Lecture 12 Homework Problems
Tidal Forces and Roche Limit

1. (a) Use the binomial expansion (general expression given in the lecture) to show that DF = F2-F1 = GMm/d2[(1 + R/d)-2 - (1 - R/d)-2], after expansion of the two terms in the square brackets and keeping the leading term in R/d, becomes DF = -4GMmR/d3. (b) Give the expression for the ratio of DFmoon/DFearth in terms of the Earth and Moon masses and radii (DF is the difference in force on the two sides of one body due to the other--be careful with the mass M) and show that this ratio is about 22 (i.e. the Moon feels a much larger tidal force than the Earth).

2. The lecture mentions that the Earth's rotation is slowing due to the tidal interaction with the Moon. The measured rate of the increase in rotational period is 0.0016 s/century. (a) How much longer is the length of the day now than when the pyramids were built 4570 years ago? (b) The Earth will be tidally locked with the Moon when its period is about 47 days. How many years in the future will this happen? (c) How far from the Earth will the Moon be at that time? Express your answer in units of the current Earth-Moon distance, dmoon.

3. Use the data from Appendix C to calculate the average density of Mars. The density of Mars' moon Phobos is 2000 kg/m3, and it orbits at a distance of 9378 km from the center of Mars. Is it within the Roche limit for the Mars-Phobos system? The Wikipedia entry for Phobos says that it is losing altitude by about 20 m/century. Using your calculation of the Roche limit and the rate of decrease of orbital distance, how long would you predict before it reaches the Roche limit? The prediction is that it will then break up and form a ring around Mars.

4. The lecture says "Incidentally, although the rings of Saturn cover a huge area, they are very thin, with a total mass of only 1016 kg.  A body of this mass, with a density of 1200 kg/m3, would have a radius of only about 13 km." However, according to this article from 2010, the A ring mass estimate is as much as 0.7x1019 kg, and the B ring mass estimate is as much as 7x1019 kg. Do the calculation with the combined A and B ring mass to find what is the radius, in km, of a body with density 1200 kg/m3 that could supply this mass. Which of Saturn's existing moons is closest to the required mass?