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Encyclopedia > Chandrasekhar limit

The Chandrasekhar limit (named after the Indian astrophysicist Subrahmanyan Chandrasekhar) is the maximum nonrotating mass which can be supported against gravitational collapse by electron degeneracy pressure. It is commonly given as being about 1.4[1][2] solar masses. Computed values for the limit will vary depending on the approximations used, the nuclear composition of the mass, and the temperature.[3] Chandrasekhar[4], eq. (36),[5], eq. (58),[6], eq. (43) gives a value of An astrophysicist is a person whose profession is astrophysics. ... Chandrasekhar redirects here. ... For other uses, see Mass (disambiguation). ... This article or section does not cite its references or sources. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... In astronomy, the solar mass is a unit of mass used to express the mass of stars and larger objects such as galaxies. ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ...

Here, μe is the average molecular weight per electron, mH is the mass of the hydrogen atom, and is a constant connected with the solution to the Lane-Emden equation. Numerically, this value is approximately (2/μe)2 · 2.85 · 1030 kg, or , where is the standard solar mass.[7] As is the Planck mass, , the limit is of the order of MPl3/mH2. The molecular mass of a substance (less accurately called molecular weight and abbreviated as MW) is the mass of one molecule of that substance, relative to the unified atomic mass unit u (equal to 1/12 the mass of one atom of carbon-12). ... This article is about the chemistry of hydrogen. ... For other uses, see Atom (disambiguation). ... In astrophysics, the Lane-Emden equation is applicable to magnetohydrodynamic fluids under the action of force-free magnetic fields. ... In astronomy, the solar mass is a unit of mass used to express the mass of stars and larger objects such as galaxies. ... The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by Ï€. â‰ˆ 1. ...

As white dwarf stars are supported by electron degeneracy pressure, this is an upper limit for the mass of a white dwarf. Main-sequence stars with a mass exceeding approximately 8 solar masses therefore cannot lose enough mass to form a stable white dwarf at the end of their lives, and instead form either a neutron star or black hole.[8][9][10] This article or section does not adequately cite its references or sources. ... This article is about the astronomical object. ... Hertzsprung-Russell diagram The main sequence of the Hertzsprung-Russell diagram is the curve where the majority of stars are located in this diagram. ... For the story by Larry Niven, see Neutron Star (story). ... For other uses, see Black hole (disambiguation). ...

## Contents

Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle. Since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energy levels. Compression of the electron gas increases the number of electrons in a given volume and raises the maximum energy level in the occupied band. Therefore, the energy of the electrons will increase upon compression, so pressure must be exerted on the electron gas to compress it. This is the origin of electron degeneracy pressure. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... For other uses, see Electron (disambiguation). ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ...

In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form P=K1ρ5/3. Solving the hydrostatic equation leads to a model white dwarf which is a polytrope of index 3/2 and therefore has radius inversely proportional to the cube root of its mass, and volume inversely proportional to its mass.[11] In physics and thermodynamics, an equation of state is a relation between state variables. ... A polytrope is a solution to the Lane-Emden equation. ...

As the mass of a model white dwarf increases, the typical energies to which degeneracy pressure forces the electrons are no longer negligible relative to their rest masses. The velocities of the electrons approach the speed of light, and special relativity must be taken into account. In the strongly relativistic limit, we find that the equation of state takes the form P=K2ρ4/3. This will yield a polytrope of index 3, which will have a total mass, Mlimit say, depending only on K2.[12] For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...

Radius versus mass for a model white dwarf.

For a fully relativistic treatment, the equation of state used will interpolate between the equations P=K1ρ5/3 for small ρ and P=K2ρ4/3 for large ρ. When this is done, the model radius still decreases with mass, but becomes zero at Mlimit. This is the Chandrasekhar limit.[5] The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored green and red, respectively. μe has been set equal to 2. Radius is measured in standard solar radii[7] and mass in standard solar masses. Image File history File links Size of this preview: 800 Ã— 600 pixelsFull resolution (3200 Ã— 2400 pixel, file size: 47 KB, MIME type: image/png) Graph plotting the radius (in standard solar radii) against the mass (in standard solar masses) for a model white dwarf star consisting of a cold Fermi... Image File history File links Size of this preview: 800 Ã— 600 pixelsFull resolution (3200 Ã— 2400 pixel, file size: 47 KB, MIME type: image/png) Graph plotting the radius (in standard solar radii) against the mass (in standard solar masses) for a model white dwarf star consisting of a cold Fermi...

