= Solution
Outside the emitting layer, take $m\simeq M_c$, $L_r\simeq L$ and $P=\mathcal R\rho T/\mu$. The <Kramers' opacity law> then reads $\kappa=(\kappa_0\mu/\mathcal R)PT^{-9/2}$. Division of the <radiative diffusion in a star> equation by the <stellar hydrostatic equation> gives
$$
\frac{dP}{dT}=\frac{16\pi acGM_c}{3\kappa L}T^3,
\qquad
P\frac{dP}{dT}=\frac{16\pi acGM_c\mathcal R}{3\kappa_0L\mu}T^{15/2}.
$$
With the outer <pressure> <constant of integration> neglected, <integration> yields
$$
\boxed{P=CT^{17/4},\qquad
C=\left(\frac{64\pi acGM_c\mathcal R}{51\kappa_0L\mu}\right)^{1/2},\qquad
\rho=\frac{\mu C}{\mathcal R}T^{13/4}.}
$$
Substitute this into <hydrostatic equilibrium>:
$$
\frac{17}4CT^{13/4}\frac{dT}{dr}
=-\frac{GM_c}{r^2}\frac{\mu C}{\mathcal R}T^{13/4},
\qquad
\frac{dT}{dr}=-\frac{4\mu GM_c}{17\mathcal Rr^2}.
$$
Define $B=4\mu GM_c/(17\mathcal R)$. The general <antiderivative> is $T=B/r+T_0$. The <Kramers radiative-zero envelope around a stellar core> sets $T_0=0$, giving
$$
\boxed{T=\frac{4\mu GM_c}{17\mathcal Rr}.}
$$
This is exact for the idealized boundary $T\to0$ as $r\to\infty$. With a finite outer <radius> $R_s$ and <temperature> $T_s$, instead $T=T_s+B(1/r-1/R_s)$; the displayed profile is the deep-envelope approximation when $R_s\gg R_c$ and $|T_s-B/R_s|\ll B/R_c$. Small outer <pressure> alone does not eliminate this <temperature> <constant of integration>.
The specific <hydrogen burning> rate becomes
$$
\epsilon=\epsilon_0\rho T^{67/4}
=\frac{\epsilon_0\mu C}{\mathcal R}T^{20}\propto r^{-20}.
$$
Consequently
$$
\boxed{\frac{\epsilon(1.05R_c)}{\epsilon(R_c)}=1.05^{-20}
=e^{-20\log(1.05)}\simeq0.377\simeq e^{-1}.}
$$
Its local radial e-folding length is $R_c/20$, so burning is concentrated within a small fraction of the <stellar core> <radius>. The volume heating is even steeper: $\rho\epsilon\propto C^2T^{93/4}$. Using the exterior envelope profile to estimate the thin shell's <luminosity> gives
$$
L\simeq4\pi\epsilon_0\left(\frac{\mu C}{\mathcal R}\right)^2B^{93/4}
\int_{R_c}^{\infty}r^{-85/4}dr
=\frac{16\pi\epsilon_0}{81}\left(\frac{\mu C}{\mathcal R}\right)^2B^{93/4}R_c^{-81/4}.
$$
A finite but very extended upper limit changes the factor by $1-(R_c/R_s)^{81/4}$. The large exponent makes this correction small and further verifies the <hydrogen burning> thin-shell approximation. Since $C^2\propto M_c/L$ and $B\propto M_c$, the <integral> implies $L^2\propto M_c^{97/4}R_c^{-81/4}$. Hence
$$
\boxed{L\propto M_c^{97/8}R_c^{-81/8},\qquad
L\propto M_c^{97/8}\ \text{along the fixed-}R_c\text{ sequence}.}
$$
This is the <fixed-radius-core shell-burning luminosity relation>. The proportionality coefficient is a thin-shell estimate: <luminosity> rises from its value beneath the shell to $L$ through the emitting layer, so treating it as constant there uses the exterior profile rather than resolving the burning region. The scaling holds while its <dimensionless> shell structure and <stellar composition> remain fixed.
For consistency, $M_c/R_c$ must provide a hot <hydrogen burning> shell while the <stellar core> remains inert to <core helium burning>; the <radiative envelope> <mass> must be much less than $M_c$, and $R_c$ must be well inside an extended <radiative envelope>. At the shell base the gas must remain a <nondegenerate gas> and dominated by <gas pressure>, in particular $aT_b^4/3\ll CT_b^{17/4}$ with $T_b=B/R_c$. The assumed <stellar core> sequence must genuinely have nearly fixed $R_c$ over the <mass> interval considered. A cold <helium> <stellar core> supported by a nonrelativistic <degenerate electron gas> instead has the mass-radius law from part (a), so substituting that law would describe a different model and change the <luminosity> exponent.
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