11institutetext: Astrophysics, Cosmology and Gravity Centre (ACGC), Astronomy
Department, University of Cape Town,
Private Bag X3, 7700, Rondebosch, Republic of South Africa 22institutetext:
Raman Research Institute, Sadashivanagar, Bangalore 560 080, India
33institutetext: Laboratoire Lagrange, UMR7293, Université de Nice Sophia-
Antipolis, CNRS, Observatoire de la Côte d’ Azur, 06300 Nice, France
44institutetext: Department of Physics and Astrophysics, Vrije Universiteit
Brussel, Pleinlaan 2, 1050 Brussels, Belgium 55institutetext: European Space
Agency (ESTEC), Keplerlaan 1, 2200 AG, Noordwijk, the Netherland

# Morphology parameters: substructure identification in X-ray galaxy clusters

Viral Parekh<EMAIL_ADDRESS>Kurt van der Heyden 11 Chiara Ferrari
33 Garry Angus 1144 Benne Holwerda 55

(Accepted)

###### Abstract

Context. In recent years multi-wavelength observations have shown the presence
of substructures related to merging events in a high fraction of galaxy
clusters. Clusters can be roughly grouped into two categories – relaxed and
non-relaxed – and a proper characterisation of the dynamical state of these
systems is of crucial importance both for astrophysical and cosmological
studies.

Aims. In this paper we investigate the use of a number of morphological
parameters (Gini, $M_{20}$, Concentration, Asymmetry, Smoothness, Ellipticity
and Gini of the second order moment, $G_{M}$) introduced to automatically
classify clusters as relaxed or dynamically disturbed systems.

Methods. We apply our method to a sample of clusters at different redshifts
extracted from the Chandra archive and we investigate possible correlations
between morphological parameters and other X-ray gas properties. We conclude
that a combination of the adopted parameters is a very useful tool to properly
characterise the X-ray cluster morphology.

Results. According to our results three parameters – Gini, $M_{20}$ and
Concentration – are very promising for identifying cluster mergers. The Gini
coefficient is a particularly powerful tool, especially at high redshift,
being independent from the choice of the position of the cluster centre. We
find that high Gini ($>$ 0.65), high Concentration ($>$ 1.55) and low $M_{20}$
($<$ -2.0) values are associated with relaxed clusters, while low Gini ($<$
0.4), low Concentration ($<$ 1.0) and high $M_{20}$ ($>$ -1.4) characterise
dynamically perturbed systems. We also estimate the X-ray cluster
morphological parameters in the case of radio loud clusters. In excellent
agreement with previous analyses we confirm that diffuse intracluster radio
sources are associated with major mergers.

###### Key Words.:

galaxies: X-ray clusters: substructure - merger: radio halo: radio relic

##  1 Introduction

It is now well proven that massive galaxy clusters form and evolve at the
intersection of cosmic web filaments through merging and accretion of smaller
mass systems (e.g. Maurogordato et al. 2011, and references therein). Huge
gravitational energy is released during cluster collisions ($\approx 10^{64}$
ergs) and several billion years are required for the cluster to re-establish a
situation of (quasi-)equilibrium after a major merger episode.

Both observations and numerical simulations have shown that merging events
deeply affect the properties of the different cluster components (e.g. Ferrari
et al. 2008, and references therein). Multiple merger episodes could for
instance be responsible for disturbing the dynamically relaxed cores of the
X-ray emitting hot intracluster medium (ICM, e.g. Burns et al. 2008), as well
as the ICM density, temperature and metallicity distribution (e.g. Kapferer et
al. 2006). In addition, it is well known that a disturbed X-ray morphology is
typical of dynamically perturbed galaxy clusters (Kapferer et al. 2006, and
references therein). The presence of substructures, a highly elliptical
cluster X-ray morphology or an X-ray centroid variation are typical features
that suggests that a cluster is not virialized. This has important
implications both for using clusters as tools for cosmology, and for studying
the complex gravitational and non-gravitational processes acting during large-
scale structure formation and evolution, since in both cases we need to know
if observed clusters are relaxed or not.

Joint X-ray and optical studies can provide detailed information about the
dynamical state of a cluster (e.g. Ferrari et al. 2006), but they are
extremely time demanding from the observational and analysis point of view.
For statistical studies of large cluster samples we need to identify robust
indicators that can somehow quantify the cluster dynamical state. Since the
morphology of clusters is deeply related to their evolutionary history,
different morphological estimators have been proposed. Jones & Forman (1992)
classified galaxy clusters observed by the Einstein X-ray satellite into six
morphological classes which include single, elliptical, offset center, primary
with small secondary, bimodal and complex. Several studies have tried to
quantify the fraction of dynamically disturbed clusters from morphological
analyses. Jones & Forman (1999) showed that around 30% of clusters observed
with the Einstein satellite contain substructure. More recently Schuecker
(2005) have noticed in ROSAT observations that $\sim$ 50$\%$ of clusters have
substructure. With high resolution telescopes such as Chandra and XMM, it has
become easier to identify subclusters, bimodality and X-ray centroid shifts in
clusters. However some mergers are too complex to be identified only from
X-ray morphological analysis and, particularly in the case of high redshift
clusters, some special techniques and statistics are required, which can
provide a more quantitative and robust measure of the degree of the cluster
disturbance.

Various techniques have been suggested to provide a more quantitative and
qualitative measure of the degree of the cluster disturbance: Power ratios
(Buote & Tsai 1995b; Jeltema et al. 2005; Böhringer et al. 2010) and the
emission centroid shift (Mohr et al. 1993; Böhringer et al. 2010) are most
commonly used to classify X-ray galaxy clusters from the morphological point
of view. Recently, Andrade-Santos et al. (2012) have used a residual flux
method; in order to calculate the substructure level in a given X-ray galaxy
cluster, they take into account the ratio between number of counts on the
residual (which they obtain by the subtraction of a surface brightness model
from the original X-ray image) and on the original cluster images. Weißmann et
al. (2012) proposed to use the maximum of the third order power ratios
calculated in annuli of fixed width and constantly increasing radius to
measure the degree of substructure. Rasia et al. (2012) have used six
different morphology parameters namely, asymmetry, fluctuation of the X-ray
brightness (smoothness), hardness ratios, concentration, the centroid-shift
method and third order power ratio to characterise simulated clusters. They
took hydrodynamical simulation of 60 clusters and passed it through a Chandra
telescope simulator with uniform exposure time (100 ks) for all clusters. Out
of all of these parameters they found that only the asymmetry and
concentration parameters could straightforwardly and clearly separate relaxed
and non-relaxed systems. The smoothness parameter is affected by the choice of
radii and smoothing kernel size. The centroid-shift parameter also works
reasonably well and leaves only a few overlapping relaxed and non-relaxed
clusters. The third order power ratio technique also depends on the choice of
radius and is limited to only detecting substructure near to the cluster
center. More recently Nurgaliev et al. (2013) have used photon asymmetry and
central Concentration parameters to quantify morphology of high-$z$ clusters
which suffer from low photon counts.

In this paper, we investigate seven morphology parameters, typically used for
galaxy classification, to study X-ray galaxy cluster morphology and we
motivate which parameters are the optimal for identifying substructure or
characterising dynamical states. The combination of morphology parameters has
been successfully used to classify different galaxy morphologies (Zamojski et
al. 2007; Scarlata et al. 2007; Holwerda et al. 2011a) and we want to
investigate their usefulness in galaxy cluster classification. Our focus is
not limited to separating galaxy clusters into relaxed and non-relaxed
categories, but to study correlations between X-ray gas properties and
morphology parameters, as well as the evolution of morphological properties of
galaxy clusters from the high redshift universe to present.

We explore the usefulness of the non-parametric morphology parameters on a
subset of the ROSAT 400 deg2 cluster sample observed by the Chandra X-ray
telescope (Vikhlinin et al. 2009, hereafter V09). We choose this sample
because it has good quality X-ray data and also a broad distribution of
redshifts. There are in total 85 (49 low-$z$ (0.02–0.3) and 36 high-$z$
(0.3–0.8)) galaxy clusters in our analysis. In addition, V09 has measured
global properties of the galaxy clusters (such as luminosity, temperature,
mass, etc.) which we use to compare with our measured morphology parameters.

As a test case, in this paper, we study the dynamical activity of clusters
hosting diffuse radio sources (radio halos), in particular we focus our
attention on clusters taken from Giovannini et al. (2009). Current results
suggest a strong link between the presence of diffuse intracluster radio
emission and cluster mergers (Ferrari et al. 2008; Feretti et al. 2012, and
references therein). Similar to what was done by Cassano et al. (2010b), but
using our set of X-ray morphological parameters, in this paper we analyze the
X-ray morphology of relaxed clusters and non-relaxed clusters (which includes
both radio quiet and radio loud mergers).

This paper is organized as follows. § 2 gives a brief introduction to the
morphology parameters. § 3 gives the sample selection and X-ray data
reduction. In § 4, we present our results. § 5 shows the systematic and
possible bias on morphology parameters. In § 6, we compare our parameter
measures with available cluster global properties. Finally, § 7 will give our
discussions and conclusions. We assumed $H_{0}$=73 km s-1 Mpc-1
$\Omega_{M}$=0.3 and $\Omega_{\Lambda}$=0.7 throughout the paper, unless
stated otherwise.

##  2 Introduction of morphology parameters

The non-parametric morphology parameters (Gini, $M_{20}$, Concentration,
Asymmetry, Smoothness, Ellipticity and Gini of the second order moment) are
widely used to automatically separate galaxies of different Hubble types. As
an example, they are used for galaxy morphology classification in the analysis
of the HST and SDSS galaxy surveys (Abraham et al. 2003; Conselice 2003; Lotz
et al. 2004; Zamojski et al. 2007; Holwerda et al. 2011a; Wang et al. 2012).

Abraham et al. (2003), Lotz et al. (2004) and Wang et al. (2012) also revealed
the inter-relation between Gini, Concentration and $M_{20}$, as well as the
possible inter-change between the Concentration and Gini parameter for
high-$z$ galaxies. This encouraged us to investigate these parameters in more
detail in order to characterise the dynamical state of galaxy clusters,
particularly at high-$z$. In this paper we adopt the definition of
Concentration, Asymmetry and Smoothness from Conselice (2003), and of the Gini
coefficient and $M_{20}$ from Lotz et al. (2004). Gini of the second order
moment was defined by Holwerda et al. (2011b). The required input parameters
for computing the morphological indicators (except the Gini parameter) are the
central position ($x_{c},y_{c}$) of the galaxy clusters, as well as a fixed
aperture size or area over which these morphology parameters are measured.

We calculate the center position by first assigning initial coordinates based
on visual observation of each cluster image and then allow the flux weighted
coordinates to iterate in a fixed aperture size of e.g. 500 kpc until they
have converged. The centre coordinates are then the unique point at the center
of the distribution of flux, essentially the light distribution equivalent to
the “centre of mass”.

###  2.1 Gini coefficient and Gini of the second order moment ($G_{M}$)

The Gini parameter is widely used in the field of economics, where it
originated as the Lorenz curve (Lorenz 1905). It describes the inequality of
wealth in a population. Here we use it as a calculation of flux distribution
in a cluster image. If the total flux is equally distributed among the pixels,
then the Gini value is equal to 0 (there is constant flux across the pixels
regardless of whether those pixels are in the projected centre or not); but if
the total flux is unevenly distributed and belongs to only a small number of
pixels, then the Gini value is equal to 1. We adopt the following definition
from Lotz et al. (2004):

$G=\frac{1}{\bar{K}n(n-1)}\sum_{i}(2i-n-1)K_{i},$ (1)

where $K_{i}$ is the pixel value in the $i^{th}$ pixel of a given image, $n$
is the total number of pixels in the image, and $\bar{K}$ is the mean pixel
value of the image.

We also apply a Gini value to the second order moment of each pixel, defining
Gini of the second order moment as:

$G_{M}=\frac{1}{\bar{F}n(n-1)}\sum_{i}(2i-n-1)F_{i},$ (2)

where $F_{i}$ is the second order moment of each pixel:

$F_{i}=K_{i}\times[(x-x_{c})^{2}+(y-y_{c})^{2}],$ (3)

where ($x$, $y$) is the pixel position with flux value $K_{i}$ in the cluster
image; and $(x_{c}$, $y_{c})$ is the coordinate of the cluster centre.

###  2.2 Moment of light, $M_{20}$

Lotz et al. (2004) define the total second order moment $F_{tot}$ as the flux
in each pixel $K_{i}$ multiplied by the squared distance to the centre of the
source, summed over all the selected pixels:

$F_{tot}=\sum_{i}F_{i}=\sum_{i}K_{i}[(x_{i}-x_{c})^{2}+(y_{i}-y_{c})^{2}],$
(4)

where ($x_{c}$, $y_{c}$) is the centre of the cluster.

The second order moment can be used to trace various properties of galaxy
clusters, such as the spatial distribution of multiple bright cores,
substructure or mergers. $M_{20}$ is defined as the normalised second order
moment of the relative contribution of the brightest 20% of the pixels. To
compute $M_{20}$, we rank-order the image pixels by flux, calculate $F_{i}$
over the brightest pixels until their sum equals 20% of the total selected
cluster flux, and then normalise by $F_{tot}$:

$M_{20}=log\left(\frac{\sum_{i}F_{i}}{F_{tot}}\right),{\rm
while}\sum_{i}K_{i}\leq 0.2K_{tot},$ (5)

where $K_{tot}$ is the total flux of the cluster image (image pixels are
selected from the segmentation map111A map which defines the chosen circular
aperture size with all pixels fixed to a value of 1.), and $K_{i}$ is the flux
value for each pixel i (where $K_{1}$ = the brightest pixel, $K_{2}$ = the
second brightest pixel, etc).

###  2.3 Concentration, Asymmetry and Smoothness (CAS)

Concentration, Asymmetry and Smoothness parameters are commonly known as CAS.

Concentration is defined by Bershady et al. (2000) and Conselice (2003) as:

$C=5\times log\left(\frac{r_{80}}{r_{20}}\right),$ (6)

where $r_{80}$ and $r_{20}$ represent the radius within which 80% and 20% of
the flux reside, respectively. Concentration is widely used in the
classification of cool core, especially among distant clusters. Santos et al.
(2008) define the surface brightness concentration parameter for finding cool
core clusters at high redshift as,

$c_{sb}=\frac{C_{r}(r<40kpc)}{C_{r}(r<400kpc)},$ (7)

where $C_{r}$ ($r$ $<$ 40 kpc) and $C_{r}$ ($r$ $<$ 400 kpc) are the
integrated surface brightness within 40 kpc and 400 kpc, respectively. Instead
of physical radii, we used the percentages of total flux within a given
aperture size. This has an advantage (atleast in low-$z$ clusters) in that the
flux is independent of angular bin size and galaxy cluster redshift. Our
sample covers the redshift range 0.02 $<$ $z$ $<$ 0.9. This means that, by
adopting a pixel size of $2^{\prime\prime}$, 250 kpc correspond to a pixel
range from $\sim$16 to $\sim$225\. In order to avoid this large deviation in
pixel spread, it is best to use various percentages of the total flux of the
galaxy clusters. For the inner radii we use 20$\%$–50$\%$, and for the outer
radii we used 80$\%$–90$\%$ of the total flux. For example, we use the
$C_{5080}$ Concentration parameter which means 50% of the flux within the
inner radii and 80% within the outer radii.

The Asymmetry value, which will give rotational symmetry around the cluster
centre, is calculated when a cluster image is rotated by 180∘ around its
centre ($x_{c},y_{c}$) and is then subtracted from its original image.

