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Deformation and differential rotation in slowly rotating young intermediate-mass stars

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Using TESS photometry, this paper reports strong radial differential rotation in TIC 307930890 and solar-like latitudinal differential rotation in TIC 408165734, plus deformation measurements in fifteen delta Scuti stars.

desk verdict A useful ensemble of a1/a2 measurements for 16 delta Scuti stars, but the two headline differential-rotation detections rest on fragile mode identifications and a ~1.3-sigma a3. read the letter →

arxiv 2505.12052 v1 pith:F2GJOCAH submitted 2025-05-17 astro-ph.SR

S. K. Panda , S. Hanasoge This is my paper · ORCID
classification astro-ph.SR
keywords asteroseismologydeltaScutistarsdifferentialrotationstellardeformationrotationalmodesplittinga-coefficientsTESSmagnetism
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 tries to show that the frequency splittings of pulsation modes in slowly rotating delta Scuti stars can be read as a clean diagnostic of two things at once: how fast the star spins at different depths and latitudes, and how far its shape departs from a sphere. Using TESS photometry, the authors report that TIC 307930890 shows a roughly 60 percent drop in rotation rate across the outer 8 percent of its radius, and that TIC 408165734 shows solar-like latitudinal shear with an $a_3/a_1$ ratio of about 10 percent. For fifteen stars they extract the even splitting coefficient $a_2$, which measures deformation: nine come out oblate and six come out prolate, with asphericity up to roughly 0.1 percent. If these mode identifications hold, the results turn delta Scuti envelopes into a testbed for angular momentum transport, meridional circulation, and magnetic fields in intermediate-mass stars.

What carries the argument

The load-bearing object is the $a$-coefficient expansion of non-radial mode splittings: projecting the frequency shifts $\nu_{n,\ell,m}-\nu_{n,\ell}$ onto the polynomial basis $P_j^{(\ell)}(m)$ separates rotation (odd $j$) from centrifugal deformation and magnetic effects (even $j$). The first odd coefficient $a_1$ measures the mean rotation once the Ledoux constant $C_L$ is removed ($a_1/(1-C_L)$); the next odd coefficient $a_3$, which requires $\ell=2$ modes, measures the equator-to-pole rotation difference; the even coefficient $a_2$, read from the position of the $m=0$ peak, measures asphericity. These coefficients are extracted by fitting power spectra with the paper's line-profile model in a nested-sampling Bayesian fitter, and the radial-order assignment for TIC 307930890 is anchored by matching the observed frequencies to a stellar-evolution model and a companion pulsation calculation. The spatial rotation profile is then obtained by discretizing the rotational kernel $K_{n,\ell}(r)$ onto three zones in the outer envelope.

What would settle it

Longer or higher-cadence photometry that resolves the $m=0$ component of any of the TIC 307930890 doublets would directly test the mode identifications: if the recovered $a_2$ is inconsistent with the centrifugal value implied by the measured rotation and the stellar model, the assumed splitting pattern is wrong. For TIC 408165734, the four quadrupole peaks must keep their relative spacings and continue to give the same $a_1$ as the dipole doublet as frequency resolution improves; if new peaks appear or the quadruplet resolves into unrelated modes, the $a_3$ detection collapses.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that mode splittings in slowly rotating delta Scuti stars are rich enough to separate rotation from shape. In TIC 307930890, four consecutive dipole ($\ell=1$) doublets give kernel-weighted rotation rates $\langle f_{\rm rot}\rangle$ that fall from $0.532$ to $0.382$ d$^{-1}$ with increasing radial order; forward-matching these four splittings on a three-shell grid at $r=0.91,\,0.95,\,0.99\,R_\star$ yields a profile that drops from $0.62$ to $0.24$ d$^{-1}$, an outward spin decline of almost 60 percent over 8 percent of the stellar radius. In TIC 408165734, a candidate quadrupole ($\ell=2$) quadruplet yields $a_3\simeq 0.033$ d$^{-1}$ with the same $a_1$ as the dipole doublet within errors, indicating solar-like latitudinal shear at the 10 percent level. Across fifteen stars, the displacement of the $m=0$ component relative to its $\pm1$ siblings gives $a_2$, and through it an asphericity $(R_{\rm eq}-R_{\rm pole})/R_{\rm eq}$ ranging from $-0.05\%$ (prolate) to $+0.10\%$ (oblate), about a hundred times the solar value.

Load-bearing premise

The load-bearing premise is that the individual peaks in the spectra are what the authors say they are: the four doublets in TIC 307930890 are $\ell=1$ modes of consecutive radial orders $n=3$ through $6$, and the four peaks in TIC 408165734 form a single $\ell=2$ quadruplet with its $m=0$ component missing, an identification made visually from echelle diagrams and model-frequency matching, with the quadrupole called 'potential' even in the paper.

