taken by the Hubble Space Telescope . Sirius B, which is a white dwarf, can be seen as a faint dot to the lower left of the much brighter Sirius A.]]
A , also called a '''degenerate dwarf''', is the kind of , April 17 , 2007 . white dwarfs appear to have been formed by mass loss in binary systems. 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. On Possible Oxygen/Neon White Dwarfs: H1504+65 and the White Dwarf Donors in Ultracompact X-ray Binaries , K. Werner, N. J. Hammer, T. Nagel, T. Rauch, and S. Dreizler, pp. 165 ff. in ''14th European Workshop on White Dwarfs; Proceedings of a meeting held at Kiel, July 19-23, 2004'', D. Koester and S. Moehler, eds., San Francisco: Astronomical Society of the Pacific, 2005. White dwarfs are thought to be the final state of over 97% of all stars in our galaxy., §1. They comprise roughly 6% of all known stars in the solar neighborhood.2
The material in a white dwarf no longer undergoes fusion reactions, so the star has no source of energy, nor is it supported against , accessed online May 3 , 2007 .
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 is the source of its faint , 2007 , even the oldest white dwarfs still radiate at temperatures of a few thousand Kelvin , and no black dwarfs are thought to exist yet. The Potential of White Dwarf Cosmochronology , G. Fontaine, P. Brassard, and P. Bergeron, ''Publications of the Astronomical Society of the Pacific'' , #782 (April 2001), pp. 409–435.
The first white dwarf discovered was in the . An A-Type Star of Very Low Luminosity , Walter S. Adams, ''Publications of the Astronomical Society of the Pacific'' , #155 (October 1914), p. 198.
The companion of , 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. The Spectrum of the Companion of Sirius , W. S. Adams, ''Publications of the Astronomical Society of the Pacific'' , #161 (December 1915), pp. 236–237.
In has found over 9,000 white dwarfs, mostly new. A Catalog of Spectroscopically Confirmed White Dwarfs from the Sloan Digital Sky Survey Data Release 4 , Daniel J. Eisenstein, James Liebert, Hugh C. Harris, S. J. Kleinman, Atsuko Nitta, Nicole Silvestri, Scott A. Anderson, J. C. Barentine, Howard J. Brewington, J. Brinkmann, Michael Harvanek, Jurek Krzesiński, Eric H. Neilsen, Jr., Dan Long, Donald P. Schneider, and Stephanie A. Snedden, ''Astrophysical Journal Supplement'' , #1 (November 2006), pp. 40–58.
Although white dwarfs are known with estimated masses as low as 0.17 The Lowest Mass White Dwarf , Mukremin Kulic, Carlos Allende Prieto, Warren R. Brown, and D. Koester, ''Astrophysical Journal'' , #2 (May 2007), pp. 1451–1461. and as high as 1.33 White dwarf mass distribution in the SDSS , S. O. Kepler, S. J. Kleinman, A. Nitta, D. Koester, B. G. Castanheira, O. Giovannini, A. F. M. Costa, and L. Althaus, ''Monthly Notices of the Royal Astronomical Society'' '''375''', #4 (March 2007), pp. 1315-1324. , 2005 ).
White dwarfs were found to be extremely dense soon after their discovery. If a star is in a . The Relativity Displacement of the Spectral Lines in the Companion of Sirius , Walter S. Adams, ''Proceedings of the National Academy of Sciences of the United States of America'' , #7 (July 1925), pp. 382–387.
Such densities are possible because white dwarf material is not composed of in 1926 by an application of the newly devised Quantum Mechanics . Since electrons were known to obey Fermi-Dirac Statistics , also introduced in 1926, the Pauli Exclusion Principle meant that no two electrons could occupy the same state. At zero temperature, therefore, electrons had to occupy a band of energy levels at the bottom of the Fermi Sea —a state called '' Degenerate ''—meaning that a star could cool to zero temperature and still possess high energy. This also meant that compression of the electrons increased the number of electrons in a given volume and therefore raised the maximum energy level occupied by an electron, 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 .
Another consequence of being supported by electron degeneracy pressure is the existence of a limiting mass which no white dwarf can exceed. These limiting masses were first published in 1929 by Wilhelm AndersonÜber die Grenzdichte der Materie und der Energie, Wilhelm Anderson, ''Zeitschrift für Physik'' , #11–12 (Nov. 1929), pp. 851–856. and in 1930 by web site, accessed May 4 , 2007 . The limiting mass is now called the '' Chandrasekhar Limit ''.
If a white dwarf were to exceed the Chandrasekhar limit, and explosion in which the white dwarf is destroyed, just prior to reaching the limiting mass. Type IA Supernova Explosion Models , Wolfgang Hillebrandt and Jens C. Niemeyer, ''Annual Review of Astronomy and Astrophysics'' (2000), pp. 191–230.
White dwarfs have low , 2007 .
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 , 2007 .
Solving for the kinetic energy per unit mass, ''E''k, we find that
::
The white dwarf will be at equilibrium when its total energy, ''E''g + ''E''k, is minimized. At this point, the kinetic and gravitational potential energies should be comparable, so we may derive a rough mass-radius relationship by equating their magnitudes:
|
Dropping ''N'', which depends only on the composition of the white dwarf, and the universal constants leaves us with a relationship between mass and radius:
::
i.e., the radius of a white dwarf is inversely proportional to the cube root of its mass.
