1. By the virial theorem, the amount of energy available for radiating (mechanical energy) is 1/2 the potential energy of a body. Equation 10.23 of the text indicates that the current mechanical energy of a planet of mass M and radius R is E = -3/10 GM²/R, which means that over its lifetime the planet would have had an equal amount, E_rad = 3/10 GM²/R available for radiating due to gravitational collapse. (a) Calculate E_rad for Jupiter to get the amount of energy it would have had to radiate over its 4.55 billion year lifetime. (b) Estimate the rate of energy (J/s, or watts) of energy output due to gravitational collapse, assuming the rate is constant over its lifetime. (c) Compare your answer to (b) with the current value of dE/dt given in the lecture. What can you conclude about the rate of Jupiter's energy output in the past?

2. Use the T_eq expression given in Lecture 14 to determine the current temperature Neptune should have if it gets its energy only from the Sun. In fact, half of the energy radiated by Neptune is due to internal energy sources (e.g. left over from gravitational collapse). Using an argument similar to that given for Jupiter in Lecture 17, what is the current temperature at the "surface" (cloud tops) of Neptune taking into account this amount of internal energy?

3. To get an idea of the longevity of the rings of Saturn, use the equation given in part (a) of problem 21.16 of the text (I am not asking you to derive it) to answer the question in problem 21.16 (b). This pertains to the E ring, and comes from the light from the Sun being radiated in all directions, which leads to deceleration of the particle (the Poynting-Robertson effect, see problem 21.15 of the text). The closer ring particles in the A and B ring should take even less time, although some parts of this estimate may be violated (they do not absorb 100% of the radiation that falls on them), but overall the rings are expected to last only 10-100 million years.