Formation of a protocluster: A virialized structure from gravoturbulent collapse: II. A two-dimensional analytical model for a rotating and accreting system

Context. Most stars are born in the gaseous protocluster environment where the gas is reprocessed after the global collapse from the diffuse molecular cloud. The knowledge of this intermediate step gives more accurate constraints on star formation characteristics. Aims. We demonstrate that a virialized globally supported structure, in which star formation happens, is formed out of a collapsing molecular cloud, and we derive a mapping from the parent cloud parameters to the protocluster to predict its properties with a view to confront analytical calculations with observations and simulations. Methods. We decomposed the virial theorem into two dimensions to account for the rotation and the flattened geometry. Equilibrium was found by balancing rotation, turbulence, and self-gravity, while turbulence was maintained through accretion driving and it dissipates in one crossing time. We estimated the angular momentum and the accretion rate of the protocluster from the parent cloud properties. Results. The two-dimensional virial model predicts the size and velocity dispersion given the mass of the protocluster and that of the parent cloud. The gaseous protoclusters lie on a sequence of equilibrium with the trend R ~ M^0.5 with limited variations, depending on the evolutionary stage, parent cloud, and parameters that are not well known, such as turbulence driving efficiency by accretion and turbulence anisotropy. The model reproduces observations and simulation results successfully. Conclusions. The properties of protoclusters follow universal relations and they can be derived from that of the parent cloud. The gaseous protocluster is an important primary stage of stellar cluster formation, and should be taken into account when studying star formation. Using simple estimates to infer the peak position of the core mass function (CMF) we find a weak dependence on the cluster mass, suggesting that the physical conditions inside protoclusters may contribute to set a CMF, and by extension an initial mass function (IMF), that appears to be independent of the environment.