Chandrasekhar limit
Encyclopedia
When a star starts running out of fuel, it usually cools off and collapses (possibly with a supernova
) into one of three compact forms, depending on its total mass:
The Chandrasekhar limit (tʃʌndrəˈʃeɪkɑr) is the maximum mass of a stable
white dwarf
star
. It was named after Subrahmanyan Chandrasekhar
, the Indian astrophysicist who predicted it in 1930. White dwarfs, unlike main sequence
stars, resist gravitational collapse
primarily through electron degeneracy pressure
, rather than 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, white dwarfs with masses greater than the limit undergo further gravitational collapse, evolving
into a different type of stellar remnant
, such as a neutron star
or black hole
. Those with masses under the limit remain stable as white dwarfs. The Chandrasekhar limit is analogous to the Tolman–Oppenheimer–Volkoff limit for neutron stars.
The currently accepted numerical value of the limit is about 1.4
, or 2.864 × 1030 kg.
effect arising from the Pauli exclusion principle
. Since electron
s are fermion
s, 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, producing electron degeneracy pressure. With sufficient compression, electrons are forced into nuclei in the process of electron capture
, relieving the pressure.
In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state
of the form (with
being the number density of the electrons) 
. 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.
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
. This will yield a polytrope of index 3, which will have a total mass, Mlimit say, depending only on K2.
For a fully relativistic treatment, the equation of state used will interpolate between the equations
for small ρ and 
for large ρ.
When this is done, the model radius still decreases with mass, but becomes zero at Mlimit. 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. μe has been set equal to 2.
Radius is measured in standard solar radii or kilometers, and mass in standard solar masses.
Calculated values for the limit will vary depending on the nuclear
composition of the mass. Chandrasekhar, eq. (36),, eq. (58),, eq. (43) gives the following expression, based on the equation of state
for an ideal Fermi gas
:
where:
As
is the Planck mass, the limit is of the order of 
.
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
.
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
. 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. Wilhelm Anderson
applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37 kg. 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 μe=2.5) 2.19 · 1030 kg. Stoner went on to derive the pressure
-density
equation of state, which he published in 1932. These equations of state were also previously published by the Soviet
physicist
Yakov Frenkel
in 1928, together with some other remarks on the physics of degenerate matter. 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. 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 Davidovich Landau, who, however, did not apply it to white dwarfs.
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 hole
s 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 P=K1ρ5/3 universally applicable, even for large ρ. 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., pp. 110–111 Through the rest of his life, Eddington held to his position in his writings, including his work on his fundamental theory. 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:
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.
If a main-sequence star is not too massive (less than approximately 8 solar mass
es), 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.) During the collapse, neutron
s are formed by the capture of electron
s by proton
s in the process of electron capture
, leading to the emission of neutrino
s., 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 joule
s (100 foes). Most of this energy is carried away by the emitted neutrinos. This process is believed to be responsible for supernovae of types Ib, Ic, and II.
Type Ia supernova
e derive their energy from runaway fusion of the nuclei in the interior of a white dwarf
. This fate may befall carbon
-oxygen
white dwarfs that accrete matter from a companion giant star
, leading to a steadily increasing mass. It has been inferred 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. This results in an increasing rate of fusion
reactions, eventually igniting a thermonuclear flame (carbon detonation
) which causes the supernova., §5.1.2
Strong indications of the reliability of Chandrasekhar's formula are:
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 University of Oklahoma astronomer 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
.
Supernova
A supernova is a stellar explosion that is more energetic than a nova. It is pronounced with the plural supernovae or supernovas. Supernovae are extremely luminous and cause a burst of radiation that often briefly outshines an entire galaxy, before fading from view over several weeks or months...
) into one of three compact forms, depending on its total mass:
- a White DwarfWhite dwarfA white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
, a big lump of CarbonCarbonCarbon is the chemical element with symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalent—making four electrons available to form covalent chemical bonds...
and OxygenOxygenOxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...
atoms, almost like one huge molecule. It glows only because it's still hot and cooling off - there's rarely any fusion going on. - a Neutron StarNeutron starA neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...
