Charged-particle thermonuclear reaction rates: I. Monte Carlo method and statistical distributions

Abstract

A method based on Monte Carlo techniques is presented for evaluating thermonuclear reaction rates. We begin by reviewing commonly applied procedures and point out that reaction rates that have been reported up to now in the literature have no rigorous statistical meaning. Subsequently, we associate each nuclear physics quantity entering in the calculation of reaction rates with a specific probability density function, including Gaussian, lognormal and chi-squared distributions. Based on these probability density functions the total reaction rate is randomly sampled many times until the required statistical precision is achieved. This procedure results in a median (Monte Carlo) rate which agrees under certain conditions with the commonly reported recommended "classical" rate. In addition, we present at each temperature a low rate and a high rate, corresponding to the 0.16 and 0.84 quantiles of the cumulative reaction rate distribution. These quantities are in general different from the statistically meaningless "minimum" (or "lower limit") and "maximum" (or "upper limit") reaction rates which are commonly reported. Furthermore, we approximate the output reaction rate probability density function by a lognormal distribution and present, at each temperature, the lognormal parameters µ and σ. The values of these quantities will be crucial for future Monte Carlo nucleosynthesis studies. Our new reaction rates, appropriate for bare nuclei in the laboratory, are tabulated in the second paper of this series (Paper II). The nuclear physics input used to derive our reaction rates is presented in the third paper of this series (Paper III). In the fourth paper of this series (Paper IV) we compare our new reaction rates to previous results.

Nuclear astrophysics in the laboratory and in the universe

“…c A first step in this regard was the recent extraction of mean reduced widths from high-resolution data measured at TUNL for target nuclei in the A = 28−40 (α-particles) and A = 34−67 (protons) mass ranges. 54 For example, a mean value of θ 2 α = 0.018, averaged over target nuclei, spin-parities, and excitation energies, was obtained for α-particles, almost a factor of two larger than the preliminary value suggested in Longland et al 48 An example for the relevance of these results is given in Fig. 10.…”

“…31,[48][49][50] The method is conceptually straightforward and follows a Monte Carlo approach. First, all of the measured nuclear physics (input) properties entering into the reaction rate calculation are randomly sampled according to their individual probability density functions.…”

“…The first determines the location of the distribution, while the second controls the width. An exhaustive account of this method can be found in Longland et al 48 As an example, we show in Fig. 9 the Monte Carlo based rates of the 22 Ne(α,n) 25 Mg reaction, which is a key neutron source for the astrophysical s-process that occurs both in AGB stars and massive helium burning stars.…”

“…Since each nuclear physics input parameter is sampled according to a physically motivated probability density function, 48 the Monte Carlo sampling provides a statistically rigorous coverage probability for each contribution. As an example, consider Fig.…”

New measurements of low-energy resonances in theNe22(p,γ)Na23reaction

“…21. The distribution resulting from the Monte Carlo post-processing calculations using the starlib rates is shown in blue and the distribution 23 Na reaction calculated using the code ratesmc [22,64] assuming the present rates with the direct-capture cross section of Ref. [53] (bottom panel) to that assuming the starlib rates (top panel).…”