Chandrasekhar limit
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The Chandrasekhar limit, is the maximum mass possible for a white dwarf (one of the end stages of stars when they cool down) and is approximately 3 × 1030 kg, around 1.44 times the mass of the Sun. This number is different in various articles, from 1.2 to 1.46 times the mass of the Sun and depends on the chemical composition of the star.
History and Development of the Chandrasekhar limit
The limit was first discovered and calculated by the Indian physicist Subrahmanyan Chandrasekhar in 1930, during his maiden voyage to Britain from India. 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 study/deduce the end stage evolution of stars and the second significance comes from the fact that it prophesied the existence of fascinating stellar phenomena, albeit not characterized further. Dr. Chandrasekhar provides an excellent review of this work in his Nobel lecture [link] 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. It was particularly harsh on the young physicist since most of 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 has now been brought out as a novel "Empire of the Stars" by Arthur I. Miller. Some suggest that the autocracy of Eddington may have delayed the progress of astrophysics by 1 or 2 decades.
Stellar Mechanics of the limit
The heat generated by a star due to nuclear fusion of atoms of lighter elements into heavier ones 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 the star has a mass above the Chandrasekhar limit it has sufficient gravity to collapse past the white dwarf stage and become a neutron star, black hole, or possibly a theoretical quark star. If a stable white dwarf is in a very close binary system with a giant star (closer to each other than the giant star's diameter), then the dwarf may accrete enough material from the giant to exceed the Chandrasekhar limit, and the dwarf collapses and becomes a type Ia supernova. The remnant of the collapse is unlikely to be a neutron star as the explosion is violent enough to rip the white dwarf apart, leaving no remnant at all.
The Chandrasekhar limit arises from taking account of the effects of quantum mechanics in considering the behaviour of the electrons providing the degeneracy pressure supporting the white dwarf. Electrons, being fermions, cannot be at equal energy levels, so that, when an electron gas is cooling down, it is impossible for them to be given all 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):
M(Ch) = ((3*sqrt(2*pi)/8) * (h-dash*c/G)^1.5 * (z/mH)**2) + M(Sun)
where:
M(Ch) is the mass of the Chandrasekhar limit measured in gram
M(Sun)is the mass of the Sun measured in gram = 1.989 E33 g
sqrt = ’square root of’
^ indicates involution (eg. 3^2 = 9)
Ex (E-x) is the exponent of 10 (eg. 3E2 = 100, 3E-1 = 0.3)
pi (approximately) = 3.141592654
h-dash = Planck's constant/(2 * pi) = 6.6260755 E-27 / (2 * pi) erg s = 1.054572669 E-27 erg s
c = is the speed of light in a vacuum = 2.99792458 E10 cm/s
G = the gravitational constant = 6.67259 E-08 dyne cm * cm /g/g
z = Z/A (the proportion of protons Z to the sum of all neucleons (protons + neutrons) A)
mH = mass of hydrogen atom = 1.673534 E-24 g
It is immediately intelligible, that in this formula is only 1 variable, namely z.
The rest are constants, three of which are universal physical constants (h-dash, c og G).
Consequently the Chandrasekhar limit varies by the proportion of protons to the sum of all neucleons.
Strong indications of the reliability of Chandrasekhar's formula are:
1. No white dwarf with mass > Chandrasekhar limit has been observed
2. Supernovae type Ia (the result of exceeding M(Ch)) have an absolute luminosity (Mv) of -19.6 ± 0.6. This interval is only a factor of 3 in luminosity.
Inserting z = 0.5, yields M(Ch) = (0.44 + 1) M(Sun) = 1.44 M(Sun) the Chandrasekhar limit
z = 1,0: 2.74 M(Sun) A proton star
z = 0,6: 2.05 M(Sun)
z = 0,4: 1.28 M(Sun)
z = 0,0: 1.00 M(Sun) A hypothetical neutron star
See also
External links
- [Chandrasekhar's Nobel Lecture, 1983]
- [White Dwarf Stars and the Chandrasekhar Limit]
- [Estimating the Chandrasekhar Limit] using simple energy arguments
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