Olbers's paradox

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In this animation depicting an infinite and homogeneous sky, successively more distant stars are revealed in each frame. As the animation progresses, the more distant stars fill the gaps between closer stars in the field of view. Eventually, the entire image is as bright as a single star.
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.

Olbers's paradox, also known as the dark night paradox, is an argument in

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]

The darkness of the night sky is one piece of evidence for a dynamic universe, such as the

microwave radiation background
has wavelengths much longer (millimeters instead of nanometers), which appears dark to the naked eye and bright for a radio receiver.

Other explanations for the paradox have been offered, but none have wide acceptance in cosmology. Although he was not the first to describe it, the paradox is popularly named after the German astronomer Heinrich Wilhelm Olbers (1758–1840).

History

The first one to address the problem of an infinite number of stars and the resulting heat in the Cosmos was

Topographia Christiana: "The crystal-made sky sustains the heat of the Sun, the moon, and the infinite number of stars; otherwise, it would have been full of fire, and it could melt or set on fire."[2]

Heinrich Wilhelm Olbers, who described it in 1823, but Harrison shows convincingly 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,[5] 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.[6]

The paradox

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]

A view of a square section of four concentric shells

To show this, we divide 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 a bright night sky.

While dark clouds could obstruct the light, these clouds would heat up, until they were as hot as the stars, and then radiate the same amount of light.

Kepler saw this as an argument for a finite

general relativity theory, it is still possible for the paradox to hold in a finite universe:[7]
Though the sky would not be infinitely bright, every point in the sky would still be like the surface of a star.

Explanation

See also: Redshift, Lambda-CDM model, and expansion of the universe

The poet

Stelliferous Era is only finitely old) and the speed of light is finite, only finitely many stars can be observed from Earth (although the whole universe can be infinite in space).[9][10]
The density of stars within this finite volume is sufficiently low that any line of sight from Earth is unlikely to reach a star.

However, the Big Bang theory seems to introduce a new problem: it states that the sky was much brighter in the past, especially at the end of the recombination era, when it first became transparent. All points of the local sky at that era were comparable in brightness to the surface of the Sun, due to the high temperature of the universe in that era; and most light rays will originate not from a star but the relic of the Big Bang.

This problem is addressed by the fact that the Big Bang theory also involves the

galaxies
.

Other factors

Steady state

The redshift hypothesised in the Big Bang model would by itself explain the darkness of the night sky even if the universe were infinitely old. In the

Steady state theory the universe is infinitely old and uniform in time as well as space. There is no Big Bang in this model, but there are stars and quasars at arbitrarily great distances. The expansion of the universe causes the light from these distant stars and quasars to redshift, so that the total light flux from the sky remains finite. Thus the observed radiation density (the sky brightness of extragalactic background light) can be independent of finiteness of the universe. Mathematically, the total electromagnetic energy density (radiation energy density) in thermodynamic equilibrium from Planck's law
is

e.g. for temperature 2.7 K it is 40 fJ/m3 ... 4.5×10−31 kg/m3 and for visible temperature 6000 K we get 1 J/m3 ... 1.1×10−17 kg/m3. But the total radiation emitted by a star (or other cosmic object) is at most equal to the total nuclear binding energy of isotopes in the star. For the density of the observable universe of about 4.6×10−28 kg/m3 and given the known abundance of the chemical elements, the corresponding maximal radiation energy density of 9.2×10−31 kg/m3, i.e. temperature 3.2 K (matching the value observed for the optical radiation temperature by Arthur Eddington[11][12]). This is close to the summed energy density of the cosmic microwave background (CMB) and the cosmic neutrino background. However, the steady-state model does not predict the angular distribution of the microwave background temperature accurately (as the standard ΛCDM paradigm does).[13]

Brightness

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. Eventually, it would reach 3000 K (corresponding to a typical photon energy of 0.3

eV and so a frequency of 7.5×1013 Hz), and the photons would begin to be absorbed by the hydrogen plasma filling most of the universe, rendering outer space opaque. This maximal radiation density corresponds to about 1.2×1017 eV/m3 = 2.1×10−19 kg/m3, which is much greater than the observed value of 4.7×10−31 kg/m3.[4] So the sky is about five hundred billion times darker than it would be if the universe was neither expanding nor too young to have reached equilibrium yet. However, recent observations increasing the lower bound on the number of galaxies suggest UV absorption by hydrogen and reemission in near-IR (not visible) wavelengths also plays a role.[14]

Fractal star distribution

A different resolution, which does not rely on the Big Bang theory, was first proposed by

Cantor dust)—the average density of any region diminishes as the region considered increases—it would not be necessary to rely on the Big Bang theory to explain Olbers's paradox. This model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.[citation needed
]

Mathematically, the light received from stars as a function of star distance in a hypothetical fractal cosmos is[citation needed]

where:

  • r0 = the distance of the nearest star, r0 > 0;
  • r = the variable measuring distance from the Earth;
  • L(r) = average luminosity per star at distance r;
  • N(r) = number of stars at distance r.

