Preferred alignments of angular momentum vectors of galaxies in six dynamically unstable Abell clusters
1 Introduction
The formation of a galaxy cluster is one of the major unsolved problems of modern astrophysics. The process by which larger structures (e.g., clusters, superclusters) are formed through the continuous merging of smaller structures (e.g., galaxies and galaxy groups) is called hierarchical clustering, which is supported by the concordance model (ΛCDM). The study of the preferred alignment of angular momentum vectors of galaxies in clusters is one of the most effective ways of testing the concordance model. Godłowski et al. ( 2003 ) described the Li ( 1998 ) model and showed the relation between angular momenta and masses of the large scale structures. This relationship was observationally tested by several authors (Godłowski et al. 2005 ; Hu et al. 2006 ; Aryal & Saurer 2006 ; Aryal et al. 2007 , 2008 ; Godłowski et al. 2010 , Godłowski 2012 ) and they found vanishing angular momenta for less massive structures but nonvanishing cases for larger structures.
Aryal et al. ( 2013 ) studied preferred alignments of angular momentum vectors of galaxies in six rotating clusters (A954, A1139, A1399, A2162, A2169 and A2366) that are dynamically stable and have a single peak in onedimensional (1D)threedimensional (3D) number density maps. These clusters have no substructures. They found a random orientation of angular momentum vectors of galaxies in all six clusters, supporting the hierarchy model (Peebles 1969 ).
In the present work we intend to study the preferred alignments of angular momentum vectors of galaxies in six Abell clusters, namely S1171, S0001, A1035, A1373, A1474 and A4053, that have multiple peaks in 1D3D number density maps with a larger value of velocity dispersion. These clusters have substructures. We intend to answer the following: (1) Does the orientation of angular momentum vectors of galaxies that have substructures favor hierarchical clustering? (2) Do the clusters with large velocity dispersion prefer a random orientation of angular momentum vectors of galaxies? And finally, (3) does substructure formation cause large velocity dispersion in the cluster? Our aim is to compare results with the concordance model. The database and methods are described in Sections
2 Database
Hwang & Lee ( 2007 , HL hereafter) identified six rotating clusters (A954, A1139, A1399, A2162, A2169 and A2366) that are in dynamical equilibrium and show a single peak in 1D3D number density maps. In addition, six dynamically unstable clusters (S1171, S0001, A1035, A1373, A1474 and A4053) that have multiple numberdensity peaks in 1D3D maps with a large velocity dispersion are present. After investigating substructure using the DresslerSchectman method, HL classified rotating clusters into two categories: (1) clusters with a single number density peak which hence are in dynamical equilibrium and (2) clusters with multiple numberdensity peaks which are dynamically unstable. In both cases, clusters have a very large value of velocity dispersion. In this paper we study the preferred alignments of angular momentum vectors of galaxies in Abell clusters S1171, S0001, A1035, A1373, A1474 and A4053.
Table


3 Method
The position angle (PA)inclination method is used to convert given twodimensional (2D) parameters (positions, diameters, PAs) into 3D ones (galaxy rotation axes: angular momentum vectors and their projections) (Flin & Godlowski 1986 ). The expected isotropic distribution curves for angular momentum vectors and their projections are determined by performing a random simulation (Aryal & Saurer 2000 ). The observed distributions are compared with the expected ones using various statistical tests.
The angular momentum vectors (θ) of galaxies and their projections (ϕ) to the galactic (G) and supergalactic (S) planes are obtained by using the method described by Flin & Godlowski ( 1986 ). For this, the SDSS/2dFGRS databases (positions, PAs and diameters) provided by Hwang (private communication in 2011) are used. In previous works (Godlowski 1993 ; Baier et al. 2003 ; Hu et al. 2006 and the references therein), authors have studied the preferred alignments of galaxies in clusters with respect to the galactic or supergalactic (or both) system. The formulae to obtain angular momentum vectors (θ) and their projection (ϕ) in the Ssystem are as follows (Flin & Godlowski 1986 ):
The inclination angle (i) is the angle between the lineofsight and normal to the plane of the galaxy. This angle can be calculated using the Holmberg (
1946
) formula
Equations (
Aryal & Saurer (
2000
) performed random simulations imposing various types of selections in the database and concluded that any selections can cause changes in the expected isotropic distribution curves for both angular momentum vectors (polar angles) and their projections (azimuthal angles). We noticed the following selection effects in our database: (1) the positions of galaxies in the clusters are inhomogeneous; (2) the PAs of faceon cases (
We apply chisquare, autocorrelation, Fourier (Godlowski
1993
), KolmogorovSmirnov (KS) (Press et al.
