UNLV Retrospective Theses & Dissertations
1-1-1999
Classification of galaxies using fractal dimensions
Sandip G Thanki University of Nevada, Las Vegas
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Repository Citation Thanki, Sandip G, "Classification of galaxies using fractal dimensions" (1999). UNLV Retrospective Theses & Dissertations. 1050. http://dx.doi.org/10.25669/8msa-x9b8
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CLASSIFICATION OF GALAXIES USING
FRACTAL DIMENSIONS
by
Sandip G. Thanki
Bachelor of Science Widener University 1997
A thesis submitted in partial fulfillment of the requirements for the
Master of Science Degree Department of Physics College of Sciences
Graduate College University of Nevada, Las Vegas August 1999
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thesis Approval The Graduate College University of Nevada, Las Vegas
July 29______.19 99
The Thesis prepared by
Sandip G Thanki
Entitled
Classification of Galaxies Using Fractal Dimensions
is approved in partial fulfillment of the requirements for the degree of
______Master of Science in Physics ______
6 X
Examination Committee Chair
Dean of the Graduate College
Examination Committee M emer
Examination Committee Member
Graduate College faculty Represematwe
11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT
Classification of Galaxies Using Fractal Dimensions
by
Sandip Thanki
Dr. George Rhee, Examination Committee Chair Assistant Professor University of Nevada, Las Vegas
The classification of galaxies is morphological. Shapes of the galaxies range from
the very simple (e.g. elliptical galaxies) to the highly complex (e.g. irregular galaxies).
Analyzing a measure of complexity for such shapes could lead to automatic classifica
tion. Fractal dimension, a quantity related to the complexity of a given shape, could
be such a measure. Capacity dimension and correlation dimension are two of the
several types of fractal dimensions. In this project, correlation dimensions and the
capacity dimensions of the contours generated around different intensity levels of the
galaxy images, versus the intensity levels were computed. It was found that elliptical
galaxies tend to have a lower value of the average correlation dimension (computed
for a selected range of intensity levels) than spirals.
Ill
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS
ABSTRACT ...... iii
LIST OF TABLES...... vi
LIST OF FIGURES......
ACKNOWLEDGMENTS...... viii
CHAPTER 1 INTRODUCTION...... 1
CHAPTER 2 GALAXY CLASSIFICATION SCHEMES ...... 3 “Tuning Fork” ...... 3 Elliptical Galaxies ...... 3 Spiral Galaxies ...... 4 Irregular Galaxies ...... 5 Other Classification Schemes...... 5
CHAPTER 3 FRACTAL DIMENSIONS...... 6 Defining Fractals and Fractal Dimensions...... 6 Capacity Dimension...... 9 Correlation Dimension...... 10 Calculating Fractal Dimension...... 11
CHAPTER 4 DATA SET ...... 12
CHAPTER 5 FRACTAL DIMENSIONS OF GALAXIES...... 17 Contour Generation...... 17 Calculation of Fractal Dimensions of the Contours...... 18 R e su lts ...... 20 Capacity Dimensions and Correlation Dimensions...... 20
IV
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparing Correlation Dimensions and Capacity Dimensions...... 23
CHAPTER 6 CONCLUSIONS ...... 40
REFERENCES...... 42
V IT A ...... 43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES
Table 1 Length of the coast of Britain...... 7 Table 2 Length of a circle with a diameter of 1000 k m ...... 8 Table 3 Data set ...... 12 Table 4 NGC 2403 to NGC 3953 (Data for the entire intensity range) 32 Table 5 NGC 4013 to NGC 4527 (Data for the entire intensity range) 33 Table 6 NGC 4535 to NGC 5746 (Data for the entire intensity range) 34 Table 7 NGC 5792 to NGC 6503 (Data for the entire intensity range) 35 Table 8 NGC 2403 to NGC 3953 (Data for a selected intensity range) 36 Table 9 NGC 4013 to NGC 4527 (Data for a selected intensity range) 37 Table 10 NGC 4535 to NGC 5746 (Data for a selected intensity range) 38 Table 11 NGC 5792 to NGC 6503 (Data for a selected intensity range) 39
VI
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES
Figure 1 Tuning Fork Diagram ...... 3 Figure 2 Koch curve ...... 6 Figure 3 The coast of Britain...... 7 Figure 4 Data Set: NGC 2403 to NGC 4242...... 14 Figure 5 D ata Set: NGC 4254 to NGC 5248...... 15 Figure 6 Data Set: NGC 5322 to NGC 6503 ...... 16 Figure 7 NGC 4374 and NGC 4303 ...... 18 Figure 8 Fractal Dimensions of NGC 4374 (Elliptical) and NGC 4303 (Spiral) 20 Figure 9 Number of spirals and ellipticals versus avg. cap. and corr. dimensions 21 Figure 10 Number of galaxies versus fractal dim. for a selected intensity range 23 Figure 11 Difference between corr. and cap. dimensions for spirals and ellipticals 24 Figure 12 Difference between dimensions for selected intensity range...... 25 Figure 13 Intensity fall off from the center of an elliptical and a spiral galaxy . 27 Figure 14 Some of the contours for NGC 2403 and its fractal dimensions 28 Figure 15 Some of the contours for NGC 5813 and its fractal dimensions 29 Figure 16 Fractal dimensions of NGC 5813 after removing the s ta...... r 30 Figure 17 Some of the contours for NGC 3031 and its fractal dimensions 31
Vll
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS
I would like to thank my advisors Dr. George Rhee and Dr. Stephen Lepp for
their constant guidance and support. I also thank my committee members Dr. Lon
Spight, Dr. Donna Weistrop and Dr. Wanda Taylor for their insight.
I thank Mr. John Kilburg for all the computer support and the knowledge that
he provided.
I would also like to express my appreciation to Diane Eggers, Greg Piet, Mark
Hancock and Anthony Zukaitis for providing me help and confidence when I needed
them.
Finally, I thank my family, especially my brother Ketan Thanki whose inspiration
and support brought me to physics.
