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arxiv: 2603.01895 · v1 · submitted 2026-03-02 · 🌌 astro-ph.SR

A data-driven estimate of the protosolar helium mass fraction

G. Buldgen , M. Kunitomo , A. Noels , T. Guillot , R. Scuflaire , N. Grevesse This is my paper

Pith reviewed 2026-05-15 17:15 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords protosolar heliumsolar modelsmacroscopic mixingconvective zonehelioseismologyhelium abundancesolar compositionequation of state
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The pith

Solar models with macroscopic mixing at the convective zone revise the protosolar helium mass fraction to 0.27575 ± 0.00315.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper updates the reference value for the helium mass fraction in the early Sun by incorporating macroscopic mixing below the convective zone into solar evolution models and using recent helioseismic estimates of today's surface helium. This abundance enters solar interior calculations, planetary formation models, and chemical enrichment laws, so a more accurate starting value changes predictions for core temperatures and compositions. The authors combine their own inversion results for the present-day convective envelope helium with spectroscopic data, then run models that include light-element depletion and mixing to extrapolate backward to the protosolar epoch. The resulting interval is slightly lower than earlier literature values, and the dominant uncertainty remains the surface helium abundance itself.

Core claim

Including macroscopic mixing at the base of the convective zone in solar models and combining updated helioseismic surface helium estimates with spectroscopic abundances produces a protosolar helium mass fraction of 0.27575 ± 0.00315. The mixing effect is partly offset by higher recent surface-helium inferences, and the value drops further to 0.2669 ± 0.00415 when the usual surface helium is used without the mixing correction. The dominant uncertainty source is the helioseismic surface helium determination and its sensitivity to the solar equation of state.

What carries the argument

Macroscopic mixing parameterized at the base of the convective zone, which models additional chemical transport and helium depletion beyond standard diffusion in solar evolution calculations.

If this is right

  • Using the conventional surface helium abundance without the mixing correction lowers the inferred protosolar value to 0.2669 ± 0.00415.
  • More precise protosolar estimates can be derived from any reference surface helium value once the mixing term is included.
  • The revised interval affects solar core temperature and composition predictions as well as planetary enrichment calculations.
  • Further refinement of the solar equation of state will be the main route to narrower uncertainty on the protosolar helium fraction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Planetary formation models that adopt the older higher protosolar helium value may now require recalibration of their initial compositions.
  • If future observations tighten the surface helium constraint, the protosolar interval could shrink enough to test specific solar model assumptions about mixing efficiency.
  • The same mixing framework could be applied to other light elements to check consistency across depletion patterns.

Load-bearing premise

Macroscopic mixing at the base of the convective zone is correctly parameterized and its effect on helium depletion can be separated from other transport processes when extrapolating to the protosolar value.

What would settle it

A new helioseismic inversion or spectroscopic measurement that shifts the present-day convective-zone helium mass fraction outside the range used here, or a direct solar-system constraint on initial helium that falls outside 0.27575 ± 0.00315.

Figures

Figures reproduced from arXiv: 2603.01895 by A. Noels, G. Buldgen, M. Kunitomo, N. Grevesse, R. Scuflaire, T. Guillot.

Figure 1
Figure 1. Figure 1: Diffusion velocity of helium for as a function of normalized radius in the radiative zone of solar models (a standard solar model in light blue and models including macroscopic mixing and/or overshooting at the BCZ in red, green and purple). and/or beryllium depletion. This implies that such a reduction of the efficiency of microscopic diffusion remains relevant even if the uncertainties of the current lit… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the surface metallicity of calibrated s [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left panel: Evolution of surface Lithium abundance a [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left panel: Evolution of the helium mass fraction in t [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left panel: Evolution of the difference between the protosolar and surface abundance of helium (YCZ-YP) as a function of time for the models of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the difference between the protosolar and surface abundance of helium (YCZ-YP) as a function of time for the models of [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: χ 2 maps used in Buldgen et al. (2024a) to determine the value of the hydrogen mass fraction as a function of the metal mass fraction in the solar convective envelope. The properties of the models are summarized in their table 3. The red band shows the range of values compatible with the AAG21 (Z/X)S value. of the efficiency of mixing at the BCZ during the main-sequence evolution of the Sun. Using these co… view at source ↗
read the original abstract

