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pretendy:

Why is the night sky dark?

This is a question that at first sounds a bit stupid, but the observation that the night sky is dark is in fact a deeply profound one that provides much of the basis for modern cosmology.

The question which has now come to be known as Olbers’ paradox goes something like this: “In an infinite and static universe with an infinite amount of stars, why is the night sky dark?”

Why?

The argument was that if you looked at any point in the sky and drew your line of sight, it would eventually reach a star. In other words, along every possible direction, there should be a star, and hence light should be coming from every point in the sky.

No, really, why?

This is a bit of a wishy-washy argument when posed in terms of words, so let’s try some maths:

Imagine that throughout the universe, the density of stars (number per cubic lightyear, say), let’s call it n, remains roughly constant. Now, imagine that we construct a series of spherical shells surrounding the Earth, and that each has a thickness dr. See the main picture to see what I mean.

What we want to do is count up the number of stars, N, in a shell. For a shell a distance r away, we multiply its volume by the star density:

Now let’s work out how bright that shell is. We can assume that each star has a total luminosity of L, but we have to take into account the fact that the further away a star is the fainter it appears. In fact, the apparent brightness, F, of any star varies like:

The brightness of a thin shell - which we’ll call dJ- is just the number of stars times the brightness of each!

Now we integrate over all space, i.e., add up the contribution from every consecutive shell all the way to infinity.

In other words, the total brightness of the sky, J, is infinite!

Okay but WHY?

The essential reason for this is the fact we said that the brightness of a star decreased by an inverse square law, but the number of stars increased by a regular square law. The two r^2 terms cancelled each other out and we found that each shell had the same brightness! Therefore when you add up an infinite number of same-brightness shells the answer you get is ∞.

Oh. So?

Well, this is obviously not true when we look up at the sky, so there must be a problem somwhere. Like most things in science, the problem lies within our initial assumptions, namely: ‘the universe is static and infinite’. We have shown that this just can’t be true! The night sky being dark forces the universe to have a finite size and age!

Edgar Allen Poe was eerily accurate when he postulated that no light reaches Earth from beyond a certain distance - corresponding to the age of the oldest stars. Cosmology caught on to this idea and introduced the concepts of the big bang, universal expansion, and the cosmic horizon in order to account for this seemingly trivial darkness problem.

Think of this next time you look at a starry sky. We see faint objects as they were hundreds, thousands, millions and billions of years ago (the time it has taken light from them to reach our eyes). At the farthest depths of what our most powerful telescopes can make out are objects from the beginning of the universe itself, and beyond that… nothing.

We can see the edge. It’s black.

Cool.

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