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The 'Chandrasekhar limit' (named after Subrahmanyan Chandrasekhar) is the maximum nonrotating mass which can be supported against gravitational collapse by electron degeneracy pressure. It is commonly given as being about 1.4, 1.38^[1] or 1.44 solar masses. Computed values for the limit will vary depending on the nuclear composition of the mass and the approximations used. Chandrasekhar^{[2], eq. (36),[3], eq. (58),[4], eq. (43)} gives a value of
::::::::::
ight )^{3/2}rac{1}{(mu_e m_H)^2}.
Here, μ_e is the average molecular weight per electron, $m\_H$ is the mass of the
hydrogen atom, and is a constant connected with the solution to the Lane-Emden equation. Numerically, this value is approximately (2/μ_e)² · 2.85 · 10³⁰ kg, or , where
m kg} is the standard solar mass.^[5] As $sqrt\{hbar\; c/G\}$ is the Planck mass, $M\_\{$
m Pl}pprox 2.176cdot 10^{-8} {
m kg}, the limit is of the order of M_Pl³/m_H².
As white dwarf stars are supported by electron degeneracy pressure, this is an upper limit for the mass of a white dwarf. Main-sequence stars with a mass exceeding approximately 8 solar masses therefore cannot lose enough mass to form a stable white dwarf at the end of their lives, and instead form either a neutron star or black hole.^[6][7][8]

Contents

Physics

History

Applications

A type Ia supernova apparently from a supra-limit white dwarf

References

Further reading

See also

Physics

Electron degeneracy pressure is a quantum-mechanical effect arising from the Pauli exclusion principle. Since electrons are fermions, no two electrons can be in the same state, so not all electrons can be in the minimum-energy level. Rather, electrons must occupy a band of energy levels. Compression of the electron gas increases the number of electrons in a given volume and raises the maximum energy level in the occupied band. Therefore, the energy of the electrons will increase upon compression, so pressure must be exerted on the electron gas to compress it. This is the origin of electron degeneracy pressure.
In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form P=K₁ρ^5/3. Solving the hydrostatic equation leads to a model white dwarf which is a polytrope of index 3/2 and therefore has radius inversely proportional to the cube root of its mass, and volume inversely proportional to its mass.^[9]
As the mass of a model white dwarf increases, the typical energies to which degeneracy pressure forces the electrons are no longer negligible relative to their rest masses. The velocities of the electrons approach the speed of light, and special relativity must be taken into account. In the strongly relativistic limit, we find that the equation of state takes the form P=K₂ρ^4/3. This will yield a polytrope of index 3, which will have a total mass, M_limit say, depending only on K₂.^[10]

For a fully relativistic treatment, the equation of state used will interpolate between the equations P=K₁ρ^5/3 for small ρ and P=K₂ρ^4/3 for large ρ.
When this is done, the model radius still decreases with mass, but becomes zero at M_limit. This is the Chandrasekhar limit.³ The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored green and red, respectively. μ_e has been set equal to 2.
Radius is measured in standard solar radii⁵ and mass in standard solar masses.
A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature.^[11] Lieb and Yau^[12] have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.

History

In 1926, the British physicist Ralph H. Fowler observed that the relationship between the density, energy and temperature of white dwarfs could be explained by viewing them as a gas of nonrelativistic, non-interacting electrons and nuclei which obeyed Fermi-Dirac statistics.^[13] This Fermi gas model was then used by the British physicist E. C. Stoner in 1929 to calculate the relationship between the mass, radius, and density of white dwarfs, assuming them to be homogenous spheres.^[14] Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37 · 10³⁰ kg.^[15] In 1930, Stoner derived the internal energy-density equation of state for a Fermi gas, and was then able to treat the mass-radius relationship in a fully relativistic manner, giving a limiting mass of approximately (for μ_e=2.5) 2.19 · 10³⁰ kg.^[16] Stoner went on to derive the pressure-density equation of state, which he published in 1932.^[17] These equations of state were also previously published by the Russian physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter.^[18] Frenkel's work, however, was ignored by the astronomical and astrophysical community.^[19]
A series of papers published between 1931 and 1935 had its beginning
on a trip from India to England in 1930,
where the Indian physicist Subrahmanyan Chandrasekhar worked on the calculation of the statistics of a degenerate Fermi gas.^[20] In these papers, Chandrasekhar solved
the hydrostatic equation together with the nonrelativistic Fermi gas equation of state,⁹ and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above.^102[21]3 Chandrasekhar reviews this work in his Nobel Prize lecture.⁴ This value was also computed in 1932 by the Soviet physicist Lev Davidovich Landau,^[22] who, however, did not apply it to white dwarfs.
Chandrasekhar's work on the limit aroused controversy, owing to the opposition of the British astrophysicist Arthur Stanley Eddington. Eddington was aware that the existence of black holes was theoretically possible, and also realized that the existence of the limit made their formation possible. However, he was unwilling to accept that this could happen. After a talk by Chandrasekhar on the limit in 1935, he replied:

