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Chandrasekhar limit

The Chandrasekhar limit is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about ($2,765,000,000,000,000,000,000,000,000,000 kg$). The limit was named after Subrahmanyan Chandrasekhar.

White dwarfs resist gravitational collapse primarily through electron degeneracy pressure, 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.

Physics
[[Image:WhiteDwarf mass-radius en.svg|thumb|upright=1.95|Radius–mass relations for a model white dwarf.

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Normal stars fuse gravitationally compressed hydrogen into helium, generating vast amounts of heat. As the hydrogen is consumed, the stars' core compresses further allowing the helium and heavier nuclei to fuse ultimately resulting in stable iron nuclei, a process called stellar evolution. The next step depends upon the mass of the star. Stars below the Chandrasekhar limit become stable white dwarf stars, remaining that way throughout the rest of the history of the universe absent external forces. Stars above the limit can become neutron stars or black holes.

The Chandrasekhar limit is a consequence of competition between gravity and electron degeneracy pressure. Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle. Since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energy levels. Compression of the electron gas increases the number of electrons in a given volume and raises the maximum energy level in the occupied band. Therefore, the energy of the electrons 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, relieving the pressure.

In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form $P = K_{1}ρ^{5/3}$, where $P$ is the pressure, $ρ$ is the mass density, and $K_{1}$ is a constant. Solving the hydrostatic equation leads to a model white dwarf that 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, the equation of state takes the form $P = K_{2}ρ^{4/3}$. This yields a polytrope of index 3, which has a total mass, $M_{limit}$, depending only on $K_{2}$.

For a fully relativistic treatment, the equation of state used interpolates between the equations $P = K_{1}ρ^{5/3}$ for small $ρ$ and $P = K_{2}ρ^{4/3}$ 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. 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 or kilometers, and mass in standard solar masses.

Calculated values for the limit vary depending on the nuclear composition of the mass. Chandrasekhar   gives the following expression, based on the equation of state for an ideal Fermi gas: $$ M_\text{limit} = \frac{\omega_3^0 \sqrt{3\pi}}{2} \left ( \frac{\hbar c}{G}\right )^\frac{3}{2} \frac{1}{(\mu_\text{e} m_\text{H})^2}$$ where: As $μ_{e}$ is the Planck mass, the limit is of the order of $$\frac{M_\text{Pl}^3}{m_\text{H}^2}$$ The limiting mass can be obtained formally from the Chandrasekhar's white dwarf equation by taking the limit of large central density.
 * $ħ$ is the reduced Planck constant
 * $c$ is the speed of light
 * $G$ is the gravitational constant
 * $m_{H}$ is the average molecular weight per electron, which depends upon the chemical composition of the star
 * $ω0 3 ≈ 2.018236$ is the mass of the hydrogen atom
 * $√ħc/G$ is a constant connected with the solution to the Lane–Emden equation

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.

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 that obey Fermi–Dirac statistics. This Fermi gas model was then used by the British physicist Edmund Clifton Stoner 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 $1,370,000,000,000,000,000,000,000,000,000 kg$. 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 $2,190,000,000,000,000,000,000,000,000,000 kg$ (for $μ_{e} = 2.5$). 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.

The existence of a related limit, based on the conceptual breakthrough of combining relativity with  Fermi degeneracy, was first established in separate  papers published by Wilhelm Anderson and E. C. Stoner for a  uniform density star in 1929. Eric G. Blackman wrote that roles of Stoner and Anderson in the discovery of mass limits were overlooked when Freeman Dyson wrote a biography of Chandrasekhar. Michael Nauenberg claims that Stoner established the mass limit first. The priority dispute has also been discussed at length by Virginia Trimble who writes that: "Chandrasekhar famously, perhaps even notoriously did his critical calculation on board ship in 1930, and ... was not aware of either Stoner's or Anderson's work at the time. His work was therefore independent, but, more to the point, he adopted Eddington's polytropes for his models which could, therefore, be in hydrostatic equilibrium, which constant density stars cannot, and real ones must be." This value was also computed in 1932 by the Soviet physicist Lev Landau, 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–Eddington dispute
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:

"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!"

Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law $P = K_{1}ρ^{5/3}$ universally applicable, even for large $ρ$. Although Niels Bohr, Fowler, Wolfgang Pauli, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar. 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:

"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."

However, Chandrasekhar chose to move on, leaving the study of stellar structure to focus on stellar dynamics. In 1983 in recognition for his work, Chandrasekhar shared a Nobel prize "for his theoretical studies of the physical processes of importance to the structure and evolution of the stars" with William Alfred Fowler.

Applications
The core of a star is kept from collapsing by the heat generated by the fusion of nuclei of lighter elements into heavier ones. At various 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 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, black hole, or, speculatively, a quark star. (For very massive, low-metallicity stars, it is also possible that instabilities destroy the star completely.)  During the collapse, neutrons are formed by the capture of electrons by protons in the process of electron capture, leading to the emission of neutrinos. The decrease in gravitational potential energy of the collapsing core releases a large amount of energy on the order of $J$ (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. This fate may befall carbon–oxygen white dwarfs that accrete matter from a companion giant star, 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 compressional heating, its temperature also increases. This eventually ignites nuclear fusion reactions, leading to an immediate carbon detonation, which disrupts the star and causes the supernova.

A strong indication of the reliability of Chandrasekhar's formula is that 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 0.3. 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 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 before exploding. They believe that the star, dubbed the "Champagne Supernova" 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 dwarfs 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
Stars sufficiently massive to pass the Chandrasekhar limit provided by electron degeneracy pressure do not become white dwarf stars. Instead they explode as supernovae. If the final mass is below the Tolman–Oppenheimer–Volkoff limit, then neutron degeneracy pressure contributes to the balance against gravity and the end result will be a neutron star; otherwise the result will be a black hole.

title: Chandrasekhar limit


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Congregational polity

Congregational polity, or congregationalist polity, often known as congregationalism, is a system of ecclesiastical polity in which every local church (congregation) is independent, ecclesiastically sovereign, or "autonomous". Its first articulation in writing is the Cambridge Platform of 1648 in New England.

Major Protestant Christian traditions that employ congregationalism include Quakerism, the Baptist churches, the Congregational Methodist Church, and Congregational churches known by the Congregationalist name and having descended from the Independent Reformed wing of the Anglo-American Puritan movement of the 17th century. More recent generations have witnessed a growing number of nondenominational churches, which are often congregationalist in their governance. Although autonomous, like minded congregations may enter into voluntary associations with other congregations, sometimes called conventions, denominations, or associations.

Congregationalism is distinguished from episcopal polity which is governance by a hierarchy of bishops, and is also distinct from presbyterian polity in which higher assemblies of congregational representatives can exercise considerable authority over individual congregations.

Congregationalism is not limited only to organization of Christian church congregations. The principles of congregationalism have been inherited by the Unitarian Universalist Association and the Canadian Unitarian Council.

Basic form
The term congregational polity describes a form of church governance that is based on the local congregation. Each local congregation is independent and self-supporting, governed by its own members. Some band into loose voluntary associations with other congregations that share similar beliefs (e.g., the Willow Creek Association and the Unitarian Universalist Association). Others join "conventions", such as the Southern Baptist Convention, the National Baptist Convention or the American Baptist Churches USA (formerly the Northern Baptist Convention). In Quaker Congregationalism, monthly meetings, which are the most basic unit of administration, may be organized into larger Quarterly meetings or Yearly Meetings. Monthly, quarterly, or yearly meetings may also be associated with large "umbrella" associations such as Friends General Conference or Friends United Meeting. These conventions generally provide stronger ties between congregations, including some doctrinal direction and pooling of financial resources. Congregations that belong to associations and conventions are still independently governed. Most non-denominational churches are organized along congregationalist lines. Many do not see these voluntary associations as "denominations", because they "believe that there is no church other than the local church, and denominations are in variance to Scripture."

