Establishing the accuracy of asteroseismic mass and radius estimates of giant stars. I. Three eclipsing systems at [Fe/H]~ -0.3 and the need for a large high-precision sample

We aim to establish and improve the accuracy level of asteroseismic estimates of mass, radius, and age of giant stars. This can be achieved by measuring independent, accurate, and precise masses, radii, effective temperatures and metallicities of long period eclipsing binary stars with a red giant component that displays solar-like oscillations. We measured precise properties of the three eclipsing binary systems KIC 7037405, KIC 9540226, and KIC 9970396 and estimated their ages be $5.3\pm0.5$, $3.1\pm0.6$, and $4.8\pm0.5$ Gyr. The measurements of the giant stars were compared to corresponding measurements of mass, radius, and age using asteroseismic scaling relations and grid modeling. We found that asteroseismic scaling relations without corrections to $\Delta\nu$ systematically overestimate the masses of the three red giants by 11.7%, 13.7%, and 18.9%, respectively. However, by applying theoretical correction factors $f_{\Delta\nu}$ according to Rodrigues et al. (2017), we reached general agreement between dynamical and asteroseismic mass estimates, and no indications of systematic differences at the precision level of the asteroseismic measurements. The larger sample investigated by Gaulme et al. (2016) showed a much more complicated situation, where some stars show agreement between the dynamical and corrected asteroseismic measures while others suggest significant overestimates of the asteroseismic measures. We found no simple explanation for this, but indications of several potential problems, some theoretical, others observational. Therefore, an extension of the present precision study to a larger sample of eclipsing systems is crucial for establishing and improving the accuracy of asteroseismology of giant stars.