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Reduced 4D oscillators and orbital elements in Keplerian systems: Cushman–Deprit coordinates

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Abstract

We study the reduction and regularization processes of perturbed Keplerian systems from an astronomical point of view. Our approach connects axially symmetric perturbed 4-DOF oscillators with Keplerian systems, including the case of rectilinear solutions. This is done through a preliminary reduction recently studied by the authors. Then, the reduction program continues by removing the Keplerian energy. For each value of the semi-major axis, we explain the astronomical meaning of the sextuples defining the orbit space \(\mathbb {S}^2\times \mathbb {S}^2\) and its connection with the orbital elements. More precisely, we present alternative sextuple coordinates for the set of bounded Keplerian orbits that ‘separate’ the node of the orbital plane from the argument of perigee giving the Laplace vector in that plane. Still, the reduction of the axial symmetry defined by the third component of the angular momentum is performed. For the thrice reduced space \(\varGamma _{0,L,H}\) we propose the Cushman–Deprit coordinates, a variant to the set given by Cushman. The main feature of these variables is that they are all with the same dimensions, which is convenient for the normalization procedure. As an application of the proposed scheme, we study the spatial lunar problem.

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Acknowledgements

We deeply appreciate comments and suggestions of the anonymous referees which contributed to the improvement and clarity of the paper. Support from Research Agencies of Spain and Chile is acknowledged. They came in the form of research Projects MTM2015-64095-P and ESP2017-87271-P, of the Ministry of Science of Spain and from the Project 11160224 of the Chilean national agency FONDECYT. The author J.L.Z. acknowledges support from CONICYT PhD/2017-21170836.

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Authors and Affiliations

  1. Grupo de Dinámica Espacial, DITEC, Facultad de Informática, Universidad Murcia, 30100, Murcia, Spain

    S. Ferrer

  2. GISDA, Dept. de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Collao 1202, Casilla 5-C, Concepción, Chile

    F. Crespo & J. L. Zapata

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  1. S. Ferrer
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  2. F. Crespo
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  3. J. L. Zapata
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Correspondence to S. Ferrer.

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Ferrer, S., Crespo, F. & Zapata, J.L. Reduced 4D oscillators and orbital elements in Keplerian systems: Cushman–Deprit coordinates. Celest Mech Dyn Astr 132, 52 (2020). https://doi.org/10.1007/s10569-020-09995-z

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