ratio between the minimum core mass and $M_e$ is larger for larger $N$. The minimum core mass is $O(1/N)$, since the number of particles in the core at the core bounce is always around 10-30. In other words, the energy production rate at the core bounce is larger than the time-averaged rate by a factor proportional to $N$. Thus, roughly speaking, the core mass has to stay at the value larger than the value at the core bounce for most of time. The typical relaxation timescale of the core not at the maximum contraction is, therefore, $N$ times longer than that at the maximum contraction. Since the relaxation time is proportional to $M_c^2$, this implies that the typical core mass is $N^{-1/2}$. This is in fact a reasonable fit for the average core mass in figure 7.

One interesting question is whether it is possible to distinguish the core in the gravothermal oscillation and the core dominated by primordial binaries. The theoretical prediction by Goodman and Hut (1989) gives $r_c/r_h \sim
0.02$. The Fokker-Planck calculation by Gao (1991) gave similar result. On the other hand, $N$-body simulations by McMillan (1990) gave $r_c/r_h \sim 0.2$. McMillan (1990) used 1136 particles. If 