, a big lump of neutrons, almost like one huge nucleus. To get here, it must have been big enough to burn carbonCarbon burning processThe carbon-burning process or carbon fusion is a set of nuclear fusion reactions that take place in massive stars that have used up the lighter elements in their cores...
and create larger nuclei, then end with a supernova. - a Black HoleBlack holeA black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
, if it's very massive.
The Chandrasekhar limit (tʃʌndrəˈʃeɪkɑr) is the maximum mass of a stable
Hydrostatic equilibrium
Hydrostatic equilibrium or hydrostatic balance is the condition in fluid mechanics where a volume of a fluid is at rest or at constant velocity. This occurs when compression due to gravity is balanced by a pressure gradient force...
white dwarf
White dwarf
A white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
. It was named after Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar, FRS ) was an Indian origin American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for Physics for key discoveries that led to the currently accepted theory on the later evolutionary stages of massive stars...
, the Indian astrophysicist who predicted it in 1930. White dwarfs, unlike main sequence
Main sequence
The main sequence is a continuous and distinctive band of stars that appears on plots of stellar color versus brightness. These color-magnitude plots are known as Hertzsprung–Russell diagrams after their co-developers, Ejnar Hertzsprung and Henry Norris Russell...
stars, resist gravitational collapse
Gravitational collapse
Gravitational collapse is the inward fall of a body due to the influence of its own gravity. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body, in the opposite direction to the force of gravity...
primarily through electron degeneracy pressure
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 half integer spin particles from occupying the same quantum state at a given time. The resulting emergent repulsive force is...
, rather than 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, white dwarfs with masses greater than the limit undergo further gravitational collapse, evolving
Stellar evolution
Stellar evolution is the process by which a star undergoes a sequence of radical changes during its lifetime. Depending on the mass of the star, this lifetime ranges from only a few million years to trillions of years .Stellar evolution is not studied by observing the life of a single...
into a different type of stellar remnant
Compact star
In astronomy, the term compact star is used to refer collectively to white dwarfs, neutron stars, other exotic dense stars, and black holes. These objects are all small for their mass...
, such as a neutron star
Neutron star
A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...
or black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
. Those with masses under the limit remain stable as white dwarfs. The Chandrasekhar limit is analogous to the Tolman–Oppenheimer–Volkoff limit for neutron stars.
The currently accepted numerical value of the limit is about 1.4
, or 2.864 × 1030 kg.Physics
Electron degeneracy pressure is a quantum-mechanicalQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
effect arising from the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
. Since electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s are fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s, 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, producing electron degeneracy pressure. With sufficient compression, electrons are forced into nuclei in the process of electron capture
Electron capture
Electron capture is a process in which a proton-rich nuclide absorbs an inner atomic electron and simultaneously emits a neutrino...
, relieving the pressure.
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
of the form (with
being the number density of the electrons) . 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.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
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
must be taken into account. In the strongly relativistic limit, we find that the equation of state takes the form
. This will yield a polytrope of index 3, which will have a total mass, Mlimit say, depending only on K2.For a fully relativistic treatment, the equation of state used will interpolate between the equations
for small ρ and for large ρ.When this is done, the model radius still decreases with mass, but becomes zero at Mlimit. 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. μe has been set equal to 2.
Radius is measured in standard solar radii or kilometers, and mass in standard solar masses.
Calculated values for the limit will vary depending on the nuclear
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
composition of the mass. Chandrasekhar, eq. (36),, eq. (58),, eq. (43) gives the following expression, based on the equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
for an ideal Fermi gas
Fermi gas
A Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
:
where:
is the reduced Planck constant- c is the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
- G is the gravitational constantGravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
- μe is the average molecular weight per electron, which depends upon the chemical composition of the star.
- mH is the mass of the hydrogenHydrogenHydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
atomAtomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
. 
is a constant connected with the solution to the Lane-Emden equationLane-Emden equationIn astrophysics, the Lane–Emden equation is Poisson's equation for the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid. It is named after the astrophysicists Jonathan Homer Lane and Robert Emden...
