BIBECHANA 18 (1) (2021) 26-32 

26 
 

BIBECHANA 
ISSN 2091-0762 (Print), 2382-5340 (Online) 

Journal homepage: http://nepjol.info/index.php/BIBECHANA 

Publisher: Department of Physics, Mahendra Morang A.M. Campus, TU, Biratnagar, Nepal 

 

 

Spatial orientations of angular momentum vectors of 

galaxies in Supercluster S [173+014+0082] 
 

 

Janak Ratna Malla1*, Walter Saurer2, Binil Aryal1 

1Central Department of Physics, Kirtipur, Kathmandu 
2Institute of Astro-particle Physics, Innsbruck University, Austria  
*Email: janak_malla@yahoo.com 
 

 

Article Information: 

Received: May 28, 2020 

Accepted: June 16, 2020 

 

Keywords:  

SDSS  

Supercluser  

Galaxy  

Redshift  

Intrinsic flatness  

 

 

 

ABSTRACT 

The spin vector orientation of 1302 SDSS (Sloan Digital Sky Survey) galaxies in 

Supercluster S[173+014+0082] having redshift 0.076 to 0.091 has been analysed. 

The positions, position angles and inclination angles of galaxies are used to 

convert two-dimensional observed parameters into three-dimensional angular 

momentum vectors of the galaxy using the `position angle-inclination' method. 

The expected isotropic distribution curves are determined performing numerical 

simulation by generating 107 virtual galaxies. The observed distribution is 

compared with the expected isotropic distribution curves using three statistical 

tools namely Chi-square test, auto-correlation test and Fourier test. Redshift map 

is studied and found that the distributions fit with the Gaussian. No preferred 

alignment of angular momentum vectors is noticed, supporting Hierarchy model 

of galaxy formation.  

DOI: https://doi.org/10.3126/bibechana.v18i1.29165 
This work is licensed under the Creative Commons CC  BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ 
 

 
1. Introduction 

 

During the path of history different ideas had been 

recommended describing the starting place of the 

universe. The most persuasive model is the 

standard Big Bang model, which remains valid. 

This model shows the creation technique of the 

universe began with an enlargement 20 billion 

years ago. Nowadays, the big Bang model is 

believed to be the beginning of space and time. The 

tiny, hot (1012K) universe started out to cool and 

fundamental symmetry damaged down. The 

breaking of this symmetry results in the inflation of 

the universe exponentially. Within a fraction of 

second, after the Big Bang, the procedure of 

Nucleosynthesis, baryongenesis & leptogenesis 

took place. After this procedure the universe 

entered into the matter dominated era. During dark 

epoch, the stars and galaxies are assumed to be 

formed. Today, how matter is distributed in the 

universe is the crucial field of study in cosmology 

and astronomy. 
 

The mutual attraction between the galaxies due to 

the gravitational force leads to the formation of 

clusters. Depending upon the number of galaxies 

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Janak Ratna Malla et al. / BIBECHANA 18 (1) (2021) 26-32  

27 
 

inside the cluster, they're labeled as poor and rich 

clusters. In case of poor clusters the force of gravity 

is weak to keep them tightly due to lower mass of 

the galaxy. Thus they seem irregular in shape. Our 

Milky Way is part of a poor cluster referred to as a 

local group which includes 54 galaxies [1]. The 

Virgo cluster is an example of a moderately rich 

cluster that is closest to us and the coma cluster is a 

rich cluster, has thousands of spiral and elliptical 

galaxies. The clustering technique of galaxy 

clusters lead to the formation of large scale 

structures known as superclusters. Thus they're 

collections of galaxy clusters, with 20-100h-1Mpc 

[2]. The scientist assumed that the variety of 

superclusters in the universe is set 10 million and it 

occupies the 10% area of the universe. As it 

contains organizations and clusters of galaxies, it 

provides the strong proof that galaxies inside the 

universe are not distributed uniformly. Abell 

catalogue is the pioneer work in the study of 

superclusters. It was proposed by George Abell in 

1958 and he called them second order cluster or 

clusters of clusters [3]. The study of superclusters 

highly benefited from the 2dF Galaxy Redshift 

Survey (2dFGRS) and the Sloan Digital Sky 

Survey (SDSS) [4]. Currently Sloan Digital Sky 

Survey (SDSS) is taking data from the deep sky. 

