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Germany Observatory ab out we can study aredivide outlining these voids voids into with smaller mean voids The diameters faintest of cal Coma and the ern Local p ervoids we have studied theinvestigate closest the example distribution the ofwhich galaxies do in not and containAbstract around rich clusters su of galaxies In order to Accepted for publication February 4 3 2 1 ronment are smaller than voids in lowdensity regions The Send of voids in dierent environments ical type and luminositylogue and of analyze voids dened mean byand diameters galaxies other of of galaxies dierent in morpholog ing two the pro jections distribution We of present p a o or cata clusters Ulrich Lindner ing largescale structureKey of words the ronment shows that voidsluminosity form of a galaxies hierarchical system as well as on the Northern diameter of up to of galaxies and by astro-ph/9503044 ies 10tains Marcovered by available surveys 1995 Your thesaurus co des are will b e inserted byAA hand manuscript later no Europ ean Southern Observatory DSpace Garching Germany Telescope Europ eanTartu Co ordinating Astrophysical Facility Observatory D EE Estonia Garching Toravere Germany University Observatory D GottingenGermany void walls The This analysis shows that sizes of voids and prop erties We nd that this sup ervoid is not empty but it con We small oprint dep endence study h Structure Sup ervoids 1 cosmology observations galaxies cluster are Mp c Geismarlandstr It is dened as the region b etween the Lo the requests 1 related Jaan Einasto bright elliptical galaxies have a mean cosmography Lo cal Voids lo cated in a highdensity envi systems of h are void 1 to Voids Mp c regions diameters and sup erclusters of Ulrich Sup ervoid dened of Faint latetype p o or 2 Sup ervoids Maret Einasto in this D of galaxies elliptical LindnerUniversity the clusters on by void largescale envi lo cal the p o or which Gottingen by type of Universe galaxies clusters analyz North is well galax and 2 Wolfram Freudling Einasto et al is lo cated al The Bo otes void iscal lo cated example at of a a distance sup ervoid of is ab out the Bo otes void Kirshner et sup ercluster al Szomoru Gorkomal and Gregg Peimbert have Szomoru TorresPeimbert b een et found Balzanocompletely empty In the Weedman Bo otes void Zmo o dy a et number of galaxies the ters and the Hercules sup erclusterComa From sup erclusters the and southern by side walls b etween these sup erclus EEG est sup ervoid thep ervoid Northern describ ed by Bahcallsup ercluster Einasto and Soneira Joeveer Saar Thened and clos voids the are su the Perseus void b ehind the PerseusPisces Hercules and EETDA The has a diameter of at least galaxies eg Joeveer b een discovered even in early redshift surveys of galaxies The Introduction rich clusters ies Gregory and Tully Fisher Further stud p ervoids cluster Figure a inition of voids sup ervoids This denition is analogous to the def I Previous Catalogues show NLV h presence h 1 Void 1 cf and were reaches that Mp c Corona Mp c sup erclusters Einasto studies and sup erclusters This void is b ordered outline of voids in the the EETDA presented from us of Einasto Tarenghi et the 3 Borealis Hierarchy clusterdened mean diameter of sup ervoids largest 4 et Klaus Fricke have the Cetus al which This regions by shown Einasto voids and wall and distribution of galaxies has Batuski h Lo cal void hereafter void consist of galaxies We call these Bo otes 1 are that in front of the of hereafter Mp c Other cluster de sup erclusters et is by the Lo cal and the sup ervoids 1 those and Erik Tago the al ASTROPHYSICS surrounded sup ervoids and of sup erclusters in rich Shapley al Tit EETDA delineated Burns AND the Szomoru et clusters hereafter A Hercules is ab out and are by sup er classi NLV and and not the su by of 2

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Fig a Distributi on of rich clusters of galaxies and strong radio galaxies in sup ergalactic co ordinates Clusters of galaxies

which are members of rich sup erclusters with at least members cf EETDA are plotted as lled circles lo cated in interval

1 1

 X h Mp c or op en circles lo cated in interval  X h Mp c dierent notation enables to

distinguish members of the Shapley sup ercluster isolated clusters and clusters in p o or sup erclusters are plotted as small dots

radio galaxies as crosses The zone of avoidance is given by dashed lines Identications of some sup erclusters are given CB

stands for AC for AquariusCapricornus Box shows the contour of the region of NLV plotted in Figure b

al The same has b een found for the void near the The Hercules region has attracted the attention of as

PerseusPisces sup ercluster Weinberg et al and tronomers since Shapley discovered the Hercules

in the Northern Lo cal Sup ervoid Freudling et al sup ercluster The spatial structure of the lowdensity re

These studies show that dwarf galaxies in large voids tend gion in front of the Hercules sup ercluster was studied by

to form small systems In particular Szomoru et al have Joeveer Einasto Zeldovich Einasto and Shan

shown that galaxies in the Bo otes void cluster to form darin Einasto et al and Freudling Haynes

groups and laments of galaxies and Giovanelli This lowdensity region was inde

p endently discovered and studied by Tully

Tully et al by the CfA team de Lapparent et

The aim of the present pap er is to investigate the de

al Geller and Huchra Vogeley Geller

tailed structure of the area of the Northern Lo cal void

and Huchra Vogeley et al ab and by

which will serve as an example of prop erties of sup ervoids

Shaver More recent redshift surveys of the area

to guide future investigations of sup ervoids We investi

include Freudling et al and Tarenghi et al

gate in detail the prop erties of voids in the Northern Lo

These studies have shown that the region of NLV is lled

cal sup ervoid and in the neighboring sup erclusters the

with a network of faint galaxy systems which divide the

Lo cal Coma and Hercules sup erclusters The closeness of

sup ervoid into smaller voids see Figure b Mean diame

this sup ervoid and the surrounding sup erclusters gives us

ters of galaxydened voids are times smaller than

a unique p ossibility to compare structures of galaxies in

diameters of sup ervoids These systems are p opulated

low and highdensity environment Dierences if found

mostly by spiral galaxies Tago et al

give imp ortant information on the galaxy formation in various environments

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Fig b Distribution of galax

ies in sup ergalactic co ordinates

1

in a sheet  X h Mp c

Bright galaxies M 

are plotted with lled circles

fainter galaxies M 

with op en circles Solid

lines show equidensity contours

found by Gaussian smo othing of

the density eld with disp ersion

1

h Mp c cf Section

The mean density is denoted

as Highdensity regions 

corresp ond to sup erclusters

lowdensity regions to

the lo cal sup ervoid intermedi

ate density regions 

to density enhancements in

the sup ervoid the sup ercluster

outskirts and the walls b etween

them We see the Lo cal the

Coma and the Hercules sup er

clusters in the lower left lower

right and upp er right part of

the gure resp ectively The in

termediate density region in the

right part of the gure b etween

the Coma and the Hercules su

p erclusters b elongs to the Great

Wall

in the direction of the NLV and Coma and Hercules sup er Recently several catalogues of galaxydened voids

