*natbibCitation

11institutetext: Department of Physics, University of Helsinki, Gustaf
Hällströmin katu 2A, Helsinki, FI-00014, Finland 22institutetext: Tartu
Observatory, University of Tartu, Observatooriumi 1, 61602 Tõravere, Estonia
33institutetext: Estonian Academy of Sciences, Kohtu 6, 10130 Tallinn, Estonia
44institutetext: Institut d’Astrophysique de Paris (UMR 7095: CNRS & Sorbonne
Université), F-75014 Paris, France

# AXES-2MRS: A new all-sky catalogue of extended X-ray galaxy groups

H. Khalil 11 A. Finoguenov 11 E. Tempel 2233 G. A. Mamon 44

(Received March 21 2024)

###### Abstract

Context. Understanding the baryonic physics on the galaxy group level is a
prerequisite for cosmological studies of large-scale structure. One poorly
understood aspect of galaxy groups is related to the properties of their hot
intragroup medium. The well-studied X-ray groups have strong cool cores by
which they were selected, so expanding the selection of groups is currently an
important venue.

Aims. We present a new all-sky catalogue of X-ray detected groups (AXES-2MRS),
based on the identification of large X-ray sources found in the ROSAT All-Sky
Survey (RASS) with the Two Micron Redshift Survey (2MRS) Bayesian Group
Catalogue. We study the basic properties of these galaxy groups to gain
insights into the effect of different group selections on the properties.

Methods. In addition to X-ray luminosity coming from shallow survey data of
RASS, we have obtained detailed X-ray properties of the groups by matching the
AXES-2MRS catalogue to archival X-ray observations by XMM-Newton and
complemented this by adding the published XMM-Newton results on galaxy
clusters in our catalogue. We analyse temperature and density to the lowest
overdensity accessible by the data, obtaining hydrostatic mass estimates and
comparing them to the velocity dispersions of the groups. We explore the
relationship between X-ray and optical properties of AXES-2MRS groups through
the $\sigma_{\mathrm{v}}-L_{\mathrm{X}}$, $\sigma_{\mathrm{v}}-kT$, $kT-
L_{\mathrm{X}}$, $\sigma_{\mathrm{v}}-M$, and $c_{200}-L_{\rm X}$ scaling
relations.

Results. We find a large spread in the central mass to virial mass ratios for
galaxy groups in the XMM-Newton subsample. This can either indicate large non-
thermal pressure of galaxy groups affecting our X-ray mass measurements, or
the effect of a diversity of halo concentrations on X-ray properties of galaxy
groups. Previous catalogues, based on detecting the peak of the X-ray emission
preferentially sample the high-concentration groups, while our new catalogue
includes many low-concentration groups.

###### Key Words.:

catalogs – galaxies: groups: general – galaxies: clusters: intracluster medium
– X-rays: galaxies: clusters

##  1 Introduction

The hot intergalactic medium of galaxy groups plays an important role in
galaxy evolution and reflects the energetics of galactic outflows and metal
production. Several studies have suggested a direct link between the baryonic
content of galaxy groups and the shape of matter power spectrum on spatial
scales below 10 Mpc (Debackere et al., 2020). Deep X-ray surveys enabled
significant advances in understanding galaxy groups, which discovered a large
population of X-ray emitting groups down to the mass below $10^{13}M_{\odot}$
and reaching redshifts above 2 on high-mass groups (Gozaliasl et al., 2019).
However, the low-redshift ($z<0.1$) population of galaxy groups still lacks a
full understanding. In contrast, this population is a main source of our
knowledge on the detailed properties of galaxy groups. Previous catalogues of
X-ray-selected local groups and clusters of galaxies were primarily based on
identifying sources encompassing the emitting zone of $2^{\prime}$. This has
been shown to account for only a fraction of galaxy groups consisting of
relaxed groups with luminous central objects (Mulchaey, 2000). A large
population of sources is omitted from these catalogues (Xu et al., 2018),
which is confirmed by the dedicated consideration of galaxy group emission by
Käfer et al. (2019). In this paper we will continue the investigation of such
sources, considering the spatially resolved X-ray emission down to the lowest
signal-to-noise ratio (SNR) of the ROSAT All-Sky Survey (RASS) and detected on
virial spatial scales.

This paper is organized as follows: in Section 2, we present the construction
and basic properties of the new X-ray source catalogue, describe the 2MRS
optical group catalogue used for the identification, and introduce a
representative subsample observed by XMM-Newton. The analysis of X-ray and
optical properties of X-ray detected groups in our catalogue is provided in
Section 3. In Section 4, we present the scaling relations including a
comparison with the literature. We summarize our results in Section 5. In this
study, we adopt a flat $\Lambda$CDM cosmology with parameters $H_{0}$ = 70 km
s-1 Mpc-1, $\Omega_{\rm m}$ = 0.3, and $\Omega_{\Lambda}=1-\Omega_{\rm m}$.
Unless otherwise stated, errors represent standard $1\sigma$ uncertainties
(drawn at the 68$\%$ confidence level). For radii, masses, and concentrations,
the suffixes 200, 500, and 10000 correspond to the mean densities relative to
the critical density of the Universe at the redshift of the group.

