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

For a nonrelativistic three-dimensional ideal electron gas at zero temperature, the pressure is $p=2n{E}_F/5$ and the electronic bulk modulus is $B=2n{E}_F/3$. This is the electronic model contribution, not automatically the total bulk modulus of a real crystal.

For a nonrelativistic zero-temperature free electron model with fixed particle number, electron degeneracy pressure obeys $P\propto (N/V{)}^{5/3}$. Consequently its bulk modulus is $K=-V(\partial P/\partial V{)}_N=5P/3=2n{E}_F/3$. At temperatures small compared with the electron Fermi temperature, the leading isothermal compressibility is $1/K$. 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 $X$, $n_e=(1+X)\rho /(2m_p)$. Filling the spin-degenerate momentum states to $p_F$ gives $n_e=8\pi p_F^3/(3h^3)$ and $P_e=(1/3){\int }_0^{p_F}(p^2/m_e)(8\pi p^2/h^3)dp=8\pi p_F^5/(15m_e{h}^3)$. Eliminating $p_F$ gives $P_e=K{\rho }^{5/3}$, whereIt is a polytropic index $3/2$ relation. Complete degeneracy suppresses thermal dependence, while the nonrelativistic assumption requires $p_F\ll m_{e}c$.

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.