The Fractal Dimension of Projected Clouds
ABSTRACT The interstellar medium seems to have an underlying fractal structure which can be characterized through its fractal dimension. However, interstellar clouds are observed as projected two-dimensional images, and the projection of a tri-dimensional fractal distorts its measured properties. Here we use simulated fractal clouds to study the relationship between the tri-dimensional fractal dimension (D_f) of modeled clouds and the dimension resulting from their projected images. We analyze different fractal dimension estimators: the correlation and mass dimensions of the clouds, and the perimeter-based dimension of their boundaries (D_per). We find the functional forms relating D_f with the projected fractal dimensions, as well as the dependence on the image resolution, which allow to estimatethe "real" D_f value of a cloud from its projection. The application of these results to Orion A indicates in a self-consistent way that 2.5 < D_f < 2.7 for this molecular cloud, a value higher than the result D_per+1 = 2.3 some times assumed in literature for interstellar clouds.
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ABSTRACT: The dynamics of very heavy particles suspended in incompressible flows is studied in the asymptotics in which their response time is much larger than any characteristic time of fluid motion. In this limit of very large Stokes numbers, particles behave as if suspended in a δ-correlated-in-time Gaussian flow. At those spatial scales where the fluid velocity field is smooth, following Piterbarg [L.I. Piterbarg, The top Lyapunov exponent for stochastic flow modeling the upper ocean turbulence, SIAM J. Appl. Math. 62 (2002) 777] and Mehlig et al. [B. Mehlig, M. Wilkinson, K. Duncan, T. Weber, M. Ljunggren, Aggregation of inertial particles in random flows, Phys. Rev. E 72 (2005) 051104], the two-particle dynamics is reduced to a nonlinear system of three stochastic differential equations with additive noise. This model is used to single out the mechanisms leading to the preferential concentration of particles. Scaling arguments are used to predict the large Stokes number behavior of the distribution of the stretching rate and of the probability distribution function of the longitudinal velocity difference between two particles. As for the fractal character of the particle distribution, strong numerical evidence is obtained in favor of saturation of the correlation dimension to the space dimension at large Stokes numbers. Numerical results at finite Stokes number values reveal that this model catches some important qualitative features of particle clustering observed in more realistic flows.Physica D Nonlinear Phenomena 02/2007; · 1.67 Impact Factor - SourceAvailable from: Nicola LoaringIdo Finkelman, Noah Brosch, Alexei Y. Kniazev, Petri Väisänen, David A. H. Buckley, Darragh O'Donoghue, Amanda Gulbis, Yas Hashimoto, Nicola Loaring, Encarni Romero-Colmenero, Ramotholo Sefako[Show abstract] [Hide abstract]
ABSTRACT: We present new results from an ongoing programme to study the dust extragalactic extinction law in E/S0 galaxies with dust lanes with the Southern African Large Telescope (SALT) during its performance verification phase. The wavelength dependence of the dust extinction for seven galaxies is derived in six spectral bands ranging from the near-ultraviolet atmospheric cut-off to the near-infrared. The derivation of an extinction law is performed by fitting model galaxies to the unextinguished parts of the image in each spectral band, and subtracting from these the actual images. We compare our results with the derived extinction law in the Galaxy and find them to run parallel to the Galactic extinction curve with a mean total-to-selective extinction value of RV= 2.71 ± 0.43. We use total optical extinction values to estimate the dust mass for each galaxy, compare these with dust masses derived from IRAS measurements, and find them to range from 104 to 107 M⊙. We study the case of the well-known dust-lane galaxy NGC 2685 for which Hubble Space Telescope/Wide Field Planetary Camera 2 (HST/WFPC2) data are available to test the dust distribution on different scales. Our results imply a scale-free dust distribution across the dust lanes, at least within ∼1 arcsec (∼60 pc) regions.Monthly Notices of the Royal Astronomical Society 11/2010; 409(2):727 - 736. · 5.52 Impact Factor - SourceAvailable from: jkas.kas.orgYoungung Lee, Miju Kang, Bong Kyu Kim, Jae Hoon Jung, Hyun Goo Kim, Insung Yim, Hyung Woo Kang, Jihoon Choi[Show abstract] [Hide abstract]
