Pith. sign in

REVIEW 3 major objections 5 minor 78 references

Linearity of Structure Kernels in Main-sequence and Subgiant Solar-like Oscillators

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Linear structure inversions work on the main sequence but break down for subgiant mixed modes.

desk verdict A useful, mostly well-executed warning about subgiant structure inversions that needs broader test coverage and full parameter reporting before its negative conclusion is taken as final. read the letter →

arxiv 2507.03083 v1 pith:HK2ROMHV submitted 2025-07-03 astro-ph.SR

Lynn Buchele , Earl P. Bellinger , Saskia Hekker , Sarbani Basu This is my paper · ORCID
classification astro-ph.SR
keywords asteroseismologyseismicstructureinversionsmixedmodessubgiantstarssolar-likeoscillationskernellinearitySOLAinversionstellarkernels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tests a hidden assumption in asteroseismic structure inversions: that the mode kernels computed from a reference model remain accurate representations of the modes' true sensitivity when the star differs from the model. Using grids of MESA models and GYRE frequencies, the authors compare model pairs on the main sequence and on the subgiant branch, first measuring per-mode 'kernel errors' and then propagating those errors through full OLA inversions. On the main sequence, kernel errors are small across mass, age, and composition, and the few larger errors roughly cancel when modes are combined, so the inversions recover known structure differences. On the subgiant branch, the rapid evolution of mixed acoustic-gravity modes through avoided crossings makes the kernels of a few g-dominated modes strongly wrong, and because those modes also receive large inversion coefficients, the inversions return erroneous structure even for test models within observational uncertainties. If this is right, current linear structure inversions of subgiant stars are not reliable probes of core structure, and subgiant inversions will need new fitting and inversion techniques.

What carries the argument

The load-bearing objects are the structure kernels $K_i^{(\hat u,Y)}$ and $K_i^{(Y,\hat u)}$, which map relative frequency differences $\delta\hat u/\hat u$ and $\delta Y$ between star and model. The paper defines a per-mode kernel error that isolates how much of the frequency difference is not captured by the reference model's kernels to first order, and a cumulative error $\varepsilon(j)=\sum_{i\le j} c_i\,\mathrm{KE}_i$ that tracks how those errors enter the inversion through the SOLA or MOLA coefficients $c_i$. The main-sequence success and subgiant failure are explained by the behavior of $\varepsilon$: on the main sequence it oscillates around zero and ends below the propagated uncertainty, while on the subgiant branch a few mixed modes contribute most of the error and push $\varepsilon(N)$ above the uncertainty. The $\lambda=1$ singularity in the $(\hat u,Y)$ kernel transformation, found earlier by Bellinger et al. (2021), is shown to shrink the linearity region drastically when the reference model lies near it.

What would settle it

Take a subgiant model pair where the reference model is away from the lambda=1 singularity and the test model lies within 1 sigma in mass, large frequency separation, and metallicity, then run the linear SOLA inversion and compare the recovered sound-speed difference with the known difference at several target radii; the paper's claim predicts failure in most such cases, so a systematic success across a grid of test models would force the conclusion to be narrowed to the specific hand-picked configurations.

Watch

Extended reading notes

Core claim

The paper's central claim is that the linearity of structure kernels, which underpins localized inversions, holds broadly for main-sequence solar-like oscillators but fails for subgiant oscillators exhibiting mixed modes. The authors define the kernel error as the difference between the true relative frequency difference and the first-order kernel prediction for a given mode, and show that on the subgiant branch these errors become large precisely for the g-dominated mixed modes that make subgiants attractive targets. In test inversions, two of three subgiant cases fail to recover the known structure difference, while all main-sequence cases, including a model with a convective core, succeed. The authors attribute the subgiant failure to a few modes with both high kernel errors and high inversion coefficients, and note that the marginal success in the third subgiant case was helped by an accidentally small coefficient on its one high-error mode.

Load-bearing premise

The subgiant verdict depends on the handful of hand-selected test models and hand-chosen SOLA trade-off parameters being representative of actual inversion practice, since a different reference model, target-kernel width, or error-suppression choice could change how kernel errors propagate.

