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

## TL;DR

This paper revises the $
u_{
m max}$ scaling relation in asteroseismology by including a mean molecular weight term, improving the accuracy of stellar property estimations especially for red giants.

## Contribution

It introduces a modified $
u_{
m max}$ scaling relation incorporating mean molecular weight, addressing deviations observed in stellar models across various types.

## Key findings

- Deviations in $
u_{
m max}$ are mainly due to mean molecular weight and $
m \Gamma_1$ terms.
- Errors in the scaling relation minimally affect $
m \log g$ estimates.
- Mass and radius estimates are significantly impacted for red giants.

## Abstract

The scaling relations that relate the average asteroseismic parameters $\Delta \nu$ and $\nu_{\max}$ to the global properties of stars are used quite extensively to determine stellar properties. While the $\Delta \nu$ scaling relation has been examined carefully and the deviations from the relation have been well documented, the $\nu_{\max}$ scaling relation has not been examined as extensively. In this paper we examine the $\nu_{\max}$ scaling relation using a set of stellar models constructed to have a wide range of mass, metallicity, and age. We find that as with $\Delta \nu$, $\nu_{\max}$ does not follow the simple scaling relation. The most visible deviation is because of a mean molecular weight term and a $\Gamma_1$ 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.