Why the Sky is Dark at Night

Here’s an extract from my book Companion to the Cosmos, in response to a recent enquiry.

Olbers’ paradox: The puzzle of why the sky is dark at night.

This is only a puzzle if the Universe is eternal and unchanging, which was widely assumed to be the case until the 1920s. The real mystery is why nobody before then turned the “paradox” on its head, and used the darkness of the night sky as a reason to argue that the Universe must have been born a finite time ago.

The term “Olbers’ paradox” was popularised as a name for this puzzle by the cosmologist Hermann Bondi in the 1950s, in honour of Heinrich Olbers, a 19th century German astronomer who wrote a landmark paper discussing the puzzle. But just as it is not really a paradox, so the puzzle did not originate with Olbers. Like Mach’s principle, discussion of the problem goes back much further than the contribution of the person whose name it bears.

The puzzle can be stated very simply. It rests upon three assumptions: that the Universe is infinitely big, that it is eternal and unchanging, and that it is filled with stars (in the modern version of the puzzle, filled with galaxies) more or less the same as the stars of the Milky Way (or galaxies like the Milky Way itself). In that case, when we look out into the Universe we ought to see stars (galaxies) in every direction. Every single “line of sight” ought to end on the surface of a star; so why do we see darkness in the gaps between the stars? The whole sky should be a blaze of light.

You can see why this is a puzzle by imagining yourself standing inside an infinitely large forest. It doesn’t matter how far apart, or close together, the trees in the forest are; if it is really infinitely big, then wherever you look in a gap between the trunks of the nearer trees you will see the trunk of a more distant tree.

This is a simple version of the puzzle, the kind astronomers refer to as a “hand waving” argument. But the puzzle can be brought on to a secure mathematical footing, when it strikes with even greater force. Imagine that the Earth is at the centre of a large sphere encompassing many stars (galaxies). We can envisage a thin shell around this sphere, like the skin of an orange, that contains a certain number of stars (galaxies), all at the same distance from us. If the numbers involved are big enough (easy to arrange if the Universe is infinite), then we can say that, in round terms, each star (galaxy) contributes the same average brightness to the appearance of the night sky here on Earth. But the apparent brightness of each star (galaxy) is inversely proportional to the square of its distance from us. So the whole shell of stars (galaxies) contributes a brightness equal to the number of stars (galaxies) in the shell, multiplied by the average brightness and divided by the square of the distance to the shell.

If individual stars (galaxies) all have about the same absolute brightness, but they each look fainter the further away they are, you might guess that shells of this kind, dotted with stars (galaxies), will appear fainter if they have a larger radius. But a bigger shell will also contain more stars (galaxies). In fact, if the stars (galaxies) are distributed uniformly throughout the Universe, the number of stars (galaxies) in each shell turns out to be proportional to the square of the distance to the shell, because the surface area of a sphere goes as the square of its radius. This exactly compensates for the diminished apparent brightness of each star (galaxy) in the shell. So every spherical shell contributes the same brightness to the night sky.

The accurate calculation is actually much stronger than the hand waving argument. With an infinite number of shells, each contributing the same amount of light, the sky should be infinitely bright. The best you can hope for is to argue that light from nearby stars blocks out some of the light from more distant stars — which still leaves the prediction that the sky should be as bright all over as the surface of a star like the Sun. The puzzle becomes not so much why the sky is dark at night, but why it is so dark even in the daytime.

The extent to which you find this paradoxical depends on how strongly you hold to the three basic assumptions — which tells us more about the culture of our recent ancestors than about the structure of the Universe. It was the Englishman Thomas Digges, writing in 1576, who discarded the Ptolemaic idea of the stars being attached to a single crystal sphere surrounding the Earth, and distributed the stars, in his imagination, into an endless infinity of space. Digges introduced the concept of infinity into the modern picture of the Universe (although in ancient times Democritus had considered the possibility of infinite space), and he also realised the need to explain why, in an infinite universe, the sky should be dark at night. He believed that the sky was dark because the more distant stars were simply too faint to be seen, an acceptable notion in the 16th century, but one which doesn’t stand up once you work out how the light from spherical shells actually adds up (the accurate calculation uses Isaac Newton‘s calculus, and Newton wasn’t even born until 1642).

In 1610, the puzzle was investigated by Johannes Kepler, who seems to have been the first person to realise (even without the aid of calculus!) that the darkness of the night sky directly conflicts with the idea of an infinite universe filled with bright stars. He saw the darkness of the night sky simply as evidence that the Universe is finite in extent — he said, in effect, that when we look through the gaps between the stars we see a dark wall that surrounds the Universe. On this picture, instead of standing in an infinite forest you are standing in a small copse, and when you look through the gaps between the nearby tree trunks you see the world outside the copse. A century later, Edmund Halley also investigated the mystery, but he went back to the incorrect notion that more distant stars are simply too faint to be seen.

The first person to formulate the puzzle of the dark night sky in more or less the form outlined here was the Swiss astronomer Jean-Phillippe Los de Chesaux, {ACCENT on the e} later in the 18th century. After a careful step-by-step calculation based on the sizes of stars and their separation, he estimated that there would be a star visible in every direction we look into space, provided that the Universe is (in modern terms) 1015 light years (a million billion light years) across. Unlike Halley and Digges, de Chesaux realised that the geometry of the situation ensures that the faintness of distant stars is exactly compensated for by their increased numbers. But his “explanation” of the puzzle was no better than theirs — he suggested that empty space simply absorbs the energy in the light from distant stars, so that the light gets fainter and fainter as it travels through the Universe.

