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| The '''Chandrasekhar limit''' {{IPAc-en|tʃ|ʌ|n|d|r|ə|ˈ|ʃ|eɪ|k|ɑr}} is the maximum mass of a [[Hydrostatic equilibrium|stable]] [[white dwarf]] [[star]]. The limit was first published by [[Wilhelm Anderson]] and [[Edmund Clifton Stoner|E. C. Stoner]], and was named after [[Subrahmanyan Chandrasekhar]], the Indian-American [[astrophysicist]] who improved upon the accuracy of the calculation in 1930, at the age of 19. This limit was initially ignored by the community of scientists because such a limit would logically require the existence of black holes, which were considered a scientific impossibility at the time. White dwarfs, unlike [[main sequence]] stars, resist [[gravitational collapse]] primarily through [[electron degeneracy pressure]], rather than [[Pressure#Pressure of an ideal gas|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, [[Stellar evolution|evolving]] into a different type of [[Compact star|stellar remnant]], such as a [[neutron star]] or [[black hole]]. Those with masses under the limit remain stable as white dwarfs.<ref name = DarkMatter> Sean Carroll, Ph.D., Cal Tech, 2007, The Teaching Company, ''Dark Matter, Dark Energy: The Dark Side of the Universe'', Guidebook Part 2 page 44, Accessed Oct. 7, 2013, “...Chandrasekhar limit: The maximum mass of a white dwarf star, about 1.4 times the mass of the Sun. Above this mass, the gravitational pull becomes too great, and the star must collapse to a neutron star or black hole...”</ref>
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| The currently accepted value of the limit is about 1.44 [[Solar mass|<math>\begin{smallmatrix}M_\odot\end{smallmatrix}</math>]] ( 2.864 × 10<sup>30</sup> kg).<ref>{{cite book|last=Israel|first=edited by S.W. Hawking, W.|title=Three hundred years of gravitation|year=1989|publisher=Cambridge University Press|location=Cambridge [Cambridgeshire]|isbn=0-521-37976-8|edition=1st pbk. ed., with corrections.}}</ref><ref>p. 55, How A Supernova Explodes, Hans A. Bethe and Gerald Brown, pp. 51–62 in ''Formation And Evolution of Black Holes in the Galaxy: Selected Papers with Commentary'', Hans Albrecht Bethe, Gerald Edward Brown, and Chang-Hwan Lee, River Edge, NJ: World Scientific: 2003. ISBN 981-238-250-X.</ref><ref>{{cite doi|10.1126/science.1136259 }}</ref>
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| ==Physics==
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| Electron degeneracy pressure is a [[quantum mechanics|quantum-mechanical]] 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.
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| [[Image:WhiteDwarf mass-radius en.svg|thumb|300px|right|Radius–mass relations for a model white dwarf. The green curve uses the general pressure law for an ideal [[Fermi gas]], while the blue curve is for a non-relativistic ideal Fermi gas. The black line marks the [[ultrarelativistic limit]].]]
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| In the nonrelativistic case, electron degeneracy pressure gives rise to an [[equation of state]] of the form <math>P = K_1 \rho^{5\over 3}</math>, where ''P'' is the [[pressure]], <math>\rho</math> is the [[mass density]], and <math>K_1</math> is a constant. Solving the hydrostatic equation then 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.<ref name="chandra3">The Density of White Dwarf Stars, S. Chandrasekhar, ''Philosophical Magazine'' (7th series) '''11''' (1931), pp. 592–596.</ref>
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| 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 <math>P = K_2 \rho^{4\over 3}</math>. This will yield a polytrope of index 3, which will have a total mass, M<sub>limit</sub> say, depending only on <var>K</var><sub>2</sub>.<ref name="chandra4">[http://adsabs.harvard.edu/abs/1931ApJ....74...81C The Maximum Mass of Ideal White Dwarfs], S. Chandrasekhar, ''Astrophysical Journal'' '''74''' (1931), pp. 81–82.</ref>
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| For a fully relativistic treatment, the equation of state used will interpolate between the equations <math>P = K_1 \rho^{5\over 3}</math> for small ρ and <math>P = K_2 \rho^{4\over 3}</math> for large ρ.
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| When this is done, the model radius still decreases with mass, but becomes zero at M<sub>limit</sub>. This is the Chandrasekhar limit.<ref name="chandra2"/> 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<sub>e</sub> has been set equal to 2.
