for the homologous expansion is $N^{-2/3}$ (Goodman 1984, Giersz and Heggie 1993). Thus we obtain $$<M_c^{-3}>^{-1/3} \propto N^{-2/3},
\label{eq:mcpower}$$ where $<x>$ means the time averaging.

Figure 7 shows the arithmetic mean and the maximum of the core mass after the collapse as functions of $N$. We also plot the quantity $ M_e
=<M_c^{-3}>^{-1/3}$, since we have the theoretical prediction only for this quantity. For maximum core mass, the FP result by Breeden (1994) are plotted for $N=5\times 10^4$ and $10^6$. These values are read by eye from their figure 5. The quantity $M_e$ indeed shows reasonable agreement with the theoretical prediction of equation (\[eq:mcpower\]).

However, the average and maximum core radii shows the dependence noticeably weaker than the theoretical prediction. It should be noted that our $N$-body results for the maximum core mass and the FP results of Breeden (1994) are on one line expressed as $M_c \propto N^{-1/3}$. From figure 7, the core mass at the maximum expansion is approximated as $$M_{c,\max} \sim 0.006\left({N \over 10^5}\right)^{-1/3}M.$$ The slope of the average core size seems to be somewhat steeper.

Qualitatively, it is natural that the ratio between the maximum core mass and $M_e$ is larger for larger $N$, since the 