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== Composition and structure == {{star nav}} Although white dwarfs are known with estimated masses as low as {{solar mass|0.17}}<ref> {{cite journal |last1=Kilic |first1=M. |last2=Allende Prieto |first2=C. |last3=Brown |first3=Warren R. |last4=Koester |first4=D. |year=2007 |title=The lowest mass white dwarf |journal=[[The Astrophysical Journal]] |volume=660 |issue=2 |pages=1451–1461 |arxiv= astro-ph/0611498 |bibcode=2007ApJ...660.1451K |doi= 10.1086/514327 |s2cid=18587748 }} </ref> and as high as {{solar mass|1.33}},<ref name="sdsswd"> {{cite journal |last1=Kepler |first1=S.O. |author1-link=Kepler de Souza Oliveira |last2=Kleinman |first2=S.J. |last3=Nitta |first3=A. |last4=Koester |first4=D. |last5=Castanheira |first5=B.G. |last6=Giovannini |first6=O. |last7=Costa |first7=A.F.M. |last8=Althaus |first8=L. |year=2007 |title=White dwarf mass distribution in the SDSS |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=375 |issue=4 |pages=1315–1324 |arxiv= astro-ph/0612277 |bibcode=2007MNRAS.375.1315K |doi=10.1111/j.1365-2966.2006.11388.x |doi-access=free |s2cid=10892288 }} </ref> the mass distribution is strongly peaked at {{solar mass|0.6}}, and the majority lie between {{solar mass|0.5 and 0.7}}.<ref name=sdsswd/> The estimated radii of observed white dwarfs are typically 0.8–2% the [[solar radius|radius of the Sun]];<ref> {{cite journal |last=Shipman |first=H.L. |year=1979 |title=Masses and radii of white-dwarf stars. III – Results for 110 hydrogen-rich and 28 helium-rich stars |journal=[[The Astrophysical Journal]] |volume=228 |page=240 |bibcode=1979ApJ...228..240S |doi=10.1086/156841 }} </ref> this is comparable to the Earth's radius of approximately 0.9% solar radius. A white dwarf, then, packs mass comparable to the Sun's into a volume that is typically a million times smaller than the Sun's; the average density of matter in a white dwarf must therefore be, very roughly, {{val|1000000}} times greater than the average density of the Sun, or approximately {{val|e=6|ul=g/cm3}}, or 1 [[tonne]] per cubic centimetre.<ref name=osln/> A typical white dwarf has a density of between 10<sup>4</sup> and {{val|e=7|u=g/cm3}}. White dwarfs are composed of one of the densest forms of matter known, surpassed only by other [[compact star]]s such as [[neutron star]]s and the hypothetical [[quark star]]s.<ref name="Saumon2022"/><ref>{{cite book|first=Andreas |last=Schmitt |title= Dense Matter in Compact Stars: A Pedagogical Introduction |series=Lecture Notes in Physics |year=2010 |volume=811 |publisher=Springer |pages=4,143 |doi=10.1007/978-3-642-12866-0 |isbn=978-3-642-12866-0 |arxiv=1001.3294}}</ref> White dwarfs were found to be extremely dense soon after their discovery. If a star is in a [[binary star|binary]] system, as is the case for Sirius B or 40 Eridani B, it is possible to estimate its mass from observations of the binary orbit. This was done for Sirius B by 1910,<ref> {{cite book |last=Boss |first=L. |year=1910 |title=Preliminary General Catalogue of 6188 stars for the epoch 1900 |publisher=[[Carnegie Institution of Washington]] |bibcode=1910pgcs.book.....B |lccn=10009645 |url=https://archive.org/details/preliminarygene00obsegoog |via=Archive.org }} </ref> yielding a mass estimate of {{solar mass|0.94}}, which compares well with a more modern estimate of {{solar mass|1.00}}.<ref name=apj_630> {{cite journal |last1=Liebert |first1=James |last2=Young |first2=P. A. |last3=Arnett |first3=D. |last4=Holberg |first4=J. B. |last5=Williams |first5=K. A. |date=2005 |title=The age and progenitor mass of Sirius B |journal=[[The Astrophysical Journal]] |volume=630 |issue=1 |page=L69 |arxiv= astro-ph/0507523 |bibcode=2005ApJ...630L..69L |doi= 10.1086/462419 |s2cid=8792889 }}</ref> Since hotter bodies radiate more energy than colder ones, a star's surface brightness can be estimated from its [[effective temperature|effective surface temperature]], and that from its [[stellar spectrum|spectrum]]. If the star's distance is known, its absolute luminosity can also be estimated. From the absolute luminosity and distance, the star's surface area and its radius can be calculated. Reasoning of this sort led to the realization, puzzling to astronomers at the time, that due to their relatively high temperature and relatively low absolute luminosity, Sirius B and 40 Eridani B must be very dense. When [[Ernst Öpik]] estimated the density of a number of visual binary stars in 1916, he found that 40 Eridani B had a density of over {{val|25000}} times that of the [[Sun]], which was so high that he called it "impossible".<ref> {{cite journal |last1=Öpik |first1=E. |date=1916 |title=The densities of visual binary stars |journal=[[The Astrophysical Journal]] |volume=44 |page=292 |bibcode=1916ApJ....44..292O |doi= 10.1086/142296 |doi-access=free }} </ref> As [[Arthur Eddington]] put it later, in 1927:<ref> {{cite book |last=Eddington |first=A.S. |author-link=Arthur Stanley Eddington |date=1927 |title=Stars and Atoms |url=https://archive.org/details/in.ernet.dli.2015.173636 |publisher=[[Clarendon Press]] |lccn=27015694 }} </ref>{{rp|page=50}} <blockquote>We learn about the stars by receiving and interpreting the messages which their light brings to us. The message of the companion of Sirius when it was decoded ran: "I am composed of material 3000 times denser than anything you have ever come across; a ton of my material would be a little nugget that you could put in a matchbox." What reply can one make to such a message? The reply which most of us made in 1914 was — "Shut up. Don't talk nonsense."</blockquote> As Eddington pointed out in 1924, densities of this order implied that, according to the theory of [[general relativity]], the light from Sirius B should be [[gravitational redshift|gravitationally redshifted]].<ref name="eddington"> {{cite journal |last1=Eddington |first1=A. S. |date=1924 |title=On the relation between the masses and luminosities of the stars |journal=Monthly Notices of the Royal Astronomical Society |volume=84 |issue=5 |pages=308–333 |bibcode=1924MNRAS..84..308E |doi=10.1093/mnras/84.5.308 |doi-access=free }}</ref> This was confirmed when Adams measured this redshift in 1925.<ref> {{cite journal |last1=Adams |first1=W. S. |date=1925 |title=The Relativity Displacement of the Spectral Lines in the Companion of Sirius |journal=Proceedings of the National Academy of Sciences |volume=11 |issue=7 |pages=382–387 |bibcode=1925PNAS...11..382A |doi= 10.1073/pnas.11.7.382 |pmid=16587023 |pmc=1086032|doi-access=free }}</ref> {| class="wikitable" style="width:50%; text-align:left; float:left; margin-right:1em;" |- ! Material !! [[Density]] [{{val|u=kg/m3}}] !! Notes |- | Supermassive black hole || {{circa}} 1000<ref name="CMS1999">{{cite journal|last1=Celotti |first1=A. |last2=Miller |first2=J.C. |last3=Sciama |first3=D.W. |title= Astrophysical evidence for the existence of black holes |date=1999|pages=A3–A21|issue=12A |journal=Class. Quantum Grav. |volume=16 |arxiv=astro-ph/9912186 |doi = 10.1088/0264-9381/16/12A/301 |bibcode=1999CQGra..16A...3C |s2cid=17677758 }}</ref> || Critical density of a black hole of around 10<sup>8</sup> solar masses. |- | Water (liquid) || 1000 || At [[Standard temperature and pressure|STP]] |- | [[Osmium]] || {{val|22610}} || Near [[room temperature]] |- | The core of the Sun || {{circa}} {{val|150000}} || |- | White dwarf || {{val|1|e=9}}<ref name="osln" /> || |- | [[Atomic nuclei]] || {{val|2.3|e=17}}<ref> {{cite web |last=Nave |first=C. R. |url=http://hyperphysics.phy-astr.gsu.edu/HBASE/Nuclear/nucuni.html |title=Nuclear Size and Density |work=[[HyperPhysics]] |publisher=[[Georgia State University]] |access-date=26 June 2009 |archive-url=https://web.archive.org/web/20090706034134/http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/nucuni.html |archive-date=6 July 2009 |url-status=live }}</ref> || Does not depend strongly on size of nucleus |- | Neutron star core || {{val|8.4|e=16}} – {{val|1|e=18}} || |- | Small black hole || {{val|2|e=30}}<ref name=adams1997> {{cite book |first1=Steve |last1=Adams |date=1997 |title=Relativity: an introduction to space-time physics |place=London; Bristol |page=240 |publisher=[[CRC Press]] |isbn=978-0-7484-0621-0 |bibcode=1997rist.book.....A }}</ref> || Critical density of an Earth-mass black hole. |} Such densities are possible because white dwarf material is not composed of [[atom]]s joined by [[chemical bond]]s, but rather consists of a [[plasma (physics)|plasma]] of unbound [[atomic nucleus|nuclei]] and [[electron]]s. There is therefore no obstacle to placing nuclei closer than normally allowed by [[atomic orbital|electron orbitals]] limited by normal matter.