View source for Chandrasekhar limit

{{Short description|Maximum mass of a stable white dwarf star}}
{{Use American English|date = February 2019}}
The '''Chandrasekhar limit''' ({{IPAc-en|ˌ|tʃ|ə|n|d|r|ə|ˈ|ʃ|eɪ|k|ər}})<ref>{{Cite web|url=https://www.youtube.com/watch?v=n-lJjR7pM7k&t=16s|title=Great Indians: Professor Subrahmanyan Chandrasekhar|date=26 January 2014 |via=[[NDTV]]}}</ref> is the maximum mass of a [[Hydrostatic equilibrium|stable]] [[white dwarf]] [[star]]. These stars resist [[gravitational collapse]] primarily through [[electron degeneracy pressure]], compared to [[main sequence]] stars, which resist collapse through [[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.<ref name="DarkMatter">Sean Carroll, Ph.D., Caltech, 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 much, and the star collapses to a neutron star or black hole..."</ref> The value of the Chandrasekhar limit depends upon the ratio of the number of electrons to nucleons (neutrons plus protons) in the star. For small stars this ratio is around 1/2 and the limit is  about {{Solar mass|1.44|link=y}} ({{val|2.765e30|u=kg}}).<ref>{{Cite journal |last1=Bethe |first1=Hans A. |last2=Brown |first2=Gerald |date=1985 |title=How a Supernova Explodes |url=https://www.jstor.org/stable/24967637 |journal=Scientific American |volume=252 |issue=5 |pages=60–69 |doi=10.1038/scientificamerican0585-60 |jstor=24967637 |bibcode=1985SciAm.252e..60B |issn=0036-8733}}</ref> The limit was named after [[Subrahmanyan Chandrasekhar]].<ref>{{Cite web |title=Chandrasekhar limit {{!}} White Dwarf, Neutron Star & Supernova {{!}} Britannica |url=https://www.britannica.com/science/Chandrasekhar-limit |access-date=2024-07-13 |website=www.britannica.com |language=en}}</ref>

==Physics==
[[Image:WhiteDwarf mass-radius en.svg|thumb|upright=1.95|Radius–mass relations for a model white dwarf. 
{{legend-line|solid green|Using the general pressure law for an ideal [[Fermi gas]]}}
{{legend-line|solid navy|Non-relativistic ideal Fermi gas}}
{{legend-line|solid black|[[Ultrarelativistic limit]]}}]]

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 star]]s, remaining that way throughout the rest of the history of the universe (assuming the absence of external forces). Stars above the limit can become [[neutron star]]s or [[black hole]]s.<ref name=Illari2019>{{Cite journal |last=Illari |first=Phyllis |date=2019 |title=Mechanisms, Models and Laws in Understanding Supernovae |journal=Journal for General Philosophy of Science |language=en |volume=50 |issue=1 |pages=63–84 |doi=10.1007/s10838-018-9435-y |issn=0925-4560|doi-access=free }}</ref>{{rp|74}}

The Chandrasekhar limit is a consequence of competition between gravity and electron degeneracy pressure.
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 [[Electronic band structure|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 {{math|1=''P'' = ''K''<sub>1</sub>''ρ''<sup>5/3</sup>}}, where {{mvar|P}} is the [[pressure]], {{mvar|ρ}} is the [[mass density]], and {{math|''K''<sub>1</sub>}} is a constant.  Solving the hydrostatic equation leads to a model white dwarf that is a [[polytrope]] of index {{sfrac|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">{{cite journal | last1 = Chandrasekhar | first1 = S. | year = 1931 | title = The Density of White Dwarf Stars | journal = Philosophical Magazine | volume = 11 | issue = 70| pages = 592–596 | doi=10.1080/14786443109461710| s2cid = 119906976 }}</ref>

