Classification of galaxies using fractal dimensions

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UNLV Retrospective Theses & Dissertations

1-1-1999

Classification of galaxies using fractal dimensions Classification of galaxies using fractal dimensions

Sandip G Thanki University of Nevada, Las Vegas

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CLASSIFICATION OF GALAXIES USING

FRACTAL DIMENSIONS

by

Sandip G. Thanki

Bachelor of Science Widener University

1997

A thesis subm itted 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


UMI Number: 1396434

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Thesis ApprovalThe 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

_______M aster o f S c ie n c e in P h y s ic s ______________________

Examination Committee M em er

Examination Committee Member

Graduate College faculty Represematwe

6 XExamination Committee Chair

Dean of the Graduate College

11


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


TABLE OF CONTENTS

ABSTRACT ..................................................................................................................... iii

LIST OF TABLES............................................................................................................ vi

LIST OF FIG URES..........................................................................................................

ACKNOWLEDGMENTS................................................................................................. viii

CHAPTER 1 INTROD UCTIO N............................................................................... 1

CHAPTER 2 GALAXY CLASSIFICATION SCHEMES .................................... 3“Tuning Fork” ............................................................................................................ 3

Elliptical Galaxies ............................................................................................ 3Spiral G alaxies................................................................................................... 4Irregular G alaxies.............................................................................................. 5

Other Classification S ch em es.................................................................................. 5

CHAPTER 3 FRACTAL DIM ENSIONS................................................................. 6Defining Fractals and Fractal Dimensions............................................................. 6

Capacity D im ension......................................................................................... 9Correlation D im ension ..................................................................................... 10

Calculating Fractal D im ension ................................................................................ 11

CHAPTER 4 DATA SET ........................................................................................... 12

CHAPTER 5 FRACTAL DIMENSIONS OF GALAXIES......................... 17Contour G enera tion ................................................................................................... 17Calculation of Fractal Dimensions of the C ontours............................................. 18R e su lts .......................................................................................................................... 20

Capacity Dimensions and Correlation D im ensions................................... 20

IV


Comparing Correlation Dimensions and Capacity Dimensions.............. 23

CHAPTER 6 CONCLUSIONS .................................................................................. 40

R E FE R E N C E S................................................................................................................. 42

V IT A .................................................................................................................................... 43


LIST OF TABLES

Table 1 Length of the coast of B r ita in ..................................................................... 7Table 2 Length of a circle with a diameter of 1000 k m ....................................... 8Table 3 Data set ........................................................................................................... 12Table 4 NGC 2403 to NGC 3953 (Data for the entire intensity r a n g e ) 32Table 5 NGC 4013 to NGC 4527 (Data for the entire intensity r a n g e ) 33Table 6 NGC 4535 to NGC 5746 (D ata for the entire intensity r a n g e ) 34Table 7 NGC 5792 to NGC 6503 (Data for the entire intensity r a n g e ) 35Table 8 NGC 2403 to NGC 3953 (Data for a selected intensity r a n g e ) 36Table 9 NGC 4013 to NGC 4527 (Data for a selected intensity r a n g e ) 37Table 10 NGC 4535 to NGC 5746 (D ata for a selected intensity r a n g e ) 38Table 11 NGC 5792 to NGC 6503 (D ata for a selected intensity r a n g e ) 39