ABSTRACT: We have estimated the fractal dimension of the molecular clouds associated with the HII region Sh 156 in the Outer Galaxy. We selected the \cocube data from the FCRAO CO Survey of the Outer Galaxy. Using a developed code within IRAF, we identified slice-clouds (2-dimensional clouds in velocity-channel maps) with two threshold temperatures to estimate the fractal dimension. With the threshold temperatures of 1.8 K, and 3 K, we identified 317 slice-clouds and 217 slice-clouds, respectively. There seems to be a turn-over location in fractional dimension slope around NP (area; number of pixel) = 40. The fractal dimensions was estimated to be D=1.5 ˜ 1.53 for NP >= 40, where P ∝ A^{D/2} (P is perimeter and A is area), which is slightly larger than other results. The sampling rate (spatial resolution) of observed data must be an important parameter when estimating fractal dimension. Fractal dimension is apparently invariant when varying the threshold temperatures applied to slice-clouds identification.Journal of the Korean Astronomical Society 01/2008; 41. · 0.91 Impact Factor
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arXiv:astro-ph/0501573v2 18 Feb 2005
THE FRACTAL DIMENSION OF PROJECTED CLOUDS
N´ estor S´ anchez,1,2Emilio J. Alfaro,1and Enrique P´ erez1
ABSTRACT
The interstellar medium seems to have an underlying fractal structure which
can be characterized through its fractal dimension. However, interstellar clouds
are observed as projected two-dimensional images, and the projection of a tri-
dimensional fractal distorts its measured properties. Here we use simulated frac-
tal clouds to study the relationship between the tri-dimensional fractal dimension
(Df) of modeled clouds and the dimension resulting from their projected images.
We analyze different fractal dimension estimators: the correlation and mass di-
mensions of the clouds, and the perimeter-based dimension of their boundaries
(Dper). We find the functional forms relating Df with the projected fractal di-
mensions, as well as the dependence on the image resolution, which allow to
estimate the “real” Df value of a cloud from its projection. The application of
these results to Orion A indicates in a self-consistent way that 2.5 ? Df ? 2.7
for this molecular cloud, a value higher than the result Dper+1 ≃ 2.3 some times
assumed in literature for interstellar clouds.
Subject headings: ISM: structure — ISM: clouds — ISM: general
1. INTRODUCTION
Maps of nearby cloud complexes have shown that the gas and dust are organized into
irregular hierarchical structures which seem similar to each other over a wide range of scales
(Scalo 1990; Falgarone et al. 1992). This self-similarity has been interpreted as a signature of
an underlying fractal geometry that may be related to the physical processes supporting the
generation of structures in the interstellar medium. The origin of this fractal-like structure
seems to be turbulence (Falgarone & Phillips 1990; Kritsuk & Norman 2004; Padoan et al.
2004), although self-gravity could also play an important role (de Vega et al. 1996). A fractal
1Instituto de Astrof´ ısica de Andaluc´ ıa, CSIC, Apdo. 3004, E-18080, Granada, Spain; nestor@iaa.es,
emilio@iaa.es, eperez@iaa.es.
2Departamento de F´ ısica, Universidad del Zulia, Maracaibo, Venezuela.
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structure for the interstellar medium is consistent with many observed features, including
the cloud size and cloud mass distribution functions (Elmegreen & Falgarone 1996), the
properties of the intercloud medium (Elmegreen 1997a), and even the stellar initial mass
function (Larson 1992; Elmegreen 1997b, 2002).
Therefore, it is important to measure the degree of fractality and/or complexity of the
ISM. There are several ways this can be done, such as structure tree methods (Houlahan &
Scalo 1992), delta variance techniques (Stutzki et al. 1998), principal components analysis
(Ghazzali et al. 1999), multifractal analysis (Chappell & Scalo 2001), metric space techniques
(Khalil et al. 2004), etc. A simple method consists of calculating the fractal dimension
characterizing structures in the ISM, but it has to be mentioned that quantifying the fractal
dimension of an object is generally not sufficient to characterize it, because objects with
different morphological properties could have the same fractal dimension (Mandelbrot 1983).
