The sensitivity and behaviour of the curvature in the échelle diagram of red-giant stars

¹ Heidelberger Institut für Theoretische Studien, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany
² Center for Astronomy (ZAH/LSW), Heidelberg University, Königstuhl 12, 69117 Heidelberg, Germany
³ School of Physics and Astronomy, University of Birmingham, Birmingham B5 2TT, UK
⁴ Department of Astronomy, Yale University, PO Box 208101 New Haven, CT 06520-8101, USA
⁵ Advanced Research Computing, University of Birmingham, Birmingham B5 2TT, UK
⁶ Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 14 October 2025
Accepted: 10 March 2026

Abstract

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.

Aims. 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.

Methods. 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.

Results. 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.

Conclusions. 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.