The dynamics of Laplace-like resonances

The three inner Galilean satellites of Jupiter-Io, Europa, and Ganymede-are observed to move in a particular dynamical configuration, which is commonly known as the Laplace resonance. These satellites are characterized by a 2:1 ratio between the mean longitudes of Io-Europa and Europa-Ganymede. Another dynamical configuration, known as the de Sitter resonance, occurs when the longitude of Ganymede is fixed, instead of rotating like in the Laplace resonance. Besides studying the Laplace and de Sitter resonances, we also consider their generalizations to the case in which the mean longitudes of the first two satellites are in a ratio k:j, while those of the second and third satellites are in a ratio m:n with k,j,m,n∈Z+ and |j-k|, |n-m|≤2. We derive a model apt to describe such resonant configurations. We make an extensive study of the structural stability of the resonances; we show that the libration of the Laplace resonant angle is deeply affected by small variations of some quantities, most notably the semimajor axes and the oblateness. A remarkable result is that in several cases, the standard Laplace resonance of the Galilean satellites displays a regular behavior in comparison to other resonances characterized by different mean longitude ratios, which instead show a rather chaotic behavior even on short time scales. This result provides a motivation to support why the Galilean satellites are found in the actual Laplace resonance.