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

TL;DR

This work addresses the challenge of constraining the low-energy $^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion $S$ factor, essential for stellar carbon burning, by developing a Bayesian smooth-fit framework. It models $y(E)=\log_{10}S^{*}(E)$ as a quadratic in energy and coherently combines direct, THM, and inverse-kinematics data within a global posterior over three coefficients. From this posterior, it yields a tightly bounded extrapolation, e.g., $S^{*}(1.5\mathrm{ MeV})=(2.13^{+0.01}_{-0.01})\times 10^{16}$ keV b, with a $68\%$ credible interval, and LO/MED/HI bands over $1.5$–$3.5$ MeV. The extrapolated result is consistent with recent inverse-kinematics and Coulomb-renormalized THM constraints, providing a transparent, reproducible basis for stellar-rate calculations and a framework that can be extended with future data or resonance-level modeling.

Abstract

A Bayesian analysis of the astrophysical $S$ factor for the $^{12}\mathrm{C}+^{12}\mathrm{C}$ fusion reaction is presented, based on available experimental information at carbon--carbon relative energies $E \gtrsim 2~\mathrm{MeV}$, including direct measurements, indirect Coulomb-renormalized Trojan Horse Method (THM) results, and recent inverse-kinematics data. The Bayesian inference is performed on the quantity $\log_{10}S^{*}(E)$ rather than on $S^{*}(E)$ itself, which naturally accommodates the wide dynamic range of the data and leads to approximately Gaussian uncertainties. The logarithm of the astrophysical factor is parametrized by a quadratic polynomial in energy, and the posterior distribution of the fit coefficients is determined using a weighted Bayesian regression. From this posterior, a global median $S^{*}(E)$ curve is constructed, and the associated covariance matrix is used to define a low/medium/high (LO/MED/HI) band corresponding to a $68\%$ credible interval. Particular emphasis is placed on the extrapolation below $E_{\mathrm{cm}}=2~\mathrm{MeV}$, where the fusion reaction rate is most relevant for stellar carbon burning. At $E_{\mathrm{cm}}=1.5~\mathrm{MeV}$, the posterior distribution yields $S_{\mathrm{global}}^{*}(1.5~\mathrm{MeV})= \left(2.13^{+0.01}_{-0.01}\right)\times10^{16}\,\mathrm{keV\,b}, $ corresponding to a $68\%$ credible interval. The extracted result is consistent with recent inverse-kinematics measurements and with Coulomb-corrected Trojan Horse Method constraints, providing a tightly constrained estimate of the $^{12}\mathrm{C}+^{12}\mathrm{C}$ fusion $S$ factor in the energy region relevant for stellar carbon burning.

Figures (2)

• Figure 1: Global posterior distribution of $\log_{10}S^{*}(E)$ obtained from the combined analysis of all experimental data. The black shaded band represents the $68\%$ credible interval derived from the propagated uncertainties of the polynomial coefficients. The median curve corresponding to the maximum-a-posteriori coefficients $\bm{a}_{\rm MAP}$ lies within this band and is visually indistinguishable from its boundaries because of the narrow width of the posterior uncertainty.
• Figure 2: Global posterior for $S^{*}(E)$ in linear space, obtained by transforming the log-space result via $S^{*}(E)=10^{\,y(E)}$. The black shaded band shows the $68\%$ uncertainty band, which is multiplicative rather than additive in linear space. As in Fig. \ref{['figpost_logS']}, the median curve lies within the band and may be difficult to distinguish when the uncertainty is small.