$A=\frac{\sum_{i,j}\mid K(i,j)-K_{180}(i,j)\mid}{\sum_{i,j}\mid K(i,j)\mid},$
(8)

where $K(i,j)$ is the value of the pixel at the image position $i,j$, and
$K_{180}(i,j)$ is the value of the pixel in the cluster’s image rotated by
180∘ around its centre. The Asymmetry value is sensitive to any region of the
cluster that is responsible for asymmetric flux distribution. Hence, if the
substructure affects the flux distribution at any scale, we can pick it up
from the Asymmetry value for that galaxy cluster.

The Smoothness parameter can be used to identify patchy flux distribution
expected in non-relaxed clusters. By smoothing a cluster image with a filter
of width $\sigma$, high frequency structures can be removed from the image. At
this point the original image is subtracted from this newly smoothed, lower
resolution image. The effect is to produce a residual map that has only high-
frequency components of the galaxy cluster’s flux distribution. The flux of
this residual image is then summed and divided by the total flux of the
original cluster image in order to find its smoothness value,

$S=\frac{\sum_{i,j}\mid K(i,j)-K_{s}(i,j)\mid}{\sum_{i,j}\mid K(i,j)\mid},$
(9)

where $K_{s}(i,j)$ is the pixel in a smoothed image. Here we choose a Gaussian
smoothing kernel of $\sigma$ = $12^{\prime\prime}$ as an arbitrary scale to
smooth the cluster image.

###  2.4 Ellipticity

Ellipticity is commonly defined by the ratio between a semi-major axis (A) and
a semi-minor axis (B) as

$E=1-\frac{B}{A},$ (10)

where A and B can be computed directly from the second order moments of the
flux in the cluster image as

$A^{2}=\frac{{\overline{x^{2}}}+{\overline{y^{2}}}}{2}+\sqrt{{\left(\frac{{\overline{x^{2}}}-{\overline{y^{2}}}}{2}\right)^{2}}+\overline{xy}^{2}}$
(11)
$B^{2}=\frac{{\overline{x^{2}}}+{\overline{y^{2}}}}{2}-\sqrt{{\left(\frac{{\overline{x^{2}}}-{\overline{y^{2}}}}{2}\right)^{2}}+\overline{xy}^{2}},$
(12)

where

$\tiny\overline{x}=\frac{\displaystyle\sum_{i\in
S}K_{i}x_{i}}{\displaystyle\sum_{i\in S}K_{i}},\hskip
14.22636pt\overline{y}=\frac{\displaystyle\sum_{i\in
S}K_{i}y_{i}}{\displaystyle\sum_{i\in S}K_{i}},$ (13)

$\displaystyle\overline{x^{2}}=\frac{\displaystyle\sum_{i\in
S}K_{i}x_{i}^{2}}{\displaystyle\sum_{i\in S}K_{i}}-\overline{x}^{2},$
$\displaystyle\hskip 14.22636pt\overline{y^{2}}=\frac{\displaystyle\sum_{i\in
S}K_{i}y_{i}^{2}}{\displaystyle\sum_{i\in S}K_{i}}-\overline{y}^{2},$
$\displaystyle\hskip 14.22636pt\overline{xy}=\frac{\displaystyle\sum_{i\in
S}K_{i}x_{i}y_{i}}{\displaystyle\sum_{i\in S}K_{i}}-\overline{x}\hskip
2.84544pt\overline{y}.$ (14)

Here ($x_{i}$, $y_{i}$) is the ($x$, $y$) coordinate of the image of a pixel
$i$ of value $K_{i}$ inside an area $S$.

###  2.5 Uncertainty estimation

There are three sources of uncertainty in the calculation of the morphology
parameters. These are: (1) shot noise in the image pixel values, (2)
uncertainties in the centre of the cluster, and (3) variation in the area over
which morphology parameters are calculated. The first two uncertainties can be
approximated using a number of iterations of the Monte Carlo method. For
estimating the third uncertainty, we used a jackknifing technique.

The shot noise effect can be approximated by replacing each pixel value with a
Poisson random variable of the mean value of each pixel for the given image,
and recalculating the parameters a number of times. After a set number of
iterations (in our case 10), the rms of the spread in parameter values is an
approximation of uncertainty in the parameters.

Resultant uncertainty from variation of the central position of a cluster is
computed by deviating the input ($x_{c}$, $y_{c}$) coordinate within a fixed
Gaussian width ($\sim$ $30^{\prime\prime}$ in our case). We then recalculate
the parameters for several ($x_{c}$, $y_{c}$) value. After a number of
iterations (10) (Holwerda et al. 2011c), we compute the rms of the spread in
parameter values as an approximate value of the uncertainty in the parameters.

The Gini coefficient is less sensitive to the shot noise than to changes in
area, as the pixels are ordered first and do not depend on the adopted central
position of the cluster in any way. We estimated its uncertainty from a shot
noise, and the rms in Gini values from a series of subsets of the pixels in
the image (by allowing the pixels to be varied using Poisson statistics for a
given area). This is known as jackknife error estimation. Yitzhaki (1991)
suggested the use of the jackknife approach to estimate an uncertainty in the
Gini coefficient. The jackknife and shot noise uncertainty estimates in the
Gini coefficient are of similar order.

We used an error propagation formula to combined the shot noise and central
position uncertainties for all parameters except the Gini coefficient. For the
Gini coefficient, we combined the uncertainties from the shot noise and the
jackknife estimation. Hence, in this work, all reported uncertainties are a
combination of those mentioned above.

##  3 Sample selection and data reduction

###  3.1 Chandra sample

For our analysis, we required high quality data, particularly for high-$z$
clusters. Burenin et al. (2007) compiled a 400 deg2 galaxy cluster catalogue
based on the ROSAT PSPC survey. They detected a large number of extended X-ray
sources with $f>$ 1.4$\times$10-13 erg s-1 cm-2 in the soft (0.5–2 keV) band.
They compared all their detections with their optical counterparts, and
confirmed 266 out of 283 as galaxy clusters. From this catalogue, a subsample
of the low (0.02–0.3)-$z$ and high (0.3–0.9)-$z$ galaxy clusters has been
observed with the $Chandra$ X-ray telescope (V09). Fig. 1a shows the
distribution of redshift, while Fig. 1b shows the luminosity distribution. In
this plot, median $z$ = 0.0853 and median luminosity = 1.8$\times$1044 (erg/s)
(0.5–2 keV). Tables 1 and 2 list the low- and high-$z$ samples of galaxy
clusters, respectively. We normalised $H$ dependent quantities (e.g. $L_{X}$
and mass) with $H_{0}$=73 km s-1 Mpc-1.

Figure 1: (a) Redshift and (b) luminosity distribution for low- and high-$z$ clusters (from the V09). Solid line = low-$z$; and dashed line = high-$z$ clusters. Table 1: Low-$z$ (0.02–0.3) cluster sample from the V09. (1) cluster name; (2) redshift; (3) total X-ray flux (0.5–2 keV); (4) total X-ray luminosity (0.5–2 keV); (5) average temperature in [0.15-1]$R_{500}$ annulus; (6) dynamical state (a) (according to the V09); (7) dynamical state (b) (combination of morphology parameters (see § 4.3)) (SR = strong relaxed, R = relaxed, NR = non-relaxed, SNR = strong non-relaxed); (8) exposure time. Name | $z$ | flux | Luminosity | Temperature | Dynamical state | Dynamical state | Exposure time
---|---|---|---|---|---|---|---
|  |  (10 -11cgs) |  (erg s-1) | (keV) | (a) | (b) | ks
A3571 | 0.0386 | 7.42 |  2.30$\times$1044 |  6.81 $\pm$ 0.10 | R | NR | 31.224
A2199 | 0.0304 | 6.43 |  1.23$\times$1044 |  3.99 $\pm$ 0.10 | R | NR | 17.787
2A 0335 | 0.0346 | 6.24 |  1.55$\times$1044 |  3.43 $\pm$ 0.10 | R | SR | 19.716
A496 | 0.0328 | 5.33 |  1.19$\times$1044 |  4.12 $\pm$ 0.07 | R | NR | 88.878
A3667 | 0.0557 | 4.64 |  3.05$\times$1044 |  6.33 $\pm$ 0.06 | M | NR | 60.403
A754 | 0.0542 | 4.35 |  2.70$\times$1044 |  8.73 $\pm$ 0.00 | M | SNR | 43.972
A85 | 0.0557 | 4.30 |  2.83$\times$1044 |  6.45 $\pm$ 0.10 | R | NR | 38.201
A2029 | 0.0779 | 4.23 |  5.56$\times$1044 |  8.22 $\pm$ 0.16 | R | SR | 10.739
A478 | 0.0881 | 4.16 |  7.04$\times$1044 |  7.96 $\pm$ 0.27 | R | SR | 42.393
A1795 | 0.0622 | 4.14 |  3.42$\times$1044 |  6.14 $\pm$ 0.10 | R | SR | 91.301
A3558 | 0.0469 | 4.11 |  1.90$\times$1044 |  4.88 $\pm$ 0.10 | M | NR | 14.024
A2142 | 0.0904 | 3.94 |  7.00$\times$1044 |  10.04 $\pm$ 0.26 | R | NR | 44.367
A2256 | 0.0581 | 3.61 |  2.58$\times$1044 |  8.37 $\pm$ 0.24 | M | SNR | 11.647
A4038 | 0.0288 | 3.48 |  6.01$\times$1043 |  2.61 $\pm$ 0.05 | R | NR | 33.523
A2147 | 0.0355 | 3.47 |  9.14$\times$1043 |  3.83 $\pm$ 0.12 | M | SNR | 17.879
A3266 | 0.0602 | 3.39 |  2.61$\times$1044 |  8.63 $\pm$ 0.18 | M | SNR | 29.289
A401 | 0.0743 | 3.19 |  3.79$\times$1044 |  7.72 $\pm$ 0.30 | R | NR | 18.005
A2052 | 0.0345 | 2.93 |  7.26$\times$1043 |  3.03 $\pm$ 0.07 | R | NR | 36.754
Hydra-A | 0.0549 | 2.91 |  1.87$\times$1044 |  3.64 $\pm$ 0.06 | R | R | 89.809
A119 | 0.0445 | 2.47 |  1.03$\times$1044 |  5.72 $\pm$ 0.00 | M | NR | 11.537
A2063 | 0.0342 | 2.39 |  5.81$\times$1043 |  3.57 $\pm$ 0.19 | R | NR | 8.777
A1644 | 0.0475 | 2.33 |  1.10$\times$1044 |  4.61 $\pm$ 0.14 | M | NR | 18.712
A3158 | 0.0583 | 2.30 |  1.67$\times$1044 |  4.67 $\pm$ 0.07 | R | NR | 30.921
MKW3s | 0.0453 | 2.08 |  9.02$\times$1043 |  3.03 $\pm$ 0.05 | R | NR | 57.123
A1736 | 0.0449 | 2.04 |  8.69$\times$1043 |  2.95 $\pm$ 0.09 | M | SNR | 14.918
EXO0422 | 0.0382 | 2.01 |  6.17$\times$1043 |  2.84 $\pm$ 0.09 | R | R | 10.001
A4059 | 0.0491 | 2.00 |  1.02$\times$1044 |  4.25 $\pm$ 0.08 | R | NR | 13.236
A3395 | 0.0506 | 1.95 |  1.06$\times$1044 |  5.10 $\pm$ 0.17 | M | SNR | 21.094
A2589 | 0.0411 | 1.94 |  6.90$\times$1043 |  3.17 $\pm$ 0.27 | R | NR | 13.478
A3112 | 0.0759 | 1.89 |  2.36$\times$1044 |  5.19 $\pm$ 0.21 | R | R | 15.466
A3562 | 0.0489 | 1.84 |  9.32$\times$1043 |  4.31 $\pm$ 0.12 | R | NR | 19.283
A1651 | 0.0853 | 1.80 |  2.85$\times$1044 |  6.41 $\pm$ 0.25 | R | NR | 9.643
A399 | 0.0713 | 1.78 |  1.95$\times$1044 |  6.49 $\pm$ 0.17 | M | NR | 48.575
A2204 | 0.1511 | 1.74 |  9.10$\times$1044 |  8.55 $\pm$ 0.58 | R | SR | 9.609
A576 | 0.0401 | 1.72 |  5.82$\times$1043 |  3.68 $\pm$ 0.11 | R | NR | 29.078
A2657 | 0.0402 | 1.62 |  5.50$\times$1043 |  3.62 $\pm$ 0.15 | R | NR | 16.148
A2634 | 0.0305 | 1.61 |  3.11$\times$1043 |  2.96 $\pm$ 0.09 | R | - | 49.528
A3391 | 0.0551 | 1.58 |  1.02$\times$1044 |  5.39 $\pm$ 0.19 | R | NR | 17.461
A2065 | 0.0723 | 1.56 |  1.77$\times$1044 |  5.44 $\pm$ 0.09 | M | NR | 49.416
A1650 | 0.0823 | 1.53 |  2.26$\times$1044 |  5.29 $\pm$ 0.17 | R | R | 27.258
A3822 | 0.0760 | 1.48 |  1.85$\times$1044 |  5.23 $\pm$ 0.30 | M | NR | 8.067
S 1101 | 0.0564 | 1.46 |  1.00$\times$1044 |  2.44 $\pm$ 0.08 | R | R | 9.946
A2163 | 0.2030 | 1.38 |  1.33$\times$1045 |  14.72 $\pm$ 0.31 | M | NR | 71.039
ZwCl1215 | 0.0767 | 1.38 |  1.75$\times$1044 |  6.54 $\pm$ 0.21 | R | NR | 11.999
RXJ1504 | 0.2169 | 1.35 |  1.51 $\times$1045 |  9.89 $\pm$ 0.53 | R | SR | 13.290
A2597 | 0.0830 | 1.35 |  2.03$\times$1044 |  3.87 $\pm$ 0.11 | R | SR | 26.414
A133 | 0.0569 | 1.35 |  9.33$\times$1043 |  4.01 $\pm$ 0.11 | R | R | 34.471
A2244 | 0.0989 | 1.34 |  2.89$\times$1044 |  5.37 $\pm$ 0.12 | R | NR | 56.964
A3376 | 0.0455 | 1.31 |  5.72$\times$1043 |  4.37 $\pm$ 0.13 | M | NR | 44.267
Table 2: As in Table 1, but for high-$z$ (0.3–0.9) clusters. Name | $z$ | flux | Luminosity | Temperature | Dynamical state | Dynamical state | Exposure time
---|---|---|---|---|---|---|---
|  |  (10 -13cgs) |  (erg s-1) | (keV) | (a) | (b) | ks
0302-0423 | 0.3501 | 15.34 |  5.09$\times$1044 |  4.78 $\pm$ 0.75 | R | SR | 100.41
1212+2733 | 0.3533 | 10.53 |  3.51$\times$1044 |  6.62 $\pm$ 0.89 | M | NR | 14.581
0350-3801 | 0.3631 | 1.68 |  6.61$\times$1043 |  2.45 $\pm$ 0.50 | M | NR | 23.733
0318-0302 | 0.3700 | 4.63 |  1.77$\times$1044 |  4.04 $\pm$ 0.63 | M | NR | 14.578
0159+0030 | 0.3860 | 3.30 |  1.38$\times$1044 |  4.25 $\pm$ 0.96 | R | R | 19.880
0958+4702 | 0.3900 | 2.22 |  1.01$\times$1044 |  3.57 $\pm$ 0.73 | R | R | 25.231
0809+2811 | 0.3990 | 5.40 |  2.43$\times$1044 |  4.17 $\pm$ 0.73 | M | SNR | 19.338
1416+4446 | 0.4000 | 4.01 |  1.88$\times$1044 |  3.26 $\pm$ 0.46 | R | NR | 29.187
1312+3900 | 0.4037 | 2.71 |  1.33$\times$1044 |  3.72 $\pm$ 1.06 | M | SNR | 26.421
1003+3253 | 0.4161 | 3.04 |  1.48$\times$1044 |  5.44 $\pm$ 1.40 | R | R | 19.859
0141-3034 | 0.4423 | 2.06 |  1.28$\times$1044 |  2.13 $\pm$ 0.38 | M | SNR | 28.273
1701+6414 | 0.4530 | 3.91 |  2.32$\times$1044 |  4.36 $\pm$ 0.46 | R | R | 49.294
1641+4001 | 0.4640 | 1.43 |  9.20$\times$1043 |  3.31 $\pm$ 0.62 | R | NR | 45.345
0522-3624 | 0.4720 | 1.47 |  1.01$\times$1044 |  3.46 $\pm$ 0.48 | M | NR | 45.999
1222+2709 | 0.4720 | 1.39 |  9.61$\times$1043 |  3.74 $\pm$ 0.61 | R | NR | 49.117
0355-3741 | 0.4730 | 2.48 |  1.71$\times$1044 |  4.61 $\pm$ 0.82 | R | R | 27.190
0853+5759 | 0.4750 | 1.22 |  8.20$\times$1043 |  3.42 $\pm$ 0.67 | M | NR | 42.179
0333-2456 | 0.4751 | 1.33 |  9.52$\times$1043 |  3.16 $\pm$ 0.58 | M | NR | 34.160
0926+1242 | 0.4890 | 2.04 |  1.45$\times$1044 |  4.74 $\pm$ 0.71 | M | NR | 18.620
0030+2618 | 0.5000 | 2.09 |  1.52$\times$1044 |  5.63 $\pm$ 1.13 | M | NR | 57.362
1002+6858 | 0.5000 | 2.19 |  1.66$\times$1044 |  4.04 $\pm$ 0.83 | M | NR | 19.786
1524+0957 | 0.5160 | 2.45 |  2.01$\times$1044 |  4.23 $\pm$ 0.51 | M | SNR | 49.886
1357+6232 | 0.5250 | 1.90 |  1.58$\times$1044 |  4.60 $\pm$ 0.69 | R | NR | 43.862
1354-0221 | 0.5460 | 1.45 |  1.36$\times$1044 |  3.77 $\pm$ 0.53 | M | NR | 55.039
1120+2326 | 0.5620 | 1.68 |  1.74$\times$1044 |  3.58 $\pm$ 0.44 | M | SNR | 70.262
0956+4107 | 0.5870 | 1.64 |  1.79$\times$1044 |  4.40 $\pm$ 0.50 | M | NR | 40.165
0328-2140 | 0.5901 | 2.09 |  2.23$\times$1044 |  5.14 $\pm$ 1.47 | R | NR | 56.192
1120+4318 | 0.6000 | 3.24 |  3.64$\times$1044 |  4.99 $\pm$ 0.30 | R | NR | 19.837
1334+5031 | 0.6200 | 1.76 |  2.16$\times$1044 |  4.31 $\pm$ 0.28 | M | NR | 19.492
0542-4100 | 0.6420 | 2.21 |  2.83$\times$1044 |  5.45 $\pm$ 0.77 | M | NR | 50.008
1202+5751 | 0.6775 | 1.34 |  2.15$\times$1044 |  4.08 $\pm$ 0.72 | M | SNR | 57.210
0405-4100 | 0.6861 | 1.33 |  2.16$\times$1044 |  3.98 $\pm$ 0.48 | M | NR | 77.163
1221+4918 | 0.7000 | 2.06 |  3.25$\times$1044 |  6.63 $\pm$ 0.75 | M | NR | 78.887
0230+1836 | 0.7990 | 1.09 |  2.48$\times$1044 |  5.50 $\pm$ 1.02 | M | SNR | 67.178
0152-1358 | 0.8325 | 2.24 |  5.31$\times$1044 |  5.40 $\pm$ 0.97 | M | NR | 36.285
1226+3332 | 0.8880 | 3.27 |  8.19$\times$1044 |  11.08 $\pm$ 1.39 | M | NR | 64.21

