# The SRG/eROSITA all-sky survey: The morphologies of clusters of galaxies I: A catalogue of morphological parameters

J.S Sanders, Y.E Bahar, E Bulbul, V Ghirardini, A Liu, N Clerc, M.E

Ramos-Ceja, T.H Reiprich, F Balzer, J Comparat, et al.

► To cite this version:

J.S Sanders, Y.E Bahar, E Bulbul, V Ghirardini, A Liu, et al. The SRG/eROSITA all-sky survey: The morphologies of clusters of galaxies I: A catalogue of morphological parameters. Astronomy & Astrophysics - A&A, 2025, 695, pp.A160. 10.1051/0004-6361/202452618. hal-04966949

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# The SRG/eROSITA all-sky survey: The morphologies of clusters of galaxies

## I. A catalogue of morphological parameters

J. S. Sanders<fnref n="1" />,*, Y. E. Bahar<fnref n="1" />, E. Bulbul<fnref n="1" />, V. Ghirardini<fnref n="1" />,<fnref n="2" />, A. Liu<fnref n="1" />, N. Clerc<fnref n="3" />, M. E. Ramos-Ceja<fnref n="1" />,
T. H. Reiprich<fnref n="4" />, F. Balzer<fnref n="1" />, J. Comparat<fnref n="1" />, M. Kluge<fnref n="1" />, F. Pacaud<fnref n="4" />, and X. Zhang<fnref n="1" />

Received 15 October 2024 / Accepted 31 January 2025

## ABSTRACT

The first SRG/eROSITA all-sky X-ray survey, eRASS1, resulted in a catalogue of over 12 000 optically confirmed galaxy groups and clusters in the western Galactic hemisphere. Using the eROSITA images of these objects, we measured and studied their morphological properties, including their concentration, central density and slope, ellipticity, power ratios, photon asymmetry, centroid shift, and Gini coefficient. We also introduced new forward-modelled parameters that take account of the instrument point spread function (PSF), namely, slosh, which measures how asymmetric the surface brightness distribution is, and multipole magnitudes, which are analogues to power ratios. Using simulations, we found that some non-forward-modelled parameters are strongly biased due to PSF and data quality. When using Chandra and previous results from XMM-Newton, we found similar values of concentration and central density compared to our results for the same clusters. The population as a whole has log concentrations that are typically around 0.3 dex larger than samples selected from the South Pole Telescope or Planck and the deeper eFEDS sample. The exposure time, detection likelihood threshold, extension likelihood threshold, and number of counts affect the concentration distribution but generally not enough to reduce the concentration to match the other samples. The concentration of clusters in the survey strongly affects whether they are detected as a function of redshift and luminosity. We introduced a combined disturbance score based on a Gaussian mixture model fit to several of the parameters. For brighter clusters, around one-fourth of the objects are classified as disturbed using this score, which may be due to our sensitivity to concentrated objects.

**Key words.** galaxies: clusters: intracluster medium – X-rays: galaxies: clusters

## 1. Introduction

The intracluster medium (ICM), the hot atmosphere that fills the potential well of clusters of galaxies, is sensitive to many of the physical processes taking place when clusters form as well as those that occur internally within clusters. This atmosphere is frequently assumed to consist of a spherically symmetric $\beta$-model, where the gas density profile follows $n(r) = n_0[1 + (r/r_c)^2]^{-3\beta/2}$ (Cavaliere & Fusco-Femiano 1978), where $n_0$ is an inner density, $r_c$ is the core radius, and $\beta$ controls the outer slope, where $\beta = 2/3$ is often assumed as a typical value.

The ICM is visible by its emission in the X-ray band, primarily through bremsstrahlung radiation, although X-ray emission lines are dominant for cool objects (e.g. Böhringer & Werner 2010). From studying this emission, we know that many clusters show profiles that deviate from this $\beta$-model form and show shapes that are non-spherical. A substantial fraction of clusters are cool core systems with steeply peaked surface brightness profiles where the ICM has very short mean radiative cooling times (e.g. Fabian 2012; McNamara & Nulsen 2012). Without the injection of energy, these systems would be rapidly radiatively cooling. Feedback by active galactic nuclei (AGN)

appears to be responsible for preventing this from happening (e.g. Bîrzan et al. 2004; Rafferty et al. 2006).

Galaxy clusters are also dynamic systems. In the hierarchical formation of structure, galaxies, groups, and clusters merge together to form larger systems (e.g. Voit 2005). These mergers can be minor or major. In a minor merger, the passage of a subcluster through another cluster can cause a displacement of the hot gas from the potential well, which can introduce motions (or sloshing) in the ICM. This is thought to produce the contact discontinuities, or cold fronts, frequently seen in clusters (e.g. Markevitch & Vikhlinin 2007), even in those that otherwise appear relaxed.

In more extreme major mergers, the whole shape of the cluster can be affected, with the most prominent example being the Bullet Cluster (Clowe et al. 2006). If a merger is strong enough, it may completely destroy a cool core (e.g. Valdarnini & Sarazin 2021). Depending on the phase of a merger and the viewing angle, the system may be detected as two or more separate clusters, an elongated or non-circular single cluster, or a circular cluster.

There are many different reasons why it is interesting to know whether a cluster or sample of clusters are disturbed. If one would like to understand merger processes, then identifying these systems is useful. The environment of the cluster may also affect its halo properties (e.g. Wechsler et al. 2002)

<footnote n="1">¹ Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching, Germany</footnote>

<footnote n="2">² INAF, Osservatorio di Astrofisica e Scienza dello Spazio, via Piero Gobetti 93/3, 40129 Bologna, Italy</footnote>

<footnote n="3">³ IRAP, CNRS, UPS, CNES, 14 Avenue Edouard Belin, 31400 Toulouse, France</footnote>

<footnote n="4">⁴ Argelander-Institut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany</footnote>

<footnote>* Corresponding author; jsanders@mpe.mpg.de</footnote>
and therefore its morphology (e.g. if it is within a supercluster). Indeed, evidence for halo assembly bias in clusters has been detected (Liu et al. 2024) by the extended ROentgen Survey with an Imaging Telescope Array (eROSITA). Another important case is identifying clusters where there is a cool core in order to study the impact of AGN feedback. One common technique for measuring cluster masses is by the assumption of hydrostatic equilibrium, which may not apply if the object is substantially out of equilibrium (Lau et al. 2009; Biffi et al. 2016). Clusters with a higher degree of disturbance have a larger range of hydrostatic mass bias scatter (Gianfagna et al. 2023).

To say whether a cluster is disturbed or not requires a quantitative measure. There are several so-called morphological parameters that characterise the shape or peakiness of a cluster and therefore its level of disturbance. We describe the parameters in our analysis in detail in Sect. 2. It is important to note that the quantities measured can be affected by either the signal to noise of the data being analysed or the angular size of the object relative to the point spread function (PSF) of the telescope. The luminosity or redshift of the cluster can also affect what is measured. Morphological parameters are also affected by projection effects. In the X-ray waveband, observations are also most sensitive to denser regions due to the bremsstrahlung emission process. Morphological parameters often cannot be straightforwardly compared between surveys, due to differences of depth and instrumentation.

The eROSITA instrument on board the Spektrum Roentgen Gamma (SRG) space observatory is an instrument optimised for surveying the sky in the X-ray waveband (Predehl et al. 2021). The first eROSITA X-ray all sky survey is described in Merloni et al. (2024), which also presents the catalogue from the western Galactic hemisphere. The survey is characterised by a full coverage without gaps on the sky and a relatively uniform exposure except for the deeply observed ecliptic pole. The ~30 arcsec PSF of the survey is also very uniform because each sky region passes through the field of view several times as the telescope scans over the sky. A catalogue of clusters identified by the survey is presented in Bulbul et al. (2024, hereafter B24). Extended sources from the main X-ray catalogue were optically confirmed (Kluge et al. 2024) to create a main eRASS1 catalogue of 12 247 clusters spanning 13 116 deg². Despite optical confirmation, the catalogue contains contaminants of falsely identified extended X-ray sources, for example, due to bright point sources and background fluctuations (Seppi et al. 2022) coincidentally lying in an overdensity of red-sequence galaxies. A contamination model estimates the purity of the main sample as 86%. Using a stronger cut on the extent likelihood of the X-ray sources, a second purer catalogue of 5263 clusters spanning 12 791 deg² has been created. This sample has an estimated purity of 95% and is intended for cosmological analyses. The integrated X-ray properties, including flux, temperature, total mass, gas mass, gas mass fraction, and mass proxy $Y_X$ were obtained for each object. This sample of clusters is unmatched by other X-ray selected cluster samples.

In this paper, we measure the morphological properties of the clusters detected in B24 in the main eRASS1 sample. These parameters are published in the form of a catalogue, and it represents the largest set of X-ray parameters collected for a cluster sample. We also study the systematic effects on the parameters due to observational effects via simulations. A combined disturbance parameter is also computed based on combining individual parameters.

Where not otherwise specified, log refers to $\log_{10}$, while $\ln$ is $\log_e$. We used the relative solar abundances of Asplund et al.

(2009). A cosmology with $H_0 = 70 \text{ km s}^{-1} \text{ Mpc}^{-1}$, $\Omega_m = 0.3$, and $\Omega_\Lambda = 0.7$ was assumed.

This paper is organised as follows. In Sect. 2, we introduce the parameters we are measuring for the cluster, including new parameters measuring ‘slosh’ and multipole magnitudes. We describe how the parameters were obtained from the data in Sect. 3. In Sect. 4, we describe the simulations made to understand the systematic uncertainties of the parameters. Section 5 presents the results from the analysis of the eROSITA data. A discussion of the various biases that affect the parameters is given in Sect. 6.1. Values for different cluster subsets are compared to previous results in Sect. 7. Finally, a combined disturbance parameter is described in Sect. 8.

## 2. Parameters

### 2.1. Introduction

In our analysis we measure a variety of parameters for each cluster. These include parameters used in previous studies in clusters and new parameters developed by us during the course of our analysis. The parameters are listed in Table 1 and described below.

### 2.2. Central density

A high central gas density, and therefore likely a short mean radiative cooling time, is often associated with the presence of a cool core (e.g. Fabian 2012). Therefore, central gas density is often used as a morphological parameter (e.g. Ghirardini et al. 2022). Cool cores are often located in regular (or ‘relaxed’) clusters, although there not a one-to-one match of regular clusters and those with a cool core (e.g. Hudson et al. 2010; Lovisari et al. 2017).

If a cool core evolves with redshift with the rest of the cluster, it makes sense to measure it a fixed fraction of $R_{500}$. In addition, as the critical density of the universe evolves it can make sense to measure the density relative to the average within a cluster, if it contains the universal baryon fraction. Therefore, we computed the scaled density from the electron density profile $n_e(r)$ as

$$n_{s,0} = \log \frac{n_e(0.02R_{500})}{\text{cm}^{-3}} - \log \left( 500 \rho_{\text{crit}} \frac{f_B}{\mu_e m_u} \frac{1}{\text{cm}^{-3}} \right), \quad (1)$$

where $\rho_{\text{crit}}$ is the critical density at the redshift, $f_B$ is the baryon fraction (fixed to 0.175 from Spergel et al. 2007), and $\mu_e m_u$ is the mass per electron for an ionised plasma (here $1.175 \times 1.661 \times 10^{-24}$ g). The choice of $0.02R_{500}$ was chosen for consistency with Ghirardini et al. (2022), although the density in that work is not relative to the critical density. Due to the wide dynamic range, we report our density parameters in log space.

However, if a cool core evolves independently from the rest of the cluster, it might be better to measure a physical electron density at a fixed physical radius. Therefore, we also measured a parameter for the log electron density at a radius of 50 kpc,

$$n_{50} = \log[n_e(50 \text{ kpc})/\text{cm}^{-3}]. \quad (2)$$

The density and uncertainties are computed from a Markov chain Monte Carlo (MCMC) analysis using MBProj2D (B24) of each cluster (Sect. 3), with the emcee sampler (Foreman-Mackey et al. 2013). We take a random sample of entries from the MCMC chain, compute the log density from each one, then compute the median and $1\sigma$ percentiles as the parameter. The assumed density functional form can have some
<table><caption>**Table 1.** Morphological parameters.</caption><thead><tr><th>Name</th><th>Description</th><th>Type</th><th>Centre</th><th>R<sub>500</sub></th></tr></thead><tbody><tr><td>n<sub>s,0</sub></td><td>Log gas density at a radius of 0.02R<sub>500</sub> relative to a scaled critical density</td><td>M</td><td>V</td><td>Y</td></tr><tr><td>n<sup>*</sup><sub>s,0</sub></td><td>Log gas density at a radius of 0.02R<sub>500</sub> relative to a scaled critical density</td><td>M</td><td>P</td><td>Y</td></tr><tr><td>n<sub>50</sub></td><td>Log gas electron density at a radius of 50 kpc</td><td>M</td><td>V</td><td>N</td></tr><tr><td>n<sup>*</sup><sub>50</sub></td><td>Log gas electron density at a radius of 50 kpc</td><td>M</td><td>P</td><td>N</td></tr><tr><td>α</td><td>Inner density slope at 0.04R<sub>500</sub> radius</td><td>M</td><td>V</td><td>Y</td></tr><tr><td>α<sup>*</sup></td><td>Inner density slope at 0.04R<sub>500</sub> radius</td><td>M</td><td>P</td><td>Y</td></tr><tr><td>α<sub>50</sub></td><td>Inner density slope at 50 kpc radius</td><td>M</td><td>V</td><td>N</td></tr><tr><td>α<sup>*</sup><sub>50</sub></td><td>Inner density slope at 50 kpc radius</td><td>M</td><td>P</td><td>N</td></tr><tr><td>c<sub>500</sub></td><td>Log ratio of integrated model surface brightness in apertures of 0.1R<sub>500</sub> and R<sub>500</sub></td><td>M</td><td>V</td><td>Y</td></tr><tr><td>c<sup>*</sup><sub>500</sub></td><td>Log ratio of integrated model surface brightness in apertures of 0.1R<sub>500</sub> and R<sub>500</sub></td><td>M</td><td>P</td><td>Y</td></tr><tr><td>c<sub>80–800</sub></td><td>Log ratio of integrated model surface brightness in apertures of 80 and 800 kpc</td><td>M</td><td>V</td><td>N</td></tr><tr><td>c<sup>*</sup><sub>80–800</sub></td><td>Log ratio of integrated model surface brightness in apertures of 80 and 800 kpc</td><td>M</td><td>P</td><td>N</td></tr><tr><td>F</td><td>Ratio of offset between best fitting cluster centre and peak and R<sub>500</sub></td><td>M/I</td><td>–</td><td>Y</td></tr><tr><td>P<sub>10</sub>, P<sub>20</sub>, P<sub>30</sub>, P<sub>40</sub></td><td>Log power ratio with orders from 1 to 4</td><td>I</td><td>F</td><td>Y</td></tr><tr><td>P<sup>*</sup><sub>10</sub>, P<sup>*</sup><sub>20</sub></td><td>Log power ratio with orders from 1 to 2</td><td>I</td><td>P</td><td>Y</td></tr><tr><td>G</td><td>Gini coefficient (0–1)</td><td>I</td><td>F</td><td>Y</td></tr><tr><td>A<sub>phot</sub></td><td>Photon asymmetry</td><td>I</td><td>F</td><td>Y</td></tr><tr><td>A<sup>*</sup><sub>phot</sub></td><td>Photon asymmetry</td><td>I</td><td>P</td><td>Y</td></tr><tr><td>w</td><td>Centroid shift</td><td>I</td><td>F</td><td>Y</td></tr><tr><td>ε</td><td>Ellipticity, the ratio of minor to major axis (0–1)</td><td>M</td><td>M</td><td>N</td></tr><tr><td>H</td><td>Slosh, the degree of sloshing factor (0–1)</td><td>M</td><td>M</td><td>N</td></tr><tr><td>M<sub>1</sub>, M<sub>2</sub>, M<sub>3</sub>, M<sub>4</sub></td><td>Multipole magnitudes (0–1) for orders 1 to 4</td><td>M</td><td>M</td><td>N</td></tr><tr><td>D<sub>shape</sub></td><td>Combined disturbance score (0–1), based on ε, H, M<sub>1</sub> to M<sub>4</sub>, and F</td><td>M/I</td><td>–</td><td>Y</td></tr><tr><td>D<sub>comb</sub></td><td>Combined disturbance score (0–1), also including c<sub>80–800</sub></td><td>M/I</td><td>–</td><td>Y</td></tr></tbody></table>

**Notes.** The Type column lists whether the parameter is the output of models (M) or computed from images (I). I values can also depend on the model fits (e.g. cluster centre, background rate, model surface brightness profile for masking). The type of cluster centre is given, including symmetric model position, where the value can vary during the analysis (V), when it is fixed to the best fit location (F), the cluster peak (P) or comes from the model being fitted, which can vary during the analysis (M). The column R<sub>500</sub> specifies whether the parameter depends on the cluster radius. The disturbance indicators combine other parameters and are discussed in Sect. 8.

impact on the measured central densities, especially for low mass or high redshift objects where the PSF size is larger than the radius the density is measured at. However, the wide prior on the central density slope we used reduces the bias due to the parametrisation and other priors.

