The fractal properties of the two-dimensional distribution of galaxy clusters

Chin. Asrron. Asrrophys. Vol. 20, No.1, PP. 9-14. 1996 A translation of Acra Astrophys. Sin. Vol. 15. No.4. pp. 297-303, 1995 Copyright 0 1996 El-&ier Science Ltd Printed in Great Britain. All rights reserved 0275-1062196 $24.00+0.00

Pergamon

SO2751062(96)00002-l

of the two-dimensional distribution of galaxy clusters t

The fractal propertks

2 Graduate 3Beijing

School,

Nanchang

University,

of Sciences tY Technology

University

Astronomical

LIU Yong-zhen2

DENG ZU-gan2*3

DENG Xin-fa’

1 Departmen t oj Basic Courses, Observatory,

Chinese Academy

Nanchang

330029

of China, Beijing 100039 of Sciences,

Beijing

100080

Abstract We investigate the fractal properties of the two-dimensional distribution of different richness and distance class subsamples for an all-sky catalog of rich clusters of galaxies. The results show that the two-dimensional distribution of the clusters can be described by a multi-level fractal structure. In the D lg4 and D = 5 + 6 subsamples, the apparent turning scale between successive fractal are similar in the northern and levels is N 42 h-’ Mpc. The fractal properties southern galactic latitudes on small and large scales. Key

words:

galaxy

clusters-fractal

1. INTRODUCTION The completion of the southern sky survey provides us with a homogeneous, all-sky catalogue enabling us to investigate more completely their law of distribution. of galaxy cluster&l, As in the investigation of the distribution of galaxies, the most widely used method in that of the galaxy clusters is still correlation analysis and the two-point correlation function has been used to investigate the clustering properties of Abel1 and AC0 catalogues and other various subsamples [2-101, and it was found that the law of clustering is basically the same in the northern and southern sky regions, that both the angular and space correlation functions can be well expressed by the power-law form, w(0) = (0/&,)-~,<(r) = (r/ro)-Q. In the range from about 5O to 10’ however, there are some appreciable “north-southern” differences[3*41. We believe that none of the methods of analysis are perfect, that nay one can only reveal certain properties of the distribution of the sample. The method of correlation analysis has its own shortcomings, such as its limited maximum scale of application. Therefore, we have considered using different from different aspects. t Supported by National Foundation Received 1994-08-27

method

Natural

of statistical

Science Foundation

9

analysis,

and Jiangxi

to tackle

Province

the law of distribution

Young Scientists

10

DENG

Xin-fa

et al.

The similarity in form of the twopoint correlation functions for galaxies, galaxy clusters and even superclusters shows that the large-scale structure of the universe possesses the property of scale invariance, which is the basic characteristic of fractal geometry. This naturally prompts us to think that the distribution of galaxy clusters could be fractal,and the basic quantity for characterizing a fractal distribution is the fractal dimension. Hence our investigation

will star with the calculation

of fractal

dimension.

Because

of the whole

cluster catalogue, only a small portion of the clusters have known redshifts, and because estimates of redshift based on the ml0 N t relation have large errors131, we have decided to restrict our analysis to the two-dimensional distribution.

2.

THE

SAMPLE

The southern catalogue of rich clusters of galaxies was the joint effort of Abell, Corwin and Olowin (“AGO Catalogue” hereafter)l’l. Together with the Abel1 cluster catalogue, they form an all-sky catalogue 9of rich clusters of galaxies. The Abel1 catalogue covers the region north of declination -27“ and contains 2712 clusters. The AC0 catalogue covers the region south of declination -17’) and contains 1364 clusters, not counting those in the common range between declinations -17“ and -27’. Here, we divided the entire catalogue into two completely symmetrical northern and southern samples, limited by galactic latitude b” 35OWe selected only those clusters of richness and distance classes in the ranges 1 5 R < 5,1 _< D 5 6. Thus, our northern galactic sample contains 1045 clusters, and our southern, 1154. In the analysis, each of the two samples is further divided either according as D 2 4 or = 5 or 6 (for investigating possible variation of clustering in depth) , or according as r+ 1 or 2 ” (possible variation with richness class). The northern subsamples will be labelled NDI, ND2, NRl, NR2, the southern ones, SDl, SD”, SRl, SR2. Table 1 gives some of the parameters of the subsamples. We shall also compared the results from the northern and southern sample, to see if there is any variation of clustering with the region of sky. Table 1 Parameters

