POSITION ANGLES AND ALIGNMENTS OF
CLUSTERS OF GALAXIES
Manolis PLIONIS
SISSA - International School for Advanced Studies,
Via Beirut 2-4, 34013 Trieste, Italy and
ICTP - International Centre for Theoretical Physics
Abstract
The position angles of a large number of Abell and Shectman clusters, identi ed in the Lick map as
surface galaxy-density enhancements, are estimated. In total I determine the major axis orientation
of 637 clusters out of which 448 are Shectman clusters (202 of which are also Abell clusters) and
189 are Abell clusters not originally detected by Shectman due to his adopted density threshold.
Using published redshifts for 277 of these clusters I have detected strong nearest neighbour alignments
over scales up to 15 h 1 Mpc at a & 2:5 3 signi cance level, while quite weak alignments are
detected even up to 60 h 1 Mpc. A more signi cant alignment signal ( 4 ) is detected among
all neighbours residing in superclusters and having separations . 10 h 1 Mpc. Again, weaker but
signi cant alignments are found when larger separations are considered. Since my cluster sample is
neither volume limited nor redshift complete, a fact that would tend to wash-out any real alignment
signal, the alignments detected should re ect a real and possibly a stronger underline e ect.
Subject headings: Cosmology - galaxies: clustering - galaxies: formation
Submitted to ApJ Supplements (December 1993)
1
1 Introduction
The issue of the elongation of clusters of galaxies (cf. Carter & Metcalfe 1980, Binggeli 1982, Plionis,
Barrow & Frenk 1991) and their tendency to be aligned with their nearest neighbour and in many
cases with the position angle of their rst ranked galaxy is well studied and well documented, although
con icting results appear occasionally in the literature.
The alignment e ect was rst noted by Binggeli (1982) who claimed that clusters tend to be aligned
with their neighbours over scales of 10-15 h 1 Mpc. Since then, a number of authors have supported
the reality of this e ect (Rhee and Katgert 1987, Flin 1987, West 1989a,b, Lambas et:al: 1990, Rhee,
van Haarlem & Katgert 1992), although doubts have been put forward about the signi cance and the
strength of the e ect (Struble & Peebles 1985; McMillan, Kowalski & Ulmer 1989; Fong, Stevenson &
Shanks 1990). Note, however, that Fong et:al: searched for alignments between neighbouring clusters
found in 2-dimensions. Any real signal could be diluted by projection and therefore one should be
cautious on how to interpret these results. In fact, as it will be shown in what follows, a signi cant
alignment signal between neighbours in 3 dimensions becomes insigni cant, although still present,
when the cluster pairs are chosen in angular space.
Further support for the reality of the alignment e ect comes from the work of Argyres et al (1986)
and Lambas, Groth & Peebles (1988) who found that the Lick galaxy counts around Abell clusters
tend to be aligned with the cluster major axis out to 15 h 1 Mpc, especially for clusters in high
density regions.
It was thought initially that the observed alignment e ects would provide a very e ective test to
discriminate among di erent models of cosmic structure formation. In the pancake scenario (Zeldovich
1970, Doroshkevich, Shandarin & Saar 1978), like the Hot Dark Matter model, where clusters and
galaxies form by fragmentation in already attened sheet- and lament-like superclusters, one expects
the clusters to be elongated and aligned. In accordance with this Dekel, West & Aarseth (1984) and
West, Dekel & Oemler (1989) found that cluster alignments occur only when the initial uctuation
spectrum has a large coherence length as that expected in the HDM model. They where unable
to reproduce the observed alignments in hierarchical clustering models, like the Cold Dark Matter
(CDM) model, where the cosmic structures form by gravitational clustering from small to large scales
(cf. Peebles 1982, Blumenthal et:al: 1984, Davis et:al: 1985, Frenk et:al: 1985, 1988). However, both
the asphericity of clusters and the alignment e ect could be produced in hierarchical models by a
di erent mechanism. Tidal e ects could in uence the shapes of protoclusters and induce alignments.
Indeed, Binney & Silk (1979) found that tidal e ects between protostructures can induce attening
and prolate shapes for clusters with a mean ellipticity before virialization is h"i 0:5, which is in good
agreement with observations (Plionis, Barrow & Frenk 1991). However this issue is a controversial one
since con icting results have been presented in the literature. For example Barnes & Efstathiou (1987)
using N-body simulations nd that tidal interactions cannot induce alignments between neighboring
protoclusters while Salvador-Sole & Solanes (1993) nd, using analytical methods, that tides can
induce the observed alignments.
2
Aside of these con icting results and contrary to the numerical work of Dekel et:al: (1984) and West
et:al: (1989), Bond (1987) has shown that within the framework of Gaussian statistics and if the clusters
form at the peaks of the eld then an alignment up to 20 h 1 Mpc should be expected even in the
CDM model. Soon after, analysis of new high-resolution CDM simulations showed that the alignment
e ect is present and even stronger than what was anticipated by the analytical work (West, Villumsen
& Dekel 1991 and references therein). In view of these results, it would seem unavoidable to conclude
that that cluster-cluster alignments is a generic feature of the cluster formation process which is
independent of the speci c model and form of the uctuation spectrum. Therefore, cluster alignments
cannot be used as an e ective discriminant between models of structure formation. However, studying
alignments and similar features of the distribution of matter on large scales, could provide interesting
clues about the details of the cluster formation process and therefore it is essential:
1. To unambiguously determine whether alignments do occur in the real universe.
2. To nd the amplitude of the e ect and the scale over which it takes place.
3. To identify details of the alignment e ect, for example whether galaxies within a cluster exhibit
any alignment (cf. Struble 1990, van Kampen & Rhee 1990, Trevese, Cirimele & Flin 1992, Rhee,
van Haarlem & Katgert 1992), which could then provide clues about the internal dynamics of
the cluster (cf. Rhee & Roos 1990).
In this paper I present new cluster position angle determinations for a sample of 637 clusters of
galaxies and I study the cluster alignment properties for a subsample of these clusters (N = 277) for
which redshift information is available.
2 Cluster Position Angles
The cluster- nding algorithm, used to identify clusters from the Lick galaxy catalogue, and the resulting cluster catalogues were presented in Plionis, Barrow & Frenk (1991) [hereafter PBF]. The
algorithm is based in identifying surface galaxy-density peaks above a given threshold and connecting
in a unique cluster all neighbouring cells that also ful ll the overdensity criteria. A thorough statistical
analysis of the distribution of these clusters was presented in Plionis & Borgani (1992), Borgani, Jing
& Plionis (1992) and Borgani, Plionis & Valdarnini (1993). I therefore refer the interested reader to
those articles as well as to Shane & Wirtanen (1967) and Seldner et:al: (1977) for issues related to
the Lick galaxy catalogue and to Plionis (1988) for details of this particular use of the catalogue. I
will just remind the reader that the Lick catalogue contains 810000 unique galaxies with mb . 18:8
which are binned in 10 10 arcmin2 cells. The catalogue covers 70% of the sky and its characteristic
depth is 210 h 1 Mpc (Groth & Peebles 1977).
