= Radiative homology with density-half inverse-five-halves opacity
{title2=$R\propto M^{2/7},\quad L\propto M^{33/7}$}
For gas-pressure-supported <stellar homology> with $\kappa\propto\rho^{1/2}T^{-5/2}$ and specific nuclear heating $\epsilon\propto\rho T^5$, the central scalings give $L_{\rm nuc}\propto M^7R^{-8}$ and <radiative diffusion in a star> gives $L_{\rm rad}\propto M^5R^{-1}$. Equating them yields the displayed laws. The <Stefan–Boltzmann law> gives $T_{\rm eff}\propto M^{29/28}$, hence $d\log L/d\log T_{\rm eff}=132/29$. Composition and dimensionless profiles are held fixed.
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