A white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. As white dwarfs have mass comparable to the Sun's and their volume is comparable to the Earth's, they are very dense. Their faint luminosity comes from the emission of stored heat. They comprise roughly 6% of all known stars in the solar neighborhood. The unusual faintness of white dwarfs was first recognized in 1910 by Henry Norris Russell, Edward Charles Pickering and Williamina Fleming;, p. 1 the name white dwarf was coined by Willem Luyten in 1922.
White dwarfs are thought to be the final evolutionary state of all stars whose mass is not too high—over 97% of the stars in our Galaxy., §1. After the hydrogen-fusing lifetime of a main-sequence star of low or medium mass ends, it will expand to a red giant which fuses helium to carbon and oxygen in its core by the triple-alpha process. If a red giant has insufficient mass to generate the core temperatures required to fuse carbon, an inert mass of carbon and oxygen will build up at its center. After shedding its outer layers to form a planetary nebula, it will leave behind this core, which forms the remnant white dwarf. Usually, therefore, white dwarfs are composed of carbon and oxygen. It is also possible that core temperatures suffice to fuse carbon but not neon, in which case an oxygen-neon-magnesium white dwarf may be formed. Also, some helium white dwarfs appear to have been formed by mass loss in binary systems.
The material in a white dwarf no longer undergoes fusion reactions, so the star has no source of energy, nor is it supported against gravitational collapse by the heat generated by fusion. It is supported only by electron degeneracy pressure, causing it to be extremely dense. The physics of degeneracy yields a maximum mass for a nonrotating white dwarf, the Chandrasekhar limit—approximately 1.4 solar masses—beyond which it cannot be supported by degeneracy pressure. A carbon-oxygen white dwarf that approaches this mass limit, typically by mass transfer from a companion star, may explode as a Type Ia supernova via a process known as carbon detonation. (SN 1006 is thought to be a famous example.)
A white dwarf is very hot when it is formed, but since it has no source of energy, it will gradually radiate away its energy and cool down. This means that its radiation, which initially has a high color temperature, will lessen and redden with time. Over a very long time, a white dwarf will cool to temperatures at which it is no longer visible and become a cold black dwarf. However, since no white dwarf can be older than the age of the Universe (approximately 13.7 billion years), even the oldest white dwarfs still radiate at temperatures of a few thousand kelvins, and no black dwarfs are thought to exist yet.
The first white dwarf discovered was in the triple star system of 40 Eridani, which contains the relatively bright main sequence star 40 Eridani A, orbited at a distance by the closer binary system of the white dwarf 40 Eridani B and the main sequence red dwarf 40 Eridani C. The pair 40 Eridani B/C was discovered by Friedrich Wilhelm Herschel on January 31, 1783;, p. 73 it was again observed by Friedrich Georg Wilhelm Struve in 1825 and by Otto Wilhelm von Struve in 1851. In 1910, it was discovered by Henry Norris Russell, Edward Charles Pickering and Williamina Fleming that despite being a dim star, 40 Eridani B was of spectral type A, or white. In 1939, Russell looked back on the discovery:, p. 1
I was visiting my friend and generous benefactor, Prof. Edward C. Pickering. With characteristic kindness, he had volunteered to have the spectra observed for all the stars—including comparison stars—which had been observed in the observations for stellar parallax which Hinks and I made at Cambridge, and I discussed. This piece of apparently routine work proved very fruitful—it led to the discovery that all the stars of very faint absolute magnitude were of spectral class M. In conversation on this subject (as I recall it), I asked Pickering about certain other faint stars, not on my list, mentioning in particular 40 Eridani B. Characteristically, he sent a note to the Observatory office and before long the answer came (I think from Mrs Fleming) that the spectrum of this star was A. I knew enough about it, even in these paleozoic days, to realize at once that there was an extreme inconsistency between what we would then have called "possible" values of the surface brightness and density. I must have shown that I was not only puzzled but crestfallen, at this exception to what looked like a very pretty rule of stellar characteristics; but Pickering smiled upon me, and said: "It is just these exceptions that lead to an advance in our knowledge", and so the white dwarfs entered the realm of study!