Editorial extensions

If this is right

  • If the radial gradient in TIC 307930890 is real, the outer envelope of at least some delta Scuti stars is not rotating rigidly, which means angular momentum transport in these stars can be inefficient enough to preserve near-surface shear.
  • If the $a_3>0$ detection in TIC 408165734 stands, solar-like latitudinal shear (equator faster than pole) exists in hot intermediate-mass envelopes, implying meridional circulation strong enough to balance angular momentum transport.
  • A population of 15 measured shapes, with 9 oblate and 6 prolate candidates, turns deformation from an isolated measurement into an ensemble constraint linking rotation rate, inclination, and possible magnetic support against centrifugal flattening.
  • Asphericity values of order $10^{-3}$ in relative radius difference are large enough that independent geometric probes, such as interferometry or precise photometric ellipticity, could check the seismic shapes.
  • The six prolate candidates, if confirmed at higher confidence, would require a non-rotational contribution to the even splitting, most naturally an equatorial toroidal magnetic field, and would thereby motivate spectropolarimetric follow-up.

Reading between the lines

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

  • The paper leaves implicit that the same fitting machinery could be applied directly to the remaining split stars in the parent sample; this is a low-cost extension because the fitting and decomposition steps are already in place.
  • The three-shell result is a summary of a model, not a unique inversion; a smoother prior on the rotation profile could change the exact 60 percent figure while preserving the qualitative outward spin-down.
  • If latitudinal shear near the 10 percent level is common in these stars, single-doublet rotation measurements would carry a systematic bias: the $m=\pm1$ spacing alone would not equal the equatorial rotation rate.
  • For the prolate candidates, the natural next test is spectropolarimetry: a confirmed $a_2>0$ pattern predicts an equatorial toroidal field strong enough to counter centrifugal flattening, which is a concrete magnetic-field signature to look for.
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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 analyzes TESS light curves of 16 slowly rotating delta Scuti stars from a larger sample and applies the a-coefficient decomposition of rotationally split non-radial modes. It reports three main results: (i) a radial gradient in the outer envelope of TIC 307930890 inferred from four dipole doublets, with the rotation rate falling from 0.62 d^-1 at r = 0.91 R to 0.24 d^-1 at r = 0.99 R; (ii) an asphericity measurement for 15 stars, of which nine are oblate and six prolate; and (iii) a latitudinal shear signature in TIC 408165734 with a3/a1 about 10%, interpreted as solar-like. The authors use standard first- and second-order perturbation formulas, Dynesty-based posterior sampling of the power spectrum, and MESA/GYRE stellar models for Ledoux corrections and mode identifications.

Significance. If the results hold, the two differential-rotation detections would be among the first such constraints for delta Scuti stars and would usefully complement the solar-like shear inference in the Sun and the few individual main-sequence cases. The deformation ensemble is a genuinely new observational sample linking a2 asymmetries to oblateness or prolateness. The analysis is transparent in several respects: the a-coefficient formulas are standard, the Bayesian fitting procedure is described in enough detail to be reproduced, and the paper is candid about the limitation that only four splittings cannot support a full inversion. The same candor, however, does not extend to the mode-identification and significance claims, which are the main weaknesses of the paper.

