On the initial mass-radius relation of stellar clusters
  Young stellar clusters across nearly five orders of magnitude in mass appear
to follow a power-law mass-radius relationship (MRR), $R_{\star} \propto
M_{\star}^{\alpha}$, with $\alpha \approx 0.2 - 0.33$. We develop a simple
analytic model for the cluster mass-radius relation. We consider a galaxy disc
in hydrostatic equilibrium, which hosts a population of molecular clouds that
fragment into clumps undergoing cluster formation and feedback-driven
expansion. The model predicts a mass-radius relation of $R_{\star} \propto
M_{\star}^{1/2}$ and a dependence on the kpc-scale gas surface density
$R_{\star} \propto \Sigma_{\rm g}^{-1/2}$, which results from the formation of
more compact clouds (and cluster-forming clumps within) at higher gas surface
densities. This environmental dependence implies that the high-pressure
environments in which the most massive clusters can form also induce the
formation of clusters with the smallest radii, thereby shallowing the observed
MRR at high-masses towards the observed $R_{\star} \propto M_{\star}^{1/3}$. At
low cluster masses, relaxation-driven expansion induces a similar shallowing of
the MRR. We combine our predicted MRR with a simple population synthesis model
and apply it to a variety of star-forming environments, finding good agreement.
Our model predicts that the high-pressure formation environments of globular
clusters at high redshift naturally led to the formation of clusters that are
considerably more compact than those in the local Universe, thereby increasing
their resilience to tidal shock-driven disruption and contributing to their
survival until the present day.
