9. Recall the general form of Kepler's Third Law: a' (in AU) pz (in Earth years) - Mtotal (in solar masses) where the orbital period, P, has units of Earth years, the semimajor axis, a, has units of AU, and the total mass of the two objects orbiting each other, Mtotal, has units of solar masses. This equation can be used to describe any two objects orbiting each other under the influence of gravity. Now, consider Saturn, which has a mass of 5.68 x 1020 kg. By expressing its mass in units of solar masses (1 Mo = 1.99 x 1030 kg), and converting the units of P and a to days and kilometres, respectively, Kepler's Third Law can instead be written as: pz (in days) = _ a (in km) Written in this form, P is the orbital period of any one of Saturn's moons (in units of days), a is the semimajor axis of that moon's orbit (in units of km), and C is a constant for a given planet. For Saturn, the value of this constant is C = 7.16 x 1015, Note that the masses of Saturn's moons are negligible compared to the mass of Saturn itself, so this equation can be used to describe the orbit of any of Saturn's moons. (8 marks) a) The values of the semimajor axes (in units of km) for Saturn's largest moons are listed in the table below. Using the above form of Kepler's Third Law for Saturn, calculate the orbital period (in days) for each of these moons. Moon a (in km) Mimas 185,520 Enceladus 238,020 Tethys 294.660 Dione 377,400 Rhea 527,040 Titan 1,221,850 Hyperion 1,481,100 lapetus 3,561,300 b) Which moon is in a 2:1 resonance with Mimas? Show this by taking the ratio of the orbital periods of the two moons. c) Which moon is in a 2:1 resonance with Enceladus? Show this by taking the ratio of the orbital periods of the two moons. d) Which moon is in a 4:3 resonance with Titan? Show this by taking the ratio of the orbital periods of the two moons