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REVIEW 4 major objections 5 minor 1 cited by

Anomalously fast core and envelope rotation in red giants

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Asteroseismic survey finds red giants whose envelopes rotate faster than their cores.

desk verdict A genuine large-sample measurement with transparent caveats, but the anomalous-rotator population is not yet validated where it lives. read the letter →

arxiv 2506.06415 v1 pith:S4HKY24K submitted 2025-06-06 astro-ph.SR

Siddharth Dhanpal , Othman Benomar , Shravan Hanasoge , Jim Fuller This is my paper
classification astro-ph.SR
keywords redgiantsclumpstarsasteroseismologystellarrotationcore-envelopedifferentialangularmomentumtransportneuralnetworkbinaryinteractions
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 seeks to establish that the standard two-zone picture of red-giant rotation, in which cores spin much faster than envelopes, is not universal. From space-based photometry of about 1,517 red giants, it reports a population of red-giant and red-clump stars whose envelopes appear to rotate faster than their cores, and a separate group of clump stars whose cores rotate 10 to 20 times faster than the typical clump core. If correct, these anomalies show that angular-momentum transport between core and envelope is not a single universal process, and that some stars either transport angular momentum very weakly or have been spun up by binary interactions. The authors favor weak internal transport for the rapid-core population, noting that their binary models reproduce the fast cores only at the price of over-fast envelopes.

What carries the argument

The load-bearing object is the two-zone rotational splitting relation $\delta\nu_{\rm rot} = \frac{1}{2}\big(\frac{\Omega_{\rm core}}{2\pi}\zeta(\nu) + \frac{\Omega_{\rm env}}{2\pi}[1-\zeta(\nu)]\big)$, where the mixing fraction $\zeta(\nu)$ interpolates between pressure-dominated envelope modes and gravity-dominated core modes. Envelope rates come mainly from $\ell=2$ p-dominated splittings and core rates from $\ell=1$ mixed-mode splittings. On top of that sits a convolutional-LSTM network trained as an ordinal classifier on five million synthetic spectra, which outputs probability distributions for the core rotation, envelope rotation, and four seismic parameters; MCMC fits verify the network's inferences for the anomalous stars, and the rotation-ratio uncertainties are computed from the two marginal distributions under a conservative negative-correlation assumption.

What would settle it

Apply the paper's own MCMC forward-fitting, without the neural network, to every claimed anomalous rotator and check whether the $\ell=2$ p-mode splittings remain larger than the core-dominated $\ell=1$ splittings; if most inverted-ratio cases disappear or shift below one, the anomaly population is an artifact of the machine-learning prior.

Watch

Extended reading notes

Core claim

The central claim is that exceptions to the seismic rotation ordering of red giants are real and form a distinct population. Using a neural network trained on synthetic oscillation spectra and confirmed with MCMC fits for representative stars, the paper infers core and envelope rotation for 1,517 red giants and finds a systematic evolution of the envelope-to-core rotation ratio: it declines along the red-giant branch and then rises to values near 0.01 to 4 in the clump phase. Within that spread sit stars with the envelope-to-core ratio above 1 and clump stars with core rotation near one microhertz, that is, 10 to 20 times the median clump core rate. The paper argues these anomalies challenge current angular-momentum-transport models, that a weaker magnetic-transport prescription can explain the fast cores, and that binary spin-up offers an alternative that is in tension with the observed slow envelopes.

Load-bearing premise

The neural network is trusted to be unbiased precisely for the unusual stars it was not validated against: the synthetic training data cover normal rotation profiles, and the paper itself warns that stars with unexpected internal rotation may receive biased or inaccurate predictions.

