The astrophysical direct nuclear capture reaction ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ is studied within the framework of a potential model. Parameters of the nuclear $p\text{−}{}^{12}\mathrm{C}$ interaction potentials of the Woods-Saxon form are adjusted to reproduce experimental $p\text{−}{}^{12}\mathrm{C}$ scattering phase shifts, as well as the binding energies and empirical values of the asymptotic normalization coefficient (ANC) for the ${}^{13}\mathrm{N}$($1/2^{-}$) ground state from the literature. The reaction rates are found to be very sensitive to the description of the value of the ANC of the ${}^{13}\mathrm{N}$($1/2^{-}$) ground state. The potential model, which yields the ANC value of 1.63 ${\mathrm{fm}}^{-1/2}$ for the ${}^{13}\mathrm{N}$($1/2^{-}$) ground state and a value of $\mathrm{\Gamma }=39$ keV for the ${}^{13}\mathrm{N}$($1/2^{+}$) resonance width, is able to reproduce the astrophysical $S$ factor in the energy interval up to 2 MeV, the empirical values of the reaction rates in the temperature region up to $T={10}^{10}\mathrm{K}$ of the LUNA Collaboration [J. Skowronski et al., Phys. Rev. Lett. 131, 162701 (2023)], and the results of the $R$-matrix fit. The astrophysical factor $S\left(0\right)=1.35$ keV b is found using the asymptotic expansion method of Baye and Brainis [Phys. Rev. C 61, 025801 (2000)]. The obtained value is in good agreement with the Solar Fusion II result. At the same time, the calculated value of 1.44 keV b of the astrophysical $S$ factor at the solar Gamow energy is consistent with the result of the $R$-matrix fit of $S\left(25\mathrm{keV}\right)=1.48\pm 0.09$ keV b by Kettner et al. [Phys. Rev. C 108, 035805 (2023)], but slightly less than the result of $S\left(25\mathrm{keV}\right)=1.53\pm 0.06$ keV b from the LUNA Collaboration [B. Acharya, M. Aliotta, A. B. Balantekin et al., arXiv:2405.06470].

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• Direct reactions
• Nuclear astrophysics
• Nuclear reaction rates
• Nuclear reactions
• Radiative capture
• S-factor
• 6 ≤ A ≤ 19