VI


LIST OF FIGURES

Figure 1 Tuning Fork Diagram .................................................................................. 3Figure 2 Koch curve....................................................................................................... 6Figure 3 The coast of B rita in ....................................................................................... 7Figure 4 D ata Set: NGC 2403 to NGC 4242... ......................................................... 14Figure 5 D ata Set: NGC 4254 to NGC 5248... ......................................................... 15Figure 6 D ata Set: NGC 5322 to NGC 6503... ......................................................... 16Figure 7 NGC 4374 and NGC 4303 ........................................................................... 18Figure 8 Fractal Dimensions of NGC 4374 (Elliptical) and NGC 4303 (Spiral) 20Figure 9 Number of spirals and ellipticals versus avg. cap. and corr. dimensions 21Figure 10 Number of galaxies versus fractal dim. for a selected intensity range 23Figure 11 Difference between corr. and cap. dimensions for spirals and ellipticals 24Figure 12 Difference between dimensions for selected intensity r a n g e ................. 25Figure 13 Intensity fall off from the center of an elliptical and a spiral galaxy . 27Figure 14 Some of the contours for NGC 2403 and its fractal dim ensions 28Figure 15 Some of the contours for NGC 5813 and its fractal dim ensions 29Figure 16 Fractal dimensions of NGC 5813 after removing the s t a r ................... 30Figure 17 Some of the contours for NGC 3031 and its fractal dim ensions 31

Vll


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. W anda 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


CHAPTER 1

INTRODUCTION

There are no bigger assemblies of stars, gas, dust and dark m atter 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 im portant information. In the last two decades the research has

reached a significant level of m aturity with galaxies routinely detected at many wave

lengths. W ith 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%.

W ith the improved technology and quality of data, the quantity of da ta 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


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.


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


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. If a and b are the m ajor 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 dm ded 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 rate as I(r) oc e"*^, where a is a constant

(Zeilik, Gregory & Smith 1992).


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.


CHAPTER 3

FRACTAL DIMENSIONS

Defining Fractals and Fractal Dimensions

Fractals are sets that appear to have complex structure no m atter 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 G reat 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


the coast. This would give a polygonal representation of the coast of B ritain as

shown in Figure 3. To obtain the length of the coast, we can count the number of

steps, multiply the number of steps w ith 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 2600258.82 3800130.53 577065.40 8640


8

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 300012 258.82 310624 130.53 313348 65.40 313996 32.72 3141192 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,


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 a t the size of the squares is small enough so that the

curve is approximated well by straight line segments a t tha t 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, and k is some constant. Now suppose tha t 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 expect N{e) to increase by a factor of four when e is halved. This

is consistent w'ith an equation of the form.


10

,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 find D from N[e) % k j e ^ we can solve the formula for D, by taking

the limit as e ^ 0. This is the capacity method of estimating D. 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 o f pairs (i, j ) with s{i, j ) < r).

C{r) has been found to follow a power law similar to the one seen in the capacity

dimension: C(r) = k r ^ . Therefore, we can find Dcorr with estimation techniques

derived from the formula:


11

c (r) can be written in a more mathematical form as

iV N

N—cx iV^CM = j E L Z » ( r - - il).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%.


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 magnitude Bt of 12.90. The sample was chosen to be suitable to test

automatic galaxy classification techniques with the idea that autom atic methods of

classifying galaxies are necessary to handle the huge amount of da ta 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 820Lowell 58 14 450 and 650

12


13

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.


14

NGC2403 NGC254L

ff- '•1 NGC290S NGC 2976

01 NGC3Ht7 NGC 3166

#NGC335L NGC 3368

T4GC35K NGC 3623

e INGC38UI NGC 3877

\NGC403A NGC 4088

NGC 4157 NGC 4178

NGC 2683 NGC 2715

%NGC 2985 NGC 3031

#

NGC 3184 NGC 3198

NGC 3377 NGC 3379

% #

NGC 2768 NGC2775

NGC 3077 NGC3079

1NGC 3319 NGC3344

NGC 3486 NGC3556

>•***

NGC 3631 NGC 3672

=î. -

NGC 3893 NGC 3938

NGC 4173 NGC 4125

*

NGC 4189 NGC 4192

# /

NGC 3675 NGC3726

1NGC 3953 NGC 4013

NGC 4136 NGC 4144

NGC 4216 NGC4242

\Figure 4 Data Set: NGC 2403 to NGC 4242


15

NGCdZSd NGC425S NGC 4303 NGC 4321 NGC 4340 NGC4365# %NGC-1374 NGC 43*4 NGC 4406 NGC 4414 NGC 442* NGC4442e ## ♦NGC4449 NGC 4450 NGC 4472 NGC 4477 NGC 4466 NGC4447

