formulasearchengine - User contributions [en]

Image (category theory)

2014-09-27T01:11:50Z

129.82.84.122: Changed "Postfix Notation" for composition to the more common notation i.e. the composition of f:X->Y and g:Y->Z is gf, rather than fg.

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Einstein coefficients

2014-02-05T16:28:55Z

129.82.80.219: /* Photo absorption */

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Electron diffraction

2014-01-31T18:40:00Z

129.82.78.196: /* Benefits */

{{Semireg polyhedra db|Semireg polyhedron stat table|tT}}
In [[geometry]], the '''truncated tetrahedron''' is an [[Archimedean solid]]. It has 4 regular [[hexagon]]al faces, 4 [[equilateral triangle]] faces, 12 vertices and 18 edges (of two types). It can be constructed by [[truncation (geometry)|truncating]] all 4 vertices of a regular [[tetrahedron]] at one third of the original edge length.

A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called [[Rectification (geometry)|rectification]]. The rectification of a tetrahedron produces an [[octahedron]].<ref>{{cite web |url=https://theory.org/geotopo/tt/html/truncatering.html |title=Truncated Trickery: Truncatering |first=Matt |last=Chisholm |first2=Jeremy |last2=Avnet |work=theory.org |year=1997 |accessdate=2013-09-02}}</ref>

A ''truncated tetrahedron'' is the [[Goldberg polyhedron]] GIII(1,1), containing triangular and hexagonal faces.

A ''truncated tetrahedron'' can be called a '''cantic cube''', with [[Coxeter diagram]], {{CDD|node_h1|4|node|3|node_1}}, having half of the vertices of the cantellated cube ([[rhombicuboctahedron]]), {{CDD|node_1|4|node|3|node_1}}. There are two dual positions of this construction, and combining them creates the uniform [[compound of two truncated tetrahedra]].

==Area and volume==
The area ''A'' and the [[volume]] ''V'' of a truncated tetrahedron of edge length ''a'' are:
:<math>A = 7\sqrt{3}a^2 \approx 12.12435565a^2</math>
:<math>V = \frac{23}{12}\sqrt{2}a^3 \approx 2.710575995a^3.</math>

==Densest Packing==
The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, as reported by two independent groups using [[Monte Carlo methods]].<ref name="Damasceno">{{cite web | url=http://arxiv.org/pdf/1109.1323v2.pdf | title=Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces | date= Dec 2011}}</ref><ref name="Jiao">{{cite web | url=http://arxiv.org/pdf/1107.2300v4.pdf | title=A Packing of Truncated Tetrahedra that Nearly Fills All of Space | date= Sep 2011}}</ref> Although no mathematical proof exists that this is the best possible packing for those shapes, the high proximity to the unity and independency of the findings make it unlikely that an even denser packing is to be found. In fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space.<ref name="Damasceno" />

==Cartesian coordinates==
[[Cartesian coordinates]] for the 12 vertices of a [[Truncation (geometry)|truncated]] [[tetrahedron]] centered at the origin, with edge length √8, are all permutations of (±1,±1,±3) with an even number of minus signs:
*(+3,+1,+1), (+1,+3,+1), (+1,+1,+3)
*(−3,−1,+1), (−1,−3,+1), (−1,−1,+3)
*(−3,+1,−1), (−1,+3,−1), (−1,+1,−3)
*(+3,−1,−1), (+1,−3,−1), (+1,−1,−3)

{|class=wikitable width=600
|- valign=top
|[[File:Truncated_tetrahedron_in_unit_cube.png|200px]]
|[[File:Triangulated truncated tetrahedron.png|200px]]
|[[Image:UC54-2 truncated tetrahedra.png|200px]]
|- valign=top
|[[Orthogonal projection]] showing Cartesian coordinates inside it [[bounding box]]: (±3,±3,±3).
|The hexagonal faces of the truncated tetrahedra can be divided into 6 coplanar equilateral triangles. The 4 new vertices have Cartesian coordinates: (-1,-1,-1), (-1,+1,+1), (+1,-1,+1), (+1,+1,1).
|The set of vertex permutations (±1,±1,±3) with an odd number of minus signs forms a complementary truncated tetrahedron, and combined they form a [[Compound polyhedron#Uniform compounds|uniform compound polyhedron]].
|}

