CHARA interferometry and <i>TESS</i> asteroseismology of the core-helium burning red giant <b>κ</b> Cyg

ABSTRACT We present a detailed study of the secondary red clump star, $\kappa$ Cyg, by combining long-baseline visible interferometry using the PAVO beam combiner at the CHARA Array with high-precision asteroseismology from TESS. This dual approach allowed for a stringent test of stellar evolutionary models in the core helium-burning phase, which remains a regime of significant theoretical uncertainty. Using the PAVO interferometric data and fitting the limb-darkened intensity profile directly, we measured $R = 8.65\pm 0.10~\rm R_\odot$. We fitted the spectral energy distribution using Phoenix model atmospheres and calculated $L = 44.46 \pm 1.09~\rm L_\odot$ and $T_{\rm eff} = 5066^{+47}_{-50}~\mathrm{K}$. Using 16 sectors of TESS photometry, we detected clear solar-like oscillations in $\kappa$ Cyg. Through comparison of oscillation frequencies with mesa grids using either predictive mixing or exponential overshooting (OS), we found that models reproducing the oscillation frequencies systematically overestimate the stellar radius, with OS models performing only marginally better. The same models also under-predict the observed dipole-mode period spacing ($\Delta \Pi _1$). By inspecting the phase offset ($\epsilon _\mathrm{ p}$), we conclude that models misrepresent the interior structure of the star. Our results demonstrate that matching envelope-dominated asteroseismic observables alone is insufficient to ensure a correct core or even global structure, and highlight the need for improved treatments of convective boundary mixing in the models of core helium-burning stars.