For noninteracting electrons at zero temperature, define $x=p_F/(m_ec)$ with <Fermi momentum> $p_F=\hbar(3\pi^2n_e)^{1/3}$. The isotropic momentum flux gives <electron degeneracy pressure>
$$
P=\frac1{3\pi^2\hbar^3}\int_0^{p_F}\frac{p^4c^2}{\sqrt{m_e^2c^4+p^2c^2}}\,dp
=\frac{m_e^4c^5}{24\pi^2\hbar^3}
\left[x(2x^2-3)\sqrt{1+x^2}+3\operatorname{arsinh}x\right].
$$
Its nonrelativistic and ultrarelativistic limits are
$$
P\sim\frac{\hbar^2(3\pi^2)^{2/3}}{5m_e}n_e^{5/3},\qquad
P\sim\frac{\hbar c(3\pi^2)^{1/3}}4n_e^{4/3},
$$
respectively. In a fully ionized stellar gas, $n_e=\rho/(\mu_em_u)$, using <mean molecular weight per electron> $\mu_e$. This neglects finite-temperature effects, interactions, and changes in composition.


 Equation of state of a cold electron gas

ID: equation-of-state-of-a-cold-electron-gas