Another simple construction exists in 4-space as cells of the [[truncated 16-cell]], with vertices as coordinate permutation of:
:(0,0,1,2)

== Orthogonal projection ==
{| class=wikitable
|+ [[Orthogonal projection]]
!Centered by
!Edge normal
!Face normal
!Edge
!Face/vertex
|- align=center
!Image
|[[File:tetrahedron t01 ae.png|100px]]
|[[File:tetrahedron t01 af36.png|100px]]
|[[File:3-simplex t01.svg|100px]]
|[[File:3-simplex t01 A2.svg|100px]]
|- align=center
!Dual image
|[[File:Dual tetrahedron t01 ae.png|100px]]
|[[File:Dual tetrahedron t01 af36.png|100px]]
|[[File:Dual tetrahedron t01.png|100px]]
|[[File:Dual tetrahedron t01 A2.png|100px]]
|- align=center
|- align=center
!Projective symmetry
![1]
![1]
![3]
![4]
|}

==Related polyhedra and tilings==

{{Tetrahedron family}}

{{Octahedral truncations}}

{{Cantic table}}

This polyhedron is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[vertex configuration]]s (3.2n.2n), and [n,3] [[Coxeter group]] symmetry.

{{Truncated figure1 table}}

=== Friauf polyhedron ===
A lower symmetry version of the truncated tetrahedron (a truncated [[tetragonal disphenoid]] with order 8 D2d symmetry) is called a Friauf polyhedron in crystals such as [[complex metallic alloys]]. This form fits 5 Friauf polyhedra around an axis, giving a 72 degree [[dihedral angle]] on a subset of 6-6 edges.<ref>http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf</ref> Its named after [[J. B. Friauf]] and his 1927 paper ''The crystal structure of the intermetallic compound MgCu2''. <ref>Friauf, J. B. ''The crystal structure of the intermetallic compound MgCu2'' (1927) J. Am. Chem. Soc. 19, 3107-3114.</ref>

==Use in architecture==
Giant truncated tetrahedra were used for the "Man the Explorer" and "Man the Producer" theme pavilions in [[Expo 67]]. They were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms. All of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada and the photo collections of tourists of the times.<ref>http://expo67.ncf.ca/man_the_producer_p1.html</ref>

==See also==
*[[Quarter cubic honeycomb]] - Fills space using truncated tetrahedra and smaller tetrahedra
*[[Truncated 5-cell]] – Similar uniform polytope in 4-dimensions
*[[Truncated triakis tetrahedron]]
*[[Triakis truncated tetrahedron]]
*[[Octahedron]] – a rectified tetrahedron

==References==
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
<references/>

==External links==
*{{mathworld2 |urlname=TruncatedTetrahedron |title=Truncated tetrahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x3o - tut}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=3cqTmfu7gdEZ8I7kRUVvji6qxBATVSp2WpmIWGx7l7pWe7bveylFxv3piHnPNZN&name=Truncated+Tetrahedron#applet Editable printable net of a truncated tetrahedron with interactive 3D view]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra

{{Archimedean solids}}
{{Polyhedron navigator}}

[[Category:Archimedean solids]]

Manning formula

2013-12-11T15:39:02Z

129.82.80.241:

The '''Mundell–Fleming model''', also known as the '''IS-LM-BoP model''', is an [[economics|economic]] model first set forth (independently) by [[Robert Mundell]] and [[Marcus Fleming]].<ref name="Mundell">{{Cite journal |last=Mundell |first=Robert A. |year=1963 |title=Capital mobility and stabilization policy under fixed and flexible exchange rates |journal=Canadian Journal of Economic and Political Science |volume=29 |issue=4 |pages=475–485 |doi=10.2307/139336 }} Reprinted in {{Cite book |last=Mundell |first=Robert A. |year=1968 |title=International Economics |location=New York |publisher=Macmillan |isbn= }}</ref><ref name="Fleming">{{Cite journal |last=Fleming |first=J. Marcus |year=1962 |title=Domestic financial policies under fixed and floating exchange rates |journal=IMF Staff Papers |volume=9 |issue= |pages=369–379 |doi= }} Reprinted in {{Cite book |editor-last=Cooper |editor-first=Richard N. |year=1969 |title=International Finance |location=New York |publisher=Penguin Books |isbn= }}</ref> The model is an extension of the [[IS-LM model|'''IS-LM''' Model]]. Whereas the traditional '''IS-LM''' Model deals with economy under [[autarky]] (or a closed economy), the Mundell–Fleming model describes an open economy.

