Everything about White Dwarf totally explained
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're very
dense. Their faint
luminosity comes from the emission of stored
heat. The unusual faintness of white dwarfs was first recognized in 1910 by
Henry Norris Russell,
Edward Charles Pickering and
Williamina Fleming; 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, which enables 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 can't 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's formed, but since it has no source of energy, it'll 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's no longer visible and become a cold
black dwarf. even the oldest white dwarfs still radiate at temperatures of a few thousand
kelvins, and no black dwarfs are thought to exist yet.
, 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'd 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'd then have called "possible" values of the surface brightness and density. I must have shown that I wasn't 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 wouldn't 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; It wasn't until
January 31,
1862 that
Alvan Graham Clark observed a previously unseen star close to Sirius, later identified as the predicted companion.
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. the term was later popularized by
Arthur Stanley Eddington. and by 1999, over 2,000 were known. Since then the
Sloan Digital Sky Survey has found over 9,000 white dwarfs, mostly new.
Composition and structure
Although white dwarfs are known with estimated masses as low as 0.17 and as high as 1.33 solar masses, the mass distribution is strongly peaked at 0.6 solar mass, and the majority lie between 0.5 to 0.7 solar mass. this is comparable to the Earth's radius of approximately 0.009 solar radius. A white dwarf, then, packs mass comparable to the Sun's into a volume that's typically a million times smaller than the Sun's; the average density of matter in a white dwarf must therefore be, very roughly, 1,000,000 times greater than the average density of the Sun, or approximately 10
6 grams (1
tonne) per
cubic centimeter.
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's 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've 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 isn't 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. 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 couldn't 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. 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. 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's 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, The limiting mass is now called the
Chandrasekhar limit.
If a white dwarf were to exceed the Chandrasekhar limit, and
nuclear reactions didn't 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 shouldn't 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.
Mass-radius relationship and mass limit
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'll be approximately
N p2/2
m, 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, Δ
p, given by the
uncertainty principle, which says that Δ
p Δ
x is on the order of the reduced
Planck constant,
ħ. Δ
x will be on the order of the average distance between electrons, which will be approximately
n−1/3, for example, 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
» :
Solving this for the radius,
R, gives 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's no limit to the mass for which it's possible for a model white dwarf to be in static equilibrium. Not all of these model stars, however, will be
dynamically stable.
Radiation and cooling
The visible radiation emitted by white dwarfs varies over a wide color range, from the blue-white color of an O-type
main sequence star to the red of a M-type
red dwarf. White dwarf
effective surface temperatures extend from over 150,000 K In accordance with the
Stefan-Boltzmann law, 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.
Unless the white dwarf
accretes matter from a companion star or other source, this radiation comes from its stored heat, which isn't replenished. White dwarfs have an extremely small surface area to radiate this heat from, so they remain hot for a long time.
, 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. 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. 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 hasn't 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.
, §5–6 This atmosphere, the only part of the white dwarf visible to us, is thought to be the top of an envelope which is a residue of the star's envelope in the
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 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.
, §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 almost
isothermal, and it's 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. and various classification schemes have been proposed and used since then. The system currently in use was introduced by Edward M. Sion and his coauthors in 1983 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 50,400 K by the
effective temperature. For example:
