Bayesian estimation of the $S$ factor and thermonuclear reaction rate for ${}^{16}\mathrm{O}(p,\gamma ){}^{17}\mathrm{F}$

The ${}^{16}\mathrm{O}(p,\gamma ){}^{17}\mathrm{F}$ reaction is the slowest hydrogen-burning process in the CNO mass region. Its thermonuclear rate sensitively impacts predictions of oxygen isotopic ratios in a number of astrophysical sites, including AGB stars. The reaction has been measured several times at low bombarding energies using a variety of techniques. The most recent evaluated experimental rates have a reported uncertainty of about 7.5% below 1 GK. However, the previous rate estimate represents a best guess only and was not based on rigorous statistical methods. We apply a Bayesian model to fit all reliable ${}^{16}\mathrm{O}(p,\gamma ){}^{17}\mathrm{F}$ cross section data, and take into account independent contributions of statistical and systematic uncertainties. The nuclear reaction model employed is a single-particle potential model involving a Woods-Saxon potential for generating the radial bound state wave function. The model has three physical parameters, the radius and diffuseness of the Woods-Saxon potential, and the asymptotic normalization coefficients (ANCs) of the final bound state in ${}^{17}\mathrm{F}$. We find that performing the Bayesian $S$-factor fit using ANCs as scaling parameters has a distinct advantage over adopting spectroscopic factors instead. Based on these results, we present the first statistically rigorous estimation of experimental ${}^{16}\mathrm{O}(p,\gamma ){}^{17}\mathrm{F}$ reaction rates, with uncertainties $(\pm 4.2\%)$ of about half the previously reported values.

Figure 1

Results of Bayesian fits to experimental ${}^{16}\mathrm{O}(p,\gamma ){}^{17}\mathrm{F}S$ factors [19, 20, 21, 22]: (top) Total $S$ factor; (middle) $S$ factor for the transition to the ${}^{17}\mathrm{F}$ ground state; (bottom) $S$ factor for the transition to the ${}^{17}\mathrm{F}$ first-excited state. Becker et al. [20] measured only the $S$ factor for the first-excited state transition at a single energy (green datum in the bottom panel). Hester, Pixley, and Lamb [19] measured only the total $S$ factor (blue data points in the top panel). The dark and light shaded regions represent coverage probabilities of 68% and 95%, respectively. Notice the logarithmic scale on the abscissa.

Figure 2

One- and two-dimensional projections of the posterior probability distributions of two physical model parameters, i.e., the ANCs, $C_{gs}^2$ and $C_{\mathit{fes}}^2$, and four data set parameters, $f_j$, when adopting broad priors in the Bayesian fit. These results correspond to those given in column 2 of Table 1 (boldface) and in Table 3. The dark and light shaded areas correspond to 68% and 95% coverage probabilities, respectively.

Figure 3

Comparison of present to previously evaluated ${}^{16}\mathrm{O}(p,\gamma ){}^{17}\mathrm{F}$ reaction rates. For better comparison, all rates have been normalized to the present median rate (see column 3 in Table 4). The magenta-shaded regions correspond to our 68% uncertainties (see column 5 in Table 4). The gray-shaded regions depict the rates and uncertainties reported in Ref. [7] (top panel) and Ref. [12] (bottom panel).