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[[Image:Olber's Paradox - All Points.gif|thumb|right|300px|Olbers' paradox in action]]
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In [[astrophysics]] and [[physical cosmology]], '''Olbers' paradox''', named after the German astronomer [[Heinrich Wilhelm Matthias Olbers|Heinrich Wilhelm Olbers]] (1758–1840) and also called the "'''dark night sky paradox'''", is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal [[static universe]]. The darkness of the night sky is one of the pieces of evidence for a non-static universe such as the [[Lambda-CDM model|Big Bang model]]. If the universe is static and populated by an infinite number of stars, any sight line from Earth must end at the (very bright) surface of a star, so the night sky should be completely bright. This contradicts the observed darkness of the night.


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[[Edward Robert Harrison|Edward Robert Harrison's]] ''Darkness at Night: A Riddle of the Universe'' (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.<ref>{{cite book|title=The Scientific Revolution: The Essential Readings|volume=7|series=Blackwell Essential Readings in History|editor-first=Marcus|editor-last=Hellyer|publisher=John Wiley & Sons|year=2008|isbn=9780470754771|page=63|url=http://books.google.com/books?id=1VhC63yV-WgC&pg=PA63|quotation=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.}}</ref> [[Johannes Kepler|Kepler]] also posed the problem in 1610, and the paradox took its mature form in the 18th century work of [[Edmond Halley|Halley]] and [[Jean-Philippe de Cheseaux|Cheseaux]].<ref name="new cosmos">{{cite book|title=The New Cosmos: An Introduction to Astronomy and Astrophysics|series=Physics and astronomy online|first1=Albrecht|last1=Unsöld|first2=Bodo|last2=Baschek|publisher=Springer|year=2001|isbn=9783540678779|page=485|url=http://books.google.com/books?id=nNnmR8ljctoC&pg=PA485|quotation=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).}}</ref> The paradox is commonly attributed to the [[Germany|German]] amateur [[astronomer]] [[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,<ref>For a key extract from this paper, see Harrison (1987), pp. 227–28.</ref> and that [[Edgar Allan Poe]]'s essay ''[[Eureka: A Prose Poem|Eureka]]'' (1848) curiously anticipated some qualitative aspects of Kelvin's argument:
 
{{quote|Were the succession of stars endless, then the background of the sky would present us a uniform luminosity, like that displayed by the Galaxy &ndash; 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.<ref name="eureka">{{cite article|title=Eureka: A Prose Poem|author=Poe, Edgar Allan|year = 1848|url=http://books.eserver.org/poetry/poe/eureka.html}}</ref>}}
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==The paradox==
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[[Image:Eye-stars.svg|thumb|220px|right|What if every line of sight ended in a star? (Infinite universe assumption #2)]]


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.
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To show this, we divide the universe into a series of concentric shells, 1 light year thick .  Thus, a certain number of stars will be in the shell 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 to 2,000,000,001 light years away.  However, the second shell is twice as far away, so each star in it would appear four times dimmer than the first shell.  Thus the total light received from the second shell is the same as the total light received from the first shell.
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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.
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Dark clouds could obstruct the light. But in that case the clouds would heat up, until they were as hot as stars, and then radiate the same amount of light.
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Kepler saw this as an argument for a finite [[observable universe]], or at least for a finite number of stars. In [[general relativity theory]], it is still possible for the paradox to hold in a finite universe:<ref>D'Inverno, Ray. ''Introducing Einstein's Relativity'', Oxford, 1992.</ref> though the sky would not be infinitely bright, every point in the sky would still be like the surface of a star.
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[[Image:Olbers' paradox.gif|right|thumb|In a universe of three dimensions with stars distributed evenly, the number of stars would be proportional to volume. If the surface of concentric sphere shells were considered, the number of stars on each shell would be proportional to the square of the radius of the shell. In the picture above, the shells are reduced to rings in two dimensions with all of the stars on them.]]
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{{See also|Redshift|Lambda-CDM model|metric expansion of space}}
 