I determine the orientation of a cluster by estimating its position angle, , measured relative to
North in the anticlockwise direction. The principal axes of each cluster and their orientation are
calculated by diagonalizing the 2 2 inertia tensor I:
3
Ikl =
Xn (rkrl)imi
(1)
i=1
where k; l = 1; 2, r1 = xi , r2 = yi , mi is the galaxy number-count of the ith cell, n is the number of
10 10 cells for each cluster, and the origin of the coordinate axis is the centre of mass of the cluster.
This method for determining cluster position angles is quite di erent from those used in most other
studies (cf. Binggeli 1982) which measure the galaxy distribution within a xed circular aperture
around each cluster. This method has the disadvantage that it does not map the same area around
each cluster but it has the advantage that it avoids any biases resulting from the use of a circular
aperture (for a detailed discussion of this see Carter & Metcalfe 1980 and Binggeli 1982.)
My primary sample consists of all C36 clusters, ie., those with =h i = 3:6 (see PBF), which sample
a large enough area to make the determination of their position angle possible, ie., cover at least ve
10 10 arcmin2 cells. Note that the overdensity threshold used is the same as that used by Shectman
(1985) and therefore these clusters correspond to Shectman clusters. In order to test the robustness
of the position angle determination the clusters are traced to lower overdensity levels (=h i=3, 2.5
and 1.8) and at each level their position angles are determined. Since, however, at lower overdensities
there is a higher probability of the clusters being a ected by projection e ects that could distort their
position angles and since projection e ects should cause the apparent position of the cluster centre to
vary from level to level, PBF attempted to minimize this problem adopting the following procedure.
At each overdensity level the cluster centre of mass is estimated and only those levels are considered
for which the cluster centre of mass, calculated at that overdensity, has shifted by < 10 arcmin from
its original C36 position (ie., the cluster centre of mass should not move out of its original 10 cell).
I also estimate the position angles of all Abell clusters, found at lower overdensity levels and which
cover enough area to make the position angle determination possible. Note that in order to eliminate
the gross e ects of Galactic extinction the samples are limited to jbj 40.
In table 1 I present the C36 cluster position angles with their corresponding Shectman (1985) and
Abell (1958) number. Where it was possible I also present the mean and standard deviation of their
values from measurements at three, at least, overdensity levels. In such cases the median value is
also listed. In table 1b I present the position angles of Abell clusters identi ed at =h i=2.5 and
determined at the same level, while in table 1c the listed position angles are of clusters identi ed at a
level =h i=1.8. Finally at table 1d I present position angles of clusters identi ed at the =h i=2.5
but estimated at =h i=1.8
i
0
i
0
0
3 Tests for systematic e ects
PBF gave a thorough discussion of the most signi cant systematic biases that could a ect the cluster
shape parameters and found that the position angles remain mostly una ected by the e ect of the
grid while when tracing the clusters to lower overdensity levels they found small variations of their
position angle. However, when the cluster sampling area is small (ie, when few 10 10 cells de ne
0
4
0
the cluster), then due to the limiting geometry of the possible cell con guration, the position angles of
such clusters tend to have a preferred range of values. Figure 1 shows the position angles as a function
of the number of cells, de ning the cluster sampling area, for clusters with only one determination of
their position angle. Especially for Ncells 7 the e ect is evident. However, the e ect is suppressed
when more than one determinations of the position angle of a cluster is possible and in these cases the
nal value of is the average over all the determinations. This can be seen in gure 2 where we plot
the position angles of clusters estimated in more than one overdensity level; panel (a) shows the values
of hi as a function of the minimum number of cells used while panel (b) the values of as a function
of the maximum number of cells used for the hi determination. I conclude that clusters with one
position angle determination and Ncells 7, have an uncertainty, due to the grid, of 30 35
and therefore the interested reader should be cautious of this e ect and should consider their position
angles rather as indicative.
To test whether this e ect is signi cant and whether it could create a systematic orientation bias
I calculate, following Struble & Peebles (1985), the following functions:
r 2 XN
Cn = N cos 2ni
i=1
r 2 XN
(2)
(3)
Sn = N sin 2ni
i=1
where n 1 and N is the total number of clusters in my sample. If the position angles are independently drawn from a uniform distribution between 0 and 180 then the Cn and Sn have zero mean
and unit standard deviation. A systematic bias in the cluster orientations will manifest itself as a
large value of jCn j and/or jSn j. For all the 637 cluster position angles I obtain jC1 j, jS1 j . 1:6, jC2 j,
jS2j . 1:2 and jC3j, jS3j . 1 indicating that the distribution of position angles has no signi cant
deviation from uniformity. However, when I use clusters with only one position angle determination
and with Ncells = 5, I get jS2j, jC2 j 2 which re ects the grid e ect mentioned before.
PBF found that when they compared the position angles of clusters in common with other studies,
the typical uncertainty between the di erent measurements is 30 35, in agreement with West
(1989). For example the position angle deviation for the clusters in common with Struble & Peebles
(77 clusters) is 0 35:5 while when I compare with 61 common clusters of Rhee et:al: (1992) I
nd 1:4 40:5 for those determined by the Fourier method while I nd slightly worse results
for their other methods. Interestingly, I nd the best agreement when I compare my cluster position
angles with their 1st rank galaxy position angles ( 0:5 39).
Note, however, that PBF found that if the comparison is restricted only to those common clusters
for which their estimate of the position angle is based on an area similar in size to the circular aperture
used in other studies then the position angle deviation is reduced by 50%. This shows that the
di erent position angle estimation methods lead to similar values when applied to similar regions
around the clusters.
5
4 Alignments of Clusters
In order to search for alignments in my sample I collected all the published cluster redshifts to end up
with 277 clusters in the North and South galactic hemisphere. For this sample I nd jC1 j, jS1j 1:2,
jC2j, jS2j 0:2 and jC3j, jS3j 0:9 which again shows that this sample is free of orientation bias. Note
that my sample is not volume limited, a major drawback in such a study, nor is it redshift complete
a fact that could hinder me from reaching any strong conclusion, especially if I would not detect any
alignments.