The star has to go on radiating and radiating and contracting and contracting until, I suppose, it gets down to a few km. radius, when gravity becomes strong enough to hold in the radiation, and the star can at last find peace. … I think there should be a law of Nature to prevent a star from behaving in this absurd way!^[23]

Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law P=K₁ρ^5/3 universally applicable, even for large ρ.^[24] Although Bohr, Fowler, Pauli, and other physicists agreed with Chandrasekhar's analysis, at the time, owing to Eddington's status, they were unwilling to publicly support Chandrasekhar.^{[25], pp. 110–111} Through the rest of his life, Eddington held to his position in his writings,^{[26][27][28][29][30]} including his work on his fundamental theory.^[31] The drama associated with this disagreement is one of the main themes of ''Empire of the Stars'', Arthur I. Miller's biography of Chandrasekhar.²⁵ In Miller's view:

Chandra's discovery might well have transformed and accelerated developments in both physics and astrophysics in the 1930s. Instead, Eddington's heavy-handed intervention lent weighty support to the conservative community astrophysicists, who steadfastly refused even to consider the idea that stars might collapse to nothing. As a result, Chandra's work was almost forgotten.^{25, p. 150}

Applications

The core of a star is kept from collapsing by the heat generated by the fusion of nuclei of lighter elements into heavier ones. At various points in a star's life, the nuclei required for this process will be exhausted, and the core will collapse, causing it to become denser and hotter. A critical situation arises when iron accumulates in the core, since iron nuclei are incapable of generating further energy through fusion. If the core becomes sufficiently dense, electron degeneracy pressure will play a significant part in stabilizing it against gravitational collapse.^[32]
If a main-sequence star is not too massive (less than approximately 8 solar masses), it will eventually shed enough mass to form a white dwarf having mass below the Chandrasekhar limit, which will consist of the former core of the star, For more massive stars, electron degeneracy pressure will not keep the iron core from collapsing to very great density, leading to formation of a neutron star, black hole, or, speculatively, a quark star. (For very massive, low-metallicity stars, it is also possible that instabilities will destroy the star completely.)^678[33] During the collapse, neutrons are formed by the capture of electrons by protons, leading to the emission of neutrinos.^{32, pp. 1046–1047.} The decrease in gravitational potential energy of the collapsing core releases a large amount of energy which is on the order of 10⁴⁶ joules (100 foes.) Most of this energy is carried away by the emitted neutrinos.^[34] This process is believed to be responsible for supernovae of types Ib, Ic, and II.³²
Type Ia supernovae derive their energy from runaway fusion of the nuclei in the interior of a white dwarf. This fate may befall carbon-oxygen white dwarfs that accrete matter from a companion giant star, leading to a steadily increasing mass. It is believed that, as the white dwarf's mass approaches the Chandrasekhar limit, its central density increases, and, as a result of compressional heating, its temperature also increases. This results in an increasing rate of fusion reactions, eventually igniting a thermonuclear flame which causes the supernova.^{[35], §5.1.2}
Strong indications of the reliability of Chandrasekhar's formula are:
# Only one white dwarf with a mass greater than Chandrasekhar's limit has ever been observed. (See below.)
# The absolute magnitudes of supernovae of Type Ia are all approximately the same; at maximum luminosity, M_V is approximately -19.3, with a standard deviation of no more than 0.3.^{35, (1)} A 1-sigma interval therefore represents a factor of less than 2 in luminosity. This seems to indicate that all type Ia supernovae convert approximately the same amount of mass to energy.