Denominational families
These Christian traditions use forms of congregational polity.

Congregational churches
Congregationalism is a Protestant tradition with roots in the Puritan and Independent movements. In congregational government, the covenanted congregation exists prior to its officers, and as such the members are equipped to call and dismiss their ministers without oversight from any higher ecclesiastical body. Their churches ordinarily have at least one pastor, but may also install ruling elders.

Statements of polity in the congregational tradition called "platforms". These include the Savoy Confession's platform, the Cambridge Platform, and the Saybrook Platform. Denominations in the congregational tradition include the UCC, CCCC, and EFCC. Denominations in the tradition support but do not govern their constituent members.

Baptist churches
Most Baptists hold that no denominational or ecclesiastical organization has inherent authority over an individual Baptist church. Churches can properly relate to each other under this polity only through voluntary cooperation, never by any sort of coercion. Furthermore, this Baptist polity calls for freedom from governmental control. Exceptions to this local form of local governance include the Episcopal Baptists that have an episcopal system.

Independent Baptist churches have no formal organizational structure above the level of the local congregation. More generally among Baptists, a variety of parachurch agencies and evangelical educational institutions may be supported generously or not at all, depending entirely upon the local congregation's customs and predilections. Usually doctrinal conformity is held as a first consideration when a church makes a decision to grant or decline financial contributions to such agencies, which are legally external and separate from the congregations they serve. These practices also find currency among non-denominational fundamentalist or charismatic fellowships, many of which derive from Baptist origins, culturally if not theologically.

Most Southern Baptist and National Baptist congregations, by contrast, generally relate more closely to external groups such as mission agencies and educational institutions than do those of independent persuasion. However, they adhere to a very similar ecclesiology, refusing to permit outside control or oversight of the affairs of the local church.

Churches of Christ
Ecclesiastical government is congregational rather than denominational. Churches of Christ purposefully have no central headquarters, councils, or other organizational structure above the local church level. Rather, the independent congregations are a network with each congregation participating at its own discretion in various means of service and fellowship with other congregations. Churches of Christ are linked by their shared commitment to restoration principles.

Congregations are generally overseen by a plurality of elders (also known in some congregations as shepherds, bishops, or pastors) who are sometimes assisted in the administration of various works by deacons. Elders are generally seen as responsible for the spiritual welfare of the congregation, while deacons are seen as responsible for the non-spiritual needs of the church. Deacons serve under the supervision of the elders, and are often assigned to direct specific ministries. Successful service as a deacon is often seen as preparation for the eldership. Elders and deacons are chosen by the congregation based on the qualifications found in Timothy 3 and Titus 1. Congregations look for elders who have a mature enough understanding of scripture to enable them to supervise the minister and to teach, as well as to perform governance functions. In lieu of willing men who meet these qualifications, congregations are sometimes overseen by an unelected committee of the congregation's men.

While the early Restoration Movement had a tradition of itinerant preachers rather than "located Preachers", during the 20th century a long-term, formally trained congregational minister became the norm among Churches of Christ. Ministers are understood to serve under the oversight of the elders. While the presence of a long-term professional minister has sometimes created "significant de facto ministerial authority" and led to conflict between the minister and the elders, the eldership has remained the "ultimate locus of authority in the congregation". There is a small group within the Churches of Christ which oppose a single preacher and, instead, rotate preaching duties among qualified elders (this group tends to overlap with groups which oppose Sunday School and also have only one cup to serve the Lord's Supper).

Churches of Christ hold to the priesthood of all believers. No special titles are used for preachers or ministers that would identify them as clergy. Churches of Christ emphasize that there is no distinction between "clergy" and "laity" and that every member has a gift and a role to play in accomplishing the work of the church.

Congregational Methodist Church
Methodists who disagreed with the episcopal polity of the Methodist Episcopal Church, South left their mother church to form the Congregational Methodist Church, which retains Wesleyan-Arminian theology but adopts congregationalist polity as a distinctive.

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title: Congregational polity