Feedback regulated star formation: II. dual constraints on the SFE and the age spread of stars in massive clusters

We show that the termination of the star formation process by winds from massive stars in protocluster forming clumps imposes dual constraints on the star formation efficiencies (SFEs) and stellar age spreads ($\Delta \tau_{*}$) in stellar clusters. We have considered two main classes of clump models. One class of models in one in which the core formation efficiency (CFE) per unit time and as a consequence the star formation rate (SFR) is constant in time and another class of models in which the CFE per unit time, and as a consequence the SFR, increases with time. Models with an increasing mode of star formation yield shorter age spreads (a few 0.1 Myrs) and typically higher SFEs than models in which star formation is uniform in time. We find that the former models reproduce remarkably well the SFE$-\Delta \tau_{*}$ values of starburst clusters such as NGC 3603 YC and Westerlund 1, while the latter describe better the star formation process in lower density environments such as in the Orion Nebula Cluster. We also show that the SFE and $\Delta \tau_{*}$ of massive clusters are expected to be higher in low metallicity environments. This could be tested with future large extragalactic surveys of stellar clusters. We advocate that placing a stellar cluster on the SFE-$\Delta \tau_{*}$ diagram is a powerful method to distinguish between different stellar clusters formation scenarios such as between generic gravitational instability of a gas cloud/clump or as the result of cloud-cloud collisions. It is also a very useful tool for testing star formation theories and numerical models versus the observations.