The spectral type of 40 Eridani B was officially described in 1914 by Walter Adams.
The companion of Sirius, Sirius B, was next to be discovered. During the nineteenth century, positional measurements of some stars became precise enough to measure small changes in their location. Friedrich Bessel used just such precise measurements to determine that the stars Sirius (α Canis Majoris) and Procyon (α Canis Minoris) were changing their positions. In 1844 he predicted that both stars had unseen companions:
If we were to regard Sirius and Procyon as double stars, the change of their motions would not surprise us; we should acknowledge them as necessary, and have only to investigate their amount by observation. But light is no real property of mass. The existence of numberless visible stars can prove nothing against the existence of numberless invisible ones.
Bessel roughly estimated the period of the companion of Sirius to be about half a century; C. H. F. Peters computed an orbit for it in 1851. It was not until January 31, 1862 that Alvan Graham Clark observed a previously unseen star close to Sirius, later identified as the predicted companion. Walter Adams announced in 1915 that he had found the spectrum of Sirius B to be similar to that of Sirius.
In 1917, Adriaan Van Maanen discovered Van Maanen's Star, an isolated white dwarf. These three white dwarfs, the first discovered, are the so-called classical white dwarfs., p. 2 Eventually, many faint white stars were found which had high proper motion, indicating that they could be suspected to be low-luminosity stars close to the Earth, and hence white dwarfs. Willem Luyten appears to have been the first to use the term white dwarf when he examined this class of stars in 1922; the term was later popularized by Arthur Stanley Eddington. Despite these suspicions, the first non-classical white dwarf was not definitely identified until the 1930s. 18 white dwarfs had been discovered by 1939., p. 3 Luyten and others continued to search for white dwarfs in the 1940s. By 1950, over a hundred were known, and by 1999, over 2,000 were known. Since then the Sloan Digital Sky Survey has found over 9,000 white dwarfs, mostly new.
White dwarfs were found to be extremely dense soon after their discovery. If a star is in a binary system, as is the case for Sirius B and 40 Eridani B, it is possible to estimate its mass from observations of the binary orbit. This was done for Sirius B by 1910, yielding a mass estimate of 0.94 solar mass. (A more modern estimate is 1.00 solar mass.) Since hotter bodies radiate more than colder ones, a star's surface brightness can be estimated from its effective surface temperature, and hence from its spectrum. If the star's distance is known, its overall luminosity can also be estimated. Comparison of the two figures yields the star's radius. Reasoning of this sort led to the realization, puzzling to astronomers at the time, that Sirius B and 40 Eridani B must be very dense. For example, 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 25,000 times the Sun's, which was so high that he called it "impossible". As Arthur Stanley Eddington put it later in 1927:, p. 50
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 3,000 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."
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 gravitationally redshifted. This was confirmed when Adams measured this redshift in 1925.
Such densities are possible because white dwarf material is not composed of atoms bound by chemical bonds, but rather consists of a plasma of unbound nuclei and electrons. There is therefore no obstacle to placing nuclei closer to each other than electron orbitals—the regions occupied by electrons bound to an atom—would normally allow. Eddington, however, wondered what would happen when this plasma cooled and the energy which kept the atoms ionized was no longer present. 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 state, and they must obey Fermi-Dirac statistics, also introduced in 1926 to determine the statistical distribution of particles which satisfy the Pauli exclusion principle. At zero temperature, therefore, electrons could not all occupy the lowest-energy, or ground, state; some of them had to occupy higher-energy states, forming a band of lowest-available energy states, the Fermi sea. This state of the electrons, called degenerate, meant that a white dwarf could cool to zero temperature and still possess high energy. Another way of deriving this result is by use of the uncertainty principle: the high density of electrons in a white dwarf means that their positions are relatively localized, creating a corresponding uncertainty in their momenta. This means that some electrons must have high momentum and hence high kinetic energy.
