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Apply the preceding temperature profile at the envelope base. Introduce the dimensionless parameterThen $R/r_b-1=q$, so the geometrical thickness of a white-dwarf radiative envelope isThe smallness to be established is therefore $\mathcal{R}{T}_c\ll GM/R$, up to the numerical factor $17/4$.

That ordering follows from the cold, strongly electron-degenerate white dwarf model. Throughout the mass-bearing core, $k{T}_c\ll E_F$, where $E_F$ is the Fermi energy. For nonrelativistic electron degeneracy pressure,The stellar hydrostatic equation gives a characteristic core $P_{\mathrm{deg}}/\rho$ of order $GM/R$. Consequentlyfor fixed composition with order-one molecular-weight ratios, and $s\ll R$. Equivalently, the matched base mass density is much lower than a characteristic core mass density, since ${\rho }_b/{\rho }_{\mathrm{core}}\sim [\mathcal{R}{T}_c/(GM/R){]}^{3/2}$. The degeneracy ordering applies in the bulk core; at the matching point degeneracy is becoming marginal.

The inequalities $T_c\gg T_{\mathrm{eff}}$ and $k{T}_c\ll E_F$ express different facts. The first justifies ignoring surface-temperature terms, while the second makes the envelope geometrically thin. If the core were hot enough that $q$ were not small, the exact expression $s/R=q/(1+q)$ would not justify a thin-layer approximation.

Compare two cooling states of the same white dwarf, holding its mass, nearly temperature-independent degenerate radius and composition fixed. 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 $s\propto T_c$ at leading thin-layer order. Eliminate $T_c$:Thus a luminosity drop makes the envelope thinner, with $s_2/s_1=(L_2/L_1{)}^{2/7}$. This is the white-dwarf envelope thickness-luminosity relation. Retaining finite thickness in the radiative-zero model gives the correctionwhich reduces to the boxed result for $q_1,q_2\ll 1$.

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 $(1+q_2)/(1+q_1)$.