clusters In x we present a void catalogue in this region have b een published Kaufmann and Fairall here

of sky In x we analyze the distribution of void diameters after KF HaqueCopilah and Basu hereafter HB

dened by p o or clusters of galaxies and by galaxies of dif and Slezak et al hereafter SLB In these catalogues

ferent morphological type and luminosity In x we discuss however no distinction has b een made b etween voids de

our results The pap er ends with a summary of principal ned by dierent types of ob jects By contrast in this

conclusions work we emphasize a comparison of prop erties of voids

dened by galaxies of various luminosity and morphol

Throughout this pap er we use a Hubble constant of

1

1

ogy lo cated in dierent environments Thereby we con

H h km s Mp c

rm previous results by EEG and Einasto et al

hereafter EEGS suggesting that the void analysis char

acterizes some prop erties of the distribution of galaxies

The data

b etter than other statistics esp ecially in lowdensity re

The Hercules sup ercluster covers a large area of the sky

gions EEG and EEGS studied the void prop erties in the

north of the celestial equator in the right ascension range

direction of the Coma sup ercluster applying mean void

h h

b etween Centers of the Lo cal and Coma

diameter statistics and the void probability function us

h h

sup erclusters lie also in the region b etween

ing redshift data available in the late s In the present

 

and Strong galactic obscuration starts at

pap er we shall use a much deep er and larger dataset and

h

Therefore we select for our study the region

combine the quantitative analysis with a cosmographic de

 h h

and The main b o dy of the Her

scription of the galaxy distribution

cules sup ercluster lies b etween and

1

km s Thus we considered only redshifts up to The pap er is organized as follows In x we describ e

1

km s We use cubic samples of clusters and galaxies up the data used In x we present a cosmographic descrip

1

to size L h Mp c in analyzing data we take into tion of the distribution of galaxies and clusters of galaxies

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

account the fact that in far corners of these cub es data ters as describ ed by Einasto et al and Einasto et

are incomplete al

Distances to galaxies derived from observed redshifts

are expressed in redshift space To compare observed prop

Galaxies

erties of galaxies with mo dels it is necessary either to con

vert observational data to real space or simulated data

For galaxies we primarily have used the redshift compi

to redshift space Our observational samples are to o lim

lations by Huchra ZCAT for the northern sky



ited in volume to make the correction to the observational

Redshift from several sources have b een added to

data In order to estimate this eect on void size statistics

improve the completeness in the region of interest These

we have p erformed the necessary analysis for mo del simu

include several redshift surveys sp ecically targeting the

lations found by Frisch et al This analysis shows

region of the Hercules sup ercluster Freudling et al

that the reduction of mo del samples to redshift space in

Tarenghi et al kindly provided as a computer le

creases mean void diameters only by a few p er cent This

by Bianca Garilli An additional compilation of published

is much smaller than statistical errors of void sizes in sam

redshifts mostly of spiral galaxies with cm line mea

ples Thus in the void size analysis we can ignore the dif

surements was kindly provided by Martha Haynes Fi

ference b etween redshift and real space

nally unpublished cm line measurements in the region

of the Hercules sup ercluster obtained by Marcio Maia and

by one of us WF were added to the sample All sources

of redshifts were carefully cross correlated to avoid dupli

cate entries from several sources in our nal sample Only

galaxies up to an apparent magnitude of have b een

included in the sample The resulting sample contains a

total of more than galaxies within the region dened

ab ove

From this combined catalogue we selected a number

of subsamples with dierent magnitude limits and mor

phological type in order to investigate separately voids

dened by all galaxies and by elliptical galaxies only de

Vaucouleurs type T A summary of the observational

samples used is given in Table for further explanations

of Table cf Section

To assess the completeness of the sample it was com

pared to the Catalogue of Galaxies and of Clusters of

Galaxies Zwicky et al It was found that

Fig The histograms of the completeness factors in POSS

the compilation is complete up to an apparent magnitude

elds expressed as the fraction of galaxies in our redshift cat

for almost the whole region In the declination zones

 

alogue in units of the number of galaxies in the Zwicky et

our data are complete up to In the declina

   

al catalogue for galaxies up to apparent magnitude In

tion zones and the data are complete

some elds the fraction is larger than this means that the

up to in approximately half of the elds in the re

Zwicky catalogue itself is incomplete

maining area the completeness varies and is on average

h

The data are less complete b etween and

h

To visualize the completeness of our data we present

in Figure histograms of the completeness for each of the

Clusters of galaxies

Palomar Observatory Sky Survey POSS elds lo cated

In this study we are interested in the distribution of galax in the direction under study Because our data are com

ies and systems of galaxies of various richness from p o or plete up to Figure shows the incompleteness in the

apparent magnitude range from to

to rich clusters of galaxies Thus we used the Zwicky et

al catalogue of clusters which contains data Since our actual sample is only approximately com

plete to mag absolute magnitude limited subsamples

ab out rich and p o or clusters We used only clusters of

we use are not strictly volume limited The impact of the

the distance class Near Near clusters are dened

1

as having distances up to h Mp c Most of these incompleteness on the void analysis is investigated in Sec

clusters contain bright galaxies listed in various compi tion

lations of galaxy redshifts Huchras ZCAT and others To derive distances from observed redshifts we have

and therefore it is p ossible to identify bright members of corrected the redshifts for the solar motion galacto centric

Zwicky clusters on galaxy catalogues and use them to de ow the virgo centric ow and the velocity disp ersion in

termine distances of clusters This study has b een carried clusters of galaxies compression from our list of clus

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Table Cubic Samples

L Type M N M hD i N M N M N M

0 0 v v v

1 1 1

h Mp c mag h Mp c h Mp c

all

E

all

E

all

E

all

E

all

E

all

E

all

E

Cluster

out by Tago and we used his catalogue of redshifts Another p ossibility to present the distribution of

of Zwicky Near clusters of galaxies galaxies and clusters is the use of slices through a rect

For ab out p er cent of Zwicky Near clusters red angular grid The co ordinate system is dened as follows

h 

shifts were extracted from earlier cluster redshift compi The p ositive xaxis p oints towards and

h 

lations or from redshift data for individual galaxies In the p ositive y axis towards and and the



other cases redshifts for more than one galaxy p er cluster p ositive z axis towards

