Chandrasekhar limit

When a star starts running out of fuel, it usually cools off and collapses (possibly with a supernova) into one of three compact forms, depending on its total mass:

• a White Dwarf, a big lump of Carbon and Oxygen atoms, almost like one huge molecule. It glows only because it's still hot and cooling off - there's rarely any fusion going on.
• a Neutron Star, a big lump of neutrons, almost like one huge nucleus. To get here, it must have been big enough to burn carbon and create larger nuclei, then end with a supernova.
• a Black Hole, if it's very massive.

The Chandrasekhar limit (pronounced /tʃʌndrəˈʃeɪkɑr/) is the maximum mass of a stable white dwarf star. It was named after Subrahmanyan Chandrasekhar, the Indian astrophysicist who predicted it in 1930. White dwarfs, unlike main sequence stars, resist gravitational collapse primarily through electron degeneracy pressure, rather than thermal pressure. The Chandrasekhar limit is the mass above which electron degeneracy pressure in the star's core is insufficient to balance the star's own gravitational self-attraction. Consequently, white dwarfs with masses greater than the limit undergo further gravitational collapse, evolving into a different type of stellar remnant, such as a neutron star or black hole. Those with masses under the limit remain stable as white dwarfs. The Chandrasekhar limit is analogous to the Tolman–Oppenheimer–Volkoff limit for neutron stars.

The currently accepted numerical value of the limit is about 1.4 $\begin{smallmatrix}M_\odot\end{smallmatrix}$, or 2.864 × 10³⁰ kg.^[1][2]

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, producing electron degeneracy pressure. With sufficient compression, electrons are forced into nuclei in the process of electron capture, relieving the pressure.

Radius–mass relations for a model white dwarf. The green curve uses the general pressure law for an ideal Fermi gas, while the blue curve is for a non-relativistic ideal Fermi gas. The black line marks the ultra-relativistic limit.

In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form $P = K_1 \rho^{5\over 3}$. 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.^[3]

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 $P = K_2 \rho^{4\over 3}$. This will yield a polytrope of index 3, which will have a total mass, M_limit say, depending only on K₂.^[4]

For a fully relativistic treatment, the equation of state used will interpolate between the equations $P = K_1 \rho^{5\over 3}$ for small ρ and $P = K_2 \rho^{4\over 3}$ 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.^[5] The curves of radius against mass for the non-relativistic and relativistic models are shown in the graph. They are colored blue and green, respectively. μ_e has been set equal to 2. Radius is measured in standard solar radii^[6] or kilometers, and mass in standard solar masses.

Calculated values for the limit will vary depending on the nuclear composition of the mass.^[7] Chandrasekhar^{[8], eq. (36),[5], eq. (58),[9], eq. (43)} gives the following expression, based on the equation of state for an ideal Fermi gas:

$M_{\rm limit} = \frac{\omega_3^0 \sqrt{3\pi}}{2}\left ( \frac{\hbar c}{G}\right )^{3/2}\frac{1}{(\mu_e m_H)^2}$

where:

• $\hbar$ is the reduced Planck constant
• c is the speed of light
• G is the gravitational constant
• μ_e is the average molecular weight per electron, which depends upon the chemical composition of the star.
• m_H is the mass of the hydrogen atom.
• $\omega_3^0 \approx 2.018236$ is a constant connected with the solution to the Lane-Emden equation.

As $\sqrt{\hbar c/G}$ is the Planck mass, the limit is of the order of $\frac{M_{Pl}^3}{m_H^2}$.

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.^[7] Lieb and Yau^[10] 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.^[11] 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.^[12] Wilhelm Anderson applied a relativistic correction to this model, giving rise to a maximum possible mass of approximately 1.37×10³⁰ kg.^[13] 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.^[14] Stoner went on to derive the pressure-density equation of state, which he published in 1932.^[15] These equations of state were also previously published by the Soviet physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter.^[16] Frenkel's work, however, was ignored by the astronomical and astrophysical community.^[17]

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.^[18] In these papers, Chandrasekhar solved the hydrostatic equation together with the nonrelativistic Fermi gas equation of state,^[3] and also treated the case of a relativistic Fermi gas, giving rise to the value of the limit shown above.^{[4][5][8][19]} Chandrasekhar reviews this work in his Nobel Prize lecture.^[9] This value was also computed in 1932 by the Soviet physicist Lev Davidovich Landau,^[20] 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!

—^[21]

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.^[22] 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.^{[23], pp. 110–111} Through the rest of his life, Eddington held to his position in his writings,^{[24][25][26][27][28]} including his work on his fundamental theory.^[29] The drama associated with this disagreement is one of the main themes of Empire of the Stars, Arthur I. Miller's biography of Chandrasekhar.^[23] 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.

—p. 150, ^[23]

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.^[30]

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.)^{[31][32][33][34]} During the collapse, neutrons are formed by the capture of electrons by protons in the process of electron capture, leading to the emission of neutrinos.^{[30], 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.^[35] This process is believed to be responsible for supernovae of types Ib, Ic, and II.^[30]

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 has been inferred 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 (carbon detonation) which causes the supernova.^{[36], §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.^{[36], (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.

