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

The oriented normal vectors to the planet and dust-belt planes areTheir inner product is the cosine of the mutual inclination, soIf ${\varphi }_d=\Omega$, this reduces to $\cos I_m=\cos (I-I_d)$; equal inclinations then give $I_m=0$. If both planes are face-on, it likewise gives $I_m=0$, independently of their undefined nodal longitudes. Reversing an observationally unidentified ascending node by $\pi$ produces the familiar orbital-plane orientation degeneracy for an axisymmetric belt.