Folding model approach to the elastic $p+^{12,13}$C scattering at low energies and radiative capture $^{12,13}$C$(p,\gamma)$ reactions

The proton radiative capture $^{12,13}$C$(p,{\gamma})$ reactions at astrophysical energies, key processes in the CNO cycle, are revisited in the potential model with the proton-nucleus potential for both the scattering and bound states obtained in the folding model, using a realistic density dependent nucleon-nucleon interaction. For the consistency, this same folding model is also used to calculate the optical potential of the elastic $p+^{12,13}$C scattering at energies around the Coulomb barrier. The folded $p+^{12,13}$C optical potentials are shown to account well for both the elastic $p+^{12,13}$C3 scattering and astrophysical $S$ factors of the radiative capture $^{12,13}$C$(p,\gamma)$ reactions.