Celestial Mechanics and Dynamical Astronomy

, Volume 60, Issue 1, pp 69–89

Expansions of elliptic motion based on elliptic function theory

  • Eugene Brumberg
  • Toshio Fukushima

DOI: 10.1007/BF00693093

Cite this article as:
Brumberg, E. & Fukushima, T. Celestial Mech Dyn Astr (1994) 60: 69. doi:10.1007/BF00693093


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 problemelliptic anomalyhypergeometric polynomialselliptic functions

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