Our work is based totally on the data provided by 

SDSS-DR7. 
 

As suggested by Weizscker & Gamow [5, 6], the 

observe of origin of angular momentum of galaxies 

might be the clue for the physical properties of the 

initial situations of galaxies. Thus, the observed 

rotation of galaxy is critical to the cosmology and 

subsequently it is essential to recognize how they 

obtained their angular momentum. Pebbles [7] put 

forward detail calculations of the acquisition of 

angular momentum in early stage of proto-galactic 

evolution. He argued that, the gravitational 

interplay of quadrapole moment of the system with 

tidal field of the matter causes transformation of 

angular momentum to the developing proto-galaxy. 

The transformation of angular momentum between 

rotation and orbital motion is a outcome of the 

gravitational instability [8]. Peebles assumed that, 

as soon as the primeval plasma has recombined and 

decoupled from the primeval fireball radiation, 

gravitation appears to be the dominant force 

transferring the angular momentum to a developing 

proto-galaxy. 
 

On the basis of orientation of angular momentum 

vectors of galaxies one can determine the evolution 

of large scale structure, i.e., Supercluster. The 

alignment of rotational axes of galaxies in the 

Supercluster provides us information about the 

formation process of galaxies in early epoch [9]. 

There had been contradictory theoretical models 

predicting various galaxy cluster formation. The 

Hierarchy model [7] predicts random orientation of 

angular momentum vectors. The Pancake model 

[10] predicts the parallel orientation whereas the 

primordial vorticity model [11] says angular 

momentum vector distributed perpendicular to the 

cluster plane. We are interested to study the 

evolution of galaxies in the Supercluster by 

analyzing their preferred alignments. 
 

2. Materials and Method 
 

Database 
 

The analyzed data set has been recorded by Sloan 

Digital Sky Survey (SDSS). With the collaboration 

of Prof. Walter Saurer's group at the Institute of 

Astro-particle Physics, Innsbruck University, 

Austria, we obtained the set of SDSS (7th Data 

release) data. This Supercluster S [173 + 014 + 

0082] contains a total number of 1302 galaxies 

having redshift ranges from 0.076 to 0.091. In this 

work, r-filter is used to study the spin vector 

orientation of the galaxies. In order to calculate 

inclination angle first of all we used intrinsic 

flatness (q*) = 0.2 (Holmberg, [12]) and, then 0.13 

for those galaxies whose inclination angle was 

undetermined from the previous value of q*. In this 

process nine galaxies were excluded whose 

intrinsic flatness is less than 0.13. Hence, finally 

there are 1293 galaxies in the data set whose 

position (right ascension and declination), position 

angle and inclination are known.  



Janak Ratna Malla et al. / BIBECHANA 18 (1) (2021) 26-32  

28 
 

Godlowskian transformation 

 

The two dimensional parameters such as position, 

position angle and diameter can be transformed into 

3 dimensional parameters using the method 

proposed by Flin and Godlowski [13] and polar (𝜃) 

and azimuthal (𝜙) angle can be calculated for each 

galaxy. The polar angle (𝜃) is the angle between 

galactic spin vector (SV) and reference plane, and 

azimuthal angle (𝜙) is the angle between projection 

of a galactic spin vector to the reference plane and 

X-axis. These angles are given below. 