For several of the parameters we examined, the quantity can be dependent on the position of the cluster centre (for discussion of this see Sect. 6.4). This is particularly important for quantities affected by the shape of the central density profile. Therefore, for these parameters we computed their value with the central position at two positions: the best fitting cluster centre (fitting the whole cluster) and using the peak of emission. The peak values are denoted with a *, while the best fit centre values are not marked in this way. The central density parameters are n<sup>*</sup><sub>s,0</sub>, n<sub>s,0</sub>, n<sup>*</sup><sub>50</sub>, and n<sub>50</sub>.

## 2.3. Inner density slope or cuspiness

Similarly, to the central density, the cuspiness parameter (Vikhlinin et al. 2007), α, is another indicator of the presence of a cool core and therefore a cluster with a regular morphology. Cool core clusters have steeply rising density profiles, while clusters with a more unrelaxed X-ray appearance have a flatter central density profile. The slope is defined at some radius, r, by

$$ \alpha(r) = -\frac{d \log n_e}{d \log r}. \qquad (3) $$

Following Vikhlinin et al. (2007), the parameter α is the slope at 0.04R<sub>500</sub>. We also calculated a slope at fixed 50 kpc physical

radius, α<sub>50</sub>. Similarly to the central density, we also computed values computed using the peak cluster position instead of the best fit position, α<sup>*</sup> and α<sup>*</sup><sub>50</sub>.

## 2.4. Concentration

The concentration is defined as the ratio of integrated surface brightness within two apertures (e.g. Santos et al. 2008). This is again a parameter that is sensitive to the presence of a cool core. We used two different sets of apertures. The first set has fixed physical apertures of 80 and 800 kpc, where

$$ c_{80-800} = \log \frac{I_X(80 \text{ kpc})}{I_X(800 \text{ kpc})}. \qquad (4) $$

Here, I<sub>X</sub>(r) is the integrated surface brightness within radius r. This definition has the advantage of not being affected by the determination of the cluster R<sub>500</sub>. In addition, if a cool cores do not evolve in time, as found previously in relaxed systems (McDonald et al. 2017), then a fixed aperture could be more appropriate.

We also measured the concentration using a fraction of the R<sub>500</sub> for the system,

$$ c_{500} = \log \frac{I_X(0.1R_{500})}{I_X(R_{500})}. \qquad (5) $$

The fixed aperture version uses a larger set of apertures than some other definitions (e.g. 40–400 kpc), but 80–800 kpc is better suited to the ~30 arcsec eROSITA PSF at moderate redshifts
(for example, at z = 0.3, a radius of 40 kpc is only 9 arcsec). $c_{80-800}$ and $c_{500}$ will be equivalent for $R_{500}$ = 800 kpc, which is close to the median value of $R_{500}$ = 740 kpc.

We note that we took the log of the values, unlike in other works. With this definition, flatter surface brightness profiles have more negative values (a completely flat profile would give a value of -2), while steeply peaked profiles tend towards a value of zero.

If we simply took surface brightness on the sky, the concentration would be affected by the PSF and background. To avoid this issue, the concentration was computed using our cluster model, before convolution with the PSF and the addition of background. We used the MCMC chains to compute the parameter and its uncertainties. For more compact clusters, the effect of the PSF is to increase the size of the uncertainties, rather than bias the results to less concentration. However, these results are dependent on how realistic the density parametrisation and its priors are. Disturbed objects may not be well fitted by our parametrisation or our priors may be inaccurate in these cases, potentially giving rise to a bias. As the concentration also depends on the central position, we also calculated the peak position concentrations, $c_{80-800}^*$ and $c_{500}^*$.

## 2.5. Fit-peak offset

As discussed above, the preferred cluster position from the symmetric model analysis and the peak X-ray emission may be offset from each other. This offset is itself a morphological parameter, as perfectly regular clusters would have their X-ray peaks at their centres. We defined the fit-peak offset, $F$, to be

$$F = \frac{\theta D_A}{R_{500}}, \qquad (6)$$

where $\theta$ is the angular separation between the fit and peak position in radians, $D_A$ is the angular diameter distance to the cluster, and $R_{500}$ is the cluster radius.

There are several ways to define the X-ray peak of a cluster. In the lower count regime of eROSITA, one of the most reliable ones is to smooth the X-ray image and find the position of the maximum pixel. We smoothed the exposure-corrected and background-subtracted image using a Gaussian with $\sigma$ = 24 arcsec, after replacing source regions of neighbouring sources with a realisation of the best fitting model of the main source and background (see Sect. 3.3). We searched for the maximum within a radius of $R_{500}$ from the best fit cluster centre and compute $F$ from its offset. Uncertainties were calculated by bootstrap resampling of the counts in the input image before smoothing, then calculating the $1\sigma$ range of $F$.

## 2.6. Power ratio

Power ratios are a method for the multipole decomposition of the surface brightness distribution (Buote & Tsai 1995). The power decomposed into a particular multipole is measured relative to a zero-th order value. The values are measured within a particular aperture, for which we used $R_{500}$. If the higher order multipoles for a cluster have relatively high power compared to the zero-th order, then this indicates an object with a non-spherical morphology.

The ratio for order $m$ is defined as

$$P_{m0} = \frac{P_m}{P_0}, \qquad (7)$$

where the zero order value is

$$P_0 = (a_0^2 \ln R_{500}^2)^2, \qquad (8)$$

and for higher orders,

$$P_m = \frac{1}{2m^2 R_{500}^{2m}} (a_m^2 + b_m^2), \qquad (9)$$

where for polar coordinate $x = (r, \phi)$ given the surface brightness $S(x)$,

$$a_m = \int_{r<R_{500}} S(x) r^m \cos m\phi \, d^2 x, \qquad (10)$$

and

$$b_m = \int_{r<R_{500}} S(x) r^m \sin m\phi \, d^2 x. \qquad (11)$$

In our results we quote the power ratio logarithm, $\log P_{m0}$, as they cover a wide dynamical range, for orders $m = 1$ to 4.

We note that the surface brightness has been background-subtracted by the best fitting background rate from the spherical MBProj2D analysis (Sect. 3). As the method does not take account of contaminating sources within $R_{500}$ these regions are filled with a Poisson realisation of the best fitting spherical MBProj2D model. The details of this filling procedure are given in Sect. 3.3. Statistical uncertainties were computed using bootstrap resampling of the pixels within $R_{500}$ and the quoted value is the median from this analysis.

We also note that the power ratios are computed directly from the X-ray image and therefore do not take account of the effect of the PSF. Blurring by the PSF will affect objects with a smaller angular size, reducing the power measured, although both terms of the ratio will be affected to different degrees. In addition, the non-Gaussian Poisson noise will affect the determined power in the low count regime (see Sect. 6.8). Previous studies have shown that there is bias in the parameters due to the number of photons (e.g. Weißmann et al. 2013). The ratio $P_{10}$ gives the relative strength of the dipole to the monopole, while $P_{20}$, $P_{30}$, and $P_{40}$ can be associated with quadrupole, hexapole, and octopole moments, respectively. $P_{20}$, the quadrupole moment, is also strongly correlated with the ellipticity of an object. The lower order power ratios are affected by the choice of the cluster centre. We therefore also computed $P_{10}^*$ and $P_{20}^*$ measured around the cluster peak.

## 2.7. Gini coefficient

The Gini coefficient is a well-known economic indicator of income inequality. Nevertheless, it can be applied to astronomical images (Abraham et al. 2003). It is primarily an indicator of how peaked the surface brightness distribution is. High Gini coefficients should be correlated with high concentrations. Following the definition used by Lotz et al. (2004), we measured the X-ray variation using

$$G = \frac{1}{|\bar{K}| n(n-1)} \sum_{i} (2i - n - 1) |K_i|, \qquad (12)$$

where the $n$ pixels in the input aperture have sorted surface brightness values $K_i$ and the mean value is $\bar{K}$.

As eROSITA images have a low number of counts per pixel (typically zero or one), the Gini coefficient is insensitive to morphology when measured from raw images because the pixels
cannot be ordered. Therefore we measured the Gini coefficient from a Gaussian-smoothed background-subtracted exposure-corrected image, using a fixed smoothing scale of $\sigma = 6$ pixels (24 arcsec). Similarly to the power ratios, we used an aperture of $R_{500}$. In addition, we filled masked areas and other source regions in images as was done in the power ratio analysis (Sect. 3.3). We used bootstrap resampling of the pixels in the aperture to obtain the median value and its $1\sigma$ uncertainties. We note that the Gini coefficient does not take account of the PSF of the instrument, as it works purely from an image. The result of blurring by the PSF will be to reduce the differences between pixels, leading to lower values of $G$ for compact or distant objects. As the Gini coefficient depends on the smoothing of the image and the instrument PSF, it cannot be compared directly with previous surveys.

## 2.8. Photon asymmetry

The photon asymmetry parameter ($A_{phot}$; Nurgaliev et al. 2013) measures the degree of symmetry in the source image. Simulations show this parameter is insensitive to the redshift of the cluster, if the instrument has a small PSF relative to the examined regions. We note that eROSITA has a significant PSF on these scales, so the photon asymmetry is not redshift independent for our clusters.

It is computed by first using Watson's test to compare the photons in an annulus with a radially symmetric cluster using

$$U_N^2[F_N, G] = N \min_{\phi_0} \int (F_N - G)^2 dG, \qquad (13)$$

where the total number of counts is given by $N$, $F_N$ is the cumulative distribution of the angle of all the photons in an annulus, $G$ is the expected distribution and $\phi_0$ is a starting angle for the distribution. If there are $C$ counts in the annulus, then the distance between these distributions is

$$\hat{d}_{N,C} = \frac{N}{C^2} \left( U_N^2 - \frac{1}{12} \right). \qquad (14)$$

The photon asymmetry is then computed using

$$A_{phot} = 100 \sum_{k=1}^{X} C_k \hat{d}_{N_k, C_k} / \sum_{k=1}^{X} C_k, \qquad (15)$$

where we computed it over $X = 4$ annuli, with edges of 0.05, 0.12, 0.2, 0.3, and 1.0 $R_{500}$. The uncertainties were obtained by repeating the calculation using a bootstrap resampled set of the photons.

The effect of PSF blurring should be to reduce the measured asymmetry for those clusters with small angular extent. Photon asymmetry is also sensitive to the choice of cluster centre. A version of this parameter with the peak position is also included as $A_{phot}^*$.

Although photon asymmetry is assumed to be insensitive to redshift, we checked this using simple symmetric models. We made a $\beta$ model cluster ($r_c = 0.17R_{500}$, $\beta = 2/3$) image with no PSF broadening or background and with the annuli are properly resolved in the image ($R_{500} = 120$ arcsec, with 0.25 arcsec pixels). If the model is scaled to have $N$ counts within $R_{500}$, and then a Poisson realisation is made, we found $A_{phot} \approx 20N^{-1}$. The reference implementation<fnref n="1" /> gives consistent results with our code. Therefore, for faint symmetric clusters of a few tens of counts, $A_{phot}$ can be relatively large (0.8 for 20

counts, 0.4 for 50 counts, and 0.2 for 100 counts) and therefore not redshift independent. This simple model does not account for the effect of background nor for the 4 arcsec pixels we used. However, more realistic simulations of the intrinsic distribution can be found in Sect. 4.

## 2.9. Centroid shift

The centroid shift parameter is a commonly applied parameter which measures the variance of the centroid of the emission with increasing apertures, although different authors use different definitions (Mohr et al. 1995; Poole et al. 2006; O'Hara et al. 2006; Böhringer et al. 2010). We defined it as

$$w = \frac{1}{R_{500}} \left( \frac{1}{N-1} \sum_{i=1}^{N} |\mathbf{C}_i - \mathbf{C}_{\text{peak}}|^2 \right)^{1/2}, \qquad (16)$$

where $\mathbf{C}_i$ is the centroid within aperture $i$ and $\mathbf{C}_{\text{peak}}$ is the peak position.

In our analysis we computed the centroid shift using the shifts from $N = 10$ linearly radially increasing apertures within $R_{500}$ about the peak position. The peak position was measured from a Gaussian-smoothed image of the cluster (smoothed using $\sigma = 24$ arcsec), with masked and fitted sources replaced by model realisations (Sect. 3.3). The value and its uncertainties were generated from the median and $1\sigma$ percentiles of $w$ values measured from Poisson realisations of the smoothed image.

We note that this quantity is calculated directly from the image and therefore does not take account of the instrument PSF. Blurring of the image means that for apertures smaller or around the PSF size, offsets from the peak will be reduced, leading to lower values of $w$. This is most important for compact objects or those at higher redshifts. Similarly to other quantities, we computed $\log w$ as its dynamic range is large.

## 2.10. Ellipticity

The ellipticity parameter ($\epsilon = b/a$) follows the definition in Ghirardini et al. (2022), of the ratio of the minor ($b$) to the major ($a$) axis ($a$). A value of one is circular and zero is the extreme elliptical case. The ellipticity of the X-ray emission of clusters may be sensitive to the physics of the ICM (Lau et al. 2012), in addition to being affected by mergers. Ellipticity is implemented within MBProj2D to produce an elliptical surface brightness distribution on the sky. An example model with ellipticity is shown in Fig. 1. It models the surface brightness on the sky in Cartesian coordinates where the projected surface brightness is

$$S'(x, y) = S(\{\epsilon[x \cos \theta_0 - y \sin \theta_0]^2 + [x \sin \theta_0 + y \cos \theta_0]^2 / \epsilon\}^{1/2}), \qquad (17)$$

where $S(r)$ is the original radial surface brightness and $\theta_0$ is the inclination angle of ellipse (between 0 and 180 deg). The radial surface brightness was parametrised in the same way as for the standard profile analysis. As is standard when modelling clusters with MBProj2D, observational effects such as PSF and background were also accounted for by the model and uncertainties were calculated from the MCMC chain.

We note that the ellipticity parameter can describe clusters which are likely physically unrealistic (where $\epsilon \lesssim 0.2$). Informative, rather than flat, priors could be useful for a future analysis. The eccentricity parameter might be better than ellipticity, as its parameter space is biased towards more circular objects.

