The Chandrasekhar limit limits the mass of bodies made from electron-degenerate matter, a dense form of matter which consists of nuclei immersed in a gas of electrons. Degenerate matter is matter which has sufficiently high Density that the dominant contribution to its Pressure rises from the Pauli Exclusion The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The limit is the maximum nonrotating mass which can be supported against gravitational collapse by electron degeneracy pressure. Electron degeneracy pressure is a consequence of the Pauli exclusion principle, which states that two Fermions cannot occupy the same Quantum state at the It is named after the astrophysicist Subrahmanyan Chandrasekhar, and is commonly given as being about 1. Astrophysics is the branch of Astronomy that deals with the Physics of the Universe, including the physical properties ( Luminosity, Padma Vibhushan Subrahmanyan Chandrasekhar, FRS ( Tamil: சுப்பிரமணியன் சந்திரசேகர் English ˌtʃʌndrəˈʃeɪkɑr( 4^[1][2] solar masses. The solar mass is a standard way to express Mass in Astronomy, used to describe the masses of other Stars and galaxies. As white dwarfs are composed of electron-degenerate matter, no nonrotating white dwarf can be heavier than the Chandrasekhar limit. A white dwarf, also called a degenerate dwarf, is a small Star composed mostly of Electron-degenerate matter.

Stars produce energy through nuclear fusion, producing heavier elements from lighter ones. A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth In Physics and Nuclear chemistry, nuclear fusion is the process by which multiple- like charged atomic nuclei join together to form a heavier nucleus A chemical element is a type of Atom that is distinguished by its Atomic number; that is by the number of Protons in its nucleus. The heat generated from these reactions prevents gravitational collapse of the star. Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity. Over time, the star builds up a central core which consists of elements which the temperature at the center of the star is not sufficient to fuse. For main-sequence stars with a mass below approximately 8 solar masses, the mass of this core will remain below the Chandrasekhar limit, and they will eventually lose mass (as planetary nebulae) until only the core, which becomes a white dwarf, remains. The main sequence is the name for a continuous and distinctive band of stars that appear on a plot of stellar color versus brightness A planetary nebula is an Emission nebula consisting of a glowing shell of Gas and plasma formed by certain types of Stars when they die A white dwarf, also called a degenerate dwarf, is a small Star composed mostly of Electron-degenerate matter. Stars with higher mass will develop a degenerate core whose mass will grow until it exceeds the limit. At this point the star will explode in a core-collapse supernova, leaving behind either a neutron star or a black hole. Type II Supernova, or core-collapse supernova, is a sub-category of cataclysmic Variable stars that results from the internal collapse and violent explosion A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e ^[3][4][5]

Computed values for the limit will vary depending on the approximations used, the nuclear composition of the mass, and the temperature. The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom ^[6] Chandrasekhar^{[7], eq. (36),[8], eq. (58),[9], eq. (43)} gives a value of

Here, μ_e is the average molecular weight per electron, m_H is the mass of the hydrogen atom, and ω₃⁰≈2. The molecular mass (abbreviated m of a substance, more commonly referred to as molecular weight and abbreviated as MW, is the Mass of one Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1 History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny 018236 is a constant connected with the solution to the Lane-Emden equation. In Astrophysics, the Lane-Emden equation is Poisson's equation for the gravitational potential of a self-gravitating spherically symmetric Polytropic Numerically, this value is approximately (2/μ_e)² · 2. 85 · 10³⁰ kg, or 1. 43 (2/μ_e)² M_☉, where M_☉=1. 989·10³⁰ kg is the standard solar mass. The solar mass is a standard way to express Mass in Astronomy, used to describe the masses of other Stars and galaxies. ^[10] As is the Planck mass, M_Pl≈2. The Planck mass is the unit of Mass, denoted by m P in the system of Natural units known as Planck units. 176·10⁻⁸ kg, the limit is of the order of M_Pl³/m_H².

Physics

Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 Since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. 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.

Radius-mass relations for a model white dwarf. The green curve uses the general pressure law for an ideal Fermi gas, while the blue curve is for a non-relativistic ideal Fermi gas. A Fermi gas, or Free electron gas, is a collection of non-interacting Fermions. The black line marks the ultra-relativistic limit.

