Probing the Cores of Subdwarf B Stars: How do They Compare to Cores in Helium Core-Burning Red Giants?

Anselmo Falorca; Juliana Amaral; Margarida S. Cunha; Pedro Avelino; Sofia Avelino; Yuri Damasceno

REVIEW 5 major objections 5 minor 30 references

Probing the Cores of Subdwarf B Stars: How do They Compare to Cores in Helium Core-Burning Red Giants?

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that all three subdwarf B stars show the same sharp structural glitch at a similar buoyancy radius near the C-O/He transition, farther out and stronger than in helium core-burning red giants, pointing to more extensive…

desk verdict A transparent and careful application of an established glitch model to three sdB stars, but the central physical interpretation is undermined by the inner/outer buoyancy degeneracy, which the paper never resolves. read the letter →

arxiv 2505.01381 v1 pith:V3G3VMGU submitted 2025-05-02 astro-ph.SR

Margarida S. Cunha , Juliana Amaral , Sofia Avelino , Anselmo Falorca , Yuri Damasceno , Pedro Avelino This is my paper · ORCID

classification astro-ph.SR

keywords asteroseismologysubdwarfBstarsgravity-modepulsationsbuoyancyglitchesperiodspacingsC-O/Hetransitioncore-boundarymixingKeplerspacetelescope

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

The paper aims to find out what lies just outside the convective cores of subdwarf B (sdB) stars, without building any stellar model, by fitting an analytical description of gravity-mode period spacings to Kepler data on three stars. It finds that a sharp structural discontinuity, most plausibly the carbon-oxygen/helium (C-O/He) transition, sits at a similar and well-constrained buoyancy radius in all three stars, near the inner turning point of the g-mode cavity. Compared with the same quantity inferred for helium core-burning red giants, the sdB glitches have larger amplitude and lie at a larger buoyancy radius. The paper argues this is direct evidence that chemical mixing beyond the adiabatically stratified core is more extensive in sdB stars, or that the stratification of that mixed region differs between the two stellar classes. These are the first model-independent constraints on sdB core-adjacent structure, and they bear on how these stripped stars form.

What carries the argument

The load-bearing object is the buoyancy glitch. The machinery is a non-perturbative analytical model in which a sharp variation of the buoyancy frequency $N$ at radius $r_*$ adds a phase $\phi$ to the asymptotic period relation, $\Pi \approx \Pi_s - (\Delta\Pi_{as}/\pi)\phi$, with $\phi$ given by an arccotangent expression that takes a step-like or Gaussian-like form. The glitch position enters through the degree-independent buoyancy radius $\tilde{W}^*_g = \int_{r_1}^{r_*} (N/r)\,dr$, measured from the inner turning point (the edge of the adiabatically stratified core), and this quantity fixes the periodicity $\Pi_{sig} = 2\pi^2/\tilde{W}^*_g$ of the dips in the period spacings. That strict periodicity is what makes the inferred position reliable while the amplitude stays poorly constrained. Fitting the predicted period spacings to the Kepler data with a nested-sampling algorithm, with the intrinsic error as a free parameter, yields $\Delta\Pi_{as}$, glitch amplitude, and $\tilde{W}^*_g$ for each star without recourse to stellar models.

What would settle it

One concrete test is to fit the same one-glitch versus two-glitch models to all three stars rather than only KIC 10001893: if the two-glitch alternative matches any other star's dips as well as the one-glitch fit does, or if additional Kepler or TESS data reveal dip spacings that are not strictly periodic, the identification of the dips with a single sharp C-O/He transition fails. Conversely, if the offset persists across a larger sample, with red giants at small buoyancy radius and sdB stars at large buoyancy radius, then the mixing difference is real.

Watch

Extended reading notes

Core claim

Fitting a glitch-induced phase perturbation to the reduced period spacings of KIC 10553698A, EPIC 211779126, and KIC 10001893, the paper finds that one step-like glitch located just outside the inner turning point of the g-mode propagation cavity reproduces the observed period-spacing dips in all three stars, with inferred buoyancy radii $\tilde{W}^*_g \approx 0.0085$–$0.0090$ rad/s and asymptotic reduced period spacings $\Delta\Pi_{as} \approx 314$–$325$ s. On the basis of the similar positions and the contrast with models of sdB cores, the paper identifies the glitch with the C-O/He transition. Comparing these values with a published analysis of 23 helium core-burning red giants, it finds that the sdB glitches are systematically stronger and, most robustly, sit at a larger buoyancy radius; because the position sets the strict periodicity of the dip pattern, this offset cannot be blamed on glitch shape. The paper concludes that the layers just beyond the convective core are either more extensively mixed in sdB stars than in helium core-burning red giants, or stratified differently.