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| Radius is measured in standard solar radii<ref name="stds">[http://vizier.u-strasbg.fr/doc/catstd-3.2.htx ''Standards for Astronomical Catalogues, Version 2.0''], section 3.2.2, web page, accessed 12-I-2007.</ref> or kilometers, and mass in standard solar masses.
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| Calculated values for the limit will vary depending on the [[Atomic nucleus|nuclear]] composition of the mass.<ref name="timmes"/> Chandrasekhar<ref name="chandra1">[http://adsabs.harvard.edu/abs/1931MNRAS..91..456C The Highly Collapsed Configurations of a Stellar Mass], S. Chandrasekhar, ''Monthly Notices of the Royal Astronomical Society'' '''91''' (1931), 456–466.</ref><sup>, eq. (36),</sup><ref name="chandra2">[http://adsabs.harvard.edu/abs/1935MNRAS..95..207C The Highly Collapsed Configurations of a Stellar Mass (second paper)], S. Chandrasekhar, ''Monthly Notices of the Royal Astronomical Society'', '''95''' (1935), pp. 207--225.</ref><sup>, eq. (58),</sup><ref name="chandranobel">[http://nobelprize.org/nobel_prizes/physics/laureates/1983/chandrasekhar-lecture.pdf ''On Stars, Their Evolution and Their Stability''], Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.</ref><sup>, eq. (43)</sup> gives the following expression, based on the [[equation of state]] for an ideal [[Fermi gas]]:
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| :<math> M_{\rm limit} = \frac{\omega_3^0 \sqrt{3\pi}}{2}\left ( \frac{\hbar c}{G}\right )^{3/2}\frac{1}{(\mu_e m_H)^2},</math>
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| where:
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| *''<math>\hbar</math>'' is the [[Planck_constant#Reduced_Planck_constant|reduced Planck constant]]
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| *''c'' is the [[speed of light]]
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| *''G'' is the [[gravitational constant]]
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| *''μ''<sub>e</sub> is the average [[molecular weight]] per electron, which depends upon the chemical composition of the star.
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| *''m<sub>H</sub>'' is the mass of the [[hydrogen]] [[atom]].
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| *<math>\omega_3^0 \approx 2.018236</math> is a constant connected with the solution to the [[Lane-Emden equation]].
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| As <math>\sqrt{\hbar c/G}</math> is the [[Planck mass]], the limit is of the order of
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| :<math>\frac{M_{Pl}^3}{m_H^2}.</math>
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| 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.<ref name="timmes">[http://adsabs.harvard.edu/abs/1996ApJ...457..834T The Neutron Star and Black Hole Initial Mass Function], F. X. Timmes, S. E. Woosley, and Thomas A. Weaver, ''Astrophysical Journal'' '''457''' (February 1, 1996), pp. 834–843.</ref> Lieb and Yau<ref>[http://adsabs.harvard.edu/abs/1987ApJ...323..140L A rigorous examination of the Chandrasekhar theory of stellar collapse], Elliott H. Lieb and Horng-Tzer Yau, ''Astrophysical Journal'' '''323''' (1987), pp. 140–144.</ref> have given a rigorous derivation of the limit from a relativistic many-particle [[Schrödinger equation]].