<footnote n="1">¹ https://github.com/ndaniyar/aphot</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Example models exhibiting the effect of the different shape parameters. Shown are an elliptical model (top left), a slosh model (top right), an $M_1$ model (centre left), an $M_2$ model (centre right), an $M_3$ model (bottom left), and an $M_4$ model (bottom right). For the models, we used an angle of $\theta = 30$ deg. The numeric values show the magnitude of the respective parameter.</figcaption></figure>

## 2.11. Slosh

Slosh ($H$) is a new parameter we designed to model how asymmetric the surface brightness is about the cluster. It has the appearance of the gas sloshing in the potential well of the cluster, as is thought to be the origin of cold fronts (see Markevitch & Vikhlinin 2007). As with ellipticity, we modelled this within MBProj2D as a modification to the projected surface model brightness. Similarly, an example model with slosh is shown in Fig. 1.

The projected surface brightness on the sky, $S'$, at a radius $r$ and angle $\theta$, is

$$S'(r, \theta) = A(H) S(r[1 + H \cos(\theta + \theta_0)]), \quad (18)$$

where the slosh factor, $H$, lies between zero and one, $S(r)$ is the original symmetric distribution and $\theta_0$ is the sloshing angle (between 0 and 360 deg). As $H$ tends towards zero, the distribution becomes the original symmetric one, while values of one give an extreme asymmetric distribution. Here, $A(H)$ is a factor to ensure that the total brightness of the cluster model is not modified by changes in $H$. By considering the relative area of a circle with the slosh transformation applied, compared to its

original area, one finds that

$$A(H) = (1 - H^2)^{3/2}. \quad (19)$$

Within MBProj2D the surface brightness is computed more easily in Cartesian coordinates $(x, y)$ around the cluster centre, where

$$S'(x, y) = A(H) S([x^2 + y^2]^{1/2} + H[x \cos \theta_0 - y \sin \theta_0]). \quad (20)$$

If there is significant disturbance of the cluster, the position of the centre is likely to be different from the outer regions or one obtained assuming the cluster is symmetric. As described in Sect. 3, the centre of the main cluster was left free in the analysis.

## 2.12. Multipole magnitude

Forward modelling of parameters has some advantages over values measured directly from images. For example, the PSF of the source and the background can be included within the modelling, and the parameter uncertainties obtained using MCMC. Therefore we created forward models of the cluster which include azimuthal variations similar to power ratios. The surface brightness of a symmetric model profile $S(r)$ was modified to produce the following distribution:

$$S'(r, \theta) = [1 + M_m \sin(m\theta + \theta_0)] S(r), \quad (21)$$

where $m$ is a multipole index, $\theta_0$ is an inclination angle that ranges from 0 to 360/m deg, and $M_m$ is the multipole magnitude that varies between zero and one. Images of models with these multipole variations are shown in Fig. 1. MBProj2D computes the model image in Cartesian coordinates before comparison with the data. Firstly, the pixel sky coordinates $(x, y)$ are rotated on the sky in reverse around the cluster centre by $\theta_0$ to compute rotated coordinates $(x_r, y_r)$. Using $\sin \theta = y_r/r$ and $\cos \theta = x_r/r$, $\sin(m\theta)$ is calculated using the standard multiple angle formulae. When $r = 0$, $\theta = 0$ is used above.

In our analysis we computed the morphological parameters $M_1$ to $M_4$ by fitting the multipole model to the data. As can be seen by the images, the parameter $M_1$ has a shape which is somewhat similar to the shape of a cluster with the slosh parameter ($H$) applied. This leads to a correlation between these parameters, particularly for objects with lower data quality. Parameter $M_2$ also looks similar to an elliptical cluster ($\epsilon$) and so these parameters are also correlated.

## 2.13. Test of new parameters

We tested our new morphological parameters on images of 83 galaxy clusters detected from South Pole Telescope (SPT) surveys (Bleem et al. 2015) using the Sunyaev-Zel'dovich (SZ) technique and observed by Chandra in the X-ray waveband (McDonald et al. 2013; Sanders et al. 2018). Section 3.4 describes these data and their analysis. These clusters show a wide variety of different morphologies and their images have high spatial resolution and good data quality. The data were analysed seven times, one for each of the morphological parameters $\epsilon$, $H$, $M_1$, $M_2$, $M_3$, and $M_4$, and also for a spherical cluster.

Figure 2 shows a corner plot of the parameters for the clusters in the sample, revealing significant variation in these parameters between the clusters. Some parameters are clearly correlated with each other, including $\epsilon$ and $M_2$, and $M_1$ and $H$.
## 3. Data analysis

### 3.1. Introduction

The inputs of our analysis were created using the same pipeline as in B24, although we only used the monochromatic X-ray images in a band of 0.2–2.3 keV. We only took data from the cameras without a light leak (i.e. the so-called TM8 data which combines data from telescope modules, TMs, 1, 2, 3, 4, and 6). These images and exposure maps were produced using the standard eSASS analysis software (Brunner et al. 2022), version eSASSUsers_211214_0_4, which is equivalent to the publicly available version eSASS4DR1. We assumed that the clusters lie at the best redshift from the cluster catalogue. For those parameters which assume a value of R$_{500}$, we took this from the cluster catalogue. The best fitting positions, for consistency, came from our MBProj2D analysis assuming a radially symmetric model, and are not necessarily the same as in B24.

### 3.2. MBProj2D analyses

Several of the parameters resulted from modelling the cluster with MBProj2D. For these analyses, we followed a common procedure which we describe here. As we only fitted monochromatic images, we assumed isothermal temperature profiles fixed to the median temperature from the MCMC chain obtained by B24 and used the Galactic absorbing column density from the catalogue.

The cluster profiles in B24 use a fairly strong prior on the inner density slope model parameter, $\alpha$. This leads to some bias on the central slope measurement and concentration. Therefore, we re-analysed the cluster images using a density model with a weak prior on $\alpha$ (Table 2). As discussed by Käfer et al. (2019), clusters typically follow a $\beta$ model with $\beta = 2/3$ in the outskirts, provided the core is otherwise modelled. The data quality in the eRASS1 survey also limits how sophisticated a model can be fitted to the data. Therefore, we used a simplified form for the density profile of Vikhlinin et al. (2009), compared to B24. We allowed $\beta$, the standard $\beta$-model outer slope, to vary over a more restricted range around $\beta = 2/3$ using a prior. The assumption of $\beta = 2/3$ could bias results in objects which deviate from that (e.g. flatter profiles in groups, see for example Johnson et al. 2009), although those parameters which depend on the density profile are sensitive to the central density. We did not allow the outermost slope in the parametrisation to vary and thus fixed $\epsilon$ to 0 and froze $r_s$. The second $\beta$ component was also not included in the model. The fitted profile was therefore a $\beta$ model with an inner power-law core of the form

$$n_e^2 = n_0^2 \frac{(r/r_c)^\alpha}{[1 + (r/r_c)^2]^{3\beta - \alpha/2}}, \quad (22)$$

where $n_0$ is the central density and $r_c$ is the core radius. We note that due to the definition being density-squared, $\alpha$ here is twice the inner density slope and is not the same as the cluster density inner slope morphological parameter, $\alpha$.

The input images used the same masking and sizes as B24. If a point source was fitted for in B24, we fitted for it here. Unlike B24, we only used monochromatic 0.2–2.3 keV band input images and used a fixed 4 arcsec bin size to ensure the best resolution. As our PSF, we took the standard eROSITA calibration 1.486 keV survey-averaged PSF, but averaged it in radius to make it symmetric. Signal outside 4 arcmin radius was also removed to match the definition of the eROSITA ancillary response matrices.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Corner plot of the median MBProj2D shape parameters for Chandra observations of clusters in the SPT cluster sample. The quantities are plotted against each other, while the rightmost panels show the probability density of the values. The Pearson correlation coefficient is shown in each panel.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Example Chandra images from the SPT sample shown in Fig. 2. Top left: SPT-CLJ0014-4952 [$\epsilon = 0.85$, $H = 0.50$, $M_1 = 0.36$, $M_2 = 0.26$, $M_3 = 0.06$, $M_4 = 0.03$]. Top right: SPT-CLJ0307-6225 [$\epsilon = 0.58$, $H = 0.75$, $M_1 = 0.88$, $M_2 = 0.37$, $M_3 = 0.27$, $M_4 = 0.28$]. Bottom left: SPT-CLJ0304-4401 [$\epsilon = 0.75$, $H = 0.14$, $M_1 = 0.64$, $M_2 = 0.26$, $M_3 = 0.33$, $M_4 = 0.11$]. Bottom right: SPT-CLJ2342-5411 [$\epsilon = 0.64$, $H = 0.08$, $M_1 = 0.71$, $M_2 = 0.38$, $M_3 = 0.07$, $M_4 = 0.07$].</figcaption></figure>

The most disturbed clusters are found by selecting objects with more extreme values of these parameters. Fig. 3 shows SPT-CLJ0014-4952 has a strong slosh and $M_2$ signal. SPT-CLJ0307-6225 has pronounced non-circular $\epsilon$, $H$ and $M_1$. SPT-CLJ0304-4401 has a large $M_1$, $M_2$ and $M_3$ signal because it has a triangular morphology. SPT-CLJ2342-5411 is elliptical, with large $\epsilon$ and $M_1$ values. The modelling appears to perform well by picking up the disturbances that the human eye can see in the images. We further compare the Chandra and eRASS1 cluster results in Sect. 7.
<table><caption>**Table 2.** Priors on the MBProj2D analyses with the different models.</caption><thead><tr><th>Description</th><th>Parameter</th><th>Prior</th></tr></thead><tbody><tr><td colspan="3">Standard (circular and other models)</td></tr><tr><td>Central density</td><td>ln(n<sub>0</sub>/cm<sup>-3</sup>)</td><td>U(-14, 5)</td></tr><tr><td>Outer slope</td><td>β</td><td>TN(2/3, 1/3, 0.2, ∞)</td></tr><tr><td>Inner slope</td><td>α</td><td>TN(0, 5, -10, 10)</td></tr><tr><td>Core radius</td><td>ln(r<sub>c</sub>/kpc)</td><td>TN(ln 400, 3, ln 5, ln 5000)</td></tr><tr><td>Cluster RA shift <sup>(*)</sup></td><td>δx/arcsec</td><td>N(0, 30)</td></tr><tr><td>Cluster Dec. shift <sup>(*)</sup></td><td>δy/arcsec</td><td>N(0, 30)</td></tr><tr><td>Background</td><td>ln(B/ct/s/arcsec<sup>2</sup>)</td><td>U(-18.4, -8.4)</td></tr><tr><td>Point source normalisation <sup>(**)</sup></td><td>ln(N/photon/keV/cm<sup>2</sup>/s)</td><td>U(-17, -12)</td></tr><tr><td colspan="3">Elliptical</td></tr><tr><td>Rotation angle <sup>(†)</sup></td><td>θ<sub>0</sub>/°</td><td>U<sup>s</sup>(θ<sub>fit</sub> - 90, θ<sub>fit</sub> + 90)</td></tr><tr><td>Ellipticity</td><td>ε</td><td>U<sup>s</sup>(0, 0.999)</td></tr><tr><td colspan="3">Slosh</td></tr><tr><td>Rotation angle <sup>(†)</sup></td><td>θ<sub>0</sub>/°</td><td>U<sup>s</sup>(θ<sub>fit</sub> - 180, θ<sub>fit</sub> + 180)</td></tr><tr><td>Slosh</td><td>H</td><td>U<sup>s</sup>(0.001, 1)</td></tr><tr><td colspan="3">Multipole 1</td></tr><tr><td>Rotation angle <sup>(†)</sup></td><td>θ<sub>0</sub>/°</td><td>U<sup>s</sup>(θ<sub>fit</sub> - 180, θ<sub>fit</sub> + 180)</td></tr><tr><td>Multipole magnitude</td><td>M<sub>1</sub></td><td>U<sup>s</sup>(0, 1)</td></tr><tr><td colspan="3">Multipole 2</td></tr><tr><td>Rotation angle<sup>†</sup></td><td>θ<sub>0</sub>/°</td><td>U<sup>s</sup>(θ<sub>fit</sub> - 90, θ<sub>fit</sub> + 90)</td></tr><tr><td>Multipole magnitude</td><td>M<sub>2</sub></td><td>U<sup>s</sup>(0, 1)</td></tr><tr><td colspan="3">Multipole 3</td></tr><tr><td>Rotation angle <sup>(†)</sup></td><td>θ<sub>0</sub>/°</td><td>U<sup>s</sup>(θ<sub>fit</sub> - 60, θ<sub>fit</sub> + 60)</td></tr><tr><td>Multipole magnitude</td><td>M<sub>3</sub></td><td>U<sup>s</sup>(0, 1)</td></tr><tr><td colspan="3">Multipole 4</td></tr><tr><td>Rotation angle<sup>†</sup></td><td>θ<sub>0</sub>/°</td><td>U<sup>s</sup>(θ<sub>fit</sub> - 45, θ<sub>fit</sub> + 45)</td></tr><tr><td>Multipole magnitude</td><td>M<sub>4</sub></td><td>U<sup>s</sup>(0, 1)</td></tr></tbody></table>

**Notes.** Here $U(a, b)$ is a uniform prior between $a$ and $b$. $N(μ, σ)$ is a normal distribution centred on $μ$ with width $σ$. $TN(μ, σ, a, b)$ is a truncated normal distribution with the peak at $μ$, with width $σ$, truncated below $a$ or above $b$. $U^s(a, b)$ is a uniform distribution with softened edges, where the log prior is $1 - e^{-(a-x)/0.01}$ if $x < a$, $1 - e^{-(x-b)/0.01}$ if $x > b$, and zero otherwise. In this case, during model evaluation, the parameter was forced to the range of $a$ to $b$. <sup>(*)</sup>Only the main cluster position was allowed to vary from the source detection position during the MCMC run. Other cluster positions were fitted for once, then fixed to these positions. If fitted using the peak position, these parameters were frozen. <sup>(**)</sup>Normalisations defined at 1 keV assuming a power-law photon index of 1.7. <sup>(††)</sup>The best fitting value of $θ_0$ was used as $θ_{fit}$ to reduce boundary issues when sampling. In the output catalogue, the value of $θ_0$ was reduced to the range about $θ_{fit} = 0$.

As in B24, we assumed that the sky background is flat across the fields examined. As most of the background in this energy band is due to X-ray sources, rather than noise intrinsic to the detector, we multiplied this background by the vignetted exposure map. Faint point sources with ML_RATE < 0.4 in the 0.2-2.3 keV band were masked out in the fit, while brighter ones were fitted as part of the model. If there were other clusters within the fitted image, these were included separately in the model. The position of the main cluster was allowed to vary in the MCMC analysis, while point source positions and positions of other clusters were fixed at their best fitting values.

MBProj2D was run with a standard spherically symmetric cluster model. We fitted the model to the data to obtain the best fitting, maximum-likelihood parameters. We then conducted an MCMC analysis to obtain chains of parameters. As described in Sect. 2, these chains were processed to calculate a random subset of density profiles. From these profiles, the distributions of the central density, inner density slope (cuspiness), and concentration were calculated. The resulting distributions were then used to compute the median values and $1σ$ percentiles. The median cluster position in the chain was used as the 'best fitting' cluster position and used to derive $F$.

