5. Power spectrum of the image of an ensemble of randomly positioned clumps with a power low mass spectrum
The complex structure of molecular clouds is often described as clumpy. This phrasing assumes that the structure is actually composed of discrete entities, i.e. clumps, making up the cloud. In projection, the many individual clumps along the line of sight overlap and it would be difficult to identify individual clumps in projected, integrated intensity maps. Due to the interclump velocity dispersion the observed molecular cloud intensity distribution can, however, be decomposed into clumps, i.e. substructures coherent in their spatial and velocity distribution. As long as the observation is done in an optically thin line, the integrated intensity of the thus identified clumps is directly proportional to the clumps mass. In this way, one can measure the clump mass distribution of a molecular cloud.
The two methods to identify clumps via automated procedures that have been published in the literature (Stutzki & Güsten 1990; Williams et al. 1994) differ in how they define clumps. They nevertheless agree, together with several other results on clump mass spectra based on eye inspection of molecular line maps, in that the observed clouds have a clump mass spectrum of the shape , with in the range 1.6 to 1.8 for most clouds (Kramer et al. 1998). There is a tendency that the higher quality data sets tend to give steeper mass spectra, i.e. closer to 1.8. This is expected as with lower signal to noise ratio the decomposition has more difficulty in separating smaller clumps that almost merge with larger ones (see Kramer et al. (1998) for a detailed comparison of the various methods for clump decomposition described in the literature).
Although the concept of dividing the observed structure into discrete subentities seems to be almost orthogonal to analysing the structure in terms of fractal dimension or power spectra , we will now show that the two concepts are actually closely connected, and that the clump mass spectral index and the power spectrum power spectral index of the 2-dimensional projected image are connected.
5.1. fBm -structure of a clump ensemble with a power law mass spectrum
In order to establish a link between the clumps mass distribution and the spatial structure of the image of the clump ensemble, one has to assume in addition a relation between the clump mass and size, , where represents clumps with on average constant clump volume density, and represents clumps with on average constant clump column density. The latter case corresponds to a clump density and is commonly referred to as one of Larson's relations (Larson 1992).
We assume the clumps to have Gaussian shape. Though actual clumps are likely to have more complex shapes, the assumption of Gaussian clumps is convenient for the mathematical derivation in the following, and is not critical for the connection between the clump mass spectral index and the power spectrum power law index to be derived, as will be discussed below.
We consider an ensemble of randomly positioned, Gaussian shaped clumps with a mass spectrum , with , a high mass cutoff , and a low mass cutoff . The restriction guarantees the mass to be dominated by the high mass clumps. Assuming , the total mass is thus determined by the high mass cutoff,
The restriction implies the total number of clumps to be dominated by the low mass clumps, so that the total number of clumps is determined by the low mass cutoff,
(note that should not be too close to either 1 or 2 in order for these approximations to be valid).
A Gaussian shaped clump, i.e. a clump with a Gaussian density distribution, has as well a Gaussian column density distribution in projection, and hence, for an optically thin line and uniform excitation conditions, a Gaussian intensity distribution on the sky. The intensity for a single clump, centered at , with peak intensity and size (FWHM) is thus given by
and leads to a Gaussian shaped Fourier transform of the intensity,
where the peak amplitude in Fourier space is the integral in the image domain, i.e. is proportional to the spatially integrated clump intensity, and hence the clump mass (as long as the clump density is above the critical density): . The constant of proportionality µ depends on the details of the molecular species and transition observed as well as on its abundance. The width of the clump in Fourier space is inversely proportional to its FWHM in the image domain: . The non zero center position of the clump results in the phase factor .
Using the above relation between clump mass and size, , we can then substitute the clump size by the clump mass and obtain for the Fourier transform of an individual clump
where we have introduced the abbreviation .