\T11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1
INTRODUCTION
There are no bigger assemblies of stars, gas, dust and dark matter known than galaxies or clusters of galaxies. Our galaxy, the Milky-way galaxy, is an average galaxy
and yet it contains more than 10^^ stars. The importance of studying galaxies can
not be over emphasized because studying them is almost synonymous with studying the universe since they contain the information about its past, present and the future. Human beings have been looking at galaxies for more than two hundred years
with telescope aided eyes. The first photograph, in the visible range of the spectrum,
came about 100 years ago but the proof of their existence, as external systems, did
not come until 1924. In the 1950s the other windows of the electromagnetic spec
trum started to give important information. In the last two decades the research has
reached a significant level of maturity with galaxies routinely detected at many wave lengths. With improvements in the computer technology, the galaxy images moved
from conventional photographic plates to computer files with digitized data. The
chemical processes on the plates were now replaced with electron counts by charge
coupled devices (CCDs) bringing the detection efficiency close to 100%. With the improved technology and quality of data, the quantity of data being archived also started to increase rapidly. The classification of this data, which is
done manually, has become tedious and time consuming and has started to call for
an automated process of classification. Fractal dimension is a mathematical quantity directly related to the complexity
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of a given data set. Classification of galaxies, discussed in chapter 2, is based on the morphology of the galaxies. It is also related to the complexity of their shapes. In
this project, fractal dimensions of galaxies were computed for a data set of galaxies
and conclusions were derived for the automated classification possibilities of galaxies
using fractal dimensions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2
GALAXY CLASSIFICATION SCHEMES
“Tuning Fork”
Edwin Hubble, in the 1920s, devised a classification scheme for galaxies based
on their appearance. According to his scheme, galaxies are distributed in three classes: elliptical, spiral and irregular. Hubble’s classification scheme follows a so called “tuning-fork” diagram as shown in Figure 1. The sphtting of the diagram is because of the barred and the ordinary spiral galaxies (Zeilik, Gregory & Smith 1992).
Irr
Figure 1 Tuning Fork Diagram
Elliptical Galaxies
Elliptical galaxies have the shape of an oblate spheroid. They appear as luminous elliptical disks. The distribution of the light is smooth and the intensity falls off from
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the center as I{r) oc where 7(r) is the intensity and r is the distance from the center of the galaxy. Elliptical galaxies are classified based on the elongation of their apparent projected images. Ifa andb are the major and minor axes of the
apparent ellipse, than 10(a —6)/a would be the expression of the observed ellipticity.
The classification is based on the apparent elongation because the true orientation and ellipticity of the galaxies are not known (Zeilik, Gregory & Smith 1992). The
classification of elliptical galaxies ranges from EO to E7. The most spherical looking
ones (with apparent ellipticity of 0) are classified EO, whereas the most flattened ones (with maximum apparent ellipticity) are designated E7 (Fix 1995).
Spiral Galaxies
Spiral galaxies are dmded into two subclasses: ordinary (designated S or SA) and barred (designated SB). Both have spiral arms, with two arms generally placed
sjTTimetrically about the center of the axis of the rotation. In an ordinary galaxy,
the arms originate directly from the nucleus of the galaxy. In the barred spirals, on
the other hand, a bar of stars cuts through the center of the galaxy and the arms originate from the ends of the bar. Both ordinary and barred galaxies are further classified, starting from ‘a’ to ‘c’ according to how tightly the arms are wound. In
‘Sa’ and ‘SBa,’ the arms are tight and they form an almost a circular pattern; in
‘Sb’ and ‘SBb,’ they are more open and in ‘Sc’ and ‘SBc’ the nuclei are small and have extended arms. The intensity of the spheroidal component of the spiral galaxies,
around their nucleus, follows the law as in elliptical galaxies, but the intensity
of the disk components falls off at a slower rateas I(r) oc e"*^, where a is a constant
(Zeilik, Gregory & Smith 1992).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Irregular Galaxies
Irregular galaxies have no symmetrical or regular structure. Their further classi fication is done depending on the types of stars that they contain (Zeilik. Gregory &
Smith 1992).
Other Classification Schemes
Even though the Hubble classification scheme is the most widely used system,
there are several other schemes for classifying galaxies. De Vaucouleurs’ T system,
Elmergeen’s classification of spiral arms, Morgan’s classification system and Van den
Bergh’s classification of galaxies are few of them. These systems are described in detail by Van den Bergh (1998). This project made use of the Hubble classification
svstem.
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FRACTAL DIMENSIONS
Defining Fractals and Fractal Dimensions
Fractals are sets that appear to have complex structure no matter what scale is
used to examine them. True fractals are infinite sets and have self-similarity across scales, so that the same quality of structure is seen as one zooms in on them. Figure
2 shows a finite set of one such well-behaved fractal called a Koch curve.
Figure 2 Koch curve
If one wants to know the length of the Koch curve, it can be derived from its
construction formula. But such computations cannot be done for fractals in nature,
such as the outline of a cloud, the outline of a leaf or coastlines. For example, there is no construction process or a formula for the coastline of Great Britain. The only way to get the length of the coastline is to measure it. We can measure the coast on a geographical map by taking rulers set at a certain length. For a scale of 1:1,000,000
meters, the ruler length of 5 cm would be 50 km. Now we can walk this ruler along
6
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the coast. This would give a polygonal representation of the coast of Britain as shown in Figure 3. To obtain the length of the coast, we can count the number of steps, multiply the number of steps with 5 cm and convert the result to km. Smaller
settings of the rulers would result in more detailed polygons and surprisingly bigger
values of the measurements. This can be clearly seen in Table 1 which lists the length
measurements of the coast of Britain for différent ruler settings (Peitgen, Jurgens & D Saupe,1992). One can also conclude from Table 1 that for ruler settings smaller
than 65.40 km the length would have even higher values.
j
Figure 3 The coast of Britain
Table 1 Length of the coast of Britain
Ruler Setting(km) Length (km) 500.00 2600 258.82 3800 130.53 5770 65.40 8640
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A similar measurement of length for a circle shows much less variation in the length when the ruler setting is changed. Table 2 shows such measurements for a circle with a diameter of 1000 km.
Table 2 Length of a circle with a diameter of 1000 km
Number of Sides Ruler Setting(km) Length(km) 6 500.00 3000 12 258.82 3106 24 130.53 3133 48 65.40 3139 96 32.72 3141 192 16.36 3141
Curves, surfaces, and volumes can be so complex that the ordinary measurements like length, area and volume become meaningless. However, one can measure the de
gree of complexity by evaluating how fast these measurements increase if we measure
with smaller and smaller scales. The fundamental idea is to assume that the mea
surement and the scale do not vary arbitrarily but are related by a power law which allows us to compute one from the other. The power law can be stated as y oc
where x is the scale used to measure the quantity y and d is a constant, d is a useful quantity in describing fractal dimensions. In the beginning of the twentieth century,
determining the meaning of dimensions was one of the major problems. Since then
the topic has become more complex because now there are many more notions of dimensions. Some of theses notions are reviewed by Mayer-Kress (1986), Schuster
(1988) and Gershenfeld (1988). Fractal dimensions are always smaller than the number of degrees of freedom
(Grassberger & Procaccia 1983) i.e. they are smaller than 2 for two dimensional
geometrical objects, smaller than 3 for three dimensional data etc. In simple terms,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. they can be looked at as indications of how much closer to a geometrical dimension a given set is. All the different dimensions are related. Sometimes they agree with each other, giving the same values, and sometimes they differ from each other depending
on the distribution of the set they are being applied to. In this project we used
capacity dimension and correlation dimension.