Champagne Supernova

Main article: Champagne Supernova

In 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 University of Oklahoma astronomer 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.^[37][38][39]

See also

• White dwarf
• Stellar evolution
• Degenerate matter
• Schönberg-Chandrasekhar limit
• Tolman-Oppenheimer-Volkoff limit

References

1. ^ p. 55, How A Supernova Explodes, Hans A. Bethe and Gerald Brown, pp. 51–62 in Formation And Evolution of Black Holes in the Galaxy: Selected Papers with Commentary, Hans Albrecht Bethe, Gerald Edward Brown, and Chang-Hwan Lee, River Edge, NJ: World Scientific: 2003. ISBN 981238250X.
2. ^ Mazzali, P. A.; Ropke, F. K.; Benetti, S.; Hillebrandt, W. (2007). "A Common Explosion Mechanism for Type Ia Supernovae" (PDF). Science 315 (5813): 825–828. doi:10.1126/science.1136259. PMID 17289993. http://arxiv.org/pdf/astro-ph/0702351v1.  edit
3. ^ ^{a b} The Density of White Dwarf Stars, S. Chandrasekhar, Philosophical Magazine (7th series) 11 (1931), pp. 592–596.
4. ^ ^{a b} The Maximum Mass of Ideal White Dwarfs, S. Chandrasekhar, Astrophysical Journal 74 (1931), pp. 81–82.
5. ^ ^{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.
6. ^ Standards for Astronomical Catalogues, Version 2.0, section 3.2.2, web page, accessed 12-I-2007.
7. ^ ^{a b} 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.
8. ^ ^{a b} The Highly Collapsed Configurations of a Stellar Mass, S. Chandrasekhar, Monthly Notices of the Royal Astronomical Society 91 (1931), 456–466.
9. ^ ^{a b} On Stars, Their Evolution and Their Stability, Nobel Prize lecture, Subrahmanyan Chandrasekhar, December 8, 1983.
10. ^ A rigorous examination of the Chandrasekhar theory of stellar collapse, Elliott H. Lieb and Horng-Tzer Yau, Astrophysical Journal 323 (1987), pp. 140–144.
11. ^ On Dense Matter, R. H. Fowler, Monthly Notices of the Royal Astronomical Society 87 (1926), pp. 114–122.
12. ^ The Limiting Density of White Dwarf Stars, Edmund C. Stoner, Philosophical Magazine (7th series) 7 (1929), pp. 63–70.
13. ^ Über die Grenzdichte der Materie und der Energie, Wilhelm Anderson, Zeitschrift für Physik 56, #11–12 (November 1929), pp. 851–856. DOI 10.1007/BF01340146.
14. ^ The Equilibrium of Dense Stars, Edmund C. Stoner, Philosophical Magazine (7th series) 9 (1930), pp. 944–963.
15. ^ The minimum pressure of a degenerate electron gas, E. C. Stoner, Monthly Notices of the Royal Astronomical Society 92 (May 1932), pp. 651–661.
16. ^ 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.
17. ^ 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.
18. ^ Chandrasekhar's biographical memoir at the National Academy of Sciences, web page, accessed 12-I-2007.
19. ^ Stellar Configurations with degenerate Cores, S. Chandrasekhar, The Observatory 57 (1934), pp. 373–377.
20. ^ 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.
21. ^ Meeting of the Royal Astronomical Society, Friday, 1935 January 11, The Observatory 58 (February 1935), pp. 33–41.
22. ^ On "Relativistic Degeneracy", Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 95 (1935), 194–206.
23. ^ ^{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.
24. ^ The International Astronomical Union meeting in Paris, 1935, The Observatory 58 (September 1935), pp. 257–265, at p. 259.
25. ^ Note on "Relativistic Degeneracy", Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 96 (November 1935), 20–21.
26. ^ 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.
27. ^ Relativity Theory of Protons and Electrons, Sir Arthur Eddington, Cambridge: Cambridge University Press, 1936, chapter 13.
28. ^ The physics of white dwarf matter, Sir A. S. Eddington, Monthly Notices of the Royal Astronomical Society 100 (June 1940), pp. 582–594.
29. ^ Fundamental Theory, Sir A. S. Eddington, Cambridge: Cambridge University Press, 1946, §43–45.
30. ^ ^{a b c} 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.
31. ^ 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.
32. ^ 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.
33. ^ 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.
34. ^ 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.
35. ^ The Physics of Neutron Stars, by J. M. Lattimer and M. Prakash, Science 304, #5670 (2004), pp. 536–542; also arXiv astro-ph/0405262.
36. ^ ^{a b} Type IA Supernova Explosion Models, Wolfgang Hillebrandt and Jens C. Niemeyer, Annual Review of Astronomy and Astrophysics 38 (2000), pp. 191–230.
37. ^ The weirdest Type Ia supernova yet, LBL press release, web page accessed 13-I-2007.
38. ^ Champagne Supernova Challenges Ideas about How Supernovae Work, web page, spacedaily.com, accessed 13-I-2007.
39. ^ 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.