= Constant-opacity radiative-zero giant envelope
{title2=$P=\Lambda T^4,\quad T\simeq\mu\beta GM_c/(4\mathcal Rr)$}
A negligible-mass <radiative envelope> surrounding a core of mass $M_c$, carrying constant luminosity and having constant <opacity>, obeys $dP/dT=16\pi acGM_cT^3/(3\kappa_0L)$. Neglecting the outer pressure integration constant gives $\Lambda=4\pi acGM_c/(3\kappa_0L)$. The <gas pressure> fraction $\beta$ is constant because $P_{\rm rad}=aT^4/3$, and $\rho=\mu\beta\Lambda T^3/\mathcal R$. Hydrostatic balance gives $T=T_s+\mu\beta GM_c(1/r-1/R_s)/(4\mathcal R)$; dropping this outer-boundary constant gives the displayed deep-envelope approximation. A positive gas-pressure fraction requires $L<4\pi cGM_c/\kappa_0$.
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