2010 Vol.XX No.X, 000–000

11institutetext: Central Department of Physics, Tribhuvan University,
Kirtipur, Nepal<EMAIL_ADDRESS>
22institutetext: Institut of Astro- and Particle Physics, Innsbruck
University, Technikerstrasse 25, A-6020 Innsbruck, Austria
Received [2010] [January] [14]; accepted [2010] [October] [23]

# Winding sense of galaxies around the Local Supercluster

B. Aryal 1122

###### Abstract

We present an analysis of the winding sense (S and Z-shaped) of 1 621 field
galaxies that have radial velocity between 3 000 km s-1 to 5 000 km s-1. The
preferred alignments of S- and Z-shaped galaxies are studied using chi-square,
auto-correlation and the Fourier tests. We classified total galaxies into 32
subsamples and noticed a good agreement between the position angle (PA)
distribution of S- and Z-shaped galaxies. The homogeneous distribution of the
S- and Z-shaped galaxies is noticed for the late-type spirals (Sc, Scd, Sd and
Sm) than that of the early-types (Sa, Sab, Sb and Sbc). A significant
dominance of S-mode galaxies is noticed in the barred spirals. A random
alignment is noticed in the PA-distribution of Z- and S-mode spirals. In
addition, homogeneous distribution of the S- and Z-shaped galaxies is found to
be invariant under the global expansion. The PA-distribution of the total
S-mode galaxies is found to be random, whereas preferred alignment is noticed
for the total Z-mode galaxies. It is found that the galactic planes of Z-mode
galaxies tend to lie in the equatorial plane.

###### keywords:

spiral galaxies – clusters: individual (Local Supercluster)

##  1 Introduction

Differential rotation in a galaxy’s disc generate density waves in the disc,
leading to spiral arms. According to gravitational theory, the spiral arms
born as leading and subsequently transform to trailing modes. With the passage
of time, the spiral pattern deteriorates gradually by the differential
rotation of the equatorial plane of the galaxy, but the bar structure persists
for a long time (Oort 1970a). This structure can again regenerate spiral
pattern in the outer region. Thus, a close relation between the origin of the
arms in the spirals and barred spirals can not be denied (Oort 1970b).

Land et al. (2008) studied the distribution of projected spin vectors of
$\sim$ 37 000 spiral galaxies taken from Solon Dizital Sky Survey. They did
not notice any evidence for overall preferred handedness of Universe. In a
similar study, Longo (2007) found evidence for a preferred axis. Sugai & Iye
(1995) used statistics and studied the winding sense of galaxies (S- and
Z-patterns) in 9 825 spirals. No significant dominance from a random
distribution is noticed. Aryal & Saurer (2005) studied the spatial
orientations of spin vectors of 4 073 galaxies in the Local Supercluster. No
preferred alignment is noticed for the total sample. These results hint that
the distribution of angular momentum of galaxies is entirely random in two-
(S- and Z-shaped) and three-dimensional (spin vector) analysis provided the
database is rich.

In order to understand true structural modes (leading or trailing) of spiral
galaxies, we need to know the direction of the spiral pattern (S- or
Z-patterns), the approaching and receding sides and the near and far parts,
since galaxies are commonly inclined in space to the line of the sight. The S
and Z-patterns can be determined from the image of the galaxy. Similarly, the
approaching and receding sides can be defined if spectroscopy data on rotation
is available. The third one is fairly hard to established. For this, Pasha
(1985) used ‘tilt’ criteria and studied the sense of winding of the arms in
132 spirals. He found 107 spirals to have trailing arm. Thomasson et al.
(1989) studied theoretically and performed $N$-body simulations in order to
understand the formation of spiral structures in retrograde galaxy encounters.
Interestingly, they noticed the importance of halo mass. They concluded that
the spirals having halos with masses larger than the disk mass exhibit leading
pattern. Thus, the makeup of galactic haloes is important to cosmology in
order to understand the evolution of galaxies.

By considering the group of transformations acting on the configuration space,
Capozziello & Lattanzi (2006) predicted that the progressive loss of
inhomogeneity in the S- and Z-shaped galaxies might have some connection with
the rotationally-supported (spirals, barred spirals) and randomized stellar
systems (lenticulars, ellipticals).

The preferred alignments of galaxies can be an indicator of initial conditions
when galaxies and clusters formed provided the angular momenta of galaxies
have not been altered too much since their formation. A useful property of
galaxies in clusters for which theories make different predictions is the
angular momentum distribution. The ‘Pancake model’ by Doroshkevich
Doroshkevich (1973), the ‘Hierarchy model’ by Peebles Peebles (1969) and, the
‘Primordial vorticity model’ by Ozernoy Ozernoy (1978) predict different
scenarios concerning the formation of large-scale structure. Thus, the study
of galaxy orientation has the potential to yield important information
regarding the formation and evolution of cosmic structures. In this work, we
present an analysis of winding sense and preferred alignments of galaxies that
have radial velocity (RV) 3 000 km s-1 to 5 000 km s-1. These are field
galaxies. We intend to study the importance of winding sense in order to
understand the true structural modes (i.e., leading and trailing arm) of the
galaxy. We expect to study the following: (1) Are the distribution of S- and
Z-shaped galaxies homogeneous in the field? (2) Is there any correlation
between the preferred alignment and the winding sense of galaxies? (3) Does
radial velocity dependence exist concerning winding sense of galaxies? and
finally, (4) What can we say about the distribution of true structural modes
(i.e., leading or trailing arm) of galaxies in the large scale structure?

This paper is organized as follows: in Sect. 2 we describe the method of data
reduction. In Sect. 3 we give a brief account of the methods and the
statistics used. Finally, a discussion of the statistical results and the
conclusions are presented in Sects. 4 and 5.

##  2 The sample: data reduction

Figure 1: (a) A sketch representing the winding sense (S or Z) of the galaxy.
(b) All-sky distribution of Z-mode ($\triangle$) and S-mode ($\circ$) galaxies
that have RVs in the range 3 000 km s-1 to 5 000 km s-1. The morphology (c),
radial velocity (d), axial ratio (e) and the magnitude (f) distribution of Z
and S-mode galaxies in our database. The statistical $\pm$1$\sigma$ error bars
are shown for the S-mode ($\bullet$) subsample. The dashed line (e) represents
the expected distribution.

