Question #43474

A self-gravitating fluid planet of constant mass $M$ orbits a star of mass $M_*$ in a circular orbit of radius $a$. The planet's internal density profile is given by $\rho(r) = \frac{M}{\pi R^4} r$, where $r$ is the radial distance from the planet's center and $R$ is its dynamic surface radius. Due to tidal interactions, the planet's radius adjusts to its orbital velocity $v$ such that $R(v) = \frac{k}{v^2}$, where $k$ is a positive constant. The gravitational self-energy of the planet is defined as $U = - \int_0^R \frac{G m(r)}{r} d m(r)$, where $m(r)$ is the mass enclosed within radius $r$. Which of the following represents the rate of change of the planet's self-energy with respect to the orbital radius $a$, $\frac{dU}{da}$?