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Last modified: January 26, 2018

Olbers' Paradox
---------------
Olber thought the sky should be light at night based on the belief at the 
time that the universe was infinitely big and infinitely old.  In every 
direction you should eventually see a star and the sum of all these sources
would cause the sky to by light.  Consider a spherical shell.

volume = 4πr2dr where r is radius and dr is the thickness of the shell.

Let N = number of stars per unit volume of space.

# of stars in shell = 4πr2Ndr

Total intensity from all stars in shell ∝ 4πr2Ndr/r2 = 4πNdr

Olbers' interpretation was that when r is small, there are fewer stars 
but their intensities are larger.  When r is large there are more stars,
but their intensities are smaller. These 2 effects cancel out, and all
shells produce the same intensity.  Now, since the number of shells is 
infinite in an infinite universe, ITotal should equal ∞ and lead to a 
light sky (and a frazzled earth!).  Noting that the night sky is 
obviously not that bright and that we are still here, begs the 
question "what is going on?".  There are several options.

1.  The universe is not infinite and there are only a finite number of stars. 
2.  The universe is of finite age so that light from stars at an great
    distances hasn't reached us yet.
3.  Space is expanding, so distant stars are red-shifted into obscurity.  
    Hubble's law states that the recessional velocity of a star, v is 
    related to its distance, D, by:

    v = H0D

    So, the further the star is away, the faster it is moving. The outermost
    stars can be travelling at > c, so we will never see them.

1. This is possible, but there still should be enough to light up the sky.

2. Possible, but suggests the universe was instantly created - not an 
   immediate answer to this question!

3. This is the generally accepted argument. The implication is that 
   there was a point in time when the universe was small enough to 
   observe constant light. In the present day, the wavelength of light 
   from this time has been stretched as space itself expands.  We now 
   see this doppler shifted light in the form of cosmic background 
   radiation which is not visible.

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