6. The relation between mass spectrum, size spectrum, mass-size relation and fractal dimension
The relations established in the preceding sections link the indices of the clump mass spectrum and the clump size spectrum, together with the power law index of the mass-size relation of the clumps, with the fractal dimension of the image of the clump ensemble. These relations will be discussed in the next two subsections, together with a critical analysis of similar relations discussed in the literature, that are based on a different fractal cloud model.
6.1. The clump mass spectrum and the fractal dimension
With the relation between the power law mass spectral index of an ensemble of clumps and the power spectral index of its 2-dimensional projection as derived above, , we can now connect the mass spectral index with the drift index H and the fractal dimension of the projected image:
We recall from Sect. 3 that for an fBm -structure with power law index to have drift behavior or a fractal dimension , the value of H is limited to the range . thus has to be in the range . The range corresponds to ranging from . For the case , ranges from . For the proper range would be . Reversely, for the case of on average constant column density, , the suitable range of , defined above in order to guarantee the total mass to be dominated by the high mass clumps and the total number of clumps by the low mass clumps, thus agrees with the full range of , guaranteeing the cloud image to be an fBm structure. For clumps with on average constant volume density, , this is not the case. In this case, for mass spectra shallower than the structure has , i.e. a smooth projected image with a fractal dimension identical to the Euclidean dimension . At the steepest spectral index it only reaches , i.e. or a fractal dimension of 2.5. It would reach the extreme value of only at the extreme value , i.e. for a structure whose mass is already dominated by the smallest clumps. The most likely case, , is in between these two extremes.
The relation between power law mass spectral index and fractal dimension of the cloud, is strictly valid for the assumed model of the cloud, i.e. an ensemble of clumps with a power law mass spectrum and a power law mass-size relation. We arrived at it by combining the relation between and the power spectrum index derived in Sect. 4, the relation between and the drift index H derived in the Appendix, and the relation between H and the fractal dimension defined as box coverage dimension of the -dimensional fractional Brownian motion surface, discussed in Sect. 3. Only the connection to the area-perimeter fractal dimension has not been derived explicitly but has been assumed to be valid following Voss (1988).
A different relation between the fractal dimension of a molecular cloud and its clump mass spectral index , namely in the notation adopted in the present paper, has been derived by Elmegreen & Falgarone (1996). This is clearly in conflict with the relation derived above. The discrepancy can be traced back to the different concepts used in deriving them and will be discussed together with the clumps size spectrum in the following subsection.
6.2. The clump size spectrum and the fractal dimension
A clump ensemble with power law mass spectrum and mass-size relation has a size spectrum ; the index of the size spectrum, defined by , is thus (this connection was actually already noted by Henriksen 1991). Inserting the relation between and its fractal dimension from above, the size spectral index is .
In contrast, Mandelbrot states that the power law index of the clump size spectrum is given simply by the thus defined fractal dimension D, (Mandelbrot 1983, page 118). This is clearly different from the above relation and the difference is due to the different concepts used. Mandelbrot derives this relation for Koch-islands generated the usual way via a generator that creates smaller islands in front of each island border line and counting the substructures thus generated. The number N of self-similar substructures on a scale greater than L then scales as .
The crucial point thus is, whether the relation can be applied to fractal structures other than the Koch-island structures it has been derived for. We claim that this is not the case. The fBm -structure discussed above, generated as a clump ensemble with given mass spectrum and mass-size relation, gives a good counter example.