For the shape parameters, $ε$, $H$, $M_1$, $M_2$, and $M_3$, we repeated the fits to the data using MBProj2D, but we allowed the shape parameters to vary. In these analyses, we took the best fitting model from the above symmetric fit, added new free parameters which vary the morphology, then refit the data. Rather than give the median parameter values from the chain for the shape parameters or angles (which tend to 0.5 for the shape parameters with low counts), we provide the maximum-likelihood values, but computed the uncertainties using the MCMC distribution about the best fit.

We also repeated the first MBProj2D analysis with a fixed cluster peak position. The peak was computed from a Gaussian smoothed image of the cluster (see Sect. 2.5). The fit started from the previous best fitting parameters and the analysis was identical, except for freezing the main cluster position to the location of its peak.

### 3.3. Image-derived parameters

Several parameters were derived directly from the X-ray images rather than fitting models. Power ratios, the Gini coefficient, and
centroid shift calculations require input images that do not con-
tain point sources or neighbouring clusters. Therefore, for these
measurements, we need to 'fill' in the regions occupied by these
neighbouring sources. The photon asymmetry parameter, how-
ever, was derived with these regions masked out.

All these images initially require a mask. Firstly we com-
puted a model of the sky from the best fitting radially symmetric
MBProj2D model, including the effect of the PSF, but not back-
ground. We masked out regions where neighbouring clusters or
point sources were at most half the contribution of the main clus-
ter. In addition we masked out regions where these objects were
more than 1/10 of the background level.

To make the filled input images where these masked regions
were replaced, we computed the best fitting model of the main
cluster and background from the MBProj2D analysis, including
PSF, and computed a Poisson realisation of this. The regions
which were masked out in the input image were replaced by this
simulated image.

3.4. South Pole Telescope cluster data analysis

For comparison with the eROSITA clusters, we studied a subset
of SPT-detected clusters (Bleem et al. 2015), observed in X-rays
using *Chandra* and previously analysed in Sanders et al. (2018).
For the *Chandra* analysis we used the same X-ray data and X-ray
bands as described in Sanders et al. (2018). However, we used
MBProj2D rather than MBProj2 as described in that paper and
therefore fitted images rather than profiles.

The results for these clusters are compared with both the
standard eRASS1 processing and those obtained from deeper
eRASS:4 data. To make a better comparison for the eRASS:4
data we used very similar analysis procedures for both *Chan-
dra* and the deeper eROSITA data. These data and procedures
are close to those described in B24 for the eROSITA-*Chandra*
flux comparison. We used a very similar mask for both datasets,
where the cluster is masked beyond a radius of 5 Mpc and outer
point sources outside a radius of 3 arcmin from the cluster core
were excluded with a mask size with a minimum of 0.5 arcmin
radius. The eRASS:4 eROSITA data were masked where they
were outside of the field of view of the *Chandra* data. In the
inner part of the cluster, due to the high spatial resolution, the
point sources were masked for the *Chandra* analysis, while the
sources are modelled within MBProj2D for the eROSITA data
with the positions fixed from *Chandra* values. The *Chandra* data
were binned to a spatial resolution of 4 pixels (2 arcsec), while
we used the standard eROSITA spatial binning.

For both analyses, the clusters were assumed to be isother-
mal. When analysing the density profile for the symmetric clus-
ter profile case for *Chandra*, we fitted multiple energy bands in
order to constrain the cluster temperature. The *Chandra* back-
ground in each energy band was assumed to have the shape of
the standard blank sky background, but with a variable log nor-
malisation factor.

For eROSITA we fitted monochromatic 0.2–2.3 keV TM8
images of the clusters, but fixed the temperature to the median
value obtained by *Chandra*. The centre of the cluster was fixed to
the *Chandra* peak. When studying the morphological parameters
which affect the shape of the cluster (see Sect. 2.13), we fitted a
single band *Chandra* image in the band 0.5–4.0 keV, again fixing
the temperature at the value obtained from the multi-band fit.

For both eRASS:4 and *Chandra* we assumed the same den-
sity parametrisation as for our eRASS1 analysis, with the same
priors on the parameters. The metallicity was fixed to be 0.3
solar, as for the eRASS1 analysis.

4. Cluster simulations

To better understand our results we undertook simplified simu-
lations of clusters on the eROSITA sky. The main aim of these
simulations was to investigate the parameter biases due to the
measurement process and the quality of data. In addition we
wanted to check whether cluster morphology can have a signifi-
cant affect on the eROSITA selection function.

Rather than making a full eROSITA simulation of the whole
sky (Comparat et al. 2020; Seppi et al. 2022) we made simula-
tions of individual clusters at random positions on the eROSITA
sky and attempted to detect and measure their properties. The
clusters were assigned luminosities and redshifts from a grid,
with 9 logarithmic-spaced values of L₅₀₀ between 10⁴³ and
10⁴⁵ erg s⁻¹ and redshift values of 0.02, 0.035, 0.05, 0.07, 0.1,
0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1.0, and 1.2. Using the
covariance matrix method described in Comparat et al. (2020),
we generated random cluster masses, temperatures, and profiles.
Clusters were assigned to a grid point by generating a mass func-
tion for the relevant redshift and assigning randomly generated
clusters nearby in mass and redshift. Those clusters with lumi-
nosities close to the grid point value were assigned to it.

To make the simulated image of a randomly selected clus-
ter for the grid point, we used MBProj2D to generate a model
image for the cluster given its density profile, temperature, and
Galactic absorption. The images were created in the band 0.2–
2.3 keV, with a dimension of 70 arcmin for z = 0.02, 60 arcmin
for z = 0.035, 50 arcmin for z = 0.05, and 40 arcmin other-
wise. The cluster location was chosen randomly to lie within the
region of sky that the cluster sample was obtained from in B24.
The image was convolved with the eROSITA PSF and multi-
plied by the eROSITA eRASS1 exposure map at that sky posi-
tion. Wavelet filtered maps of the eROSITA sky were used as a
background, where structures below scales of around 30 arcmin
were removed. No AGN were included in the simulations. We
split the eROSITA TMs into two sets (TMs 5 and 7, and the
rest), which were simulated separately. Poisson realisations of
the two sets of model images were created. The summed image
of the two sets of TMs was used for detection, while only the
one excluding TMs 5 and 7 was used for characterisation.

To tell whether a cluster would be detected, we applied the
eSASS pipeline to the simulated image. Initially, erbox was
used on the simulated image to make a list of sources (with
likemin=6, nruns=3, boxsize=4, and bkima_flag=N). This
source list was supplied to erbackmap to make an initial back-
ground map (using scut=0.00005, mlmin=6, maxcut=0.5,
smoothval=15, snr=40, and smoothmax=360). erbox was run
a second time using the background map (with likemin=4 and
bkima_flag=Y). We run erbackmap a second time to produce
a new background map based on the generated source list.
erbox was run again using the previous background map and a
new background map was generated using erbackmap. We then
ran the maximum-likelihood source detection based on the pre-
vious source list, using ermldet with parameters likemin=5,
extlikemin=3, cutrad=15, multrad=15, extmin=2,
extmax=15, nmaxfit=4, nmulsou=2, shapelet_flag=no,
and photon_flag=no. These parameters are the same as used
for the main eROSITA catalogue. A cluster was detected if
there was an object in the output catalogue with non-zero extent
within 2 arcmin of the input position.

One of the major differences between this simplified detec-
tion pipeline and the standard one is that we used image based
rather than photon based detection. Photon based detection uses
a separate PSF for each photon, making use of smaller PSF at the
centre of the telescopes relative to the outskirts as sources scan across the field of view. Image based detection uses an average survey PSF and is significantly faster, but should be less sensitive close to the detection threshold.

The detected clusters were analysed in a very similar way to the real clusters. The images of the clusters, excluding TMs 5 and 7, were analysed with the MBProj2D and image based routines. The background component was assumed to have a flat count rate over the 40 arcmin region. To obtain the density-based parameters, such as concentration, we fitted a Vikhlinin model using MBProj2D with the same priors as used for the real data here. Rather than run a full MCMC analysis for each simulated cluster, we just found the maximum-likelihood parameters. The forward-modelled shape parameters (ε, *H*, *M*₁, *M*₂, *M*₃, and *M*₄) were obtained by fitting in MBProj2D using the same density profile priors as for the density analysis. We note that the centre of the cluster was allowed to be free during these fits. The power ratios, photon asymmetry, centroid shift, and Gini parameters were calculated from the simulated X-ray image but assuming the input *R*₅₀₀ rather than determining it from the data.

In our input models, we simulated a variety of cluster shapes, including spherical, elliptical (with $\epsilon = 0.7$), sloshed (with $H = 0.3$), or with $M_1, M_2, M_3$, or $M_4$ equal to 0.3. This allowed us to test our ability to recover the input parameter shape values or examine how cluster shape affects the obtained image-based parameters (such as power ratios or photon asymmetry).

## 5. Results

In this section we present the morphological parameters obtained from the real eROSITA cluster sample. Detailed information about the catalogue is given in Appendix A. In Fig. 4 we plot each of the parameters against the redshift of the source. For this plot we split the clusters into two subsamples (Table 3). The first is a bright subsample of clusters, where there are more than 300 counts in an 800 kpc radius aperture and the number of counts is greater than three times the lower uncertainty on the number of counts. For these points we colour markers using the X-ray luminosity within the 800 kpc aperture and show the uncertainties on the values. A fainter subsample with at least 25 significant counts are plotted with smaller markers.

Figure 5 shows the parameters plotted as a function of the number of counts in the cluster, computed within an aperture of 800 kpc. As some of our measurements are sensitive to the quality of the data, this plot is ideal for identifying biases due to this. The colour scale indicates the redshift of the cluster. The same results are plotted in Fig. 6, but showing them as a function of X-ray luminosity within an aperture of 800 kpc. The colour scale in this figure shows the redshift of the clusters.

Plotting the parameters as a function of the number of counts, we see that many of them are clearly increasingly biased towards the low count regime. This is particularly true for those parameters which are not based on the MBProj2D forward modelling method (e.g. power ratios and photon asymmetry). In addition we see that when plotting these values as a function of luminosity, the clusters are clearly separated in redshift for these parameters. The same parameters are also separated as a function of luminosity, if plotted as a function of redshift.

In theory, it could be possible that there are intrinsically strong variations in these parameters as a function of redshift and luminosity. However, in Sect. 6.8 our simulations show that these strong variations with luminosity and redshift (and therefore counts) are what is expected based on the level of noise in the survey and the PSF size.

The density parameters show increasingly higher values for more luminous or higher redshift clusters. However, there is no clear evolution with the number of counts. The scaled central density shows a weaker evolution than the physical density at fixed radius. The concentration parameters do not show any clear evolution with redshift of luminosity. For the shape parameters, $\epsilon, H$, and $M_1$ to $M_4$, the results become roughly randomly distributed in the lower count regime below 100–200 counts.

Similarly to the Chandra observed SPT sample (Fig. 2) we plot a corner plot of the various shape parameters. Figure 7 shows the parameters $\epsilon, H, M_1, M_2, M_3$, and $M_4$ for the subset of the brightest clusters (≥300 counts). The parameters show a very similar distribution to the SPT sample results, although they are noisier due to the lower number of counts. We observed that some of the parameters are correlated with each other, such as for Chandra, with $\epsilon$ and $M_2$ being inversely correlated and $H$ and $M_1$ correlated.

In Fig. 8 we show a correlation matrix between the various parameters for a subset of bright clusters. We only include the bright clusters, as the parameters become noise dominated when a cluster is faint, leading to a reduction in any correlation. In this plot we also include the peak and best fit versions of those parameters when appropriate. It can be seen that there are generally strong correlations between a high central density, concentration and central slope. There are some weaker correlations between $n_{50}^*$ and the central slope and concentrations, compared to $n_{50}$. The concentrations measured in physical or scaled apertures, or measured around the peak and best fit positions are all strongly correlated.

There is some mild correlation between the ellipticity (noting that higher ellipticities are rounder clusters) and the central density and concentration. The parameters $H, M_1$ to $M_4$, power ratios, $F$, and $w$ are generally inversely correlated with the central density, concentration and central slope. The ellipticity and $M_2$ are strongly inversely correlated, while $H$ and $M_1$ show correlation. The Gini coefficient is correlated with central density, concentration and inner slope. Several of the parameters measured from images (power ratios, photon asymmetry and centroid shift) are negatively correlated with the number of counts (see Sect. 6.8).

In Fig. 9 we show example brighter clusters with a similar number of counts, showing a diversity of morphology. Examples include a cluster with a high concentration and central density (1eRASS J124009.9-482612) and another with low concentration and central density (1eRASS J124116.4+183310). Elliptical clusters (low $\epsilon$) include 1eRASS J043817.8-541917, 1eRASS J053128.2-751032, and 1eRASS J124116.4+183310, while 1eRASS J124009.9-482612 has a low ellipticity. The elliptical clusters generally have larger $M_2$ values. 1eRASS J043817.8-541917 has a high slosh ($H$) because of its offset central peak and similarly with 1eRASS J053128.2-751032, 1eRASS J055252.8-210407, 1eRASS J124116.4+183310, and 1eRASS J224956.6-642532 has a large $M_1$ value. 1eRASS J224956.6-642532 has the largest $M_3$ value because there is bright triangular shape feature around the central peak. 1eRASS J051016.7-451911, 1eRASS J124009.9-482612, and 1eRASS J124116.4+183310 have the largest $M_4$ values in this subsample. Those clusters with the largest peak-fit offsets are 1eRASS J051636.6-543031, 1eRASS J053128.2-751032 and 1eRASS J124116.4+183310, while 1eRASS J124009.9-482612, 1eRASS J140755.5-505938, and 1eRASS J155821.9-140957 show small offsets.

Although for the most disturbed appearing clusters (e.g. 1eRASS J124116.4+183310) several of the parameters indicate
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Morphological parameters against redshift. The parameters are plotted for the bright cluster subset (≥ 300 counts) with larger markers. The small markers show clusters with at least 25 counts, without plotting uncertainties. The colour scale shows the X-ray luminosity (log erg s⁻¹ in the 0.2–2.3 keV band) inside an aperture of radius 800 kpc. The parameters are the central scaled gas density (n_s,0), central electron density (n_50), concentration from 80–800 kpc (c_80–800), concentration from 0.1R_500 to R_500 (c_500), central density slope (α), slope at fixed radius (α_50), ellipticity (ϵ), slosh (H), multiple magnitudes (M_1 to M_4), power ratios (P_10 to P_40), photon asymmetry (A_phot), centroid shift (w), Gini coefficient (G), and fit-peak offset (F).</figcaption></figure>

their disturbance, there are cases where only one or a few of the
parameters indicates some disturbance (e.g. 1eRASS J124009.9-
482612). One interesting example is J043817.8-541917, where
visual inspection would likely miss the disturbance-indicating
values of ϵ, H, M₁, M₂, and M₃.

6. Parameter biases

6.1. Introduction

There are several biases which can affect the obtained parame-
ters. It is important to understand these biases before using the
parameters blindly, as they can be a function of other cluster
properties. For example, some parameters show strong evolution

with redshift or luminosity with no change in cluster morphol-
ogy. The morphology of the cluster can also affect how likely it is
to be selected, presenting a biased subset of clusters with respect
to certain characteristics. We summarise the various biases in
Table 4.