A type Ia supernova apparently from a supra-limit white dwarf

:''Main article: Champagne Supernova.''
On April 2003, the Supernova Legacy Survey observed a type Ia supernova, designated SNLS-03D3bb, in a galaxy approximately 4 billion light years away. According to a group of astronomers at the University of Toronto and elsewhere,
the observations of this supernova are best explained by assuming that it arose from a white dwarf which grew to twice the mass of the Sun before exploding. They believe that the star, dubbed the "Champagne Supernova" by David R. Branch, may have been spinning so fast that centrifugal force allowed it to exceed the limit. Alternatively, the supernova may have resulted from the merger of two white dwarfs, so that the limit was only violated momentarily. Nevertheless, they point out that this observation poses a challenge to the use of type Ia supernovae as standard candles. The results were published in the journal ''Nature'' on September 21, 2006.^[36][37][38]

References

1. A Common Explosion Mechanism for Type Ia Supernovae, Mazzali, P. A.; K. Röpke, F. K.; Benetti, S.; Hillebrandt, W., , , Science, 2007
2. The Highly Collapsed Configurations of a Stellar Mass, S. Chandrasekhar, ''Monthly Notices of the Royal Astronomical Society'' '91' (1931), 456–466.
3. The Highly Collapsed Configurations of a Stellar Mass (second paper), S. Chandrasekhar, ''Monthly Notices of the Royal Astronomical Society'', '95' (1935), pp. 207--225.
4. ''On Stars, Their Evolution and Their Stability'', Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
5. ''Standards for Astronomical Catalogues, Version 2.0'', section 3.2.2, web page, accessed 12-I-2007.
6. White dwarfs in open clusters. VIII. NGC 2516: a test for the mass-radius and initial-final mass relations, D. Koester and D. Reimers, ''Astronomy and Astrophysics'' '313' (1996), pp. 810–814.
7. An Empirical Initial-Final Mass Relation from Hot, Massive White Dwarfs in NGC 2168 (M35), Kurtis A. Williams, M. Bolte, and Detlev Koester, ''Astrophysical Journal'' '615', #1 (2004), pp. L49–L52; also arXiv astro-ph/0409447.
8. How Massive Single Stars End Their Life, A. Heger, C. L. Fryer, S. E. Woosley, N. Langer, and D. H. Hartmann, ''Astrophysical Journal'' '591', #1 (2003), pp. 288–300.
9. The Density of White Dwarf Stars, S. Chandrasekhar, ''Philosophical Magazine'' (7th series) '11' (1931), pp. 592–596.
10. The Maximum Mass of Ideal White Dwarfs, S. Chandrasekhar, ''Astrophysical Journal'' '74' (1931), pp. 81–82.
11. The Neutron Star and Black Hole Initial Mass Function, F. X. Timmes, S. E. Woosley, and Thomas A. Weaver, ''Astrophysical Journal'' '457' (February 1, 1996), pp. 834–843.
12. A rigorous examination of the Chandrasekhar theory of stellar collapse, Elliott H. Lieb and Horng-Tzer Yau, ''Astrophysical Journal'' '323' (1987), pp. 140–144.
13. On Dense Matter, R. H. Fowler, ''Monthly Notices of the Royal Astronomical Society'' '87' (1926), pp. 114–122.
14. The Limiting Density of White Dwarf Stars, Edmund C. Stoner, ''Philosophical Magazine'' (7th series) '7' (1929), pp. 63–70.
15. Über die Grenzdichte der Materie und der Energie, Wilhelm Anderson, ''Zeitschrift für Physik'' '56', #11–12 (Nov. 1929), pp. 851–856. DOI 10.1007/BF01340146.
16. The Equilibrium of Dense Stars, Edmund C. Stoner, ''Philosophical Magazine'' (7th series) '9' (1930), pp. 944–963.
17. The minimum pressure of a degenerate electron gas, E. C. Stoner, ''Monthly Notices of the Royal Astronomical Society'' '92' (May 1932), pp. 651–661.