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| ==History==
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| In 1926, the [[United Kingdom|British]] [[physicist]] [[Ralph H. Fowler]] observed that the relationship among 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]].<ref>[http://adsabs.harvard.edu/abs/1926MNRAS..87..114F On Dense Matter], R. H. Fowler, ''Monthly Notices of the Royal Astronomical Society'' '''87''' (1926), pp. 114–122.</ref> This [[Fermi gas]] model was then used by the British physicist [[Edmund Clifton Stoner|E. C. Stoner]] in 1929 to calculate the relationship among the mass, radius, and density of white dwarfs, assuming them to be homogenous spheres.<ref>The Limiting Density of White Dwarf Stars, Edmund C. Stoner, ''Philosophical Magazine'' (7th series) '''7''' (1929), pp. 63–70.</ref> [[Wilhelm Anderson]] applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37{{e|30}} kg.<ref>[http://dx.doi.org/10.1007/BF01340146 Über die Grenzdichte der Materie und der Energie], Wilhelm Anderson, ''Zeitschrift für Physik'' '''56''', #11–12 (November 1929), pp. 851–856. DOI 10.1007/BF01340146.</ref> In 1930, Stoner derived the [[internal energy]]-[[density]] [[equation of state]] for a Fermi gas, and was then able to treat the mass-radius relationship in a fully relativistic manner, giving a limiting mass of approximately (for μ<sub>e</sub>=2.5) 2.19 · 10<sup>30</sup> kg.<ref>The Equilibrium of Dense Stars, Edmund C. Stoner, ''Philosophical Magazine'' (7th series) '''9''' (1930), pp. 944–963.</ref> Stoner went on to derive the [[pressure]]-[[density]] equation of state, which he published in 1932.<ref>[http://adsabs.harvard.edu/abs/1932MNRAS..92..651S The minimum pressure of a degenerate electron gas], E. C. Stoner, ''Monthly Notices of the Royal Astronomical Society'' '''92''' (May 1932), pp. 651–661.</ref> These equations of state were also previously published by the [[Soviet Union|Soviet]] [[physicist]] [[Yakov Frenkel]] in 1928, together with some other remarks on the physics of [[degenerate matter]].<ref>[http://dx.doi.org/10.1007/BF01328867 Anwendung der Pauli-Fermischen Elektronengastheorie auf das Problem der Kohäsionskräfte], J. Frenkel, ''Zeitschrift für Physik'' '''50''', #3–4 (March 1928), pp. 234–248. DOI 10.1007/BF01328867.</ref> Frenkel's work, however, was ignored by the astronomical and astrophysical community.<ref>[http://adsabs.harvard.edu/abs/1994UsFiN..37..609Y The article by Ya I Frenkel' on `binding forces' and the theory of white dwarfs], D. G. Yakovlev, ''Physics Uspekhi'' '''37''', #6 (1994), pp. 609–612.</ref>
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| A series of papers published between 1931 and 1935 had its beginning on a trip from [[India]] to [[England]] in 1930,
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| where the [[ethnic Indian|Indian]] [[physicist]] [[Subrahmanyan Chandrasekhar]] worked on the calculation of the statistics of a degenerate Fermi gas.<ref name="nasbio">[http://www.nap.edu/readingroom/books/biomems/schandrasekhar.html Chandrasekhar's biographical memoir at the National Academy of Sciences], web page, accessed 12-I-2007.</ref> In these papers, Chandrasekhar solved
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| the [[hydrostatic equation]] together with the nonrelativistic Fermi gas [[equation of state]],<ref name="chandra3"/> and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above.<ref name="chandra4"/><ref name="chandra2"/><ref name="chandra1"/><ref>[http://adsabs.harvard.edu/abs/1934Obs....57..373C Stellar Configurations with degenerate Cores], S. Chandrasekhar, ''The Observatory'' '''57''' (1934), pp. 373–377.</ref> Chandrasekhar reviews this work in his Nobel Prize lecture.<ref name="chandranobel"/> This value was also computed in 1932 by the Soviet physicist [[Lev Davidovich Landau]],<ref>On the Theory of Stars, in ''Collected Papers of L. D. Landau'', ed. and with an introduction by D. ter Haar, New York: Gordon and Breach, 1965; originally published in ''Phys. Z. Sowjet.'' '''1''' (1932), 285.</ref> who, however, did not apply it to white dwarfs. | |
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| Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the [[United Kingdom|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:
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| {{quote|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!|<ref>[http://adsabs.harvard.edu/abs/1935Obs....58...33. Meeting of the Royal Astronomical Society, Friday, 1935 January 11], ''The Observatory'' '''58''' (February 1935), pp. 33–41.</ref>}}