6.3. fBm -model versus Koch-island model structures
In this context it is important to note that a fractal structure with a given fractal dimension does not necessarily have a well defined size spectrum. Though this is clearly the case for the Koch-island structures analyzed by Mandelbrot and other hierarchically nested fractal structures, being generated by self similar replication of the basic structure at subsequently smaller scales, it is not the case in general. Again, the fractional Brownian motion structures give a good counter example: the fractal dimension is fixed once the power spectrum spectral index is given. But an fBm structure as such has no size spectrum defined. The fact that the power spectrum has a certain amplitude at a given spatial frequency, i.e. that the image has a certain power on the corresponding length scale, still leaves open, whether this power is due to a few bright substructures of that size, or whether it is contributed by many weak substructures. The size distribution thus has to be specified separately, either directly or by a clump mass spectrum and a mass size relation, as was actually done with the ensemble of clumps which we showed to have fractional Brownian motion structure and a well defined fractal dimension.
In reverse, specifying a size spectrum of an ensemble of self similarly nested building blocks (such as the Gaussian clump ensemble discussed above) is not sufficient to fully characterize the fractal properties of its image. One needs, in addition, a link between the intensity resulting from a particular structure at its given size scale. In the case of the molecular cloud clump ensemble studied above, this is given by the mass-size relation for the clumps, , and the fact that the mass, resp. column density, determines the emitted intensity for an optically thin species 2.
Also, it is not at all obvious how to relate the hypothetical Koch-island structure, i.e. the set of boundary lines, for which the relation has been derived, to a physical molecular cloud structure, i.e. a density or rather column density distribution. Elmegreen & Falgarone (1996) implicitly assume that the fractal dimension of the Koch-island structure is related to e.g. the fractal dimension defined via the area-perimeter relation of the iso-intensity contours of molecular clouds. The arguments given illustrate that this does not apply, at least for the fBm -structure generated as a clump ensemble with given mass spectrum and mass-size relation.
The important new idea in the Elmegreen & Falgarone (1996) paper is the suggestion that the mass distribution in molecular clouds is the results of fractal gas structure . They derive the relation , which is essentially the basic connection between the mass spectral index , size spectral index and mass-size relation index as derived above, , plus the additional assumption, that the size spectral index is related to the fractal dimension D via the Koch-island relation . Our cloud structure analysis, based on a fractional Brownian motion cloud model, also shows that the mass distribution is determined by the fractal structure, and we derive a similar, but different, relation: .
It is, of course, of interest to check how well the observed values agree with either model. In this context, it is important to note, that the agreement between and , both for the individual cloud surveys and for the ensemble distribution (Elmegreen & Falgarone 1996) cannot be taken to support the Koch-island model. This is, because D was actually not determined independently. It rather is derived from the size spectral index . The fact that the fitted values agree with the trivial relation simply confirms that the observed clump masses and sizes are derived in a consistent way. This is true both for the individual cloud data sets with their larger scatter of the particular values for (respectively D in the context of that paper), , and , as well as for the total ensemble of the data sets. It has nothing to do with the cloud structure or its fractal characteristics.
The close numerical agreement between the average value of derived from the various fits to the clump size spectra by Elmegreen & Falgarone, and the range of values expected from fractal analysis via area-perimeter studies, giving for the fractal dimension of the iso-intensity contours, and hence 2.3-2.5 for the fractal dimension of the 2-dimensional image, thus has to be regarded as coincidental. In fact, the detailed case study of the Polaris Flare presented in Sects. 2 and 4 shows that the area-perimeter fractal dimension in this case is , well consistent with the value of derived independently, and hence also consistent with the relation and the fitted values of and . The size spectral index derived with the fBm -model, , of course has to and does agrees with the value fitted to the observed size spectrum (Heithausen et al. 1998). In contrast, the relation (according to Elmegreen & Falgarone 1996) together with the measured values of and (and hence ) for the Polaris Flare would result in , i.e. formally , even outside the range of allowed values ( to 2) for the area-perimeter fractal dimension of the contour lines.
Our analysis supports the Elmegreen & Falgarone (1996) result that the mass distribution of molecular clouds is closely connected to the fractal structure. Clearly, the different concepts used to describe the cloud structure result in different relations between the various indices involved. Based on the arguments presented in the preceding paragraphs, we prefer the fBm -structure concept to describe molecular clouds.
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998