Completing the uncertainty-aware assessment of updated proton–proton reaction rates: uncertainty-weighted compilation intercomparison and stellar-evolution phases beyond central hydrogen burning

The p(p,e⁺ν)²H fusion reaction is the bottleneck of the proton–proton (pp) chain and a key input of stellar-evolution and solar models. Tognelli et al. (2015) quantified the impact of adopting different p-p reaction-rate evaluations on cluster ages and standard solar models, finding maximum rate differences of 3–4%, sub-1% effects on cluster ages, and up to ~8% (worst case) on solar neutrino fluxes; they concluded that, with the chiral-effective-field-theory evaluation of Marcucci, Schiavilla and Viviani (2013) carrying a few-permil uncertainty, the p-p rate is no longer a significant uncertainty source. Two parts of that assessment were explicitly left incomplete: the comparison with the JINA rate compilation could not be made uncertainty-aware because JINA documents no uncertainty for its p-p rate, and evolutionary phases beyond central hydrogen exhaustion were not examined. This problem asks to close both gaps: establish (document or defensibly reconstruct) uncertainty information for compilation p-p rates — in particular the JINA REACLIB entry — and perform the resulting uncertainty-aware intercomparison of the widely used p-p rate compilations over the stellar temperature range; and extend the differential stellar-evolution assessment of p-p rate choices to post-main-sequence phases, to determine whether the impact found on the main sequence persists or to quantify residual effects.

Background

In main-sequence stars like the Sun, hydrogen burning proceeds mainly through the pp chain, whose first and slowest step is the weak fusion of two protons into deuterium, p(p,e⁺ν)²H. Because this reaction cannot be measured at stellar energies, its cross section is purely theoretical. It is customary to quote it as the astrophysical S-factor, $S(E)$, defined by dividing the cross section by the Coulomb-barrier (Gamow) penetration factor, so that $S(E)$ varies slowly with the center-of-mass energy $E$. The thermonuclear reaction rate $N_A\langle\sigma v\rangle(T)$ (per amu of stellar material) is then obtained by Maxwellian averaging of the cross section over the stellar temperature $T$; for the pp bottleneck the relevant regime is roughly $5\times10^6$–$3\times10^7$ K. Stellar-evolution codes take this rate from tabulated compilations, e.g. NACRE (Angulo et al. 1999), the Solar Fusion II review (Adelberger et al. 2011, which recommended $S_{11}(0)=4.01(1\pm0.009)\times10^{-25}$ MeV b), and the JINA REACLIB database (Cyburt et al. 2010), a widely used library of fitted rate coefficients that does not publish per-rate uncertainties. In 2013 Marcucci, Schiavilla and Viviani recomputed $S(E)$ for pp fusion within chiral effective field theory (an expansion in powers of a typical low momentum over the chiral symmetry-breaking scale, with systematically improvable nuclear forces and weak currents), including for the first time the P-wave contributions of the initial p-p state; they obtained $S_{11}(0)=(4.030\pm0.006)\times10^{-25}$ MeV b with an estimated uncertainty of a few per mil (referred to as MSV13 in Tognelli et al. 2015). A 2019 erratum correcting one P-wave reduced matrix element revised the central value to $S_{11}(0)=4.008(1\pm0.005)\times10^{-25}$ MeV b, but the 2015 stellar implications were computed with the original value. Tognelli, Degl'Innocenti, Marcucci and Prada Moroni (2015) then redid the stellar implications in a strictly differential way — recomputing stellar models with each rate while keeping all other input physics fixed — for the age of 1–12 Gyr stellar clusters (via the luminosity of the turn-off point, where central hydrogen is exhausted) and for standard solar models and their neutrino fluxes. They found that the compilation rates (NACRE, Solar Fusion II, JINA) differ from MSV13 by at most about 3–4% in rate, shifting cluster ages by less than 1% and solar neutrino fluxes by at most ~8% in the worst case, and concluded that the pp rate uncertainty no longer matters for stellar modelling. Two comparisons, however, were left explicitly incomplete. First, the differences between the JINA and MSV13 rates (about 2–3%) could not be judged for significance, because no uncertainty for the JINA p-p rate is documented on the JINA web page, so no uncertainty-weighted intercomparison of the compilations was performed. Second, the analysis stopped at central hydrogen exhaustion: post-main-sequence phases (subgiant and red-giant stages, where hydrogen burns in a shell chiefly through the CNO cycle — the carbon–nitrogen–oxygen catalytic burning chain — which then dominates over the pp chain) were excluded on the physical expectation, not yet tested by explicit computation, that the p-p contribution becomes negligible there, so the implications of the updated p-p rate for those later phases were never computed or quantified.

Problem Statement

Complete the uncertainty-aware assessment of the updated proton–proton reaction rate that Tognelli et al. (2015) left unfinished, in two jointly supporting parts with the common objective of closing the uncertainty budget of the p(p,e⁺ν)²H rate in stellar evolution. (1) Rate-compilation intercomparison: establish defensible uncertainty information for the p-p rate entries of the widely used compilations — in particular the JINA REACLIB entry, whose rate is currently published without documented uncertainty — whether by tracing its fitted coefficients back to the underlying published S-factor evaluation and propagating that evaluation's uncertainty, by documenting the database maintainers' assessment, or by an equivalent justified reconstruction; then perform the full uncertainty-aware intercomparison of the available p-p rate evaluations (JINA REACLIB, NACRE, Solar Fusion II, the modern effective-field-theory evaluations, and the current Solar Fusion III recommendation) over the stellar temperature range relevant to hydrogen burning, stating which differences are statistically significant given modern uncertainty estimates. (2) Evolutionary-phase extension: extend the differential stellar-evolution assessment — recomputing stellar models with each p-p rate while all other input physics is held fixed — beyond central hydrogen exhaustion, through the subgiant and shell-hydrogen-burning (red giant) phases for the stellar models covered by the 2015 assessment (cluster turn-off models across its 1–12 Gyr age range and the standard solar model), to determine whether the main-sequence impact found in 2015 persists or to quantify residual effects on post-main-sequence model properties and observables (e.g. subgiant-branch and red-giant-branch track positions, evolutionary timescales, and isochrone features).

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