The night sky is not entirely dark because of the thousands of stars that shine. Some stars are closer to us, while others are farther away, which, along with other factors, affects their brightness. For simplicity, we assume that stars are uniformly

The night sky is not entirely dark because of the thousands of stars that shine. Some stars are closer to us, while others are farther away, which, along with other factors, affects their brightness. For simplicity, we assume that stars are uniformly distributed with a density of 0.004 stars per cubic light-year, and that their luminosity is comparable to that of the Sun ( $3.8 \times 10^{26}$ watts). \nConsider a thin spherical shell of radius $r$ and thickness $\Delta r$ with the Earth at its center: (a) Find an expression for the number of stars $\Delta N(r)$ in this shell and for the total radiation power $\Delta P(r)$ reaching the Earth from this shell. (b) Find an expression for the total power $P(r)$ received from all stars up to a distance $r$. (c) The universe is infinite. What is the total power received for $r \rightarrow \infty$ ? (d) Explain, why modern cosmology is relevant for correctly interpreting this result.