  Most stars are formed as star clusters in galaxies, which then disperse into
galactic disks. Upcoming exascale supercomputational facilities will enable
performing simulations of galaxies and their formation by resolving individual
stars (star-by-star simulations). This will substantially advance our
understanding of star formation in galaxies, star cluster formation, and
assembly histories of galaxies. In previous galaxy simulations, a simple
stellar population approximation was used. It is, however, difficult to improve
the mass resolution with this approximation. Therefore, a model for forming
individual stars that can be used in simulations of galaxies must be
established. In this first paper of a series of the SIRIUS (SImulations
Resolving IndividUal Stars) project, we demonstrate a stochastic star formation
model for star-by-star simulations. An assumed stellar initial mass function
(IMF) is randomly assigned to newly formed stars. We introduce a maximum search
radius to assemble the mass from surrounding gas particles to form star
particles. In this study, we perform a series of N-body/smoothed particle
hydrodynamics simulations of star cluster formations from turbulent molecular
clouds and ultra-faint dwarf galaxies as test cases. The IMF can be correctly
sampled if a maximum search radius that is larger than the value estimated from
the threshold density for star formation is adopted. In small clouds, the
formation of massive stars is highly stochastic because of the small number of
stars. We confirm that the star formation efficiency and threshold density do
not strongly affect the results. We find that our model can naturally reproduce
the relationship between the most massive stars and the total stellar mass of
star clusters. Herein, we demonstrate that our models can be applied to
simulations varying from star clusters to galaxies for a wide range of
resolutions.
