The Laplace Relation among the Galilean Satellites of Jupiter

The orbits of the inner three Galilean satellites of Jupiter (J), Io (I), Europa (E), and Ganymede (G), are in a set of resonances, with the orbital periods nearly in the ratios 1:2:4. The following sequence of figures illustrates the positions of these satellites as Ganymede orbits around Jupiter once. The ellipses are the orbits of the satellites, with the periapses (where the satellites are closest to Jupiter) marked by dashes. (The eccentricities of the orbits shown are much larger than the real ones so that the positions of the periapses are more visible.)

Panel (a) is a configuration with Europa and Ganymede in conjunction. Europa is at the periapse of its orbit, and Ganymede can be anywhere in its orbit relative to its periapse. Io is on the opposite side of Jupiter.

Panels (c) and (e) are configurations with Io and Europa in conjunction. Io is at its periapse, and Europa is at the apoapse (the most distant point from Jupiter) of its orbit.

Panel (d) is a configuration with Io and Ganymede in conjunction and Europa on the opposite side of Jupiter. Io is at its periapse, and Ganymede can be anywhere in its orbit relative to its periapse.