Astron. Nachr. 306 (1985) 5, 283--300

New results on large-scale structures

11. KALISKOV, K . STAVREV, and I. KUNEVA, Sofia

Departmcnt of .4stronomical Astronomy and National Observatory

With 1 3 figures (Received 1985 January 30)

A new version of the magnetic-tape catalog of ABELL clustcrs of galaxies is used to obtain redshift estimators and to gcnerate two samples of clusters. A procedure for searching for superclusters of galaxies is applied and the results are given in tabular and graphic form. For a limited homogeneous saniple (distance 60-275 hIpc, galactic latitude B > 35’). I 2 multiplets, having member clusters with known redshifts, are found. I t is shown that the spatial covariance function for rich clustcm has tlw forin E 7 (v0 /y)? with Y,, .-- 22 .4 1.8 Mpc and y = 1.90 f 0.25 for 3 Mpc 5 Y 5 80 Mpc.

Es wird cine neuc Version IICS Katalogs tlrr Gal;txienhaufcn yon ABELL, tler auf Magnetband aufgcnommcn ist, vcrwendct, um Zusammen- h8nge dcr liotvcrschicbung zu bckomnwn und zwci Stichprobcn dcr I iaufen xu scldfcn. lis wird cine ncuc Suchprozrdur \‘on Gala- xiensupcrhaufcn angcwendct und die Ergcbnissc als Tabellcn und graphisch d:trgcstellt. Fur die begrenztc homogene Stichprobc (Ab- stand 60- 275 Mpc, galaktische Breitr B > 35’) wurdcn 12 Multiplette Kcfunden. deren Mitglicder bckanntr Hotverschicbungcn haben. Es wird gezcigt, daO die raumlichc Kovarianzfunktion fur reiche Galasicnhaufcn in der Form f =- (r,,/r)r mit yo = Z Z . . ~ 1.8 Mpc und 7 T 1.90 f 0.25 IUr 3 Mpc 5 Y 5 80 Mpc dargestellt wcrden kann.

1. Introduction

The interest in studying the large-scale structure of the Universe has tremendously incrcascd in recent years. This interest is equally maintained by the theoretical investigations and the direct processing of fresh observational data. Several international meetings, dedicated wholly or partially to large-scale structure problems have taken placc in less than a decade. Many new results have been published, e.g. in the Proceedings edited by LONCAIR and EISASTO (1978), ABELL and CHISCAHISI (1983). By the way, the present paper is also initiated by a similar meeting. Along with this the flow of newest results is not decreasing: it is enough t o compare the two excellent reviews of SHANDARIN, DOROSHKEVECH and ZELDOVICH (1983) and of OORT (1983) with the current literature (the end of 1984) in order to establish the quick “aging” of the reviews. Few are the branches in contemporary astronomy and astro- physics, subjected to such a quick renovation.

At present one can definetely state, that the dispute for the existence of second-order clusters of galaxies already belongs to the history of extragalactic astronomy. I t should be noted, however, that not only the theoretical stu- dies on cosmogony and evolution of large-scale structures, not only the new optical, radio and X-ray observations and the intensive redshift determinations, contributed to end the dispute, but the bias in terminology as well.

I t is intuitively clear, without any specific definition, that “a cluster of galaxies” should contain “many” galaxies. A double, triple, in general multiple galaxy is not yet a cluster of galaxies. I t appeared intuitively clear that a super- cluster of galaxies, or a second-order cluster, should contain “many” clusters of galaxies, too. For better or worse this naive terminology shifted as early as the fact, that A1656j1367 is something more than a double cluster, has been established. With the addition of several not large groups of galaxies and according to the modern termi- nology, it is the famous Coma supercluster. Thus, according to the already accepted terminology, a supercluster contains “small”. number of clusters of galaxies. Of course, here we have in mind only rich clusters, according to ABELL (1958). Taking advantage of the freedom in the term szcpercluster, we shall use as a synonym the term multz- $let (of clusters), as well as doublef, traplet, . . ., but in general we mean by supercluster only a configuration of rich A-clusters, and not group-of groups of galaxies. I t may be suggested next t o this, and in agreement with the not yet sufficient observational data, that it is hardly probable that superclusters with more than a dozen very rich clusters exist. To finish with terminology let us add that sometimes besides a supercluster and a multiplet, other terms have been introduced - e.g. class 2 cluster (MURRAY et al. 1978), but they have not been adopted.

Our knowledge for the spatial structure of the large agglomerations of galaxies steeply decreases with the distance. Thus, the Local supercluster, whose existence has been established due, most of all, to DE VAUCOULEURS (1953, 1956, 1960, 1970, 1971, 1978), cf. also the references cited therein, is wellstudied in respect to its spatialstructure (TULLY 1982). This is true, t o a somewhat less degree, for the Coma (GREGORY and THOMPSOS 1978) and Perseus (GREGORY, THOMPSON, and TIFFT 1981) superclusters, as well as for the Hercules supercluster (TAREKGHI et al. 1979, 1980; CHINCARINI;~<OOD, and THOMPSON 1981). As for the considerably most distant superclusters, some in- formation is available only for SCls I451 + 22 and 1615 f 43 (FORD et al. 1981; HARMS et al. 1981; CIARDULLO e t al. 1983).

We present here lists of superclusters, composed only of rich clusters of galaxies, and results, concerning the spa- tial structural features of the superclusters.

19’

284 Astron. Nachr. 306 (1985) 5

Lists of superclusters have been published repeatedly. The first list belongs to ABELL (1961). I t is based mainly on the apparent (2-D) surface distribution of clusters on the sky. The information in depth, contained in m,,, the magnitude of the tenth rank galaxy, has been partially used too. Two incidental lists have been given by Roon and SASTRY (1971) and KALIXKOV (1976), based mainly on the 2 D case, and for A-clusters, too. ROOD (1976) has used data for the radial velocities to compose a list of multiplets. An extensive list of possible doublets and triplets of rich clusters has been published by KARACHENTSEV and SHCHERBAKOVSKY (1978). The successfull identification of rich clusters with X-ray sources provoked the creation of a new list of superclusters by MURRAY et al. (1978), based on the investigation by KARACHENTSEV, TSAREVSKAYA, and SHCHERBANOVSKY (1975). The first list of super- clusters, extracted from a 3-D sample of rich clusters with measured redshifts, is published by THUAK (1980) and repeatedly used later (e.g. by WAGNER and PERRENOD 1981).

In 1982 we presented a t the IAU symposium No. 104 (Crete) a catalogue of A-clusters, possible members of super- clusters (KALINKOV, STAVREV, and KUNEVA 1983a). In that paper we used not measured redshift, but the calibrated distance, and a special procedure for search for enhancements in the 3-D case, exceeding 10 and 50 times the mean spatial density of the A-clusters. More defined results have been given later by KALINKOV, STAVREV, and KUSEVA (1984a, 1984b). Another catalog of superclusters has been published by BAHCALL and SOSEIRA (1983b, 1984) - for various density enhancements, also in the redshift space.

The most extensive catalogs of superclusters for the 3-D case, wliich is in fact a guasi - three-dimensional red- shift-space case, are presented in this paper.

2. Distance calibration to Abell clusters

We use here the last version (KAIJXKOV, STAVREV, and KUNEVA 1983b) of our machine-readable catalog of clusters of galaxies (KALIXKOV, STAVREV, and KUNEVA 1976). This version contains numerous data for the clusters in ABELL (1958) and ZWICKY et al. (1961-1968) catalogs. Here we shall consider only the A-clusters. The magnetic-tape catalog contains: i) coordinates (equatorial, epoch 1950.0. galactic, supergalactic, Lambert-equivalent) ; ii) magni- tudes m,, mlo, Am = m,, - m,, M , , Mlo , m3, m,; iii) apparent ARELL radius of the clusters, and the area covered by the first-rank galaxy; iv) distance group, richness group; v) types according to BAUTZ and MORGAN (1970). ROOD and SASTRY (I971), STRURLE and ROOD (1982, 1984); vi) redshift data - results of measurements and some estimates by different regressors. The magnetic tape catalog contains additional information on detailed photometric data for cluster galaxies, as well as results from radio and X-ray observations, references for the A-clusters, and other data.

The last version of the catalog of rich Abell clusters of galaxies is a result of the merging of a variety of large and small catalogs or lists of A-clusters, and numerous single observations or investigations of galaxies, members of A-clusters. Let us note, that the cataloge of LEIR (1976), LEIR and VAS DEN BERGH (1977), further on referred to as LB, HAYNES (1980), STRUBLE and ROOD (1982), etc. are included in this version after a critical evaluation. The redshift data are from the well known lists of NOONAN (1973, 1981), FETISOVA (1981, 1983), SARAZIN, ROOD, and STRUBLE (1982) and many other published papers, as well as from private communications. Let us also note, that all cosmologically dependent cluster characteristics in our catalog are calculated for H , = 100 km s-1 Mpc-1 and qo = +I.

