New constraints on sodium production in globular clusters from the ${}^{23}\mathrm{Na}({}^3\mathrm{He},\mathrm{d}){}^{24}\mathrm{Mg}$ 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 ${}^{23}\mathrm{Na}(p,\gamma )$ and ${}^{23}\mathrm{Na}(p,\alpha )$. In this paper, we report the results of a ${}^{23}\mathrm{Na}{({}^3\mathrm{He},d)}^{24}\mathrm{Mg}$ transfer reaction carried out at Triangle Universities Nuclear Laboratory using a $21\mathrm{MeV}{}^3\mathrm{He}$ beam. Astrophysically relevant states in ${}^{24}\mathrm{Mg}$ between $11<E_x<12\mathrm{MeV}$ 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, intramodel variations on sodium production from the ${}^{23}\mathrm{Na}(p,\gamma )$ and ${}^{23}\mathrm{Na}(p,\alpha )$ reaction channels are a lingering source of uncertainty.

Figure 1

Full and partial focal plane position spectrum at ${\theta }_{\text{lab}}={11}^{\circ }$ after 6 h (${10}^{15}$ particles) of data accumulation. The top panel () shows the entire focal plane spectrum with the calibration states (given in keV) highlighted in orange and the astrophysical region of interest is between the dashed black lines. The energy values for the states below the proton threshold ($11692.69\mathrm{keV}$) are taken from Ref. [21], while the rest are from Table 1. All values in the top panel are rounded to the nearest integer. The bottom panel () is zoomed in on the astrophysical region of interest. Peaks from ${}^{24}\mathrm{Mg}$ have been identified with their final weighted average energy value in keV.

Figure 2

Pair correlation plot of the posterior samples for the nested sampling run. The discrete ambiguity is prominent in the ${}^3\mathrm{He}+{}^{23}\mathrm{Na}$ data, posing a significant challenge in estimating the optical model parameter posteriors.

Figure 3

The discrete ambiguity as seen in the $V$ versus $r_0$ correlations between the histogrammed samples of the nested sampling calculation shown in black. The colored lines show the description of the correlation based on the analytic form $V{r}_0^n=c$. The value of $c$ provides a way to distinguish these modes.

Figure 4

Values from the volume integral of the real potential as calculated using Eq. (21) and the samples from the nested sampling calculation. The discrete ambiguity causes two well separated peaks to appear. The black dashed line shows the value of $J_R$ for the global potential.

Figure 5

The credibility intervals obtained for the elastic scattering fit compared to the measured yields relative to Rutherford scattering ($Y_R/{\sigma }_R$). The dark and light purple bands show the $68\%$ and $95\%$ credibility intervals, respectively. The measured error bars are smaller than the points, while the adjusted uncertainty of Eq. (19) that is inferred from the data is not shown.

Figure 6

DWBA calculations for the states of ${}^{24}\mathrm{Mg}$. The $68\%$ and $95\%$ credibility intervals are shown in purple and light purple, respectively. Only the data points shown in black were considered in each calculation, with the triangles being excluded based on the cross section increasing after the minimum value was reached. For the $11826$ keV state, the $68\%$ bands are shown for all of the $\ell$ transfers between 0 and 3.

Figure 7

The distributions from the nested sampling algorithm for the most likely $\ell$ values for the $11826\mathrm{keV}$ state.

Figure 8

Pair correlation plot for the MCMC posterior samples of ${\mathrm{\Gamma }}_{\text{sp}}$ and $(2J_f+1)C^{2}S$ for the $12051\mathrm{keV}$ state. A strong anticorrelation exists when the same bound-state parameters are used to calculate both quantities, resulting in ${\mathrm{\Gamma }}_p$ having less sensitivity to these parameters.

Figure 9

New1 reaction rates normalized to their median. (Left) The $(p,\gamma )$ reaction rate taken from Ref. [17]. The blue contours show the relative uncertainty as a function of temperature. The gray contour is the recommend rate of Ref. [15] normalized to the updated rate's median. Both contours show $68\%$ coverage. (Right) The reaction rate ratio plot for the $(p,\alpha )$ rate.

Figure 10

New2 reaction rates normalized to their median. The gray contour is the recommend rate of Ref. [15] normalized to the New2 rate's median. (Left) The $(p,\gamma )$ reaction rate ratio. Large variations are seen around 80 MK due to the uncertain spin-parity assignment for the $133\mathrm{keV}$ resonance. (Right) The reaction rate ratio plot for the $(p,\alpha )$ rate.

Figure 11

${}^{23}\mathrm{Na}$ surface abundance for median (solid line) and low/high (dashed lines) reaction rates for LUNA, New1, and New2 (see text for details). The high rates for the ${}^{23}\mathrm{Na}(p,\gamma )$ and ${}^{23}\mathrm{Na}(p,\alpha )$ lead to lower ${}^{23}\mathrm{Na}$ surface abundances and vice versa for the low rates. Model number is a proxy for time. The impact of rate uncertainties shown here is small.