Determination of the direct capture contribution for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">N</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:mmultiscripts><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">O</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:mmultiscripts></mml:mrow></mml:math>from the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">O</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:mmultiscripts><mml:mo>→</mml:mo><mml:mmultiscripts><mml:mi mathvariant="normal">N</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:mmultiscripts><mml:mo>+</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:math>asymptotic normalization coefficient

$^{13}\mathrm{N}(p,\ensuremath{\gamma})^{14}\mathrm{O}$ is one of the key reactions which trigger the onset of the hot CNO cycle. This transition occurs when the proton capture rate on $^{13}\mathrm{N}$ is faster, due to increasing stellar temperature $(\ensuremath{\geqslant}{10}^{8}\phantom{\rule{0.3em}{0ex}}\mathrm{K})$, than the $^{13}\mathrm{N}$ $\ensuremath{\beta}$-decay rate. The rate of this reaction is dominated by the resonant capture through the first excited state of $^{14}\mathrm{O}$ $({E}_{r}=0.528\phantom{\rule{0.3em}{0ex}}\text{MeV})$. However, through constructive interference, direct capture below the resonance makes a non-negligible contribution to the reaction rate. We have determined this direct contribution by measuring the asymptotic normalization coefficient for $^{14}\mathrm{O}\ensuremath{\rightarrow}^{13}\mathrm{N}+p$. In our experiment, an $11.8\phantom{\rule{0.3em}{0ex}}\text{MeV}∕\text{nucleon}$ $^{13}\mathrm{N}$ radioactive beam was used to study the $^{14}\mathrm{N}(^{13}\mathrm{N},^{14}\mathrm{O})^{13}\mathrm{C}$ peripheral transfer reaction, and the asymptotic normalization coefficient, ${({C}_{{p}_{1∕2}}^{^{14}\mathrm{O}})}^{2}=29.0\ifmmode\pm\else\textpm\fi{}4.3\phantom{\rule{0.3em}{0ex}}{\text{fm}}^{\ensuremath{-}1}$, was extracted from the measured cross section. The radiative capture cross section was estimated using an $R$-matrix approach with the measured asymptotic normalization coefficient and the latest resonance parameters. We find the $S$ factor for $^{13}\mathrm{N}(p,\ensuremath{\gamma})^{14}\mathrm{O}$ to be larger than previous estimates. Consequently, the transition from the cold to hot CNO cycle for novae would be controlled by the slowest proton capture reaction $^{14}\mathrm{N}(p,\ensuremath{\gamma})^{15}\mathrm{O}$.

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