Diffuse H in a Fractal Interstellar Medium

Implications for Interstellar Clouds

Fractal cloud structure showed up first in the form of fractal cloud edges (see reviews in Scalo 1990; Falgarone 1989; Zimmerman & Stutzki 1993; Pfenniger 1996). This gave a projected fractal dimension equal to 1.3. Now fractal structure shows up in two other observations: the size distribution for clouds or clumps in CO surveys, and the mass-size relation for clouds that are at about the same temperature (Elmegreen & Falgarone 1996). This second study gives a fractal dimension for whole clouds equal to about 2.3. This dimension also applies to laboratory turbulence (Meneveau & Sreenivasan 1990), and to such common structures as clouds in the Earth's atmosphere and smoke swirls. Interstellar clouds look like transparent atmospheric clouds because both get their structure from convected material undergoing turbulent motions. Elmegreen & Falgarone (1996) also showed that the mass distribution function for interstellar clouds may result from turbulence.

If the fractal dimension of interstellar gas is D, then some basic correlations and distribution functions follow (Elmegreen 1997a; hereafter E97a):

1. A distribution function for structures of size S:

2. A mass-size relation for clouds that have the same temperature: (temperature matters because the inferred M is obtained from the cloud luminosity).

3. A mass distribution function for clouds that have the same temperature:

4. A mass distribution function for clouds that have an arbitrary mass-size relation, , giving,

5. A density-mass relation:

6. A density-filling factor relation:

7. A column-density (), angular filling factor () relation: .

8. A distribution function for density:
For all of these relations, we should take . Many properties of turbulent interstellar gas can be obtained from these relations, or conversely, D can be measured from observed gas properties using them.

A fractal cloud with dimension D=2.3 that is made from hierarchically clustered points is shown in figure 1. Each point in the cluster is divided into N=5 other points, all within a distance for hierarchical level h and geometric factor L=2. To make it random, we choose random positions for these points within the distance . The point-clusters continue to subdivide down to H=6 total levels in the hierarchy, giving a maximum density contrast and a total number of points equal to . The fractal dimension is defined to be , which makes it .

  
Figure 1: Model fractal cloud with D=2.3

The resulting cloud is clumpy on a wide range of scales, like an interstellar cloud, and there are numerous empty regions inside of it. The fraction of the total volume that is empty (the porosity), is given by

for maximum density contrast C and fractal dimension D (E97a). This is about 95% for the fractal in figure 1 and slightly less for interstellar gas since the maximum density contrast for either atomic or molecular regions is around . The fraction of the projected area of the cloud that is empty in Figure 1 is about 1/2, as obtained from the distribution function of total empty area in numerous fractal models. We have proposed that such holes in the interstellar fractal produce much of the low density intercloud medium, as reviewed in the next section.

© Copyright Astronomical Society of Australia 1997