Editorial extensions

If this is right

  • If the anomalous rotators are real, the usual assumption that red-giant cores always rotate faster than their envelopes must give way to a picture with at least two rotation channels: normal strong-transport stars and anomalous weak-transport or spun-up stars.
  • The fast clump cores, near $\Omega_{\rm core}/2\pi\sim1\,\mu$Hz, lie close to original magnetic-dynamo predictions and about ten times above enhanced-transport models, implying that some stars avoid efficient internal angular-momentum transport.
  • Binary tidal spin-up and merger models reproduce fast cores but predict envelopes rotating far faster than observed, so the data favor weak transport over binarity for the rapid-core population.
  • Anomalous rotators show no unusual lithium, carbon-to-nitrogen, or binarity indicators, so binarity does not obviously identify them, and larger samples will be needed to detect any weak abundance or activity correlation.
  • If the fast cores persist into later evolutionary phases, they would naturally produce rapidly rotating white dwarfs and, in more massive stars, could yield energetic supernovae and gamma-ray bursts.

Reading between the lines

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

  • A sharper test of the neural-network interpretation would be to retrain it on synthetic spectra with the prior support for $\Omega_{\rm env}>\Omega_{\rm core}$ removed and check whether the anomalous population still emerges; without such a test, part of the signal could in principle be a regression-to-prior artifact.
  • The paper computes ratio uncertainties from independent core and envelope marginals, but the joint posterior is likely correlated, so the reported probabilities $P(\Omega_{\rm env}/\Omega_{\rm core}>1)$ may overstate or understate confidence for individual stars.
  • Applying the same network to TESS and PLATO red giants would provide a population-level check of how the incidence of envelope-super-rotation varies with evolutionary state, independent of any single Kepler spectrum.
  • The anomalous-rotator list could be cross-matched with asteroseismic catalogs of subgiants and early red-giant-branch stars to see whether the inverted ratio is acquired at a specific evolutionary transition or inherited from main-sequence binaries.
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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

4 major / 5 minor

Summary. The paper develops a convolutional neural network that infers core and envelope rotation rates, along with other seismic parameters, directly from Kepler power spectra of red giants. The network is trained on ~5 million synthetic spectra built from asymptotic theory, applied to ~21,000 Kepler targets, and a 'confident' subset of 1,517 stars is analyzed. The authors report that most measurements reproduce existing core-rotation catalogs, but they also identify two claimed anomaly populations: red giants and clump stars with envelope rotation faster than core rotation, and clump stars with rapidly rotating cores (Ω_core/2π near 1 μHz). They verify three individual cases with MCMC fits and visual splitting inspection, and they propose weak angular-momentum transport or binary interactions as explanations, supported by MESA single-star and binary models. The central claim is that these anomalies challenge standard angular-momentum transport theory and have implications for compact remnant spins.

Significance. If the anomalies are real, the paper would be significant: it would extend the established two-zone rotation picture for red giants (Ω_env < Ω_core in essentially all published stars) to include inverted rotation profiles and clump cores rotating 10-20 times faster than the median, with consequences for angular-momentum transport prescriptions and the spins of white dwarfs and neutron stars. The paper has several concrete strengths: the training simulator is available, the MCMC follow-up of three anomalous stars is a good-faith attempt at external verification, the visual inspection of splittings adds transparency, and the comparison with the Gehan et al. (2018) catalog provides a quantitative benchmark for the bulk of the sample. These strengths make the paper a useful contribution even if the anomaly population requires further validation. However, the population-level claims rest entirely on the reliability of the CNN in a regime where the authors themselves state it may be biased, and the external validation does not cover that regime; therefore the significance is conditional on an unresolved validation gap.