A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature.[3] Lieb and Yau[13] have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation. For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

## History

A series of papers published between 1931 and 1935 had its beginning on a trip from India to England in 1930, where the Indian physicist Subrahmanyan Chandrasekhar worked on the calculation of the statistics of a degenerate Fermi gas.[21] In these papers, Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state,[11] and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above.[12][4][22][5] Chandrasekhar reviews this work in his Nobel Prize lecture.[6] This value was also computed in 1932 by the Soviet physicist Lev Davidovich Landau,[23] who, however, did not apply it to white dwarfs. Year 1931 (MCMXXXI) was a common year starting on Thursday (link will display full 1931 calendar) of the Gregorian calendar. ... 1935 (MCMXXXV) was a common year starting on Tuesday (link will display full calendar). ... For other uses, see England (disambiguation). ... Year 1930 (MCMXXX) was a common year starting on Wednesday (link will display 1930 calendar) of the Gregorian calendar. ... A non-resident Indian (NRI) is an Indian citizen who has migrated to another country. ... Not to be confused with physician, a person who practices medicine. ... Chandrasekhar redirects here. ... Table of Hydraulics and Hydrostatics, from the 1728 Cyclopaedia. ... In physics and thermodynamics, an equation of state is a relation between state variables. ... Year 1932 (MCMXXXII) was a leap year starting on Friday (the link will display full 1932 calendar) of the Gregorian calendar. ... Lev Davidovich Landau (Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ...

Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the British astrophysicist Arthur Stanley Eddington. Eddington was aware that the existence of black holes was theoretically possible, and also realized that the existence of the limit made their formation possible. However, he was unwilling to accept that this could happen. After a talk by Chandrasekhar on the limit in 1935, he replied: An astrophysicist is a person whose profession is astrophysics. ... One of Sir Arthur Stanley Eddingtons papers announced Einsteins theory of general relativity to the English-speaking world. ... For other uses, see Black hole (disambiguation). ...

The star has to go on radiating and radiating and contracting and contracting until, I suppose, it gets down to a few km. radius, when gravity becomes strong enough to hold in the radiation, and the star can at last find peace. … I think there should be a law of Nature to prevent a star from behaving in this absurd way![24]

Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law P=K1ρ5/3 universally applicable, even for large ρ.[25] Although Bohr, Fowler, Pauli, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar.[26], pp. 110–111 Through the rest of his life, Eddington held to his position in his writings,[27][28][29][30][31] including his work on his fundamental theory.[32] The drama associated with this disagreement is one of the main themes of Empire of the Stars, Arthur I. Miller's biography of Chandrasekhar.[26] In Miller's view: Niels Henrik David Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... This article is about the Austrian-Swiss physicist. ... One of Sir Arthur Stanley Eddingtons papers announced Einsteins theory of general relativity to the English-speaking world. ...

Chandra's discovery might well have transformed and accelerated developments in both physics and astrophysics in the 1930s. Instead, Eddington's heavy-handed intervention lent weighty support to the conservative community astrophysicists, who steadfastly refused even to consider the idea that stars might collapse to nothing. As a result, Chandra's work was almost forgotten.[26], p. 150

## Applications

The core of a star is kept from collapsing by the heat generated by the fusion of nuclei of lighter elements into heavier ones. At various points in a star's life, the nuclei required for this process will be exhausted, and the core will collapse, causing it to become denser and hotter. A critical situation arises when iron accumulates in the core, since iron nuclei are incapable of generating further energy through fusion. If the core becomes sufficiently dense, electron degeneracy pressure will play a significant part in stabilizing it against gravitational collapse.[33] The deuterium-tritium (D-T) fusion reaction is considered the most promising for producing fusion power. ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ... The periodic table of the chemical elements A chemical element, or element, is a type of atom that is defined by its atomic number; that is, by the number of protons in its nucleus. ... General Name, symbol, number iron, Fe, 26 Chemical series transition metals Group, period, block 8, 4, d Appearance lustrous metallic with a grayish tinge Standard atomic weight 55. ...