Editorial extensions

If this is right

  • Published and future subgiant structure inversions that rely on current linear OLA techniques should include kernel nonlinearity, not just frequency uncertainties, in their error budgets.
  • Main-sequence structure inversions remain viable across the tested mass range of 1.05 to 1.15 solar masses, large frequency separations, and composition differences, including a 1.35 solar mass convective-core case.
  • Mixed modes cannot simply be added to the p-mode set in a structure inversion, because their character changes too quickly with age for a single reference model to represent them linearly.
  • Removing the offending high-kernel-error modes could in principle restore the subgiant inversion, but observations do not currently tell which modes to remove, since frequency residuals correlate only weakly with kernel errors.
  • Reliable subgiant inversions will require forward modeling that matches mixed-mode character explicitly and inversion techniques that go beyond the linear variational approach, such as non-linear iterative schemes or higher-order terms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the same kernel-error test could be applied to other rapidly evolving mode sets, such as red-giant mixed modes or higher-mass early subgiants, where avoided crossings occur even faster than in the cases studied here.
  • Editorial extension: the main-sequence error suppression suggests a practical diagnostic: compute the cumulative kernel-error curve during any inversion and flag cases where a single mode dominates the total error.
  • Editorial extension: if the subgiant failure is driven by a few high-kernel-error modes, an inversion that deliberately down-weights g-dominated modes should recover the outer structure while sacrificing exactly the core information that made subgiants attractive targets.
  • Editorial extension: with next-generation data, the limiting factor for subgiant structure inversions will shift from frequency precision to kernel linearity, so model-development effort should go into nonlinear or iteratively perturbed inversion schemes rather than simply adding more modes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper tests whether linear structure inversion techniques developed for pure acoustic modes of the Sun and main-sequence stars remain valid for mixed acoustic-buoyancy modes in subgiant stars. The authors define a "kernel error" (Eq. 6) as the difference between the true dimensionless frequency difference and the first-order kernel integral, compute these errors over two MESA/GYRE model grids (main-sequence and early subgiant), and perform model-model OLA inversions against known structure differences. They find that main-sequence inversions are reliable across a broad range of masses, ages, and compositions, including for a convective-core test case, while two of three subgiant test inversions fail. The failures are attributed to large kernel errors in a few mixed dipole modes whose inversion coefficients are large; Appendix A shows that cumulative kernel errors cancel on the main sequence but not on the subgiant branch. The paper concludes that reliable subgiant structure inversions with current methods will require improvements to forward modeling and to the inversion techniques themselves.

Significance. If the conclusion holds, the paper materially challenges the use of linear structure inversions with mixed modes for subgiant stars, with direct implications for published subgiant inversion results and for planned PLATO-era analyses. The kernel-error definition (Eq. 6) is direct and falsifiable, and the Appendix A cumulative-error mechanism is a useful diagnostic for understanding why main-sequence errors cancel while subgiant errors do not. The grids cover sensible ranges in mass and large frequency separation, and the main-sequence tests include useful robustness checks against composition changes, abundance-scale changes, and convective cores. The main limitation is that the subgiant branch conclusion rests on only three hand-selected test models and one manually chosen set of SOLA trade-off parameters, with no code or data shipped; these gaps leave the quantitative robustness of the central claim unverified.

major comments (3)
  1. [Section 2 (SOLA parameter choice) and Section 4.2 (Table 2)] The central subgiant conclusion depends on a single hand-chosen set of SOLA trade-off parameters ("we choose by hand values of the trade-off parameters which result in nicely localized averaging kernels"), and the resulting inversion coefficients enter the cumulative kernel error directly through Eq. (A1). Because the successful 'Match g-dominated Dipole Modes' case works only because a high-kernel-error mode receives an accidentally small coefficient, it is possible that other reasonable trade-off choices, or different target-kernel widths, would suppress the offending modes and make the two failing cases succeed. The paper should report the actual SOLA parameters used and demonstrate robustness by scanning the trade-off parameters and target width over a plausible range, showing that the failures persist for at least a representative subset. Without this, the abstract's claim that current inversion techniques are unreliable for subgiants is not established.
  2. [Section 4.2 (test-model selection)] The three subgiant test models were chosen because their frequency agreement with the reference model is "qualitatively similar" to best-fit models in Noll et al. (2021), Bellinger et al. (2021), and Lindsay et al. (2024), but no quantitative metric (e.g., RMS frequency residuals normalized by observational uncertainties, or a reduced chi-squared against the adopted µ Her mode set) is provided. Two of the three hand-picked models fail, and one succeeds only under an "extremely optimistic" fit. To support the claim that kernel errors make subgiant inversions unreliable in practice, the paper should quantify the frequency residuals of the test models against the adopted µ Her mode set and, ideally, test with reference models that are genuine best fits from the cited studies. Otherwise the failure could be an artifact of unusually poor mixed-mode agreement in the chosen test models.
  3. [Sections 3.2 and 4.2 (inversion methods)] The main-sequence tests use MOLA while the subgiant tests use SOLA, so the comparison across evolutionary stage conflates the inversion method with the underlying kernel behavior. This matters because the key mechanism invoked in Appendix A (failure of cumulative-error cancellation) depends on the inversion coefficients, which are method-dependent. The authors should either run both methods on both evolutionary stages or explicitly argue that the MOLA/SOLA distinction cannot explain the subgiant failures. As written, a reader cannot separate the hypothesis "mixed modes are intrinsically nonlinear" from the alternative that "the chosen SOLA implementation with hand-tuned parameters fails on these test models."
minor comments (5)
  1. [Section 2 (Eq. 6)] The term "kernel error" is defined as a residual between the true frequency difference and the first-order kernel integral; since the kernels themselves are not claimed to be in error, a name such as "first-order kernel residual" would avoid confusion.
  2. [Section 4.2 (mode set)] The observed mode set and uncertainties for µ Her are attributed to "Kjelsen et al., in prep."; for reproducibility, please provide the mode-set table or a stable reference with details on the membership and uncertainties of the included modes.
  3. [Tables 1 and 2] The SOLA and MOLA trade-off parameter values are never reported; please include them in or near the model tables, since the inversion coefficients and hence the Appendix A conclusions depend on them.
  4. [Table 2] In the "Different Z" subgiant test model, only [Fe/H] is listed; since Y is one of the inversion variables, the actual changes in Y and Z should be given explicitly.
  5. [Figure 5 (Eq. 9)] The acoustic mode inertia Ep/E is displayed as the color scale, but the text does not state the normalization of Eq. (9) beyond the integration domains; please clarify whether the denominator is the total mode inertia.

