Olbers' paradox

Olbers' paradox

Discussion
Ask a question about 'Olbers' paradox'
Start a new discussion about 'Olbers' paradox'
Answer questions from other users
Full Discussion Forum
 
Encyclopedia

In astrophysics
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...

 and physical cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

, Olbers' paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe
Static universe
A static universe, also referred to as a "stationary" or "Einstein" universe, is a model in which space is neither expanding nor contracting. Albert Einstein proposed such a model as his preferred cosmology in 1917...

. It is one of the pieces of evidence for a non-static universe such as the current Big Bang model
Lambda-CDM model
ΛCDM or Lambda-CDM is an abbreviation for Lambda-Cold Dark Matter, which is also known as the cold dark matter model with dark energy...

. The argument is also referred to as the "dark night sky paradox". The paradox states that at any angle from the Earth the sight line will end at the surface of a star, so the night sky should be completely white. This contradicts the darkness of the night sky and leads many to wonder why we do not see only light from stars in the night sky (see physical paradox
Physical paradox
A physical paradox is an apparent contradiction in physical descriptions of the universe. While many physical paradoxes have accepted resolutions, others defy resolution and may indicate flaws in theory...

).

History


Harrison's
Edward Robert Harrison
Edward R. Harrison was a British astronomer and cosmologist, who spent much of his career at the University of Massachusetts and University of Arizona...

 (1987) is the definitive account to date 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
Thomas Digges
Sir Thomas Digges was an English mathematician and astronomer. He was the first to expound the Copernican system in English but discarded the notion of a fixed shell of immoveable stars to postulate infinitely many stars at varying distances; he was also first to postulate the "dark night sky...

, who was also the first to expound the Copernican system in English and may have been the first to postulate an infinite universe with infinitely many stars. Kepler also posed the problem in 1610, and the paradox took its mature form in the 18th century work of Halley
Edmond Halley
Edmond Halley FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley's Comet. He was the second Astronomer Royal in Britain, following in the footsteps of John Flamsteed.-Biography and career:Halley...

 and Cheseaux
Jean-Philippe de Cheseaux
Jean-Philippe Loys de Chéseaux was an astronomer from Lausanne in Switzerland. In 1746 he presented a list of nebulae, eight of which were his own new discoveries, to the Académie Française des Sciences. The list was noted privately by Le Gentil in 1759, but only made public in 1892 by Guillaume...

. The paradox is commonly attributed to the German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 amateur astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

 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, and that Edgar Allan Poe
Edgar Allan Poe
Edgar Allan Poe was an American author, poet, editor and literary critic, considered part of the American Romantic Movement. Best known for his tales of mystery and the macabre, Poe was one of the earliest American practitioners of the short story and is considered the inventor of the detective...

's essay Eureka (1848) curiously anticipated some qualitative aspects of Kelvin's argument:

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.

To show this we divide the universe in to a series of concentric shells, 1 light year thick (say). Thus a certain number of stars will be in the shell 1,000,000,000 to 1,000,000,001 light years away, say. 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.

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.

Dark clouds could obstruct the light. But in that case the clouds would heat up, until they were as hot as stars, and then radiated the same amount of light.

Kepler saw this as an argument for a finite observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...

, 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: though the sky would not be infinitely bright, every point in the sky would still be like the surface of a star.


The mainstream explanation



In order to explain Olbers' paradox, it is necessary to account for the relatively low brightness of the night sky in relation to the circle of our sun. The universe is only finitely old
Age of the universe
The age of the universe is the time elapsed since the Big Bang posited by the most widely accepted scientific model of cosmology. The best current estimate of the age of the universe is 13.75 ± 0.13 billion years within the Lambda-CDM concordance model...

, and stars have existed only for part of that time. So, as Poe suggested, the Earth receives no starlight from beyond a certain distance, corresponding to the age of the oldest stars. Space is sufficiently rarefied that most lines from the Earth do not touch any star within this distance of Earth.

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
Recombination (cosmology)
In cosmology, recombination refers to the epoch at which charged electrons and protons first became bound to form electrically neutral hydrogen atoms.Note that the term recombination is a misnomer, considering that it represents the first time that electrically neutral hydrogen formed. After the...

 era, when it first became transparent. All points of the local sky at that era were brighter than the circle of the sun, due to the high temperature of the universe in that prehistoric era
Graphical timeline of the Big Bang
This timeline of the Big Bang shows the sequence of events as predicted by the Big Bang theory, from the beginning of time to the end of the Dark Ages....

; and we have seen that most light rays will terminate not in a star but in the relic of the Big Bang.

This paradox is explained by the fact that the Big Bang Theory also involves the expansion of the "fabric" of space itself (not just the distance of objects in that space) that can cause the energy of emitted light to be reduced via redshift
Redshift
In physics , redshift happens when light seen coming from an object is proportionally increased in wavelength, or shifted to the red end of the spectrum...

. More specifically, the extreme levels of radiation from the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

 have been redshifted to microwave wavelengths (1100 times lower than its original wavelength) as a result of the cosmic expansion, and thus form the cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....

. 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
Quasar
A quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...

s, but the diminution is only an order of magnitude or so, since the most distant galaxies and quasars have redshifts of only around 5 to 8.6.

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. The steady state
Steady State theory
In cosmology, the Steady State theory is a model developed in 1948 by Fred Hoyle, Thomas Gold, Hermann Bondi and others as an alternative to the Big Bang theory...

 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 thermalisation
Thermalisation
In physics, thermalisation is the process of particles reaching thermal equilibrium through mutual interaction....

