Observational constraints on star cluster formation theory - I. The mass-radius relation

Stars form predominantly in groups usually denoted as clusters or associations. The observed stellar groups display a broad spectrum of masses, sizes and other properties, so it is often assumed that there is no underlying structure in this diversity. Here we show that the assumption of an unstructured multitude of cluster or association types might be misleading. Current data compilations of clusters show correlations between cluster mass, size, age, maximum stellar mass etc. In this first paper we take a closer look at the correlation of cluster mass and radius. We use literature data to explore relations in cluster and molecular core properties in the solar neighborhood. We show that for embedded clusters in the solar neighborhood there exists a clear correlation between cluster mass and half-mass radius of the form $M_c = C R_c^{\gamma}$ with gamma = 1.7 +/-0.2. This correlation holds for infra red K band data as well as X-ray sources and for clusters containing a hundred stars up to those consisting of a few tens of thousands of stars. The correlation is difficult to verify for clusters containing <30 stars due to low-number statistics. Dense clumps of gas are the progenitors of the embedded clusters. We find a similar slope for the mass-size relation of dense, massive clumps as for the embedded star clusters. This might point at a direct translation from gas to stellar mass: however, it is difficult to relate size measurements for clusters (stars) to those for gas profiles. Taking into account multiple paths for clump mass into cluster mass, we obtain an average star-formation efficiency of 18%{+9.3}{-5.7} for the embedded clusters in the solar neighborhood. The derived mass-radius relation gives constraints for the theory of clustered star formation. Analytical models and simulations of clustered star formation have to reproduce this relation in order to be realistic (abridged)