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

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1. Introduction

Observations of molecular clouds have shown that with each step in increasing spatial resolution we see the structure to break up into substructure down to the new resolution limit. This holds down to the highest linear resolution achievable (Langer et al. 1995, Falgarone & Phillips 1996). The presence of such clumpy structure has been inferred indirectly since early times of molecular cloud observations (see Stutzki (1993) for a review) from the small area filling factors of typically 10-20% (NH3: Ho & Townes 1983; CS: Snell et al. 1984), or from the similarity of 12CO and 13CO line profiles despite of their large difference in optical depth (Martin et al. 1984).

The study of photo dissociation regions (PDRs) has given independent evidence for the clumpy structure of molecular clouds: only clumpiness with a clump- interclump density contrast of several factors of 10, allowing FUV penetration through the much less opaque interclump material much deeper into the cloud, can explain the large spatial extent and coexistence of several PDR tracers. (Stutzki et al. 1988; Howe et al. 1991). The low-J CO lines very commonly show line ratios in conflict with the emission originating in a homogeneous density and temperature structure: PDR models of UV illuminated clumps give a natural explanation of the observed line ratios over a large range of UV intensities and clump densities (Castets et al. 1990; Gierens et al. 1991; Köster et al. 1994; Störzer et al. 1996; and Störzer et al. 1997).

The observed line widths of molecular clouds are highly supra-thermal and the internal cloud motions would lead to very fast energy dissipation unless cushioned by coupled magnetic fields. Again, this scenario implies the presence of substructure, i.e. higher density clumps with intrinsically narrow lines moving around with a larger interclump velocity dispersion. The smoothness of the observed, close to Gaussian, line profiles at lower spatial resolution allows to infer a very large number of very small clumps (Tauber et al. 1991; Tauber 1996). Observations of the nearest molecular cloud L1457 (Zimmermann 1993) show that the line profiles indeed break up into many individual components at high spatial resolution.

Molecular cloud structure is obviously closely related to the star formation process, likely controlling the overall star formation efficiency and possibly the mass spectrum of the newly formed stars (Zinnecker 1989; Larson 1992). Also, the clumpy structure drastically enhances the back reaction of newly formed stars with the molecular material of their parent cloud, thus affecting the overall cloud evolution (Bertoldi & McKee 1996). The importance of molecular cloud structure has resulted in many attempts over the last few years to characterize the structure observed in a quantitative way, e.g. by means of autocorrelation analysis (Dickman & Kleiner 1985; Kleiner & Dickman 1987; Perault et al. 1986), measuring the area-perimeter relation for iso-intensity contours of 100 µm dust emission maps, dust extinction maps, and HI or CO integrated line intensity maps (Bazell & Desert 1988; Scalo 1990; Dickman et al. 1990; Falgarone et al. 1991; Zimmermann & Stutzki 1993; Vogelaar & Wakker 1994), wavelet analysis (Langer et al. 1993), structure tree methods (Houlahan & Scalo 1992) or by clump decomposition and the determination of clump mass spectra (Stutzki & Güsten 1990; Williams et al. 1994; Kramer et al. 1998). In a recent paper Elmegreen & Falgarone (1996) suggest that the mass distribution is the result of the fractal structure of the molecular cloud gas, and that the power law index of the clump mass spectrum and the fractal dimension of the cloud are thus related.

In this paper we investigate the structure of molecular cloud images, i.e. the spatial distribution of the line integrated intensity of a particular molecular transition used as a tracer for the molecular cloud material. In the optically thin limit and for uniform, thermalized excitation conditions the line integrated intensity basically measures the clouds column density, i.e. the 2-dimensional projection of the clouds density structure onto the plane of the sky. Obvious extensions of the present work, which will be presented in follow up papers, are for one to study the structure of individual velocity channel maps and the variation between those (Bensch et al., in prep. ), and secondly to investigate the influence of optical depth effects on the structure of the observed integrated intensity image resulting from a given 3-dimensional density distribution (Ossenkopf et al., in prep. ).

We discuss in Sect. 2 a new method for analyzing molecular cloud structure which is basically a 2-dimensional generalization of the Allan variance method used in the time stability analysis of sensitive equipment, e.g. atomic clocks (Allan 1966; Barnes et al. 1971) or radio astronomy receivers (Schieder et al. 1989). This analysis applied to observed molecular cloud images shows that the power spectrum of their images is a power law with the spectral index in a narrow range. Moreover, the phases in the Fourier-transformed image are distributed randomly. Molecular cloud structure thus may well be described as having a power law power spectrum and completely random phases, a structure that is called fractional Brownian motion structure in the theory of fractal images. In Sect. 3 we discuss the connection between the various methods commonly used for structure analysis of molecular clouds and show that they can be viewed in an unique way, connecting the various indices measured, like the fractal dimension from area-perimeter relations or from box coverage methods, power spectrum power law index or drift behavior. We show in Sect. 4 that artificial cloud images generated as fractional Brownian motion images give indeed a good simulation of real molecular cloud images. On this background, we derive in Sect. 5 that an ensemble of clumps with a given power law mass spectrum also gives a fractional Brownian motion image in its 2-dimensional projection, where the power spectrum spectral index is determined by the mass spectral index and the power law index in the mass-size relation of the clumps. In Sect. 6 we critically review the connection of the various fractal indices presented in the literature and derived in the present paper, with particular emphasis on the connection between the clump mass and size spectra, the mass-size relation and the fractal dimension of the cloud. The discussion in Sect. 7 addresses the implications of the observed structural parameters of the 2-dimensional observed cloud images for the intrinsic 3-dimensional cloud structure, the implications of the results for further observations down to yet higher angular resolution, and the limitation of the present analysis within the framework of more complex models for the cloud structure. The results are summarized in Sect. 8.

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998