Properties of each cluster provided by the V09 are useful in comparison with
their X-ray morphology. The total X-ray luminosity is calculated over the
0.5–2 keV band from accurate $Chandra$ flux. The average temperature is
measured from the spectrum integrated over a [0.15-1]$R_{500}$222$R_{500}$
being the radius defined as that enclosing a region with an over-density
$\bigtriangleup$ = 500 times the critical density at the cluster redshift.
annulus.

In agreement with V09, we used the ‘cuspiness’ parameter defined by Vikhlinin
et al. (2007) to discriminate between cooling flow (relaxed) and non-cooling
flow (unrelaxed) clusters. Based on this classification, we used 49 (34
relaxed + 15 non-relaxed) low-$z$ clusters and 36 (12 relaxed + 24 non-
relaxed) high-$z$ clusters.

There are three high-$z$ and two low-$z$ clusters that have multiple
components in their images:

  * •

There are two subclumps visible in the high redshift 1701+6414 cluster. We
used the North clump in our calculation. The other clump (in the S-W) is a
foreground low-$z$ X-ray cluster, confirmed by the NED database.

  * •

In the case of the high redshift system 1641+4001, there is a small clump
(foreground galaxy) in the S-W, which we excluded from our analysis.

  * •

Double components are visible in the 0328-2140 system. One is in the East and
the other in the West, the latter being a low redshift interloper. We used the
high redshift East cluster.

  * •

In A85, we focused on only the main North relaxed cluster, excluding the small
southern clump (Kempner et al. 2002).

  * •

In A1644 we excluded the small North component, and focused on the main larger
southern cluster only (Johnson et al. 2010; Reiprich et al. 2004).

###  3.2  Chandra data reduction and image preparation

In our sample, each cluster had at least 1500–2000 photons (V09), which
ensured good S/N. We visually checked all these clusters, and verified that
there was no cluster peak emission or centroid too close to the CCD edges.
There are still a number of cases where cluster emission falls within CCD gaps
and this can potential influence our results. We divided the counts image by
the corresponding exposure map and applied light smoothing (as mentioned
below) so as to minimise the effects of CCD gaps. This inspection revealed
that the relaxed cluster A2634 (Obs-Id 4816, ACIS-S) alone had low total
counts. It was therefore excluded from our sample and now we have total of 84
clusters in sample.

We processed all Chandra archival data with CIAO version 4.3 and CALDB version
4.4.6. We first used the `chandra_repro` task to reprocess all ACIS imaging
data. This script creates a new second level event file as well as a bad pixel
file by reading the data from the standard Chandra data distribution. After
reprocessing, we removed any high background flares (3$\sigma$ clipping) with
the task `lc_sigma_clip` routine and then attached the good time interval
(GTI) file to the events. All of our event files included the 0.3–7 keV broad
energy band. We binned each event file with $\sim$ 2” pixels. We detected
point sources in observation using the `wavdetect`
task333http://cxc.harvard.edu/ciao4.3/threads/wavdetect/ with scale = 1, 2, 4,
8 and 16, which would be a reasonable default for Chandra data. This scale
parameter refers the wavelet radii in image pixels. Then a wavelet transform
is performed for each scale in the given radii list to estimate source
properties. We also supplied information on the size of the PSF at each pixel
in the image using energy value of 1.49. We then selected elliptical
background annuli (twice the point source detected region) around each
detected point source. Then we removed the point sources using the `dmfilth`
task444http://cxc.harvard.edu/ciao4.3/threads/diffuse emission/, which
replaces pixel values in selected source regions of an image with values
interpolated from the surrounding background region. We used the POISSON
interpolation method, which randomly samples values in the selected source
regions from the Poisson distribution, whose mean is that of the pixel values
in the background region. We took care not to remove point source which was
detected close to the cluster centre, because in most cases, this point source
was simply the peak of the surface brightness distribution rather than an
actual X-ray point source. We also aimed to study the effect of cluster
surrounded point sources on parameters, hence we calculated morphology
parameters for both images, i.e. the one with all point sources, the other
without point sources (except central point source). For creating the exposure
map we used the `fluximage`
task555http://cxc.harvard.edu/ciao/ahelp/fluximage.html with broad band energy
(0.3–7 keV), then divided the counts image by the exposure map to remove the
CCD gaps, vignetting, and telescope effects. This task generates the aspect
histogram, instrument and exposure maps, automatically. We also used the
`dmimgthresh` task to perform a 5$\%$ total counts cut to enable uniform
exposure everywhere in the image. Two or more of the available multiple
observations were combined to make a single image. Finally we smoothed the
cluster images with a $\sigma$ = 10” Gaussian width. Smoothing is very
important in calculating the Gini coefficient, because there will be zero
count pixels in the unsmoothed count image, which will not contribute to the
flux distribution calculation and will change the Gini value greatly. For
calculating the Smoothness parameter, we initially used unsmoothed cluster
images.

Because of limited FOV of Chandra, in the study of nearby clusters, we decided
to use a fixed circular aperture size over which morphology parameters are
calculated for individual clusters (200 kpc for $z$ $<$ 0.05 and 500 kpc for
$z$ $>$ 0.05) with sufficient area, and to retain consistency with our
morphology parameter calculations on the same relative scale. The possible
effects of aperture size are discussed in § 5.2.

##  4 Morphology parameter results

###  4.1 Distribution of morphology parameters

Fig. 2 shows the distribution of parameters for our sample of 84 clusters.
Except for three parameters, Gini, $M_{20}$ and Concentration, most of the
parameters show a similar distribution for relaxed and non-relaxed clusters
based on the classification described in § 3.1. The Smoothness parameter
separates the two peaks of relaxed and non-relaxed clusters towards low
Smoothness and high Smoothness, respectively, but there are large numbers of
clusters in the overlap region. To further investigate the Smoothness
parameter, we (1) varied the $\sigma$ value between $0.5^{\prime}$ to
$1^{\prime}$, and (2) varied the smoothing size of input cluster images
(initial smoothing) on the Smoothness parameter (see § 4.4). For the Gini and
Concentration parameters, non-relaxed clusters are distributed towards low
values of Gini and Concentration and, oppositely, relaxed clusters are
characterised by high Gini and Concentration values. A similar trend is
visible for the $M_{20}$, but the extreme of the relaxed clusters is on the
left hand side (low $M_{20}$ values) and the extreme of the non-relaxed
clusters is on the right hand side (high $M_{20}$ values). Tables 10 and 11
list all parameter values together with 1$\sigma$ uncertainty (§ 2.5) for the
V09 low- and high-$z$ sample clusters.

Figure 2: Seven morphology parameter distributions: solid line = relaxed cluster; dashed line = non-relaxed cluster. All four definitions of the Concentration parameter are displayed. Galaxy cluster separation (i.e. relaxed vs non-relaxed systems) is based on the V09. Table 3: Statistics for the subsample of relaxed (R) and non-relaxed (N-R) clusters, and the combined (C) cluster sample from the V09. | mean | median | K-S probability | R-S probability
---|---|---|---|---
| R N-R C | R N-R C |  |
Gini | 0.55 0.39 0.47 | 0.54 0.39 0.44 |  1.42$\times$10-07 |  8.73$\times$10-09
$M_{20}$ | -1.96 -1.44 -1.72 | -1.96 -1.44 -1.73 |  8.65$\times$10-09 |  8.63$\times$10-11
Concentration | 1.38 0.95 1.18 | 1.33 0.92 1.11 |  8.87$\times$10-08 |  2.26$\times$10-9
Asymmetry | 0.39 0.44 0.42 | 0.38 0.44 0.40 | 0.12 | 0.027
Smoothness | 0.68 1.05 0.85 | 0.61 1.09 0.94 | 0.0020 |  6.7$\times$10-5
$G_{M}$ | 0.27 0.31 0.29 | 0.28 0.30 0.28 | 0.012 | 0.003
Ellipticity | 0.08 0.09 0.09 | 0.08 0.09 0.08 | 0.95 | 0.66

We performed the Kolmogorov-Smirnov (K-S) test to investigate whether or not
the relaxed and non relaxed cluster distributions are drawn from the same
parent distribution. The results are given for each parameter in Table 3,
where a threshold of 1% means that the probability is $<$ 0.01, implying a
possible rejection of the null hypothesis. In the table we include the
Wilcoxon rank sum (R-S) test which establishes the probability of whether the
two samples with the null hypothesis have the same mean. A threshold of 0.1%
implies that we reject the null hypothesis (that they have the same mean) if
the probability is $<$ 0.001. In addition, the mean and median values for all
morphology parameters are supplied. From the statistical tests, we observed a
significant separation between the two distributions (relaxed and non-relaxed
clusters) for the Gini, $M_{20}$ and Concentration, which rejects the null
hypotheses that the two samples (relaxed and non-relaxed clusters) are the
same and they have the same mean values. The K-S and R-S probabilities are $<$
0.01 and 0.001, respectively, for the Smoothness parameter; but we observed a
high incidence of overlapping between relaxed and non-relaxed clusters for the
Smoothness distribution. We found that the right side peak of non-relaxed
clusters (dotted line) in the Smoothness distribution corresponds mainly to
the high-$z$ clusters which have low S/N compared with nearby clusters (which
fall mainly on the left side). Further details concerning the Smoothness
parameter is given in § 4.4. For the remaining parameters, we cannot reject
the above null hypotheses, which means that Asymmetry, Gini of the second
order moment and Ellipticity are not useful in separating relaxed and non-
relaxed clusters.

###  4.2 Parameter vs Parameter planes

Using combinations of morphology parameters we investigated relaxed vs non-
relaxed clusters in the parameter-parameter plane to study the dynamical
states of galaxy clusters and the correlation between each morphology
parameter. In Fig. 3 we plotted our results in parameter-parameter planes.
Three parameters, Gini, Concentration and $M_{20}$, look particularly powerful
after combining with other parameters to differentiate between non-relaxed and
relaxed clusters. Clusters with different dynamical states (as classified by
the V09) occupy distinct regions in the parameter-parameter planes; for e.g.,
in the Concentration vs Asymmetry plot, all relaxed clusters occupy the upper
region, while the lower region is occupied by non-relaxed clusters. In our
analysis we did not observe any correlation between cluster dynamical states
and Ellipticity or Asymmetry. As seen in Fig. 3, galaxy clusters within our
sample show a range of different morphologies and are not concentrated in
particular positions of the parameter-parameter space. This is probably
because of the hierarchical cluster formation process, indicating that
clusters pass through multiple (merger) phases in their evolution. Each phase
is dynamically important, and traces the cluster properties. This could help
in the understanding of large and complex structure formations in the standard
cosmological model.

We used the Spearman rank-order correlation coefficient, $\rho$, to quantify
any correlation between parameter pairs. This resulted in a correlation
coefficient between ranks of a group of individuals for a given pair of
attributes. In order to calculate $\rho$, it is necessary to assign ranks (low
to high or high to low) to a given set of variables, individually. The next
step is to measure the deviation between the ranks of variable pairs, square
this rank difference, and sum up. The value of the rank order correlation
coefficient varies between -1 and 1. If the variables are anti-correlated then
$\rho$ falls between -1 and 0, while a value for $\rho$ between 0 and 1
implies a positive correlation between given variables. $\rho$ = 0 implies no
correlation between variables.

