# American Institute of Mathematical Sciences

## A Galilean dance 1:2:4 resonant periodic motions and their librations of Jupiter and his Galilean moons

 1 Bernoulli Instituut, Rijksuniversiteit Groningen, Postbus 407, NL 9700 AK Groningen, The Netherlands 2 Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, NL 3508 TA Utrecht, The Netherlands

Received  November 2017 Revised  March 2018 Published  April 2019

The four Galilean moons of Jupiter were discovered by Galileo in the early 17th century, and their motion was first seen as a miniature solar system. Around 1800 Laplace discovered that the Galilean motion is subjected to an orbital $1{:}2{:}4$-resonance of the inner three moons Io, Europa and Ganymedes. In the early 20th century De Sitter gave a mathematical explanation for this in a Newtonian framework. In fact, he found a family of stable periodic solutions by using the seminal work of Poincaré, which at the time was quite new. In this paper we review and summarize recent results of Broer, Hanßmann and Zhao on the motion of the entire Galilean system, so including the fourth moon Callisto. To this purpose we use a version of parametrised Kolmogorov-Arnol'd-Moser theory where a family of multi-periodic isotropic invariant three-dimensional tori is found that combines the periodic motions of De Sitter and Callisto. The $3$-tori are normally elliptic and excite a family of invariant Lagrangean $8$-tori that project down to librational motions. Both the $3$- and the $8$-tori occur for an almost full Hausdorff measure set in the product of corresponding dimension in phase space and a parameter space, where the external parameters are given by the masses of the moons.

Citation: Henk W. Broer, Heinz Hanssmann. A Galilean dance 1:2:4 resonant periodic motions and their librations of Jupiter and his Galilean moons. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020062
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##### References:
Left: Galileo Galilei 1564-1642. Right: The Galilean moons of Jupiter
Left: Pierre Simon de Laplace 1749-1827. Right: Willem de Sitter 1872-1934
Delaunay angles: 'mean anomaly' $\ell$ and 'argument' $g$ of the pericenter of a Keplerian motion
Poincaré's Ansatz: look near the 16 possible collinearities of Jupiter-Io-Europa-Ganymedes. It turns out that the one in the last column in the first row from above corresponds to the stable situation, where all moons are in their perijoves and the ellipse of Europa is $\pi$-rotated with respect to those of Io and Ganymedes
Sketch of the set $(\mathbb{R}^2)_{\tau,\gamma}$
Sketch of $\Gamma^\gamma_{\tau,\gamma} \subseteq \Gamma^\gamma \subseteq \Gamma$
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