# The relation between the true and observed fractal dimensions of turbulent clouds

## TL;DR

This paper introduces an empirical relation between the 2D projected and 3D fractal dimensions of turbulent interstellar clouds, derived from simulations across a wide Mach number range, aiding the interpretation of observational data.

## Contribution

The study presents a new empirical formula linking 2D and 3D fractal dimensions of turbulent clouds, based on extensive simulations, improving understanding of cloud structures from 2D observations.

## Key findings

- Derived a new empirical relation between $\, ext{D}_ ext{p}$ and $\, ext{D}_ ext{3D}$.
- Found minimum 3D fractal dimension $\, ext{D}_ ext{3D,min} \,= 2.06 \, ext{±} \,0.35$.
- Showed the relation's potential for interpreting observational data of molecular clouds.

## Abstract

Observations of interstellar gas clouds are typically limited to two-dimensional (2D) projections of the intrinsically three-dimensional (3D) structure of the clouds. In this study, we present a novel method for relating the 2D projected fractal dimension ($\mathcal{D}_{\text{p}}$) to the 3D fractal dimension ($\mathcal{D}_{\text{3D}}$) of turbulent clouds. We do this by computing the fractal dimension of clouds over two orders of magnitude in turbulent Mach number $(\mathcal{M} = 1-100)$, corresponding to seven orders of magnitude in spatial scales within the clouds. This provides us with the data to create a new empirical relation between $\mathcal{D}_{\text{p}}$ and $\mathcal{D}_{\text{3D}}$. The proposed relation is $\mathcal{D}_{\text{3D}}(\mathcal{D}_{\text{p}}) = \Omega_1 erfc ( \xi_1 erfc^{-1}[ (\mathcal{D}_{\text{p}} - \mathcal{D}_{\text{p,min}})/\Omega_2 ] + \xi_2 ) + \mathcal{D}_{\text{3D,min}}$, where the minimum 3D fractal dimension, $\mathcal{D}_{\text{3D,min}} = 2.06 \pm 0.35$, the minimum projected fractal dimension, $\mathcal{D}_{\text{p,min}} = 1.55 \pm 0.13$, $\Omega_1 = 0.47 \pm 0.18$, $\Omega_2 = 0.22 \pm 0.07$, $\xi_1 = 0.80 \pm 0.18$ and $\xi_2 = 0.26 \pm 0.19$. The minimum 3D fractal dimension, $\mathcal{D}_{\text{3D,min}} = 2.06 \pm 0.35$, indicates that in the high $\mathcal{M}$ limit the 3D clouds are dominated by planar shocks. The relation between $\mathcal{D}_{\text{p}}$ and $\mathcal{D}_{\text{3D}}$ of molecular clouds may be a useful tool for those who are seeking to understand the 3D structures of molecular clouds, purely based upon 2D projected data and shows promise for relating the physics of the turbulent clouds to the fractal dimension.