NGC 44% NGC 4501 NGC4S26 NGC 4527 NGC 4535 NGC 451»#NGC4S59 NGC 4564 NGC 456* NGC 4571 NGC 4579 NGC4593%% #NGC4S94 NGC 4621 NGC 4636 NGC 4651 NGC 4651 NGC4689

■ *NGC 4710 NGC 4725 NGC 4731 NGC4% 4 NGC 4426 NGC4861rv # %N G C 4«6 NGC 5005 NGC 5033 NGC 5 0 5 NGC 5201 NGC5248i-

Figure 5 Data Set: NGC 4254 to NGC 5248


16

NGC5322 NGC 5334 NGC 5364

NGC5669 NGC 5701 NGC 5746

t 1

NGC5985 NGC 6015 NGC 6118

%

N C C 537L

NGC 57*2

NGC 6384

NGC 5337 NGC5585

%NGC 5813 NGC5850mNGC 6503

✓Figure 6 Data Set: NGC 5322 to NGC 6503


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 w riting data files in FITS format. This package was

written to provide an interface w ith FITS files without having to deal directly with

the complicated internal details o f 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 th a n the desired contour intensity value were set to zero.

17


18

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 tha t 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.


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^, where d 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


20

1.5

£3 I

Object:'NCC4374

M agnitude: 10.09

F ilte rr’R1.5 - '

0bject:-NGC4303

M agnitude: 10.18

, -F ilte r : '/ ■

\/-Ya I -

0.5avg. cap . d im (soIid line): 1.232812

avg. co r. d im (d a sh ed line): 1.146107

avg. ( c o r - c a p ) : -0 .08670

2 00 400In ten sity

600

0.5 -avg. cap . d im (so lid line): 1.122010

avg. c o r d im (d a sh e d line): 1.272104

avg (c o r—cap): 0.15009

1000 1500 2000In te n s ity

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.

ResultsCapacity 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


21

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S 6 r

■4

T n“

00.6 0 .8 1 1.2 1.4 1.6

A verage c a p a c ity d im en sion

.2 8 r 5 r â Lf 6 F

14*o r a 2 “I r

, I T,-0.6 0 .8 1 1.2 1.4 1.6A verage c o r r e la tio n d im en sion

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


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.


23

CQO

cCo-5 30 r-CLC/]

oE3Z

20 ru

10 h

, , I .

0.6 0.8 1 1.2 1.4 1.6Average c a p a c ity d im e n sio n

moX

cao

30 Pr n I I I I I i 1 i—!—!—r“ T*T“ i—1—;—1—1—r i r

I

« 20 -CL72

10 -

0.6 0.8 1 1.2 1.4 1.6Average co rre la tio n d im e n s io n

s n "X t".2 8 -

g 6 r

- 4

.i 2 ^E

' I I I I I I I

_ I

0.6 0.8 1 1.2 1.4 1.6Average cap acity d im en sion

S

f a ro -

i 4 h

b 2 ^^ -I Lz

0.6 0.8 1 1.2 1.4 1.6A verage correla tion d im en sion

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


24

30 h

XS

u 20

t : -

o_o

= 1 0 -

.0.2 0 0 2 0.4A vg c o r r . d i m . - Avg. c a p . d im .

| 3 h

0 2^

E3Z

" 4 1K).2 0 0 2 0.4

A vg c o r r . d i m . - Avg. c a p . d im .

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 a t 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


25

4 -

30

71OXsGo 20

cc

o10 —

0 .2 0 02. 0.4A vg c o r r . d i m . - A vg. c a p . d im .