Load-bearing premise

The comparison rests on treating each star's period-spacing dips as the signature of a single sharp, step-like structural discontinuity, identified as the C-O/He transition just beyond the convective core, so that the fitted buoyancy radius really measures the extent of the mixed region; if the dips come from the He/H transition or from a more complex structure, as the alternative two-glitch fit for KIC 10001893 allows, the inferred position does not measure core-boundary mixing.

Editorial extensions

If this is right

The inferences provide the first model-independent constraints on the structure of the layers immediately outside the convective core in sdB stars, fixing the sharpness and buoyancy position of the C-O/He transition.
If the C-O/He identification is correct, the larger buoyancy radius in sdB stars means that mixing beyond the adiabatically stratified core is more extensive there than in helium core-burning red giants, or that the stratification of the mixed region differs.
A comparison with recent sdB structural models favours evolved stars with a low core-helium abundance and a relatively small core mass, giving a concrete target for evolutionary calculations of stripped stars.
Because the sdB pulsation periods are roughly five times shorter than the red-giant mixed-mode periods, part of the inferred amplitude difference could come from a finite glitch width being absorbed into the step-like amplitude, a caveat the paper states explicitly.
The robust position offset provides a direct, model-independent test for non-canonical sdB formation channels and for mixing prescriptions in stellar evolution codes.

Reading between the lines

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

If the buoyancy-radius offset holds up in a larger sample, the inferred glitch position could serve as a calibration point for convective overshoot and boundary-mixing prescriptions, because it measures the integral of $N/r$ between the core edge and the chemical transition directly from data.
The step-versus-Gaussian ambiguity for EPIC 211779126 is testable: a step-like glitch predicts dips of constant depth across periods, while a Gaussian glitch predicts dips that fade at longer periods, so longer or denser monitoring of that star would distinguish the two.
Some sdB stars may show both the C-O/He and the He/H transition as separate glitches, as the two-glitch fit for KIC 10001893 suggests; resolving both would give a model-independent measure of the helium-layer mass and probe the stripped envelope directly.
Applying the same pipeline to g-mode pulsators of different masses or envelope properties could map how the post-core mixing region depends on the formation channel, turning a three-star result into a population diagnostic.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper fits Cunha et al.'s glitch formalism to three sdB stars and finds a consistent glitch position at buoyancy radius ~0.008 rad/s, which it identifies with the C-O/He transition. The fits are transparent, the data tables and corner plots are in the appendix, and the glitch position is formally well constrained in the favored cases. The sdB application is new, and the comparison with the red-giant results of Vrard et al. (2022) is a genuinely fresh empirical step. If the interpretation is right, the systematic difference in buoyancy radius between sdB stars and helium-burning red giants is a real constraint on convective-boundary mixing.

The soft spots are not hidden, but they are substantial. The main one is the inner/outer degeneracy. Eq. (8) shows that a step-like glitch at buoyancy radius x produces the same period-spacing signature as a glitch at buoyancy depth x, up to a phase shift. The fits return only the periodicity Pi_sig, so they cannot tell whether the glitch sits just outside the convective core or just inside the outer turning point. The paper labels all positions as buoyancy radius and, in Sec. 5.1, identifies them with the C-O/He transition by comparing with Guyot et al. (2025) models. But the He/H transition near the surface would give exactly the same signature if its buoyancy depth matched. Given Wg ~ 0.06 rad/s here, a buoyancy depth of 0.008 is a glitch very close to the outer turning point—plausible for He/H. So the cross-class comparison with red giants rests on a model-based identification, which also undercuts the 'model-independent' framing in the abstract.

