In consultation with the other members of your group, solve the problem outlined below. If you finish your assigned problem quickly, try one of the other problems as well.
1. Mars has an average density of 3900 kg/m3. Assuming a mantle density of 3800 kg/m3 and a core radius of 0.44xRM, what is the core density of Mars?
MT = MM + MC
rhoa 4/3 pi R3 = rhoM[4/3 pi R3 -
4/3 pi RC3] + rhoC 4/3 pi RC3
rhoa = rhoM[ 1 - (0.44)3] + rhoC (0.44)3
rhoC = (rhoa - rhoM)/(0.44)3 + rhoM = 5000 kg/m3
2. Mercury has an average density of 5420 kg/m3. Assuming a mantle density of 4500 kg/m3 and a core density of 8000 kg/m3, what is the radius of Mercury's core relative to the total radius?
MT = MM + MC
rhoa 4/3 pi R3 = rhoM 4/3 pi R3 +
4/3 pi RC3 [rhoC - rhoM]
rhoa = rhoM + (RC/R)3[rhoC -rhoM]
RC/R = [(rhoa - rhoM)/(rhoC - rhoM)]1/3 = 0.64
3. Suppose you discover a new planet/moon system in the inner solar system. You determine that the moon orbits the planet with a period of 42.5 days, at a distance of 4.18 x 105 km. Based on the angular size of the planet, you measure its equitorial radius to be 5100 km. What is its average density? If it has a mantle density of 4500 kg/m3 and a core density of 8000 kg/m3, what is the radius of the core relative to the total radius?
P2 = 4 pi a3 / G M
M = 4 pi a3/ G P2 = 3.20 x 1024 kg
rhoa = M/V = [3/(4 pi)] M / R3 = 5800 kg/m3
MT = MM + MC
rhoa 4/3 pi R3 = rhoM 4/3 pi R3 +
4/3 pi RC3 [rhoC - rhoM]
rhoa = rhoM + (RC/R)3[rhoC -rhoM]
RC/R = [(rhoa - rhoM)/(rhoC - rhoM)]1/3 = 0.72