Question #43512

A satellite of instantaneous mass $m$ is in a nearly circular orbit around a planet of mass $M$ ($M \gg m$). The satellite passes through a uniform, stationary dust cloud, accreting dust at a constant rate $\frac{dm}{dt} = \lambda$. Due to this accretion, the satellite's instantaneous equilibrium circular orbit radius $r_c$ slowly shrinks. The satellite undergoes small radial oscillations $x$ such that its radial distance is $r(t) = r_c + x(t)$, where $x \ll r_c$. Assume the fractional change in mass during one oscillation is negligible. The differential equation governing these small radial oscillations can be mapped to the equation of a damped L-C-R circuit: $L_e \frac{d^2q}{dt^2} + R_e \frac{dq}{dt} + \frac{q}{C_e} = 0$. If the radial displacement $x$ is analogous to the charge $q$, and the instantaneous mass $m$ is analogous to the inductance $L_e$, what are the equivalent resistance $R_e$ and capacitance $C_e$ of the circuit?