The sensitivity and behaviour of the curvature in the \'echelle diagram of red-giant stars

In the convective envelopes of relatively cool 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. In this work, we investigate the second-order deviation from uniform spacing, referred to as curvature. We aim to provide homegeneous 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. 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 to which the curvature is most sensitive and performed a glitch analysis. We found that the curvature is sensitive to evolutionary phase and mass. The observed values and values from models show discrepancies. The glitch analysis shows that in theory this provides information on the location and strength of the HeI and HI ionisation layers. The curvature provides a probe into the near-surface structure of the star. The deviations between the curvature derived from observations and models call henceforth for improvements in the near-surface layers of stellar models.