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*Astron. Astrophys. 336, 697-720 (1998)*
## 8. Summary
We have introduced a new method, the
*-variance* analysis, to study the structure
of molecular cloud images. The drift behavior, measured by the
*-variance* , is linked to the shape of the
*power spectrum* of the cloud image. Application to the observed
CO-maps of the Polaris Flare and the subset of the FCRAO outer galaxy
survey shows, that at least for these two examples, the *power
spectrum* has power law shape and the power law index is close to
in both cases. Analysis of the phase
distribution of the images shows them to be completely random. The
application of these concepts to a more complete sample of observed
cloud images will be studied in a future paper (Bensch et al., *in
prep.* ).
We have shown that other parameters derived via independent ways to
measure and characterize fractal cloud structure, such as the
traditional area-perimeter relation, are related to the drift behavior
measured via the *-variance* method, or the
power law index of the *power spectrum* . This also includes
other, at first sight independent, properties such as the mass
spectral index derived from clump decomposition of observed cloud
images. We have shown that an ensemble of randomly positioned clumps
with a given power law index of the clump mass spectrum and a given
power law mass-size relation has a *fractional Brownian motion*
structure of its projected image. The clump mass spectral index of
molecular clouds, derived by clump decomposition of the observed
intensity and velocity distribution, thus, together with the derived
index of the mass-size relation, determines the power law index of the
image *power spectrum* , . The observed
values for the mass spectral index and the
mass-size index of the Polaris Flare, both
derived from a clump decomposition of the observed cloud image over a
large range of spatial scales, as well as the power law index of the
*power spectrum* , , independently derived
via the *-variance* analysis agree with each
other along this relation.
These results show that, similar to the result by Elmegreen &
Falgarone (1996), the mass spectrum of molecular cloud clumps is
closely linked to the fractal structure of the gas. The relation
between clump mass and clump size spectrum and the fractal dimension
of the cloud image derived within the *fBm* concept agrees with
the observed values, but is in conflict with their relation based on
the *Koch-island* model for the fractal structure.
The above results suggest that the basic characteristics of
molecular cloud structure might well be described in a unified way as
a *fractional Brownian motion* structure, characterized by a
single parameter, e.g. the power law index of the *power
spectrum* . We show that images synthesized along these rules as
*fractional Brownian motion* images indeed look very much like
observed molecular cloud maps. Such synthesis thus provides a
potentially very useful tool to generate artificial structures well
representing real molecular clouds, e.g. for radiative transfer
modeling (Ossenkopf et al., *in prep.* ). Also, hydrodynamic
modeling of molecular clouds has to meet the structural
characteristics of such *fBm* -structures.
One should not forget, however, that molecular cloud structure is
likely to be much more complex than the simple concept of
*fractional Brownian motion* , which nevertheless applies well to
the basic characteristics of observed, 2-dimensional projected cloud
images. The clouds themselves are 3-dimensional and it might well be
that the 3-dimensional structure is much more complex than a simple
*fBm* structure, which only emerges in projection. Also, the
turbulent velocity fields within molecular clouds (providing pseudo
3-dimensional information from molecular line maps which only makes
such analysis as clump decomposition methods possible) are an
important ingredient and have to be included into a full treatment and
understanding of molecular cloud structure. Nevertheless, the
characteristics derived for the 2-dimensional projected images already
give certain constraints on the 3-dimensional structure. If the
3-dimensional phases are essentially as randomly distributed as is the
case for the phases of the 2-dimension image, the measured power law
index of the 2-dimensional image implies that the surface grows
proportional to volume for the 3-dim cloud structure, and that hence
most of the material is surface material. This is in accordance with
the well established fact that even ^{12}CO, though being a
completely optically thick tracer, measures cloud mass, as well as the
recently emerging view, that most line emission from molecular cloud
tracers is largely dominated by surface effects.
Examining the applicability of the concepts presented to a larger
sample of observed molecular images is certainly one important goal of
future work. Another one will be, to extend the observations to much
larger spatial coverage and higher signal to noise. The discussion
shows that, due to the steep power law decrease observed in the
*power spectrum* of cloud images, this will be very difficult
observationally even with large focal plane single dish arrays and
large size future interferometers.
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998
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