CNO cycle

The CNO cycle catalyses hydrogen fusion through carbon, nitrogen, and oxygen nuclei. Near $2\times 10^7\mathrm{K}$, its specific energy generation is approximately proportional to $\rho T^{17}$ and is therefore strongly concentrated near a star's centre.

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.

A hotter stellar core can favor a higher-barrier route despite fewer catalyst nuclei. To isolate this competition, prescribe a local hydrogen burning toy model at fixed mass density $\rho$, with reference $T_0=15\mathrm{MK}$ andChoose $E_{G,pp}=493\mathrm{keV}$ and $E_{G,\mathrm{CNO}}=45000\mathrm{keV}$, rounded barrier parameters for a proton pair and proton capture on nitrogen-14. These prescribe the temperature dependence, while the following freely chosen heating normalization incorporates effective nuclear factors and deposited energy:Here $X$ is hydrogen mass fraction, $X_{\mathrm{CNO}}$ is CNO mass fraction, and ${ϵ}_{\ast }$ is an arbitrary positive unit of stellar energy-generation rate. For $X=0.70$ and $X_{\mathrm{CNO}}=0.01$, the reference ratio is $0.01$ by construction. It is not a measured solar ratio. The CNO cycle bottleneck is represented by one effective rate, without evolving the existing proton-proton chain branches or catalyst network.

At $T_0$, the logarithmic temperature slopes are $({\tau }_{pp,0}-2)/3=3.90$ and $({\tau }_{\mathrm{CNO},0}-2)/3=19.90$. The larger Coulomb barrier gives a steeper fractional response. Because these slopes change with temperature, this example retains the exponential rather than extending two fixed power laws. Writing $R_0=0.01(X_{\mathrm{CNO}}/0.01)(0.70/X)$ and $D={\tau }_{\mathrm{CNO},0}-{\tau }_{pp,0}$ givesprovided the bracket is positive. With the chosen composition, $T_{\ast }=20.30\mathrm{MK}$. Reducing $X_{\mathrm{CNO}}$ to $0.001$ at unchanged $X$ shifts it to $23.91\mathrm{MK}$. For $X_{\mathrm{CNO}}=0$, this catalytic route supplies no heat in this model. These are local toy crossovers: a whole star's luminosity requires integrating its stellar energy-generation rate through its stellar composition profile and stellar structure. Nuclear features, plasma screening and changes of the dominant bottleneck limit the model's extrapolation.

A branch of the CNO cycle proceeds through proton captures on nitrogen-15 and oxygen-16, beta-plus decay of fluorine-17, and ${}^{17}\mathrm{O}(p,\alpha ){}^{14}\mathrm{N}$. Returning through oxygen-15 and nitrogen-15 closes the cycle. It also catalyzes the conversion of four protons into helium-4, with branch-dependent neutrino loss.

The principal CNO cycle follows ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$, beta-plus decay to carbon-13, proton capture to nitrogen-14, proton capture to oxygen-15, beta-plus decay to nitrogen-15, and ${}^{15}\mathrm{N}(p,\alpha ){}^{12}\mathrm{C}$. The carbon seed is regenerated. The ordinary bottleneck is ${}^{14}\mathrm{N}(p,\gamma ){}^{15}\mathrm{O}$, so nitrogen-14 accumulates in cycle equilibrium.