major comments (3)
  1. [Section 3.3, Fig. 6, Eq. (26)] The latitudinal shear detection for TIC 408165734 is not statistically secure. The four peaks between the radial and dipole ridges are assigned to an ell=2 quadruplet with m = +/-1, +/-2 and an unseen m = 0; the manuscript itself calls this a 'potential' presence and says the authors 'interpreted' the peaks. No independent mode-identification test is given. With four peaks and thirteen fitted parameters (nu, a1, a2, a3, a4, four heights, four widths), the quadruplet model is not overdetermined, and the agreement of the fitted a1 with the dipole a1 is not independent evidence because a1 is itself fitted from the same quadruplet. The reported a3 = 0.033 (+0.011, -0.026) d^-1 is consistent with zero at about 1.3 sigma when the larger lower error is used, so the a3/a1 about 10% claim and the solar-like shear conclusion need either a quantitative mode-identification test (e.g., asymptotic spacing, amplitude ratios, or model-predicted frequencies) or a downgrade to a tentative upper limit.
  2. [Section 3.1.2, Fig. 3, Eqs. (22)-(24)] The radial differential rotation of TIC 307930890 is a property of an assumed three-shell model rather than a direct measurement. The radii r = 0.91, 0.95, 0.99 R are 'arbitrarily chosen' with Delta r = 0.04 R, and the inferred drop from 0.62 d^-1 to 0.24 d^-1 over 8% of the stellar radius is the best-fit value on this grid. The text acknowledges that a full inversion is impossible with only four splittings, but it does not show that the decreasing trend is robust to changes in the grid or to a continuous profile parameterization, nor does it propagate the uncertainty in the Ledoux constants, which come from a single MESA/GYRE model. The four doublets are also assumed to be ell=1 modes of consecutive radial orders n = 3 to 6 without quantified evidence; a misidentification of even one doublet changes both the kernel weighting and the Ledoux correction. Adding a grid-robustness test and a model-uncertainty term is necessary to support the abstract's 'significant radial shear' claim.
  3. [Section 3.2, Table 3] The prolate interpretation rests on 1-sigma sign determinations. The text says the six prolate candidates are identified 'with 1 sigma (thus about 68%) confidence,' and inspection of Table 3 shows that not all positive-a2 entries are significantly nonzero: for example, TIC 30624832 has a2 = 0.0174 (+0.0112, -0.0321) and TIC 423159418 has a2 = 0.0086 (+0.0074, -0.0173), both consistent with zero at the 1-sigma level. A positive median is not a detection of prolateness. The authors should report the significance of each a2 sign, or explicitly label the six prolate stars as tentative candidates rather than measurements.
minor comments (5)
  1. [Section 2, Eq. (11)] The text following Eq. (11) misspells 'equatorial' as 'equatrial'; please proofread the manuscript.
  2. [Section 3.2, Fig. 5] The Pearson correlation R = -0.84 is quoted for only 9 stars without a p-value or confidence interval; add a significance estimate or bootstrap interval to support the claim of a 'faint correlation'.
  3. [Section 3.1.2] The sentence 'Solving four linear equations comprising three independent degrees of freedom ensures the uniqueness of the obtained solution' is confusing; with three unknown rotation rates and four splittings the system is overdetermined in a least-squares sense, and the likelihood in Eq. (24) treats it as such. Please clarify.
  4. [Sections 1 and 3] The stellar parameters and mean rotation rates for most of the sample are attributed to 'Singh et al. (under review)' and 'Singh et al. (in preparation)'; because the Ledoux corrections and mode identifications depend on these values, the paper should either include the relevant values or describe how the reader can access the companion work.
  5. [Fig. 6, panels (h) and (i)] The best-fit titles in panels (h) and (i) report only nu, a1, and a3, while the fits also include a2 and a4; please report the marginalized values of all coefficients or state that they were treated as nuisance parameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the a-coefficient measurements, asphericity conversion, and radial-profile fit are direct data reductions, with mode-identification uncertainty being a correctness risk rather than a circular step.

full rationale

The paper's derivations are self-contained against external standards. The a-coefficients a1, a2, and a3 are obtained by direct spectral fits to observed splittings using the standard polynomial expansion (Eq. 7) and the explicit estimators (Eqs. 13–15); no fitted parameter is renamed as a prediction. The asphericity conversion (Eq. 11) is an algebraic rearrangement of Bazot et al. (2019) formulas and does not use the paper's own measurements to define the target. The Ledoux correction for TIC 307930890 is computed independently from MESA/GYRE and is small (about 0.7–1.2%), so the radial differential rotation signal is dominated by the measured a1 sequence, not by the model correction. The forward-model inversion on the arbitrarily chosen grid is underdetermined and model-dependent, but it is an honest mapping of four measured splittings to a coarse profile; choosing a different grid would change the profile but not circularly force the conclusion. The main vulnerabilities—the tentative ell=2 assignment for TIC 408165734 and the modest significance of a3—are mode-identification and significance concerns, not circular derivations. Citations to Bedding et al. (2020), Steindl et al. (2022), and Bazot et al. (2019) supply external mode-identification conventions and formulas, and no load-bearing argument reduces to a self-citation by the present authors. The only self-citation (Hanasoge 2022, for meridional circulation) is contextual rather than load-bearing. The paper explicitly flags its own limitations, calling the quadrupole 'potential' and the inversion grid 'arbitrarily chosen,' which further supports that these are model-dependence caveats rather than hidden circularities.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claims rest on direct frequency-splitting measurements but depend on mode identification, a hand-chosen model grid, and unpublished stellar parameters. No new physical entities are introduced; the prolate and magnetic-field discussion is interpretive.

