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

On the fractal structure of molecular clouds

J. Stutzki, F. Bensch, A. Heithausen, V. Ossenkopf and M. Zielinsky

Physikalisches Institut der Universität zu Köln, Zülpicher Strasse 77, D-50937 Köln, Germany (lastname@ph1.uni-koeln.de)

Received 5 January 1998 / Accepted 5 May 1998

Abstract

We present a new method to analyze the structure of observed molecular cloud images which is the generalization of the Allan-variance method traditionally used in the stability and drift analysis of instrumentation and electronic devices. Applied to integrated intensity maps of two molecular cloud data sets, the method shows, together with an analysis of the phases of the cloud images, the observed structures to be well characterized by what is called a fractional Brownian motion (fBm) -structure in the context of fractal images. An fBm -structure results from a power law power spectrum of the image and a completely random distribution of the image phases. The power law index of the power spectrum derived for two sample clouds turns out to be close to 2.8. For an fBm -structure, the power spectral index determines other fractal measures such as the traditionally used box-counting dimension and the fractal dimension describing iso-intensity contours via their area-perimeter relation. We use a large data set covering observations at both large and small angular scales available for the Polaris Flare (Heithausen et al. 1998) as the sample cloud to test these concepts. The area-perimeter dimension independently measured for this cloud is consistent with . The fBm -concept allows easy generation of realistic density representations for model clouds, to be used in radiative transfer and other cloud simulations.

In a second step, we show that an ensemble of randomly positioned clumps with a power law mass spectrum gives an fBm -image. The power spectral index , the mass spectral index , and the power law index of the mass-size relation turn out to be related: . The value of derived via this relation and the independently determined values for and is consistent with the value directly determined for the sample cloud. Our analysis confirms the recent suggestion by Elmegreen & Falgarone (1996) that the mass distribution in molecular clouds is closely connected with their fractal structure, although the detailed form of the relation depends on the fractal structure model used.

We discuss the implications of these results, obtained for the 2-dimensional observed images, for the underlying 3-dimensional cloud density structure. With some extrapolating assumptions on the 3-dim structure, they imply that the 3-dimensional structure is very much broken up, with the surface growing proportional to the volume. Clearly, additional information on the velocity structure, and in particular its physical link to the assumed fBm -density structure, is needed to describe the relevant properties of molecular cloud line shapes and line radiative transfer.

The fBm -structure model allows an estimate on the observability of molecular cloud structure down to much smaller angular scales than presently reachable, e.g. with interferometric observations. It turns out that, due to the steepness of the image power spectrum, these will be extremely difficult. Only the next generation large mm-wave array will bring such observations into the regime of the feasible.

Key words: ISM: structure ISM: clouds ISM: general

Present address: Radioastronomisches Institut der Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany

Send offprint requests to: J. Stutzki

This article contains no SIMBAD objects.

Contents

• 1. Introduction
• 2. Power spectrum and phase distribution of molecular cloud images
  • 2.1. The -variance analysis: a generalized Allan-variance method
  • 2.2. Application to observed cloud images
  • 2.3. Phase distribution of observed cloud images
  • 2.4. Discussion
• 3. Connection with the theory of fractal images
• 4. Generating artificial molecular cloud images
• 5. Power spectrum of the image of an ensemble of randomly positioned clumps with a power low mass spectrum
  • 5.1. fBm -structure of a clump ensemble with a power law mass spectrum
  • 5.2. Comparison with observations
• 6. The relation between mass spectrum, size spectrum, mass-size relation and fractal dimension
  • 6.1. The clump mass spectrum and the fractal dimension
  • 6.2. The clump size spectrum and the fractal dimension
  • 6.3. fBm -model versus Koch-island model structures
• 7. Discussion
  • 7.1. Implications for the underlying 3-dimensional structure
  • 7.2. Observational limits
  • 7.3. Beyond fractional Brownian motion structure
• 8. Summary
• Acknowledgements
• Appendix
  • Appendix A: statistical properties of 1-dimensional random functions: average, variance, autocorrelation function, power spectrum, Allan-variance and -variance
  • Appendix B: the definition of the -variance as the analogue of the Allan Variance in higher dimensions
  • Appendix C: relation between the power law index of the power spectrum and the drift behavior
• References

© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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