Chandrasekhar limit

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The Chandrasekhar limit is the maximum mass possible for a white dwarf star supported by electron degeneracy pressure, and is approximately 3 × 10³⁰ kg, around 1.44 times the mass of the Sun. If a white dwarf (normally formed with about 0.6 solar masses) were to exceed this mass through accretion, it would begin to collapse under gravity. It was once believed that this mechanism triggered Type Ia supernova explosions, but this idea fell out of favor in the 1960s. However, when iron cores of aging massive stars pass this limit, they do collapse, and this process is believed to initiate supernovae of Types Ib, Ic and II, releasing a vast amount of energy (10⁴⁶ joules, or 100 foes) and a flood of neutrinos. The Chandrasekhar limit mass <math>M_{Ch}</math> is defined as:

<math>M_{Ch} \approx \left ( \frac{\hbar c}{G}\right )^{3/2}\frac{1}{m_{p}^{2}}</math>

where <math>\hbar</math> is the reduced Planck constant, <math>c</math> is the speed of light, <math>G</math> is the gravitational constant and <math>m_{p}</math> is the mass of a proton.

The precise value depends on the chemical composition of the star.

1 History and Development of the Chandrasekhar limit
2 Stellar Mechanics of the Limit
3 A Challenge to the Limit?
4 See also
5 External links

[edit] History and Development of the Chandrasekhar limit

The limit was first discovered and calculated by the Indian physicist Subrahmanyan Chandrasekhar in 1930, during his first visit to Great Britain. At that time Chandrasekhar had just completed his undergraduate work and was on his way to Cambridge to pursue graduate studies.

The first scientific significance of this limit comes from the fact that he introduced/applied Einstein's special theory of relativity to deduce the end stage evolution of stars. The second significance comes from the fact that it predicted the existence of fascinating stellar phenomena, albeit not characterized further. Dr. Chandrasekhar provides an excellent review of this work in his Nobel lecture [1] with references to his papers published between 1931 and 1936. In this lecture paper he shows how he deviated from the earlier work of British physicists Arthur Eddington and Ralph H. Fowler (not William Alfred Fowler who won the Nobel prize with Chandrasekhar) which had concluded that white dwarfs represent the last stages in the evolution of *all* stars.

When Chandrasekhar eventually presented this work in a Royal Society meeting in 1935, it was ridiculed and put down by Arthur Eddington. Particularly harsh on the young physicist was the fact that the senior English and European physicists were not willing to openly support his work although many of them approved of it privately. This embittered him and eventually led to his moving to the United States where he remained at the University of Chicago for the rest of his career. The drama associated with this episode is one of the main themes of Empire of the Stars, Arthur I. Miller's biography of Chandrasekhar. It has been suggested that the autocracy of Eddington may have delayed the progress of black hole research by "at least 30 years"[2].

[edit] Stellar Mechanics of the Limit

The heat generated by nuclear fusion of atoms of lighter elements into heavier ones in a star's core pushes the atmosphere of the star out. As the star runs out of fuel the atmosphere collapses back on the star's core, pulled by the star's own gravity. At this stage, if the star has a mass below the Chandrasekhar limit this collapse is limited by electron degeneracy pressure, which results in a stable white dwarf. If a star not capable of producing further energy (generally not the case for white dwarfs) had a mass above the Chandrasekhar limit, the pressure exerted by electrons would be unable to resist the force of gravity, and collapse would ensue. The star's density would increase far beyond that of a white dwarf, leading to formation of a neutron star, black hole, or possibly a theoretical quark star. For each neutron formed by the merger of a proton and an electron during the collapse, a neutrino would be released (to conserve lepton number).

The Chandrasekhar limit arises from taking account of the effects of quantum mechanics in considering the behavior of the electrons providing the degeneracy pressure supporting the white dwarf. Electrons, being fermions, cannot be at equal energy levels. So when an electron gas is cooling down, it is impossible for all electrons to be given minimal energy. Plenty of electrons will have to stay at higher energy levels and will thus give a certain pressure, which is purely quantum mechanical in its nature.