), so that the total light flux from the sky remains finite. Thus the observed radiation density (the sky brightness of extragalactic background light
Extragalactic background light
The Extragalactic Background Light or simply the "extragalactic background" is the faint diffuse light of the night sky, consisting of the combined flux of all extragalactic sources...

) can be independent of finiteness of the Universe. Mathematically, the total electromagnetic energy density (radiation energy density) in thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

 from Planck's law is


e.g. for temperature 2.7K it is 40 fJ/m3 ... 4.5×10−31 kg/m3 and for visible 6000K 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
Nuclear binding energy
Nuclear binding energy is the energy required to split a nucleus of an atom into its component parts. The component parts are neutrons and protons, which are collectively called nucleons...

 of isotope
Isotope
Isotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...

s in the star. For the density of the observable universe
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...

 of about 4.6×10−28 kg/m3 and given the known abundance of the chemical elements
Abundance of the chemical elements
The abundance of a chemical element measures how relatively common the element is, or how much of the element is present in a given environment by comparison to all other elements...

, the corresponding maximal radiation energy density of 9.2×10−31 kg/m3, i.e. temperature 3.2K. This is close to the summed energy density of the cosmic microwave background and the cosmic neutrino background
Cosmic neutrino background
The cosmic neutrino background is the universe's background particle radiation composed of neutrinos.Like the cosmic microwave background radiation , the CνB is a relic of the big bang, and while the CMB dates from when the universe was 379,000 years old, the CνB decoupled from matter when the...

. The Big Bang hypothesis, by contrast, predicts that the CBR should have the same energy density as the binding energy density of the primordial helium
Big Bang nucleosynthesis
In physical cosmology, Big Bang nucleosynthesis refers to the production of nuclei other than those of H-1 during the early phases of the universe...

, 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. Nevertheless the isotropy
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...

 is very probable in steady state as in the kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

.

Thus, Olbers' paradox can not decide between a finite (e.g. some variants of the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

 model) and an infinite (Steady State theory
Steady State theory
In cosmology, the Steady State theory is a model developed in 1948 by Fred Hoyle, Thomas Gold, Hermann Bondi and others as an alternative to the Big Bang theory...

 or static universe
Static universe
A static universe, also referred to as a "stationary" or "Einstein" universe, is a model in which space is neither expanding nor contracting. Albert Einstein proposed such a model as his preferred cosmology in 1917...

) solution.

Finite age of stars


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.

Absorption


An alternative explanation is that the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

 is not transparent, and the light from distant stars is blocked by intermediate dark stars or absorbed by dust or gas, so that there is a bound on the distance from which light can reach the observer.

However, this reasoning alone would not resolve the paradox given the following argument: According to the second law of thermodynamics
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

, there can be no material hotter than its surroundings that does not give off radiation and at the same time be uniformly distributed through space. Energy must be conserved, per the first law of thermodynamics
First law of thermodynamics
The first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...

. Therefore, the intermediate matter would heat up and soon reradiate the energy (possibly at different wavelengths). This would again result in intense uniform radiation.

How bright would the sky be?


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 3000K (corresponding to a typical photon energy of 0.3 eV and so a frequency of 7.5×1013 Hz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

), 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 eV/m3 = , which is nearly eleven orders of magnitude greater than the observed value of . 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.

Fractal star distribution


A different resolution, which does not rely on the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...

 theory, was first proposed by Carl Charlier
Carl Charlier
Carl Vilhelm Ludwig Charlier was a Swedish astronomer.He received his Ph.D. from Uppsala University in 1887, later worked there and at the Stockholm Observatory and was Professor of Astronomy and Director of the Observatory at Lund University from 1897.He made extensive statistical studies of the...

 in 1908 and later rediscovered by Benoît Mandelbrot
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...

 in 1974. They both postulated that if the stars in the universe were distributed in a hierarchical fractal cosmology
Fractal cosmology
In physical cosmology, fractal cosmology is a set of minority cosmological theories which state that the distribution of matter in the Universe, or the structure of the universe itself, is a fractal. More generally, it relates to the usage or appearance of fractals in the study of the universe and...

 (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:


where:

r0 = the distance of the nearest star. r0 > 0;

r = the variable measuring distance from the Earth;

L(r) = average luminosity
Luminosity
Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...

 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
Fractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...

 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. Moreover, the majority of cosmologists take the cosmological principle
Cosmological Principle
In modern physical cosmology, the cosmological principle is the working assumption that observers on Earth do not occupy an unusual or privileged location within the universe as a whole, judged as observers of the physical phenomena produced by uniform and universal laws of physics...

 as a given, which assumes that matter at the scale of billions of light years is distributed isotropically. Contrasting this, fractal cosmology requires anisotropic
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...

 matter distribution at the largest scales.

Light incoherence


In May 2011, Pierre Blanc suggested an explanation which does not require any assumption on the size or age of the universe.

Any observation apparatus is restricted in the number of stars it can detect. If a device can detect only one star in a given (very small) cone, a better apparatus would see two stars; a still better one would see ten and so on. The first device “sees” one star as well as the combined light coming out of "unseen" or "undetected" stars. But the light signals sent by these “subliminar” stars are incoherent, in phase as well as in polarisation. Moreover, they are in large part reflected by less distant astronomical objects. Blanc shows that their combination results in a “background noise” with very little brightness, although the sky would appear less dark with more powerful instruments.

Further reading

  • Edward Robert Harrison
    Edward Robert Harrison
    Edward R. Harrison was a British astronomer and cosmologist, who spent much of his career at the University of Massachusetts and University of Arizona...

     (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

External links