Eighteen catalogues were used for the data compilation. A list of the
catalogues and their references are given in Table 1. The abbreviations given
in the first column of Table 1 are as follows: NGC - New General Catalogue,
UGC - Uppsala General Catalogue of Galaxies, ESO - ESO/Uppsala Survey of the
ESO (B) Atlas, IC - Index Catalogue, MCG - Morphological Galaxy Catalogue,
UGCA - Uppsala obs. General Catalogue, Addendum, CGCG - Catalogue of Galaxies
and Clusters of Galaxies, KUG - Kiso Ultraviolet Galaxy Catalogue, MRK -
Markarian Galaxy Catalogue, MESSIER - Catalogue des nebuleuses et des amas
d’etoiles, BCG - Brandner+Grebel+Chu Catalogue, LSBG - Low Surface Brightness
Galaxies, SBS - Second Byurakan Survey, LCRS - Las Companas Red Shift Survey,
DDO - David Dunlap Observatory Publications, IRAS - Infrared Astronomical
Satellite, SGC - Southern Galaxy Catalogue and UM - University of Michigan:
Curtis Schmidt-thin prism survey for extragalactic emission-line objects: List
I-V.

The NASA/IPAC extragalactic database (NED, http://nedwww.ipac.caltech.
edu/) was used to compile these catalogues. The main editing process was as
follows: first, galaxies having RVs in the range 3 000 km s-1 to 5 000 km s-1
were collected. We downloaded the image of all these galaxies from NED in FITS
format. The second step was to compile the morphology of these galaxies from
the catalog. A galaxy with doubtful morphology (eg., ‘S?’, ‘S0’ or ‘Sa’) is
omitted. Finally, the position angles (PAs) of galaxies were added from the
UGC, ESO, and Third Reference Catalogue of Bright Galaxies (de Vaucouleurs et
al. 1991).

Table 1: The list of catalogues used for the data compilation. The first
column lists the abbreviation of the catalogue. The second column gives the
total number of galaxies. The references are listed in the last column.
$\begin{array}[]{p{0.12\linewidth}rll}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr Catalogue&$N$&$References$\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr NGC&623&$Dreyer (1895,
1908) $\\\ UGC&276&$Nilson (1973)$\\\ ESO&123&$Lauberts (1982)$\\\
IC&93&$Dreyer (1895, 1908) $\\\ MCG&88&$Vorontsov-Vel'Yaminov et al.$\\\
&&$(1962-74)$\\\ UGCA&84&$Nilson (1974)$\\\ CGCG&75&$Zwicky et al.
(1961-68)$\\\ KUG&44&$Takase (1980-2000)$\\\ MRK&40&$Markarian (1967)$\\\
MESSIER&41&$Messier (1784)$\\\ BCG&28&$Brandner et al. (2000)$\\\
LSBG&23&$Impey et al. (1996)$\\\ SBS&17&$Markarian et al. (1983)$\\\
LCRS&16&$Shectman et al. (1996)$\\\ DDO&15&$Bergh (1959, 1966)$\\\
IRAS&15&$Infrared Astronomical Satellite$\\\ &&$(1983)$\\\ SGC&10&$Corwin et
al. (1985)$\\\ UM&10&$MacAlpine et al.$\\\ &&$(1977a,b,c; 1978; 1981)$\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$

There were two clusters Abell 0426 ($\alpha$(J2000) = 03h18m 36.4s,
$\delta$(J2000) = +41∘30’54”) and Abell 3627 ($\alpha$(J2000) = 16h15m32.8s,
$\delta$(J2000) = –60∘54’30”) in our region. These clusters have mean RVs 5
366 km s-1 (75 $\pm$ 5 Mpc) and 4 881 km s-1 (63 $\pm$ 4 Mpc), respectively
(Abell, Corwin & Olowin 1989, Struble & Rodd 1999). We removed the galaxies
belong to the cluster Abell 0426 using the catalog established by Brunzendorf
& Meusinger (1999). For the cluster Abell 3627 galaxies, we used Photometric
Atlas of Northern Bright Galaxies (Kodaira, Okamura & Ichikawa 1990) and
Uppsala Galaxy Catalogue (Nilson 1973). There were 174 galaxies belongs to
these clusters in our database. We remove these galaxies.

The RVs were compiled from Las Campanas Redshift Survey (Shectman 1996). The
PAs and the diameters of galaxies were added from the Uppsala Galaxy Catalogue
(Nilson 1973), Uppsala obs. General Catalogue, Addendum (Nilson 1974),
Photometric Atlas of Northern Bright Galaxies (Kodaira, Okamura, & Ichikawa
1990), ESO/Uppsala Survey of the European Southern Observatory (Lauberts
1982), Southern Galaxy Catalogue (Corwin et al. 1985) and Third Reference
Catalogue of Bright Galaxies (de Vaucouleurs et al. 1991).

In the NED, 6 493 galaxies having RVs 3 000 km s-1 to 5 000 km s-1 were listed
until the cutoff date. Morphological information was given in the catalogues
for 3 276 (50%) galaxies. We visually inspected all these galaxies using
ALADIN2.5 software.

The arm patterns (S- or Z-type) of the galaxies were studied visually by the
author in order to maintain homogeneity. The contour maps of the galaxies were
studied in order to identify their structural modes. For this, we used
ALADIN2.5 software. The Z-mode is one whose outer tip points towards the anti-
clockwise direction (see Fig. 1a). Similarly, the outer tip of S-mode directs
in the clockwise direction. These two patterns are obviously the two
dimensional projections of three dimensional galaxy. The re-examination of the
S- and Z-modes using MIDAS software resulted the rejection of more than 17% of
the objects. These rejected galaxies were nearly edge-on spiral and barred
spiral galaxies. As expected, it was relatively easier to identify the
structural modes of nearly face-on than that of nearly edge-on galaxies.

In this way, we compiled a database of 1 621 galaxies showing either S- or
Z-structural mode. There were 807 Z-mode and 821 S-mode galaxies in our
database. All sky distribution of Z- and S-mode galaxies is shown in Fig. 1b.
The symbols “$\circ$” and “$\triangle$” represent the positions of the S-mode
and Z-mode galaxies, respectively. Several groups and aggregations of the
galaxies can be seen in the figure.

The morphology, radial velocity, axial ratio and the magnitude distributions
of S- and Z-mode galaxies are shown in Figs. 1c,d,e,f. The spirals (47%)
dominate our database (Fig. 1c). However, a significant dominance of S-modes
are noticed in the barred spirals whereas a weak dominance of Z-modes are
found in the spirals. The population of galaxies in the RV distribution
($\Delta$RV = 5 00 km s-1) were nearly equal (Fig. 1d). The axial ratio
distribution shows a good agreement with the expected cosine curve in the
limit 0.2 $<$ $b$/$a$ $\leq$ 0.9 (Fig. 1e). The values of absolute magnitude
lie between 13.0 and 16.0 for 82% galaxies in our database (Fig. 1f).