<ref name="eddington" /> Eddington wondered what would happen when this plasma cooled and the energy to keep the atoms ionized was no longer sufficient.<ref name="fowler"> {{cite journal |last1=Fowler |first1=R. H. |date=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 paradox was resolved by [[R. H. Fowler]] in 1926 by an application of the newly devised [[quantum mechanics]]. Since electrons obey the [[Pauli exclusion principle]], no two electrons can occupy the same [[quantum state|state]], and they must obey [[Fermi–Dirac statistics]], also introduced in 1926 to determine the statistical distribution of particles that satisfy the Pauli exclusion principle.<ref> {{cite journal |last1=Hoddeson |first1=L. H. |last2=Baym |first2=G. |date=1980 |title=The Development of the Quantum Mechanical Electron Theory of Metals: 1900–28 |journal=Proceedings of the Royal Society of London |volume=371 |issue=1744 |pages=8–23 |doi=10.1098/rspa.1980.0051 |jstor=2990270 |bibcode = 1980RSPSA.371....8H |s2cid=120476662 }}</ref> At zero temperature, therefore, electrons cannot all occupy the lowest-energy, or ''[[ground state|ground]]'', state; some of them would have to occupy higher-energy states, forming a band of lowest-available energy states, the ''[[Fermi sea]]''. This state of the electrons, called ''[[degenerate matter|degenerate]]'', meant that a white dwarf could cool to zero temperature and still possess high energy.<ref name="fowler" /><ref name="scibits"> {{cite web |title=Estimating Stellar Parameters from Energy Equipartition |url=http://www.sciencebits.com/StellarEquipartition |website=ScienceBits |author=Nir Shaviv |access-date=9 May 2007 |archive-url=https://web.archive.org/web/20120522041201/http://www.sciencebits.com/StellarEquipartition |archive-date=22 May 2012 |url-status=live }}</ref> Compression of a white dwarf will increase the number of electrons in a given volume. Applying the Pauli exclusion principle, this will increase the kinetic energy of the electrons, thereby increasing the pressure.<ref name="fowler" /><ref> {{cite web |last1=Bean |first1=R. |author-link1=Rachel Bean |title=Lecture 12 – Degeneracy pressure |url=http://www.astro.cornell.edu/~rbean/a211/211_notes_lec_12.pdf |series=Lecture notes, Astronomy 211 |publisher=[[Cornell University]] |access-date=21 September 2007 |archive-url=https://web.archive.org/web/20070925204454/http://www.astro.cornell.edu/~rbean/a211/211_notes_lec_12.pdf |archive-date=2007-09-25 }}</ref> This ''[[electron degeneracy pressure]]'' supports a white dwarf against gravitational collapse. The pressure depends only on density and not on temperature. Degenerate matter is relatively compressible; this means that the density of a high-mass white dwarf is much greater than that of a low-mass white dwarf and that the radius of a white dwarf decreases as its mass increases.<ref name="osln" /> The existence of a limiting mass that no white dwarf can exceed without collapsing to a neutron star is another consequence of being supported by electron degeneracy pressure. Such limiting masses were calculated for cases of an idealized, constant density star in 1929 by [[Wilhelm Anderson]]<ref> {{cite journal |last1=Anderson |first1=W. |author-link=Wilhelm Anderson |date=1929 |title=Über die Grenzdichte der Materie und der Energie |journal=Zeitschrift für Physik |language=de |volume=56 |issue=11–12 |pages=851–856 |bibcode=1929ZPhy...56..851A |doi=10.1007/BF01340146 |s2cid=122576829 }}</ref> and in 1930 by [[Edmund C. Stoner]].<ref name="stoner"> {{cite journal |last1=Stoner |first1=C. |date=1930 |title=The Equilibrium of Dense Stars |journal=[[Philosophical Magazine]] |volume=9 |issue=60 |page=944 |bibcode=1930LEDPM...9..944S }}</ref> This value was corrected by considering hydrostatic equilibrium for the density profile, and the presently known value of the limit was first published in 1931 by [[Subrahmanyan Chandrasekhar]] in his paper "The Maximum Mass of Ideal White Dwarfs".<ref name="chandra4"> {{cite journal |last1=Chandrasekhar |first1=S. |date=1931 |title=The Maximum Mass of Ideal White Dwarfs |journal=The Astrophysical Journal |volume=74 |page=81 |bibcode=1931ApJ....74...81C |doi= 10.1086/143324 |doi-access=free }}</ref> For a non-rotating white dwarf, it is equal to approximately {{math|{{solar mass|5.7}} / ''μ''<sub>e</sub><sup>2</sup>}}, where {{math|''μ''<sub>e</sub>}} is the average molecular weight per electron of the star.<ref name="chandra2"> {{cite journal |last1=Chandrasekhar |first1=S. |date=1935 |title=The highly collapsed configurations of a stellar mass (Second paper) |volume=95 |issue=3 |pages=207–225 |journal=Monthly Notices of the Royal Astronomical Society |bibcode=1935MNRAS..95..207C |doi=10.1093/mnras/95.3.207 |doi-access=free }}</ref>{{rp|eqn.(63)}} As the carbon-12 and oxygen-16 that predominantly compose a carbon–oxygen white dwarf both have [[atomic number]]s equal to half their [[atomic weight]], one should take {{math|''μ''<sub>e</sub>}} equal to 2 for such a star,<ref name="scibits" /> leading to the commonly quoted value of {{solar mass|1.4}}. (Near the beginning of the 20th century, there was reason to believe that stars were composed chiefly of heavy elements,<ref name="stoner" />{{rp|page=955}} so, in his 1931 paper, Chandrasekhar set the average molecular weight per electron, {{math|''μ''<sub>e</sub>}}, equal to 2.5, giving a limit of {{solar mass|0.91}}.) Together with [[William Alfred Fowler]], Chandrasekhar received the [[Nobel Prize in Physics|Nobel Prize]] for this and other work in 1983.<ref> {{cite web |title=The Nobel Prize in Physics 1983 |url=http://nobelprize.org/nobel_prizes/physics/laureates/1983/ |publisher=[[The Nobel Foundation]] |access-date=4 May 2007 |archive-url=https://web.archive.org/web/20070506154131/http://nobelprize.org/nobel_prizes/physics/laureates/1983/ |archive-date=6 May 2007 |url-status=live }}</ref> The limiting mass is now called the ''[[Chandrasekhar limit]]''.<ref> {{cite journal |doi=10.1088/1361-6552/acdbb0 |last=Low |first=Andrew M. |title=The Chandrasekhar limit: a simplified approach |journal=Physics Education |year=2023 |volume=58 |issue=4 |page=045008|bibcode=2023PhyEd..58d5008L |doi-access=free }}</ref> If a carbon-oxygen white dwarf accreted enough matter to reach the [[Chandrasekhar limit]] of about 1.44 [[solar mass]]es (for a non-rotating star), it would no longer be able to support the bulk of its mass through electron degeneracy pressure<ref name=collapse/> and, in the absence of nuclear reactions, would begin to collapse.<ref name="Chandrasekhar"> {{cite journal |last1=Lieb |first1=E. H. |last2=Yau |first2=H.-T. |date=1987 |title=A rigorous examination of the Chandrasekhar theory of stellar collapse |journal=[[The Astrophysical Journal]] |volume=323 |issue=1 |pages=140–144 |bibcode=1987ApJ...323..140L |doi=10.1086/165813 |url=https://dash.harvard.edu/handle/1/32706795 |access-date=20 March 2020 }}</ref><ref name="Mazzali2007"> {{cite journal |last1=Mazzali |first1=P. A. |last2=Röpke |first2=F. K. |last3=Benetti |first3=S. |last4=Hillebrandt |first4=W. |date=2007 |title=A Common Explosion Mechanism for Type Ia Supernovae |journal=[[Science (journal)|Science]] |volume=315 |issue=5813 |pages=825–828 |arxiv=astro-ph/0702351 |bibcode=2007Sci...315..825M |doi=10.1126/science.1136259 |pmid=17289993 }}</ref> However, the current view is that this limit is not normally attained; increasing temperature and density inside the core ignite carbon fusion as the star approaches the limit (to within about 1%) before collapse is initiated.