1

have b een obtained from ZCAT In these cases the un

In Figures and cub es with L h Mp c and

1

weighted mean of redshifts of individual galaxies in clus

L h Mp c are shown resp ectively In each Figure

ters was calculated In some cases there are several density

the z co ordinate is divided into six equal parts and the

concentrations in the cluster in these cases all individual

resulting sheets are plotted separately In Figure the

concentrations have were considered as separate clusters

distributions of clusters elliptical galaxies and all galaxies

lo cated at dierent distances

are given in separate panels In the cluster panels Ab ell

Our sample of the region considered contains

clusters are marked with lled circles and Zwicky clusters

Zwicky Near clusters of galaxies of them are also

with op en circles In Figure we plot only the galaxies

Ab ell clusters ie they b elong to the class of rich clus

and use dierent symbols for bins in absolute magnitudes

ters

The highest z panels are not plotted in Figures and

they contain much fewer galaxies than other panels and

no additional imp ortant information

Cosmography of Clusters and Galaxies in the

In Figures and and in the subsequent void size

NLV region

analysis we use absolute magnitude limited data Over the

Visual presentations of the dimensional

whole cub e the absolute magnitude limit is the same and

distribution

regions lo cated at dierent distances from us are repre

sented equally The absolute magnitude limit corresp onds

Before trying to apply statistical measures to the three

to an apparent magnitude limit of m mag at a dis

dimensional distribution of galaxies it is useful to get

tance L the cub e sidelength According to the formula

a visual impression rst Therefore in this section we

present two series of plots which show the structure in

m M logv sinb

slices through the region of the NLV

The rst series of plots shows the distribution of

galaxies in distance slices in equatorial co ordinates Fig we get M and for L

1

ure Dierent symbols are used for dierent morpho and h Mp c resp ectively In this formula the

1

logical types Plots are shown for redshift intervals of redshift v cz is given in km s The term sinb

1

km s takes into account the galactic extinction where b denotes

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Fig Distribution of galaxies in equatorial co ordinates at dierent distance intervals corresp onding to redshift intervals

1

cz km s Elliptical galaxies are plotted with lled circles spiral and irregular galaxies with op en circles

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

1

Fig Distribution of clusters and galaxies in rectangular equatorial co ordinates with cub e size h Mp c Cluster distri

bution is given in left panels that of elliptical galaxies in middle panels and of all galaxies in right panels The z co ordinate of

1

the cub e is divided into intervals of h Mp c In cluster panels Zwicky clusters are plotted with op en circles Ab ell clusters

with lled circles In the rightmost panels we show also voids from our void catalogue in Table a The circles are intersections

1

of the spherical voids with planes dened by z and h Mp c resp ectively

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

1

Fig Distributio n of galaxies in rectangular equatorial co ordinates with cub e size h Mp c The z co ordinate of the cub e

1

is divided into intervals of h Mp c In left panels only bright galaxies M  are plotted in middle panels bright

and intermediate bright galaxies  M  in right panels galaxies of all three luminosity class are plotted faint

galaxies  M  Bright intermediate and faint galaxies are plotted with lled large and small op en circles resp ectively

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

the galactic latitude In these gures we plot galaxies lo Wal ls between

cated in the whole cub e including those in the far corners

Figures and show a number of systems of galaxies

Because of this magnitude limit the relevant galaxy data

which form walls b etween sup erclusters

are almost complete in the sphere inside the cub e with

1 1

In the km s cz km s panel of Fig

the center at the origin of the co ordinate system and with

ure a long chain of galaxies can b e seen extending from

radius equal to the cub e size L

h  h 

and to ab out and This chain forms a con

Data on cubic samples built up according to the

nection b etween the Lo cal and Coma sup erclusters Tago

metho d describ ed ab ove are compiled in Table L is

Einasto and Saar

1

the side length of the cub e given in h Mp c and M the

0

1 1

In the km s cz km s panels of

corresp onding absolute magnitude limit M according to

Figure chains of galaxies extend from the

the formula ab ove N is the number of galaxies in the re

to left and up increasing and these galaxies b elong

sp ective cubic sample This number dep ends on M in

0

to the Great Wall Geller and Huchra The Great

the following analysis we use samples of galaxies brighter

Wall is clearly seen in Figure b as a strong chain of

than the limiting absolute magnitudes and

galaxies b etween the Coma and Hercules sup erclusters

The mean void diameter hD i will b e explained in

v

1

On distance intervals b etween km s and

Section

1

km s of Figure the Hercules sup ercluster re

gion clusters of galaxies are connected by systems of

giant galaxies of various richness and shap e

Superclusters of galaxies

In the lower left panel z of Figure we

see a very distinct chain of p o or clusters of galaxies ex

In the volume under study there are three sup erclusters

tending from the Lo cal Sup ercluster at the origin of the

the Virgo Coma and Hercules sup erclusters

1

co ordinate system to the Ab ell Cluster A a member

In the panel cz km s of Figure

of the Hercules Sup ercluster

we see in the lower right corner the the

These gures suggest that walls b etween sup erclusters

center of the Lo cal Sup ercluster The Virgo cluster is so

do not form large structureless clouds of galaxies but

large on the sky that it has not b een included into the

consist of numerous smaller systems of galaxies Ramella

Zwicky or Ab ell catalogue of clusters of galaxies In the

et al came to a similar conclusion These systems

next distance interval there are much fewer galaxies This

are less rich than similar systems in sup erclusters but

marks the outskirts of the Lo cal sup ercluster The Lo cal

contain more galaxies than systems in the sup ervoid region

sup ercluster is also well recognizable in the lower z panels

discussed in the next section

of Figures and near the co ordinates x and y

1

h Mp c

1 1

In the km s cz km s and

Systems of galaxies in the NLV

1 1

km s cz km s panels of Figure we

can recognize the Coma Sup ercluster The core of the su

In the region of the NLV b etween the Hercules the Lo cal

p ercluster is formed by the rich Ab ell cluster A which

and the Coma sup erclusters and the walls b etween these

is visible as an elongated cluster in the panel cz

sup erclusters there is a signicant number of small sys

In Figures and the Coma Sup ercluster is seen

tems of galaxies and p o or clusters The distribution of

1 1

at co ordinates x h Mp c and y h Mp c in

galaxies in the sup ervoid region is discussed in the fol

z and z panels

lowing

Galaxy enhancements cover most of the sky in pan

In the rst panels of Figure up to redshifts

1

1

els on distance intervals b etween km s and

km s we see the main b o dy of the Virgo cluster

1

1

km s of Figure In these distance intervals we

Starting from redshifts km s a number of small

see the Hercules sup ercluster It covers a very large area

systems app ear which consist mainly of faint galaxies

on the sky and is characterized by a number of rich clus

M brighter galaxies can b e seen in the group

h 

ters of galaxies cf the list of sup ercluster members in

of galaxies at and redshift interval

EETDA In Figure the Hercules sup ercluster is seen at

mentioned divide the region into a number

1 1

x h Mp c and y h Mp cpanel z

of voids in which we do not see any galaxies

here Ab ell clusters A A and A are lo cated

At the lower right corner of the panel

1

as well as a number of galaxy systems

km s of Figure we see galaxies forming a wall

b etween two voids see corresp onding region on the neigh In Figure a we also show the distribution of bright

b oring panels This wall b egins at the Coma sup ercluster radio galaxies These galaxies are lo cated only in high



area and crosses with another cloud of galaxies at density environment indicating the presence of sup erclus

