Abstract
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.
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Notes
Readers comparing this article with de Sitter (1909) should be careful that De Sitter took \(R=-F_{pert}\).
We shall eventually only consider persistence of invariant objects under \(O(\varepsilon )\)-perturbations. The restriction is made to adapt to this.
Note that, in de Sitter (1909), De Sitter took (for small eccentricities) \(\epsilon _{i}=e_{i}/2, \, i=1,2,3\) instead of \(e_{i}\) in the expression of \(F_{res}\).
In de Sitter (1909), De Sitter stated this as a necessity. This is not evident at all.
We need to impose the assumption that these roots are distinct in order to guarantee that linearly stable periodic orbits can be continued to linearly stable ones for small parameters. De Sitter seems to have overlooked this point, though it does not change the result.
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Acknowledgments
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.
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Broer, H., Zhao, L. De Sitter’s theory of Galilean satellites. Celest Mech Dyn Astr 127, 95–119 (2017). https://doi.org/10.1007/s10569-016-9718-8
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DOI: https://doi.org/10.1007/s10569-016-9718-8