<ref name="Mazzali2007"/><ref> {{cite book |last=Wheeler |first=J. C. |date=2000 |title=Cosmic Catastrophes: Supernovae, Gamma-Ray Bursts, and Adventures in Hyperspace |url=https://books.google.com/books?id=s3SFQgAACAAJ |page=96 |publisher=[[Cambridge University Press]] |isbn=978-0-521-65195-0 }}</ref> In contrast, for a core primarily composed of oxygen, neon and magnesium, the collapsing white dwarf will typically form a [[neutron star]]. In this case, only a fraction of the star's mass will be ejected during the collapse.<ref name="collapse"> {{cite book |last1=Canal |first1=R. |last2=Gutierrez |first2=J. |date=1997 |chapter=The Possible White Dwarf-Neutron Star Connection |title=White Dwarfs |arxiv=astro-ph/9701225 |doi=10.1007/978-94-011-5542-7_7 |volume=214 |pages=49–55 |series=Astrophysics and Space Science Library |isbn=978-94-010-6334-0 |bibcode=1997ASSL..214...49C |s2cid=9288287 }}</ref> If a white dwarf star accumulates sufficient material from a stellar companion to raise its core temperature enough to [[Carbon detonation|ignite]] [[Carbon burning process|carbon fusion]], it will undergo [[Thermal runaway|runaway]] nuclear fusion, completely disrupting it. There are three avenues by which this detonation is theorised to happen: stable [[accretion (astrophysics)|accretion]] of material from a companion, the collision of two white dwarfs, or accretion that causes ignition in a shell that then ignites the core. The dominant mechanism by which type Ia supernovae are produced remains unclear.<ref name="Piro2014"> {{cite journal |last1=Piro |first1=A. L. |last2=Thompson |first2=T. A. |last3=Kochanek |first3=C. S. |year=2014 |title=Reconciling 56Ni production in Type Ia supernovae with double degenerate scenarios |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=438 |issue=4 |pages=3456 |arxiv=1308.0334 |bibcode=2014MNRAS.438.3456P |doi=10.1093/mnras/stt2451 |doi-access=free |s2cid=27316605 }}</ref> Despite this uncertainty in how type Ia supernovae are produced, type Ia supernovae have very uniform properties and are useful [[Cosmic distance ladder|standard candles]] over intergalactic distances. Some calibrations are required to compensate for the gradual change in properties or different frequencies of abnormal luminosity supernovae at high redshift, and for small variations in brightness identified by light curve shape or spectrum.<ref name="chen"> {{cite journal |last1=Chen |first1=W.-C. |last2=Li |first2=X.-D. |year=2009 |title=On the Progenitors of Super-Chandrasekhar Mass Type Ia Supernovae |journal=[[The Astrophysical Journal]] |volume=702 |issue=1|pages=686–691 |arxiv=0907.0057 |bibcode=2009ApJ...702..686C |doi=10.1088/0004-637X/702/1/686 |s2cid=14301164 }}</ref><ref> {{cite journal |last1=Howell |first1=D. A. |last2=Sullivan |first2=M. |last3=Conley |first3=A. J. |last4=Carlberg |first4=R. G. |date=2007 |title=Predicted and Observed Evolution in the Mean Properties of Type Ia Supernovae with Redshift |journal=[[Astrophysical Journal Letters]] |volume=667 |issue=1 |pages=L37–L40 |arxiv=astro-ph/0701912 |bibcode=2007ApJ...667L..37H |doi=10.1086/522030 |s2cid=16667595 }}</ref><ref>{{cite journal|title=Standardization of type Ia supernovae |first1=Rodrigo C. V. |last1=Coelho |first2=Maurício O. |last2=Calvão |first3=Ribamar R. R. |last3=Reis |first4=Beatriz B. |last4=Siffert |arxiv=1411.3596 |journal=European Journal of Physics |volume=36 |year=2015 |issue=1 |page=015007 |doi=10.1088/0143-0807/36/1/015007|bibcode=2015EJPh...36a5007C }}</ref> White dwarfs have low [[luminosity]] and therefore occupy a strip at the bottom of the [[Hertzsprung–Russell diagram]], a graph of stellar luminosity versus color or temperature. They should not be confused with low-luminosity objects at the low-mass end of the main sequence, such as the [[hydrogen fusion|hydrogen-fusing]]<!-- it is THIS place where a hyphen must stay, oh typographers from hell --> [[red dwarf]]s, whose cores are supported in part by thermal pressure,<ref> {{cite journal |last1=Chabrier |first1=G. |last2=Baraffe |first2=I. |date=2000 |title=Theory of low-Mass stars and substellar objects |journal=Annual Review of Astronomy and Astrophysics |volume=38 |pages=337–377 |arxiv= astro-ph/0006383 |bibcode=2000ARA&A..38..337C |doi= 10.1146/annurev.astro.38.1.337 |s2cid=59325115 }}</ref> or the even lower-temperature [[brown dwarf]]s.<ref> {{cite web |last=Kaler |first=J. |title=The Hertzsprung-Russell (HR) diagram |url=http://stars.astro.illinois.edu/sow/hrd.html |access-date=5 May 2007 |archive-url=https://web.archive.org/web/20090831174414/http://stars.astro.illinois.edu/sow/hrd.html |archive-date=31 August 2009 |url-status=live }}</ref> === Mass–radius relationship === {{See also|Chandrasekhar's white dwarf equation|Neutron star#Gravity and equation of state}} The relationship between the mass and radius of white dwarfs can be estimated using the nonrelativistic [[Fermi gas]] equation of state, which gives<ref name="kawaler">{{cite book |last1=Kawaler |first1=S. D. |chapter=White Dwarf Stars |editor1-last=Kawaler |editor1-first=S. D. |editor2-last=Novikov |editor2-first=I. |editor3-last=Srinivasan |editor3-first=G. |date=1997 |title=Stellar remnants |publisher=1997 |isbn=978-3-540-61520-0 }}</ref>{{rp|25}} <math display="block"> \frac{R}{R_\odot} \approx 0.012\left ( \frac{M}{M_\odot}\right )^{-1/3} \left (\frac{\mu_e}{2}\right)^{-5/3},</math> where {{mvar|R}} is the radius, {{mvar|M}} is the mass of the white dwarf, and the subscript <math>\odot</math> indicates relative to the Sun. The [[chemical potential]], <math>\mu_e</math> is a thermodynamic property giving the change in energy as one electron is added or removed; it relates to the composition of the star. Numerical treatment of more complete models have been tested against observational data with good agreement.<ref>{{Cite journal |last1=Bédard |first1=A. |last2=Bergeron |first2=P. |last3=Fontaine |first3=G. |date=October 2017 |title=Measurements of Physical Parameters of White Dwarfs: A Test of the Mass–Radius Relation |journal=The Astrophysical Journal |language=en |volume=848 |issue=1 |pages=11 |doi=10.3847/1538-4357/aa8bb6 |doi-access=free |arxiv=1709.02324 |bibcode=2017ApJ...848...11B |issn=0004-637X}}</ref> Since this analysis uses the non-relativistic formula {{math|1= ''p''<sup>2</sup> / 2''m''}} for the kinetic energy, it is non-relativistic. When the electron velocity in a white dwarf is close to the [[speed of light]], the kinetic energy formula approaches {{math|1=''pc''}} where {{math|''c''}} is the speed of light, and it can be shown that the Fermi gas model has no stable equilibrium in the [[ultrarelativistic limit]]. In particular, this analysis yields the maximum mass of a white dwarf, which is:<ref name="kawaler"/> <math display="block">M_{\rm limit} \approx 1.46\left (\frac{\mu_e}{2}\right)^{-2}</math> The observation of many white dwarf stars implies that either they started with masses similar to the Sun or something dramatic happened to reduce their mass.<ref name="kawaler"/> [[File:ChandrasekharLimitGraph.svg|thumb|upright=1.2|right|Radius–mass relations for a model white dwarf. {{math|''M''<sub>limit</sub>}} is denoted as ''M''<sub>Ch</sub>.]] For a more accurate computation of the mass-radius relationship and limiting mass of a white dwarf, one must compute the [[equation of state]] that describes the relationship between density and pressure in the white dwarf material. If the density and pressure are both set equal to functions of the radius from the center of the star, the system of equations consisting of the [[hydrostatic equation]] together with the equation of state can then be solved to find the structure of the white dwarf at equilibrium. In the non-relativistic case, the radius is inversely proportional to the cube root of the mass.<ref name="chandra2" />{{rp|eqn.(80)}} Relativistic corrections will alter the result so that the radius becomes zero at a finite value of the mass. This is the limiting value of the mass – called the ''[[Chandrasekhar limit]]'' – at which the white dwarf can no longer be supported by electron degeneracy pressure. The graph on the right shows the result of such a computation. It shows how radius varies with mass for non-relativistic (blue curve) and relativistic (green curve) models of a white dwarf. Both models treat the white dwarf as a cold [[Fermi gas]] in hydrostatic equilibrium. The average molecular weight per electron, {{math|''μ''<sub>e</sub>}}, has been set equal to 2. Radius is measured in standard solar radii and mass in standard solar masses.