Fig. 3 indicates that the Concentration is tightly correlated with the Gini,
while anti-correlated with the $M_{20}$ (see also Table 4). The Concentration
(Santos et al. 2008; Hudson et al. 2010; Cassano et al. 2010b) is a useful
parameter in separating non-relaxed from relaxed clusters with almost all
morphology parameters. Fig. 3 illustrates our plot of the $C_{5080}$
Concentration parameter. As seen in Fig. 3, relaxed clusters occupy the upper
left corner, while non-relaxed clusters occupy the bottom-right corner of the
$M_{20}$ vs Gini and Concentration planes. In the Gini vs Concentration plane,
relaxed clusters fall in the upper-right corner, while the bottom-left corner
is occupied by non-relaxed clusters. In this study we found that the Gini
coefficient was also useful in separating non-relaxed clusters from relaxed
clusters when plotted against most other parameters. The Gini, Concentration
and $M_{20}$ parameters are all inter-related as well (Table 4); and the Gini
coefficient could be useful to use as a proxy of the Concentration for
detecting substructure in high-$z$ clusters. The advantage of using the Gini
coefficient is that it is independent of the precise location of a galaxy
cluster’s centre. The Asymmetry parameter is correlated with Gini of the
second order moment and Smoothness parameters (Okabe et al. 2010; Zhang et al.
2010; Rasia et al. 2012). We did not observe any correlation between the other
six parameters and Ellipticity. In general, three parameters - Gini, $M_{20}$
and Concentration - are very promising tools in the morphological
classification of clusters.

Despite this reasonable division between relaxed and non-relaxed clusters, we
observed an overlap between some clusters, such as A401, A3571, A1651, A3158,
A3562, A576, A2063, ZwCl1215, A2657, A2589, A3391, 0355-3741, 1641+4001,
1120+4318, 1222+2709, 0328-2140 and 1357+6232. These are identified as relaxed
clusters in the V09, but in our parameter space they fall into the non-relaxed
region. A401 (Murgia et al. 2010) and A3562 (Venturi et al. 2003) host radio
halos (indicating possible signature of merger, see § 6.4) which are clearly
identified as non-relaxed clusters by our parameters (although they are
classified as relaxed in the V09). Dupke et al. (2007) predicted a line of
sight bullet-like cluster in A576, using a combination of $Chandra$ and $XMM$
observations. This gave us confidence that our parameter set provided a good
indication of cluster dynamical states. The remaining clusters may be weak
mergers (pre- or post-merger) or in intermediate state.

In cluster A85 (Kempner et al. 2002), in addition to the main cool core
cluster, two subclusters are visible, in the far South and NW, respectively.
We calculated parameters for the main relaxed cluster, and found that it falls
into the relaxed category, although very close to the boundary of the non-
relaxed side, which could indicate a weak interaction between the subcluster
and the main cool core cluster. Based on our measurements, we categorised most
relaxed clusters by Concentration $>$ 1.55 and $M_{20}$ $<$ -2.0, while
intermediate clusters were categorised by 1.0 $<$ Concentration $<$ 1.55 and
-2.0 $<$ $M_{20}$ $<$ -1.4, and distorted and non-relaxed clusters were
categorised by Concentration $<$ 1.0 and $M_{20}$ $>$ -1.4. According to the
Gini coefficient, most relaxed clusters have a Gini value $>$ 0.65 and non-
relaxed have Gini value $<$ 0.40. We chose these boundaries based on visual
identification of each cluster morphology (Fig. 3). Mean and median values
(Table 3) produce a significant overlap due to the presence of a large number
of intermediate clusters in our sample.

Figure 3: Seven morphology parameters plotted in the parameter-parameter
planes. $C_{5080}$ was plotted as the Concentration parameter. $\circ$ =
relaxed cluster;  $\times$ = non-relaxed cluster. Galaxy cluster separation is
based on the V09.

###  4.3 Combination of morphology parameters

We decided to further classify dynamical states of each cluster based on the
combination of Gini, $M_{20}$ and Concentration. In § 4.2, we defined three
parameter boundaries. Based on the combination of these three parameters, we
categorised the V09 sample of clusters into four stages - strong relaxed,
relaxed, non-relaxed and strong non-relaxed clusters. We selected the
following criteria for identifying the dynamical state of each cluster:

  * •

If the Gini, $M_{20}$ and Concentration all indicate relaxed state, then the
cluster will be “strong relaxed”.

  * •

The intermediate state of clusters is further divided into two classes: if one
or two parameters fall into the intermediate state and another is relaxed (or
non-relaxed), then the cluster will be “relaxed (or non-relaxed)”.

  * •

If all three parameters indicate a non-relaxed state, then the cluster will be
“strong non-relaxed”.

Based on these categories, in the entire V09 cluster sample, 8 ($\sim$ 10%)
clusters were strong relaxed, 11 ($\sim$ 13%) were relaxed, 52 ($\sim$ 62%)
were non-relaxed, and 13 ($\sim$ 15%) were strong non-relaxed. In the low-$z$
sample, there were 7 ($\sim$ 15%) strong relaxed, 6 ($\sim$ 12.5%) relaxed, 29
($\sim$ 60%) non-relaxed, and 6 ($\sim$ 12.5%) strong non-relaxed clusters. In
the high-$z$ sample, there were 1 ($\sim$ 3%) strong relaxed, 5 ($\sim$ 14%)
relaxed, 23 ($\sim$ 64%) non-relaxed, and 7 ($\sim$ 19%) strong non-relaxed
clusters. This could suggest that the highest fraction of clusters (in the V09
sample) are evolving or showing some substructure activity (particularly in
the high-$z$ sample) and that fewer clusters are fully evolved. Fig. 4 shows
the 3-D plot of the Gini, $M_{20}$ and Concentration parameters. A combination
of these three parameters classified galaxy clusters as strong relaxed
($\bullet$), relaxed ($\diamond$), non-relaxed ( $+$) and strong non-relaxed
clusters ( $\times$).

Figure 4: Gini, $M_{20}$ and Concentration parameters plotted in the 3-D plot.
We plotted $C_{5080}$ as the Concentration parameter. $\bullet$ = strong
relaxed clusters, $\vartriangle$ = relaxed clusters,  $+$ = non-relaxed
clusters, and  $\times$ = strong non-relaxed clusters.

###  4.4 Smoothness and Asymmetry parameters

In this section we will test different possible smoothing and angular size
issues relating to the Smoothness and Asymmetry parameters. Okabe et al.
(2010) used a $2^{\prime}$ smoothing scale for calculating the fluctuation
parameter. They, however, used it for $R_{500}$ radius and the data of XMM-
Newton which has a large FOV compared with Chandra. We used $0.5^{\prime}$ (15
pix) and $1^{\prime}$ (30 pix) smoothing scales, $\sigma$, with initial
$4^{\prime\prime}$ (2 pix) and $10^{\prime\prime}$ (5 pix) smoothing of input
cluster images (which is sufficient to not washed out any underlying
substructure features). Results are shown in Fig. 5. We find that the
different smoothing scales ($\sigma$) had little affect on the Smoothness
parameter. In this analysis, we used a fixed angular size of
$2^{\prime\prime}$/pix to bin each cluster image and the same angular
smoothing scale ($\sigma$) for low- and high-$z$ clusters. This angular
scaling could affect the Smoothness parameter. To overcome this, and instead
of using a fixed angular size, we scaled each cluster image (of low- and
high-$z$) in terms of fixed physical size of 10 kpc/pix. We then used 50 and
150 kpc smoothing scales ($\sigma$) to smooth cluster images in order to
calculate the Smoothness parameter. Both of these $\sigma$ values were chosen
based on the available aperture size (segmentation map) of cluster for which
we calculated the Smoothness parameter. We also smoothed cluster input images
with (1) 15 (1.5 pix) and (2) 30 (3 pix) kpc Gaussian kernel sizes (which is
sufficient to not washed out any underlying substructure features). Fig. 6
gives the calculated smoothness parameter for fixed physical smoothing kernel
size. The Smoothness parameter had barely changed from the previous results,
and had still not separated the two classes of cluster, i.e. relaxed and non-
relaxed. Furthermore, most clusters (mainly high-$z$) do not have $\gg$ 1
count in each (binned) pixel. So, in general, the Smoothness is not a good
parameter for a large number of clusters in which each has a different
exposure time. In § 5.3, further we show that the Smoothness parameter depends
highly on S/N.

We also investigated the Asymmetry parameter using various smoothing Gaussian
kernel sizes to smooth cluster input images. We rotated this smoothed image by
180∘, subtracted it from the input image, and normalised it. Our results for
the fixed angular and physical scale cluster images are given in Fig. 7. None
of the results indicates the separation of relaxed and non-relaxed clusters.
As for the Smoothness parameter, Asymmetry parameter is not useful because the
S/N is inadequate in a given large sample of clusters (see § 5.3).

Figure 5: Calculation of the Smoothness parameter for two different Gaussian kernel sizes of fixed angular scale: solid line = relaxed cluster; dashed line = non-relaxed cluster. N shows smoothing size for input cluster image. Galaxy cluster separation is based on the V09. Figure 6: Calculation of the Smoothness parameter for two different Gaussian kernel sizes of fixed physical scale: solid line = relaxed cluster; dashed line = non-relaxed cluster. N shows smoothing size for input cluster image. Galaxy cluster separation is based on the V09. Figure 7: Calculation of the Asymmetry parameter for two different Gaussian kernel sizes of fixed angular and physical scale: solid line = relaxed cluster; dashed line = non-relaxed cluster. N shows smoothing size for input cluster image. Galaxy cluster separation is based on the V09. Table 4: Spearman coefficient, $\rho$, for the subsample of relaxed, non-relaxed, and combined clusters, between each morphology parameter. | Gini | $M_{20}$ | Concentration | Asymmetry | Smoothness | $G_{M}$ | Ellipticity
---|---|---|---|---|---|---|---
| R N-R C | R N-R C | R N-R C | R N-R C | R N-R C | R N-R C | R N-R C
Gini | - | -0.75 -0.58 -0.83 | 0.87 0.81 0.93 | 0.05 0.34 -0.02 | -0.40 0.14 -0.38 | 0.11 0.14 -0.12 | 0.25 -0.04 0.05
$M_{20}$ | -0.75 -0.58 -0.83 | - | -0.91 -0.80 -0.92 | 0.13 0.12 0.24 | 0.18 0.15 0.40 | 0.07 0.21 0.29 | -0.15 0.18 0.04
Concentration | 0.87 0.81 0.93 | -0.91 -0.80 -0.92 | - | -0.06 0.04 -0.16 | -0.29 -0.10 -0.41 | -0.05 -0.20 -0.28 | 0.13 -0.10 -0.01
Asymmetry | 0.05 0.34 -0.02 | 0.13 0.12 0.24 | -0.06 0.04 -0.16 | - | 0.38 0.76 0.56 | 0.68 0.83 0.72 | -0.22 -0.05 -0.12
Smoothness | -0.40 0.14 -0.38 | 0.18 0.15 0.40 | -0.29 -0.10 -0.41 | 0.38 0.76 0.56 | - | 0.42 0.85 0.65 | -0.33 -0.01 -0.13
$G_{M}$ | 0.11 0.14 -0.12 | 0.07 0.21 0.29 | -0.05 -0.20 -0.28 | 0.68 0.83 0.72 | 0.42 0.85 0.65 | - | -0.20 -0.03 -0.06
Ellipticity | 0.25 -0.04 0.05 | -0.15 0.18 0.04 | 0.13 -0.10 -0.017 | -0.22 -0.05 -0.12 | -0.33 -0.019 -0.13 | -0.20 -0.03 -0.06 | -

##  5 Systematics

We investigated a number of possible systematic effects, discussed below, to
study how robust these parameters are in various conditions.

###  5.1 Effect of point sources

In order to test the effect of point sources detected around cluster on the
calculation of parameters, we calculated morphology parameters on (exposure
corrected and smoothed) cluster images without removing any of the detected
point source. We noticed that parameters are fairly robust against the
inclusion (or exclusion) of point sources into the parameter calculation. Fig.
8 indicates how we plotted the offset for four parameters (Gini, $M_{20}$,
Concentration and Smoothness) calculated with and without point sources. As we
observed, less offset between parameters (with or without point sources)
suggests that they are quite robust. The Gini and $M_{20}$, however, each show
a small extended tail on the right hand side (Fig. 8). This result is obvious,
because the Gini coefficient includes bright pixels from point sources in the
calculation, and thus increases by a small value. In the case of the $M_{20}$,
if the point sources are very close to the centre, its value will decrease by
a small factor.

Figure 8: Four parameters calculated with and without point sources, with
plotted offset between them. Here we plotted $C_{5080}$ as the Concentration
parameter. Less offset suggests that parameters are quite robust.

###  5.2 Aperture size effect

Aperture radii have an important effect when measuring morphology parameters.
We chose to use a fixed physical size radius rather than a fixed overdensity
($R_{500}$) radius, because it is difficult to measure $R_{500}$ accurately
for non-relaxed clusters. Although a study of the possible bias effects of
this choice goes beyond the purpose of this paper, we note that possible bias
effects could be introduced when comparing clusters at the same redshift, but
characterised by different luminosities (masses). The large variation in
aperture size does affect the parameters because they are related to surface
brightness. To check the effect of various aperture radii on the parameters,
we chose a subsample that included both distant and nearby clusters and
calculated the morphology parameters in the radius sequence of 100 kpc to 1
Mpc for distant clusters ($z$ $>$ 0.05), while 100 kpc to 500 kpc for nearby
clusters ($z$ $<$ 0.05). Fig. 9 illustrates our result. In Fig. 9, the distant
clusters are plotted with a continuous line, while the nearby clusters are
plotted with a dotted line. As per Fig. 9, in the case of distant clusters, a
few parameters (Asymmetry, Smoothness, $G_{M}$, and Ellipticity) remain
constant in spite of aperture size, while the Gini, $M_{20}$ and Concentration
parameters are sensitive to the aperture radius within which they are
calculated, while tending to be stable for aperture radii greater than 400
kpc. The Gini parameter increases with radius as more (faint) sky pixels are
included in the extraction aperture. Parameter values for nearby clusters (in
particular $z$ $<$ 0.05) are very sensitive to the chosen aperture size.

Figure 9: Morphology parameters calculated for a range of aperture sizes from
100 kpc to 1 Mpc for distant clusters. Nearby clusters are calculated in a
range of radii from 100 kpc to 500 kpc only. Solid lines with $\bullet$ symbol
are plotted for distant clusters, and dashed lines with $\times$ symbol are
plotted for nearby clusters.

###  5.3 Exposure time effect

It is important to check the consistency of the parameters over different
exposure times. Observations with a shorter exposure time are likely to have
lower S/N. Consequently, we were able to check the robustness of the
parameters for cluster images with lower S/N. However, to rescale the real
data by exposure time and then add Poisson noise gives an image which has an
excessive amount of Poisson noise, in addition to the intrinsic noise present
in the real observation. The simplest solution is to simulate a cluster image
with no intrinsic noise, rather than using real data. To achieve this, we
needed to estimate the different complex characteristics of a model to
simulate galaxy clusters. This task is considered difficult in the cases of
non-relaxed and disturbed clusters. To overcome this problem, Hashimoto et al.
(2007) suggested a novel technique for galaxy cluster simulations called
“adaptive scalings”, using real data and adding noise to the rescaled image.
We refer the reader to Hashimoto et al. (2007), for more details about this
technique.

An example of this adaptive method is given in Fig. 10; an example of the low
exposure time effect on a cluster. Subsequently we calculated our morphology
parameters for each short to long exposure cluster image.

