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Radial differential rotation is common in massive β Cep stars and often non-monotonic, from consistent modelling of 36 pulsators.

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2026-07-14 08:54 UTC pith:N5PKHMDN

load-bearing objection Solid first population-level β Cep modelling with consistent second-order rotation; the differential/non-monotonic claim is a useful lower-limit indication, not a finished profile map. the 2 major comments →

arxiv 2607.10834 v1 pith:N5PKHMDN submitted 2026-07-12 astro-ph.SR

Asteroseismic forward modelling of 36 β Cep pulsators and inferences on their internal differential rotation

Mathijs Vanrespaille , Dario J. Fritzewski , Vincent Vanlaer , Conny Aerts This is my paper
classification astro-ph.SR
keywords asteroseismologyβ Cephei starsdifferential rotationangular momentum transportstellar interiorsmain-sequence evolutionmode identificationrotational splitting
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Asteroseismology has shown that angular momentum is transported far more efficiently inside main-sequence stars than classical models predict, yet the responsible mechanisms remain unclear. Massive β Cephei pulsators are especially useful because several already show clear radial differential rotation, but until now fewer than ten had been modelled in detail. This paper builds a homogeneous sample of 36 β Cep stars whose modes can be identified, then applies a new forward-modelling pipeline that includes second-order rotational effects. The resulting parameters reveal that the stars’ interior rotation rates decrease as they evolve, just as intermediate-mass stars do. In the 17 stars that possess more than one rotationally split multiplet, the rotation rate changes by more than 10 percent in at least 14 of them; most slow outward from the core, but four speed up, pointing to non-monotonic profiles. These empirical rotation maps supply the population-level constraints needed to test which transport processes actually operate in high-mass stars.

Core claim

Forward modelling of 36 β Cep stars, using a grid that consistently includes second-order rotation, shows that radial differential rotation is common: among the 17 stars with multiple multiplets, the rotation rate varies by more than 10 percent in at least 14, with ten decreasing from core to surface and four increasing, implying that the typical β Cep rotation profile may be non-monotonic.

What carries the argument

A five-parameter MESA–StORM grid in which age and rigid-body rotation frequency are fixed for each evolutionary track by matching one identified “fixed mode” and the observed multiplet splittings, then refined by a χ^{2} that includes both zonal frequencies and second-order asymmetric splittings.

Load-bearing premise

The oscillation frequencies are computed under the assumption of rigid-body rotation, yet differences among those same rigid models are later read as evidence of radial differential rotation.

What would settle it

Independent inversion or multi-dimensional modelling of any of the 14 stars that show >10 percent multiplet-to-multiplet f_rot differences would recover a flat rotation profile, or would reverse the reported core-to-envelope gradients.

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

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper presents the first homogeneous asteroseismic forward-modelling sample of 36 β Cep stars. Mode degrees from multi-colour photometry are refined with rotational multiplets; a MESA grid is combined with StORM frequencies that include second-order rotational effects (deformation and Coriolis coupling). A fixed-mode procedure optimises age and f_rot per evolutionary track, then yields statistical estimates of M, f_ov, log D_mix,0, X_c and f_rot. Validation on six literature stars recovers masses, ages and radial orders within ~1σ once Z differences are allowed for. The sample shows that internal rotation declines along the main sequence and that M_cc/M is well described by a bivariate fit in M and X_c. For 17 stars with ≥2 multiplets, multiplet-by-multiplet re-optimisation of f_rot yields contrasts >10% in 14 stars (10 core-faster, 4 envelope-faster), from which the authors conclude that radial differential rotation is common and may be non-monotonic.

Significance. A population-level β Cep modelling sample is a genuine advance: previous detailed models numbered fewer than ten and were never treated uniformly. Consistent inclusion of second-order rotation, public code and electronic tables, and explicit validation against literature stars are clear strengths. The calibrated M_cc/M(M,X_c) relations and the demonstration that second-order effects matter above ~10% f_crit are immediately useful. The differential-rotation census more than triples the number of purely asteroseismic constraints and supplies concrete targets for angular-momentum-transport theory, even if the profiles themselves remain lower limits.