 

sin𝜃 = −cos𝑖 sin𝛼 ± sin𝑖 sin𝑝 cos𝛿  (1) 

 
sin𝜙 = (cos𝜃)−1[−cos𝑖 cos𝛿sin𝛼 +
sin𝑖(∓sin𝑝 sin𝛿 sin𝛼 ∓ cos𝑝cos𝛼)]   

  (2) 

 

Here, i, δ, α, and 𝜌 represent the inclination angle, 

declination, right ascension and position angles 

respectively and ∓  and  ±  indicate the two 

possible solution. 
 

The inclination angle (i) defined the angle between 

normal to the galaxy plane and line of sight of  the 

observer. This is given by following relation: 

𝑐𝑜𝑠2𝑖 =
(

𝑏

𝑎
)

2
−𝑞∗2

1−𝑞∗2   (3) 
 

Here b/a is axial ratio and q*represents the intrinsic 

flatness factor of the galaxy. For the disk galaxies 

its value depends upon the morphological type. 

Holmberg [12] showed that the values of q*varies 

from 0.083 for Sd spirals to 0.3 for elliptical. At 

first, q*= 0.20 for morphologically unidentified 

galaxies which leads to the removal of a certain 

number of galaxies that have axial ratio (b/a) is less 

than 0.20. Then, we used q*= 0.13 for those 

galaxies that have b/a < 0.20. 
 

Method of analysis 
 

Aryal & Saurer [14], proposed a method for the 

removal the selection effects and the expected 

isotropic distribution curves (θ and ϕ) are 

determined using the numerical simulation. For 

this, a true spatial distribution of the galaxy rotation 

axis is assumed to be isotropic. Then, due to the 

projection effects, i can be distributed ∝sin i, 

latitude can be distributed ∝cos B, the variables 

longitude () and position angle (p) can be 

distributed randomly and formulae (1) and (2) can 

be used to simulate (numerically) the 

corresponding distribution of θ and ϕ. At first, the 

isotropic distribution curves are based on 

simulations including 107 virtual galaxies Aryal & 

Saurer [14]. We use these numbers to make an 

input file and the predicted distribution by using a 

running simulation in MATLAB 15.0. 
 

Finally, the observed and expected distributions are 

compared with the help of appropriate statistical 

tests. In order to use ORIGIN 5.0 to determine the 

all sky distribution of galaxies where as ORIGIN 

8.0 used for the redshift distribution of galaxy. 
 

3. Results and Discussion 
 

All sky distribution is the pictorial representation of 

distribution of galaxies within a certain range of 

right ascension and declination in the sky. The all 

sky distribution plot for our Supercluster S 

[173++014+0082]  is shown in Fig. 1. The diagram 

shows that galaxies are distributed within the range 

of right ascension from 1700 to 1810 and declination 

from 60 to 180. 
 

Figure 1 shows that galaxies are distributed 

inhomogenously. To study the preferred alignment 

of each galaxy, we plotted the various parameters 

of galaxies. For each parameter the proper bin size 

is calculated using ORIGIN 5.0. These plots are 

presented in following Fig. 2. From these histogram 

plots of galaxies it is seen that the distribution of 

observed numbers of galaxies are clearly 

inhomogeneous. 
 

Figure 3 shows redshift distribution of the galaxy in 

the supercluster S [173 + 014 + 0082] [15]. In the 

map, low redshift galaxies are found in the 

substructure region. High redshifted galaxies are 

found to be distributed randomly [16]. The color 

map shows that the velocity dispersion is minimum 

in the region where the number density of the 



Janak Ratna Malla et al. / BIBECHANA 18 (1) (2021) 26-32  

29 
 

galaxy is minimum. In the substructure region, at 80 

declination, the redshift dispersion is found to be 

Δz = 0.084, whereas the lower substructure shows 

Δz= 0.076 and 0.080, respectively. It seems that 

there is Gaussian distribution between redshift 

dispersion and number density of galaxies in the 

substructures. It shows that the Superclusters 

evolve naturally. 

