Local-Density Driven Clustered Star Formation
  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.