Three lesser issues. First, the KIC 10553698A range cut at 11100 s is post-hoc; including the longer periods shifts the inferred position by more than the formal error (case E in Table 1). Second, the one-glitch preference for KIC 10001893 is argued mainly from consistency with the other stars rather than from a quantitative model comparison; the two-glitch alternative changes the position by nearly a factor of two (case K in Table 4). Third, in EPIC 211779126 the l=1 and l=2 fits disagree by far more than their individual uncertainties, and that systematic scatter is not propagated into the reported values. Also, the Vrard et al. (2022) comparison values appear only in figures, which makes reproduction harder.

None of this makes the paper unserious. It is honest about its assumptions and the formal constraints are well characterized. But the central physics conclusion is conditional, not as secure as the abstract suggests. A referee should ask for a treatment of the inner/outer degeneracy and a model comparison that tests C-O/He against He/H explicitly before the sdB/red-giant difference is accepted.

Referee Report

5 major / 5 minor

Summary. The paper applies the analytical buoyancy-glitch formalism of Cunha et al. (2019, 2024) to g-mode period spacings of three subdwarf B stars observed with Kepler/K2, inferring the asymptotic reduced period spacing, the amplitude, and the buoyancy position of a sharp structural variation in each star. The authors then compare these quantities with the values previously inferred for helium core-burning red giants by Vrard et al. (2022), finding that the sdB glitches have larger amplitudes and are located at larger buoyancy radii. They interpret the common glitch as the C-O/He transition near the edge of the g-mode cavity and conclude that mixing beyond the convective core may be more extensive, or differently stratified, in sdB stars than in red giants. The manuscript is clearly written, the data are tabulated in full, and the fitting procedure, including posterior corner plots, is transparent.

Significance. If the physical identification is correct, the paper would provide the first model-independent asteroseismic constraints on the core-edge structure of sdB stars and a novel comparison with helium core-burning red giants. The authors deserve credit for using a published analytical model that was already benchmarked on red giants, for presenting all input periods in an appendix, and for openly discussing degeneracies in amplitude and in glitch-model choice. However, the central physical claim currently rests on an untested side-of-cavity identification and on several consequential data-selection and model-selection choices. The comparison with red giants therefore remains suggestive rather than established, and the main conclusion should be correspondingly hedged.

major comments (5)

[Sect. 3.1, Eq. (8)] The glitch phase entering Eq. (6) is invariant under replacing an inner glitch at buoyancy radius W~*g = x by an outer glitch at buoyancy depth W*g = x. Since the fits return only the periodicity Pi_sig = 2 pi^2 / (W~*g or W*g), the data cannot distinguish a sharp feature just beyond the inner turning point (e.g., the C-O/He transition) from a sharp feature at the same buoyancy distance from the outer turning point (e.g., the He/H transition). The paper explicitly states this symmetry in Sect. 3.1, but then reports all positions as W~*g and, in Sect. 5.1, identifies the glitch with the C-O/He transition based on a comparison with Guyot et al. (2025) models. The model-independent comparison with red giants in Sect. 5.2 therefore rests on an assumption that the period-spacing data themselves do not test. I ask the authors to fit the outer-half formulation as an explicit alternative and to state, if possible, what observable discriminates the two interpretations.
[Sect. 4.1, Table 1, cases C and E] For KIC 10553698A, restricting the fit to reduced periods below 11100 s gives W~*g = 0.00785 +/- 0.00012 rad/s (case C), whereas including all listed periods gives W~*g = 0.00707 +/- 0.00020 rad/s (case E), a shift of roughly 3-4 sigma. The exclusion of the high-period data is motivated by mode-identification concerns, but the value used in the cross-star comparison in Fig. 6 depends on this cut. The impact of this cut on the comparison with the red-giant sample should be quantified, and a less ad hoc criterion for excluding the high-period data should be provided.
[Sect. 4.2, Table 2, cases F and G] For EPIC 211779126, the l=1 and l=2 fits give W~*g = 0.00898 +/- 0.00014 and 0.00783 +/- 0.00008 rad/s, respectively, which differ by more than 7 sigma even though the adopted asymptotic model predicts the same glitch position for both degrees. Fitting the two degrees separately and then using only the l=1 result in Fig. 6 discards a direct inconsistency that the model cannot accommodate. The paper should either identify a cause, such as misidentified modes in the original catalogue, or treat the l=1/l=2 disagreement as a dominant systematic uncertainty in the inferred position.
[Sect. 4.3, Table 4, cases J and K; Sect. 5.1] For KIC 10001893, the one-glitch fit to the overlapping l=1/l=2 region (case J) gives W~*g = 0.00849 rad/s, while the two-glitch fit to all data (case K) places a glitch at W~*g = 0.00473 rad/s, nearly half the single-glitch value. The paper prefers case J in Fig. 6 partly because it yields positions comparable to the other two stars; this reasoning is circular when the goal is to demonstrate a common phenomenon. A quantitative model comparison, such as Bayesian evidence or an information criterion, should be reported, and the physical consequences of the two-glitch solution should be discussed rather than set aside.
[Sect. 5.2] The comparison of glitch amplitudes between sdB stars and red giants is weakened by the finite-width effect that the authors themselves describe. If the sdB glitch is sharp and the red-giant glitch has a finite width, fitting both samples with a step-like model would translate the width into a smaller inferred amplitude at the longer red-giant periods. Since the paper's own discussion allows this explanation, the abstract's statement that the sdB structural variations have 'larger amplitudes' should be qualified, or a quantitative estimate of the effect should be added, for example by fitting a Gaussian-glitch model to the red-giant data.