In spite of this a number of studies have been done to estimate the fractal dimension of
interstellar clouds. Most of these studies use the so-called perimeter-area method: if the iso-
contours exhibit a power-law perimeter-area relation with a noninteger exponent over certain
range of scales, this exponent may be interpreted in terms of a fractal dimension (Dper) that
characterizes the manner these curves fill space (Mandelbrot 1983). This method was used
by Beech (1987) on a group of selected dark clouds to estimate a fractal dimension of 1.4,
whereas Bazell & Desert (1988) calculated an average fractal dimension of 1.26 for cirrus
regions discovered by IRAS. Dickman et al. (1990) studied five different molecular cloud
complexes (Chamaeleon, R CrA, ρ Oph, Taurus and the Lynds 134/183/1778 group) finding
1.17 ? Dper ? 1.28. Also for the Taurus complex, Scalo (1990) found Dper = 1.4 and
Falgarone et al. (1991) found Dper= 1.36 over a very wide range of sizes. Hetem & Lepine
(1993) reported 1.3 ? Dper? 1.52 for the Chamaeleon complex with an average of 1.44, a
value notoriously higher than the one of Dickman et al. (1990). Vogelaar & Wakker (1994)
also estimate relatively high values for Dperwhich generally range between 1.35 and 1.5 for
high-velocity clouds and IRAS cirrus.
The observational evidence can be summarized by saying that the boundaries of molecu-
lar clouds appear to be fractal curves with dimension Dper∼ 1.35 (or some value in the range
1.2 ? Dper? 1.5). But the clouds are necessarily recorded as two-dimensional images pro-
jected onto a plane perpendicular to the line-of-sight. The connection between Dperand the
fractal dimension of the tri-dimensional clouds is still an open question. The fractal nature
of the projected boundaries suggests that these clouds may have fractal surfaces with dimen-
sions given by Dper+1 (Mandelbrot 1983). Then, it is usually assumed that Dper+1 ≃ 2.35
should be the fractal dimension of interstellar clouds (e.g. Beech 1992). However, although
it is possible that much of their internal structure is directly reflected in surface features,
there is not necessarily any simple relation between the fractal dimension of the surface of
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a cloud and that of its internal structure. On the other hand, it has been proven that the
intersection of a fractal with dimension Dfand a plane gives another fractal with dimension
given by Df−1 (Mandelbrot 1983), but a projection is a totally different operation and the
result Dpro= Df− 1 does not have to be expected for the projected fractal dimension. In
fact, if an object of fractal dimension Dfembedded in a tri-dimensional Euclidean space, is
projected on a plane, it is possible to show that the projection has dimension Dpro= Dfor
Dpro= 2 depending on whether Df< 2 or Df> 2, respectively (Falconer 1990). Thus, the
projection of a cloud with Df≃ 2.35 would give rise to a compact shadow that may not be
a fractal, but we want to emphasize that the relationship between the fractal dimension of
this projection and that of its boundary is not clearly established.
The purpose of this work is to investigate the relationship between the fractal dimen-
sion of a tri-dimensional cloud and the fractal dimension of its projection, both for the whole
projected image and for its boundary. To do this we use artificial fractal clouds generated by
using an algorithm which mimics in some way the hierarchical fragmentation process occur-
ring in molecular cloud complexes. Our main goal is to estimate the real fractal dimension of
interstellar clouds from observed images. In § 2, we explain the manner in which we simulate
the clouds and, after a brief review on some definitions of fractal dimension, we calculate
the fractal (correlation and mass) dimensions of the projected clouds. In § 3, we address the
perimeter-area relation of the projected images and its dependence on the image resolution.
This analysis is applied to the Orion A molecular cloud in § 4 and, finally, the main results
are discussed in § 5.
2. FRACTAL DIMENSION OF SIMULATED CLOUDS
We first have to generate tri-dimensional fractal clouds. For convenience we use simple
recursive construction rules based on the Soneira & Peebles (1978) method. Within a sphere
of radius R we place randomly N spheres of radius R/L with L > 1, in each of these spheres
we place again N smaller spheres with radius R/L2, and so on up to level H of hierarchy.
The set of NHpoints (actually little spherical particles of radius R/LH) in the last level
forms an object with fractal dimension given by Df= logN/logL. Most simulations were
made using N = 3 fragments in a total of H = 9 levels of hierarchy (∼ 2 × 104particles
in the last level) with the fractal dimension in the range 1 < Df < 3. However, when the
fractal dimension is too high (Df? 2.6) the filling factor tends to 1 and it becomes difficult
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to place randomly the spheres through all the volume without they being superposed1. In
these cases we use N = 8 fragments in H = 5 levels (∼ 3×104particles) and each fragment
is placed randomly within the available volume in the direction of each of the eight octants.