6.2. Morphological detection bias

The morphology of a cluster may affect how easy it is for a
cluster to be detected in the X-ray waveband. For example, cool
core clusters are known to have higher luminosities for the same
mass compared to non-cool core clusters (e.g. Edge & Stewart
1991; Pratt et al. 2009; Hudson et al. 2010; Mittal et al. 2011).
<table><caption>**Table 3.** eRASS1 cluster subsamples.</caption><thead><tr><th>Sample</th><th>Number</th></tr></thead><tbody><tr><td>All clusters</td><td>12075</td></tr><tr><td>≥25 counts, L<sub>ext</sub> > 6</td><td>5387</td></tr><tr><td>≥300 counts, L<sub>ext</sub> > 6</td><td>541</td></tr><tr><td>≥400 counts, L<sub>ext</sub> > 6</td><td>395</td></tr><tr><td>≥800 counts, L<sub>ext</sub> > 6</td><td>175</td></tr><tr><td>L<sub>ext</sub> > 6, L<sub>det</sub> > 20, cosmology</td><td>4160</td></tr><tr><td>L<sub>ext</sub> > 6, L<sub>det</sub> > 40, cosmology</td><td>2484</td></tr><tr><td>L<sub>ext</sub> > 6, L<sub>det</sub> > 80, cosmology</td><td>1193</td></tr><tr><td>L<sub>ext</sub> > 6, L<sub>det</sub> > 320, cosmology</td><td>360</td></tr><tr><td>eRASS1/Planck comparison</td><td>66</td></tr><tr><td>eRASS1/SPT comparison</td><td>66</td></tr><tr><td>eRASS4/SPT comparison</td><td>83</td></tr><tr><td>eRASS1/eFEDS comparison</td><td>69</td></tr></tbody></table>

**Notes.** The subsamples marked cosmology were additionally selected to be part of the cosmology subsample from B24. Count selection is done within 800 kpc in the 0.2–2.3 keV band.

Clusters with disturbed morphologies are known to have lower luminosities. Therefore, for a fixed detection threshold it may be easier to detect cool core clusters and less easy to detect disturbed clusters (Eckert et al. 2011), although there are methods which can reduce this bias (e.g. Käfer et al. 2020). However, as a result of the PSF fitting maximum likelihood detection approach used to make the eROSITA catalogues, there may be objects in which the cool core makes the cluster more likely to be detected as a point source and missed in the cluster catalogue (e.g. Bulbul et al. 2022), particularly towards higher redshift where the PSF is more important. At low redshifts the PSF fitting detection method may miss objects which have flat cores in surface brightness, particularly if the core size becomes larger than the fitting radius (Xu et al. 2022). Large groups or clusters with flat cores may be confused with background variations. This bias will not change the parameters for an individual object, but as cool core clusters are more regular, the parameters obtained for the population near the flux detection limit or towards higher redshift may be biased by this effect.

The easiest to measure indicator of a cool core is the surface brightness concentration of a cluster. Cool cores also have higher central densities than non cool core clusters and steeper inner surface brightness profiles. In the eROSITA detection pipeline (Merloni et al. 2024), the significance of sources is measured by two parameters, firstly the detection likelihood ($L_{det}$; DET_LIKE_0 in the catalogue) and secondly the likelihood for the source to be extended ($L_{ext}$; EXT_LIKE in the catalogue). When computing the extension likelihood, the source is parametrised as a $\beta$-model with $\beta = 2/3$ and a variable extent.

Figure 10 shows these parameters for the whole cluster sample, as a function of the cluster detection likelihood or extension likelihood, and the total number of counts in the cluster in an 800 kpc aperture. It can be seen that for a fixed number of counts in a cluster, those with a higher detection likelihood have a higher concentration, a more dense core and a steeper density slope. For a fixed detection likelihood, those with the fewest counts are more concentrated, more dense and have steeper profiles. The pattern of extension likelihood is less clear. A selection in count space is less affected by the cluster morphology, particular above the threshold of ~40 counts.

The effect of concentration on selection is shown in further detail in Fig. 11, where the cumulative distribution of concentration

is shown for subsamples with different cuts on exposure, counts, detection likelihood and extension likelihood. The effect of increased exposure time is to reduce the peak of the distribution to lower values of $c_{500}$, but the distribution of $c_{80-800}$ is relatively unchanged. Lower numbers of counts is associated with larger values of both concentration values, although above 80 counts, the distributions do not change further. Increasing the detection likelihood threshold leads to distributions of $c_{500}$ with a peak at higher values, but a narrower width. The effect of $L_{det}$ on $c_{80-800}$ is mostly to narrow the distribution peak. The effect of extension likelihood cuts on $c_{500}$ is less clear, with low and high extension likelihoods showing a peak at higher values. For $c_{80-800}$, higher $L_{ext}$ thresholds lead typically to lower concentration values, although this effect seems to saturate near values of around 12. We also note that for the low count regime, the distributions will be broadened out due to statistical noise.

We measured concentration with apertures with angular sizes that vary with redshift, and for $c_{500}$ also vary with mass. Choosing different cluster subsamples, for example according to exposure time, will select clusters with different redshift and mass distributions. Therefore, the interactions between how likely a cluster is detected, concentration, exposure time, number of counts, and detection threshold cuts are rather complex. However, it might be noted that the maximum shift for the median concentration is 0.2 dex for $c_{80-800}$ between the high and low count cluster subsamples.

We examined the effect of concentration on selection in more detail with the aid of simulations. Figure 12 shows the fraction of clusters detected as a function of concentration at different redshifts and luminosities. However, we note that the concentration in this plot is that computed from the input and not from the reconstructed profiles. It can be seen at the lowest redshifts bins we would not detect groups or clusters if they have a flat surface brightness profiles ($c_{80-800} \le -1$). The effect is stronger for the less luminous (fainter) objects. This is likely due to the fixed size of the detection aperture and the background map generation process.

At higher redshifts we would not detect clusters if they have very concentrated surface brightnesses, for example due to a cool core. At a redshift of 0.4, clusters with $c_{80-800} \ge -0.2$ (or around 60% of their flux within 80 kpc) are lost, even if they have luminosities as high as $10^{45}$ erg s$^{-1}$ (roughly a $10^{15}$ M$_{\odot}$ cluster). This threshold reduces to $-0.4$ at $z = 0.6$, $-0.5$ at $z = 0.8$ and $-0.6$ at $z = 1.2$ (with 25% of their flux within 80 kpc). For the SPT-selected clusters observed by Chandra (Sect. 3.4), we found 4% of objects have $c_{80-800} \ge -0.4$, 9% are at value greater than $-0.5$ and 13% greater than $-0.6$. This selection effect is due to these objects becoming more point-like and the eSASS detection pipeline not detecting the extension of the object, because of a low extension likelihood. Therefore, extreme cool core objects at high redshifts may be missing from the cluster sample. One example is the extreme cool core Phoenix cluster (McDonald et al. 2012), which is not detected as an extended source in the eRASS1 catalogue (also see Bulbul et al. 2022).

The selection effect seen above can also be seen in observed clusters by plotting the average cluster luminosity as a function of redshift in bins of concentration (Fig. 13). At lower redshift the least luminous clusters are the most concentrated. Clusters with similar luminosities but flatter cores would not be detected by the source detection. The opposite effect is present at high redshifts for $c_{500}$, where the least luminous clusters have the flat-test cores, as peaked clusters that look similar to point sources are not detected as extended clusters.
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Parameters against number of counts in an 800 kpc aperture. These parameters are plotted for the bright cluster subset (≥300 counts) with larger markers. The small markers show clusters with at least 25 counts, without plotting uncertainties. The colour scale shows the cluster redshift.</figcaption></figure>

## 6.3. Cool core mass and radius bias

Parameters measured at or within radii scaled with $R_{500}$ are sensitive to the accuracy of that measurement. Cool core clusters have high central gas densities and therefore X-ray bright cores. As the cluster mass and radius is obtained using the cluster luminosity in the eRASS1 cluster catalogue, the presence of a cool core can bias the cluster radius and parameters which are measured at a scaled radius. A cool core will increase the cluster luminosity and $R_{500}$, leading to a measurement at fixed scale radius being measured at a higher physical radius than expected. However, the cluster radius only scales weakly with the mass with a one-third power. In cool core clusters, $n_{s,0}$, will be reduced by this effect. Similarly, non-cool core clusters may have boosted central densities, although the effect will be smaller, as they have a flat core in their density profile.

Many of the parameters are affected by this bias in radius determination including $n_{s,0}$, $\alpha$, $c_{500}$, power ratios, $w$, and $A_{phot}$. Parameters unaffected include $n_{50}$, $c_{80-800}$, $\epsilon$, $H$, and multipole magnitudes. Those parameters which are affected by this issue are indicated in Table 1. For example, if we artificially increase $R_{500}$ by 10%, the median $c_{500}$ increases by 0.035, $c_{500}^*$ by 0.026, $n_{s,0}$ by -0.022, $n_{s,0}^*$ by -0.036, $\alpha$ by 0.038, and $\alpha^*$ by 0.033. Similar changes in the opposite direction are seen if $R_{500}$ is decreased by 10%.

## 6.4. Position bias

The central positions of our clusters are determined by the fitting of a model for the surface brightness to an image. For those parameters obtained using MBProj2D, the central cluster position was allowed to vary during the analysis, while other
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Parameters against X-ray luminosity. These parameters are plotted for the bright cluster subset (≥300 counts) with larger markers. The small markers show clusters with at least 25 counts, without plotting uncertainties. The X-ray luminosity is inside an 800 kpc radius aperture in the 0.2–2.3 keV band. The colour scale shows the cluster redshift.</figcaption></figure>

clusters and modelled point sources in the field were frozen at their best fit location. For those parameters not measured using MBProj2D, we used the median chain value from the MBProj2D symmetric modelling as the cluster centre. Some of these positions can differ from the standard cluster catalogue location, due to the slightly different modelling procedure. The cluster positions obtained from the MBProj2D analyses are the locations of the centre of the model which fit the cluster best overall, in a statistical manner.

As stated previously, for some parameters where the choice of centre makes a difference, we provide fitted centre and peak centre values. Figure 14 shows average quantities in bins of off-

set between the peak and best fitting positions (F). The central density parameters, concentration, central slope, photon asymmetry, and power ratio $P_{10}$ are all significantly different if there is an offset between the peak and best fitting positions. We also see lower dispersions for some quantities when using the peak position.

For the MBProj2D-derived shape parameters, the cluster centre was also allowed to be free in the analysis. For parameters $\epsilon$, $H$, and $M_1$ the size of the offset between the standard fit position and the one with varying 2D shape is clearly correlated with the shape parameter. At more extreme parameter values, this shift can be 0.1 to $0.2R_{500}$.
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Corner plot of the median MBProj2D shape parameters for the bright cluster (≥300 count) subset. The quantities are plotted against each other, while the rightmost panels show the probability density distribution of each value.</figcaption></figure>

## 6.5. Point spread function bias

When looking at objects towards higher redshifts (or more precisely, angular diameter distance) or lower masses, the finite size of the PSF of eROSITA becomes increasingly important and makes it harder to measure morphological parameters. Those quantities measured using the forward-modelling procedure of MBProj2D already take account of the PSF. For more distant objects, the uncertainties on the measured parameters grow due to the increased importance of the PSF size, but there should be no bias on the measured parameters if the functional form of the fitted profile is realistic (however, see Sect. 6.6).

There are several parameters, however, that do not consider the PSF in their determination. These include power ratios, $w$, $A_{phot}$, and $G$. Clusters at increasing distances with the same degree of disturbance will look less disturbed with these parameters, as the structure is smoothed out. In Sect. 6.8 we present the intrinsic variations of these parameters due to both the effects of noise and the PSF, as a function of redshift and luminosity.

## 6.6. Fixed parameter range bias

Some parameters, including $\epsilon$, $H$, the multipole magnitudes, and $G$, are only defined for a range of zero to one. The definition of concentration restricts values to the range 0 to -2. In addition, for the very faintest clusters or most distant clusters, model

priors may implicitly restrict the range on some of the forward-modelling obtained parameters.

The finite range means that as the uncertainty on a value increases, the obtained median values from an MCMC chain tend towards the centre of the allowed parameter range. If there is no constraint on a value, its $1\sigma$ error ranges, obtained from the $1\sigma$ percentiles of the chain will also tend towards encompassing 68% of the parameter range. For those parameters obtained using MCMC, the full posterior probability distribution can still be used as this effect increases, although model priors could become increasingly important.

To reduce this effect for the shape parameters, $\epsilon$, $H$, and $M_1$ to $M_4$, rather than quoting the median chain value, we instead give the maximum likelihood value. The best fitting values are more evenly distributed over the parameter range when the number of counts is low, rather than being clustered around 0.5. This also makes comparisons with simulations easier, as we cannot run a full MCMC analysis on each simulation. Figure 5 shows that for objects with fewer than 200-300 counts, it is difficult to trust these individual shape parameters for single objects.

Using our simulations we also checked how well the fixed-range shape parameters can be recovered from the maximum likelihood fit. Figure 15 shows the range of parameter values obtained for the input cluster shapes given. At high luminosities and low redshifts, the input parameter can be recovered. Surprisingly, we found that the average best fit parameter does not tend towards the input parameter in the low count regime. The
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Correlation matrix between different parameters for the bright cluster subset. Values were measured for clusters with more than 300 counts to reduce the effect of the statistical errors. Log values were taken for the photon asymmetry, power ratios and centroid shift. We also show the correlation with log cluster luminosity ($L_{500}$), log number of counts in $R_{500}$ (cts$_{500}$), and log redshift.</figcaption></figure>

mean values tend towards an extreme (e.g. zero for $\epsilon$ and one for $M_1$ to $M_4$). The $H$ parameter does not seem so badly affected by this bias. In these extreme cases seen in low photon count clusters, we found often the maximum likelihood model is a very elongated ellipse which passes through the brightest pixels. Therefore, one should take care if using the maximum likelihood values for these parameters from our catalogue of parameters in the very low count regime. The maximum likelihood parameters chosen for objects with low counts are obviously unphysical. In this case, an informative prior on these parameters could prevent these poor results, although this could make interpretation more difficult. More detail about the count range where this issue occurs is discussed in Sect. 6.12.

## 6.7. Cluster shape affecting selection

In addition to how concentrated a cluster is, its 2D shape may affect whether it is selected as an extended source or not. The emldet task assumes that a cluster has a symmetric shape with a $\beta$ model profile.

In our simulations we tried to assess how much the cluster shape affects the likelihood that a cluster is selected as a function of redshift and luminosity. We computed this for a limited number of cluster shapes, including spherical, elliptical ($\epsilon = 0.7$), sloshed ($H = 0.3$), first multipole ($M_1 = 0.3$), and second multipole ($M_2 = 0.3$).

Figure 16 shows the fraction detected for these different cluster shapes for different luminosities as a function of redshift. The results show that for these degrees of non-spherical morphology there are no significant changes in the detection efficiency of clusters. The largest effect seems to be for low luminosity objects at lower redshifts, where a slosh reduces the detection efficiency by ~5%.

