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The Chandrasekhar limit () is the maximum mass of a stable
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes ...
star. The currently accepted value of the Chandrasekhar limit is about (). White dwarfs resist gravitational collapse primarily through
electron degeneracy pressure Electron degeneracy pressure is a particular manifestation of the more general phenomenon of quantum degeneracy pressure. The Pauli exclusion principle disallows two identical half-integer spin particles (electrons and all other fermions) from si ...
, compared to main sequence stars, which resist collapse through thermal pressure. The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction. Consequently, a white dwarf with a mass greater than the limit is subject to further gravitational collapse,
evolving Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation t ...
into a different type of stellar remnant, such as a
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
or black hole. Those with masses up to the limit remain stable as white dwarfs.Sean Carroll, Ph.D., Caltech, 2007, The Teaching Company, ''Dark Matter, Dark Energy: The Dark Side of the Universe'', Guidebook Part 2 page 44, Accessed Oct. 7, 2013, "...Chandrasekhar limit: The maximum mass of a white dwarf star, about 1.4 times the mass of the Sun. Above this mass, the gravitational pull becomes too great, and the star must collapse to a neutron star or black hole..." Tolman–Oppenheimer–Volkoff limit is theoretically a next level to reach in order for a neutron star to collapse into a denser form such as a black hole. The limit was named after Subrahmanyan Chandrasekhar. Chandrasekhar improved upon the accuracy of the calculation in 1930 by calculating the limit for a polytrope model of a star in hydrostatic equilibrium, and comparing his limit to the earlier limit found by E. C. Stoner for a uniform density star. Importantly, the existence of a limit, based on the conceptual breakthrough of combining relativity with Fermi degeneracy, was indeed first established in separate papers published by Wilhelm Anderson and E. C. Stoner in 1929. The limit was initially ignored by the community of scientists because such a limit would logically require the existence of black holes, which were considered a scientific impossibility at the time. The fact that the roles of Stoner and Anderson are often overlooked in the astronomy community has been noted.


Physics

Electron degeneracy pressure is a quantum-mechanical effect arising from the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
. Since
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s 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 Band or BAND may refer to: Places *Bánd, a village in Hungary *Band, Iran, a village in Urmia County, West Azerbaijan Province, Iran * Band, Mureș, a commune in Romania *Band-e Majid Khan, a village in Bukan County, West Azerbaijan Province, I ...
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 increases on compression, so pressure must be exerted on the electron gas to compress it, producing electron degeneracy pressure. With sufficient compression, electrons are forced into nuclei in the process of
electron capture Electron capture (K-electron capture, also K-capture, or L-electron capture, L-capture) is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shells. Thi ...
, relieving the pressure. In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form , where is the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, is the mass density, and is a constant. Solving the hydrostatic equation leads to a model white dwarf that is a polytrope of index – and therefore has radius inversely proportional to the cube root of its mass, and volume inversely proportional to its mass. 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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
must be taken into account. In the strongly relativistic limit, the equation of state takes the form . This yields a polytrope of index 3, which has a total mass, , depending only on . For a fully relativistic treatment, the equation of state used interpolates between the equations for small and for large . When this is done, the model radius still decreases with mass, but becomes zero at . This is the Chandrasekhar limit. The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored blue and green, respectively. has been set equal to 2. Radius is measured in standard solar radii''Standards for Astronomical Catalogues, Version 2.0''
, section 3.2.2, web page, accessed 12-I-2007.
or kilometers, and mass in standard solar masses. Calculated values for the limit vary depending on the nuclear composition of the mass. Chandrasekhar, eq. (36),, eq. (58),''On Stars, Their Evolution and Their Stability''
, Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
, eq. (43) gives the following expression, based on the equation of state for an ideal
Fermi gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer sp ...
: M_\text = \frac \left ( \frac\right )^\frac \frac where: * is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
* is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
* is the gravitational constant * is the average molecular weight per electron, which depends upon the chemical composition of the star * is the mass of the
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
atom * is a constant connected with the solution to the Lane–Emden equation As is the Planck mass, the limit is of the order of \frac The limiting mass can be obtained formally from the Chandrasekhar's white dwarf equation by taking the limit of large central density. 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. Lieb and Yau have given a rigorous derivation of the limit from a relativistic many-particle
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
.


