Composition-dependent CNO homology ($L\propto {\mu }^4{M}^3/\kappa ,R\propto (X\kappa {\mu }^9{)}^{1/16}{M}^{3/4}$)

For homologous, chemically uniform, gas-pressure-supported radiative stars with specific energy generation $ϵ\propto X\rho T^{13}$ and composition-dependent constant opacity, stellar homology gives $L_{\mathrm{nuc}}\propto X{\mu }^{13}{M}^{15}{R}^{-16}$ and $L_{\mathrm{rad}}\propto {\mu }^4{M}^3/\kappa$. Equating them gives the displayed scalings. Hydrogen depletion changes both mean molecular weight and electron-scattering opacity; holding composition fixed gives $R\propto M^{3/4}$ and $L\propto M^3$. The power-law burning approximation applies only while hydrogen fuel remains.

With negligible heavy-element contribution to the mean molecular weight, $\mu =4/(3+5X)$ and $\kappa \propto 1+X$. At fixed mass, composition-dependent CNO homology gives $L\propto (1+X{)}^{-1}(3+5X{)}^{-4}$ and $R\propto [X(1+X){]}^{1/16}(3+5X{)}^{-9/16}$. The Stefan–Boltzmann law then gives $T_e\propto X^{-1/32}(1+X{)}^{-9/32}(3+5X{)}^{-23/32}$. From the formal $X=1$ reference, declining $X$ initially increases luminosity and effective temperature, unlike the usual redward main-sequence turn-off. A trace CNO catalyst abundance is implicit in the burning coefficient; an exactly metal-free pure-hydrogen star cannot burn through that cycle.