Normal stars fuse gravitationally compressed hydrogen into helium, generating vast amounts of heat. As the hydrogen is consumed, the stars' core compresses further allowing the helium and heavier nuclei to fuse ultimately resulting in stable iron nuclei, a process called
stellar evolution. The next step depends upon the mass of the star. Stars below the Chandrasekhar limit become stable
white dwarf stars, remaining that way throughout the rest of the history of the universe (assuming the absence of external forces). Stars above the limit can become
neutron stars or
black holes. The Chandrasekhar limit is a consequence of competition between gravity and electron degeneracy pressure. 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 , where is the
pressure, is the
mass density, and is a constant. Solving the hydrostatic equation leads to a model white dwarf that is a
polytrope of index – 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 speeds 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 yields a polytrope of index 3, which has a total mass, , depending only on . For a fully relativistic treatment, the equation of state used interpolates between the equations for small and for large . When this is done, the model radius still decreases with mass, but becomes zero at . This is the Chandrasekhar limit. or kilometers, and mass in standard solar masses. Calculated values for the limit vary depending on the
nuclear composition of the mass. gives the following expression, based on the
equation of state for an ideal
Fermi gas: M_\text{limit} = \frac{\omega_3^0 \sqrt{3\pi}}{2} \left ( \frac{\hbar c}{G}\right )^\frac{3}{2} \frac{1}{(\mu_\text{e} m_\text{H})^2} where: • is the
reduced Planck constant • is the
speed of light • is the
gravitational constant • is the average
molecular weight per electron, which depends upon the chemical composition of the star • is the mass of the
hydrogen atom • 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_\text{Pl}^3}{m_\text{H}^2} The limiting mass can be obtained formally from the
Chandrasekhar's white dwarf equation by taking the limit of large central density. 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. == History ==