White Dwarfs and the Chandrasekhar Limit

A white dwarf is the remnant of a main-sequence star of mass (less than about four times the mass of the Sun) that has exhausted its hydrogen fuel by fusion into helium. Such a star will first expand to form a red giant as it fuses helium in its core to carbon and oxygen by triple-alpha processes. After the star sheds its outer layers, ejecting a planetary nebula, the remnant will be composed mainly of carbon and oxygen, incapable of further fusion reactions. The surface temperature initially lies in the range 8000 to 40,000 K, which implies a white color, hence the designation white dwarf. The gravitational field of the white dwarf causes a collapse to a body about the size of the Earth. Thereby a mass comparable to the Sun's, , is compressed to a radius comparable to that of the Earth, . Further collapse is resisted by the electrons of the carbon and oxygen atoms, which form a degenerate electron gas following a Fermi–Dirac distribution. The outward pressure of the electrons, countering the gravitational compression, is thus a purely quantum-mechanical effect, which can be attributed to the exclusion principle.
Among the first identified white dwarfs, in 1915, is Sirius B, the companion to Sirius.
S. Chandrasekhar proposed in 1931 that in a stellar remnant with mass greater than approximately 1.44 , known as the Chandrasekhar limit, gravitation overcomes the electron degeneracy pressure and the white dwarf collapses into a fraction of its volume to form a neutron star. This is associated with the electrons near the Fermi level becoming ultra-relativistic, with energies approaching the electron rest energy . A neutron star is also a degenerate fermionic quantum system of neutrons, into which the carbon and oxygen nuclei collapse. In a neutron star, a stellar mass is compressed to a radius of the order of 10 km. Some neutron stars can emit beams of electromagnetic radiation, which makes them detectable as pulsars.
It is sometimes said that white dwarfs have densities of the order of tonnes per teaspoon, while neutron stars have densities of billions of tonnes per teaspoon (1 tonne, or metric ton, equals 1000 kg).
If the remnant star has a mass exceeding the Tolman–Oppenheimer–Volkoff limit of around , the combination of degeneracy pressure and nuclear forces becomes insufficient to support the neutron star and it continues collapsing to form a black hole.

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The derivation of the radius versus mass relationship for a white dwarf is given in the references. The inward gravitational pressure must be balanced by the outward pressure of a relativistic Fermi gas of electrons. This results in a differential equation that can only be solved numerically. We have obtained a reasonably accurate analytic approximation to this result in the form
,
where is the radius of the Earth and is the mass of the Sun. The radius reduces to zero at the Chandrasekhar mass , which is given in our model by
,
where is the proton mass and is the gravitational constant. Remarkably, represents the Planck mass, showing that quantum mechanics, relativity and gravitation are all involved in the theory of white dwarfs.
References
[1] S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects, New York: Wiley, 1983 pp. 51–81.
[2] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, New York: Dover, 1957 pp. 412 ff.
[3] D. Garfinkle, "The Planck Mass and the Chandrasekhar Limit," American Journal of Physics, 77(8), 2009 pp. 683–687. doi:10.1119/1.3110884.
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