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In the radiative layer, take the enclosed mass to be $M_1$, the transported luminosity to be $L$, and the mean molecular weight to be constant. The envelope's own gravity is neglected. Divide the hydrostatic and radiative temperature equations, and then eliminate density using the gas equation of state:Put $p=n+1$ and $b=4+m+n$. For $p,b\ne 0$, integration givesDefining $C^p=pB/b$ and $K=K_P/C^p$ yieldsIn the relevant positive-exponent case, writing $K=T_0^b$ produces the stated form whenever this integration constant is nonnegative. The exceptional values excluded above lead to logarithms: $p=0$ gives $\log P=B{T}^b/b+\mathrm{const}$, $b=0$ gives $P^p=pB\log T+\mathrm{const}$, and both zero give $\log P=B\log T+\mathrm{const}$.

For radiative-envelope convection matching, compute the logarithmic gradient of the pressure relation:At the marginal boundary, the Schwarzschild criterion equates this with the ideal-gas adiabatic gradient ${\nabla }_{\mathrm{ad}}=(\gamma -1)/\gamma$. HenceThe positive real $T_0$ notation requires a nonnegative bracket; a general signed $K$ is the safer integration constant otherwise. For the later case $n=1,m=3$ and a monatomic gas, this bracket is $3/5$, so $T_0$ is of the same order as $T_b$.

Deep below the convective boundary, with the temperature-power term dominating $K$, the pressure law becomes $P\simeq C{T}^{b/p}$. Differentiating this approximation and using hydrostatic balance givesThe general temperature is a term proportional to $1/r$ plus a boundary-dependent constant. Well inside the boundary radius, where that term dominates, it reduces toThe inverse-radius profile is an inner approximation, not an assertion that the boundary term vanishes everywhere.

Now take $n=1,m=3$, so $p=2$, $b=8$. Then $P\simeq C{T}^4$, $\rho \simeq (\mu C/\mathcal{R})T^3$, and define $A_T=\mu G{M}_1/(4\mathcal{R})$ so that $T\simeq A_T/r$. The luminosity supplied by the thin burning layer isBecause the integral is strongly concentrated near $R_1$ and $r_b\gg R_1$, the outer term is negligible. ThusThe factor $\rho ϵ$, rather than merely $ϵ$, is essential since the specified energy-generation rate is per unit mass. This integration uses the deep radiative profile as a leading thin-shell approximation.

Crucially, $C$ is not independent of luminosity. Its defining relation givesSubstitution into the shell integral therefore yields $L^2\propto M_1^{17}{R}_1^{-13}$. The degenerate-core relation gives $R_1\propto M_1^{-1/3}$, soThis degenerate-core shell-burning mass-luminosity relation is specific to the adopted opacity, burning exponent and structural approximations.

Removing some envelope mass while retaining the hydrogen-burning shell and the deep-envelope regime leaves the leading luminosity approximately unchanged at fixed core mass. The envelope can instead change the radius and effective temperature. This independence is not valid for arbitrarily complete stripping: removing the fuel or the pressure-confined shell structure quenches or modifies shell burning, and the assumed giant model then fails.