The total image is then the sum of the images of the individual clumps. As the Fourier transform is linear, the same holds for the Fourier transform of the clumpy cloud image:
The power spectrum of the clumpy cloud image is the square of the Fourier transform :
Due to the randomness of the clump positions, i.e. the randomness of the phases in the Fourier transform , the mixed terms, i.e. those with in this sum average out to zero. This approximation is the better, the more clumps we average over. It only holds for sufficiently large f, a condition guaranteed anyway by the limitations for f given below. Only the terms with , i.e. those with the square of the Fourier transform of an individual clump, have zero phase and thus remain. To a good approximation, we can then replace the square of the sums by the sum of the squares:
The sum can now be approximated by an integral over clump masses by introducing the mass distribution of the clumps . This gives
Substituting gives an expression, where the f-dependence is completely outside the integral, except for the dependence through the lower and upper mass cutoff:
One can easily see that the f-dependence through the integration boundaries can be neglected for spatial frequencies f well above the shortest, and well below the largest spatial frequencies, corresponding to the high and low mass cutoff respectively. At the upper integration boundary , the error made in moving the boundary towards approaches zero as long as the argument of the exponential function in the integrand is much larger than unity, i.e. , thus . In the range of the integral rapidly converges toward a constant value at the lower boundary as long as the argument of the exponential function in the integrand is small there, i.e. . Extending the integration boundaries to 0 and infinity is thus a good approximation as long as . The resulting integral converges and can actually be evaluated in closed form. The complete frequency-dependence of the power spectrum is in the trailing term:
We see that the power spectrum is indeed of power law shape. The spectral exponent is .
Whereas we have assumed an exact relation between clump mass and size in the above derivation, one can easily see that it holds as well for a distribution of clumps which satisfy the mass-size relation only on average: the individual clump properties simply have to be replaced by their average properties in each mass bin. Also, though we derived the result for spherical clumps, it extends straightforwardly to randomly oriented elliptical clumps. We should also note that, although we assumed Gaussian shaped clumps, the relation will hold as well for other shapes, as long as the power spectrum of an individual clump image drops with a power law exponent higher than , so that the upper integration boundary can be extended to infinity.
Repeating the derivation above for the 3-dimensional density distribution shows that its power spectrum power law index, rather than that of the 2-dimensional projection derived above, also turns out to be . This result is immediately obvious considering the fact derived in Sect. 3 that the power law spectral index of the power spectrum does not change on projection.
5.2. Comparison with observations
Let us now compare and discuss the numerical values derived for the various parameters from observational data and their agreement with the relation derived. For the case , i.e. assuming on average constant clump column density, we obtain ; for the case , i.e. assuming on average constant volume density, the relation would be . For the typical range of between 1.6 to 1.8 observed in molecular clouds (as discussed above), we obtain a range of values for between 3.8 to 2.4 for the case , and between 4.2 to 3.6 for the case . The former is in accordance with the observed value of around 2.8, the latter is clearly too high, thus favoring a value of closer to 2.
The value of is usually determined observationally by a fit to the mass and size values of the clumps identified in a given data set. Clump sizes are difficult to determine and the actual values will depend on the method used to identify clumps. This is different from the determination of clump masses, which, being derived as the integral over the spatial and velocity coordinate of the observed intensity distribution are a much more robust quantity and in particular independent of the resolution of the observations (Kramer et al. 1998). In addition the relatively small coverage of linear spatial scales even for relatively large molecular cloud maps and the systematic effects coming in close to the resolution limit make the determination of the mass-size index marginal for each single set of observations. Values for determined from individual data sets typically range from 2.5 up to 3.3 (Elmegreen & Falgarone 1996) and the mass-size plots often show systematic curvatures beyond a simple power law behavior. A smaller value and a better correlation is obtained when combining various observations covering a larger range of angular scales for a particular source. This has been done for the Polaris Flare data (Heithausen et al. 1998), which gives (and for the clump mass spectral index), presumably providing the most consistent determination up to date of these parameters within a single molecular cloud complex. An identical value of is obtained by combining a large set of data from different sources in the recent analysis by Elmegreen & Falgarone (1996). The values determined from individual data sets are thus systematically larger than the ones from data sets combining observations with different angular resolution and source distances. This indicates that the larger values obtained for the individual surveys are likely to be biased by resolution effects and other systematic errors of the clump size determination in the clump identifying procedure.
The relation derived above between the mass spectral index , the fBm -index and the mass-size relation index can be used to determine from the measured values of and , independently and without relying on the actual values of the sizes derived for individual clumps. For the Polaris Flare, used as the prime example within this paper, we then get , in perfect agreement with the value derived by Heithausen et al. (1998) from the combination of small and large scale observations. To summarize, there is good support for the concept, that the different methods of analyzing molecular cloud structure, including the determination of clump mass spectra and their spectral index, simply describe different aspects of the same underlying physical structure, which is characterized by a power law power spectrum of otherwise random density fluctuations. The derived relation between mass spectral index and fBm -index , can be used to independently determine the power law index of the mass-size relation.
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998