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In a hierarchically structured region, the average density increases as you go down the levels of the hierarchy to smaller and smaller scales. If there are dense star-forming cores at the bottom of the hierarchy, where the densities are largest and the sizes are smallest, then the fractional mass in the form of these cores increases as their level is approached. This is because more and more interclump gas is removed from the scale of interest as the densest substructures are approached. The fractional mass of cores is proportional to the instantaneous efficiency of star formation if the cores form stars. Therefore the local efficiency of star formation in a hierarchical cloud increases as the average density increases. The efficiency on the scale of a galaxy where the average density is low is ~ 1%; on the scale of an OB association it is ~ 5%, and in a cloud core where a bound cluster forms, it is ~ 40%. Bound cluster formation requires a high efficiency so there is a significant gravitating mass of stars remaining after the gas leaves. It follows that in hierarchical clouds, the probability of forming a bound cluster is automatically highest where the density is highest. Star clusters are the inner bound regions of a hierarchy of stellar and gaseous structures (Elmegreen 2008).

Outside the inner region, stars that form are not as likely to be bound to each other after the gas leaves. Then there are loose stellar groups, unbound OB subgroups, OB associations, and so on up to star complexes. Flocculent spiral arms and giant spiral-arm clouds are the largest scale on which gravitational instabilities drive the hierarchy of cloud and star-formation structures.

The hierarchy of young stellar structure continues inside bound clusters as well. Smith et al. (2005) found several levels of stellar subclustering inside the rho-Ophiuchus cloud, and Dahm & Simon (2005) found 4 subclusters with slightly different ages ( ± 1 Myr) in NGC 2264. Feigelson et al. (2009) observed X-rays from young stars in NGC 6334. The x-ray maps are nearly complete to stars more massive than 1 Modot and their distribution is hierarchical, with clusters of clusters inside this region. Gutermuth et al. (2005) studied azimuthal profiles of clusters and found that they have intensity fluctuations that are much larger than what would be expected from the randomness of stellar positions; the stars are sub-clustered in a statistically significant way. Sánchez & Alfaro (2009) measured the fractal dimension and hierarchical-Q parameter for 16 Milky Way clusters, using the ratio of cluster age to size as a measure of youth. They found that stars in younger and larger clusters are more clumped than stars in older and smaller clusters. Greater clumping means they have lower Q and lower fractal dimension. Schmeja et al. (2008) measured Q for several young clusters. For IC 348, NGC 1333, and Ophiuchus, Q is lower (more clumpy) for class 0/1 objects (young) than for class 2/3 objects (old). Among four of the subclumps in Ophiuchus, Q is lower and the region is more gassy where class 0/1 dominates; Q is also lower for class 0/1 alone than it is for class 2/3 in Ophiuchus.

Pretellar cores are spatially correlated too. Johnstone et al. (2000) derived a power-law 2-point correlation function from 103.8 AU to 104.6 AU for 850 µm sources in Ophiuchus, which means they are spatially correlated in a hierarchical fashion. Johnstone et al. (2001) found a similar power-law from 103.6 AU to 105.1 AU for 850 µm sources in Orion. Enoch et al. (2006) showed that 1.1 µm pre-stellar clumps in Perseus have a power-law 2-point correlation function from 104.2 AU to 105.4 AU. Young et al. (2006) found similar correlated structure for pre-stellar cores from 103.6 AU to 105 AU in Ophiuchus. These structures could go to larger scales, but the surveys end there.

In summary, clusters form in the cores of the hierarchy of interstellar structures and they are themselves the cores of the stellar hierarchy that follows this gas. Presumably, this hierarchy comes from self-gravity and turbulence. Gas structure continues to sub-stellar scales. The densest regions, which are where individual stars form, are always clustered into the next-densest regions. Stars form in the densest regions, some independently and some with competition for gas, and then they move around, possibly interact a little, and ultimately mix together inside the next-lower density region. That mixture is the cluster. More and more sub-clusters mix over time until the cloud disrupts. Simulations of such hierarchical merging have been done by many groups, such as Bonnell & Bate (2006) and Maschberger et al. (2010). Because of hierarchical structure, the efficiency is automatically high on small scales where the gas is dense.

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