For example, we have for the absolute magnitudes

Mio,i = myo 1 - 5 log 2 - 42.52 ,

~ t ~ ~ , , = mlo,l - K(z) - 0.136,

K ( : 5 0.2) = 2 ,

K(0.2 < z 5 0.36) = 1.1252 - 0.005 ,

(1)

( 2 )

where the apparent magnitudes are corrected for the galactic absorption and the K-effect according to the relation

and the correction K(z ) , following SANDAGE (1973) and the literature cited therein, is

(3) K(o.36 5 z -= 0.4) = 22 - 0.32 .

We also define a mean population P of the clusters of galaxies

P = q n r : : , (4)

IA = 0.0286(1 + i ) z / ~ , ( 5 )

where = 40, 65, 105 ... galaxies for richness group p = 0, I, 2, ..., and the Abell radius is

in degrees. Our catalog contains observational data for the redshifts of more than 600 A-clusters. The data for 70 of them

became available too late to be included in the present analysis. Anyhow, at present redshifts are available for more than 20% of all A-clusters and this reassures us that conclusions for the spatial distribution of a representative part of the rich clusters of galaxies might be drawn. Unfortunately, the extreme nonhomogcneity, and even ani- sotropy, of the sample of clusters with measured redshifts would not allow us using this sample only. Therefore, we have decided to evaluate the redshifts for all Abell clusters by some calibration, and the 3-D analysis has been carried out in the following way: the clusters with measured radial velocities participate with the directly evaluated distance R = V , / H , and for the rest clusters the distance is estimated by calibration.

KALIFKOV, M. e t al. : N e w results on large-scale structures 285

Calibration of distances to A-clusters has been repeatedly done, the first one, using m,, as a regressor, Ilelonging to ABELL (1958), and based only on 18 redshifts. The relation, used by ABELI., is

as FULLERTON and HOOVER lrad shown (12 = 17, because A465 is excluded for reasons given by ABELL). It is worth noting, that later processings have not indicated a nonlinear term. ROWAN-ROBINSON (1972) uses a linear relation for 31 clusters (3 clusters, deviating by more than 3a, being excluded - AIg4, A465, and A189o). A similar relation for 75 clusters has been determined by KALINKOV, STAVKI-V, and KUSEVA (1g75), showing, that the cali- bration regressions log z on inlO do not significantly differ one from another for the clusters belonging and not be- longing to the homogeneous ABELL’S sample.

We would remind here the remark in this paper about the danger, connected with the determination of the indi- vidual distances by calibration, especially when only a single regressor is uscd, e.g. nz,,. Another linear calibration has been determined by CORWIS (1974) for III A-clusters. With 101 radial velocities for A-clusters, referred to their catalog, 1-B present 3 linear regressions for log z , each one with two regressors, namely m: and BhI, in& and f ~ , p and log R,, where R, is the cluster radius, measured by I-EIR. According to LB, the final estimate of P is a combination of these three estimates.

Another approach is suggested by SARAZIN, Roon, and STRUBLE (1982) - they derive the photometric redshift estimate with four regressors, namely myo, my, n A , and BM, where i i A is the ABELL’S population of clusters.

With radial velocities for 342 A-clusters KALINKOV, STAVREV, and KCKEVA (1983a) have dcducctl the bivariatc linear regression

log z : -8.5532 -+ 0.72121n1, - o . o 1 5 8 3 4 m ~ ~ , (6)

log z = -3.637 + 0.135m,, 4- 0.179 log P , 14 30

wliere P is determined according to (4)- ( 5 ) , but with the preliminary estimate z1 = -4.568 -t 0.216m1,. (8)

The values below the coefficients in (7) are the corresponding standard deviations and for all next foimulas the s.d. will be shown in this way.

The newly supplemented version of Abell catalog (KALIXKOV, STAVREV, and KUKEVA 1983b) allows a more com- plete study of the problem of the distance calibration t o A-clusters. The Hubble diagram for 533 A-clusters with measured redshift is shown in Figure I, m,, and the distance group DG I , ..., 6, being marked on the abscissa. A distinction is made for the belonging to ABELL’S homogeneous statistical sample. Thus we have

logz = -1.0361 -1- 0 . 2 1 1 1 ( m ~ ~ - 16.43) = -4.5054 + 0 . 2 1 1 1 m ~ ~ 52 52 858

(9)

for n =. 533, with correlation coefficient Y = 0.87 and standard deviation for the best fit u = 0.121. Actually, log z

in (9) and further on, is the estimate log z . Only two clusters, A480 and A2084, deviate by more than 3u from (9). -4.4 and +3.2 respectively.

Figure 2 presents the normalized deviations from regression (I), i.e. d = (log z, - log z,)/0.121. Figures I and 2 show a peculiarity in the magnitudescale: the regression for m,, (= 16.4 definitely differs from the regression for m,, 2 16.5. Indeed, for the clusters with m,, 5 16.4 we have

n

I\

log z = -1.2191 + o.2076(m1, - 15.50) = -4.4373 + 0.2076nz,, 74 I04 1620

for n = 243, with r = 0.79 and u = 0.115, and for In,, 2 16.5

log z = -0.8829 + 0.3377(m10 - 17.21) = -6.6934 4- o.3377m1, (11) 68 I73 2986

for n = 290, with Y = 0.75 and u = 0.116. The hypothesis Ho tha t regressions (10) and (11) are parallel, tested with the statistic T, e.g. JOHNSOW and LEONE

(1g77), is rejected, since Prob ( t529 2 6.44) = 5 . 8 . 10-l~. Excluding all clusters with Id1 > zu, we determine new regressions instead of (10) and (11). However, H , is also rejected, with Prob (tbos 2 8.05) = 4.1 . 10-l~.

I t should be noted that not a single A-cluster from DG 0-4 deviates by more than 3u from the regression (IO), while clusters 4 6 5 , 480, 680, and 2084 deviate from (11) by -3.0, -5.0, -3.3 and f 3 . 3 respectively. These 4 clusters are excluded from further processing, since the corresponding galaxies with measured rcdshifts are either back- ground objects, or m,, for these clusters is wrongly evaluated. Thus, instead of (11), we have

log z = -0.8814 + 0.3503(m,, - 17.20)’= -6.9076 + o.3503m10 62 158 2715

n for n = 286, with Y = 0.80 and CT = 0.105. The normalized deviations from (12), i e . , A = (log zi - log zi)/o.105, are presented in Figure 3 and they do not show a linear trend as in Figure 2 . Instead of (9) we have

logz = -1.0365 + 0.2132 (mIo - 16.42) = -4,5372 + 0.2132m,, 51 50 829

for n = 529, with Y = 0.88 and u = 0.117. Regressions (IO), ( IZ) , and (13) are plotted in Figure I.

286

- 2 0 1 I I I I 1 1 I I I I " 14 16 m10 18

Astron. Nachr. 306 (1985) 5

3 01

-15

533 A - C / S

0 non-romple romple

L

0 .

A 2084 *.. . I,

" t

I I

N

I I I I ; I 1 * ' 1 A 2 Q U ,

533 A - C I S 1 - L k 5 - 6 3

somplc

o non-somple . I

, I 8 . 2 I . . .*

ALeQ

I I 1 I :r I , I **"f a .

11 mlQ

1L 16 -3

Fig. 2 . Normalized dcviations .I from regression (9). A linear trend of the deviations with the mla is evident for DG 5-6 .

The peculiarity of the ml0 magnitude scale, established here, is not unexpected, although previous investigations (e .g . , CONVIY 1974) have found systematic effects of somewhat different character. I t should be assumed that ABELL'S rrt,,-scalc IS composed of two linear parts, delimited just by mlo = 16.45, i.e. the boundary between DG 4 and DG 5. Tlw regressions adducedhere allow reduction of the scale for m,o 2 16.5 to the scale for ml,, 5 16.4, but a correction of the observed peculiarity is possible only after the accumulation of statistics from a detailed photo- metry of the roth rank galaxies. However, if a similar peculiarity is confirmed for m,, mz, m3, ...., then it will be easy to develop a correction procedure from the data of SAXDAGI' and HARDY (1073) and SCHSI~IDER, CUSS, and HOESSEL (1983)

KALINKOV, M. et al. : New results on Iarge-scalc structurcs 287

2

1

a

0

-1

- 2

I

A 208L ' t '

. . . . c . .

