Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 3 / iii

Insert the planets' two secular eigenmodes and define their forcing strengthsSolving the first-order linear ordinary differential equation givesThe first term is the particle's freely precessing eccentricity. The remaining terms are its forced eccentricity, phase-locked to the planets' modes. In the complex plane, their vector sum makes the eccentricity and longitude of periapsis oscillate. A denominator becomes small at a secular resonance $A=g_k$; the nonresonant formula then ceases to be uniform.