Dynamical equivalence, the origin of the Galactic field stellar and
  binary population, and the initial radius--mass relation of embedded clusters
  In order to allow a better understanding of the origin of Galactic field
populations, dynamical equivalence of stellar-dynamical systems has been
postulated by Kroupa and Belloni et al. to allow mapping of solutions of the
initial conditions of embedded clusters such that they yield, after a period of
dynamical processing, the Galactic field population. Dynamically equivalent
systems are defined to initially and finally have the same distribution
functions of periods, mass ratios and eccentricities of binary stars. Here we
search for dynamically equivalent clusters using the {\sc mocca} code. The
simulations confirm that dynamically equivalent solutions indeed exist. The
result is that the solution space is next to identical to the radius--mass
relation of Marks \& Kroupa, $\left( r_h/{\rm pc} \right)=
0.1^{+0.07}_{-0.04}\, \left( M_{\rm ecl}/{\rm M}_\odot \right)^{0.13\pm0.04}$.
This relation is in good agreement with the observed density of molecular cloud
clumps. According to the solutions, the time-scale to reach dynamical
equivalence is about 0.5~Myr which is, interestingly, consistent with the
lifetime of ultra-compact HII regions and the time-scale needed for gas
expulsion to be active in observed very young clusters as based on their
dynamical modelling.
