Chandrasekhar limit

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The Chandrasekhar limit 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 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[1], eq. (36),[2], eq. (58),[3], eq. (43) gives a value of

\frac{\omega_3^0 \sqrt{3\pi}}{2}\left ( \frac{\hbar c}{G}\right )^{3/2}\frac{1}{(\mu_e m_H)^2}.

Here, μe is the average molecular weight per electron, mH is the mass of the hydrogen atom, and \omega_3^0 \approx 2.018236 is a constant connected with the solution to the Lane-Emden equation. Numerically, this value is approximately (2/μe)2 · 2.85 · 1030 kg, or 1.43 (2/\mu_e)^2 M_{\bigodot}, where M_{\bigodot}=1.989\cdot 10^{30} \ {\rm kg} is the standard solar mass.[4] As \sqrt{\hbar c/G} is the Planck mass, M_{\rm Pl}\approx 2.176\cdot 10^{-8}\ {\rm kg}, the limit is of the order of MPl3/mH2.

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.[5][6][7]

Contents

[edit] 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, and it is therefore impossible for all electrons to be in a minimum-energy level. Electrons must fill higher-energy levels. Compression of the electron gas increases the number of electrons in a given volume and therefore raises the maximum energy level occupied by an electron, causing the pressure. In the nonrelativistic case, this gives rise to an equation of state of the form P=K1ρ5/3. 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.[8]

As the mass of this 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=K2ρ4/3. This will yield a polytrope of index 3, which will have a total mass, Mlimit say, depending only on K2.[9]

Radius versus mass for a model white dwarf.
Radius versus mass for a model white dwarf.

For a fully relativistic treatment, the equation of state used will interpolate between the equations P=K1ρ5/3 for small ρ and P=K2ρ4/3 for large ρ. When this is done, the model radius still decreases with mass, but becomes zero at Mlimit. This is the Chandrasekhar limit.[2] 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. μe has been set equal to 2. Radius is measured in standard solar radii[4] 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.[10] Lieb and Yau[11] have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.

[edit] 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.[12] 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.[13] Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37 · 1030 kg,[14] and Stoner then treated phase space in a fully relativistic manner, giving a limiting mass of approximately (for μe=2.5) 2.19 · 1030 kg.[15] 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.[16] In these papers, Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state,[8] and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above.[9][1],[17][2] Chandrasekhar reviews this work in his Nobel Prize lecture.[3] This value was also computed in 1932 by the Soviet physicist Lev Davidovich Landau,[18] 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![19]

Eddington's proposed solution to the perceived problem was to modify relativistic mechanics so as to make the law P=K1ρ5/3 universally applicable, even for large ρ.[20] 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.[21], pp. 110–111 Through the rest of his life, Eddington held to his position in his writings,[22][23][24][25][26] including his work on his fundamental theory.[27] The drama associated with this disagreement is one of the main themes of Empire of the Stars, Arthur I. Miller's biography of Chandrasekhar.[21] 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.[21], p. 150

[edit] 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.

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 stars, it is also possible that instabilities will destroy the star completely.)[5][6],[7][28] During the collapse, neutrons are formed by the capture of electrons by protons, leading to the emission of neutrinos.[29], 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.[30] This process is believed to be responsible for supernovae of types Ib, Ic, and II.[29]

Supernovae of type Ia derive their energy from runaway fusion of the carbon and oxygen nuclei in the interior of a white dwarf. This fate befalls white dwarfs that accrete matter from a companion giant star, leading to a steadily increasing mass. It was once believed that fusion was triggered when the white dwarf reached the Chandrasekhar limit, but this idea fell out of favor in the 1960s. The current view is that fusion is initiated in accreting white dwarfs somewhat before the limiting mass is attained.

Strong indications of the reliability of Chandrasekhar's formula are:

  1. Only one white dwarf with a mass greater than Chandrasekhar's limit has ever been observed. (See below.)
  2. Supernovae of Type Ia have an absolute luminosity (MV) of -19.6 ± 0.6. This interval represents only a factor of 3 in luminosity. This seems to indicate that all type Ia supernovae convert approximately the same amount of mass to energy.