<ref name="chandra2" /><ref name="stds"> {{cite web |title=Basic symbols |url=http://vizier.u-strasbg.fr/doc/catstd-3.2.htx |work=Standards for Astronomical Catalogues, Version 2.0 |access-date=12 January 2007 |publisher=[[VizieR]] |archive-url=https://web.archive.org/web/20170508162629/http://vizier.u-strasbg.fr/doc/catstd-3.2.htx |archive-date=8 May 2017 |url-status=live }}</ref> These computations all assume that the white dwarf is non-rotating. If the white dwarf is rotating, the equation of hydrostatic equilibrium must be modified to take into account the [[centrifugal pseudo-force]] arising from working in a [[rotating frame]].<ref> {{cite web |last1=Tohline |first1=J. E. |author-link=Joel E. Tohline |url=http://www.phys.lsu.edu/astro/H_Book.current/H_Book.html |title=The Structure, Stability, and Dynamics of Self-Gravitating Systems |access-date=30 May 2007 |archive-url=https://web.archive.org/web/20100627133917/http://www.phys.lsu.edu/astro/H_Book.current/H_Book.html |archive-date=27 June 2010 |url-status=live }}</ref> For a uniformly rotating white dwarf, the limiting mass increases only slightly. If the star is allowed to rotate nonuniformly, and [[viscosity]] is neglected, then, as was pointed out by [[Fred Hoyle]] in 1947,<ref> {{cite journal |last1=Hoyle |first1=F. |date=1947 |title=Stars, Distribution and Motions of, Note on equilibrium configurations for rotating white dwarfs |volume=107 |issue=2 |pages=231–236 |journal=Monthly Notices of the Royal Astronomical Society |bibcode=1947MNRAS.107..231H |doi=10.1093/mnras/107.2.231 |doi-access=free }}</ref> there is no limit to the mass for which it is possible for a model white dwarf to be in static equilibrium. Not all of these model stars will be [[dynamics (mechanics)|dynamically]] stable.<ref> {{cite journal |last1=Ostriker |first1=J. P. |last2=Bodenheimer |first2=P. |date=1968 |title=Rapidly Rotating Stars. II. Massive White Dwarfs |journal=The Astrophysical Journal |volume=151 |page=1089 |bibcode=1968ApJ...151.1089O |doi= 10.1086/149507 |doi-access=free }}</ref> Rotating white dwarfs and the estimates of their diameter in terms of the angular velocity of rotation has been treated in the rigorous mathematical literature.<ref>{{cite journal |bibcode=1994CMaPh.166..417C |title=On diameters of uniformly rotating stars |last1=Chanillo |first1=Sagun |last2=Li |first2=Yan Yan |journal=Communications in Mathematical Physics |year=1994 |volume=166 |issue=2 |page=417 |doi=10.1007/BF02112323 |s2cid=8372549 |url=http://projecteuclid.org/euclid.cmp/1104271617 }}</ref> The fine structure of the free boundary of white dwarfs has also been analysed mathematically rigorously.<ref>{{cite journal |bibcode=2012JDE...253..553C |title=A remark on the geometry of uniformly rotating stars |last1=Chanillo |first1=Sagun |last2=Weiss |first2=Georg S. |journal=Journal of Differential Equations |year=2012 |volume=253 |issue=2 |page=553 |doi=10.1016/j.jde.2012.04.011 |arxiv=1109.3046 |s2cid=144301 }}</ref> === Radiation and cooling === The degenerate matter that makes up the bulk of a white dwarf has a very low [[opacity (optics)|opacity]], because any absorption of a photon requires that an electron must transition to a higher empty state, which may not be possible as the energy of the photon may not be a match for the possible quantum states available to that electron, hence radiative heat transfer within a white dwarf is low; it does, however, have a high [[thermal conductivity]]. As a result, the interior of the white dwarf maintains an almost uniform temperature as it cools down, starting at approximately 10<sup>8</sup> K shortly after the formation of the white dwarf and reaching less than 10<sup>6</sup> K for the coolest known white dwarfs.<ref name="Saumon2022">{{cite journal |last1=Saumon |first1=Didier |last2=Blouin |first2=Simon |last3=Tremblay |first3=Pier-Emmanuel |title=Current challenges in the physics of white dwarf stars |journal=Physics Reports |date=November 2022 |volume=988 |pages=1–63 |doi=10.1016/j.physrep.2022.09.001|arxiv=2209.02846 |bibcode=2022PhR...988....1S |s2cid=252111027 }}</ref> An outer shell of non-degenerate matter sits on top of the degenerate core. The outermost layers, which are cooler than the interior, radiate roughly as a [[black body]]. A white dwarf remains visible for a long time, as its tenuous outer atmosphere slowly radiates the thermal content of the degenerate interior.<ref name="Saumon2022"/> The visible radiation emitted by white dwarfs varies over a wide color range, from the whitish-blue color of an O-, B- or A-type main sequence star to the yellow-orange of a [[late-type star|late]] K- or early M-type star.<ref name="sionspectra"> {{cite journal |last1=Sion |first1=E. M. |last2=Greenstein |first2=J. L. |last3=Landstreet |first3=J. D. |last4=Liebert |first4=James |last5=Shipman |first5=H. L. |last6=Wegner |first6=G. A. |date=1983 |title=A proposed new white dwarf spectral classification system |journal=The Astrophysical Journal |volume=269 |page=253 |bibcode=1983ApJ...269..253S |doi= 10.1086/161036 |doi-access=free }}</ref> White dwarf effective surface temperatures extend from over {{val|150000}} K<ref name="villanovar4" /> to barely under 4000 K.<ref name="cool">{{cite journal |last1=Hambly |first1=N. C. |last2=Smartt |first2=S. J. |last3=Hodgkin |first3=S. T. |date=1997 |title=WD 0346+246: A Very Low Luminosity, Cool Degenerate in Taurus |journal=The Astrophysical Journal |volume=489 |issue=2 |pages=L157 |bibcode=1997ApJ...489L.157H |doi=10.1086/316797 |doi-access=free}}</ref><ref name="wden"> {{cite encyclopedia |last1=Fontaine |first1=G. |last2=Wesemael |first2=F. |title=White dwarfs |editor1-last=Murdin |editor1-first=P. |date=2001 |encyclopedia=Encyclopedia of Astronomy and Astrophysics |publisher=[[IOP Publishing]]/[[Nature Publishing Group]] |isbn=978-0-333-75088-9 }}</ref> In accordance with the [[Stefan–Boltzmann law]], luminosity increases with increasing surface temperature (proportional to ''T''{{sup|4}}); this surface temperature range corresponds to a luminosity from over 100 times that of the Sun to under {{frac|1|{{val|10000}}}} that of the Sun.<ref name="wden" /> Hot white dwarfs, with surface temperatures in excess of {{val|30000|u=K}}, have been observed to be sources of soft (i.e., lower-energy) [[X-ray]]s. This enables the composition and structure of their atmospheres to be studied by soft [[X-ray astronomy|X-ray]] and [[UV astronomy|extreme ultraviolet observations]].<ref> {{cite journal |last1=Heise |first1=J. |date=1985 |title=X-ray emission from isolated hot white dwarfs |journal=Space Science Reviews |volume=40 |issue=1–2 |pages=79–90 |bibcode=1985SSRv...40...79H |doi= 10.1007/BF00212870 |s2cid=120431159 }}</ref> White dwarfs also radiate [[neutrino]]s through the [[Urca process]].<ref>{{cite journal |bibcode=2005MNRAS.356..131L |title=A two-stream formalism for the convective Urca process |journal=Monthly Notices of the Royal Astronomical Society |volume=356 |issue=1 |pages=131–144 |last1=Lesaffre |first1=P. |last2=Podsiadlowski |first2=Ph. |last3=Tout |first3=C. A. |year=2005 |arxiv=astro-ph/0411016 |doi=10.1111/j.1365-2966.2004.08428.x |doi-access=free |s2cid=15797437 }}</ref> This process has more effect on hotter and younger white dwarfs. Because neutrinos can pass easily through stellar plasma, they can drain energy directly from the dwarf's interior; this mechanism is the dominant contribution to cooling for approximately the first 20 million years of a white dwarf's existence.<ref name="Saumon2022"/> [[File:Size IK Peg.png|left|upright=1.2|thumb|A comparison between the white dwarf [[IK Pegasi]] B (center), its A-class companion IK Pegasi A (left) and the Sun (right). This white dwarf has a surface temperature of {{val|35500|u=K}}.]] As was explained by [[Leon Mestel]] in 1952, unless the white dwarf [[accretion (astrophysics)|accretes]] matter from a companion star or other source, its radiation comes from its stored heat, which is not replenished.