Solar measurement of proton-proton chain nuclear reaction rates under propagated solar-model systematics

The ignition reaction of the proton-proton (pp) chain, p+p -> d+e+nu, cannot be measured in the laboratory at solar energies; its astrophysical S-factor $S_{pp}(0)$ is known only at the about-1% level from theory, an uncertainty that is generally left unpropagated in the ages of low-mass stars. Bellinger & Christensen-Dalsgaard (2022) showed that, conditional on the standard solar model being otherwise correct, present helioseismic and solar-neutrino data could in principle improve $S_{pp}(0)$ by a factor of 7 (to 0.14%), with gains of about 6 for the 3He-3He rate and 5 for the 7Be-p rate - but they held fixed the radiative opacities, the equation of state, and the heavy-element composition, reported a 4.1-sigma tension in the 8B neutrino flux, and found a strong pp / 7Be-p posterior degeneracy. The open problem is whether a genuine solar measurement of the pp-chain S-factors survives when the solar-model systematics whose uncertainties the source left unpropagated - radiative opacities, the equation of state, and the heavy-element composition - together with the fixed element-diffusion treatment and electron-screening enhancement highlighted by later literature, in the presence of the unresolved base-of-convection-zone discrepancy, are treated as nuisance parameters and propagated.

Background

Nuclear energy generation in the Sun proceeds mainly through the proton-proton (pp) chain of fusion reactions, ignited by the reaction p+p -> deuterium + positron + electron neutrino (the pp reaction). Because this is a weak-interaction process with an exceedingly small cross section at solar energies (a few keV), it cannot be measured directly in the laboratory; its rate must be computed from theory. Fusion rates for charged particles are conventionally encoded as the astrophysical S-factor, $S(E)=sigma(E) E exp(2 pi eta)$, where $sigma(E)$ is the fusion cross section and the Gamow factor $exp(2 pi eta)$ removes the dominant exponential Coulomb-barrier energy dependence; the zero-energy limit $S(0)$ is the standard quoted datum. The recommended compilation of Adelberger et al. (2011) gives $S_{pp}(0)$ with an uncertainty of about 1%, which propagates into a systematic error of similar order in the theoretical main-sequence turn-off ages of low-mass stars - an uncertainty usually not propagated at all in stellar-chronology applications. A standard solar model (SSM) is a one-dimensional stellar-evolution model calibrated to reproduce the present solar mass, radius, luminosity, age, and photospheric metal-to-hydrogen ratio, with microphysics - equation of state (EOS), radiative opacities, nuclear network, and element diffusion (the gravitational settling of helium and heavier elements) - taken from tables and approximations. Two classes of observation probe its deep interior. First, helioseismology: the Sun oscillates in millions of global acoustic (pressure) modes, and for low-degree modes the frequency separation ratios $r_{02}$ and $r_{13}$ (combinations of mode-frequency differences constructed to cancel the poorly modelled near-surface layers) are sensitive probes of the core. Second, solar neutrinos, produced directly by the fusion reactions, with the fluxes of the pp, pep, 7Be, 8B, and CNO reactions measured by detectors after correction for neutrino flavour-conversion (oscillation) probabilities. Bellinger & Christensen-Dalsgaard (2022) quantified how precisely present solar data could measure the pp-chain rates. Under the explicit assumption that the SSM is otherwise correct - adopting the classical GS98 composition (the abundance scale in closest agreement with helioseismic inference), fixed EOS and opacity tables, and a fixed diffusion treatment - and combining Birmingham Solar Oscillations Network (BiSON) separation ratios with the neutrino-flux determination of Bergström et al. (2016) in a Markov-chain Monte Carlo (MCMC) analysis, they found that current data could in principle reduce the uncertainty on $S_{pp}(0)$ by a factor of 7 (to 0.14%), on the 3He-3He S-factor $S_{33}(0)$ by a factor of 6, and on the 7Be-p S-factor $S_{17}(0)$ by a factor of 5. The inference is limited by a strong posterior correlation (coefficient 0.66) between the pp and 7Be-p rates: raising the pp rate lowers the central temperature required to sustain the observed luminosity, which must be compensated by a higher 7Be-p rate to hold the observed 8B flux. They noted that future measurements of solar gravity modes (g modes - buoyancy-restored oscillations that probe the energy-generating core and are not yet detected at confirmation level) could further improve the constraints. The promise is, however, conditional on exactly the ingredients that remain unsettled. The analysis did not vary uncertainties in radiative opacities, the EOS, or the heavy-element mixture (the last only partially through a metallicity prior), and it worked with a reference model whose predicted 8B flux exceeded the measured flux by 4.1 sigma. Meanwhile the largest known discrepancy between the Sun and SSMs sits at the base of the convection zone and is entangled with the decades-old solar abundance (or solar modeling) problem: three-dimensional photospheric spectroscopy lowered the C, N, O abundances and destroyed the earlier agreement with helioseismology; the revised scale of Magg et al. (2022) restores agreement in some seismic indicators but is contested by Buldgen et al. (2023); and radiative-opacity calculations remain in tension with Z-pinch and laser opacity measurements, as reviewed by Pain (2024). An additional unpropagated systematic is the electron-screening enhancement of fusion rates in a plasma, whose laboratory-versus-stellar magnitude is still debated (Däppen 2024). This motivates asking how much solar data actually determine the pp-chain rates once these model systematics are treated as unknowns rather than assumptions.