Capacity Dimension
In order to find the capacity dimension of a set, we assume that the number of elements covering a data set is inversely proportional to e^, where e is the scale of covering elements and D is a constant. For example, we have a line segment and
we try to cover the segment with squares of a certain size, and find that we need
three squares. If we then tried to see how many squares of half the original size were required to cover the segment, it could be expected to have six squares covering the
segment, which is twice the number of squares needed when the squares were at their original size. Thus, the number of squares required to cover the segment is inversely proportional to the size of the squares. The covering of any smooth, continuous curve
works the same way, provided th at the size of the squares is small enough so that the curve is approximated well by straight line segments at that scale.
Thus, for one-dimensional objects, we see that
N{e) % - , e where e is the side of the square, N{e) is the number of squares of that size required to cover the set, andk is some constant. Now suppose that we are covering a scrap of paper with little squares. In this case, if we halve the size of the squares, it would take four of the smaller squares to cover what one of the larger squares would cover, and so we would expectN{e) to increase by a factor of four whene is halved. This
is consistent w'ith an equation of the form.
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,V(e) % A
It seems reasonable to say that for more arbitrary sets,
where D is the dimension of the set. In other words, we can hope to measure how much of two-dimensional space some subset of it comes near by examining how efficiently
the set can be covered by cells of different size. In order to findD from N[e) % kje^ ^ we can solve the formula for D, by taking
the limit as e ^ 0. This is the capacity method of estimatingD. If we further assume that the set is scaled so that it fits into a square with side 1, then we get iV(l) = A: = 1. This yields the formula,
■
Correlation Dimension
Correlation dimension can be calculated using the distances between each pair of
points in the set of N number of points,
A correlation function, C(r), is then calculated using,
C{r) = X {number of pairs (i, j) with s{i, j) < r).
C{r) has been foundto follow a power law similar to the one seen in the capacity
dimension:C(r) = k r^ . Therefore, we can findDcorr with estimation techniques
derived from the formula:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11
c (r) can be written in a more mathematical form as
iV N CM = jELZ » ( r - - ^il). N—^cx iV^ j = l i = j + l
where 9 is the Heaviside step function described as,
Calculating Fractal Dimension
Fd3, a program written by John Sarraille and Peter DiFalco, estimates fractal dimension for finite sets of data. The program was created using ideas from Liebovitch
and Toth (1989). It uses the box-counting method of estimating dimensions. Fd3,
estimates capacity dimension, and the correlation dimension which we used for this project. The uncertainties in the computed fractal dimensions are of order 5%.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4
DATA SET
A data set of 113 galaxies was presented by Zsolt Frei, Puragra Guhathakurta, and James E. Gunn at Princeton University and J. Anthony Tyson at AT&T BeU Lab
oratories (1996). The galaxies were chosen to span the Hubble classification classes.
All the galaxies in the set are nearby, well resolved and bright with the faintest hav
ing the total magnitudeB t of 12.90. The sample was chosen to be suitable to test automatic galaxy classification techniques with the idea that automatic methods of
classifying galaxies are necessary to handle the huge amount of data that will soon be available from large survey projects, such as the Sloan Digital Sky Survey. All the images of the set were recorded with charge coupled devices (CCDs) at the Palomar
Observatory with the 1.5 meter telescope and at the Lowell Observatory with the
1.1 meter telescope. The images were stored in FITS (Flexible Image Transport Sys tem) format with important data on these galaxies published in the Third Reference
Catalog of Bright Galaxies (de Vaucouleurs et ai, 1991). Table 3 shows the number of spiral and elliptical galaxies observed at both of the observatories along with the
broad band pass wavelengths of the filters through which they were observed.
Table 3 Data set
Observatory Spiral Galaxies Elliptical Galaxies Bands (nanometer) Palomar 31 0 500, 650 and 820 Lowell 58 14 450 and 650
12
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The images were processed to a point where the flat field and bias corrections were made and stars were removed from them. It can be seen from Figures 4 through 6, which contain scaled down versions of the all the images from the catalog, that the
collection contains a wide range of galaxy classes.
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NGC2403 NGC254L NGC 2683 NGC 2715 NGC 2768 NGC2775
ff- '• %
1 NGC290S NGC 2976 NGC 2985 NGC 3031 NGC 3077 NGC3079 0 # 1 1 NGC3Ht7 NGC 3166 NGC 3184 NGC 3198 NGC 3319 NGC3344 #
NGC335L NGC 3368 NGC 3377 NGC 3379 NGC 3486 NGC3556
>•*** % #
T4GC35K NGC 3623 NGC 3631 NGC 3672 NGC 3675 NGC3726 =î. - e I 1 NGC38UI NGC 3877 NGC 3893 NGC 3938 NGC 3953 NGC 4013 \ NGC403A NGC 4088 NGC 4173 NGC 4125 NGC 4136 NGC 4144
*
NGC 4157 NGC 4178 NGC 4189 NGC 4192 NGC 4216 NGC4242 # / \
Figure 4 Data Set: NGC 2403 to NGC 4242
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 NGCdZSd N G C 425S NGC# 4303 NGC 4321 NGC% 4340 NG C4365 NGe C -1374 NGC 43*4 NGC## 4406 NGC 4414 NGC 442* NG♦ C4442 N G C 4449 NGC 4450 NGC 4472 NGC 4477 NGC 4466 NGC4447 NGC# 44% NGC 4501 NGC4S26 NGC 4527 NGC 4535 NGC 451» NG C4S59 NGC% 4564 NGC%# 456* NGC 4571 NGC 4579 NG C4593 N G C4S94 NGC 4621 NGC 4636 NGC 4651 NGC 4651 NG C4689 ■ * NGC 4710 NGC 4725 NGC 4731 N G# C 4% 4 NGC 4426 NG% C4861 rvN G C 4 « 6 NGC 5005 NGC 5033 NG C 5 0 5 NGC 5201 NG C 5248 i-
Figure 5 Data Set: NGC 4254 to NGC 5248
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NGC5322 NGC 5334 NGC 5364 N C C 537L NGC 5337 NGC5585 %
NGC5669 NGC 5701 NGC 5746 NG C 57*2 NGC 5813 NGC5850
t 1 m
NGC5985 NGC 6015 NGC 6118 NGC 6384 NGC 6503
% ✓
Figure 6 Data Set: NGC 5322 to NGC 6503
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5
FRACTAL DIMENSIONS OF GALAXIES
This project concentrated on separating elliptical galaxies from spiral galaxies. As discussed in chapter 2, the intensity of the galaxies falls off as one moves away
from the center. Therefore, in order to find the fractal dimensions for a galaxy, one could divide the entire intensity range of the galaxy into finite steps, extract contours around each of these steps and find fractal dimensions for each contour. A plot of
fractal dimensions versus the intensity step can then be generated which would be the fractal dimensions for the entire galaxy. As the elliptical galaxies, in general, have
relatively less complex structure, one could expect such a plot to have lower values
compared to a plot for a spiral galaxy.