The distance to each cluster is determined by the usual relation (Mattig 1958):
(4)
R = H q 2 (1c + z)2 [qz + (q 1)[(1 + 2q z) 21 1]]
with q = 0:2. The cluster coordinate system is transformed to a Cartesian one using the transformations: x = R cos b sin l, y = R cos b cos l and z = R sin b. In this coordinate system the relative distance
Dij of each cluster pair are found by Dij = (x2ij + yij2 + zij2 )1=2. I determine the position angle ij
of each cluster, i, relative to the direction of a neighbour, j , as the acute angle between the major
axis of the cluster and the great circle connecting the two clusters. Firstly the position angle of the
cluster-pair separation, is , at the position of the primary cluster is found by using straightforward
spherical trigonometry and nally ij = i is , where i is the position angle of the primary cluster.
The mean ij in an isotropic distribution with large N would be equal to 45. A deviation from
this value, in a bias free sample, would be an indication of an alignment/misalignment e ect. A useful
measure of such an e ect is given by Struble & Peebles (1985):
=
XN ij
i=1
N
45
(5)
If the values of ij are isotropically distributed between 0 and 90 then for large N , h i = 0 and the
standard deviation is:
= p90
(6)
12N
A negative value of would indicate alignment and a positive one a misalignment. We must consider
though that a comparison of observations with these formulas should take place only if the determination of position angles does not entail systematic errors. If there is a systematic bias of the cluster
position angles towards a preferred direction it could produce a false alignment or misalignment signal.
This, however, is not the case for my samples since it was found, in section 3, that such a bias is not
present. I searched for two types of cluster alignments:
1. Nearest-Neighbour alignments (N-N)
2. All neighbour alignments within superclusters (A-N)
6
4.1 Nearest-Neighbour Alignments
To study this type of alignments I found for each cluster i a neighbour j for which their distance is
min[Dij ]. For such pairs we then calculate ij ( nn ) and estimate the alignment signal and its
deviation . Note that hmin[Dij ]i is 25 h 1 Mpc, a rather large value but consider that the
sample is not volume limited and there are a few clusters with very large redshifts. If we limit the
N-N separations to a maximum value of 30 or 50 h 1 Mpc then 14 and 20 h 1 Mpc, respectively.
In Table 2 we present our results for di erent limiting cluster pair separations, Dlim . A quite
signi cant signal (2:5 e ect) is found when Dlim = 15 h 1 Mpc which persists, although having a
smaller value of , even when very large separations are considered (Dlim = 50 - 60 h 1 Mpc). Figure
3 shows the frequency distribution of the nn angles. Figure 3a shows the results for Dlim = 10
and 15 h 1 Mpc (solid and dashed lines respectively) while gure 3b shows the results when all the
N-N separations are considered. A clear excess of small nn values is present in all cases, being more
pronounced at smaller values of Dlim .
I attempted to further test the signi cance of my result by testing whether the observed nn
distribution could have been drawn from a uniform one. To this end we perform a KolmogorovSmirnov two-sided test and we nd that the probability of the observed distribution being drawn from
a uniform parent distribution is 4 10 2, 3 10 2 and 5 10 2 for Dlim = 10, 15 and 60 h 1
Mpc respectively.
4.2 All-Neighbour Alignments within Superclusters
I have used the above procedure to study alignments between all possible cluster-pairs that lie within
the same supercluster. Binggeli (1982) has claimed a positive alignment signal of this type, although
at a low signi cance level. Similarly, West (1989b) has found that such alignments occur over scales
of at least 30 h 1 Mpc and maybe over even larger distances, while Plionis, Valdarnini & Jing (1992),
using Abell clusters and a subset of the position angles presented here found that indeed there is a
strong alignment signal between all neighbours within superclusters which seems also to be correlated
with the shape of the supercluster, being stronger in prolate con gurations.
I de ne superclusters using a friend of friend algorithm and two percolation radii, Rp, of 35 h 1
and 25 h 1 Mpc length. Since my main results seem to be insensitive to these two choices of Rp, I will
present only those for the Rp = 35 h 1 case. For this value of Rp I obtain 33 superclusters (with more
than two members) containing 209 out of the 277 clusters. The alignment signal (eq. 5) is used but
note that I used clusters that belong into superclusters of any membership number above unity, while
Plionis et:al: (1992) used supercluster with more than 8 members (because they wanted to determine
also their shapes).
In table 3 I present the values of the alignment signal and its signi cance for di erent cluster
separation ranges. Strong alignments occur in my sample only for Dlim = 10 h 1 Mpc while weaker
but still signi cant alignments are present even when Dlim 150 h 1 Mpc. Remember, however, that
my samples are not volume limited nor redshift complete and therefore even such an alignment signal
7
should be considered as an indication of a probably larger and more signi cant alignment e ect. In
gure 4 I present the frequency distribution of the ij angles at four cluster separation ranges. The
strong alignment e ect at the D 10 h 1 Mpc range is evident. Our results are in agreement with
those of West (1989b) and Plionis et:al: (1992) even more so if we take into account that our samples
are not as complete, a fact that would tend wash-out any inherent alignment signal.
4.3 Possible systematic biases
The observed alignments could be arti cial, in principle, resulting from the fact that clusters which are
near in space are in many occasions also near on the sky and therefore their galaxy density envelopes
could overlap in the angular projection. This could produce a directional bias when determining their
position angles and thus a preferred orientation along the direction of a neighbour, in particular of the
nearest neighbour. A manifestation of such an e ect would be to nd a stronger and more signi cant
alignment signal between nearest neighbours in angular space rather than in real 3-D space. Since
however, a number of clusters that are neighbours in real space are neighbours also in angular space
some weak alignment signal should survive the projection. To test whether this e ect is responsible
for the alignment signal found, I searched for nearest neighbour alignments in my sample but using
angular separations instead of spatial ones. I found for all separations a signi cantly weaker alignment
signal. For example when the angular separations are 1:4 (for which I have in total 66 separation,
a value similar to that for D 10 h 1 Mpc) I obtain a weaker and less signi cant alignment signal
( = 4:1 3:2) with a 0.15 probability of the distribution of nn being drawn from a uniform one,
con rming that the observed alignment signal in 3-D is not caused by the above mentioned bias.
Furthermore, to test whether the grid e ect, mentioned in section 3, has induced the detected
cluster alignments, I repeated the N-N analysis but after having reshued the cluster position angles.
I performed a 100 such reshuings and I nd, for all separations considered, h i = 0 with a standard
deviation equal to that obtained from eq. 6, which indicates that the grid e ect is not responsible for
the detected cluster alignment signal.
