1. Imagine you are looking at main sequence stars in the Milky Way, which all have an absolute magnitude of M=4.2. Assume that the density of these stars in the local region is constant. For this case, the number of stars closer than \"r\" is N(r)=4mdpi'3, where p is the density of the stars. (a) Show that the number of these stars brighter than apparent magnitude m is given by: log N{m) = 0.6 m + C. (b) Draw a sketch of log N(m) versus m. (c) Assume that the absolute magnitude of the stars counted is M = 4.2. How would your sketch change if there were a strongly absorbing cloud (1 magnitude of extinction) of interstellar dust localized at a distance of 10 kpe'? Draw the new sketch. (Hint: Does the cloud affect how we see objects closer than it? Farther away?)