A comparison of dierent panels shows that this wall con ters A high concentration of radio galaxies to nearby

sists of a thin net of laments of faint galaxies These l sup erclusters was noticed by Joeveer Einasto and Tago

aments are thinner than the kms slices shown here and Shaver

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

and therefore can b e seen with high contrast when slices tems to sup erclusters in which the morphology and lu

of kms depth are plotted minosity of galaxies are related to the prop erties of the

systems in which they are lo cated The void sizes in this

In Figure we plot the brightest galaxies M

hierarchy dep end on the ob jects which have b een used to

and the elliptical galaxies with the same absolute magni

detect the void voids dened by clusters are larger than

tude limit Giant galaxies show the strongest features in

galaxy dened voids and voids formed by elliptical galax

the distribution of galaxies The distribution of elliptical

ies are larger than voids dened by galaxies of all types

galaxies is similar to that of giant galaxies elliptical galax

ies are lo cated in denser regions and are absent from low

These observations motivated the more quantitative

density regions which consist mainly of fainter later type

approach of Section where we strengthen these impres

galaxies sions by calculations of mean void diameters The void

hierarchy will b e quantitatively conrmed by generating

In Figure the cub e size is half the cub e size of Figure

void catalogues for galaxy samples of dierent magnitude

and consequently the apparent magnitude limit allows to

limit

see fainter galaxies In Figure galaxies of all morpho

logical types with M are plotted in three panels

for each z sheet In the left panel galaxies are restricted

to absolute magnitudes brighter than M Those

Distribution of Voids

panels corresp ond to the right panels in Figure apart

from the dierent thickness of the z sheets In the mid

Method to nd voids

dle and right panels galaxies for fainter magnitude limits

The purp ose of our void catalogue is to help the study of

are plotted Dierent symbols are used for dierent abso

the dep endence of void parameters on the type of ob jects

lute magnitude intervals A comparison of dierent panels

used in the void denition The catalogue could also b e

for the same z sheet shows that not all of the faint galax

useful as a basis of searching faint dwarf galaxies in the

ies are members of structures formed by bright galaxies

vicinity of voids Voids can b e dened in a number of

Instead some of the fainter galaxies form new structures

ways and their prop erties dep end on this denition For

which consist only of faint galaxies A go o d example is a

this investigation we dene voids as regions which are

system of galaxies around the void in the z

1

completely empty of certain types of ob jects We used

panel with its center at ab out x and y h Mp c

a computerbased voidsearching algorithm to determine

This structure is completely absent in the leftmost panel

voids ob jectively Topological voids are connected here we

galaxies with M of the Figure Many other less

consider geometrical voids

pronounced but clearly recognizable voids and galaxy sys

tems can b e found by comparing corresp onding panels in

Three dierent algorithms have b een used in the liter

Figure This is a direct indication that structures de

ature to determine voids namely the empty cub e metho d

p end on the luminosity of galaxies p o or systems consists

by KF the empty sphere metho d by EEG and the

only of faint galaxies

smo othed density eld metho d using a wavelet technique

All these plots show that the NLV region contains by SLB The rst metho d nds completely empty cu

bic regions the third metho d nds low density regions

a network of galaxies and p o or clusters of galaxies We

Empty sphere and cub e metho ds yield very similar re

see b oth luminosity segregation b etween giant and nor

sults only the sizes of voids and void center co ordinates

mal galaxies and largescale morphology segregation b e

are slightly dierent The empty sphere metho d is more

tween elliptical and later type galaxies in systems far

from sup ercluster regions Einasto Mo et al sensitive to ne details of the underdense regions and

Faint galaxies tend to lie either in structures already out numerical results reect the void structure b etter The

smo othed density eld metho d is less sensitive to ne

lined by brighter galaxies or form new structures They

details in the distribution of galaxies in lowdensity re

do not form a smo oth p opulation of isolated galaxies in

gions In particular relatively isolated galaxies inuence

the voids Our gures do not show the distribution of ex

tremely faint dwarf galaxies but other studies Szomoru the smo othed density eld much less than parameters of

et al Hopp and references therein have shown completely empty voids thus the preference dep ends on

the purp ose of the study either the study of underdense

that up to now the search of dwarf galaxies in voids has

or of empty regions In our attempt to conrm the visual

given negative results

impression describ ed in Section we are mostly interested

To summarize this section the comparison of the dis

in completely empty regions and thus give preference to

tribution of all galaxies elliptical galaxies and Zwicky

the empty sphere metho d

clusters shows that most voids dened by clusters are seen

in the distribution of galaxies as well Clusters form knots The principal dierence of the present study from pre

in galaxy systems The richest clusters are lo cated in the vious ones lies in the fact that we take into account the lu

region of the Hercules and Coma sup erclusters galaxy minosity morphological type and environment of galaxies

systems inside the sup ervoid contain only p o or clusters while only little attention has b een given to these asp ects

We see a hierarchy of systems of galaxies from p o or sys b efore

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Visual insp ection of Figures shows that the consideration are isolated or touching An overlapping de

shap es of voids resemble irregular ellipsoids For simplicity gree means that the smaller sphere is completely

we identify voids by searching for empty spheres In the inside the larger one The normalization by the factor of

following we use the term void synonymous with empty minr r the diameter of the smaller sphere is used

1 2

sphere keeping in mind that our voids are only a more or to get values b etween and as a characteristic number

less rough approximation to the real or physical voids describing the degree of overlapping We give overlapping

which are very complex and irregular empty regions in in p er cent ie as numbers from to For example an

space To nd voids in a cubic sample we use the empty overlapping of means that the center of the smaller

sphere metho d by EEG In this metho d we divide our sphere lies exactly on the surface of the larger sphere