History

In 1926, the British
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
Ralph H. Fowler observed that the relationship between the density, energy, and temperature of white dwarfs could be explained by viewing them as a gas of nonrelativistic, non-interacting electrons and nuclei that obey Fermi–Dirac statistics. This
Fermi gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer sp ...
model was then used by the British physicist
Edmund Clifton Stoner Edmund Clifton Stoner FRS (2 October 1899 – 27 December 1968) was a British theoretical physicist. He is principally known for his work on the origin and nature of itinerant ferromagnetism (the type of ferromagnetic behaviour associated with ...
in 1929 to calculate the relationship among the mass, radius, and density of white dwarfs, assuming they were homogeneous spheres. Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately . In 1930, Stoner derived the internal energy
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
equation of state for a Fermi gas, and was then able to treat the mass–radius relationship in a fully relativistic manner, giving a limiting mass of approximately (for ). Stoner went on to derive the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
equation of state, which he published in 1932. These equations of state were also previously published by the Soviet
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
Yakov Frenkel __NOTOC__ Yakov Il'ich Frenkel (russian: Яков Ильич Френкель; 10 February 1894 – 23 January 1952) was a Soviet physicist renowned for his works in the field of condensed matter physics. He is also known as Jacov Frenkel, frequ ...
in 1928, together with some other remarks on the physics of
degenerate matter Degenerate matter is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant pressure in addition to, or in lieu of, thermal pressure. The description applies to matter composed of electrons, protons, n ...
. Frenkel's work, however, was ignored by the astronomical and astrophysical community. 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.Chandrasekhar's biographical memoir at the National Academy of Sciences
, web page, accessed 12-01-2007.
In these papers, Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state, and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above. Chandrasekhar reviews this work in his Nobel Prize lecture. This value was also computed in 1932 by the Soviet physicist
Lev Landau Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet-Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics. His ac ...
, who, however, did not apply it to white dwarfs and concluded that quantum laws might be invalid for stars heavier than 1.5 solar mass. Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the British astrophysicist Arthur 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: Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law universally applicable, even for large . Although
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
, Fowler,
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics ...
, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar.''Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes'', Arthur I. Miller, Boston, New York: Houghton Mifflin, 2005, ; reviewed at ''The Guardian''
The battle of black holes
.
, pp. 110–111 Through the rest of his life, Eddington held to his position in his writings, including his work on his
fundamental theory Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
. The drama associated with this disagreement is one of the main themes of ''Empire of the Stars'', Arthur I. Miller's biography of Chandrasekhar. In Miller's view:


Applications

The core of a star is kept from collapsing by the heat generated by the
fusion Fusion, or synthesis, is the process of combining two or more distinct entities into a new whole. Fusion may also refer to: Science and technology Physics *Nuclear fusion, multiple atomic nuclei combining to form one or more different atomic nucl ...
of nuclei of lighter elements into heavier ones. At various stages of stellar evolution, the nuclei required for this process are exhausted, and the core collapses, causing it to become denser and hotter. A critical situation arises when
iron Iron () is a chemical element with Symbol (chemistry), symbol Fe (from la, Wikt:ferrum, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 element, group 8 of the periodic table. It is, Abundanc ...
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. If a main-sequence star is not too massive (less than approximately 8 solar masses), it eventually sheds enough mass to form a white dwarf having mass below the Chandrasekhar limit, which consists of the former core of the star. For more-massive stars, electron degeneracy pressure does not keep the iron core from collapsing to very great density, leading to formation of a
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
, black hole, or, speculatively, a
quark star A quark star is a hypothetical type of compact, exotic star, where extremely high core temperature and pressure has forced nuclear particles to form quark matter, a continuous state of matter consisting of free quarks. Background Some massive ...
. (For very massive, low- metallicity stars, it is also possible that instabilities destroy the star completely.)Kurtis A. Williams, M. Bolte, and Detlev Koester 200
An Empirical Initial-Final Mass Relation from Hot, Massive White Dwarfs in NGC 2168 (M35)
, ''Astrophysical Journal'' 615, pp. L49–L5
arXiv astro-ph/0409447
.
During the collapse,
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
s are formed by the capture of
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s by protons in the process of
electron capture Electron capture (K-electron capture, also K-capture, or L-electron capture, L-capture) is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shells. Thi ...
, leading to the emission of
neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s., pp. 1046–1047. The decrease in
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
of the collapsing core releases a large amount of energy on the order of 1046 
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applie ...
s (100  foes). Most of this energy is carried away by the emitted neutrinos and the kinetic energy of the expanding shell of gas; only about 1% is emitted as optical light. This process is believed responsible for supernovae of types Ib, Ic, and II. Type Ia supernovae derive their energy from runaway fusion of the nuclei in the interior of a
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes ...
. This fate may befall
carbon Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent—its atom making four electrons available to form covalent chemical bonds. It belongs to group 14 of the periodic table. Carbon mak ...
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as ...
white dwarfs that accrete matter from a companion
giant star A giant star is a star with substantially larger radius and luminosity than a main-sequence (or ''dwarf'') star of the same surface temperature.Giant star, entry in ''Astronomy Encyclopedia'', ed. Patrick Moore, New York: Oxford University Press ...
, leading to a steadily increasing mass. As the white dwarf's mass approaches the Chandrasekhar limit, its central density increases, and, as a result of
compression Compression may refer to: Physical science *Compression (physics), size reduction due to forces *Compression member, a structural element such as a column *Compressibility, susceptibility to compression * Gas compression *Compression ratio, of a ...
al heating, its temperature also increases. This eventually ignites
nuclear fusion Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is manife ...
reactions, leading to an immediate
carbon detonation Carbon detonation or carbon deflagration is the violent reignition of thermonuclear fusion in a white dwarf star that was previously slowly cooling. It involves a runaway thermonuclear process which spreads through the white dwarf in a matter of se ...
, which disrupts the star and causes the supernova., §5.1.2 A strong indication of the reliability of Chandrasekhar's formula is that the
absolute magnitude Absolute magnitude () is a measure of the luminosity of a celestial object on an inverse logarithmic astronomical magnitude scale. An object's absolute magnitude is defined to be equal to the apparent magnitude that the object would have if it ...
s of supernovae of Type Ia are all approximately the same; at maximum luminosity, is approximately −19.3, with a standard deviation of no more than 0.3., (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.