The Mundell–Fleming model portrays the short-run relationship between an economy's nominal exchange rate, interest rate, and output (in contrast to the closed-economy '''IS-LM''' model, which focuses only on the relationship between the interest rate and output). The Mundell–Fleming model has been used to argue that an economy cannot simultaneously maintain a [[fixed exchange rate]], [[free movement of capital and payments|free capital movement]], and an independent [[monetary policy]]. This principle is frequently called the "[[impossible trinity]]," "unholy trinity," "irreconcilable trinity," "inconsistent trinity" or the "Mundell–Fleming [[trilemma]]."

==Basic set up==

===Variables===
This model uses the following variables:
* ''Y'' is [[GDP]]
* ''C'' is [[Consumption (economics)|consumption]]
* ''I'' is [[fixed investment|physical investment]]
* ''G'' is [[government spending]] (an [[exogenous variable]])
* ''M'' is the nominal [[money supply]]
* ''P'' is the [[price level]]
* ''i'' is the nominal [[interest rate]]
* ''L'' is liquidity preference (real [[money demand]])
* ''T'' is [[taxes]]

===Equations===
The Mundell–Fleming model is based on the following equations.

The '''IS''' curve:
:<math>Y=C+I+G+NX \, </math>
where ''NX'' is net [[exports]].

The '''LM''' curve:
:<math>\frac{M}{P}=L(i, Y)</math>
A higher interest rate or a lower income (GDP) level leads to lower money demand.

The '''BoP''' (Balance of Payments) Curve:
:<math>BoP=CA+KA \, </math>
where '''BoP''' is the [[balance of payments]] surplus, ''CA'' is the [[current account]] surplus, and ''KA'' is the [[capital account]] surplus.

===IS components===
:<math>C=C (Y-T(Y), i-E( \pi )) \, </math>
where ''E''(π) is the expected rate of [[inflation]]. Higher disposable income or a lower real interest rate (nominal interest rate minus expected inflation) leads to higher consumption spending.

:<math>I=I(i-E(\pi), Y_{-1}) \, </math>
where ''Y''-1 is GDP in the previous period. Higher lagged income or a lower real interest rate leads to higher investment spending.

:<math>NX=NX (e, Y, Y*)</math>
where ''NX'' is [[net exports]], ''e'' is the nominal [[exchange rate]] (the price of domestic currency in terms of units of the foreign currency), ''Y'' is GDP, and ''Y*'' is the combined GDP of countries that are foreign trading partners. Higher domestic income (GDP) leads to more spending on imports and hence lower net exports; higher foreign income leads to higher spending by foreigners on the country's exports and thus higher net exports. A higher ''e''leads to higher net exports.

===Balance of payments (BoP) components===
*<math>CA=NX</math>
**where ''CA'' is the current account and ''NX'' is net exports. That is, the current account is viewed as consisting solely of imports and exports.

*<math>KA=z(i-i*)+k</math>
**where <math>i*</math> is the foreign interest rate, ''k'' is the exogenous component of financial capital flows, ''z'' is the interest-sensitive component of capital flows, and the derivative of the function ''z'' is the degree of capital mobility (the effect of differences between domestic and foreign interest rates upon capital flows ''KA'').

===Variables determined by the model===
After the subsequent equations are substituted into the first three equations above, one has a system of three equations in three unknowns, two of which are GDP and the domestic interest rate. Under [[Exchange rate flexibility|flexible exchange rate]]s, the exchange rate is the third [[Endogeny#Economics and finance|endogenous variable]] while '''BoP''' is set equal to zero. In contrast, under [[fixed exchange rate]]s ''e'' is exogenous and the balance of payments surplus is determined by the model.

Under both types of exchange rate regime, the nominal domestic money supply ''M'' is exogenous, but for different reasons. Under flexible exchange rates, the nominal money supply is completely under the control of the central bank. But under fixed exchange rates, the money supply in the short run (at a given point in time) is fixed based on past international money flows, while as the economy evolves over time these international flows cause future points in time to inherit higher or lower (but pre-determined) values of the money supply.