71a3 r-

aZJao

o

1 r

0 '—

L j0.4-0.2 0 0.2 0.4

A vg c o r r . d i m .— A vg. c a p . d im .

Figure 12 Difference between dimensions for selected intensity range


26

less than 80. The dashed lines show 20% of the intensity range around the center of

the above mentioned range. It can be seen tha t the intensity curve for the elliptical

galaxy in Figure 13 has a smoother faU-off than tha t 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

a t 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. W hen 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 th a t 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 tha t 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


27

2000

i NGC 4374 (Blipfcal)1500

o 1000

500

160 170 180130 140 150

5000

NGC 4303 (Spiral)4000

3000

oc

2000

1000

280 300 340320

Figure 13 Intensity fall off from the center of an elliptical and a spiral galaxy

(Figure 16), we find th a t (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.


28

Contour around Intensity close to 20ÔO

Contour around Intensity close to 3000

< 3 - . 5 » ^ -


aÜS 0-5

Obi'ect ’NGC2403 Magnitude: 8.93 Fdtif: T

avg. cap. dim(solid line): 1 .0 l5 9 e i- \ javg. cor. dim(dashed line): 1.265813 ~ avg. (cor-cap): 0.24985 j

2000 3000Intensity

4000

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.


29


oContour around Intensity close to 1200

I Contour aroundI Intensity close to 1400

o

2 r

aÇ

s"5

Object: ’NGC5813 Magnitude: 11.45 Filter r

[ avg. cap. dim(solid line): 1.1957402 [T avg. cor. dim(dashed line): 1.314487

~ L avg. (cor-cap): 0.11874

1000 1100 1200 1300In ten sity

1400



30

Contour around Intensity d o se to 1000


o


o

2 —Object; ’NGC5813’ Magnitude: 11.45 Alter: 'R

[ avg. cap. dim(solid line): 1.276239avg. cor. dim(dashed line): 1.243038 J

[ avg. (cor-cap): -0.03320

1000 1100 1200 1300In ten sity

1400

Figure 16 Fractal dimensions of NGC 5813 after removing the star


31



o

Contour around Intensity close to 8000 c

coE5

2Object: ’NGC3031 Magnitude: 7.89 Filter: 'r.5

1

0.5avg. cap. dim(solid line): 1.155579 Iavg. cor. dim(dashed line): 1.150331 avg. (cor-cap): -0.00524 Zj

2000 4000 6000 8000Intensity



32

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 0.25NGC 2541 S 12.26 0.90 1.26 0.36NGC 2683 S 10.64 1.19 1.23 0.04NGC 2715 S 11.79 1.17 1.24 0.07NGC 2768 E 10.84 1.13 1.18 0.05NGC 2775 S 11.03 1.26 1.20 -0.06NGC 2903 S 9.68 1.21 1.28 0.08NGC 2976 S 10.82 1.23 1.30 0.07NGC 2985 s 11.18 1.27 1.28 0.01NGC 3031 s 7.89 1.16 1.15 -0.01NGC 3079 s 11.54 0.97 1.13 0.16NGC 3147 s 11.43 1.20 1.22 0.02NGC 3166 s 11.32 1.18 1.17 -0.01NGC 3184 s 10.36 1.17 1.35 0.18NGC 3198 s 10.87 1.11 1.30 0.19NGC 3319 s 11.48 0.92 1.23 0.31NGC 3344 s 10.45 1.22 1.34 0.12NGC 3351 s 10.53 1.16 1.22 0-06NGC 3368 s 10.11 1.18 1.17 -0.01NGC 3377 E 11.24 1.27 1.29 0.02NGC 3379 E 10.24 1.32 1.26 -0.05NGC 3486 S 11.05 1.13 1.18 0.05NGC 3556 S 10.69 1.13 1.19 0.06NGC 3596 S 11.95 1.22 1.21 -0.01NGC 3623 S 10.25 1.13 1.14 0.01NGC 3631 S 11.01 1.18 1.27 0.10NGC 3672 s 12.09 1.20 1.20 -0.01NGC 3675 s 11.00 1.28 1.25 -0.03NGC 3726 s 10.91 1.24 1.37 0.14NGC 3810 s 11.35 1.13 1.18 0.05NGC 3877 s 11.79 1.16 1.19 0.02NGC 3893 s 11.16 1.15 1.20 0.05NGC 3938 s 10.90 1.15 1.27 0.12NGC 3953 s 10.84 1.19 1.28 0.09