: . . ' a msamplr A L65

A680 t B sample .,

I I I I

17 18 mw

Fig. 3 . Normalized deviations A from regression (12), for DG 5-6. There is no linear trend with ml0.

Froin regression (13) we can get an approximate estimate of the retlshift of any A-cluster; tliis estimate can be improved by the implicit partial participation of a second regressor DG, using regressions (10) and (IZ), but one should be careful since the peculiarity of ABELL'S m,,-scale causes transposition with regard to distance for the clusters with GI, = 16.4-16.5. Eg., for m,, = 16.4 regression (10) gives a distance of 278 Mpc, and for ml0 = 16.5 regression (12) gives 227 I'vlpc. Moreover, as it is seen from Figure I, the conjunction of.regressions (10) and (12) around nil, = 16.4-16.5 for near and distant clusters is practically impossible. Therefore, relation (10) should be used for the near clusters, relation (12) for the distant clusters, and (13) - for the intermediate ones.

An improved redshift estimate can be obtained by introducing additional regressions for other cluster parameters. Let us write again the regression in the form

log2 = b, +- bm,, = a + b(m,, - El,) , (14) -

where a = log z and m,, are the mean values and the coefficients arc further on tabulated. For the various cases of the richness class, according to Abell, the regressions of type (14) are given as formulas

(15)-(19) in Table I , w11ere the standard deviations of the coefficients b,, b , and logz are presented, as will as Y , u, and tlie size tz of the corresponding sample. Let us note that tlie standard deviation of log z is in fact tlie s.d. of the coefficient ain regression (14). The last two formulas in Table I , (zo)-(21) , refer to the clusters in and out of the homogeneous sample of A B E L L (1958).

Let Z, be estimates by different regressions: i = I by (13) , i = 2 by (10) and (IZ), i = 3 by (15)-(19), and i = 4 by (zo) - (zI ) . 'lhcn, the improved cstiniate will be

- -

= 2 i i / 4 , i = I , .... 4 (22 )

Our grounds for applying tliis averaging are the relatively small differences between u for the various cases l'he smallest s.d. r~ for the best fit is for the case RG 4 + 5 , although the number of clusters is only 9.

Table I

Simplc regrcssions log z = b, + bnt,, ~ . .~ ~ . __- -______ - ___ . . ....

-log 2 I a U ~ . ~ . _.__ .

b . ~. __ Formula For - bo

. __- ~

(15) KGO 4.42 0.2038 1.1738 0.81 0. I 23 I 66 1 8 1 1 5 96

RG I 4 .14 0.1875 1.0067 0.85 0 . 1 1 3 I 9 9 82 81

(16) 14

(1 7) l iG 2 4.12 0.1goo 0.9331 0.89 0.107. 98 17 99 104

(18) 1 x 3 5'34 0.2631 0.7598 0 .78 0.085 57 50 2 90 114

(19) I1G4 + 5 4.34 0.2072 0.7191 0.75 0.06~ 9 1.20 685 5

( 2 0 ) A-sample 4.43 0.2059 1.5537 0.84 0. I 26 200

I4 82 89

288 Astroii. Nachr. 306 (1985) 5

According to our investigations the application of two regressors, namely m,, and log f‘, is quite cffectivc., and a still better estimate may be attained by tlie regression

10s 2 = -2.837 i - 0.0677tnn,, + 0.3140 106 I’ 53 96

for I I == 529, r = 0.962, and u = 0.067. However, the mean population P in (23) has been calculated from the meas- ured redshifts, using (4) and (5). When computing calibrated values, P in (23) has to be determined by (4)-(5), using zI, and then the final estimate 211 is determined by the bivariate regression (23).

In order to improve the redshift estimate, specially for the clusters, included in the LB-catalog, the addition of two more regressors will be reasonable. Then, if we write the multiple regression in the form

log z = b, -t b, log P 4- b,m,, $- b,m, 4- b, (BM) , (24) with coded designation of BM-class, we get formulas (25)-(30) in Table 2 for 420 A-clusters witli measured z from tlle LB-catalog. Tlic standard deviation for log 3 is again for tlie casc when log z is regarded as thc free term (L

in the regression

-

log z = z -i , b,(log 1’ - log P ) . I - ... + b,((BJI) - (BM)) . (31 1 ‘11ie standard deviation for the best fit in Table 2 is extremely small, but this i.s due not so much to tlic increased

number of regressors - 4 explicit and I implicit, as to the fact that the mcan population P is directly tlcterniined from the measured redshifts. In spite of that , all our investigations with the introduction of otlicr (and more) re- gressors, e.g. T&,, v ~ : , logs, log S, ..., definitely show that the best estimate 2111 for the redsliift of clusters, which 1iave data for i n , and Bhl, is obtaincd by the rcgrcssions from Table 2, the mean population I? being dctermined on the basis of tlic estimate 21, considered here preliminary. By the way, formula (30) has been used ior the clusters of RG 4 and 5 (that is A665), because it is dangerous t o use a relation with four regressors if 12 = 8.

Let us remind that r in l a b l c 2 is a multiple correlation coefficient and r2 defines the part of the variance of tlle de1”:iident variable (logz, in our case), which is “explained” by regression (25). On the other hand, the simple coefficient of Correlation betwcen the observed value and the estimates of the dependent value is equivalent to ;I multiple correlation coefficient (an excellent introduction in multiple linear regression and correlation with appli- cations, is given by AI;II;I and AZEX 1979).

I n the present paper we use three types of distances to the A-clusters: (i) determined by direct measurement of the redshift, (ii) by calibration - estimate 211, and (iii) by calibration - estimate ,2111, applicable only for A-clusters, included in LB-catalog. Systematic differences between these three types of distances have not been found, but tlie standard deviations of tlie calibrated distances arc not small. For instance, tlie study of tlie residuals (zOb - - z ~ , , ~ , , ) shows, that the standard deviation of tliecalibrated distance in the range of 200-300 Mpc is 61 hfpc (about 24%), for 400-500 hIpc the s.d. is 60 Mpc (13%), for 500-600 Mpc is 87 Mpc (16%), and for 600-700 hIpc it is 63 Mpc (9,70/). Let us note that for the very large distances, which are not uscd here, the standard deviation is 16% for 700-8800 hlpc, and 8.776 for ~ O O - ~ O O Mpc.

Cl’e should conclude, that even the most careful and detailed calibration of distances to A-clustcrs of galaxies is uncapable to assure sufficient accuracy for tlie determination of the membership of a given cluster in a super- cluster.

In fact , this problem is mucli more serious, since even when measured redshifts are available the membership of a clustcr in a supercluster cannot be reliably determined if the number of measured redshifts is small. l l iere are two reasons for this: first, tlic measured redshift may belong to a background galaxy, and second, an inaccurate knowledge of the velocity dispersion for the clusters of galaxies. Beyond dispute, tlie first reason niay lead to corn- pletely incorrect conclusions. Nost of all, this concerns those clusters, whose distance is dcteriniiml from ;1 single measured redshift. The only consolation is that not so often a background galaxy is nicasurcd instead of a cluster mcniber. Tlie second reason may affect the mcan densities and dimensions of tlie supcrclusters, which is of secondary importancc, concerning the establishing of the existence itself of the superclusters.

The upper reasoning is necessary only when the distances to the clusters can be accurately tlctcrmined. As was inentioned before, we assume for dcfiniteness q, = -1-1, H , = 100, in which case K = cz/H,.

Tablr : XIultiple rcgressions log L

Formula For 1IG

6, + 6, log P + b2ml0 + b p i , + b, (BAT)

-60 b, 62 b, -106 Z r (I n

.- . . -. .. ~ .. - . . . . . ~. . . .-

b, -

(25) 0 2.070 0.5415 -0.0053 0.0034 -0.0011 1.2103 0.998 0.011 78 73 68 33 32 I3 13

(26) I 2.291 0.5983 -0.0007 . -0.0027 0.0018 1.0606 0.998 0.014 189 52 57 24 IS I 0 I 0

(27) 2,406 0.6048 -0.0039 -0.0006 0.0016 0.9268 0.998 0.014 92 86 97 40 28 ’4 I 5

3 2.740 0.6975 -0.0080 0.0015 -0.0026 0.7614 0.996 0.013 5 2 (28) I57 I 62 76 40 I8 18

(29) 4 -i 5. 4.636 0.2915 0.0814 0.0950 -0.0119 0.7026 0.887 0.053 8

(30) 3 -1- 4 + 5 3.238 0.4720 0.0416 0.0199 -0.0026 0.7537 0.948 0.044 61 3.315 4010 1550 797 512 82 I

. . .~ I33 60 448 . .. .- . 2 4 2 . . 58 ~

. 4 94 ._ ..