major comments (4)
  1. [Section 2.2] The load-bearing population-level claims require the CNN to be accurate for the anomalous stars, but the paper itself states that the network performs reliably only within the simulated parameter space and that 'unexpected internal rotation' may yield biased predictions. The anomalous rotators are precisely the 'unexpected internal rotation' cases, and Figure 33 shows that none of the envelope-super-rotation stars overlap with the Gehan et al. (2018) catalog, so there is no external validation for them. The synthetic validation in Figure 2 and Appendix D is averaged over the training distribution, not conditioned on the anomalous region, and the stress test in Appendix D uses Δν = 16 μHz, ΔΠ = 85 s, which is the RGB regime rather than the clump regime. The authors should validate the network on synthetic spectra drawn from the anomalous parameter space (low Δν, ΔΠ > 150 s, Ω_env > Ω_core, Ω_core/2π near 1 μHz) and report the accuracy and calibration in that region; without this, the anomaly population may be an artifact of extrapolation.
  2. [Section 3.1] The degeneracy between rotational splitting and mixed-mode spacing, described in Section 3.1, is a concrete mechanism by which normal stars could be mapped into the anomalous region. The authors note that a spectrum with splitting equal to one-quarter of the mixed-mode spacing at i=90° can look like one-third splitting at i=55°, and they use this to explain discrepancies with Gehan et al. (2018). However, the same degeneracy could affect the anomalous stars, whose splittings are large relative to the mixed-mode spacing, and for which the network is trained on synthetic mode patterns that may not cover all inclination-splitting combinations. The MCMC fits to three stars do not resolve this at the population level, since MCMC uses the same asymptotic model and the same priors. I request a dedicated synthetic test that quantifies how often normal rotation configurations are misclassified into the anomalous region as a function of SNR, inclination, and mixed-mode spacing.
  3. [Section 3.1] The validation against Gehan et al. (2018) shows that only 59.2% of the 426 overlapping confident stars agree within 20% of the 1:1 relation, while 25.3% fall in no defined proximity zone. This is a substantial disagreement rate, and the paper attributes most discrepancies to the network's use of inclination-dependent amplitudes. Yet this explanation is not demonstrated for the 25.3% category. More importantly, the anomalous stars are not present in this overlap sample at all (Figure 33), so the external validation does not establish reliability for the anomaly population. The authors should show whether the anomalous candidates, when they do have any published counterpart or when re-analyzed with an independent method (e.g., a classical peak-bagging fit), still retain Ω_env > Ω_core and fast clump cores.
  4. [Section 2.2] The selection criteria (pmax thresholds and i > 45°) and the bin sizes for the ordinal classification are presented as choices, but there is no sensitivity analysis. For instance, the minimum pmax(Ω_env/2π) > 0.2 is a low bar for a network that outputs probabilities over bins, and the inclination cut at 45° may interact with the degeneracy noted in Section 3.1. I request that the authors demonstrate that the anomalous population is stable under reasonable variations of these thresholds and bin sizes, e.g., by showing the number of anomalous stars as a function of the pmax thresholds. Without such a test, it is unclear whether the anomaly population is robust or a threshold artifact.
minor comments (5)
  1. [Abstract] The phrase 'anomalously fast' is used for both envelope-super-rotation and fast clump cores, but the two phenomena are physically distinct; using separate terms (e.g., 'inverted rotation ratio' and 'fast clump cores') would improve clarity.
  2. [Section 2.2] The sentence 'The network’s distributions, depicted in gray, were juxtaposed against the posterior distributions obtained from MCMC in red' in Figure 1 uses 'gray' and 'red' but the figure is reproduced in black and white in the arXiv version; please ensure the figure caption and text are consistent with the actual rendering.
  3. [Section 2.1] The training range for Ω_core/2π is given as 0.005-2.8 μHz and for Ω_env/2π as 0.005-0.4 μHz. The paper should state whether the training distribution is uniform in these ranges and whether any samples were generated with Ω_env > Ω_core, since the anomaly population requires that configuration.
  4. [Appendix C] The computation of the rotation-rate ratio assumes independent distributions for Ω_core and Ω_env to obtain the largest uncertainty interval, as shown in Figure 27. This is a conservative choice, but the paper should note that the actual ratio distribution and the probability P(Ω_env/Ω_core > 1) may be different if a physical correlation exists.
  5. [Table 3] Several entries in Table 3 have very large asymmetric uncertainties (e.g., KIC 10219075 with Ω_core/2π = 0.11 +1.88 -0.07 μHz), and the dagger flag for Nyquist proximity is helpful but only applied to a few rows. The paper should consider flagging or excluding stars with relative uncertainties exceeding a threshold, or at least clearly list them separately.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation by construction; the anomalous-rotator claim rests on network extrapolation beyond its externally validated regime, which is an accuracy risk rather than circularity.

