Celestial Mechanics and Dynamical Astronomy

, Volume 60, Issue 1, pp 69–89 | Cite as

Expansions of elliptic motion based on elliptic function theory

  • Eugene Brumberg
  • Toshio Fukushima


New expansions of elliptic motion based on considering the eccentricitye as the modulusk of elliptic functions and introducing the new anomalyw (a sort of elliptic anomaly) defined bywu/2K−π/2,g=amu−π/2 (g being the eccentric anomaly) are compared with the classic (e, M), (e, v) and (e, g) expansions in multiples of mean, true and eccentric anomalies, respectively. These (q,w) expansions turn out to be in general more compact than the classical ones. The coefficients of the (e,v) and (e,g) expansions are expressed as the hypergeometric series, which may be reduced to the hypergeometric polynomials. The coefficients of the (q,w) expansions may be presented in closed (rational function) form with respect toq, k, k′=(1−k2)1/2,K andE, q being the Jacobi nome relatedk whileK andE are the complete elliptic integrals of the first and second kind respectively. Recurrence relations to compute these coefficients have been derived.

Key words

Elliptic two-body problem elliptic anomaly hypergeometric polynomials elliptic functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Broucke, R. and Cefola, P.: 1973, ‘A Note on the Relation Between True and Eccentric Anomalies in the Two-Body Problem’,Celes. Mech. 7, 388Google Scholar
  2. Brown, E.W. and Shook, C.: 1933,Planetary Theory, Cambridge Univ. Press, CambridgeGoogle Scholar
  3. Brumberg, E.V.: 1993, ‘Perturbed Two-Body Problem with Elliptic Functions’, in: H. Kinoshita and H. Nakai (eds.),Proc. 25th Symp. on Celes. Mech., NAO, Tokyo,in press Google Scholar
  4. Brumberg, V.A.: 1980,Analytical Algorithms of Celestial Mechanics, Nauka, Moscow,in Russian Google Scholar
  5. Brumberg, V.A.: 1993, ‘General Planetary Theory Revisited with the Aid of Elliptic Functions’, in: H. Kinoshita and H. Nakai (eds.),Proc. 25th Symp. on Celes. Mech., NAO, Tokyo,in press Google Scholar
  6. Erdélyi, A. (ed.): 1953,Higher Transcendental Functions, vol. I, McGraw-Hill Book Company, New York-Toronto-LondonGoogle Scholar
  7. Fukushima, T.: 1991, ‘Numerical Computation of Elliptic Integrals and Functions’, in: H. Kinoshita and H. Yoshida (eds.),Proc. 24th Symp. on Celes. Mech., NAO, Tokyo, 158Google Scholar
  8. Giacaglia, G.E.O.: 1976, ‘A Note on Hansen's Coefficients in Satellite Theory’,Celes. Mech. 14, 515Google Scholar
  9. Gradshteyn, I.S. and Ryzhik, I.M.: 1965,Table of Integrals, Series, and Products, Academic Press, New York and LondonGoogle Scholar
  10. Hughes, S.: 1981, ‘The Computation of Tables of Hansen Coefficients’,Celes. Mech. 25, 101Google Scholar
  11. Nacozy, P.: 1977, ‘The Intermediate Anomaly’,Celes. Mech. 16, 309Google Scholar
  12. Vinh, N.X.: 1969, ‘Recurrence Formulae for the Hansen's Developments’,Celes. Mech. 2, 64Google Scholar
  13. Wolfram, S.: 1991,Mathematica: a System for Doing Mathematics by Computer, Addison-Wesley Publishing Co., Redbook City, California, 2nd ed.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Eugene Brumberg
    • 1
  • Toshio Fukushima
    • 2
  1. 1.National Astronomical ObservatoryMitaka, TokyoJapan
  2. 2.National Astronomical ObservatoryMitaka, TokyoJapan

Personalised recommendations