Multifractal Scaling, Geometrical Diversity, and Hierarchical Structure in the Cool Interstellar Medium

Abstract

Multifractal scaling (MFS) refers to structures that can be described as a collection of interwoven fractal subsets which exhibit power-law spatial scaling behavior with a range of scaling exponents (concentration, or singularity, strengths) and dimensions. The existence of MFS implies an underlying multiplicative (or hierarchical, or cascade) process. Panoramic column density images of several nearby star-forming cloud complexes, constructed from IRAS data, are shown to exhibit such multifractal scaling, which we interpret as indirect but quantitative evidence for nested hierarchical structure. The relation between the dimensions of the subsets and their concentration strengths (the "multifractal spectrum") appears to satisfactorily order the observed regions in terms of the mixture of geometries present, from strong point-like concentrations, to line-like filaments or fronts, to space-filling diffuse structures. This multifractal spectrum is a global property of the regions studied, and does not rely on any operational definition of "clouds." The range of forms of the multifractal spectrum among the regions studied implies that the column density structures do not form a universality class, in contrast to indications for velocity and passive scalar fields in incompressible turbulence, providing another indication that the physics of highly compressible interstellar gas dynamics differs fundamentally from incompressible turbulence. There is no correlation between the geometrical properties of the regions studied and their level of internal star formation activity, a result that is also apparent from visual inspection. We discuss the viability of the multifractal spectrum as a measure of the structural "complexity" of the regions studied, and emphasize the problematic dependence of all structural descriptors on the subjective pre-selection of the region to be described. A comparison of IRAS 100 µm column density (not intensity) images with 13 CO, visual extinction, and C 18 O data suggests that structural details are captured by IRAS up to at least 30 magnitudes of visual extinction, except in the vicinity of embedded stars, and that lower-column density connective structure not seen by other methods is revealed.

Interstellar Turbulence I: Observations and Processes

“…If the perimeter-area dimension of a projected 3D structure is the same as the perimeter-area dimension of a slice, then the ISM value of ∼ 1.4 for projected contours is consistent with analogous measures in laboratory turbulence (Sreenivasan 1991). Chappell & Scalo (2001) determined the multifractal spectrum, f (α), for column density maps of several regions constructed from IRAS 60 µ and 100 µ images. The parameter α is the slope of the increase of integrated intensity with scale, F (L) ∝ L α .…”

H I Power Spectra and the Turbulent Interstellar Medium of Dwarf Irregular Galaxies

“…They found that regions with the highest sonic Mach number lie around the bar. Chappell & Scalo (2001) arrived at a similar conclusion using a multifractal spectrum analysis of low-mass star-forming cloud complexes: there is little correlation between the geometrical properties of the gas and the level of internal SF.…”

Magnetic evolution of superactive regions

“…For a bi-dimensional image 0 ≤ α q ≤ 2, where α q = 0 in the case of a sharp spike and α q = 2 for a homogeneous image. Therefore, low values of α q indicate denser regions, while high values indicate more rarefied regions, and the value of α q describes the degree of "spikiness" of an image (Chappell & Scalo 2001). The set of points of the domain with the same value of α q has fractal dimension F(α q ).…”