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| Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law P=K<sub>1</sub>ρ<sup>5/3</sup> universally applicable, even for large ρ.<ref>[http://adsabs.harvard.edu/abs/1935MNRAS..95..194E On "Relativistic Degeneracy"], Sir A. S. Eddington, ''Monthly Notices of the Royal Astronomical Society'' '''95''' (1935), 194–206.</ref> Although [[Niels Bohr|Bohr]], Fowler, [[Wolfgang Pauli|Pauli]], and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar.<ref name="eos">''Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes'', Arthur I. Miller, Boston, New York: Houghton Mifflin, 2005, ISBN 0-618-34151-X; reviewed at ''The Guardian'': [http://books.guardian.co.uk/reviews/scienceandnature/0,,1472561,00.html The battle of black holes].</ref><sup>, pp. 110–111</sup> Through the rest of his life, Eddington held to his position in his writings,<ref>[http://adsabs.harvard.edu/abs/1935Obs....58..257. The International Astronomical Union meeting in Paris, 1935], ''The Observatory'' '''58''' (September 1935), pp. 257–265, at p. 259.</ref><ref>[http://adsabs.harvard.edu/abs/1935MNRAS..96...20E Note on "Relativistic Degeneracy"], Sir A. S. Eddington, ''Monthly Notices of the Royal Astronomical Society'' '''96''' (November 1935), 20–21.</ref><ref>[http://links.jstor.org/sici?sici=0080-4630%2819351101%29152%3A876%3C253%3ATPOADE%3E2.0.CO%3B2-Z The Pressure of a Degenerate Electron Gas and Related Problems], Arthur Eddington, ''Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences'' '''152''' (November 1, 1935), pp. 253–272.</ref><ref>''Relativity Theory of Protons and Electrons'', Sir Arthur Eddington, Cambridge: Cambridge University Press, 1936, chapter 13.</ref><ref>[http://adsabs.harvard.edu/abs/1940MNRAS.100..582E The physics of white dwarf matter], Sir A. S. Eddington, ''Monthly Notices of the Royal Astronomical Society'' '''100''' (June 1940), pp. 582–594.</ref> including his work on his [[Arthur Stanley Eddington#Fundamental theory|fundamental theory]].<ref>''Fundamental Theory'', Sir A. S. Eddington, Cambridge: Cambridge University Press, 1946, §43–45.</ref> The drama associated with this disagreement is one of the main themes of ''[[Empire of the Stars]]'', [[Arthur I. Miller]]'s biography of Chandrasekhar.<ref name="eos"/> In Miller's view:
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| {{quote|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.|p. 150|<ref name="eos"/>}}
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| ==Applications==
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| The core of a star is kept from collapsing by the heat generated by the [[nuclear fusion|fusion]] of [[Atomic nucleus|nuclei]] of lighter [[chemical element|elements]] into heavier ones. At various stages of [[stellar evolution]], 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.<ref name="evo2">[http://adsabs.harvard.edu/abs/2002RvMP...74.1015W The evolution and explosion of massive stars], S. E. Woosley, A. Heger, and T. A. Weaver, ''Reviews of Modern Physics'' '''74''', #4 (October 2002), pp. 1015–1071.</ref>
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| 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.)<ref name="ifmr1">[http://adsabs.harvard.edu/abs/1996A%26A...313..810K White dwarfs in open clusters. VIII. NGC 2516: a test for the mass-radius and initial-final mass relations], D. Koester and D. Reimers, ''Astronomy and Astrophysics'' '''313''' (1996), pp. 810–814.</ref><ref name="ifmr2">[http://adsabs.harvard.edu/abs/2004ApJ...615L..49W An Empirical Initial-Final Mass Relation from Hot, Massive White Dwarfs in NGC 2168 (M35)], Kurtis A. Williams, M. Bolte, and Detlev Koester, ''Astrophysical Journal'' '''615''', #1 (2004), pp. L49–L52; also [http://arxiv.org/abs/astro-ph/0409447 arXiv astro-ph/0409447].</ref><ref name="evo">[http://adsabs.harvard.edu/abs/2003ApJ...591..288H How Massive Single Stars End Their Life], A. Heger, C. L. Fryer, S. E. Woosley, N. Langer, and D. H. Hartmann, ''Astrophysical Journal'' '''591''', #1 (2003), pp. 288–300.</ref><ref>[http://adsabs.harvard.edu/abs/2005JPhG...31S.651S Strange quark matter in stars: a general overview], Jürgen Schaffner-Bielich, ''Journal of Physics G: Nuclear and Particle Physics'' '''31''', #6 (2005), pp. S651–S657; also [http://arxiv.org/abs/astro-ph/0412215 arXiv astro-ph/0412215].</ref> 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.<ref name="evo2"/><sup>, pp. 1046–1047.</sup> The decrease in gravitational potential energy of the collapsing core releases a large amount of energy which is on the order of 10<sup>46</sup> [[joule]]s (100 [[foe (unit)|foe]]s). Most of this energy is carried away by the emitted neutrinos.<ref name="physns">[http://adsabs.harvard.edu/abs/2004astro.ph..5262L The Physics of Neutron Stars], by J. M. Lattimer and M. Prakash, ''Science'' '''304''', #5670 (2004), pp. 536–542; also [http://arxiv.org/abs/astro-ph/0405262 arXiv astro-ph/0405262].</ref> This process is believed to be responsible for [[core-collapse supernova|supernovae of types Ib, Ic, and II]].<ref name="evo2"/>
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| [[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 [[compression (physical)|compression]]al heating, its temperature also increases. This results in an increasing rate of [[nuclear fusion|fusion]] reactions, eventually igniting a [[thermonuclear]] flame ([[carbon detonation]]) which causes the supernova.<ref name="sniamodels">[http://adsabs.harvard.edu/abs/2000ARA&A..38..191H Type IA Supernova Explosion Models], Wolfgang Hillebrandt and Jens C. Niemeyer, ''Annual Review of Astronomy and Astrophysics'' '''38''' (2000), pp. 191–230.</ref><sup>, §5.1.2</sup>
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| 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<sub>V</sub> is approximately -19.3, with a [[standard deviation]] of no more than 0.3.<ref name="sniamodels"/><sup>, (1)</sup> A [[Confidence interval|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.