## 6.8. Noise and distance

Several parameters do not take account of the PSF of the telescope. In addition, they may be affected by noise in the low count regime. Figure 17 shows measured non-forward-modelled morphological parameters for simulated clusters. Here we simulated spheri-
<table><caption>**Table 4.** Potential biases.</caption><thead><tr><th>Name</th><th>Section</th><th>Description</th><th>Effect</th></tr></thead><tbody><tr><td>Morphological detection bias</td><td>6.2</td><td>Morphology of cluster, in particular the peakiness of its profile, affects how easy an object is to be detected.</td><td>Produces a biased sample, giving biased distributions of parameters, in particular those associated with detection (e.g. concentration).</td></tr><tr><td>Cool core mass and radius bias</td><td>6.3</td><td>Presence of a cool core gives higher luminosity and therefore biased mass or radius, affecting parameter measurements.</td><td>Affects parameters measured at scaled radii, see Table 1, e.g. c<sub>500</sub>, n<sub>s,0</sub>, α, P<sub>10</sub>, A<sub>phot</sub>, G, and w.</td></tr><tr><td>Position bias</td><td>6.4</td><td>Choice of cluster centre affects parameter</td><td>Parameters sensitive to core, e.g. n<sub>s,0</sub>, n<sub>50</sub>, α, c<sub>500</sub>, c<sub>80–800</sub>, P<sub>10</sub>, and A<sub>phot</sub>.</td></tr><tr><td>PSF bias</td><td>6.5</td><td>PSF blurs out features affecting parameters which are not model based and increasingly strongly for higher redshifts.</td><td>Affects image measured parameters, including power ratios, G, A<sub>phot</sub>, and w.</td></tr><tr><td>Fixed parameter range bias</td><td>6.6</td><td>Parameter range is small compared to uncertainties, leading to biases in median values from MCMC.</td><td>Affects ε, H, M<sub>1</sub> to M<sub>4</sub>, and to a lesser extent, concentrations.</td></tr><tr><td>Cluster shape selection</td><td>6.7</td><td>Related to the Morphological detection bias, a non-circular cluster shape can affect how easy a cluster is to be detected.</td><td>A sample biased towards more relaxed clusters, affecting the distribution of all parameters</td></tr><tr><td>Noise and distance bias</td><td>6.8</td><td>More distant clusters produce fewer counts and more blurred images, giving rise to biased parameters.</td><td>Affects many image-derived parameters, including power ratios, G, w, and A<sub>phot</sub>.</td></tr><tr><td>Resolution bias</td><td>6.9</td><td>Ability to resolve inner parts of cluster depends on its profile, distance and brightness.</td><td>Not important, if parametric profile is realistic.</td></tr><tr><td>Temperature and metallicity bias</td><td>6.10</td><td>Non-isothermal temperature profile or varying metallicity could affect density estimates.</td><td>For groups could affect n<sub>50</sub>, n<sub>s,0</sub>, and α.</td></tr><tr><td>Unresolved point sources</td><td>6.11</td><td>Point sources may be unresolved within the cluster.</td><td>Could affect all parameters and also lead to biases in the sample selection.</td></tr></tbody></table>

dominant over bremsstrahlung emission for X-ray temperatures below around 1 keV. A substantial fraction of eROSITA clusters are lower mass objects with lower temperatures. Of those with temperature measurements, there are 550 objects with temperatures less than 1 keV and 2900 with temperatures less than 3 keV.

## 6.11. Unresolved point sources

Detected point sources are masked out in our analysis or simultaneously fitted. Due to the ~30 arcsec survey PSF, its limited sensitivity and the source detection algorithm, we may miss point sources within clusters, or resolve parts of clusters as point sources. In addition, some fraction of clusters in the sample are contaminants, where single or multiple point sources could be confused to be extended sources. The purity of the main cluster catalogue is 86% (B24).

Unresolved point sources could give rise to artificial substructure within an object and so affect the morphological parameters. In addition, very peaked substructure could be detected as a point source and therefore removed. The strength of this effect could be redshift and luminosity-dependent. To better assess this in a future work would likely require the analysis of more realistic simulations incorporating a full range of cluster morphologies and source populations. Our existing simulations do not include point sources. If our simulations were to include a point source population, it would likely give rise to additional bias in the parameters, particularly in low luminosity and higher redshift objects.

## 6.12. Recovered parameter biases and uncertainties

Some of the listed biases affect whether a cluster is detected and others affect the measured parameters. Here we examine the

combined effect of biases on the measured parameters using our simulations, ignoring selection effects. These simulations took account of the PSF, the data quality, the fixed parameter range and the choice of centre (fitted or peak), although point sources were not included. We also computed the statistical uncertainty of the parameters. These biases and uncertainties are studied as a function of redshift and number of cluster counts. Counts are more reliable than luminosity in predicting the data quality because it takes account of survey depth and absorption.

Figure 18 shows the difference between the recovered parameter value and its input. We emphasise that we are just using the median obtained parameter (or maximum likelihood in the case of the shape parameters) and do not include the probability density function for the parameter. For the density parameters, the input value was obtained by deprojecting the emissivity surface brightness profile (noting that this is not valid in all cases). For the shape parameters (ε, H and M<sub>1</sub> to M<sub>4</sub>), we simulated clusters with known fixed parameter values and looked at the difference between the input and output. For the image based parameters (w, A<sub>phot</sub>, and P<sub>10</sub> to P<sub>40</sub>), we simulated spherical clusters and just plot the average value, noting that given perfect signal to noise the value should be -∞ in the log space used.

The largest biases are seen for fainter objects in the very lowest redshift bin, where the extent of the source means that the background level varies significantly over the region of the object. At these redshifts, the background level means that for the same number of counts the signal to noise ratio is lower. For higher redshifts, we see fairly moderate evolution in bias with redshift. The concentration measurements show some positive bias for low count clusters, although it is rather flat and small when there are more than 50–100 counts, except at very
<figure><img src="image_10.png" /><figcaption>**Fig. 10.** Relationship between detection likelihood ($\mathcal{L}_{\text{det}}$), extension likelihood ($\mathcal{L}_{\text{ext}}$), counts, and morphological parameters. (Top panels) The median value of the concentration ($c_{80-800}$; left), central scaled density ($n_{s,0}$; centre), and central slope ($\alpha$; right), in grid points of $\mathcal{L}_{\text{ext}}$ and $\mathcal{L}_{\text{det}}$. (Bottom panels) The median morphological parameter values for grid points of counts in an 800 kpc aperture and $\mathcal{L}_{\text{det}}$ or $\mathcal{L}_{\text{ext}}$.</figcaption></figure>

low redshifts. The peak centred concentrations show larger bias, although these simulated clusters are unrealistic by being perfectly spherical. The central density shows some bias of the order of 0.05–0.15 dex, although the sign and magnitude depends on whether the density at fixed radius is measured. Some positive bias in the central slope is seen, of between 0 and 0.5. For the shape parameters, there is some bias below 100 counts or so, due to the finite range of the parameter (Sect. 6.6). For the image based parameters, we see the same biases as seen in Fig. 17 and described in Sect. 6.8.

The uncertainties on the density profile parameters (Fig. 19) generally decrease with the number of counts and increase with redshift. There is not such a strong decrease in uncertainty on the central density with increasing counts if the peak position is used, which is likely because the central density is very sensitive to the central position and we are only using a subset of the data to determine this position. The statistical uncertainty on the cuspiness is generally large and redshift dependent. The forward-modelled shape parameters show declining uncertainties with increasing number of counts, and relatively little redshift evolution.
<figure><img src="image_11.png" /><figcaption>**Fig. 11.** Cumulative distribution of concentration for different subsamples of the eRASS1 catalogue. The cumulative distributions are shown as a function of exposure time ($t_{exp}$), number of counts in a 800 kpc aperture, detection likelihood ($\mathcal{L}_{det}$), and extension likelihood ($\mathcal{L}_{ext}$). The left panels show the distribution of $c_{500}$, while the right panels show the distribution of $c_{80-800}$.</figcaption></figure>

# 7. Comparisons with previous results

## 7.1. Matched subsamples

To check for consistency, we compare the $c_{500}$ concentrations and $n_{8,0}$ central densities for the same clusters with different analyses in Fig. 20. This comparison will be unaffected by selection effects. For three of these comparisons we computed using MCMC a median linear relation between eROSITA and the other values, assuming that the distribution is normal about the relation. We note that B24 report 15% lower fluxes in eROSITA first data release (DR1) processing compared to *Chandra*, which likely will give rise to a difference in central density of roughly half this value.

The first comparison is to an *XMM-Newton* analysis of the Planck-detected early SZ (ESZ) sample (Lovisari et al. 2017). These clusters were matched to counterparts in the eRASS1 catalogue within a radius of 4 arcmin to make a jointly detected sample of 66 clusters, with a median eROSITA mass of $M_{500} = 10^{14.92}$ M$_{\odot}$. We note that the values of $R_{500}$ used in the two analyses are different, which could lead to systematic differences, although $R_{500}$ scales fairly weakly with mass. The cluster centres could also differ, although both used X-ray peak centres. Despite these possible differences, there is a reasonable agreement between the sets of concentrations, although there appears to be an offset increasing with lower concentration. The median cluster has a 0.1 dex higher concentration in eRASS1. If we examine the central density, there is scatter above the relation, which may be due to the assumed density profile form being too simple for these very bright clusters. There is evidence for a non-unity slope or offset between the relations, with low central densities for eROSITA found higher than *XMM-Newton*, and high central densities lower, although the small uncertainties on the bright objects are likely driving this fit. The median value, how-

ever, is exactly the same between the two sets of measurements for the same clusters. The differences between the two analyses could be the result of the $R_{500}$ differences, background modelling, particularly due to the large field of view of eROSITA, point sources, and the differences in the assumed functional form of the density or surface brightness.

We also examined a sample of clusters detected by SPT (Bleem et al. 2015), in particular those previously studied in X-rays using *Chandra* by Sanders et al. (2018). We matched an eROSITA to SPT cluster if its distance is less than 4 arcmin, to make a sample of 66 objects, with a median eROSITA mass of $M_{500} = 10^{14.14}$ M$_{\odot}$. Section 3.4 describes the *Chandra* data analysis, which uses the same parametrisation for the density profile as for the eROSITA data. For both these analyses, we used the concentration as defined by the peak of the cluster. However, $R_{500}$ for the *Chandra* analysis was from the SPT catalogue, while the eROSITA $R_{500}$ was the value in the eRASS1 catalogue. The best fitting relation between eROSITA and *Chandra* is similar to the *Planck* clusters, with an increasing offset towards lower concentrations. The median cluster again shows a 0.1 dex higher concentration for eRASS1, although the median concentration is lower than for the *Planck* sample. For the central densities, again the median values are exactly the same between the two different sets of analyses. However, there is a similar non one-to-one relation found as with the *Planck* data, if fitting the datapoints.

In addition to matching the eRASS1 and *Chandra* SPT objects, we did a detailed analysis of eRASS:4 data for the whole *Chandra* SPT sample, using as similar a methodology as possible (see Sect. 3.4). In this case, we used the SPT $R_{500}$ for both analyses and also studied objects which were not detected as clusters in eRASS1, producing a sample of 83 objects. The best fitting relation between the two sets of values is consistent with 1:1. The increased agreement compared to the eRASS1 sample
<figure><img src="image_12.png" /></figure>

**Fig. 12.** Selection function of clusters with different redshifts and luminosities as a function of concentration that were calculated from simulated observations of spherical clusters. The panels show the detected fraction of clusters at different redshift. For each redshift, we show the detection fraction as a function of concentration for different luminosities (erg s⁻¹). The log concentration (c₈₀-800) is computed from the input model profile and is binned into bins of 0.1 width.

is likely a combination of using exactly the same extraction regions and cluster centre and masking or modelling the same point source regions. However, the median cluster again shows a ~0.1 dex higher concentration for eROSITA. The central densities show a good agreement between the two analyses, with a relation which is consistent with 1:1. The median density values are also very close to each other.

Finally, we compared two different eROSITA surveys, eRASS1 and the deeper eROSITA Final Equatorial-Depth Survey (eFEDS; Ghirardini et al. 2022), for clusters which match with a counterpart within 2 arcmin of the other survey, finding a sample of 69 objects. The median eROSITA DR1 mass of this sample is M₅₀₀ = 10¹⁴.⁴¹ M☉. eFEDS shows a rather narrower concentration distribution for the same clusters, with a median around 0.1 dex lower than eRASS1. The width of the concentration distribution is also narrower than the XMM and Chandra results, which may be due to the prior on the inner slope of the density parametrisation, α > 0, used by Ghirardini et al. (2022). The median central densities are similar for the two different analyses and datasets.

<figure><img src="image_13.png" /></figure>

**Fig. 13.** Median X-ray luminosity of clusters in bins of concentration. The left panel uses bins of c₈₀-800, while the right panel shows c₅₀₀. The uncertainties shown are calculated with bootstrap resampling. All eRASS1 clusters are included in this analysis.

<figure><img src="image_14.png" /></figure>

**Fig. 14.** Fit-centred and peak-centred (*) quantities as a function of the distance between the cluster peak and the best fit position for a cluster subsample with $L_{\text{ext}} > 6$ and $L_{\text{det}} > 20$. The median quantities in bins of F are shown with bootstrap resampling uncertainties. The shaded regions show the 1σ percentile width of the data points within each bin.

Overall, there is a reasonable agreement between the different analyses for the same cluster samples, considering that
<figure><img src="image_15.png" /></figure>

**Fig. 15.** Recovery of forward modelled parameters as a function of redshift and luminosity using maximum likelihood. The plots show the recovered parameter values as a function of redshift, for clusters with an input shape compared to an input spherical cluster. The error bar shows the average best fitting parameter and the uncertainty on the mean. The shaded region shows the 1σ range of the recovered best fitting parameters. The panels show shaped models with $\epsilon = 0.7$, $H = 0.3$, $M_1 = 0.3$, $M_2 = 0.3$, $M_3 = 0.3$, and $M_4 = 0.3$, respectively, compared to the spherical models.

the datasets are different and some of the analyses are independent. However, there are suggestions for the analyses using the eRASS1 clusters combined with Planck ESZ or SPT, of an offset for the lowest concentration or lowest density objects. There is good agreement between the matched analysis procedures for the eRASS:4 and Chandra data. The scatter seen in the eRASS data is affected by the number of counts. Planck clusters are X-ray bright (mean in log space of 740 counts in eRASS1) compared to the SPT sample (log-space mean of 100 counts). The eFEDS sample is fainter than the SPT objects, with a log-space mean of 35 eRASS1 counts. The differences in the other comparisons could be due to different analysis procedures, the locations of the cluster centres, unresolved point sources in eRASS1, the spatial masks for the datasets, background modelling and the 15% offset in flux found in the DR1 processing (B24).

## 7.2. Whole samples

We also compared whole samples, although it must be noted that selection effects may dominate any differences between them. In Paper II we investigate the underlying distributions, taking account of selection effects. Figure 21 shows cumulative distributions of parameters for the Chandra observed SPT sample, the XMM-Newton observed Planck ESZ sample (Lovisari et al. 2017), and the eFEDS sample (Ghirardini et al. 2022). For

<figure><img src="image_16.png" /></figure>

**Fig. 16.** Detection fraction as a function of redshift for different luminosities and shapes.

eFEDS1, we show distributions of parameters for subsamples with increasing cuts on the detection likelihood. High detection likelihood objects are the most significant in the survey and have the smallest statistical uncertainties on the parameter values. For comparison with eFEDS, we also show an eRASS1 subsample with the same cuts in detection and extension likelihood and with a similar observation depth, although we do not match the background levels. Where appropriate, we also show the distributions for both the parameters measured with a free cluster centre and those measured where the peak position is used.

We see that the SPT and Planck samples show similar distributions of concentration. eFEDS shows a median concentration close to the SPT value, although the wings are broader. The eRASS1 clusters have systematically higher concentrations than the other samples. The $c_{500}^*$ values are larger than $c_{500}$, as expected. The eRASS1 median concentration is around 0.3 dex larger than SPT or eFEDS medians. If we examine subsamples with larger $L_{\text{det}}$ thresholds, the median shifts to higher concentration values, as previously seen in Fig. 11. However, as we discussed for Figs. 10 and 11, cuts in count space above ~40 do not substantially change the concentration distribution.