Poet [[Edgar Allan Poe]] suggested that the finite size of the observable universe resolves the apparent paradox.<ref>{{cite web|url=http://xroads.virginia.edu/~hyper/poe/eureka.html |title=Poe: Eureka |publisher=Xroads.virginia.edu |date= |accessdate=2013-05-09}}</ref> More specifically, because the universe is [[Age of the Universe|finitely old]] and the speed of light is finite, only finitely many stars can be observed within a given volume of space visible from Earth. The density of stars within this finite volume is sufficiently low that any line of sight from Earth is unlikely to reach a star.
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However, the Big Bang theory introduces a new paradox: it states that the sky was much brighter in the past, especially at the end of the [[recombination (cosmology)|recombination]] era, when it first became transparent. All points of the local sky at that era were brighter than the surface of the sun, due to the high temperature of the universe in that [[Graphical timeline of the Big Bang|era]]; and most light rays will terminate not in a star but in the relic of the Big Bang.
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This paradox is explained by the fact that the Big Bang theory also involves the expansion of space which can cause the energy of emitted light to be reduced via [[redshift]]. More specifically, the extreme levels of radiation from the [[Big Bang]] have been redshifted to microwave wavelengths (1100 times longer than its original wavelength) as a result of the cosmic expansion, and thus form the [[cosmic microwave background radiation]]. This explains the relatively low light densities present in most of our sky despite the assumed bright nature of the Big Bang. The redshift also affects light from distant stars and [[quasar]]s, but the diminution is minor, since [[List of most distant astronomical object record holders|the most distant galaxies and quasars]] have redshifts of only around 5 to 8.6.
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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. The [[Steady state theory|steady state]] cosmological model assumed that 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 light from these distant stars and quasars will be redshifted accordingly (by the [[Doppler effect]] and [[thermalisation]]<ref>http://physicsessays.org/doi/abs/10.4006/1.3028952?journalCode=phes - Gilles Corriveau: The 3 K Background Emission, the Formation of Galaxies, and the Large‐Scale Structure of the Universe</ref>), 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
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e.g. for temperature 2.7&nbsp;K it is 40&nbsp;fJ/m<sup>3</sup>&nbsp;...&nbsp;4.5&times;10<sup>&minus;31</sup>&nbsp;kg/m<sup>3</sup> and for visible temperature 6000&nbsp;K we get 1&nbsp;J/m<sup>3</sup>&nbsp;...&nbsp;1.1&times;10<sup>&minus;17</sup>&nbsp;kg/m<sup>3</sup>. But the total radiation emitted by a star (or other cosmic object) is at most equal to the total [[nuclear binding energy]] of [[isotope]]s in the star. For the density of the [[observable universe]] of about 4.6&times;10<sup>&minus;28</sup>&nbsp;kg/m<sup>3</sup> and given the known [[abundance of the chemical elements]], the corresponding maximal radiation energy density of 9.2&times;10<sup>&minus;31</sup> kg/m<sup>3</sup>, i.e. temperature 3.2&nbsp;K.<ref>{{cite web|url=http://www.astro.ucla.edu/~wright/Eddington-T0.html| title=Eddington's Temperature of Space|last=Wright|first=Edward L.|date=23 Oct 2006|accessdate=10 July 2013}}</ref><ref>{{cite book|url=http://webdev.archive.org/stream/TheInternalConstitutionOfTheStars/Eddington-TheInternalConstitutionOfTheStars#page/n379/mode/2up| title=Eddington's 3.18°K "Temperature of Interstellar Space"|pages=371-372|work=The Internal Constitution of the Stars| publisher=Cambridge University Press|last=Eddington|first=A.S.|year=1926|accessdate=10 July 2013}}</ref> This is close to the summed energy density of the [[cosmic microwave background]] and the [[cosmic neutrino background]]. The Big Bang hypothesis, by contrast, predicts that the CBR should have the same energy density as the binding energy density of [[Big Bang nucleosynthesis|the primordial helium]], which is much greater than the binding energy density of the non-primordial elements; so it gives almost the same result. But (neglecting quantum fluctuations in the early universe) the Big Bang would also predict a uniform distribution of CBR, while the steady-state model predicts nothing about its distribution{{Citation needed|date=September 2010}}. Nevertheless the [[isotropy]] is very probable in steady state as in the [[kinetic theory]].
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Since the speed of light is a constant value, regardless of the shift towards infrared frequencies, the universe is still sharply constrained to finite sizes in space as well as in time. Some models of an infinite ([[Steady State theory]] or [[static universe]]) solution of the universe are still viable, and Olber's paradox cannot sharply distinguish between them from some variants of the [[Big Bang]] model.
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Stars have a finite age and a finite power, thereby implying that each star has a finite impact on a sky's light field density.  But if the universe were infinitely old, there would be infinitely many other stars in the same angular direction, with an infinite total impact.