We classified the database into 32 subsamples for both the S- and Z-modes on
the basis of the morphology, radial velocity, area and the group of the
galaxies. The galaxies with doubtful morphology are omitted in the spiral and
barred spiral subsamples. The total number of early- and late-type spirals or
barred spirals is much less than that of the total spiral or total barred
spirals. It is because of the fact that the galaxy with incomplete morphology,
say, simply ‘S’ or ‘SB’ can not be included in the subsamples. In other words,
the galaxy with morphology Sa, Sab, Sb, Sbc are included in the early spirals
whereas the galaxies with morphology Sc, Scd, Sd and Sdm are classified as
late type spirals. The galaxies having morphology other than Sa, Sab, Sb, Sbc,
Sc, Scd, Sd and Sm can not be included in the early and late subsamples. A
statistical study of these subsamples are given in Table 1 and discussed in
Sect. 4.1.

##  3 Method of analysis

Basic statistics is used to study the dominance of Z- or S-mode galaxies. At
first, morphology and RV dependence are studied. Secondly, sky is divided into
16 equal parts in order to observe deviation from the homogeneity. Several
galaxy groups are identified in the all-sky map where the structural dominance
are noticed. Finally, we study the dominance of Z- or S-mode galaxies in these
groups.

We assume isotropic distribution as a theoretical reference and studied the
equatorial PA-distribution in the total sample and subsamples. In order to
measure the deviation from isotropic distribution we have carried out three
statistical tests: chi-square, auto-correlation and the Fourier.

We set the chi-square probability P($>\chi^{2}$) = 0.050 as the critical value
to discriminate isotropy from anisotropy, this corresponds to a deviation from
isotropy at the 2$\sigma$ level (Godlowski 1993). Auto correlation test takes
account the correlation between the number of galaxies in adjoining angular
bins. We expect, auto correlation coefficient C$\rightarrow$0 for an isotropic
distribution. The critical limit is the standard deviation of the correlation
coefficient C.

If the deviation from isotropy is only slowly varying with angles (in our
case: PA) the Fourier test can be applied (Godlowski 1993). A method of
expanding a function by expressing it as an infinite series of periodic
functions (sine and cosine) is called Fourier series. Let $N$ denote the total
number of solutions for galaxies in the sample, $N$k the number of solutions
in the kth bin, $N$0 the mean number of solutions per bin, and $N$0k the
expected number of solutions in the kth bin. Then the Fourier series is given
by (taking first order Fourier mode),

$\begin{array}[]{l}N_{k}=N_{k}(1+\Delta_{11}\cos 2\beta_{k}+\Delta_{21}\sin
2\beta_{k}+......)\\\ \end{array}$ (1)

Here the angle $\beta$k represents the polar angle in the kth bin. The Fourier
coefficients $\Delta_{11}$ and $\Delta_{21}$ are the parameters of the
distributions. We obtain the following expressions for the Fourier
coefficients $\Delta_{11}$ and $\Delta_{21}$,

$\begin{array}[]{l}\Delta_{11}=\sum(N_{k}-N_{0k})\cos 2\beta_{k}/\sum
N_{0k}\cos^{2}2\beta_{k}\\\ \end{array}$ (2)
$\begin{array}[]{l}\Delta_{21}=\sum(N_{k}-N_{0k})\sin 2\beta_{k}/\sum
N_{0k}\sin^{2}2\beta_{k}\\\ \end{array}$ (3)

The standard deviations ($\sigma$($\Delta_{11}$)) and
($\sigma$($\Delta_{21}$)) can be estimated using the expressions,

$\begin{array}[]{l}\sigma(\Delta_{11})=(\sum
N_{0k}\cos^{2}2\beta_{k})^{-1/2}\\\ \end{array}$ (4)
$\begin{array}[]{l}\sigma(\Delta_{21})=(\sum
N_{0k}\sin^{2}2\beta_{k})^{-1/2}\\\ \end{array}$ (5)

The probability that the amplitude

$\begin{array}[]{l}\Delta_{1}=(\Delta_{11}^{2}+\Delta_{21}^{2})^{1/2}\\\
\end{array}$ (6)

greater than a certain chosen value is given by the formula

$\begin{array}[]{l}P(>\Delta_{1})=\exp(-nN_{0}\Delta_{1}^{2}/4)\\\
\end{array}$ (7)

with standard deviation

$\begin{array}[]{l}\sigma(\Delta_{1})=(2/nN_{0})^{1/2}\\\ \end{array}$ (8)

The Fourier coefficient $\Delta_{11}$ gives the direction of departure from
isotropy. The first order Fourier probability function $P$($>$$\Delta_{1}$)
estimates whether (smaller value of $P$($>$$\Delta_{1}$) or not (higher value
of $P$($>$$\Delta_{1}$) a pronounced preferred orientation occurs in the
sample.

##  4 Results

First we present the statistical result concerning the distribution of Z- and
S-mode galaxies in the total sample and subsamples. Second, we study the
distribution of Z- and S-mode galaxies in the unit area of the sky and in the
groups. Then, the equatorial PA-distribution of galaxies in the total sample
and subsamples are discussed. At the end, a general discussion and a
comparison with the previous results will be presented.

###  4.1 Distribution of Z and S mode galaxies

A statistical comparison between the total sample and subsamples of the Z- and
S-modes of galaxies is given in Table 2. Fig. 2 shows this comparison in the
histogram. The $\Delta$(%) in Table 1 and Fig. 2 represent the percentage
difference between the number Z- and S-mode galaxies. We studied the standard
deviation of the major diameters ($a$) of galaxies in the total sample and
subsamples for both the Z- and S-modes. In Table 2, $\Delta(a\,$sde$)$
represents the difference between the standard deviation of the major
diameters of Z- and S-mode galaxies.

An insignificant difference (0.4% $\pm$ 0.2%) between the total number of Z-
and S-mode galaxies are found (Table 2). The difference between the standard
deviation of the major diameters ($\Delta(a\,$sde$)$) of the Z- and S-mode
galaxies is found less than 0.019 (eighth column, Table 2). Interestingly, the
sum of the major diameters of total Z- and S-mode galaxies coincide. This
result suggests the homogeneous distribution of Z- and S-mode field galaxies
that have RV in the range 3 000 km s-1 to 4 000 km s-1.

