Multi-harmonic Hamiltonian models with applications to first-order resonances

In this work, two multi-harmonic Hamiltonian models for mean motion resonances are formulated and their applications to first-order resonances are discussed. For the $k_p$:$k$ resonance, the usual critical argument $\varphi = k \lambda - k_p \lambda_p + (k_p - k) \varpi$ is taken as the resonant angle in the first model, while the second model is characterized by a new critical argument $\sigma = \varphi / k_p$. Based on canonical transformations, the resonant Hamiltonians associated with these two models are formulated. It is found that the second Hamiltonian model holds two advantages in comparison to the first model: (a) providing a direct correspondence between phase portraits and Poincar\'e sections, and (b) presenting new phase-phase structures where the zero-eccentricity point is a visible saddle point. Then, the second Hamiltonian model is applied to the first-order inner and outer resonances, including the 2:1, 3:2, 4:3, 2:3 and 3:4 resonances. The phase-space structures of these first-order resonances are discussed in detail and then the libration centers and associated resonant widths are identified analytically. Simulation results show that there are pericentric and apocentric libration zones where the libration centers diverge away from the nominal resonance location as the eccentricity approaches zero and, in particular, the resonance separatrices do not vanish at arbitrary eccentricities for both the inner and outer (first-order) resonances.