==Mechanics of the model==
The model's workings can be described in terms of an '''IS-LM-BoP''' graph with the domestic interest rate plotted vertically and real GDP plotted horizontally. The '''IS''' curve is downward sloped and the '''LM''' curve is upward sloped, as in the closed economy '''IS-LM''' analysis; the '''BoP''' curve is upward sloped unless there is perfect capital mobility, in which case it is horizontal at the level of the world interest rate.

In this graph, under less than perfect capital mobility the positions of both the '''IS''' curve and the '''BoP''' curve depend on the exchange rate (as discussed below), since the '''IS-LM''' graph is actually a two-dimensional [[cross-section (geometry)|cross-section]] of a three-dimensional space involving all of the interest rate, income, and the exchange rate. However, under perfect capital mobility the '''BoP''' curve is simply horizontal at a level of the domestic interest rate equal to the level of the world interest rate.

===Under a flexible exchange rate regime===
In a system of flexible exchange rates, central banks allow the exchange rate to be determined by market forces alone.

==== Changes in the money supply ====
An increase in money supply shifts the '''LM''' curve to the right. This directly reduces the local interest rate relative to the global interest rate.A decrease in the money supply causes the exact opposite process.

==== Changes in government spending ====
An increase in government expenditure shifts the '''IS''' curve to the right. The shift causes both the local interest rate and income (GDP) to rise.A decrease in government expenditure reverses the process.

==== Changes in the global interest rate ====
An increase in the global interest rate shifts the '''BoP''' curve upward and causes capital flows out of the local economy. This depreciates the local currency and boosts net exports, shifting the '''IS''' curve to the right. Under less than perfect capital mobility, the depreciated exchange rate shifts the '''BoP''' curve somewhat back down.Under perfect capital mobility, the '''BoP''' curve is always horizontal at the level of the world interest rate. When the latter goes up, the '''BoP''' curve shifts upward by the same amount, and stays there. The exchange rate changes enough to shift the '''IS''' curve to the location where it crosses the new '''BoP''' curve at its intersection with the unchanged '''LM''' curve; now the domestic interest rate equals the new level of the global interest rate.

A decrease in the global interest rate causes the reverse to occur.

===Under a fixed exchange rate regime===
In a system of fixed exchange rates, central banks announce an exchange rate (the parity rate) at which they are prepared to buy or sell any amount of domestic currency. Thus net payments flows into or out of the country need not equal zero; the exchange rate ''e'' is exogenously given, while the variable '''BoP''' is endogenous.

Under the fixed exchange rate system, the central bank operates in the [[foreign exchange market]] to maintain a specific exchange rate. If there is pressure to depreciate the domestic currency's exchange rate because the supply of domestic currency exceeds its demand in foreign exchange markets, the local authority buys domestic currency with foreign currency to decrease the domestic currency's supply in the foreign exchange market. This keeps the domestic currency's exchange rate at its targeted level. If there is pressure to appreciate the domestic currency's exchange rate because the currency's demand exceeds its supply in the foreign exchange market, the local authority buys foreign currency with domestic currency to increase the domestic currency's supply in the foreign exchange market. Again,this keeps the exchange rate at its targeted level.

==== Changes in the money supply ====
In the very short run the money supply is normally predetermined by the past history of international payments flows. If the central bank is maintaining an exchange rate that is consistent with a balance of payments surplus, over time money will flow into the country and the money supply will rise (and vice versa for a payments deficit). If the central bank were to conduct open market operations in the domestic bond market in order to offset these balance-of-payments-induced changes in the money supply — a process called [[sterilization (economics)|sterilization]], it would absorb newly arrived money by decreasing its holdings of domestic bonds (or the opposite if money were flowing out of the country). But under perfect capital mobility, any such sterilization would be met by further offsetting international flows.

