4) This problem attempts to account for the universe's helium. In virtually everything in the universe (old stars, young stars, interstellar matter, cosmic rays,...) the abundance of helium by mass is 25%. We first determine if we can blame the origin of the helium on fusion in stars. The mass of the galaxy is reasonably well known:
M_gal = 4 x 10⁴¹ kg. Assume that the galaxy's mass initially consisted entirely of hydrogen, and further assume that all of the galaxy's luminosity (assumed to have been constant with time: L_gal = 10³⁷ W) is and always has been the result of hydrogen-to-helium fusion in stars. The galaxy's age is known from stellar-evolution data to be 13 billion years.

a) Calculate the total amount of light energy that has been released by the galaxy's stars up until the present time.

b) Calculate the total mass of hydrogen that the galaxy must have fused into He in order to have released the light energy found in part a.

c) Compare the fraction of the galaxy's mass that should be He at present (under the problem premises) and compare this with the present fraction of He in the galaxy.

d) Suggest 2 ways out of the discrepancy posed in the previous question. Be fairly specific in explaining how each of your methods could eliminate the discrepancy between the He abundance predicted (in part c) and that observed. You cannot blame it on an error in the given values of the mass, present luminosity, age of the galaxy, or the observed He mass fraction; each of these numbers is fairly well fixed by observation. Anything else, I think, is up for grabs.