5 Conclusions
I have presented a large number of new cluster position angles. In total 637 position angles were
estimated. Using available redshifts for a subset of these clusters I found a strong alignment signal
( 2:5 3 ) between nearest cluster neighbours when their separations are 15 h 1 Mpc, as well
as a stronger ( 4 ) signal between all neighbours residing in superclusters when separations 10
h 1 Mpc are considered. A quite weak but signi cant alignment signal persists even when considering
larger separations (up to 150 h 1 Mpc).
8
: Mean position angles (column 5) of Abell and Shectman clusters (identi ed at the 3.6
level). We present the number of overdensity levels used to estimate the position angles (column 3),
the minimum and maximum number of 10 10 cells, Ncells , de ning the cluster sampling area [at the
di erent overdensity levels] (column 4), while for the cases were the position angle () was determined
in 3 or more overdensity levels we list also (column 6) and the median value (column 7) [in those
cases where 4 levels were used we averge the two central values].
Table 1a
0
0
9
Shectman Abell No of levels min-max Ncells
1
2
4
5
6
8
9
10
11
12
16
18
21
22
23
24
25
26
27
28
30
31
33
35
37
38
39
42
43
45
46
47
48
50
724
727
757
819
858
879
912
921
949
952
978
986
993
1020
1033
1035
1050
1
2
2
2
2
1
1
2
1
1
1
2
2
3
2
3
3
3
2
2
2
2
3
3
2
1
4
2
1
1
1
4
2
3
6
10-15
7-12
5-9
6-14
7
7
7-10
8
6
6
5-9
7-9
8-20
7-10
5-13
6-13
8-13
5-7
5-8
7-11
8-15
9-27
6-19
6-14
6
10-40
5-28
7
6
8
5-84
5-10
5-13
177.2
65.8
80.4
79.3
36.5
111.0
96.1
103.1
132.4
166.1
0.6
118.1
40.5
165.9
119.8
115.4
19.4
169.3
47.0
119.9
97.8
157.2
114.9
133.7
22.0
109.0
137.1
102.5
60.9
44.5
43.9
93.8
131.6
76.6
10
Med[]
...
...
...
...
...
...
...
...
...
...
...
...
...
14.6
...
5.7
19.2
3.5
...
...
...
...
3.4
7.1
...
...
19.2
...
...
...
...
21.1
...
6.5
...
...
...
...
...
...
...
...
...
...
...
...
...
157.5
...
114.4
20.1
167.7
...
...
...
...
113.8
131.8
...
...
144.6
...
...
...
...
90.6
...
74.2
Shectman Abell No of levels min-max Ncells
52
53
54
55
57
59
60
61
62
63
65
66
67
68
69
70
71
72
73
74
75
76
77
78
82
83
84
85
86
87
88
89
90
91
1066
1067
1085
1100
1126
1139
1149
1169
1168
1173
1177
1185
1190
1187
1205
1213
1235
4
1
1
2
1
2
2
3
2
1
2
2
1
2
2
2
4
2
2
2
3
1
1
1
3
1
1
2
3
1
3
4
3
2
6-27
8
10
6-10
8
5-10
13-25
5-11
7-9
5
6-11
6-14
10
7-9
6-9
5-11
9-89
5-11
5-9
8-16
14-51
13
5
11
8-16
7
9
11-19
12-29
5
5-16
8-26
8-20
10-11
9.6
45.0
157.4
116.9
176.6
101.4
155.2
102.2
119.7
161.8
91.5
112.5
44.5
138.2
61.5
135.6
52.1
44.0
102.5
13.9
169.4
168.5
180.0
156.7
89.3
61.5
21.3
54.3
96.0
57.6
116.0
50.8
6.4
73.6
11
Med[]
24.0
...
...
...
...
...
...
6.4
...
...
...
...
...
...
...
...
33.8
...
...
...
47.4
...
...
...
7.4
...
...
...
10.4
...
49.4
29.0
53.5
...
6.7
...
...
...
...
...
...
103.6
...
...
...
...
...
...
...
...
57.5
...
...
...
145.4
...
...
...
91.4
...
...
...
90.2
...
107.1
43.2
2.5
...
Shectman Abell No of levels min-max Ncells
97
99
100
102
103
104
105
106
107
108
109
110
111
112
113
114
115
117
118
119
121
122
124
125
126
131
128
129
130
134
135
136
137
138
1267
1291
1307
1308
1314
1317
1318
1332
1336
1341
1346
1364
1362
1367
1365
1371
1377
1380
1
1
2
1
2
1
3
2
3
1
1
4
2
3
2
1
1
1
1
2
4
2
1
3
1
3
1
2
3
1
2
3
3
1
6
8
6-9
7
8-13
15
7-15
5-7
5-12
7
7
10-26
8-18
6-13
7-13
8
5
5
7
8-11
13-45
17-26
12
9-26
7
21-46
6
9-14
8-12
5
8-17
9-20
10-19
7
30.8
135.0
93.3
7.7
33.7
107.6
172.6
103.2
6.9
80.2
135.8
75.0
97.2
42.6
85.2
1.6
105.5
18.0
135.8
120.4
159.6
164.8
120.9
149.8
78.3
156.8
90.0
118.2
11.7
20.4
100.1
126.5
165.3
173.9
12
Med[]
...
...
...
...
...
...
21.5
...
9.3
...
...
2.0
...
38.2
...
...
...
...
...
...
20.9
...
...
5.7
...
20.9
...
...
12.4
...
...
1.4
13.4
...
...
...
...
...
...
...
1.6
...
5.0
...
...
74.4
...
21.2
...
...
...
...
...
...
160.1
...
...
148.2
...
150.6
...
...
7.3
...
...
126.4
165.5
...
Shectman Abell No of levels min-max Ncells
139
140
142
144
145
148
151
153
154
155
156
157
158
159
161
162
163
164
165
167
169
170
171
174
175
176
177
178
180
181
183
184
185
186
1383
1399
1407
1413
1416
1424
1436
1448
1452
1468
1502
1507
1516
1517
1520
1535
1
1
2
2
1
3
1
1
2
2
1
1
2
1
3
3
3
2
1
2
2
2
3
2
3
3
1
3
2
2
4
1
3
3
5
5
5-8
8-12
6
5-17
9
6
5-9
7-17
9
6
9-23
5
5-13
5-12
10-14
5-6
5
8-11
5-7
5-7
9-32
7-11
10-24
7-10
7
8-18
9-20
9-20
17-52
5
5-13
7-13
17.4
72.2
78.9
148.8
90.0
152.1
149.4
0.8
92.8
135.0
117.1
2.2
107.0
73.9
89.3
57.0
29.5
121.3
109.4
106.7
165.8
37.4
4.7
7.1
143.6
60.7
60.3
16.3
89.6
158.0
8.6
19.5
173.6
141.2
13
Med[]
...