Subsamples

6”

of Subsamples

N

I) -.-__

Number of Abell Clusters

Number of Ace Clusters

NDI

<4

67

1

ND2

5+6

954

23

NRI

31

772

21

NR2

>l

249

3

2350

B4

-

31

37

5+6

546

540

SRI

>:I

404

416

SR2

s=l

173

159

SD1 SD2

G-350

Fractal Galaxy Clusters

AND

2. METHOD

RESULTS

11

OF ANALYSIS

Take the position of each cluster as centre and a solid angle R (aperture size, and calculate the average number of neighbouring clusters within We then examine the relation between structure, then we should have

N and Sz. If the sample

angle 8) of a fixed the solid angle, N.

has a two-dimensional

NCXilD

(1)

where D is the fractal

dimension:

D = d(log N)/d(logR) Obviously,

(2)

for a uniform

random

D = 1 + d(log N/N,)/d(logQ) where N is the average number corresponding umber in a random generated samples.

sample,

D = 1. Hence we can rewrite

(2) as

(3) of clusters within s1 in the actual sample, and N,, the sample. In our calculation, N, was based on 50 randomly

1.0

0.4

0.9

I

I

I

I

4

0.9

I

/-

(b) ND2

0.3 -

0.7 I? > z G c

I2

0.6

$ 0.5

0.2-

0

0.4 0.1 -

0.3 0.2

".O

0.1 -2.5

-2.0

-1.5

-1.0

-0.5

0.0

I

- 3.5

hw)

0.3

I2 2

fractal

0.2

0

I

-2.5

“\..-

,

,

- 1.5

,

-0.5

MQ )

/

I/I’

r’ /i

0.7

I

IZ

0.6

2 s

0.5

=

0.4 0.3

0. I

0.0 - 3.0

-2.0 IoeW)

Fig. 1

-1.0

0.0

- 3.0

- 2.0

- 1.0

0.0

log(l))

Analysis of results for the northeru Galactic latitude subsamples

(a) ND1 (D 5 4) (b) ND2 (D = 5 + 6) (c) NRl (R = 1) (d) NR2 (R >_ 2)

DENG

12

Xin-fa,et

al.

The random samples were generated to have the sample total number of clusters, for the In this way, boundary and selection effects same boundary region and selection function. in the results of our calculation of D will be effectively reduced. This device enables us to carry our analysis up to where abnormal variation begins to appear. Of the various selection effects in the cluster sample, we consider only the latitude selection. The latitude selection function is usually expressed in the form log P(b”) = ff log(1 - csc lP1) + const..

Bahcall

et a1.121averaging

the result

from subsamples

of Abel1

clusters, found o = 0.3. Bataski et a1.151for the AC0 clusters, found (Y= 0.2. In this paper, we calculate the effect for both our northern and southern subsamples,’ then on taking the average of the northern and southern subsamples of the same type we found subsamples ND1 and SDl: o = 0.24; s u b samples ND2 and SD2: Q = 0.46 subsamples NRl and SRl: Q = 0.46; subsamples NR2 and SR2: Q = 0.46 In the calculation for subsample NDl, we excluded the region of the void around I” 180°[“l.

