Canceling Effects of Conjunctions Render Higher-order Mean Motion Resonances Weak

Abstract Mean motion resonances (MMRs) are a key phenomenon in orbital dynamics. The traditional disturbing function expansion in celestial mechanics shows that, for coplanar systems at low eccentricities, p : p − q MMRs exhibit a clear hierarchy of strengths, scaling as e q , where q is the order of the resonance. This explains why first-order MMRs (e.g., 3:2 and 4:3) are important, while the infinite number of higher-order integer ratios are not. However, this relationship, derived from a technical perturbation series expansion, provides little physical intuition. In this paper, we provide a simple physical explanation of this result for closely spaced orbits in coplanar systems. In this limit, interplanetary interactions are negligible except during close encounters at conjunction, where the planets impart a gravitational “kick” to each other’s mean motion. We show that while first-order MMRs involve a single conjunction before the configuration repeats, higher-order MMRs involve multiple conjunctions per cycle, whose effects cancel out more precisely the higher the order of the resonance. Starting from the effects of a single conjunction, we provide an alternate, physically motivated derivation of MMRs’ e q strength scaling.