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A study about the integration of the elliptical orbital motion based on a special one-parametric family of anomalies. (English) Zbl 1470.70022

Summary: This paper aimed to address the study of a new family of anomalies, called natural anomalies, defined as a one-parameter convex linear combination of the true and secondary anomalies, measured from the primary and the secondary focus of the ellipse, and its use in the study of analytical and numerical solutions of perturbed two-body problem. We take two approaches: first, the study of the analytical development of the basic quantities of the two-body problem to be used in the analytical theories of the planetary motion and second, the study of the minimization of the errors in the numerical integration by an appropriate choice of parameters in our family for each value of the eccentricity. The use of an appropriate value of the parameter can improve the length of the developments in the analytical theories and reduce the errors in the case of the numerical integration.

MSC:

70F15 Celestial mechanics
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[1] Bretagnon, P.; Francou, G., Variations seculaires des orbites planetaires. Théorie VSOP87, Astronomy & Astrophysics, 114, 69-75 (1988)
[2] Simon, J. L., Computation of the first and second derivatives of the Lagrange equations by harmonic analysis, Astronomy & Astrophysics, 17, 661-692 (1982)
[3] Simon, J. L.; Francou, G.; Fienga, A.; Manche, H., New analytical planetary theories VSOP2013 and TOP2013, Astronomy & Astrophysics, 557, article A49 (2013)
[4] Tisserand, F. F., Traité de Mecanique Celeste (1986), Paris, France: Gauthier-Villars, Paris, France
[5] Brumberg, V. A., Analytical Techniques of Celestial Mechanics (1995), Berlin, Germany: Springer, Berlin, Germany · Zbl 0830.70001
[6] Nacozy, P., Hansen’s method of partial anomalies: an application, The Astronomical Journal, 74, 544-550 (1969)
[7] Brumberg, E.; Fukushima, T., Expansions of elliptic motion based on elliptic function theory, Celestial Mechanics and Dynamical Astronomy, 60, 1, 69-89 (1994) · Zbl 0819.70006
[8] López, J. A.; Barreda, M., A formulation to obtain semi-analytical planetary theories using true anomalies as temporal variables, Journal of Computational and Applied Mathematics, 204, 1, 77-83 (2007) · Zbl 1110.70016
[9] López Ortí, J. A.; Martínez Usó, M. J.; Marco Castillo, F. J., Semi-analytical integration algorithms based on the use of several kinds of anomalies as temporal variable, Planetary and Space Science, 56, 14, 1862-1868 (2008)
[10] Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration (2006), Heidelberg, Germany: Springer, Heidelberg, Germany
[11] Sundman, K., Memoire sur le probleme des trois corps, Acta Mathematica, 36, 1, 105-179 (1913) · JFM 43.0826.01
[12] Nacozy, P., The intermediate anomaly, Celestial Mechanics, 16, 3, 309-313 (1977) · Zbl 0376.70013
[13] Janin, G., Accurate computation of highly eccentric satellite orbits, Celestial Mechanics, 10, 4, 451-467 (1974) · Zbl 0303.70045
[14] Janin, G.; Bond, V. R., The elliptic anomaly, NASA Technical Memorandum, 58228 (1980)
[15] Velez, C. E.; Hilinski, S., Time transformations and cowell’s method, Celestial Mechanics, 17, 1, 83-99 (1978) · Zbl 0377.70021
[16] Ferrándiz, J. M.; Ferrer, S.; Sein-Echaluce, M. L., Generalized elliptic anomalies, Celestial Mechanics, 40, 3-4, 315-328 (1987) · Zbl 0652.70007
[17] Brumberg, E. V., Length of arc as independent argument for highly eccentric orbits, Celestial Mechanics and Dynamical Astronomy, 53, 4, 323-328 (1992) · Zbl 0759.70008
[18] Brouwer, D.; Clemence, G. M., Celestial Mechanics (1965), New York, NY, USA: Academic Press, New York, NY, USA
[19] Levallois, J. J.; Kovalevsky, J., Géodésie Générale, 4 (1971), Paris, France: Eyrolles, Paris, France
[20] Hagihara, Y., Celestial Mechanics (1970), Cambridge, Mass, USA: MIT Press, Cambridge, Mass, USA · JFM 66.0407.02
[21] Kovalewsky, J., Introduction to Celestial Mechanics (1967), Dodrecht, The Netherlands: D. Reidel, Dodrecht, The Netherlands
[22] López, J. A.; Agost, V.; Barreda, M., A note on the use of the generalized sundman transformations as temporal variables in celestial mechanics, International Journal of Computer Mathematics, 89, 433-442 (2012) · Zbl 1238.70004
[23] Deprit, A., Note on Lagrange’s inversion formula, Celestial Mechanics, 20, 4, 325-327 (1979) · Zbl 0454.30002
[24] López, J. A.; Barreda, M.; Artes, J., Integration algorithms to construct semi-analytical planetary theories, Wseas Transactions on Mathematics, 5, 6, 609-614 (2006)
[25] Chapront, J.; Bretagnon, P.; Mehl, M., Un formulaire pour le calcul des perturbations d’ordres élevés dans les problèmes planétaires, Celestial Mechanics, 11, 3, 379-399 (1975) · Zbl 0311.70014
[26] Simon, J. L.; Bretagnon, P.; Chapront, P.; Chapront-Touzé, J.; Francou, G.; Laskar, J., Numerical expressions for precession formulae and mean elements for the Moon and the planets, Astronomy & Astrophysics, 282, 663-683 (1994)
[27] Fehlberg, E.; Marsall, G. C., Classical fifth, sixth, seventh and eighth runge-kutta formulas with stepsize control, NASA Technical Report, R-287 (1968)
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