The puzzle of the cluster-forming core mass-radius relation and why it matters

We highlight how the mass-radius relation of cluster-forming cores combined with an external tidal field can influence infant weight-loss and disruption likelihood of clusters at the end of their violent relaxation, namely, when their dynamical response to the expulsion of their residual star-forming gas is over. Specifically, building on the cluster N-body model grid of Baumgardt & Kroupa (2007), we study how the relation between the bound fraction of stars staying in clusters at the end of violent relaxation and the cluster-forming core mass is affected by the slope and normalization of the core mass-radius relation. Assuming mass-independent star formation efficiency and gas-expulsion time-scale tau(GExp)/tau(cross) and a given external tidal field, it is found that constant surface density cores and constant radius cores have the potential to lead to the preferential removal of high- and low-mass clusters, respectively. In contrast, constant volume density cores result in mass-independent cluster infant weight-loss, as suggested by some observations. These trends result from how core volume density and core mass scale with each other. Infant weight-loss is quantified for cluster-forming cores with number density n(H2,core) similar or equal to 6 x 10(4) cm(-3) , surface density Sigma(core) similar or equal to 0.5 g cm(-2) or radius r(core) = 0.3 pc. Our modelling includes predictions about the evolution of high-mass cluster-forming cores (say m(core) > 10(5) M-circle dot), a regime not yet covered by the observations. We show how, for a given external tidal field, the core mass-radius diagram constitutes a straightforward diagnostic tool to assess whether the tidal field influences the fate of clusters after gas expulsion. An overview of various issues directly affected by the nature of the core mass-radius relation is presented. In relation to the tidal field impact, these are the evolution of the cluster mass function at young ages (i.e. over the first similar or equal to 30 Myr), and our ability to reconstruct the star formation history of galaxies from their cluster age distribution. Independently of the tidal field impact, the slope and/or normalization of the cluster-forming core mass-radius relation also influences the mass-metallicity relation of old globular clusters predicted by self-enrichment models, and the duration of cluster violent relaxation. Finally, we emphasize that observational mass-radius data sets of dense gas regions must be handled with caution as they may be the imprint of the molecular tracer used to map them, rather than reflecting cluster formation conditions.