The reason for the different concentration distribution for eRASS1 compared to the other surveys could be due to its X-ray selection and relatively shallow depth, although we note that there are differences in the measured parameters for the same clusters which could contribute (Sect. 7.1). X-ray selection is more sensitive to concentrated clusters (Sect. 6.2). Figure 11 also shows that increasing $L_{\text{ext}}$ would not reduce the median concentration to bring it in agreement with the SZ-selected samples. Increasing the depth of the survey decreases the median concentration, although the effect is relatively modest up to
<figure><img src="image_17.png" /><figcaption>**Fig. 17.** Measured non-forward-modelled morphological parameters for simulated model clusters. Clusters are simulated with different luminosities and redshifts, after sampling from the mass function. The shaded region shows the $1\sigma$ range of recovered parameters of detected clusters simulated from spherical cluster models. The dotted and dashed lines show the median obtained parameters for non-spherical detected clusters with the parameters shown.</figcaption></figure>

800s. Lovisari et al. (2017) give a median concentration for the REXCESS sample of representative X-ray selected clusters (Böhringer et al. 2007) as $c_{500} \sim -0.6$, which is consistent with what we found for eRASS1 (Fig. 21).

eFEDS contains less concentrated clusters, despite its X-ray selection, which may be due to the survey depth. We check this using the ‘eFEDS-like’ eRASS1 subsample, noting that the eFEDS cluster positions are fitted for. Comparing the eFEDS-like concentrations when the central position is fitted for, we see a similar distribution to eFEDS. The cumulative fraction at the most and least concentrated peaks are very similar to eFEDS. However, eFEDS-like is more concentrated by around 0.1 dex at the median. This may be due to the difference in the fitting method, including the functional form used, or the average background level.

The median $\alpha^*$ is similar to eFEDS, although eFEDS does not use peak cluster positions. The difference in slope between eFEDS and eFEDS-like is likely due to the positive central density slope prior used by Ghirardini et al. (2022) compared to the wide prior used here. The SPT and Planck clusters do

not show such steep inner profiles, with median values around 0.25 smaller than eRASS1. Larger $\mathcal{L}_{\text{det}}$ thresholds increase $\alpha^*$, although the effect is small for modest increases.

The median central scaled densities in eRASS1 clusters with a lower detection threshold are similar to those found in the SPT and Planck clusters, although the central density for eRASS1 strongly increases with $\mathcal{L}_{\text{det}}$. eFEDS clusters show a density distribution with a median reduced by around 0.3 dex from eRASS1 with $\mathcal{L}_{\text{det}} > 20$. The differences between eFEDS and eFEDS-like is again likely affected by the different priors on the central density slope assumed in the analyses.

The raw ellipticity distribution for eRASS1 is dominated by measurement errors for lower detection thresholds. As the values can only lie between zero and one, and clusters are more circular than elliptical, values are biased low for low numbers of counts. As the detection threshold increases, the distribution approaches the SPT distribution, with a median of $\epsilon = 0.77$. eFEDS values are lower, but they also have significant measurement errors. It is unclear why the eFEDS and eFEDS-like samples show different ellipticity distributions. Planck clusters, show
<figure><img src="image_18.png" /><figcaption>**Fig. 18.** Recovered parameter bias from simulated clusters as a function of the redshift and number of counts inside an 800 kpc radius circle. For the concentration, central density and cuspiness, shown is the difference between the recovered values and the input profile, after deprojecting the emissivity profile. For the shape parameters, $\epsilon$, $H$ and $M_1$ to $M_4$, we show the difference between the recovered and input values (0.7 for $\epsilon$ and 0.3 for the others), for simulations with non-spherical clusters. For the parameters $w$, $A_{\text{phot}}$ and $P_{10}$ to $P_{40}$, we show the value for a spherical cluster.</figcaption></figure>

significantly larger ellipticity values, indicating they are more circular. However, we can compare directly the eRASS1 and XMM-Newton measured Planck values for the matched sample. The XMM-Newton values are significantly larger, despite the low bias in the values expected in the eROSITA high count regime (Fig. 18). Fitting a Gaussian linear model, we found a relation between the two of $\epsilon_{eRASS1} = 0.51\epsilon_{XMM} + 0.32$. It is unclear what the origin of these differences is, as both should be the ratio of the minor to major axis, although it may be because the two measurements might be more sensitive to different parts of the cluster due to the field of view and PSF. In

addition, there may be statistical differences due to the parameter range and the larger number of counts in the XMM-Newton data.

## 8. Combined disturbance indicators

Different morphological parameters can be combined together to produce a common disturbance parameter, as proposed in Rasia et al. (2013) and also demonstrated in Ghirardini et al. (2022) and Campitiello et al. (2022), and Yuan et al. (2022). We used a different method to those papers to characterise the
<figure><img src="image_19.png" /><figcaption>**Fig. 19.** Recovered statistical uncertainty in measured parameters from simulated clusters. This is the standard deviation on the difference between the input and recovered values, as in Fig. 18.</figcaption></figure>

disturbance of a cluster. Many of the parameters measure similar aspects of the morphology of a cluster and they can have strong biases. Therefore, it is probably not a good idea to combine all possible parameters. Our method is to fit a Gaussian mixture model (GMM) to the parameters of brightest galaxy clusters. In such a model the data values are fitted by a mixture of multi-dimensional Gaussian distributions. In our analysis we decide to use two such components, to separate the clusters into regular and disturbed populations. The underlying assumption is that two components are sufficient for distinguishing these populations. The model can then be applied to the whole cluster sample to say whether a set of parameters for a cluster is better repre-

sented by the distribution given by the regular component or the disturbed one.

The choice of fitted parameters is important for how well this technique works. To avoid the issues of strong bias with red-shift or luminosity, we restrict ourselves to the forward-modelled parameters. In addition, the shape of the cluster may vary independently from the presence of a bright core. We therefore created two disturbance indicators, one of which is only dependent on shape and the other also includes the concentration. The term $D_{\text{shape}}$ is the disturbance score for shape-based measurements, fitted to $\epsilon, H, M_1, M_2, M_3$, and $M_4$, and the peak-fit offset $F$, whereas $D_{\text{comb}}$ is a combined disturbance indicator, fitted to
<figure><img src="image_20.png" /><figcaption>**Fig. 20.** Comparison of concentration and scaled central density measurements for the same cluster samples between eRASS and other measurements. The two left columns compare $c_{500}$ and its distributions, while the two right columns show $n_{s,0}$. Median values are plotted as vertical lines. The number of counts for the cluster in the eRASS1 or eRASS:4 data is indicated by the colour. Top row: 66 Planck ESZ and eRASS1-detected clusters, with XMM-Newton results from Lovisari et al. (2017) and using the eRASS1 peak values. Second row: 66 SPT and eRASS1-detected clusters, analysed using Chandra data and taking the peak values from eRASS1. Third row: 83 SPT-detected clusters, extracting the values from Chandra eRASS:4 data using similar analysis procedures. Bottom row: 69 clusters detected in both eFEDS and eRASS1. The eFEDS survey results are from Ghirardini et al. (2022), while the eRASS1 results are the best fit position values. For the top three rows, we compute the best fitting linear relation between the results, assuming a normal distribution around the relation.</figcaption></figure>

the same parameters as $D_{\text{shape}}$ but adding $c_{80-800}$ to include the effects of bright cores.

These parameters were fitted using the `pyGMMis` package (Melchior & Goulding 2018), which has the advantage of being able to account for measurement errors by allowing a covariance

matrix for each input data point. As we measured our parameters independently, we do not know the covariance between the parameters for each cluster; we only know their uncertainties. We therefore used a diagonal covariance matrix for each cluster, which does not take account of the likely covariance between
<figure><img src="image_21.png" /><figcaption>**Fig. 21.** Cumulative distribution of parameters for SPT, *Planck* ESZ, and eFEDS clusters compared to eRASS1 for different detection thresholds. For eROSITA the solid lines show the values fixing the cluster position at the peak, while the dotted lines show the distribution for when the cluster position varies during the fit. Shown are the concentration (top left), cuspiness (top right), central scaled density (bottom left), and ellipticity (bottom right). *Planck* values are taken from Lovisari et al. (2017) and eFEDS values are obtained by Ghirardini et al. (2022). The eFEDS-like curves are for eRASS1 clusters with $\mathcal{L}_{\text{det}} > 5$, $\mathcal{L}_{\text{ext}} > 6$ and an exposure time between 1000 and 1400s, to match eFEDS. The median published log masses of the SPT, *Planck* and eFEDS samples are 14.7, 14.8, and 14.0, respectively. The eROSITA sample median log $M_{\odot}$ masses are 14.4, 14.5, 14.6, and 14.8, for detection likelihood thresholds of 20, 40, 80, and 320, respectively.</figcaption></figure>

parameters (for example $\epsilon$ and $M_2$). This may have the effect of double counting some aspects of the morphology in the model. The parameter error bars were symmetrised by computing the root mean squares to make the matrix (noting that is not strictly valid as fixed range parameters may be close to a limit). The two-component GMM was fitted to the data and covariances from clusters, only using those 175 with more than 800 counts to not be dominated by measurement errors. We assume that one of these Gaussians is closer to disturbed clusters and one is closer to relaxed clusters. Table 5 shows the parameters of the two GMMs, one for $D_{\text{comb}}$ and one for $D_{\text{shape}}$, fitted to the bright clusters.

To classify the clusters, we took this model and computed the likelihood of each of the Gaussians given the cluster parameters and their uncertainties. To better take account of the model uncertainties, we repeated this for a random selection of model parameters taken from the MCMC chains (or bootstrap values for $F$). The score for a cluster ($D$) is the fraction of times the Gaussian corresponding to disturbed clusters has a higher likelihood compared to the one for relaxed clusters.

In Fig. 22 we show the parameters plotted as a function of number of counts, as in Fig. 5, but coloured by the disturbance

score. The plot shows that the method is preferentially detecting relaxed clusters as having a high central density, high concentration, steep central density profile, higher $\epsilon$, lower slosh, lower $M_1$, lower $M_2$, low $w$, higher $G$, and low $F$.

The fraction of clusters which have a high disturbance score $D_{\text{comb}}$ is relatively constant with the number of counts in the object. Using thresholds of 0, 25, 100, and 300 counts there are 27, 31, 27, and 25% of objects with scores greater than 0.5, respectively (see Fig. 23). 20% of clusters with more than 300 counts reach the higher disturbance threshold of $D_{\text{comb}} > 0.8$, while 66% have a $D_{\text{comb}} < 0.2$. Looking at clusters with more than 25 counts, there are 14% with $D_{\text{comb}} > 0.8$ and 46% with $D_{\text{comb}} < 0.2$. Faintier clusters have $D_{\text{shape}}$ which lie more towards the centre of the range. This is likely because the shape parameters have large error bars when the number of counts is low, compared to the concentration, and therefore we know less about the cluster shape. In contrast, $D_{\text{comb}}$ is driven largely by concentration in the fainter objects.

As mentioned above, it matters which parameters were used to make the mixture model. In Fig. 23 are plotted the $D_{\text{shape}}$ and $D_{\text{comb}}$ scores against each other for a sample of bright and fainter
<table><caption>**Table 5.** Best fitting parameters of the GMM for the morphological parameters.</caption><thead><tr><th rowspan="2">Model</th><th colspan="4">D<sub>comb</sub></th><th colspan="4">D<sub>shape</sub></th></tr><tr><th>Mean R</th><th>Width R</th><th>Mean D</th><th>Width D</th><th>Mean R</th><th>Width R</th><th>Mean D</th><th>Width D</th></tr></thead><tbody><tr><td>Amplitude</td><td>0.739</td><td></td><td>0.261</td><td></td><td>0.769</td><td></td><td>0.231</td><td></td></tr><tr><td>ε</td><td>0.813</td><td>0.079</td><td>0.746</td><td>0.097</td><td>0.818</td><td>0.075</td><td>0.722</td><td>0.093</td></tr><tr><td>H</td><td>0.113</td><td>0.066</td><td>0.241</td><td>0.140</td><td>0.109</td><td>0.059</td><td>0.259</td><td>0.141</td></tr><tr><td>M<sub>1</sub></td><td>0.238</td><td>0.123</td><td>0.522</td><td>0.238</td><td>0.227</td><td>0.107</td><td>0.588</td><td>0.209</td></tr><tr><td>M<sub>2</sub></td><td>0.204</td><td>0.104</td><td>0.286</td><td>0.129</td><td>0.192</td><td>0.096</td><td>0.327</td><td>0.114</td></tr><tr><td>M<sub>3</sub></td><td>0.076</td><td>0.024</td><td>0.117</td><td>0.062</td><td>0.071</td><td>0.024</td><td>0.136</td><td>0.062</td></tr><tr><td>M<sub>4</sub></td><td>0.074</td><td>0.029</td><td>0.114</td><td>0.063</td><td>0.070</td><td>0.024</td><td>0.130</td><td>0.063</td></tr><tr><td>F</td><td>0.007</td><td>0.004</td><td>0.107</td><td>0.153</td><td>0.004</td><td>0.003</td><td>0.106</td><td>0.159</td></tr><tr><td>c<sub>80–800</sub></td><td>-0.656</td><td>0.244</td><td>-1.175</td><td>0.179</td><td>—</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

**Notes.** Parameters for the two GMM for the disturbance based on shape and concentration ($D_{comb}$) and purely based on the shape ($D_{shape}$). Shown are the amplitudes for the two Gaussian components (relaxed, R, and disturbed, D) in each of the models, and the mean and width of these Gaussians in the dimension of the parameter given. The width of the Gaussian is the square-root of the diagonal components of the covariance matrix. We do not list the non-diagonal entries in the matrix.

clusters. Although the scores are close for a substantial number of clusters, there are cases where they lie at opposite sides of the scale. The statistical uncertainty on concentration is typically much smaller than the other parameters, so it can drive the disturbance score when included. If $D_{shape}$ is large, but $D_{comb}$ is small, this may indicate a cluster which has a disturbed shape but also a bright core. If $D_{comb}$ is large, but $D_{shape}$ is small, this may indicate a symmetric cluster with a very flat surface brightness profile.

There is also some choice of how many clusters are initially fitted by the GMMs. We chose a threshold of 800 counts to ensure that statistical uncertainties on the input parameters were relatively small. We also reran the analysis using a threshold of 400 counts. The results were fairly similar, except the D scores were generally lower, with around 20% of objects classified as disturbed. If we increased the number of GGM components to three and fitted the combined parameters, we found that the third component has a low 4% amplitude compared to the others (64 and 32% for 800 counts) and lay at a more extreme position in the irregular cluster parameter space. Fitting the 400 count clusters yielded a similar result. If we added a fourth component, the modelling preferred this to have a low amplitude, with around 13% in the smallest two components. Therefore, the GGM model does not show strong evidence for more than two components.

Some examples of clusters with extreme low and high disturbance scores are shown in Fig. 24. In this brighter cluster regime, the parameter appears to be able to easily separate relaxed and disturbed clusters. The clusters include well known relaxed and disturbed objects.

We find a large fraction of relaxed objects. X-ray selection is better at finding clusters with dense central cores, likely associated with relaxed cool core clusters, while SZ selection finds a larger number of disturbed systems. As in some previous X-ray surveys, we found a large fraction of relaxed objects.

Rossetti et al. (2017) studied the morphology of clusters selected using the SZ effect by Planck, using Chandra and some XMM-Newton data. Based on the concentration in 40 and 400 kpc apertures, they find 29 ± 4% of their clusters are cool-core objects. They conducted the same measurement on the ROSAT all-sky X-ray selected MACS sample (Mann & Ebeling 2012), finding 59 ± 5% could be classified as cool-core.