Circularity Check

0 steps flagged · score 1.0 of 10

Kernel errors are defined as residual misfit and inversion tests use known model differences; no fitted parameter or self-citation forces the subgiant conclusion.

full rationale

The paper's central measure, the kernel error of Equation 6, is by definition the difference between a true frequency difference and its first-order kernel integral (δν/ν minus the kernel integral). That is a residual diagnostic for nonlinearity, not a fitted quantity. The subsequent subgiant test inversions compare the inversion output against the known structure difference δu/u of the test model, so the success or failure of the inversion is an externally grounded outcome rather than an input. No parameter is fitted to the data whose value then reappears as the prediction, and the main-sequence versus subgiant contrast is obtained from the same diagnostic applied to two grids. The authors' self-citation to Bellinger et al. (2021) for the (u,Y)-kernel singularity is not load-bearing here because the paper re-derives the consequence by computing kernel errors with a reference model passing through that singularity (Appendix C.2, Figure 12), giving an independent demonstration. The hand-selected SOLA trade-off parameters and the choice of test models to resemble literature fits are legitimate concerns for robustness and external validity, but they are not circularity in the derivation chain. The paper is self-contained against model-model benchmarks and its conclusions follow from the reported computations; overall circularity is minimal.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on first-order linear perturbation theory with reference-model kernels, plus representative grid and mode-set choices. The paper introduces no new physical entities or fitted constants; the main unquantified free choice is the hand-selected inversion trade-off parameters, which could affect the subgiant inversion failures.

free parameters (1)
  • MOLA/SOLA trade-off parameters and target kernel width = not reported; chosen by hand
    Section 2 states 'we choose by hand values of the trade-off parameters which result in nicely localized averaging kernels.' These choices affect which modes receive high coefficients and therefore how kernel errors propagate in the OLA test inversions.
assumptions (4)
  • domain assumption Adiabatic variational principle and first-order perturbation relation (Eq. 1) accurately describe mode frequency differences.
    Section 2, Eq. 1. This is the linearity assumption the paper tests; the definition of kernel error uses it as the baseline.
  • domain assumption Change of variables to dimensionless u = P/rho is valid to first order.
    Section 3.1: the paper notes du/u = dP/P - drho/rho 'is only true to first-order', so second-order effects in the variable transformation contribute to kernel errors.
  • domain assumption Reference-model kernels represent observed mode sensitivities to first order.
    Section 2: 'The key assumption of a structure inversion is that the mode kernels, calculated using a reference model, accurately represent the sensitivity of the observed modes to first-order.'
  • domain assumption The MESA model grids with fixed composition and input physics are representative of main-sequence and subgiant stars observed by Kepler and SONG.
    Section 2: grid covers 1.05-1.15 solar masses, Yinit=0.28, Zinit=0.02, Grevesse and Sauval abundances. If real stars differ, for example in mixing or mass range, kernel-error magnitudes could change.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Linearity of Structure Kernels in Main-sequence and Subgiant Solar-like Oscillators." pith.science (2026). https://pith.science/paper/HK2ROMHV

