  The study of CNO cycle involves the examination of the proton radiative
capture, or the $(p,\gamma)$ reactions below 2 MeV. The astrophysical
$\mathcal{S}$ factor characterizing the $(p,\gamma)$ reaction is usually
reduced to the electric dipole transition $E1$ from the scattering state to the
bound state. In this work, the partial scattering and the single-particle bound
wave functions in the reduced matrix element of the transition are obtained
from the single self-consistent mean-field potential deduced from the Skyrme
Hartree-Fock calculation. The astrophysical $\mathcal{S}$ factors of the
$(p,\gamma)$ reactions in the CNO cycle were successfully reproduced. The
self-consistent Hartree-Fock calculation from the discrete to the continuum is
a promising approach for the microscopic analysis of the nucleon-induced
reactions in nuclear astrophysics.
