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 \( ^{12}\textrm{C} \) , which crucially influences the isotopic ratio of \( ^{12}\textrm{C} \) to \( ^{13}\textrm{C} \) observed in celestial bodies, including our Solar System. To address this, we applied the astrophysical \(R\) -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 \( 1.34 \pm 0.10 \, \mathrm {keV \, barn} \) . Our investigation sheds light on its implications for nuclear reaction rates, suggesting that at low temperatures in hydrogen-burning sites, the conversion of \( ^{12}\textrm{C} \) to \( ^{13}\textrm{C} \) 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 \((M \le 2 \, M_\odot )\) 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.