==== Changes in government expenditure ====
[[File:Mundell-Fleming example fixed exchange rate .png|thumb|250px|right|An increase in government spending forces the monetary authority to supply the market with local currency to keep the exchange rate unchanged. Shown here is the case of perfect capital mobility, in which the '''BoP''' curve (or, as denoted here, the FE curve) is horizontal.]]
Increased government expenditure shifts the '''IS''' curve to the right. The shift results in an incipient rise in the interest rate, and hence upward pressure on the exchange rate (value of the domestic currency) as foreign funds start to flow in, attracted by the higher interest rate. However, the exchange rate is controlled by the local monetary authority in the framework of a fixed exchange rate system. To maintain the exchange rate and eliminate pressure on it, the monetary authority purchases foreign currency using domestic funds in order to shift the '''LM''' curve to the right. In the end, the interest rate stays the same but the general income in the economy increases. In the '''IS-LM-BoP''' graph, the '''IS''' curve has been shifted exogenously by the fiscal authority, and the '''IS''' and '''BoP''' curves determine the final resting place of the system; the '''LM''' curve merely passively reacts.

The reverse process applies when government expenditure decreases.

==== Changes in the global interest rate ====
To maintain the fixed exchange rate, the central bank must accommodate the capital flows (in or out) which are caused by a change of the global interest rate, in order to offset pressure on the exchange rate.

If the global interest rate increases, shifting the '''BoP''' curve upward, capital flows out to take advantage of the opportunity. This puts pressure on the home currency to depreciate, so the central bank must buy the home currency — that is, sell some of its foreign currency reserves — to accommodate this outflow. The decrease in the money supply resulting from the outflow, shifts the '''LM''' curve to the left until it intersect the '''IS''' and '''BoP''' curves at their intersection. Once again, the '''LM''' curve plays a passive role, and the outcomes are determined by the '''IS-BoP''' interaction.

Under perfect capital mobility, the new '''BoP''' curve will be horizontal at the new world interest rate, so the equilibrium domestic interest rate will equal the world interest rate.

If the global interest rate declines below the domestic rate, the opposite occurs. The '''BoP''' curve shifts down, foreign money flows in and the home currency is pressured to appreciate, so the central bank offsets the pressure by selling domestic currency (equivalently, buying foreign currency). The inflow of money causes the '''LM''' curve to shift to the right, and the domestic interest rate becomes lower (as low as the world interest rate if there is perfect capital mobility).

==Differences from IS-LM==
It is worth noting that some of the results from this model differ from those of the '''IS-LM''' model because of the [[open economy]] assumption. Results for a large open economy, on the other hand, can be consistent with those predicted by the '''IS-LM''' model. The reason is that a large open economy has the characteristics of both an autarky and a small open economy. In particular, it may not face perfect capital mobility, thus allowing internal policy measures to affect the domestic interest rate, and it may be able to sterilize balance-of-payments-induced changes in the money supply (as discussed above).

In the '''IS-LM''' model, the domestic interest rate is a key component in keeping both the money market and the goods market in equilibrium. Under the Mundell–Fleming framework of a small economy facing perfect capital mobility, the domestic interest rate is fixed and equilibrium in both markets can only be maintained by adjustments of the nominal exchange rate or the money supply (by international funds flows).

===Example===
The Mundell–Fleming model applied to a small open economy facing perfect capital mobility, in which the domestic interest rate is exogenously determined by the world interest rate, shows stark differences from the closed economy model.

Consider an exogenous increase in government expenditure. Under the '''IS-LM''' model, the '''IS''' curve shifts upward, with the '''LM''' curve intact, causing the interest rate and output to rise. But for a small open economy with perfect capital mobility and a flexible exchange rate, the domestic interest rate is predetermined by the horizontal '''BoP''' curve, and so by the '''LM''' equation given previously there is exactly one level of output that can make the money market be in equilibrium at that interest rate. Any exogenous changes affecting the '''IS''' curve (such as government spending changes) will be exactly offset by resulting exchange rate changes, and the '''IS''' curve will end up in its original position, still intersecting the '''LM''' and '''BoP''' curves at their intersection point.

The Mundell–Fleming model under a fixed exchange rate regime also has completely different implications from those of the closed economy '''IS-LM''' model. In the closed economy model, if the central bank expands the money supply the '''LM''' curve shifts out, and as a result income goes up and the domestic interest rate goes down. But in the Mundell–Fleming open economy model with perfect capital mobility, monetary policy becomes ineffective. An expansionary monetary policy resulting in an incipient outward shift of the '''LM''' curve would make capital flow out of the economy. The central bank under a fixed exchange rate system would have to instantaneously intervene by selling foreign money in exchange for domestic money to maintain the exchange rate. The accommodated monetary outflows exactly offset the intended rise in the domestic money supply, completely offsetting the tendency of the '''LM''' curve to shift to the right, and the interest rate remains equal to the world rate of interest.

