= Degenerate-electron pressure crossover
{title2=$x=p_F/(m_ec)$}
Consider an ideal <degenerate electron gas> at zero <temperature>. Counting two spin states per <phase space> cell gives <number density> $n_e=p_F^3/(3\pi^2\hbar^3)$. The isotropic <momentum> flux gives <pressure>
$$
P=\frac{1}{3\pi^2\hbar^3}\int_0^{p_F}p^3v(p)\,dp
=P_*I(x),\qquad
P_*=\frac{m_e^4c^5}{3\pi^2\hbar^3},\qquad
I(x)=\int_0^x\frac{t^4}{\sqrt{1+t^2}}\,dt.
$$
Here $m_e$ is <Electron> <mass>, $c$ is the <speed of light>, $\hbar=h/(2\pi)$ with <Planck constant> $h$, and <special relativity> gives $v(p)=pc/\sqrt{p^2+m_e^2c^2}$. The factor $1/3$ averages the directional <momentum> flux, and the spin factor is already included. This normalization makes $P_*$ a <pressure>, not an <energy> per <Electron>.
For $x\ll1$, expand the denominator: $I(x)\sim x^5/5$ and $P\simeq n_ep_F^2/(5m_e)$. For $x\gg1$, the integrand approaches $t^3$: $I(x)\sim x^4/4$ and $P\simeq n_ep_Fc/4$. At fixed composition, <mass density> is $\rho=\mu_em_un_e$, with $\mu_e$ the <mean molecular weight per electron> and $m_u$ the <unified atomic mass unit>. Since $n_e\propto x^3$, the logarithmic stiffness is
$$
\gamma_{\rm eff}=\frac{d\log P}{d\log\rho}
=\frac{x^5}{3\sqrt{1+x^2}I(x)},\qquad
\gamma_{\rm eff}\longrightarrow\begin{cases}5/3&x\ll1,\\4/3&x\gg1.\end{cases}
$$
<Pressure> keeps rising; its fractional gain per fractional compression becomes smaller. Relativistic <Electrons> approach a speed ceiling while their <momenta> keep increasing. Thus the high-density <equation of state> loses the extra compression leverage of a <nonrelativistic white dwarf>. The <Chandrasekhar mass from central polytropic constants> supplies the corresponding global <mass> calculation.
For a calculated local example with $\mu_e=2$, $x=1$ gives $\rho\simeq1.95\times10^9\,\mathrm{kg\,m^{-3}}$, $I(1)=0.1537384$ and $P\simeq7.38\times10^{21}\,\mathrm{Pa}$. Its stiffness is $1.53314$, intermediate between the two limits. These are ideal <equation of state> values at one point, not a measured central <mass density> or a whole <white dwarf> model. Finite <temperature> and interactions are excluded.
\Image[/white-dwarf-degenerate-electron-crossover.png]
{title=How relativistic electron degeneracy softens white-dwarf support}
{description=Original calculation of the cold ideal pressure integral and its logarithmic density slope. The two asymptotes have their actual coefficients, with a common pressure normalization. The vertical line marks Fermi momentum equal to electron rest mass times light speed; it is a smooth crossover, not a discontinuous transition.}
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