(a) Make a rough estimate of the mass contained in Saturn's rings. Assume that the rings have a constant mass density and that the disk is 30 m thick with an inner radius of 1.5 RS and an outer radius of 3 RS (neglect the E ring). Assume also that all of the ring particles are water-ice spheres of radius 1 cm and that the optical depth of the disk is unity. The density of the particles is approximately 1000 kg m−3. Hint: Refer to Eq. (9.12) to estimate the number density of water-ice spheres.
(b) If all of the material in Saturn's rings were contained in a sphere having an average density of 1000 kg m−3, what would the radius of the sphere be? For comparison, the radius and mass of Mimas are 196 km and 4.55×1019 kg, respectively, and it has an average density of 1440 kg m−3.