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