Proton-Capture Rates on Carbon Isotopes and Their Impact on the Astrophysical ${}^{12}\mathrm{C}/{}^{13}\mathrm{C}$ Ratio

The ${}^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio is a significant indicator of nucleosynthesis and mixing processes during hydrogen burning in stars. Its value mainly depends on the relative rates of the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ and ${}^{13}\mathrm{C}(p,\gamma ){}^{14}\mathrm{N}$ reactions. Both reactions have been studied at the Laboratory for Underground Nuclear Astrophysics (LUNA) in Italy down to the lowest energies to date ($E_{\mathrm{c}.\mathrm{m}.}=60\mathrm{keV}$) reaching for the first time the high energy tail of hydrogen burning in the shell of giant stars. Our cross sections, obtained with both prompt $\gamma$-ray detection and activation measurements, are the most precise to date with overall systematic uncertainties of 7%–8%. Compared with most of the literature, our results are systematically lower, by 25% for the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction and by 30% for ${}^{13}\mathrm{C}(p,\gamma ){}^{14}\mathrm{N}$. We provide the most precise value up to now of $3.6\pm 0.4$ in the 20–140 MK range for the lowest possible ${}^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio that can be produced during H burning in giant stars.

Figure 1

$R$-matrix fit to available $S$-factor data for the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction (top panel) and the ${}^{13}\mathrm{C}(p,\gamma ){}^{14}\mathrm{N}$ (bottom panel). The red solid line shows the best fit obtained with fixed LUNA data (black symbols) and normalized literature data (coefficients $n$ in the legend). The blue dashed line shows the fit obtained with free normalization also for the LUNA data. Uncertainties are statistical only. The brown and yellow regions represent the energy range relevant to RGB [53] and AGB stars [54, 55], respectively.

Figure 2

Reaction rate ratio to NACRE and NACRE2 values for the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ (upper panel) and the ${}^{13}\mathrm{C}(p,\gamma ){}^{14}\mathrm{N}$ (lower panel). The brown bar represents the temperature range of interest for RGB and AGB stars.