Radiative homology with density-half inverse-five-halves opacity ($R\propto M^{2/7},L\propto M^{33/7}$)

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