##  2 Data

Figure 1: Cumulative distribution for the X-ray flux in the 0.5–2.0 keV band
per sky area ($\log N-\log S$) for AXES X-ray sources (black), and AXES-2MRS
groups (blue). AXES X-ray sources in the extragalactic region (nH $<5\times
10^{20}$ cm-2) are shown in purple, while extragalactic AXES-2MRS groups are
shown in red. The flux of the AXES-2MRS sources has been extrapolated to
$R_{500}$, while for AXES we show the measured flux.

###  2.1 AXES: a new catalogue of X-ray sources from ROSAT All Sky Survey

The ROSAT all-sky survey (RASS) has been an enormous legacy for X-ray
astronomy (see Truemper, 1993, for a review). Of particular importance are the
all-sky catalogues of sources (Voges et al., 1999), which formed the base of
X-ray studies in the last three decades. Exploration of the RASS data down to
its faint limits has recently become an active field (e.g., Finoguenov et al.,
2020). In the present paper, we report a new study of RASS data. We have
produced a new catalogue of RASS sources, All-sky X-ray Extended Sources
(AXES), found using 0.5–2.0 keV band images. For the source detection and
determination of the flux extraction regions, we employ the wavelet scales of
12 or 24 arcmin after removing the emission detected on scales of 6 arcmin and
below. This is a multiscale detection, unaffected by the emission on scales
smaller than the scale of interest. We construct the experiment to scale with
the baryonic content of galaxy groups near $R_{500}$, which is different from
the point of finding spatially resolved X-ray sources on scales of the point
spread function, which forms a base of the Xu et al. (2018) catalogue. By
reducing the dependence of the detection on the shape of X-ray emission in the
centre, the modelling of the source detection becomes feasible through the use
of currently available hydrodynamical simulations, which reproduce group
outskirts, but not the group cores. In Fig. 1, we show the normalized
cumulative number count of AXES sources and AXES-2MRS groups (see Section 2.2)
as a function of the flux in the 0.5–2.0 keV band ($\log N-\log S$). We do not
attempt to restore the original $\log N-\log S$, but rather to look for
indications of the catalogue completeness. On the adopted spatial scales the
detection is background limited and completeness as a function of flux
indicates a completeness limit of $2\times 10^{-12}$ ergs s-1 cm-2 (see Fig.
1), which is characteristic of the extragalactic areas. The 2MRS survey (see
Section 2.2) stops where X-ray sensitivity drops by a factor of two due to
foreground absorption. When comparing the $\log N-\log S$ distribution inside
and outside the zone of avoidance, we see an excess of the bright sources in
the zone of avoidance, which we attribute to additional sources coming from
the galactic plane. We also note that at fluxes fainter than $10^{-11}$ erg
s-1 cm2 the two curves converge, which indicates a predominance of
extragalactic sources at those fluxes even in the galactic plane. AXES
contains over six thousand unique X-ray sources, with a large concentration of
sources towards the galactic centre, with many of them identified with
supernova remnants. Therefore, to report on a new galaxy group, external
identification of sources is required.

Figure 2: Distribution of AXES-2MRS X-ray sources and full 2MRS optical
sources represented with supergalactic coordinates on the sky. The small
points are 2MRS optical groups bar coloured according to their (CMB rest frame
corrected) redshift, while larger black circles denote the full AXES-2MRS
X-ray catalogue of 558 groups.

###  2.2 Optical Group Catalogue

The choice of the angular scales for our X-ray detection is designed to cover
the virial radius of groups at $z<0.1$. To identify the sources we find with
AXES in this work, we consider a group catalogue from the 2MASS spectroscopic
survey (Tempel et al., 2018, 2MRS), selecting the groups that contain at least
three spectroscopic members. The advantage of this catalogue is that it is
all-sky and extends into the galactic disk, allowing us to improve on the
studies of the local dynamics. In assigning the X-ray sources to the optical
group, we compute the radius of 200 kpc using the redshift of the group and
use it to find an X-ray counterpart within this radius. Using the redshift of
the group and the H i absorption-corrected flux of the X-ray source, we then
compute the source rest-frame X-ray luminosity in the 0.1–2.4 keV band, with
K-corrections obtained iteratively using the $L-T$ relation. Our choice of
using 2MRS groups down to 3 spectroscopic members is an attempt to improve the
completeness of the 2MRS towards $10^{13}M_{\odot}$ groups expected to emit
X-rays while avoiding the inclusion of a large population of low-mass groups
(with masses extending down to $10^{12}M_{\odot}$), present in the two-member
catalogue (for a discussion of tracing group mass with a few members see e.g.
Knobel et al., 2009). Our choice of a minimum of 3 members compares well with
the results of REFLEX spectroscopic identification (Böhringer et al., 2004),
which made the largest contribution to the exhaustive X-ray cluster catalogue
MCXC (Piffaretti et al., 2011). Given the small number of sources and the high
fraction of matches, AXES-2MRS has a high level of purity of 97%. Given the
importance of the low-z systems to the studies of the local dynamics, we do
not cut the catalogue to the extragalactic areas, which is uniquely possible
given our choice of source identification using the 2MRS catalogue. In Fig.2
we show the sky distribution of the groups in AXES-2MRS using a supergalactic
coordinate system. We use the symbol’s colour to illustrate the group’s
redshift. X-ray sources identified with the groups are marked with large
black-filled circles. Most groups in the 2MRS catalogue have no mass
estimates, given they have just a few members. Our catalogue improves this
situation by providing an X-ray luminosity estimate, which is a mass proxy,
and marks the massive parts of the local cosmic web. In the application of
X-ray studies to the Galactic areas, it is important to acknowledge
technological differences in X-ray detectors. ROSAT PSPC detector, used in
RASS, is based on the photon interaction with gas, which is not sensitive to
stellar light, and is a serious problem for X-ray CCDs. On XMM-Newton, this
problem is addressed by selecting an appropriate filter for the observation.
However, this is not done for eROSITA, resulting in an additional source of
contamination, absent in RASS. Thus, the AXES catalogue provides a reliable
list of sources, against which to compare CCD detections on comparable spatial
and flux scales.

