Astrophysical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> factor and rate of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mmultiscripts><mml:mi>Be</mml:mi><mml:mprescripts/><mml:none/><mml:mn>7</mml:mn></mml:mmultiscripts><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>)</mml:mo><mml:mmultiscripts><mml:mi mathvariant="normal">B</mml:mi><mml:mprescripts/><mml:none/><mml:mn>8</mml:mn></mml:mmultiscripts></mml:mrow></mml:math> direct capture reaction in a potential model

The astrophysical $^{7}\mathrm{Be}(p,\ensuremath{\gamma})\phantom{\rule{0.16em}{0ex}}^{8}\mathrm{B}$ direct capture process is studied in the framework of a two-body single-channel model with potentials of the Gaussian form. A modified potential is constructed to reproduce the new experimental value of the $S$-wave-scattering length and the known astrophysical $S$ factor at the Gamow energy, extracted from the solar neutrino flux. The resulting potential is consistent with the theory developed by Baye [Phys. Rev. C 62, 065803 (2000)] according to which the $S$-wave scattering length and the astrophysical $S$ factor at zero energy divided by the square of the asymptotic normalization coefficient are related. The obtained results for the astrophysical $S$ factor at intermediate energies are in good agreement with the two data sets of Hammache et al. [Phys. Rev. Lett. 86, 3985 (2001); Phys. Rev. Lett.80, 928 (1998)]. Linear extrapolation to zero energy yields ${S}_{17}(0)\ensuremath{\approx}20.{51}_{\ensuremath{-}1.85}^{+2.02}\phantom{\rule{0.28em}{0ex}}\mathrm{eV}\phantom{\rule{0.16em}{0ex}}\mathrm{b}$ consistent with the Solar Fusion II estimate. The calculated reaction rates are substantially lower than the results of the NACRE II Collaboration.

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