@misc{pith2026250703083,
  author       = {Pith},
  title        = {Pith review of: Linearity of Structure Kernels in Main-sequence and Subgiant Solar-like Oscillators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HK2ROMHV}},
  note         = {Machine review of arXiv:2507.03083}
}
read the original abstract

Seismic structure inversions have been used to study the solar interior for decades. With the high-precision frequencies obtained using data from the Kepler mission, it has now become possible to study other solar-like oscillators using structure inversions, including both main-sequence and subgiant stars. Subgiant stars are particularly interesting because they exhibit modes of mixed acoustic-buoyancy nature, which provide the opportunity to probe the deeper region of stellar cores. This work examines whether the structure inversion techniques developed for the pure acoustic modes of the Sun and other main-sequence stars are still valid for mixed modes observed in subgiant stars. We construct two grids of models: one of main-sequence stars and one of early subgiant stars. Using these grids, we examine two different parts of the inversion procedure. First, we examine what we call the "kernel errors", which measure how well the mode sensitivity functions can recover known frequency differences between two models. Second, we test how these kernel errors affect the ability of an inversion to infer known structure differences. On the main sequence, we find that reliable structure inversion results can be obtained across the entire range of masses and large frequency separations we consider. On the subgiant branch, however, the rapid evolution of mixed modes leads to large kernel errors and hence difficulty recovering known structure differences. Our tests show that using mixed modes to infer the structure of subgiant stars reliably will require improvements to current fitting approaches and modifications to the structure inversion techniques.

Figures

Figures reproduced from arXiv: 2507.03083 by the authors.

Figure 1
Figure 1. Hertzsprung–Russell diagram showing the evolutionary tracks used in this work. For reference, we provide the masses of three tracks marked with darker gray lines. The colored points indicate the values of ∆ν for models used to calculate the kernel errors, with each color map corresponding to a different evolutionary stage. The reference model used for each evolutionary stage is indicated with a white star. This allo… view at source ↗
Figure 2
Figure 2. Contour plots of the kernel errors, scaled by a representative uncertainty (σ = 10−4 ), of a main sequence star for the modes with frequencies nearest νmax. The x-axis, which shows ∆ν as a proxy for stellar age, is reversed so that a model of a given mass evolves horizontally from left to right through the plot. The point in the center of the plot shows the location of the reference model used and the error bars rep… view at source ↗
Figure 3
Figure 3. Results of several model-model test inversions. In all cases the same reference model and inversion parameters were used, and only the test model was varied. The properties of each test model are given in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Contour plots of kernel errors for several radial modes of our subgiant grid. The reference model is indicated with a black point and the error bars correspond to the uncertainty of mass and ∆ν of µHer given in Grundahl et al. (2017). We also show, in the bottom row, t…
Figure 5
Figure 5. Figure 5: Left column: same as the top row of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Results of several model-model test inversions for the subgiant case. In all cases the same reference model and inversion parameters were used, and only the test model was varied. The properties of each test model are given in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Cumulative values of the kernel error, ε(j), defined in Equation A1, for our test inversions on the main sequence (left) and subgiant branch (right). The rightmost point in each panel represents ε(N), the total error in the inversion due to the underlying kernel errors…
Figure 8
Figure 8. Figure 8: Kernel errors for a main-sequence star with a convective core. All colors and symbols have the same meaning as in [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Results of a set of model-model test inversions for a model with a convective core. All symbols have the same meaning as in [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Kernel errors for the quadrupole modes of the subgiant stars. All colors and symbols have the same meaning as in [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Kernel errors for the octopole modes of the subgiant stars. All colors and symbols have the same meaning as in [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Kernel errors for several modes when the reference model used is passing through the singularity discussed in Bellinger et al. (2021). The symbols have the same meaning as in [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

78 extracted references · 22 canonical work pages

  1. [1]

    2021, Reviews of Modern Physics, 93, 015001, doi: 10.1103/RevModPhys.93.015001

    Aerts, C. 2021, Reviews of Modern Physics, 93, 015001, doi: 10.1103/RevModPhys.93.015001

    doi:10.1103/revmodphys.93.015001 2021

  2. [2]

    Angelou, G. C. 2020, A&A, 639, A98, doi: 10.1051/0004-6361/201936947

    work page doi:10.1051/0004-6361/201936947 2020

  3. [3]

