The effect of the adiabatic assumption on asteroseismic scaling relations for luminous red giants

Although stellar radii from asteroseismic scaling relations agree at the percent level with independent estimates for main sequence and most first-ascent red giant branch stars, the scaling relations over-predict radii at the tens of percent level for the most luminous stars ($R \gtrsim 30 R_{\odot}$). These evolved stars have significantly superadiabatic envelopes, and the extent of these regions increase with increasing radius. However, adiabaticity is assumed in the theoretical derivation of the scaling relations as well as in corrections to the large frequency separation. Here, we show that a part of the scaling relation radius inflation may arise from this assumption of adiabaticity. With a new reduction of Kepler asteroseismic data, we find that scaling relation radii and Gaia radii agree to within at least $2\%$ for stars with $R \lesssim 30 R_{\odot}$, when treated under the adiabatic assumption. The accuracy of scaling relation radii for stars with $50 R_{\odot} \lesssim R \lesssim 100 R_{\odot}$, however, is not better than $10\%-15\%$ using adiabatic large frequency separation corrections. We find that up to one third of this disagreement for stars with $R \approx 100 R_{\odot}$ could be caused by the adiabatic assumption, and that this adiabatic error increases with radius to reach $10\%$ at the tip of the red giant branch. We demonstrate that, unlike the solar case, the superadiabatic gradient remains large very deep in luminous stars. A large fraction of the acoustic cavity is also in the optically thin atmosphere. The observed discrepancies may therefore reflect the simplified treatment of convection and atmospheres.