.
As
is the Planck mass, the limit is of the order of .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
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
.
History
In 1926, the BritishUnited Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Ralph H. Fowler
Ralph H. Fowler
Sir Ralph Howard Fowler OBE FRS was a British physicist and astronomer.-Education:Fowler was initially educated at home but then attended Evans' preparatory school at Horris Hill and Winchester College...
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
Fermi-Dirac statistics
Fermi–Dirac statistics is a part of the science of physics that describes the energies of single particles in a system comprising many identical particles that obey the Pauli Exclusion Principle...
. This Fermi gas
Fermi gas
A Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
model was then used by the British physicist E. C. Stoner
Edmund Clifton Stoner
Edmund Clifton Stoner was a British theoretical physicist. He is principally known for his work on the origin and nature of itinerant ferromagnetism , including the collective electron theory of ferromagnetism...
in 1929 to calculate the relationship between the mass, radius, and density of white dwarfs, assuming them to be homogenous spheres. Wilhelm Anderson
Wilhelm Anderson
Wilhelm Robert Karl Anderson was a German-Estonian astrophysicist who studied the physical structure of the stars.- Life :...
applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37 kg. In 1930, Stoner derived the internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
-density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
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 μe=2.5) 2.19 · 1030 kg. Stoner went on to derive the pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
-density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
equation of state, which he published in 1932. These equations of state were also previously published by the Soviet
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Yakov Frenkel
Yakov Frenkel
Yakov Il'ich Frenkel, was a Soviet physicist renowned for his works in the field of solid-state physics. He is also known as Jacov Frenkel....
in 1928, together with some other remarks on the physics of degenerate matter. 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
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
to England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
in 1930,
where the Indian physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar, FRS ) was an Indian origin American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for Physics for key discoveries that led to the currently accepted theory on the later evolutionary stages of massive stars...
worked on the calculation of the statistics of a degenerate Fermi gas. In these papers, Chandrasekhar solved
the hydrostatic equation together with the nonrelativistic Fermi gas equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
, 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 Davidovich Landau, who, however, did not apply it to white dwarfs.
Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the British
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
astrophysicist Arthur Stanley Eddington
Arthur Stanley Eddington
Sir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...
. Eddington was aware that the existence of black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
s 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 P=K1ρ5/3 universally applicable, even for large ρ. Although Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...
, Fowler, Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...
, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar., pp. 110–111 Through the rest of his life, Eddington held to his position in his writings, including his work on his fundamental theory. 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 fusionNuclear fusion
Nuclear fusion is the process by which two or more atomic nuclei join together, or "fuse", to form a single heavier nucleus. This is usually accompanied by the release or absorption of large quantities of energy...
of nuclei
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
of lighter elements
Chemical element
A chemical element is a pure chemical substance consisting of one type of atom distinguished by its atomic number, which is the number of protons in its nucleus. Familiar examples of elements include carbon, oxygen, aluminum, iron, copper, gold, mercury, and lead.As of November 2011, 118 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
Iron
Iron is a chemical element with the symbol Fe and atomic number 26. It is a metal in the first transition series. It is the most common element forming the planet Earth as a whole, forming much of Earth's outer and inner core. It is the fourth most common element in the Earth's crust...
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 mass
Solar mass
The solar mass , , is a standard unit of mass in astronomy, used to indicate the masses of other stars and galaxies...
es), 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
Neutron star
A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...
, black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
, or, speculatively, a quark star
Quark star
A quark star or strange star is a hypothetical type of exotic star composed of quark matter, or strange matter. These are ultra-dense phases of degenerate matter theorized to form inside particularly massive neutron stars....
. (For very massive, low-metallicity
Metallicity
In astronomy and physical cosmology, the metallicity of an object is the proportion of its matter made up of chemical elements other than hydrogen and helium...
stars, it is also possible that instabilities will destroy the star completely.) During the collapse, neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s are formed by the capture of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s by proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s in the process of electron capture
Electron capture
Electron capture is a process in which a proton-rich nuclide absorbs an inner atomic electron and simultaneously emits a neutrino...