Figure 3: Full AXES-2MRS catalogue (558 groups, red circles) compared to
overlapping systems between MCXC and AXES-2MRS (blue triangles), and unique
MCXC systems (grey circles) in the $L_{\rm X}-z$ plane. The XMM-Newton-
observed subset with at least eight members selected from AXES-2MRS is shown
with neon green squares.

To demonstrate the state of our new catalogue relative to the literature, we
compare it with the MCXC in the $L_{\rm X}-z$ plane in Fig. 3. The MCXC
consists of 1743 unique cluster detections representing a compilation of
different catalogues based on RASS data, and ROSAT pointed data. Using MCXC
mass estimates we find 905 AXES sources to be located inside $R_{200}$ of 838
unique clusters. In Fig.3 we plot the full MCXC catalogue, zooming on the low-
redshift ($z<0.2$) subspace, its overlap with AXES sources, and present the
AXES-2MRS catalogue. MCXC, being a literature compilation catalogue, reveals
clear signatures of different completeness below and above $10^{44}$ ergs s-1.
Nevertheless, for our purpose of illustrating the limitations imposed by using
2MRS as follow-up data, it is sufficient. The AXES-2MRS sources are limited to
$z<0.08$ with incompleteness signatures appearing at $z>0.04$, which match the
expectations on the completeness of the 2MRS catalogue. At $z>0.1$, the extent
of X-ray emission reaches AXES detection scales only for the most massive
clusters, so the selection effects become noticeable close to a redshift of
0.2. At $z<0.04$ AXES-2MRS luminosities are located well on the extrapolation
of the MCXC trends towards lower redshift, while MCXC itself does not have
many systems in the same redshift range. We also cross-matched AXES-2MRS with
the newly released X-ray-selected extended galaxy cluster catalogue (RXGCC, Xu
et al. 2022), which is based on RASS and contains 944 systems. Within a 5′
radius (positional uncertainty of large X-ray sources at lowest detection
significance), we identified 162 overlapping systems, out of a total number of
558 AXES-2MRS groups. To further illustrate the role of AXES-2MRS in enhancing
the completeness of X-ray group catalogues, we compared our catalogue to the
extensive eROSITA-based catalogue of galaxy clusters and groups (Bulbul et
al., 2024, eRASS1). With over 12,200 systems, eRASS1 covers a total of 13,116
deg2 in the western Galactic hemisphere of the sky and has a redshift range of
$0.003<z<1.32$. Despite the majority (68$\%$) of eRASS1 systems being new
identifications with no counterparts in the literature, we could match only 73
overlapping groups with AXES-2MRS within a 5′ radius and a redshift tolerance
of 0.01. The published eRASS1 catalogues reach much fainter fluxes, but the
source detection is limited to $3^{\prime}$, which is not optimal for low-
redshift groups (Käfer et al., 2019). In addition, the red sequence
identification, employed in the identification of eRASS1 sources, is
incomplete for groups (Rykoff et al., 2014).

The all-sky distribution of the identified AXES-2MRS X-ray groups overlaid
with the galaxy compilation used to create the 2MRS optical catalogue is shown
in Fig. 5. The figure is split about the equator into two declination regions:
the top panel shows the northern [$0^{\circ}<\text{Dec}\leq 90^{\circ}$] range
with 316 AXES-2MRS groups and 21484 galaxies. While the bottom panel shows the
southern [$-90^{\circ}\leq\text{Dec}\leq 0^{\circ}$] range with 242 AXES-2MRS
groups and 20727 galaxies. The maps show the location of the largest local
structures in the Universe, on which 2MRS-AXES sources mark the most massive
virialized systems. In addition, we mark the locations of the local
underdensity regions (voids) taken from Plionis & Basilakos (2002); Tully et
al. (2019). The former work detected voids in the PCSz redshift survey
(Saunders et al., 2000), while the latter modelled the morphology of the local
voids using peculiar velocities.