...
...
...
...
11.8
...
...
...
...
...
...
...
...
13.1
9.8
3.3
...
...
...
...
...
7.4
...
15.8
11.2
...
22.3
...
...
19.0
...
45.7
41.2
...
...
...
...
...
158.4
...
...
...
...
...
...
...
...
95.8
58.6
31.4
...
...
...
...
...
8.1
...
138.0
64.8
...
13.7
...
...
0.5
...
162.8
129.9
Shectman Abell No of levels min-max Ncells
187
189
190
192
193
194
195
196
197
198
199
201
203
204
208
209
210
211
212
213
215
216
218
219
221
222
224
226
227
228
229
230
231
232
1541
1552
1553
1555
1564
1569
1589
1620
1631
1638
1644
1650
1651
1656
1663
1668
-
4
2
4
1
3
2
2
3
1
1
3
2
1
3
2
3
2
1
3
3
3
3
1
4
2
3
3
2
3
2
1
2
2
2
8-44
6-14
5-40
8
6-20
5-12
5-10
7-18
5
15
7-15
7-22
8
8-23
7-14
29-73
5-8
6
5-14
5-10
20-48
21-45
9
6-17
7-13
32-67
7-14
6-12
6-46
5-13
6
5-7
9-36
6-12
117.0
160.4
163.3
137.1
96.0
113.3
111.1
104.1
162.0
148.6
142.0
142.9
177.8
80.5
48.5
142.5
120.6
64.1
26.9
91.3
86.4
128.8
105.7
145.6
59.3
71.5
30.7
73.8
91.7
113.1
48.4
5.8
101.1
37.6
14
Med[]
9.3
...
42.6
...
7.9
...
...
4.7
...
...
34.5
...
...
10.0
...
5.7
...
...
5.1
3.8
42.8
1.2
...
6.2
...
1.0
20.0
...
41.0
...
...
...
...
...
114.2
...
165.0
...
94.7
...
...
104.6
...
...
152.9
...
...
78.9
...
142.6
...
...
29.4
90.8
91.8
128.1
...
144.6
...
71.8
39.0
...
109.1
...
...
...
...
...
Shectman Abell No of levels min-max Ncells
234
235
236
237
238
240
241
243
244
246
247
248
249
250
253
254
256
257
258
259
261
262
263
264
265
266
267
268
270
271
272
273
274
275
1691
1706
1709
1711
1738
1749
1764
1767
1775
1773
1778
1783
1780
1793
1795
1800
1797
3
3
2
1
2
1
4
1
3
2
3
3
4
3
4
2
1
1
1
1
2
1
3
3
3
4
4
1
1
2
3
3
2
2
14-26
6-14
10-16
6
5-8
5
6-75
9
6-16
6-12
5-27
5-17
8-19
10-27
7-19
5-8
8
5
9
8
6-8
7
12-41
5-14
11-44
6-37
6-18
5
6
8-10
11-21
9-18
6-12
9-16
121.2
130.9
121.8
45.5
158.0
72.2
127.2
158.9
106.9
22.5
69.2
113.0
125.1
168.2
79.7
35.5
50.0
161.5
20.8
33.4
132.8
59.0
88.3
61.1
89.0
86.8
77.0
109.2
92.6
124.7
13.8
48.9
62.2
135.4
15
Med[]
0.7
23.9
...
...
...
...
22.0
...
25.7
...
18.9
16.0
9.6
13.6
22.2
...
...
...
...
...
...
...
6.0
28.4
21.5
12.1
36.3
...
...
...
12.6
22.3
...
...
120.8
138.0
...
...
...
...
125.9
...
104.8
...
71.4
108.2
122.4
169.4
90.0
...
...
...
...
...
...
...
90.9
71.1
97.5
84.2
65.3
...
...
...
16.5
57.1
...
...
Shectman Abell No of levels min-max Ncells
276
277
278
279
280
282
283
285
286
287
288
289
290
291
293
294
296
297
298
300
301
304
305
306
307
308
309
310
312
313
314
315
316
317
1809
1812
1825
1834
1831
1836
1852
1873
1882
1890
1899
1904
1906
1
3
2
3
4
4
3
3
1
2
1
3
4
3
3
4
1
1
1
1
2
2
1
2
2
3
3
2
3
2
3
1
4
2
6
6-25
6-9
5-14
8-12
5-37
5-11
6-28
5
6-10
5
5-14
11-74
10-25
6-15
6-35
8
6
6
8
5-27
5-6
6
8-16
7-14
8-11
11-18
5-12
6-12
6-11
9-14
6
7-20
5-8
179.2
26.9
94.8
85.9
83.8
22.0
14.7
164.9
158.9
81.4
161.8
157.2
130.9
18.3
1.2
88.9
70.7
4.3
91.6
178.5
171.9
29.4
91.5
90.7
71.3
26.2
11.9
32.3
61.1
31.9
111.2
91.3
25.2
63.3
16
Med[]
...
8.4
...
13.4
6.5
10.8
5.4
12.6
...
...
...
18.1
35.4
15.9
10.1
9.6
...
...
...
...
...
...
...
...
...
34.9
7.2
...
15.9
...
4.4
...
14.8
...
...
23.5
...
93.0
86.8
21.1
16.6
162.2
...
...
...
161.6
124.9
25.0
...
88.0
...
...
...
...
...
...
...
...
...
46.3
14.8
...
53.0
...
110.0
...
23.7
...
Shectman Abell No of levels min-max Ncells
318
319
321
322
323
324
327
329
331
332
333
334
335
339
340
341
342
343
345
346
347
348
350
351
352
353
354
362
365
366
368
369
370
371
1908
1913
1964
1983
1986
1991
2020
2022
2028
2029
2048
2052
2055
-
2
1
4
1
2
1
3
1
3
3
4
1
1
3
1
4
2
2
3
1
4
2
1
1
2
3
1
3
1
2
1
1
3
1
11-18
6
14-91
5
5-7
12
5-20
6
5-15
5-16
5-13
9
7
11-24
5
9-44
6-9
5-8
5-13
5
7-34
8-10
5
15
7-14
5-37
23
10-19
45
8-12
9
6
5-34
31
20.7
88.4
141.9
69.7
10.0
101.0
99.8
59.0
21.2
58.3
53.3
108.6
81.7
157.4
20.8
5.0
50.5
120.3
17.6
161.9
112.3
53.3
20.1
136.5
19.1
93.4
5.1
124.6
106.2
46.4
3.0
139.4
89.9
138.4
17
Med[]
...