<ref> {{cite journal |last1=Mestel |first1=L. |date=1952 |title=On the theory of white dwarf stars. I. The energy sources of white dwarfs |journal=Monthly Notices of the Royal Astronomical Society |volume=112 |issue=6 |pages=583–597 |bibcode=1952MNRAS.112..583M |doi=10.1093/mnras/112.6.583 |doi-access=free }}</ref><ref> {{cite conference |last1=Kawaler |first1=S. D. |date=1998 |title=White Dwarf Stars and the Hubble Deep Field |conference=The Hubble Deep Field: Proceedings of the Space Telescope Science Institute Symposium |page=252 |arxiv=astro-ph/9802217 |bibcode=1998hdf..symp..252K |isbn=978-0-521-63097-9 }}</ref>{{rp|§2.1}} White dwarfs have an extremely small surface area to radiate this heat from, so they cool gradually, remaining hot for a long time.<ref name="rln" /> As a white dwarf cools, its surface temperature decreases, the radiation that it emits reddens, and its luminosity decreases. Since the white dwarf has no [[Sources and sinks|energy sink]] other than radiation, it follows that its cooling slows with time. The rate of cooling has been estimated for a [[carbon]] white dwarf of {{solar mass|0.59}} with a [[hydrogen]] atmosphere. After initially taking approximately 1.5 billion years to cool to a surface temperature of 7140 K, cooling approximately 500 more kelvins to 6590 K takes around 0.3 billion years, but the next two steps of around 500 kelvins (to 6030 K and 5550 K) take first 0.4 and then 1.1 billion years.<ref> {{cite journal |last1=Bergeron |first1=P. |last2=Ruiz |first2=M. T. |last3=Leggett |first3=S. K. |date=1997 |title=The Chemical Evolution of Cool White Dwarfs and the Age of the Local Galactic Disk |journal=The Astrophysical Journal Supplement Series |volume=108 |issue=1 |pages=339–387 |bibcode=1997ApJS..108..339B |doi= 10.1086/312955 |doi-access=free }}</ref>{{rp|Table 2}} Most observed white dwarfs have relatively high surface temperatures, between 8000 K and {{val|40000|u=K}}.<ref name="sdssr4" /><ref name="villanovar4" /> A white dwarf, though, spends more of its lifetime at cooler temperatures than at hotter temperatures, so we should expect that there are more cool white dwarfs than hot white dwarfs. Once we adjust for the [[selection effect]] that hotter, more luminous white dwarfs are easier to observe, we do find that decreasing the temperature range examined results in finding more white dwarfs.<ref name="disklf"> {{cite journal |last1=Leggett |first1=S. K. |last2=Ruiz |first2=M. T. |last3=Bergeron |first3=P. |date=1998 |title=The Cool White Dwarf Luminosity Function and the Age of the Galactic Disk |journal=The Astrophysical Journal |volume=497 |issue=1 |pages=294–302 |bibcode=1998ApJ...497..294L |doi= 10.1086/305463 |doi-access=free }}</ref> This trend stops when we reach extremely cool white dwarfs; few white dwarfs are observed with surface temperatures below {{val|4000|u=K}},<ref> {{cite journal |last1=Gates |first1=E. |last2=Gyuk |first2=G. |last3=Harris |first3=H. C. |last4=Subbarao |first4=M. |last5=Anderson |first5=S. |last6=Kleinman |first6=S. J. |last7=Liebert |first7=James |last8=Brewington |first8=H. |last9=Brinkmann |first9=J. | display-authors = 6 |date=2004 |title=Discovery of New Ultracool White Dwarfs in the Sloan Digital Sky Survey |journal=The Astrophysical Journal |volume=612 |issue=2 |pages=L129 |arxiv= astro-ph/0405566 |bibcode=2004ApJ...612L.129G |doi= 10.1086/424568 |s2cid=7570539 }}</ref> and one of the coolest so far observed, [[WD J2147–4035]], has a surface temperature of approximately 3050 K.<ref name="Elms2022">{{cite journal |last1=Elms |first1=Abbigail K. |last2=Tremblay |first2=Pier-Emmanuel |last3=Gänsicke |first3=Boris T. |last4=Koester |first4=Detlev |last5=Hollands |first5=Mark A. |last6=Gentile Fusillo |first6=Nicola Pietro |last7=Cunningham |first7=Tim |last8=Apps |first8=Kevin |date=2022-12-01 |title=Spectral analysis of ultra-cool white dwarfs polluted by planetary debris |journal=Monthly Notices of the Royal Astronomical Society |volume=517 |issue=3 |pages=4557–4574 |arxiv=2206.05258 |bibcode=2022MNRAS.517.4557E |doi=10.1093/mnras/stac2908 |issn=0035-8711 |doi-access=free}}</ref> The reason for this is that the Universe's age is finite;<ref> {{cite journal |last1=Winget |first1=D. E. |last2=Hansen |first2=C. J. |last3=Liebert |first3=James |last4=Van Horn |first4=H. M. |last5=Fontaine |first5=G. |last6=Nather |first6=R. E. |last7=Kepler |first7=S. O. |last8=Lamb |first8=D. Q. |date=1987 |title=An independent method for determining the age of the universe |journal=The Astrophysical Journal |volume=315 |pages=L77 |bibcode=1987ApJ...315L..77W |doi=10.1086/184864 |hdl=10183/108730 |doi-access=free |hdl-access=free }}</ref><ref> {{cite book |last1=Trefil |first1=J. S. |date=2004 |title=The Moment of Creation: Big Bang Physics from Before the First Millisecond to the Present Universe |publisher=[[Dover Publications]] |isbn=978-0-486-43813-9 }}</ref> there has not been enough time for white dwarfs to cool below this temperature. The [[white dwarf luminosity function]] can therefore be used to find the time when stars started to form in a region; an estimate for the age of our [[galactic disk]] found in this way is 8 billion years.<ref name="disklf" /> A white dwarf will eventually, in many trillions of years, cool and become a non-radiating ''[[black dwarf]]'' in approximate thermal equilibrium with its surroundings and with the [[cosmic background radiation]]. No black dwarfs are thought to exist yet.<ref name="osln" /> At very low temperatures (<4000 K) white dwarfs with hydrogen in their atmosphere will be affected by [[Collision-induced absorption and emission|collision induced absorption]] (CIA) of hydrogen molecules colliding with helium atoms. This affects the optical red and infrared brightness of white dwarfs with a hydrogen or mixed hydrogen-helium atmosphere. This makes old white dwarfs with this kind of atmosphere bluer than the main cooling sequence. White dwarfs with hydrogen-poor atmospheres, such as WD J2147–4035, are less affected by CIA and therefore have a yellow to orange color.<ref name="Bergeron2022">{{cite journal |last1=Bergeron |first1=P. |last2=Kilic |first2=Mukremin |last3=Blouin |first3=Simon |last4=Bédard |first4=A. |last5=Leggett |first5=S. K. |last6=Brown |first6=Warren R. |date=2022-07-01 |title=On the Nature of Ultracool White Dwarfs: Not so Cool after All |journal=The Astrophysical Journal |volume=934 |issue=1 |pages=36 |arxiv=2206.03174 |bibcode=2022ApJ...934...36B |doi=10.3847/1538-4357/ac76c7 |issn=0004-637X |doi-access=free}}</ref><ref name="Elms2022" /> [[File:Gaia hrd wds2.png|thumb|The white dwarf cooling sequence seen by ESA's [[Gaia (spacecraft)|Gaia mission]]. The axes are [[absolute magnitude]] in the [[Photometric system|G-band]] vs. a [[color index]] G-band magnitude minus RP ([[Gaia (spacecraft)|Gaia]] red photometer) magnitude]] White dwarf core material is a completely [[Ionization|ionized]] [[plasma (physics)|plasma]] – a mixture of [[atomic nucleus|nuclei]] and [[electron]]s – that is initially in a fluid state. It was theoretically predicted in the 1960s that at a late stage of cooling, it should [[crystallize]] into a solid state, starting at its center.<ref>{{cite journal |last1=van Horn |first1=H. M. |title=Crystallization of White Dwarfs |journal=The Astrophysical Journal |date=January 1968 |volume=151 |page=227 |doi=10.1086/149432|bibcode=1968ApJ...151..227V }}</ref> The crystal structure is thought to be a [[body-centered cubic]] lattice.<ref name="cosmochronology" /><ref> {{cite journal |last1=Barrat |first1=J. L. |last2=Hansen |first2=J. P. |last3=Mochkovitch |first3=R. |date=1988 |title=Crystallization of carbon-oxygen mixtures in white dwarfs |journal=Astronomy and Astrophysics |volume=199 |issue=1–2 |pages=L15 |bibcode=1988A&A...199L..15B }}</ref> In 1995 it was suggested that [[asteroseismology|asteroseismological]] observations of [[#Variability|pulsating white dwarfs]] yielded a potential test of the crystallization theory,<ref> {{cite journal |last1=Winget |first1=D. E. |date=1995 |title=The Status of White Dwarf Asteroseismology and a Glimpse of the Road Ahead |volume=4 |issue=2 |page=129 |journal=Baltic Astronomy |bibcode=1995BaltA...4..129W |doi=10.1515/astro-1995-0209|doi-access=free }}</ref> and in 2004, observations were made that suggested approximately 90% of the mass of [[BPM 37093]] had crystallized.