New Constraints on Sodium Production in Globular Clusters From the $^{23}$Na$(^3$He$, \textbf{d})^{24}$Mg Reaction

The star to star anticorrelation of sodium and oxygen {is} a defining feature of globular clusters, but, to date, the astrophysical site responsible for this unique chemical signature remains unknown. Sodium enrichment within these clusters depends sensitively on reaction rate of the sodium destroying reactions $^{23}$Na$(p, \gamma)$ and $^{23}$Na$(p, \alpha)$. In this paper, we report the results of a $^{23}$Na$(^3\text{He}, d)^{24}$Mg transfer reaction carried out at Triangle Universities Nuclear Laboratory using a $21$ MeV $^3$He beam. Astrophysically relevant states {in $^{24}$Mg} between $11 < E_x < 12$ MeV were studied using high resolution magnetic spectroscopy, thereby allowing the extraction of excitation energies and spectroscopic factors. Bayesian methods are combined with the distorted wave Born approximation to assign statistically meaningful uncertainties to the extracted spectroscopic factors. For the first time, these uncertainties are propagated through to the estimation of proton partial widths. Our experimental data are used to calculate the reaction rate. The impact of the new rates are investigated using asymptotic giant branch star models. It is found that while the astrophysical conditions still dominate the total uncertainty, intra-model variations on sodium production from the $^{23}$Na$(p, \gamma)$ and $^{23}$Na$(p, \alpha)$ reaction channels are a lingering source of uncertainty.