Modelling the high-eccentricity planetary three-body problem. Application to the GJ876 planetary system

The discovery of extrasolar planets located in the vicinity of mean-motion commensurabilities has opened a new arena for the study of resonance capture and its possible role in the dynamical evolution and long-term stability of planetary systems. Contrary to our own Solar System, many of these planets have highly eccentric orbits (∼0.1–0.6), making the use of usual analytical perturbative models very limited. Even so, several attempts have been made to apply classical expansions of the resonant Hamiltonian to these cases, leading to results which are, at best, extrapolations of the low-eccentricity resonant structure, and not necessarily precise. In this paper we present a new analytical expansion for the Hamiltonian of the planetary three-body problem which does not suffer these restrictions, and is even valid for crossing orbits. The only limitation is its applicability to planar motions. The resulting model can be applied to resonant and non-resonant configurations alike. We show examples of this expansion in different resonances and we compare the results with numerical determinations of the exact Hamiltonian. Finally, we apply the developed model to the case of two planets in the 2/1 mean-motion commensurability (such as the Gliese 876 system), and we analyse its periodic orbits and general structure of the resonant phase space at low and high eccentricities.