The initial mass function of star clusters that form in turbulent molecular clouds

We simulate the formation and evolution of young star clusters using the combination of smoothed particle hydrodynamics (SPH) simulations and direct N-body simulations. We start by performing SPH simulations of the giant molecular cloud (GMC) with a turbulent velocity field, a mass of 4 × 10⁴ to 5 × 10⁶ M_⊙, and a density between ρ ∼ 1.7 × 10³ and 170 cm^-3. We continue the hydrodynamical simulations for a free-fall time-scale (t_ff ≃ 0.83 and 2.5 Myr), and analyse the resulting structure of the collapsed cloud. We subsequently replace a density-selected subset of SPH particles with stars by adopting a local star formation efficiency proportional to ρ^1/2. As a consequence, the local star formation efficiency exceeds 30 per cent, whereas globally only a few per cent of the gas is converted to stars. The stellar distribution by the time gas is converted to stars is very clumpy, with typically a dozen bound conglomerates that consist of 100-10⁴ stars. We continue to evolve the stars dynamically using the collisional N-body method, which accurately treats all pairwise interactions, stellar collisions and stellar evolution. We analyse the results of the N-body simulations when the stars have an age of 2 and 10 Myr. During the dynamical simulations, massive clusters grow via hierarchical merging of smaller clusters. The shape of the cluster mass function that originates from an individual molecular cloud is consistent with a Schechter function with a power-law slope of β = -1.73 at 2 Myr and β = -1.67 at 10 Myr, which fits to observed cluster mass function of the Carina region. The superposition of mass functions have a power-law slope of ≲ -2, which fits the observed mass function of star clusters in the Milky Way, M31 and M83. We further find that the mass of the most massive cluster formed in a single molecular cloud with a mass of M_g scales with 6.1 M_g^{0.51} which also agrees with recent observation of the GMC and young clusters in M51.