Asteroseismic Diagnostics for Red Giants with Kepler: Measuring epsilon and Small Frequency Separations in 16,000 Stars

Abstract

Asteroseismic studies of red giants have primarily relied on two global parameters: the large frequency separation (Dnu) and the frequency of maximum power (numax). Meanwhile, the p-mode phase shift (epsilon) and small frequency separations (dnu01, dnu02), which offer additional constraints on stellar interiors, remain underexplored due to measurement challenges. Here we develop an automated pipeline based on collapsed echelle diagrams and apply it to about 16,000 Kepler red giants, jointly measuring Dnu, epsilon, dnu01, and dnu02 and assembling the largest homogeneous catalogue of these quantities to date, together with updated Dnu values and formal internal uncertainties. Using this catalogue, we quantify evolutionary trends across the red-giant branch and core-helium-burning phase. We find that dnu02/Dnu stays nearly constant for RGB stars and, for core-helium-burning stars, organises into two sequences that are systematically offset but partially overlap, broadly separating stars in the red-clump and secondary-clump regimes. We also trace the mass- and metallicity-dependent helium-flash transition. Meanwhile, epsilon follows a single Dnu-epsilon relation common to both evolutionary phases. Comparisons with stellar-evolution models reveal systematic offsets in epsilon and dnu01, which we interpret as signatures of near-surface and outer-envelope modelling deficiencies. These comparisons further suggest that dipole-mode small separations are sensitive to mode-dependent surface terms in evolved stars. Overall, our results demonstrate that epsilon and the small separations provide important diagnostics of core structure, convective-boundary mixing, and helium ignition that are complementary to those provided by Dnu and numax alone. The resulting catalogue offers a reference for testing and calibrating future stellar-evolution models.

Figures (8)

• Figure 1: Illustration of our collapsed-échelle method for a representative Kepler red giant (KIC 3530333). (a) Pre-processed PDS (black), super-Gaussian window function (yellow), and the filtered spectrum used for the analysis (red). (b) Échelle diagram constructed from the filtered spectrum using the adopted large frequency separation $\Delta\nu$, shown over a frequency range of $2\Delta\nu$ to reduce edge effects. (c) Metric (see Eq. \ref{['metric']}) as a function of trial $\Delta\nu$ values; open circles show the metric for each trial, and the red cross marks the adopted value corresponding to the maximum metric. (d) Collapsed échelle diagram for the adopted $\Delta\nu$. Each peak corresponds to a mode ridge in panel (b), and the symbols indicate the quantities measured from this diagram ($\epsilon$ and the small separations).
• Figure 2: Frequency separation ratio diagrams and the $\varepsilon$--$\Delta\nu$ relation. (a) $\delta\nu_{01}/\Delta\nu$ versus $\Delta\nu$; (b) $\delta\nu_{02}/\Delta\nu$ versus $\Delta\nu$, and (c) $\varepsilon$ versus $\Delta\nu$. Blue and red points represent RGB and CHeB stars, respectively.
• Figure 3: Distributions of the formal $1\sigma$ uncertainties on the seismic parameters. Panels (a)–(d) show histograms of $\sigma_{\Delta\nu}$, $\sigma_{\varepsilon}$, $\sigma_{\delta\nu_{01}}$, and $\sigma_{\delta\nu_{02}}$ respectively, for RGB (blue) and core-helium-burning (CHeB; red) stars. In each panel, the vertical lines indicate the median uncertainty for each evolutionary phase, and the numerical labels in the upper-right corner list the corresponding median values for the RGB and CHeB samples. The horizontal axis is logarithmic in all panels.
• Figure 4: Comparison of $\Delta\nu$ values derived in this work with those from Yu2018 and Pinsonneault2025. The horizontal axis shows $\Delta\nu_{\text{this work}}$. (a) The ratio $\Delta\nu_{\text{Yu18}} / \Delta\nu_{\text{this work}}$, with RGB stars in blue and CHeB stars in red. (b) The ratio $\Delta\nu_{\text{APOKASC-3}} / \Delta\nu_{\text{this work}}$.
• Figure 5: Comparison between $\varepsilon$ and $\Delta\nu$ for observations and models, colour-coded by stellar mass. Panels (a) and (b) show RGB stars, while panels (c) and (d) show CHeB stars. In each row, the left panel displays the observed values and the right panel shows the corresponding $\pi$-mode model predictions; in the right-hand panels, grey points in the background reproduce the observations for reference.