Coplanarity requires both $I=I_d$ and $\Omega=\phi_d$, up to the <orbital-plane orientation degeneracy>. Under that hypothesis the measured planet angle must obey
$$
\tan(\phi-\phi_d)=\cos I_d\tan f.
$$
For a nearly edge-on belt this offset is small except close to the projected crossings where $\cos f=0$. Measurements at several <orbital phases> can therefore test the entire projected ellipse: a systematic offset from the predicted relation implies a nonzero <mutual inclination>. A single projected location generally cannot establish coplanarity because $f$, the front-back branch, and the nodal orientation can be degenerate.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 1 / vii / Solution

Orbital phase specifies where a body lies within its orbital cycle relative to a chosen reference event.


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