    2022, A&A, 668, A98, doi: 10.1051/0004-6361/202142510 —

    Mokrytska, D. 2022, A&A, 668, A98, doi: 10.1051/0004-6361/202142510 —. 2025, A&A, 693, A274, doi: 10.1051/0004-6361/202449901

    doi:10.1051/0004-6361/202142510 2022

  4. [4]

    1977, A&A, 58, 41

    Aizenman, M., Smeyers, P., & Weigert, A. 1977, A&A, 58, 41

    work page 1977

  5. [5]

    J., & Scott, P

    Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481, doi: 10.1146/annurev.astro.46.060407.145222

    arXiv 2009

  6. [6]

    2009, A&A, 506, 411, doi: 10.1051/0004-6361/200810860

    Auvergne, M., Bodin, P., Boisnard, L., et al. 2009, A&A, 506, 411, doi: 10.1051/0004-6361/200810860

    doi:10.1051/0004-6361/200810860 2009

  7. [7]

    1968, Geophysical Journal, 16, 169, doi: 10.1111/j.1365-246X.1968.tb00216.x —

    Backus, G., & Gilbert, F. 1968, Geophysical Journal, 16, 169, doi: 10.1111/j.1365-246X.1968.tb00216.x —. 1970, Philosophical Transactions of the Royal Society of London Series A, 266, 123, doi: 10.1098/rsta.1970.0005

    arXiv 1968

  8. [8]

    2009, in IAU

    Baglin, A., Auvergne, M., Barge, P., et al. 2009, in IAU

    work page 2009

Show all 78 references
  1. [9]

    253, Transiting Planets, ed

    Symposium, Vol. 253, Transiting Planets, ed. F. Pont, D. Sasselov, & M. J. Holman, 71–81, doi: 10.1017/S1743921308026252

    doi

  2. [10]

    2003, Ap&SS, 284, 153, doi: 10.1023/A:1023223115165 —

    Basu, S. 2003, Ap&SS, 284, 153, doi: 10.1023/A:1023223115165 —. 2016, Living Reviews in Solar Physics, 13, 2, doi: 10.1007/s41116-016-0003-4

    2003 doi

  3. [11]

    Basu, S., & Chaplin, W. J. 2017, Asteroseismic Data Analysis: Foundations and Techniques

    2017

  4. [12]

    1997, A&A, 322, L5, doi: 10.48550/arXiv.astro-ph/9702162

    Basu, S., & Christensen-Dalsgaard, J. 1997, A&A, 322, L5, doi: 10.48550/arXiv.astro-ph/9702162

    1997 doi arXiv

  5. [13]

    Basu, S., Christensen-Dalsgaard, J., & Thompson, M. J. 2002, in ESA Special Publication, Vol. 485, Stellar Structure and Habitable Planet Finding, ed. B. Battrick, F. Favata, I. W. Roxburgh, & D. Galadi, 249–252, doi: 10.48550/arXiv.astro-ph/0110446

    2002 doi arXiv

  6. [14]

    H., & Bahcall, J

    Basu, S., Pinsonneault, M. H., & Bahcall, J. N. 2000, ApJ, 529, 1084, doi: 10.1086/308302

    2000 doi

  7. [15]

    R., & Kjeldsen, H

    Bedding, T. R., & Kjeldsen, H. 2007, Communications in Asteroseismology, 150, 106, doi: 10.1553/cia150s106

    2007 doi

  8. [16]

    R., Butler, R

    Bedding, T. R., Butler, R. P., Kjeldsen, H., et al. 2001, ApJL, 549, L105, doi: 10.1086/319139

    2001 doi

  9. [17]

    P., Basu, S., Hekker, S., & Ball, W

    Bellinger, E. P., Basu, S., Hekker, S., & Ball, W. H. 2017, ApJ, 851, 80, doi: 10.3847/1538-4357/aa9848

    2017 doi

  10. [18]

    2019, ApJ, 885, 143, doi: 10.3847/1538-4357/ab4a0d

    Christensen-Dalsgaard, J. 2019, ApJ, 885, 143, doi: 10.3847/1538-4357/ab4a0d

    2019 doi

  11. [19]

    Christensen-Dalsgaard, J., & Ball, W. H. 2021, ApJ, 915, 100, doi: 10.3847/1538-4357/ac0051

    2021 doi

  12. [20]

    Benomar, O., Takata, M., Shibahashi, H., Ceillier, T., & Garc ´ ıa, R. A. 2015, MNRAS, 452, 2654, doi: 10.1093/mnras/stv1493

    2015 doi

  13. [21]

    J., Koch, D., Basri, G., et al

    Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977, doi: 10.1126/science.1185402

    2010 doi

  14. [22]