KALISKOV, hI. et al. : New results on large scale structures 289

Because of the mentioned difficulties, we will search only for superclusters, whose members have measured

In this way we shall use the A-clusters with calibrated distances only as background objects, which create the redshifts.

decor for the clustering of the true performers - the A-clusters with measured distances.

3.

The lists of superclusters in $4 of the present paper are extracted out of two samples of A-clusters. Here we describe the procedure for the formation of the first sample and give the boundaries for the second one.

Sample I. This sample contains A-clusters from the region defined by B > 35" and R < 300 hlpc, but it includes only clusters, satisfying the condition that the corrected population n, 2 50 galaxies. This needs some explanations.

As it is known, ABELL'S procedure for search and cataloging of rich clusters of galaxies comprises the determina- tion of the apparent angular cluster radius (the spatial radius being assumed constant, 1.5 Mpc), based on the ori- ginal calibration of ABELL (1958), presented analytically by (6), according to FULLERTON and HOOVER (1972). I t is not suprirsing that in some cases ABELL'S calibration, derived from the data for 17 clusters only, differs signifi- cantly from the measured redshifts or the present calibration. Therefore, we can try to correct the populations, determined by ABELL, for the sample, containing near clusters in general, the majority of them with measured redshift. Actually, it is not easy to make such a correction, because it requires that the surface density law for the clusters of galaxies be known. Assuming a very simple law u, we may operate without any free parameters and test a t first approximation the significance of the expected effect. We assume

Two samples of Abell clusters

u cc Y - 1

and then, as i t may be easily shown, the corrected population is given by (321

f tc = nA(RA/Rz;cal) 8 (33) where n A is the population according to ABELL (1999, R A is the distance, calibrated according to ABELL (in fact, by (6) ) , and R z ; d is the observed or calibrated distance. ABELL (1958) has assumed that clusters with population n A ;1 50 have richness P A 2 I. We use the same assumption, but for the corrcted population, i.e. for n, 2 50 the richness is 9, 2 I.

Thus, sample I includes all A-clusters with 8 > 30°, K < 300 Mpc, having p , ;2 I. Table 3 contains some data for the entire ABELL catalog: nt,,, distance and richness groups, RA - distance,

according to ABELL'S calibration, R, - by redshift, nA - ABELL'S population, n, - corrected population from (33), and s - indication of membership in ABELL'S sample (sA = I), or in the sample with n, 2 50 (s, = I).

The first 6 clusters in Table 3, with B < 0, are given here only for completeness. Thus, A75 should be added to the more homogeneous sample. There is no change in the status of A126, although the error from the initial cali- bration is about twice as large, since nA = 51. The error in the calibrated distance to A326 is even larger, leading to a corrected population 2.5 times larger. The error for A2634 is in the opposite direction, but this cluster is not from ABELL'S homogeneous sample, because of the low value of B.

Table 3 Hichiiess correction for sonic Abell clusters

A %o - - _ _

75 16.5 I 2 6 16.6 326 I 7.0 419 15.7 496 15.3

2634 13.8 634 14.9 999 15.6

1169 16.6 1177 15.7 1213 '4.5 1228 13.8 1836 '5.7 I837 '5.7 2147 13.8 2151 13.8 2152 13.8 1186 16.5 1187 15.6 1190 16.6 I474 16.0 1954 17.6 2108 15.7 2178 17.1

43 15.9 2001 17.2 2 2 2 0 1 7 ? - _____ - -

5 0 341 I79 42 5 I 357 165 51 5 0 425 167 34 4 0 23 1 122 32 3 I 187 95 50 1 I 76 96 52 3 3 5 4 2 1

4 4 I I I

0 0 I 0 I I 0 I

I 2 I

I49 219 357 231 I18

76 231 231 76 76 76

40 33 73 32 51 50 41 50 52

60 87

5 2 341 237 107 3 I 219 237 55 5 2 357 238 87 4 I 269 237 70 6 2 540 276 I 2 0

4 0 231 276 45 5 I 443 278 51 4 0 256 334 37 5 I 462 333 57

520 332 40- - 6 0 _ _ _ _ _. ._ ._

nc gal 80

86 G I 99 4 1 75 76

I49 78 43 37 87

- .-

I 1 0

102

37 59 40

754 51

130 79

235 38 81

28 79 63 -.

SA

0 I 0 0

I 0

0 0 I 0 I I 0 I

I

I I

I I I I

I 0 I

0

I 0 ~-

sc .

I I I I I 0

I I I I 0 0

I I

0 I 0

I I I I

I 0

I

0 I I ._

10 Astron. Kochr.. Bd. 306, H. J

290 hstron. Kathr. 306 (1985) 3

The 8 clusters, beginning with A634, fall into the examined region. A634, A999, A1177 and A1836 enter our sample, although they do not participate in ABELL'S homogeneous sample, while A1213 and A1228, with SA = I, are ex- cluded from our analysis, since they have S, = 0.

The last three blocks of clusters in Table 3 show the extent to which the results from the spatial investigation, based only on Abell's calibration, may be affected. Thus, A1187, AI474, A1186, and A I I ~ O , which fall into three different distance groups, are practically a t the same distance (237-238 Mpc). The same is valid for A2108, A2178, and A1954, as well as for the last three clusters in the Table. Let us note, that A1169, A1186, A1190, and A2178 are included in our sample, because R, < 300 Mpc, while RA > 300 Mpc. Our sample contains also A1954 with R A = 540 Mpc, DG 6, since R, = 276 Mpc.

I t should be noted, that out of the three A-clusters, connected with the Hercules supercluster only A2151 was retained in sample I.

Naturally, the assumption of another surface density law for the clusters would be lead to another order of the clusters, according to their correctedpopoulation n,. This question needs additional investigations, and the correc- tions introduced here should be regarded only as a very rough approximation.

The corrected population n, is calculated for DG 0-4 with n~ from STRUBLE and ROOD (1982), and for DG 5-6 the mean populations Z = 40, 65, 105, ... for

The distribution of A-clusters from our sample by galactic latitude B and distance R is given in Table 4. The column, marked with t, contains the clusters with measured redshift, and the column cal - the clusters with cali- brated distances, as explained in 9 2. The total number of clusters is 239, 129 (54%) of them are with measured z. The region, defined by B > 35" and R < 275 Mpc, marked in Table 43 and for which the corresponding sums are given in parentheses, contains 138 A-clusters, 1x2 (81%) of them are with measured z. This is just the region, chosen for detailed examination in 4, to minimize the boundary effects. The assumed mean density of clusters in this region is DA = 7.5 x I O - ~ NPC-~. The nearest cluster is A1367, R = 64 Mpc. In this same region up to a distance of 225 Mpc 5 clusters, namely Ax334 ( R = 171 Mpc), A1332 (194 Mpc), A1297 (203 Mpc), A1171 (212 Mpc), and .41336 (221 hlpc), have calibrated distances. Only one cluster, A628 (221 Mpc), in the region with B = 30-35" and R 5 225 Xlpc, has a calibrated distance.

Sanzple 11. This sample includes all 1786 A-clusters with B > o", but here we shall present some data for a com- paratively more homogeneous sample with B > 40'. Table 5 shows the distribution of 1289 A-clusters by B and by distance up to R = 650 Mpc, the clusters with measured and calibrated distances being separated. I t may be supposed, mainly from the space density variation with the distance, that the distribution of clusters may be regarded as not strongly influences by selective effects up to 450 Mpc. We adopt mean spatial density 0:: = 1.22 x x I O - ~ Mpc for B > 40" and R = 60-450 Mpc.

The space density variations for B > 40' and step 50 Mpc (only the first interval is smaller, 60-100 Mpc) are presented in Figure 4. With ALL are denoted all A-clusters with measured or calibrated distances, and z,ALL in- dicates the space density only for the clusters with measured redshift. In the same Figure RICH marks the space density of the A-clusters from Sample I , with measured or calibrated distances, also plotted up to 600 Mpc. The bars of the latter histogram refer to I standard deviation, calculated for a simple Poisson statistics. As it is seen, up to 300 Mpc, the space density for ALL is larger than this for RICH. But for larger distances these densities do not differ, which is probably due t o a selection effect. A t larger distances mainly clusters with pA > I are observed and the probability toexclude someof theseclusters because p , = o is very small (this can happen only for an excep- tionally large error in Abell's calibrated distance R.4, or, all the same in ml,,.

Sample I is composed of selected A-clusters, which are assumed in the present paper as really rich clusters. There- fore, the conclusions drawn from the investigation of this sample should be regarded as more reliable than from Sample 11. On the other hand Sample I1 would allow to find superclusters a t much larger depth.

= 0, I, 2, ... have been used.

