THE FRACTAL DIMENSION OF PROJECTED CLOUDS
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1
N. Sánchez , E. Alfaro , E. Pérez
Instituto de Astrofísica de Andalucía, CSIC, España
2
Departamento de Física, Universidad del Zulia, Venezuela
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GENERAT ION OF FRAC TAL CLOUDS
INTRODUCTION
Maps of nearby cloud complexes show that the gas and dust have a fractal structure that may be related to the
Within a sphere of radius R we place randomly N spheres of radius R/L with L>1, in each of these
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physical processes supporting the generation of structures in the ISM (Falgarone et al. 1992). Most of studies measure
spheres we place again N smaller spheres with radius R/L , and so on up to level H of hierarchy. The set
the degree of fractality of the ISM using the so-called perimeter-area method: if the iso-contours exhibit a power-law
H
H
of N spherical particles (of radius R/L ) in the last level forms an object with fractal dimension given
perimeter-area relation with a noninteger exponent, this exponent may be interpreted in terms of a fractal dimension
by Df = log N / log L. Most of simulations were made using N=3 fragments in a total of H=9 levels of
(Dper) that characterizes the manner these curves fill space (Mandelbrot 1983). The observational evidence can be
hierarchy (~ 2 x 104 particles in the last level) with the fractal dimension in the range 1 < Df < 3.
summarized by saying that the boundaries of molecular clouds appear to be fractal curves with dimension Dper ~ 1.35
(e.g. Dickman et al. 1990, Scalo 1990, Falgarone et al. 1991, Hetem and Lepine 1993, Vogelaar and Wakker 1994).
Df = 1.5
Df = 2.5
Df = 2.0
But the clouds are necessarily recorded as two-dimensional images projected onto a plane perpendicular to the lineof-sight and the connection between Dper and the fractal dimension of the tri-dimensional clouds (Df) is still an
open question. The fractal nature of the projected boundaries suggests that clouds may have fractal surfaces with
dimensions given by Dper+1 (Mandelbrot 1983), then it is usually assumed that Dper+1 ~ 2.35 should be the fractal
dimension of interstellar clouds (Beech 1992).
The purpose of this work is to investigate the relationship between the fractal dimension of a tri-dimensional cloud
and the fractal dimension of its projection, both for the whole projected image and for its boundary.
Figure 1: examples of projections of tri-dimensional fractals generated by using
three different values of fractal dimension Df.
FRACTAL DIMENSIONS
In general, a fractal quantity is a number which is connected to some
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3.5
Da
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length L in a manner like A ~ L , where Da would be the (constant)
2.5
DM
DC
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dimension associated to the quantity A. In practice Da corresponds with
the slope of the best fit in a logA-logL plot.
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2.5
DM
DC
Box-counting dimension (Db): if we cover the fractal with a grid and
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1.5
2
Db
count the number N(r) of occupied cells of size r, then N(r) ~ r
DC , DM
DC
DM
.
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Correlation dimension (Dc): given a set of points, then C(r) ~ rDc
where the correlation integral C(r) is proportional to the number of pairs
1.5
0.5
1
1.5
2
Df
for which the distance is less than r.
Mass dimension (Dm): in this case the number of particles M(r) inside a
sphere of radius r obeys the relation M(r) ~ rDm.
Perimeter-area based dimension (Dper): in a plane the perimeter (P)
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1
2.5
3
1
1.5
2
Df
2.5
3
Figure 2: the calculated correlation (Dc) and mass (Dm)
Figure 3: the average correlation and mass dimensions for the
dimensions as a function of the fractal dimension Df used to
projected fractals as a funciont of the tri-dimensional fractal
generate the cloud. Each point is the average of ten different
dimension Df. The line shows the theoretical result given by
realizations and the bars show the standard deviation.
Dpro=min{2,Df} (Falconer 1990).
and the area (A) are related by P ~ ADper/2.
PROJECTED IMAGES
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We project the simulated clouds on random planes, after that we place grids with various pixel sizes
Df=1.2
Df=1.6
Df=2.0
Df=2.4
Df=2.8
Npix=50
Npix=100
Npix=200
Npix=400
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1.8
1.6
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and the pixel size (i.e. the resolution is the maximum object size in pixel units). Finally we calculate the
perimeter-based dimension...
Dper
define the image "resolution" (Npix) as the ratio between the maximum two-pixel distance in the image
Dper
(resolutions) whose "brightness" is assigned by counting the number of particles inside each pixel. We
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10
APPLICATION TO ORION A
100
1
1.2
1.4
1.6
Npix
MOLECULAR CLOUD
We use a CS(1-0) integrated intensity map
obtained from high-resoltuion observations
1.8
2
Df
2.2
2.4
2.6
2.8
Figure 4: the perimeter based dimension as a function
Figure 5: the perimeter based dimension as a function
of the projected image resolution for different cloud
of the cloud fractal dimension for different image
fractal dimension Df.
resolutions Npix.
(Tatematsu et al. 1993).
MAIN RESULTS
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1) The calculated values for the projected fractal (correlation and mass) dimensions (Figure 3) are always below
0.67 ± 0.01
perimeter
1000
the theoretical result given by Falconer (1990).
2) Dper decreases as Df increases (Figure 5) because higher Df values generate clouds with more round-shaped
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boundaries (as can be seen in Figure 1).
10
10
100
1000
10000
100000
3) There is a tendency of Dper to decrease as Npix decreases (Figure 4) because when the pixel size is bigger the
area
Figure 6: the perimeter as a function of the
details of the roughness of the boundary disappear and the objects tend to have smoother boundaries. At
area (in pixel units) for Orion A.
relatively high resolution values Dper converges toward some value which we associate with the “real“ value.
4) If the perimeter-area relation of interstellar clouds yields Dper ~ 1.35 then the ISM fractal dimension should
1.5
Orion A
Df=2.5
Df=2.6
Df=2.7
100000
1.4
0.98
10000
be in the range 2.5 < Df < 2.7, higher than the result Df ~ 2.3 usually assumed.
5) For Orion A we obtained Dper = 1.34 (Figure 6) and therefore Df = 2.6 (Figure 5). When we decreased the
Dper
M(r)
1.3
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image resolution we verified that 2.5 < Df < 2.7 (Figure 7) for this molecular cloud.
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6) In order to test these results we estimated the mass dimension of Orion A calculating the total intensity of cells
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1.77
1.1
10
1
10
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of different side sizes (Figure 8). The index ~1.8 in the range of little cell size can be associated with the
1
100
r (pixels)
1000
Npix
estimated mass dimension and from Figure 3 we see that Df = 2.5-2.6. The index ~1 for high r values
Figure 8: the “mass“ (total intensity) as a
Figure 7: the perimeter-based dimension as a
corresponds to the bevavior expected for a line in the plane: this behavior is due to the overall geometrical shape
function of cell size for Orion A.
function of resolution for Orion A and for
of Orion A.
simulated clouds.
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