---
DOAJ_listed: '1'
OA_place: publisher
OA_type: diamond
PlanS_conform: '1'
_id: '23045'
abstract:
- lang: eng
  text: "Context. In the convective envelopes of relatively cool (surface temperature
    ≲6700 K) stars, oscillations are excited by turbulent convection. In these so-called
    solar-like oscillators, radial oscillation modes appear at nearly equally spaced
    frequencies. This spacing is referred to as the ‘large-frequency separation’.
    Deviations from equally spaced frequencies are a result of the internal structure
    of a star being different from a sphere of ideal gas at constant temperature.
    Hence, these deviations provide information on the internal structure of the star.\r\n\r\nAims.
    In this work, we investigate the second-order (quadratic) deviation from uniform
    spacing, referred to as curvature. We aim to provide homogeneous values for observed
    red-giant stars, understand differences between the results from observations
    and predictions from stellar models, and reveal the connection between curvature
    and stellar structure.\r\n\r\nMethods. We used Kepler data of red-giant stars
    and computed the curvature for several thousand stars. We compared these to the
    curvature derived from MESA models. We subsequently investigated the trends and
    differences between results from observations and models. Finally, we computed
    sensitivity kernels to identify the stellar region(s) to which the curvature is
    most sensitive and performed a glitch analysis.\r\n\r\nResults. We found that
    the curvature is sensitive to evolutionary phase and mass. Interestingly, the
    observed values and values from models show some discrepancies. Including the
    surface effect in the model frequencies reduces the discrepancies, though it introduces
    a frequency-dependent over- or under-estimation of the curvature from the models
    compared to the observations. From the kernels, we confirmed that the curvature
    is mostly sensitive to the near-surface layers of the star. The glitch analysis
    shows that in theory this provides information on the location and strength of
    the He I and H I ionisation layers.\r\n\r\nConclusions. The curvature provides
    a probe into the near-surface structure of the star. The deviations between the
    curvature derived from observations and models call for improvements in the near-surface
    layers of stellar models."
acknowledgement: "We thank the referee and editor for their comments\r\nand work,
  which improved the manuscript significantly. We acknowledge funding from the ERC
  Consolidator Grant ‘DipolarSound’ (grant agreement #\r\n101000296). SB is partially
  supported by NSF grant AST-2205026. YE acknowledges funding from the Science and
  Technology Facilities Council (STFC)"
article_number: A84
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: S.
  full_name: Hekker, S.
  last_name: Hekker
- first_name: Y.
  full_name: Elsworth, Y.
  last_name: Elsworth
- first_name: S.
  full_name: Basu, S.
  last_name: Basu
- first_name: F.
  full_name: Ahlborn, F.
  last_name: Ahlborn
- first_name: W. H.
  full_name: Ball, W. H.
  last_name: Ball
- first_name: E. P.
  full_name: Bellinger, E. P.
  last_name: Bellinger
- first_name: Lynn
  full_name: Buchele, Lynn
  id: 158c74c1-eed5-11ef-8c05-e45f52925cdc
  last_name: Buchele
- first_name: F.
  full_name: Espinoza-Rojas, F.
  last_name: Espinoza-Rojas
citation:
  ama: Hekker S, Elsworth Y, Basu S, et al. The sensitivity and behaviour of the curvature
    in the échelle diagram of red-giant stars. <i>Astronomy and Astrophysics</i>.
    2026;709. doi:<a href="https://doi.org/10.1051/0004-6361/202557690">10.1051/0004-6361/202557690</a>
  apa: Hekker, S., Elsworth, Y., Basu, S., Ahlborn, F., Ball, W. H., Bellinger, E.
    P., … Espinoza-Rojas, F. (2026). The sensitivity and behaviour of the curvature
    in the échelle diagram of red-giant stars. <i>Astronomy and Astrophysics</i>.
    EDP Sciences. <a href="https://doi.org/10.1051/0004-6361/202557690">https://doi.org/10.1051/0004-6361/202557690</a>
  chicago: Hekker, S., Y. Elsworth, S. Basu, F. Ahlborn, W. H. Ball, E. P. Bellinger,
    Lynn Buchele, and F. Espinoza-Rojas. “The Sensitivity and Behaviour of the Curvature
    in the Échelle Diagram of Red-Giant Stars.” <i>Astronomy and Astrophysics</i>.
    EDP Sciences, 2026. <a href="https://doi.org/10.1051/0004-6361/202557690">https://doi.org/10.1051/0004-6361/202557690</a>.
  ieee: S. Hekker <i>et al.</i>, “The sensitivity and behaviour of the curvature in
    the échelle diagram of red-giant stars,” <i>Astronomy and Astrophysics</i>, vol.
    709. EDP Sciences, 2026.
  ista: Hekker S, Elsworth Y, Basu S, Ahlborn F, Ball WH, Bellinger EP, Buchele L,
    Espinoza-Rojas F. 2026. The sensitivity and behaviour of the curvature in the
    échelle diagram of red-giant stars. Astronomy and Astrophysics. 709, A84.
  mla: Hekker, S., et al. “The Sensitivity and Behaviour of the Curvature in the Échelle
    Diagram of Red-Giant Stars.” <i>Astronomy and Astrophysics</i>, vol. 709, A84,
    EDP Sciences, 2026, doi:<a href="https://doi.org/10.1051/0004-6361/202557690">10.1051/0004-6361/202557690</a>.
  short: S. Hekker, Y. Elsworth, S. Basu, F. Ahlborn, W.H. Ball, E.P. Bellinger, L.
    Buchele, F. Espinoza-Rojas, Astronomy and Astrophysics 709 (2026).
das_tickbox: '0'
date_created: 2026-10-04T22:01:54Z
date_published: 2026-05-01T00:00:00Z
date_updated: 2026-10-07T12:49:31Z
day: '01'
ddc:
- '520'
department:
- _id: LiBu
doi: 10.1051/0004-6361/202557690
external_id:
  arxiv:
  - '2603.16597'
file:
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file_date_updated: 2026-10-07T12:35:39Z
fulldoi: https://doi.org/10.1051/0004-6361/202557690
has_accepted_license: '1'
intvolume: '       709'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
publication: Astronomy and Astrophysics
publication_identifier:
  eissn:
  - 1432-0746
  issn:
  - 0004-6361
publication_status: published
publisher: EDP Sciences
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: yes
title: The sensitivity and behaviour of the curvature in the échelle diagram of red-giant
  stars
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 709
year: '2026'
...