...
26.9
...
...
...
8.5
...
6.1
42.5
13.4
...
...
7.1
...
25.8
...
...
8.7
...
22.1
...
...
...
...
19.4
...
3.7
...
...
...
...
21.6
...
...
...
132.8
...
...
...
96.9
...
18.5
70.6
49.3
...
...
154.6
...
4.5
...
...
18.8
...
112.1
...
...
...
...
103.7
...
126.4
...
...
...
...
83.8
...
Shectman Abell No of levels min-max Ncells
372
374
375
376
377
378
380
383
384
385
386
389
390
391
392
393
395
396
397
398
400
401
403
404
407
409
411
414
415
416
417
418
427
428
2062
2061
2063
2065
2069
2079
2083
2089
2092
2107
2122
2142
2149
2151
2162
2199
-
1
1
1
3
3
1
2
1
1
1
4
3
1
3
3
4
2
2
3
1
2
2
2
3
3
3
4
3
1
1
1
3
2
2
6
8
8
5-8
20-39
9
25-39
16
6
10
6-20
6-22
6
8-29
7-14
5-20
6-7
5-10
6-16
11
14-25
7-17
5-8
46-74
21-34
8-17
7-23
7-22
8
17
8
5-14
5-10
5-11
137.1
135.5
90.5
83.9
35.2
45.1
176.1
116.4
89.5
37.0
39.9
17.9
87.8
24.9
52.4
150.9
124.7
177.7
90.8
13.8
145.1
139.1
111.4
126.7
169.4
1.7
82.2
54.1
124.6
47.2
124.6
68.8
41.9
72.2
18
Med[]
...
...
...
44.0
6.7
...
...
...
...
...
2.8
15.3
...
18.8
24.0
16.1
...
...
2.9
...
...
...
...
25.1
29.8
10.2
8.3
46.5
...
...
...
10.4
...
...
...
...
...
72.1
32.8
...
...
...
...
...
39.4
23.4
...
32.9
46.5
156.2
...
...
89.4
...
...
...
...
112.9
3.6
7.3
83.3
72.0
...
...
...
69.7
...
...
Shectman Abell No of levels min-max Ncells
429
431
432
434
435
437
438
439
440
441
443
444
445
447
450
452
453
454
456
457
461
463
464
467
470
472
473
475
478
479
486
487
488
490
2366
2372
2377
2382
2384
2399
2401
2410
2415
2420
2426
2448
2457
2459
2529
2554
2569
2593
2592
2657
-
2
2
4
2
3
1
1
3
4
3
2
2
2
1
1
1
3
4
1
1
1
3
1
4
3
2
3
1
3
2
3
3
2
1
8
5-6
14-41
7-9
6-10
6
5
16-23
5-18
11-22
7-11
6-7
6-9
6
15
7
7-14
7-21
6
11
8
5-20
5
8-15
5-13
8-10
8-19
9
10-26
6-7
5-16
5-11
5-8
6
133.6
32.8
122.7
33.9
132.3
142.1
0.1
94.4
138.8
121.9
46.8
85.5
59.7
23.8
37.9
59.6
25.5
144.5
134.2
61.1
34.6
90.2
16.7
52.3
110.0
66.8
148.6
59.1
179.0
74.6
84.7
130.3
21.5
0.1
19
Med[]
...
...
14.7
...
39.3
...
...
9.2
19.6
4.1
...
...
...
...
...
...
3.6
5.5
...
...
...
1.3
...
17.1
4.6
...
4.8
...
4.4
...
14.6
17.5
...
...
...
...
127.2
...
126.9
...
...
99.6
144.0
123.9
...
...
...
...
...
...
25.3
145.1
...
...
...
90.0
...
43.7
112.5
...
149.2
...
178.5
...
82.9
135.6
...
...
Shectman Abell No of levels min-max Ncells
491
492
494
495
496
498
500
502
503
504
505
508
510
511
512
513
514
517
518
520
522
524
525
526
529
530
532
533
534
536
539
541
542
543
2670
2686
13
16
23
27
44
76
84
85
93
95
98
112
114
3
4
3
1
3
3
3
3
3
3
1
2
2
4
4
4
1
2
3
2
2
4
1
4
1
1
1
2
4
1
2
1
1
1
8-12
8-36
7-19
8
8-22
6-12
7-14
5-10
5-11
8-22
5
14-19
5-9
5-40
7-34
6-20
8
5-11
18-44
7-14
5-10
5-23
6
12-34
6
5
12
10-20
6-11
9
6-9
6
6
6
33.2
39.1
178.1
92.9
124.9
151.8
83.1
49.6
149.6
133.3
70.4
63.5
145.2
65.2
131.0
153.0
31.6
54.1
165.8
145.6
100.2
17.9
133.3
163.1
42.2
161.0
18.2
45.7
3.3
53.6
124.8
46.0
2.6
84.6
20
Med[]
4.9
12.8
18.4
...
11.2
25.1
28.9
41.7
38.4
11.9
...
...
...
39.2
38.7
18.2
...
...
2.6
...
...
14.9
...
4.3
...
...
...
...
34.5
...
...
...
...
...
32.7
37.2
3.0
...
124.6
152.9
89.2
70.1
163.3
135.7
...
...
...
48.6
134.9
147.8
...
...
166.1
...
...
20.5
...
162.9
...
...
...
...
6.4
...
...
...
...
...
Shectman Abell No of levels min-max Ncells
544
546
547
548
549
550
551
552
553
555
557
559
560
561
563
565
567
568
569
570
571
572
576
578
580
581
582
583
584
585
586
587
588
590
116
117
119
120
126
147
150
151
154
160
168
171
175
193
1
2
2
3
4
3
1
1
2
3
2
1
1
1
1
3
2
2
4
3
3
1
2
1
2
2
1
2
1
2
2
3
2
3
12
10-13
5-10
6-28
15-84
22-45
6
8
6-12
5-14
5-12
12
5
17
12
7-28
7-22
7-12
10-23
5-10
29-47
9
6-9
10
7-9
13-19
5
9-13
8
6-19
5-13
7-14
6-14
9-14
131.8
40.8
36.9
145.9
35.2
28.3
12.1
66.4
15.4
97.3
12.6
15.1
81.9
47.1
150.6
94.2
87.7
77.2
78.8
164.2
40.5
172.2
174.1
0.8
22.8
149.1
95.0
177.3
35.3
78.8
95.9
95.3
104.8
80.1
21
Med[]
...
...
...
44.0
17.1
1.0
...
...
...
10.3
...
...
...
...
...
34.3
...
