{"id":"2119207b-7d72-4f19-adc3-1852df904439","arxiv_id":"2505.12052","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Asteroseismic splittings of 16 TESS delta Scuti stars yield oblate and prolate deformation estimates and evidence for radial and latitudinal differential rotation in two stars.","lead":"Using TESS asteroseismology, the authors measure the deformation and internal rotation of 16 slowly rotating delta Scuti stars. They report outward-decreasing rotation in one star and solar-like latitudinal shear in another, plus mostly oblate shapes with six prolate candidates.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The latitudinal-shear detection in TIC 408165734 rests on an unconfirmed ell=2 quadruplet assignment and a ~1.3-sigma a3; if the mode assignment is wrong, the solar-like shear claim vanishes.","rationale":"The reader's weakest-assumption analysis and my independent read converge on the same point: the latitudinal-shear claim is the least secure of the paper's headline results. The radial shear in TIC 307930890 is supported by four resolved splittings with a monotonic trend in radial order, so the qualitative outward decrease is likely robust even if the quantitative three-zone profile is arbitrary. The latitudinal result, by contrast, rests on a single 'potential' quadruplet, a marginal a3 significance of roughly 1.3 sigma, and an underdetermined fit. If the mode assignment is wrong, the a3 coefficient, the a3/a1 ratio, and the solar-like shear interpretation all disappear. The deformation measurements and the a1/a2 coefficients are more straightforwardly tied to the spectra and are probably acceptable, which is why the paper need not be rejected outright. The concrete test I propose would settle whether the quadruplet interpretation is actually preferred over the null hypothesis of four unrelated peaks; until that test is done, the paper should be cited with the condition that the latitudinal differential rotation is an unconfirmed candidate rather than an established detection. This does not change the reader's CONDITIONAL verdict, so the verdict remains unchanged.","tokens_in":19598,"tokens_out":8757,"duration_ms":94670,"concrete_test":"On the TIC 408165734 spectrum in the 44-46 d^-1 band: (i) fit the four peaks with the assumed ell=2 multiplet model of Eq. 26; (ii) fit the same four peaks as four independent mode profiles with no imposed multiplet relation; (iii) compute the Bayesian evidence ratio (or BIC) and the resulting posterior probability of the ell=2 assignment, and report the fully marginalized posterior of a3. If the ell=2 assignment is not strongly favored (e.g., Delta ln Z > 5) or the marginalized a3 remains within 2 sigma of zero, the latitudinal-shear result should be downgraded from a detection to an upper limit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline latitudinal-differential-rotation result for TIC 408165734 (a3/a1 ~ 10%, solar-like shear) depends on one interpretive step: four peaks between the radial and dipole ridges are assigned to a complete ell=2 multiplet with m = +/-1, +/-2 and no visible m=0 (Section 3.3). The paper itself calls this a 'potential presence' and says the authors 'interpreted' the peaks. No independent confirmation is provided: no asymptotic spacing test, no amplitude-ratio check, and no resolved rotational sidelobe pattern. The only supporting evidence is consistency between the a1 from the assumed quadruplet and the dipole doublet, but a1 is a fitted parameter of the quadruplet model; with four peaks and five parameters (nu0, a1, a2, a3, a4), the pattern is not overdetermined unless a1 is fixed externally. The fitted a3 = 0.033 (+0.011, -0.026) d^-1 is only about 1.3 sigma above zero using the larger lower error bar. Replacing any one of the four peaks with a different mode, or allowing the m=0 component to be present but blended, changes or removes a3. Because this detection is one of the two showcased seismic inferences, the central claim is not secure without a quantitative mode-identification test and a significance statement that includes model uncertainty.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19850,"tokens_out":7732,"duration_ms":76319,"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":[{"comment":"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.","section":"Section 3.3, Fig. 6, Eq. (26)"},{"comment":"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.","section":"Section 3.1.2, Fig. 3, Eqs. (22)-(24)"},{"comment":"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.","section":"Section 3.2, Table 3"}],"minor_comments":[{"comment":"The text following Eq. (11) misspells 'equatorial' as 'equatrial'; please proofread the manuscript.","section":"Section 2, Eq. (11)"},{"comment":"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'.","section":"Section 3.2, Fig. 5"},{"comment":"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.","section":"Section 3.1.2"},{"comment":"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.","section":"Sections 1 and 3"},{"comment":"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.","section":"Fig. 6, panels (h) and (i)"}],"recommendation":"major_revision","confidential_remarks":"The two headline detections are intriguing but rest on mode identifications that are not yet independently verified: the a3 detection is marginal, and the radial shear depends on an arbitrary grid. These concerns are fixable in revision through quantitative mode-identification tests and significance statements, so I do not recommend rejection. Please ensure the companion papers by Singh et al. are available to the referee, since several numerical inputs come from them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the useful part: this is a clean application of the a-coefficient formalism to a TESS sample of 16 slowly rotating delta Scuti stars. The a1 and a2 measurements for the 15 stars with visible m=0 components are the kind of result that can anchor future work. The asphericity values, even with the prolate caveats, are a new ensemble constraint. The paper is honest about its own limits; it explicitly labels the quadrupole 'potential' and describes the three-zone grid as 'arbitrarily chosen'. That honesty earns credit.