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|1=''P'' = ''K''<sub>2</sub>''ρ''<sup>4/3</sup>}}. This yields a polytrope of index 3, which has a total mass, {{math|''M''<sub>limit</sub>}}, depending only on {{math|''K''<sub>2</sub>}}.<ref name="chandra4">{{cite journal | year = 1931 | title =  The Maximum Mass of Ideal White Dwarfs| doi = 10.1086/143324 | journal = Astrophysical Journal | volume = 74 | pages = 81–82 | bibcode = 1931ApJ....74...81C | last1 = Chandrasekhar | first1 = S.| doi-access = free }}</ref>

For a fully relativistic treatment, the equation of state used interpolates between the equations {{math|1=''P'' = ''K''<sub>1</sub>''ρ''<sup>5/3</sup>}} for small {{mvar|ρ}} and {{math|1=''P'' = ''K''<sub>2</sub>''ρ''<sup>4/3</sup>}} for large {{mvar|ρ}}. When this is done, the model radius still decreases with mass, but becomes zero at {{math|''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. {{math|''μ''<sub>e</sub>}} has been set equal to 2. 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''] {{Webarchive|url=https://web.archive.org/web/20170508162629/http://vizier.u-strasbg.fr/doc/catstd-3.2.htx |date=2017-05-08 }}, section 3.2.2, web page, accessed 12-I-2007.</ref> or kilometers, and mass in standard solar masses.

Calculated values for the limit vary depending on the [[Atomic nucleus|nuclear]] composition of the mass.<ref name="timmes"/>  Chandrasekhar<ref name="chandra1">{{cite journal | last1 = Chandrasekhar | first1 = S. | year = 1931 | title = The Highly Collapsed Configurations of a Stellar Mass | journal = [[Monthly Notices of the Royal Astronomical Society]] | volume = 91 | issue = 5| pages = 456–466 | bibcode=1931MNRAS..91..456C | doi = 10.1093/mnras/91.5.456| doi-access = free }}</ref>{{rp|loc=eq. (36)}}<ref name="chandra2">{{cite journal | last1 = Chandrasekhar | first1 = S. | year = 1935 | title = The Highly Collapsed Configurations of a Stellar Mass (second paper) | journal = Monthly Notices of the Royal Astronomical Society | volume = 95 | issue = 3| pages = 207–225 | bibcode=1935MNRAS..95..207C | doi=10.1093/mnras/95.3.207| doi-access = free }}</ref>{{rp|loc=eq. (58)}}<ref name="chandranobel">[https://www.nobelprize.org/prizes/physics/1983/chandrasekhar/lecture/ ''On Stars, Their Evolution and Their Stability''] {{Webarchive|url=https://web.archive.org/web/20101215092618/http://nobelprize.org/nobel_prizes/physics/laureates/1983/chandrasekhar-lecture.pdf |date=2010-12-15 }}, Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.</ref>{{rp|loc=eq. (43)}} gives the following expression, based on the [[equation of state]] for an ideal [[Fermi gas]]:
<math display="block"> 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}</math>
where:
*{{mvar|ħ}} is the [[reduced Planck constant]]
*{{mvar|c}} is the [[speed of light]]
*{{mvar|G}} is the [[gravitational constant]]
*{{math|''μ''<sub>e</sub>}} is the average [[molecular weight]] per electron, which depends upon the chemical composition of the star
*{{math|''m''<sub>H</sub>}} is the mass of the [[hydrogen]] atom
*{{math|''ω''{{su|b=3|p=0|lh=0.9}} ≈ 2.018236}} is a constant connected with the solution to the [[Lane–Emden equation]]
As <math>\sqrt{\hbar c/G}</math> is the [[Planck mass]], the limit is of the order of
<math display="block">\frac{M_\text{Pl}^3}{m_\text{H}^2}</math>
The limiting mass can be obtained formally from the [[Chandrasekhar's white dwarf equation]] by taking the limit of large central density.

