An Integrable Model for the Dynamics of Planetary Mean Motion Resonances

Abstract: I consider the dynamics of mean motion resonances between pairs of co-planar planets and derive a new integrable Hamiltonian model for planets' resonant motion. The new model generalizes previously-derived integrable Hamiltonians for first-order resonances to treat higher-order resonances by exploiting a surprising near-symmetry of the full, non-integrable Hamiltonians of higher-order resonances. Whereas past works have frequently relied on truncated disturbing function expansions to derive integrable approximations to resonant motion, I show that no such expansion is necessary, thus enabling the new model to accurately capture the dynamics of both first- and higher-order resonances for eccentricities up to orbit-crossing. I demonstrate that predictions of the new integrable model agree well with numerical integrations of resonant planet pairs. Finally, I explore the secular evolution of resonant planets' eccentricities. I show that the secular dynamics are governed by conservation of an AMD-like quantity. I also demonstrate that secular frequencies depend on planets' resonant libration amplitude and this generally gives rise to a secular resonance inside the mean motion resonance at large libration amplitudes. Outside of the secular resonance the long-term dynamics are characterized small adiabatic modulations of the resonant motion while inside the secular resonance planets can experience large variations of the resonant trajectory over secular timescales. The integrable model derived in this work can serve as a framework for analyzing the dynamics of planetary MMRs in a wide variety of contexts.