Table 4 Distribution for Sample I

Table 5 Distribution for a part of Sainple IT

4. Catalog of superclusters

Our procedure for supercluster search has been briefly described by KALINKOV, STAVREV, KUNEVA (1983a, 1984b), and in detail by KUNEVA (1985).

Generally speaking, the procedure consists of localization of the regions with space density of the clusters D > > kD,, where Do is the mean space density for the whole examined sample or catalog, and k is the enhancement

KALINROV, M. et al. : New results oxi large-scale structures

100 300 500 Mpc I O - ~

Fig. 4. Space density variations for clusters with 9, 2 I (RICH), with PA 2 o (ALL), and with measured redshift (z, ALL).

factor. Besides, we introduce the concept of “closed” supercluster - this is auch a configuration, which will be detected by the procedure independently of the cluster,. member of the multiplet, which will be chosen to start the procedure with. On the contrary, “open” are Such configurations, whose membership and richness depend on the choice of initial cluster. Here only “closed” multiplets of A-clusters are examined.

Let us first adduce the results for Sample I.

Table 6 Superclusters (sample I) . R < 275 Mpc, B > 35O, 2): = 7.5 X I O - ~ Mp@, Iz = 10

- _ _ _ ~ _______ ___ ~ . ~~ ~ _ _ _ __ A R Y x Y A R Y x Y z D/D$

.____ -_____ ~- z DID,’

-__ - -~ - ___

1040 + 09, 238 + 54, DIST = 104 (-31, -5r, 84) Mpc I344 + 04, 335 + 63, DIST = 233 (96, -46, 208) MPC 1016 96.2 10.9 -5.2 5.0 -8.1 25 999 95.3 12.0 2.9 1.1 11.6 37

1139 112.7 17.8 -9.7 -14.0 5.2 22

1108 + 39> 177 + 66, DIST = 231 (-94, 5, 210) M ~ c 1203 227.8 6.5 2.1 5.5 -2.6 IIS

1142 111.8 13.8 -7.4 7.4 -8.7 36

1187 237.1 6.7 -2.4 2.6 5.7 211 1190 238.0 10.4 -4.2 7.6 5.7 86 1155 221.2 18.5 4.4 -15.7 -8.8 20

1113 + 28, 202 + 68, DIST = 96 (-32, -13. 89) Mpc 1185 91.1 4.5 0.1 0.0 -4.5 350 1177 94.7 11.9 2.8 --II.4 -2.2 38 1257 101.6 13.5 -2.8 11.4 6.7 39

I275 180.8 16.6 -0.4 -12.9 10.4 7 1169 174.5 16.8 -16.0 3.3 -33.9 I 3

1132 i- 41, I66 + 68, DIST = 172 (-60, 14, 160) Mpc

1461 161.3 20.1 16.4 9.6 -6.5 12

1143 + 53, 144 + 62, DIST = 241 (-93, 68, 212) Mpc 1400 233.2 12.6 3.2 7.3 -9.8 16 1218 237.4 18.0 -12.4 -11.4 -6.3 11

1468 253.0 19.0 9.1 4.1 16.1 14

1149 + 56, 140 + 59, DIST = 177 (-68, 58. 152) Mpc

1780 230.8 3.8 -1.4 -2.5 -2.5 1773 232.6 b.7 -1.3 -6.4 -1.5

I349 + 24. 24 3- 76, DIST = 195 (43, 18, 189) Mpc

1809 236.2 10.2 2.7 9.0 4.0

1827 200.3 10.8 10.0 -2.1 3.5 1795 186.2 12.5 - 8 8 4 7 -7.4 1825 1808 18.1 6.7 - 6 4 -15.6 1775 212.6 21.4 - 7 9 3.7 19.5

141j + 27, 36 + 71, DIST = 230 (62, 44, 217) Mpc

1898 233.8 6.9 6.3 -1.1 2.5 1873 232.6 9.3 -7.8 2.4 4.4 I927 221.8 14.7 7.6 4.1 -11.9 I909 242.8 15.5 11.8 . 2.0 9.8 1831 219.8 20.0 -17.9 -7.3 -4.9

1432 + 18, 16 + 64, DIST = 165 (68, 21, 149) Mpc

1899 160.7 10.1 -7.7 -6.5 -0.7 I913 159.8 8.1 -3.8 -6.3 -3.4

1991 175.7 17.7 11.5 12.8 4.0

1524 + 30, 46 + 56, DIST = 217 (84, 87. 180) Mpc 2065 21G.2 6.6 3.4 -5.7 0.6 2069 219.4 6.8 -4.5 3.6 3.8 2089 222.8 10.0 9.4 3.2 I 4 2056 228.7 14.0 5.2 -1.6 12.9 2061 230.2 15.1 -0.6 6.2 13.7 - .

I383 179.3 4.4 -1.6 -2.6 3.2 360 2079 198.5 18.2 -4.7 -7.1 -16.2 1291 175.7 8.8 -7.5 -2.8 -3.7 94 2092 200.6 18.3 -8.2 1.4 -16.3 1318 169.7 9.9 -2.6 -6.7 -6.8 97 1507 177.5 16.6 4.0 15.6 -4.1 28 1554 + 22, 36 + 48, DIST = 123 (66, 49. 91) Mpc

2107 126.2 8.7 -1.6 -4.3 7.4 1436 193.1 16.9 -3.5 8.8 14.0 33 17.4 30 2148 132.5 12.9 0.5 11.5 5.9

2151 111.2 15.2 1.1 -7.2 -13.4 27 1452 189.2 18.4 4.3 -4.3 1377 154.1 22.7 6.9 -8.0 -20.0 I9

____~_---- - - _ _ _ ~ - _ . ~ -

.561 208

90

25 35 16 I 3

98 80 30 34 20

60 62 I7

I08

95 198

46 46 31 36

48 29

10’

292 Astron. Nachr. 306 (1985) 5

Table 6 contains superclusters. all cluster members of which are with measured I in the region restricted by B > 35", R = 60 to 275 Mpc. The tables with the multiplets and their member clusters have the following structure. The - headline for each supercluster contains i ts center in equatorial coordinates - we have for the first case in Table & = I O ~ 40m, 6 -- fog", in galactic coordinates ( L = 238'. = +54"), and in Cartesian galactic coordinate system, with DIST being thc distance HI= 104 Mpc with its components

-51 Mpc, y= 84 Mpc. The X axis is aligned toward L = o", & = 0". the Y axis is aligned toward L = goo, B = o", and the 2 axis is defined to point in the direction B = +goo. Therefore, the components are

- = -31 Mpc, Y

X = R cos R cos L Y = R c o s B s i n L Z = R s i n B .

(34)

The first column of the table shows the A-number, and the second - thc distance H to the corresponding cluster; r is the distance from the supercluster center, and x . y, I are the corresponding components in Cartesian coordinate system with origin in the center of multiplet. and axes, parallel to X, Y, Z. Everywhere R. r, x , y , and E are in Mpc. The last column presents the running relative density excess, determined by the number of clusters up to a given r. The adopted value for the enhancement factor for Table 6 is k = 10. Of course, D/D, for the cluster, which is nearest to the multiplet center, is a very rough estimate: this ratio is more accurate for the most distant clusters. We must point out here, that our searching procedure in the version, uscd here, ascribes equal weights to all sample clusters and in this way the center of a given multiplet may be regarded as a mass center of all clusters, supposed with equal masses.

We have detected two septets, one quintuplet, three quadruplets, and six triplets. Table 6 contains 49 A-clusters, i.e. 44% of all 112 A-clusters with measured I in our sample participate in superclusters. This percentage should be higher since the ordinary doublets have to added, too. All closed doublets are given in Table 7, Ar being the distance between their components. In other words, A Z I W is the nearest neighbour of Azxgg. By the way, this is the closest doublet, and the widest is A957/978. Supposedly, that the physical doublets have components separated up to about 25 Mpc, i.e. almost as much as the distance bctneen A1367 and A1656. But a t the same time around two clusters, separated by a distance of 37 Mpc, a sphere, with diameter 37 Mpc, in,which the density formally ex- ceeds xoD& can be circumscribed. So, about 70% of the clusters with measured z can be regarded as members of superclusters.

The first column of Table 8 contains clusters, whose nearest neighbours are such, that the condition D > IOD; is satisfied. We call these doublets open - e.g., the nearest neighbour of A2124 is Azogz, the separation between them being 19.2 Mix. but the nearest neighbour of A2092 is not A2124: there is a tliirtl cluster a t a distance of less than 19.2 Mpc from A2og2. As a matter of fact, i t may be definitely asserted, that 86 A-clusters out of 1x2 clusters with measured z (i.e., 77%) in the examined volume are located in regions with D > IoD,,.