3

cubic samples into k cubic cells where k is the resolu

tion parameter The size l of the edge of such a cell is

The void catalogue

l Lk where L is the size of the whole cubic sam

ple First we calculate the distance to the nearest ob ject

With our void catalogue we intend to cover as much empty

galaxy or cluster for the centers of all cells and then nd

space with a reasonable small number of empty spheres as

lo cal maxima in this distance eld Maxima are lo cated in

p ossible According to our denition of voids this empty

centers of empty spheres resp ective distances corresp ond

space dep ends on the type of ob jects considered For our

to radii of empty spheres Void diameters determined with

further studies we use galaxies of all morphological types

this metho d describ e the extent of the smallest axis of a

but of three dierent absolute magnitude limits to dene

void

voids The void catalogue was constructed in several steps

Several problems connected with the application of the

as follows

empty sphere metho d have to b e taken into account First

In the rst step voids were determined with the empty

empty spheres found near the b orders of the cubic sample

sphere method applied to cubic samples of size L

1

may not corresp ond to centers of real voids b ecause ob

and h Mp c and corresp onding absolute magnitude

jects lo cated outside the cub e are not taken into account

limit M and Resolution parame

For this reason we dene a reduced cubic sample where

ters k and have b een used All

all those empty spheres that lie nearer to any cub e b or

the voids found for dierent L and k but with identical

der than the radius of the particular void are discarded

M were assembled into three lists one for each absolute

Also we discard all voids with centers outside the sphere

magnitude limit The purp ose of this pro cedure is to b e

of radius L around the observer

come indep endent of the resolution parameter The voids

We investigated the dep endence of our results on the

in those compilations were sorted according to their size

resolution parameter k The number center p osition and

in descending order and thus we get three void search lists

diameter of voids found dep ends on this parameter Typ

for our further pro ceeding Note that this list contains a

1

ical cell sizes are in the range of ab out to h Mp c

large number of more or less strongly overlapping voids

and typical void radii are in the range of ab out to

In the next step we use this list to nd a catalogue of

1

h Mp c Voids with radii much greater than the cell size

nonoverlapping voids which we will call parent voids

may b e counted more than once esp ecially in the case of

to simplify terminology for further explanations The rst

voids of irregular shap e we may obtain several voids with

and largest void in the void search list is taken as the

close centers The smaller the cell size l Lk the more

rst parent void Then we go through this list until we

often the same physical void may b e counted To reduce

meet the next void which do es not overlap with the rst

multiple void counting we apply an elimination algorithm

parent void this is the second parent void Then we

using the overlapping degree dened in the next section

go further through the list until we meet the next void

which do es not overlap with the two preceding parent

voids and so on With this pro cedure we compile a cat

The overlapping degree

alogue of all largest nonoverlapping voids from the void

search list

For further studies we dene the overlapping degree of

All the remaining voids are overlapping with one or

spheres as follows

more parents with dierent degree We call these voids

children and each of them will b e assigned to that par

r r a

1 2

ent void where the overlapping degree is largest Thus

r r a

1 2

parent voids together with their children are forming

r r a minr r

1 2 1 2

minr r

1 2

families of voids Parent voids do not cover the whole

empty space part of it is covered only by children voids

r r a minr r

1 2 1 2

To increase the volume of empty space covered by our void

where r and r are the radii of the resp ective spheres catalogue we include some children voids to the cata

1 2

and a is the distance b etween their centers An overlap logue by the following pro cedure The largest void among

ping degree equal to zero means that the spheres under the children of each family was identied and in case

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Table a Void Catalogue M Table b Void Catalogue M

No R D e No R D e

v v

1 1 1 1

h m h Mp c h Mp c h m h Mp c h Mp c

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

A B

B

B

B

B

when its diameter was considerably larger by a factor of

or more than the diameter of the parent void and if

the overlapping with the parent void was not very strong

The last step in the construction of the void catalogue less than this largest child has b een considered

is the determination of void parameters We take the cen as a new parent and was added to the void catalogue

ter and diameter of voids found by the previous steps as Note that these additional voids may overlap with several

the center and diameter of the whole void The ellipticity parent voids but with a smaller All the other chil

of the void is determined by the maximum value of all dren of the family were taken into account through the

segments drawn through the center of the parent void ellipticity of the parent void see b elow

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Table c Void Catalogue M Table Overlapping Voids Column void number

in the sample of brighter galaxies cf Table Column

void number in the sample of fainter galaxies The

No R D e

v

1 1

overlapping degree see Section with the void from

h m h Mp c h Mp c

brighter sample is given in parenthesis

C

C

C

A B

C

A B B

C

A B B

C

A B

C

A B

C

A B

C

A B B B B

C

A B B

C

A B B B B B

C

A B

C

A B B B B B

C

B

C

A B

A B

C

C

A C C C

C

A C

C

A C

C

A C C C

C

B C

C

B C

C

B C C

C

B C

C

B C

B C C

and the centers of all children voids The ellipticity e is

B C C

calculated by dividing the length of this maximal axis by

B C C

the diameter of the parent void

B C C

B C

B C

maxr r a

0

e

B C C

r

0

r is the radius of the parent void r is the radius of

0

any other void and a is the distance of centers b etween shown are intersections of the spherical voids with inter

1

this void and the parent void The ellipticity parame mediate planes dened by z and h Mp c

ter describ es the deviation of the real void from an exact resp ectively Due to pro jection eects those voids are

not completely empty

spherical shap e

The nal void catalogue is presented in Table Voids

are listed separately for three absolute magnitude limits

The environment of voids

of galaxies used in our void denition Voids found for the

brightest galaxy limit are lab eled A through A voids To determine the density of the environment of voids we

found for the next two magnitude limits B through B calculated the density eld for the cubic samples used in

and C through C We give equatorial co ordinates of our void determination As input data we used our abso

void centers distance R void diameter D ellipticity e lute magnitude limited galaxy samples The density eld

v

and the mean density at the lo cation of the void center number density of galaxies of all magnitudes in the re

The latter will b e explained in the next section The lim sp ective magnitude interval was calculated using Gaus

1 3

iting magnitude of galaxies used in the void denition is sian smo othing with a disp ersion of h Mp c for a

given in the header of Table Some voids from our void grid Exp eriments with numerical mo dels Frisch et al

catalogue Table a are plotted in Figure The circles show that this smo othing length characterizes the

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Table a Mean void diameters Dep endence on luminosity Diameters are given with rms errors is the disp ersion

v

of void diameters in each sample Mean diameters typed with b old face at the end of each column of absolute magnitude

limits corresp ond to complete samples for a given L

L Type M M M

hD i hD i hD i

v v v v v v v v v

1 1 1 1 1 1 1

h Mp c h Mp c h Mp c h Mp c h Mp c h Mp c h Mp c

all

E

all

E

all

E

all

E

all

E

all

E

all

E

Table b lies completely inside the larger void A D density eld on sup ercluster scales In Table we give the

v

1

h Mp c of the brighter sample Other examples density value at the lo cation of the void center in units