free parameters (4)
  • f_rot at r=0.91 R_star = ~0.62 d^-1
    Fitted in the three-zone forward model for TIC 307930890 (Section 3.1.2); this value anchors the inner edge of the inferred shear.
  • f_rot at r=0.95 R_star = ~0.40 d^-1
    Same fit; intermediate grid point.
  • f_rot at r=0.99 R_star = ~0.24 d^-1
    Same fit; outer grid point. The three values together produce the reported 60% drop.
  • grid radii r=0.91,0.95,0.99 R_star = chosen by hand
    The paper calls the grid 'arbitrarily chosen positions' (Section 3.1.2); the inferred gradient depends on this choice.
assumptions (5)
  • domain assumption First- and second-order perturbation theory for rotation and centrifugal deformation is adequate for these stars because they rotate below 10% of Keplerian breakup.
    Invoked in Section 3 with references to Ballot et al. 2010 and Aerts & Tkachenko 2024.
  • domain assumption The observed frequency splittings are caused only by rotation, centrifugal deformation, and possibly magnetic activity, with no unidentified mode coupling or misidentification effects.
    Assumed in the multiplet fitting throughout Sections 3.1 to 3.3.
  • standard math The a-coefficient decomposition (equation 7) and the asphericity formula (equation 11) from Ritzwoller & Lavely 1991, Bazot et al. 2019, and Benomar et al. 2023 are valid.
    Used as the theoretical basis; not re-derived in the paper.
  • domain assumption The MESA/GYRE stellar model for TIC 307930890 with mass 1.7 solar masses, Z=0.018, and age 17 Myr is accurate enough to identify radial orders and compute Ledoux constants.
    Section 3.1.1; the Ledoux correction is small, but the radial order identification relies on this model.
  • domain assumption The solar comparison values in Table 3 are correctly transcribed and consistent with known solar oblateness.
    The table's Sun row appears inconsistent with the text value of 8.19e-4%, so this assumption may be violated.

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Cite this review

Pith. "Pith review of Deformation and differential rotation in slowly rotating young intermediate-mass stars." pith.science (2026). https://pith.science/paper/F2GJOCAH

@misc{pith2026250512052,
  author       = {Pith},
  title        = {Pith review of: Deformation and differential rotation in slowly rotating young intermediate-mass stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F2GJOCAH}},
  note         = {Machine review of arXiv:2505.12052}
}
abstract

Asteroseismology, the study of stellar vibrations, is a method which can probe the structure deformation and internal rotation of stars. Salient among the seismic inferences of rotation from TESS observations are TIC 408165734, whose equatorial rotation rate is 10\% faster than the pole, and TIC 307930890, which has significant radial shear and shows a decreasing spin rate outward through its envelope. We also measure structural deformation in fifteen stars, nine of which are oblate, a finding consistent with expectations for relatively fast-rotating, non-magnetic stars. The difference between polar and equatorial radii in TIC 47639058 is 130 times larger than that for the Sun. The remaining six stars display splittings consistent with a prolate shape (surprisingly), possibly indicating the presence of equatorial toroidal magnetic fields. These inferences provide constraints for numerical simulations and new insights to guide theories of $\delta$ Scuti structure and rotation.

Figures

Figures reproduced from arXiv: 2505.12052 by the authors.

Figure 1
Figure 1. Radial differential rotation in a δ Scuti star. (a) Power spectrum of TIC 307930890 with rotationally split dipole￾mode (ℓ = 1) doublets at four radial orders. The m = −1 (+1) components are marked with dotted (dashed) lines. (b) The ´echelle diagram associated with this spectrum, obtained by vertically stacking equal-width segments of oscillation spectra, which aligns the modes of a given harmonic degree (ℓ) into s… view at source ↗
Figure 2
Figure 2. Fitting the first mode splitting of TIC 307930890. (a) Sequences of parameter νn,1, the unperturbed mode frequency, as traced by the sampler with subsequent iterations. The sampler, after wandering through the parameter space over thousands of epochs, eventually converges, allowing estimation of the best-fit parameters with uncertainty. (b) The similarity trace plot for parameter a1, which relates to the mean rotati… view at source ↗
Figure 3
Figure 3. Optimizing the spatial rotation of TIC 307930890 over a three zone grid within its outer envelope. (a,b,c) Trace plots of the rotation rates at r = 0.91, 0.95, 0.99R⋆, as a function of iterations made by the sampler. (d) Best-fit rotation rates and uncertainties shown over the radial coordinates of the assumed spatial grid. (a) (c) (d) (e) (b) (f) [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Analyzing the asymmetric dipole mode splitting of TIC 14172135. (a) ´echelle diagram of pulsation spectrum of the star, with the star symbols marking the dipole triplet we used for calculation of the a2 coefficients. (b) A zoomed-in power spectrum of the star, with ver…
Figure 5
Figure 5. Figure 5: Asphericity, the relative difference between equa￾torial and polar radii, plotted against the line-of-sight pro￾jected rotational velocity (v sin i). The latter was available in literature for only 9 stars of our sample. compute the 1σ uncertainties. The a1 and a2 coef…
Figure 6
Figure 6. Figure 6: Inferring latitudinal differential rotation in a δ Scuti star. (a) ´echelle diagram of TIC 408165734, a star potentially harboring quadrupole (ℓ = 2) modes. The ridge on the right comprises radial (ℓ = 0) modes, and the one on the left appears to be formed by dipole (ℓ…