In Fig. 2, the slanting-line (grey-shaded) region corresponds to the region
showing $\leq$ 10% (5%) $\Delta$ value. Almost all subsamples lie in this
region, suggesting the homogeneous distribution of Z- and S-mode galaxies
within 10% error limit. Now, we present the distribution Z- and S-mode
galaxies in the subsamples classified according as their morphology, RVs, area
and the groups below.

####  4.1.1 Morphology

In the spirals, Z-mode galaxies are found 3.7% ($\pm$1.8%) more than that of
S-mode. The homogeneous distribution of Z- and S-modes is found for the late-
type spirals (Sc, Scd, Sd and Sm) than that of early-type (Sa, Sab, Sb and
Sbc): $\Delta$ value turned out to be 9.5% ($\pm$4.8%) and 1.8% ($\pm$1.0%)
for early- and late-types (Table 2). Thus, no preferred winding pattern is
noticed in the late-type spirals than that of early-types.

Figure 2: The basic statistics of the Z and S-mode of galaxies in the total
sample and subsamples. The full form of the abbreviations (X-axis) are given
in Table 2 (first column). $\Delta$(%)=$S$$-$$Z$, where $S$ and $Z$ represent
the number of S- and Z-mode galaxies, respectively. The statistical error bars
$\sigma$(%) shown in the figure are calculated as: $\sigma$(%) =
$\sigma$/($\sqrt{S}$+$\sqrt{Z}$)$\times$100, where $\sigma$ =
($\sqrt{S}$-$\sqrt{Z}$). The grey-shaded and the slanting-line region
represent the $\leq$$\pm$5% and $\leq$$\pm$10% $\Delta$ value, respectively.
Table 2: Statistics of leading (column 3) and trailing arm (column 4) galaxies
in the total sample and subsamples. The fifth and sixth column give the
numeral and percentage difference ($\Delta$ = $S$–$Z$) between the S- ($S$)
and the Z- ($Z$) modes. The next two columns give the error: $\sigma$ =
($\sqrt{S}$–$\sqrt{Z}$) and $\sigma$(%) =
$\sigma$/($\sqrt{S}$+$\sqrt{Z}$)$\times$100\. The eighth column gives the
difference between the standard deviation (in arcmin) of the major diameters
($a$) of the S- and Z-modes galaxies ($\Delta$($a\,sde)$). The difference
between the sum of the major diameters ($\Delta$($a$)%) are listed in the last
column. The sample/subsample and their abbreviations are given in first two
columns. $\begin{array}[]{p{0.30\linewidth}rccrrrrrr}\hline\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr
sample/subsample&$symbol$&$Z$&$S$&$$\Delta$$&$$\Delta$(\%)$&$$\sigma$(\%)$&$$\Delta$($a$\,sde)$&$$\Delta$($a$)(\%)$\\\
\hline\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
Total&$T$&814&807&-7&-0.4&-0.2&0.019&0.0\\\
Spiral&$S$&395&367&-28&-3.7&-1.8&0.031&3.0\\\ Spiral (early-
type)&$SE$&150&124&-26&-9.5&-4.8&0.058&8.1\\\ Spiral (late-
type)&$SL$&131&126&-5&-1.9&-1.0&0.031&0.3\\\ Barred
Spiral&$SB$&191&269&78&17.0&8.5&0.062&15.2\\\ Barred Spiral (early-
type)&$SBE$&97&140&43&18.1&9.1&0.091&14.6\\\ Barred Spiral (late-
type)&$SBL$&68&83&15&9.9&5.0&0.066&9.2\\\ 3\,000$<$RV (km
s${}^{-1}$)$\leq$3\,500&$RV1$&208&194&-14&-3.5&-1.7&0.046&2.6\\\ 3\,500$<$RV
(km s${}^{-1}$)$\leq$4\,000&$RV2$&201&201&0&0.0&0.0&0.031&3.1\\\ 4\,000$<$RV
(km s${}^{-1}$)$\leq$4\,500&$RV3$&172&182&10&2.8&1.4&0.034&0.6\\\ 4\,500$<$RV
(km s${}^{-1}$)$\leq$5\,000&$RV4$&226&237&11&2.4&1.2&0.041&1.0\\\ Grid
1&$G1$&20&21&1&2.4&1.2&0.068&6.8\\\ Grid
2&$G2$&121&116&-5&-2.1&-1.1&0.014&1.6\\\ Grid
3&$G3$&88&112&24&12.0&6.0&0.076&9.3\\\ Grid
4&$G4$&12&14&2&7.7&3.9&0.647&9.7\\\ Grid
5&$G5$&11&8&-3&-15.8&-7.9&0.042&22.3\\\ Grid
6&$G6$&75&80&5&3.2&1.6&0.081&4.5\\\ Grid
7&$G7$&62&56&-6&-5.1&-2.5&0.095&8.9\\\ Grid
8&$G8$&22&33&11&20.0&10.1&0.073&12.9\\\ Grid
9&$G9$&20&31&11&21.6&10.9&0.028&18.1\\\ Grid
10&$G10$&124&108&-16&-6.9&-3.5&0.004&5.4\\\ Grid
11&$G11$&61&52&-9&-8.0&-4.0&0.025&6.2\\\ Grid
12&$G12$&20&20&0&0.0&0.0&0.409&7.4\\\ Grid
13&$G13$&78&66&-12&-8.3&-4.2&0.039&2.9\\\ Grid
14&$G14$&37&44&7&8.6&4.3&0.356&10.3\\\ Grid
15&$G15$&47&44&-3&-3.3&-1.6&0.050&5.2\\\ Grid
16&$G16$&9&9&0&0.0&0.0&0.191&8.6\\\ Group
1&$Gr1$&37&30&-7&-10.4&-5.2&0.032&7.1\\\ Group
2&$Gr2$&48&70&22&18.6&9.4&0.097&12.6\\\ Group
3&$Gr3$&31&37&6&8.8&4.4&0.027&4.3\\\ Group
4&$Gr4$&34&40&6&8.1&4.1&0.031&3.9\\\ Group
5&$Gr5$&107&85&-22&-11.5&-5.7&0.089&11.2\\\ Group
6&$Gr6$&42&45&3&3.4&1.7&0.024&1.6\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\hline\cr\end{array}$

A significant dominance of S-mode galaxies are noticed (17%$\pm$8.5%) in
spiral barred galaxies. The $\Delta$ value is found $>$ 9% for both early-
(SBa, SBab, SBb and SBbc) and late-type (SBc, SBcd, SBd, SBm) barred spirals.
Similar result (i.e., $\Delta$ $>$ 8%) is found for the irregulars and the
morphologically unidentified galaxies.

