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>.

\Image[../../../paper-316-inclination-resonance.svg]
{title=Geometry of the inclination-type resonance}
{description=The first two panels show a conjunction just after the outer body crosses its ascending node. The third shows the representative figure-eight projection of the q equals two resonant orbit for p equals one in the frame rotating with the inner body.}

Solution

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

An inclination-type mean-motion resonance has a leading <resonant argument> containing a <longitude of ascending node>. Reflection symmetry forces the total nodal coefficient in each disturbing-function harmonic to be even, so the leading resonant strength is an even power of $\sin(I/2)$.

= Inclination resonance
{synonym}

Codex @codex  0

 Incoming links: Inclination-type mean-motion resonance