Compression of a white dwarf will increase the number of electrons in a given volume. Applying either the Pauli exclusion principle or the uncertainty principle, we can see that this will increase the kinetic energy of the electrons, causing pressure. This electron degeneracy pressure is what supports a white dwarf against gravitational collapse. It 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 so much greater than that of a low-mass white dwarf that the radius of a white dwarf decreases as its mass increases.
The existence of a limiting mass that no white dwarf can exceed is another consequence of being supported by electron degeneracy pressure. These masses were first published in 1929 by Wilhelm Anderson and in 1930 by Edmund C. Stoner. The modern value of the limit was first published in 1931 by Subrahmanyan Chandrasekhar in his paper "The Maximum Mass of Ideal White Dwarfs". For a nonrotating white dwarf, it is equal to approximately 5.7/μe2 solar masses, where μe is the average molecular weight per electron of the star., eq. (63) As the carbon-12 and oxygen-16 which predominantly compose a carbon-oxygen white dwarf both have atomic number equal to half their atomic weight, one should take μe equal to 2 for such a star, leading to the commonly-quoted value of 1.4 solar masses. (Near the beginning of the 20th century, there was reason to believe that stars were composed chiefly of heavy elements,, p. 955 so, in his 1931 paper, Chandrasekhar set the average molecular weight per electron, μe, equal to 2.5, giving a limit of 0.91 solar mass.) Together with William Alfred Fowler, Chandrasekhar received the Nobel prize for this and other work in 1983. The limiting mass is now called the Chandrasekhar limit.
If a white dwarf were to exceed the Chandrasekhar limit, and nuclear reactions did not take place, the pressure exerted by electrons would no longer be able to balance the force of gravity, and it would collapse into a denser object such as a neutron star or black hole. However, carbon-oxygen white dwarfs accreting mass from a neighboring star undergo a runaway nuclear fusion reaction, which leads to a Type Ia supernova explosion in which the white dwarf is destroyed, just before reaching the limiting mass.
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-fusing red dwarfs, whose cores are supported in part by thermal pressure, or the even lower-temperature brown dwarfs.
It is simple to derive a rough relationship between the mass and radii of white dwarfs using an energy minimization argument. The energy of the white dwarf can be approximated by taking it to be the sum of its gravitational potential energy and kinetic energy. The gravitational potential energy of a unit mass piece of white dwarf, Eg, will be on the order of −GM/R, where G is the gravitational constant, M is the mass of the white dwarf, and R is its radius. The kinetic energy of the unit mass, Ek, will primarily come from the motion of electrons, so it will be approximately N p2/2m, where p is the average electron momentum, m is the electron mass, and N is the number of electrons per unit mass. Since the electrons are degenerate, we can estimate p to be on the order of the uncertainty in momentum, Δp, given by the uncertainty principle, which says that Δp Δx is on the order of the reduced Planck constant, ħ. Δx will be on the order of the average distance between electrons, which will be approximately n−1/3, i.e., the reciprocal of the cube root of the number density, n, of electrons per unit volume. Since there are N M electrons in the white dwarf and its volume is on the order of R3, n will be on the order of N M / R3.
Solving for the kinetic energy per unit mass, Ek, we find that
Since this analysis uses the non-relativistic formula p2/2m for the kinetic energy, it is non-relativistic. If we wish to analyze the situation where the electron velocity in a white dwarf is close to the speed of light, c, we should replace p2/2m by the extreme relativistic approximation p c for the kinetic energy. With this substitution, we find
To interpret this result, observe that as we add mass to a white dwarf, its radius will decrease, so, by the uncertainty principle, the momentum, and hence the velocity, of its electrons will increase. As this velocity approaches c, the extreme relativistic analysis becomes more exact, meaning that the mass M of the white dwarf must approach Mlimit. Therefore, no white dwarf can be heavier than the limiting mass Mlimit.