This allows us to reach high fractal dimension values (and filling factors) easily. In any case,
several tests showed that our results do not depend (at least within the error bars) on N
and/or H (as long as we have a sufficiently high number of particles in the last level). In
order to illustrate the appearance of the point distributions, Figure 1 shows examples of
projections of tri-dimensional fractals generated by using N = 3, H = 9 and three different
values of fractal dimension (Df= 1.5, 2.0 and 2.5).
2.1.Fractal Dimension Estimation
In the following, we will focus our attention on the empirical determination of the
fractal dimension both for the simulated clouds and for their two-dimensional projections. In
general, a fractal can be defined as a set of points whose Hausdorff dimension DHis strictly
larger than its topological dimension DT (Mandelbrot 1983). To estimate the Hausdorff
dimension authors normally use working definitions which fit their method and needs, and
thus there is not a unique definition of fractal dimension. In general, a fractal quantity is
a number which is connected to some length l in a manner like A ∼ lDA, where DAwould
be the (constant) dimension associated to the quantity A. The most straightforward way
to estimate the Hausdorff dimension of a fractal set is through the box-counting method,
based on the fact that the number of boxes, N(r), having side r needed to cover a fractal
object varies as r−DB, being DB an estimation of DH (Mandelbrot 1983) which is called
the “box-counting dimension” (or “capacity”). Then, if we cover the object with a grid and
count the number N(r) of occupied cells of size r, we can compute
DB= −lim
r→0
logN(r)
logr
. (1)
In practice DBcorresponds with the slope of the best fit in a logN−logr plot, but obviously
the range where this scale law is valid in real (finite) fractals is limited to the range between
the smallest measurable region (the resolution) and the object size.
The major problem with the box-counting method is that it leads to imprecise results
because of its sensitivity to, for example, the placement and orientation of the grid and
1We avoid superposition because the tests made allowing superposition showed that the properties are
modified in such a way that it appears a multifractal behavior and a unique fractal dimension can not be
defined in a wide scale range
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the range and sequence of values of box side length (Buczkowski et al. 1998; Gonzato et
al. 2000). A more useful method when dealing with a point distribution in space has been
introduced by Grassberger & Procaccia (1983): given a set of Nppoints with positions x,
one first computes the correlation integral C(r) defined as
C(r) =
2
Np(Np− 1)
?
1≤i<j≤Np
H (r − |xi− xj|) , (2)
where H is the Heaviside step function. The summation counts the number of pairs for
which the distance |xi−xj| is less than r. For a fractal set the correlation integral scales at
small r like rDC, being DCthe “correlation dimension” of the set, so that
DC= lim
r→0
logC(r)
logr
. (3)
The dimension DCcan be identified with the slope of the log-log plot of C(r) versus r, but
usually this power-law behavior is valid only in a finite range. Different (larger) slopes can be
observed at very small scales which are associated with random (experimental) errors, noise
and with the lack of statistics of finite samples at very small scales; and there are deviations
due to nonlinear effects for too large r values (Theiler 1987).
If one now regards the mass (number of particles) inside a sphere of radius r we can define
the “mass dimension” (DM) if the mass obeys the self-similar scaling relation (Mandelbrot
1983):
M(r) ∼ rDM. (4)
A simple algorithm to estimate DMconsists in generating a sequence of spheres with different
radii, count the number of points inside each one, and calculate the slope in a logM versus
logr plot. An obvious problem is where to place the spheres, because large fluctuations in
M (lower values) can appear if the spheres lie by chance in or near the voids that exist
in the distribution. To avoid this under-sampling problem we modified this algorithm in
the following way. We choose randomly a position within the fractal cloud where we place
spheres having progressively greater radii and we calculate the density (number of points per
unit volume) in each case. Usually the density shows an absolute maximum corresponding
to the sphere which gives a better sampling in that point, then we take only this sphere
and reject all the other ones. This procedure is repeated many times at random positions
through the whole fractal volume. Finally, if several spheres have the same radius we only
consider the densest sphere (i.e., the “best” sampling) to calculate the mass corresponding to
that radius value. With this strategy we improve the sampling taking the spheres containing
more information throughout all the structure.