Figure 10: Simulated clusters for different exposure times. Horizontally, from
top left to bottom right, the images were arranged from 2 ks to 18 ks,
respectively, in order of 2 ks exposure time. Figure 11: Robustness of
morphology parameters calculated for several clusters against various
simulated exposure lengths.

Fig. 11 shows how morphology parameters behave with different exposure
lengths. We simulated a few original clusters with various exposure times, as
described above, and re-calculated the morphology parameters for each
simulated cluster. As depicted in Fig. 11, all parameters are robust against
different exposure times, with the exception of the Smoothness parameter. We
noted, however, that the Gini, Gini of the second order moment and Asymmetry
were not reliable when exposure time was $\lesssim$ 5 ks. In the cases of the
Gini and Gini of the second order moment, low exposure means low S/N,
indicating that the Gini and Gini of the second order moment values are high
for low exposure observations, although fairly consistent for exposure times
$\gtrsim$ 5 ks. For the short exposure times, there are few bright pixels
within the given aperture radius (the rest are scattered to very low $\sim$
0). In other words, low S/N causes broader flux distribution in the faintest
pixels, resulting in strong variation in the flux distribution, for which we
calculated the Gini coefficient; and hence the Gini coefficient gives a high
value compared with the long exposure time. The value of the Smoothness
decreases continuously with increasing exposure time. This is to be expected
because, for high S/N and exposure time, the flux distribution in any given
cluster image becomes smooth and less patchy. Care must therefore be taken
when calculating the Smoothness parameter. We found that the Smoothness is
weakly correlated with the Gini of the second order moment and Asymmetry
parameters, yet it is not established whether we can use these latter
parameters as a substitute for the Smoothness. The remainder of the parameters
were relatively constant over different exposure lengths.

###  5.4 Redshift effect

It is hoped that these morphology parameters could be used to trace the
evolution of galaxy clusters with redshifts. It is therefore important to
understand the robustness as a function of redshift and, to this end, it is
necessary to check the following criteria:

  1. 1.

The effect of various angular bin sizes on morphology parameters.

  2. 2.

The surface brightness dimming effect on morphology parameters.

We simulated several observations of galaxy clusters at a higher redshift
($z_{1}$) than its real redshift ($z_{0}$) using real data from our sample
clusters. Again, we adopted the procedure describe by Hashimoto et al. (2007).

In Fig. 12 we give an example of clusters simulated via the method described
above that illustrates how a cluster appears at high redshift. Subsequent to
the simulation, we calculated morphology parameters for each cluster redshift
and plotted them (Fig. 13).

Figure 12: Simulated clusters for different redshifts. Figures are read from
top left to bottom right in order of low- to high-$z$, respectively. Figure
13: Robustness of morphology parameters which are calculated for several
clusters against simulated clusters at different redshifts.

We simulated a few original clusters with various redshifts, using the method
described above, and re-calculated morphology parameters for each simulated
cluster. Fig. 13 illustrates that the Smoothness is systematically increased
with redshift. The reason could be that high-$z$ clusters are noisy and appear
patchy. The Ellipticity parameter appears particularly noisy. Except for a few
clusters, and after $z$ $\gtrsim$ 0.5, the Gini, $M_{20}$, Concentration,
Asymmetry and Gini of the second order moment parameters are fairly constant,
and the systematics caused by redshifts are small.

##  6 Comparison of parameters with X-ray gas properties

We expect that the dynamical states of clusters to be influence by factors
like merger histories, which in turn could influence a number of properties
like luminosity, temperature structure, cooling flows etc. In order to test
the potential of our parameters as probes of the physical conditions in
clusters, we investigated the parameter correlations with source redshift,
global X-ray properties, cooling times and the presence of diffuse radio
continuum emission.

###  6.1 Redshift evolution

According to the concordance model, massive galaxy clusters start to form
around $z$ $\sim$ 1 and have continue to evolve up to the present epoch. To
look into the evolutionary effect on the distribution of the parameters, we
divided our entire sample into low-$z$ (0.02–0.3) and high-$z$ (0.3–0.9)
clusters and classified the samples based on our morphology parameters. We
also performed the K-S and R-S tests on each parameter distribution to observe
the difference between the two redshift bins. Table LABEL:kstest_redshift
lists the mean, median and statistical test results for the two redshift bins.

Below are more descriptions for the results of our three most promising
parameters (viz. Gini, $M_{20}$ & Concentration). The combination of the three
parameters are plotted in Fig. 14, with boundaries between relaxed and non-
relaxed clusters.

  1. 1.

Based on the Gini, we found that 9 clusters are relaxed, 48 are intermediate
and 27 are non-relaxed. In the low-$z$ subsample there are 8 ($\sim$ 17%)
relaxed, 25 intermediate ($\sim$ 52%) and 15 ($\sim$ 31%) non-relaxed
clusters; in the high-$z$ subsample there are 1 ($\sim$ 3%) relaxed, 23
($\sim$ 64%) intermediate and 12 ($\sim$ 33%) non-relaxed clusters.

  2. 2.

Based on the $M_{20}$, we found that 18 clusters are relaxed, 50 are
intermediate and 16 are non-relaxed. In the low-$z$ subsample there are 12
($\sim$ 25%) relaxed, 28 intermediate ($\sim$ 58%) and 8 ($\sim$ 17%) non-
relaxed clusters; in the high-$z$ subsample there are 6 ($\sim$ 17%) relaxed,
22 ($\sim$ 61%) intermediate and 8 ($\sim$ 22%) non-relaxed clusters.

  3. 3.

Based on the Concentration, we found 12 clusters are relaxed, 41 clusters are
intermediate and 31 clusters are non-relaxed. In the low-$z$ subsample there
are 10 ($\sim$ 21%) relaxed, 20 intermediate ($\sim$ 42%) and 18 ($\sim$ 37%)
non-relaxed clusters; in the high-$z$ subsample there are 2 ($\sim$ 6%)
relaxed, 21 ($\sim$ 58%) intermediate and 13 ($\sim$ 36%) non-relaxed
clusters.

As seen in Table LABEL:kstest_redshift, the K-S and R-S probabilities are $<$
1% and $<$ 0.1%, respectively, for the Asymmetry and Smoothness parameters
implying that we can reject both null hypotheses mentioned in § 4.1. Clearly
most of the clusters in our sample are in the intermediate stage ($\sim$
57$\%$ based on the Gini coefficient, $\sim$ 60$\%$ based on the $M_{20}$, and
$\sim$ 49$\%$ based on the Concentration parameter). Weak evolution is visible
in the Gini and Gini of the second order moment, which indicates the
possibility that high redshift clusters are more extended (which could mean
that they do not have a density peak at the cluster centre) compared with
those of low redshift. The $M_{20}$, Concentration and Ellipticity do not show
significant evolution. From our results, there is indication that relaxed
clusters are more dominant within the low-$z$ sample, which could indicate
that, in the current epoch, clusters show less substructure and are (fully)
evolved as compared to distant clusters, but these results are marginal.

Table 5: Statistics for two redshift bins: low-$z$ (0.02–0.3) and high-$z$ (0.3–0.9). Parameters |  mean of low-$z$ |  mean of high-$z$ |  median of low-$z$ |  median of high-$z$ | K-S probability | R-S probability
---|---|---|---|---|---|---
Gini | 0.50 | 0.44 | 0.47 | 0.43 | 0.08 | 0.09
$M_{20}$ | -1.75 | -1.68 | -1.76 | -1.67 | 0.74 | 0.35
Concentration | 1.21 | 1.14 | 1.14 | 1.06 | 0.57 | 0.47
Asymmetry | 0.38 | 0.46 | 0.36 | 0.46 | 0.0002 | 0.0008
Smoothness | 0.65 | 1.11 | 0.53 | 1.16 |  1.57 $\times$ 10-7 |  2 $\times$ 10-6
$G_{M}$ | 0.28 | 0.31 | 0.28 | 0.30 | 0.0016 | 0.004
Ellipticity | 0.10 | 0.08 | 0.09 | 0.07 | 0.08 | 0.024
Figure 14: Three parameters plotted in the parameter-parameter planes to show
cluster evolution with redshift. Here we plotted $C_{5080}$ as the
Concentration parameter.  $\circ$ = low-$z$ cluster (0.02–0.3);  $\times$ =
high-$z$ cluster (0.3–0.9). The dashed lines represent the boundaries between
relaxed and non-relaxed clusters. Boundary values for Gini = 0.65, $M_{20}$ =
-2.0 and Concentration = 1.55. R indicates relaxed clusters and N-R indicates
non-relaxed clusters.

###  6.2 X-ray luminosity, temperature and mass

We compared seven morphology parameters with three global cluster properties
(luminosity, temperature and mass) taken from the V09 to search for any
possible correlation of these global properties with cluster morphology. Fig.
15 shows a comparison, while Table 6 lists the Spearman coefficient values
calculated for the clusters’ global properties and galaxy cluster
morphologies. Fig. 15 defines each clusters’ dynamical state according to the
combination of the Gini, $M_{20}$ and Concentration morphology parameters (§
4.3).

No obvious correlation between cluster morphology and X-ray global properties
was found during our analysis. This may be due to the fact that quantities
such as X-ray luminosity and temperature are not solely related to the
dynamical state of clusters (Hashimoto et al. 2007). These properties also
depends strongly on cluster mass and on non-gravitational processes such as
supernovae feedback, central AGN heating, etc. (Donnelly et al. 1999; Tozzi &
Norman 2001; Neumann et al. 2003). Buote & Tsai (1996) also compared their
power ratio measurements for X-ray clusters with the ICM temperature and
luminosity, without finding any strong correlation between subclustering and
global X-ray properties. In agreement with our conclusion, they pointed out
that this lack of correlation is reasonable, since power ratios are a measure
of cluster evolution that do not take into account the cluster mass, to which
all the other X-ray quantities (e.g. luminosity and temperature) are
sensitive. We observed that all relaxed clusters occupy high Gini, low
$M_{20}$ and high Concentration values in all three plots. We therefore
concluded that, to identify the dynamical state of a galaxy cluster, X-ray
global properties are not as useful as the surface brightness distribution or
cluster morphology, quantifiable using morphology parameters.

Figure 15: Comparison of Left Luminosity, Middle Temperature and Right Mass (estimated from the $Y_{X}$ parameter) with morphology parameters. $\circ$ = strong relaxed clusters; $\lozenge$ = relaxed clusters;  $+$ = non-relaxed clusters; and  $\times$ = strong non-relaxed clusters. We plotted $C_{5080}$ as the Concentration parameter. We take all these global properties (luminosity, temperature and mass) values from the V09.  Table 6: Spearman coefficient, $\rho$, for morphology parameters and X-ray global properties (luminosity, temperature and mass).  | Luminosity | Temperature | Mass
---|---|---|---
Gini | 0.19 | 0.03 | 0.03
$M_{20}$ | -0.13 | 0.03 | 0.07
Concentration | 0.18 | -0.03 | -0.05
Asymmetry | -0.22 | -0.23 | -0.25
Smoothness | -0.33 | -0.26 | -0.30
$G_{M}$ | -0.25 | -0.17 | -0.21
Ellipticity | 0.26 | 0.30 | 0.35

###  6.3 X-ray cluster cooling time

X-ray emission is considered to be the primary cooling process for the ICM.
The cooling time is much shorter than the Hubble time at the centre of a
cluster, allowing gas to cool from a high ICM temperature $\sim$ 108 K down to
$\sim$ 107 K. Numerical simulations suggest that cluster mergers may disturb
gas cooling (Ritchie & Thomas 2002; Burns et al. 2008; Hudson et al. 2010). We
aimed to investigate the relationship between the degree of substructure and
cooling times or rates of a cluster. We used the cooling time information
supplied by Hudson et al. (2010). Table LABEL:cool_list lists the cooling time
values for low-$z$ clusters.

Table 7: Available cooling time, $t_{cool}$, values from (Hudson et al. 2010) for the low-$z$ clusters. Cluster name |  $t_{cool}$ (Gyr) | Cluster name |  $t_{cool}$ (Gyr)
---|---|---|---
A3571 | 2.13${}_{0.71}^{-0.43}$ | A2597 | 0.42${}_{0.04}^{-0.03}$
A2199 | 0.6${}_{0.07}^{-0.06}$ | A133 | 0.47${}_{0.03}^{-0.03}$
2A 0335 | 0.31${}_{0.01}^{-0.01}$ | A2244 | 1.53${}_{0.27}^{-0.2}$
A496 | 0.47${}_{0.02}^{-0.02}$ | RXJ1504 | 0.59${}_{0.02}^{-0.02}$
A85 | 0.51${}_{0.04}^{-0.04}$ | A2204 | 0.25${}_{0.01}^{-0.01}$
A478 | 0.43${}_{0.1}^{-0.07}$ | A2029 | 0.53${}_{0.04}^{-0.04}$
A1795 | 0.61${}_{0.02}^{-0.02}$ | A2142 | 1.94${}_{0.16}^{-0.14}$
A4038 | 1.68${}_{0.12}^{-0.11}$ | A3562 | 5.15${}_{0.72}^{-0.57}$
A2052 | 0.51${}_{0.02}^{-0.02}$ | A401 | 8.81${}_{1.41}^{-1.08}$
Hydra-A | 0.41${}_{0.02}^{-0.02}$ | A3558 | 1.69${}_{1.15}^{-0.5}$
A2063 | 2.36${}_{0.14}^{-0.13}$ | A2147 | 17.04${}_{3.64}^{-2.72}$
A3158 | 8.22${}_{0.54}^{-0.47}$ | A3266 | 7.62${}_{2.63}^{-1.6}$
MKW3s | 0.86${}_{0.18}^{-0.13}$ | A119 | 14.03${}_{5.95}^{-3.43}$
EXO0422 | 0.47${}_{0.07}^{-0.05}$ | A1644 | 0.84${}_{0.32}^{-0.18}$
A4059 | 0.7${}_{0.07}^{-0.06}$ | A1736 | 16.59${}_{13.17}^{-5.52}$
A2589 | 1.18${}_{0.33}^{-0.22}$ | A3395 | 12.66${}_{3.04}^{-2.18}$
A3112 | 0.37${}_{0.08}^{-0.05}$ | A2065 | 1.34${}_{0.37}^{-0.24}$
A1651 | 3.63${}_{0.43}^{-0.37}$ | A3667 | 6.14${}_{0.52}^{-0.45}$
A576 | 3.62${}_{0.84}^{-0.59}$ | A754 | 9.53${}_{2.37}^{-1.64}$
A2657 | 2.68${}_{1.2}^{-0.66}$ | A2256 | 11.56${}_{2.43}^{-1.81}$
A3391 | 12.46${}_{2.49}^{-1.89}$ | A399 | 12.13${}_{1.44}^{-1.22}$
A1650 | 1.25${}_{0.29}^{-0.2}$ | A2163 | 9.65${}_{0.73}^{-0.78}$
S 1101 | 0.88${}_{0.2}^{-0.14}$ | A3376 | 16.47${}_{3.1}^{-2.35}$
ZwCl1215 | 10.99${}_{2.09}^{-1.61}$ |  |

Of the relaxed systems (as identified by the V09), A3158, A3391, ZwCl1215,
A3562 and A401 have relatively high cooling time values. These systems were
classified as non-relaxed clusters by our combination of morphology
parameters. In the non-relaxed sample (again from the V09), only three
clusters - A3558, A1644, and A2065 have relatively short cooling times. The
short cooling times may suggest that the cores of these clusters might not yet
be disturbed by cluster merger. Rossetti et al. (2007) analysed the Chandra
and XMM observations of A3558 and found that its cool core had survived a
merger. The Chandra observations show bright cluster nuclei, which also
supports this idea. Chatzikos et al. (2006) showed that A2065 is an unequal
mass merger, which could be a reason behind the survival of one of its cool
cores. According to the classical definition of a cooling flow, $t_{cool}$ $<$
$t_{age}$, where $t_{age}$ (age of galaxy cluster) $\sim$ 10 Gyr. We found the
mean $t_{cool}$ value for strong relaxed clusters to be 0.44 Gyr, for relaxed
clusters 0.64 Gyr, for non-relaxed clusters 4.72 Gyr, and for strong non-
relaxed clusters 12.5 Gyr. This implies that the cooling mechanism is
completely disturbed in strong non-relaxed clusters, while only $\sim$ 17% are
completely disturbed among non-relaxed clusters (five non-relaxed clusters
have $t_{cool}$ $>$ $t_{age}$). Unfortunately, we did not have cooling time
information for the high-$z$ clusters in our sample.