Contour Generation
The galaxy data set contains galaxy image-files in the FITS format. To be able to create contours around different intensity levels of a galaxy, one needs to extract the intensity information for each pixel for the image. FITSIO, a subroutine, is an interface for reading or writing data files in FITS format. This package was written to provide an interface with FITS files without having to deal directly with
the complicated internal details of the FITS files. Using FITSIO the pixel values were
extracted from the galaxy image files in array format. The following method was used to extract contours around different intensity
values. All the pixel values greater than the desired value of contour intensity were set to one, and the values less than the desired contour intensity value were set to zero.
17
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This divided the image into two groups, where one group contained the pixels having intensity higher than the desired contour intensity and the second group contained the pixels having intensity lower than the desired contour intensity. Now, all the pixels
having the value of one that were surrounded by at least one pixel of value zero were
retained and all the other pixels were discarded. The only pixels that remained were
the pixels at the boundaries of the groups constructing contours around the desired
intensity value. The pixel coordinates were stored in a file. Figure 7 shows examples of contours generated using the method described above. Each example contains one of the 50 contours generated for different intensity values
for an elliptical galaxy (NGC 4374) and a spiral galaxy^ (NGC 4303).
Figure 7 NGC 4374 and NGC 4303
Calculation of Fractal Dimensions of the Contours
Running the program Fd3 (described in chapter 3) on the contour data files cal
culated capacity dimensions and correlation dimensions of the contours. 50 contours were created for a range of intensities for each galaxy. The range was determined by the sky value and the number of points required by the fractal dimension program.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19
As the background of a galaxy image is the sky, at the intensity value of the sky most of the image would be filled with the contours around this value. This would make the fractal dimensions very high (close to 2) for all the images and would be useless
for classification being common in both elliptical and spiral galaxies. The intensity
values around the mean sky value, therefore, could be considered noise. As one moves
to higher intensity values from the mean sky value, this noise starts dropping and
one begins to get contours from the galaxy. Therefore, the lower side of the range for contour generation was set to the mean intensity level of the sky plus four times
the standard deviation around the mean sky value. The minimum number of data
points required to obtain reliable fractal dimensions is 10^, whered is the true fractal
dimensions of the object (Liebovitch and Toth 1989). Since we are dealing with the
geometrical objects with unknown fractal dimensions, deciding the minimum number of data points is difidcult. Therefore, knowing that the fractal dimension of a two
dimensional object cannot be greater than 2, a minimum of 200 (significantly greater
than 10^ = 100) points were initially used. Running the program on several galaxies led to the conclusion that their fractal dimension, on average, is significantly less
than 1.7. This gives us the minimum of 10^ ' % 50. To be on the conservative side, a minimum of 80 was chosen. At high intensity levels, closer to the center of a galaxy,
the contours start to become smaller, containing fewer data points. At one point, the
contours run out of the minimum required data points (80) setting the higher side of the intensity range for generating contours. This range was divided into 50 intensity
levels and fractal dimensions were computed for contours around each level.
Figure 8 shows example of fractal dimensions for two of the galaxies computed us
ing the Fd3 program. The examples show fractal dimensions of 50 contours generated for diflferent intensity values for an elliptical galaxy (NGC 4374) and a spiral galaxy (NGC4303). The average correlation dimension is significantly lower for the elliptical
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20
Object:'NCC4374 0bject:-NGC4303
M agnitude: 10.09 Magnitude: 10.18 1.5 1.5 - ' Filterr’R ,-Filter:'/ ■ \/-Y
£3 I a I -
avg. cap. dim(soIid line): 1.232812 avg. cap. dim(solid line): 1.122010 0.5 0.5 - avg. cor. dim(dashed line): 1.146107 avg. cor dim(dashed line): 1.272104
avg. (cor-cap): -0.08670 avg (cor—cap): 0.15009
2 00 400 600 1000 1500 2000 Inten sity In te n sity
Figure 8 Fractal Dimensions of NGC 4374 (Elliptical) and NGC 4303 (Spiral)
galaxy than the spiral relating to its less complex structure. The average capacity
dimension on the other hand is higher for the elliptical than the spiral galaxy, which
contradicts our expectation.
Results Capacity Dimensions and Correlation Dimensions
Fractal dimensions for all the 89 spiral galaxies and 14 elliptical galaxies were com
puted and data similar to Figure 8 were obtained. For all the computations, R filter
(centered around 650 nanometer) images were used. The R filter was chosen for the
following reasons. Different filter images contain different magnitude distributions, depending on the material contained within the galaxies. Therefore, the contours
created around the range of intensity values have slightly different shapes in one filter
image than the other for the same galaxy. This gives different fractal dimensions
for the same galaxy for a different filter image. Since our goal is to compare fractal dimensions of various classes of galaxies, it would be advisable to use galaxy images from only one frequency band. R was a common band in all the images taken at the
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p i I I I : 1 . I V PH" . T| I I I I T n“ L .2 8 S3 HU !_ a a !- a r a [ a 30 F S 6 r - t t ^ r o 2 0 r ■4 « r .n 10 L_ H Ol h 0 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6 Average capacity dimension Average capacity dimension
Vn 50 .2 8 r X 5 r a H â L oa 40 f 6 F a .b 30 a. VI o 20 14*o r a : XI a 2 “ 3E 10 L^ I r L , I T,- 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6 Average correlation dimension Average correlation dimension
Figure 9 Number of spirals and ellipticals versus avg. cap. and corr. dimensions
Palomar Observatory and the Lowell Observatory. Fractal dimensions, found for the R filter images, were averaged over the intensity
range in each galaxy for comparison. Histograms in Figure 9 show average frac
tal dimensions (capacity dimension and correlation dimension) for all the spiral and
elliptical galaxies. As mentioned earlier, one would expect the average fractal dimension for the el liptical galaxies to be lower than the average fractal dimension for the spiral galaxies
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22
because of their less complex structure. Histograms of the number of galaxies versus the capacity dimensions, in Figure 9, show a tendency opposite to that expectation, showing higher average capacity dimensions for ellipticals than spirals. Histograms of
number of galaxies versus the correlation dimensions, on the other hand, show an over
lap in peaks of the histograms for both classes of galaxies. The Kolmogorov-Smimov
test on these histograms confirms that the differences between the distributions for
spirals and ellipticals are not statistically significant. Although in the example of Figure 8, correlation dimension seems to be working as a separator between the two
classes, the histograms show that, in general, we cannot rely on either of the two
average fractal dimensions computed for the intensity range selected here for classifi
cation.