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We have calculated the correlation and mass dimensions as a function of Df for the
simulated tri-dimensional fractal objects. Figure 2 shows the correlation integral as a func-
tion of distance (normalized to the maximum distance between two particles in the fractal
rmax) for the same three fractal clouds shown in Figure 1. The solid lines do not show the
best fit but the expected slope for each fractal dimension value Df. As mentioned above,
the correlation integral has a well-defined power-law dependence but over a limited range:
for too high r values the slope becomes smaller as C approaches the asymptotical value
C = 1, while for too small r values statistical fluctuations produce a steeper slope. This
effect is more notorious for low fractal dimensions because the low filling factor values allow
a bigger random component when placing the spheres in each level of hierarchy. Figure 3
shows the sphere mass M(r) as a function of its radius for the same fractal clouds shown in
Figure 1. Again, the lines show the expected (theoretical) slope value. In this case a low
fractal dimension value (Df = 1.5) shows more fluctuations because the fractal has more
empty regions (see Figure 1) and there is a higher probability of underestimating the mass.
Notwithstanding the errors being larger than with the correlation integral, a good estimation
of the fractal dimension can be done with this method as it will be shown next. For a given
(theoretical) fractal dimension Df, we have calculated C(r) and M(r) for ten different ran-
dom clouds and we have estimated the average correlation (DC) and mass (DM) dimensions.
The results are plotted in Figure 4, where we can see that both DCand DMare very similar
to the theoretical value Df in the analyzed range (1 < Df < 3), although the error bars
are higher for DMmainly for very low Dfvalues where there are more empty spaces in the
simulated clouds.
2.2.Projected Fractal Dimension
We are interested in knowing what happens with the estimated values of fractal di-
mensions when the measurements are made on the two-dimensional projected images of the
fractal clouds. It has been proven analytically that the projection of a fractal with tri-
dimensional dimension Df yields a projected dimension given by (Falconer 1990; Hunt &
Kaloshin 1997):
Dpro= min{2,Df} .(5)
Thus, Dfand Dproare equal if Df< 2 and Dpro= 2 if Df> 2 (in this last case it would not
be possible to estimate Df from the projected image). However, equation (5) was deduced
for indefinitely extensive (mathematical) fractals and its validity for real finite fractals with
certain random component is not obvious. At least for random fractal aggregates the equa-
tion (5) can not be applied (Nelson et al. 1990; Jullien et al. 1994; Maggi & Winterwerp
2004). We have made projections of the simulated clouds on random planes and we have
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calculated the average correlation and mass dimensions for the resulting projections. An
example is shown in Figure 5, where we have plotted the correlation integral for the original
tri-dimensional fractal and for its two-dimensional projection using the cloud with dimen-
sion Df = 1.5 shown in Figure 1. The line indicates the expected slope (1.5) for both the
tri-dimensional and bi-dimensional fractals (according to eq. 5). For the assumed values of
N, H and Dfthe smallest possible distance between two particles in the simulated fractal is
rmin∼ 10−3, and all the distances smaller than rminin the projected fractal arise from the
random projection itself. Thus, for r > rminwe find that the projected fractal has a slope in
the log-log plot similar to (although a little smaller than) the expected one, but for r < rmin
the slope increases tending to ∼ 2 (the value corresponding to a random distribution of
particles on the plane). When estimating the fractal dimension of the projected fractals we
have considered only distances greater than rmin. Figure 6 shows the measured correlation
dimension (DC,2D) and mass dimension (DM,2D) for the projected clouds as a function of the
tri-dimensional fractal dimension (Df). Each point is the result of calculating the average
of ten different realizations (random fractals) and the bars indicate the standard deviations.
We can see that the calculated values are always below the theoretical result given by equa-
tion (5), except for very low (Df∼ 1) or very high (Df∼ 3) fractal dimension values. This
result looks similar to the one of Maggi & Winterwerp (2004) obtained for the capacity di-
mension of projected fractal aggregates. Since DC,2Dvaries continuously with Df, one might
generally be able to extract the fractal dimension of a three-dimensional cloud by analyzing
the correlation dimension of its projection, even when Df> 2, but DC,2Dchanges too slowly
with Df(within the error bars) for Df? 2.5. In spite of this degeneration both DC,2Dand
DM,2Dcould be used for doing a rough estimation of the fractal dimension Df.