Lovisari et al. (2017) studied the morphology of the Planck ESZ clusters using XMM-Newton. By visual inspection, they

classify 32% of objects as relaxed, 39% disturbed and 29% were of an intermediate state. All of their clusters which are visually identified as relaxed have a concentration of more than 0.15 or $c_{500} > -0.82$ in our log units. Ignoring the statistical uncertainties on our values, this threshold would imply 80% of our eRASS1 objects are relaxed, similar to the fraction identified by our GMM.

Campitiello et al. (2022) studied the morphology of 188 clusters observed by XMM-Newton and selected using Planck. Using a combination of visual inspection and combining morphological parameters, they find 13% were robustly classified as relaxed, while 21% are disturbed.

Ghirardini et al. (2022) find that the X-ray selected eROSITA eFEDS cluster sample have a similar fraction of cool cores to SZ selected surveys. They constructed a relaxation score based on a combination of several morphological parameters, finding that concentration probes the inner part of the cluster but is not overall a good measure of relaxation. Around 70% of their clusters at the lowest redshifts are found to be relaxed, similar to what is found by us. However, this decreases rapidly to 35–50% at $z > 0.2$, becoming closer to the SZ-selected samples. This difference in relaxation over redshift could be evolution or selection effects.

## 9. Conclusions

We measured 29 morphological parameters for over 12 000 clusters found in the eROSITA survey by B24. The parameters we measured include concentration, central density, inner density slope, peak offset, power ratios, Gini coefficient, photon asymmetry, centroid shift, ellipticity, slosh, and multipole magnitudes. In our work, we have introduced new slosh and multipole magnitude forward modelled parameters. This paper provides a catalogue of the largest X-ray derived set of cluster morphological parameters to date.

Several of the measured quantities (e.g. power ratios, centroid shift, Gini coefficient, photon asymmetry) show values that correlate strongly with the number of counts, luminosity, or redshift. When we conducted simulations of clusters observed in the survey, we found similar relations. These correlations are caused by biases due to noise or PSF, and therefore these parameters are unreliable for eROSITA clusters. We discussed the various biases that can be present in the parameters and provided
<figure><img src="image_22.png" /><figcaption>**Fig. 22.** Parameter values as a function of counts. The values are coloured according to the combined shape and concentration disturbance score, $D_{\text{comb}}$.</figcaption></figure>

the typical biases and uncertainties as a function of redshift and the number of counts.

Using both simulations and data, we observed that the con- centration of clusters strongly affects whether they are detected in the eROSITA survey. At low redshifts ($z \le 0.2$), we pref- erentially detected more highly concentrated objects and did not detect low luminosity groups and clusters with flat sur- face brightness profiles. At high redshifts ($z \ge 0.4$), we missed the most concentrated clusters. Alternative detection algorithms (e.g. wavelets) may find more flat surface brightness objects at low redshifts.

We made comparisons of our results for matched cluster subsamples for eRASS and other analyses. There is reasonable agreement in the concentration and density with matched *Planck*

ESZ and SPT subsamples. The median values are close to those we obtained using eROSITA, although there may be mild offsets or differences in slopes between the analyses. These differences may be due to different analysis procedures, data quality, choice of cluster centre, and parametrisation or unresolved AGN in the datasets.

Comparing whole cluster samples, we found that eRASS1 clusters have higher concentrations than SPT, *Planck* ESZ, and eFEDS. The difference is around 0.3 dex in our log units, implying that the ratio of flux inside $0.1R_{500}$ to $R_{500}$ is twice what is found in these other samples.

We constructed combined parameters based on a Gaussian mixture model for measuring the disturbance of clusters ($D_{\text{shape}}$ and $D_{\text{comb}}$), and we found that one-fourth of the objects are
<figure><img src="image_23.png" /><figcaption>**Fig. 23.** Disturbance scores for the sample for clusters with greater than 25 and greater than 300 counts. Left: Histograms of the distributions of $D_{\text{comb}}$ (including concentration) and the purely 2D shape based $D_{\text{shape}}$. Right: $D_{\text{comb}}$ plotted against $D_{\text{shape}}$ for the two count thresholds.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 24.** Examples of eight relaxed and eight disturbed clusters. Clusters are randomly selected from the sample with more than 300 counts, where $D_{\text{comb}} = 0$ (top panels) or $D_{\text{comb}} = 1$ (bottom panels). The exposure corrected images in the 0.2–2.3 keV band have been smoothed with a Gaussian of $\sigma = 8$ arcsec or 64 arcsec in the case of 1eRASS J000842.4-570213. The bars show a fixed angular scale of 5 arcmin.</figcaption></figure>
classified as disturbed. The high relaxation fraction is likely due to X-ray selection selecting highly concentrated clusters that are more likely to be relaxed.

## Data availability

The table described in Appendix A is available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/695/A160

**Acknowledgements.** This work is based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint Russian-German science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with support from the Max Planck Institute for Extraterrestrial Physics (MPE). The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The eROSITA data shown here were processed using the eSASS/NRTA software system developed by the German eROSITA consortium. E.B., V.G., A.L. and X.Z. acknowledge financial support from the European Research Council (ERC) Consolidator Grant under the European Union's Horizon 2020 research and innovation program (grant agreement CoG DarkQuest No 101002585). Funded in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 450861021.

## References

* Abraham, R. G., van den Bergh, S., & Nair, P. 2003, ApJ, 588, 218

* Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481

* Biffi, V., Borgani, S., Murante, G., et al. 2016, ApJ, 827, 112

* Birzan, L., Rafferty, D. A., McNamara, B. R., Wise, M. W., & Nulsen, P. E. J. 2004, ApJ, 607, 800

* Bleem, L. E., Stalder, B., de Haan, T., et al. 2015, ApJS, 216, 27

* Böhringer, H., & Werner, N. 2010, A&ARv, 18, 127

* Böhringer, H., Schuecker, P., Pratt, G. W., et al. 2007, A&A, 469, 363

* Böhringer, H., Pratt, G. W., Arnaud, M., et al. 2010, A&A, 514, A32

* Brunner, H., Liu, T., Lamer, G., et al. 2022, A&A, 661, A1

* Bulbul, E., Liu, A., Pasini, T., et al. 2022, A&A, 661, A10

* Bulbul, E., Liu, A., Kluge, M., et al. 2024, A&A, 685, A106

* Buote, D. A., & Tsai, J. C. 1995, ApJ, 452, 522

* Campitiello, M. G., Ettori, S., Lovisari, L., et al. 2022, A&A, 665, A117

* Cavaliere, A., & Fusco-Femiano, R. 1978, A&A, 70, 677

* Clowe, D., Bradač, M., Gonzalez, A. H., et al. 2006, ApJ, 648, L109

* Comparat, J., Eckert, D., Finoguenov, A., et al. 2020, Open J. Astrophys., 3, 13

* Eckert, D., Molendi, S., & Paltani, S. 2011, A&A, 526, A79

* Edge, A. C., & Stewart, G. C. 1991, MNRAS, 252, 414

* Fabian, A. C. 2012, ARA&A, 50, 455

* Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306

* Ghirardini, V., Bahar, Y. E., Bulbul, E., et al. 2022, A&A, 661, A12

* Gianfagna, G., Rasia, E., Cui, W., et al. 2023, MNRAS, 518, 4238

* Hudson, D. S., Mittal, R., Reiprich, T. H., et al. 2010, A&A, 513, A37

* Johnson, R., Ponman, T. J., & Finoguenov, A. 2009, MNRAS, 395, 1287

* Käfer, F., Finoguenov, A., Eckert, D., et al. 2019, A&A, 628, A43

* Käfer, F., Finoguenov, A., Eckert, D., et al. 2020, A&A, 634, A8

* Kluge, M., Comparat, J., Liu, A., et al. 2024, A&A, 688, A210

* Lau, E. T., Kravtsov, A. V., & Nagai, D. 2009, ApJ, 705, 1129

* Lau, E. T., Nagai, D., Kravtsov, A. V., Vikhlinin, A., & Zentner, A. R. 2012, ApJ, 755, 116

* Liu, A., Bulbul, E., Shin, T., et al. 2024, A&A, 688, A186

* Lotz, J. M., Primack, J., & Madau, P. 2004, AJ, 128, 163

* Lovisari, L., Forman, W. R., Jones, C., et al. 2017, ApJ, 846, 51

* Mann, A. W., & Ebeling, H. 2012, MNRAS, 420, 2120

* Markevitch, M., & Vikhlinin, A. 2007, Phys. Rep., 443, 1

* McDonald, M., Bayliss, M., Benson, B. A., et al. 2012, Nature, 488, 349

* McDonald, M., Benson, B. A., Vikhlinin, A., et al. 2013, ApJ, 774, 23

* McDonald, M., Allen, S. W., Bayliss, M., et al. 2017, ApJ, 843, 28

* McNamara, B. R., & Nulsen, P. E. J. 2012, New J. Phys., 14, 055023

* Melchior, P., & Goulding, A. D. 2018, Astron. Comput., 25, 183

* Merloni, A., Lamer, G., Liu, T., et al. 2024, A&A, 682, A34

* Mittal, R., Hicks, A., Reiprich, T. H., & Jaritz, V. 2011, A&A, 532, A133

* Mohr, J. J., Evrard, A. E., Fabricant, D. G., & Geller, M. J. 1995, ApJ, 447, 8

* Nurgaliev, D., McDonald, M., Benson, B. A., et al. 2013, ApJ, 779, 112

* O'Hara, T. B., Mohr, J. J., Bialek, J. J., & Evrard, A. E. 2006, ApJ, 639, 64

* Panagoulia, E. K., Fabian, A. C., & Sanders, J. S. 2014, MNRAS, 438, 2341

* Poole, G. B., Fardal, M. A., Babul, A., et al. 2006, MNRAS, 373, 881

* Pratt, G. W., Croston, J. H., Arnaud, M., & Böhringer, H. 2009, A&A, 498, 361

* Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1

* Rafferty, D. A., McNamara, B. R., Nulsen, P. E. J., & Wise, M. W. 2006, ApJ, 652, 216

* Rasia, E., Meneghetti, M., & Ettori, S. 2013, Astron. Rev., 8, 40

* Rossetti, M., Gastaldello, F., Eckert, D., et al. 2017, MNRAS, 468, 1917

* Sanders, J. S., Fabian, A. C., Russell, H. R., & Walker, S. A. 2018, MNRAS, 474, 1065

* Santos, J. S., Rosati, P., Tozzi, P., et al. 2008, A&A, 483, 35

* Seppi, R., Comparat, J., Bulbul, E., et al. 2022, A&A, 665, A78

* Spergel, D. N., Bean, R., Doré, O., et al. 2007, ApJS, 170, 377

* Valdarnini, R., & Sarazin, C. L. 2021, MNRAS, 504, 5409

* Vikhlinin, A., Burenin, R., Forman, W. R., et al. 2007, in Heating versus Cooling in Galaxies and Clusters of Galaxies, eds. H. Böhringer, G. W. Pratt, A. Finoguenov, & P. Schuecker, 48

* Vikhlinin, A., Burenin, R. A., Ebeling, H., et al. 2009, ApJ, 692, 1033

* Voit, G. M. 2005, Rev. Mod. Phys., 77, 207

* Wechsler, R. H., Bullock, J. S., Primack, J. R., Kravtsov, A. V., & Dekel, A. 2002, ApJ, 568, 52

* Weißmann, A., Böhringer, H., Šuhada, R., & Ameglio, S. 2013, A&A, 549, A19

* Xu, W., Ramos-Ceja, M. E., Pacaud, F., Reiprich, T. H., & Erben, T. 2022, A&A, 658, A59

* Yuan, Z. S., Han, J. L., & Wen, Z. L. 2022, MNRAS, 513, 3013
## Appendix A: Catalogue description

Table A.1 describes the columns in the morphology catalogue, giving the column name and respective morphological parameter. Most of the parameters are dimensionless, except for the inner physical gas density.

<table><caption>**Table A.1.** Column names in morphology catalogue.</caption><thead><tr><th>Column</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>NAME</td><td>IAU Cluster name</td><td></td></tr><tr><td>DETUID</td><td>eROSITA detection ID</td><td></td></tr><tr><td>CTS_800</td><td>Number of counts</td><td></td></tr><tr><td>NS_0</td><td>n<sub>s,0</sub></td><td></td></tr><tr><td>NS_0_P</td><td>n<sup>*</sup><sub>s,0</sub></td><td></td></tr><tr><td>NE_50</td><td>n<sub>50</sub></td><td>log cm<sup>-3</sup></td></tr><tr><td>NE_50_P</td><td>n<sup>*</sup><sub>50</sub></td><td>log cm<sup>-3</sup></td></tr><tr><td>ALPHA</td><td>α</td><td></td></tr><tr><td>ALPHA_P</td><td>α<sup>*</sup></td><td></td></tr><tr><td>ALPHA_50</td><td>α<sub>50</sub></td><td></td></tr><tr><td>ALPHA_50_P</td><td>α<sup>*</sup><sub>50</sub></td><td></td></tr><tr><td>C_R500</td><td>c<sub>500</sub></td><td></td></tr><tr><td>C_R500_P</td><td>c<sup>*</sup><sub>500</sub></td><td></td></tr><tr><td>C_80_800</td><td>c<sub>80-800</sub></td><td></td></tr><tr><td>C_80_800_P</td><td>c<sup>*</sup><sub>80-800</sub></td><td></td></tr><tr><td>F</td><td>F</td><td></td></tr><tr><td>P10</td><td>P<sub>10</sub></td><td></td></tr><tr><td>P10_P</td><td>P<sup>*</sup><sub>10</sub></td><td></td></tr><tr><td>P20</td><td>P<sub>20</sub></td><td></td></tr><tr><td>P20_P</td><td>P<sup>*</sup><sub>20</sub></td><td></td></tr><tr><td>P30</td><td>P<sub>30</sub></td><td></td></tr><tr><td>P40</td><td>P<sub>40</sub></td><td></td></tr><tr><td>GINI</td><td>G</td><td></td></tr><tr><td>APHOT</td><td>A<sub>phot</sub></td><td></td></tr><tr><td>APHOT_P</td><td>A<sup>*</sup><sub>phot</sub></td><td></td></tr><tr><td>CENTROID</td><td>w</td><td></td></tr><tr><td>ELLIPTICITY</td><td>ε</td><td></td></tr><tr><td>ELLIPTICITY_THETA</td><td>θ<sub>0</sub> for ε fit</td><td>deg</td></tr><tr><td>SLOSH</td><td>H</td><td></td></tr><tr><td>SLOSH_THETA</td><td>θ<sub>0</sub> for H fit</td><td>deg</td></tr><tr><td>M1</td><td>M<sub>1</sub></td><td></td></tr><tr><td>M1_THETA</td><td>θ<sub>0</sub> for M<sub>1</sub> fit</td><td>deg</td></tr><tr><td>M2</td><td>M<sub>2</sub></td><td></td></tr><tr><td>M2_THETA</td><td>θ<sub>0</sub> for M<sub>2</sub> fit</td><td>deg</td></tr><tr><td>M3</td><td>M<sub>3</sub></td><td></td></tr><tr><td>M3_THETA</td><td>θ<sub>0</sub> for M<sub>3</sub> fit</td><td>deg</td></tr><tr><td>M4</td><td>M<sub>4</sub></td><td></td></tr><tr><td>M4_THETA</td><td>θ<sub>0</sub> for M<sub>4</sub> fit</td><td>deg</td></tr><tr><td>D Shape</td><td>D<sub>shape</sub></td><td></td></tr><tr><td>D Comb</td><td>D<sub>comb</sub></td><td></td></tr></tbody></table>

**Notes.** For each numerical column there are corresponding columns with the suffixes _POS and _NEG which denote the 1σ uncertainties. The number of counts is that within 800 kpc radius in the 0.2-2.3 keV band.