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solutions-ps06 - Physics 341 Problem Set#6 Solutions 1 Mars...

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Physics 341: Problem Set #6 Solutions 1. Mars has a mass of 6 . 4 × 10 26 g (about one tenth M ) and a radius of 3400 km (about half R ). Its small moon Phobos has a mass of 1 . 1 × 10 19 g and a radius of just 11 km . Phobos orbits Mars with a semimajor axis of 9380 km . (a) What are the mean densities of Mars and Phobos, in g cm - 3 ? The average densities are just ρ Mars = 3 M Mars 4 πR 3 Mars = 3(6 . 4 × 10 26 g) 4 π (3 . 4 × 10 8 cm) 3 = 3 . 9 g cm - 3 ρ Phobos = 3(1 . 1 × 10 19 g) 4 π (1 . 1 × 10 6 cm) 3 = 2 . 0 g cm - 3 (b) What is the Roche limit for the Mars/Phobos system? Is Phobos currently inside or outside of the Roche limit? The Roche limit is r T = f ρ planet ρ moon 1 / 3 R planet where f 2 . 5. Plugging in the numbers from part (a) we find: r T = 2 . 5 3 . 9 g cm - 3 2 . 0 g cm - 3 1 / 3 R Mars = 3 . 1 R Mars = 10 , 500 km That means that Phobos’ orbit ( a = 9380 km) is inside the Roche limit today! (c) Use Kepler’s Third Law to calculate the orbital period of Phobos, in hours. We have P = 2 π a 3 GM 1 / 2 = 2 π (9 . 38 × 10 8 cm) 3 (6 . 67 × 10 - 8 cm 3 g - 1 s - 2 )(6 . 4 × 10 26 g) 1 / 2 = 2 . 8 × 10 4 s = 7 . 7 hours (d) Recall in class we said that tidal forces are causing the Moon’s orbit to recede from the Earth. Because Phobos orbits Mars faster than the rotation period of Mars, unlike the Moon, tidal forces cause Phobos’ orbit to shrink. The semimajor axis is decreasing at a rate of 20 cm yr - 1 . At that rate, how long is it until Phobos hits the surface of Mars?
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