A similarity is noticed between the late-type spirals and barred spirals: the
$\Delta$ value for both the late-types are found to be less than that of
early-types (see Table 2).

The difference between the standard deviation of the major diameters
($\Delta(a\,$sde$)$) for S- and Z-mode galaxies is found less than 0.050 arc
minute for the total sample, spirals and the late-type spirals (eighth column,
Table 1). These samples showed $\Delta$ value $<$ 5% (grey-shaded region, Fig.
2a). Thus, we noticed a good correlation between the $\Delta$(%) and
$\Delta(a\,$sde$)$ value.

The difference between the sum of the major diameters (in percentage) are
found greater than 10% for the barred spirals and early-type barred spirals.
Interestingly, these two subsamples showed $\Delta$ value greater than 15%
(Fig. 2a). Thus, inhomogeniety in the distribution of S- and Z-mode galaxies
is noticed for barred spirals.

####  4.1.2 Radial velocity

A very good correlation between the number of S- and Z-mode can be seen in the
RV classifications (Fig. 1d). All 4 subsamples show the $\Delta$ and
$\Delta(a\,$sde$)$ values less than 5% and 0.050, respectively (Table 2). In
addition, $\Delta$($a$) is found to be $<$ 5%. This result is important in the
sense that the statistics in these subsamples is rich (number of galaxies $>$
170) enough. Thus, we could not observe preference structural modes (S or Z)
in the low and high RV galaxies in our database.

A difference is noticed: dominance of Z- and S- modes, respectively in low
(RV1) and high (RV3, RV4) RV subsamples. However, this dominance is not
significant (i.e., $\Delta$ $<$ 5%). An equal number of S- and Z-mode galaxies
are found in the subsample RV2 (3 500 $<$ RV (km s-1) $\leq$ 4 000) (Table 1).
In order to check the binning effect, we further classify the total galaxies
in 6 ($\Delta$RV = 333 km s-1) and 8 bins ($\Delta$RV = 250 km s-1) and study
the statistics. No significant dominance of either S- and Z-modes are noticed.

Thus, it is found that the homogeneous distribution of S- and Z-mode galaxies
remain invariant with the global expansion (i.e, expansion of the Universe).
We further discuss this result below.

####  4.1.3 Area

We study the distribution of S- and Z-mode galaxies by dividing the sky into
16 equal parts (Fig. 3a). The area of the grid (G) is 90∘ $\times$ 45∘ (RA
$\times$ Dec). The area distribution of S- and Z-mode galaxies are plotted,
that can be seen in Fig. 3a’. The statistical parameters are given in Table 2.

Figure 3: (a) All sky distribution of Z-mode (hollow circle) and S-mode
(hollow triangle) galaxies in 16 area grids. (a’) The histogram showing the
distribution of the Z- and S-mode galaxies in the grids G1 to G16. (b) Six
groups of the galaxies, represented by the grey-shaded region. (b’) The
distribution of Z- and S-modes in 6 groups. The statistical error bar
$\pm$1$\sigma$ is shown. The positions of the clusters Abell 0426 and Abell
3627 are shown by the symbol “$\times$” (a,b).

A significant dominance ($>$2$\sigma$) of S-mode is noticed in grid 3 (RA: 0∘
to 90∘, Dec: 0∘ to 45∘ (J2000)) (Fig. 3a,a’). An elongated group of galaxies
can be seen in this grid. In this grid, $\Delta$, $\Delta(a\,$sde$)$ and
$\Delta(a)\%$ are found to be 12% $\pm$ 6%, 0.076 and 9.3%, respectively.
These figures suggest that the distribution of S- and Z-mode galaxies in G3 is
not homogeneous. Probably, this is due to the apparent subgroupings or
subclusterings of the galaxies.

The S-mode galaxies dominate in the grids G8 and G9 (Fig. 3a’). However, the
statistics is poor ($<$ 40) in these grids (Table 2). In addition, no
groupings or subclustering are noticed.

A dominance ($\sim$1.5$\sigma$) of Z-mode is noticed in G10 (RA: 180∘ to 270∘,
Dec: –45∘ to 0∘ (J2000)) and G13 (RA: 270∘ to 360∘, Dec: –90∘ to –45∘ (J2000))
(Fig. 3a,a’). In both the grids, a large aggregation of the galaxies can be
seen. A subcluster-like aggregation can be seen in G10. An elongated structure
can be seen in G13. In both the grids, $\Delta$ value is found to be greater
than 5% (Table 2).

No dominance of either S- and Z-mode galaxies is noticed in the groups G1, G2,
G4, G5, G6, G7, G11, G12, G14, G15 and G16. Thus, homogeneous distribution of
S- and Z-mode galaxies is found intact in $\sim$ 80% area of the sky. We
suspect that the groupings or subclusterings of the galaxies lead the
preference structural modes (S or Z) in G3, G10 and G13.

####  4.1.4 Galaxy groups

In all-sky map, several groups of galaxies can be seen (Fig. 3a). It is
interesting to study the distribution of structural modes (S or Z) of galaxies
in these groups. For this, we systematically searched for the groups
fulfilling following selection criteria: (a) major diameter $>$ 30∘, (b)
cutoff diameter $<$ 2 times the background galaxies, (c) number of galaxies
$>$ 50\. We found 6 groups fulfilling these criteria (Fig. 3b). All 6 groups
(Gr) are inspected carefully. In 3 groups (Gr2, Gr5 and Gr6), subgroups can be
seen. The number of galaxies in the groups Gr2 and Gr5 are found more than
100.

The clusters Abell 0426 and Abell 3627 are located close to the groups Gr2 and
Gr6. The symbol “$\times$” represents the position of the cluster center in
Fig. 3b. The mean radial velocities of these clusters are 5 366 km s-1 and 4
881 km s-1, respectively. However, we have removed the member galaxies of
these clusters from our database.

A significant dominance ($>$2$\sigma$) of S-mode galaxies is noticed in the
group Gr2 (Fig. 3b,b’). The $\Delta$, $\Delta(a\,$sde$)$ and $\Delta(a)\%$
values are found to be 18.6% ($\pm$9.4%), 0.097 and 12.6%, respectively (Table
2). We suspect that the galaxies in this group is under the influence of the
cluster Abell 0426, due to which apparent subclustering of the galaxies is
seen.