W

Fig. 2

Analysis

of results

for the Southern

Galactic

(a) SD1 (D 5 4) (b) SD2 (D = 5 + 6) (c) SRI

(R=

latitude

SD2

subsamples

1) (d) SR2

(R 2

2)

13

Fractal Galaxy Clusters

Table

Subsamples ND1

ND2

NRI

NR2

SD1

SD2

SRI

SR2

2

Values of Fractal Dimension for All Subsamples

Ranges of Angular Radiu (0)

Fhctal Dimensions

Correlation Coefficients

--

2.3-11.5

0.591

0.998

12.9-25..9

0.958

0.726

0.6-6.5

0.871

0.998

7.2-18.3

0.950

0.998

20.5-32.8

0.975

0.994

1.0-6.5

0.882

0.996

7.2-18.3

0.967

0.991

20’.5-32.8

0.980

0.966

0.6-6.5

0.742

0.994

7.2-29.1

0.857

--

--

0.997

2.3-11.5

0.646

0.972

12.9-32.8

0.878

0.977

0.7-6.5

0.840

0.998

7.2-18.3

0.955

0..978

20.5-32.8

0.983

0.939

0.8-6.5

0.863

0.996 0.976

7.2-18.3

0.960

20.5-32.8

0.981

0.965

0.6-6.5

0.747

0.998

7.2-29.1

0.915

0.992

Figures 1 and 2 give separately the calculated log(N/N,) - log0 curves for all the subsamples in the northern and southern regions. flu error bars are shown. The x-axis is log 52. In later discussion we shall find it more convenient to use the aperture angle 0 instead of R.

In the subsamples ND1 and SDl, because error of analysis is relatively large, especially calculation

have become

statistically

the number of clusters is rather small, the for log0 5 -2.3 (0 5 2.3’), the results of

meaningless.

In contrast,

in the subsamples

ND2 and

SD2, the analysis could be pushed down to about 0.6’, not far from the minimum scale reached in the correlation analyses. The maximum scale analyzed for the all the subsamples is much greater than that in the correlation analyses, showing that fractal analysis is indeed capable of investigating the distribution of galaxy clusters on a much larger scale. In all the subsamples, we have a linear relationship between log(N/flr) and log D only within certain ranges, and moreover, the fractal dimension is different in different ranges. All the calculated values of fractal dimension and their relevant ranges are given in Table 2. This means that the distribution of galaxy clusters is not a simple fractal, rather, it is a multi-level fractal. WEN Zheng et al.l12l analyzing the fractal structure in CfA galaxy sample, found that the distribution of galaxies is also a multi-level fractal. Our result here shows that the large scale structure of the universe is not a simple fractal, and so the process of formation of the large scale structure could not have been a pure scalefree, self-similar process.

14

DENG Xin-fa et al

Beside

the multi-level

character,

all the subsamples

show this feature:

the fractal

di-

mension for larger scales is always greater that on the smaller scales galaxy clusters

than that for smaller scales. This results shows form “clumps”, while on the larger scales, they have a bubbly structure. This is in qualitative agreement with the distribution of IRAS and optically bright galaxies112~131. Both for the northern

and the southern

galactic

samples,

and over the whole range

of

all scales, the fractal dimension for the R > 2 subsample is always smaller than that of the R = 1 subsample. In the R = 1 subsample, the distribution already approaches random distribution at 0 E 18’) whereas this does not happen in the R 2 2 subsample until 0 = 29’. This result shows that the distribution of galaxy clusters is dependent on the richness class. An important characteristic of multi-level fractal is the existence of a typical scale-the scale of transitions between the levels. For the Dl subsamples, the transition scale is - 13’, we analyzed, while for the D2 subsamples,its is - 7”. This shows that in the subsamples the clustering characteristic is not produced by projection effects. According to the average depths of these subsamples estimated by Bahcall et al.121, we find that the linear scales corresponding to the transition scales in the two cases are basically equal, about 42 h-’ Mpc, and this also the scale given by the R = 1 and R > 2 subsamples.

4. CONCLUSIONS Using the method of fractal analysis, we investigated the law of distribution subsamples of the all-sky galaxy cluster catalogue and we found the following

in different conclusions:

1) The distribution of galaxy clusters is like the distribution of galaxies,and is a multilevel fractal structure. 2) The law of distribution of galaxy clusters is similar in different parts of the sky. 3) The fractal property of galaxy clusters depends on the richness class of the clusters. 4) The D 5 4 and D = 5 or 6 subsamples have the same linear projected transition scale of about 42 he1 Mpc. ACKNOWLEDGMENT useful discussion.

We thank

XIA Xiao-yang

for providing

the cluster

catalogue

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