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">{{cite journal | last1 = Timmes | first1 = F. X. | last2 = Woosley | first2 = S. E. | last3 = Weaver | first3 = Thomas A. | year = 1996 | title = The Neutron Star and Black Hole Initial Mass Function | journal = Astrophysical Journal | volume = 457 | pages = 834–843 | bibcode = 1996ApJ...457..834T | doi= 10.1086/176778 | arxiv = astro-ph/9510136 | s2cid = 12451588 }}</ref> Lieb and Yau<ref>{{cite journal | last1 = Lieb | first1 = Elliott H. | last2 = Yau | first2 = Horng-Tzer | year = 1987 | title = A rigorous examination of the Chandrasekhar theory of stellar collapse | journal = Astrophysical Journal | volume = 323 | pages = 140–144 | bibcode = 1987ApJ...323..140L | doi = 10.1086/165813 | url = https://dash.harvard.edu/bitstream/1/32706795/1/1987ApJ___323__140L.pdf | access-date = 2019-09-04 | archive-date = 2022-01-25 | archive-url = https://web.archive.org/web/20220125032717/https://dash.harvard.edu/bitstream/handle/1/32706795/1987ApJ___323__140L.pdf;jsessionid=78877ABA942CDCAA7AC77557A12D3D9D?sequence=1 | url-status = live }}</ref> 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]].<ref>{{cite journal | last1 = Fowler | first1 = R. H. | year = 1926 | title = On Dense Matter | journal = Monthly Notices of the Royal Astronomical Society | volume = 87 | issue = 2| pages = 114–122 | bibcode=1926MNRAS..87..114F | doi=10.1093/mnras/87.2.114| doi-access = free }}</ref>  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.<ref>{{cite journal | last1 = Stoner | first1 = Edmund C. | year = 1929 | title = The Limiting Density of White Dwarf Stars | journal = Philosophical Magazine | volume = 7 | issue = 41| pages = 63–70 | doi=10.1080/14786440108564713}}</ref> [[Wilhelm Anderson]] applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately {{val|1.37e30|u=kg}}.<ref>{{cite journal | doi = 10.1007/BF01340146 | volume=56 | issue=11–12 | title=Uber die Grenzdichte der Materie und der Energie | year=1929 | journal=Zeitschrift für Physik | pages=851–856 | last1 = Anderson | first1 = Wilhelm|bibcode = 1929ZPhy...56..851A | s2cid=122576829 }}</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 {{val|2.19e30|u=kg}} (for {{math|''μ''<sub>e</sub> {{=}} 2.5}}).<ref>{{cite journal | last1 = Stoner | first1 = Edmund C. | year = 1930 | title = The Equilibrium of Dense Stars | journal = Philosophical Magazine | volume = 9 | issue = 60 | pages = 944–963 | bibcode = 1930LEDPM...9..944S }}</ref>  Stoner went on to derive the [[pressure]]–[[density]] equation of state, which he published in 1932.<ref>{{cite journal | last1 = Stoner | first1 = E. C. | year = 1932 | title = The minimum pressure of a degenerate electron gas | journal = Monthly Notices of the Royal Astronomical Society | volume = 92 | issue = 7| pages = 651–661 | bibcode=1932MNRAS..92..651S | doi=10.1093/mnras/92.7.651| doi-access = free }}</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>{{cite journal | doi = 10.1007/BF01328867 | volume=50 | issue=3–4 | title=Anwendung der Pauli-Fermischen Elektronengastheorie auf das Problem der Kohäsionskräfte | year=1928 | journal=Zeitschrift für Physik | pages=234–248 | last1 = Frenkel | first1 = J.|bibcode = 1928ZPhy...50..234F | s2cid=120252049 }}.</ref>  Frenkel's work, however, was ignored by the astronomical and astrophysical community.<ref>{{cite journal | last1 = Yakovlev | first1 = D. G. | year = 1994 | title = The article by Ya I Frenkel' on 'binding forces' and the theory of white dwarfs | journal = Physics-Uspekhi | volume = 37 | issue = 6| pages = 609–612 | bibcode=1994PhyU...37..609Y | doi=10.1070/pu1994v037n06abeh000031| s2cid = 122454024 }}</ref>