 

 
 

Fig. 1: All sky distribution of galaxies in the SDSS 

Supercluster S[173 + 014 + 0082] in equatorial 

coordinate system. Right ascension (R.A.) and 

Declination (Dec.) are given in degree. 

 
 

In order to identify the preferred alignment of the 

spin vector of galaxies we have performed three 

different statistical tests [17]. For anisotropy, the 

limit of chi-square probability P (> 𝜒2) is < 0.050, 

auto correlation coefficient (C/C (𝜎)) is > 1.0, first 

order Fourier coefficient (Δ11/𝜎(Δ11)) is >1.5 and 

Fourier probability P( >Δ11) is < 0.150 

respectively. 
 

For  θ distribution (Table 1), the result of these 

tests is found as: the chi square probability P (> χ2) 

is 0.653 which is greater than critical value (0.05) 

for anisotropy. In addition to this, the auto-

correlation coefficient C/C (σ)) is 0.410 which is 

less than critical value; this result is in favor of 

isotropic distribution. Similarly, the first order 

Fourier coefficient  (Δ11/σ(Δ11))  and Fourier 

probability (P >∆1) found to be 0.860 and 0.643 

respectively. These two results also support the 

isotropic distribution of 𝜃. Hence, all three 

statistical tests show isotropic distribution of 

Supercluster S[173 + 014 + 0082] indicating 

random orientation of spin vectors of galaxies. 

From this result we can conclude that the galaxy 

formation model of Supercluster S [173 + 014 + 

0082] follows the Hierarchy model. According to 

this model galaxies are believed to be formed from 

the clustering. 
 

Fig. 4(a) shows the polar (𝜃) distribution of 

galaxies of the Supercluster S[173 + 014 + 0082]. 

In the figure, solid and dotted curves represent 

expected isotropic and cosine distribution. In the 

small angle (𝜃< 450) angle region i.e. shaded region 

no significant hump and dip are observed. While, in 

a larger angle (𝜃 > 450) a significant dip is observed 

at an angle 750 with 1𝜎 error limit. In the shaded 

region the observed number of solutions is 1674 

and expected number of solutions is 1652. Thus the 

observed number of solutions exceeds the expected 

solution by 22. This means that there are 11 more 

galaxies observed. On the other hand, in a larger 

angle region the observed solution is less than 

expected solution by 24. This causes deficiency of 

12 galaxies in that region. Furthermore, at the 

bimodal region one extra galaxy is observed. Thus 

these humps at small angle regions are totally 

nullified by the dip at the larger angle. The excess 

number of galaxies in the smaller angle can be 

regarded as transfer of the galaxies from the larger 

angle region. This may happen due to the 

gravitational tidal effect. 
 

In 𝜙 distribution, the statistics (Table 1) support the 

isotropic distribution. The chi square probability  ( 

P > 𝜒2) and auto-correlation coefficient (C/C(𝜎) 

found to be 0.610 (greater than 0.05) and -0.481 

(less than 1𝜎). Similarly, the value of first order 

Fourier coefficient (∆11/𝜎 (∆11)) and first order 

Fourier probability (P > ∆1) is -0.358 (less than 

1.5𝜎) and 0.883 (greater than 0.15). These results 

are strong evidence to advocate the Hierarchy 

model of galaxy evolution in the Supercluster 

S[173 + 014 + 0082].  



Janak Ratna Malla et al. / BIBECHANA 18 (1) (2021) 26-32  

30 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 
 
 
 

 
 
 
 
 

 
 

 
 
 
 
 
 

 
 

 
 
 
 
 
 
 
 
 

 
Fig. 2: The distribution of right ascension (𝛼), declination (𝛿), position angle (P) and inclination angle (i) of galaxies 

in the Supercluster S [173 + 014 + 0082]. The Y-axis represents the number of observed galaxies. 

The inhomogeneous distribution of galaxies can be seen. 

 
 

 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 

Fig. 3: (a) red shift map and (b) the distribution of galaxies in the Supercluster region. Solid curves 

represent Gaussian fit. The statistical error bars are shown. 