==See also==
*[[Optimum currency area]]

==References==
{{Reflist}}

==Further reading==
*{{Citation |last=Young |first=Warren |last2=Darity |first2=William, Jr. |authorlink2=Sandy Darity, Jr. |year=2004 |title=IS-LM-BP: An Inquest |journal=History of Political Economy |volume=36 |issue=Suppl 1 |pages=127–164 |url=http://www.econ.ucdavis.edu/faculty/kdhoover/pdf/Hope/Young.pdf |doi=10.1215/00182702-36-Suppl_1-127 }} (Tells the difference between the IS-LM-BP model and the Mundell–Fleming model.)
*{{citation |last=Carlin |first=Wendy |last2=Soskice |first2=David W. |title=Macroeconomics and the Wage Bargain |year=1990 |publisher=Oxford University Press |location=New York |isbn=0-19-877245-9 }}
*{{citation |last=Mankiw |first=N. Gregory |authorlink=N. Gregory Mankiw |title=Macroeconomics |year=2007 |edition=6th |publisher=Worth |location=New York |isbn=978-0-7167-6213-3 }}
*{{citation |last=Blanchard |first=Olivier |title=Macroeconomics |year=2006 |publisher=Prentice Hall |location=Upper Saddle River, NJ |edition=4th |isbn=0-13-186026-7 }}
*{{citation |last=DeGrauwe |first=Paul |title=Economics of Monetary Union |year=2000 |publisher=Oxford University Press |location=New York |edition=4th |isbn=0-19-877632-2 }}

{{Macroeconomics}}

{{DEFAULTSORT:Mundell-Fleming model}}
[[Category:Economics models]]
[[Category:International economics]]
[[Category:Open economy macroeconomics]]
[[Category:Macroeconomics]]

Virtual temperature

2013-09-11T19:31:03Z

129.82.49.88: Previous edit gave an incorrect approximation for Tv

[[Image:Sn2006gy collapse ill.jpg|thumb|right|400px|This illustration explains the pair-instability supernova process that astronomers think triggered the explosion in SN 2006gy. When a star is very massive, the [[gamma-ray]]s produced in its core can become so energetic that some of their energy is drained away into production of particle and [[anti-particle]] pairs. The resulting drop in pressure causes the star to partially collapse under its own huge gravity. After this violent collapse, runaway thermonuclear reactions (not shown here) ensue and the star explodes, spewing the remains into space.]]

A '''pair-instability supernova''' occurs when [[pair production]], the production of free [[electron]]s and [[positron]]s in the collision between [[atomic nuclei]] and energetic [[gamma-ray|gamma rays]], reduces thermal pressure inside a [[hypergiant|supermassive star]]'s core. This pressure drop leads to a partial collapse, then greatly accelerated burning in a [[thermal runaway|runaway]] thermonuclear explosion which blows the star completely apart without leaving a [[black hole]] remnant behind.<ref>{{cite journal|last=Fraley|first=Gary S.|title=Supernovae Explosions Induced by Pair-Production Instability|journal=[[Astrophysics and Space Science]]|year=1968|volume=2|issue=1|pages=96–114|doi=10.1007/BF00651498|url=http://adsabs.harvard.edu/abs/1968Ap%26SS...2...96F|bibcode = 1968Ap&SS...2...96F }}</ref> Pair-instability [[supernova]]e can only happen in stars with a mass range from around 130 to 250 [[solar mass]]es and low to moderate [[metallicity]] (low abundance of elements other than hydrogen and helium, a situation common in [[Population III stars]]). The recently observed objects [[SN 2006gy]], [[SN 2007bi]],<ref name="SN2007bi">{{Citation |last=Gal-Yam |first=A. |last2=Mazzali |first2=P. |last3=Ofek |first3=E. O. |last4=''et al.'' |title=Supernova 2007bi as a pair-instability explosion |journal=[[Nature (journal)|Nature]]|date=3 December 2009 |volume=462 |pages=624–627 |doi=10.1038/nature08579|url=http://adsabs.harvard.edu/abs/2009Natur.462..624G|bibcode = 2009Natur.462..624G |arxiv = 1001.1156 }}</ref> [[SN 2213-1745]] and [[SN 1000+0216]]<ref name=cooke>{{cite doi|10.1038/nature11521}}</ref> are hypothesized to have been pair-instability supernovae.