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.<ref name="nasbio">[http://www.nap.edu/readingroom/books/biomems/schandrasekhar.html Chandrasekhar's biographical memoir at the National Academy of Sciences] {{Webarchive|url=https://web.archive.org/web/19991008143159/http://www.nap.edu/readingroom/books/biomems/schandrasekhar.html |date=1999-10-08 }}, web page, accessed 12-01-2007.</ref> In these papers, Chandrasekhar solved 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>{{cite journal | last1 = Chandrasekhar | first1 = S. | year = 1934 | title = Stellar Configurations with degenerate Cores | journal = The Observatory | volume = 57 | pages = 373–377 | bibcode = 1934Obs....57..373C}}</ref>  Chandrasekhar reviews this work in his Nobel Prize lecture.<ref name="chandranobel"/>

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 [[Edmund Clifton Stoner|E. C. Stoner]] for a  uniform density star in 1929.  Eric G. Blackman wrote that the roles of Stoner and Anderson in the discovery of mass limits were overlooked when [[Freeman Dyson]] wrote a biography of Chandrasekhar.<ref>Eric G. Blackman, "Giants of physics found white-dwarf mass limits", [http://adsabs.harvard.edu/abs/2006Natur.440..148B ''Nature'' 440, 148 (2006)]</ref> Michael Nauenberg claims that Stoner established the mass limit first.<ref>Michael Nauenberg, "Edmund C. Stoner and the Discovery of the Maximum Mass of White Dwarfs," [http://adsabs.harvard.edu/abs/2008JHA....39..297N ''Journal for the History of Astronomy'', Vol. 39, p. 297–312, (2008)] {{Webarchive|url=https://web.archive.org/web/20220125032719/https://ui.adsabs.harvard.edu/abs/2008JHA....39..297N/abstract |date=2022-01-25 }}</ref>
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."<ref name=Trimble2011>Virginia Trimble, "Chandrasekhar and the history of astronomy", [https://www.worldscientific.com/doi/10.1142/9789814374774_0005 ''Fluid Flows to Black Holes'', pp. 49–50 (2011)]</ref>  This value was also computed in 1932 by the Soviet physicist [[Lev 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 and concluded that quantum laws might be invalid for stars heavier than 1.5 solar mass.

=== Chandrasekhar–Eddington dispute ===
{{Main|Chandrasekhar–Eddington dispute}}
In 1935, Chandrasekhar presented his work on the limit at a scientific conference. It was immediately opposed by the established British [[astrophysicist]] [[Arthur Eddington]].<ref name="eos"/>{{rp|12}} 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 the talk by Chandrasekhar on the limit, Eddington remarked:

{{Blockquote|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>{{cite journal | year = 1935 | title = Meeting of the Royal Astronomical Society, Friday, 1935 January 11 | journal = The Observatory | volume = 58 | pages = 33–41 | bibcode=1935Obs....58...33.}}</ref>}}

Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law {{math|''P'' {{=}} ''K''<sub>1</sub>''ρ''<sup>5/3</sup>}} universally applicable, even for large {{mvar|ρ}}.<ref>{{cite journal | last1 = Eddington | first1 = A. S. | year = 1935 | title = On "Relativistic Degeneracy" | journal = Monthly Notices of the Royal Astronomical Society | volume = 95 | issue = 3| pages = 194–206 | bibcode=1935MNRAS..95..194E | doi=10.1093/mnras/95.3.194a| doi-access = free }}</ref>  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.<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'': [https://www.theguardian.com/books/2005/apr/30/featuresreviews.guardianreview24 The battle of black holes] .</ref>{{rp|pp.=110–111}}  Through the rest of his life, Eddington held to his position in his writings,<ref>{{cite journal | year = 1935 | title = The International Astronomical Union meeting in Paris, 1935 | journal = The Observatory | volume = 58 | pages = 257–265 [259] | bibcode=1935Obs....58..257.}}</ref><ref>{{cite journal | last1 = Eddington | first1 = A. S. | year = 1935 | title = Note on "Relativistic Degeneracy" | journal = Monthly Notices of the Royal Astronomical Society | volume = 96 | pages = 20–21 | bibcode=1935MNRAS..96...20E|doi = 10.1093/mnras/96.1.20 | doi-access = free }}</ref><ref>{{cite journal | last1 = Eddington | first1 = Arthur | year = 1935| title = The Pressure of a Degenerate Electron Gas and Related Problems | journal = Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences | volume = 152 | issue = 876| pages = 253–272 | jstor=96515 | doi=10.1098/rspa.1935.0190|bibcode = 1935RSPSA.152..253E | doi-access = free }}</ref><ref>''Relativity Theory of Protons and Electrons'', Sir Arthur Eddington, Cambridge: Cambridge University Press, 1936, chapter 13.</ref><ref>{{cite journal | last1 = Eddington | first1 = A. S. | year = 1940 | title = The physics of white dwarf matter | journal = Monthly Notices of the Royal Astronomical Society | volume = 100 | issue = 8| pages = 582–594 | bibcode=1940MNRAS.100..582E | doi=10.1093/mnras/100.8.582| doi-access = free }}</ref> including his work on his [[Arthur Stanley Eddington#Fundamental theory and the Eddington number|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:

{{blockquote|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.<ref name="eos"/>{{rp|p=150}}}}

However, Chandrasekhar chose to move on, leaving the study of stellar structure to focus on stellar dynamics.<ref name=Trimble2011/>{{rp|51}} 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]].<ref>{{Cite web |title=The Nobel Prize in Physics 1983 |url=https://www.nobelprize.org/prizes/physics/1983/summary/ |access-date=2023-10-03 |website=NobelPrize.org |language=en-US}}</ref>

==Applications==
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 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.<ref name="evo2">{{cite journal | last1 = Woosley | first1 = S. E. | last2 = Heger | first2 = A. | last3 = Weaver | first3 = T. A. | s2cid = 55932331 | year = 2002 | title = The evolution and explosion of massive stars | journal = Reviews of Modern Physics | volume = 74 | issue = 4| pages = 1015–1071 | bibcode=2002RvMP...74.1015W | doi=10.1103/revmodphys.74.1015}}</ref>

