SOLUTION TO EXERCISE 8


According to the problem, the average galaxy in the universe has 15%
of the mass of the Milky Way.  According to the text, there are about
100 billion galaxies in the observable universe.  We can estimate that
a typical star's mass is that of the Sun.  That means, including dark 
matter, the mass of the observable universe from these galaxies is roughly

  M = (10^11 galaxies) * (0.15) * (10) * (10^11 stars * 2 * 10^30 kg/star) 
    = 3.0 * 10^52 kg.

The size of the observable universe is roughly 13 billion light-years (ly) 
in every direction.  So if we approximate the size and shape of the observable
universe as a sphere of radius 13 billion light-years,

  r = (13 * 10^9 ly) * (3 * 10^8 m/s) * (3.15 * 10^7 s/yr) = 1.22 * 10^26 m

  V = (4/3) * pi * r^3 = 7.6 * 10^78 m^3 

and thus the density of the observable universe is

  D = M/V = (3.0 * 10^52 kg)/(7.6 * 10^78 m^3) = 3.9 * 10^-27 kg/m^3

Or, since the mass of a hydrogen atom is almost exactly the mass of a proton

  D = (3.9 * 10^-27 kg/m^3) / (1.67 * 10^-27 kg/atom) = 2.3 h-atoms/m^3.