= Solution
Write $\mathcal R=k/(\mu m_H)$ for the specific gas constant, with the printed $m_H$ denoting the atomic mass unit, and $a_{\rm r}=4\sigma/c$ for the <radiation constant>. In the low-mass envelope the <enclosed mass> is approximately $M$ and the outward <luminosity> is approximately $L$. The <stellar hydrostatic equation>, <stellar radiative diffusion>, and <ideal gas> equation give
$$
\frac{dP}{dr}=-\frac{GM\rho}{r^2},\qquad
\frac{dT}{dr}=-\frac{3\kappa\rho L}{16\pi a_{\rm r}c r^2T^3},\qquad
P=\mathcal R\rho T.
$$
The <Kramers' opacity law> becomes $\kappa=(\kappa_0/\mathcal R)PT^{-9/2}$. Dividing the two differential equations eliminates radius and <mass density>:
$$
P\frac{dP}{dT}=\frac{16\pi a_{\rm r}cGM\mathcal R}{3\kappa_0L}T^{15/2}.
$$
Integrating from a surface matching point $(P_s,T_s)$ gives
$$
P^2=P_s^2+\frac{64\pi a_{\rm r}cGM\mathcal R}{51\kappa_0L}
\left(T^{17/2}-T_s^{17/2}\right).
$$
Deep compared with the <photosphere>, the surface terms are negligible. The <radiative-zero white-dwarf envelope> therefore has
$$
\boxed{P(T)=\left(\frac{64\pi a_{\rm r}cGM\mathcal R}{51\kappa_0L}\right)^{1/2}T^{17/4},\qquad
\rho\propto T^{13/4}.}
$$
This is the leading envelope approximation used in the later parts, rather than an exact atmospheric boundary condition at $T_{\rm eff}$.
At the base, <temperature>, <mass density> and <pressure> match the isothermal degenerate core. Let $K_{\rm NR}=K_1\mu_e^{-5/3}$, where $\mu_e$ is the <mean molecular weight per electron>. Equating the gas and <electron degeneracy pressure> gives
$$
\mathcal R\rho_bT_c=K_{\rm NR}\rho_b^{5/3},\qquad
\boxed{\rho_b=\left(\frac{\mathcal RT_c}{K_{\rm NR}}\right)^{3/2}
=\mu_e^{5/2}\left(\frac{\mathcal R}{K_1}\right)^{3/2}T_c^{3/2}.}
$$
Hence $P_b=\mu_e^{5/2}\mathcal R^{5/2}K_1^{-3/2}T_c^{5/2}$. Substituting this into the envelope relation at $T=T_c$ yields
$$
\boxed{\frac LM=\frac{64\pi a_{\rm r}cG}{51\kappa_0}
\frac{K_1^3}{\mu_e^5\mathcal R^4}T_c^{7/2}
=\frac{256\pi\sigma G}{51\kappa_0}\frac{K_1^3}{\mu_e^5}
\left(\frac{\mu m_H}{k}\right)^4T_c^{7/2}.}
$$
The <matching of a white-dwarf envelope to a degenerate core> thus gives \b[$L/M\propto T_c^{7/2}$], the envelope relation underlying <Mestel's cooling law>. It depends on <pressure> as well as <mass density> continuity at the interface; otherwise the supplied core equation would not determine the matching <mass density>.
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