Table 7 Closed doublets ______- - -

A A r A Mpc

2197 2 .3 2199 1836 4 0 I837 2052 4.7 2063

1982 12.8 2022

__ __ +-- _ _ _

2028 7.4 2029 1650 11.5 1651

1541 2 0 6 1552 1939 21.2 I954 2x49 22.6 2169 1367 22.8 1656 1100 23 9 1267 1767 24.8 1783 1983 25.0 2040 957 3 0 8 978

Table 8 Open pairs

2124 19.2

1474 30.3 I564 31.4 1035 31.5 2020 33.4 1904 34.6 I793 35.3

1279 21.2

1371 36.8

A

2092

1254 1552 I 650 I 190 1991 I783 I873 I474

The applied searching procedure registers only sphereshaped configurations. This may be regarded as an ad- vantage if the central parts of the superclusters show spherical symmetry, or as a disadvantage if filamentary structures prevail. Of course, the procedure is not sensitive if the search aims to find prolonged configurations. The most important objection to the construction of catalogs of the type shown in Table 6, with 12 SCls, is formu- lated by h l A T E m E (1978), actually on another occasion, for groups of galaxies, and accepted, with insignificant modification, by PEHRENOD and LESSER (1980). The objection is that the distance R = cz/H in the redshift space is obtained by a too naive procedure. The distance components, defined by (34), should also be avoided because of the velocity dispersions in the clusters or for the errors in the redshift measurements, which elongate the observed configuration along the radial direction. This objection should be taken into account for smaller groups of ga- laxies and, maybe, for clusters of galaxies. However, for superclusters it is not serious or a t least it may be tested.

The idea how to verify the statement that redshift distance may be regarded as a reliable estimate for the dis- tance is very simple, which is evident from Figure 5.

Let OXYZ is a Cartesian galactic coordinate system, and O'xyz is a coordinate system, with an origin in the mass center 0' of the multiplet. Then, the components of the distance Y from 0' can be easily calculated from the com- ponents X, Y, 2 of the distance R to each cluster, defined by (34):

- x = x - x . _- y = Y - Y , __ z = z - 2 .

(35)

Let us rotate the coordinate systemo'xyzin such a way, that the new axisz' isacontinuation of the radius vector R, and y' lies in the plane O'zz'. Then the components of the radius vector r in the new coordinate system O'x'y'z'

KALISKOV, M. et al. : New results on large-scalc structures 293

are

. Fig. 5. The transformation problem for the root-mean-square dimen- sions. The observer is a t 0 and 0' is the supercluster centre.

x' = x sin L - y cos L , y' = x sin B cos L + y sin B sin L - z cos B , z' = x cos B cos L + y cos B sin L -+ z sin B.

In this way the elongation, if it really exists, will be carried over and concentration along the axis 2'. In other words, the root-mean-square dimensions a,. of the multiplets have to satisfy the inequality

a,, > a,,, a,,, 6,. a,, a, . (37) The results are given in Table 9, where the superclusters are identified by the central clusters, i.e. A-clusters,

located most closely to the mass center. The superclusters are arranged by multiplicity Y , the mean galactic latitude B also being given. The first row for each configuration contains the RMS dimensions s X , s,, sz in the coordinate system O'xyz, and the second row - sd, syt, sd in O'x'y'z'. The last column contains s, the RMS dimension of each multiplet. (Actually, 2s may be regarded as a characteristic size of the supercluster, the mean value for the 12 con- figurations being 31.7 Mpc, with standard deviation 6.9 Mpc.)

As i t is seen from Table 9, the inequality (37) is satisfied only for CrB supercluster. The inequality a,, > a, is valid for 6 multiplets, while for the rest 6 multiplets 6, > a,,. Table 9 apparently proves that the naive procedure for the determination of R = V / H works for the superclusters. Oneshouldkeep in mind, however, that our analysis totally excludes a possible influence from the velocity dispersion of the superclusters, since distance only is ascribed to the radial velocity. On the opposite, all sI, ..., s,, can be regarded as velocity dispersions, if multiplied by 100. Then the mean velocity dispersion of the examined superclusters will be 3170 km/s. More correctly, this is the upper limit for the velocity dispersion since a part of it, interpreted as distance, should explain the 3-D, and not the z-D structure of the superclusters.

-

Table g Root-mean-square dimensions in Mpc for the superclusters from Table 6. - ___ ~ -. ~

A v B s x SY SZ s (centre) SX' SY' SZ'

( U W 2065 7 56 6.3 5 .o (CrB)

_ _ _ ~ .____ -_ I383 7 59O f5.1 j-8.8 f12.8 16.3

12.2 14.6

I 898 5 71 12.4 4.5 8.5 15.7

10x6 4 54 7.8 9-7 10.0 16.0

7.4 10.3 10.3

5.6 4.6 12.7

5.3 11.2 9.6

11.6 7.9 7.7

10.5 6.9 4.7

8.4 8.8 14.3

1203 4 66 4.0 10.7 7.0 13.4

1827 4 76 9.7 5.2 15.2 18.8

1185 3 68 2.8 11.4 5.9 13.1 11.6 1.5 6.0

12.9 14.5 10.2

14.3 3.3 14.3

8.2 2.8 2.6

I275 3 68 16.2 11.6 9.1 21.9

1400 3 61 rr.1 10.0 14.0 20.5

1780 3 63 2.3 8.0 3.5 9.0

1913 3 64 10.1 11.1 3.8 15.5 7.8 10.2 8.7

2107 3 48 1.1 10.0 11.6 15.4 j-8.0 j-7.3 j-11.0

294 Astron. Nachr. 306 (1985) 5

The next figures show an isometric representation of some clusters in two coordinate systems, O'xyi and O'x'y'z ' . The scale, which is the same for the three axes, is denoted a t 10 Mpc; the cluster dimensions are everywhere 3 Mpc (two ABELL radii), independent of the perspective effects.

Figure 6 represents the superclusters UMa (SCI 1149 + 56), and Figure 7 - the supercluster CrB (SCI 1524 + 30). The central part

SCI 1415 + 27, a quintet, is shown in Figure 8a-h, and SCl 1108 + 39, a quartet, in Figure ga-b. We will make here a short diversion. Our searching procedure has been applicd to two well-studied configurations, SCI 1451 + 22

and SCI 1615 + 43 (CIARDULLO et al. 1983). We have used only the A-cluster with measured redshifts (Table 2 of CIARDULLO et al. 1983) and since the above mentioned lists contain some clusters, which are not from ABELL'S homogeneous sample, we assume the mean density Dhl = 1.22 x I O ~ Mpc-J. Our results are presented in Table 10. which is same as Table 6, and in Table 11, same as Table 9. Our criteria do not incorporate A2036 to SCI 1451 + 2 2 . whose multiplicity is by I lower than in the cited paper, but for SCI 1615 + 43 the procedure leads again to a sextet. Table I I definitely confirms the conclusions made for our 12 SCls; here again the inequality (37) is not satisfied, moreover, for SCI 1615 + 43 we have s:' < SX, sy. .... s y ' . The characteristic dimensions do not differ significantly from those of the I L SCls.

of UMa contains three close clusters, A1383/1zg1/1318.

' fie two additional SCls are shown in Figum 10 and I I . Let u g now go back to Sample I. Let us assume k = 50. Then, as it is seen from Table 12, only one quartet and four triplets remain.

Tliecorresponding RXS dimensions are given in Table 13 - the situationwith S I ' , is tile same as it was before. As it should be expected, the RMS dimensions are smaller than in Table 9.

Let uc now turn to Sample 11. The main catalog of multiplets is represented in Table 14, and the data for the RYS dimensions - in Table 15. One should keep in mind, that all processings for Sample I[ refer to k = 50, configurations with R > 450 Mpc and B < 30' being added too. This catalog is not so homogeneous, as the one presented in Table 6, but, of course, all clusters in Table 14 have measur- ed redshifts.

Table 6 contains one of the richest superclusters, UMa, SCI I 149 + 56. With k = 50 UMa becomes a triplet already as SCI 1136 -t- 55 (Table 12) . There is a triplet A1452/1383/1436. for the same sample and the same k = 50, which is open with respect to A1383, and that is why i t is not included in Table I L . The corresponding relative densities are DID: = 46, 86, 125. The same configuration is detected with Sample 11, but then = 28. 53, 77. As it is seen from Figure 6, the disintegration of UMa for large k is expected: two tri- plets form, and one object, Ar383, is a bridge.

UMa

Fig. 6. SCI 1x49 + 56, UMa. Isometric projections in O'xya, (a), and in O'x'y'z', (b), coordinate

2

CrB

2056

. .

systems (cf. Fig. 5).

