 Inspiral time of circumplanetary dust

ID: inspiral-time-of-circumplanetary-dust

For coplanar circular dust, constant

A = βG M_{⋆} / c

, fixed stellar radius

a_{p}

and initial planetocentric radius

a_{0} > R_{p}

, integrating

\overset{a}{˙} = - 3 A a / a_{p}^{2}

gives

a (t) = a_{0} e^{- 3 A t / a_{p}^{2}}, t_{cp} = \frac{a _{p}^{2}}{3 A} lo g \frac{a _{0}}{R _{p}} .

t_{cs} = (a_{p}^{2} - R_{⋆}^{2}) / (4 A)

. 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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