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Works this paper leans on

76 extracted references · 18 canonical work pages

  1. [1]

    2010, Asteroseismology, Astronomy and Astrophysics Library (Springer Netherlands)

    Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. 2010, Asteroseismology, Astronomy and Astrophysics Library (Springer Netherlands). https://ui.adsabs.harvard.edu/abs/2010aste.book.....A

    work page 2010

  2. [2]

    Aerts, C., Mathis, S., & Rogers, T. M. 2019, Annual Review of Astronomy and Astrophysics, 57, 35, https://doi.org/10.1146/annurev-astro-091918-104359

    doi:10.1146/annurev-astro-091918-104359 2019

  3. [3]

    2024, A&A, 692, R1, 10.1051/0004-6361/202348575

    Aerts, C., & Tkachenko, A. 2024, A&A, 692, R1, 10.1051/0004-6361/202348575

    doi:10.1051/0004-6361/202348575 2024

  4. [4]

    M., Benomar, O., et al

    Appourchaux, T., Antia, H. M., Benomar, O., et al. 2014, A&A, 566, A20, 10.1051/0004-6361/201323317

    doi:10.1051/0004-6361/201323317 2014

  5. [5]

    A., Silvester, J., et al

    Aurière, M., Wade, G. A., Silvester, J., et al. 2007, A&A, 475, 1053, 10.1051/0004-6361:20078189

    doi:10.1051/0004-6361:20078189 2007

  6. [6]

    R andRieutord, M

    Ballot, J., Lignières, F., & Reese, D. R andRieutord, M. 2010, A&A, 518, A30, 10.1051/0004-6361/201014426

    doi:10.1051/0004-6361/201014426 2010

  7. [7]

    Balona, L. A. 2019, Monthly Notices of the Royal Astronomical Society, 490, 2112, 10.1093/mnras/stz2808

    work page doi:10.1093/mnras/stz2808 2019

  8. [8]

    1984, , 93, 219, 10.1007/BF02270836

    Balthasar , H. 1984, , 93, 219, 10.1007/BF02270836

    work page doi:10.1007/bf02270836 1984

Show all 76 references
  1. [9]

    2019, A&A, 623, A125, 10.1051/0004-6361/201834594

    Bazot, M., Benomar, O., Christensen-Dalsgaard, J., et al. 2019, A&A, 623, A125, 10.1051/0004-6361/201834594

    2019 doi

  2. [10]

    R., Murphy , S

    Bedding , T. R., Murphy , S. J., Hey , D. R., et al. 2020, , 581, 147, 10.1038/s41586-020-2226-8

    2020 doi

  3. [11]

    P., Goupil, M

    Belkacem, K., Marques, J. P., Goupil, M. J., et al. 2015, A&A, 579, A30, 10.1051/0004-6361/201526042

    2015 doi

  4. [12]

    2023, Astronomy & Astrophysics, 680, A27, 10.1051/0004-6361/202347095

    Benomar, O., Takata, M., Bazot, M., et al. 2023, Astronomy & Astrophysics, 680, A27, 10.1051/0004-6361/202347095

    2023 doi

  5. [13]

    Benomar, O., Takata, M., Shibahashi, H., Ceillier, T., & García, R. A. 2015, Monthly Notices of the Royal Astronomical Society, 452, 2654, 10.1093/mnras/stv1493

    2015 doi

  6. [14]

    B., et al

    Benomar, O., Bazot, M., Nielsen, M. B., et al. 2018, Science, 361, 1231–1234, 10.1126/science.aao6571

    2018 doi

  7. [15]

    Bowman, D. M. 2020, Frontiers in Astronomy and Space Sciences, 7, 10.3389/fspas.2020.578584

    2020

  8. [16]

    2014, The Astrophysical Journal, 788, 93, 10.1088/0004-637X/788/1/93

    Cantiello, M., Mankovich, C., Bildsten, L., Christensen-Dalsgaard, J., & Paxton, B. 2014, The Astrophysical Journal, 788, 93, 10.1088/0004-637X/788/1/93

    2014 doi

  9. [17]

    2018, A&A, 618, A90, 10.1051/0004-6361/201732538

    Damiani, C., & Mathis, S. 2018, A&A, 618, A90, 10.1051/0004-6361/201732538

    2018 doi

  10. [18]

