Exploring effects of IMF sampling and SN feedback injection on star formation and metallicity in ultra-faint dwarf galaxies

We examine the impact of various Initial Mass Function (IMF) sampling and supernova (SN) feedback injection methods on the star formation and metal enrichment histories of Ultra-Faint Dwarf (UFD) galaxy analogs. These analogs, characterized by M_vir~10^8 solar mass and M_star < 10^4.5 solar mass at z=0, are simulated using high-resolution cosmological hydrodynamic zoom-in simulations with a gas particle mass resolution of ~63 solar mass. We compare three approaches: the burst model, stochastic IMF sampling, and individual IMF sampling. These methods differ not only in how star particles are sampled following the IMF, but also in how SN feedback energy is injected -- specifically in the degree of temporal and spatial discreteness, with the individual IMF sampling method being the most discrete SN feedback and thus the most physically realistic. Our findings indicate that, despite variations in sampling and SN feedback injection, the final stellar masses across methods are generally similar. However, star formation is notably more continuous in the individual sampling runs due to the weaker suppression from discrete SN events, which enables star formation in denser environments. This sustained star formation leads to more frequent self-enrichment of star-forming gas, resulting in stellar metallicities that are 0.2 to 0.5 dex higher in individual sampling runs compared to burst and stochastic models. These findings highlight the importance of considering both IMF sampling and SN feedback implementation when modeling the star formation and chemical evolution of UFD galaxies.