= Solution
Let $\mu_p=GM_p$, $A=\beta GM_\star/c$ and $\kappa=A/a_p^2$. A tightly bound <circumplanetary orbit> lies well inside the <Hill sphere>, so $a/a_p\ll1$ and its <orbital period> is short compared with the <planet>'s year. Treat the stellar flux and stellar direction $\widehat{\boldsymbol R}$ as constant during one dust <orbit>. With stellar-frame <velocity> $\boldsymbol V_p+\boldsymbol u$, the velocity-dependent <acceleration> is
$$
\boldsymbol a_{\rm PR}=-\kappa[\boldsymbol V_p+\boldsymbol u+
((\boldsymbol V_p+\boldsymbol u)\cdot\widehat{\boldsymbol R})\widehat{\boldsymbol R}].
$$
The terms independent of $\boldsymbol u$, including the leading static <radiation pressure>, do no net <work> on an unperturbed closed circular <orbit>. Hence
$$
\langle\dot\varepsilon_p\rangle=-\kappa\langle u^2+(\boldsymbol u\cdot\widehat{\boldsymbol R})^2\rangle.
$$
For a coplanar circular <orbit>, $u^2=\mu_p/a$ and the component along the stellar direction has mean square $u^2/2$. Thus $\dot a=2a^2\dot\varepsilon_p/\mu_p$ yields the <Poynting–Robertson decay of a circumplanetary orbit>
$$
\boxed{\langle\dot a\rangle_{\rm cp}=-3\frac{\beta\mu}{c}\frac a{a_p^2}}.
$$
\b[The coefficient three assumes coplanarity, which the PDF does not state]. For orbital normal $\widehat{\boldsymbol k}$, the general circular covariance is $\langle u_i u_j\rangle=(u^2/2)(\delta_{ij}-k_i k_j)$, giving
$$
\boxed{\langle\dot a\rangle_{\rm cp}=-\kappa a[3-(\widehat{\boldsymbol k}\cdot\widehat{\boldsymbol R})^2]}.
$$
An <orbital plane> normal to the stellar radial direction has coefficient two during that <orbit>, a counterexample to an orientation-independent coefficient three. Averaging also over the <planet>'s circular stellar <orbit>, with fixed dust-plane <orbital inclination> $I$ to it, gives coefficient $3-\tfrac12\sin^2I$. The conservative <forces> must be weak enough for the assumed approximately circular, planet-bound <orbit> to persist; the small planet-to-star <mass> ratio alone does not ensure this for arbitrary grain $\beta$.
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