= Poynting–Robertson decay of a circumplanetary orbit
{c}
{title2=$\langle\dot a\rangle_{\rm cp}$}
Consider a circular <circumplanetary orbit> well inside the <Hill sphere>, with dust speed $u^2=GM_p/a$ and unit plane normal $\widehat{\boldsymbol k}$. During one dust <orbital period>, take the stellar direction $\widehat{\boldsymbol R}$ and flux at the <planet>'s stellar radius $a_p$ as constant. The velocity-dependent <Poynting–Robertson drag> has relative <work>
$$
\langle\dot\varepsilon_p\rangle=-\frac A{a_p^2}
\langle u^2+(\boldsymbol u\cdot\widehat{\boldsymbol R})^2\rangle,\qquad
A=\frac{\beta GM_\star}c.
$$
The constant terms have zero <work> over the closed circular <orbit>. Since $\langle u_i u_j\rangle=(u^2/2)(\delta_{ij}-k_i k_j)$,
$$
\langle\dot a\rangle=-\frac A{a_p^2}a
[3-(\widehat{\boldsymbol k}\cdot\widehat{\boldsymbol R})^2].
$$
The coefficient is three for coplanar stellar and dust <orbits>; it is two for a dust plane normal to the instantaneous stellar direction. Averaging over a circular planetary year at fixed dust <orbital inclination> $I$ gives $3-\sin^2I/2$. Conservative perturbations must remain small enough for this circular-orbit average.
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