Since this analysis uses the non-relativistic formula ''p''
2/2''m'' 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 ''p''
2/2''m'' by the extreme relativistic approximation ''p'' ''c'' for the kinetic energy. With this substitution, we find
::
If we equate this to the magnitude of ''E''
g, we find that ''R'' drops out and the mass, ''M'', is forced to be
::
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 ''M''
limit. Therefore, no white dwarf can be heavier than the limiting mass ''M''
limit.
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 at the right shows the result of such a computation. It shows how radius varies with mass for non-relativistic (green curve) and relativistic (red 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, μ
e, has been set equal to 2. Radius is measured in standard solar radii and mass in standard solar masses.
''Standards for Astronomical Catalogues, Version 2.0'' , section 3.2.2, web page, accessed 12-I-2007.
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 stable.
Rapidly Rotating Stars. II. Massive White Dwarfs , Jeremiah P. Ostriker and Peter Bodenheimer, ''Astrophysical Journal'' (March 1968), pp. 1089–1098.
The visible radiation emitted by white dwarfs varies over a wide color range, from the blue-white color of an O-type , luminosity increases with increasing surface temperature; this surface temperature range corresponds to a luminosity from over 100 times the Sun's to under 1/10,000th that of the Sun's. Hot white dwarfs, with surface temperatures in excess of 30,000K, 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 and
Extreme Ultraviolet Observations .
X-ray emission from isolated hot white dwarfs , J. Heise, ''Space Science Reviews'' (February 1985), pp. 79–90.
B (center), its A-class companion IK Pegasi A (left) and the Sun (right). This white dwarf has a surface temperature of 35,500 K.]]
Unless the white dwarf estimated, on the basis of such observations, that approximately 90% of the mass of
BPM 37093 had crystallized.
4 Press release , 2004, Harvard-Smithsonian Center for Astrophysics.
Whole Earth Telescope observations of BPM 37093: a seismological test of crystallization theory in white dwarfs , A. Kanaan, A. Nitta, D. E. Winget, S. O. Kepler, M. H. Montgomery, T. S. Metcalfe, et al., arXiv:astro-ph/0411199 Other work gives a crystallized mass fraction of between 32% and 82%.
Asteroseismology of the Crystallized ZZ Ceti Star BPM 37093: A Different View , P. Brassard and G. Fontaine, ''Astrophysical Journal'' , #1 (March 2005), pp. 572–576.
Most observed white dwarfs have relatively high surface temperatures, between 8,000K and 40,000K.
III/235A: A Catalogue of Spectroscopically Identified White Dwarfs , G.P. McCook and E.M. Sion, on line at the 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 most white dwarfs are thought to be composed of carbon and oxygen, 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 1/100th of the star's total mass, which, if the atmosphere is hydrogen-dominated, is overlain by a hydrogen-rich layer with mass approximately 1/10,000th of the stars total mass.White Dwarf Stars, Steven D. Kawaler, in ''Stellar remnants'', S. D. Kawaler, I. Novikov, and G. Srinivasan, edited by Georges Meynet and Daniel Schaerer, Berlin: Springer, 1997. Lecture notes for Saas-Fee advanced course number 25. ISBN 3540615202.
, §4–5.
Although thin, these outer layers determine the thermal evolution of the white dwarf. The degenerate
Electron s 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 8,000K and 16,000K will have a core temperature between approximately 5,000,000K and 20,000,000K. The white dwarf is kept from cooling very quickly only by its outer layers' opacity to radiation.
The first attempt to classify white dwarf spectra appears to have been by and has been subsequently revised several times. It classifies a spectrum by a symbol which 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 table to the right), and a temperature index number, computed by dividing 50,400K by the
Effective Temperature . For example:
- A white dwarf with only He I lines in its spectrum and an effective temperature of 15,000 K could be given the classification of DB3, or, if warranted by the precision of the temperature measurement, DB3.5.
- A white dwarf with a polarized Magnetic Field , an effective temperature of 17,000 K, and a spectrum domainated by He I lines which also had Hydrogen features could be given the classification of DBAP3.
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. The classifiable remainder (DB, DC, DO, DZ, and 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 100,000K to 45,000K, the spectrum will be classified DO, dominated by singly ionized helium. From 30,000K to 12,000K, the spectrum will be DB, showing neutral helium lines, and below about 12,000K, 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 30,000K and 45,000K, 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.
'', was never generally accepted, and by the 1950s even Blackett felt it had been refuted.
Patrick Maynard Stuart Blackett, Baron Blackett, of Chelsea, 18 November 1897-13 July 1974 , Bernard Lovell,
''Biographical Memoirs of Fellows of the Royal Society'' (November 1975), pp. 1–115.
, pp. 39–43 In the 1960s, it was proposed that white dwarfs might have magnetic fields because of conservation of total surface ,
2007 ), pp. 1097–1101.
The True Incidence of Magnetism Among Field White Dwarfs , James Liebert, P. Bergeron, and J. B. Holberg, ''Astronomical Journal'' '''125''', #1 (January 2003), pp. 348–353.