3. PERIMETER-AREA RELATION
The fractal dimension of a bounding contour can be determined via the perimeter-area
relation (Mandelbrot 1983). In a plane, the perimeter (P) and the area (A) are related by
P ∼ ADper/2, (6)
where the dimension Dper characterizes the degree of perimeter contortion. Objects with
smoothly varying contours (e.g. circles) will have Dper = 1 (i.e. P ∼ A1/2), whereas ex-
tremely convoluted plane-filling contours will have Dper= 2 (P ∼ A). Thus, the irregularity
of a bounding contour is characterized by 1 ≤ Dper≤ 2. The same argument applies for
three-dimensional surfaces but in this case the dimension is 2 for smooth surfaces tending
toward 3 for a convoluted surface which fills the space. The contour of a projected object is a
subset of its surface and the measure of the contour Dperis not related (at least in an obvious
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way) to, for example, the capacity DB. The perimeter segmentation reflects the roughness of
the boundary while the capacity measures the space-filling ability. Both of them, however,
give information on the fractal structure of the object.
We wish to investigate to which extent the information of the tri-dimensional structure
can be found from the projected perimeter-based fractal dimension Dper. In order to do this
we first generate two-dimensional images applying the following procedure: we project the
simulated clouds on random planes, then we place grids with various pixel sizes (resolutions)
whose “brightness” is assigned by counting the number of particles inside each pixel. We
define the image “resolution” (Npix) as the ratio between the maximum two-pixel distance in
the image and the pixel size (i.e. the resolution is the maximum object size in pixel units).
The calculation of Dper begins fixing a threshold brightness level ncri and defining each
“object” into the image as a set of connected pixels (having a side or a corner in common)
whose brightness value is above ncri. The area of each object is found by counting the total
number of pixels and the perimeter is calculated by summing the lengths of the sides of
pixels along the edge of the object2. Then, we calculate the best linear fit in a logP −logA
plot using the objects measured for different brightness levels and the slope of this fit would
be Dper/2. To increase considerably the number of data points in the linear fit we have taken
∼ 20 brightness levels equally spaced between the minimum and the maximum brightness.
If an object is relatively small (containing less than a few pixels) its perimeter is not well
defined and the structural details will disappear, we therefore demanded that objects contain
a minimum (∼ 15) number of pixels. Of course, several variations of the above procedure are
possible: the definition of connectivity for neighbouring pixels (4-connected or 8-connected),
the way for calculating the perimeter (for example, the length of the diagonal could be
considered for the pixels at corners instead of the sum of the two sides), to take (or not) into
account the inner holes, or the minimum number of pixels that can constitute one object
in the fit. It has been proven that the estimation of Dper depends very weakly on these
criteria (Vogelaar & Wakker 1994). In any case we tested the influence of all these criteria
and verified that the result were not affected as long as objects too little were not considered
in the procedure.
Figure 7 shows the perimeter-area based dimension Dperas a function of resolution Npix
for different values of the cloud fractal dimension Df(for clarity, only five values of Dfare
shown). The bars indicate the standard deviations resulting from 10 different realizations for
each Dfvalues, the mean errors of the best linear fits in the logP −logA plots always were
less than these error bars. The first feature we note is that Dperdecreases as Df increases.
2In this computation, we consider the sum of both the external and the internal perimeter, i.e. the
perimeter of the internal voids are also computed
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This is an expected result because bigger Dfvalues generate clouds with more round-shaped
boundaries, otherwise when Df decreases the clouds have more irregular boundaries, as it
can be seen in Figure 1. For a given fractal dimension Df there is a tendency of Dperto
decrease as Npixdecreases, and this is because when the pixel size is bigger the details of the
roughness of the boundary disappear and the objects tend to have smoother boundaries (and
Dper→ 1). Moreover, a very low number of pixels (and consequently number of objects)
and/or very low Dfvalues (i.e. very fragmented structures) increase errors notoriously. On
the other hand, at relatively high resolution values Dperconverges toward some value which
we associate with the “real” Dpervalue resulting from the projection. The dependence of
Dperon Df can be seen more clearly in Figure 8 for different fixed resolutions (only a few
Npixvalues are shown for clarity, but for Npix? 400 the curves overlap each other within
the error bars). In general Dperdecreases (from ∼ 2 to ∼ 1) as Df increases (from ∼ 1 to
∼ 3), as can be seen in Figure 8a, but as resolution decreases the curves move down mainly
at relatively high Dfvalues. Figure 8b shows Dperin a smaller range of Dfvalues but using
50 different random clouds instead of 10, so that the random fluctuations become smoother
(these data are also shown in Table 1). We can estimate that if the perimeter-area relation
of interstellar clouds images yields Dper≃ 1.35 (this value is indicated with a horizontal line
in Figure 8b) then the fractal dimension of the tri-dimensional cloud should be some value
in the range 2.5 ? Df? 2.7. This is higher than the result Df≃ 2.3 that could be inferred
when a relation of the form Df= Dper+ 1 is assumed.
