Sensitivity of the ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}S$ factor to the uncertainty in the level parameters of the near-threshold state

The ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ reaction is the main source of neutrons for the $s$ and $i$ processes in asymptotic giant branch stars and carbon-enhanced metal-poor stars, respectively. The reaction rate over the relevant temperature range from 0.1 to 0.3 GK translates into a center-of-mass energy range of 150 to 540 keV. Current measurements extend down to 300 keV, still requiring an extrapolation of the cross section. At these low energies, the high-energy tail of a $1/2^{+}$ state near the reaction threshold makes a significant contribution to the cross section, but its amplitude is still highly uncertain. In this paper the uncertainties associated with the low-energy cross-section extrapolation are investigated, in particular the sensitivity to the energy, width, and Coulomb renormalized asymptotic normalization coefficient of the near-threshold resonance. Recently it has been suggested that the energy of the near-threshold level may have a large impact on the extrapolation, but this is not found to be the case compared with the other sources of uncertainty.

Figure 1

Comparison of $R$-matrix evaluations for the ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ [5] and ${}^{13}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reactions [6]. The arrow represents the energy region of interest for the $s$ and $i$ processes.

Figure 2

Astrophysical $S$ factor calculated by using $C$ from Avila et al. [33] and the $R$-matrix level parameters from Leal et al. [5] (red solid line). The calculation is compared with the data of Drotleff et al. [8], Heil et al. [9], and Harissopulos et al. [10].

Figure 3

Same as Fig. 2, but the uncertainty reflects the uncertainty in the level energy when ${\tilde{\gamma }}_{\alpha }$ is held constant.

Figure 4

Same as Fig. 2, but the calculations now use the values for $C$ calculated from ${\tilde{C}}^2$ from Keeley et al. [29]. See text for details.

Figure 5

Same as Fig. 2, but the calculations now reflect the ${\tilde{C}}^2$ and their uncertainties from La Cognata et al. [31] and Mezhevych et al. [42].

Figure 6

Same as Fig. 2, but the calculations now reflect the ${\tilde{C}}^2$ and their uncertainties from Pellegriti et al. [43] and Guo et al. [44].

Figure 7

Variation in the extrapolation of the low-energy $S$ factor from different widths of the near-threshold state previously reported in the literature (solid lines) [5, 14, 30, 31, 32]. The $S$ factor calculated with ${\tilde{C}}^2$ from Avila et al. [33] and ${\mathrm{\Gamma }}_n$ from Leal et al. [5] is used as the reference calculation (red line). The data shown are the same as those in Fig. 2.

Figure 8

$S$ factors (solid lines) and their corresponding range of uncertainty (bounded by the dashed lines) for the ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ reaction using the ${\tilde{C}}^2$ of La Cognata et al. [31] (blue) and Avila et al. [33] (red). The $S$ factor with no near-threshold state contribution is shown by the gray dashed-dotted line. The black arrow indicates the energy range of astrophysical interest.

Figure 9

Uncertainty in the rate stemming from the uncertainty and inconsistency between the values of ${\tilde{C}}^2$ for the near-threshold state reported by Avila et al. [33] and La Cognata et al. [31] as a function of temperature for the ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ reaction. The blue dotted-dashed line represents the discrepancy on the scale of the evaluated cross section between the recent evaluations [49, 51].

Figure 10

Fraction of the reaction rate stemming from the high-energy tail of the near-threshold state as a function of temperature. The black circles indicate the calculation using the central value of the $S$ factor determined by using the ${\tilde{C}}^2$ of La Cognata et al. [31] while the blue squares correspond to that of Avila et al. [33].