For a more accurate computation of the mass-radius relationship and limiting mass of a white dwarf, one must compute the equation of state which 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, we will still find that the radius is inversely proportional to the cube root of the mass., eq. (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, μe, has been set equal to 2. Radius is measured in standard solar radii and mass in standard solar masses.
These computations all assume that the white dwarf is nonrotating. 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. For a uniformly rotating white dwarf, the limiting mass increases only slightly. However, if the star is allowed to rotate nonuniformly, and viscosity is neglected, then, as was pointed out by Fred Hoyle in 1947, 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, however, will be dynamically stable.
Unless the white dwarf accretes matter from a companion star or other source, this radiation comes from its stored heat, which is not replenished. White dwarfs have an extremely small surface area to radiate this heat from, so they remain hot for a long time. As a white dwarf cools, its surface temperature decreases, the radiation which it emits reddens, and its luminosity decreases. Since the white dwarf has no energy sink other than radiation, it follows that its cooling slows with time. Bergeron, Ruiz, and Leggett, for example, estimate that after a carbon white dwarf of 0.59 solar mass with a hydrogen atmosphere has cooled to a surface temperature of 7,140 K, taking approximately 1.5 billion years, cooling approximately 500 more kelvins to 6,590 K takes around 0.3 billion years, but the next two steps of around 500 kelvins (to 6,030 K and 5,550 K) take first 0.4 and then 1.1 billion years., Table 2. Although white dwarf material is initially plasma—a fluid composed of nuclei and electrons—it was theoretically predicted in the 1960s that at a late stage of cooling, it should crystallize, starting at the center of the star. The crystal structure is thought to be a body-centered cubic lattice. In 1995 it was pointed out that asteroseismological observations of pulsating white dwarfs yielded a potential test of the crystallization theory, and in 2004, Travis Metcalfe and a team of researchers at the Harvard-Smithsonian Center for Astrophysics estimated, on the basis of such observations, that approximately 90% of the mass of BPM 37093 had crystallized. Other work gives a crystallized mass fraction of between 32% and 82%.
Most observed white dwarfs have relatively high surface temperatures, between 8,000 K and 40,000 K. 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. This trend stops when we reach extremely cool white dwarfs; few white dwarfs are observed with surface temperatures below 4,000 K, and one of the coolest so far observed, WD 0346+246, has a surface temperature of approximately 3,900 K. The reason for this is that, as the Universe's age is finite, there has not been time for white dwarfs to cool down 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 the Galactic disk found in this way is 8 billion years.
A white dwarf will eventually cool and become a non-radiating black dwarf in approximate thermal equilibrium with its surroundings and with the cosmic background radiation. However, no black dwarfs are thought to exist yet.
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 almost isothermal, and it is also hot: a white dwarf with surface temperature between 8,000 K and 16,000 K will have a core temperature between approximately 5,000,000 K and 20,000,000 K. The white dwarf is kept from cooling very quickly only by its outer layers' opacity to radiation.
|Primary and secondary features|
|A||H lines present; no He I or metal lines|
|B||He I lines; no H or metal lines|
|C||Continuous spectrum; no lines|
|O||He II lines, accompanied by He I or H lines|
|Z||Metal lines; no H or He I lines|
|Q||Carbon lines present|
|X||Unclear or unclassifiable spectrum|
|Secondary features only|
|P||Magnetic white dwarf with detectable polarization|
|H||Magnetic white dwarf without detectable polarization|
|E||Emission lines present|
The symbols ? and : may also be used if the correct classification is uncertain.