In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form P=K₁ρ^5/3. In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic 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. In Astrophysics, a polytrope refers to a solution of the Lane-Emden equation in which the Pressure depends upon the Density in the form P ^[11]

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. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In the strongly relativistic limit, we find that the equation of state takes the form P=K₂ρ^4/3. This will yield a polytrope of index 3, which will have a total mass, M_limit say, depending only on K₂. ^[12]

For a fully relativistic treatment, the equation of state used will interpolate between the equations P=K₁ρ^5/3 for small ρ and P=K₂ρ^4/3 for large ρ. When this is done, the model radius still decreases with mass, but becomes zero at M_limit. This is the Chandrasekhar limit. ^[8] 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. μ_e has been set equal to 2. Radius is measured in standard solar radii^[10] or kilometers, and mass in standard solar masses. The kilometre ( American spelling: kilometer) symbol km is a unit of Length in the Metric system, equal to one thousand

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. ^[6] Lieb and Yau^[13] have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system

History

In 1926, the British physicist 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 which obeyed Fermi-Dirac statistics. The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning Sir Ralph Howard Fowler OBE FRS ( January 17 1889 &ndash July 28 1944) was a British Physicist and Astronomer In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that ^[14] This Fermi gas model was then used by the British physicist E. C. Stoner in 1929 to calculate the relationship between the mass, radius, and density of white dwarfs, assuming them to be homogenous spheres. A Fermi gas, or Free electron gas, is a collection of non-interacting Fermions. Edmund Clifton Stoner ( October 2, 1899, in Surrey, England – December 27, 1968 in Leeds, England ^[15] Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1. Wilhelm Robert Karl Anderson ( 29 October 1880, Minsk, Belarus - 26 March 1940, Międzyrzecz) was an Estonian 37×10³⁰ kg. ^[16] In 1930, Stoner derived the internal energy-density 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 μ_e=2. In Thermodynamics, the internal energy of a Thermodynamic system, or a body with well-defined boundaries, denoted by  U, or sometimes  The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic 5) 2. 19 · 10³⁰ kg. ^[17] Stoner went on to derive the pressure-density equation of state, which he published in 1932. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different ^[18] These equations of state were also previously published by the Russian physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter. Russia (Россия Rossiya) or the Russian Federation ( Rossiyskaya Federatsiya) is a transcontinental Country extending A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning Yakov Il'ich Frenkel, Яков Ильич Френкель ( February 10, 1894, Rostov-on-Don – January 23, 1952, Leningrad ^[19] Frenkel's work, however, was ignored by the astronomical and astrophysical community. ^[20]

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. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland A non-resident Indian (NRI is an Indian citizen who has migrated to another country a person of Indian origin who is born outside India or a person of A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning Padma Vibhushan Subrahmanyan Chandrasekhar, FRS ( Tamil: சுப்பிரமணியன் சந்திரசேகர் English ˌtʃʌndrəˈʃeɪkɑr( ^[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. Fluid statics (also called hydrostatics) is the Science of Fluids at rest and is a sub-field within Fluid mechanics. In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic ^{[12][7][22][8]} Chandrasekhar reviews this work in his Nobel Prize lecture. ^[9] This value was also computed in 1932 by the Soviet physicist Lev Davidovich Landau,^[23] who, however, did not apply it to white dwarfs. Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968

Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the British astrophysicist Arthur Stanley Eddington. The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located Astrophysics is the branch of Astronomy that deals with the Physics of the Universe, including the physical properties ( Luminosity, Sir Arthur Stanley Eddington, OM (28 December 1882 – 22 November 1944 was an English Astrophysicist of the early 20th century 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. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e However, he was unwilling to accept that this could happen. After a talk by Chandrasekhar on the limit in 1935, he replied:

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=K₁ρ^5/3 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. Niels Henrik David Bohr (nels ˈb̥oɐ̯ˀ in Danish 7 October 1885 – 18 November 1962 was a Danish Physicist who made fundamental contributions to understanding ^{[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. Sir Arthur Stanley Eddington, OM (28 December 1882 – 22 November 1944 was an English Astrophysicist of the early 20th century ^[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:

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. In Physics and Nuclear chemistry, nuclear fusion is the process by which multiple- like charged atomic nuclei join together to form a heavier nucleus The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom A chemical element is a type of Atom that is distinguished by its Atomic number; that is by the number of Protons in its nucleus. 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. Iron (ˈаɪɚn is a Chemical element with the symbol Fe (ferrum and Atomic number 26 If the core becomes sufficiently dense, electron degeneracy pressure will play a significant part in stabilizing it against gravitational collapse. ^[33]

