= Solution
In <resonant trapping of dust>, the resonant torque must compensate the inward <Poynting–Robertson drag>. At the nearly fixed resonant <semi-major axis>, $\dot a_{\rm res}+\dot a_{\rm pr}=0$ requires
$$
C e^q\sin\phi=-\frac{A}{a^2(p+q)}(1+3e^2).
$$
Since $|\sin\phi|\le1$, 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 gives
$$
\boxed{e_{\min}\simeq\left(\frac{A}{|C|a^2(p+q)}\right)^{1/q}.}
$$
Equality allows only the extreme phase $|\sin\phi|=1$, 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.
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