White dwarfs whose primary spectral classification is DA have hydrogen-dominated atmospheres. They make up the majority (approximately three-quarters) of all observed white dwarfs. A small fraction (roughly 0.1%) have carbon-dominated atmospheres, the hot (above 15,000 K) DQ class. The classifiable remainder (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 100,000 K to 45,000 K, the spectrum will be classified DO, dominated by singly ionized helium. From 30,000 K to 12,000 K, the spectrum will be DB, showing neutral helium lines, and below about 12,000 K, the spectrum will be featureless and classified DC.,§ 2.4 The reason for the absence of white dwarfs with helium-dominated atmospheres and effective temperatures between 30,000 K and 45,000 K, called the DB gap, is not clear. It is suspected to be due to competing atmospheric evolutionary processes, such as gravitational separation and convective mixing.
|DAV (GCVS: ZZA)||DA spectral type, having only hydrogen absorption lines in its spectrum|
|DBV (GCVS: ZZB)||DB spectral type, having only helium absorption lines in its spectrum|
|GW Vir (GCVS: ZZO)|| Atmosphere mostly C, He and O; |
may be divided into DOV and PNNV stars
|Types of pulsating white dwarf, §1.1, 1.2.|
Early calculations suggested that there might be white dwarfs whose luminosity varied with a period of around 10 seconds, but searches in the 1960s failed to observe this., § 7.1.1; The first variable white dwarf found was HL Tau 76; in 1965 and 1966, Arlo U. Landolt observed it to vary with a period of approximately 12.5 minutes. The reason for this period being longer than predicted is that the variability of HL Tau 76, like that of the other pulsating variable white dwarfs known, arises from non-radial gravity wave pulsations., § 7. Known types of pulsating white dwarf include the DAV, or ZZ Ceti, stars, including HL Tau 76, with hydrogen-dominated atmospheres and the spectral type DA;, pp. 891, 895 DBV, or V777 Her, stars, with helium-dominated atmospheres and the spectral type DB;, p. 3525 and GW Vir stars (sometimes subdivided into DOV and PNNV stars), with atmospheres dominated by helium, carbon, and oxygen.,§1.1, 1.2;,§1. GW Vir stars are not, strictly speaking, white dwarfs, but are stars which are in a position on the Hertzsprung-Russell diagram between the asymptotic giant branch and the white dwarf region. They may be called pre-white dwarfs., § 1.1; These variables all exhibit small (1%–30%) variations in light output, arising from a superposition of vibrational modes with periods of hundreds to thousands of seconds. Observation of these variations gives asteroseismological evidence about the interiors of white dwarfs.
The mass of an isolated, nonrotating white dwarf cannot exceed the Chandrasekhar limit of ~1.4 solar masses. (This limit may increase if the white dwarf is rotating rapidly and nonuniformly.) White dwarfs in binary systems, however, can accrete material from a companion star, increasing both their mass and their density. As their mass approaches the Chandrasekhar limit, this could theoretically lead to either the explosive ignition of fusion in the white dwarf or its collapse into a neutron star.
Accretion provides the currently favored mechanism, the single-degenerate model, for type Ia supernovae. In this model, a carbon-oxygen white dwarf accretes material from a companion star,, p. 14. increasing its mass and compressing its core. It is believed that compressional heating of the core leads to ignition of carbon fusion as the mass approaches the Chandrasekhar limit. Because the white dwarf is supported against gravity by quantum degeneracy pressure instead of by thermal pressure, adding heat to the star's interior increases its temperature but not its pressure, so the white dwarf does not expand and cool in response. Rather, the increased temperature accelerates the rate of the fusion reaction, in a runaway process that feeds on itself. The thermonuclear flame consumes much of the white dwarf in a few seconds, causing a type Ia supernova explosion that obliterates the star. In another possible mechanism for type Ia supernovae, the double-degenerate model, two carbon-oxygen white dwarfs in a binary system merge, creating an object with mass greater than the Chandrasekhar limit in which carbon fusion is then ignited., p. 14.
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