Long-term resonances between two Jovian exoplanets

Within the plane planetary problem we present two new approaches for the determination of purely resonant eccentricity and semimajor axis variations in terms of simple, closed algebraic relationships. We consider the motion of two Jovian exoplanets in 2:1, 3:1, and 7:4 resonance. Even with initial eccentricities of 0.05, we have found two numerical examples of purely resonant motion of two Jovian exoplanets in 2:1 and 3:1 resonance, fitting throughout the theoretical relationships for over 105 revolutions of the outer exoplanet. The maximum eccentricities of the two Jovian exoplanets are <0.15, if the initial ratio of semimajor axes is <0.6992 and the initial eccentricities are ≤0.05. During intervals of negligible secular perturbations, the agreement between theoretical and numerical maximum resonant eccentricity variations is generally much better than within a factor of 2. The theoretical and calculated maximum eccentricity of a Plutino in 2:3 resonance with Neptune is >0.053.