4. APPLICATION TO ORION A MOLECULAR CLOUD
In this section we use the results obtained to estimate the fractal dimension of Orion
A giant molecular cloud3. We use a13CO (1-0) integrated intensity map obtained from
high-resolution observations (15” beam and 40” grid spacing) with the 45 m telescope of the
Nobeyama Radio Observatory (Tatematsu et al. 1993). To calculate Dperwe run exactly the
same algorithm used in § 3 for simulated clouds. Figure 9 shows the perimeter-area log-log
plot for Orion A (in pixel size units). Each point represents different objects (clumps) for
different intensity thresholds and the line shows the best linear fit including all the points
plotted. The resulting dimension (twice the slope) is Dper = 1.32 ± 0.02, a value within
the range of measured dimensions in the interstellar clouds. As concluded before for the
simulated clouds, from Figure 8b (see also Table 1) one obtains that the tri-dimensional
fractal dimension of Orion A should be around 2.6 − 2.7.
3It has to be mentioned that Orion A has a very elongated morphology, and the analysis presented applies,
in principle, to homogeneous and isotropic fractals
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In order to verify this result we have decreased the resolution of the Orion A image in
successive steps, defining new larger pixels whose intensities are given by the total intensities
of the neighbouring pixels. The left side panel in Figure 10 is an image with the original
resolution given by Tatematsu et al. (1993) while in the right side panel the size of each
pixel is ten times the original one. The comparison of these images shows clearly that
progressively worse resolutions smooth the contours decreasing its degree of convolutedness,
because the details blend with each other. Thus, Dpershould decrease when the resolution
is degraded beyond a critical value which depends on the fractal dimension (as shown in
Figure 7). In each step we have estimated the resolution Npix(number of pixels along the
maximum distance) and we have repeated the calculation of the perimeter-based dimension
Dper. The results are shown in Figure 11, where for comparison we have also plotted the
calculated results for the clouds simulated with Df= 2.5, 2.6 and 2.7. The bars indicate the
standard errors of the fits and, although they become relatively high at low resolutions, it is
clear from this Figure that the way in which Dperdepends on Npixfor Orion A is consistent
with a fractal dimension in the range 2.5 ? Df? 2.7. The dependence of Dperon Npixcould
be a better method to estimate Dfthat one single Dpermeasure, mainly for low resolution
observations.
A final test we can do is to estimate the mass dimension of Orion A by placing cells of
different side sizes (r) on the image and calculating the total intensity of the cells. We call
this quantity the “mass” M(r) assuming that the intensity of each pixel is proportional to the
total column mass along the line-of-sight. We have used the same algorithm explained in § 2
to calculate the mass as a function of the cell size, doing random sampling of relatively dense
regions on the Orion A image. We first generate ∼ 5000 random positions along the rectangle
of ∼ 1000×1300 pixels containing the Orion A image (see Figure 12). The positions located
outside the image (∼ 80%) are automatically rejected as well as those positions where no
relative density maximum is found when increasing the size of the cells (see § 2). At the
end of this procedure, only ∼ 40 points remained available to be used in the calculation of
the mass dimension. Open circles in Figure 12 indicate the position of the final sampling
on the CO map of Orion A used to calculate M(r). We see that the algorithm avoids to
select low density regions where noise could introduce a higher error in the determination
of the mass dimension. Figure 13 is the resulting plot of M(r) versus r for the Orion A
cloud. The slope of the best fit (1.82 ± 0.03) can be associated with the mass dimension of
the projected image (DM,2D) from which we can obtain, using the results shown in Figure 6,
that the tri-dimensional cloud would have a fractal dimension of Df≃ 2.5−2.7. This result
is in agreement with the one obtained above from the perimeter-area relation. We tested
the reliability of this result varying the sampling (both changing and increasing the number
of random positions on the image), and the estimated mass dimension remained unchanged
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within the error bar resulting from the linear fit.
