= Nonrelativistic electron-degeneracy pressure in a hydrogen-helium mixture
{title2=$P_e=K\rho^{5/3}$}
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 $P_e=(1/3)\int_0^{p_F}(p^2/m_e)(8\pi p^2/h^3)dp=8\pi p_F^5/(15m_eh^3)$. Eliminating $p_F$ gives $P_e=K\rho^{5/3}$, where
$$
K=\left(\frac3{2\pi}\right)^{2/3}\frac{h^2(1+X)^{5/3}}{40m_em_p^{5/3}}.
$$
It is a <polytropic index> $3/2$ relation. Complete degeneracy suppresses thermal dependence, while the nonrelativistic assumption requires $p_F\ll m_ec$.
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