If a main-sequence star is not too massive (less than approximately 8 [[solar mass]]es), 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.)<ref name="ifmr1">{{cite journal | last1 = Koester | first1 = D. | last2 = Reimers | first2 = D. | year = 1996 | title = White dwarfs in open clusters. VIII. NGC 2516: a test for the mass-radius and initial-final mass relations | journal = Astronomy and Astrophysics | volume = 313 | pages = 810–814 | bibcode=1996A&A...313..810K}}</ref><ref name="ifmr2">Kurtis A. Williams, M. Bolte, and Detlev Koester 2004 [http://adsabs.harvard.edu/abs/2004ApJ...615L..49W An Empirical Initial-Final Mass Relation from Hot, Massive White Dwarfs in NGC 2168 (M35)] {{Webarchive|url=https://web.archive.org/web/20070819215754/http://adsabs.harvard.edu/abs/2004ApJ...615L..49W |date=2007-08-19 }},  ''Astrophysical Journal'' '''615''', pp. L49–L52 [https://arxiv.org/abs/astro-ph/0409447 arXiv astro-ph/0409447] .</ref><ref name="evo">{{cite journal | last1 = Heger | first1 = A. | last2 = Fryer | first2 = C. L. | last3 = Woosley | first3 = S. E. | last4 = Langer | first4 = N. | last5 = Hartmann | first5 = D. H. | year = 2003 | title = How Massive Single Stars End Their Life | journal = Astrophysical Journal | volume = 591 | issue = 1| pages = 288–300 | bibcode=2003ApJ...591..288H | doi=10.1086/375341|arxiv = astro-ph/0212469 | s2cid = 59065632 }}</ref><ref>{{cite journal | last1 = Schaffner-Bielich | first1 = Jürgen | year = 2005 | title = Strange quark matter in stars: a general overview] | arxiv = astro-ph/0412215 | journal = Journal of Physics G: Nuclear and Particle Physics | volume = 31 | issue = 6| pages = S651–S657 | doi=10.1088/0954-3899/31/6/004|bibcode = 2005JPhG...31S.651S | s2cid = 118886040 }}</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" />{{rp|pp=1046–1047}} The decrease in [[gravitational energy|gravitational potential energy]] of the collapsing core releases a large amount of energy on the order of {{val|e=46|ul=J}} (100&nbsp;[[foe (unit)|foe]]s). Most of this energy is carried away by the emitted neutrinos<ref name="physns">{{cite journal |last1=Lattimer |first1=James M. |last2=Prakash |first2=Madappa |year=2004 |title=The Physics of Neutron Stars |arxiv=astro-ph/0405262 |journal=Science |volume=304 |issue=5670 |pages=536–542 |doi=10.1126/science.1090720 |pmid=15105490 |bibcode=2004Sci...304..536L |s2cid=10769030 }}</ref> and the kinetic energy of the expanding shell of gas; only about 1% is emitted as optical light.<ref>Schneider, Stephen E.; and Arny, Thomas T.; [http://abyss.uoregon.edu/~js/ast122/lectures/lec18.html ''Readings: Unit 66: End of a star's life''] {{Webarchive|url=https://web.archive.org/web/20200214212338/http://abyss.uoregon.edu/~js/ast122/lectures/lec18.html |date=2020-02-14 }}, Astronomy 122: ''Birth and Death of Stars'', University of Oregon</ref> This process is believed responsible for [[core-collapse supernova|supernovae of types Ib, Ic, and II]].<ref name="evo2" />

[[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. 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 eventually ignites [[nuclear fusion]] reactions, leading to an immediate [[carbon detonation]], which disrupts the star and causes the supernova.<ref name="sniamodels">{{cite journal |last1=Hillebrandt |first1=Wolfgang |last2=Niemeyer |first2=Jens C. |year=2000 |title=Type IA Supernova Explosion Models |journal=Annual Review of Astronomy and Astrophysics |volume=38 |pages=191–230 |doi=10.1146/annurev.astro.38.1.191 |bibcode=2000ARA&A..38..191H |arxiv = astro-ph/0006305 |s2cid=10210550 }}</ref>{{rp|loc=§5.1.2}}

A strong indication of the reliability of Chandrasekhar's formula is that the [[absolute magnitude]]s of supernovae of Type Ia are all approximately the same; at maximum luminosity, {{math|''M''<sub>V</sub>}} is approximately −19.3, with a [[standard deviation]] of no more than 0.3.<ref name="sniamodels"/>{{rp|loc=eq. (1)}}  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.

