De Sitter’s theory of Galilean satellites
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In this article, we investigate the mathematical part of De Sitter’s theory on the Galilean satellites, and further extend this theory by showing the existence of some quasi-periodic librating orbits by application of KAM theorems. After showing the existence of De Sitter’s family of linearly stable periodic orbits in the Jupiter–Io–Europa–Ganymede model by averaging and reduction techniques in the Hamiltonian framework, we further discuss the possible extension of this theory to include a fourth satellite Callisto, and establish the existence of a set of positive measure of quasi-periodic librating orbits in both models for almost all choices of masses among which one sufficiently dominates the others.
KeywordsGalilean satellites Secular systems KAM theory Normal forms
The manuscript was prepared during the stay of LZ in Johann Bernoulli Institute, University of Groningen as a postdoc. We thank the anonymous referees, Sylvio Ferraz Mello and Heinz Hanssmann for their careful reading of the manuscript and for their suggestions of improvements.
- Broer, H.W., Hanssmann, H.: On Jupiter and his Galilean satellites: librations of De Sitter’s periodic motion. Submitted (2016)Google Scholar
- de Sitter, W.: On the periodic solutions of a particular case of the problem of four bodies. KNAW Proc. 11, 682–698 (1909)Google Scholar
- de Sitter, W.: Outlines of a new mathematical theory of Jupiter’s satellites. Annalen van de sterrewacht te Leiden 12, 1–55 (1925)Google Scholar
- Féjoz, J.: Mouvements périodiques et quasi-périodiques dans le problème des n corps. Habilitation à diriger des recherches. Université Pierre et Marie Curie-Paris VI, Paris (2010)Google Scholar
- Féjoz, J.: The normal form of Moser and applications. preprint (in revision), (2011)Google Scholar
- Féjoz, J.: Introduction to KAM theory. (2013)Google Scholar
- Ferraz-Mello, S.: Dynamics of the Galilean Satellites-an Introductory Treatise. Mathematical and Dynamical Astronomy Series. Universidade de Sao Paulo, Instituto Astronomico e Geofisico, São Paulo (1979)Google Scholar