A numerical study of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in planetary systems

Abstract

We present a numerical study on the stability of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in the three-body problem composed of a solar mass star, a Jupiter mass planet, and an additional body with zero mass (elliptic restricted three-body problem) or masses corresponding to either Neptune, Saturn, or Jupiter (planetary three-body problem). For each system, we obtain stability maps using the n-body numerical integrator REBOUND and computing the chaos indicator mean exponential growth factor of nearby orbits (MEGNO). We show that families of periodic orbits exist in all configurations and they correspond to the libration of either a single resonant argument or all resonant arguments (fixed points). We compare the results obtained in the elliptic restricted three-body problem with previous results in the literature, and we show the differences and similarities between the phase space topology for these retrograde resonances in the circular restricted, elliptic restricted, and planetary three-body problems.