33


Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.NGC 4013 S 12.19 1 1.30 1.43 0.13NGC 4030 S 11.42 1.20 1.24 0.04NGC 4088 s 11.15 1.15 1.26 0.11NGC 4123 s 11.98 1.25 1.37 0.13NGC 4125 E 10.65 1.13 1.22 0.10NGC 4136 S 11.69 1.18 1.28 0.10NGC 4144 S 12.05 1.20 1.28 0.08NGC 4157 S 12.66 1.20 1.27 0.07NGC 4178 S 11.90 1.11 1.22 0.11NGC 4189 s 12.51 1.28 1.34 0.07NGC 4192 s 10.95 1.23 1.23 0.01NGC 4216 s 10.99 1.17 1.21 0.04NGC 4242 s 11.37 1.31 1.41 0.09NGC 4254 s 10.44 1.04 1.20 0.17NGC 4258 s 9.10 1.16 1.29 0.13NGC 4303 s 10.18 1.12 1.27 0.15NGC 4321 s 10.05 1.01 1.22 0.21NGC 4365 E 10.52 1.24 1.20 -0.04NGC 4374 E 10.09 1.23 1.15 -0.09NGC 4394 S 11.73 1.15 1.23 0.07NGC 4406 E 9.83 1.26 1.20 -0.05NGC 4414 S 10.96 1.17 1.20 0.03NGC 4450 S 10.90 1.16 1.22 0.06NGC 4472 E 9.37 1.21 1.12 -0.09NGC 4486 E 9.59 1.21 1.10 -0.10NGC 4487 S 11.63 1.17 1.30 0.12NGC 4498 S 12.79 1.22 1.26 0.04NGC 4501 S 10.36 1.23 1.31 0.08NGC 4527 S 11.38 1.20 1.27 0.07


34


Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.NGC 4535 S 10.59 1.25 1.39 0.14NGC 4548 S 10.96 1.19 1.31 0.11NGC 4559 S 10.46 1.18 1.36 0.18NGC 4564 E 12.05 1.20 1.26 0.06NGC 4569 S 10.26 1.20 1.26 0.06NGC 4571 S 11.82 1.32 1.39 0.07NGC 4579 S 10.48 1.14 1.14 0.01NGC 4593 S 11.67 1.12 1.20 0.08NGC 4594 s 8.98 1.21 1.14 -0.07NGC 4621 E 10.57 1.20 1.23 0.03NGC 4636 E 10.43 1.23 1.14 -0.09NGC 4651 S 11.39 1.08 1.16 0.08NGC 4654 S 11.10 1.24 1.33 0.09NGC 4689 S 11.60 1.33 1.39 0.06NGC 4725 S 10.11 1.12 1.22 0.10NGC 4731 S 11.90 1.13 1.25 0.13NGC 4826 S 9.36 1.30 1.32 0.02NGC 4861 S 12.90 1.20 1.35 0.16NGC 5005 S 10.61 1.21 1.27 0.06NGC 5033 S 10.75 1.21 1.24 0.03NGC 5055 S 9.31 1.19 1.29 0.10NGC 5204 S 11.73 1.21 1.26 0.05NGC 5248 S 10.97 1.20 1.26 0.06NGC 5322 E 11.14 1.20 1.27 0.06NGC 5334 s 11.99 1.41 1.50 0.09NGC 5364 s 11.17 1.13 1.22 0.10NGC 5371 s 11.32 1.22 1.35 0.13NGC 5377 s 12.24 1.19 1.26 0.08NGC 5585 s 11.20 1.10 1.22 0.11NGC 5669 s 12.03 1.20 1.25 0.05NGC 5701 s 11.76 1.07 1.13 0.06NGC 5746 s 11.29 1.18 1.18 0.00


