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

In resonant trapping of dust, the resonant torque must compensate the inward Poynting–Robertson drag. At the nearly fixed resonant semi-major axis, ${\dot{a}}_{\mathrm{res}}+{\dot{a}}_{\mathrm{pr}}=0$ requiresSince $|\sin \varphi |\le 1$, a necessary torque-balance condition is $|C|e^q\ge A(1+3e^2)/[a^2(p+q)]$. When the limiting orbital eccentricity is small, this givesEquality allows only the extreme phase $|\sin \varphi |=1$, with no margin for restoring torque; a robust trapped state needs $e>e_{\min }$. This is a necessary condition for resonance capture, not a sufficient probability-of-capture criterion for a migrating particle.