<ref>{{cite journal |last1=Metcalfe |first1=T. S. |last2=Montgomery |first2=M. H. |last3=Kanaan |first3=A. |title=Testing White Dwarf Crystallization Theory with Asteroseismology of the Massive Pulsating DA Star BPM 37093 |journal=The Astrophysical Journal |date=20 April 2004 |volume=605 |issue=2 |pages=L133–L136 |doi=10.1086/420884|arxiv=astro-ph/0402046 |bibcode=2004ApJ...605L.133M |s2cid=119378552 }}</ref><ref name="lucy">{{cite news |url=http://news.bbc.co.uk/2/hi/science/nature/3492919.stm |title=Diamond star thrills astronomers |archive-url=https://web.archive.org/web/20070205114340/http://news.bbc.co.uk/2/hi/science/nature/3492919.stm |archive-date=5 February 2007 |first=David |last=Whitehouse |work=BBC News |date=16 February 2004 |access-date=6 January 2007}}</ref><ref> {{cite journal |last1=Kanaan |first1=A. |last2=Nitta |first2=A. |last3=Winget |first3=D. E. |last4=Kepler |first4=S. O. |last5=Montgomery |first5=M. H. |last6=Metcalfe |first6=T. S. |last7=Oliveira |first7=H. |last8=Fraga |first8=L. |last9=Da Costa |first9=A. F. M.| display-authors = 6 |date=2005 |title=Whole Earth Telescope observations of BPM 37093: A seismological test of crystallization theory in white dwarfs |journal=Astronomy and Astrophysics |volume=432 |issue=1 |pages=219–224 |arxiv=astro-ph/0411199 |bibcode= 2005A&A...432..219K |doi=10.1051/0004-6361:20041125 |s2cid=7297628 }}</ref> Other work gives a crystallized mass fraction of between 32% and 82%.<ref name="Brassard"> {{cite journal |last1=Brassard |first1=P. |last2=Fontaine |first2=G. |date=2005 |title=Asteroseismology of the Crystallized ZZ Ceti Star BPM 37093: A Different View |journal=The Astrophysical Journal |volume=622 |issue=1 |pages=572–576 |bibcode=2005ApJ...622..572B |doi= 10.1086/428116 |doi-access=free }}</ref> As a white dwarf core undergoes crystallization into a solid phase, [[latent heat]] is released, which provides a source of thermal energy that delays its cooling.<ref>{{cite journal |first1=B. M. S. |last1=Hansen |first2=James |last2=Liebert |title=Cool White Dwarfs |journal=Annual Review of Astronomy and Astrophysics |volume=41 |page=465 |year=2003|doi=10.1146/annurev.astro.41.081401.155117 |bibcode=2003ARA&A..41..465H }}</ref> Another possible mechanism that was suggested to explain this [[White dwarf cooling anomaly|cooling anomaly]] in some types of white dwarfs is a solid–liquid distillation process: the crystals formed in the core are buoyant and float up, thereby displacing heavier liquid downward, thus causing a net release of gravitational energy.<ref>{{cite journal |last1=Antoine |first1=Bédard |last2=Simon |first2=Blouin |last3=Sihao |first3=Cheng |date=2024 |title=Buoyant crystals halt the cooling of white dwarf stars |url=https://www.nature.com/articles/s41586-024-07102-y.epdf |journal=Nature |language=en |volume=627 |issue=8003 |pages=286–288 |doi= 10.1038/s41586-024-07102-y|pmid=38448597 |arxiv=2409.04419 |bibcode=2024Natur.627..286B |issn=1476-4687}}</ref> Chemical [[fractionation]] between the ionic species in the plasma mixture can release a similar or even greater amount of energy.<ref>{{cite journal |last1=Althaus |first1=L. G. |last2=García-Berro |first2=E. |last3=Isern |first3=J. |last4=Córsico |first4=A. H. |last5=Miller Bertolami |first5=M. M. |title=New phase diagrams for dense carbon-oxygen mixtures and white dwarf evolution |journal=Astronomy & Astrophysics |date=January 2012 |volume=537 |pages=A33 |doi=10.1051/0004-6361/201117902|arxiv=1110.5665 |bibcode=2012A&A...537A..33A |s2cid=119279832 }}</ref><ref>{{cite journal |last1=Blouin |first1=Simon |last2=Daligault |first2=Jérôme |last3=Saumon |first3=Didier |title=22 Ne Phase Separation as a Solution to the Ultramassive White Dwarf Cooling Anomaly |journal=The Astrophysical Journal Letters |date=1 April 2021 |volume=911 |issue=1 |pages=L5 |doi=10.3847/2041-8213/abf14b|arxiv=2103.12892 |bibcode=2021ApJ...911L...5B |s2cid=232335433 |doi-access=free }}</ref><ref>{{cite journal |last1=Blouin |first1=Simon |last2=Daligault |first2=Jérôme |last3=Saumon |first3=Didier |last4=Bédard |first4=Antoine |last5=Brassard |first5=Pierre |title=Toward precision cosmochronology: A new C/O phase diagram for white dwarfs |journal=Astronomy & Astrophysics |date=August 2020 |volume=640 |pages=L11 |doi=10.1051/0004-6361/202038879|arxiv=2007.13669 |bibcode=2020A&A...640L..11B |s2cid=220793255 }}</ref> This energy release was first confirmed in 2019 after the identification of a pile up in the cooling sequence of more than {{val|15000}} white dwarfs observed with the ''Gaia'' satellite.<ref> {{cite journal |last1=Tremblay |first1=P.-E. |last2=Fontaine |first2=G. |last3=Fusillo |first3=N. P. G. |last4=Dunlap |first4=B. H. |last5=Gänsicke |first5=B. T. |last6=Hollands |first6=M. H. |last7=Hermes |first7=J. J. |last8=Marsh |first8=T. R. |last9=Cukanovaite |first9=E. |last10=Cunningham |first10=T. |display-authors=6 |date=2019 |title=Core crystallization and pile-up in the cooling sequence of evolving white dwarfs |journal=Nature |volume=565 |issue=7738 |pages=202–205 |bibcode=2019Natur.565..202T |doi=10.1038/s41586-018-0791-x |pmid=30626942 |url=http://wrap.warwick.ac.uk/112800/7/WRAP-core-crystallization-pile-up-cooling-sequence-evolving-white-dwarfs-Tremblay-2019.pdf |arxiv=1908.00370 |s2cid=58004893 |access-date=23 July 2019 |archive-url=https://web.archive.org/web/20190723202013/http://wrap.warwick.ac.uk/112800/7/WRAP-core-crystallization-pile-up-cooling-sequence-evolving-white-dwarfs-Tremblay-2019.pdf |archive-date=23 July 2019 |url-status=live }}</ref> Low-mass helium white dwarfs (mass {{solar mass|< 0.20}}), often referred to as extremely low-mass white dwarfs (ELM WDs), are formed in binary systems. As a result of their hydrogen-rich envelopes, residual hydrogen burning via the CNO cycle may keep these white dwarfs hot for hundreds of millions of years.<ref>{{cite journal|last1=Chen |first1=J. |display-authors=etal |year=2021 |title=Slowly cooling white dwarfs in M13 from stable hydrogen burning |journal=Nature Astronomy |volume=5 |number=11 |pages=1170–1177 |doi=10.1038/s41550-021-01445-6 |arxiv=2109.02306|bibcode=2021NatAs...5.1170C }}</ref> In addition, they remain in a bloated proto-white dwarf stage for up to 2 Gyr before they reach the cooling track.<ref>{{cite journal |first1=A. G. |last1=Istrate |first2=T. M. |last2=Tauris |first3=N. |last3=Langer |first4=J. |last4=Antoniadis |year=2014 |title=The timescale of low-mass proto-helium white dwarf evolution |journal=Astronomy and Astrophysics|volume=571 |page=L3 |bibcode=2014A&A...571L...3I |arxiv=1410.5471 |doi=10.1051/0004-6361/201424681 |s2cid=55152203 }}</ref> === Atmosphere and spectra === [[File:Artist’s impression of the WDJ0914+1914 system.tif|thumb|Artist's impression of the [[WD J0914+1914]] system<ref>{{cite web |title=First Giant Planet around White Dwarf Found – ESO observations indicate the Neptune-like exoplanet is evaporating |url=https://www.eso.org/public/news/eso1919/ |website=www.eso.org |access-date=4 December 2019 |language=en |archive-url=https://web.archive.org/web/20191204214723/https://www.eso.org/public/news/eso1919/ |archive-date=4 December 2019 |url-status=live}}</ref>]] Although most white dwarfs are thought to be composed of carbon and oxygen, [[spectroscopy]] typically shows that their emitted light comes from an atmosphere that is observed to be either hydrogen or [[helium]] dominated. The dominant element is usually at least 1000 times more abundant than all other elements. As explained by [[Evry Schatzman|Schatzman]] in the 1940s, the high [[surface gravity]] is thought to cause this purity by gravitationally separating the atmosphere so that heavy elements are below and the lighter above.<ref> {{cite journal |last1=Schatzman |first1=E. |date=1945 |title=Théorie du débit d'énergie des naines blanches |volume=8 |page=143 |journal=Annales d'Astrophysique |bibcode=1945AnAp....8..143S }}</ref><ref name="physrev"> {{cite journal |last1=Koester |first1=D. |last2=Chanmugam |first2=G. |date=1990 |title=Physics of white dwarf stars |journal=Reports on Progress in Physics |volume=53 |issue=7 |pages=837–915 |bibcode=1990RPPh...53..837K |doi= 10.1088/0034-4885/53/7/001 |s2cid=122582479 }}</ref>{{rp|§§5–6}} This atmosphere, the only part of the white dwarf visible to us, is thought to be the top of an envelope that is a residue of the star's envelope in the [[asymptotic giant branch|AGB]] phase and may also contain material accreted from the [[interstellar medium]]. The envelope is believed to consist of a helium-rich layer with mass no more than {{frac|1|100}} of the star's total mass, which, if the atmosphere is hydrogen-dominated, is overlain by a hydrogen-rich layer with mass approximately {{frac|1|{{val|10000}}}} of the star's total mass.