Changing the $ν_{\max}$ Scaling Relation: The Need For a Mean Molecular Weight Term

The scaling relations that relate the average asteroseismic parameters $Δν$ and $ν_{\max}$ to the global properties of stars are used quite extensively to determine stellar properties. While the $Δν$ scaling relation has been examined carefully and the deviations from the relation have been well documented, the $ν_{\max}$ scaling relation has not been examined as extensively. In this paper we examine the $ν_{\max}$ scaling relation using a set of stellar models constructed to have a wide range of mass, metallicity, and age. We find that as with $Δν$, $ν_{\max}$ does not follow the simple scaling relation. The most visible deviation is because of a mean molecular weight term and a $Γ_1$ term that are commonly ignored. The remaining deviation is more difficult to address. We find that the influence of the scaling relation errors on asteroseismically derived values of $\log g$ are well within uncertainties. The influence of the errors on mass and radius estimates is small for main sequence and subgiants, but can be quite large for red giants.