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REVIEW 3 major objections 6 minor 48 references

Six mixed f/g modes can invert solar-core rotation at 0.07 and 0.2 solar radii and separate solid-body from fast-core angular-momentum scenarios.

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2026-07-11 12:50 UTC pith:N4QYJGAG

load-bearing objection Clean feasibility study: six mixed f/g modes plus modified MOLA kernels can localise solar-core rotation at two radii and separate the two leading transport scenarios, provided the modes are eventually measured. the 3 major comments →

arxiv 2607.04830 v1 pith:N4QYJGAG submitted 2026-07-06 astro-ph.SR

On the feasibility of inverting the rotation of the solar core with mixed f/g modes

Armand Leclerc , Arthur Le Saux , J. M. Joel Ong , Rafael A. Garcia This is my paper
classification astro-ph.SR
keywords helioseismologysolar core rotationmixed f/g modesangular momentum transportOLA inversionrotational kernelssolar oscillations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Helioseismology already maps solar rotation down to about 0.2 solar radii, but the core itself remains inaccessible because ordinary p-modes do not reach it and pure g-modes are too weak at the surface to detect. This paper shows that a small set of mixed f/g modes—gravity waves trapped in the radiative interior that couple to surface f-modes—simultaneously feel the core’s rotation and still have enough surface amplitude to be observed. Using a solar model and a modified optimally-localised-average inversion that incorporates their mixing fractions, the authors construct two independent averaging kernels that peak near 0.07 and 0.2 solar radii. With the six usable mixed modes (angular degrees 3–8), the expected measurement uncertainty is small enough to distinguish a solid-body core from a core spinning several times faster, thereby deciding between two leading theories of angular-momentum transport. Even a single high-frequency detection at ℓ=7 would already be decisive if measured to roughly 25 nHz precision.

Core claim

The Sun’s oscillation spectrum contains six mixed f/g modes (ℓ = 3–8) whose rotational kernels, when combined with existing p-mode kernels in a modified MOLA inversion, produce localised averages of the rotation rate at r ≈ 0.07 R⊙ and r ≈ 0.2 R⊙. The resulting uncertainties are small enough to discriminate the solid-body core predicted by one angular-momentum-transport model from the rapidly rotating core predicted by the competing model.

What carries the argument

Modified method of optimally localised averages (MOLA) that folds the g-mode mixing fraction ζ of each mixed mode into the inversion coefficients, allowing the construction of two independent radial averaging kernels that separately sample the deep core and the base of the radiative zone.

Load-bearing premise

That the mixed f/g modes will actually be detectable at the surface with frequency precision of tens of nanohertz despite the strong convective noise that rises at low frequencies.

What would settle it

A multi-year, multi-wavelength search that either recovers the predicted ℓ=7 multiplet near 305 µHz with a splitting measured to ~25 nHz (or better) or definitively rules it out at that precision.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Detection of the six mixed modes would yield two independent core-rotation measurements and immediately decide between solid-body and fast-core transport models.
  • Even a single high-frequency mixed-mode splitting at the 25 nHz level would already separate the two competing scenarios.
  • The same mixed-mode kernels can be used to test any future theoretical rotation profile that differs in the inner 0.2 R⊙.
  • Multi-wavelength chromospheric observations that suppress granulation noise become a high-priority route to core seismology.

Where Pith is reading between the lines

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

  • If the modes are found, the same technique could be applied to solar-like stars with mixed f/g or p/g modes, giving a new observational handle on core rotation beyond the Sun.
  • A null result after decades of multi-wavelength data would force a re-examination either of the predicted coupling strengths or of the assumed surface amplitudes of the mixed modes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper examines whether mixed f/g modes—arising from coupling of surface f modes with core-trapped g modes—can be used to invert the solar core rotation profile without detecting pure g modes. Using a published MESA solar model and GYRE eigenfunctions, the authors identify candidate mixed modes for ℓ=3–8 via period-spacing dips, resonance widths Γ, mixing fractions ζ, core sensitivities s, and inertias Enorm (Table 1, Figs. 1–2). They construct modified MOLA averaging kernels that combine these modes with existing p-mode kernels, obtaining localised sensitivity near r≈0.07 R⊙ and r≈0.2 R⊙ (Fig. 3, Appendix B). Simulated inversions of Fuller (solid-body) and Eggenberger (fast-core) profiles show that the two kernels can discriminate the scenarios under deliberately inflated (10×) splitting uncertainties. A single high-frequency mode (ℓ=7, ~305 µHz) is also argued to suffice for a coarse discrimination. Detectability challenges from granulation noise are acknowledged.