The galaxies in Gr5 shows an opposite preference: a significant dominance of
the Z-mode galaxies ($>$2$\sigma$) (Fig. 3b,b’). In this group, $\Delta$,
$\Delta(a\,sde)$ and $\Delta(a)\%$ are found to be 11.5% $\pm$ 5.7%, 0.089 and
11.2%, suggesting inhomogeneous distribution of structural modes (Table 2).

No humps or dips can be seen in the groups Gr1, Gr3, Gr4 and Gr6 (Fig.
3.2b,b’). Thus, the distribution of S- and Z-mode galaxies in these groups are
found to be homogeneous. The number of galaxies in these groups are less than
100.

In the group 6, we could not notice the influence of the cluster Abell 3627.
This might be due to the off location of the cluster center from the group
center.

###  4.2 Anisotropy in the position angle distribution

Figure 4: The equatorial position angle (PA) distribution of total Z- and
S-mode galaxies plotted in 9 (a) and 18 (b) bins. The solid and the dashed
line represent the expected isotropic distribution for S- and Z-mode galaxies,
respectively. The observed counts with statistical $\pm$1$\sigma$ error bars
are shown. PA = 90∘$\pm$45∘ (grey-shaded region) corresponds to the galactic
rotation axes tend to be oriented perpendicular with respect to the equatorial
plane.

We study the equatorial position angle (PA) distribution of S- and Z-mode
galaxies in the total sample and subsamples. A spatially isotropic
distribution is assumed in order to examine non-random effects in the PA-
distribution. In order to discriminate the deviation from the randomness, we
use three statistical tests: chi-square, auto correlation and the Fourier. The
bin size was chosen to be 20∘ (9 bins) in all these tests. The statistically
poor bins (number of solution $<$ 5) are omitted in the analysis. The
conditions for anisotropy are the following: the chi-square probability
P($>\chi^{2}$) $<$ 0.050, correlation coefficient $C$/$\sigma(C)$ $>$ 1, first
order Fourier coefficient $\Delta_{11}$/$\sigma(\Delta_{11}$) $>$ 1 and the
first order Fourier probability P($>\Delta_{1}$)$<$0.150 as used by Godlowski
(1993). Table 3 lists the statistical parameters for the total samples and
subsamples.

In the Fourier test, $\Delta_{11}$ $<$ 0 (i.e., negative) indicates an excess
of galaxies with the galactic planes parallel to the equatorial plane. In
other words, a negative $\Delta_{11}$ suggests that the rotation axis of
galaxies tend to be oriented perpendicular with respect to the equatorial
plane. Similarly, $\Delta_{11}$ $>$ 0 (i.e., positive) indicates that the
rotation axis of galaxies tend to lie in the equatorial plane.

In the histograms (see Figs. 4-7), a hump at 90∘$\pm$45∘ (grey-shaded region)
suggests that the galactic planes of galaxies tend to lie in the equatorial
plane. In other words, the rotation axes of galaxies tend to be oriented
perpendicular with respect to the equatorial plane when there is excess number
of solutions in the grey-shaded region in the histogram.

All three statistical tests show isotropy in the total S-mode galaxies. Thus,
no preferred alignment is noticed for the total S-mode galaxies (solid circles
in Fig. 4a). Interestingly, all three statistical tests show anisotropy in the
total Z-mode galaxies. The chi-square and Fourier probabilities
(P$(>\chi^{2})$, P($>\Delta_{1}$)) are found 1.5% ($<$ 5% limit) and 8.5% ($<$
15% limit), respectively (Table 2). The auto correlation coefficient
(C/C($\sigma$)) turned –3.2 ($>>$1). The $\Delta_{11}$/$\sigma(\Delta_{11}$)
value is found to be negative at $\sim$ 2$\sigma$ level, suggesting that the
rotation axes of Z-mode galaxies tend to be oriented perpendicular the
equatorial plane. Three humps at 50∘ ($>$1.5$\sigma$), 90∘ ($>$2$\sigma$) and
130∘ (1.5$\sigma$) support this result (Fig. 4a). We checked the biasness in
the results due to bin size by increasing and decreasing the number of bins. A
similar statistical result is found for both structural modes. Fig. 4b shows
the PA-distribution histogram for the total sample in 18 bins. The Z-mode
galaxies show three significant humps in the grey-shaded region, supporting
the results mentioned above.

Thus, we conclude isotropy for S-mode whereas anisotropy for Z-mode galaxies
in the total sample.

Table 3: Statistics of the PA-distribution of galaxies in the total sample and
subsamples (first column). The second, third, fourth and fifth columns give
the chi-square probability (P$(>\chi^{2})$), correlation coefficient
(C/C($\sigma$)), first order Fourier coefficient
($\Delta_{11}$/$\sigma$($\Delta_{11}$)), and first order Fourier probability
P($>\Delta_{1}$), respectively. The last four columns repeats the previous
columns. $\begin{array}[]{p{0.1\linewidth}ccccccccc}\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr sample&&$S-mode$&&&&$Z-mode$&&\\\ \vskip 3.0pt plus 1.0pt
minus
1.0pt\cr&$P$(>\chi^{2})$$&$C/C($\sigma$)$&$$\Delta_{11}$/${\sigma}$($\Delta_{11}$)$&$P(${>}\Delta_{1}$)$&$P$(>\chi^{2})$$&$C/C($\sigma$)$&$$\Delta_{11}$/${\sigma}$($\Delta_{11}$)$&$P(${>}\Delta_{1}$)$\\\
\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
total&0.666&+0.0&-0.9&0.381&0.015&-3.2&-1.9&0.085\\\
S&0.511&-0.7&-1.2&0.434&0.225&+0.4&+0.8&0.383\\\
SE&0.973&+0.1&-0.9&0.569&0.031&+2.0&+2.8&0.015\\\
SL&0.234&+0.5&+0.8&0.209&0.460&-0.1&-0.5&0.345\\\
SB&0.729&+0.3&+1.0&0.454&0.285&-1.0&-0.2&0.497\\\
SBE&0.739&+0.1&-0.5&0.566&0.230&-0.7&+0.1&0.521\\\
SBL&0.043&+1.8&+1.7&0.046&0.620&-0.9&-0.2&0.872\\\
RV1&0.910&+0.3&+0.8&0.362&0.369&-0.9&-0.6&0.285\\\
RV2&0.790&+0.3&-1.0&0.496&0.925&-0.4&-0.2&0.887\\\
RV3&0.050&+1.6&-1.5&0.083&0.033&-1.8&-2.3&0.046\\\
RV4&0.043&-2.3&-1.5&0.116&0.636&+0.2&-0.7&0.692\\\
Gr2&0.455&+0.6&+0.8&0.861&0.033&-1.8&+1.7&0.116\\\
Gr5&0.033&-1.4&-2.0&0.085&0.516&+0.4&-0.4&0.548\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\end{array}$

