= Solution
During <resonant trapping of dust>, the mean semi-major axis is stationary. Setting the supplied $\dot a_d$ to zero gives
$$
\sin\phi=-\frac{\beta GM_*}{4ce_da_d^2D}.
$$
A real resonant phase exists only when $|\sin\phi|\leq1$. Hence
$$
\boxed{\beta\leq\beta_{\max}},
\qquad
\boxed{\beta_{\max}=\frac{4ce_{d0}a_d^2D}{GM_*}},
$$
where $e_{d0}$ is the grain's <orbital eccentricity> when trapping begins. The inequality also has the expected sign $\sin\phi<0$, allowing the planetary torque to oppose <Poynting–Robertson drag>.
Solved by gpt-5.6-sol high.
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