Abstract
A two-parameter time transformationdt=r 3/2(α0+α1 r)−1/2 dτ is proposed, where τ is the radial distance while α0 and α1 are, if not constants, at least conservative functions of positions and velocities. In Keplerian systems, the quadrature implied by the transformation may by carried out by elliptic functions. When α0=0, τ is the eccentric anomaly; if α1=0, then τ is the intermediate or elliptic anomaly. Considering several values of α0 and α1, numerical examples of the relation of thegeneralized elliptic anomaly τ with the classical and elliptic anomalies are given. Application of this transformation to some perturbed Kepler problems is briefly outlined.
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References
Belen'kii, I. M.: 1981,Celest. Mech. 23, 9.
Bohlin, M. K.: 1911,Bull. Astron. 28, 113.
Brown, E. W. and Shook, C. A.: 1933,Planetary Theory, Cambridge University Press.
Bulirsh, R.: 1965,Num. Math. 7, 78.
Byrd, P. and Friedman, M.: 1971,Handbook of Elliptic Integrals for Engineers and Scientists, Springer, New York.
Cid, R., Ferrer, S. and Elipe, A.: 1983,Celest. Mech. 31, 73.
Cid, R., Ferrer, S. and Sein-Echaluce, M. L.: 1986,Celest. Mech. 38, 191.
Ferrándiz, J. M.: 1986,Celest. Mech. 39, 23.
Ferrándiz, J. M. and Ferrer, S.: 1986,Bull. Astron. Inst. Czechosl. 37, 226.
Ferrer, S. and Sein-Echaluce, M. L.: 1986,Homenaje al Profesor Cid, Secretariado de Publicaciones, Universidad de Zaragoza, 107.
Janin, G. and Bond, V.: 1980,The Elliptic Anomaly, NASA, T.M. 58228.
Levi-Cività, T.: 1906,Acta Math. 30, 305.
McGehee, R.: 1974,Inventiones Math. 27, 191.
Nacozy, P.: 1976,Celest. Mech. 13, 495.
Nacozy, P.: 1977,Celest. Mech. 16, 309.
Nacozy, P.: 1981,Celest. Mech. 23, 173.
Scheifele, G. and Graf, O.: 1974,AIAA Paper, 74-838.
Sein-Echaluce, M. L.: 1986,Estudio comparativo de intermediarios radiales y su aplicación a la teoría del satélite artificial zonal. Ph. D. dissertation, University of Zaragoza, available at Secretariado de Publicaciones, Universidad de Zaragoza.
Stiefel, E. and Scheifele, G.: 1971,Linear and Regular Celestial Mechanics, Springer, New York.
Sundman, K.: 1912,Acta Math. 36, 105.
Szebehely, V.: 1976,Celest. Mech. 14, 499.
Szebehely, V.: 1977,Long-Time Predictions in Dynamics, ASI Reidel, 17.
Tapley, B. D. and Szebehely, V.: 1967,JPL Space Programs Summary,4, 37.
Velez, C.: 1974,Celest. Mech. 10, 405.
Velez, C. and Hilinski, S.: 1978,Celest. Mech. 17, 83.
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Ferrándiz, J.M., Ferrer, S. & Sein-Echaluce, M.L. Generalized elliptic anomalies. Celestial Mechanics 40, 315–328 (1987). https://doi.org/10.1007/BF01235849
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DOI: https://doi.org/10.1007/BF01235849