1

are B D h Mp c which is almost completely of the mean density These density values were used to

v

1

inside void A D h Mp c and and divide the voids into three environmental density classes

v

1

C D h Mp c which is almost inside B of high medium and low

v

1

D h Mp c and In these cases sys density

v

tems of faint galaxies enter into the void and make its

Examples of equidensity contours for sup erclusters high

diameter smaller

density regions with and the Northern Lo cal Su

In other cases a large void in sample A has b een split

p ervoid low density region with are shown in Fig

into several smaller voids in samples B or C For exam

ure b

1

ple A D h Mp c is split into C D

v v

1 1

h Mp c and and C D h Mp c and

v

The analysis of the void catalogue

Those two voids dened by the faintest galax

ies also show strong overlapping with void B D

v

In this section we compare voids dened by galaxies of

1

h Mp c B itself lies partly inside void A

dierent magnitude limits Ab out one third of the voids in

Many further examples of such an overlapping hierarchy

sample A can b e identied also in sample B To make such

can b e found from Table In these cases one or more

comparison easier we give in Table a list of overlapping

faint galaxy systems split the void into several subvoids

voids Because barely overlapping voids are of no interest

Here we see a full analogy with the structure of sup ervoids

void pairs with overlapping degrees less than p er cent

which are split by systems of p o or clusters and galaxies

are not included in the Table In the rst part of the Table

into smaller interconnecting subvoids

we list the names of voids of sample B overlapping with

voids of sample A in the next two parts we give similar

Diameters of Voids

information for voids of sample C In parenthesis we give

the overlapping degree as dened ab ove Only one pair of

Determination of void diameters

voids in Table overlaps p er cent indicating that void

B lies completely inside of void A

For a more detailed investigation of the dep endence of

the void prop erties sp ecically of their diameter on the Tables and show that in sample B the mean diam

type of ob jects used to dene the void the same void eter of voids is smaller than in sample A and in sample C

analysis as describ ed in Section was applied to samples it is smaller than in sample B In the example mentioned

1

of clusters of galaxies galaxies of all morphological types ab ove the smaller void B with D h Mp c see v

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

and separately to elliptical galaxies We used cub e sizes essary to use data obtained with identical void denition

1

L and h Mp c and the corresp onding algorithms

absolute magnitude limits M and

To investigate the sensitivity of the results to the

Results

resolution parameter k we used k

and with resulting cell sizes in the range of to

Mean void diameters of galaxy samples for all cub e sizes

1 1 1

h Mp c for L h Mp c to h Mp c for

used are given in Table separately for voids dened by

1 1

L h Mp c and to h Mp c for L

all galaxies and by elliptical galaxies as well as by p o or

1

h Mp c Additionally we used subsamples with L

clusters of galaxies Nearby galaxy samples contain b oth

1

and h Mp c and resp ective absolute magnitude

faint and bright galaxies thus for nearby samples it is

limits

p ossible to calculate void diameters using several absolute

magnitude limits In Table a the mean diameters of voids

For each galaxy sample we determined the void sam

hD i are given for various absolute magnitude limits M

ple using the same void selection pro cedure as discussed

v 0

and for dierent cub e sizes L We computed the scatter

ab ove We kept only voids lying inside a sphere with the

of mean void diameters From this we estimated errors

center at the origin of the co ordinate system and radius

v

of the mean void diameters as

equal to the cub e size L Voids lo cated closer to the sam

v

ple b oundaries than the radius of the void were eliminated p

N

v v v

Mean void diameters were calculated in two steps First

the mean value was calculated from voids found with the

where N is the number of voids used in the determina

v

empty sphere metho d using the corresp onding resolution

tion of the mean value Void diameter scatter and mean

parameter k In the second step the mean value for all k

diameter errors are also given in Tables and

was calculated

In Figure we show the cumulative distribution of void

To study the sensitivity of our results to details of the

diameters for some samples In Figure a the cumulative

elimination pro cedure we calculated also mean void diam

distribution of diameters of voids dened by p o or clusters

1 eters using a constant distance from the sample b oundary

within the h Mp c cubic sample is plotted for dif

In rst approximation we to ok the mean radius of the

ferent resolutions k We see that the scatter of individual

1

void to b e h Mp c the mean void radius of galaxy

curves is rather small and there is no systematic dep en

samples We found that mean void diameters are almost

dence of the distribution on k Similar plots for galaxy

the same for variable and constant distant from sample

samples show no systematic dierences for k Thus we can

b oundaries used to reduce void catalogues We carried

use the mean value for all k as a reliable distribution curve

out analogous studies for a cluster sample with cub e size

of void diameters

1

L h Mp c and three dierent distances

The cumulative frequency distribution of void diame

1

and h Mp c from the cub e b orders to eliminate

ters in all galaxy and samples are shown

voids The results do not show any signicant dierences

in Figures b and c resp ectively The void diameter dis

in the number and mean diameter of voids dep ending on

tribution is calculated for galaxy samples with absolute

the distance to the cub e b orders

magnitude limits and

To reduce multiple counting of voids we used the fol

lowing pro cedure lo ok for all pairs of empty spheres cal

Dependence on morphology and luminosity

culate the overlapping degree and if then

the sphere with the smaller radius is discarded For com

Mean void diameters of samples of various cub e sizes L for

parison we also calculated the mean diameter of all voids

galaxies of dierent morphological type and dierent lu

including all overlapping voids The results show that the

minosity are listed in Tables and In some samples the

distribution of void diameters in b oth samples are rather

number of elliptical galaxies was to o small to calculate the

similar and thus the elimination of overlapping voids only

mean void diameter in these cases no void diameter value

1

slightly reduces the mean void diameter The deviation in

is given In the case of the cub e sizes L h Mp c

1 1

creases with increasing resolution parameter k and reaches

h Mp c and h Mp c the values with the abso

up to This deviation is much smaller than the spread

lute magnitude limit corresp onding to these distances are

of mean diameters inside a particular void sample

printed b old face in Table a A graphical presentation of

the results is given in Figure

The overlapping void elimination pro cedure used in

The comparison of diameters of voids dened by galax this section is dierent from the resp ective pro cedure used

ies of dierent morphological type and magnitude shows to construct our void catalogues in Section For this

that voids dened by luminous galaxies and galaxies of reason mean void diameters for full samples given in Ta

early morphological type tend to b e larger than voids de ble a do not exactly coincide with resp ective data for

ned by fainter galaxies and galaxies of all types The all galaxies given in Table b These dierences although

reason for this can b e seen in Figure which shows that small demonstrate that in comparing void sizes it is nec

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Fig Panel a shows the distribution of diameters of voids dened by clusters of galaxies for dierent resolution parameter

k Panels b and c show the distribution of diameters mean for dierent resolution parameter k of voids dened by all

galaxies and elliptica l galaxies dierent lines give the distribution of void diameters dened by galaxies of absolute magnitude

brighter than M and In panels d f we give the distribution of diameters of voids lo cated in dierent

environment for classicati on criteria we refer to Section separately for galaxies of dierent absolute magnitude limit

we recognize that the distribution of void diameters for intrinsically fainter galaxies form faint systems in voids

the bright elliptical galaxies resembles strongly that for delineated by bright galaxies Also faint galaxies p opu