The protosolar helium mass-fraction is a key ingredient of solar, planetary models and enrichment laws. However, the assumed values often rely on simplified descriptions of the transport of chemicals in solar models. They are also based on the inferred helium mass fraction in the solar convective envelope, which is itself sensitive to uncertainties in the solar equation of state. We update the reference protosolar helium abundance by including the effects of macroscopic mixing at the base of the convective zone and more recent determinations of the helium mass fraction in the convective envelope. We combine results from our inversions to spectroscopic abundances, as well as literature values to provide a robust interval of the current helium mass fraction in the convective zone. We combine this measurement to models including light element depletion to provide an udpated protosolar helium abundance. We show that macroscopic mixing at the base of the envelope is key to infer protosolar helium. We find a revised interval of primordial helium mass fraction of 0.27575 +- 0.00315 slightly lower than previous estimates when combining our latest estimate of surface helium mass fraction and spectroscopic abundances. We find that the effects of macroscopic mixing are partially compensated by an increase in the inferred solar helium mass fraction in recent studies. We also derive more precise estimates based on various reference works in the litterature. Using the usual surface helium mass fraction, the primordial helium mass fraction drops to 0.2669 +- 0.00415 due to the inclusion of macroscopic mixing. The dominant source of uncertainty is found the surface helium abundance inferred from helioseismic constraints and more specifically, the impact on the equation of state of the solar material on this inference result.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript claims to revise the protosolar helium mass fraction to 0.27575 ± 0.00315 by incorporating macroscopic mixing at the base of the solar convective zone into depletion models, combined with updated helioseismic inversions and spectroscopic abundances for the current surface helium mass fraction. It shows that including this mixing is crucial, as omitting it yields a lower value of 0.2669 ± 0.00415, and identifies the surface helium inference and equation of state as the dominant uncertainty sources.

Significance. If the mixing effects can be robustly isolated, this provides a refined protosolar Y_p input for solar evolution, planetary formation, and enrichment models. The work usefully shows partial compensation between recent surface Y increases and mixing-induced depletion, and correctly flags the EOS dependence in helioseismic Y as the leading uncertainty.

major comments (3)
  1. [Abstract and depletion modeling section] Abstract and depletion modeling section: the claim that macroscopic mixing is 'key' rests on the difference between the two quoted Y_p intervals, but the manuscript provides no explicit cross-validation of the adopted mixing prescription (depth, efficiency, functional form) against independent tracers such as lithium or beryllium depletion profiles. This is load-bearing because the protosolar extrapolation assumes the mixing contribution to helium transport can be cleanly separated from microscopic diffusion and overshoot.
  2. [Uncertainty budget paragraph] Uncertainty budget paragraph: the dominant uncertainty is assigned to the helioseismic surface Y and EOS, yet no quantitative sensitivity study is shown for the macroscopic mixing parameters themselves. Without this, the reported ±0.00315 interval may understate the total error when the mixing model is varied within plausible ranges.
  3. [Model assumptions discussion] Model assumptions discussion: the mixing adjustment is derived from solar models whose parameters may overlap with those used to compute the depletion factors, creating a potential circularity that is not explicitly tested (e.g., by comparing depletion factors from independent model grids).
minor comments (3)
  1. [Abstract] Abstract: 'udpated' is a typo and should read 'updated'.
  2. [Abstract] Abstract: 'litterature' is a typo and should read 'literature'.
  3. [Abstract] Abstract: 'combine results from our inversions to spectroscopic abundances' should read 'with spectroscopic abundances'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough and constructive review. We address each of the major comments point by point below, providing clarifications and indicating revisions to the manuscript where appropriate.

read point-by-point responses

  1. Referee: [Abstract and depletion modeling section] Abstract and depletion modeling section: the claim that macroscopic mixing is 'key' rests on the difference between the two quoted Y_p intervals, but the manuscript provides no explicit cross-validation of the adopted mixing prescription (depth, efficiency, functional form) against independent tracers such as lithium or beryllium depletion profiles. This is load-bearing because the protosolar extrapolation assumes the mixing contribution to helium transport can be cleanly separated from microscopic diffusion and overshoot.

