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Most stars are born in the gaseous proto-cluster environment. The knowledgeof this intermediate stage gives more accurate constraints on star formationcharacteristics. We demonstrate that a virialized globally supported structure,in which star formation happens, is formed out of a collapsing molecular cloud,and derive a mapping from the parent cloud parameters to the proto-cluster topredict its properties, with a view to confront analytical calculations withobservations and simulations. The virial theorem is decomposed into twodimensions to account for the rotation and the flattened geometry. Equilibriumis found by balancing rotation, turbulence and self-gravity, while turbulenceis maintained by accretion driving and dissipates in one crossing time. Theangular momentum and the accretion rate of the proto-cluster are estimated fromthe parent cloud properties. The two-dimensional virial model predicts the sizeand velocity dispersion given the mass of the proto-cluster and that of theparent cloud. The gaseous proto-clusters lie on a sequence of equilibrium withthe trend $R \sim M^{0.5}$, with limited variations depending on theevolutionary stage, the parent cloud, and the parameters not well known liketurbulence driving efficiency by accretion and the turbulence anisotropy. Themodel reproduces successfully observations and simulation results. Theproperties of proto-clusters follow universal relations and they can be derivedfrom that of the parent cloud. Using simple estimates to infer the peakposition of the core mass function (CMF) we find a weak dependence on thecluster mass suggesting that the physical conditions inside proto-clusters maycontribute to set a CMF, and by extension an IMF, that looks independent of theenvironment.

astronomy and astrophysics

Formation of proto-cluster: a virialized structure from gravo-turbulent  collapse II. A two-dimensional analytical model for rotating and accreting  system