Question #42868

Two identical satellites, $S_1$ and $S_2$, are located in deep space at $0\text{ K}$. Each contains an identical open organ pipe filled with an ideal diatomic gas. The pipes are continuously sounded at their fundamental frequencies. The walls of the pipes have a zero coefficient of thermal expansion, and the heat capacity of the gas is negligible. Satellite $S_1$ is maintained at a constant temperature $T_0$, emitting sound of frequency $f_0$ and constant acoustic intensity $I_0$. Satellite $S_2$ is initially at $T_0$. At $t=0$, its heating system fails, and it cools by radiating as a perfect black body. $S_2$ has mass $M$, specific heat capacity $c_s$, and surface area $A$. As $S_2$ cools, the local speed of sound drops, modulating the frequency $f(t)$ of its pipe. Furthermore, the acoustic intensity $I(t)$ of the pipe in $S_2$ is directly proportional to the absolute temperature of the gas ($I(t) \propto T$). An observer counts the acoustic beats between the two pipes. However, the beat detection system automatically shuts off when the intensity $I(t)$ from $S_2$ drops to $\frac{I_0}{4}$. Find the total number of beats registered by the observer from $t=0$ until the detection system shuts off. (Assume $\sigma$ is the Stefan-Boltzmann constant).