The number of galaxies in the histograms in Figure 9, is plotted versus the average
of the fractal dimensions for the entire intensity range starting from the sky value
plus 4 times the standard deviation around the sky value, to the point where the
contours run out of a minimum number of points required for the fractal dimension program. When we reexamine Figure 8, we notice that both capacity and correlation
dimensions are higher for the spiral galaxy than the elliptical galaxy around the center
of the intensity range. We therefore expect that if we have a more selective range,
a fraction of the entire range around the center, different results for the averages of
the fractal dimensions would be obtained. Figure 10 shows histograms for 20% of the intensity range around the center of the entire range. The Kolmogorov-Smirnov
test on these histograms confirms that the difference between the distributions for
spirals and ellipticals are statistically significant for average correlation dimension.
We conclude that the average correlation dimension, for a selected intensity range
around the center of the entire intensity range, could have possible use for galaxy
classification.
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, , I . ' I I I I I I I CQ s n " O X t" .2 8 - cC o -5 30 r- g 6 r
CL C/] _ I 20 r - 4 u o 10 h .i 2 ^ 3E E Z
0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6 Average capacity dimension Average capacity dimension
Pr n I I I I I i 1 i—!—!—r“ T*T“ i—1—;—1—1—ri r S om 30 X I ca ofar - o « 20 -
CL i 4 h 72
10 - ^b 2 -^ zI L
0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 1.6 Average correlation dimension Average correlation dimension
Figure 10 Number of galaxies versus fractal dim. for a selected intensity range
Comparing Correlation Dimensions and Capacity Dimensions
If we now compare the capacity dimensions for elliptical galaxies with their corre
lation dimensions, for the entire range of intensities (Figure 9), not much difference is
seen, but spiral galaxies show a noticeable amount of difference between both dimen
sions. Figure 11 shows histograms for the number of spiral galaxies and the number
of elliptical galaxies versus the difference between the average correlation dimension
and the average capacity dimension for the entire intensity range. Figure 12 shows
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30 h
XS | 3 h u 20
t : -
o 0 2^ _o
= 1 0 - E Z3
" 4 1 .0.2 0 0 2 0.4 K).2 0 0 2 0.4 Avg corr. dim .- Avg. cap. dim. Avg corr. dim .- Avg. cap. dim.
Figure 11 Difference between corr. and cap. dimensions for spirals and ellipticals
similar histograms for 20% of the entire intensity range around the center. For ellip
tical galaxies, one can see a tendency towards negative values. This inspires one to
reexamine the methods of calculating the dimensions. W'Ten we look at the methods described in chapter 3, we see that the correlation dimension takes the distances between two pairs of points into account for the calcu
lation of the correlation summation C'(r). If more points are distributed at greater
distances, C(r) would have higher values resulting in higher values of correlation
dimensions,Dcorr- Figure 13 shows intensity fall-offs for an elliptical galaxy and a spiral galaxy as
we move away from the center. The horizontal solid lines show the intensity range from the mean sky value plus 4 times the standard deviation around the mean sky
value to the point were the number of points on the contours around the galaxy are
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4 - 30
71 71 a O 3 r- X a sG ZJ o 20 oa cc
o o 10 — 1 r
0 '— L j 0.2 0 02. 0.4 -0.2 0 0.2 0.4 Avg corr. dim .- Avg. cap. dim. Avg corr. dim .— Avg. cap. dim.
Figure 12 Difference between dimensions for selected intensity range
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less than 80. The dashed lines show 20% of the intensity range around the center of
the above mentioned range. It can be seen that the intensity curve for the elliptical galaxy in Figure 13 has a smoother faU-off than that of the spiral galaxy. The increase
in the intensity for the spiral galaxy for greater distances from the center is because
of its spiral arms. For such increases in the intensity, the contours around a range of
intensities will break into smaller contours. For such ranges, one could expect more
pairs of points at greater distances in their images. Figure 14 is good example for
such an effect. When the contours are drawn around NGC 2403 for various values of magnitudes, they break into many small contours creating a distribution of points
at greater distances. When we examine the fractal dimensions for NGC 2403, we can clearly see the correlation dimension curve above the capacity dimension curve.
This is also true for the spiral galaxy NGC 4303, one of whose contours and fractal
dimensions are shown in Figure 7 and Figure 8 respectively. When we look at the
elliptical galaxy NGC 4374 also described by Figure 7 and Figure 8, we notice that the correlation dimension curve is below^ the capacity dimension curve for the most
part. NGC 5813 is an elliptical galaxy. From our discussion, we would expect the above
mentioned difference to be negative but it turns out to be a positive number, making
the average correlation dimension higher than the average capacity dimension. When
we examine the contours around different intensity counts, we see that its image
(Figure 15) contains a star that did not get removed by the reduction process. Because of this, we can expect the correlation summation C(r) to increase as we move to
higher intensity levels. The fractal dimension plot of NGC 5813, also shown in Figure
15, confirms this explanation. It was found that the point where the correlation dimensions curve drops below the capacity dimension curve is the point where the star in the lower left corner disappears. Now, if we remove the star from the image
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2000 5000
4000 NGC 4303 (Spiral) i NGC 4374 (Blipfcal) 1500
3000
o 1000 o c
2000
500 1000
130 140 150 160 170 180 280 300 340320
Figure 13 Intensity fall off from the center of an elliptical and a spiral galaxy
(Figure 16), we find that (on average) the correlation dimension curve is below the
capacity dimension curve. NGC 3031 is classified as a spiral galaxy. From our discussion, we would expect the difference between the averages of the correlation dimension and the capacity
dimension to be positive but the difference turns out to be a negative number. When
we look at its contours around various intensity levels (Figure 17), we find that the
most of its contours are smooth and very much unlike the average spiral galaxies and
do not break in to smaller contours.
