= CNO crossover temperature
{c}
{title2=$T_*/T_{*,0}=(Z_{\mathrm{CNO},0}/Z_{\mathrm{CNO}})^{1/(\nu_{\mathrm{CNO}}-\nu_{pp})}$}
If <proton–proton chain> heating scales as $X_H^2\rho T^{\nu_{pp}}$ and <CNO cycle> heating as $X_HZ_{\mathrm{CNO}}\rho T^{\nu_{\mathrm{CNO}}}$, 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.
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