Abstract
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 byw=πu/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−k 2)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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Broucke, R. and Cefola, P.: 1973, ‘A Note on the Relation Between True and Eccentric Anomalies in the Two-Body Problem’,Celes. Mech. 7, 388
Brown, E.W. and Shook, C.: 1933,Planetary Theory, Cambridge Univ. Press, Cambridge
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
Brumberg, V.A.: 1980,Analytical Algorithms of Celestial Mechanics, Nauka, Moscow,in Russian
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
Erdélyi, A. (ed.): 1953,Higher Transcendental Functions, vol. I, McGraw-Hill Book Company, New York-Toronto-London
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, 158
Giacaglia, G.E.O.: 1976, ‘A Note on Hansen's Coefficients in Satellite Theory’,Celes. Mech. 14, 515
Gradshteyn, I.S. and Ryzhik, I.M.: 1965,Table of Integrals, Series, and Products, Academic Press, New York and London
Hughes, S.: 1981, ‘The Computation of Tables of Hansen Coefficients’,Celes. Mech. 25, 101
Nacozy, P.: 1977, ‘The Intermediate Anomaly’,Celes. Mech. 16, 309
Vinh, N.X.: 1969, ‘Recurrence Formulae for the Hansen's Developments’,Celes. Mech. 2, 64
Wolfram, S.: 1991,Mathematica: a System for Doing Mathematics by Computer, Addison-Wesley Publishing Co., Redbook City, California, 2nd ed.
Author information
Authors and Affiliations
Additional information
on leave from Institute of Applied Astronomy, St.-Petersburg 197042, Russia
Rights and permissions
About this article
Cite this article
Brumberg, E., Fukushima, T. Expansions of elliptic motion based on elliptic function theory. Celestial Mech Dyn Astr 60, 69–89 (1994). https://doi.org/10.1007/BF00693093
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00693093