Level curves of the Hamiltonian on the (ecosϖ,esinϖ) plane for the 2:1 mean motion resonance with an outer perturber for low values of e′ (=0.05 and 0.1, both with ϖ′ = 0). The General Relativity effect is not included. Lighter colours indicate a higher value of the Hamiltonian. The white shaded disks centred at the origin indicate the regions where the adiabatic method is not valid (see Sect. 3), i.e. where our calculations do not necessarily reflect the true dynamics of the system. The dark dashed line indicates a set of critical orbits which separate the phase space into a circulation zone near the origin, a libration zone near the stable equilibrium point at ϖ = 0, and an outer circulation region. All orbits with initially low eccentricities do not experience an appreciable increase in e.