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This would not resolve the paradox given the following argument: According to the [[laws of thermodynamics]], the intermediate matter must eventually heat up (or cool down, if it was initially hotter) until it is in [[thermal equilibrium]] with the surrounding stars. Once this happens, the matter would then radiate the energy it receives from the stars at the same (average) temperature. So the sky would still appear uniformly bright.
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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&nbsp;K (corresponding to a typical photon energy of 0.3 [[electron-volts|eV]] and so a frequency of 7.5&times;10<sup>13</sup> [[Hertz|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 {{val|1.2|e=17}} eV/m<sup>3</sup> = {{val|2.1|e=-19|u=kg/m3}}, which is much greater than the observed value of {{val|4.7|e=-31|u=kg/m3}}.<ref name="new cosmos"/> So the sky is about fifty billion times darker than it would be if the universe were neither expanding nor too young to have reached equilibrium yet.
 
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A different resolution, which does not rely on the [[Big Bang]] theory, was first proposed by [[Carl Charlier]] in 1908 and later rediscovered by [[Benoit Mandelbrot|Benoît Mandelbrot]] in 1974. They both postulated that if the stars in the universe were distributed in a hierarchical [[fractal cosmology]] (e.g., similar to [[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' 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.


Mathematically, the light received from stars as a function of star distance in a hypothetical fractal cosmos is:
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:<math>\text{light}=\int_{r_0}^\infty L(r) N(r)\,dr</math>
  <li>[http://tskgaming.net16.net/index.php?site=forum_topic&topic=3208 http://tskgaming.net16.net/index.php?site=forum_topic&topic=3208]</li>
 
 
where:
  <li>[http://13123188787.com/news/html/?23375.html http://13123188787.com/news/html/?23375.html]</li>
 
 
''r''<sub>0</sub> = the distance of the nearest star. ''r''<sub>0</sub> > 0;
  <li>[http://www.9combo.com/forum.php?mod=viewthread&tid=146147&fromuid=3799 http://www.9combo.com/forum.php?mod=viewthread&tid=146147&fromuid=3799]</li>
 
 
''r'' = the variable measuring distance from the Earth;
  <li>[http://www.daixiec.com/bbs/boke.asp?tcsnwsso.showtopic.112597.html http://www.daixiec.com/bbs/boke.asp?tcsnwsso.showtopic.112597.html]</li>
 
 
''L''(''r'') = average [[luminosity]] per star at distance ''r'';
  <li>[http://www.ym0921.com/news/html/?23822.html http://www.ym0921.com/news/html/?23822.html]</li>
 
