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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 = 4πr^{2}dr
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 = 4πr^{2}Ndr
Total intensity from all stars in shell is
proportional 4πr^{2}Ndr/r^{2} = 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, I_{Total} 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 = H_{0}D
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.