= Kramers radiative-zero envelope around a stellar core
{c}
{title2=$P=CT^{17/4},\quad T\simeq\frac{4\mu GM_c}{17\mathcal Rr}$}
For a negligible-mass ideal-gas envelope above a core of <mass> $M_c$, with constant <luminosity> $L$ and <Kramers' opacity law>, division of <hydrostatic equilibrium> by radiative diffusion gives $P\,dP/dT=16\pi acGM_c\mathcal R\,T^{15/2}/(3\kappa_0L\mu)$. Taking the outer <pressure> constant to be negligible yields $P=CT^{17/4}$, where $C^2=64\pi acGM_c\mathcal R/(51\kappa_0L\mu)$, and $\rho=(\mu C/\mathcal R)T^{13/4}$. <Hydrostatic equilibrium> then gives $dT/dr=-4\mu GM_c/(17\mathcal Rr^2)$, so $T=B/r+T_0$ with $B=4\mu GM_c/(17\mathcal R)$. The radiative-zero choice $T_0=0$ is exact for a vanishing-temperature boundary at infinity and a deep-envelope approximation for a very extended finite star. It is not implied by small surface <temperature> alone at a specified finite <radius>.
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