Electron degeneracy pressure supports dense objects when electrons fill momentum states up to a Fermi surface. In the nonrelativistic limit, a fixed-mass self-gravitating object has the characteristic degenerate scaling .
For a nonrelativistic zero-temperature free electron model with fixed particle number, electron degeneracy pressure obeys . Consequently its bulk modulus is . At temperatures small compared with the electron Fermi temperature, the leading isothermal compressibility is . This electronic contribution alone omits ionic binding and interaction corrections in a real metal.
For completely degenerate, nonrelativistic electrons in fully ionized hydrogen and helium with hydrogen mass fraction , . Filling the spin-degenerate momentum states to gives and . Eliminating gives , whereIt is a polytropic index relation. Complete degeneracy suppresses thermal dependence, while the nonrelativistic assumption requires .
The condition gives . Degenerate Electrons are nonrelativistic for and ultrarelativistic for . This density threshold differs from thermal relativity in a nondegenerate gas.
For noninteracting electrons at zero temperature, define with Fermi momentum . The isotropic momentum flux gives electron degeneracy pressureIts nonrelativistic and ultrarelativistic limits arerespectively. In a fully ionized stellar gas, , using mean molecular weight per electron . This neglects finite-temperature effects, interactions, and changes in composition.
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Electron degeneracy pressure is a quantum mechanical phenomenon that arises from the Pauli exclusion principle, which states that no two fermions (particles with half-integer spin, such as electrons) can occupy the same quantum state simultaneously. This principle plays a crucial role in the behavior of electrons in dense materials. In astrophysics, electron degeneracy pressure is especially significant in the context of white dwarfs, which are the remnants of stars that have exhausted their nuclear fuel and collapsed under gravity.
