Fundamental properties of stars from Kepler and Gaia data: parallax offset and revised scaling relations

Data from the space missions {\it Gaia}, {\it Kepler}, {\it CoRoT} and {\it TESS}, make it possible to compare parallax and asteroseismic distances. From the ratio of two densities \(\rho_{\rm sca}/\rho_{\pi}\), we obtain an empirical relation \(f_{\Delta \nu}\) between the asteroseismic large frequency separation and mean density, which is important for more accurate stellar mass and radius. This expression for main-sequence (MS) and subgiant stars with \(K\)-band magnitude is very close to the one obtained from interior MS models by Yıldız, Çelik \& Kayhan. We also discuss the effects of effective temperature and parallax offset as the source of the difference between asteroseismic and non-asteroseismic stellar parameters. We have obtained our best results for about 3500 red giants (RGs) by using 2MASS data and model values for \(f_{\Delta \nu}\) from Sharma et al. Another unknown scaling parameter \(f_{\nu_{\rm max}}\) comes from the relationship between the frequency of maximum amplitude and gravity. Using different combinations of \(f_{\nu_{\rm max}}\) and the parallax offset, we find that the parallax offset is generally a function of distance. The situation where this slope disappears is accepted as the most reasonable solution. By a very careful comparison of asteroseismic and non-asteroseismic parameters, we obtain very precise values for the parallax offset and \(f_{\nu_{\rm max}}\) for RGs of \(-0.0463\pm0.0007\) mas and \(1.003\pm0.001\), respectively. Our results for mass and radius are in perfect agreement with those of APOKASC-2: the mass and radius of \(\sim\)3500 RGs are in the range of about 0.8-1.8 M\(_{\odot}\) (96 per cent) and 3.8-38 R\(_{\odot}\), respectively.