Radiative-zero white-dwarf envelope ($P\propto T^{17/4},\rho \propto T^{13/4}$)

In a negligible-mass white dwarf envelope, take enclosed mass and luminosity constant, neglect radiation pressure, and use an ideal gas with the Kramers opacity law. Dividing hydrostatic pressure support equation by radiative diffusion in a star gives $PdP\propto (M/L)T^{15/2}dT$. Neglecting the surface integration constant produces the displayed pressure and mass density laws. It is a deep-envelope approximation, not a finite-temperature photospheric boundary condition.

Integrating the hydrostatic pressure support equation with the radiative-zero white-dwarf envelope law yields $T=4GM(1/r-1/R)/(17\mathcal{R})$. Matching $T=T_c$ at $r_b$ gives $s=R-r_b$ as displayed. A strongly degenerate bulk core has thermal energy small compared with its Fermi energy, making $q\ll 1$ and hence $s\ll R$. The inequality $T_c\gg T_{\mathrm{eff}}$ alone does not establish geometrical thinness.

For fixed mass, degenerate radius and composition, the matching of a white-dwarf envelope to a degenerate core gives $L\propto T_c^{7/2}$, while the geometrical thickness of a white-dwarf radiative envelope gives $s\propto T_c$ to leading thin-layer order. Cooling therefore thins this idealized envelope. At finite thickness the ratio additionally contains $(1+q_2)/(1+q_1)$.

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.