Element history of the Laplace resonance: a dynamical approach

Abstract

Context. We consider the three-body mean motion resonance defined by the Jovian moons Io, Europa, and Ganymede, which is commonly known as the Laplace resonance. In terms of the moons’ mean longitudes λ1 (Io), λ2 (Europa), and λ3 (Ganymede), this resonance is described by the librating argument φL ≡ λ1 − 3λ2 + 2λ3 ≈ 180°, which is the sum of φ12 ≡ λ1 − 2λ2 + ϖ2 ≈ 180° and φ23 ≡ λ2 − 2λ3 + ϖ2 ≈ 0°, where ϖ2 denotes Europa’s longitude of perijove. Aims. In particular, we construct approximate models for the evolution of the librating argument φL over the period of 100 yr, focusing on its principal amplitude and frequency, and on the observed mean motion combinations n1 − 2n2 and n2 − 2n3 associated with the quasi-resonant interactions above. Methods. First, we numerically propagated the Cartesian equations of motion of the Jovian system for the period under examination, and by comparing the results with a suitable set of ephemerides, we derived the main dynamical effects on the target quantities. Using these effects, we built an alternative Hamiltonian formulation and used the normal forms theory to precisely locate the resonance and to semi-analytically compute its main amplitude and frequency. Results. From the Cartesian model we observe that on the timescale considered and with ephemerides as initial conditions, both φL and the diagnostics n1 − 2n2 and n2 − 2n3 are well approximated by considering the mutual gravitational interactions of Jupiter and the Galilean moons (including Callisto), and the effect of Jupiter’s J2 harmonic. Under the same initial conditions, the Hamiltonian formulation in which Callisto and J2 are reduced to their secular contributions achieves larger errors for the quantities above, particularly for φL. By introducing appropriate resonant variables, we show that these errors can be reduced by moving in a certain action-angle phase plane, which in turn implies the necessity of a tradeoff in the selection of the initial conditions. Conclusions. In addition to being a good starting point for a deeper understanding of the Laplace resonance, the models and methods described are easily generalizable to different types of multi-body mean motion resonances. Thus, they are also prime tools for studying the dynamics of extrasolar systems.