
Consider a shell of stars of thickness T and radius R.
F = f x N = L x n x T
===> a simple and pretty (and interesting) result.
The key point is that the amount of light we receive from the shell does not depend upon how far away is the shell. We receive the same amount of light from distant shells as we do from nearby shells. Hmmmmmm. So, if there are million such shells in the Universe, then we simply multiply the contribution of 1 shell by million to get the total amount of energy we receive from the Universe. Further, we should see this light at all times, even at night, since the shells completely surround us. This type of reasoning gave rise to Olbers's Paradox
The fraction of the surface area of a sphere of radius r covered by such a star is then
so that
Here, I measured the stellar density n as the number of stars per cubic parsec and the thickness of the shell in parsecs. Recall that 1 parsec = 3.26 light years. These are convenient units because in our Galaxy, there is roughly 1 star per cubic parsec and the average separation between stars is on the order of 1 parsec.
The fraction of the shell blocked out by the stars in the shell does not depend upon the radius of the shell (how far away the shell lives) ===> Olbers's Paradox if the Universe is big enough.
Okay, so what's the way out? Something must be wrong with one (or more) of the original assumptions, or some physics has not been considered. Possibilities:
Expansion of the Universe 2 ===> Imagine that the star (galaxy) produces 1 photon every second. If there is no relative motion between the star and us, then we will also see photons (energy) fly by at the same rate (===>same luminosity). However, if the star is moving away from us then we will see the photons fly by a slower rate than 1 per second. This means that the energy will arrive at the Earth at a slower rate ===> a lower luminosity.
The preceding effects conspire to make distant objects in an expanding universe have apparent brightnesses which fall off faster than the inverse square law. This decreases the contributions from distant shells. The expanding universe effects partially explain Olbers's Paradox.
To calculate the number of shells, we note that there is roughly 1 star per cubic parsec in our galaxy ===> average distance between stars in our Galaxy (shell thickness, T) is ~ 1 parsec (explain why).
===> number of shells ~ 1 / [ 5 x 10^{16} ] ~ 2 x 10^{15}
Because each shell is ~ 1 parsec thick ===> Universe needs to be at least 2 x 10^{15} parsecs in radius. Recall that 1 parsec = 3.3 light years and so the Universe must be at least 6.6 x 10^{15} light years in size in order to make the night sky as bright as the surface of the Sun.
It is interesting that in asking and answering the seemingly trivial question, "Why is the night sky dark?" one could have inferred that the Universe was expanding and that the Universe had a finite age (or at the least the stars and galaxies had finite ages).