De Sitter’s theory of Galilean satellites

Broer, Hendrik; Zhao, Lei

doi:10.1007/s10569-016-9718-8

Cited by 8 publications

(31 citation statements)

References 10 publications

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“…It turns out that the inner 1:2:4 resonance greatly affects the dynamics of the entire system. Summarizing from [15,29] we have the following result.…”

Section: The De Sitter Periodic Orbitsmentioning

confidence: 67%

“…In a recent paper by Broer and Zhao [15] De Sitter's work was retold and updated regarding the periodic orbit. Also librations of the family of De Sitter periodic orbits were found with help of Kolmogorov-Arnold-Moser (or KAM) theory, mainly based on the work of Arnold, Féjoz and Zhao [1,5,16,17,34].…”

Section: Setting Of the Problemmentioning

confidence: 99%

“…On the corresponding covering space we have to keep the associated discrete group of deck transformations into account, which is automatically taken care of in our approach. In this respect the present paper forms both a simplification and an improvement of [15]. The Lagrangean tori arise by the excitation of normal modes of a family of normally elliptic isotropic invariant 2-tori, where the latter arise from superposing the De Sitter periodic orbits with the periodic motion of Callisto.…”

Section: Setting Of the Problemmentioning

confidence: 99%

“…-It turns out that for a qualitative explanation of De Sitter's theory [29][30][31] in the resonant normalization mentioned before only one normalizing step has to be performed. For a discussion and an update of this theory, comparing this with the effects of the 2:5-resonance of Jupiter and Saturn and with the influence of the Sun, see [15].…”

Section: The De Sitter Periodic Orbitsmentioning

confidence: 99%

“…Both Theorems 2 and 3 were proven in [15], based on methods of Kolmogorov-Arnold-Moser (KAM) theory, largely inspired by [1,5,16,17,34]. In this paper we present the proofs of both theorems as an adaptation of one central parametrised isotropic Hamiltonian KAM-theorem that keeps track of the deck group of the covering space.…”

Section: Librationsmentioning

confidence: 99%

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On Jupiter and his Galilean satellites: Librations of De Sitter’s periodic motions

Broer

Hanßmann

2016

Indagationes Mathematicae

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The motion of Jupiter's four Galilean satellites Io-Europa-Ganymedes-Callisto is subjected to an orbital 1:2:4-resonance of the former (and inner) three. Willem de Sitter in the early 20th century gave a mathematical explanation of this in a Newtonian framework. He found a family of stable periodic solutions by using the work of Poincaré. This paper briefly reviews De Sitter's theory, and focuses on the underlying geometry of a suitable covering space, where we develop Kolmogorov-Arnold-Moser theory to find Lagrangean invariant tori excited by the normal modes of the De Sitter periodic orbits. In this way we find many librations near these periodic orbits that may well offer a more realistic explanation of the observations.

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“…It turns out that the inner 1:2:4 resonance greatly affects the dynamics of the entire system. Summarizing from [15,29] we have the following result.…”

Section: The De Sitter Periodic Orbitsmentioning

confidence: 67%

Section: Setting Of the Problemmentioning

confidence: 99%

Section: Setting Of the Problemmentioning

confidence: 99%

Section: The De Sitter Periodic Orbitsmentioning

confidence: 99%

Section: Librationsmentioning

confidence: 99%

See 3 more Smart Citations

On Jupiter and his Galilean satellites: Librations of De Sitter’s periodic motions

Broer

Hanßmann

2016

Indagationes Mathematicae

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The dynamics of the de Sitter resonance

Celletti

Paita

Pucacco

2018

Celest Mech Dyn Astr

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Abstract. We study the dynamics of the de Sitter resonance, namely the stable equilibrium configuration of the first three Galilean satellites. We clarify the relation between this family of configurations and the more general Laplace resonant states. In order to describe the dynamics around the de Sitter stable equilibrium, a one-degree of freedom Hamiltonian normal form is constructed and exploited to identify initial conditions leading to the two families.The normal form Hamiltonian is used to check the accuracy in the location of the equilibrium positions. Besides, it gives a measure of how sensitive it is with respect to the different perturbations acting on the system. By looking at the phase-plane of the normal form, we can identify a Laplace-like configuration, which highlights many substantial aspects of the observed one.

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Normal forms for the Laplace resonance

Pucacco

2021

Celest Mech Dyn Astr

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We describe a comprehensive model for systems locked in the Laplace resonance. The framework is based on the simplest possible dynamical structure provided by the Keplerian problem perturbed by the resonant coupling truncated at second order in the eccentricities. The reduced Hamiltonian, constructed by a transformation to resonant coordinates, is then submitted to a suitable ordering of the terms and to the study of its equilibria. Henceforth, resonant normal forms are computed. The main result is the identification of two different classes of equilibria. In the first class, only one kind of stable equilibrium is present: the paradigmatic case is that of the Galilean system. In the second class, three kinds of stable equilibria are possible and at least one of them is characterised by a high value of the forced eccentricity for the ‘first planet’: here, the paradigmatic case is the exo-planetary system GJ-876, in which the combination of libration centres admits triple conjunctions otherwise not possible in the Galilean case. The normal form obtained by averaging with respect to the free eccentricity oscillations describes the libration of the Laplace argument for arbitrary amplitudes and allows us to determine the libration width of the resonance. The agreement of the analytic predictions with the numerical integration of the toy models is very good.

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De Sitter’s theory of Galilean satellites

Cited by 8 publications

References 10 publications

On Jupiter and his Galilean satellites: Librations of De Sitter’s periodic motions

On Jupiter and his Galilean satellites: Librations of De Sitter’s periodic motions

The dynamics of the de Sitter resonance

Normal forms for the Laplace resonance

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