The relation between the true and observed fractal dimensions of turbulent 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 ${\mathcal{D}}_{\text{3D}}({\mathcal{D}}_{\text{p}})={\Omega }_{1}erfc({\xi }_{1}erf{c}^{-1}[({\mathcal{D}}_{\text{p}}-{\mathcal{D}}_{\text{p,min}})/{\Omega }_2]+{\xi }_2)+{\mathcal{D}}_{\text{3D,min}}$, 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.

14 pages, 7 figures. Accepted 2019 May 13

References in corpus (10)

• The Statistics of Supersonic Isothermal Turbulence
• What determines the density structure of molecular clouds ? A case study of Orion B with Herschel
• The Fractal Density Structure in Supersonic Isothermal Turbulence: Solenoidal versus Compressive Energy Injection
• Unfolding the Laws of Star Formation: The Density Distribution of Molecular Clouds
• A cluster in the making: ALMA reveals the initial conditions for high-mass cluster formation
• Scaling Relations of Compressible MHD Turbulence
• Interstellar filaments and star formation
• An Observational Method to Measure the Relative Fractions of Solenoidal and Compressible Modes in Interstellar Clouds
• Statistical link between the structure of molecular clouds and their density distribution
• The world's largest turbulence simulations

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