Question #43513

A spherically symmetric planet of radius $R$ has a non-uniform mass density given by $\rho(r) = \frac{\rho_0 R}{r}$ for $r \le R$, where $\rho_0$ is a positive constant and $r$ is the radial distance from the center. A satellite of mass $m$ is in a circular orbit around this planet at a radial distance of $r = 2R$. A particle, also of mass $m$, is projected from the center of the planet ($r=0$) through a narrow, smooth radial tunnel. If the particle's speed when it reaches $r = 2R$ is exactly equal to the orbital speed of the satellite, what is the required projection speed $v_0$ at the center? (Assume no other celestial bodies are present and $G$ is the universal gravitational constant).