, leading to the emission of neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
s., 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 joule
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
s (100 foes). Most of this energy is carried away by the emitted neutrinos. This process is believed to be responsible for supernovae of types Ib, Ic, and II.
Type Ia supernova
Type Ia supernova
A Type Ia supernova is a sub-category of supernovae, which in turn are a sub-category of cataclysmic variable stars, that results from the violent explosion of a white dwarf star. A white dwarf is the remnant of a star that has completed its normal life cycle and has ceased nuclear fusion...
e derive their energy from runaway fusion of the nuclei in the interior of a white dwarf
White dwarf
A white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
. This fate may befall carbon
Carbon
Carbon is the chemical element with symbol C and atomic number 6. As a member of group 14 on the periodic table, it is nonmetallic and tetravalent—making four electrons available to form covalent chemical bonds...
-oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...
white dwarfs that accrete matter from a companion giant star
Giant star
A giant star is a star with substantially larger radius and luminosity than a main sequence star of the same surface temperature. Typically, giant stars have radii between 10 and 100 solar radii and luminosities between 10 and 1,000 times that of the Sun. Stars still more luminous than giants are...
, leading to a steadily increasing mass. It has been inferred 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. This results in an increasing rate of fusion
Nuclear fusion
Nuclear fusion is the process by which two or more atomic nuclei join together, or "fuse", to form a single heavier nucleus. This is usually accompanied by the release or absorption of large quantities of energy...
reactions, eventually igniting a thermonuclear flame (carbon detonation
Carbon detonation
Carbon detonation is the violent re-ignition of thermonuclear fusion in a dead star, which produces a Type Ia supernova. A white dwarf undergoes carbon detonation only if it has a normal binary companion which is close enough to the dwarf star to dump sufficient amounts of matter onto the dwarf,...
) which causes the supernova., §5.1.2
Strong indications of the reliability of Chandrasekhar's formula are:
- Only one white dwarf with a mass greater than Chandrasekhar's limit has ever been observed. (See below.)
- The absolute magnitudes of supernovae of Type Ia are all approximately the same; at maximum luminosity, MV is approximately -19.3, with a standard deviationStandard deviationStandard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of no more than 0.3., (1) A 1-sigma intervalConfidence intervalIn statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...
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.
Champagne Supernova
In April 2003, the Supernova Legacy SurveySupernova Legacy Survey
The Supernova Legacy Survey Program 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. It also carries out detailed spectroscopy...
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
University of Toronto
The University of Toronto is a public research university in Toronto, Ontario, Canada, situated on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution of higher learning in Upper Canada...
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
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
before exploding. They believe that the star, dubbed the "Champagne Supernova" by University of Oklahoma astronomer 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
Standard Candles
Standard Candles is a compilation of short stories by American science fiction author Jack McDevitt. The sixteen stories in the anthology were originally published in various magazines from 1982 to 1996...
.
See also
- White dwarfWhite dwarfA white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
- Stellar evolutionStellar evolutionStellar evolution is the process by which a star undergoes a sequence of radical changes during its lifetime. Depending on the mass of the star, this lifetime ranges from only a few million years to trillions of years .Stellar evolution is not studied by observing the life of a single...
- Degenerate matterDegenerate matterDegenerate matter is matter that has such extraordinarily high density that the dominant contribution to its pressure is attributable to the Pauli exclusion principle. The pressure maintained by a body of degenerate matter is called the degeneracy pressure, and arises because the Pauli principle...
- Schönberg–Chandrasekhar limit
- Tolman–Oppenheimer–Volkoff limit
Further reading
- On Stars, Their Evolution and Their Stability, Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
- White dwarf stars and the Chandrasekhar limit, Masters' thesis, Dave Gentile, DePaul UniversityDePaul UniversityDePaul University is a private institution of higher education and research in Chicago, Illinois. Founded by the Vincentians in 1898, the university takes its name from the 17th century French priest Saint Vincent de Paul...
, 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.