Olbers's paradox

Not to be confused with Oberth paradox.

Olbers's paradox, also known as the dark night paradox or Olbers and Cheseaux's paradox, is an argument in astrophysics and physical cosmology that says the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. In the hypothetical case that the universe is static, homogeneous at a large scale, and populated by an infinite number of stars, any line of sight from Earth must end at the surface of a star and hence the night sky should be completely illuminated and very bright. This contradicts the observed darkness and non-uniformity of the night sky.^[1]

As more distant stars are revealed in this animation depicting an infinite, homogeneous, and static universe, they fill the gaps between closer stars. Olbers's paradox says that because the night sky is dark, at least one of these three assumptions must be false.

The darkness of the night sky is one piece of evidence for a dynamic universe, such as the Big Bang model. That model explains the observed darkness by invoking expansion of the universe, which increases the wavelength of visible light originating from the Big Bang to microwave scale via a process known as redshift. The resulting microwave radiation background has wavelengths much longer (millimeters instead of nanometers), which appear dark to the naked eye. Although he was not the first to describe it, the paradox is popularly named after the German astronomer Heinrich Wilhelm Olbers (1758–1840).

History

Edward Robert Harrison's Darkness at Night: A Riddle of the Universe^[2] (1987) gives an account of the dark night sky paradox, seen as a problem in the history of science. According to Harrison, the first to conceive of anything like the paradox was Thomas Digges, who was also the first to expound the Copernican system in English and also postulated an infinite universe with infinitely many stars.^[3] Kepler also posed the problem in 1610, and the paradox took its mature form in the 18th-century work of Halley and Cheseaux.^[4] The paradox is commonly attributed to the German amateur astronomer Heinrich Wilhelm Olbers, who described it in 1823, but Harrison points out that Olbers was far from the first to pose the problem, nor was his thinking about it particularly valuable. Harrison argues that the first to set out a satisfactory resolution of the paradox was Lord Kelvin, in a little known 1901 paper,^[2]: 227 and that Edgar Allan Poe's essay Eureka (1848) curiously anticipated some qualitative aspects of Kelvin's argument:^[1]

Were the succession of stars endless, then the background of the sky would present us a 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.^[5]

The paradox and resolution

See also: Redshift and expansion of the universe

The paradox is that a static, infinitely old universe with an infinite number of stars distributed in an infinitely large space would be bright rather than dark.^[1] The paradox comes in two forms: flux within the universe and the brightness along any line of sight. The two forms have different resolutions.^[6]: 354

A view of a square section of four concentric shells

Flux form

The flux form can be shown by dividing the universe into a series of concentric shells, 1 light year thick. A certain number of stars will be in the shell, say, 1,000,000,000 to 1,000,000,001 light years away. If the universe is homogeneous at a large scale, then there would be four times as many stars in a second shell between 2,000,000,000 and 2,000,000,001 light years away. However, the second shell is twice as far away, so each star in it would appear one quarter as bright as the stars in the first shell. Thus the total light received from the second shell is the same as the total light received from the first shell. Thus each shell of a given thickness will produce the same net amount of light regardless of how far away it is. That is, the light of each shell adds to the total amount. Thus the more shells, the more light; and with infinitely many shells, there would be an infinitely bright night sky.^[7]

If intervening gas is added to this infinite model, the light from distant stars will be absorbed. However, that absorption will heat the gas, and over time the gas itself will begin to radiate. With this added feature, the sky would not be infinitely bright, but every point in the sky would still be like the surface of a star.^[8]

The flux form assumes a static universe age. With a finite age, only a limited volume of the universe contributes to light on Earth, resolving the paradox.^[6]: 355

Another way to describe the flux version is to suppose that the universe were not expanding and always had the same stellar density; then the temperature of the universe would continually increase as the stars put out more radiation. After something like 10²³ years, the universe would reach the average surface temperature of a star. However, the universe is only 13.8 billion (10¹⁰) years old, eliminating the paradox.^[4]: 486

Line-of-sight form

The line-of-sight version of the paradox starts by imagining a line in any direction in an infinite Euclidean universe. In such universe, the line would terminate on a star, and thus all of the night sky should be filled with light. This version is known to be correct, but the result is different in our expanding universe governed by general relativity. The termination point is on the surface of last scattering where light from the Big Bang first emerged. This light is dramatically redshifted from the energy similar to star surfaces down to 2.73 K. Such light is invisible to human observers on Earth.^[6]: 355

Recent observations suggest that the estimated number of galaxies based on direct observations is too low by a factor of ten. However this does not materially alter the resolution of the paradox. A full explanation involves a combination of finite age and redshifts, and that UV absorption by hydrogen followed reemission in near-IR wavelengths also plays a role.^[9]

See also

• Heat death paradox
• List of paradoxes
• Horizon problem