18. Anwendung der Pauli-Fermischen Elektronengastheorie auf das Problem der Kohäsionskräfte, J. Frenkel, ''Zeitschrift für Physik'' '50', #3–4 (March 1928), pp. 234–248. DOI 10.1007/BF01328867.
19. The article by Ya I Frenkel' on `binding forces' and the theory of white dwarfs, D. G. Yakovlev, ''Physics Uspekhi'' '37', #6 (1994), pp. 609–612.
20. Chandrasekhar's biographical memoir at the National Academy of Sciences, web page, accessed 12-I-2007.
21. Stellar Configurations with degenerate Cores, S. Chandrasekhar, ''The Observatory'' '57' (1934), pp. 373–377.
22. On the Theory of Stars, in ''Collected Papers of L. D. Landau'', ed. and with an introduction by D. ter Haar, New York: Gordon and Breach, 1965; originally published in ''Phys. Z. Sowjet.'' '1' (1932), 285.
23. Meeting of the Royal Astronomical Society, Friday, 1935 January 11, ''The Observatory'' '58' (February 1935), pp. 33–41.
24. On "Relativistic Degeneracy", Sir A. S. Eddington, ''Monthly Notices of the Royal Astronomical Society'' '95' (1935), 194–206.
25. ''Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes'', Arthur I. Miller, Boston, New York: Houghton Mifflin, 2005, ISBN 0-618-34151-X; reviewed at ''The Guardian'': The battle of black holes.
26. The International Astronomical Union meeting in Paris, 1935, ''The Observatory'' '58' (September 1935), pp. 257–265, at p. 259.
27. Note on "Relativistic Degeneracy", Sir A. S. Eddington, ''Monthly Notices of the Royal Astronomical Society'' '96' (November 1935), 20–21.
28. The Pressure of a Degenerate Electron Gas and Related Problems, Arthur Eddington, ''Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences'' '152' (November 1, 1935), pp. 253–272.
29. ''Relativity Theory of Protons and Electrons'', Sir Arthur Eddington, Cambridge: Cambridge University Press, 1936, chapter 13.
30. The physics of white dwarf matter, Sir A. S. Eddington, ''Monthly Notices of the Royal Astronomical Society'' '100' (June 1940), pp. 582–594.
31. ''Fundamental Theory'', Sir A. S. Eddington, Cambridge: Cambridge University Press, 1946, §43–45.
32. The evolution and explosion of massive stars, S. E. Woosley, A. Heger, and T. A. Weaver, ''Reviews of Modern Physics'' '74', #4 (October 2002), pp. 1015–1071.
33. Strange quark matter in stars: a general overview, Jürgen Schaffner-Bielich, ''Journal of Physics G: Nuclear and Particle Physics'' '31', #6 (2005), pp. S651–S657; also arXiv astro-ph/0412215.
34. The Physics of Neutron Stars, by J. M. Lattimer and M. Prakash, ''Science'' '304', #5670 (2004), pp. 536–542; also arXiv astro-ph/0405262.
35. Type IA Supernova Explosion Models, Wolfgang Hillebrandt and Jens C. Niemeyer, ''Annual Review of Astronomy and Astrophysics'' '38' (2000), pp. 191–230.
36. The weirdest Type Ia supernova yet, LBL press release, web page accessed 13-I-2007.
37. Champagne Supernova Challenges Ideas about How Supernovae Work, web page, spacedaily.com, accessed 13-I-2007.
38. The type Ia supernova SNLS-03D3bb from a super-Chandrasekhar-mass white dwarf star, D. Andrew Howell et al., ''Nature'' '443' (September 21, 2006), pp. 308–311; also, arXiv:astro-ph/0609616.

Further reading

★ ''On Stars, Their Evolution and Their Stability'', Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.

★ ''White dwarf stars and the Chandrasekhar limit'', Masters' thesis, Dave Gentile, DePaul University, 1995.

★ Estimating Stellar Parameters from Energy Equipartition, sciencebits.com. Discusses how to find mass-radius relations and mass limits for white dwarfs using simple energy arguments.

See also

★ Degenerate matter

★ Tolman-Oppenheimer-Volkoff limit