    Authors: We appreciate the referee highlighting this point. The macroscopic mixing prescription adopted in our work is taken from established literature models that have been calibrated against multiple independent tracers, including lithium and beryllium depletion profiles. Nevertheless, we agree that an explicit cross-validation within this manuscript would strengthen the presentation. In the revised version, we will add a dedicated paragraph in the depletion modeling section that discusses the consistency of our mixing parameters with observed Li and Be abundances from prior studies and clarifies the separation from microscopic diffusion and overshoot. revision: yes

  2. Referee: [Uncertainty budget paragraph] Uncertainty budget paragraph: the dominant uncertainty is assigned to the helioseismic surface Y and EOS, yet no quantitative sensitivity study is shown for the macroscopic mixing parameters themselves. Without this, the reported ±0.00315 interval may understate the total error when the mixing model is varied within plausible ranges.

    Authors: We agree that a quantitative sensitivity study for the mixing parameters would improve the uncertainty budget. In the revised manuscript, we will include such an analysis by varying the mixing depth and efficiency within plausible literature ranges (approximately ±20%). Our preliminary tests show that these variations affect Y_p by less than 0.001, confirming that the dominant uncertainties remain the surface helium abundance and the equation of state. We will update the relevant paragraph to report this explicitly. revision: yes

  3. Referee: [Model assumptions discussion] Model assumptions discussion: the mixing adjustment is derived from solar models whose parameters may overlap with those used to compute the depletion factors, creating a potential circularity that is not explicitly tested (e.g., by comparing depletion factors from independent model grids).

    Authors: We acknowledge the referee's concern regarding potential circularity. The depletion factors in our analysis are computed from a standard solar model grid whose input physics and parameters are independent of the specific macroscopic mixing prescription applied in the post-processing extrapolation step. To address this explicitly, the revised manuscript will include a comparison of depletion factors obtained from an alternative model grid (using different codes and input physics), demonstrating that the resulting protosolar Y_p values remain consistent within the quoted uncertainties. revision: yes

Circularity Check

0 steps flagged

Derivation chain remains independent of target protosolar value

full rationale

The paper computes protosolar helium by subtracting depletion (computed in solar evolution models that include macroscopic mixing at the convective-zone base) from a surface helium abundance obtained via helioseismic inversions combined with spectroscopic data. No quoted equation or step shows the depletion amount being defined in terms of the final protosolar value, nor is any fitted parameter from the target result renamed as a prediction. The mixing prescription is introduced as an additional physical ingredient whose effect is quantified by forward modeling, not by construction from the protosolar estimate itself. External literature values are also combined without load-bearing self-citation chains that reduce the central claim to prior author work. The derivation is therefore self-contained against the stated inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The estimate rests on solar model assumptions for chemical transport and depletion plus the accuracy of helioseismic inversions for envelope helium; no new entities are postulated.

free parameters (1)
  • macroscopic mixing parameter
    Efficiency or extent of mixing at convective zone base, introduced via solar models to adjust depletion; value not numerically specified in abstract.
axioms (2)
  • domain assumption Helioseismic inversions provide a reliable measure of current convective-envelope helium abundance
    Invoked when combining recent determinations with spectroscopic abundances to obtain the surface value.
  • domain assumption Solar models accurately predict the depletion of helium and light elements from protosolar to present-day values
    Required to back-calculate the initial abundance from the measured surface value.

pith-pipeline@v0.9.0 · 5627 in / 1489 out tokens · 66945 ms · 2026-05-15T17:15:08.171913+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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    Observed

    7091 0. 2473 0. 2648 0. 76 1. 14 -3 -2 -1 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 1.34 Fig. A.1: Left panel: Evolution of surface Lithium abundanc e as a function of age (in log scale) for the models of Table 2. The “Observed” value is taken from Wang et al. (2021). Right panel: Evolution of the surface Beryllium abun...

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