Dynamics versus structure: breaking the density degeneracy in star formation

The initial density of individual star-forming regions (and by extension the birth environment of planetary systems) is difficult to constrain due to the "density degeneracy problem": an initially dense region expands faster than a more quiescent region due to two-body relaxation and so two regions with the same observed present-day density may have had very different initial densities. We constrain the initial densities of seven nearby star-forming regions by folding in information on their spatial structure from the $\mathcal{Q}$-parameter and comparing the structure and present-day density to the results of $N$-body simulations. This in turn places strong constraints on the possible effects of dynamical interactions and radiation fields from massive stars on multiple systems and protoplanetary discs. We apply our method to constrain the initial binary population in each of these seven regions and show that the populations in only three - the Orion Nebula Cluster, $\rho$ Oph and Corona Australis - are consistent with having evolved from the Kroupa universal initial period distribution and a binary fraction of unity.