(a) $0^{\circ}<\text{Dec}\leq 90^{\circ}$

(b) $-90^{\circ}\leq\text{Dec}\leq 0^{\circ}$

Figure 5: Projection of the 3-dimensional distribution of local cosmic
structure onto the RA-redshift plane, illustrating the correspondence of the
AXES-2MRS groups (red ellipses) to large-scale structure traced by the 2MRS
galaxies (grey dots), with the location of the largest local voids marked. The
coloured regions denote void centres, but their sizes do not reflect void
dimensions. Pink regions/labels are density minima in the Local Void, orange
is the Hercules Void, while green and brown represent the Sculptor and
Eridanus Voids, respectively. Blue colour denotes the PSCz-IRAS survey data
(Plionis & Basilakos, 2002), while the other voids are based on Tully et al.
(2019). (a) Northern hemisphere (positive Dec values). (b) Southern hemisphere
(negative Dec values).

###  2.3 XMM-Newton Data

For a subset of AXES-2MRS groups, we obtained a much more detailed picture of
the group emission using XMM-Newton observations. We have searched the
archival XMM-Newton data on the sample, limiting the study to the groups
having at least 8 spectroscopic members, retained after cleaning the
membership using Clean (Mamon et al., 2013). Our XMM-Newton data reduction
pipeline is described in Finoguenov et al. (2007). We have used the XMMSAS
version 21.0.0. In the imaging analysis, we used the 0.5–2 keV band, removing
instrument lines, as described in Finoguenov et al. (2007). Point source
detection and removal have been performed following Finoguenov et al. (2010).
We perform the wavelet decomposition (Vikhlinin et al., 1998,
111https://github.com/avikhlinin/wvdecomp) of the extended emission on scales
from half an arcminute to 4 arcmin. These scales provide insight into the
central part of the object. Larger scales are analyzed using a symmetrical
beta model. In cases where several optical groups are present, the
availability of XMM-Newton data helps to select the correct counterpart of the
emission and often finds more than one extended X-ray source. Spectroscopic
group membership for galaxies is also complicated in those cases, and the
catalogues list the probability of being a member of several adjacent groups.
In these cases, we considered all the galaxies associated with the main
optical counterpart with a probability above 10%. Looking at the target
selection in the archival data, we see that some studies are follow-ups of
radio sources, and some are follow-ups of X-ray sources. The optically-driven
survey with X-ray follow-up, CLOGS (O’Sullivan et al., 2017), which is
analogous to our approach here, occupies a smaller redshift range compared to
our data.

After screening the data to eliminate failed observations and removing
repeated fields of view (FOVs) using exposure maps, we retained 25 distinct
and usable XMM observations, detailed in Table 4 and Table 5. The uniquely
identified X-ray groups are shown in Fig. 19, where we compare the wavelet-
filtered X-ray images cleaned from background and point sources with the
location of group galaxies. The spatial scales shown in Fig.19 range from 0.5
to 8 arcmin. In most cases, X-ray emission can be unambiguously identified
with a single galaxy group.

We detect a merging behaviour in three systems (Group IDs: 361, 5089, 6407,
superscripted ”M” in Table 5), where the X-ray emission could be linked to
more than one galaxy group (see Fig. 19), while positional and velocity
difference between the optical groups is small. We established a
correspondence of X-ray emission to optical groups in these merging systems
based on the dominant representation of optical sources near the centre of the
X-ray emission regions. Nonetheless, in one case (Group ID: 5089), an extreme
merging behaviour (with Group ID: 5084) was observed in which galaxies from
the two optical sources were tightly packed in the centre. We selected the
optical system 5089 because it had the closest galaxy member to the core of
X-ray emission. This system was omitted from the scaling relations analysis
due to its notably non-relaxed behaviour, significantly surpassing the other
systems in the XMM subsample in terms of dynamical complexity. Furthermore,
four systems were identified as over-split (superscripted ”OS” in Table 5),
having double X-ray emission components. These double X-ray component groups
were examined separately, and a single component was selected as a
representative for each system determined by the X-ray peak best associated
with the optical group. The coordinates of those X-ray peaks are listed in
Table 4.

##  3 Results

###  3.1 X-ray Temperatures

For the spectral extraction, we applied the evselect task in XMMSAS to filter
out bright pixels and hot columns (FLAG == 0). We selected only single and
double patterns (PATTERN $<=$ 4) for the EPIC pn camera. Source and background
spectra were created from the same FOV using the same criteria. Redistribution
matrix files (rmfs) and ancillary files (arfs) were generated using XMMSAS’s
rmfgen and arfgen tasks, respectively. Point sources identified in detector
images were visually checked and excluded from the event files. To ensure
Gaussian statistics in both background and source spectra for the $\chi^{2}$
minimization used in temperature modelling, a channel binning scheme was
applied using the grppha task from HEASARC’s FTOOLS package. Background
spectra were re-binned to achieve at least 30, 60, or 200 counts per bin,
depending on the source brightness and the observation’s SNR.

