Bayesian Smooth-Fit Extrapolation of the $^{12}\mathrm{C}+{}^{12}\mathrm{C}$ Astrophysical $S$ Factor

Abstract

A Bayesian analysis of the astrophysical \(S\) factor for the \(^{12}{\rm C}+{}^{12}{\rm C}\) fusion reaction is presented using available low-energy information at carbon--carbon relative energies \(E <3.5~{\rm MeV}\), including direct measurements and recent inverse-kinematics data. The goal of the global Bayesian fit is not to reproduce the local resonance-by-resonance structure of the \(^{12}{\rm C}+{}^{12}{\rm C}\) system, but rather to constrain the smooth global component of \(S^{*}(E)\) using the direct and inverse-kinematics constraints. The resulting posterior \(S^{*}(E)\) lies systematically below the traditional FCZ75 reference normalization over the energy interval considered. For example, at \(E=1.5~{\rm MeV}\) the median posterior value is \(S^{*}(1.5~{\rm MeV})=1.20\times10^{16}~{\rm keV\,b}\), whereas the Fowler--Caughlan--Zimmerman reference value is \(S^{*}_{\rm FCZ75}=3.0\times10^{16}~{\rm keV\,b}\) [W.~A. Fowler, G.~R. Caughlan, and B.~A. Zimmerman, Annu. Rev. Astron. Astrophys. \textbf{13}, 69 (1975)]. Thus, the updated posterior favors a lower smooth low-energy trend and does not support a sharp increase of \(S^{*}(E)\) as the energy decreases. The astrophysical consequence of the analysis is assessed through the thermonuclear reaction rate \(N_A\langle\sigma v\rangle\). The resulting median rate is lower than the CF88 analytic rate [G.~R. Caughlan and W.~A. Fowler, At. Data Nucl. Data Tables \textbf{40}, 283 (1988)] over the temperature interval considered, with \(R_{\rm present}/R_{\rm CF88}\simeq0.33\)--\(0.46\) for \(0.3\leq T_9\leq1.2\). At higher temperatures, where the Gamow window extends above the upper end of the adopted integration interval, the calculated rate should be regarded as a truncated rate rather than a complete stellar rate.