A scientific probe is sent to orbit a newly discovered spherical asteroid of uniform density ρ\rhoρ. The probe is placed into a circular orbit at an altitude very close to the asteroid's surface, so that its orbital radius can be assumed to be equal to the radius of the asteroid, RRR.
Show that the orbital period T T\,T of the probe is independent of the radius RRR, and is given by:
T=3πGρ T = \sqrt{\frac{3\pi}{G\rho}} T=Gρ3πwhere G G\,G is the gravitational constant.