    M., & Bugnet, L

    Bowman, D. M., & Bugnet, L. 2024, arXiv e-prints, arXiv:2410.01715, doi: 10.48550/arXiv.2410.01715

    2024 doi arXiv

  15. [23]

    M., Gilliland, R

    Brown, T. M., Gilliland, R. L., Noyes, R. W., & Ramsey, L. W. 1991, ApJ, 368, 599, doi: 10.1086/169725

    1991 doi

  16. [24]

    P., Hekker, S., & Basu, S

    Buchele, L., Bellinger, E. P., Hekker, S., & Basu, S. 2024a, arXiv e-prints, arXiv:2412.05094, doi: 10.48550/arXiv.2412.05094

    doi

  17. [25]

    P., Hekker, S., et al

    Buchele, L., Bellinger, E. P., Hekker, S., et al. 2024b, ApJ, 961, 198, doi: 10.3847/1538-4357/ad1680

    doi

  18. [26]

    2022, A&A, 661, A143, doi: 10.1051/0004-6361/202142001

    Buldgen, G., Farnir, M., Eggenberger, P., et al. 2022, A&A, 661, A143, doi: 10.1051/0004-6361/202142001

    2022 doi

  19. [27]

    R., & Dupret, M

    Buldgen, G., Reese, D. R., & Dupret, M. A. 2015a, A&A, 583, A62, doi: 10.1051/0004-6361/201526390 —. 2017, A&A, 598, A21, doi: 10.1051/0004-6361/201629485 —. 2018, A&A, 609, A95, doi: 10.1051/0004-6361/201730693

    2017 doi

  20. [28]

    R., Dupret, M

    Buldgen, G., Reese, D. R., Dupret, M. A., & Samadi, R. 2015b, A&A, 574, A42, doi: 10.1051/0004-6361/201424613

    doi

  21. [29]

    2002, Reviews of Modern Physics, 74, 1073, doi: 10.1103/RevModPhys.74.1073

    Christensen-Dalsgaard, J. 2002, Reviews of Modern Physics, 74, 1073, doi: 10.1103/RevModPhys.74.1073

    2002 doi

  22. [30]

    R., Silva Aguirre, V., Bedding, T

    Davies, G. R., Silva Aguirre, V., Bedding, T. R., et al. 2016, MNRAS, 456, 2183, doi: 10.1093/mnras/stv2593

    2016 doi

  23. [31]

    2011, A&A, 535, A91, doi: 10.1051/0004-6361/201117232

    Deheuvels, S., & Michel, E. 2011, A&A, 535, A91, doi: 10.1051/0004-6361/201117232

    2011 doi

  24. [32]

    J., et al

    Deheuvels, S., Do˘ gan, G., Goupil, M. J., et al. 2014, A&A, 564, A27, doi: 10.1051/0004-6361/201322779

    2014 doi

  25. [33]

    A., Pamyatnykh, A

    Dziembowski, W. A., Pamyatnykh, A. A., & Sienkiewicz, R. 1990, MNRAS, 244, 542

    1990

  26. [34]

    Elliott, J. R. 1996, Monthly Notices of the Royal Astronomical Society, 280, 1244, doi: 10.1093/mnras/280.4.1244

    1996 doi

  27. [35]

    2024, A&A, 686, A267, doi: 10.1051/0004-6361/202449430

    Farrell, E., Buldgen, G., Meynet, G., et al. 2024, A&A, 686, A267, doi: 10.1051/0004-6361/202449430

    2024 doi

  28. [36]

    R., Cochran, W

    Furlan, E., Ciardi, D. R., Cochran, W. D., et al. 2018, The Astrophysical Journal, 861, 149, doi: 10.3847/1538-4357/aaca34

    2018 doi

  29. [37]

    2018, Nature, 554, 73, doi: 10.1038/nature25136 20

    Giammichele, N., Charpinet, S., Fontaine, G., et al. 2018, Nature, 554, 73, doi: 10.1038/nature25136 20

    2018 doi

  30. [38]

    O., & Kosovichev, A

    Gough, D. O., & Kosovichev, A. G. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 40, IAU Colloq. 137: Inside the Stars, ed. W. W. Weiss & A. Baglin, 541

    1993

  31. [39]

    O., & Thompson, M

    Gough, D. O., & Thompson, M. J. 1991, in Solar Interior and Atmosphere, ed. A. N. Cox, W. C. Livingston, & M. S. Matthews, 519–561

    1991

  32. [40]

    Grevesse, N., & Sauval, A. J. 1998, SSRv, 85, 161, doi: 10.1023/A:1005161325181

    1998 doi

  33. [41]

