A positive power-law trend between the local surface densities of molecular gas, $Sigma_{gas}$, and young stellar objects, $Sigma_{star}$, in molecular clouds of the Solar Neighbourhood has been identified by Gutermuth et al. How it relates to the properties of embedded clusters, in particular to the recently established radius-density relation, has so far not been investigated. In this paper, we model the development of the stellar component of molecular clumps as a function of time and initial local volume density so as to provide a coherent framework able to explain both the molecular-cloud and embedded-cluster relations quoted above. To do so, we associate the observed volume density gradient of molecular clumps to a density-dependent free-fall time. The molecular clump star formation history is obtained by applying a constant SFE per free-fall time, $eff$. For volume density profiles typical of observed molecular clumps (i.e. power-law slope $simeq -1.7$), our model gives a star-gas surface-density relation $Sigma_{star} propto Sigma_{gas}^2$, in very good agreement with the Gutermuth et al relation. Taking the case of a molecular clump of mass $M_0 simeq 10^4 Msun$ and radius $R simeq 6 pc$ experiencing star formation during 2 Myr, we derive what SFE per free-fall time matches best the normalizations of the observed and predicted ($Sigma_{star}$, $Sigma_{gas}$) relations. We find $eff simeq 0.1$. We show that the observed growth of embedded clusters, embodied by their radius-density relation, corresponds to a surface density threshold being applied to developing star-forming regions. The consequences of our model in terms of cluster survivability after residual star-forming gas expulsion are that due to the locally high SFE in the inner part of star-forming regions, global SFE as low as 10% can lead to the formation of bound gas-free star clusters.
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