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At night, streetlights aside, the sky gets dark. This seems obvious - at night the side of the Earth that you happen to be on is facing away from the Sun - where would light come from? However, this question has been worrying astronomers and cosmologists for around 250 years.
What is Wrong With the Dark?
Every good scientist makes assumptions. For the sake of this paradox, we will assume that the universe is homogeneous and isotropic. This assumption means that where we are in the universe is in no way special - the same substances can be found everywhere else as well as here, and the same laws of physics apply everywhere else as well as here. This also means that in any direction, when averaged over a large enough amount of space, there is around the same number of stars. We will also assume that the universe has always existed so that it is infinite in time, and is infinite in space. Finally, we will assume the universe to be Euclidean. This means that space is flat - for example, if you have two parallel lines they will stay the same distance apart.
So, because the universe is infinite, and there should be stars in all directions, which ever direction we choose to look in we should be able to see stars. Any line of sight should eventually intersect with a star, so all the sky should be bright.
In fact, because there should be infinitely many stars in all directions, all the sky should be infinitely bright, which it clearly is not.
The 'Maths' Bit
No one will think any the worse of you if you skip this section - it is included here for the sake of completeness.
The flux (or power per unit area) we get from one star, f is
where L is the luminosity (power output) of the star, and r is the distance that the star is away from us. The total flux we would receive from all stars, ftotal is the flux from one star f, multiplied by the number density of stars n integrated over all space, in this case using spherical polar co-ordinates r, q and f.
A Brief History of the Paradox
Although called Olbers' paradox, the problem originated with Johannes Kepler, who is best known for his work on the orbits of the planets around the Sun. It was subsequently taken up by Jean Philippe Leys de Cheseaux of Lausanne, who thought that all of the sky should be as bright as the Sun.
Heinrich Wilhelm Mathäus Olbers was only an astronomer in his spare time - his day job was physician. However, he studied comets and worked out a method for calculating their orbits, as well as formulating what would subsequently be called Olbers' Paradox in 1826. Olbers realised that actually the light reaching us should be infinitely bright, rather than as bright as the sun as de Cheseaux had proposed.
We have shown that it ought to be the case that the night sky is infinitely bright, but experience shows us that it isn't - how do we explain that?
Explanations that Don't Work
de Chesaux and Lord Kelvin (who gave his name to the temperature scale) suggested that there might be dust in between us and many of the stars, blocking out the light that we receive from them. However, absorbing dust would eventually come to equilibrium, and emit as much radiation as it absorbed. Even if it was at a different wavelength, we would still receive the same amount of light as before.
Some astronomers thought that if the universe was expanding (as Hubble showed that it is) light from distant stars could be redshifted by the Doppler effect. While this effect provides a contribution, it doesn't account for enough light to darken the entire sky; the remainder should still be detectable1.
Explanations that Do Work
This means that at least one of our initial assumptions was wrong - the Universe is not Euclidean, isotropic, homogeneous and infinite in space and time. Here are a couple of popular explanations that do work, although there are others:
Benoit Mandelbrot suggested that there is fractal distribution of galaxies. This would mean that there was a lot of empty spaces between stars, and galaxies, so the light coming from each direction would not add up to infinity.
Maybe the universe is not infinite in time - because the speed of light is finite, perhaps there hasn't been time for the light from stars further than a certain distance to reach us yet. Surprisingly, according to some scholars, this argument was first put forward in Edgar Allen Poe's prose-poem Eureka.
Currently, the finite age solution of the paradox is preferred because it supports Big Bang theory, which says that the universe started a few billion years ago, and has been expanding ever since. This alone (with a contribution from the expanding universe explanation) is enough to account for the sky being as dark as it is2.
The Importance of the Paradox
Olbers' paradox is important because it brings Cosmology into contact with the observable world. By simply looking at the night sky and seeing that it is dark, you can work out that the universe cannot be infinite in both space and time, supporting things like Big Bang theory.