full rationale

The paper's derivation chain is not circular. The CNN is trained on synthetic spectra generated from an asymptotic forward model (Eqs. 1-2) and then applied to Kepler spectra; the reported core and envelope rotation rates are outputs of this inversion, not quantities fitted to the observed anomalies. The network is validated on held-out synthetic data (Figure 2) and, more importantly, against external catalogs: 59.2% of confident core-rotation measurements agree with Gehan et al. (2018) within 20% (Section 3.1), and magnetic-star comparisons with Li et al. (2023) show no systematic bias. Individual anomalous stars are re-examined with MCMC fits and by visual inspection of rotational splittings (Figures 6-10), so the anomalies are not read off the network alone. The self-citations to Dhanpal et al. (2022, 2023) and Benomar (2023) provide provenance for the simulator rather than a forced conclusion, and the central claim does not reduce to those citations. The main caveat is explicitly stated in Section 2.2: the network "performs reliably only for stars that fall within this simulated parameter space" and "unexpected internal rotation" may bias predictions. Appendix E (Figure 33) admits that none of the anomalous rotators overlap the Gehan et al. catalog, so the anomaly population lacks direct external validation. This is a genuine extrapolation and robustness concern, but it is not circularity: the predictions are not equivalent to the training inputs by construction, and the paper does not fit a parameter to a subset and then rename it a prediction. Therefore the circularity score is low.

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

The central measurement rests on the synthetic training distribution and the two-zone rotation ansatz; every claimed rotation rate inherits these assumptions. No new physical entity is introduced; the 'anomalous rotator' label is a classification of measurements, not a new object.

free parameters (5)
  • Confidence thresholds (pmax) and inclination cut = pmax(dnu)>0.3, pmax(q)>0.15, q_pred>0.05, pmax(dPi)>0.2, pmax(Omega_core)>0.2, pmax(Omega_env)>0.2, iota>45 deg
    These thresholds select the 1517-star 'confident' sample; they are chosen by hand and directly determine which stars enter the anomaly statistics.
  • Output bin sizes for ordinal classification = 0.1 uHz (dnu), 2.5/7 s (dPi), 0.02 (q), 0.025 uHz (Omega_env), 0.095 uHz (Omega_core), 5 deg (iota)
    Bin sizes set the resolution of the classification; chosen so the network precision matches classical uncertainties (Section 2.2).
  • Training rotation ranges = core 0.005-2.8 uHz; envelope 0.005-0.4 uHz
    Uniform ranges chosen to cover published results (Table 1). If real stars fall outside these ranges, the network cannot represent them, which directly bounds the anomaly search space.
  • Low/high frequency split of training data = DeltaNu = 9 uHz
    Heuristic classification of red giants vs clump stars for the two training datasets (Section 2.1).
  • MESA binary model inputs = Spin-up to Omega_env/2pi = 1.5e-2 uHz near RGB tip; specific angular momentum j = 1e17 cm2/s for sdB merger; Rappaport…
    Exploratory parameters used to test whether binary scenarios can explain the anomalies; the models fail to match observed envelope rates, weakening the binary hypothesis.
assumptions (5)
  • standard math Mixed-mode frequencies follow asymptotic theory (Eq. 2, tan product equation)
    Used to generate all synthetic training spectra and MCMC models; standard in asteroseismology, cited to Mosser et al. 2015, Farnir et al. 2021, Ong and Gehan 2023.
  • domain assumption Two-zone rigid rotation with mixing fraction zeta (Eq. 1)
    Assumes uniform core and envelope rotation with no latitudinal differential rotation or magnetic asymmetry in splittings; the entire core/envelope decomposition rests on this.
  • domain assumption ell=2 splittings measure envelope rotation with minor deep-layer contamination
    Section 2.1, citing Ahlborn et al. 2020. If the contamination is not minor, envelope-faster-than-core cases could be artifacts of core sensitivity.
  • domain assumption Softmax outputs are Bayesian posteriors
    Section 2.2, citing Richard and Lippmann 1991; pmax is used as a confidence metric. Calibration is tested only on the same synthetic simulator family (Appendix D).
  • domain assumption Noise is chi-squared with 2 dof and mode heights follow the visibility model (Eq. 3)
    Used to build synthetic noise realizations and the MCMC likelihood (Section 2.3).

