[Paper Review] Constraints on proton-proton fusion from helioseismology

Kevin I. T. Brown, M. N. Butler+1 more|ArXiv.org|Jul 3, 2002

Solar and Space Plasma Dynamics2 references3 citations

TL;DR

This paper uses helioseismic observations of solar p-mode frequencies to constrain the unknown two-nucleon effect parameter $ L_{1,A} $ in proton-proton fusion via effective field theory. By comparing a fully evolved Standard Solar Model with BiSON observational data, it determines $ L_{1,A} = 7.0 \pm 5.4 \, \text{fm}^3 $, consistent with theoretical estimates and confirming the model's accuracy in reproducing solar interior structure and $^8$B neutrino flux.

ABSTRACT

The proton-proton ($pp$) fusion cross-section found at the heart of solar models is unconstrained experimentally and relies solely on theoretical calculations. Effective field theory provides an opportunity to constrain the $pp$ cross-section experimentally, however, this method is complicated by the appearance of two-nucleon effects in the form of an unknown parameter $L_{1,A}$. We present a method to constrain $L_{1,A}$ using the Standard Solar Model and helioseismology. Using this method, we determine a value of $L_{1,A}$ = 7.0 fm$^{3}$ with a range of 1.6 to 12.4 fm$^{3}$. These results are consistent with theoretical estimates of $L_{1,A} \approx$ 6 fm$^{3}$.

Motivation & Objective

• To constrain the unknown two-nucleon effect parameter $ L_{1,A} $ in the proton-proton fusion cross-section, which is not accessible via direct laboratory experiments.
• To use helioseismic data—specifically low-$ l $ p-mode frequencies and small spacing frequencies—from the BiSON network to test the accuracy of the Standard Solar Model (SSM) in reproducing the Sun's internal structure.
• To determine the range of $ L_{1,A} $ values that yield SSM-predicted oscillation frequencies within the observational uncertainties of the Sun.
• To validate the consistency of the SSM with observed $^8$B neutrino fluxes and assess the reliability of theoretical nuclear physics inputs in solar modeling.

Proposed method

• Employ a fully developed stellar evolution code to compute a reference Standard Solar Model (SSM) with $ L_{1,A} = 7.0 \, \text{fm}^3 $, matching observed p-mode frequencies to within 0.3%.
• Systematically vary $ L_{1,A} $ across a range of values (from 1.6 to 12.4 fm³) to generate alternative SSMs and compute their corresponding p-mode frequency spectra.
• Compare the computed small spacing frequencies $ \delta\nu(n,l) $ from each model to observed BiSON data in the 2500–3000 $\mu$Hz frequency range, where the model–observation agreement is best.
• Use a linear fit between $ \delta\nu(n,l) $ differences and $ L_{1,A} $ values to determine the upper and lower bounds of $ L_{1,A} $ that remain within the observed error bars of ±0.066 $\mu$Hz.
• Apply a statistical approach to quantify the uncertainty in $ L_{1,A} $, resulting in the final range $ 1.6 \leq L_{1,A} \leq 12.4 \, \text{fm}^3 $, centered at 7.0 fm³.
• Validate the model by comparing the predicted total $^8$B neutrino flux of $ 4.93 \times 10^6 \, \text{cm}^{-2}\text{s}^{-1} $ with the SNO observation of $ (5.09^{+0.44}_{-0.43}) \times 10^6 \, \text{cm}^{-2}\text{s}^{-1} $.

Experimental results

Research questions

• RQ1 What is the range of values for the unknown two-nucleon parameter $ L_{1,A} $ that are consistent with helioseismic observations of solar p-mode frequencies?
• RQ2 How well does the Standard Solar Model, with varying $ L_{1,A} $, reproduce the observed small spacing frequencies from the BiSON network?
• RQ3 To what extent does the helioseismic data constrain the theoretical uncertainty in the proton-proton fusion cross-section due to two-nucleon effects?
• RQ4 Is the value of $ L_{1,A} $ derived from helioseismology consistent with independent theoretical estimates based on dimensional analysis or other nuclear physics models?
• RQ5 Does the SSM with the constrained $ L_{1,A} $ value produce a $^8$B neutrino flux consistent with the SNO measurement?

Key findings

• The value of the unknown two-nucleon parameter $ L_{1,A} $ is constrained to $ 7.0 \pm 5.4 \, \text{fm}^3 $, with a 95% confidence interval of 1.6 to 12.4 fm³, based on helioseismic data from the BiSON network.
• The reference SSM with $ L_{1,A} = 7.0 \, \text{fm}^3 $ shows excellent agreement with observed p-mode frequencies, with differences within 0.3% and small spacing frequencies lying within observational error bars.
• The linear relationship between $ \delta\nu(n,l) $ differences and $ L_{1,A} $ values allows precise determination of the upper and lower bounds of the parameter range.
• The theoretical total $^8$B neutrino flux of $ 4.93 \times 10^6 \, \text{cm}^{-2}\text{s}^{-1} $ is in excellent agreement with the SNO-measured flux of $ (5.09^{+0.44}_{-0.43}) \times 10^6 \, \text{cm}^{-2}\text{s}^{-1} $, supporting the model's consistency.
• The derived $ L_{1,A} $ value is consistent with theoretical estimates of $ \approx 6 \, \text{fm}^3 $ from dimensional analysis and with other evaluations such as 5.6 ± 2.0 fm³ and 6.5 ± 2.4 fm³ from independent studies.
• The study demonstrates that helioseismology provides a powerful, model-independent method to constrain nuclear physics parameters in the solar core, particularly those not accessible through terrestrial experiments.