minor comments (5)

[Sect. 4.1] The text 'restricting the data to reduced periods smaller than 111000 s' appears to contain a typo; the context and Table 1 indicate 11100 s.
[Fig. 6] The caption does not define what the blue shaded region represents; it should state whether it is the full range, the 68 per cent interval, or the interquartile range of the Vrard et al. (2022) sample, and should give the number of red giants and the corresponding numerical values.
[Appendix C] The table headers in Appendix C reuse case labels A-D that do not correspond to the case labels in the main text, which makes cross-referencing difficult.
[Abstract] The phrase 'model-independent' should be qualified as 'independent of stellar-structure models' to match the content of the paper, since the analysis still assumes a specific glitch shape and a specific choice of which structural transition produces the observed signature.
[Sect. 4.1 and 4.2] There are minor typographical errors, including 'pyhton' for 'Python' and 'struture' for 'structure'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytical model is stated in full, the fit parameters are inferred, and the comparison uses an independent red-giant sample.

full rationale

The paper's derivation chain is an inference from data, not a fit-then-prediction loop. The glitch model (Eqs. 5-11) is written out in the text, and the two-glitch extension is re-derived from the wave equation in Appendix B, so the appeal to Cunha et al. (2019, 2024) is attribution rather than unverified load-bearing authority. The quantities ΔΠas, A_st (or A_G), and the glitch position are free parameters determined by nested-sampling fits to observed period spacings; they are not constructed from the paper's conclusions. The comparison with helium-core-burning red giants uses the previously published Vrard et al. (2022) results, and the C-O/He identification is presented as a model comparison with Guyot et al. (2025), not as a consequence of the fitting equations. The paper explicitly flags the inner/outer cavity degeneracy in Sect. 3.1 ('the expression for the phase φ is independent of the side of the cavity in which the glitch is located') and the difficulty constraining the second glitch in Sect. 4.3; these are identifiability and interpretation caveats, not circular reductions. No fitted parameter is renamed as a prediction, and no conclusion is built into the definition of an input. Score 0.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper uses an existing analytical glitch model with several fitted parameters per star, and no new physical entities are introduced. The physical interpretation relies on identifying the glitch as the C-O/He transition and on the inner turning point lying at the edge of the adiabatic core; both are domain assumptions rather than independently verified facts.

free parameters (6)

Delta Pi_as (asymptotic reduced period spacing) = 306-336 s depending on star and case
Fitted per star; controls the baseline period spacing in Eq. (1) and scales the glitch perturbation.
W~*g (glitch buoyancy radius) = 0.0078 to 0.0090 rad/s for preferred one-glitch fits
Fitted per star; the central inferred quantity used for the comparison with red giants.
A_st or A_G (glitch amplitude) = 17-96 (step) or 0.06-0.12 rad/s (Gaussian)
Fitted per star; poorly constrained in several fits, especially when period-spacing dips are sparsely sampled.
delta (phase) = 0.7 to 3.3
Fitted per star; absorbs turning-point approximations and possible sign changes of the structural variation.
sigma (noise/error scale) = median values around 19-60 s in the discussed cases
Fitted as a free parameter because the original data papers do not provide observational errors.
Pi_s,min (first reduced period) = e.g., 3616 s for KIC 10553698A period fits
Only appears in fits to reduced periods (cases C-E); sets the zero-point of the period series.

