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NSC111: Physics/Earth/Space
Resource page: Cosmology and Olbers' Paradox


Olbers' Paradox

I am indebted to Professor Peter Saulson for the artistic comparisons I use below.
While "his" paradox had been considered by other astronomers all the way back to Kepler, Heinrich Olbers (1926) gave the question its final shape:
Why is the night sky black?
Gustave Courbet, 'Deer in the forest'
Claude Monet, 'Poppy fields near Giverny'
If we assume that
  1. the universe is infinite and
  2. stars are evenly distributed throughout the universe
we ought to see a star in any direction we look. We can call this "Courbet's Law" because it is exemplified by Gustave Courbet's painting "Deer in the forest." No matter where you look, you see greenery because the forest goes on far enough in all directions to block out any view of the sky.

We can safely assume that all stars have the same surface brightness on the average as the Sun; it is simply that they are so far away that they appear very small. We can call this "Monet's Law" because it's the same principle as when Monet used larger and smaller dots of the same paint to represent near and distant poppies.

Consider a 100-watt light bulb, on the table next to you. It's too bright to look at, right? Now move the same light bulb to the other end of a football field (about 110 meters). It's a lot easier to look at directly, isn't it? That's why we can look directly at distant stars, but NOT at the one we have nearby! The distant stars have the same surface brightness as the Sun, but we see much less of their surfaces because most of their light goes somewhere else.

Now consider this: the light from an individual star falls off as the square of its distance r. But suppose we have spherical shells, spaced at some specified distance apart; each shell has its own distance r. Suppose we have stars evenly distributed on each shell (so that each star occupies a specific area on its shell, the same area from shell to shell). The intensity of light from each individual star will be proportional to 1/r². But the number of stars in each shell will go up according to Area = 4πr², that is, the number of stars (each with brightness proportional to 1/r²) goes up according to r²! If the shells are spaced evenly, then, we are led to the conclusion that each shell contributes the same amount of light to our eyeballs! (The number of stars goes up in proportion to r²; the intensity of each star goes down in proportion to r².)

If Courbet's Law and Monet's Law are true, then every part of the sky should be as bright as the surface of the Sun--in fact, we ought to be fried because everywhere we look the sky should appear as the picture to the left! But instead what we see (or what van Gogh saw, anyhow) is the picture to the right:

Vincent van Gogh, 'The Starry Night'

how the Milky Way Galaxy might look edge-on
how the Milky Way Galaxy might look face-on
For this to be true, one of the assumptions must fail: either
  1. The universe is not infinite in extent or
  2. Stars are not evenly distributed in the universe.

During the mid-to-late 19th Century, it became apparent that stars did not go on forever: objects which were readily identifiable as stars appeared to get fewer and fewer as you went deeper and deeper into space. The distributions were mapped, and we found that we appear to live in a galaxy about 100,000 light-years across, rather disk-shaped, and the Sun is about 2/3 of the way out from the center along the disk. Viewed from above, the Milky Way Galaxy takes on a spiral shape. Within the galaxy are not only stars, but clouds of gas of various shapes (nebulae) which will eventually form new stars. But stars do not go on forever.

Or so it seemed.

In the 1920s, using a powerful new telescope, Edwin Hubble was able to resolve individual stars within a spiral nebula (the Andromeda Galaxy). This definitively proved that some nebulae were actually galaxies outside our own. Furthermore, as telescopes improved it became apparent that there were galaxies everywhere you looked...

The three main types of galaxies are shown in the pictures below: spiral galaxies like the Milky Way have spiral structure, and look like disks when seen edge-on. Elliptical galaxies are more globular in shape; irregular galaxies have no shape at all.

the Whirlpool Galaxy
the Whirlpool Galaxy has an irregular companion M32 is an elliptical galaxy

the Virgo cluster of galaxies Just as stars collect into galaxies under their mutual gravitational attraction, so do galaxies collect into clusters. Some clusters are rather small, like the Local Group, which contains the Milky Way and Andromeda galaxies and a few small satellite galaxies; or very large, like the giant cluster in Virgo (shown at right).

Astronomers have identified clusters of clusters ("superclusters") and even larger structures, called "walls" and "voids". The universe appears to have structure on as large a scale as we can examine it.

But what does this do for Olbers' Paradox? If there are galaxies everywhere you look, why is the night sky black?

Let's digress a little...

When astronomers take the spectrum of the Sun or other stars, we find the spectral lines of various elements, where they have absorbed light from the star. These lines allow us to identify the components of the stellar atmosphere (cooler than the surface behind it). When we look at some stars, we find that the spectral lines are shifted from their normal positions because of the star's motion: a Doppler Effect for light! If the wavelengths are shortened (a blueshift) the star is approaching us; if the wavelengths are lengthened (a redshift) the star is receding from us.

When astronomers looked at the spectra of galaxies, they found that while a few were blueshifted (notably the Andromeda Galaxy, which will eventually collide with the Milky Way), almost all galaxies are redshifted--they are receding from us! In fact, when Edwin Hubble started establishing distances to various galaxies, he found a relationship between distance and redshift: the further away the galaxy is, the faster it is receding from us! This relationship is linear, and described by the Hubble Constant.

The lovely thing about the Hubble Constant is that its reciprocal is a good estimate for the age of the Universe; the current best values lie between 60 and 80 km/s/Mpc, giving the Universe an age of between 12 and 16 billion years.
Incidentally, we shouldn't think that we're at a central position just because all the galaxies are receding from us. Think of each galaxy as a dot on a balloon. Now blow up the balloon. Each dot will recede from its neighbors, and vice versa. Furthermore, the neighbors of the neighbors will recede faster, and so on and so on. But this is true for every dot, and no dot is in a privileged position.
But if the Universe is expanding, it must have begun at some finite time in the past (current best estimates are 12-16 billion years). This led Father Georges Lemaître, a Jesuit priest and cosmologist, to propose what we now know as the Big Bang Theory. This theory, now dominant in cosmology, asserts that the Universe began in a gigantic explosion which synthesized all the matter of the Universe, in the form of hydrogen (~80%), helium (~20%) and smatterings of lithium, beryllium and boron. All the higher elements were synthesized in the cores of stars.

At last we have a solution to Olbers' Paradox. If the Universe has a finite age, it does not go on forever: we only see the light which has had time to reach us! Even if there were something physical "outside the universe", we couldn't see it because the light from it has not had enough time to reach us! If the universe is a finite age, it isn't "infinite in extent", and the first assumption of Olbers' Paradox fails.

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Copyright © 2001 by Daniel J. Berger. This work may be copied without limit if its use is to be for non-profit educational purposes. Such copies may be by any method, present or future. The author requests only that this statement accompany all such copies. All rights to publication for profit are retained by the author.