An Analytical Model for Galaxy Metallicity: What Do Metallicity Relations Tell Us About Star Formation and Outflow?

We develop a simple analytical model that tracks galactic metallicities governed by star formation and feedback to gain insight from the observed galaxy stellar mass–metallicity relations over a large range of stellar masses and redshifts. The model reveals the following implications of star formation and feedback processes in galaxy formation. First, the observed metallicity relations provide a stringent upper limit for the averaged outflow mass-loading factors of local galaxies, which are ∼20 for galaxies and monotonically decrease to ∼1 for galaxies. Second, the inferred upper limit for the outflow mass-loading factor sensitively depends on whether the outflow is metal-enriched with respect to the interstellar medium metallicity. If half of the metals ejected from supernovae leave the galaxy in metal-enriched winds, the outflow mass-loading factor for galaxies at any mass can barely be higher than ∼10, which puts strong constraints on galaxy formation models. Third, the relatively lower stellar-phase to gas-phase metallicity ratio for lower-mass galaxies indicates that low-mass galaxies are still rapidly enriching their metallicities in recent times, while high-mass galaxies are more settled, which seems to show a downsizing effect in the metallicity evolution of galaxies. The analysis presented in the paper demonstrates the importance of accurate measurements of galaxy metallicities and the cold gas fraction of galaxies at different redshifts for constraining star formation and feedback processes, and demonstrates the power of these relations for constraining the physics of galaxy formation.