1992
) and KuiperV (Kuiper
1960
) tests to discriminate anisotropy from isotropy in the observed and expected distributions. Details about these statistical tests are given in the appendix of Aryal et al. (
2007
). The limits for anisotropy are as follows:
chisquare probability (
autocorrelation coefficient (C/C(
first order Fourier coefficient (
Fourier probability (
KS=1,
KuiperV = 1.
In the last two statistical tests, the null hypothesis (isotropy) is represented by “0” which is not rejected at the chosen significance level, whereas the value “1” designates that the null hypothesis is rejected (anisotropy).
The first order Fourier coefficient (
4 Results
Table


Abell S1171 is the spherically shaped nearby (
Figure
In the ϕdistribution, 0° corresponds to the projections of angular momentum vectors that tend to point radially towards the center of the reference coordinate system (center of the Milky Way in Gsystem and Virgo Cluster center in Ssystem). No humps or dips are observed in Figure
HL noticed substructures in 2D and 3D number density maps. A lowvelocity tail at
All six statistical parameters show isotropy in both the θ and ϕdistributions (Table
This cluster has the largest velocity dispersion with two subclusterings along north and south (Fig.
In the present study we used spectroscopic databases (SDSS and 2dFGRS) and verified the prediction made by HL by observing a significant hump at
This cluster is the most distant (
All statistical tests suggest isotropy in both the θ and ϕdistributions, advocating the hierarchy model (Peebles
1969
) of galaxy evolution. In the θdistribution, a significant hump at 75° (
Einasto et al. (
2001
) studied the VirgoComa Supercluster and concluded that Abell 1474 is a member cluster of that supercluster. HL noticed three substructures in the number density map (Fig.
The chisquared and autocorrelation tests show anisotropy in the θdistribution, whereas isotropy is noticed in the Fourier, KS and KuiperV tests. A hump at 25° (
Porter & Raychaudhury (
2005
) reported that the cluster A4053 is a member of the PiscesCetus Supercluster. HL found that the cluster shows substructures in 1D and 3D maps. The number density map (Fig.
In the histogram of polar (θ) angle distribution, a good agreement between the expected and observed distribution is noticed, suggesting no preferred alignments (Fig.
5 Conclusions
The preferred alignments of angular momentum vectors of galaxies in six clusters having multiple numberdensity peaks with a spatial segregation of high and lowvelocity galaxies are studied. We adopted the ‘position angle  inclination’ method (Flin & Godlowski 1986 ) to compute 3D parameters (polar and azimuthal angles of galaxy rotation axes) using 2D observed parameters (e.g., positions, diameters, PA). To remove selection effects from the database, a numerical simulation is performed, as proposed by Aryal & Saurer ( 2000 ). The observed and expected isotropic distributions are compared using five statistical tests, namely chisquare, autocorrelation, Fourier, KS and KuiperV.
In general, no preferred alignment is noticed for all six clusters, supporting the hierarchy model as predicted by Peebles ( 1969 ). However, local effects are noticed in the clusters that have substructures in 1D, 2D and 3D analysis (HL). Therefore, a large value of velocity dispersion with substructures in the clusters does not lead their galaxies to support the pancake (Doroshkevich 1973 ) or primordial vorticity theory (Ozernoi 1978 ). A very good correlation between the hierarchy (Peebles 1969 ) and Li model (Li 1998 ) is found, as in our previous work (Aryal et al. 2013 ). Therefore, vanishing angularmomenta favor the formation of substructures in clusters that have large velocity dispersion. The preferred alignment is found to increase with richness of the cluster. Therefore, it can be interpreted as an effect from the mechanism of tidal forces (Heavens & Peacock 1988 ; Catelan & Theuns 1996 ; Stephanovich & Godłowski 2015 ), but also is in agreement with the Li ( 1998 ) model in which galaxies form in the rotating universe.
Tidal torque naturally arises in the hierarchical clustering scenario and hence the distribution of angular momentum vectors of galaxies becomes random. However, a tidal torque shear tensor (due to the effect of gravity) can cause a local preference in angular momentum vectors as predicted by Lee ( 2004 ) and Trujillo et al. ( 2006 ).
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