Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 4 / i / Solution

At the exterior $(p+q):p$ mean-motion resonance,Using the Kepler third law for $M_{\ast }\gg M_1$ givesThe disturbing function is a Fourier series in integer combinations of the orbital angles. The D'Alembert characteristic permits the eccentric termAway from resonance, terms with rapidly circulating angles average away. Here, however,so ${\varphi }_1$ is a slow resonant argument. Successive astronomical conjunctions then act coherently, making this term dominate the long-period resonant dynamics even though a $q$th-order resonance has coefficient proportional to $e^q$ at small eccentricity.