Degeneracy pressure

Degeneracy pressure is the pressure of a fermionic system that remains at zero temperature because the Pauli exclusion principle forces particles to occupy states with nonzero momentum.

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 $R\propto M^{-1/3}$.

The condition $p_F=m_{e}c$ gives ${\rho }_{\ast }=[{\mu }_e{m}_u/(3{\pi }^2)](m_{e}c/\hslash {)}^3\simeq 9.74\times 10^5{\mu }_e\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. Degenerate Electrons are nonrelativistic for $\rho \ll {\rho }_{\ast }$ and ultrarelativistic for $\rho \gg {\rho }_{\ast }$. This density threshold differs from thermal relativity in a nondegenerate gas.

For noninteracting electrons at zero temperature, define $x=p_F/(m_{e}c)$ with Fermi momentum $p_F=\hslash (3{\pi }^2{n}_e{)}^{1/3}$. The isotropic momentum flux gives electron degeneracy pressureIts nonrelativistic and ultrarelativistic limits arerespectively. In a fully ionized stellar gas, $n_e=\rho /({\mu }_e{m}_u)$, using mean molecular weight per electron ${\mu }_e$. This neglects finite-temperature effects, interactions, and changes in composition.