Stars grow by accreting gas that has an evolving composition owing to the growth and inward drift of dust (pebble wave), the formation of planetesimals and planets, and the selective removal of hydrogen and helium by disk winds. We investigated how the formation of the Solar System may have affected the composition and structure of the Sun, and whether it plays any role in solving the solar abundance problem. We simulated the evolution of the Sun from the protostellar phase to the present age and attempted to reproduce spectroscopic and helioseismic constraints. We performed chi-squared tests to optimize our input parameters. We confirmed that, for realistic models, planet formation occurs when the solar convective zone is still massive; thus, the overall changes due to planet formation are too small to significantly improve the chi-square fits. We found that solar models with up-to-date abundances require an opacity increase of 12% to 18% centered at $T=10^{6.4}$ K to reproduce the available observational constraints. This is slightly higher than, but is qualitatively in good agreement with, recent measurements of higher iron opacities. These models result in better fits to the observations than those using old abundances; therefore, they are a promising solution to the solar abundance problem. Using these improved models, we found that planet formation processes leave a small imprint in the solar core, whose metallicity is enhanced by up to 5%. This result can be tested by accurately measuring the solar neutrino flux. In the improved models, the protosolar molecular cloud core is characterized by a primordial metallicity in the range 0.0127-0.0157 and a helium mass fraction in the range 0.268-0.274. (abridged)
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