# A Possible Solution for the $M/L-\mathrm{[Fe/H]}$ Relation of Globular Clusters in M31. II. the Age-Metallicity Relation

## TL;DR

This paper proposes a model that explains the observed mass-to-light ratio and metallicity relation of M31 globular clusters by considering an age-metallicity anti-correlation, evolving stellar populations, and variable initial mass functions.

## Contribution

It introduces an empirical age-metallicity relation and combines it with dynamical SSP models and variable IMFs to match observed $M/L$ ratios of M31 GCs.

## Key findings

- The $M/L- ext{[Fe/H]}$ relation aligns with observations when considering an anti-correlation between age and metallicity.
- A density- and metallicity-dependent top-heavy IMF improves model accuracy.
- Combining stellar evolution, dynamical evolution, and variable IMFs explains the observed trends.

## Abstract

This is the second of a series of papers in which we present a new solution to reconcile the prediction of single stellar population (SSP) models with the observed stellar mass-to-light ($M/L$) ratios of globular clusters (GCs) in M31 and its trend with respect to $\mathrm{[Fe/H]}$. In the present work our focus is on the empirical relation between age and metallicity for GCs and its effect on the $M/L$ ratio. Assuming that there is an anti-correlation between the age of M31 GCs and their metallicity, we evolve dynamical SSP models of GCs to establish a relation between the $M/L$ ratio (in the $V$ and $K$ band) and metallicity. We then demonstrate that the established $M/L-\mathrm{[Fe/H]}$ relation is in perfect agreement with that of M31 GCs. In our models we consider both the canonical initial mass function (IMF) and the top-heavy IMF depending on cluster birth density and metallicity as derived independently from Galactic GCs and ultra-compact dwarf galaxies by Marks et al. Our results signify that the combination of the density- and metallicity-dependent top-heavy IMF, the anti-correlation between age and metallicity, stellar evolution and standard dynamical evolution yields the best possible agreement with the observed trend of $M/L-\mathrm{[Fe/H]}$ for M31 GCs.