Unified iterative methods in orbit determination

  • C. Dumoulin


The classical problem of Keplerian orbit determination from only three measurements of time and angular coordinates (ti, αi, δi) has been solved here numerically in two different ways, using Newton's method for non-linear equations in both cases. The first method (Perov, 1989) is based on KS variables, whereas the second emphasizes the fundamental part played by the unified Lambert's equation and the related formulae in that kind of applications. These two methods have been compared and put into practice in various numerical tests based on real asteroid orbits and ficitious Keplerian asteroid, comet and artificial satellite orbits in order to try the stability of these methods for peculiar orbits.

Key words

Lambert's equation orbit determination 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Callandreau, O.: 1902,Ann. Obs. Paris, Mémoires t. XXIII, 7.Google Scholar
  2. Danby, J. M. A.: 1988,Fundamental of Celestial Mechanics, William-Bell.Google Scholar
  3. Gauss, C. F.: 1809,Theoria Motus Corporum Cælestium in Sectionibus Conicis Solem Ambientium, Hamburg-Perthes und Besser, English translation by C. H. Davis, 1857, reprinted 1963, Dover, NY.Google Scholar
  4. Goldberg, M. A.: 1990,Numerical Solution of Integral Equations, Plenum Press, New York, London, pp. 147–157.Google Scholar
  5. Herget, P.: 1962,The Computation of Orbits, published privately by the author, first edition, 1948, Cincinnati, pp. 66–68.Google Scholar
  6. Jaquel, R.: 1977,Le Savant et Philosophe Mulhousien Jean-Henri Lambert (1728–1777), Etudes Critiques et Documentaires, éditions Ophrys, pp. 61–77.Google Scholar
  7. Marsden, B. G.: 1991,Astron. J. 102, 1539.Google Scholar
  8. Ortega, J. M. and Rheinboldt, W. C.: 1970,Iterative Solution of Non-Linear Equations with Several Variables, Academic Press, New York, pp. 243–245 and 421–431.Google Scholar
  9. Perov, N. I.: 1989,Soviet Astron. 33, 561.Google Scholar
  10. Picart, L.: 1913,Calcul des Orbites et des Ephémérides, Doin Ed. Paris, pp. 96–101.Google Scholar
  11. Plummer, H. C.: 1909,Monthly Notices Roy. Astron. Soc. 69, 181.Google Scholar
  12. Plummer, H. C.: 1918,An Introductory Treatise on Dynamical Astronomy, Cambridge University Press, London, pp. 49–56.Google Scholar
  13. Stiefel, E. L. and Scheifele, G.: 1971,Linear and Regular Celestial Mechanics, Springer-Verlag, Berlin.Google Scholar
  14. Subbotin, M.: 1922a,Monthly Notices Roy. Astron. Soc. 82, 383.Google Scholar
  15. Subbotin, M.: 1922b,Monthly Notices Roy. Astron. Soc. 82, 419.Google Scholar
  16. Taff, L. G.: 1984,Astron. J. 89, 1426.Google Scholar
  17. Taff, L. G.: 1985,Celestial Mechanics: A Computational Guide for the Practitioner, Wiley-Interscience Publication, pp. 217–287.Google Scholar
  18. Taff, L. G. and Randall, P. M. S.: 1985,Celest. Mech. 37, 149.Google Scholar
  19. Volk, O.: 1980,Celest. Mech. 21, 237.Google Scholar
  20. Wintner, A.: 1947,The Analytical Foundations of Celestial Mechanics, Princeton University Press, pp. 178–191.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • C. Dumoulin
    • 1
  1. 1.Observatoire de l'Université Bordeaux IFloiracFrance

Personalised recommendations