Intragroup medium (IGrM) X-ray temperatures were estimated based on fitting
the spectra with an absorbed APEC thermal plasma model using the redshift of
each group (zmed in Table 5), and allowing the normalization and metal
abundances to vary. The Galactic absorption component was fixed using the
emission centres coordinates and the online HEASARC’s web tool
nH222https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl which is based
on the HI4PI Survey (HI4PI Collaboration et al., 2016). To ensure consistent
temperature measurements, we used a fixed spectral fitting range of [0.4-3.0]
keV. The lower limit is set to avoid energies with a high background-to-signal
ratio. The [1.45-1.6] keV range was excluded due to the strong instrumental Al
line, which dominates the background and its exclusion improves the SNR of the
data. We checked that the number of spectral bins left after channel trimming
was larger than the degrees of freedom of the model. In one case (Group ID:
5089), an inspection of the background spectrum showed strong peaks in the
soft band ($>0.6$ keV) indicating the presence of an extra background
component due to soft protons (Kuntz & Snowden, 2000). In that case, we
increased the lower fitting limit to 0.6 keV.

To better constrain our analysis, the temperature modelling was repeated using
a wider energy range of [0.4-7.0] keV. The systematic error, defined as the
difference in $kT$ between the two energy ranges, was always checked and found
to be less than 10$\%$. In all of the observations analyzed, the reduced
$\chi^{2}$ values were in the range of [0.7-1.8]. The obtained temperature
estimates for the XMM-Newton subsample are summarised in Table 5.

###  3.2 Surface Brightness Profiles

The Chandra Interactive Analysis of Observations (CIAO) v4.15.1 was used in
the surface brightness profile extraction and fitting. Assuming circular
symmetry, a set of concentric circular annuli centred on the temperature
extraction region was used for each observation. Depending on the location of
each centre of extended emission on the detector, the outer radius of the
annuli RkT was set to fully encompass the outskirts of the galaxy groups and
reach the background level. In the four over-split systems, the same component
used for temperature estimation was also used in the surface brightness
profile extraction, and the other component was manually masked. In one case
(Group ID: 6116), the centre of the source was on the edge of the detector,
and since we have no data available on scales outside XMM-Newton’s FOV, no
usable radial profile fit could be obtained. Thus, it is excluded from the
surface brightness and mass analyses, however, we kept it in the $\sigma_{\rm
v}-kT$ study as its temperature and velocity dispersion are well-constrained.
Point sources were given special attention due to the faint nature of the
extended X-ray emission from our group sample. Therefore, any left-over
emission from interlopers, failure to detect faint-emission interlopers, or
both, was found to affect the radial profile fit to a large extent.
Accordingly, point source regions were re-examined by eye and often modified
manually to fully cover their emission area. CIAO’s dmcopy and dmextract tasks
were used in the point source subtraction and radial profile extraction,
respectively. A variance map ($\sigma^{2}$) was produced, by squaring the
error on counts and fed to dmextract along with the background subtracted and
point source corrected image. Surface brightness profiles were then calculated
by dividing the number of counts in each concentric ring by its area, and
errors were estimated by dividing the variance of each ring by its area.

A one-dimensional beta model was fitted to the radial profiles of each galaxy
group using the beta1d model of the Sherpa package. The beta1d model has the
form:

$\displaystyle\Sigma(r)=\Sigma_{0}\left[1+\left(\frac{r}{r_{c}}\right)^{2}\right]^{-3\beta+0.5}$
(1)

where $\Sigma(r)$ is the surface brightness at radius $r$, rc is the core
radius, and $\beta$ is the slope parameter of the profile.

We are only interested in the slope of the profiles at large radii, hence, a
simple power-law relation with one slope parameter $\beta$ fitting the
outskirts of the groups was our goal. Good fits, with reduced $\chi^{2}$ in
the range $0.8<\chi^{2}<2$, were obtained in which the core radius parameter
was fixed at artificially small values. The extreme merger case (Group ID:
5089) was the only system that did not have robust statistical results
($\chi^{2}>2$). The parameters used in the radial profile extraction and
fitting are listed in Table 6.

Figure 6: Cumulative distributions (_thick histograms_) and frequency
distributions (_shaded and thin histograms_ of the number of member galaxies
(which also traces the group richnesses) for the full AXES-2MRS groups (grey),
and the XMM subsample (red).

