  Core collapse is a prominent evolutionary stage of self-gravitating systems.
In an idealised collisionless approximation, the region around the cluster core
evolves in a self-similar way prior to the core collapse. Thus, its radial
density profile outside the core can be described by a power law, $\rho \propto
r^{-\alpha}$. We aim to find the characteristics of core collapse in $N$-body
models. In such systems, a complete collapse is prevented by transferring the
binding energy of the cluster to binary stars. The contraction is, therefore,
more difficult to identify. We developed a method that identifies the core
collapse in $N$-body models of star clusters based on the assumption of their
homologous evolution. We analysed different models (equal- and multi-mass),
most of which exhibit patterns of homologous evolution, yet with significantly
different values of $\alpha$: the equal-mass models have $\alpha \approx 2.3$,
which agrees with theoretical expectations, the multi-mass models have $\alpha
\approx 1.5$ (yet with larger uncertainty). Furthermore, most models usually
show sequences of separated homologous collapses with similar properties.
Finally, we investigated a correlation between the time of core collapse and
the time of formation of the first hard binary star. The binding energy of such
a binary usually depends on the depth of the collapse in which it forms, for
example from $100\,kT$ to $10^4\,kT$ in the smallest equal-mass to the largest
multi-mass model, respectively. However, not all major hardenings of binaries
happened during the core collapse. In the multi-mass models, we see large
transfers of binding energy of $\sim 10^4\,kT$ to binaries that occur on the
crossing timescale and outside of the periods of the homologous collapses.
