- Lew Thomas Revised 3-20-01
Why is the nighttime sky dark? This a question which has bothered astronomers for centuries. We shall discuss it now. But first an historical perspective.
In 55 BC, Lucretius in his poem, The Nature of the Universe, advanced the idea of an infinite universe. In 1576, Thomas Gigges proposed that, although the universe may be infinite, the distant stars are too faint to be seen and, whereas the stars are everywhere, the faintest of some produce dark gaps in our nighttime sky. We shall see that this thought is incorrect. In 1610, Kepler in his response to The Starry Messenger by Galileo, noted that if the universe were infinite, then, in all directions, we should see stars and the nighttime sky should never be dark. He therefore concluded that the universe must be finite since the sky is observed to be dark. Edmund Halley was the first to discuss the dark nighttime sky from a mathematical standpoint. Halley kept an infinite universe but indicated that the most remote stars were too faint to be recorded even in the greatest of telescopes and, therefore, dark gaps must exist between the stars as observed. This argument cannot be true because the collective light from the faint stars, if these stars be infinite in number, must render the night sky brilliant. It is akin to the fact that we cannot observe one electron dropping to a lower energy level in a single atom but in a multitude of atoms the light is easily visible.
In 1744, the Swiss astronomer, Jean-Phillippe Loys de Cheseaux, took up the problem of the dark sky. He stated that the area of the entire sky is 180,000 times greater than the observed sun, He concluded that if all the starlight from the infinite numbers of stars reached the Earth, then the night sky should be 180,000 times brighter than the Sun! He believed that the darkness of the night sky was due to absorbing material between us and the distant stars. We shall find that this cannot be true for such absorbing material would collect enough energy to glow in its own right and thereby render the night sky brilliant.
It appears that Lord Kelvin also took a crack at this problem, but I can find no good reference to his work.
Finally, in 1826, the Viennese astronomer, Heinrich Olbers, again asked this very fundamental and far reaching question: why is the night sky dark? This question now bears his name. At first, such a query seems ridiculous and begs the obvious answer, "because at night we face away from the Sun." However, the question is profound and the answer is far from obvious.
The immediate answer stated above would be correct if one could demonstrate that the light from all the nighttime stars would be sufficiently low in intensity to render the sky dark. To his surprise, Olbers found that this was not the case! Here is his reasoning.
Olbers assumed the following:
1) the universe extends infinitely far into space,
2) the universe is infinitely old,
3) the universe contains stars which are evenly distributed and are of about equal brightness,
4) that there is no obscuring matter between us and the stars.
To find the brightness of the night sky, one has to calculate the amount of light received from all luminous objects. Let there be "n" such objects in a unit volume of space (the density is n) and let each of these objects emit "E" units of light energy. We shall next divide space into a number of thin, concentric spherical shells, each of thickness, "t", and each centered on the Earth. Any particular shell will then have an inner radius of "r" and an outer radius of "r+t". Such a shell is illustrated below.
Since the area of a sphere of radius r is
A = 4p r2 (1)
the volume of such a shell is
V = 4p r2t (2)
If the density of each of the luminous objects within the shell is "n", then the total number of these objects in the shell must be
N = 4p r2nt (3)
Now let us ask just what amount of energy such a shell will send to the Earth. Since the shell's thickness is small, it is reasonable to assume that the entire shell is at a distance "r" from the earth. The energy, E, emitted by any source at distance r, produces an intensity, "I", over a given area, A, on the Earth of (inverse square law)
I = E/4p r2 (4)
The total intensity received on the Earth from all the sources in the shell r units away must then be the intensity produced by each source times the total number of sources or
T = IN (5)
Substituting the value of N previously calculated into the above, we find that
T = tnE (6)
We notice at once that the total energy received from any chosen shell does not depend upon its distance from us (no r in the above equation). The total energy received from all the shells is the sum of the contributions of each shell. If there are M shells this total is
S = tnEM (7)
But there is an infinite number of shells and so the total intensity on the earth must be infinite. Therefore, the nighttime sky should be blindingly bright!
This conclusion puzzled scientists for over a century. Clearly something must be wrong with one or more of the hypotheses used to reach this result. It was suggested that one could not simply sum the contributions of each of the concentric shells since stars are not precisely points of light and some of the nearer ones must surely occult those which are farther away. If this be true, then the night time sky would only be as bright as the average surface brightness of a typical star. This helps, but not much. Now the sky drops from being infinitely bright to the brightness of a typical star, say, the Sun.
Lets look at the last assumption by Olbers. Certainly we know that there is obscuring material in the universe and, at first, one might argue that this material could reduce the calculated sky brightness. This idea is in trouble since any obscuring material, where ever it be located, would be heated by the energy it received from more distance sources and would, in time, begin to radiate energy itself. If the universe is not infinite in extent, then the shells would not be infinite in number and the sky brightness could be reduced. There is no evidence, however, to suggest that the universe is bounded and finite. In the absence of such observations, we cannot be sure that Olbers' first assumption is untrue.
Suppose that the universe is not infinitely old. Suppose that it came into being X years ago. Then it is age bounded. The energy we receive could not have come to us from a distance beyond the limits of this expansion. Before the concept of expanding space, this limit was found to be cX, where c = the speed of light. In any event, this limits the total amount of energy, but calculations show that even this would still produce a bright nighttime sky.
The solution to Olbers' paradox came with the discovery that the universe is expanding: that distant galactic groups are receding from us. This recession results in a diminution of the light from these distant sources over and above that forecast by the inverse square law. In addition, light from a receding source is red shifted and it can be shown that red light possesses less energy than blue light of the same intensity. Therefore, because of this red shift, not only does less visible light reach us, but the total energy is also less.
These last two effects combine to reduce the light contributed by distant galaxies to our nighttime skies to insignificance, leaving only the nearby stars which we see as points of light in a darkened sky. To answer our original question: why is the night time sky dark; it is dark because the universe is expanding. I think we must admit that this is a rather startling answer to what initially seemed to be an age old and trivial question.
Yet in a larger sense the night sky is not really dark at all wavelengths. The background radiation, representing light stretched into the microwave region, is truly omnidirectional and floods the entire sky. However, it stands at a temperature of 2.8O Kelvin, By Wein’s Law this amounts to less energy (as we have said) and so we are not fried to a crisp. Isn't it fortunate that this universe expands!
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