35


Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.NGC 5792 S 12.08 1.21 1.35 0.14NGC 5813 E 11.45 1.20 1.31 0.12NGC 5850 S 11.54 1.21 1.27 0.06NGC 5985 S 11.87 1.25 1.31 0.06NGC 6015 S 11.69 1.14 1.23 0.09NGC 6118 S 12.42 1.23 1.30 0.07NGC 6384 s 11.14 1.17 1.24 0.07NGC 6503 s 10.91 1.11 1.15 0.04


36

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 0.25NGC 2541 S 12.26 0.90 1.31 0.41NGC 2683 S 10.64 1.12 1.13 0.00NGC 2715 S 11.79 1.04 1.21 0.16NGC 2768 E 10.84 1.03 1.07 0.05NGC 2775 S 11.03 1.24 1.15 -0.09NGC 2903 S 9.68 1.28 1.35 0.07NGC 2976 S 10.82 1.25 1.33 0.07NGC 2985 S 11.18 1.22 1.17 -0.05NGC 3031 S 7.89 1.21 1.30 0.09NGC 3079 s 11.54 1.08 1.26 0.18NGC 3147 s 11.43 1.26 1.26 0.00NGC 3166 s 11.32 1.13 1.07 -0.05NGC 3184 s 10.36 1.14 1.37 0.23NGC 3198 s 10.87 1.18 1.39 0.21NGC 3319 s 11.48 0.87 1.10 0.22NGC 3344 s 10.45 1.16 1.29 0.13NGC 3351 s 10.53 1.32 1.40 0.08NGC 3368 s 10.11 1.24 1.22 -0.02NGC 3377 E 11.24 1.37 1.33 -0.04NGC 3379 E 10.24 1.33 1.23 -0.11NGC 3486 S 11.05 1.07 1.14 0.07NGC 3556 S 10.69 1.17 1.24 0.07NGC 3596 S 11.95 1.37 1.34 -0.03NGC 3623 S 10.25 1.20 1.24 0.04NGC 3631 S 11.01 1.23 1.35 0.12NGC 3672 S 12.09 1.13 1.18 0.05NGC 3675 S 11.00 1.28 1.24 -0.03NGC 3726 S 10.91 1.25 1.35 0.10NGC 3810 S 11.35 1.06 1.16 0.10NGC 3877 s 11.79 1.18 1.15 -0.02NGC 3893 s 11.16 1.24 1.29 0.05NGC 3938 s 10.90 1.19 1.25 0.06NGC 3953 s 10.84 1.15 1.24 0.09


37


1 Galaxy TjTe Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.NGC 4013 S 12.19 1.35 1.52 0.17NGC 4030 S 11.42 1.21 1.27 0.06NGC 4088 S 11.15 1.16 1.23 0.07NGC 4123 S 11.98 1.16 1.31 0.15NGC 4125 E 10.65 1.10 1.16 0.06NGC 4136 S 11.69 1.11 1.17 0.06NGC 4144 S 12.05 1.14 1.19 0.05NGC 4157 S 12.66 1.27 1.29 0.02NGC 4178 S 11.90 1.12 1.19 0.07NGC 4189 S 12.51 1.36 1.41 0.05NGC 4192 S 10.95 1.27 1.31 0.05NGC 4216 S 10.99 1.20 1.27 0.07NGC 4242 S 11.37 1.31 1.39 0.08NGC 4254 S 10.44 1.18 1.35 0.18NGC 4258 s 9.10 1.31 1.34 0.04NGC 4303 s 10.18 1.24 1.36 0.12NGC 4321 s 10.05 1.06 1.30 0.25NGC 4365 E 10.52 1.22 1.14 -0.08NGC 4374 E 10.09 1.20 1.09 -0.11NGC 4394 S 11.73 1.14 1.19 0.05NGC 4406 E 9.83 1.19 1.13 -0.07NGC 4414 S 10.96 1.22 1.27 0.05NGC 4450 S 10.90 1.06 1.13 0.08NGC 4472 E 9.37 1.22 1.13 -0.09NGC 4486 E 9.59 1.23 1.11 -0.12NGC 4487 S 11.63 1.17 1.28 0.10NGC 4498 S 12.79 1.21 1.32 0.11NGC 4501 S 10.36 1.24 1.31 0.07NGC 4527 S 11.38 1.21 1.32 0.11


