Assume that you are in an infinitely old, infinitely large, static, Euclidean universe. The average density of stars is n∗ = 10^9 Mpc−3 and the average stellar radius is similar to the Sun’s, R∗ = 7×10^8 m. a) We can define a distance r_1/2 such that stars closer than r_1/2 will cover half of the sky. Calculate r_1/2 in Mpc. Hint: start by computing the number of stars in a thin spherical shell of width dr and the fraction of the sky blocked by these stars. You can make the simplifying assumption that stars don’t overlap.

Assume that you are in an infinitely old, infinitely large, static, Euclidean universe. The average density of stars is n∗ = 10^9 Mpc−3 and the average stellar radius is similar to the Sun’s, R∗ = 7×10^8 m.
a) We can define a distance r_1/2 such that stars closer than r_1/2 will cover half of the sky. Calculate r_1/2 in Mpc. Hint: start by computing the number of stars in a thin spherical shell of width dr and the fraction of the sky blocked by these stars. You can make the simplifying assumption that stars don’t overlap.
b) Now suppose these stars have only been shining for 13 billion years. What fraction of the sky is covered by stars in this case?