Janak Ratna Malla et al. / BIBECHANA 18 (1) (2021) 26-32  

31 
 

Fig. 4(b) represents the distribution of azimuthal 

angle of Supercluster S[173 + 014 + 0082]. The 

solid and dotted line indicate the expected and 

average distribution. The observed number of 

solutions in the small angle region (-900 < 𝜙 < -450) 

is 19 more than expected solution; however, no 

significant humps are observed in the distribution 

curve. Similar result is obtained at a larger angle 

region (450 < 𝜙 < 900) also. The shaded region 

represents the bimodal region (-450 < 𝜙 < 450). In 

this region the expected and observed numbers of 

solutions are 1405 and 1396 respectively. From the 

figure, two crystal clear humps are observed at -50 

and 50and dip is observed at -150within error limit 

1𝜎. 

 
Table 1: Statistics of the polar and azimuthal angle 

distributions of galaxies in the Supercluster 

S[173+014+0082]. The first column represents the 

statistics used, P ( > 𝜒2)  represents the chi-square 

probability (second row). Similarly, C/C(𝜎) 

represents the auto-correlation coefficient(third 

row). The last two rows give the first order Fourier 

coefficient (∆11/𝜎 (∆11)) and first order Fourier 

probability P (>Δ1). 

 

 
 
 

 
Fig. 4: (a) The polar (𝜃) and (b) azimuthal (𝜑) 

angle distributions of galaxies in the Supercluster 

S[173+014+0082]. The solid line represents the 

expected isotropic distributions. The dash lines 

represent the cosine and average distributions 

respectively. The solid circles with ±1  error bars 

are represent the observed distribution. 
 

 

4. Conclusions 
 
 

The Supercluster S[173+014+0082] consists of 1302galaxies of 

which position angles, right ascension, declination, 

and axial ratio are known. These are obtained from 

two dimensional projections on the celestial sphere by 

SDSS DR 12 Survey. The conversion of two 

dimensional observed parameters into three 

dimensional is carried out by using `position angle-

inclination' method. In order to find out the theoretical 

distribution of galaxy rotation axes we ran random 

simulation by generating 107 virtual galaxies as 

proposed by Aryal & Saurer. Here we assumed 

cosmological principle: distribution of galaxies in the 

large scale structure is homogeneous and isotropic. 

The simulated results were compared with the 

observed distribution using the statistical tools namely 

chi-square test, auto-correlation test and Fourier test. 

The distribution of spin vector and spin vector 

projections of total SDSS galaxies that have redshift 

in the range 0.076 to 0.091 are found to be random in 

all samples. We conclude our results as follows: 

Statistics  Polar angle 

(θ)  

Azimuthal angle 

() 

P ( > 𝜒2)   0.653 0.610 

C/C(𝜎) 0.410 -0.481 

(∆11/𝜎 (∆11)) 0.860 -0.358 

P (>Δ1). 0.643 0.883 



Janak Ratna Malla et al. / BIBECHANA 18 (1) (2021) 26-32  

32 
 

(a) The polar and azimuthal angles of supercluster 

galaxies shows isotropic isotropic distribution 

suggesting no preferred alignment of angular 

momentum vectors as well as its projection. In 

other words we observed vanishing angular 

momentum in the Supercluster. This result 

advocates the Hierarchy model of galaxy 

evolution as suggested by Peebles. 

(b) The redshift distribution of galaxies in 

Supercluster showed Gaussian-like nature, 

suggesting the distribution of galaxies’ Hubble 

recessional velocity is isotropic, satisfying 

cosmological principle. 

 

Acknowledgements  

One of the authors (J.R.Malla) is thankful to the 

Department of Astro-Particle Physics, Innsbruck 

University, Austria, for providing data access . One 

of them (J.R.Malla ) is grateful to the authorities of 

the  SDSS  for providing the database. 

 
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