The gravothermal catastrophe. This is a variant on a classic celestial me- chanics result. We're going to start with the result of the last problem, and with the stellar heat capacity problem from homework 2. We start out with a star with mass M and size R. For the moment, let's say it's made up of hydrogen. (a) What is the temperature of the star, T? (b) We surround the star with a cloud of ideal gas (stars are born in clouds of ideal gas) of hydrogen. For simplicity, we assume that it has the same mass as the star. Its temperature Tg is lower than that of the star. Draw a plot of entropy vs. energy for both the star and the gas. (c) With this assumption about the mass and content of the two systems, what is the relationship between the heat capacity of the gas and the heat capacity of the star? (d) Let a tiny amount of heat, AU flow from one system to the other. What are the temperatures T and T of the star and gas after this amount of energy transfers? Which way does heat flow? Hints: use the heat capacities to relate AU to temperatures. Let the second law guide you! (e) Imagine that heat keeps flowing from one part of your system to the other. Do you reach thermal equilibrium? If so, at what temperature? (f) you have a relationship between the energy of a star, its size, and its temperature. How does the radius of the star change with the energy and temperature of the star, as the system evolves?