The moment of core collapse in star clusters with a mass function
  Star clusters with multi-mass components dynamically evolve faster than those
modeled with equal-mass components. Using a series of direct $N$-body
simulations, we investigate the dynamical evolution of star clusters with mass
functions, especially their core collapse time. Multi-mass clusters tend to
behave like systems with a smaller number of particles, which we call the
effective number of particles ($N_{eff}$) and for which $N_{eff} = M/m_{max}$
(here $M$ and $m_{max}$ are the total cluster mass and the mass of the most
massive star in the cluster, respectively). We find that the time of core
collapse is inversely proportional to the mass of the most massive star in the
cluster and analytically confirm that this is because the core collapse of
clusters with a mass function proceeds on the dynamical friction timescale of
the most massive stars. As the mass of the most massive star increases,
however, the core-collapse time, which is observed as a core bounce of the
cluster core from the evolution of the core density or core radius, becomes
ambiguous. We find that in that case the total binding energy of the hard
binaries gives a good diagnosis for determining the moment of core collapses.
Based on the results of our simulations, we argue that the core bounce becomes
ambiguous when the mass of the most massive star exceeds 0.1\% of the total
mass of the cluster.