==Physics==

===Photon pressure===
Light in thermal equilibrium has a [[black body spectrum]] with an energy density proportional to the fourth power of the temperature (hence the [[Stefan-Boltzmann law]]). The wavelength of maximum emission from a blackbody is inversely proportional to its temperature. That is, the frequency, and the energy, of the greatest population of photons of black body radiation is directly proportional to the temperature, and reaches the gamma ray energy range at temperatures above {{val|3|e=8}} K.

In very large hot stars, pressure from gamma rays in the stellar core keeps the upper layers of the star supported against gravitational pull from the core. If the energy density of gamma rays is suddenly reduced, then the outer layers of the star will collapse inwards. The sudden heating and compression of the core generates gamma rays energetic enough to be converted into an avalanche of electron-positron pairs, further reducing the pressure. When the collapse stops, the positrons find electrons and the pressure from gamma rays is driven up, again. The population of positrons provides a brief reservoir of new gamma rays as the expanding supernova's core pressure drops.

===Pair creation and annihilation===
Sufficiently energetic gamma rays can interact with nuclei, electrons, or one another to produce electron-positron pairs, and electron-positron pairs can annihilate, producing gamma rays. From Einstein's equation <math>E = mc^2</math>, gamma rays must have more energy than the mass of the electron–positron pairs to produce these pairs.

At the high densities of a stellar core, pair production and annihilation occur rapidly, thereby keeping gamma rays, electrons, and positrons in thermal equilibrium. The higher the temperature, the higher the gamma ray energies, and the larger the amount of energy transferred.

===Pair-instability===
As temperatures and gamma ray energies increase, more and more gamma ray energy is absorbed in creating electron-positron pairs. This reduction in gamma ray energy density reduces the radiation pressure that supports the outer layers of the star. The star contracts, compressing and heating the core, thereby increasing the proportion of energy absorbed by pair creation. Pressure nonetheless increases, but in a pair-instability collapse, the increase in pressure is not enough to resist the increase in gravitational forces as the star becomes denser.

==Stellar susceptibility==
For a star to undergo pair-instability supernova, the loss in total outward pressure resulting from the increased creation of positron/electron pairs by gamma ray collisions must be sufficiently great to allow the inward gravitational pressure to overwhelm the remaining outward pressure. Among stellar mechanisms not responsive to the reduction in outward pressure effected by pair creation, rotational speed and [[metallicity]] are the most important.

Stars exhibiting these characteristics still contract as gravity's inward pressure increases relative to the star's total outward pressure. Unlike their slower or less metal-rich cousins, however, these stars continue to exert outward pressure sufficient to prevent contractions so great that gravity entirely overwhelms its opposition and collapses the star.

Stars formed by collision mergers having a metallicity Z between 0.02 and 0.001 may end their lives as pair-instability supernovae if their mass is in the appropriate range.<ref>{{cite journal|last=Belkus|first=H.|coauthors=Van Bever, J.; Vanbeveren, D.|title=The Evolution of Very Massive Stars|journal=[[The Astrophysical Journal]]|year=2007|volume=659|issue=2|pages=1576–1581|doi=10.1086/512181|url=http://adsabs.harvard.edu/abs/2007ApJ...659.1576B|arxiv = astro-ph/0701334 |bibcode = 2007ApJ...659.1576B }}</ref>

Very large high metallicity stars are probably unstable due to the [[Eddington limit]], and would tend to shed mass during the formation process.

