Complete Zonal Problem of the Artificial Satellite: Generic Compact Analytic First Order in Closed Form

  • Bernard De Saedeleer


This paper is a contribution to the Theory of the Artificial Satellite, within the frame of the Lie Transform as canonical perturbation technique (elimination of the short period terms). We consider the perturbation by any zonal harmonic J n (n ≥ 2) of the primary on the satellite, what we call here the complete zonal problem of the artificial satellite. This is quite useful for primaries with symmetry of revolution. We give an analytical formula to compute directly the first order averaged Hamiltonian. The computation is carried out in closed form for all terms, avoiding therefore tedious expansions in the eccentricity or in any anomaly; this feature makes the averaging process, not only valid for all kind of elliptic trajectories but at the same time it yields the averaged Hamiltonian in a very short and compact way. The formula allows us to now skip the averaging process, which means an asymptotic gain of a factor 3n/2 regarding the computational cost of the n th zonal. Our analytical formulae have been widely checked, by comparison on one hand with published works (Brouwer, 1959) (which contained results for particular zonal harmonics, let’s say typically from J2 to J8), and on the other hand with the results of 3 symbolic manipulation software, among which the MM (standing for ‘Moon’s series Manipulator’), which has already been used and described in (De Saedeleer B., 2004). Additionally, the first order generator associated with this transformation is given into the same closed form, and has also been validated.


artificial satellite theory Lie Hamiltonian zonal harmonic first order closed form 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Boccaletti, D. and Pucacco, G.: 1999, Theory of Orbits, 2: Perturbative and Geometrical Methods, Springer-Verlag.Google Scholar
  2. Brouwer, D. 1959‘Solution of the problem of artificial satellite theory without air drag’Astron. J.64378397CrossRefGoogle Scholar
  3. Coffey, S., Deprit, A. 1982‘Third order solution to the main problem in satellite theory’J. Guid. Control. Dyn.5366371Google Scholar
  4. De Saedeleer, B.: 2004, ‘Analytical theory of an artificial satellite of the moon’, In: E. Belbruno and P. Gurfil (eds.), Astrodynamics, Space Missions, and Chaos, To appear in Annals of the New York Academy of Sciences.Google Scholar
  5. Deprit, A. 1969‘Canonical transformations depending on a small parameter’Celest. Mech. Dyn. Astr.11230CrossRefGoogle Scholar
  6. Gradshteyn, I. and Ryzhik I.: 1980, Table of Integrals, Series, and products, Academic Press.Google Scholar
  7. Henrard, J.: 1986, ‘Algebraic manipulation on computers for lunar and planetary theories’, In: J. Kovalevsky and V. Brumberg (eds.), IAU Symposium 114, 59–62.Google Scholar
  8. Hori, G. 1966‘Theory of general perturbations with unspecified canonical variables’Publ. Astron. Soc. Japan18287296Google Scholar
  9. Jefferys, W. 1971‘Automated, closed form integration of formulas in elliptic motion’Celest. Mech. Dyn. Astr.3390394CrossRefGoogle Scholar
  10. Kelly, T. 1989‘A Note on first-order normalizations of perturbed Keplerian systems’Celest. Mech. Dyn. Astr.461925Google Scholar
  11. Kozai, Y. 1962‘Second-order solution of artificial satellite theory without air drag’Astron. J.67446461CrossRefGoogle Scholar
  12. Osácar, C., Palacián, J. 1994‘Decomposition of functions for elliptic orbits’Celest. Mech. Dyn. Astr.60207223CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Département de MathématiqueUniversity of NamurNamurBelgium

Personalised recommendations