; , . * ,.: .' 2092 a)

Fig. 7a. b. SCI 1524 + 30, CrR. (Cf. Fig. 6).

I<.ALISKOV, 31. rt al. : Sew results 011 large-scale structures

I : I

1927.+., ~

, I

. . . . . a)

. , I '. ... I

Fig. 8. a, b. SC1 1415 + 27, central cluster ArRg8. (Cf. Fig 6).

Fig. 9 a , b. SC1 1108 + 39. central cluster A1203. (Cf. Fig. 6).

Table 10

Two rich superclusters

A R r x Y t DID,!,'

1451 + 22, 29 + 62. DIST = 349 (144, 79. 308) Mpc 1986 354.4 5.5 1.6 I .5 5.1 I I G 1988A 3473 6.4 3.9 -4.6 -2 .0 151 1980 345.4 6.7 -6.5 1.1 -1.2 192 1976 350.5 6.9 1.7 -6.2 2.5 238 2001A 339.7 12.0. -4.6 5.3 -9.7 57 2006 349.0 15.8 14.2 -3.4 -6.1 30 1792 356.4 16.7 -10.3 6.4 11.4 30 - ___

Table 11

RMS dimensions for the two rich superclusters

A R r x Y 2

1615 + 43. 67 + 45. DIST = 407 (114, 264, 288) Mpc 2179 407.7 2.4 -1.8 0.2 I .6 2183 409.2 4.5 -2.8 3.2 1.4

2.4 10.6 2172 415.8 10.9 -1.0

2196 399.3 1 2 . 0 4.6 -2 .1 -10.9 2158A 404.0 18.3 -11.7 -10.1 9.7 2211 306.2 18.9 12.7 6.4 -12.5

_ _ ~ _ ~

-

ss SY Sr SX' SY' Sr'

- S v B A

(centre)

DID:'

1415 443

45 45 16 17

1986 7 62O k8.1 k 4 . 9 k7 .1 11.8 1451 -k 22 7.6 7.1 5.6

k5.9 f11.4 5 5 . 5 2179 6 45 8.1 5.7 9.9 14.0 16x5 + 43 -

Another rich supercluster in Table 6 is CrB, SCI 1524 + 30. With k = 50 its multiplicity decreases to 5 (Table 12) . leading to SCI 1522 + 29, a configuration which is entirely the Same for Sample 11, but with relative densities D / D f = 46, 78. 88. 106, 1x7, instead of D/Di = 75, ..., 191 from Table 12.

The uintet SCI 1415 + 27 in Table 6 changes into triplet SCl 1417 + 26 in Table 12, but i t does not appear as a multiplet in Table 14,

The quartet SC1 1x08 + 39 in Table 6 is a tri let SCI 1 1 x 0 + 40 in Table 12. It is preserved without any changes in Table 14, for Sample 11, but with other relative densities D/D?I - 315. 429. 2 0 5 . ~

The triplet SCl 1344 + 04 enters all three Tabfesd, 12, and 14. The last triplet in Table 6, SCI 1554 + 2 2 , includes A2151, a member of the famous Hercules multiplet. It participates in the conven-

tional Her supercluster, SC1 1602 + IG, including A2151/2152/~1.+7, for Sample 11, which is free pf any restriction for the cluster popu- lation.

since Do' 3 > D:, and its mean relative density (for Sample 11) is 40 against 65 in Table 12.

296 Astron. Kachr. 306 (1985) 5

. +

. : . .

SCL lL51.22

0 2 1 7 2

:m, I 2006

Fig. 10 a, b. SCI 1451 + 22, central cluster Arg8h. (Cf. Fig. 6 and Ciardullo c t al.. 1983)

Fig. 11 a, b. SCI 1615 t 3 , central cluster A217g. (Cf. Fig. 6 and Ciardullo e t al., 1983).

Table 12

Superclusters (sample I) k = 50 . .

I x 2 DID; - ~- -~

Y ~ ~-

A

1 1 1 0 + 40, 174 + 66, DIST = 234 (-96, 10. 214) Mpc 1187 4.0 -0.9 -2.7 2.8 513 1x90 4.5 -2.7 2.4 2.7 698 1203 6.6 3.6 0 . 3 -5 .5 334 1136 4- 55 , 143 + 59. DIST = 175 (-72. 54, 150) MPC 1291 4.0 -3.6 1.2 -1.2 499 1318 5.3 1.3 -2.7 -4.4 417 1383 6.3 2.2 1.4 5.7 388 1344 -1 - 0 4 , .. 1417 -+ 26, 35 + 70, DIST = 236 (65, 45, 223) Mpc I898 4.7 2.8 - 2 . 2 -3.0 302 I909 8.4 0.9 4.2 76 1873 11.4 9'4 -11.2 1.3 -1.2 65

1522 1 - 2 9 , 45 + 56, DIST --- 223 (86, 88, 186) Mpc.

2067 7.9 -7.0 2.4 -2.7 127

2065 9.0 0.8 -6.8 -5.9 172 2061 9.4 -3.2 5.1 7.2 XQI

2056 7.5 2.6 -2.8 6.4 75

2089 8.7 6.8 2 .0 -5.1 143

. .- .____ .-

KALINKOV, hI. e t al. : New results on large-scale structure 297

Table 13 RMS dimensions for the superclusters from Table 12

A (centre)

I 187 (1203) 1291

(UMa) I 780

I 898

2056 (CrB)

V

3

3

3

3

5

SX

SX'

- B

. --

66' f3 .2 2.5

59 3.1

63 2.3 8.2

70 10.1

6.5 56 5.3

f5.8

2.2

SY SZ SY' s d

t2.5 +4.8 4.9 3.2 2.3 5.2 4 .o 4.6 8.0 3.5 2.8 2.6 1.9 3.8 6.8 5.6 4.8 6.4

f4 .7 f 6 . o

--

Table 14 Superclusters (Sample 11). DO" = 1.22 x I O - ~ M ~ c - ~ , k = 50

- .- _____- A R Y x Y D/D;I -_ -- --

0919 + 74, 139 + 36, DIST = 402 (-244, 213. 238) MPc

765 398.7 3.8 1.1 - I .2 -3.2 349

787 405.6 . 6.4 1.1 6.2 1.5 220 762 399.3 5.5 1.4 1.9 -4.9 235

788 404.7 10.4 -3.7 -7.0 6.9 69

0934 + 74. 137 -+ 37. DIST = 365 (-215. 199, 218) MPc,

848 362.8 4.8 4.7 0.8 0.4 180 818 359.8 6.3 2.0 -5.8 -1.2 160 786 372.0 8.4 -0.7 5.0 0.9 99

1018 - 06, 2 5 0 + 40, DIST = 161 (-41, -115, 104) Mpc -2.1 816 978 -158.0 2.9 0.6 1.9

993 159.8 4.4 -1.3 3.8 1.9 456 979 164.9 5.7 0.8 -5.7 0.2 315

1019 + 50, 164 + 53, DIST = 468 (-267, 78, 376) Mpc 965 467.7 4.6 -3.4 -2.2 -2.0 207 980 473.7 6.0 -3.0 I .6 5.0 176

-3.0 162 1000 461.7 7.1 6.4 0.7

1 1 1 0 + 40, . . . 1136 + 5 5 , . . . 1140 + XI, 255 + 67, DIST = 338 (-34, -128, 312) Mpc

I354 338.2 3.4 0.7 -2.9 -1.4 518 1372 337.6 6.3 1.1 6.0 1.6 160 1345 328.3 10.4 -1.0 2.8 -10.0 52 1356 349.9 11.5 -0.9 -5.9 9.9 51

1348 + 27. 38 + 77. DIST = 216 (39. 30, 211) Mpc 1800 217.0 2 4 -1.7 1.4 0.8 I477 1831 219.8 8.4 5.0 6.6 1.5 66 1775 212.6 9.0 -3.2 -8.1 -2.3 80 __ ~ - - - -

S _ _ _

6.3

6.5

9.0

11.0

9.6 .--

______

Y X Y A R 2

I453 -! 29, 42 -t 63, DIST = 376 (128, 115, 334) Mpc - ~ - - - - __

1990 380.4 4.8 -0.2 0.8 4.8

2005 375.0 4.6 3.5 2.7 -3.4 1984 372.6 5.0 -3.3 -3.5 -1.3

1508 + 06, 7 + 51, DIST = 235 (146. 19, 183) Mpc 2028 231.4 5.3 -5.0 1.9 -0.6 2029 229.9 5.8 -1.1 -2.5 -5.1 2033 243.1 8.4 6.1 0.6 5.8

1522 + 29,. . .

