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

## TL;DR

This paper introduces a method to derive the density scaling relation in star formation regions from column-density distributions, revealing that observed relations may be artifacts of the $N$-pdf shape rather than physical conditions.

## Contribution

The study proposes a simple binning method to connect column-density PDFs with density scaling relations, comparing different PDF types and challenging the physical interpretation of the density-size relation.

## Key findings

- The method reflects local $N$-pdf morphology, not physical conditions.
- Derived slopes are roughly consistent with Larson's relation, suggesting an artifact.
- Weak correlation found when applying alternative methods.

## Abstract

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 ($7/5\le\alpha\le5/3$), lognormal ($0.7\lesssim\alpha\lesssim1.4$) 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 $\alpha\sim1.1$ is suggested to support claims that the density-size relation in molecular clouds is indeed an artifact of the observed $N$-pdf.