Significance. If the mixed f/g modes exist at the predicted frequencies and can be measured at tens-of-nHz precision, the work would open a practical path to the solar core rotation rate—one of the main open problems in helioseismology and a key discriminant between competing angular-momentum transport theories. Strengths include use of standard public codes (MESA, GYRE), a transparent multi-mode census (Table 1), explicit validation of the φ/γ decomposition (Appendix A), a documented modification of MOLA that incorporates mixing fractions (Appendix B), and falsifiable predictions (mode list, target radii, and a quantitative uncertainty threshold for distinguishing ~440 nHz vs ≳1000 nHz). The study is framed as a feasibility analysis rather than a detection claim, which is appropriate.

major comments (3)
  1. [Section 2, Table 1, Fig. 2] All eigenfrequencies, ζ values, Γ widths, and kernels are derived from a single calibrated MESA solar model (Le Saux et al. 2025). Resonance widths Γ and the number of usable mixed modes (especially the borderline ℓ=7–8 cases in Table 1 and Fig. 2) depend on the core Brunt–Väisälä profile and the precise f-mode frequencies. A short robustness check—or at least a quantitative discussion—against modest structural variations (e.g. different core N, surface-term prescriptions, or an independent standard solar model) is needed to support the claim that “6 mixed f/g modes” are generically available for inversion.
  2. [Section 4] The single-mode discrimination argument assigns σ_Ωcore/2π ≈ 420 nHz from s=0.06 and a 25 nHz splitting error. Fuller et al. predict ~440 nHz while Eggenberger et al. predict ≳1000 nHz. A measurement of 440±420 nHz is statistically consistent with a wide range of intermediate profiles; only a high measured rate would cleanly rule out solid-body rotation. The text should state more carefully what measurement outcome would constitute a decisive discrimination (e.g. a one-sided test or required significance), rather than asserting that 420 nHz “would suffice.”
  3. [Abstract; Section 3; Table 1] Abstract and Results state that the spectrum “should present 6 mixed f/g modes,” while Section 5 and Table 1 discuss up to 12 modes for ℓ=3–8 (two most f-dominated modes per degree). Clarify which six modes enter the inversion kernels of Fig. 3 (e.g. only the more sensitive member of each pair, or only modes with s>0.2 and intermediate ζ), and make the abstract count consistent with the body.
minor comments (6)
  1. [Global] Throughout the manuscript, compound terms such as “mixedf/g”, “pmode”, “gmodes”, and “fmode” lack spaces or hyphens; standardise to “mixed f/g”, “p mode”, etc.
  2. [Abstract] Abstract: “r = 0.07 and 0.2R” is missing the solar symbol; write 0.2 R⊙ for consistency with the body.
  3. [Figure 3] Figure 3 caption and bottom panel: state explicitly which solid/dashed rotation curves correspond to Fuller vs Eggenberger, and what the filled vs open circles represent, so the panel is self-contained.
  4. [Section 2, Eq. (1)] Equation (1) for Γ and the definition of α should briefly point the reader to the exact expression used from Ong & Gehan (2023), since α is not redefined in the main text.
  5. [Sections 2–3] The sensitivity threshold s>0.2 and the 10× inflation of mixed-mode splitting errors are free analysis choices; a one-sentence justification or a brief sensitivity test (e.g. 5× vs 10×) would help readers assess how fragile the Fig. 3 localisation is.
  6. [Appendix B; Section 3] Appendix B: the weight function W(x)=(x−x0)² for MOLA is standard, but stating the numerical value of µ chosen for the 100 nHz target uncertainty would aid reproducibility.

Circularity Check

1 steps flagged

Minor self-citation to authors' prior mixed-mode prediction and solar model; inversion kernels, multi-mode census and discrimination argument are independent new calculations that do not reduce by construction.

specific steps
  1. self citation load bearing [Sect. 1 (Introduction) and Sect. 2 (model description)]
    "It is based on a recent prediction that g modes in the radiative interior couple with f modes in the outer parts of the star. ... The 1D model we use for this study is the calibrated solar model introduced in Le Saux et al. (2025)."

    Existence of mixed f/g modes and the underlying solar model are taken from overlapping-author prior work rather than re-derived from first principles here. However the citation is not load-bearing for the new multi-mode census, kernel construction or discrimination argument, which are independent numerical results; hence only minor circularity.

full rationale

The paper's central feasibility claim (six mixed f/g modes for ℓ=3–8 supply enough kernel information, via modified MOLA, to localise Ω at r≈0.07 R⊙ and 0.2 R⊙ and thereby discriminate Fuller solid-body from Eggenberger fast-core) is obtained by direct GYRE eigenmode computation on a MESA solar model, construction of rotational kernels, and numerical OLA averaging (Table 1, Fig. 3, Appendices A–B). These steps do not algebraically or statistically reduce to the inputs of Le Saux et al. (2025). That citation supplies only the calibrated 1-D model and the initial motivation that one mixed mode exists; the present work recomputes the full period-spacing pattern, resonance widths Γ, mixing fractions ζ, inertias and kernels, then builds new averaging kernels that include existing p-mode data. No free parameter is fitted to data and re-labelled a prediction, no uniqueness theorem is imported to forbid alternatives, and no ansatz is smuggled. The single soft self-citation is therefore non-load-bearing for the inversion result itself, warranting only a score of 2.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The claim rests on a calibrated 1-D solar model, linear first-order rotational perturbation theory, the φ/γ isolation procedure of Ong & Basu (2020), and an adversarial but arbitrary assignment of measurement errors. No new physical entities are postulated; free parameters are limited to the OLA trade-off and the error inflation factor.

