The Aspects of 12C(p,γ)13N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{12}\textrm{C}(p, \gamma )^{13}\textrm{N}$$\end{document} Reaction in Astrophysical Regime

The carbon-nitrogen-oxygen (CNO) cycle is fundamental to the process of hydrogen burning in stars, serving as a pivotal mechanism. At its core, the primary reaction involves the radiative capture of a proton by 12C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ^{12}\textrm{C} $$\end{document}, which crucially influences the isotopic ratio of 12C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ^{12}\textrm{C} $$\end{document} to 13C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ^{13}\textrm{C} $$\end{document} observed in celestial bodies, including our Solar System. To address this, we applied the astrophysical R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R$$\end{document}-matrix approach to extrapolate low-energy cross sections and S-factors, thereby improving the precision of nuclear reaction rates. At a proton energy of around 25 keV (C.M. system), the extrapolated value of the astrophysical S-factor is determined to be 1.34±0.10keVbarn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 1.34 \pm 0.10 \, \mathrm {keV \, barn} $$\end{document}. Our investigation sheds light on its implications for nuclear reaction rates, suggesting that at low temperatures in hydrogen-burning sites, the conversion of 12C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ^{12}\textrm{C} $$\end{document} to 13C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ^{13}\textrm{C} $$\end{document} via proton capture is relatively slow, thereby influencing the abundance ratios in the cosmic environment. This slow conversion affects stellar nucleosynthesis and isotopic evolution, particularly in low-mass stars (M≤2M⊙)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M \le 2 \, M_\odot )$$\end{document} where hydrogen burning proceeds at relatively low temperatures. Unlike previous analyses with large uncertainties at low energies, our approach refines the S-factor determination by incorporating improved ANC (Asymptotic Normalization Constant) values, reducing extrapolation uncertainties.