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Cite this review

Pith. "Pith review of Anomalously fast core and envelope rotation in red giants." pith.science (2026). https://pith.science/paper/S4HKY24K

@misc{pith2026250606415,
  author       = {Pith},
  title        = {Pith review of: Anomalously fast core and envelope rotation in red giants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4HKY24K}},
  note         = {Machine review of arXiv:2506.06415}
}
abstract

Red giants undergo dramatic and complex structural transformations as they evolve. Angular momentum is transported between the core and envelope during this epoch, a poorly understood process. Here, we infer envelope and core rotation rates from Kepler observations of $\sim$1517 red giants. While many measurements are consistent with the existing studies, our investigation reveals systematic changes in the envelope-to-core rotation ratio and we report the discovery of anomalies such as clump stars with rapidly rotating cores, and red giants with envelopes rotating faster than their cores. We propose binary interactions as a possible mechanism by which some of these cores and envelopes are spun up. These results pose challenges to current theoretical expectations and can have major implications for compact remnants born from stellar cores.

Figures

Figures reproduced from arXiv: 2506.06415 by the authors.

Figure 1
Figure 1. Diagrammatic representation of the neural network applied to analyze the oscillation spectrum KIC 6191190. The trained neural network takes as input the normalized power spectrum and outputs the probability distributions of various seismic parameters, such as the large-frequency separation (∆ν), large-period separation (∆Π), coupling constant (q), core rotation (Ωcore/2π), envelope rotation (Ωenv/2π), and inclinatio… view at source ↗
Figure 2
Figure 2. Results on synthetic spectra. Shown in panels (a) and (c) are the inferred values of Ωcore/2π and Ωenv/2π plotted against the actual injected values for red giants (R) and clumps (C). Each point is associated with grey lines representing 1-σ uncertainties. In panel (a), the red-solid, blue-dotted, and green-dashed lines indicate the 1:1, 1:2, and 2:1 ratios between actual and inferred values, respectively. Some 96% … view at source ↗
Figure 3
Figure 3. Comparison of core rotation rate measurements from this work with those of Gehan et al. (2018) for 842 Kepler red giants. Confident neural-network measurements are highlighted in yellow; the solid red line marks one-to-one agreement, while blue dotted and green dashed lines denote 2:1 and 1:2 ratios, respectively. Dark-yellow circles indicate stars with three rotational components detected by Gehan et al. (2018); li… view at source ↗
Figures from the paper (31 more)
Figure 4
Figure 4. Figure 4: (a) Same as [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Evolution of core and envelope rotation. Plotted are (a) core rotation as a function of large frequency separation (∆ν); (b) envelope rotation as a function of ∆ν; (c) envelope-to-core rotation ratio versus ∆ν; and (d) envelope-to-core rotation ratio versus stellar rad…
Figure 6
Figure 6. Figure 6: MCMC fit to KIC 11615944. The best-fit model is shown in comparison with the observations in panel (a). Panels (b), (c), (d), (e), (f), and (g) show the posterior distributions of Ωcore, Ωenv, ι, ∆Π, q and Ωenv/Ωcore, respectively, in red-dashed lines. In addition to t…
Figure 7
Figure 7. Figure 7: Comparison between splittings of an ℓ = 2 pressure-dominated mode and a gravity-dominated ℓ = 1 mixed mode in KIC 11615944. In panel (a), the ℓ = 2 pressure-dominated mode is shown, while panel (b) depicts the gravity-dominated ℓ = 1 mixed mode. The red line depicts th…
