Direct capture astrophysicalSfactors at low energy

We investigate the energy dependence of the astrophysical $S$ factors for the reactions ${}^7\mathrm{Be}(p,\gamma {)}^8\mathrm{B},$ the primary source of high-energy solar neutrinos in the solar $\mathrm{pp}$ chain, and ${}^{16}\mathrm{O}(p,\gamma {)}^{17}\mathrm{F},$ an important reaction in the CNO cycle. Both of these reactions have predicted $S$ factors which rise at low energies; we find the source of this behavior to be a pole in the $S$ factor at a center-of-mass energy $E=-E_{\mathrm{B}},$ the point where the energy of the emitted photon vanishes. The pole arises from a divergence of the radial integrals.