Problem Statement

Determine what present-day solar observations - low-degree oscillation frequency separation ratios and precision solar-neutrino fluxes (pp, pep, 7Be, 8B, CNO) - actually constrain about the proton-proton-chain and CNO-cycle astrophysical S-factors, above all the laboratory-unmeasurable ignition rate $S_{pp}(0)$ and jointly $S_{33}(0)$ (3He+3He), $S_{34}(0)$ (3He+4He), $S_{17}(0)$ (7Be+p), and $S_{114}(0)$ (14N+p), when the solar-model systematics whose uncertainties Bellinger & Christensen-Dalsgaard (2022) left unpropagated - radiative opacities, the equation of state, and the heavy-element mixture - are instead treated as nuisance parameters and marginalized over. Concretely, the answer should: (i) build the inference with uncertainty parameterizations for radiative opacities, the equation of state, and the heavy-element mixture (spanning the classical and revised three-dimensional-spectroscopy abundance scales), with ranges justified against current opacity calculations and experiments; (ii) report the marginalized posterior uncertainties on, and correlations among, the S-factors - including the pp / 7Be-p degeneracy - and quantify how the conditional projections (a factor of about 7 on $S_{pp}(0)$, to 0.14%; about 6 on $S_{33}(0)$; about 5 on $S_{17}(0)$; about 2 on $S_{114}(0)$) degrade or collapse, with the 8B and other flux comparisons treated consistently with current measurements and neutrino-oscillation-parameter uncertainties; and (iii) state which additional observations (a confirmed solar g-mode detection and its period spacing, improved high-frequency p-mode precision, or next-generation neutrino-flux measurements) could enable a sub-percent solar determination. The fixed element-diffusion treatment and the electron-screening enhancement, which later literature highlights as further unpropagated systematics, are natural additional nuisance directions beyond the source's own caveat list. The problem is complete when either a solar measurement of $S_{pp}(0)$ (and its companion rates) is delivered whose quoted uncertainties are posterior uncertainties marginalized over the clause-(i) nuisance-parameter space rather than conditional on fixed solar-model microphysics, or it is quantitatively demonstrated - over the same stated nuisance-parameter space - that solar data cannot reduce the uncertainties on these rates below the adopted prior determinations of Adelberger et al. (2011) (about 1% on $S_{pp}(0)$) while the solar modeling problem remains unresolved.

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