<ref name="wden" /><ref name="kawaler"/>{{rp|§§4–5}} Although thin, these outer layers determine the thermal evolution of the white dwarf. The degenerate electrons in the bulk of a white dwarf conduct heat well. Most of a white dwarf's mass is therefore at almost the same temperature ([[isothermal]]), and it is also hot: a white dwarf with surface temperature between {{val|8000|u=K}} and {{val|16000|u=K}} will have a core temperature between approximately {{val|5000000|u=K}} and {{val|20000000|u=K}}. The white dwarf is kept from cooling very quickly only by its outer layers' opacity to radiation.<ref name="wden" /> {| class="wikitable" style="float: right" |+ White dwarf spectral types<ref name="villanovar4" /> |- ! colspan="2" | Primary and secondary features |- | A | H lines present |- | B | He I lines |- | C | Continuous spectrum; no lines |- | O | He II lines, accompanied by He I or H lines |- | Z | Metal lines |- | Q | Carbon lines present |- | X | Unclear or unclassifiable spectrum |- ! colspan="2" | Secondary features only |- | P | Magnetic white dwarf with detectable polarization |- | H | Magnetic white dwarf without detectable polarization |- | E | Emission lines present |- | V | Variable |} The first attempt to [[Stellar classification#White dwarf classifications|classify white dwarf spectra]] appears to have been by [[G. P. Kuiper]] in 1941,<ref name="sionspectra" /><ref> {{cite journal |last1=Kuiper |first1=G. P. |date=1941 |title=List of Known White Dwarfs |journal=Publications of the Astronomical Society of the Pacific |volume=53 |issue=314 |page=248 |bibcode=1941PASP...53..248K |doi= 10.1086/125335 |doi-access=free }}</ref> and various classification schemes have been proposed and used since then.<ref> {{cite journal |last1=Luyten |first1=W. J. |date=1952 |title=The Spectra and Luminosities of White Dwarfs |journal=The Astrophysical Journal |volume=116 |page=283 |bibcode=1952ApJ...116..283L |doi= 10.1086/145612 }}</ref><ref> {{cite book |last1=Greenstein |first1=J. L. |date=1960 |title=Stellar atmospheres |url=https://archive.org/details/stellaratmospher0000gree |url-access=registration |publisher=[[University of Chicago Press]] |bibcode=1960stat.book.....G |lccn=61-9138 }}</ref> The system currently in use was introduced by [[Edward M. Sion]], Jesse L. Greenstein and their coauthors in 1983 and has been subsequently revised several times. It classifies a spectrum by a symbol that consists of an initial D, a letter describing the primary feature of the spectrum followed by an optional sequence of letters describing secondary features of the spectrum (as shown in the adjacent table), and a temperature index number, computed by dividing {{val|50400|u=K}} by the [[effective temperature]]. For example, a white dwarf with only [[Spectroscopic notation|He I]] lines in its spectrum and an effective temperature of {{val|15000|u=K}} could be given the classification of DB3, or, if warranted by the precision of the temperature measurement, DB3.5. Likewise, a white dwarf with a polarized [[magnetic field]], an effective temperature of {{val|17000|u=K}}, and a spectrum dominated by [[Spectroscopic notation|He I]] lines that also had hydrogen features could be given the classification of DBAP3. The symbols "?" and ":" may also be used if the correct classification is uncertain.<ref name="villanovar4" /><ref name="sionspectra" /> White dwarfs whose primary spectral classification is DA have hydrogen-dominated atmospheres. They make up the majority, approximately 80%, of all observed white dwarfs.<ref name="wden" /> The next class in number is of DBs, approximately 16%.<ref name="sdsswd" /> The hot, above {{val|15000|u=K}}, DQ class (roughly 0.1%) have carbon-dominated atmospheres.<ref> {{cite journal |last1=Dufour |first1=P. |last2=Liebert |first2=James |last3=Fontaine |first3=G. |last4=Behara |first4=N. |date=2007 |title=White dwarf stars with carbon atmospheres |journal=Nature |volume=450 |issue=7169 |pages=522–4|pmid=18033290 |bibcode=2007Natur.450..522D |doi=10.1038/nature06318 |arxiv = 0711.3227 |s2cid=4398697 }}</ref> Those classified as DB, DC, DO, DZ, and cool DQ have helium-dominated atmospheres. Assuming that carbon and metals are not present, which spectral classification is seen depends on the effective temperature. Between approximately {{val|100000|u=K}} to {{val|45000|u=K}}, the spectrum will be classified DO, dominated by singly ionized helium. From {{val|30000|u=K}} to {{val|12000|u=K}}, the spectrum will be DB, showing neutral helium lines, and below about {{val|12000|u=K}}, the spectrum will be featureless and classified DC.<ref name="kawaler"/>{{rp|§2.4}}<ref name="wden" /> [[Molecules in stars|Molecular]] hydrogen ([[Molecular hydrogen|H<sub>2</sub>]]) has been detected in spectra of the atmospheres of some white dwarfs.<ref name="dwarf">{{cite journal |bibcode=2013ApJ...766L..18X |title=Discovery of Molecular Hydrogen in White Dwarf Atmospheres |last1=Xu |first1=S. |last2=Jura |first2=M. |last3=Koester |first3=D. |last4=Klein |first4=B. |last5=Zuckerman |first5=B. |journal=The Astrophysical Journal |year=2013 |volume=766 |issue=2 |pages=L18 |doi=10.1088/2041-8205/766/2/L18 |arxiv=1302.6619 |s2cid=119248244 }}</ref> While theoretical work suggests that some types of white dwarfs may have [[stellar corona]], searches at X-ray and radio wavelengths, where coronae are most easily detected, have been unsuccessful.<ref name=Weisskopf2007>{{cite journal |last1=Weisskopf |first1=Martin C. |last2=Wu |first2=Kinwah |last3=Trimble |first3=Virginia |last4=O’Dell |first4=Stephen L. |last5=Elsner |first5=Ronald F. |last6=Zavlin |first6=Vyacheslav E. |last7=Kouveliotou |first7=Chryssa |date=2007-03-10 |title=A Chandra Search for Coronal X-Rays from the Cool White Dwarf GD 356 |url=https://iopscience.iop.org/article/10.1086/510776 |journal=The Astrophysical Journal |language=en |volume=657 |issue=2 |pages=1026–1036 |doi=10.1086/510776 |issn=0004-637X|arxiv=astro-ph/0609585 |bibcode=2007ApJ...657.1026W }}</ref><ref name=Route2024>{{cite journal|last1=Route|first1=Matthew|title=The Decline and Fall of ROME. V. A Preliminary Search for Star-disrupted Planet Interactions and Coronal Activity at 5 GHz among White Dwarfs within 25 pc|journal=The Astrophysical Journal|date=20 December 2024|volume=977|issue=1|page=261|doi=10.3847/1538-4357/ad9567|arxiv=2411.13718|doi-access=free |bibcode=2024ApJ...977..261R }}</ref> A few white dwarves have been observed to have inhomogeneous atmosphere with one side dominated by hydrogen and the other side dominated by helium.<ref>{{Cite journal |last1=Moss |first1=Adam |last2=Kilic |first2=Mukremin |last3=Bergeron |first3=Pierre |last4=Jewett |first4=Gracyn |last5=Brown |first5=Warren R. |date=2025-04-01 |title=The Emerging Class of Double-faced White Dwarfs |journal=The Astrophysical Journal |volume=983 |issue=1 |pages=14 |doi=10.3847/1538-4357/adbd3a |doi-access=free |issn=0004-637X|arxiv=2501.05649 }}</ref> ==== Metal-rich white dwarfs ==== [[File:Periodic Table White Dwarfs.png|thumb|Elements discovered in the atmosphere of white dwarfs colder than {{val|25000|u=K}}.]] Around 25–33% of white dwarfs have metal lines in their spectra, which is notable because any heavy elements in a white dwarf should sink into the star's interior in just a small fraction of the star's lifetime.<ref name=":0">{{cite journal |title=Extrasolar Cosmochemistry |journal=Annual Review of Earth and Planetary Sciences |date=2014-01-01|pages=45–67 |volume=42 |issue=1 |doi=10.1146/annurev-earth-060313-054740 |first1=M. |last1=Jura |first2=E.D. |last2=Young |bibcode=2014AREPS..42...45J|doi-access=free }}</ref> The prevailing explanation for metal-rich white dwarfs is that they have recently accreted rocky [[planetesimal]]s.