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21. Why is it Dark at Night?

The Question:

During the day, the sun shines brightly to light up the earth. And during the night?

We may stipulate that the universe is infinite in all directions. In this infinite universe, infinitely many stars glitter in the sky. Of course, the light of a star diminishes the further away it is from the earth. But there are infinitely many stars, and the combined light of an infinite number of stars should become sufficiently strong on its way to the earth to make the sky bright, even at night, when the sun does not shine.

Correct?

Obviously not, as we observe each night.

First of all, what is the argument that the night sky should be bright?

The further away a light source, the larger the lit-up area but the weaker its brightness. One can check that by holding a light to a wall. As the lamp moves away from the wall, the area of the lit-up circle increases with the square of the distance to the wall. And the brightness of the lit-up circle becomes weaker. Weaker by how much?

Since the light source emits a finite amount of energy, the same amount of light must be spread over a larger area. Since the lit-up area increases by the square of the distance, the brightness of the lit-up circle decreases by the square root of the distance.

Now let’s consider the number of stars. We split the universe into thin, concentric layers around the earth, like the skins of an onion. In each layer there are a certain number of stars. Archimedes’ findings about the surfaces of spheres imply that the number in each layer increases with the square of the layer’s distance to the earth. For example, in a layer that is twice as far from the earth than another one, there are four times as many stars.

Taken together, this would mean the following: (a) The brightness of each star decreases with the square root of the distance. (b) The number of stars increases with the square of the distance of the layer from the earth. (c) Hence, each layer should produce the same amount of light. (d) And with an infinite number of layers, the combined light that hits the earth should be…infinite.

So, why is the night sky dark?

Background:

The paradox was fomulated in 1823 by the German astronomer and medical doctor Heinrich Wilhelm Olbers in a paper entitled “On the transparency of the universe”. An explanation was of importance in order to settle the question whether the universe is, in fact infinitely large and whether there are infinitely many stars distributed homogenousely within that infinite space. The fact that the night sky is dark would speak against an infinite universe.

But Olbers did not believe that. He thought that nights are dark because intergalactic dust absorbs the light. It was not a good explanation, since the absorbed energy would have heated the dust until it emitted as much light as it absorbed. To be fair, in Olbers’ time it was not yet known that heat and light were different forms of the same kind of energy and that one could be transformed into the other.

In 1884 Lord Kelvin gave a lecture in Baltimore on the issue and suggested a correct solution. (See below.) Originally, his paper carried the title “Note on the possible density of the luminiferous medium and on the mechanical value of a cubic mile of sunlight”, which prompted him to lament the “brain-wasting perversity of the insular inertia which still condemns British Engineers to reckonings of miles and yards and feet and inches and grains and pounds and ounces and acres ….” and to suggest the use of cubic kilometer instead of cubic mile in the title and the use of the French metric system throughout.

Dénouement:

There are two reasons why the sky is dark at night, in spite of Olbers’ very reasonable sounding argument. The first is the the one put forth by Lord Kelvin.

1. We stipulated that the universe is infinitely large in all directions but we made no assertion about its age. Indeed, it all started with the Big Bang, 14 billion years ago. (What was before the Big Bang? Well, that’s a question for another book….)

Since light travels with a very, very fast, albeit a finite speed it takes time for the stars’ light to reach the earth. Hence, the light of only a finite part of the universe can reach the earth. Light that emanates from sources that are more than 14 billion lightyears distant from the earth has not reached us yet. Albert Einstein predicted, and cosmologists since then have verified, that the universe expands in all directions: everything moves away from everything else. When stars travel away from the earth the so-called Doppler effect kicks in: the wavelengths of sound and light are stretched, like the pitch of an ambulace’s siren that increases as the ambulance moves closer, but then diminishes as it moves away. In the same manner, the wavelengths of the lights that emanate from the stars that move away from the earth increase, shifting them into the infrared part of the spectrum which is invisible to the human eye.

That’s why Olbers’ argument, though plausible, is incomplete and leads to a faulty conclusion.

Technical supplement:

Surprisingly, one of the first persons to identify the possible reason for the dark sky was not a scientist but a writer, the American author Edgar Allan Poe (1809-1849). In the prose poem Eureka, actually a 150-page treatise, he suggested that light from far-away stars may not have reached the earth yet. The oft-quoted part of the poem reads as follows:

“…Were the succession of stars endless, then the background of the sky would present us an uniform luminosity, like that displayed by the Galaxy -–since there could be absolutely no point, in all that background, at which would not exist a star. The only mode, therefore, in which, under such a state of affairs, 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.”

But it is not quite clear whether Poe actually believed that the universe is infinite. He may have given the above explanation simply as a sort of tongue-in-cheek rationalization, even slightly ridiculing it. To reach this conclusion one must read the paragraph that precedes the above quote:

“No astronomical fallacy is more untenable, and none has been more pertinaciously adhered to, than that of the absolute illimitation of the Universe of Stars. The reasons for limitation, as I have already assigned them, a priori, seem to me unanswerable; but, not to speak of these, observation assures us that there is, in numerous directions around us, certainly, if not in all, a positive limit -- or, at the very least, affords us no basis whatever for thinking otherwise….”

Deep down Poe apparently believed that the infinite universe is a fallacy.