What is Chandrasekhar limit?
Answers
Explanation:
The Chandrasekhar limit (/tʃʌndrəˈseɪkər/) is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M☉ (2.765×1030 kg).[1][2][3]
White dwarfs resist gravitational collapse primarily through electron degeneracy pressure (compare main sequence stars, which resist collapse through 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, a white dwarf with a mass greater than the limit is subject to 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.[4]
The limit was named after Subrahmanyan Chandrasekhar, an Indian astrophysicist who improved upon the accuracy of the calculation in 1930, at the age of 20, in India by calculating the limit for a polytrope model of a star in hydrostatic equilibrium, and comparing his limit to the earlier limit found by E. C. Stoner for a uniform density star. Importantly, the existence of a limit, based on the conceptual breakthrough of combining relativity with Fermi degeneracy, was indeed first established in separate papers published by Wilhelm Anderson and E. C. Stoner in 1929. The limit was initially ignored by the community of scientists because such a limit would logically require the existence of black holes, which were considered a scientific impossibility at the time. That the roles of Stoner and Anderson are often forgotten in the astronomy community has been noted.
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 increases on 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.
In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form P = K1ρ
5
/
3
, where P is the pressure, ρ is the mass density, and K1 is a constant. Solving the hydrostatic equation leads to a model white dwarf that 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.[7]
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, the equation of state takes the form P = K2ρ
4
/
3
. This yields a polytrope of index 3, which has a total mass, Mlimit say, depending only on K2.[8]
For a fully relativistic treatment, the equation of state used interpolates 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 Mlimit. This is the Chandrasekhar limit.[9] 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. μe has been set equal to 2. Radius is measured in standard solar radii[10] or kilometers, and mass in standard solar masses.
Calculated values for the limit vary depending on the nuclear composition of the mass.[11] Chandrasekhar[12], eq. (36),[9], eq. (58),[13], eq. (43) gives the following expression, based on the equation of state for an ideal
ħ is the reduced Planck constant
c is the speed of light
G is the gravitational constant
μe is the average molecular weight per electron, which depends upon the chemical composition of the star.
mH is the mass of the hydrogen atom.
ω0
3 ≈ 2.018236 is a constant connected with the solution to the Lane–Emden equation.
As is the Planck mass, the limit is of
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[14] have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.