In March 2006, two small satellites were discovered orbiting Pluto, one at a distance of 48,000 km and the other at 64,000 km. Pluto already was known to have a large satellite Charon, orbiting at 19,600 km with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital periods of the two small satellite? ithout? using the mass of Pluto.

Solution 26E Step 1 Gravitational force between pluto and charon = Centripetal force exerted by charon 2 2 That is, GMm/R = m R Where, R - Radius of charon orbiting around pluto M - mass of pluto m - Mass of charon - angular speed of charon 3 2 Or, GM/R = We know that, = 2 / T 3 2 2 That is, GM/R = 4 / T 2 2 3 Or, we can write, T = 4 R / GM 2 3 Since, G and M are constants for Pluto, we can write, T R Step 2 We can write, the square of the orbital period of Charon is directly proportional to the cube of the orbital radius of Charon around Pluto. 2 3 Or, T /R = a constant For the other satellite 1 having an orbital radius of 48000 km, the ratio, 2 3 T1R 1 = a constant 2 3 2 3 So, we can write, T /R = T /R 1 1 Provided, orbital radius of charon, R = 19600 km and orbital radius of satellite 1, R = 48000 km 1 2 3 3 2 3 3 Therefore, T /19600 km = T /48000 km 1 Provided, orbital period of charon is, T = 6.39 days Therefore, 6.39 days /19600 km = T /48000 km 3 2 3 3 1 Or, T 2 = (48000 km / 19600 km ) 6.39 days 3 2 2 1 T 2= 14.688 × 40.832 days 2 1 2 2 T1 = 600 days Taking square roots on both sides, T = 24.5 days 1