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Contour around Contour around Intensity close to 20ÔO Intensity close to 3000
< 3 - . 5 » ^ -
Contour around Obi'ect ’NGC2403 Intensity close to 4000 Magnitude: 8.93 Fdtif: T
a Ü avg. cap. dim(solid line): 1 .0 l5 9 e i-\ j S 0-5 avg. cor. dim(dashed line): 1.265813 ~ avg. (cor-cap): 0.24985 j
2000 3000 4000 Intensity
Figure 14 Some of the contours for NGC 2403 and its fractal dimensions
All the final results for the entire intensity range are tabulated in Table 4 to Table
7 and the results for the selected intensity range are tabulated in Table 8 to Table 11.
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Contour around Contour around Intensity close to 1000 Intensity close to 1200 o
2 r I Contour around Object: ’NGC5813 I Intensity close to 1400 Magnitude: 11.45 Filter r a Ç o s "5 [ avg. cap. dim(solid line): 1.195740 2 [T avg. cor. dim(dashed line): 1.314487 ~ L avg. (cor-cap): 0.11874
1000 1100 1200 1300 1400 In ten sity
Figure 15 Some of the contours for NGC 5813 and its fractal dimensions
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Contour around Contour around Intensity dose to 1000 Intensity close to 1200
o
2 — Contour around Object; ’NGC5813’ Intensity close to 1400 Magnitude: 11.45 Alter: 'R o
[ avg. cap. dim(solid line): 1.276239 avg. cor. dim(dashed line): 1.243038 J [ avg. (cor-cap): -0.03320
1000 1100 1200 1300 1400 In ten sity
Figure 16 Fractal dimensions ofNGC 5813 after removing the star
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Contour around Contour around Intensity close to 2000 Intensity close to 5000
o
2 Contour around Object: ’NGC3031 Intensity close to 8000 c Magnitude: 7.89 .5 Filter: 'r oc E 5 1
avg. cap. dim(solid line): 1.155579 I 0.5 avg. cor. dim(dashed line): 1.150331 avg. (cor-cap): -0.00524 Zj
2000 4000 6000 8000 Intensity
Figure 17 Some of the contours for NGC 3031 and its fractal dimensions
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Table 4 NGC 2403 to NGC 3953 (Data for the entire intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim. NGC 2403 S 8.93 1.02 1.27 0.25 NGC 2541 S 12.26 0.90 1.26 0.36 NGC 2683 S 10.64 1.19 1.23 0.04 NGC 2715 S 11.79 1.17 1.24 0.07 NGC 2768 E 10.84 1.13 1.18 0.05 NGC 2775 S 11.03 1.26 1.20 -0.06 NGC 2903 S 9.68 1.21 1.28 0.08 NGC 2976 S 10.82 1.23 1.30 0.07 NGC 2985 s 11.18 1.27 1.28 0.01 NGC 3031 s 7.89 1.16 1.15 -0.01 NGC 3079 s 11.54 0.97 1.13 0.16 NGC 3147 s 11.43 1.20 1.22 0.02 NGC 3166 s 11.32 1.18 1.17 -0.01 NGC 3184 s 10.36 1.17 1.35 0.18 NGC 3198 s 10.87 1.11 1.30 0.19 NGC 3319 s 11.48 0.92 1.23 0.31 NGC 3344 s 10.45 1.22 1.34 0.12 NGC 3351 s 10.53 1.16 1.22 0-06 NGC 3368 s 10.11 1.18 1.17 -0.01 NGC 3377 E 11.24 1.27 1.29 0.02 NGC 3379 E 10.24 1.32 1.26 -0.05 NGC 3486 S 11.05 1.13 1.18 0.05 NGC 3556 S 10.69 1.13 1.19 0.06 NGC 3596 S 11.95 1.22 1.21 -0.01 NGC 3623 S 10.25 1.13 1.14 0.01 NGC 3631 S 11.01 1.18 1.27 0.10 NGC 3672 s 12.09 1.20 1.20 -0.01 NGC 3675 s 11.00 1.28 1.25 -0.03 NGC 3726 s 10.91 1.24 1.37 0.14 NGC 3810 s 11.35 1.13 1.18 0.05 NGC 3877 s 11.79 1.16 1.19 0.02 NGC 3893 s 11.16 1.15 1.20 0.05 NGC 3938 s 10.90 1.15 1.27 0.12 NGC 3953 s 10.84 1.19 1.28 0.09
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Table 5 NGC 4013 to NGC 4527 (Data for the entire intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim. NGC 4013 S 12.19 1 1.30 1.43 0.13 NGC 4030 S 11.42 1.20 1.24 0.04 NGC 4088 s 11.15 1.15 1.26 0.11 NGC 4123 s 11.98 1.25 1.37 0.13 NGC 4125 E 10.65 1.13 1.22 0.10 NGC 4136 S 11.69 1.18 1.28 0.10 NGC 4144 S 12.05 1.20 1.28 0.08 NGC 4157 S 12.66 1.20 1.27 0.07 NGC 4178 S 11.90 1.11 1.22 0.11 NGC 4189 s 12.51 1.28 1.34 0.07 NGC 4192 s 10.95 1.23 1.23 0.01 NGC 4216 s 10.99 1.17 1.21 0.04 NGC 4242 s 11.37 1.31 1.41 0.09 NGC 4254 s 10.44 1.04 1.20 0.17 NGC 4258 s 9.10 1.16 1.29 0.13 NGC 4303 s 10.18 1.12 1.27 0.15 NGC 4321 s 10.05 1.01 1.22 0.21 NGC 4365 E 10.52 1.24 1.20 -0.04 NGC 4374 E 10.09 1.23 1.15 -0.09 NGC 4394 S 11.73 1.15 1.23 0.07 NGC 4406 E 9.83 1.26 1.20 -0.05 NGC 4414 S 10.96 1.17 1.20 0.03 NGC 4450 S 10.90 1.16 1.22 0.06 NGC 4472 E 9.37 1.21 1.12 -0.09 NGC 4486 E 9.59 1.21 1.10 -0.10 NGC 4487 S 11.63 1.17 1.30 0.12 NGC 4498 S 12.79 1.22 1.26 0.04 NGC 4501 S 10.36 1.23 1.31 0.08 NGC 4527 S 11.38 1.20 1.27 0.07