Figure 8
Figure 8. Figure 8: MCMC fit to KIC 11546972. The best-fit model is shown in comparison with the data in panel (a). Panels (b), (c), (d), (e), (f), and (g) show in red dashed lines the posterior distributions of Ωcore, Ωenv, ι, ∆Π, q and Ωenv/Ωcore, respectively. In addition to these dist…
Figure 9
Figure 9. Figure 9: Comparison between the splittings of an ℓ = 2 pressure dominated mode and a gravity-dominated mixed mode in KIC 11546972. In panel (a), the ℓ = 2 pressure dominated mode is shown, while panel (b) depicts the gravity-dominated mixed mode. The red line depicts the best-f…
Figure 10
Figure 10. Figure 10: MCMC fit to KIC 6695665. The best-fit model is shown in comparison with the observations in panel (a). Panels (b), (c), (d), (e), and (f) show in red-dashed lines the posterior distributions of Ωcore, Ωenv, ∆Π, q, and ι, respectively. In addition to these distribution…
Figure 11
Figure 11. Figure 11: Comparison between the inferred rotation rates from the neural network (red Cs and gray Rs) and several theoretical models (indicated in the legend). In panel (a), the red-giant and sub-dwarf binary merger models potentially account for the population of rapid clump r…
Figure 12
Figure 12. Figure 12: Variation of core rotation (left) and rotation rate ratio (right) with RUWE (Re-normalized unit-weight error) parameter obtained from GAIA. RUWE significantly greater than 1 can potentially indicate non-single systems. The color indicates the number of resolved compan…
Figure 13
Figure 13. Figure 13: (a) Core rotation rate vs. Lithium abundance distribution in 29 red clump stars. This plot focuses exclusively on clump stars since we identify anomalous core rotators mainly in the clump phase. (b) Rotation rate ratio vs. Lithium abundance distribution in 166 stars. …
Figure 14
Figure 14. Figure 14: (a) Core rotation rate vs. [C/N] abundance distribution in 72 red clump stars. (b) Rotation-rate ratio vs. [C/N] abundance distribution in 358 stars. These plots indicate that anomalous rotators do not exhibit unusual [C/N]. port prescriptions, but typically lie withi…
Figure 15
Figure 15. Figure 15: (a) Predicted mass using [C/N] abundance Martig et al. (2016) vs. estimated mass using scaling relations in 72 clump stars, where the colors of the points indicate core rotation rates. (b) Predicted mass (Martig et al. 2016) vs. estimated mass using scaling relations …
Figure 16
Figure 16. Figure 16: Comparison between core rotation-rate trends derived from the neural network (1471 stars plotted in panel (a)) and those from established catalogs (1190 stars shown in panel (b); (Mosser et al. 2017; Gehan et al. 2018; Tayar et al. 2019)). The rotation rates are plott…
Figure 17
Figure 17. Figure 17: Comparison of envelope rotation-rate trends derived from the neural network (1471 stars, shown in panel (a)) and those available in established catalogs (361 stars displayed in panel (b); (Ceillier et al. 2017; Tayar et al. 2019)). The rotation rates are plotted as a …
Figure 18
Figure 18. Figure 18: Comparison of rotation-rate-ratio trends from a neural network (1471 stars, plotted in panel (a)) and an existing catalog (33 stars shown in panel (b); (Tayar et al. 2019)). Rotation rates are plotted as a function of ∆ν. from the best-fit model. The ratio of splittin…
Figure 19
Figure 19. Figure 19: A comprehensive comparison of the best-fit model (depicted in brown) and the observed power (depicted in gray) for the star KIC 11615944. Each panel in the analysis corresponds to a different frequency range. The dark segments in the data indicate ℓ = 0 modes identifi…
Figure 20
Figure 20. Figure 20: Splitting and width as a function of ℓ = 1 mode frequencies in the range 60.5-64.5 µHz, as obtained in the MCMC fit for the star KIC 11615944 as detailed in [PITH_FULL_IMAGE:figures/full_fig_p024_20.png]