==Super-Chandrasekhar mass supernovas==
{{main|SN 2003fg|l1=Champagne Supernova}}
In April&nbsp;2003, the [[Supernova Legacy Survey]] observed a type&nbsp;Ia supernova, designated [[SNLS-03D3bb]], in a galaxy approximately 4&nbsp;billion [[light year]]s away. According to 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 "[[SN 2003fg|Champagne Supernova]]"<ref name="journal-nature">{{cite journal
 | last = Branch
 | first = David
 | date = 21 September 2006
 | title = Astronomy: Champagne supernova
 | journal = [[Nature (journal)|Nature]]
 | volume = 443
 | pages = 283–284
 | doi = 10.1038/443283a
 | bibcode = 2006Natur.443..283B
 | issue=7109
 | pmid=1698869
| doi-access = free
 }}</ref> 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. Another 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" /> Nevertheless, they point out that this observation poses a challenge to the use of type&nbsp;Ia supernovae as [[standard candles]].<ref>{{cite press release |title=The weirdest type&nbsp;Ia supernova yet |publisher=LBL |url=http://www.eurekalert.org/pub_releases/2006-09/dbnl-twt092006.php |access-date=13 January 2007 |archive-date=6 July 2017 |archive-url=https://web.archive.org/web/20170706233212/https://www.eurekalert.org/pub_releases/2006-09/dbnl-twt092006.php |url-status=live}}</ref><ref>{{cite press release
 |title=Champagne supernova challenges ideas about how supernovae work
 |website=spacedaily.com
 |url=http://www.spacedaily.com/reports/Champagne_Supernova_Challenges_Ideas_about_How_Supernovae_Work_999.html
 |access-date=13 January 2007
 |archive-date=1 July 2017
 |archive-url=https://web.archive.org/web/20170701113924/http://www.spacedaily.com/reports/Champagne_Supernova_Challenges_Ideas_about_How_Supernovae_Work_999.html
 |url-status=live
 }}</ref><ref>{{cite journal
 |last1=Howell |first1=D. Andrew
 |year=2006
 |title=The type&nbsp;Ia supernova SNLS-03D3bb from a super-Chandrasekhar-mass white dwarf star
 |journal=Nature
 |volume=443 |issue=7109 |pages=308–311
 |arxiv=astro-ph/0609616 |doi=10.1038/nature05103
 |pmid=16988705 |bibcode=2006Natur.443..308H
|s2cid=4419069
 }}</ref>

Since the observation of the Champagne Supernova in 2003, several more [[type Ia supernova|type&nbsp;Ia supernovae]] have been observed that are very bright, and thought to have originated from [[white dwarf]]s whose masses exceeded the Chandrasekhar limit. These include [[SN 2006gz]], [[SN 2007if]], and [[SN 2009dc]].<ref name="Machisu">{{cite journal
 |last1=Hachisu |first1=Izumi
 |last2=Kato |first2=M.
 |display-authors=etal
 |year=2012
 |title=A single degenerate progenitor model for type Ia supernovae highly exceeding the Chandrasekhar mass limit
 |journal=The Astrophysical Journal
 |volume=744 |issue=1 |pages=76–79
 |id=Article 69
 |bibcode=2012ApJ...744...69H
 |doi=10.1088/0004-637X/744/1/69 |arxiv=1106.3510
|s2cid=119264873
 }}</ref> These super-Chandrasekhar mass white dwarfs are believed to have had masses up to 2.4–2.8&nbsp;[[solar mass]]es.<ref name="Machisu" />

Theoretical studies of white dwarf stars with very high magnetic fields predict [[Landau quantization]] of the electron gas and under these conditions the white dwarf can form from a mass exceeding the Chandrasekhar limit.<ref name="Kundu2012">{{cite journal
 |last1=Kundu |first1=Aritra
 |last2=Mukhopadhyay |first2=Banibrata
 |year=2012
 |title=Mass of highly magnetized white dwarfs exceeding the Chandrasekhar limit: An analytical view
 |journal=Modern Physics Letters A
 |volume=27 |issue=15 |pages=1250084
 |doi=10.1142/S0217732312500845
 |doi-broken-date=24 October 2025
 }}</ref>

==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 result will be a [[neutron star]]; but if the total mass is above the Tolman-Oppenheimer-Volkhoff limit, the result will be a [[black hole]].<ref name=Illari2019/>{{rp|74}}

==See also==
*[[Bekenstein bound]]
*[[Chandrasekhar's white dwarf equation]]
*[[Schönberg–Chandrasekhar limit]]
*[[Tolman–Oppenheimer–Volkoff limit]]

==References==
{{Reflist|30em}}

==Further reading==
*[https://www.nobelprize.org/uploads/2018/06/chandrasekhar-lecture.pdf ''On Stars, Their Evolution and Their Stability''], Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
*[http://www.davegentile.com/thesis/white_dwarfs.html ''White dwarf stars and the Chandrasekhar limit''], Masters' thesis, Dave Gentile, [[DePaul University]], 1995.
*[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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