###  3.3 Hydrostatic Mass Estimates

The total X-ray-deduced mass inside a radius $r$ assuming hydrostatic
equilibrium, a $\beta$-model shape of the gas density, and a polytropic
temperature profile ($T(r)\propto N_{\rm gas}^{\gamma-1}$) is (Finoguenov et
al., 2001):

$\displaystyle M(r)=3.7\times 10^{13}\ T(r)\ r\ \frac{3\ \beta\ \gamma\
x^{2}}{1+x^{2}}\,M_{\odot}\ ,$ (2)

where $x=r/r_{c}$ is the ratio between the radius $r$ and the core radius of
the $\beta$-model $r_{c}$ and $\gamma=1.1$. X-ray masses were calculated
inside the temperature extraction radius RkT, and thus $T(r)$ is just the
spectroscopic temperature. Elliptical regions were used in the temperature
extraction, while we assume a circular symmetry in the mass analysis. We
employ the ellipsoidal quadratic mean radius to associate a radius with the
temperature extraction region:

$\displaystyle R_{kT}=\sqrt{\frac{a_{\rm spec}^{2}+b_{\rm spec}^{2}}{2}},$ (3)

The overdensity of the measurement was estimated as:

$\displaystyle\Delta=\frac{M(r)}{4\pi/3\,r^{3}\rho_{c}(z)},$ (4)

where $\rho_{c}(z)$ = $1.37\times 10^{11}E^{2}(z)M_{\odot}/(\rm Mpc)^{3}$ is
the critical density of the universe at redshift $z$. Table 6 includes the
calculated X-ray masses and their overdensities as well as the rescaled masses
at $\Delta$ = 500 and 10000. In rescaling the masses, we used the mean
expected concentration following the prescription of Hu & Kravtsov (2014). In
addition, we explore our constraints on the concentration of individual halos
coming from comparing direct mass estimates at two different overdensities.

###  3.4 Velocity Dispersion

We choose the gapper velocity dispersion estimator (Beers et al., 1990) as it
is preferred in case of a low number of member galaxies. In particular, Beers
et al. (1990) favours this method for clusters with less than 15 members. Fig.
6 shows the cumulative distribution of the number of groups ($N_{\rm grp}$) in
the full AXES-2MRS, and the XMM subsample, as a function of the number of
member galaxies $N_{\rm gal}$, as well as the corresponding frequency
distribution.

The velocity dispersion for the groups is calculated as:

$\displaystyle\sigma_{\rm gap}=\frac{c}{{1+\langle
z\rangle}}\,\left(\frac{\sqrt{\pi}}{N(N-1)}\sum_{i=1}^{N-1}w_{i}g_{i}\right),$
(5)

where $c$ is the speed of light, $N$ is the number of members, $w_{i}$ =
$i(N-i)$, $g_{i}$ = $z_{i+1}-z_{i}$ is the pairwise difference between member
redshifts, and $\langle z\rangle$ is the mean redshift. The uncertainty on
$\sigma_{\rm gap}$ is calculated as:

$\displaystyle\Delta\sigma_{gap}=\frac{C\sigma_{gap}}{\sqrt{N-1}},$ (6)

where C=0.91, based on the modelling of Ruel et al. (2014). We adopted the use
of the gapper velocity dispersion within the Clean, which removes interlopers
and we only use the galaxies inside the $R_{200}$ in the calculation. We
report our velocity dispersion measurements of the XMM-Newton subsample in
Table 4, and for the full AXES-2MRS catalogue in Table 7.

###  3.5 Halo Concentration

Our X-ray mass measurements and velocity dispersions constrain the masses of
the groups at very different overdensities: the X-ray measurements cover the
central part of the group, while spectroscopic members used for the velocity
dispersion estimates extend to the virial radius. We take the optical mass
measurement from the calibrations of Munari et al. (2013), which linked the
$M_{200}$ to the measured $\sigma_{v}$. We seek a value of concentration that
describes both X-ray and optical mass measurements. The largest contribution
to the error is the statistical error on the velocity dispersion, which is of
the order 30% for many groups. This is much larger than any bias associated
with the assumption of the hydrostatic equilibrium (HSE, Fabricant &
Gorenstein, 1983;Fabricant et al., 1980; Fabricant et al., 1984), especially
taking into account the previously found good performance of the HSE technique
at the overdensities we use for X-ray mass measurements. The resulting
constraints on the concentration are rather uncertain with no group clearly
showing deviations from the expected concentration (see Section 4.7 and Table
6).

###  3.6 Velocity Substructure

We used the Anderson-Darling (AD) normality test (Anderson & Darling, 1954) to
split the AXES-2MRS catalogue and the XMM subsample into Gaussian (G) and non-
Gaussian (NG) groups and study the effect of velocity substructure on the
scaling relations. We followed the procedure outlined in Hou et al. (2009) for
applying the AD test. In particular, we calculated the test statistic $A^{2}$
and its (sample size) weighted modification $A^{2*}$, from the ordered
velocities of the galaxy group members $x_{i}$, as:

$\displaystyle
A^{2}=-N-\frac{1}{N}\sum_{i=1}^{N}(2i-1)\,\left\\{\ln\Phi(x_{i})+\ln\left[1-\Phi\left(x_{N+1-i}\right)\right]\right\\}$
(7) $\displaystyle
A^{2*}=A^{2}\left(1+\frac{0.75}{N}+\frac{2.25}{N^{2}}\right)\ ,$ (8)

where $x_{i}\leq x<x_{i+1}$, and $\Phi(x_{i})$ is the cumulative distribution
function of the hypothetical underlying Gaussian distribution given as:

