A comprehensive model for the formation and evolution of the faintest Milky Way dwarf satellites

In this study, we modify the semi-analytic model Galacticus in order to accurately reproduce the observed properties of dwarf galaxies in the Milky Way. We find that reproducing observational determinations of the halo occupation fraction and mass-metallicity relation for dwarf galaxies requires us to include H$_2$ cooling, an updated UV background radiation model, and to introduce a model for the metal content of the intergalactic medium. By fine-tuning various model parameters and incorporating empirical constraints, we have tailored the model to match the statistical properties of Milky Way dwarf galaxies, such as their luminosity function and size$-$mass relation. We have validated our modified semi-analytic framework by undertaking a comparative analysis of the resulting galaxy-halo connection. We predict a total of $300 ^{+75} _{-99}$ satellites with an absolute $V$-band magnitude (M$_{V}$) less than $0$ within $300$ kpc from our Milky Way-analogs. The fraction of subhalos that host a galaxy at least this bright drops to $50\%$ by a halo peak mass of $\sim 8.9 \times 10^{7}$ M$_{\odot}$, consistent with the occupation fraction inferred from the latest observations of Milky Way satellite population.