  The existence of ultra-faint dwarf (UFD) galaxies highlights the need to push
our theoretical understanding of galaxies to extremely low mass. We examine the
formation of UFDs by twice running a fully cosmological simulations of dwarf
galaxies, but varying star formation. One run uses a temperature-density
threshold for star formation, while the other uses an H$_{2}$-based sub-grid
star formation model. The total number of dwarf galaxies that forms is
different by a factor of 2 between the two runs, but most of these are
satellites, leading to a factor of 5 difference in the number of luminous UFD
companions around more massive, isolated dwarfs. The first run yields a 47\%
chance of finding a satellite around a M$_{halo}$ $\sim 10^{10}$ M$_{\odot}$
host, while the H$_2$ run predicts only a 16\% chance. Metallicity is the
primary physical parameter that creates this difference. As metallicity
decreases, the formation of H$_2$ is slowed and relegated to higher-density
material. Thus, our H$_2$ run is unable to form many (and often, any) stars
before reionization removes gas. These results emphasize that predictions for
UFD properties made using hydrodynamic simulations, in particular regarding the
frequency of satellites around dwarf galaxies, the slope of the stellar mass
function at low masses, as well as the properties of ultra-faint galaxies
occupying the smallest halos, are extremely sensitive to the subgrid physics of
star formation contained within the simulation. However, upcoming discoveries
of ultra-faint dwarfs will provide invaluable constraining power on the physics
of the first star formation.
