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Olbers' Paradox: Why Is The Night Sky Dark?

Easy really - the side of the Earth that I'm standing on is turned away from the Sun, end of story... Or is it? Let's consider these commonly held beliefs:

OK, if all of the above are true, why is the night sky not a blaze of bright starlight with light from each star contributing to the general illumination? Well, this apparently simple question has perplexed the greatest thinkers through the ages including Kepler (1610), Halley (1720) and de Cheseaux (1744) [1]. However, the question gained its own special name when Olbers in 1826 gave it more than a passing thought. Strangely, a poem by Edgar Allan Poe called Eureka, written in 1848 touched on the truth [1]. So let's consider the evidence...

Olbers thought that dust was the reason for the lack of all-over starlight. But space is an empty vacuum isn't it? In the context of space, dust is micron size particles of carbon, silicates and iron at a density of one per million cubic metres and making up 1% of the total mass of interstellar matter. However as can be observed in areas of star formation, when there is a higher proportion of dust, it warms up and re-radiates in the infra-red spectrum. In fact, dust does have an effect, but it is to obscure at visible wavelengths the centre of our own galaxy, the Milky Way: in directions away from the plane of the galactic disk, space is relatively dust free.

Yes, there are more stars in the Milky Way than grains of sand on the beach, about 10E11 of the former. How far away are they, and how old? And we should also consider the light given out from other objects - other galaxies or island universes as Edwin Hubble termed them in the 1920s. Many of these galaxies dwarf ours in size and star density. There are about as many galaxies as there are stars in our Milky Way and each one of them has its own enormous quota of stars. So, to develop a simplified model, let's assume that the Universe is finite in size, is filled with "standard" stars and is isotropic (the same in all directions). The stars are uniformly spaced at a density of about one per cubic parsec and each star has a brightness or luminosity L. The amount of light or flux f that reaches us from a standard star at distance D is given by the formula f = L / (4 * Pi * D2) i.e. the flux is proportional to 1 / D2, the inverse square law. Now consider an imaginary shell of standard stars round us with a finite thickness and at a fixed distance from us - how much of the light emitted by this shell reaches us? Well the total number of stars in the shell is given by the number of stars per unit volume multiplied by the thickness of the shell and then multiplied by the surface area of the shell. As D, the distance to the shell of stars, increases, so too does the number of stars in the shell contributing to the total light output (because the surface area of the shell is proportional to D2), countering the effect of the flux loss due to the increase in distance. So, and this is the key point, the total light reaching us is independent of distance [4].

Considering the nature of light, early astronomers had observed discrepancies in the positions of the Jovian satellites which they correlated to the distance between the Earth and Jupiter. Ole Rømer (1644-1710), a Danish astronomer, was able to estimate the velocity of light by timing the eclipses of Jupiter's satellites [1] and relating the timings to those predicted by a model of the Solar System based on an infinite velocity of light; he obtained an estimate very close to the currently accepted value which is a remarkable achievement for the late 1600's.

Later, Michelson and others managed to estimate the speed of light much more accurately. Einstein suspected that space, time and matter were all interrelated leading to the famous e=mc2 equation in the 1920's and the eventual birth of the nuclear age. He predicted that massive bodies could affect light purely by gravity. It was confirmed during the solar eclipse of 1919 that light was indeed bent due to the gravity of the Sun. Subsequently it has been observed that extremely distant objects can be seen by an effect called gravitation lensing whereby the light which they emit is bent by the effect of the gravity of a galaxy-size mass acting like a giant magnifying glass.

Is the age and, for that matter the size, of the Universe infinite then? As recently as the 1930s there were two leading schools of thought. In one corner was the steady state group who argued that the Universe was as it always had been and would continue forever. In the other corner were the expansionists who argued that the Universe was born of a massive explosive event that occurred at the very dawn of time. Researchers Penzias and Wilson working at Bell Laboratories in 1965 detected an unexplained radio signal at a wavelength of 7.35 cm. The signal was received from every direction, never varied and seemed to emanate from outside the Milky Way. Theorists had already calculated that if there had been a massive explosive event a very, very long time ago, then by now the temperature would have dropped to about 3 K and radiation would be detectable as Cosmic Microwave Background Radiation. Penzias and Wilson confirmed this theory and as a result were awarded a Nobel prize. A further milestone was passed when Edwin Hubble, using the 2.5 m Mount Wilson telescope, observed a bright outburst in one of our nearest galactic neighbours, the Andromeda galaxy. It had been known for some time that a certain type of star, known as a Cepheid variable, exhibited a very predictable variation in luminosity with oscillations in the magnitude of the star related to its periodicity. Astronomers estimated directly, by parallax methods, the distance to the nearest Cepheid variables enabling calibration of the relationship between periodicity and the absolute magnitudes of the stars. They then used Cepheids as standard candles to provide a new measuring tool to probe greater distances. Hubble found by use of standard Cepheid variables that the distance to the Andromeda galaxy was far greater than anybody had expected.

Now when light is split to form a spectrum, if certain elements are present in the source of the light, bright emission lines or dark absorption lines are seen, an effect termed Fraunhoffer lines after the 18th century chemist who discovered and explained them. When Hubble used a spectrograph on light from the Andromeda galaxy, he found that the Fraunhoffer lines were shifted towards the red, lower frequency end of the visible electromagnetic spectrum. In the well known effect of an approaching train the pitch of the noise first rises as it moves closer and then, as it recedes, falls - the Doppler effect. The spectrographic shift of the lines towards the red indicated that the Andromeda galaxy was moving away from us. Hubble was able to determine the rate of recesssion by the degree of offset of the lines in the spectrum. He and others subsequently were able to detect movement in many other galaxies and so concluded that the universe as a whole was expanding. The rate of expansion became known as the Hubble constant.

Now try a thought experiment. Light travels at a finite speed of 300,000 km per second and the distance to our Sun is approximately 149,600,000 km; this means the sunlight we see now is about eight minutes old. Imagine if the Sun were switched off: we would have no knowledge of the event until eight minutes later [5]. And so it is with cosmology. We only have knowledge of the Universe from the region within which light has reached us. Why is the sky black then? The Universe is too young at only 15 billion years for Olbers' expected effect to have happened yet! If we could just hang around for a few more billion years eventually more and more light will arrive to brighten the night sky.

References

[1]

The Routledge Companion to the New Cosmology, editor Peter Coles, 2001, ISBN 0-415-24312-2.

[2] 

Universe, 5th edition, Kaufmann & Freedman, 1998, ISBN 0-7167-3495-8.

[3] 

Introductory Astronomy & Astrophysics, 4th edition, Zeilik & Gregory, 1998, ISBN 0-03-006228-4.

[4]

Cosmology, editor Barrie Jones, 1994, ISBN 074925128X.

[5]

http://zebu.uoregon.edu/~imamura/123/lecture-5/olbers.html from published lecture notes.

[6]

http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html by Scott I Chase.


Dave McCracken