...
6.3
24.1
13.3
...
...
...
...
...
...
...
...
...
...
16.4
...
8.4
...
...
...
161.2
32.6
28.8
...
...
...
95.4
...
...
...
...
...
83.3
...
...
77.1
176.7
47.6
...
...
...
...
...
...
...
...
...
...
87.1
...
82.5
Shectman Abell No of levels min-max Ncells
591
592
593
594
596
598
599
600
601
603
604
608
609
612
615
616
618
619
620
622
623
624
625
627
628
629
630
631
632
633
635
638
639
640
194
256
257
274
295
367
400
415
-
3
3
3
3
1
3
1
1
3
3
2
1
3
3
2
3
3
1
2
3
1
1
3
3
2
2
3
1
2
4
1
3
3
3
9-31
6-22
9-37
5-18
7
5-29
13
5
7-20
8-51
5-8
5
5-17
6-12
8-13
6-31
5-63
5
6-11
6-12
6
6
8-21
7-11
5-7
11-17
5-14
6
6-10
5-42
10
6-21
6-9
10-27
59.7
83.8
25.6
64.7
68.7
67.8
110.4
20.4
170.7
52.0
100.3
134.3
163.2
73.9
89.9
53.9
144.4
74.9
46.2
35.1
12.4
132.2
121.4
43.8
59.1
139.6
12.6
135.4
67.0
90.1
56.8
1.7
48.7
95.3
22
Med[]
5.1
28.3
24.8
47.2
...
22.0
...
...
29.5
29.3
...
...
41.4
2.1
...
1.5
13.7
...
...
5.8
...
...
13.1
27.7
...
...
30.4
...
...
13.1
...
2.3
15.7
10.8
56.8
86.7
12.4
63.8
...
56.3
...
...
174.6
66.9
...
...
173.5
72.7
...
54.1
139.1
...
...
38.1
...
...
123.6
44.8
...
...
12.7
...
...
96.1
...
0.5
43.7
96.8
Shectman Abell No of levels min-max Ncells
641
642
646
647
648
649
420
423
-
1
1
2
2
1
2
6
6
6-8
5-11
8
5-10
Med[]
134.6
91.7
43.3
74.1
159.1
131.8
23
...
...
...
...
...
...
...
...
...
...
...
...
: Position angles of Abell clusters, identi ed at the 3.6 level but estimated at the 2.5 level.
Table 1b
Abell Ncells
779
1069
1170
1174
1189
1227
1278
1344
1411
1427
1515
5
8
5
6
6
9
5
10
5
5
15
72.2
138.7
158.7
147.5
167.3
145.0
118.9
127.8
44.0
160.4
161.8
Abell Ncells
1518
1524
1729
1796
1818
1921
1926
1953
1982
2120
2153
5
10
14
5
10
8
6
14
13
5
6
69.9
8.8
50.5
109.1
107.5
179.9
47.9
111.0
59.9
18.1
119.6
Abell Ncells
148
152
225
403
2400
2412
2421
2490
2525
2665
8
10
5
5
9
5
6
6
5
5
56.2
110.8
72.2
69.6
142.9
89.7
1.3
18.2
17.0
108.6
: Position angles of Abell clusters, identi ed at the 3.6 level but estimated at the 1.8 level.
Table 1c
Abell Ncells
865
903
985
987
1092
1201
1242
1262
1292
1327
1387
1441
1625
1688
6
33
11
7
5
23
6
5
7
5
8
9
8
5
12.5
117.5
105.5
171.6
109.0
166.8
17.9
71.1
177.4
146.3
51.7
153.0
150.2
89.7
Abell Ncells
1741
1768
1845
1861
1870
1936
1956
1990
2021
2025
2100
2106
2195
2196
5
7
7
5
10
6
6
5
6
7
6
6
13
5
109.3
133.6
35.5
18.7
72.1
18.9
133.6
70.2
13.1
83.1
14.2
93.6
85.6
70.6
24
Abell Ncells
2208
65
94
144
211
212
240
243
2403
2477
2638
2676
2703
5
10
6
5
8
5
9
5
11
6
8
8
5
20.0
87.0
71.4
18.0
163.6
84.7
36.0
160.8
36.4
91.5
32.0
157.7
71.2
: Position angles of Abell clusters, identi ed at the 2.5 level but estimated at the 1.8 level.
Note that there are 12 clusters in common with table 1b, which for comparison reasons we have not
averaged their estimated position angles.
Table 1d
25
Abell Ncells
716
733
779
795
803
866
878
929
967
992
1003
1022
1051
1069
1097
1098
1108
1109
1118
1132
1135
1141
1143
1152
1170
1179
1189
1198
1216
1218
1257
1264
1375
1390
5
6
13
6
8
7
17
9
6
370
17
11
6
12
11
16
10
30
6
10
14
6
42
7
11
11
32
11
7
9
12
5
5
5
69.8
91.0
77.6
129.4
47.7
82.3
97.1
84.8
109.1
170.8
119.8
58.4
90.4
179.1
86.1
121.3
48.1
7.1
40.5
110.2
44.9
88.1
13.4
58.7
162.2
49.1
10.9
164.3
179.4
175.4
25.0
74.7
0.7
95.4
Abell Ncells
1444
1466
1474
1495
1518
1526
1534
1573
1581
1583
1595
1599
1606
1616
1630
1684
1690
1693
1696
1715
1729
1769
1808
1823
1844
1850
1860
1891
1909
1944
1960
1976
1999
2019
5
8
9
21
10
8
9
9
5
5
14
7
10
8
5
5
9
11
11
5
31
15
6
6
14
18
8
6
7
13
16
7
9
8
71.6
115.2
67.2
144.9
90.8
138.7
12.6
119.9
163.1
162.4
22.3
177.2
78.7
131.5
160.1
19.2
101.2
155.7
6.3
162.2
47.9
75.3
88.4
42.8
12.7
54.4
134.7
133.5
80.8
42.7
108.9
46.8
21.0
170.8
26
Abell Ncells
2148
2169
2175
2177
48
53
103
111
13
172
179
195
207
225
229
267
292
358
395
403
410
428
2346
2361
2365
2412
2490
2495
2511
2525
2528
2549
2583
2589
8
16
9
13
6
10
10
10
6
15
12
5
8
21
8
26
7
10
16
12
6
11
7
6
6
13
13
6
7
12
10
20
13
11
90.5
50.2
103.9
53.2
130.0
119.6
72.3
3.3
145.3
87.0
136.2
159.9
12.1
41.5
22.4
94.7
84.5
9.7
19.6
59.5
91.9
123.4
45.6
87.1
46.2
117.0
16.0
42.5
136.9
144.5
116.8
165.6
11.7
14.5
Abell Ncells
1408
1409
1412
1419
1423
1433
1437
6
9
5
14
11
8
5
89.7
36.2
20.7
33.2
130.4
5.6
58.2
Abell Ncells
2026
2030
2034
2064
2101
2120
2141
6
6
11
6
6
14
8
46.3
179.5
48.8
134.5
91.4
174.0
89.2
27
Abell Ncells
2614
2630
2654
2656
2665
2678
2698
9
7
10
14
15
5
9
84.4
167.2
24.1
26.3
122.1
70.5
19.1
Nearest-neighbour alignment signal and its standard deviation (column 2) as a function of
limiting maximum cluster separation (column 1). Column 3 shows the signal to noise ratio, column
4 the number of separations considered and column 5 the probability, estimated from a KolmogorovSmirnov test, that the distribution of nn 's is drawn from a uniform one.
