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Astron. Astrophys. 336, 697-720 (1998)

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8. Summary

We have introduced a new method, the [FORMULA]-variance analysis, to study the structure of molecular cloud images. The drift behavior, measured by the [FORMULA]-variance , is linked to the shape of the power spectrum of the cloud image. Application to the observed CO-maps of the Polaris Flare and the subset of the FCRAO outer galaxy survey shows, that at least for these two examples, the power spectrum has power law shape and the power law index is close to [FORMULA] in both cases. Analysis of the phase distribution of the images shows them to be completely random. The application of these concepts to a more complete sample of observed cloud images will be studied in a future paper (Bensch et al., in prep. ).

We have shown that other parameters derived via independent ways to measure and characterize fractal cloud structure, such as the traditional area-perimeter relation, are related to the drift behavior measured via the [FORMULA]-variance method, or the power law index of the power spectrum . This also includes other, at first sight independent, properties such as the mass spectral index derived from clump decomposition of observed cloud images. We have shown that an ensemble of randomly positioned clumps with a given power law index of the clump mass spectrum and a given power law mass-size relation has a fractional Brownian motion structure of its projected image. The clump mass spectral index of molecular clouds, derived by clump decomposition of the observed intensity and velocity distribution, thus, together with the derived index of the mass-size relation, determines the power law index of the image power spectrum , [FORMULA]. The observed values for the mass spectral index [FORMULA] and the mass-size index [FORMULA] of the Polaris Flare, both derived from a clump decomposition of the observed cloud image over a large range of spatial scales, as well as the power law index of the power spectrum , [FORMULA], independently derived via the [FORMULA]-variance analysis agree with each other along this relation.

These results show that, similar to the result by Elmegreen & Falgarone (1996), the mass spectrum of molecular cloud clumps is closely linked to the fractal structure of the gas. The relation between clump mass and clump size spectrum and the fractal dimension of the cloud image derived within the fBm concept agrees with the observed values, but is in conflict with their relation based on the Koch-island model for the fractal structure.

The above results suggest that the basic characteristics of molecular cloud structure might well be described in a unified way as a fractional Brownian motion structure, characterized by a single parameter, e.g. the power law index of the power spectrum . We show that images synthesized along these rules as fractional Brownian motion images indeed look very much like observed molecular cloud maps. Such synthesis thus provides a potentially very useful tool to generate artificial structures well representing real molecular clouds, e.g. for radiative transfer modeling (Ossenkopf et al., in prep. ). Also, hydrodynamic modeling of molecular clouds has to meet the structural characteristics of such fBm -structures.

One should not forget, however, that molecular cloud structure is likely to be much more complex than the simple concept of fractional Brownian motion , which nevertheless applies well to the basic characteristics of observed, 2-dimensional projected cloud images. The clouds themselves are 3-dimensional and it might well be that the 3-dimensional structure is much more complex than a simple fBm structure, which only emerges in projection. Also, the turbulent velocity fields within molecular clouds (providing pseudo 3-dimensional information from molecular line maps which only makes such analysis as clump decomposition methods possible) are an important ingredient and have to be included into a full treatment and understanding of molecular cloud structure. Nevertheless, the characteristics derived for the 2-dimensional projected images already give certain constraints on the 3-dimensional structure. If the 3-dimensional phases are essentially as randomly distributed as is the case for the phases of the 2-dimension image, the measured power law index of the 2-dimensional image implies that the surface grows proportional to volume for the 3-dim cloud structure, and that hence most of the material is surface material. This is in accordance with the well established fact that even 12CO, though being a completely optically thick tracer, measures cloud mass, as well as the recently emerging view, that most line emission from molecular cloud tracers is largely dominated by surface effects.

Examining the applicability of the concepts presented to a larger sample of observed molecular images is certainly one important goal of future work. Another one will be, to extend the observations to much larger spatial coverage and higher signal to noise. The discussion shows that, due to the steep power law decrease observed in the power spectrum of cloud images, this will be very difficult observationally even with large focal plane single dish arrays and large size future interferometers.

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998