Zwicky clusters late p eripheral regions of galaxy systems Therefore faint

galaxies form structures with smaller voids Similarly we

Figure shows that the scatter of void diameters also

can see in Figure that elliptical galaxies are concen

dep ends on the type of ob ject used in the void determina

trated to the centers of galaxy clusters or groups and to

tion The cumulative void diameter distribution for galax

the ridges of systems of galaxies and are absent in low

ies is steep er than for clusters indicating that the range

density p eripheral regions of systems of galaxies This

of void diameters is smaller than for clusters This result

eect is known from earlier studies as the largescale mor

is conrmed by the corresp onding values for L

v

1

phological segregation of galaxies Mo et al

and h Mp c in Table The relative rms scatter

calculated as over hD i from Table of void diameters

v v

A comparison of void distributions determined by

for clusters amounts to whereas the mean relative

galaxies Figures b and c and by clusters of galaxies

scatter of galaxy dened voids for all galaxy samples in

Figure a suggests a dep endence of mean void diameters

Table is

on the absolute magnitude and morphology of galaxies

The overall range of mean void diameters dened by The distribution curves for elliptical galaxies are shifted

1

all galaxies ranges from to h Mp c and from to to the right compared with the curves for galaxies of all

1

h Mp c for voids dened by elliptical galaxies This types indicating that diameters of voids outlined by el

range is much larger than the statistical error of mean liptical galaxies are larger Comparing Figure a and c

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Fig Panel a shows the dep endence of mean diameters of voids dened by all galaxies with absolute magnitude limits

M and on the cub e size of the sample Panel b shows the dep endence of mean diameters of voids dened

by elliptical galaxies with absolute magnitude limits M and on the cub e size of the sample In the case

of lled symbols the luminosity limit corresp onds to a complete sample for a given size Error bars indicate the rms error of

void diameters

void diameters thus the eect is statistically signicant absolute magnitude limit simultaneously then we see a

When we compare voids dened by galaxies of dierent more rapid increase of void sizes with sample size This

absolute magnitude inside the same parent sample ie eect the selfsimilarity of voids has b een describ ed by

values in the same row of Table a we get a smaller range several authors EEG KF HB

But again the range of void diameters is much larger than

Due to the dep endence of the void size on the luminos

the statistical errors In section we will argue that the

ity of galaxies an increase of the void size with increasing

dierences in mean void diameters are not only due to

distance is exp ected since samples taken at larger distance

dierences in number density of galaxies in samples

use a higher luminosity limit Our analysis shows that the

luminosity eect only partly explains the observed self

similarity of voids Another p ossibility is that the eect is

Dependence on the sample size

due to biases b ecause in samples taken in cub es of larger

size voids of larger diameter can b e detected In this case

Next we consider the dep endence of the mean void size

the void diameter distribution must have a tail towards

on the sample size Each column in Table a gives mean

large diameters Figures b and c show that this is not

void diameters for some subsamples of a given absolute

the case We are left with the p ossibility that void sizes

magnitude limited sample The results listed in Table a

dep end on the largescale environment nearby samples

are plotted in Figure for all galaxies in Figure a and

are lo cated in the Lo cal sup ercluster and with increasing

for elliptical galaxies in Figure b Vertical bars indicate

distance our samples enter more into the sup ervoid region

the mean statistical error of void diameters

which is a region of larger voids as explained in the next

Figure demonstrates that there is a slow but contin

section

uous increase of the mean void size with the sample size

In Figure void diameters of samples taken at absolute

magnitude corresp onding to the completeness limit for a

Dependence on the environment

given sample size are plotted with solid symbols in Ta

ble a resp ective void diameters are printed b oldface We To investigate the dep endence of void sizes on the envi

see that if we change the two parameters sample size and ronment we have calculated the smo othed density eld

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

1

Gaussian smo othing with disp ersion h Mp c for the Table b Mean void diameters Dep endence on environ

1

three cubic samples used to generate the void catalogues ment hD iand are given in units h Mp c

v v

in Table a b and c cf Section To calculate the

cumulative distribution of void diameters we have divided

Env M M M

the voids in our void catalogues into three classes voids

hD i hD i hD i

v v v v v v v v v

lo cated in high medium and low density environment The

h

limiting density values b etween high and medium density

and medium and low density environment are and in

m

units of the mean density resp ectively In Figure the

l

void diameters are plotted as a function of the smo othed

all

density at the lo cation of the void center Mean diameters

of voids hD ifor these three environment classes are given

v

in Table b we also give the scatter and statistical er

v

the denition of highdensity regions but also systems of

ror of the diameters The cumulative distribution of void

galaxies are spaced more closely

diameters for these environmental classes is plotted in the

panels d e and f of Figure

Comparison with Poisson data

Our calculations show that voids lo cated in a highdensity

environment are smaller than voids lo cated in a low

A dep endence of void diameters on the absolute luminos

density environment The mean diameters dier by a fac

ity of galaxies is exp ected since the number density of

tor of Fainter galaxies determine smaller voids in all

bright galaxies is lower than that of fainter ones There

environments the smallest voids dened by faint galaxy

fore in order to investigate the signicance of the results

systems in highdensity environments have a mean diam

of the previous sections we have generated a number of

1

eter of only h Mp c The eect is much larger than the

random samples of dierent size and number of particles

statistical errors of void diameters or from the change in

which cover the range of sizes and number densities of real

number densities see next section

samples We computed the void size distribution and its

scatter using the same algorithm as for real samples Our

exp eriments show that the mean diameter of voids in a

Poisson sample is

D R

P oiss 0

where the Poisson radius R for a cubic sample of size L

0

is dened as

13

R L

0

N

where N is the number of ob jects in the cubic sample

The void diameters for a number of samples are plot

ted in Figure as a function of the number of particles

1

in cub es of size and h Mp c As a compari

son we show the equivalent void diameters as a function

of the number of particles for Poissonian samples In the

log N log D diagram of Figure the dep endence for

v

Poisson samples is given by straight lines of slop e

It can b e seen from Figure that for full samples sam

Fig The dep endence of void diameters D on the density

v

ples with faintest galaxies included and therefore maxi

at the void center Voids from our catalogue in Table a

mum N the mean void diameter exceeds the mean di

A A are marked with op en squares voids from cata

ameter of voids in Poisson samples by a factor of

logue Table b B B with op en triangles and voids from

catalogue Table c C C with op en circles

Galaxies of increasingly higher luminosity ie smaller N

dene voids of larger diameter Elliptical galaxy samples

and samples of brightest galaxies ie minimum N dene

The conclusion on the dep endence of mean void di voids with diameters approximately equal to diameters of

ameters on the largescale environment is an imp ortant voids in Poisson samples The dep endence of void diam

result of this study It means that in highdensity regions eters on the number of particles has however a dierent

not only systems of galaxies are richer this follows from slop e considerably lower than for Poisson samples