If a main-sequence star is not too massive (less than approximately 8 solar masses), it will eventually shed enough mass to form a white dwarf having mass below the Chandrasekhar limit, which will consist of the former core of the star, For more massive stars, electron degeneracy pressure will not keep the iron core from collapsing to very great density, leading to formation of a neutron star, black hole, or, speculatively, a quark star. (For very massive, low-metallicity stars, it is also possible that instabilities will destroy the star completely.)[8][9][10][34] During the collapse, neutrons are formed by the capture of electrons by protons, leading to the emission of neutrinos.[33], pp. 1046–1047. The decrease in gravitational potential energy of the collapsing core releases a large amount of energy which is on the order of 1046 joules (100 foes.) Most of this energy is carried away by the emitted neutrinos.[35] This process is believed to be responsible for supernovae of types Ib, Ic, and II.[33] In astronomy, the solar mass is a unit of mass used to express the mass of stars and larger objects such as galaxies. ... For the story by Larry Niven, see Neutron Star (story). ... For other uses, see Black hole (disambiguation). ... A strange star or quark star is a hypothetical type of star composed of strange matter, or quark matter. ... The globular cluster M80. ... This article or section does not adequately cite its references or sources. ... For other uses, see Electron (disambiguation). ... For other uses, see Proton (disambiguation). ... For other uses, see Neutrino (disambiguation). ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... A foe is a unit of energy equal to 1044 joules. ... The expanding remnant of SN 1987A, a Type II-P supernova in the Large Magellanic Cloud. ...

Strong indications of the reliability of Chandrasekhar's formula are:

1. Only one white dwarf with a mass greater than Chandrasekhar's limit has ever been observed. (See below.)
2. The absolute magnitudes of supernovae of Type Ia are all approximately the same; at maximum luminosity, MV is approximately -19.3, with a standard deviation of no more than 0.3.[36], (1) A 1-sigma interval therefore represents a factor of less than 2 in luminosity. This seems to indicate that all type Ia supernovae convert approximately the same amount of mass to energy.

In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In this diagram, the bars represent observation means and the red lines represent the confidence intervals surrounding them. ...

## A type Ia supernova apparently from a supra-limit white dwarf

Main article: Champagne Supernova.

On April 2003, the Supernova Legacy Survey observed a type Ia supernova, designated SNLS-03D3bb, in a galaxy approximately 4 billion light years away. According to a group of astronomers at the University of Toronto and elsewhere, the observations of this supernova are best explained by assuming that it arose from a white dwarf which grew to twice the mass of the Sun before exploding. They believe that the star, dubbed the "Champagne Supernova" by David R. Branch, may have been spinning so fast that centrifugal force allowed it to exceed the limit. Alternatively, the supernova may have resulted from the merger of two white dwarfs, so that the limit was only violated momentarily. Nevertheless, they point out that this observation poses a challenge to the use of type Ia supernovae as standard candles.[37][38][39] The Champagne Supernova, SN 2003fg (designated SNLS-03D3bb by the Canada-France-Hawaii Supernova Legacy Survey, which discovered it), was an aberrant type Ia supernova discovered in 2003 and described in the journal Nature on September 21 of 2006. ... Year 2003 (MMIII) was a common year starting on Wednesday of the Gregorian calendar. ... The Supernova Legacy Survey Program[1] is a project designed to investigate dark energy, by detecting and monitoring approximately 2000 high-redshift supernovae between 2003 and 2008, using MegaPrime, a large [CCD]] mosaic at the Canada-France-Hawaii Telescope. ... A light-year or lightyear (symbol: ly) is a unit of measurement of length, specifically the distance light travels in vacuum in one year. ... The University of Toronto (U of T) is a public research university in the city of Toronto, Ontario, Canada. ... Sol redirects here. ... The Champagne Supernova, SN 2003fg (designated SNLS-03D3bb by the Canada-France-Hawaii Supernova Legacy Survey, which discovered it), was an aberrant type Ia supernova discovered in 2003 and described in the journal Nature on September 21 of 2006. ... Centrifugal force (from Latin centrum centre and fugere to flee) is a term which may refer to two different forces which are related to rotation. ... Standard Candles is a compilation of short stories by American science fiction author Jack McDevitt. ...