And then along came Olbers, who discussed the problem in the 19th century, and came to a similar conclusion, arguing that the light from distant stars is absorbed in a thin gruel of material between the stars. What he failed to appreciate was that this would heat the gruel up, until it was radiating as much energy as it received — until, in fact, it was as bright as the stars themselves. So — what is the solution to the puzzle?

Clearly, at least one of the three basic assumptions must be wrong. We still cannot say definitely whether or not the Universe is infinite in extent; it probably is full of galaxies like the ones we can see; but we do know that it is not eternal and unchanging. The Universe as we know it began in a Big Bang some 15 billion years ago, and it is changing as spacetime expands, carrying the galaxies ever further apart from one another. A partial explanation of why the sky doesn’t blaze with light is that the Universe is expanding and evolving as time passes. This produces the redshift in the light from distant galaxies, which in a very modest way does cause a weakening of the light on its journey superficially like the dimming proposed by de Chesaux.

Alas, if the Universe were infinitely old and had been expanding forever, with new galaxies being created to fill the gaps between the old ones as they move apart (as required by early versions of the Steady State hypothesis), the redshift alone would not save us from the implications of Olbers’ paradox. But what do we see when we look (with radio “eyes”) into the gaps between the stars and galaxies? What we detect there is the faint hiss of the cosmic background radiation, the equivalent of “light” with a temperature of 2.7 K. That is the highly redshifted electromagnetic radiation from the time when the Universe was about 300,000 years old and was filled with radiation as hot as the surface of the Sun is today. If the Universe had not expanded since then, all of space would still be as hot as that, and would blaze as brightly as the surface of a star — so the redshift associated with the expanding Universe is one reason why the sky is dark at night, even though Olbers and his predecessors had no inkling of the existence of the Big Bang fireball.

But the reason why starlight has not been able to fill the Universe with energy (the original “Olbers’ paradox”) is simply that there hasn’t been enough time for it to do the job. Light from a galaxy 50,000 light years away, say, takes 50,000 years to reach us, and in a Universe 15 billion years old we can only “see” galaxies out to a distance of 15 billion light years (even if they formed immediately the Universe was born). Even if the Universe is infinite, light from more distant galaxies has not had time to reach us. In round terms, de Chesaux’s calculation says that the Universe would have to be at least 1015 light years across and be at least 1015 light years old in order for every line of sight to end on a star (the numbers are bigger still if you take account of the way stars are clumped together in galaxies). That is roughly a million times older than the time that has elapsed since the Big Bang. If there was a definite moment of creation, and it was recent enough, then there is no puzzle about the darkness of the night sky.

The puzzle is why Newton and his contemporaries didn’t hit on this resolution of the “paradox”. The finite speed of light had been determined by the Dane Ole Rmer in 1676, and was well known to Newton, who mentioned it in his Opticks, published in 1704. When Halley read two papers on the puzzle of the dark night sky to the Royal Society in 1721, Newton was in the chair. Yet neither he nor anyone else pointed out that the puzzle could be resolved by assuming that no stars existed until relatively recently. This oversight is all the more baffling because at the time the Church taught that the Creation had taken place in 4004 BC. Any astronomer of Newton’s day could have immediately calculated that no light from stars more than (1721 + 4,004) light years away had had time to reach the Earth — and a sphere of space with a radius of less than 6,000 light years is far too small (as de Chesaux’s calculation shows) to hold enough stars to make the night sky bright. Perhaps the failure of Newton, Halley and their contemporaries to point this out actually indicates how little faith they had in the official date for the Creation.

So who was the great astronomer who first realised that by looking out into space and back in time we see the darkness that existed before the stars were born? None other than Edgar Allan Poe (1809-1849), best known today for his poetry, who was also a keen amateur scientist and who delivered a lecture setting out the resolution to Olbers’ paradox in February 1848, just a year before he died at the age of 40. The lecture was published, later in 1848, as an essay entitled Eureka, in which he wrote of the dark gaps between the stars (which he called voids):

The only mode, therefore, in which . . . we could comprehend the voids which our telescopes find in innumerable directions, would be by supposing the distance of the invisible background so immense that no ray from it has yet been able to reach us at all.

Nobody took much notice of these amateur speculations from a poet who died before he had time to promote his argument. Nor did anyone take much notice when the problem of the dark night sky was discussed in print by the Irish scientist Fournier d’Albe in 1907, who specifically said “if the world was created 100,000 years ago, then no light from bodies more than 100,000 light years away could possibly have reached us up to the present”. D’Albe himself had drawn on the ideas not of Poe but of Lord Kelvin, who had published his thoughts on the darkness of the night sky in 1904 in a volume of lectures where they laid forgotten until they were dug out by Edward Harrison, of the University of Massachusetts, in the 1980s.

Even after the discovery that the Universe is expanding and must have had an origin at a definite moment in the past, nobody took full account of the importance of the insight provided by Olbers’ paradox until Harrison became intrigued by the history of the idea, and dug out the full story of it over a period of many years. But it is still of more than just historical interest. Anybody from Newton to Edwin Hubble could have used the evidence of their own eyes to tell them that the Universe was born a finite time ago. And it is still true that one of the clearest pieces of evidence that there really was a Big Bang can be seen (literally!) with your own eyes, by looking up at the dark night sky.

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Further reading: Edward Harrison, Darkness at Night.