5. DISCUSSION AND CONCLUSIONS
In this work we have generated tri-dimensional clouds with given fractal dimensions
(Df) which we projected on a plane to calculate the fractal dimension of the projected image.
The main results derived from this empirical approach concern the functional forms relating
the tri-dimensional and the projected fractal dimensions. Both the correlation and the mass
dimensions increase as Dfincreases (Figure 6), but at least for the hierarchical, finite, random
fractals we generated, the projected dimensions are always below the theoretical (expected)
values given by equation (5). We also analyzed the perimeter-based fractal dimension (Dper)
and its dependence on both Df and the image resolution (Npix). In general, Dper is a
decreasing function of Df, and it tends to decrease as Npixdecreases (Figures 7 and 8).
The results obtained with this method allow to estimate the “real” fractal dimension of
a cloud from its projection using different fractal dimension estimators. The application to
Orion A yields Dper≃ 1.32 for this molecular cloud which implies, according to our results,
a tri-dimensional fractal dimension around 2.6−2.7. Moreover, both the dependence of Dper
on the resolution (Figure 11) and the projected mass dimension (Figure 13) confirm that
Df∼ 2.6 for Orion A. This value is clearly higher than the result Dper+1 ≃ 2.3 some times
assumed in literature for interstellar clouds (e.g. Beech 1992). Elmegreen & Falgarone (1996)
concluded that the size and mass spectra of interstellar clouds can be the result of an ISM
with tri-dimensional fractal dimension around 2.3, but Stutzki et al. (1998) have argued that
this numerical agreement could be coincidental. In fact, Stutzki et al. (1998) used fractional
Brownian motion structures to analyze the observed ISM properties concluding that the
index of the power spectrum that best reproduces the observed characteristics is ≃ 2.8,
which would correspond to a cloud surface fractal dimension of Dsur≃ 2.6. Although this
quantity is not necessarily equal to Df, it is interesting to note the agreement with our results.
The only result Stutzki et al. (1998) did not derive explicitly was Dper= Dsur− 1 = 1.6 for
the corresponding perimeter-area dimension.
The relationship between Dfand the physical processes determining the ISM structure
is still an open issue. Turbulent diffusion in an incompressible fluid gives Df∼ 2.3 (Meneveau
& Sreenivasan 1990) and this fact has been considered as evidence favoring the turbulent
origin for the observed fractal structure (Elmegreen & Falgarone 1996). Our results seem
to be in conflict with this interpretation, but one should not forget that ISM is a complex
system where other effects (such as self-gravity, or compressibility) have to be taken into
account. However, the results of the present study show a clear evidence favoring fractal
Page 12
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dimensions for interstellar clouds higher than the values more commonly assumed.
We want to thank Bruce Elmegreen for his valuable comments. We also thank the
referee for his/her helpful comments and criticisms which improved this paper. N. S. would
like to acknowledge the funding provided by CONDES (Universidad del Zulia), FONACIT
(Venezuela), and the Secretaria de Estado de Universidades e Investigacion (Spain). E. J.
A. acknowledges the financial support from MECyD through grants AYA-2001-1696 and
AYA-2001-3939-C03-01, and from Consejeria de Educacion y Ciencia (Junta de Andalucia)
through the group TIC 101. E.P. acknowledges financial support from spanish grants AYA-
2001-3939-C03-01 and AYA-2001-2089.
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This preprint was prepared with the AAS LATEX macros v5.2.
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Fig. 1.— Examples of clouds simulated with N = 3 fragments per step in H = 9 levels of
hierarchy for three different values of fractal dimension: 1.5, 2.0 and 2.5.
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10-4
10-3
10-2
10-1
100
10-3
10-2
10-1
100
C(r)
r/rmax
1.5
2.0
2.5
Fig. 2.— The correlation integral C(r) for the three fractals shown in Figure 1: Df= 1.5
(open squares), Df= 2.0 (open circles) and Df= 2.5 (open triangles). The lines show the
expected slope for each fractal dimension value.
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