$\displaystyle\Phi(x_{i})=\frac{1}{2}\left[1+\text{erf}\left(\frac{x_{i}-\mu}{\sqrt{2}\sigma_{\rm
v}}\right)\right]\ ,$ (9)

where $\mu$ is the mean velocity of the group, and $\sigma_{\rm v}$ is the
velocity dispersion. $A^{2*}$ is then used to compute the significance level
$\alpha_{AD}$, used to asses the Gaussianity assumption, as:

$\displaystyle\alpha_{AD}=a\exp{\left(-\frac{A^{2*}}{b}\right)},$ (10)

where $a=3.6789468$ and $b=0.1749916$ are numerical fitting parameters taken
from Nelson (1998). The percentage of the G groups in AXES-2MRS with at least
8 spectroscopic members is $\sim{77\%}$, consistent with Damsted et al.
(2023), who studied dynamics of CODEX clusters, which overlap in X-ray
luminosity, but are located at higher redshift.

##  4 Scaling Relations

###  4.1 Statistical methods

Primarily, the Python package linmix333https://github.com/jmeyers314/linmix
(Kelly, 2007) was used in the scaling relations of interest in this paper. It
was shown by Kelly (2007) that it performs better than other regression
estimators (e.g., OLS, BCES(Y—X), and FITEXY) when the measurement errors are
large and the sample size is low. Optical $\sigma_{\rm v}$ values naturally
exhibit a relatively large uncertainty ($\sim$20-30$\%$), and we perform the
scaling relations on a limited-size group sample. The linmix package is based
on a hierarchical Bayesian model that approximates a distribution function of
the input data points using a mixture of several Gaussian components ($K$).
Except for $c_{200}-L_{\rm X}$ relation which has $K=4$ (see Section 4.7), all
the scaling relations presented in this section use $K=3$. To better
illustrate linmix’s structure, we take the $\sigma_{\rm v}-kT$ relation as an
example (see Section 4.4 and figures therein). Essentially, $kT$ and
$\sigma_{\rm v}$ are supposed to follow a bivariate log-normal distribution
$\mathcal{N}_{2}(\mu,\Sigma)$ with the mean $\mu=(\xi,\eta)$. These values
$(\xi,\eta)$ represent the true, yet unobservable means of $\ln{kT}$ and
$\ln{\sigma_{\rm v}}$, and the covariance matrix $\Sigma$ contains the errors
observed in the logarithmic values of kT and $\sigma_{\rm v}$. The connection
between $\xi$ and $\eta$ is established through the conditional probability
distribution $P(\eta\,|\,\xi)$ = $\mathcal{N}(A+B\,\xi,S^{2})$. Here, $A$ is
the intercept, $B$ is the slope, and $S$ is the Gaussian intrinsic scatter of
$\eta$ around the regression line. The exact equation used in the fitting is:

$\displaystyle\eta=A+B\,x_{i}+S,$ (11)

where $x=x_{i}+x_{\rm err}$ is the predictor vector of the data points +
errors, and $y_{i}=\eta+y_{\rm err}$ is the target vector. The exact fitting
formulas used in the scaling relations are listed in Table 2. We use 100 000
Markov chain Monte Carlo (MCMC) iterations within linmix and report the mean
of the posterior distribution of the best-fit parameters in Table 3.

Figure 7: Comparison of AXES-2MRS X-ray luminosity measurements between XMM-
Newton and RASS. The dashed line is the best fit. The one-to-one relation
between XMM-Newton and RASS is given by the solid line. The shaded region is
the 1$\sigma$ uncertainty of the slope and intercept.  Figure 8: MCMC fitting
results with one and two-dimensional projections of the posteriors of the
XMM-$L_{\rm X}$–RASS $L_{\rm X}$ scaling relation. The vertical line
represents the one-to-one relation and corresponds to the _red_ line in Fig.
7. Dark and light contours represent 68$\%$ and 95$\%$ confidence levels,
respectively.

Given that we introduce a new sample of galaxy groups and we use Bayesian
methods for the analysis of the scaling relations, it is important to separate
the contribution of the sample from the difference in the analysis. So, we
also apply the orthogonal distance regression (Boggs et al., 1989, ODR). The
ODR method fits the data points by minimizing their squared orthogonal
distances to the best-fit line. The ODR accounts for measurement errors in the
two variables, however, it does not estimate the intrinsic scatter of the
relation. For that reason, and the fact that Bayesian methods typically
produce wider confidence intervals than frequentist ones, uncertainties of the
linmix parameters are somewhat larger. In the following, we present the
scaling relation between the XMM-Newton and ROSAT X-ray luminosities ($L_{\rm
XMM}-L_{\rm RASS}$). We also present the relation between optical velocity
dispersion and each of the X-ray luminosity ($\sigma_{\rm v}-L_{\rm X}$), the
X-ray gas temperature ($\sigma_{\rm v}-kT$), and the mass ($\sigma_{\rm
v}-M_{10000}$). As well as between the X-ray temperature and luminosity ($kT-
L_{\rm X}$) and between the dark matter-halo concentration and luminosity
($c_{200}-L_{\rm X}$).

