The catalog reported by Lada & Lada (2003) for embedded clusters is consistent with the mass-size relation R ~ M1/3 − M1/2 for low-mass clusters as pointed out by Murray (2009), where R and M are the gas radius and mass, respectively. While the stellar mass of an embedded cluster is not directly observable and is obtained by assuming an underlying universal initial mass function (IMF), Adams et al. (2006) compiled data from Lada & Lada (2003) and Carpenter (2000) and showed a number-size relation \hbox{$R_* \sim N_*^{1/2}$}R∗~N∗1/2 between the radius of the cluster and the number of objects it contains. The results of Gutermuth et al. (2009) are also compatible with this relation. The number-size relation is reasonably equivalent to the mass-size relation if we adopt a universal IMF and thus similar averaged mass among clusters. Despite the slight uncertainty in the power-law exponent, varying from 1/3 to 1/2, and the scatter of the data, it is clear that embedded clusters follow some mass-size relation. Larger data sets of star-forming clumps, identified as gaseous protoclusters, also exhibit a mass-size relation. Fall et al. (2010) compiled the observations of Shirley et al. (2003), Faúndez et al. (2004), and Fontani et al. (2005) in their Fig. 1, and found via least-squares regression the relation R ∝ M0.38. A regression fit on the ATLASGAL survey (Urquhart et al. 2014) data gives a dependence of R ∝ M0.50. In their work, they fitted log (M) to log (R) and found M ∝ R1.67, of which the power-law exponent is not the inverse of what we inferred. This indicates that the clump properties are differently dispersed in size and mass, and thus there exist some uncertainties in the power-law dependence. Meanwhile, both studies exhibit mass scatter of about 1 dex and are compatible with constant gas surface density, that is, R ∝ M0.5. These observations are performed with molecular lines and dust continuum of the star-forming gas, suggesting that the mass-size relation and probably some other properties of the stellar cluster are established as early as the gas-dominated phase. Pfalzner et al. (2016) pointed out that the mass-size relation for embedded clusters and gaseous protoclusters follow the same trend with different absolute value, which could be explained with star formation efficiency or cluster expansion. One interesting question to ask would be what physical processes actually determine this mass-size relation. Its existence suggests that this primary phase of cluster formation, the gaseous protocluster, is very likely in some equilibrium state governed by the molecular cloud environment in which it resides, and is crucial for understanding the nature of the cluster and, more generally, star formation. An analytical study by Hennebelle (2012) yielded, by linking the gaseous protocluster to properties of the parent cloud, R ~ M1/3 or R ~ M1/2 for protoclusters with different accretion schemes. We stress that we are emphasizing a global equilibrium, which does not imply that the structure is not locally collapsing. More precisely, we propose that the large scales are supported by a combination of rotation and turbulence that sets the M-R relation.
Citation: Fall et al. (2010)
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