Table 2:
Dlim (h 1 Mpc)
10
15
30
50
60
-9.4 3.3
-6.3 2.5
-3.9 1.9
-4.3 1.6
-4.4 1.6
Table 3:
= No of pairs PKS
2.9
2.5
2.0
2.7
2.8
61
106
192
254
260
0.04
0.03
0.22
0.06
0.05
All-neighbour alignment signal.
Dlim (h 1 Mpc)
0 - 10
10 - 30
30 - 50
50 - 150
0 - 150
-10.9 2.9
-2.45 1.13
-2 0.85
-1.5 0.41
-1.8 0.35
= No of pairs
3.8
2.2
2.3
3.6
5.1
80
528
904
3944
5456
28
PKS
2 10 3
0.06
0.015
0.01
10 4
References
Argyres, P.C., Groth, E.J. & Peebles, P.J.E. 1986, AJ, 91, 471
Barnes, J. & Efstathiou, G. 1987, ApJ, 270, 20
Binggeli, B. 1982, A&A, 107, 338
Bond, J.R. 1987, in Nearly Normal Galaxies, ed. S.Faber (NY:Springer), 388
Borgani, S., Jing, Y.P. & Plionis, M. 1992, ApJ, 395, 339
Borgani, S., Plionis, M. & Valdarnini, R. 1993, ApJ, 404, 21
Binney, J. & Silk, J. 1979, MNRAS, 188, 273
Blumenthal, G.R., Faber, S.M., Primack, J.R. & Rees, M.J. 1984, Nature, 311, 517
Carter, D. & Metcalfe, J. 1980, MNRAS, 191, 325
Davis, M., Efstathiou, G., Frenk, C.S.& White, S.D.M. 1985, ApJ, 292, 371
Doroshkevich, A,G., Shandarin, S. & Saar E. 1978, MNRAS, 184, 64
Dekel, A., West, M.J., & Aarseth, S.J. 1984, ApJ 279, 1
Dekel, A., & West, M.J. 1985, ApJ 288, 411
Efstathiou, G., Frenk, C.S., White, S.D.M., Davis, M. 1988, MNRAS, 235, 715
Frenk, C.S., White, S.D.M., Efstathiou, G. & Davis, M. 1985, Nature, 317, 595
Frenk, C.S., White, S.D.M., Davis, M. & Efstathiou, G. 1988, ApJ, 327, 507
Flin, P. 1987, MNRAS, 228, 941
Fong, R., Stevenson, P.R.F. & Shanks, T. 1990, MNRAS, 242, 146
Groth, E.J. & Peebles, J.E., 1977, ApJ, 217, 385
van Kampen, E. & Rhee, G.F.R. 1990, A&A, 237, 283
Lambas, D.G., Groth, E.J. & Peebles, P.J.E. 1988, AJ, 95, 996
Lambas, D.G., Nicotra, M., Muriel, H. & Ruiz, L. 1990, AJ, 100, 1006
Mattig, W. 1958, Astr. Nach. 284, 109
McMillan, S.L.W., Kowalski, M.P. & Ulmer, M.P., 1989, ApJS, 70, 723
Muriel, H., Nicotra, M., Lambas, D.G. 1991, AJ, 101, 1997
Peebles, P.J.E., 1982, ApJ, 263, L1
Plionis, M. 1988, MNRAS, 234, 401
Plionis, M., Barrow, J.D. & Frenk, C.S.F. 1991, MNRAS, 249, 622
Plionis, M. & Borgani, S. 1992, MNRAS, 254, 306
Plionis, M., Valdarnini, R. & Jing, Y.P. 1992, ApJ, 398, 12
Rhee, G.F.R. & Katgert, P. 1987, A&A, 183, 217
Rhee, G.F.R., van Haarlem, M. & Katgert, P. 1992, AJ, 103, 1721
Rhee, G.F.R. & Roos, N. 1990, MNRAS, 243, 629
Rood, H.J. 1988, ARAA 26, 245
Salvador-Sole, E. & Solanes, J.M. 1993, ApJ, 417, 427
Seldner, M., Siebers, B., Groth, E.J. & Peebles, J.E. 1977, AJ, 84, 249
Shane, C.D. & Wirtanen, C.A., 1967 Publs Lick Obs:, XXII, Part 1
29
Shectman, S., 1985, ApJS, 57, 77
Struble, M.F. & Peebles, P.J.E. 1985, AJ, 90, 582
Struble, M.F. 1990, AJ, 99, 743
Trevese, D., Cirimele, G. & Flin, P. 1992, AJ, 104, 935
West, M.J., Dekel, A. & Oemler, A.Jr. 1989, ApJ 336, 46
West, M.J. 1989a, ApJ 344, 535
West, M.J. 1989b, ApJ 347, 610
West, M.J., Villumsen, J.V. & Dekel, A. 1991, ApJ, 369, 287
Zeldovich, Ya. B., 1970, A&A, 5, 84
30
Figure Captions
Cluster position angles, , as a function of the number of 10 10 cells, de ning the cluster
sampling area, used in their estimation. The position angles estimated only at one overdensity level,
are plotted.
0
Figure 1.
0
As in Figure 1 but for clusters with more than one position angle determination: (a.) The
average position angle, hi as a function of minimum number of 10 10 cells used in its estimation.
(b.) hi as a function of the maximum number of cells used.
Figure 2.
0
0
The frequency distribution of nn : (a.) Solid and broken lines corresponds to nearestneighbour separations of D 15 and 10 h 1 Mpc, respectively. (b.) nn distribution for all
nearest-neighbour separations.
Figure 3.
The frequency distribution of all-neighbour relative angles, ij , at the four indicated
separation ranges.
Figure 4.
31