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| ==Super-Chandrasekhar mass Supernovae==
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| {{main|SN 2003fg|l1=Champagne Supernova}}
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| In April 2003, the [[Supernova Legacy Survey]] observed a type Ia supernova, designated [[SNLS-03D3bb]], in a galaxy approximately 4 billion [[light year]]s 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 "[[SN 2003fg|Champagne Supernova]]" by University of Oklahoma astronomer David R. Branch, may have been spinning so fast that [[Centrifugal force (fictitious)|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]].<ref>[http://www.eurekalert.org/pub_releases/2006-09/dbnl-twt092006.php The weirdest Type Ia supernova yet], LBL press release, web page accessed 13-I-2007.</ref><ref>[http://www.spacedaily.com/reports/Champagne_Supernova_Challenges_Ideas_about_How_Supernovae_Work_999.html Champagne Supernova Challenges Ideas about How Supernovae Work], web page, spacedaily.com, accessed 13-I-2007.</ref><ref>[http://www.nature.com/nature/journal/v443/n7109/abs/nature05103.html The type Ia supernova SNLS-03D3bb from a super-Chandrasekhar-mass white dwarf star], D. Andrew Howell et al., ''Nature'' '''443''' (September 21, 2006), pp. 308–311; also, [http://arxiv.org/abs/astro-ph/0609616 arXiv:astro-ph/0609616].</ref>
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| Since the observation of the Champagne Supernova in 2003, more very bright type Ia [[supernova]]e are thought to have originated by [[white dwarf]]s whose masses exceeded the Chandrasekhar limit. These include [[SN 2006gz]], [[SN 2007if]] and [[SN 2009dc]].<ref name="Machisu">{{cite journal|last=Hachisu|first=Izumi|coauthors=Kato, M. et al.|title=A single degenerate progenitor model for type Ia supernovae highly exceeding the Chandrasekhar mass limit|bibcode=2012ApJ...744...69H|doi=10.1088/0004-637X/744/1/69|arxiv=1106.3510|volume=744|issue=1|year=2012|journal=The Astrophysical Journal|at=Article ID 69|pages=76-79}}</ref> The super-Chandrasekhar mass white dwarfs that have originated these supernovae are believed to have had masses up to 2.4–2.8 [[solar mass]]es.<ref name="Machisu" /> 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 (waves)|polarization]] smaller than 0.3, making the large asphericity theory unlikely.<ref name="Machisu" />
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| ==Tolman–Oppenheimer–Volkoff limit==
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| After a supernova explosion, a [[neutron star]] may be left behind. Like white dwarfs these objects are extremely compact and are supported by degeneracy pressure, but a neutron star is so massive and compressed that electrons and protons have combined to form neutrons, and the star is thus supported by neutron degeneracy pressure instead of electron degeneracy pressure. The limit of neutron degeneracy pressure, analogous to the Chandrasekhar limit, is known as the [[Tolman–Oppenheimer–Volkoff limit]].
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| ==References==
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| {{Reflist|colwidth=30em}}
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| ==Further reading==
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| *[http://nobelprize.org/nobel_prizes/physics/laureates/1983/chandrasekhar-lecture.pdf ''On Stars, Their Evolution and Their Stability''], Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
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| *[http://www.davegentile.com/thesis/white_dwarfs.html ''White dwarf stars and the Chandrasekhar limit''], Masters' thesis, Dave Gentile, [[DePaul University]], 1995.
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| *[http://www.sciencebits.com/StellarEquipartition 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.
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| {{white dwarf}}
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| {{DEFAULTSORT:Chandrasekhar Limit}}
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| [[Category:Astrophysics]]
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| [[Category:White dwarfs]]
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