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The galaxy-wide stellar initial mass function (gwIMF) of a galaxy independence of its metallicity and star formation rate (SFR) can be calculatedby the integrated galactic IMF (IGIMF) theory. Lacchin et al. (2019) apply theIGIMF theory for the first time to study the chemical evolution of theultra-faint dwarf (UFD) satellite galaxies and failed to reproduce the data.Here, we find that the IGIMF theory is naturally consistent with the data. Weapply the time-evolving gwIMF calculated at each timestep. The number of typeIa supernova explosions per unit stellar mass formed is renormalised accordingto the gwIMF. The chemical evolution of Bo\"otes I, one of the best observedUFD, is calculated. Our calculation suggests a mildly bottom-light andtop-light gwIMF for Bo\"otes I, and that this UFD has the same gas-consumptiontimescale as other dwarfs but was quenched about 0.1 Gyr after formation, beingconsistent with independent estimations and similar to Dragonfly 44. Therecovered best fitting input parameters in this work are not covered in thework of Lacchin et al. (2019), creating the discrepancy between ourconclusions. In addition, a detailed discussion of uncertainties is presentedaddressing how the results of chemical evolution models depend on appliedassumptions. This study demonstrates the power of the IGIMF theory inunderstanding the star-formation in extreme environments and shows that UDFsare a promising pathway to constrain the variation of the low-mass stellar IMF.

Chemical evolution of ultra-faint dwarf galaxies in the self-consistently calculated IGIMF theory