The correlation of morphology parameters with cooling time is plotted in Fig.
16 for the Concentration, Gini and $M_{20}$, respectively, on a log-log scale.
Figs. 16a, 16b and 16c show that two of our parameters, the Concentration and
Gini, are anti-correlated, while the $M_{20}$ is correlated with the cooling
time of clusters. This indicates the possibility that surface brightness
imaging data could be useful in deriving the cooling time information of the
central intracluster gas using simple morphology parameters. Table 8 lists the
Spearman coefficient, $\rho$, between the Concentration, Gini and $M_{20}$
parameters and cluster cooling time.

Table 8: Spearman coefficient, $\rho$, between morphology parameters and cluster cooling time.  Morphology parameters | Cooling time (Gyr)
---|---
Concentration | -0.82
Gini | -0.71
$M_{20}$ | 0.83

Figure 16: (a) Concentration ($C_{5080}$) parameter negatively correlated with
cooling time for fixed radii. (b) Gini coefficient negatively correlated with
cooling time. (c) $M_{20}$ correlated with cooling time. $\circ$ = strong
relaxed clusters; $\lozenge$ = relaxed clusters;  $+$ = non-relaxed clusters;
and  $\times$ = strong non-relaxed clusters. We used the power-law fitting to
show the linear correlation.

In order to investigate linear fitting, we simply used the power-law model to
establish a relationship between the morphology parameters and $t_{cool}$
(Fig. 16). For all three parameters, compositely, we constituted $t_{cool}$
$\propto$ Conc-3.42±0.55, $t_{cool}$ $\propto$ Gini-3.0±0.35 and $t_{cool}$
$\propto$ $M_{20}^{4.23\pm 0.32}$.

###  6.4 Radio halo cluster dynamical states

Previous studies have shown the importance of joint X-ray and radio data to
study the origin of non-thermal radio emission from galaxy clusters, in the
form of a cluster wide radio halo which is generally situated at the cluster
center (Feretti et al. 2012, and references therein). Current results indicate
the presence of diffuse intracluster radio sources only in dynamically
disturbed clusters and it is expected that future radio-surveys will reveal
diffuse radio emission from a large fraction of major and minor mergers
(Cassano et al. 2010a).

The relations between cluster properties derived from X-ray observations
(luminosity ($L_{x}$), temperature (T) and mass (M)) and radio halo luminosity
have been widely studied in the last few years (Cassano et al. 2007;
Giovannini et al. 2009; Brunetti et al. 2009; Venturi 2011; Cassano et al.
2011). They all show a strong correlation between radio halo and X-ray
emission in galaxy clusters. Buote (2001), Schuecker et al. (2001) and Cassano
et al. (2010b) showed a relation between non-thermal radio sources and X-ray
cluster morphology. Buote (2001) noted the linear relation between radio power
($P_{1.4GHz}$) and power ratio ($P_{1}$/$P_{0}$) (Buote & Tsai 1995a, 1996)
for ROSAT observed X-ray clusters. He concluded that approximately
$P_{1.4GHz}$ $\propto$ $P_{1}$/$P_{0}$, which means the clusters that host the
powerful radio halos are experiencing the largest departures from a virialized
state. Recently, Cassano et al. (2010b) used three parameters, namely centroid
shift (Mohr et al. 1993; Poole et al. 2006; Maughan et al. 2008; Böhringer et
al. 2010), third order power ratio ($P_{3}$/$P_{0}$) and Concentration (Santos
et al. 2008) to demonstrate a relation between cluster mergers and the
presence of a radio halo.

Table 9: Radio halo sample clusters. (1) cluster name; (2) redshift; (3)
conversion factor (angular size to linear size); (4) radio flux density (1.4
GHz); (5) error in estimated radio flux density; (6) total radio power (1.4
GHz); (7) radio largest linear size; (8) total X-ray luminosity (0.1–2.4 keV);
(9) Temperature (keV); (9) Exposure time; (10) References. We normalised $H$
dependent quantities (LLS and $L_{X}$) with $H_{0}$=73 km s-1 Mpc-1.

Name | $z$ |  kpc/′′ | S (1.4) |  $\Delta$S | log P(1.4) | LLS | Luminosity | Temperature | Exposure time | Ref.
---|---|---|---|---|---|---|---|---|---|---
|  |  | mJy | mJy | W/Hz | Mpc |  (1044 erg/s) | (keV) | ks |
A1914 | 0.1712 | 2.80 | 64.0 | 3.0 | 24.04 | 1.01 | 9.86 | 10.50 | 19 | 1,2
A2218 | 0.1756 | 2.86 | 4.7 | 0.1 | 22.96 | 0.37 | 5.46 | 6.70 | 59 | 1,2
A665 | 0.1819 | 2.95 | 43.1 | 2.2 | 23.92 | 1.77 | 9.13 | 8.30 | 39 | 1,2
A520 | 0.1990 | 3.16 | 34.4 | 1.5 | 23.91 | 1.08 | 7.85 | 7.40 | 9 | 1,2
A773 | 0.2170 | 3.38 | 12.7 | 1.3 | 23.57 | 1.21 | 7.52 | 8.53 | 20 | 1,2
IE0657-56 | 0.2960 | 4.26 | 78.0 | 5.0 | 24.64 | 2.0 | 21.37 | 11.64 | 84 | 1,2
A2255 | 0.0806 | 1.46 | 56.0 | 3.0 | 23.28 | 0.8 | 2.50 | 6.42 | 39 | 1,2
A2319 | 0.0557 | 1.04 | 153.0 | 8.0 | 23.38 | 1.0 | 8.00 | 9.49 | 14 | 1,2
A754 | 0.0542 | 1.01 | 86.0 | 4.0 | 23.12 | 0.96 | 2.10 | 9.94 | 44 | 1,2
A2256 | 0.0581 | 1.08 | 103.4 | 1.1 | 23.26 | 0.79 | 3.55 | 6.90 | 12 | 1,2
A401 | 0.0737 | 1.34 | 17.0 | 1.0 | 22.70 | 0.50 | 6.17 | 8.07 | 18 | 1,2
A3562 | 0.0490 | 0.92 | 20.0 | 2.0 | 22.41 | 0.27 | 1.48 | 3.80 | 19 | 1,2
A399 | 0.0718 | 1.31 | 16.0 | 0 | 22.67 | 0.55 | 3.60 | 5.80 | 48 | 3,2
A2163 | 0.2030 | 3.22 | 155.0 | 2.0 | 24.57 | 2.21 | 21.50 | 12.12 | 71 | 1,2
A1300 | 0.3072 | 4.37 | 20.0 | 2.0 | 24.10 | 1.26 | 13.0 | 9.2 | 14 | 1,4
A1758 | 0.2790 | 4.08 | 16.7 | 0.8 | 23.93 | 1.47 | 6.70 | 7.95 | 7 | 1,2
A1995 | 0.3186 | 4.48 | 4.1 | 0.7 | 23.47 | 0.80 | 8.35 | 8.60 | 56 | 1,2
A2034 | 0.1130 | 1.97 | 13.6 | 1.0 | 23.00 | 0.59 | 3.60 | 7.15 | 195 | 1,2
A209 | 0.2060 | 3.25 | 16.9 | 1.0 | 23.65 | 1.36 | 5.84 | 8.28 | 20 | 1,2
A2219 | 0.2256 | 3.48 | 81.0 | 4.0 | 24.40 | 1.67 | 11.53 | 9.81 | 42 | 1,2
A2294 | 0.1780 | 2.90 | 5.8 | 0.5 | 23.06 | 0.52 | 3.70 | 7.10 | 10 | 1,2
A2744 | 0.3080 | 4.38 | 57.1 | 2.9 | 24.55 | 1.84 | 12.16 | 9.61 | 24 | 1,2
A521 | 0.2533 | 3.80 | 5.9 | 0.5 | 23.40 | 1.14 | 8.01 | 6.74 | 37 | 1,2
A697 | 0.2820 | 4.11 | 7.8 | 1.0 | 23.62 | 0.63 | 9.84 | 9.06 | 27 | 1,2
RXCJ2003.5-2323 | 0.3171 | 4.46 | 35.0 | 2.0 | 24.40 | 1.36 | 8.63 | 9.1 | 50 | 1,5

References: 1 = (Giovannini et al. 2009, and references therein); 2 =
Cavagnolo et al. (2009); 3 = (Feretti et al. 2012, and references therein); 4
= Ziparo et al. (2012); 5 = Giacintucci et al. (2009).

Figure 17: Bottom left and right plots show the $M_{20}$ and Gini vs
Concentration, respectively. Top left and right plots show the $M_{20}$ and
Gini vs temperature, respectively. $\circ$ = relaxed cluster; and  $\times$ =
non-relaxed cluster. Radio halo clusters are identified with a  “+” symbol.
Galaxy cluster separation is based on the V09.

Since we have shown that our morphology parameters are useful to characterise
the dynamical state of galaxy clusters, we investigate any possible
correlation between our set of parameters and the presence of diffuse
intracluster radio emission. We take 25 halo clusters from Giovannini et al.
(2009) (as listed in Table 9) for which some of them are already present in
the V09 cluster sample (A754, A2256, A401, A3562, A399, and A2163). We reduced
the X-ray data of the sample of Giovannini et al. (2009) in a similar way to
that described in § 3.2. We subsequently calculated the morphology parameters
(Gini, $M_{20}$ and Concentration) for each cluster (see Table 12).

In Fig. 17 we show that some parameters are useful for studying the dynamical
state of radio halo clusters. In the bottom left and right plots, we can see
that the radio halo clusters are separated from the relaxed clusters and
overlap with non-relaxed clusters in $M_{20}$ and Gini vs. Concentration
parameter planes, respectively (similar results were observed by Cassano et
al. (2010b), Buote (2001)). In these plots, we can roughly separate relaxed
and non-relaxed clusters based on $M_{20}$ and Concentration, where the region
contained by $M_{20}$ $<$ -2.0 and Concentration $>$ 1.55 gives exclusively
relaxed clusters. Based on the Gini and Concentration plot, all non-relaxed
clusters have Gini $<$ 0.65 and Concentration $<$ 1.55. The upper left and
right plots in Fig. 17 show $M_{20}$ vs. Temperature and Gini vs. Temperature,
respectively. We subdivided the $M_{20}$ vs. Temperature plot into three
regions: (1) the $M_{20}$ $<$ -2.0 region has all dynamically relaxed
clusters, (2) the $M_{20}$ $>$ -2.0 and Temperature $<$ 6 keV region has radio
quiet merger clusters, and (3) the $M_{20}$ $>$ -2.0 and Temperature $>$ 6 keV
region has radio loud merger clusters. Similarly, the Gini vs. Temperature
plot has three regions defined as (1) the Gini $>$ 0.65 region has only
dynamically relaxed clusters, (2) the Gini $<$ 0.65 and Temperature $<$ 6 keV
region has radio quiet merger clusters, and (3) the Gini $<$ 0.65 and
Temperature $>$ 6 keV region has radio loud merger clusters.

##  7 Discussion and Conclusion

In this work we used a set of morphological parameters, among which some have
to date been applied only to quantifying galaxy morphologies, to obtain
constraints on the dynamical state of clusters. These parameters do not depend
on any model parameter fits, such as the Beta model or power-law, and can
therefore be applied to both disturbed and regular clusters. In principle, the
Gini, Concentration and $M_{20}$ are promising for the detection of
disturbances in clusters at any scale. In order to measure the Concentration,
it was necessary to fix the cluster centre position (either optical position
of the BCG, X-ray peak or barycentre) and then define the inner and outer
regions around the chosen centre. This is not always possible in many of the
high-$z$ ($\sim$ 1) clusters due to low photon counts and distorted cluster
morphology. In this work, we showed a tight correlation between the
Concentration and the Gini coefficient for both the low- and high-$z$
clusters. This suggested that the Gini coefficient could be used as a
replacement for the Concentration mainly for high-$z$ clusters. The chief
benefit of using the Gini coefficient is that we do not need a well-defined
cluster centre. It works well for low photon counts as well as accurately
quantifying substructure. It is completely independent of cluster shape, it
does not depend on any symmetry underlying the cluster, and is independent of
the location of projected (bright) pixels (whether they be at the centre or
the edge) in the given aperture.

The Gini coefficient was also correlated with the $M_{20}$, which could
suggest that, in a relaxed system, 20$\%$ of the X-ray flux is centrally
located where the electron density is high, while this is not possible for
distorted and non-relaxed clusters where mergers may change the cluster
density distribution. Most relaxed and cool-core clusters have Gini $>$ 0.65,
which means that most of the flux comes from only a small number of bright
pixels, mainly found at the cluster centre. Weak cool-core clusters were found
between 0.4 $<$ Gini $<$ 0.65, and most disturbed clusters had Gini $<$ 0.4.
This is obvious because, in non-relaxed clusters, the X-ray emission (or
bright pixels) is not compact but equally distributed in the given aperture
radius which gives a low Gini value.

We found that, unlike the Gini, $M_{20}$ and Concentration parameters, the
Smoothness and Asymmetry parameters did not show any reliable signs of being
able to classify galaxy clusters based on their dynamical state. This was
unexpected as these parameters trace the 2-D structure and should be sensitive
to a range of substructure types. In addition, Rasia et al. (2012) found that
the Smoothness $\&$ Asymmetry were promising as indicators to probe the
presence of substructure. Rasia et al. (2012), however, tested their
parameters on simulated cluster data and all the simulations had a uniform
exposure time. We found that the value of these two parameters largely depends
on cluster exposure time or S/N ratio. We have also shown that, in clusters
sample that have heterogeneous exposure times, as in our situation, the
Asymmetry and Smoothness are not reliable parameters. Our sample does not have
uniform exposure time for all clusters, which largely affects S/N and photon
counts between clusters.

We did not find any relationship between the morphology parameters and X-ray
global properties of galaxy clusters. This is most likely due to the fact that
morphology parameters are estimated without regard for the mass of clusters,
to which other quantities, such as X-ray luminosity and temperature, are
instead related. The Gini, Concentration and $M_{20}$ appear to be correlated
with the cooling time of clusters. X-ray imaging and surface brightness maps
are therefore useful in investigating the cooling time of relaxed clusters.

The morphology parameters studied in this paper were robust in various
physical or observational conditions (high redshift, low exposure, etc). They
helped us to investigate, with sufficient accuracy, cluster morphology and
dynamical state. We noticed, however, that the Gini, Gini of the second order
moment and Asymmetry are fairly weak parameters at $<$ 5 ks exposure time. The
Smoothness parameter systematically decreases with increasing exposure time
and is therefore not suitable when comparing objects with a range of redshifts
and/or exposure times. The Concentration and $M_{20}$ are reasonably robust at
low exposure times. Systematics associated with redshift effects are low, but
some caution is required in the interpretation of the results based on
simulated clusters. We concluded that the Concentration and $M_{20}$ are more
robust than the other parameters. We also tested (Gini, $M_{20}$ and
Concentration) parameters against different background level and found that
these parameters are robust.

