Turnoff mass of a homogeneously mixed proton-proton-burning population ($M_{\mathrm{to}}\propto Z^{7/30}{t}^{-1/5}$)

Let a fixed-mass, fully mixed stellar model evolve by hydrogen burning, with $L=C{M}^6{Z}^{-7/6}H(X)$ 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.