= Solution
A stellar <equation of state> supplies <pressure> and <internal energy> as functions of density, <temperature> and composition, together with thermodynamic derivatives needed for stability and transport. <Hydrostatic equilibrium> fixes the <pressure gradient>, but does not determine which microscopic components provide the <pressure>. In ordinary dense interiors <local thermodynamic equilibrium> is a useful starting point. A consistent mixture is
$$
\boxed{P=P_i+P_e(\rho,T,\{X_A\})+P_\gamma+P_{\rm int},}
$$
where ions, <Electrons>, radiation and interaction corrections are distinguished. \b[The classical <Electron> <pressure> and <electron degeneracy pressure> are two limits of the same <Electron> contribution], and must not be added as if they belonged to different particles. The <finite-temperature electron equation of state> interpolates between them.
In a fully ionized, nondegenerate, nonrelativistic gas, $P_g=\rho k_BT/(\mu m_u)$ and specific thermal energy is $u_g=3k_BT/(2\mu m_u)$. With nuclear mass fractions $X_A$, charges $Z_A$ and mass numbers $A_A$, the <mean molecular weight> satisfies $\mu^{-1}=\sum_A X_A(1+Z_A)/A_A$, and $\mu_e^{-1}=\sum_A X_AZ_A/A_A$ is the <mean molecular weight per electron>. This regime describes much of an ordinary <main sequence> interior. Toward cooler layers, <ionization> and molecular dissociation change particle numbers and consume heat. The <Saha equation> relates <ionization> to both <temperature> and <Electron> density: there is no universal horizontal <ionization> boundary. These regions have larger <heat capacity> and can have a reduced <stellar adiabatic exponent>. The simple fully ionized formula is then insufficient.
Equilibrium <photons> give <radiation pressure> $P_\gamma=a_rT^4/3$ and energy per volume $a_rT^4$, or specific energy $u_\gamma=a_rT^4/\rho$. In the nondegenerate gas regime, equality with gas <pressure> gives the <radiation-to-gas pressure boundary>
$$
\boxed{T^3=\frac{3k_B\rho}{a_r\mu m_u},}
$$
a line of slope $1/3$ on a $(\log\rho,\log T)$ plot. Higher temperatures at fixed <mass density> favor <photon> support. A monatomic gas has <stellar adiabatic exponent> $5/3$, while radiation alone has $4/3$; their <adiabatic exponents of a monatomic gas-radiation mixture> are not obtained by assuming a fixed <pressure> fraction during compression. Radiation support is particularly important in massive stars.
For <Electrons>, <Pauli exclusion principle> and the <Fermi-Dirac distribution> determine occupation numbers. The net <Electron> density is $n_e=\rho/(\mu_em_u)$ and the <Fermi momentum> is $p_F=\hbar(3\pi^2n_e)^{1/3}$. Define the kinetic <electron Fermi temperature>
$$
\boxed{k_BT_F=\sqrt{m_e^2c^4+p_F^2c^2}-m_ec^2.}
$$
For $T\gg T_F$ the <Electrons> are nearly classical; for $T\ll T_F$ they are strongly degenerate and their <pressure> depends primarily on density. The intermediate region requires <finite-temperature electron equation of state> integrals, not a discontinuous switch of formulas.
The <equation of state of a cold electron gas> gives, in its two limits,
$$
\boxed{P_e\simeq\frac{\hbar^2(3\pi^2)^{2/3}}{5m_e}n_e^{5/3}
\quad(p_F\ll m_ec),\qquad
P_e\simeq\frac{\hbar c(3\pi^2)^{1/3}}4n_e^{4/3}
\quad(p_F\gg m_ec).}
$$
These are respectively the $n=3/2$ and $n=3$ pressure-density powers, explaining the approximate <white dwarf> <polytropic mass-radius relation> sequence and the <Chandrasekhar limit>. The kinetic energy per volume is $3P_e/2$ in the nonrelativistic limit and $3P_e$ in the ultrarelativistic limit. Ions can still supply much of the <heat capacity> even when the <Electron> <pressure> supplies the mechanical support.
The <Electron> degeneracy crossover $T\sim T_F$ has slope $2/3$ at low <mass density> and $1/3$ at high <mass density>. The <electron relativistic density threshold> is
$$
\rho_*=\frac{\mu_em_u}{3\pi^2}\left(\frac{m_ec}{\hbar}\right)^3
\simeq9.74\times10^5\mu_e\,\mathrm{g\,cm^{-3}},
$$
a vertical marker where $p_F=m_ec$. It is different from the <thermal electron relativistic threshold> $k_BT\sim m_ec^2$, near $5.93\times10^9\,\mathrm K$, a horizontal <temperature> scale. Hot dilute matter can have relativistic thermal <Electrons> without degeneracy; cold dense matter can have relativistic degenerate <Electrons> without reaching that <temperature>.
<Electron> degeneracy also does not automatically imply that <Electrons> dominate the total <pressure>. Comparing the cold <Electron> limit with radiation gives the <radiation-to-degeneracy pressure boundary>
$$
\boxed{T=\left(\frac{3P_e(\rho,0)}{a_r}\right)^{1/4}.}
$$
Its logarithmic slopes are $5/12$ for nonrelativistic <Electrons> and $1/3$ for ultrarelativistic <Electrons>. One must compare the pressures separately from the degeneracy criterion; extrapolating the classical gas-radiation line into a degenerate region is incorrect.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-55-eos-regimes.png]
{title=Density-temperature crossover diagram for classical gas, radiation and electron degeneracy, illustrated for fully ionized carbon}
{height=640}
The <stellar equation-of-state regime diagram> uses an illustrative fully ionized <carbon> composition, $\mu_e=2$, $\mu=12/7$. It displays the electron-degeneracy crossover, both radiation-pressure comparisons, and distinct thermal and density-driven relativity scales. The curves are limiting-model comparisons, not sharp phase boundaries or a calibrated complete equation of state. Partial <ionization>, molecular physics and interactions modify the low-temperature regions indicated on the plot.
At sufficiently high <temperature>, <electron-positron thermal pair abundance> can become important. The pair abundance depends on density and <chemical potential> as well as <temperature>; $k_BT=m_ec^2$ is not a universal onset line. In the dilute ultrarelativistic limit, both pair species together add energy density $7a_rT^4/4$ to the <photons>' $a_rT^4$, and have <pressure> one third of their energy density. While pairs are being created, thermal energy is spent on rest mass, which can reduce the <stellar adiabatic exponent> below $4/3$ and contribute to <pair-instability supernova> physics.
At high <mass density> and low <temperature>, Interactions governed by <Coulomb's law> invalidate the noninteracting-ion approximation. The <ionic Coulomb coupling parameter> $\Gamma=Z^2e^2/(4\pi\epsilon_0a_i k_BT)$, where $a_i=(3/(4\pi n_i))^{1/3}$, grows as $\rho^{1/3}/T$. Corrections become significant when $\Gamma$ is of order one, and a sufficiently strongly coupled plasma can crystallize. At still greater <mass density>, <electron capture> alters $\mu_e$ and nuclear matter replaces the ideal electron-ion model; <neutron star> interiors require strong-interaction and relativistic equations of state. These further regimes lie beyond the simple <pressure> curves plotted here. \b[A useful stellar EOS is thermodynamically consistent across the crossovers], rather than just the maximum of unrelated <pressure> laws.
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