We have taken 25 radio loud clusters, which are well-known non-relaxed
clusters which host radio halos. Our morphology parameters are quite useful to
study their dynamical activities. Based on Gini, $M_{20}$ and Concentration
parameters and in agreement with previous results (Cassano et al. 2010b), we
find that only – but not all – merging systems host radio halos. We can
approximately separate radio halo clusters from relaxed clusters in parameter
space with Concentration $<$ 1.5, $M_{20}$ $>$ -2.0 and Gini $<$ 0.65.

In conclusion, we have shown that the combination of Gini, Concentration and
$M_{20}$ show good potential for identifying substructure and perturbed
dynamical states. In future we will also investigate scaling relations between
cluster mass and luminosity or temperature by dividing clusters based on
parameter boundary values.

###### Acknowledgements.

V. Parekh acknowledges financial support from the South African Square
Kilometer Array Project. C. Ferrari acknowledges financial support by the
“Agence Nationale de la Recherche” through grant ANR-09-JCJC-0001-01. We are
grateful to Prof. Alexei Vikhlinin (CFA) for providing the Chandra archival
data. G. Angus acknowledges support from the Claude-Leon foundation.

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## Appendix A

Table 10: Morphology parameters value of relaxed clusters (V09). Values are listed with appropriate uncertainties of 1$\sigma$ for each parameter. Cluster name | Gini | $M_{20}$ | Concentration | Asymmetry | Smoothness | $G_{M}$ | Ellipticity
---|---|---|---|---|---|---|---
A3571 |  0.357 $\pm$ 0.002 |  -1.492 $\pm$ 0.217 |  0.870 $\pm$ 0.091 |  0.293 $\pm$ 0.232 |  0.284 $\pm$ 0.004 |  0.259 $\pm$ 0.001 |  0.084 $\pm$ 0.043
A2199 |  0.612 $\pm$ 0.001 |  -1.897 $\pm$ 0.188 |  1.165 $\pm$ 0.097 |  0.502 $\pm$ 0.078 |  0.329 $\pm$ 0.005 |  0.378 $\pm$ 0.002 |  0.083 $\pm$ 0.037
2A 0335 |  0.666 $\pm$ 0.001 |  -2.123 $\pm$ 0.258 |  1.817 $\pm$ 0.237 |  0.341 $\pm$ 0.128 |  0.321 $\pm$ 0.008 |  0.209 $\pm$ 0.001 |  0.082 $\pm$ 0.041
A496 |  0.564 $\pm$ 0.002 |  -1.667 $\pm$ 0.320 |  1.227 $\pm$ 0.086 |  0.577 $\pm$ 0.081 |  0.205 $\pm$ 0.005 |  0.284 $\pm$ 0.001 |  0.063 $\pm$ 0.036
A3667 |  0.435 $\pm$ 0.001 |  -1.590 $\pm$ 0.109 |  0.918 $\pm$ 0.087 |  0.302 $\pm$ 0.132 |  0.536 $\pm$ 0.002 |  0.285 $\pm$ 0.001 |  0.079 $\pm$ 0.053
A754 |  0.341 $\pm$ 0.001 |  -1.053 $\pm$ 0.076 |  0.798 $\pm$ 0.052 |  0.406 $\pm$ 0.058 |  0.524 $\pm$ 0.003 |  0.308 $\pm$ 0.001 |  0.202 $\pm$ 0.006
A85 |  0.624 $\pm$ 0.001 |  -1.779 $\pm$ 0.260 |  1.447 $\pm$ 0.067 |  0.633 $\pm$ 0.117 |  0.464 $\pm$ 0.003 |  0.279 $\pm$ 0.001 |  0.026 $\pm$ 0.006
A2029 |  0.718 $\pm$ 0.002 |  -2.088 $\pm$ 0.315 |  1.603 $\pm$ 0.192 |  0.445 $\pm$ 0.165 |  0.163 $\pm$ 0.009 |  0.280 $\pm$ 0.001 |  0.113 $\pm$ 0.023
A478 |  0.703 $\pm$ 0.003 |  -2.145 $\pm$ 0.255 |  1.613 $\pm$ 0.162 |  0.286 $\pm$ 0.165 |  0.248 $\pm$ 0.008 |  0.279 $\pm$ 0.001 |  0.137 $\pm$ 0.062
A1795 |  0.779 $\pm$ 0.002 |  -2.148 $\pm$ 0.296 |  1.615 $\pm$ 0.097 |  0.351 $\pm$ 0.053 |  0.143 $\pm$ 0.012 |  0.389 $\pm$ 0.001 |  0.107 $\pm$ 0.029
A3558 |  0.327 $\pm$ 0.001 |  -1.448 $\pm$ 0.226 |  0.866 $\pm$ 0.087 |  0.358 $\pm$ 0.153 |  0.520 $\pm$ 0.002 |  0.276 $\pm$ 0.003 |  0.117 $\pm$ 0.068
A2142 |  0.564 $\pm$ 0.002 |  -1.959 $\pm$ 0.218 |  1.322 $\pm$ 0.199 |  0.365 $\pm$ 0.263 |  0.329 $\pm$ 0.003 |  0.239 $\pm$ 0.000 |  0.174 $\pm$ 0.044
A2256 |  0.373 $\pm$ 0.000 |  -1.273 $\pm$ 0.064 |  0.896 $\pm$ 0.047 |  0.333 $\pm$ 0.119 |  1.024 $\pm$ 0.002 |  0.277 $\pm$ 0.000 |  0.133 $\pm$ 0.041
A4038 |  0.503 $\pm$ 0.002 |  -1.929 $\pm$ 0.193 |  1.291 $\pm$ 0.181 |  0.297 $\pm$ 0.191 |  0.563 $\pm$ 0.001 |  0.206 $\pm$ 0.000 |  0.140 $\pm$ 0.038
A2147 |  0.268 $\pm$ 0.001 |  -1.243 $\pm$ 0.102 |  0.753 $\pm$ 0.178 |  0.359 $\pm$ 0.150 |  1.188 $\pm$ 0.002 |  0.305 $\pm$ 0.000 |  0.083 $\pm$ 0.057
A3266 |  0.220 $\pm$ 0.001 |  -1.196 $\pm$ 0.232 |  0.717 $\pm$ 0.172 |  0.276 $\pm$ 0.256 |  0.514 $\pm$ 0.005 |  0.279 $\pm$ 0.002 |  0.103 $\pm$ 0.069
A401 |  0.439 $\pm$ 0.000 |  -1.734 $\pm$ 0.121 |  1.109 $\pm$ 0.122 |  0.222 $\pm$ 0.207 |  0.739 $\pm$ 0.002 |  0.205 $\pm$ 0.000 |  0.102 $\pm$ 0.040
A2052 |  0.622 $\pm$ 0.004 |  -1.934 $\pm$ 0.237 |  1.330 $\pm$ 0.195 |  0.649 $\pm$ 0.035 |  0.308 $\pm$ 0.003 |  0.332 $\pm$ 0.003 |  0.086 $\pm$ 0.046
Hydra-A |  0.543 $\pm$ 0.004 |  -2.087 $\pm$ 0.383 |  1.625 $\pm$ 0.248 |  0.258 $\pm$ 0.293 |  0.152 $\pm$ 0.013 |  0.149 $\pm$ 0.002 |  0.060 $\pm$ 0.065
A119 |  0.417 $\pm$ 0.000 |  -1.158 $\pm$ 0.066 |  0.774 $\pm$ 0.085 |  0.574 $\pm$ 0.022 |  1.542 $\pm$ 0.001 |  0.434 $\pm$ 0.000 |  0.063 $\pm$ 0.057
A2063 |  0.376 $\pm$ 0.001 |  -1.729 $\pm$ 0.169 |  0.999 $\pm$ 0.156 |  0.403 $\pm$ 0.132 |  1.156 $\pm$ 0.001 |  0.281 $\pm$ 0.001 |  0.046 $\pm$ 0.057
A1644 |  0.352 $\pm$ 0.003 |  -1.843 $\pm$ 0.183 |  0.960 $\pm$ 0.286 |  0.381 $\pm$ 0.282 |  0.843 $\pm$ 0.003 |  0.273 $\pm$ 0.001 |  0.043 $\pm$ 0.067
A3158 |  0.470 $\pm$ 0.001 |  -1.767 $\pm$ 0.104 |  1.091 $\pm$ 0.138 |  0.267 $\pm$ 0.221 |  0.788 $\pm$ 0.001 |  0.232 $\pm$ 0.000 |  0.118 $\pm$ 0.043
MKW3s |  0.509 $\pm$ 0.003 |  -1.988 $\pm$ 0.273 |  1.287 $\pm$ 0.193 |  0.222 $\pm$ 0.338 |  0.317 $\pm$ 0.006 |  0.173 $\pm$ 0.001 |  0.134 $\pm$ 0.056
A1736 |  0.215 $\pm$ 0.001 |  -0.932 $\pm$ 0.101 |  0.613 $\pm$ 0.208 |  0.415 $\pm$ 0.109 |  1.379 $\pm$ 0.001 |  0.352 $\pm$ 0.001 |  0.011 $\pm$ 0.080
EXO0422 |  0.622 $\pm$ 0.002 |  -2.304 $\pm$ 0.228 |  1.630 $\pm$ 0.268 |  0.355 $\pm$ 0.183 |  0.911 $\pm$ 0.005 |  0.254 $\pm$ 0.000 |  0.071 $\pm$ 0.071
A4059 |  0.424 $\pm$ 0.002 |  -1.747 $\pm$ 0.267 |  1.110 $\pm$ 0.107 |  0.211 $\pm$ 0.170 |  0.135 $\pm$ 0.005 |  0.167 $\pm$ 0.001 |  0.093 $\pm$ 0.044
A3395 |  0.394 $\pm$ 0.001 |  -0.801 $\pm$ 0.098 |  0.593 $\pm$ 0.093 |  0.734 $\pm$ 0.071 |  1.411 $\pm$ 0.001 |  0.473 $\pm$ 0.001 |  0.228 $\pm$ 0.090
A2589 |  0.381 $\pm$ 0.001 |  -1.415 $\pm$ 0.230 |  0.862 $\pm$ 0.166 |  0.471 $\pm$ 0.055 |  0.669 $\pm$ 0.003 |  0.317 $\pm$ 0.001 |  0.045 $\pm$ 0.037
A3112 |  0.544 $\pm$ 0.004 |  -2.121 $\pm$ 0.406 |  1.420 $\pm$ 0.204 |  0.276 $\pm$ 0.340 |  0.309 $\pm$ 0.013 |  0.170 $\pm$ 0.004 |  0.143 $\pm$ 0.072
A3562 |  0.541 $\pm$ 0.001 |  -1.870 $\pm$ 0.099 |  1.204 $\pm$ 0.175 |  0.542 $\pm$ 0.019 |  1.211 $\pm$ 0.001 |  0.384 $\pm$ 0.000 |  0.107 $\pm$ 0.051
A1651 |  0.569 $\pm$ 0.001 |  -1.957 $\pm$ 0.207 |  1.321 $\pm$ 0.180 |  0.354 $\pm$ 0.228 |  1.001 $\pm$ 0.002 |  0.266 $\pm$ 0.000 |  0.075 $\pm$ 0.049
A399 |  0.389 $\pm$ 0.001 |  -1.439 $\pm$ 0.178 |  1.013 $\pm$ 0.169 |  0.372 $\pm$ 0.137 |  0.698 $\pm$ 0.001 |  0.229 $\pm$ 0.001 |  0.080 $\pm$ 0.079
A2204 |  0.732 $\pm$ 0.005 |  -2.611 $\pm$ 0.580 |  2.054 $\pm$ 0.485 |  0.361 $\pm$ 0.159 |  0.545 $\pm$ 0.033 |  0.208 $\pm$ 0.002 |  0.051 $\pm$ 0.062
A576 |  0.308 $\pm$ 0.001 |  -1.651 $\pm$ 0.162 |  0.866 $\pm$ 0.147 |  0.267 $\pm$ 0.064 |  0.606 $\pm$ 0.002 |  0.251 $\pm$ 0.001 |  0.050 $\pm$ 0.068
A2657 |  0.380 $\pm$ 0.001 |  -1.607 $\pm$ 0.194 |  0.998 $\pm$ 0.222 |  0.402 $\pm$ 0.157 |  1.034 $\pm$ 0.003 |  0.250 $\pm$ 0.001 |  0.048 $\pm$ 0.051
A3391 |  0.408 $\pm$ 0.001 |  -1.549 $\pm$ 0.079 |  0.952 $\pm$ 0.123 |  0.431 $\pm$ 0.096 |  1.335 $\pm$ 0.001 |  0.332 $\pm$ 0.000 |  0.115 $\pm$ 0.051
A2065 |  0.477 $\pm$ 0.001 |  -1.697 $\pm$ 0.166 |  1.218 $\pm$ 0.091 |  0.402 $\pm$ 0.155 |  0.549 $\pm$ 0.003 |  0.230 $\pm$ 0.001 |  0.149 $\pm$ 0.049
A1650 |  0.673 $\pm$ 0.001 |  -1.869 $\pm$ 0.318 |  1.371 $\pm$ 0.165 |  0.429 $\pm$ 0.087 |  0.476 $\pm$ 0.004 |  0.379 $\pm$ 0.002 |  0.122 $\pm$ 0.093
A3822 |  0.450 $\pm$ 0.001 |  -1.542 $\pm$ 0.146 |  1.045 $\pm$ 0.197 |  0.536 $\pm$ 0.071 |  1.450 $\pm$ 0.002 |  0.370 $\pm$ 0.000 |  0.119 $\pm$ 0.066
S 1101 |  0.645 $\pm$ 0.002 |  -2.111 $\pm$ 0.214 |  1.561 $\pm$ 0.241 |  0.238 $\pm$ 0.199 |  0.461 $\pm$ 0.011 |  0.172 $\pm$ 0.002 |  0.110 $\pm$ 0.062
A2163 |  0.378 $\pm$ 0.002 |  -1.443 $\pm$ 0.179 |  0.943 $\pm$ 0.168 |  0.281 $\pm$ 0.158 |  0.272 $\pm$ 0.004 |  0.196 $\pm$ 0.002 |  0.082 $\pm$ 0.061
ZwCl1215 |  0.439 $\pm$ 0.001 |  -1.625 $\pm$ 0.109 |  1.040 $\pm$ 0.103 |  0.369 $\pm$ 0.117 |  1.204 $\pm$ 0.002 |  0.294 $\pm$ 0.000 |  0.120 $\pm$ 0.086
RXJ1504 |  0.807 $\pm$ 0.004 |  -2.471 $\pm$ 0.400 |  2.128 $\pm$ 0.322 |  0.397 $\pm$ 0.188 |  0.370 $\pm$ 0.038 |  0.242 $\pm$ 0.003 |  0.124 $\pm$ 0.067
A2597 |  0.781 $\pm$ 0.003 |  -2.426 $\pm$ 0.558 |  2.277 $\pm$ 0.389 |  0.397 $\pm$ 0.266 |  0.295 $\pm$ 0.010 |  0.277 $\pm$ 0.001 |  0.152 $\pm$ 0.077
