On the link between column density distribution and density scaling relation in star formation regions

We present a method to derive the density scaling relation $\langle n\rangle \propto L^{-\alpha}$ in regions of star formation or in their turbulent vicinities from straightforward binning of the column-density distribution ($N$-pdf). The outcome of the method is studied for three types of $N$-pdf: power law ($7\mathrm{/}5\le \alpha \le 5\mathrm{/}3$), lognormal ($0.7\lesssim \alpha \lesssim 1.4$) and combination of lognormals. In the last case, the method of Stanchev et al. (2015) was also applied for comparison and a very weak (or close to zero) correlation was found. We conclude that the considered `binning approach’ reflects rather the local morphology of the $N$-pdf with no reference to the physical conditions in a considered region. The rough consistency of the derived slopes with the widely adopted Larson’s (1981) value $\alpha \sim 1.1$ is suggested to support claims that the density-size relation in molecular clouds is indeed an artifact of the observed $N$-pdf.