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

lows In elds where data are complete up to we have

randomly removed part of the galaxies in the apparent

magnitude interval to get samples which are

complete relative to full samples and

Using these diluted samples we have rep eated the void

analysis Results of the void analysis show that for the

absolute magnitude interval M for all cub e sizes

1

L h Mp c there is no systematic change of

mean void diameters for any of the dilution degree ap

plied For brighter absolute magnitude limits M

and there exists a marginal increase of mean void

1

diameters for larger samples L h Mp c by

and for dilution levels and of the

full sample resp ectively In samples of smaller size there

is practically no change of mean diameters

These calculations show that the incompleteness inu

ences our void diameters insignicantly Since we are in

terested basically in comparing of void diameters for sam

ples of dierent absolute magnitude we conclude that the

Fig Void diameters plotted as a function of the number of

dierential eect is negligible

particles N for samples taken in cub es of size and

1

h Mp c Straight lines are for Poisson samples of resp ective

size Op en symbols are for samples of galaxies of all types lled

Discussion

symbols for samples of elliptica l galaxies

In the classical treatment of voids the dep endence of void

The second dierence of real voids from Poissonian

sizes on the morphology and luminosity of galaxies is ig

ones lies in the distribution of void diameters the mean

nored This approach is equivalent to the assumption that

spread of void diameters in Poisson samples is in

the void distribution is nonhierarchical as in the HDM

units of the mean void diameters whereas in real samples

scenarios of The analysis of voids in

the spread is much larger for voids dened by galax

numerical mo dels by Frisch et al has shown that

ies and for clusterdened voids cf and

in this case voids dened by simulated galaxies have mean

determined in section The spread is largest in bright

sizes comparable to sizes of clusterdened voids in most

galaxy and cluster samples cf Table where mean diam

simulated and observed cluster samples This sort of void

eters of voids are close to the diameters of voids in Poisson

distribution has b een tacitly adopted by most previous

samples with the same number density We therefore con

void searches

clude that statistical prop erties of the identied voids are

In this classical approach voids dene a certain scale

dierent from voids found in Poissonian samples We

in the Universe that is almost indep endent on the type

note that the relative spread of void diameters is one of

of ob jects used in void denition Our study has shown

the statistics used by Dekel et al in the comparison

that in the observed samples there is no preferred scale

of real data with simulations and toy mo dels

1

of galaxydened voids all scales b etween h Mp c

1

Similar dierences b etween real void diameters and

and h Mp c are present in galaxydened voids

diameters of voids in Poisson samples were found ear

dep ending on the chosen absolute magnitude limit and the

lier by EEGS for characteristic void diameters dened by

largescale environment see Figure This gure shows

void probability functions For our study it is imp ortant

that the mean void diameters in observed samples increase

that these dierences increase when we consider samples

continuously with decreasing number of particles

of fainter galaxies We conclude that the void diameter

Similarly clusters of galaxies of dierent richness de

statistics as well as the void probability function charac

1

ne voids of dierent scale from h Mp c for Zwicky

terize certain prop erties of the galaxy distribution this

1

clusters of galaxies to h Mp c for Ab ell clusters

analysis is esp ecially sensitive to prop erties of the distri

of all richness classes

bution of galaxies in lowdensity regions

The concentration of bright galaxies esp ecially bright

elliptical galaxies to central parts of galaxy systems is

wellknown from earlier studies In particular the study

The inuence of the incompleteness of observational

of the correlation function and the p ower sp ectrum of

data

galaxies has shown that dierences in the spatial distri

In order to investigate the impact of the incompleteness bution of galaxies exist for intrinsically bright galaxies

of data on our void diameter statistics we pro ceed as fol M Hamilton Einasto Gramann and

U Lindner et al Void Hierarchy in the Northern Lo cal Sup ervoid

Einasto Our study of void diameters has demon We demonstrated that the whole voidlament struc

strated that dierences exist in the distribution of fainter ture is hierarchical voids dened by p o or clusters of

galaxies to o Vogeley et al b came to a similar con galaxies and by bright galaxies contain smaller voids

clusion dened by fainter galaxies Void sizes dep end crucially

on the prop erties of galaxies considered Therefore the

The principal mechanism which creates the hierarchy

type of ob jects dening voids must b e sp ecied if void

in void distribution is the presence of multibranching sys

prop erties are investigated

tems in the galaxy distribution Bright galaxies and p o or

Voids lo cated in highdensity environment in sup er

clusters form systems which split a large clusterdened

clusters are smaller than voids lo cated in lowdensity

void into smaller units These systems have smaller

branches made of fainter galaxies which dene voids of environment

even smaller scale The fainter the galaxies we considered

These are observed prop erties of the distribution of

the ner are systems they form Thus galaxies of dierent

galaxies and eventually a realistic galaxy formation sce

magnitude and morphological type dene voids of dierent

nario has to explain them

sizes Smallest sizes have voids dened by dwarf galaxies

in highdensity regions of sup erclusters their mean diam Acknowledgements We thank Drs Martha Haynes and

1

eters are only h Mp c Table Bianca Garilli for providing computer les of redshift mea

surements Marcio Maia for providing unpublished red

The galaxy sample used in this pap er is not complete

shift data and Peter Shaver for providing unpublished

to galaxies fainter than M From the ab ove dis

redshift data on radio galaxies JE and ME thank the

cussion one could extrap olate that the hierarchy of voids

sta of Europ ean Southern Observatory Potsdam Insti

continues to fainter galaxies and that fainter galaxies de

tute for Astrophysics and Gottingen University Obser

ne voids of still smaller sizes Available data on the dis

vatory for supp ort and very stimulating atmosphere that

tribution of dwarf galaxies do not supp ort this hypothesis

made this collab oration p ossible Fruitful discussions with

Bothun et al Szomoru et al Hopp and

Neta Bahcall Mirt Gramann and Enn Saar are acknowl

references therein ab out the search of dwarf galaxies in

edged This study was supp orted by Estonian Science

voids Very faint galaxies do not form a smo oth p opu

Foundation grant International Science Foundation

lation in voids delineated by brighter galaxies but are

grant LDC and a grant of the GottingenAcademy of

lo cated in the outskirts of systems of brighter galaxies or

Sciences We thank the anonymous referee for constructive

form systems consisting only of faint galaxies These sys

criticism

tems determine the lower limit of the hierarchical struc

ture

On the other hand structures formed by sup erclus

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