[edit] 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.[31][32][33]

[edit] See also

[edit] References

  1. ^ a b The Highly Collapsed Configurations of a Stellar Mass, S. Chandrasekhar, Monthly Notices of the Royal Astronomical Society 91 (1931), 456–466. Paper at ADSABS
  2. ^ a b c The Highly Collapsed Configurations of a Stellar Mass (second paper), S. Chandrasekhar, Monthly Notices of the Royal Astronomical Society, 95 (1935), pp. 207--225. Paper at ADSABS
  3. ^ a b On Stars, Their Evolution and Their Stability, Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
  4. ^ a b Standards for Astronomical Catalogues, Version 2.0, section 3.2.2, web page, accessed 12-I-2007.
  5. ^ a b 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. Paper at ADSABS
  6. ^ a b 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. Paper at ADSABS; also arXiv astro-ph/0409447.
  7. ^ a b 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. Paper at ADSABS
  8. ^ a b The Density of White Dwarf Stars, S. Chandrasekhar, Philosophical Magazine (7th series) 11 (1931), pp. 592–596.
  9. ^ a b The Maximum Mass of Ideal White Dwarfs, S. Chandrasekhar, Astrophysical Journal 74 (1931), pp. 81–82. Paper at ADSABS
  10. ^ 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. Paper at ADSABS
  11. ^ A rigorous examination of the Chandrasekhar theory of stellar collapse, Elliott H. Lieb and Horng-Tzer Yau, Astrophysical Journal 323 (1987), pp. 140–144. Paper at ADSABS
  12. ^ On Dense Matter, R. H. Fowler, Monthly Notices of the Royal Astronomical Society 87 (1926), pp. 114–122. Paper at ADSABS
  13. ^ The Limiting Density of White Dwarf Stars, Edmund C. Stoner, Philosophical Magazine (7th series) 7 (1929), pp. 63–70.
  14. ^ Über die Grenzdichte der Materie und der Energie, Wilhelm Anderson, Zeitschrift für Physik 56, #11–12 (Nov. 1929), pp. 851–856.
  15. ^ The Equilibrium of Dense Stars, Edmund C. Stoner, Philosophical Magazine (7th series) 9 (1930), pp. 944–963.
  16. ^ Chandrasekhar's biographical memoir at the National Academy of Sciences, web page, accessed 12-I-2007.
  17. ^ Stellar Configurations with degenerate Cores, S. Chandrasekhar, The Observatory 57 (1934), pp. 373–377. Paper at ADSABS
  18. ^ 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.
  19. ^ Meeting of the Royal Astronomical Society, Friday, 1935 January 11, The Observatory 58 (February 1935), pp. 33–41. Paper at ADSABS
  20. ^ On "Relativistic Degeneracy", Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 95 (1935), 194–206. Paper at ADSABS
  21. ^ a b c 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.
  22. ^ The International Astronomical Union meeting in Paris, 1935, The Observatory 58 (September 1935), pp. 257–265, at p. 259. Paper at ADSABS
  23. ^ Note on "Relativistic Degeneracy", Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 96 (November 1935), 20–21. Paper at ADSABS
  24. ^ 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. Paper at JSTOR
  25. ^ Relativity Theory of Protons and Electrons, Sir Arthur Eddington, Cambridge: Cambridge University Press, 1936, chapter 13.
  26. ^ The physics of white dwarf matter, Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 100 (June 1940), pp. 582–594. Paper at ADSABS
  27. ^ Fundamental Theory, Sir A. S. Eddington, Cambridge: Cambridge University Press, 1946, §43–45.
  28. ^ 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. Paper at ADSABS; also arXiv astro-ph/0412215.
  29. ^ a b 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. Paper at ADSABS
  30. ^ The Physics of Neutron Stars, by J. M. Lattimer and M. Prakash, Science 304, #5670 (2004), pp. 536–542. Paper at ADSABS; also arXiv astro-ph/0405262.
  31. ^ The weirdest Type Ia supernova yet, LBL press release, web page accessed 13-I-2007.
  32. ^ Champagne Supernova Challenges Ideas about How Supernovae Work, web page, spacedaily.com, accessed 13-I-2007.
  33. ^ 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.

[edit] External links

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