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Table 6 NGC 4535 to NGC 5746 (Data for the entire intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim. NGC 4535 S 10.59 1.25 1.39 0.14 NGC 4548 S 10.96 1.19 1.31 0.11 NGC 4559 S 10.46 1.18 1.36 0.18 NGC 4564 E 12.05 1.20 1.26 0.06 NGC 4569 S 10.26 1.20 1.26 0.06 NGC 4571 S 11.82 1.32 1.39 0.07 NGC 4579 S 10.48 1.14 1.14 0.01 NGC 4593 S 11.67 1.12 1.20 0.08 NGC 4594 s 8.98 1.21 1.14 -0.07 NGC 4621 E 10.57 1.20 1.23 0.03 NGC 4636 E 10.43 1.23 1.14 -0.09 NGC 4651 S 11.39 1.08 1.16 0.08 NGC 4654 S 11.10 1.24 1.33 0.09 NGC 4689 S 11.60 1.33 1.39 0.06 NGC 4725 S 10.11 1.12 1.22 0.10 NGC 4731 S 11.90 1.13 1.25 0.13 NGC 4826 S 9.36 1.30 1.32 0.02 NGC 4861 S 12.90 1.20 1.35 0.16 NGC 5005 S 10.61 1.21 1.27 0.06 NGC 5033 S 10.75 1.21 1.24 0.03 NGC 5055 S 9.31 1.19 1.29 0.10 NGC 5204 S 11.73 1.21 1.26 0.05 NGC 5248 S 10.97 1.20 1.26 0.06 NGC 5322 E 11.14 1.20 1.27 0.06 NGC 5334 s 11.99 1.41 1.50 0.09 NGC 5364 s 11.17 1.13 1.22 0.10 NGC 5371 s 11.32 1.22 1.35 0.13 NGC 5377 s 12.24 1.19 1.26 0.08 NGC 5585 s 11.20 1.10 1.22 0.11 NGC 5669 s 12.03 1.20 1.25 0.05 NGC 5701 s 11.76 1.07 1.13 0.06 NGC 5746 s 11.29 1.18 1.18 0.00
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Table 7 NGC 5792 to NGC 6503 (Data for the entire intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim. NGC 5792 S 12.08 1.21 1.35 0.14 NGC 5813 E 11.45 1.20 1.31 0.12 NGC 5850 S 11.54 1.21 1.27 0.06 NGC 5985 S 11.87 1.25 1.31 0.06 NGC 6015 S 11.69 1.14 1.23 0.09 NGC 6118 S 12.42 1.23 1.30 0.07 NGC 6384 s 11.14 1.17 1.24 0.07 NGC 6503 s 10.91 1.11 1.15 0.04
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Table 8 NGC 2403 to NGC 3953 (Data for a selected intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim. NGC 2403 S 8.93 1.11 1.36 0.25 NGC 2541 S 12.26 0.90 1.31 0.41 NGC 2683 S 10.64 1.12 1.13 0.00 NGC 2715 S 11.79 1.04 1.21 0.16 NGC 2768 E 10.84 1.03 1.07 0.05 NGC 2775 S 11.03 1.24 1.15 -0.09 NGC 2903 S 9.68 1.28 1.35 0.07 NGC 2976 S 10.82 1.25 1.33 0.07 NGC 2985 S 11.18 1.22 1.17 -0.05 NGC 3031 S 7.89 1.21 1.30 0.09 NGC 3079 s 11.54 1.08 1.26 0.18 NGC 3147 s 11.43 1.26 1.26 0.00 NGC 3166 s 11.32 1.13 1.07 -0.05 NGC 3184 s 10.36 1.14 1.37 0.23 NGC 3198 s 10.87 1.18 1.39 0.21 NGC 3319 s 11.48 0.87 1.10 0.22 NGC 3344 s 10.45 1.16 1.29 0.13 NGC 3351 s 10.53 1.32 1.40 0.08 NGC 3368 s 10.11 1.24 1.22 -0.02 NGC 3377 E 11.24 1.37 1.33 -0.04 NGC 3379 E 10.24 1.33 1.23 -0.11 NGC 3486 S 11.05 1.07 1.14 0.07 NGC 3556 S 10.69 1.17 1.24 0.07 NGC 3596 S 11.95 1.37 1.34 -0.03 NGC 3623 S 10.25 1.20 1.24 0.04 NGC 3631 S 11.01 1.23 1.35 0.12 NGC 3672 S 12.09 1.13 1.18 0.05 NGC 3675 S 11.00 1.28 1.24 -0.03 NGC 3726 S 10.91 1.25 1.35 0.10 NGC 3810 S 11.35 1.06 1.16 0.10 NGC 3877 s 11.79 1.18 1.15 -0.02 NGC 3893 s 11.16 1.24 1.29 0.05 NGC 3938 s 10.90 1.19 1.25 0.06 NGC 3953 s 10.84 1.15 1.24 0.09
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Table 9 NGC 4013 to NGC 4527 (Data for a selected intensity range)
1 Galaxy TjTe Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim. NGC 4013 S 12.19 1.35 1.52 0.17 NGC 4030 S 11.42 1.21 1.27 0.06 NGC 4088 S 11.15 1.16 1.23 0.07 NGC 4123 S 11.98 1.16 1.31 0.15 NGC 4125 E 10.65 1.10 1.16 0.06 NGC 4136 S 11.69 1.11 1.17 0.06 NGC 4144 S 12.05 1.14 1.19 0.05 NGC 4157 S 12.66 1.27 1.29 0.02 NGC 4178 S 11.90 1.12 1.19 0.07 NGC 4189 S 12.51 1.36 1.41 0.05 NGC 4192 S 10.95 1.27 1.31 0.05 NGC 4216 S 10.99 1.20 1.27 0.07 NGC 4242 S 11.37 1.31 1.39 0.08 NGC 4254 S 10.44 1.18 1.35 0.18 NGC 4258 s 9.10 1.31 1.34 0.04 NGC 4303 s 10.18 1.24 1.36 0.12 NGC 4321 s 10.05 1.06 1.30 0.25 NGC 4365 E 10.52 1.22 1.14 -0.08 NGC 4374 E 10.09 1.20 1.09 -0.11 NGC 4394 S 11.73 1.14 1.19 0.05 NGC 4406 E 9.83 1.19 1.13 -0.07 NGC 4414 S 10.96 1.22 1.27 0.05 NGC 4450 S 10.90 1.06 1.13 0.08 NGC 4472 E 9.37 1.22 1.13 -0.09 NGC 4486 E 9.59 1.23 1.11 -0.12 NGC 4487 S 11.63 1.17 1.28 0.10 NGC 4498 S 12.79 1.21 1.32 0.11 NGC 4501 S 10.36 1.24 1.31 0.07 NGC 4527 S 11.38 1.21 1.32 0.11
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Table 10 NGC 4535 to NGC 5746 (Data for a selected intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim. NGC 4535 S 10.59 1.23 1.32 0.09 NGC 4548 S 10.96 1.17 1.18 0.01 NGC 4559 S 10.46 1.15 1.24 0.09 NGC 4564 E 12.05 1.18 1.22 0.05 NGC 4569 S 10.26 1.28 1.30 0.02 NGC 4571 S 11.82 1.26 1.43 0.16 NGC 4579 s 10.48 1.15 1.14 -0.01 NGC 4593 s 11.67 1.06 1.10 0.04 NGC 4594 s 8.98 1.33 1.31 -0.02 NGC 4621 E 10.57 1.14 1.13 -0.01 NGC 4636 E 10.43 1.23 1.12 -0.11 NGC 4651 S 11.39 1.02 1.10 0.08 NGC 4654 S 11.10 1.28 1.34 0.06 NGC 4689 S 11.60 1.19 1.32 0.13 NGC 4725 S 10.11 1.06 1.21 0.15 NGC 4731 S 11.90 1.14 1.18 0.04 NGC 4826 S 9.36 1.25 1.29 0.04 NGC 4861 s 12.90 1.14 1.32 0.17 NGC 5005 s 10.61 1.18 1.25 0.07 NGC 5033 s 10.75 1.26 1.19 -0.08 NGC 5055 s 9.31 1.21 1.36 0.14 NGC 5204 s 11.73 1.21 1.24 0.03 NGC 5248 s 10.97 1.31 1.38 0.07 NGC 5322 E 11.14 1.16 1.24 0.08 NGC 5334 S 11.99 1.49 1.60 0.11 NGC 5364 S 11.17 0.93 1.06 0.13 NGC 5371 s 11.32 1.31 1.41 0.10 NGC 5377 s 12.24 1.15 1.20 0.05 NGC 5585 s 11.20 1.03 1.18 0.14 NGC 5669 s 12.03 1.19 1.19 0.00 NGC 5701 s 11.76 1.09 1.11 0.02 NGC 5746 s 11.29 1.21 1.21 0.01
