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

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 (Dp) to the 3D fractal dimension (D_{3D}) of turbulent clouds. We do this by computing the fractal dimension of clouds over two orders of magnitude in turbulent Mach number (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 Dp and D_{3D}. The proposed relation is D_{3D}(Dp) = Ω 1 \operatorname{erfc}(ξ 1 \operatorname{erfc}^{-1}[ (Dp - D{p,min})/Ω 2 ] + ξ 2) + D{3D,min}, where the minimum 3D fractal dimension, D{3D,min} = 2.06 ± 0.35, the minimum projected fractal dimension, D{p,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, D_{3D,min} = 2.06 ± 0.35, indicates that in the high M limit the 3D clouds are dominated by planar shocks. The relation between Dp and D_{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.