Super-Chandrasekhar mass supernovas

In 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 The University of Toronto (UToronto or U of T) is a public university, public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park (Toronto), Queen's Park. It was founded by royal charter in 1827 ...
and elsewhere, the observations of this supernova are best explained by assuming that it arose from a white dwarf that had grown to twice the mass of the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
before exploding. They believe that the star, dubbed the "
Champagne Supernova "Champagne Supernova" is a song by English rock band Oasis, written by Noel Gallagher. It is the closing track on the band's second studio album, ''(What's the Story) Morning Glory?'' (1995), and was released as the sixth and final single from ...
" may have been spinning so fast that a centrifugal tendency 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. Since the observation of the Champagne Supernova in 2003, several more type Ia supernovae have been observed that are very bright, and thought to have originated from
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes ...
s whose masses exceeded the Chandrasekhar limit. These include SN 2006gz, SN 2007if, and SN 2009dc. The super-Chandrasekhar mass white dwarfs that gave rise to these supernovae are believed to have had masses up to 2.4–2.8  solar masses. One way to potentially explain the problem of the Champagne Supernova was considering it the result of an aspherical explosion of a white dwarf. However, spectropolarimetric observations of SN 2009dc showed it had a polarization smaller than 0.3, making the large asphericity theory unlikely.


Tolman–Oppenheimer–Volkoff limit

After a supernova explosion, a
neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
may be left behind (except Ia type supernova explosion, which never leaves any remnants behind). These objects are even more compact than white dwarfs and are also supported, in part, by degeneracy pressure. A neutron star, however, is so massive and compressed that electrons and protons have combined to form neutrons, and the star is thus supported by neutron degeneracy pressure (as well as short-range repulsive neutron-neutron interactions mediated by the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
) instead of electron degeneracy pressure. The limiting value for neutron star mass, analogous to the Chandrasekhar limit, is known as the Tolman–Oppenheimer–Volkoff limit.


See also

* Bekenstein bound * Chandrasekhar's white dwarf equation * Schönberg–Chandrasekhar limit


References


Further reading


''On Stars, Their Evolution and Their Stability''
Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.

Masters' thesis, Dave Gentile, DePaul University, 1995.
Estimating Stellar Parameters from Energy Equipartition
sciencebits.com. Discusses how to find mass-radius relations and mass limits for white dwarfs using simple energy arguments. {{DEFAULTSORT:Chandrasekhar Limit Astrophysics White dwarfs Neutron stars Stellar dynamics