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Last modified: January 17, 2022 ✓

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 be 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 is 
proportional 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 
    traveling at greater than the speed of light
    so we will never see them.  The possible 
    responses to this are:

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

2.  Possible, but this suggests the universe was 
    instantly created.  There is no 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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