    P., Ventura, R., Cardini, D., et al

    Di Mauro, M. P., Ventura, R., Cardini, D., et al. 2016, The Astrophysical Journal, 817, 65, 10.3847/0004-637X/817/1/65

    2016 doi

  11. [19]

    2010, Monthly Notices of the Royal Astronomical Society, 402, 271, 10.1111/j.1365-2966.2009.15955.x

    Duez, V., Mathis, S., & Turck-Chièze, S. 2010, Monthly Notices of the Royal Astronomical Society, 402, 271, 10.1111/j.1365-2966.2009.15955.x

    2010

  12. [20]

    Eggenberger , P., Buldgen , G., Salmon , S. J. A. J., et al. 2022, Nature Astronomy, 6, 788, 10.1038/s41550-022-01677-0

    2022 doi

  13. [21]

    2011, A&A, 533, A43, 10.1051/0004-6361/201117252

    Espinosa Lara, F., & Rieutord, M. 2011, A&A, 533, A43, 10.1051/0004-6361/201117252

    2011 doi

  14. [22]

    2013, A&A, 552, A35, 10.1051/0004-6361/201220844

    ---. 2013, A&A, 552, A35, 10.1051/0004-6361/201220844

    2013 doi

  15. [23]

    L., & Jermyn, A

    Fuller, J., Piro, A. L., & Jermyn, A. S. 2019, Monthly Notices of the Royal Astronomical Society, 485, 3661, 10.1093/mnras/stz514

    2019 doi

  16. [24]

    2019, A&A, 625, A88, 10.1051/0004-6361/201834599

    Gagnier, D., Rieutord, M., Charbonnel, C., Putigny, B., & Espinosa Lara, F. 2019, A&A, 625, A88, 10.1051/0004-6361/201834599

    2019 doi

  17. [25]

    2015, A&A, 580, A103, 10.1051/0004-6361/201526125

    Gaurat, M., Jouve, L., Lignières, F., & Gastine, T. 2015, A&A, 580, A103, 10.1051/0004-6361/201526125

    2015 doi

  18. [26]

    2002, Astronomische Nachrichten, 323, 251

    Gizon , L. 2002, Astronomische Nachrichten, 323, 251. https://ui.adsabs.harvard.edu/abs/2002AN....323..251G

    2002

  19. [27]

    2016, Science Advances, 2, 10.1126/sciadv.1601777

    Gizon, L., Sekii, T., Takata, M., et al. 2016, Science Advances, 2, 10.1126/sciadv.1601777

    2016 doi

  20. [28]

    Goldstein, J., & Townsend, R. H. D. 2020, The Astrophysical Journal, 899, 116, 10.3847/1538-4357/aba748

    2020 doi

  21. [29]

    O., & Thompson, M

    Gough, D. O., & Thompson, M. J. 1990, Monthly Notices of the Royal Astronomical Society, 242, 25, 10.1093/mnras/242.1.25

    1990 doi

  22. [30]

    R., Pamyatnykh, A

    Guo, Z., Bedding, T. R., Pamyatnykh, A. A., et al. 2024, Monthly Notices of the Royal Astronomical Society, 535, 2927, 10.1093/mnras/stae2423

    2024 doi

  23. [31]

    Guzik, J. A. 2021, Frontiers in Astronomy and Space Sciences, 8, 10.3389/fspas.2021.653558

    2021

  24. [32]

    Hanasoge, S. M. 2022, Living Reviews in Solar Physics, 19, 10.1007/s41116-022-00034-7

    2022 doi

  25. [33]

    J., Cox , J

    Hansen , C. J., Cox , J. P., & van Horn , H. M. 1977, , 217, 151, 10.1086/155564

    1977 doi

  26. [34]

    2023, A&A, 672, A60, 10.1051/0004-6361/202245281

    Hastings, B., Langer, N., & Puls, J. 2023, A&A, 672, A60, 10.1051/0004-6361/202245281

    2023 doi

  27. [35]

    Hastings, B., Langer, N., Wang, C., Schootemeijer, A., & Milone, A. P. 2021, A&A, 653, A144, 10.1051/0004-6361/202141269

    2021 doi

  28. [36]

    2021, A&A, 653, A127, 10.1051/0004-6361/202039369

    Koenigsberger, G., Moreno, E., & Langer, N. 2021, A&A, 653, A127, 10.1051/0004-6361/202039369

    2021 doi

  29. [37]

    2024, joshspeagle/dynesty: v2.1.4, v2.1.4, Zenodo, 10.5281/zenodo.12537467

    Koposov, S., Speagle, J., Barbary, K., et al. 2024, joshspeagle/dynesty: v2.1.4, v2.1.4, Zenodo, 10.5281/zenodo.12537467