## References

1. ^ p. 55, How A Supernova Explodes, Hans A. Bethe and Gerald Brown, pp. 51–62 in Formation And Evolution of Black Holes in the Galaxy: Selected Papers with Commentary, Hans Albrecht Bethe, Gerald Edward Brown, and Chang-Hwan Lee, River Edge, NJ: World Scientific: 2003. ISBN 981238250X.
2. ^ Mazzali, P. A.; K. Röpke, F. K.; Benetti, S.; Hillebrandt, W. (2007). "A Common Explosion Mechanism for Type Ia Supernovae". Science 315 (5813): 825-828. doi:10.1126/science.1136259. Retrieved on 2007-05-24.
3. ^ a b The Neutron Star and Black Hole Initial Mass Function, F. X. Timmes, S. E. Woosley, and Thomas A. Weaver, Astrophysical Journal 457 (February 1, 1996), pp. 834–843.
4. ^ a b The Highly Collapsed Configurations of a Stellar Mass, S. Chandrasekhar, Monthly Notices of the Royal Astronomical Society 91 (1931), 456–466.
5. ^ a b c The Highly Collapsed Configurations of a Stellar Mass (second paper), S. Chandrasekhar, Monthly Notices of the Royal Astronomical Society, 95 (1935), pp. 207--225.
6. ^ a b On Stars, Their Evolution and Their Stability, Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
7. ^ a b Standards for Astronomical Catalogues, Version 2.0, section 3.2.2, web page, accessed 12-I-2007.
8. ^ a b White dwarfs in open clusters. VIII. NGC 2516: a test for the mass-radius and initial-final mass relations, D. Koester and D. Reimers, Astronomy and Astrophysics 313 (1996), pp. 810–814.
9. ^ a b An Empirical Initial-Final Mass Relation from Hot, Massive White Dwarfs in NGC 2168 (M35), Kurtis A. Williams, M. Bolte, and Detlev Koester, Astrophysical Journal 615, #1 (2004), pp. L49–L52; also arXiv astro-ph/0409447.
10. ^ a b How Massive Single Stars End Their Life, A. Heger, C. L. Fryer, S. E. Woosley, N. Langer, and D. H. Hartmann, Astrophysical Journal 591, #1 (2003), pp. 288–300.
11. ^ a b The Density of White Dwarf Stars, S. Chandrasekhar, Philosophical Magazine (7th series) 11 (1931), pp. 592–596.
12. ^ a b The Maximum Mass of Ideal White Dwarfs, S. Chandrasekhar, Astrophysical Journal 74 (1931), pp. 81–82.
13. ^ A rigorous examination of the Chandrasekhar theory of stellar collapse, Elliott H. Lieb and Horng-Tzer Yau, Astrophysical Journal 323 (1987), pp. 140–144.
14. ^ On Dense Matter, R. H. Fowler, Monthly Notices of the Royal Astronomical Society 87 (1926), pp. 114–122.
15. ^ The Limiting Density of White Dwarf Stars, Edmund C. Stoner, Philosophical Magazine (7th series) 7 (1929), pp. 63–70.
16. ^ Über die Grenzdichte der Materie und der Energie, Wilhelm Anderson, Zeitschrift für Physik 56, #11–12 (November 1929), pp. 851–856. DOI 10.1007/BF01340146.
17. ^ The Equilibrium of Dense Stars, Edmund C. Stoner, Philosophical Magazine (7th series) 9 (1930), pp. 944–963.
18. ^ The minimum pressure of a degenerate electron gas, E. C. Stoner, Monthly Notices of the Royal Astronomical Society 92 (May 1932), pp. 651–661.
19. ^ Anwendung der Pauli-Fermischen Elektronengastheorie auf das Problem der Kohäsionskräfte, J. Frenkel, Zeitschrift für Physik 50, #3–4 (March 1928), pp. 234–248. DOI 10.1007/BF01328867.
20. ^ The article by Ya I Frenkel' on `binding forces' and the theory of white dwarfs, D. G. Yakovlev, Physics Uspekhi 37, #6 (1994), pp. 609–612.
21. ^ Chandrasekhar's biographical memoir at the National Academy of Sciences, web page, accessed 12-I-2007.
22. ^ Stellar Configurations with degenerate Cores, S. Chandrasekhar, The Observatory 57 (1934), pp. 373–377.