Figure 21
Figure 21. Figure 21: Same as [PITH_FULL_IMAGE:figures/full_fig_p025_21.png]
Figure 22
Figure 22. Figure 22: Same as Figures 19 and 21, but for KIC 6695665. we may draw samples from these distributions and cal￾culate the ratios of these samples. As shown in [PITH_FULL_IMAGE:figures/full_fig_p026_22.png]
Figure 23
Figure 23. Figure 23: This plot shows different ℓ = 1 modes obtained by best-fit model compared to the relative SNR of mixed modes for the star KIC 6695665. The power spectrum is divided by the sum of noise profile, ℓ = 0 and ℓ = 2 modes obtained by the best fit model to calculate relative…
Figure 24
Figure 24. Figure 24: Comparison of three models with different rotation rates to the data for the star KIC 6695665. Models (a) and (b) show core rotation rates of 0.084 and 0.3 µHz respectively whereas panel (c) compares best-fit model for which the rotation rate is 1.16 µHz. All the para…
Figure 25
Figure 25. Figure 25: Comparison of three models with different inclination angles to the data for the star KIC 6695665. Models (a) and (b) show core rotation rates of 30◦ and 50◦ respectively whereas panel (c) compares best-fit model for which the inclination angle is 85.4 ◦ . All the par…
Figure 26
Figure 26. Figure 26: Comparison of nine models with lower inclination angles and core rotation rates with observations for star KIC 6695665. The data is shown in grey, while models are marked red. These models do not align as closely with the data as the best-fit model shown in figures 22…
Figure 27
Figure 27. Figure 27: Distributions of the ratios Ωenv/Ωcore for KIC 11600442 in three different cases of correlation (a), (b), and (c). The distributions of Ωenv and Ωcore are similar to each other, as inferred by the machine, as shown in plots (e) and (f). However, the distributions of t…
Figure 28
Figure 28. Figure 28: This plot presents results from an uncertainty calibration test, illustrating the frequency with which the ground truth lies within the x% confidence interval of inferred values for both core and envelope rotation rates, where x ranges from 0 to 100. The plot includes…
Figure 29
Figure 29. Figure 29: (a) The true distribution of core rotation rates across 2000 simulated samples, alongside the distribution of inferred rotation rates for these same samples.(b) Inferred core rotation rates plotted against the actual core rotation rates for all 2000 samples. (c) Distr…
Figure 30
Figure 30. Figure 30: Plots (a) and (c) display neural network inferences on a synthetic example, compared to a Gaussian distribution centered around ground truth. The widths of these Gaussian distributions are derived from the standard distribution of errors shown in [PITH_FULL_IMAGE:fig…
Figure 31
Figure 31. Figure 31: (a) The distributions of z-scores for core and envelope rotation rate inferences relative to Gaussian distributions centered on the injected simulation values. Over 80% of the inferences have z-scores less than 1. (b) Distribution of z-scores for Kepler red giants, ba…
Figure 32
Figure 32. Figure 32: Inferred seismic parameters against the ground truth values for the simulations presented in [PITH_FULL_IMAGE:figures/full_fig_p034_32.png]
Figure 33
Figure 33. Figure 33: Distribution of rotation rate ratios for all stars listed in [PITH_FULL_IMAGE:figures/full_fig_p035_33.png]
Figure 34
Figure 34. Figure 34: Comparison of the inferences derived from the neural network with published measurements Vrard et al. (2016) for global seismic parameters ∆ν, ∆Π, and νmax in the anomalous rotators. The black squares in panels (a), (c), and (e), along with the black-dashed histograms…

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