New 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 (\(E_\mathrm{c.m.} = 60\,\mathrm{keV}\)) 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 \(20-140\,\mathrm{MK}\) range for the lowest possible \({}^{12}\mathrm{C}/{}^{13}\mathrm{C}\) ratio that can be produced during H burning in giant stars.