 Solution

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 $q_1,q_2\ll 1$.