In the non-relativistic approximation a white dwarf may be arbitrarily massive with its volume inversely proportional to its mass. As the mass increases the typical energies to which degeneracy pressure forces the electrons in a massive white dwarf are non-negligible relative to their rest masses. The velocities of the electrons approach the speed of light, and special relativity must be taken into account. The classical approximation is no longer appropriate. The result is that a limiting mass emerges for a self-gravitating, spherically symmetric body supported by degeneracy pressure.

Chandrasekhar's formula (modified by adding mass of the Sun):

<math>M_{Ch} = \left ( \left ( \frac{3 \sqrt{2\pi}}{8} \right ) \left ( \frac{\hbar c}{G} \right )^{1.5} \left ( \frac{z}{m_H} \right ) ^2 \right ) + M_{\bigodot} </math>

where:

<math>M_{Ch} \,</math> is the mass of the Chandrasekhar limit,

<math>M_{\bigodot} = 1.9891 \times 10^{30} \; \mathrm{kg}</math> is the mass of the Sun,

<math>\pi \approx 3.141592654</math> is the mathematical constant pi,

<math>\hbar \approx 1.054571596 \times 10^{-34} \; \mathrm{J s}</math> Dirac's constant (also known as the reduced Planck constant, and referred to as "h-bar" when you are unable to use the symbol),

<math>c = 2.99792458 \times 10^{8} \; \mathrm{m s}^{-1}</math> is the speed of light in vacuum,

<math>G \approx 6.673 \times 10^{-11} \; \mathrm{m}^3\mathrm{kg}^{-1}\mathrm{s}^{-2}</math> is the Newtonian constant of gravitation,

<math>z = Z/A \,</math> is the proportion of protons <math>Z \,</math> to the sum of all nucleons (protons + neutrons) <math>A \,</math>,

and <math>m_H \approx 1.673534 \times 10^{-27} \mathrm{kg}</math> is the mass of a hydrogen atom.

It is immediately obvious that in this formula there is only 1 variable, namely <math>z \,</math>. The rest are constants, three of which are universal physical constants (<math>\hbar</math>, <math>c \,</math> and <math>G \,</math>). Consequently, the Chandrasekhar limit varies by the proportion of protons to the sum of all nucleons.

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.)
Supernovae of Type Ia (the result of a white dwarf approaching <math>M_{Ch} \,</math>) have an absolute luminosity (<math>M_v \,</math>) of -19.6 ± 0.6. This interval is only a factor of 3 in luminosity. This seems to indicate that all SN Ia convert approximately the same amount of mass to energy, allowing for a slight variation of <math>z \,</math>.

Inserting <math>z = 0.5 \,</math> yields

<math>M_{Ch} = (0.44 + 1) M_{\bigodot} = 1.44 M_{\bigodot} \,</math>,

the Chandrasekhar limit. Other examples include:

A hypothetical proton star: <math>z = 1.0 \to 2.74 M_{\bigodot} \,</math>

<math>z = 0.6 \to 2.05 M_{\bigodot} \,</math>

<math>z = 0.4 \to 1.28 M_{\bigodot} \,</math>

A hypothetical neutron star: <math>z = 0.0 \to 1.00 M_{\bigodot} \,</math>

[edit] A Challenge to the Limit?

In 2003, scientists at the University of Toronto observed a distant white dwarf which grew to twice the mass of the Sun before exploding into a supernova in a galaxy 4 billion light years away. Scientists think that perhaps the star, dubbed the "Champagne Supernova", may have been spinning so fast that centrifugal force allowed it to go past the limit. Alternatively, the supernova may have resulted from the merger of two white dwarfs, so that the limit was only violated momentarily. Nevertheless, scientists warn that others should be careful when incorporating the Chandrasekhar limit in their work. The results were published in the journal Nature on September 21, 2006. <ref>'Champagne supernova' challenges understanding of how supernovae work - University of Toronto</ref>

Assuming that the proportion of protons to neutrons differs even slightly from 0.5, exceptionally high luminosity and low kinetic energy is easily explained.