Canceling Effects of Conjunctions Render Higher Order Mean Motion Resonances Weak

TL;DR

This paper offers a physical explanation for the weakness of higher order mean motion resonances, showing that multiple conjunctions in these resonances lead to cancellation effects that diminish their strength.

Contribution

It introduces a simple, physically motivated derivation of the $e^q$ scaling law for MMR strengths, clarifying the underlying physical mechanism.

Findings

Higher order MMRs involve multiple conjunctions per cycle.

Conjunction effects cancel more precisely at higher order, weakening the resonance.

Provides an intuitive physical explanation for the hierarchy of MMR strengths.

Abstract

Mean motion resonances (MMRs) are a key phenomenon in orbital dynamics. The traditional disturbing function expansion in celestial mechanics shows that, 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 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.