1534 + 37. 60 + 54, DIST = 457 (734, 232, 370) Mpc 2096 455.7 1.3 -0.4 -0.6 -1.1 2093 455.7 3.2 0.8 -3.1 0.0 2100 459.6 3.9 -0.4 3.8 I .o

1602 + 16, 30 +- 44, DIST = 111 (69, 40, 78) Mpc

2151 111.2 2.1 -1.1 1.7 0.5

2x47 106.7 4.6 -1.9 -3.1 -2.7 2152 114.8 4.0 3.0 7.4 2.3

1728 + 78, 110 + 31, DIST = 177 (-52, 143, 91) MPC

2256 180.2 4.9 -2.9 0.2 4.0 2296 181.1 6.2 0.5 6.0 -1.6 2271 170.3 7.0 2.4 -6.2 -2.3

1817 + 70, IOO + 28, DIST = 252 (-40, 218, 120) Mpc

2295 246.7 9.5 4.6 -6.7 2.7

2301 202.0 10.3 -0.0 8.9 5.1

2311 269.8 4.4 -3.2 -1.7 2.4

2312 275.8 8.3 4.6 6.6 2.0

2308 247.0 9.3 -4.6 -2.3 -7.8

1855 + 69, IOO + 25, DIST = 270 (-44, 240, 114) Mpc

2315 263.8 6.7 -1.4 -4.9 -4.4

'73 308 330

129 198 99

9049 1165 976

2180

624 624

169 161 170

32 49 54

236 130 102 --

It is seen from Table 6, that there are multiplets with high relative density, as well as multiplets with low relative density, as e.g. SCI 1132 + 41, which should be excluded by a more precisely elaborated procedure for search for superclusters. There is a similar case in Table 14, the triplet SC1 1817 + 70, regardless of the high value of k for this Table.

5. Discussion

The multiplets of clusters of galaxies, cataloged in 0 4, lead us to the conclusion, that the membership of A-clusters in superclusters is by no means an exception. However, i t is hard to make some quantitative conclusions, since our lists contain only multiplets, whose members have measured redshifts. More reliable conclusions may be drawn after a considerable increase in the number of measured redshifts of A-clusters, or after such an improvement of the distance calibration that the standard deviation of the regression is smaller than the characteristic dimension of the selected multiplets (< 25-30 Mpc).

298 Astron. Nachr. 306 (1985) 5

Table 15 RMS dimensions for the superclusters from Table 14

- A v B sz SY SI S

si ~

(centre) Sx' SY'

765 4

848 3

965 3

978 3

1187 3

1291 3

1354 4

1780 3

1800 3

36' 52.4 5.5

37 6.0 3.4

53 5.6 2.8

40 I .2 2.6

66 3.2 2.5

59 3.1 2.2

67 1.1

I .4 63 2.3

8.2 77 4.4

3.8

k5.5 k5.4 8.1 4.4 3.9 5.4 1.1 8.1 5.1 5.3 2.0 4.4 7.4 6.2 2.8 5 .o 2.0 5.5 3.2 3.6 2.5 4.8 6.3 4.9 3.2 2.3 5.2 6.5 4.0 4.6 5.4 8.2 9.9 7.1 6.7 8.0 3.5 9.0 2.8 2.6 7.5 2.0 8.9 7.1 3.7 - -_______.

A v B sz SY

(centre) Sr' SY'

1990 3 63 3.4 3.2 0.6 4.9

2028 3 51 5.6 2.3 2.5 2.9

2056 5 56 5.3 4.8 5.8 4.7

2096 3 54 0.7 3.5 2.3 I .9

2151 3 44 2.6 2.7 I .9 0.6

2256 3 31 2.6 6. I 2.6 4.8

2295 3 28 4.6 8.0 4.3 8. I

2 3 1 1 3 2 5 4.1 6.0 25.0 k5.2

__ _ _ _ _ _ _ _ _ -

- - __

SZ S

SZ' __ ___ _ _

4.3 6.3 4 .o 5.5 8.2

7'2 6.4 9.6 6 0 I .o 3.7

2.5 4.5 4.0 3.4 7.5 5 . I 6.8 11.5

6.9 3.8 8.2

I 3.8

2.2

___ ___ -

The main purpose of the catalogs in Tables 6, 7, 12, and 14 is to bind the so found multiplets with a variety of investigations of observational and theoretical character - photometry, dynamics, statistics, etc.

A comparison between the superclusters, found by us, and those by other authors - most of all by THUAN (1980), BAHCALL and SONEIRA (1983b, 1984), as well as ABELL (1961), ROOD and SASTRY (Ig71), ROOD (1976), MURRAY et al. (1978), KARACHEKTSEV and SHCHERBANOVSKY (1978), will be given in a following paper.

During the search for superclusters we have come across A-clusters, which may be assumed either as isolated clusters, or as indicators of sui generis voids between (or around) the clusters. E.g., the nearest neighbour of A1630 ( R = 195 Mpc, B = 67") is A1564 at a distance dr = 51.9 Mpc. This means that the sphere with diameter 103 Mpc, circumscribed around A1630, contains no other cluster. This statement is not completely correct, since the sphere does not contain any more clusters with measured or calibrated z, and as we have seen, our calibration may lead to large errors. If this error is removed, in the indicated sphere may fall, with small probability, clusters which we now regard as foreground or background. I t is worthy of note than A1654 is the nearest neighbour of A1630 both in Sample I , and in Sample 11. Such is the case also for A1691 ( R = 216 Mpc, B == 77"), whose nearest neighbour is A1749 a t AY = 41.8 Mpc.

There are some other similar cases, but they should be regarded as doubtful because of the too large distance or the low galactic latitude. E.g., the nearest neighbour of A2224 ( R = 451 Mpc, B = 34') is A2204, with AY = 65.2Mpc and the nearest neighbour of A595 ( R = 118 Mpc, B = 31") is A553, wjth rA = 54.6 Mpc.

Our procedure for supercluster search includes calculations of the distances to the first, second, etc. nearest neighbours. Figure 12 shows the comparison between theoretical and observed distributions, computed for the most representative part of Sample I. The excess of the observed over the theoretical frequencies for smaller distances is uqdoubtedly caused by the superclustering phenomenon. According to PEEBLES (1980) and referencies therein about earlier papers of PEEBLES on this subject, a direct

estimate of the spatial covariance function may be derived also from the nearest-neighbour distance distribution. Thus we have

where nt is the number of neighbours in the ib bin with volume I/(, and Y is the radius, halving Yt. The spatial covariance function is presented in Figure 13. For the range of 3 Mpc Y & 50 Mpc we obtain

which is applicable up to I z 80 Mpc. The standard deviation for the supercluster length is f1.8 Mpc, and for the power i t is fo.25. Thus, the covariance function is in accordance with the expected values of the parameters and does not significantly differ from the values of BAHCALL and SONEIRA (1983a). Written in the form of (39), their covariance function is

A similar function is obtained by KLYPIN and KOPYLOV (1983), but the power is y = 1.6. A covariance function with a supercluster length of Y, = 22.7 Mpc, with y = 2.1 has been obtained by KALIKKOV, DERMENDJIEV, and KUNEVA (1979) for A-clusters, without using redshift measurements but applying the old calibration method of KALINKOV, STAVREV, and KUNEVA (1975).

KALINROV, ill. et al.: Ncw resdts on large-scale structures

I I 299

50 I00 Npc r -

Fig. 12. Theoretical and obscrved (or computed) density distributions of distances to different neighbours n = I, 2, ..., 5 , 10, ..., 25 for the sample of 138 rich A-clusters (R = 60-275 Mpc, B > 35'). The distances are calculated up to 100 Mpc. The theoretical density distributions are computed byfn = (3u"exp (-u)/(~(n - I)!), with u = 4nDp/3 .

5 ! I

2 t

l L

i

1 O O r - - ~ - 7 7 - , , , , , , , , -

I ...... \ s;

\

lo r 5 !

l L

I

2 t

i pp; 1 0.01L- II ' ' ' ' " ' I , , , !,ill1 Fig. 13. The spatial covariance function for rich Aclusters.

lo r WDC 1

Acknowledgements. Wewish to thankN. A. BAHCALL, R. A. SUNYAEV, Y. B. ZELDOVICH, J. MATERNE and G. CHIN- CARINI for the stimulating discussions. Our thanks are due also to A. I. KOPYLOV, T. S. FETISOVA and V. F. SHVARTS- MAN for the permission to use their redshifts of galaxies in A-clusters before they have published them, and to D. STEFANOVA for help in processing the data.

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Address of the authors:

M. KALINKOV, I. KUNEVA, K. STAVREV Department of Astronomy and National Astronomical Observatory Bulgarian Academy of Sciences 72 Lenin Blvd. Sofia 1784 Bulgaria