Chandrasekhar Limit - Physics

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.

In the nonrelativistic case, electron degeneracy pressure gives rise to an equation of state of the form, where P is the pressure, is the mass density, and is a constant. Solving the hydrostatic equation then 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.

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 . This will yield a polytrope of index 3, which will have a total mass, M_limit say, depending only on K₂.

For a fully relativistic treatment, the equation of state used will interpolate between the equations for small ρ and 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. 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 or kilometers, and mass in standard solar masses.

Calculated values for the limit will vary depending on the nuclear composition of the mass. Chandrasekhar, eq. (36),, eq. (58),, eq. (43) gives the following expression, based on the equation of state for an ideal Fermi gas:

where:

• 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.
• is a constant connected with the solution to the Lane-Emden equation.

As is the Planck mass, the limit is of the order 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. Lieb and Yau have given a rigorous derivation of the limit from a relativistic many-particle Schrödinger equation.