 
''N''(''r'') = number of stars at distance ''r''.
</ul>
 
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 ''r''<sup>''a''</sup>, <math>\text{light}</math> is infinite for ''a''&nbsp;≥&nbsp;&minus;1 but finite for ''a''&nbsp;<&nbsp;&minus;1. So if ''L''(''r'') is proportional to ''r''<sup>&minus;2</sup>, then for <math>\text{light}</math> to be finite, ''N''(''r'') must be proportional to ''r''<sup>''b''</sup>, where ''b''&nbsp;<&nbsp;1. For ''b''&nbsp;=&nbsp;1, the numbers of stars at a given radius is proportional to that radius. When integrated over the radius, this implies that for ''b''&nbsp;=&nbsp;1, the ''total'' number of stars is proportional to ''r''<sup>2</sup>. 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 fractal dimension of the universe is at least 2.<ref>{{cite journal |author=Joyce, M.; Labini, F.S.; Gabrielli, A.; Montouri, M.; Pietronero, L. |year=2005 |title=Basic Properties of Galaxy Clustering in the light of recent results from the Sloan Digital Sky Survey |journal=Astronomy and Astrophysics |volume=443 |issue=11 |arxiv=astro-ph/0501583 | doi=  10.1051/0004-6361:20053658 |pages=11–16 |bibcode=2005A&A...443...11J}}</ref><ref>{{cite journal |author=Labini, F.S.; Vasilyev, N.L.; Pietronero, L.; Baryshev, Y. |year=2009 |title=Absence of sedlf-averaging and of homogeneity in the large scale galaxy distribution |journal=Europhys.Lett. |volume=86 |issue= 4|page= 49001|arxiv=0805.1132 |doi=  10.1209/0295-5075/86/49001 |pmid= |pmc= |bibcode = 2009EL.....8649001S }}</ref><ref>{{cite journal |author=Hogg, David W.; Eisenstein, Daniel J.; Blanton, Michael R.; Bahcall, Neta A.; Brinkmann, J.; Gunn, James E.; Schneider, Donald P. |year=2005 |title=Cosmic homogeneity demonstrated with luminous red galaxies |journal=[[The Astrophysical Journal]] |volume=624 |issue= |pages=54–58 |arxiv=astro-ph/0411197 |doi=  10.1086/429084 |pmid= |pmc= |bibcode=2005ApJ...624...54H}}</ref> Moreover, the majority of cosmologists take the [[cosmological principle]] as a given, which assumes that matter at the scale of billions of light years is distributed [[isotropic]]ally. Contrasting this, fractal cosmology requires [[anisotropy|anisotropic]] matter distribution at the largest scales.
 
== See also ==
* [[Heat death paradox]]
* [[List of paradoxes]]
 
==References==
{{reflist}}
 
==Further reading==
* [[Edward Robert Harrison]] (1987) ''Darkness at Night: A Riddle of the Universe'', Harvard University Press. Very readable.
*-------- (2000) ''Cosmology'', 2nd ed. Cambridge Univ. Press. Chpt. 24.
* Taylor Mattie, ''Fundamentals of Heat Transfer''. MAHS
* {{cite journal | last1 = Wesson |first1= Paul | year = 1991 | title = Olbers' paradox and the spectral intensity of the extragalactic background light | bibcode=1991ApJ...367..399W | journal = The [[Astrophysical Journal]] | volume = 367 | issue = | pages = 399–406 |doi = 10.1086/169638 }}
 
==External links==
* [http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html Relativity FAQ about Olbers' paradox]
* [http://www.faqs.org/faqs/astronomy/faq/part9/section-17.html Astronomy FAQ about Olbers' paradox]
* [http://supernova.lbl.gov/~evlinder/umass/faqm.html#olber Cosmology FAQ about Olbers' paradox]
* {{MathPages|id=home/kmath141/kmath141|title=On Olber's Paradox}}
* [http://www.physics.org/facts/sand-dark.asp Why is the sky dark?] physics.org page about Olbers' paradox
* [http://q2cfestival.com/play.php?lecture_id=8250&talk=alice Why is it dark at night?] A 60-second animation from the [[Perimeter Institute]] exploring the question with Alice and Bob in Wonderland
 
{{DEFAULTSORT:Olbers' Paradox}}
<!-- [[Category:Astronomy]] and [[Category:Astrophysics]] are too general. -->
[[Category:Physical cosmology]]
[[Category:Physical paradoxes]]

Latest revision as of 23:14, 3 January 2015

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