprinted. 1960–1962.Google Scholar
  11. Wnuk, E., Wytrzyszczak, I. 1988The inclination function in terms of non singularCelest. Mech.42251261ADSGoogle Scholar
  12. Wnuk, E. 1999Recent progress in analytical orbit theoriesAdv. Space Res.23677687ADSGoogle Scholar
  13. Wytrzyszczak, I. 1986Non singular elements in description of the motion of small eccentricity and inclination satellitesCelest. Mech.38101109ADSMATHGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Florent Deleflie
    • 1
  • Gilles Métris
    • 1
  • Pierre Exertier
    • 1
  1. 1.Observatoire de la Côte d’AzurUMR GEMINIGrasseFrance

Personalised recommendations