####  4.2.1 Morphology

In the spirals, the chi-square and auto correlation tests show isotropy for
both the S- and Z-modes. The first order Fourier probability is found greater
than 35%, suggesting no preferred alignment. However, the $\Delta_{11}$ value
exceeds 1$\sigma$ limit (–1.2$\sigma$) in the S-mode spirals. A hump at 90∘ is
not enough to turn the $\Delta_{11}$/$\sigma(\Delta_{11}$) $>$ 1.5 (Fig. 5a).
Similarly, a hump at 150∘ is not enough to make the
$\Delta_{11}$/$\sigma(\Delta_{11}$) $>$ 1.5 in the Z-mode spirals. Hence, the
preferred alignment is not profounded in both the S- and Z-mode spirals. Thus,
we conclude a random orientation of S- and Z-mode spirals.

Figure 5: The equatorial PA-distribution of Z- and S-mode galaxies in the
spirals (a), early-type spirals (b), late-type spirals (c), barred spirals
(d), early-type barred spirals (e) and late-type barred spirals (f). The
symbols, error bars, dashed lines and the explanations are analogous to Fig.
4.

Early- and late-type S-mode spirals show isotropy in all three statistical
tests (Table 3). No humps and the dips are seen in the histograms (solid
circles in Fig. 5b,c). Thus, the S-mode spirals show a random alignment in the
PA-distribution. In the subsample SE, all three statistical tests show
anisotropy (Table 2). Two significant humps at $>$ 150∘ cause the first order
Fourier coefficient ($\Delta_{11}$) $>$ +2.5$\sigma$ (hollow circle in Fig.
5b). Thus, a preferred alignment is noticed in the early-type Z-mode spirals:
the galactic rotation axes tend to lie in the equatorial plane. The late-type
Z-mode spirals show a random alignment.

The spiral barred galaxies show a random alignment in both the S- and Z-modes.
In Fig. 5d, no deviation from the expected distribution can be seen. All three
statistical tests support this result (Table 2). A similar result is found for
the early-type SB galaxies in both structural modes (Table 3, Fig. 5e).

The P$(>\chi^{2})$ and P($>\Delta_{1}$) are found less than 5%, suggesting a
preferred alignment for the late-type S-mode SB galaxies (Table 3). The auto
correlation coefficient (C/C($\sigma$)) and the hump at $>$ 150∘ support this
result (Fig. 5f). The $\Delta_{11}$/$\sigma(\Delta_{11}$) is found to be
positive at 1.7$\sigma$ level, suggesting that the S-mode SBL galaxies tend to
lie in the equatorial plane. Thus, the late-type S- and Z-mode SB galaxies
show preferred and random alignments, respectively.

####  4.2.2 Radial velocity

The subsamples RV1 and RV2 show isotropy in all three statistical tests (Table
2). No humps or dips can be seen in Figs. 6a,b. Thus, the galaxies having
radial velocity in the range 3 000 km s-1 to 4 000 km s-1 show a random
alignment for both the S- and Z-mode galaxies.

The humps at 90∘ ($>$2$\sigma$) and 110∘ ($>$2$\sigma$) are found in the Z-
and S-mode RV3 galaxies, respectively (Fig. 6c). These two significant humps
lead the subsample to show anisotropy in the statistical tests (Table 2). The
$\Delta_{11}$ values are found negative at $\geq$1.5 level, suggesting a
similar preferred alignment for both modes: the galaxy rotation axes tend to
be directed perpendicular to the equatorial plane.

Figure 6: The equatorial PA-distribution of Z- and S-mode galaxies in RV1 (a),
RV2 (b), RV3 (c) and RV4 (d). The abbreviations are listed in Table 1. The
symbols, error bars, dashed lines and the explanations are analogous to Fig.
4.

A hump at 70∘ ($>$1.5$\sigma$) and a dip at 150∘ ($\sim$2$\sigma$) cause the
S-mode RV4 galaxies to show anisotropy in all three statistical tests (Fig.
6d). Thus, the S-mode galaxies having radial velocity in the range 4 500 km
s-1 to 5 000 km s-1 show a similar preferred alignments as shown by the
subsample RV3: galactic planes of galaxies tend to lie in the equatorial
plane. The leading arm galaxies in the subsample RV4 show a random alignment
(Table 3, Fig. 6d).

####  4.2.3 Groups

We do not study PA-distribution of S- and Z-mode galaxies in the groups Gr1,
Gr3, Gr4 and Gr6 because of poor statistics (number $<$ 50).

We study the PA-distribution of S- and Z-mode galaxies in the groups Gr2 and
Gr5, where the dominance of either Z- or S-mode is noticed. In addition, the
statistics is relatively better in these groups.

Figure 7: The equatorial PA-distribution of Z- and S-mode galaxies in the
groups Gr2 and Gr5. The abbreviations are listed in Table 1. The symbols,
error bars, dashed lines and the explanations are analogous to Fig. 4.

In the group Gr2, Z-mode galaxies dominate the S-mode galaxies. In this group,
the Z-mode galaxies show a preferred alignment whereas S-mode galaxies show a
random alignment in the PA-distribution. All three statistical tests suggest
anisotropy in the Z-mode galaxies (Table 3). The humps at $>$ 150∘ cause the
$\Delta_{11}$ value to be positive at $>$ 1.5$\sigma$ level (Fig. 7a),
suggesting that the rotation axes of Z-mode galaxies in Gr2 tend to be
oriented parallel the equatorial plane.

The S-mode galaxies dominate in the group Gr5. Interestingly, a preferred
alignment of S-mode galaxies is noticed in the PA-distribution. In Fig. 7b,
two significant humps at 90∘ ($\sim$2$\sigma$) and 110∘ ($>$2$\sigma$) can be
seen. These humps lead the subsample (S-mode Gr5) to show anisotropy in the
statistical tests (Table 3). No preferred alignment is noticed in the Z-mode
galaxies of this group.

Thus, the dominating structural modes (Z or S) show a preferred alignment in
the PA-distribution. This is noticeable result.