    2024 doi

  30. [38]

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

    Kurtz, D. W., Saio, H., Takata, M., et al. 2014, Monthly Notices of the Royal Astronomical Society, 444, 102, 10.1093/mnras/stu1329

    2014 doi

  31. [39]

    1951, , 114, 373, 10.1086/145477

    Ledoux , P. 1951, , 114, 373, 10.1086/145477

    1951 doi

  32. [40]

    2018, Lightkurve: Kepler and TESS time series analysis in Python , Astrophysics Source Code Library

    Lightkurve Collaboration . 2018, Lightkurve: Kepler and TESS time series analysis in Python , Astrophysics Source Code Library. 1812.013

    2018

  33. [41]

    2009, A&A, 500, L41, 10.1051/0004-6361/200911996

    Lignières, F., Petit, P., Böhm, T., & Aurière, M. 2009, A&A, 500, L41, 10.1051/0004-6361/200911996

    2009 doi

  34. [42]

    2006, A&A, 455, 607–620, 10.1051/0004-6361:20065015

    Lignières, F., Rieutord, M., & Reese, D. 2006, A&A, 455, 607–620, 10.1051/0004-6361:20065015

    2006 doi

  35. [43]

    2021, Astronomy & Astrophysics, 647, A122, 10.1051/0004-6361/202039180

    Mathis, S., Bugnet, L., Prat, V., et al. 2021, Astronomy & Astrophysics, 647, A122, 10.1051/0004-6361/202039180

    2021 doi

  36. [44]

    2005, A&A, 440, 653, 10.1051/0004-6361:20052640

    Mathis, S., & Zahn, J.-P. 2005, A&A, 440, 653, 10.1051/0004-6361:20052640

    2005 doi

  37. [45]

    2015, Solar Physics, 290, 673–687, 10.1007/s11207-015-0655-6

    Meftah, M., Irbah, A., Hauchecorne, A., et al. 2015, Solar Physics, 290, 673–687, 10.1007/s11207-015-0655-6

    2015 doi

  38. [46]

    Mestel, L., & Weiss, N. O. 1987, Monthly Notices of the Royal Astronomical Society, 226, 123, 10.1093/mnras/226.1.123

    1987 doi

  39. [47]

    J., Belkacem, K., et al

    Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012, A&A, 548, A10, 10.1051/0004-6361/201220106

    2012 doi

  40. [48]

    2015, Monthly Notices of the Royal Astronomical Society: Letters, 454, L86, 10.1093/mnrasl/slv130

    Neiner, C., & Lampens, P. 2015, Monthly Notices of the Royal Astronomical Society: Letters, 454, L86, 10.1093/mnrasl/slv130

    2015 doi

  41. [49]

    A., & Sikora, J

    Neiner, C., Wade, G. A., & Sikora, J. 2017, Monthly Notices of the Royal Astronomical Society: Letters, 468, L46, 10.1093/mnrasl/slx023

    2017 doi

  42. [50]

    2010, The Astrophysical Journal Supplement Series, 192, 3, 10.1088/0067-0049/192/1/3

    Paxton, B., Bildsten, L., Dotter, A., et al. 2010, The Astrophysical Journal Supplement Series, 192, 3, 10.1088/0067-0049/192/1/3

    2010 doi

  43. [51]

    2013, The Astrophysical Journal Supplement Series, 208, 4, 10.1088/0067-0049/208/1/4

    Paxton, B., Cantiello, M., Arras, P., et al. 2013, The Astrophysical Journal Supplement Series, 208, 4, 10.1088/0067-0049/208/1/4

    2013 doi

  44. [52]

    2015, The Astrophysical Journal Supplement Series, 220, 15, 10.1088/0067-0049/220/1/15

    Paxton, B., Marchant, P., Schwab, J., et al. 2015, The Astrophysical Journal Supplement Series, 220, 15, 10.1088/0067-0049/220/1/15

    2015 doi

  45. [53]

    B., et al

    Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, The Astrophysical Journal Supplement Series, 234, 34, 10.3847/1538-4365/aaa5a8

    2018 doi

  46. [54]

    2019, The Astrophysical Journal Supplement Series, 243, 10, 10.3847/1538-4365/ab2241

    Paxton, B., Smolec, R., Schwab, J., et al. 2019, The Astrophysical Journal Supplement Series, 243, 10, 10.3847/1538-4365/ab2241

    2019 doi

  47. [55]

    2006, A&A, 455, 621–637, 10.1051/0004-6361:20065269

    Reese, D., Lignières, F., & Rieutord, M. 2006, A&A, 455, 621–637, 10.1051/0004-6361:20065269