<ref name=":0" /> The bulk composition of the accreted object can be measured from the strengths of the metal lines. For example, a 2015 study of the white dwarf Ton 345 concluded that its metal abundances were consistent with those of a [[Planetary differentiation|differentiated]], rocky planet whose mantle had been eroded by the host star's wind during its [[asymptotic giant branch]] phase.<ref>{{cite journal |title=The composition of a disrupted extrasolar planetesimal at SDSS J0845+2257 (Ton 345) |journal = Monthly Notices of the Royal Astronomical Society |date=2015-08-11|pages=3237–3248 |volume=451 |issue=3 |doi=10.1093/mnras/stv1201 |language=en |first1=D.J. |last1=Wilson |first2=B.T. |last2=Gänsicke |first3=D. |last3=Koester |first4=O. |last4=Toloza |first5=A. F. |last5=Pala |first6=E. |last6=Breedt |first7=S.G. |last7=Parsons |doi-access = free |arxiv=1505.07466 |bibcode=2015MNRAS.451.3237W|s2cid=54049842 }}</ref> === Magnetic field === Magnetic fields in white dwarfs with a strength at the surface of {{circa}} 1 million [[Gauss (unit)|gauss]] (100 [[tesla (unit)|teslas]]) were predicted by [[P. M. S. Blackett]] in 1947 as a consequence of a physical law he had proposed, which stated that an uncharged, rotating body should generate a magnetic field proportional to its [[angular momentum]].<ref> {{cite journal |last1=Blackett |first1=P. M. S. |date=1947 |title=The Magnetic Field of Massive Rotating Bodies |journal=Nature |volume=159 |issue=4046 |pages=658–66 |bibcode=1947Natur.159..658B |doi= 10.1038/159658a0 |pmid=20239729 |s2cid=4133416 }}</ref> This putative law, sometimes called the ''[[Blackett effect]]'', was never generally accepted, and by the 1950s even Blackett felt it had been refuted.<ref> {{cite journal |last1=Lovell |first1=B. |date=1975 |title=Patrick Maynard Stuart Blackett, Baron Blackett, of Chelsea. 18 November 1897 – 13 July 1974 |journal=Biographical Memoirs of Fellows of the Royal Society |volume=21 |pages=1–115 |doi=10.1098/rsbm.1975.0001 |jstor=769678 |s2cid=74674634 }}</ref>{{rp|page=39–43}} In the 1960s, it was proposed that white dwarfs might have magnetic fields due to conservation of total surface [[magnetic flux]] that existed in its progenitor star phase.<ref> {{cite journal |last1=Landstreet |first1=John D. |date=1967 |title=Synchrotron radiation of neutrinos and its astrophysical significance |journal=Physical Review |volume=153 |issue=5 |pages=1372–1377 |bibcode=1967PhRv..153.1372L |doi= 10.1103/PhysRev.153.1372 }}</ref> A surface magnetic field of {{circa}} 100 gauss (0.01 T) in the progenitor star would thus become a surface magnetic field of {{circa}} 100 × 100<sup>2</sup> = 1 million gauss (100 T) once the star's radius had shrunk by a factor of 100.<ref name="physrev" />{{rp|§8}}<ref> {{cite journal |last1=Ginzburg |first1=V. L. |last2=Zheleznyakov |first2=V. V. |last3=Zaitsev |first3=V. V. |date=1969 |title=Coherent mechanisms of radio emission and magnetic models of pulsars |journal=Astrophysics and Space Science |volume=4 |issue=4 |pages=464–504 |bibcode=1969Ap&SS...4..464G |doi= 10.1007/BF00651351 |s2cid=119003761 }}</ref>{{rp|page=484}} The first magnetic white dwarf to be discovered was [[GJ 742]] (also known as {{nowrap|GRW +70 8247}}), which was identified by James Kemp, John Swedlund, John Landstreet and [[Roger Angel]] in 1970 to host a magnetic field by its emission of [[circularly polarized]] light.<ref> {{cite journal |last1=Kemp |first1=J.C. |last2=Swedlund |first2=J.B. |last3=Landstreet |first3=J.D. |last4=Angel |first4=J.R.P. |date=1970 |title=Discovery of circularly polarized light from a white dwarf |journal=[[The Astrophysical Journal]] |volume=161 |page=L77 |bibcode=1970ApJ...161L..77K |doi=10.1086/180574 |doi-access=free }} </ref> It is thought to have a surface field of approximately 300 million gauss (30 kT).<ref name="physrev" />{{rp|§8}} Since 1970, magnetic fields have been discovered in well over 200 white dwarfs, ranging from {{val|2|e=3}} to {{val|e=9}} gauss (0.2 T to 100 kT).<ref> {{cite journal |last1=Ferrario |first1=Lilia |last2=de Martino |first2=Domtilla |last3=Gaensicke |first3=Boris |date=2015 |title=Magnetic white dwarfs |journal=[[Space Science Reviews]] |volume=191 |issue=1–4 |pages=111–169 |bibcode=2015SSRv..191..111F |doi= 10.1007/s11214-015-0152-0 |arxiv=1504.08072|s2cid=119057870 }} </ref> Many of the presently known magnetic white dwarfs are identified by low-resolution spectroscopy, which is able to reveal the presence of a magnetic field of 1 megagauss or more. Thus the basic identification process also sometimes results in discovery of magnetic fields.<ref> {{cite journal |last1=Kepler |first1=S.O. |last2=Pelisoli |first2=I. |last3=Jordan |first3=S. |last4=Kleinman |first4=S.J. |last5=Koester |first5=D. |last6=Kuelebi |first6=B. |last7=Pecanha |first7=V. |last8=Castanhiera |first8=B.G. |last9=Nitta |first9=A. |last10=Costa |first10=J.E.S. |last11=Winget |first11=D.E. |last12=Kanaan |first12=A. |last13=Fraga |first13=L. |date=2013 |title=Magnetic white dwarf stars in the Sloan Digital Sky Survey |journal=Monthly Notices of the Royal Astronomical Society |volume=429 |issue=4 |pages=2934–2944 |bibcode=2013MNRAS.429.2934K |doi= 10.1093/mnras/sts522 |doi-access=free |arxiv=1211.5709|s2cid=53316287 }} </ref> White dwarf magnetic fields may also be measured without spectral lines, using the techniques of broadband circular [[polarimetry]], or maybe through measurement of their frequencies of radio emission via the [[Solar radio emission#Electron-cyclotron maser emission|electron cyclotron maser]].<ref name=Route2024/> It has been estimated that at least 10% of white dwarfs have fields in excess of 1 million gauss (100 T).<ref> {{cite journal |last1=Landstreet |first1=J.D. |last2=Bagnulo |first2=S. |last3=Valyavin |first3=G.G. |last4=Fossati |first4=L. |last5=Jordan |first5=S. |last6=Monin |first6=D. |last7=Wade |first7=G.A. |date=2012 |title=On the incidence of weak magnetic fields in DA white dwarfs |journal=Astronomy and Astrophysics |volume=545 |issue=A30 |pages=9pp |bibcode=2012A&A...545A..30L |doi=10.1051/0004-6361/201219829 |arxiv=1208.3650|s2cid=55153825 }} </ref><ref> {{cite journal |last1=Liebert |first1=James |last2=Bergeron |first2=P. |last3=Holberg |first3=J. B. |title=The True Incidence of Magnetism Among Field White Dwarfs |date=2003 |journal=The Astronomical Journal |volume=125 |issue=1 |pages=348–353 |arxiv=astro-ph/0210319 |bibcode=2003AJ....125..348L |doi=10.1086/345573 |s2cid=9005227 }}</ref> The magnetic fields in a white dwarf may allow for the existence of a new type of [[chemical bond]], [[perpendicular paramagnetic bond]]ing, in addition to [[ionic bond|ionic]] and [[covalent bond]]s, though detecting molecules bonded in this way is expected to be difficult.<ref> {{cite news |first=Zeeya |last=Merali |date=19 July 2012 |title=Stars draw atoms closer together |department=Nature News & Comment |journal=[[Nature (journal)|Nature]] |doi=10.1038/nature.2012.11045 |doi-access=free |url=http://www.nature.com/news/stars-draw-atoms-closer-together-1.11045 |access-date=21 July 2012 |url-status=live |archive-url=https://web.archive.org/web/20120720200709/http://www.nature.com/news/stars-draw-atoms-closer-together-1.11045 |archive-date=20 July 2012 }} </ref> The highly magnetized white dwarf in the binary system [[AR Scorpii]] was identified in 2016 as the first [[pulsar]] in which the compact object is a white dwarf instead of a neutron star.<ref> {{cite journal |last1=Buckley |first1=D.A.H. |last2=Meintjes |first2=P.J. |last3=Potter |first3=S.B. |last4=Marsh |first4=T.R. |last5=Gänsicke |first5=B.T. |date=2017-01-23 |title=Polarimetric evidence of a white dwarf pulsar in the binary system AR Scorpii |language=en |journal=[[Nature Astronomy]] |volume=1 |issue=2 |page=0029 |doi=10.1038/s41550-016-0029 |s2cid=15683792 |arxiv=1612.03185 |bibcode=2017NatAs...1E..29B }} </ref> A second white dwarf pulsar was discovered in 2023.<ref>{{cite journal|first1=Ingrid |last1=Pelisoli |display-authors=etal |title=A 5.3-min-period pulsing white dwarf in a binary detected from radio to X-rays |journal=Nature Astronomy |volume=7 |pages=931–942 |year=2023 |issue=8 |doi=10.1038/s41550-023-01995-x |arxiv=2306.09272|bibcode=2023NatAs...7..931P }}</ref>
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