Dynamical friction and resonance trapping in planetary systems

A restricted planar circular three-body system, consisting of the Sun and two planets, is studied as a simple model for a planetary system. The mass of the inner planet is considered to be larger, and the system is assumed to be moving in a uniform interplanetary medium with constant density. Numerical integrations of this system indicate a resonance capture when the dynamical friction of the interplanetary medium is taken into account. As a result of this resonance trapping, the ratio of orbital periods of the two planets becomes nearly commensurable, and the eccentricity and semimajor axis of the orbit of the outer planet, and also its angular momentum and total energy, become constant. It appears from the numerical work that the resulting commensurability and also the resonant values of the orbital elements of the outer planet are essentially independent of the initial relative positions of the two bodies. The results of numerical integrations of this system are presented, and the first-order partially averaged equations are studied in order to elucidate the behaviour of the system while captured in resonance.