= 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 $T_c$:
$$
\boxed{\frac{s_1}{s_2}=\frac{T_{c,1}}{T_{c,2}}
=\left(\frac{L_1}{L_2}\right)^{2/7}.}
$$
Thus \b[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 correction
$$
\frac{s_1}{s_2}=\left(\frac{L_1}{L_2}\right)^{2/7}
\frac{1+q_2}{1+q_1},\qquad
q_1/q_2=(L_1/L_2)^{2/7},
$$
which reduces to the boxed result for $q_1,q_2\ll1$.
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