major comments (2)
  1. §3.2 and §7: Oscillation frequencies and mean multiplet splittings are computed under rigid rotation, yet the same rigid models are re-optimised multiplet-by-multiplet and the resulting f_rot differences are plotted at the peaks of K_nl and interpreted as radial differential rotation (Fig. 10). Overlapping broad kernels (Fig. C.1) and the selection rule that discards multiplet pairs whose Δf differ by more than a factor of two (§2.2 condition 3b) systematically suppress large contrasts. The paper notes these biases but still presents residual >10% contrasts as affirmative evidence that non-monotonic profiles are typical. The claim should be restated as a lower-limit indication of differential rotation, not a robust profile measurement, and the abstract and conclusions softened accordingly.
  2. §5 and Fig. 6: StORM systematically over-predicts the asymmetry of l=1 multiplets above ~10% f_crit, leading to under-estimated mean Δf and a unidirectional correction on f_rot. Because the primary scientific product is the set of rotation rates and their radial contrasts, the authors should quantify how this bias propagates into the 14/17 differential-rotation detections (e.g., by repeating the multiplet-by-multiplet analysis with the GYRE first-order f_rot values already computed in §5) and report whether the >10% threshold remains robust.
minor comments (5)
  1. Table 2 notes: the comparison for ν Eri uses only eight frequencies while the main results use ten; a one-sentence clarification of which set enters the population statistics would avoid confusion.
  2. Fig. 10 right panel: the colour bar is sorted by X_c but the legend lists star names; adding the corresponding X_c values (or a second colour scale) would make the evolutionary trend easier to read.
  3. §4.1.5: the metallicity-driven mass uncertainty σ_M,Z is defined as max_j(ΔM_j/M)·M; a brief statement of the typical size of this term relative to the grid-step floor would help the reader gauge the dominant error source.
  4. Data-availability statement still contains placeholder GitHub/Zenodo/CDS links; these must be finalised before publication.
  5. A few typographical inconsistencies remain (e.g., “StORMoscillation”, “βCep” without space in places); a light copy-edit pass is recommended.

Circularity Check

1 steps flagged

Mild selection bias against large multiplet Δf contrasts, but differential-rotation claim is an empirical lower-limit inference from external frequencies, not a definitional or fitted-input tautology.

specific steps
  1. other [Sect. 2.2, condition 3b (and re-used in Sect. 7)]
    "If the dominant mode belongs to a multiplet other than the candidate multiplet, we compared Δf of and the candidate multiplet. Since rotational splitting in either multiplet can be reduced by up to 50% due to the Ledoux constant, we demanded Δf in each multiplet to differ less than a factor 2."

    Candidate multiplets are accepted only when their mean rotational splittings already agree to within a factor of two. This pre-filters the sample against the strongest radial differentials before any multiplet-by-multiplet re-optimisation is performed. The subsequent claim that differentials >10% are common is therefore measured only inside a sample that has already been restricted to modest Δf contrasts; the selection rule and the differential-rotation inference are not fully independent. The paper itself lists this as one of three biases that reduce measured differentials, so the circularity is mild and acknowledged rather than hidden.

full rationale

The paper's central claim (radial differential rotation common in 14/17 stars, possibly non-monotonic) is obtained by re-optimising a single free parameter f_rot per multiplet against observed splittings that are external data, then plotting those f_rot at the peak of each K_nl. The models themselves assume rigid rotation, so the recovered contrasts are only lower limits; the paper states this explicitly and lists three biases that suppress differentials. That is a modelling limitation and an interpretive over-reach, not circularity of the kinds enumerated (self-definitional, fitted-input-called-prediction, load-bearing self-citation uniqueness, etc.). Frequencies, Gaia Teff/L, and multiplet candidates are independent of the final claim; the MESA-StORM grid is not defined by the differential-rotation result. The only mild circularity is the pre-selection rule that discards multiplet pairs whose mean Δf differ by more than a factor of two, which already assumes limited differential rotation before the differential analysis is performed. That selection step is acknowledged by the authors and does not force the >10% contrasts that remain. Score 2 is therefore appropriate: one minor, non-load-bearing selection circularity; the derivation chain is otherwise self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

The central differential-rotation and population claims rest on five free stellar parameters per star, fixed microphysics (Z, α_MLT, overshoot form, IGW mixing law), second-order rigid-rotation oscillation theory, and multiplet identification heuristics. No new physical entities are postulated; StORM and MESA are tools. The largest modelling leaps are rigid rotation for frequency computation and the multiplet selection rules that feed the differential analysis.

