TimeThe BIG BANG! Todayt0
Distance to the earth
0
s0
s1
MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 1: Olbers' Black Sky Paradox
Heinrich Olbers was an astronomer who, in 1826, drew an interesting conclusion about the darkness of
the nighttime sky. Olbers assumed that the universe is infinite in size, that it is populated uniformly with
stars and galaxies, and that it is static in time and space (so that nothing moves or evolves). Using the fact
that the light intensity I at a distance  from a light source is proportional to the inverse of the square of :
I = I() =
k
2
for some positive constant k, Olbers concluded that the total amount of light seen by an observer standing
on the face of the earth should be infinite; that is,
R3
I dV = .
Prove this result by writing the triple integral as an improper iterated triple integral in spherical coordinates,
and evaluating it.
The nighttime sky is not bright, however: it is dark. Thus at
least one of the assumptions must be wrong. Which? Well, today
astronomers believe it is a combination of factors. On one hand,
since the universe is expanding, distant stars contribute less light
than near stars. More importantly, when we look into the sky, we
are looking back in time, and we can look back in time only so long
(see the figure to the right): If you see a star in the nighttime sky
tonight, a star that is and always has been at a fixed distance s0
from the earth then you will be looking at light emitted by the star
at some time t0 in the past. You might also be able to see light
emitted by a star prior to t0, a star at a distance of between s0 and
s1 from the earth, but you could not see light from a star at any
further distance: light from a star at such a distance would have
had to have been emitted prior to the big bang.
Contributor: Dr. F. Kleinhans
Department of Physics, IUPUI