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Table 11 NGC 5792 to NGC 6503 (Data for a selected intensity range)
Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim. NGC 5792 S 12.08 1.34 1.51 0.17 NGC 5813 E 11.45 1.09 1.21 0.12 NGC 5850 S 11.54 1.24 1.33 0.09 NGC 5985 S 11.87 1.18 1.25 0.07 NGC 6015 s 11.69 1.18 1.23 0.05 NGC 6118 s 12.42 1.22 1.30 0.08 NGC 6384 s 11.14 1.05 1.14 0.09 NGC 6503 s 10.91 1.03 1.08 0.04
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CONCLUSIONS
The classification of the galaxies and a mathematical quantity fractal dimension
are both, at some level, related to the complexity of shapes. In the expectation of
devising an automated scheme of classifying galaxies, fractal dimensions of 89 spiral
galaxies and 14 elliptical galaxies were studied in this project. Two of the fractal dimensions, the capacity dimension and the correlation dimension were calculated
for the contours generated around different intensity levels of the galaxy images.
Average fractal dimensions for the elliptical galaxies were expected to have lower
values compared to the average fractal dimensions for the spiral galaxies because of
their less complex shapes. It was found that neither the capacity dimension nor the correlation dimension
can be used for a reliable automation of galaxy classification when one computes their averages for an entire possible range of intensity contours around the galaxies. Computing the average of the correlation dimension for a selected range of intensities
around the center of the entire intensity range, however, could be useful for galaxy
classification. In recent years, the use of Artificial Neural Networks has grown significantly for classifications. They are also being used for classifying galaxies. When constructing
an Artificial Neural Network, one has to specify a number of input parameters using
which the network is designed to generate outputs. For galaxy classification, the num
ber of input parameters depends on how we choose to describe a galaxy. Correlation
40
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dimension could be used as one such parameter. The correlation dimension is com
puted using the correlation integral C'(r), as described in Chapter 3. Computation
of C{r) is very sensitive to the presence of foreground stars in the galaxy images. It
is very crucial that proper care is taken in the data reduction process to ensure that
the only data remaining in the image is from the galaxy itself. It would be interesting to derive additional parameters from the function C(r)
which could be used as inputs to an Artificial Neural Network.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES
de Vaucouleurs, G., de Vaucouleurs, A., Crowin, H., Buta, R., Paturel, G., & Fouqué, P., 1991, Third Referene Catalog of Bright Galaxies (Springer-Verlag) Astronomical Journal 111, 174
Fix, J., 1995, Astronomy (Mosby), p.524
Frei, Z., Guhathakurta, P., Gunn, J. & Tyson, J., 1996, Astronomical Journal111, 174
Gershenfeld, N., 1988, Directions in chaos Vol 1 &: Vol 2 (World Scientific)
Grassberger, P. & Procaccia, I., 1983, Phys. Rev. Lett., 50, 346
Liebovitch L. & Toth T., 1989, Phys. Lett. A 141 386
Mayer-Kress, G., ed., 1986, Dimensions and entropies in chaotic systems (Springer- Verlag)
Peitgen, H., Jurgens H. k. Saupe D., 1992, Chaos and Fractals (Springer-Verlag), p. 192
Schuster H., 1988, Deterministic chaos (VCH)
Van den Bergh, S., 1998, Galaxy Morphology and Classification (Cambridge Univer sity Press)
Zeilik, M., Gregory, S. & Smith, E., 1992, Astronomy and Astrophysics (Saunders College Publishing), p. 412-417
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Graduate College University of Nevada, Las Vegas
Sandip G. Thanld
Local Address: 650 Sierra Vista, Apartment 305 Las Vegas, NV 89109
Degree: Bachelor of Science, Physics and Electrical Engineering, 1997 Widener University, PA
Thesis Title: Classification of Galaxies Using Fractal Dimensions
Thesis Committee: Chairpersons, Dr. George Rhee, Ph.D., Dr. Stephen Lepp, Ph.D. Committee Member, Dr. Donna Weistrop, Ph.D. Committee Member, Dr. Lon Spight, Ph.D. Graduate Faculty Representative, Dr. Wanda Taylor, Ph.D.
43
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