the most expanded state, and that the structure of the cluster at the maximum expansion does not depend on the total number of particles. However, it is clear from figure 6 that the cumulative time is roughly proportional to the core radius, and that the coefficient is larger for larger $N$. In other words, the size of the core depends on $N$.

Simple theoretical argument suggests that the core size must depend on $N$. The time-averaged energy production in the core would be the same as that for the stable expansion ([@Goodman1987], [@Heggie1989]). This requirement pose a constraint on the time-averaged core size. The energy production rate is expressed as $${dE  \over dt} \propto M_c \rho_c^2,$$ where $M_c$ and $\rho_c$ are the core mass and core density. Here we ignored the dependence on the velocity dispersion, since the inner part is almost isothermal. For $M_c$ and and the core radius, $r_c$, we have the relation $M_c \propto r_c$, again because the inner part of the cluster is almost isothermal. Thus we have $${dE  \over dt} \propto M_c^{-3}.
\label{eqn:erateonmc}$$ The time-averaged energy production rate must be equal to that of the homologous expansion solution obtained by (Goodman 1984). The core size 