A133 |  0.508 $\pm$ 0.003 |  -2.201 $\pm$ 0.396 |  1.438 $\pm$ 0.301 |  0.254 $\pm$ 0.292 |  0.320 $\pm$ 0.008 |  0.179 $\pm$ 0.002 |  0.094 $\pm$ 0.079
A2244 |  0.638 $\pm$ 0.001 |  -1.983 $\pm$ 0.287 |  1.445 $\pm$ 0.232 |  0.266 $\pm$ 0.151 |  0.314 $\pm$ 0.003 |  0.219 $\pm$ 0.005 |  0.030 $\pm$ 0.059
A3376 |  0.413 $\pm$ 0.000 |  -1.183 $\pm$ 0.063 |  0.803 $\pm$ 0.153 |  0.564 $\pm$ 0.045 |  1.159 $\pm$ 0.001 |  0.397 $\pm$ 0.001 |  0.072 $\pm$ 0.0869
Table 11: Morphology parameters value of (V09) non-relaxed clusters. Values are listed with appropriate uncertainties of 1$\sigma$ for each parameter. Cluster name | Gini | $M_{20}$ | Concentration | Asymmetry | Smoothness | $G_{M}$ | Ellipticity
---|---|---|---|---|---|---|---
0302-0423 |  0.796 $\pm$ 0.010 |  -2.511 $\pm$ 0.750 |  2.186 $\pm$ 0.769 |  0.322 $\pm$ 0.281 |  0.767 $\pm$ 0.027 |  0.345 $\pm$ 0.006 |  0.071 $\pm$ 0.231
1212+2733 |  0.417 $\pm$ 0.004 |  -1.670 $\pm$ 0.349 |  0.993 $\pm$ 0.553 |  0.388 $\pm$ 0.377 |  1.206 $\pm$ 0.003 |  0.293 $\pm$ 0.003 |  0.102 $\pm$ 0.212
0350-3801 |  0.465 $\pm$ 0.003 |  -1.734 $\pm$ 0.538 |  1.117 $\pm$ 1.046 |  0.591 $\pm$ 0.288 |  1.453 $\pm$ 0.000 |  0.386 $\pm$ 0.001 |  0.082 $\pm$ 0.000
0318-0302 |  0.554 $\pm$ 0.003 |  -1.631 $\pm$ 0.675 |  1.197 $\pm$ 0.900 |  0.532 $\pm$ 0.474 |  1.306 $\pm$ 0.002 |  0.339 $\pm$ 0.003 |  0.073 $\pm$ 0.000
0159+0030 |  0.559 $\pm$ 0.008 |  -2.288 $\pm$ 1.139 |  1.531 $\pm$ 0.628 |  0.597 $\pm$ 0.400 |  1.198 $\pm$ 0.002 |  0.334 $\pm$ 0.002 |  0.022 $\pm$ 0.000
0958+4702 |  0.550 $\pm$ 0.007 |  -2.165 $\pm$ 0.557 |  1.442 $\pm$ 1.034 |  0.416 $\pm$ 0.354 |  1.310 $\pm$ 0.000 |  0.354 $\pm$ 0.001 |  0.015 $\pm$ 0.000
0809+2811 |  0.366 $\pm$ 0.003 |  -1.334 $\pm$ 0.738 |  0.914 $\pm$ 1.160 |  0.449 $\pm$ 0.370 |  1.291 $\pm$ 0.002 |  0.309 $\pm$ 0.003 |  0.115 $\pm$ 0.307
1416+4446 |  0.528 $\pm$ 0.010 |  -1.972 $\pm$ 0.466 |  1.442 $\pm$ 1.005 |  0.553 $\pm$ 0.472 |  1.009 $\pm$ 0.013 |  0.274 $\pm$ 0.003 |  0.103 $\pm$ 0.000
1312+3900 |  0.322 $\pm$ 0.002 |  -1.359 $\pm$ 0.279 |  0.837 $\pm$ 0.612 |  0.507 $\pm$ 0.453 |  1.443 $\pm$ 0.000 |  0.354 $\pm$ 0.001 |  0.030 $\pm$ 0.000
1003+3253 |  0.595 $\pm$ 0.008 |  -2.369 $\pm$ 0.779 |  1.748 $\pm$ 1.035 |  0.451 $\pm$ 0.259 |  1.166 $\pm$ 0.000 |  0.353 $\pm$ 0.002 |  0.100 $\pm$ 0.242
0141-3034 |  0.347 $\pm$ 0.003 |  -0.961 $\pm$ 0.170 |  0.759 $\pm$ 0.293 |  0.623 $\pm$ 0.303 |  1.591 $\pm$ 0.000 |  0.432 $\pm$ 0.002 |  0.048 $\pm$ 0.000
1701+6414 |  0.479 $\pm$ 0.008 |  -2.017 $\pm$ 0.817 |  1.281 $\pm$ 0.855 |  0.513 $\pm$ 0.384 |  0.878 $\pm$ 0.005 |  0.240 $\pm$ 0.002 |  0.096 $\pm$ 0.000
1641+4001 |  0.481 $\pm$ 0.005 |  -1.913 $\pm$ 0.483 |  1.348 $\pm$ 0.203 |  0.349 $\pm$ 0.384 |  1.166 $\pm$ 0.000 |  0.282 $\pm$ 0.002 |  0.042 $\pm$ 0.000
0522-3624 |  0.444 $\pm$ 0.008 |  -1.429 $\pm$ 0.356 |  1.236 $\pm$ 1.353 |  0.471 $\pm$ 0.415 |  1.013 $\pm$ 0.003 |  0.307 $\pm$ 0.002 |  0.087 $\pm$ 0.351
1222+2709 |  0.444 $\pm$ 0.005 |  -1.983 $\pm$ 0.241 |  1.213 $\pm$ 0.135 |  0.412 $\pm$ 0.299 |  1.200 $\pm$ 0.000 |  0.301 $\pm$ 0.001 |  0.036 $\pm$ 0.000
0355-3741 |  0.497 $\pm$ 0.007 |  -2.031 $\pm$ 0.239 |  1.397 $\pm$ 1.198 |  0.429 $\pm$ 0.357 |  1.182 $\pm$ 0.000 |  0.303 $\pm$ 0.002 |  0.058 $\pm$ 0.000
0853+5759 |  0.418 $\pm$ 0.004 |  -1.294 $\pm$ 0.432 |  1.034 $\pm$ 0.746 |  0.715 $\pm$ 0.361 |  1.075 $\pm$ 0.000 |  0.331 $\pm$ 0.004 |  0.030 $\pm$ 0.000
0333-2456 |  0.414 $\pm$ 0.002 |  -1.435 $\pm$ 0.218 |  0.947 $\pm$ 0.199 |  0.462 $\pm$ 0.315 |  1.363 $\pm$ 0.000 |  0.341 $\pm$ 0.001 |  0.117 $\pm$ 0.000
0926+1242 |  0.506 $\pm$ 0.003 |  -1.764 $\pm$ 0.584 |  1.212 $\pm$ 1.019 |  0.558 $\pm$ 0.336 |  1.420 $\pm$ 0.000 |  0.399 $\pm$ 0.002 |  0.158 $\pm$ 0.000
0030+2618 |  0.258 $\pm$ 0.004 |  -1.639 $\pm$ 0.336 |  0.833 $\pm$ 0.392 |  0.258 $\pm$ 0.502 |  0.578 $\pm$ 0.006 |  0.282 $\pm$ 0.002 |  0.045 $\pm$ 0.130
1002+6858 |  0.474 $\pm$ 0.005 |  -1.914 $\pm$ 0.625 |  1.243 $\pm$ 0.861 |  0.475 $\pm$ 0.217 |  1.388 $\pm$ 0.000 |  0.362 $\pm$ 0.001 |  0.065 $\pm$ 0.000
1524+0957 |  0.343 $\pm$ 0.003 |  -1.343 $\pm$ 0.380 |  0.918 $\pm$ 0.429 |  0.366 $\pm$ 0.495 |  1.065 $\pm$ 0.000 |  0.262 $\pm$ 0.005 |  0.102 $\pm$ 0.000
1357+6232 |  0.419 $\pm$ 0.004 |  -1.696 $\pm$ 0.400 |  1.045 $\pm$ 1.131 |  0.385 $\pm$ 0.394 |  0.945 $\pm$ 0.015 |  0.262 $\pm$ 0.003 |  0.019 $\pm$ 0.154
1354-0221 |  0.343 $\pm$ 0.003 |  -1.481 $\pm$ 0.344 |  0.842 $\pm$ 0.756 |  0.477 $\pm$ 0.333 |  0.977 $\pm$ 0.009 |  0.308 $\pm$ 0.001 |  0.028 $\pm$ 0.000
1120+2326 |  0.367 $\pm$ 0.002 |  -1.237 $\pm$ 0.424 |  0.842 $\pm$ 0.323 |  0.439 $\pm$ 0.448 |  0.964 $\pm$ 0.000 |  0.272 $\pm$ 0.002 |  0.087 $\pm$ 0.000
0956+4107 |  0.416 $\pm$ 0.002 |  -1.450 $\pm$ 0.251 |  1.010 $\pm$ 0.107 |  0.388 $\pm$ 0.365 |  1.156 $\pm$ 0.000 |  0.300 $\pm$ 0.002 |  0.147 $\pm$ 0.000
0328-2140 |  0.430 $\pm$ 0.004 |  -1.767 $\pm$ 0.689 |  1.068 $\pm$ 1.208 |  0.480 $\pm$ 0.360 |  0.944 $\pm$ 0.000 |  0.281 $\pm$ 0.003 |  0.073 $\pm$ 0.000
1120+4318 |  0.506 $\pm$ 0.006 |  -1.665 $\pm$ 0.428 |  1.201 $\pm$ 0.851 |  0.506 $\pm$ 0.429 |  1.149 $\pm$ 0.000 |  0.287 $\pm$ 0.006 |  0.061 $\pm$ 0.000
1334+5031 |  0.428 $\pm$ 0.005 |  -1.815 $\pm$ 0.319 |  1.045 $\pm$ 0.595 |  0.517 $\pm$ 0.322 |  1.350 $\pm$ 0.000 |  0.376 $\pm$ 0.001 |  0.068 $\pm$ 0.000
0542-4100 |  0.389 $\pm$ 0.005 |  -1.508 $\pm$ 0.429 |  1.034 $\pm$ 0.332 |  0.464 $\pm$ 0.499 |  0.935 $\pm$ 0.000 |  0.264 $\pm$ 0.004 |  0.099 $\pm$ 0.000
1202+5751 |  0.303 $\pm$ 0.002 |  -1.231 $\pm$ 0.326 |  0.807 $\pm$ 0.113 |  0.409 $\pm$ 0.440 |  1.191 $\pm$ 0.000 |  0.297 $\pm$ 0.002 |  0.106 $\pm$ 0.000
0405-4100 |  0.489 $\pm$ 0.004 |  -1.976 $\pm$ 0.381 |  1.346 $\pm$ 0.845 |  0.448 $\pm$ 0.369 |  0.903 $\pm$ 0.000 |  0.291 $\pm$ 0.002 |  0.014 $\pm$ 0.000
1221+4918 |  0.362 $\pm$ 0.002 |  -1.488 $\pm$ 0.550 |  0.971 $\pm$ 1.002 |  0.282 $\pm$ 0.444 |  0.811 $\pm$ 0.000 |  0.230 $\pm$ 0.003 |  0.091 $\pm$ 0.000
0230+1836 |  0.310 $\pm$ 0.002 |  -1.079 $\pm$ 0.588 |  0.772 $\pm$ 0.158 |  0.442 $\pm$ 0.459 |  1.093 $\pm$ 0.000 |  0.296 $\pm$ 0.002 |  0.130 $\pm$ 0.000
0152-1358 |  0.337 $\pm$ 0.003 |  -1.532 $\pm$ 0.268 |  0.818 $\pm$ 0.107 |  0.510 $\pm$ 0.378 |  1.183 $\pm$ 0.000 |  0.348 $\pm$ 0.003 |  0.169 $\pm$ 0.000
1226+3332 |  0.585 $\pm$ 0.006 |  -1.859 $\pm$ 0.929 |  1.435 $\pm$ 0.799 |  0.277 $\pm$ 0.151 |  0.411 $\pm$ 0.018 |  0.168 $\pm$ 0.007 |  0.035 $\pm$ 0.246
Table 12: Three morphology parameters value of radio halo clusters, except for the V09 radio halo clusters (A754, A2256, A401, A3562, A399, and A2163). Values are listed with appropriate uncertainties of 1$\sigma$ for each parameter. Cluster name | Gini | $M_{20}$ | Concentration
---|---|---|---
A1914 | 0.621 $\pm$ 0.002 | -1.567 $\pm$ 0.247 | 1.282 $\pm$ 0.266
A2218 | 0.472 $\pm$ 0.001 | -1.720 $\pm$ 0.264 | 1.122 $\pm$ 0.156
A665 | 0.427 $\pm$ 0.002 | -1.320 $\pm$ 0.269 | 1.09 $\pm$ 0.229
A520 | 0.357 $\pm$ 0.001 | -1.090 $\pm$ 0.154 | 0.814 $\pm$ 0.322
A773 | 0.493 $\pm$ 0.003 | -1.697 $\pm$ 0.249 | 1.215 $\pm$ 0.255
IE 0657-56 | 0.358 $\pm$ 0.003 | -1.073 $\pm$ 0.195 | 0.783 $\pm$ 0.156
A2255 | 0.270 $\pm$ 0.000 | -1.441 $\pm$ 0.110 | 0.798 $\pm$ 0.126
A2319 | 0.420 $\pm$ 0.001 | -1.491 $\pm$ 0.141 | 1.041 $\pm$ 0.088
A1300 | 0.477 $\pm$ 0.004 | -1.516 $\pm$ 0.412 | 1.114 $\pm$ 0.335
A1758 | 0.371 $\pm$ 0.002 | -0.883 $\pm$ 0.184 | 0.741 $\pm$ 0.414
A1995 | 0.520 $\pm$ 0.002 | -1.807 $\pm$ 0.286 | 1.172 $\pm$ 0.405
A2034 | 0.420 $\pm$ 0.001 | -1.645 $\pm$ 0.149 | 0.930 $\pm$ 0.101
A209 | 0.449 $\pm$ 0.002 | -1.754 $\pm$ 0.248 | 1.110 $\pm$ 0.208
A2219 | 0.407 $\pm$ 0.002 | -1.642 $\pm$ 0.254 | 0.986 $\pm$ 0.125
A2294 | 0.463 $\pm$ 0.002 | -1.681 $\pm$ 0.305 | 1.172 $\pm$ 0.420
A2744 | 0.300 $\pm$ 0.002 | -1.370 $\pm$ 0.222 | 0.821 $\pm$ 0.271
A521 | 0.231 $\pm$ 0.002 | -1.055 $\pm$ 0.176 | 0.670 $\pm$ 0.417
A697 | 0.382 $\pm$ 0.002 | -1.720 $\pm$ 0.265 | 1.07 $\pm$ 0.186
RXCJ2003.5-2323 | 0.240 $\pm$ 0.012 | -1.023 $\pm$ 0.223 | 0.730 $\pm$ 0.267
Figure 18: Low-$z$ clusters of the V09. Cluster names of panels from top left
to bottom right are listed as in Table 1. Each image has the same color, scale
(log) and length. Figure 19: High-$z$ clusters of the V09. Cluster names of
panels from top left to bottom right are listed as in Table 2. Each image has
the same color, scale (log) and length. Figure 20: Radio halo clusters
(Giovannini et al. 2009). Cluster names of panels from top left to bottom
right are listed as in Table 9, except for the V09 radio halo clusters (A754,
A2256, A401, A3562, A399, and A2163). Each image has the same color, scale
(log) and length.