A Possible Solution for the M/L-[Fe/H] Relation of Globular Clusters in M31. II. The Age-Metallicity Relation

This is the second paper in a series 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 their trend with respect to[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-[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-[Fe/H] for M31 GCs.