Inspiral time of circumplanetary dust ($t_{\mathrm{cp}}$)

For coplanar circular dust, constant $A=\beta G{M}_{\star }/c$, fixed stellar radius $a_p$ and initial planetocentric radius $a_0>R_p$, integrating $\dot{a}=-3Aa/a_p^2$ givesCompare the inspiral time under Poynting–Robertson drag for a circular circumstellar orbit, $t_{\mathrm{cs}}=(a_p^2-R_{\star }^2)/(4A)$. 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.