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White Dwarf

White dwarfs are stars of about one solar mass with characteristic radii of about $5000\ km$ and mean densities of around $10^{6}\ gcm^{3}$. These stars no longer burn nuclear fuel. Instead, they are slowly cooling as they radiate away their residual thermal energy. We know today that white dwarfs support themselves against gravity by the pressure of degenerate electrons. This fact was not always clear to astronomers, although the compact nature of white dwarfs was readily apparent from early observations. For example, the mass of Sirius B, the binary companion to Sirius and the best known white dwarf, was determined by applying Kepler's third law of the binary star orbit.


What is a White Dwarf?

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The first white dwarf to be discovered historically is a companion of Sirius. White dwarfs typically have masses on the order of one solar mass, and sizes comparable to that of the Earth (which is about $100$ times smaller than the sun). White dwarfs therefore have average densities which are about $100^{3}$ = $10^{6}$, a million times greater than that of the sun:

mean density of white dwarf = $\frac{mass}{volume}$ = $10^{6}\ gm/cm^{3}$.

Composition and Structure

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To say that white dwarfs are strange is an understatement. An Earth sized white dwarfs has a density of $1\ \times\ 10^{9}\ kg/m^{3}$. In comparison, the Earth itself has an average density of only $5.4\ \times\ 103\ kg/m^{3}$. That means a white dwarf is a million times as dense. Under normal circumstances, identical electrons are not allowed to occupy the same energy level. Since there are only two ways an electron can spin, only two electrons can occupy a single energy level. This is what is known in physics as the Pauli exclusion principle.
When the white dwarf shrinks down to its final size, it will no longer have any fuel available for nuclear fusion. It will, however, still have a very hot core, and a large reservoir of residual heat. Time will pass, and with it, the white dwarf will cool down as it radiates its heat into space. As it does so, it will also grow dimmer. The more massive a white dwarf star the smaller its surface area than those white dwarfs which are less massive. This means that massive white dwarfs are less luminous for a given temperature, so that their evolutionary tracks are below those of the less massive white dwarf stars.

Characteristics of White Dwarfs

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White dwarfs have been observed with surface temperatures from 4000 to 85000 K, but computer models predict that it is possible for them to have even higher temperatures. The masses of white dwarfs ranges from perhaps 0.02 solar masses up to 1.4 solar masses. Because the typical white dwarf is comparable in size to the earth, the density of these stars is very high, about $10^{6}$ grams per cubic centimeter. A teaspoon of white dwarf material would weigh 5 tons! When the sun becomes a white dwarf, its mass will be about $0.6$ of its present mass. The remainder of its material will have been puffed away during its red giant stage and blown away during its planetary nebula stage. It will be nearly as small as the earth, with luminosity about a 10th of its current luminosity. As time passes, our sun's luminosity will keep decreasing, and it will simply fade away. To get the white dwarf stage, sun like stars must go through the following stages: protostar, main sequence star, red giant, and planetary nebula. Protostars produce energy from gravitational contraction. Main sequence stars produce energy from nuclear fusion. Red giants produce energy from gravitational contraction of their cores and from fusion within shells around the cores. White dwarfs, however, do not produce energy. They are hot because of left over energy, and as they radiate this energy away, they get cooler. After billions of years, a white dwarf will have cooled enough that it no longer radiates in the visible region of the spectrum. It will appear on the HR diagram at a position below and to the right of the bottom of the main sequence. As billions more years pass, it will cool further to become a black dwarf, the burned out cinder of a once proud star. It is unlikely that the universe is old enough for many, if any, black dwarfs to have formed. 

White Dwarf vs Neutron Star

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White dwarfs are stars of about one solar mass with a characteristic radius of 5000 km, corresponding to a mean density of $10^{6}\ g/cm^{-3}$. They are no longer burning nuclear fuel, but  are steadily cooling away their internal heat. In 1926, only three white dwarfs were firmly detected. Actual models of white dwarf stars, taking into account the special relativistic effects in the degenerate electron equation of state were then constructed in 1930 by Chandrasekhar. He made the fundamental discovery of a maximum mass of $1.4\ Mo$ for white dwarfs- the exact value somewhat depends on the chemical composition. 

The prediction of the existence of neutron stars as a possible end point of stellar evolution was independent of observations. Following the discovery of the neutron by Chadwick, it was realized by many people that at very high densities electrons would react with protons to form neutrons via inverse beta decay. Neutron stars had been found at the end of the 1960s as radio pulsars and in the beginning of the 1970s as X ray stars. A firm upper limit for the mass of neutron stars was then seen as evidence for the existence of even more exotic objects- black holes.

White Dwarf and Black Hole

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There are three ways a star can die. If it is of the same size as our sun, it will exhaust its hydrogen fuel, expand to a red giant, and then slowly shrink to a white dwarf. Eventually it will cool to a black dwarf, a permanently embalmed corpse that never changes. 
If a star is moderately more massive than the sun, it is likely to explode into a supernova leaving a still more dense residue of a neutron star astronomers call pulsar, because it sends out absolutely regular pulses or beeps of radio waves, sometimes beeps of visible light. 
Finally, if a star is much more massive than the sun, it is expected to expire in a manner so bizarre that its ultimate fate is still an unresolved mystery. On expiry it undergoes a catastrophic implosion, a sort of explosion in reverse, so that even neutrons are unable to withstand its enormous gravitational compression. All particles are crushed out of existence and the laws of physics cease to have any meaning. It has entered a singular state called black hole. 

What is Chandrasekhar's Limit?

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Even though electron degeneracy supports the white dwarf against collapsing completely, there is a limit to the amount of pressure that degenerate electrons can withstand. In 1930, a 19 years old student on his way from native India to graduate school in England calculated this limit, showing it to be 1.4 solar masses. Subrahmanyan Chandrasekhar was awarded a share of the 1983 Nobel prize in physics for this discovery, and the limit is known as the Chandrasekhar limit. Chandrasekhar's calculations showed that if a white dwarf becomes more massive than 1.4 solar masses, the pressure caused by gravity at its surface is greater than the maximum pressure a degenerate gas can support. This means that white dwarfs must have masses less than 1.4 times the Sun's mass. Main sequence stars with masses of up to about 4 solar masses can end up as white dwarfs only because they lose mass during the red giant and planetary nebula stage. Stars more massive than these stars retain cores whose mass exceeds the Chadrasekhar limit and cannot form white dwarfs. Such moderately massive and massive stars end their lives in a dramatically different way than do less massive stars. 
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