free parameters (3)
  • OLA uncertainty trade-off µ = set for 100 nHz
    Chosen so that every kernel yields a formal 100 nHz uncertainty on Ω/2π; different µ would change localization versus error.
  • mixed-mode splitting error inflation = 10×
    Assigned 10× the lowest-frequency multiplet of the same degree to simulate detection difficulty; directly controls whether the single-mode ℓ=7 case remains decisive.
  • core-sensitivity threshold s>0.2 = 0.2
    Hand-chosen so that δΩc ≲ 50 nHz given BiSON-like 10 nHz frequency errors; modes below threshold are discarded.
axioms (4)
  • domain assumption First-order linear rotational splitting formula (Eq. 2–4) remains valid for mixed f/g modes whose resonance widths Γ are larger than the splitting itself.
    Invoked throughout Sect. 2–3; the paper notes that large Γ can invalidate the linear approximation but proceeds with it for ℓ≤8.
  • domain assumption The calibrated MESA solar model of Le Saux et al. (2025) is sufficiently close to the real Sun that mixed-mode frequencies, ζ and kernels are representative.
    All numerical results (Table 1, Figs. 1–3) are computed on this single model.
  • domain assumption φ/γ isolation of Ong & Basu (2020) correctly decomposes mixed f/g eigenfunctions and yields the pure-f frequency and coupling strength α.
    Appendix A; used to locate dips in period spacing and to compute ζ and Γ.
  • standard math Standard MOLA/SOLA averaging-kernel formalism (Backus & Gilbert 1968; Pijpers & Thompson 1994) applies after the ζ-weighted modification of Appendix B.
    Core of Sect. 3 and Appendix B.

pith-pipeline@v1.1.0-grok45 · 20189 in / 2870 out tokens · 27318 ms · 2026-07-11T12:50:03.582013+00:00 · methodology

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read the original abstract

Context: Thanks to helioseismology, the rotation profile of the Sun has been measured with great precision down to 20% of its total radius. This rotation profile is used as a calibration to infer the rotation of other stars as well as a test of angular momentum transport theory in stellar interiors. However, the deepest 20% of the layers remain out of reach of current observations, preventing astronomers to discriminate between currently competing angular momentum transport mechanisms. Aims: The main obstacle is that no global oscillations modes sensitive to rotation (non-zero degree l) reaching the solar core have been detected yet, as nonradial p modes cannot reach it and g modes are evanescent at the surface and still elude detection. In this work, we propose and examine a new method to constrain the rotation of the core of the Sun, which does not require direct observation of solar g modes. Methods: It is based on a recent prediction that g modes in the radiative interior couple with f modes in the outer parts of the star. These mixed f /g are at the same time sensitive to the rotation of the core and able to reach the surface. These modes can be used together to build average inversion kernels and perform an inversion of the rotation of the solar core. Results: We find that the oscillations' spectrum of the Sun should present 6 mixed f /g modes that can be used to measure the rotation rate of the Sun at r = 0.07 and 0.2R. We estimate that the uncertainty on the measurements should be small enough to distinguish between competing scenarios of angular momentum transport in the Sun.

Figures

Figures reproduced from arXiv: 2607.04830 by Armand Leclerc, Arthur Le Saux, J. M. Joel Ong, Rafael A. Garcia.

Figure 1
Figure 1. Figure 1: Consecutive period differences ∆P as a function of frequency ν for solar oscillations modes with angular degrees between ℓ = 1 and 10. The frequency of the φ mode is identified by the vertical black lines. related to the integral over the internal solar rotation rate Ω(r, θ) weighted by the rotational kernel (Howe 2009) as ∆ωℓ,m,n = Z Kℓ,m,n(r, θ)Ω(r, θ)drdθ, (3) with r and θ the radial and latitudinal coo… view at source ↗
Figure 2
Figure 2. Figure 2: as a function of ℓ. The value of Γ decreases with ℓ, and it appears that for ℓ ≥ 9, the resonance width is smaller than the fre￾quency splittings and surface term corrections, which also sug￾gests that these modes are unlikely to be mixed in the Sun. This is because, for a given coupling strength, only g modes within Γ of an f mode will couple significantly to it, and vice versa. The resonance width determ… view at source ↗
Figure 3
Figure 3. Figure 3: Supplementing p mode rotational kernels with those of a se￾lect few f /g modes qualitatively improves our capability to measure near-core rotation rates. Top: Centroids ⟨r/R⊙⟩ of MOLA rotational kernels (Eq. (8), solid curves), plotted as a function of target locations x0 = r0/R⊙. Shaded regions indicate the spread σ of the rotational ker￾nels, per Eq. (9). Orange curves and shaded areas show quantities ca… view at source ↗

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