2019/05/20 by James R. Beattie, Christoph Federrath, Ralf S. Klessen
Agricultural and Biological Sciences · Mathematics · Physics and Astronomy · #Astrophysics and Star Formation Studies #Combinatorics #Dimension (graph theory) #Fractal #Fractal dimension #Galaxies: Formation, Evolution, Phenomena #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Omega #Physics #Plant Parasitism and Resistance #Quantum mechanics #astro-ph.GA
paper · pdf · doi:10.1093/mnras/stz1416
14 pages, 7 figures. Accepted 2019 May 13
arxiv created 2019/05/20 · openalex publication_date 2019/05/23 · arxiv updated 2019/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 Dp|) to the 3D fractal dimension (|\mathcal D3D|) 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\rm-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 Dp| and |\mathcal D3D|. The proposed relation is |\mathcal D3D(\mathcal Dp) = Ω 1 erfc(ξ 1 erfc-1[ (\mathcal Dp - \mathcal Dp,min)/Ω 2 ] + ξ 2) + \mathcal D3D,min|, where the minimum 3D fractal dimension, |\mathcal D3D,min = 2.06 ± 0.35|, the minimum projected fractal dimension, |\mathcal Dp,min = 1.55 ± 0.13|, Ω1 = 0.47 ± 0.18, Ω2 = 0.22 ± 0.07, ξ1 = 0.80 ± 0.18, and ξ2 = 0.26 ± 0.19. The minimum 3D fractal dimension, |\mathcal D3D,min = 2.06 ± 0.35|, indicates that in the high |\mathcal M| limit the 3D clouds are dominated by planar shocks. The relation between |\mathcal Dp| and |\mathcal D3D| 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.