Dynamics Of 2/1 Resonant Extrasolar Systems Application to HD82943 and GLIESE876

A complete study is made of the resonant motion of two planets revolving around a star, in the model of the general planar three body problem. The resonant motion corresponds to periodic motion of the two planets, in a rotating frame, and the position and stability properties of the periodic orbits determine the topology of the phase space and consequently play an important role in the evolution of the system. Several families of symmetric periodic orbits are computed numerically, for the 2/1 resonance, and for the masses of some observed extrasolar planetary systems. In this way we obtain a global view of all the possible stable configurations of a system of two planets. These define the regions of the phase space where a resonant extrasolar system could be trapped, if it had followed in the past a migration process.The factors that affect the stability of a resonant system are studied. For the same resonance and the same planetary masses, a large value of the eccentricities may stabilize the system, even in the case where the two planetary orbits intersect. The phase of the two planets (position at perihelion or aphelion when the star and the two planets are aligned) plays an important role, and the change of the phase, other things being the same, may destabilize the system. Also, the ratio of the planetary masses, for the same total mass of the two planets, plays an important role and the system, at some resonances and some phases, is destabilized when this ratio changes.The above results are applied to the observed extrasolar planetary systems HD 82943, Gliese 876 and also to some preliminary results of HD 160691. It is shown that the observed configurations are close to stable periodic motion.