= Radiative homology with proton-proton burning and inverse-fourth-power opacity
{title2=$L\propto M^6\mu^{26/3}X^{-1/3}[Z(1+X)]^{-7/6}$}
For a chemically uniform, ideal-gas-supported <stellar homology> sequence with nuclear energy generation $\epsilon=\epsilon_0X^2\rho T^4$ and <opacity> $\kappa=\kappa_0Z(1+X)\rho T^{-4}$, <radiative diffusion in a star> fixes $L R Z(1+X)/(\mu^8M^6)$ and nuclear heating fixes $X^2\mu^4M^6/(R^7L)$. Eliminating $L$ gives $R^6\propto X^2Z(1+X)\mu^{-4}$, hence
$$
R\propto X^{1/3}[Z(1+X)]^{1/6}\mu^{-2/3},\qquad L\propto M^6\mu^{26/3}X^{-1/3}[Z(1+X)]^{-7/6}.
$$
The absence of $M$ in the radius law is a consequence of these particular power laws; it is not a universal relation for real stars. At fixed initial $X$ and $\mu$, the <Stefan–Boltzmann law> gives $T_e\propto Z^{-1/12}$ at fixed <luminosity>.
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