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

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 \begin{document}$ 1{:}2{:}4 $\end{document} -resonance of the inner three moons Io, Europa and Ganymedes. In the early 20th century De Sitter gave a mathematical explanation for this in a Newtonian framework. In fact, he found a family of stable periodic solutions by using the seminal work of Poincare, which at the time was quite new. In this paper we review and summarize recent results of Broer, Hansmann and Zhao on the motion of the entire Galilean system, so including the fourth moon Callisto. To this purpose we use a version of parametrised Kolmogorov-Arnol'd-Moser theory where a family of multi-periodic isotropic invariant three-dimensional tori is found that combines the periodic motions of De Sitter and Callisto. The \begin{document}$ 3 $\end{document} -tori are normally elliptic and excite a family of invariant Lagrangean \begin{document}$ 8 $\end{document} -tori that project down to librational motions. Both the \begin{document}$ 3 $\end{document} - and the \begin{document}$ 8 $\end{document} -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.