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 Y{\i}ld{\i}z, \c{C}elik \& 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.