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. The solar mass is a standard way to express Mass in Astronomy, used to describe the masses of other Stars and galaxies. A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e A quark star or strange star is a hypothetical type of Exotic star composed of Quark matter, or Strange matter. (For very massive, low-metallicity stars, it is also possible that instabilities will destroy the star completely. In Astronomy and Physical cosmology, the metallicity of an object is the proportion of its matter made up of Chemical elements other than Hydrogen )^{[3][4][5][34]} During the collapse, neutrons are formed by the capture of electrons by protons, leading to the emission of neutrinos. This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive Neutrinos are Elementary particles that travel close to the Speed of light, lack an Electric charge, are able to pass through ordinary matter almost ^{[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. The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity A foe is a unit of Energy equal to 1044 Joules or 1051 Ergs used to measure the large amount of energy produced by a Supernova ) 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. Type II Supernova, or core-collapse supernova, is a sub-category of cataclysmic Variable stars that results from the internal collapse and violent explosion ^[33]

Type Ia supernovae derive their energy from runaway fusion of the nuclei in the interior of a white dwarf. A Type Ia supernova is a sub-category of cataclysmic Variable A white dwarf, also called a degenerate dwarf, is a small Star composed mostly of Electron-degenerate matter. This fate may befall carbon-oxygen white dwarfs that accrete matter from a companion giant star, leading to a steadily increasing mass. Carbon (kɑɹbən is a Chemical element with the symbol C and its Atomic number is 6 Oxygen (from the Greek roots ὀξύς (oxys (acid literally "sharp" from the taste of acids and -γενής (-genēs (producer literally begetteris the A giant star is a Star with substantially larger Radius and Luminosity than a Main sequence star of the same surface temperature. It is believed that, as the white dwarf's mass approaches the Chandrasekhar limit, its central density increases, and, as a result of compressional heating, its temperature also increases. Physical compression is the result of the subjection of a material to Compressive stress, resulting in reduction of Volume. This results in an increasing rate of fusion reactions, eventually igniting a thermonuclear flame which causes the supernova. In Physics and Nuclear chemistry, nuclear fusion is the process by which multiple- like charged atomic nuclei join together to form a heavier nucleus In Physics and Nuclear chemistry, nuclear fusion is the process by which multiple- like charged atomic nuclei join together to form a heavier nucleus ^{[36], §5. 1. 2}

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, M_V is approximately -19. 3, with a standard deviation of no more than 0. In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values 3. ^{[36], (1)} A 1-sigma interval therefore represents a factor of less than 2 in luminosity. In Statistics, a confidence interval (CI is an interval estimate of a Population parameter. This seems to indicate that all type Ia supernovae convert approximately the same amount of mass to energy.

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

Main article: Champagne Supernova. The SN 2003fg (designated SNLS-03D3bb by the Canada-France-Hawaii Supernova Legacy Survey which discovered it and sometimes called the "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. The Supernova Legacy Survey Program is a project designed to investigate Dark energy, by detecting and monitoring approximately 2000 high- Redshift Supernovae The SN 2003fg (designated SNLS-03D3bb by the Canada-France-Hawaii Supernova Legacy Survey which discovered it and sometimes called the "Champagne Supernova" A light-year or light year (symbol ly) is a unit of Length, equal to just under ten trillion Kilometres As defined by 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. This article is about the University of Toronto's St George Campus The Sun (Sol is the Star at the center of the Solar System. They believe that the star, dubbed the "Champagne Supernova" by David R. The SN 2003fg (designated SNLS-03D3bb by the Canada-France-Hawaii Supernova Legacy Survey which discovered it and sometimes called the "Champagne Supernova" 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. A standard candle is an astronomical object that has a known Luminosity. ^[37][38][39]

References

Further reading

• On Stars, Their Evolution and Their Stability, Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983. Events 1609 - Biblioteca Ambrosiana opens its reading room the second public library of Europe. Year 1983 ( MCMLXXXIII) was a Common year starting on Saturday (link displays the 1983 Gregorian calendar)
• White dwarf stars and the Chandrasekhar limit, Masters' thesis, Dave Gentile, DePaul University, 1995. DePaul University is a private institution of Higher education and Research in Chicago, Illinois, U
• 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.

See also

• Degenerate matter
• Schönberg-Chandrasekhar limit
• Tolman-Oppenheimer-Volkoff limit