38


Galaxy Type Magnitude Cap. Dim. Corr. Dim. Corr. Dim.-Cap. Dim.NGC 4535 S 10.59 1.23 1.32 0.09NGC 4548 S 10.96 1.17 1.18 0.01NGC 4559 S 10.46 1.15 1.24 0.09NGC 4564 E 12.05 1.18 1.22 0.05NGC 4569 S 10.26 1.28 1.30 0.02NGC 4571 S 11.82 1.26 1.43 0.16NGC 4579 s 10.48 1.15 1.14 -0.01NGC 4593 s 11.67 1.06 1.10 0.04NGC 4594 s 8.98 1.33 1.31 -0.02NGC 4621 E 10.57 1.14 1.13 -0.01NGC 4636 E 10.43 1.23 1.12 -0.11NGC 4651 S 11.39 1.02 1.10 0.08NGC 4654 S 11.10 1.28 1.34 0.06NGC 4689 S 11.60 1.19 1.32 0.13NGC 4725 S 10.11 1.06 1.21 0.15NGC 4731 S 11.90 1.14 1.18 0.04NGC 4826 S 9.36 1.25 1.29 0.04NGC 4861 s 12.90 1.14 1.32 0.17NGC 5005 s 10.61 1.18 1.25 0.07NGC 5033 s 10.75 1.26 1.19 -0.08NGC 5055 s 9.31 1.21 1.36 0.14NGC 5204 s 11.73 1.21 1.24 0.03NGC 5248 s 10.97 1.31 1.38 0.07NGC 5322 E 11.14 1.16 1.24 0.08NGC 5334 S 11.99 1.49 1.60 0.11NGC 5364 S 11.17 0.93 1.06 0.13NGC 5371 s 11.32 1.31 1.41 0.10NGC 5377 s 12.24 1.15 1.20 0.05NGC 5585 s 11.20 1.03 1.18 0.14NGC 5669 s 12.03 1.19 1.19 0.00NGC 5701 s 11.76 1.09 1.11 0.02NGC 5746 s 11.29 1.21 1.21 0.01


39

Table 11 NGC 5792 to NGC 6503 (D ata 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 0.17NGC 5813 E 11.45 1.09 1.21 0.12NGC 5850 S 11.54 1.24 1.33 0.09NGC 5985 S 11.87 1.18 1.25 0.07NGC 6015 s 11.69 1.18 1.23 0.05NGC 6118 s 12.42 1.22 1.30 0.08NGC 6384 s 11.14 1.05 1.14 0.09NGC 6503 s 10.91 1.03 1.08 0.04


CHAPTER 6

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 autom ated 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 th a t neither the capacity dimension nor the correlation dimension

can be used for a reliable autom ation 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 param eters depends on how we choose to describe a galaxy. Correlation

40


41

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 th a t

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.


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 Journal 111, 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 University Press)

Zeilik, M., Gregory, S. & Smith, E., 1992, Astronomy and Astrophysics (Saunders College Publishing), p. 412-417

42


VITA

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. W anda Taylor, Ph.D.

43


Documents

Classification of galaxies using fractal dimensions