    2007, Communications in Asteroseismology, 150, 300, doi: 10.1553/cia150s300

    Arentoft, T., & Frandsen, S. 2007, Communications in Asteroseismology, 150, 300, doi: 10.1553/cia150s300

    2007 doi

  34. [42]

    2017, ApJ, 836, 142, doi: 10.3847/1538-4357/836/1/142

    Christensen-Dalsgaard, J., et al. 2017, ApJ, 836, 142, doi: 10.3847/1538-4357/836/1/142

    2017 doi

  35. [43]

    Kuszlewicz, J. S. 2020, MNRAS, 499, 2445, doi: 10.1093/mnras/staa2853

    2020 doi

  36. [44]

    A., & Rogers, F

    Iglesias, C. A., & Rogers, F. J. 1993, ApJ, 412, 752, doi: 10.1086/172958 —. 1996, ApJ, 464, 943, doi: 10.1086/177381

    1993 doi

  37. [45]

    S., Bauer, E

    Jermyn, A. S., Bauer, E. B., Schwab, J., et al. 2023, ApJS, 265, 15, doi: 10.3847/1538-4365/acae8d

    2023 doi

  38. [46]

    R., Viskum, M., & Frandsen, S

    Kjeldsen, H., Bedding, T. R., Viskum, M., & Frandsen, S. 1995, AJ, 109, 1313, doi: 10.1086/117363

    1995 doi

  39. [47]

    G., Borucki, W

    Koch, D. G., Borucki, W. J., Basri, G., et al. 2010, ApJL, 713, L79, doi: 10.1088/2041-8205/713/2/L79

    2010 doi

  40. [48]

    Kosovichev, A. G. 1999, Journal of Computational and Applied Mathematics, 109, 1 —. 2011, in Lecture Notes in Physics, Berlin Springer Verlag, ed. J.-P. Rozelot & C. Neiner, Vol. 832, 3, doi: 10.1007/978-3-642-19928-8 1

    1999 doi

  41. [49]

    G., & Kitiashvili, I

    Kosovichev, A. G., & Kitiashvili, I. N. 2020, in IAU

    2020

  42. [50]

    354, Solar and Stellar Magnetic Fields: Origins and Manifestations, ed

    Symposium, Vol. 354, Solar and Stellar Magnetic Fields: Origins and Manifestations, ed. A. Kosovichev, S. Strassmeier, & M. Jardine, 107–115, doi: 10.1017/S1743921320001416

    doi

  43. [51]

    W., Saio, H., Takata, M., et al

    Kurtz, D. W., Saio, H., Takata, M., et al. 2014, MNRAS, 444, 102, doi: 10.1093/mnras/stu1329

    2014 doi

  44. [52]

    1958, Handbuch der Physik, 51, 353, doi: 10.1007/978-3-642-45908-5 6

    Ledoux, P., & Walraven, T. 1958, Handbuch der Physik, 51, 353, doi: 10.1007/978-3-642-45908-5 6

    1958 doi

  45. [53]

    2022, Nature, 610, 43, doi: 10.1038/s41586-022-05176-0

    Li, G., Deheuvels, S., Ballot, J., & Ligni` eres, F. 2022, Nature, 610, 43, doi: 10.1038/s41586-022-05176-0

    2022 doi

  46. [54]

    2023, A&A, 680, A26, doi: 10.1051/0004-6361/202347260

    Li, G., Deheuvels, S., Li, T., Ballot, J., & Ligni` eres, F. 2023, A&A, 680, A26, doi: 10.1051/0004-6361/202347260

    2023 doi

  47. [55]

    R., Christensen-Dalsgaard, J., et al

    Li, T., Bedding, T. R., Christensen-Dalsgaard, J., et al. 2020, MNRAS, 495, 3431, doi: 10.1093/mnras/staa1350

    2020 doi

  48. [56]

    J., Ong, J

    Lindsay, C. J., Ong, J. M. J., & Basu, S. 2024, ApJ, 965, 171, doi: 10.3847/1538-4357/ad2ae5

    2024 doi

  49. [57]

    N., Silva Aguirre, V., Davies, G

    Lund, M. N., Silva Aguirre, V., Davies, G. R., et al. 2017, ApJ, 835, 172, doi: 10.3847/1538-4357/835/2/172

    2017 doi

  50. [58]

    M., et al

    Mathur, S., Huber, D., Batalha, N. M., et al. 2017, The Astrophysical Journal Supplement Series, 229, 30, doi: 10.3847/1538-4365/229/2/30

    2017 doi

  51. [59]

