Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 1 / iii

Write $u=e^2$ and use the torque balance from part (ii) to eliminate the resonant argument. The resonant contribution isAdding the Poynting–Robertson drag contribution gives the dissipative resonant eccentricity equilibrium equationwhereFor $5p>q$, integrating with $u(0)=e_0^2$ givesHere $a$ is the fixed resonant semi-major axis and the expression describes averaged evolution, neglecting the small resonant-argument libration. The physical bound-orbit interpretation also requires $B<1$; a quantitatively controlled second-order calculation needs $B\ll 1$.