\n\nThe soft spots are where the reader and stress-test put them. The latitudinal shear detection in TIC 408165734 rests on a visually identified ell=2 quadruplet with no m=0 component, no asymptotic test, and a fitted a3 about 1.3 sigma above zero with asymmetric error bars. The a1 agreement between the dipole doublet and the assumed quadruplet is suggestive, not confirmatory, because a1 is a fitted parameter of the quadruplet model. If one of the four peaks is a different mode, the a3 vanishes. The paper's own 'potential presence' language is accurate, but the headline result is not secure.\n\nThe radial differential rotation in TIC 307930890 is on somewhat firmer ground, but still model-limited. Four dipole doublets at consecutive radial orders give a1 values that decline with n. The 60% drop over 8% of the radius is the output of a three-zone fit at r=0.91, 0.95, 0.99 R with the radii chosen by hand. The authors are upfront that this is a limitation of having only four splittings, and the inference is plausible. Still, the quoted gradient is not a unique inversion; a robustness check against different grid choices and a significance statement that includes model uncertainty would strengthen it considerably.\n\nA smaller issue: Table 3's solar reference row looks internally inconsistent. The a2 sign and the asphericity value do not line up with an oblate Sun; it is a minor presentational error, but it will trip up readers. Also, the stellar parameters come from two unpublished papers by the same group, which makes independent checking harder but is not itself a flaw.\n\nBottom line: this paper deserves a serious referee. The basic a1/a2 measurements are likely sound, the interpretation is honestly caveated, and the two differential-rotation claims are exactly the kind of results the field wants to see tested. A referee should ask for a quantitative mode-identification test for the quadruplet and a robustness check on the three-zone model, but the paper has a real core.","headline":"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.","tokens_in":20468,"tokens_out":2973,"would_cite":false,"duration_ms":28996,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["asteroseismology","delta Scuti stars","differential rotation","stellar deformation","rotational mode splitting","a-coefficients","TESS","stellar magnetism"],"falsifier":"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.","tokens_in":19312,"feed_emoji":"🌀","tokens_out":13883,"duration_ms":121093,"temperature":0.7,"pith_summary":"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.","feed_headline":"Seismic data reveal uneven spin inside two young stars","feed_subtitle":"One star spins about 60 percent slower near its surface; the other's equator outruns its pole by 10 percent.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the parent sample of 38 delta Scuti stars with detected rotational splittings and the masses, ages, and mean envelope rotation rates used throughout.","marker":"Singh et al. (under review)"},{"why":"Supplies the a-coefficient formalism, the expression for the centrifugal part of a2, and the method for extracting a1, a2, a3 from multiplet frequencies.","marker":"Benomar et al. 2023"},{"why":"Introduces the projection of mode splittings onto a-coefficients and the separation of rotation (odd) from deformation and magnetism (even) terms.","marker":"Ritzwoller & Lavely 1991"},{"why":"Defines the Ledoux constant C_L used to convert each fitted a1 into a kernel-weighted rotation rate.","marker":"Ledoux 1951"},{"why":"Demonstrates how a2 measures asphericity in a slowly rotating A-type star and gives the comparison value for KIC 11145123.","marker":"Gizon et al. 2016"},{"why":"Provides the relation between the a2 coefficient and (R_eq - R_pole)/R_eq and the prolate precedent in 16 Cyg A.","marker":"Bazot et al. 2019"},{"why":"Provides the MESA stellar models used to match the frequency range of TIC 307930890 and compute its Ledoux constants.","marker":"Paxton et al. 2019"},{"why":"Provides the GYRE pulsation calculation used to identify the radial orders of the observed dipole modes.","marker":"Townsend & Teitler 2013"},{"why":"Supplies the Dynesty nested-sampling algorithm used for all power-spectrum fits and for the three-shell rotation-profile optimization.","marker":"Speagle 2020"},{"why":"Supplies the TESS Input Catalog radii used to compute Keplerian break-up rotation rates.","marker":"Stassun et al. 2019"}],"fun_headline_variants":["Two young stars show hidden spin differences","Seismic data expose uneven spin in slow-rotating stars","One star's equator beats pole; another's surface lags","Stellar seismology reveals spin gradients in delta Scuti stars","Slow-rotating stars hide dramatic internal spin shifts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two young stars show hidden spin differences","Seismic data expose uneven spin in slow-rotating stars","One star's equator beats pole; another's surface lags","Stellar seismology reveals spin gradients in delta Scuti stars","Slow-rotating stars hide dramatic internal spin shifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000228,"raw_usage":{"total_tokens":1511,"prompt_tokens":1015,"completion_tokens":496,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":417}},"tokens_in":631,"tokens_out":496,"duration_ms":5596,"temperature":1.0,"reasoning_tokens":417,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:42:12.934730+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"H., & Lavely , E","cited_arxiv_id":null,"evidence_quote":"Introduces the projection of mode splittings onto a-coefficients and the separation of rotation (odd) from deformation and magnetism (even) terms."},{"cited_title":"2016, Science Advances, 2, 10.1126/sciadv.1601777","cited_arxiv_id":null,"evidence_quote":"Demonstrates how a2 measures asphericity in a slowly rotating A-type star and gives the comparison value for KIC 11145123."},{"cited_title":"2019, A&A, 623, A125, 10.1051/0004-6361/201834594","cited_arxiv_id":null,"evidence_quote":"Provides the relation between the a2 coefficient and (R_eq - R_pole)/R_eq and the prolate precedent in 16 Cyg A."}],"review_version":1}