= Uniform gas-pressure fraction and constant-luminosity radiative-envelope incompatibility
{title2=$\beta,\kappa\text{ constant}\ \Longrightarrow\ L_r/m=4\pi cG(1-\beta)/\kappa$}
= Constant-beta radiative-envelope obstruction
{synonym}
For a radiative envelope, $dP_{\rm rad}/dP=\kappa L_r/(4\pi cGm)$. Exactly uniform <stellar gas-pressure fraction> instead requires this derivative to be $1-\beta$. Constant opacity therefore requires $L_r$ to increase in proportion to enclosed mass. A positive-density finite-mass envelope with no local energy generation has constant $L_r$ and cannot satisfy all these conditions pointwise. The <Eddington-model convective-core mass fraction> uses constant $\beta$ as a global approximation, rather than an exact detailed envelope constraint.
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