    2021, A&A, 647, A187, doi: 10.1051/0004-6361/202040055

    Noll, A., Deheuvels, S., & Ballot, J. 2021, A&A, 647, A187, doi: 10.1051/0004-6361/202040055

    2021 doi

  52. [60]

    2011, ApJS, 192, 3, doi: 10.1088/0067-0049/192/1/3

    Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3, doi: 10.1088/0067-0049/192/1/3

    2011 doi

  53. [61]

    2013, ApJS, 208, 4, doi: 10.1088/0067-0049/208/1/4

    Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4, doi: 10.1088/0067-0049/208/1/4

    2013 doi

  54. [62]

    2015, ApJS, 220, 15, doi: 10.1088/0067-0049/220/1/15

    Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15, doi: 10.1088/0067-0049/220/1/15

    2015 doi

  55. [63]

    2019, ApJS, 243, 10, doi: 10.3847/1538-4365/ab2241

    Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10, doi: 10.3847/1538-4365/ab2241

    2019 doi

  56. [64]

    P., & Thompson, M

    Pijpers, F. P., & Thompson, M. J. 1992, A&A, 262, L33 —. 1994, A&A, 281, 231

    1992

  57. [65]

    C., Basu, S., & Christensen-Dalsgaard, J

    Rabello-Soares, M. C., Basu, S., & Christensen-Dalsgaard, J. 1999, MNRAS, 309, 35, doi: 10.1046/j.1365-8711.1999.02785.x

    1999

  58. [66]

    2014, Experimental Astronomy, 38, 249, doi: 10.1007/s10686-014-9383-4

    Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249, doi: 10.1007/s10686-014-9383-4

    2014 doi

  59. [67]

    2024, arXiv e-prints, arXiv:2406.05447, doi: 10.48550/arXiv.2406.05447

    Rauer, H., Aerts, C., Cabrera, J., et al. 2024, arXiv e-prints, arXiv:2406.05447, doi: 10.48550/arXiv.2406.05447

    2024 doi arXiv

  60. [68]

    R., Marques, J

    Reese, D. R., Marques, J. P., Goupil, M. J., Thompson, M. J., & Deheuvels, S. 2012, A&A, 539, A63, doi: 10.1051/0004-6361/201118156

    2012 doi

  61. [69]

    1998, in Proc

    Roxburgh, I., Audard, N., Basu, S., et al. 1998, in Proc. IAU Symp. 181: Sounding Solar and Stellar Interiors (poster vol.), ed. J. Provost & F. Schmider (Observatoire de la Cˆ ote d’Azur, Universit´ e de Nice), 245

    1998

  62. [70]

    Roxburgh, I. W. 2017, A&A, 604, A42, doi: 10.1051/0004-6361/201731057

    2017 doi

  63. [71]

    Seaton, M. J. 2005, MNRAS, 362, L1, doi: 10.1111/j.1365-2966.2005.00019.x

    2005

  64. [72]

    E., Silva Aguirre, V., et al

    Stokholm, A., Nissen, P. E., Silva Aguirre, V., et al. 2019, MNRAS, 489, 928, doi: 10.1093/mnras/stz2222

    2019 doi

  65. [73]

    2006, PASJ, 58, 893, doi: 10.1093/pasj/58.5.893

    Takata, M. 2006, PASJ, 58, 893, doi: 10.1093/pasj/58.5.893

    2006 doi

  66. [74]

    Takata, M., & Gough, D. O. 2024, MNRAS, 527, 1283, doi: 10.1093/mnras/stad3206

    2024 doi

  67. [75]

    J., & Christensen-Dalsgaard, J

    Thompson, M. J., & Christensen-Dalsgaard, J. 2002, in ESA Special Publication, Vol. 485, Stellar Structure and Habitable Planet Finding, ed. B. Battrick, F. Favata, I. W. Roxburgh, & D. Galadi, 95–101, doi: 10.48550/arXiv.astro-ph/0110447

    2002 doi arXiv

  68. [76]

    Townsend, R. H. D., & Teitler, S. A. 2013, MNRAS, 435, 3406, doi: 10.1093/mnras/stt1533

    2013 doi

  69. [77]

    A., Corsaro, E., De Ridder, J., et al

    Triana, S. A., Corsaro, E., De Ridder, J., et al. 2017, A&A, 602, A62, doi: 10.1051/0004-6361/201629186 21

    2017 doi

  70. [78]

    2023, A&A, 675, A17, doi: 10.1051/0004-6361/202245597

    Christensen-Dalsgaard, J. 2023, A&A, 675, A17, doi: 10.1051/0004-6361/202245597

    2023 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.