Question #43518

A massive space station is in a circular orbit of radius $r_0$ (in $\text{m}$) around a planet of mass $M$ (in $\text{kg}$). The station is tidally locked, rotating with a constant orbital angular velocity $\omega_0 = \sqrt{\frac{GM}{r_0^3}}$ (in $\text{rad/s}$) to keep the same face towards the planet. Inside the station, a smooth straight tube is aligned along the radial direction. A payload of mass $m$ (in $\text{kg}$) is placed inside the tube at the equilibrium position $r = r_0$. It is connected to the station by a spring of force constant $k$ (in $\text{N/m}$) and a damper providing a resistive force $F_d = -b v_{rel}$, where $v_{rel}$ (in $\text{m/s}$) is the radial velocity of the payload relative to the tube and $b$ (in $\text{kg/s}$) is the damping constant.

An engineer maps the small radial displacements $x$ of the payload to an equivalent electrical $L$-$C$-$R$ circuit equation $L_e \frac{d^2q}{dt^2} + R_e \frac{dq}{dt} + \frac{1}{C_e} q = 0$. In this analogy, the displacement $x$ maps to charge $q$, and the mass $m$ maps to inductance $L_e$. Assuming $k > 3m\omega_0^2$, what is the expression for the equivalent capacitance $C_e$?