A numerical study of the 2:1 planetary resonance

We numerically explore the long-term stability of planetary orbits locked in a 2:1 mean motion resonance for a wide range of planetary mass ratios and orbital parameters. Our major tool is Laskar's frequency map analysis. Regions of low diffusion rate are outlined in a phase space defined by the two planetary eccentricities and the libration amplitude of a critical resonance argument. Resonant systems that are dynamically stable on a long timescale must lie within these regions. The resonance locking between planets in high eccentric orbits may be destroyed by mutual close encounters. We discuss various dynamical protection mechanisms related to the resonant configuration, among which is the well-known apsidal corotation. In the case of moderate-to-low eccentricities, we find that apsidal circulators, little discussed till now, are very common among stable orbits. We also map the different types of resonant behaviour predicted by analytical theories in the phase space.