Asteroseismology of 3,642 Kepler Red Giants: correcting the scaling relations based on detailed modeling

This paper presents a correction to the scaling relations for red giant stars using model-based masses and radii. We measure radial-mode frequencies from Kepler observations for 3642 solar-like oscillators on the red giant branch and use them to characterize the stars with grid-based modeling. We determine fundamental stellar parameters with good precision: the typical uncertainty is 4.5% for mass, 16% for age, 0.006 dex for surface gravity, and 1.7% for radius. We also achieve good accuracy for estimated masses and radii, based on a comparison with those determined for eclipsing binaries. We find a systematic offset of ∼15% in mass and ∼7% in radius between the modeling solutions and the scaling relations. Further investigation indicates that these offsets are mainly caused by a systematic bias in the Δν scaling relation: the original scaling relation underestimates the Δν value by ∼4%, on average, and it is important to correct for the surface term in the calibration. We find no significant offset in the ${\nu }_{\max }$ scaling relation, although a clear metallicity dependence is seen and we suggest including a metallicity term in the formulae. Lastly, we calibrate new scaling relations for red giant stars based on observed global seismic parameters, spectroscopic effective temperatures and metallicities, and modeling-inferred masses and radii.