Testing the radius scaling relation with ${\it Gaia}$ DR2 in the ${\it Kepler}$ field

We compare radii based on ${\it Gaia}$ parallaxes to asteroseismic scaling relation-based radii of $\sim 300$ dwarfs $\&$ subgiants and $\sim 3600$ first-ascent giants from the ${\it Kepler}$ mission. Systematics due to temperature, bolometric correction, extinction, asteroseismic radius, and the spatially-correlated ${\it Gaia}$ parallax zero-point, contribute to a $2\%$ systematic uncertainty on the ${\it Gaia}$-asteroseismic radius agreement. We find that dwarf and giant scaling radii are on a parallactic scale at the $-2.1 \% \pm 0.5 \% {\rm \ (rand.)} \pm 2.0\% {\rm \ (syst.)}$ level (dwarfs) and $+1.7\% \pm 0.3\% {\rm \ (rand.)} \pm 2.0\% {\rm \ (syst.)}$ level (giants), supporting the accuracy and precision of scaling relations in this domain. In total, the $2\%$ agreement that we find holds for stars spanning radii between $0.8R_{\odot}$ and $30 R_{\odot}$. We do, however, see evidence for $\textit{relative}$ errors in scaling radii between dwarfs and giants at the $4\% \pm 0.6\%$ level, and find evidence of departures from simple scaling relations for radii above $30 R_{\odot}$. Asteroseismic masses for very metal-poor stars are still overestimated relative to astrophysical priors, but at a reduced level. We see no trend with metallicity in radius agreement for stars with $-0.5 <$ [Fe/H] $< +0.5$. We quantify the spatially-correlated parallax errors in the ${\it Kepler}$ field, which globally agree with the ${\it Gaia}$ team's published covariance model. We provide ${\it Gaia}$ radii, corrected for extinction and the ${\it Gaia}$ parallax zero-point for our full sample of $\sim 3900$ stars, including dwarfs, subgiants, and first-ascent giants.