Stellar equation-of-state regime diagram

A temperature versus density diagram separates classical, degenerate, nonrelativistic and relativistic stellar-matter limits. The Electron degeneracy boundary is $T\sim T_F(\rho )$, with slope $2/3$ at low density and $1/3$ at high density. Thermal relativity is controlled by $k_{B}T\sim m_e{c}^2$, while relativity of degenerate Electrons is controlled by $p_F\sim m_{e}c$. Photon pressure and thermal electron-positron pairs add further crossovers; all depend on composition and are gradual, not thermodynamic phase boundaries.

For one ionic species of number density $n_i$, the ion-sphere radius is $a_i=(3/(4\pi n_i){)}^{1/3}$. The ionic Coulomb coupling parameter compares the interaction energy from Coulomb's law at that separation with thermal energy $k_{B}T$. At fixed composition, $\Gamma \propto {\rho }^{1/3}/T$. Values near one mark significant corrections to the ideal ion gas; strong coupling can lead to an ordered solid. Screening, mixtures and quantum motion modify detailed transitions, so this estimate is not a universal crystallization boundary. It is separate from the electron degeneracy criterion based on the electron Fermi temperature.

Equating radiation pressure to cold electron degeneracy pressure gives $T=[3P_e(\rho )/a_{\mathrm{rad}}{]}^{1/4}$. Because the two Electron pressure limits scale as ${\rho }^{5/3}$ and ${\rho }^{4/3}$, this boundary has logarithmic slopes $5/12$ and $1/3$, respectively. Above it, photon pressure can dominate even when the net Electrons remain degenerate; degeneracy of one component and dominance of the total pressure are different criteria. Thermal pairs modify the high-temperature pressure comparison.

For a nondegenerate fully ionized gas, equating $P_{\mathrm{gas}}=\rho k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ with $P_{\gamma }=a_{\mathrm{rad}}{T}^4/3$ gives $T^3=3k_B\rho /(a_{\mathrm{rad}}{\mu }_{\mathrm{mol}}{m}_u)$. Radiation dominates at higher temperature for fixed density. A degenerate Electron gas requires its degeneracy pressure instead of extrapolating this ideal-gas formula.

In a dilute Electron gas, thermal kinetic energies become relativistic near $k_{B}T=m_e{c}^2$, corresponding to $T_{\ast }\simeq 5.93\times 10^9\mathrm{K}$. This is a crossover in the momentum distribution, not an abrupt temperature transition. Degenerate Electrons can instead be relativistic at much lower temperature if their Fermi momentum exceeds $m_{e}c$.