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tokens: in their terms) appearing at approximately Σ ≳ 100 ⟨Σ⟩ with a slope ≈−1 ± 0.2. Similarly, Lin et al. ( 2017 ) see in the survey of clouds that the PDF becomes systematically shallower, approaching −1, as clouds (or cloud regions) become more actively star forming. We should also note that similar results have been seen in other numerical simulations (e.g. Burkhart, Collins Lazarian 2015 ). 4.4 Cluster mass profile Let us assume that some fraction of objects formed remain gravitationally bound to each other. We expect that the clustered substructures that formed from fragmentation will eventually merge together into a cluster with a density profile that decreases monotonically. Let us derive the power-law index of this profile. Using our model we can calculate the relation between the densities and the survivor masses before the objects rearrange themselves into clusters. For density, take their at-formation value (ρ n ). We can express the index n as which leads to Let us assume (for now) that after formation the objects rearrange themselves to form clusters, while preserving the local volume density – i.e. the local density (of stars) around a star does not change dramatically before/after the re-arrangement. This is motivated by the fact that during mergers, tidal shredding of an object with density ρ occurs at an orbital radius R , where the mean density enclosed within R is approximately ρ. This means that the amount of mass at different density levels must be the same as before the rearrangement, thus Let us assume the relaxed cluster has a power-law density profile: ρ cl ∝ R β − 3 . Also, ε ≪ 1 thus the right hand side of equation ( 25 ) is zero. This leads to thus β = 0, so the mass profile of a bound cluster that results from the assembly of substructures formed in a scale-free fragmentation cascade is ρ cl ∝ R −3 . We can repeat the same exercise while assuming that phase space density (ρ p ) is conserved instead of real space density, as per Liouville’s theorem. This is only true however if ρ p is resolved on infinitely fine scales, as elements of higher phase space density effectively get stretched out and diluted in phase space so that the final observed coarse-grained phase space density is generally lesser than the initial (Lynden-Bell 1967 ). However, we may still suppose that our self-similarity condition means that the evolution operator on the coarse-grained phase space density can only map an initially flat (d M /dln ρ p ∼ 0) phase space distribution into another flat distribution. We can approximate the phase space density as where σ is the velocity dispersion. Assuming that the collapsing clouds are virialized, we can write which leads to From here we can formulate the surviving mass per phase density (similar to equation 25 ): where, in the last step, we used the assumption that ε ≪ 1. Since we are interested in the asymptotic case at large radii. the mass enclosed is approximately converged ( M ≈ const.), thus so we get