Consider a newly formed globular cluster, with a total mass 10^6 Solar Masses, and an initial mass function dN/dm = am^-2.35 in the mass range 0.1 - 20 Solar Masses, where m = M/1 Solar mass.

Find the constant a

Find the total luminosity of the cluster, assuming that all its stars are on the main sequence, and a mass-luinosity relation L~M^4. What fraction of the luminosity is contributed by stars more massive than 5 solar Masses?

Find the mean mass of a star in the cluster.

Assume that the main sequence lifetime of a 1 solar mass star is 10 Gyr, and main sequence lifetime scales with mass as M^-2. What is the mass of the most massive main sequence stars in the cluster after 1 Gyr? What is the total luminosity of the cluster at thtat time?