The function of luminosity from a given distance L(r)N(r) determines whether the light received is finite or infinite. For any luminosity from a given distance L(r)N(r) proportional to ra, is infinite for a ≥ −1 but finite for a < −1. So if L(r) is proportional to r−2, then for to be finite, N(r) must be proportional to rb, where b < 1. For b = 1, the numbers of stars at a given radius is proportional to that radius. When integrated over the radius, this implies that for b = 1, the total number of stars is proportional to r2. This would correspond to a fractal dimension of 2. Thus the fractal dimension of the universe would need to be less than 2 for this explanation to work.

This explanation is not widely accepted among cosmologists, since the evidence suggests that the

isotropically. Contrarily, fractal cosmology requires anisotropic
matter distribution at the largest scales.

See also

References

  1. ^ a b c Overbye, Dennis (3 August 2015). "The Flip Side of Optimism About Life on Other Planets". The New York Times. Retrieved 29 October 2015.
  2. ^ "Cosmas Indicopleustès. Topographie chrétienne, 3 vols.", Ed. Wolska–Conus, W.Paris: Cerf, 1:1968; 2:1970; 3:1973; Sources chrétiennes, Book 10, section 27, line 7 "Cosmas Indicopleustès. Topographia Christiana (4061: 002) Topographie chrétienne, 3 vols.", Ed. Wolska–Conus, W. Paris: Cerf, 1:1968; 2:1970; 3:1973; Sources chrétiennes 141, 159, 197. Book 10, section 27, line 7 (Κρυσταλλώδης ἦν ὁ οὐρανὸς ἀπὸ ὑδάτων παγείς· ἐπειδὴ δὲ ἔμελλε δέχεσθαι ἡλίου φλόγα καὶ σελήνης καὶ ἄστρων ἄπειρα πλήθη, καὶ ἦν ὅλος πυρὸς πεπληρωμένος, ἵνα μὴ οὕτως ὑπὸ τῆς θερμότητος λυθῇ ἢ φλεχθῇ ἄστρων ἄπειρα πλήθη, καὶ ἦν ὅλος πυρὸς πεπληρωμένος, ἵνα μὴ οὕτως ὑπὸ τῆς θερμότητος λυθῇ ἢ φλεχθῇ.)
  3. ISBN 9780470754771
    . The Puritan Thomas Digges (1546–1595?) was the earliest Englishman to offer a defense of the Copernican theory. ... Accompanying Digges's account is a diagram of the universe portraying the heliocentric system surrounded by the orb of fixed stars, described by Digges as infinitely extended in all dimensions.
  4. ^
    ISBN 9783540678779
    . The simple observation that the night sky is dark allows far-reaching conclusions to be drawn about the large-scale structure of the universe. This was already realized by J. Kepler (1610), E. Halley (1720), J.-P. Loy de Chesaux (1744), and H. W. M. Olbers (1826).
  5. ^ For a key extract from this paper, see Harrison (1987), pp. 227–28.
  6. ^ Poe, Edgar Allan (1848). "Eureka: A Prose Poem". Archived from the original on 26 April 2008.
  7. ISBN 9780198596868
    .
  8. ^ "Poe: Eureka". Xroads.virginia.edu. Retrieved 9 May 2013.
  9. ^ "Brief Answers to Cosmic Questions". Universe Forum. Retrieved 27 January 2023 – via harvard.edu.
  10. ISBN 3110258781
    .
  11. ^ Wright, Edward L. (23 October 2006). "Eddington's Temperature of Space". Retrieved 10 July 2013.
  12. ^ Eddington, A.S. (1926). Eddington's 3.18K "Temperature of Interstellar Space". Cambridge University Press. pp. 371–372. Retrieved 10 July 2013. {{cite book}}: |work= ignored (help)
  13. ^ Wright, E. L., E. L. "Errors in the Steady State and Quasi-SS Models". UCLA, Physics and Astronomy Department. Retrieved 28 May 2015.
  14. S2CID 17424588
    .
  15. S2CID 14466810
    .
  16. S2CID 15259697
    .
  17. S2CID 15957886
    .

Further reading

External links

Library resources about
Olbers's paradox
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