free parameters (7)
  • Initial mass M
    Free parameter in MESA grid (7–29.85 M⊙, 43 values); statistically estimated per star from frequency and HRD fit.
  • Core overshoot f_ov
    Exponential overshoot parameter 0.005–0.035 in steps of 0.005; free in forward modelling.
  • Envelope mixing log D_mix,0
    Base envelope mixing 10–10^6 cm² s⁻¹ (mass-dependent bounds); free parameter representing IGW-driven mixing.
  • Central hydrogen X_c (age proxy)
    118 values from 0.701 to 0.0001; age fixed per track by matching the fixed-mode frequency.
  • Internal rotation f_rot / f_crit
    0–40% of critical in 1% steps; optimised from multiplet splittings under rigid rotation, then re-optimised per multiplet for differential analysis.
  • Fixed initial metallicity Z=0.014
    Held fixed for all models; systematic mass uncertainty estimated from Z=0.011 and 0.017 side grids but not freely fitted per star.
  • Mixing-length α_MLT=2.0
    Fixed by hand for all MESA models; not optimised.
axioms (6)
  • domain assumption Oscillation frequencies may be computed from 1D spherical models with Chandrasekhar–Milne second-order rotational deformation and Coriolis coupling (StORM), assumed adequate up to ~20–40% f_crit.
    §3.2; Mombarg et al. (2025a) cited for reliability below 20% f_crit; sample includes faster rotators with acknowledged approximate treatment.
  • ad hoc to paper Rigid-body rotation is an adequate assumption for computing mode frequencies and mean multiplet splittings used in the primary fit.
    §3.2 explicitly states rigid rotation is assumed because profiles are unknown a priori; later §7 interprets multiplet-to-multiplet f_rot differences as differential rotation.
  • domain assumption Core-boundary mixing is exponential overshoot starting 0.005 H_P into the core; envelope mixing follows D_mix = D_mix,0 (ρ0/ρ) from IGW simulations.
    §3.1; Rogers & McElwaine (2017), Varghese et al. (2023) cited as motivation.
  • ad hoc to paper A candidate multiplet is secure if it satisfies at least four of five observational conditions (full 2l+1, Gaia f_rot sini compatibility, multi-colour degree consistency, Δf<0.75 d⁻¹, |A_m|<0.10).
    §2.2; these heuristics define the sample of identified modes that enter all modelling.
  • domain assumption Ledoux criterion for convection; solar Asplund et al. (2009) mixture; OP/Ferguson opacities; JINA REACLIB rates.
    §3.1 standard MESA microphysics choices for this mass range.
  • domain assumption Theoretical frequency uncertainty floor of 10⁻³ d⁻¹ for β Cep modes (Aerts et al. 2018) replaces smaller observational errors in χ².
    §4.1.5; used so that one grid model does not dominate solely due to tiny observational σ.

pith-pipeline@v1.1.0-grok45 · 39225 in / 4173 out tokens · 52962 ms · 2026-07-14T08:54:17.515901+00:00 · methodology

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read the original abstract

Asteroseismic observations of the interior rotation of main sequence stars have shown that angular momentum transport is much more efficient than expected. Which transport mechanisms are responsible for this is still unclear. Detections of radial differential rotation provide valuable constraints on these transport mechanisms. This has been detected in several massive main sequence $\beta$ Cep pulsators, even though fewer than ten $\beta$ Cep stars have been asteroseismically modelled in detail so far. We aim to expand the sample of asteroseismically forward modelled $\beta$ Cep pulsators to maximally exploit their potential to observationally constrain angular momentum transport mechanisms. To that end, we seek to constrain their rotation profiles. We searched for rotational splitting of non-radial modes in a large $\beta$ Cep sample with identified mode degrees. These were subjected to a novel forward modelling approach, which consistently accounts for second-order rotation effects using the state-of-the-art StORM oscillation code. We successfully modelled 36 $\beta$ Cep stars and constrained crucial parameters such as their initial mass, internal rotation frequency, convective core mass, and age. Like in intermediate-mass main sequence stars, the internal rotation rate globally decreases in $\beta$ Cep stars as they evolve along the main sequence. Radial differential rotation is constrained in 17 $\beta$ Cep stars. The rotation rate in at least 14 stars varies by more than 10%. Of these 14 stars, ten have their rotation rate decreasing from the core to the surface while it is the opposite in four of them. We affirm that radial differential rotation is common in $\beta$ Cep stars. Moreover, our constrained rotation profiles suggest that the typical $\beta$ Cep rotation profile may be non-monotonic.

Figures

Figures reproduced from arXiv: 2607.10834 by Conny Aerts, Dario J. Fritzewski, Mathijs Vanrespaille, Vincent Vanlaer.

Figure 1
Figure 1. Figure 1: Hertzsprung-Russell diagram with the targets in our sam [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the radial (green), dipole (blue), and [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparing the observed (top) and best model’s frequencies (bottom) for the six validation stars. The x-axis is di [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Difference in rotation frequency from each identified mul￾tiplet by our self-consistent modelling using StORM and by the a posteriori step with GYRE against the relative rotation rate. Mul￾tiplets belonging to a validation stars are outlined in black. The grey dotted line shows where the two estimates agree. modelling, as demonstrated in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convective core mass relative to total stellar mass against the total mass ( [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Specific angular momentum of each star against its mass (left) and core hydrogen mass fraction (right). On the left panel, the grey dashed line indicates the upper limit on the spe￾cific angular momentum for stars more massive than 2.5 M⊙ derived by Aerts (2025). Boxes show the Pearson correlation coefficients and the p-values of the associated t-tests . 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 frot, interior [d 1 … view at source ↗
Figure 9
Figure 9. Figure 9: Projected surface rotation frequency from [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Interior differential rotation in 17 stars with several rotationally split multiplets. The left panel shows the rotation frequency from a rotationally split multiplet at the radius where that multiplet is most sensitive. Rotation frequencies extracted from l = 1 and l = 2 multiplets are marked as triangles and pentagons, respectively. The right panel shows these rotation rates normalised by the rotation r… view at source ↗

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