= Turnoff mass of a homogeneously mixed proton-proton-burning population
{title2=$M_{\mathrm{to}}\propto Z^{7/30}t^{-1/5}$}
Let a fixed-mass, fully mixed stellar model evolve by <hydrogen burning>, with $L=C M^6Z^{-7/6}H(X)$ and $H(X)=\mu(X)^{26/3}X^{-1/3}(1+X)^{-7/6}$. Energy conservation gives $\dot X=-L/(ME_H)$. The time to reach $X=fX_0$ is therefore $t=(E_H/C)M^{-5}Z^{7/6}\int_{fX_0}^{X_0}H(X)^{-1}dX$. 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.
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