###  4.3 Discussion

Fig. 8 shows a comparison between the number ($\Delta$) and position angle
($\Delta_{11}$/$\sigma(\Delta_{11}$)) distribution of S- and Z-mode galaxies
in the total sample and subsamples. This plot deals the correlation between
the homogeneity in the structural modes and the random alignment in the
subsamples. The grey-shaded region represents the region of isotropy and
homogeneity for the $\Delta_{11}$/$\sigma(\Delta_{11}$) and $\Delta$(%),
respectively.

Figure 8: A comparison between the number ($\Delta$%) and the position angle
($\Delta_{11}$/$\sigma(\Delta_{11}$)) distribution of Z- and S-mode galaxies
in the total sample and subsamples.

Twenty five (out of 39, 64%) subsamples lie in the grey-shaded region (Fig.
8a), suggesting a good agreement between the homogeneous distribution of S-
and Z-mode galaxies and the random alignment of the rotation axes of galaxies.
In four subsamples (SE, SBL, Gr2 and Gr5), a good correlation between the
preferred alignment and the dominance of either S- or Z-mode galaxies is
noticed (Fig. 8a). Thus, it is noticed that the random alignment of the PAs of
galaxies hint the existence of inhomogeneity in the structural modes.

Aryal & Saurer (2006) and Aryal, Paudel & Saurer (2007) studied the spatial
orientation of galaxies in 32 Abell clusters of BM type I, II, II-III and III
and found a significant preferred alignment in the late-type cluster (BM type
II-III, III). They concluded that the randomness decreases systematically in
galaxy alignments from early-type (BM type I, II) to late-type (BM type II-
III, III) clusters.

We noticed a very good correlation between the random alignments and the
homogeneity in the structural modes. Probably, this result reveals the fact
that the progressive loss of homogeneity in the structural modes might have
some connection with the rotationally supported (spirals, barred spirals) to
the randomized (lenticulars, ellipticals) system. Thus, we suspect that the
dynamical processes in the cluster evolution (such as late-type clusters) give
rise to a dynamical loss of homogeneity in the structural modes. It would be
interesting to test this prediction by studying the S- and Z-type spirals in
the late-type clusters in the future.

As 60% of galaxies in the nearby universe are rotationally supported discs,
understanding angular-momentum acquisition is obviously a crucial part of
understanding galaxy evolution. The winding sense of spiral arm patterns
(morphological feature) allows us to infer the orientation of the angular-
momentum vector of the disc galaxy. The expected distribution of spin vectors
of galaxies shows markedly different trends according to the galaxy formation
scenarios. One can suspect the possibility that the actual distribution of
galaxy spin shows a dipole or a quadrupole component depending on the
scenarios of galaxy formation. If galaxy spins were generated according to the
primordial whirl scenario, a strong bias in either the S or Z patterns would
be seen in a face-on sample of galaxies. We did not find this trend in our
sample. A quadrupole distribution of S/Z might be observed if the primary
process was the generation of spins due to the pancake shock scenario or the
explosion scenario. On the other hand, if the galaxy spins were produced by
the tidal spin-up process, there would be no global anisotropy as we noticed
in many cases, unless galaxy-cluster tidal interaction rather than galaxy-
galaxy tidal interaction were the primary process. No significant correlation,
however, was identified in any ensemble.

##  5 Conclusions

We studied the winding sense of 1 621 field galaxies around the Local
Supercluster. These galaxies have radial velocity (RV) in the range 3 000 km
s-1 to 5 000 km s-1. The distribution of Z- and S-mode galaxies is studied in
the total sample and 32 subsamples. To examine non-random effects, the
equatorial position angle (PA) distribution of galaxies in the total sample
and subsamples are studied. In order to discriminate anisotropy from the
isotropy we have performed three statistical tests: chi-square, auto-
correlation and the Fourier. Our results are as follows:

  1. 1.

The homogeneous distribution of the total Z- and S-mode galaxies is found,
suggesting the homogeneous distribution of winding sense (S or Z) of galaxies
having RVs 3 000 km s-1 to 5 000 km s-1. The PA-distribution of S-mode
galaxies is found to be random, whereas preferred alignment is noticed for
Z-mode galaxies. It is found that the galactic rotation axes of Z-mode
galaxies tend to be oriented perpendicular the equatorial plane.

  2. 2.

Z-mode are found 3.7% ($\pm$1.8%) more than that of the S-mode in the spirals
whereas a significant dominance (17% $\pm$ 8.5%) of S-mode is noticed in the
barred spirals. This difference is found $>$ 8% for the irregulars and the
morphologically unidentified galaxies. A random alignment is noticed in the
PA-distribution of Z- and S-mode spirals. Thus, it is noticed that the random
alignment of the PAs of galaxies lead the existence of inhomogeneity in the
structural modes of galaxies.

  3. 3.

The inhomogeneity in the structural modes is found stronger for the late-type
spirals (Sc, Scd, Sd and Sm) than that of early-type (Sa, Sab, Sb and Sbc).
Similar result is found for the late-type barred spirals.

  4. 4.

A very good correlation between the number of Z- and S-mode galaxies are found
in the RV subsamples. All 4 subsamples show the $\Delta$ value less than 5%.
Thus, we conclude that the homogeneous distribution of structural modes of
field galaxies remain invariant with the global expansion.

  5. 5.

The galaxies having RVs 3 000 km s-1 to 4 000 km s-1 show a random alignment
for both the Z- and S-modes. The rotation axes of Z- and S-mode galaxies
having 4 000 $<$ RV (km s-1) $\leq$ 4 500 tend to be oriented perpendicular
the equatorial plane.

  6. 6.

The distribution of the winding sense of galaxies is found homogeneous in
$\sim$ 80% area of the sky. This property is found to be violated in few
groups of galaxies. Two such groups (Gr2 and Gr8) are identified. In these
groups, the structural dominance and the preferred alignments of galaxies are
found to oppose each other.

The true structural mode of a galaxy must involve a determination of which
side of the galaxy is closer to the observer (Binney and Tremaine 1987).
Three-dimensional determination of the leading and the trailing arm patterns
in the galaxies is a very important problem. We intend to address this problem
in the future.

###### Acknowledgements.

We are indebted to the referee for his/her constructive criticism and useful
comments. I acknowledge Profs. R. Weinberger and W. Saurer of Innsbruck
University, Austria for insightful discussions. I am thankful to Tribhuvan
University, Nepal and Innsbruck University, Austria for providing financial
assistance to visit Innsbruck University during Jan-March 2009.

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