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