Super stellar clusters with a bimodal hydrodynamic solution: an Approximate Analytic Approach

Aims. We look for a simple analytic model to distinguish between stellar clusters undergoing a bimodal hydrodynamic solution from those able to drive only a stationary wind. Clusters in the bimodal regime undergo strong radiative cooling within their densest inner regions, which results in the accumulation of the matter injected by supernovae and stellar winds and eventually in the formation of further stellar generations, while their outer regions sustain a stationary wind. Methods. The analytic formulae are derived from the basic hydrodynamic equations. Our main assumption, that the density at the star cluster surface scales almost linearly with that at the stagnation radius, is based on results from semi-analytic and full numerical calculations. Results. The analytic formulation allows for the determination of the threshold mechanical luminosity that separates clusters evolving in either of the two solutions. It is possible to fix the stagnation radius by simple analytic expressions and thus to determine the fractions of the deposited matter that clusters evolving in the bimodal regime blow out as a wind or recycle into further stellar generations.