==Stellar behavior==
Several sources describe the stellar behavior for large stars in pair-instability conditions.<ref name=Smith>{{cite journal|last=Smith|first=Nathan|coauthors=Li, Weidong; Foley, Ryan J.; Wheeler, J. Craig; Pooley, David; Chornock, Ryan; Filippenko, Alexei V.; Silverman, Jeffrey M.; Quimby, Robert; Bloom, Joshua S.; Hansen, Charles|title=SN 2006gy: Discovery of the Most Luminous Supernova Ever Recorded, Powered by the Death of an Extremely Massive Star like η Carinae|journal=The Astrophysical Journal|year=2007|volume=666|issue=2|pages=1116–1128|doi=10.1086/519949|url=http://adsabs.harvard.edu/abs/2007ApJ...666.1116S|arxiv = astro-ph/0612617 |bibcode = 2007ApJ...666.1116S }}</ref><ref name=Fryer>{{cite journal|last=Fryer|first=C.L.|coauthors=Woosley, S. E.; Heger, A.|title=Pair-Instability Supernovae, Gravity Waves, and Gamma-Ray Transients|journal=The Astrophysical Journal|year=2001|volume=550|issue=1|doi=10.1086/319719|url=http://adsabs.harvard.edu/abs/2001ApJ...550..372F|arxiv = astro-ph/0007176 |bibcode = 2001ApJ...550..372F }}</ref>

===Below 100 solar masses===
Gamma rays produced by stars of fewer than 100 or so solar masses are not energetic enough to produce electron-positron pairs. Some of these stars will undergo supernovae at the end of their lives, but the causative mechanisms are unrelated to pair-instability.

===100 to 130 solar masses===
These stars are large enough to produce gamma rays with enough energy to create electron-positron pairs, but the resulting net reduction in counter-gravitational pressure is insufficient to cause the core-overpressure required for supernova. Instead, the contraction caused by pair-creation provokes increased thermonuclear activity within the star that repulses the inward pressure and returns the star to equilibrium. It is thought that stars of this size undergo a series of these pulses until they shed sufficient mass to drop below 100 solar masses, at which point they are no longer hot enough to support pair-creation. Pulsing of this nature may have been responsible for the variations in brightness experienced by [[Eta Carinae#Brightness variations|Eta Carinae in 1843]], though this explanation is not universally accepted.

===130 to 250 solar masses===
For very high mass stars, with mass at least 130 and up to perhaps roughly 250 solar masses, a true pair-instability supernova can occur. In these stars, the first time that conditions support pair creation instability, the situation runs out of control. The collapse proceeds to efficiently compress the star's core; the overpressure is sufficient to allow runaway nuclear fusion to burn it in a few seconds, creating a thermonuclear explosion.<ref name=Fryer/> With more thermal energy released than the star's [[gravitational binding energy]], it is completely disrupted; no [[black hole]] or other remnant is left behind.

In addition to the immediate energy release, a large fraction of the star's core is transformed to [[Isotopes of nickel|nickel-56]], a radioactive [[isotope]] which decays with a half-life of 6.1 days into [[Isotopes of cobalt|cobalt-56]]. [[Isotopes of cobalt|Cobalt-56]] has a half-life of 77 days and then further decays to the stable isotope [[Isotopes of iron|iron-56]] (see [[Supernova nucleosynthesis]]). For the [[hypernova]] [[SN 2006gy]], studies indicate that perhaps 40 solar masses of the original star were released as Ni-56, almost the entire mass of the star's core regions.<ref name=Smith /> Collision between the exploding star core and gas it ejected earlier, and radioactive decay, release most of the visible light.

===250 solar masses or more===
A different reaction mechanism, [[photodisintegration]], results after collapse starts in stars of at least 250 solar masses. This endothermic (energy-absorbing) reaction causes the star to continue collapse into a black hole rather than exploding due to thermonuclear reactions.

==Appearance==

===Luminosity===
Pair instability supernovae are popularly thought to be highly luminous. This is only the case for the most massive progenitors, which can have peak luminosities of over 1037 joules/sec, brighter than type Ia supernovae. However at lower masses, peak luminosities are less than 1035 joules/sec, comparable to or less than typical type II supernovae.<ref name="appearance">{{cite doi|10.1088/0004-637X/734/2/102}}</ref>

===Spectrum===
The spectra of pair instability supernovae depend on the nature of the progenitor star. Thus they can appear as type II or type Ib/c supernova spectra.<ref name = "appearance"/>

===Light curves===
In contrast to the spectra, the light curves are quite different from the common types of supernova. The light curves are highly extended, with peak luminosity occurring months after onset.<ref name = "appearance"/> This is due to the extreme amounts of 56Ni expelled, and the optically dense ejecta, as the star is entirely disrupted.

==References==
{{Reflist}}

{{Supernovae}}

[[Category:Supernovae]]
[[Category:Hypernovae]]

[[de:Supernova#Paarinstabilitätssupernova]]

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