  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.
