 Composition-dependent temperature selection of hydrogen burning

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.