CNO crossover temperature ($T_{\ast }/T_{\ast ,0}=(Z_{\mathrm{CNO},0}/Z_{\mathrm{CNO}}{)}^{1/({\nu }_{\mathrm{CNO}}-{\nu }_{pp})}$)

If proton–proton chain heating scales as $X_H^2\rho T^{{\nu }_{pp}}$ and CNO cycle heating as $X_H{Z}_{\mathrm{CNO}}\rho T^{{\nu }_{\mathrm{CNO}}}$, their ratio at fixed hydrogen abundance scales as $Z_{\mathrm{CNO}}{T}^{{\nu }_{\mathrm{CNO}}-{\nu }_{pp}}$. Equating the rates proves the displayed crossover-temperature relation. It applies to a local power-law approximation, with unchanged coefficient normalizations apart from the catalyst abundance.