    2006 doi

  48. [56]

    V., Tkachenko, A., Aerts, C., et al

    Reeth, T. V., Tkachenko, A., Aerts, C., et al. 2015, The Astrophysical Journal Supplement Series, 218, 27, 10.1088/0067-0049/218/2/27

    2015 doi

  49. [57]

    H., & Lavely , E

    Ritzwoller , M. H., & Lavely , E. M. 1991, , 369, 557, 10.1086/169785

    1991 doi

  50. [58]

    E., & Zorec, J

    Royer, F., Grenier, S., Baylac, M.-O., Gómez, A. E., & Zorec, J. 2002, A&A, 393, 897–911, 10.1051/0004-6361:20020943

    2002 doi

  51. [59]

    2023, A&A, 674, A6, 10.1051/0004-6361/202243615

    Sartoretti, P., Marchal, O., Babusiaux, C., et al. 2023, A&A, 674, A6, 10.1051/0004-6361/202243615

    2023 doi

  52. [60]

    Schou , J., Christensen-Dalsgaard , J., & Thompson , M. J. 1994, , 433, 389, 10.1086/174653

    1994 doi

  53. [61]

    F., Wang, J

    Song, H. F., Wang, J. Z., Song, F., & Wang, J. T. 2017, A&A, 600, A42, 10.1051/0004-6361/201629784

    2017 doi

  54. [62]

    Speagle, J. S. 2020, Monthly Notices of the Royal Astronomical Society, 493, 3132–3158, 10.1093/mnras/staa278

    2020 doi

  55. [63]

    Spruit , H. C. 1999, , 349, 189, 10.48550/arXiv.astro-ph/9907138

    1999 doi arXiv

  56. [64]

    Spruit, H. C. 2002, A&A, 381, 923, 10.1051/0004-6361:20011465

    2002 doi

  57. [65]

    G., Oelkers, R

    Stassun, K. G., Oelkers, R. J., Paegert, M., et al. 2019, The Astronomical Journal, 158, 138, 10.3847/1538-3881/ab3467

    2019 doi

  58. [66]

    2022, A&A, 664, A32, 10.1051/0004-6361/202243242

    Steindl, T., Zwintz, K., & Müllner, M. 2022, A&A, 664, A32, 10.1051/0004-6361/202243242

    2022 doi

  59. [67]

    C., Goupil, M

    Su\'arez, J. C., Goupil, M. J., & Morel, P. 2006, Astronomy & Astrophysics, 449, 673–685, 10.1051/0004-6361:20054181

    2006 doi

  60. [68]

    C., García Hernández, A., Moya, A., et al

    Suárez, J. C., García Hernández, A., Moya, A., et al. 2014, A&A, 563, A7, 10.1051/0004-6361/201322270

    2014 doi

  61. [69]

    2005, A&A, 440, 981, 10.1051/0004-6361:20053020

    Talon, S., & Charbonnel, C. 2005, A&A, 440, 981, 10.1051/0004-6361:20053020

    2005 doi

  62. [70]

    2023, Monthly Notices of the Royal Astronomical Society, 526, 1728, 10.1093/mnras/stad2798

    Thomson-Paressant, K., Neiner, C., Lampens, P., et al. 2023, Monthly Notices of the Royal Astronomical Society, 526, 1728, 10.1093/mnras/stad2798

    2023 doi

  63. [71]

    2020, Monthly Notices of the Royal Astronomical Society, 500, 1992, 10.1093/mnras/staa3442

    Thomson-Paressant, K., Neiner, C., Zwintz, K., & Escorza, A. 2020, Monthly Notices of the Royal Astronomical Society, 500, 1992, 10.1093/mnras/staa3442

    2020 doi

  64. [72]

    Townsend, R. H. D., Goldstein, J., & Zweibel, E. G. 2017, Monthly Notices of the Royal Astronomical Society, 475, 879, 10.1093/mnras/stx3142

    2017 doi

  65. [73]

    Townsend, R. H. D., & Teitler, S. A. 2013, Monthly Notices of the Royal Astronomical Society, 435, 3406, 10.1093/mnras/stt1533

    2013 doi

  66. [74]

    1924, mnras, 84, 665, 10.1093/mnras/84.9.665

    von Zeipel , H. 1924, mnras, 84, 665, 10.1093/mnras/84.9.665

    1924 doi

  67. [75]

    Wentzel , D. G. 1961, , 133, 170, 10.1086/147014

    1961 doi

  68. [76]

    2020, A&A, 643, A110, 10.1051/0004-6361/202038210

    Zwintz, K., Neiner, C., Kochukhov, O., et al. 2020, A&A, 643, A110, 10.1051/0004-6361/202038210

    2020 doi

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