assumptions (5)

standard math High-radial-order g modes in a spherically symmetric star with a convective core are asymptotically equally spaced in reduced period (Tassoul 1980, Eq. 1).
Starting point of the period-spacing analysis; standard asymptotic pulsation theory.
domain assumption The glitch phase functions in Eqs (5)-(7) correctly describe period perturbations for step-like and Gaussian-like buoyancy glitches (from Cunha et al. 2019, 2024; Appendix B extends to two glitches).
The inference rests on these published analytical expressions; no independent verification is provided in this paper.
domain assumption The listed mode identifications (degree and radial order) from the original Kepler analyses are correct.
Mode misidentification is discussed as a known issue in KIC 10553698A and affects the construction of period spacings.
domain assumption The inner turning point of the g-mode cavity is at the edge of the adiabatically stratified convective core, so the inferred buoyancy radius W~*g measures the distance from that edge to the glitch.
Required for the physical interpretation of the glitch location as the extent of mixing beyond the convective core (Sect. 5.1).
domain assumption Rotation and other departures from spherical symmetry have a negligible effect on the reduced period spacings in the fitted range.
The model assumes spherical symmetry; the paper notes that reduced periods of l=1 and l=2 are generally aligned, but does not quantify rotation effects.

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Cite this review

Pith. "Pith review of Probing the Cores of Subdwarf B Stars: How do They Compare to Cores in Helium Core-Burning Red Giants?." pith.science (2026). https://pith.science/paper/V3G3VMGU

@misc{pith2026250501381,
  author       = {Pith},
  title        = {Pith review of: Probing the Cores of Subdwarf B Stars: How do They Compare to Cores in Helium Core-Burning Red Giants?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3G3VMGU}},
  note         = {Machine review of arXiv:2505.01381}
}

read the original abstract

The mixing of material from stellar convective cores into their adjacent radiative layers has been a matter of long-standing debate. Pulsating subdwarf B stars offer excellent conditions to advance our understanding of this problem. In this work we use a model-independent approach to infer information about the cores of three subdwarf B stars and compare it with similar inferences from earlier analysis of red giants in the helium core-burning phase. This is achieved by fitting an analytical description of the gravity-mode pulsation periods to pulsation data collected by the Kepler satellite. From the fits we infer the reduced asymptotic period spacings and the amplitude and position of sharp structural variations associated with chemical discontinuities in the stellar interiors. Our results indicate the presence of sharp structural variations with similar properties in all three stars, located near the edge of the gravity-mode propagation cavity and likely associated with the C-O/He transition. We find that these structural variations differ systematically from those of helium core-burning red giant stars, having larger amplitudes and being located at a larger buoyancy radius. This suggests that chemical mixing beyond the adiabatically stratified core into the radiatively stratified layers may be more extensive in subdwarf B stars than in helium core-burning red giants. Alternatively, the stratification of the mixing region beyond the adiabatically stratified core may differ significantly between the two types of stars. The model-independent constraints set on the structural variations inside these three stars are the first of a kind and will be key to enhance the modelling of layers adjacent to stellar convective cores and to test non-canonical stellar evolution channels leading to the formation of hot subdwarf stars.

Figures

Figures reproduced from arXiv: 2505.01381 by the authors.

**Figure 1.** Continuous reduced period spacing signal predicted by the analytical model (Eq. (11)), considering a single step-like glitch [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** KIC 10553698A reduced period spacings as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** KIC 10553698A echelle diagrams displaying the data for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: EPIC 211779126 reduced period spacings as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: KIC 10001893 reduced period spacings as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Physical parameters inferred from the fits with a one step-like glitch model. The stars are identified in the top axis as [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...
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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

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[2] [2]

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[3] [3]

& Sanjayan , S

Baran , A. & Sanjayan , S. 2023, , 73, 21

work page 2023

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