Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 4 / v / Solution

Just after $M_2$ crosses its ascending node, it lies above the nearly fixed orbital plane of $M_1$. At a nearby conjunction, $M_1$ is inward of $M_2$, because $r_1<r_2$. The perturbing acceleration toward $M_1$ therefore has an inward component and a downward normal component. In the view along the node toward the star, the downward component opposes $M_2$'s upward vertical velocity and reduces its orbital inclination; in the face-on view, the inward component perturbs its orbital energy and shifts the conjunction phase. A conjunction before the ascending node reverses the relation between the normal force and vertical velocity. Repeated phase-correlated impulses therefore provide the restoring dynamics of an inclination-type mean-motion resonance.