23. ^ On the Theory of Stars, in Collected Papers of L. D. Landau, ed. and with an introduction by D. ter Haar, New York: Gordon and Breach, 1965; originally published in Phys. Z. Sowjet. 1 (1932), 285.
24. ^ Meeting of the Royal Astronomical Society, Friday, 1935 January 11, The Observatory 58 (February 1935), pp. 33–41.
25. ^ On "Relativistic Degeneracy", Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 95 (1935), 194–206.
26. ^ a b c Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes, Arthur I. Miller, Boston, New York: Houghton Mifflin, 2005, ISBN 0-618-34151-X; reviewed at The Guardian: The battle of black holes.
27. ^ The International Astronomical Union meeting in Paris, 1935, The Observatory 58 (September 1935), pp. 257–265, at p. 259.
28. ^ Note on "Relativistic Degeneracy", Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 96 (November 1935), 20–21.
29. ^ The Pressure of a Degenerate Electron Gas and Related Problems, Arthur Eddington, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 152 (November 1, 1935), pp. 253–272.
30. ^ Relativity Theory of Protons and Electrons, Sir Arthur Eddington, Cambridge: Cambridge University Press, 1936, chapter 13.
31. ^ The physics of white dwarf matter, Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 100 (June 1940), pp. 582–594.
32. ^ Fundamental Theory, Sir A. S. Eddington, Cambridge: Cambridge University Press, 1946, §43–45.
33. ^ a b c The evolution and explosion of massive stars, S. E. Woosley, A. Heger, and T. A. Weaver, Reviews of Modern Physics 74, #4 (October 2002), pp. 1015–1071.
34. ^ Strange quark matter in stars: a general overview, Jürgen Schaffner-Bielich, Journal of Physics G: Nuclear and Particle Physics 31, #6 (2005), pp. S651–S657; also arXiv astro-ph/0412215.
35. ^ The Physics of Neutron Stars, by J. M. Lattimer and M. Prakash, Science 304, #5670 (2004), pp. 536–542; also arXiv astro-ph/0405262.
36. ^ a b Type IA Supernova Explosion Models, Wolfgang Hillebrandt and Jens C. Niemeyer, Annual Review of Astronomy and Astrophysics 38 (2000), pp. 191–230.
37. ^ The weirdest Type Ia supernova yet, LBL press release, web page accessed 13-I-2007.
38. ^ Champagne Supernova Challenges Ideas about How Supernovae Work, web page, spacedaily.com, accessed 13-I-2007.
39. ^ The type Ia supernova SNLS-03D3bb from a super-Chandrasekhar-mass white dwarf star, D. Andrew Howell et al., Nature 443 (September 21, 2006), pp. 308–311; also, arXiv:astro-ph/0609616.

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Degenerate matter is matter which has sufficiently high density that the dominant contribution to its pressure arises from the Pauli exclusion principle. ... This article is in need of attention from an expert on the subject. ...

Results from FactBites:

 Chandrasekhar limit - Wikipedia, the free encyclopedia (745 words) The limit was first discovered and calculated by the Indian physicist Subrahmanyan Chandrasekhar in 1930, during his maiden voyage to Britain from India. The first scientific significance of this limit comes from the fact that he introduced/applied Einstein's special theory of relativity to study/deduce the end stage evolution of stars and the second significance comes from the fact that it prophesied the existence of fascinating stellar phenomena, albeit not characterized further. The Chandrasekhar limit arises from taking account of the effects of quantum mechanics in considering the behaviour of the electrons providing the degeneracy pressure supporting the white dwarf.
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