= Inspiral time of circumplanetary dust
{title2=$t_{\rm cp}$}
For coplanar circular dust, constant $A=\beta GM_\star/c$, fixed stellar radius $a_p$ and initial planetocentric radius $a_0>R_p$, integrating $\dot a=-3Aa/a_p^2$ gives
$$
a(t)=a_0e^{-3At/a_p^2},\qquad
t_{\rm cp}=\frac{a_p^2}{3A}\log\frac{a_0}{R_p}.
$$
Compare the <inspiral time under Poynting–Robertson drag> for a circular <circumstellar orbit>, $t_{\rm cs}=(a_p^2-R_\star^2)/(4A)$. Planetary arrival is not automatically faster; in the point-star limit it is faster only for $a_0/R_p<e^{3/4}$. A different constant orientation coefficient replaces three.
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