Matching of a white-dwarf envelope to a degenerate core (${\rho }_b=(\mathcal{R}{T}_c/K_{\mathrm{NR}}{)}^{3/2},L/M\propto T_c^{7/2}$)

Continuity of temperature, pressure and mass density matches the ideal gas envelope pressure $\mathcal{R}{\rho }_b{T}_c$ to the nonrelativistic electron degeneracy pressure $K_{\mathrm{NR}}{\rho }_b^{5/3}$. Substituting this base mass density into the radiative-zero white-dwarf envelope law gives the luminosity-temperature scaling behind Mestel's cooling law. The matched point is a transition approximation; the cold degeneracy law does not represent finite-temperature corrections there exactly.