Radiative homology with proton-proton burning and inverse-fourth-power opacity ($L\propto M^6{\mu }^{26/3}{X}^{-1/3}[Z(1+X){]}^{-7/6}$)

For a chemically uniform, ideal-gas-supported stellar homology sequence with nuclear energy generation $ϵ={ϵ}_0{X}^2\rho T^4$ and opacity $\kappa ={\kappa }_{0}Z(1+X)\rho T^{-4}$, radiative diffusion in a star fixes $LRZ(1+X)/({\mu }^8{M}^6)$ and nuclear heating fixes $X^2{\mu }^4{M}^6/(R^{7}L)$. Eliminating $L$ gives $R^6\propto X^{2}Z(1+X){\mu }^{-4}$, henceThe 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.

Let a fixed-mass, fully mixed stellar model evolve by hydrogen burning, with $L=C{M}^6{Z}^{-7/6}H(X)$ and $H(X)=\mu (X{)}^{26/3}{X}^{-1/3}(1+X{)}^{-7/6}$. Energy conservation gives $\dot{X}=-L/(M{E}_H)$. The time to reach $X=f{X}_0$ is therefore $t=(E_H/C)M^{-5}{Z}^{7/6}{\int }_{f{X}_0}^{X_0}H(X{)}^{-1}dX$. If $X_0,f$ and the function $\mu (X)$ are common to the population, the integral is constant and $M_{\mathrm{to}}\propto Z^{7/30}{t}^{-1/5}$. This includes the change of luminosity during fuel depletion rather than replacing the luminosity by its initial value.