###  4.2 Characterisation of AXES X-ray Luminosities

Figure 9: Left panel: Velocity dispersion vs. X-ray luminosity for the full
AXES-2MRS systems with at least 5 members and our XMM-Newton subsample.
Gaussian and non-Gaussian groups are marked with blue circles and red crosses,
respectively. The XMM subsample is marked with filled green squares. Shaded
regions are 1$\sigma$ uncertainties. Right panel: Slope comparison for the
$\sigma_{\rm v}-L_{\rm X}$ relation with the literature. The grey-shaded
region is the self-similar expectation calibrated according to the X-ray
emissivity in the band-limited ($0.1-2.4$ keV) low-temperature ($0.7-3.0$ keV)
range. While the blue-shaded region is for the bolometric luminosity. (*
refers to relations originally expressed as $L_{\rm X}-\sigma_{\rm v}$ and was
inverted for comparability).  Figure 10: MCMC fitting results with one and
two-dimensional projections of the posteriors of the $\sigma_{\rm v}-L_{\rm
X}$ scaling relation. Other details are the same as Fig. 8.

Since AXES presents new estimates of X-ray luminosities, measured using large
sky areas, it is important to assess the quality of these estimates, using
better X-ray observations available with XMM-Newton. In the computation of
X-ray luminosities, we select the band to report the luminosity to be
consistent with other studies, 0.1–2.4 keV. The flux measurement is performed
using the 0.5–2.0 keV band for both ROSAT and XMM-Newton data. In the
calculation of the luminosity, a K-correction using the redshift of the
source, temperature estimate using the same $L-T$ relation, and a band
difference between observed 0.5–2 keV (used for the flux measurements) and the
rest-frame 0.1-2.4 keV (used for $L_{\rm X}$) is made. There are some subtle
differences, as in RASS data we use the full flux, and in XMM-Newton, we
remove the contribution of point sources. Also, with XMM-Newton data, the flux
apertures capture more precisely the contribution of the group, thus avoiding
an extra source of scatter due to confusion. Thus, even with a smaller sample
size, XMM-Newton data on luminosities deliver improvements on the association
of X-ray emission with the galaxy group. Both RASS and XMM-Newton fluxes are
extrapolated to the estimated $R_{500}$ radius following the procedure
described in Finoguenov et al. (2007). RASS data requires no extrapolation,
while these aperture corrections to XMM-Newton fluxes are within 20%. However,
a difference in the spatial coverage of the source introduces an additional
source of scatter. The correspondence between the X-ray luminosity
measurements between XMM-Newton and RASS is illustrated in Fig. 7, Fig. 8, and
detailed in Table 3. The relation is close to the one-to-one relation, and the
main effect we see is a 40% scatter in the luminosity estimates, which we
attribute to be a characteristic of the quality of survey-type measurement of
RASS.

###  4.3 Velocity Dispersion – X-ray Luminosity Relation

Figure 11: Left panel: Velocity dispersion vs. temperature for the XMM-Newton
subsample used in this work in comparison with the literature. Gaussian and
non-Gaussian groups in our sample are represented by purple squares and red
crosses, respectively. The grey transparent band is 1$\sigma$ confidence
interval for the slope and intercept. Right panel: Comparison of the slope of
the $\sigma_{\rm v}-kT$ relation with the literature. The grey solid line is
the theoretical self-similar expectation.* refers to works using Bayesian
linear regression while $\dagger$ refers to works using ODR. The slope of
Wilson et al. (2016) is subject to a potential bias discussed in Farahi et al.
(2018). Figure 12: MCMC fitting results with one and two-dimensional
projections of the posteriors of the $\sigma_{\rm v}-kT$ scaling relation.
Other details are the same as Fig. 8.

Since no selection criteria were applied on $\sigma_{\rm v}$ in our XMM-Newton
subsample, we examine the $\sigma_{\rm v}-L_{\rm X}$ relation, instead of
$L_{\rm X}-\sigma_{\rm v}$, and compare it to the relation of the full
AXES-2MRS catalogue (for a detailed discussion on the choice, see Kelly,
2007). We are particularly interested in comparing the constrained intrinsic
scatters to assess the effect of sources of contamination (point sources,
nearby groups, etc.) on the $\sigma_{\rm v}-L_{\rm X}$ relation, due to XMM-
Newton’s higher sensitivity to those compared to ROSAT. We present our scaling
relation work and compare it to the literature results on the group scale in
Fig. 9, with the one and two-dimensional projections of the posteriors of the
parameters shown in Fig. 10. Our result for the full XMM-sample ($\sigma_{\rm
v}\propto$ $L_{\rm X}^{0.22\pm 0.11}$) is in agreement with the calibration of
the self-similar expectation based on the X-ray emissivity behaviour in the
0.1-2.4 keV energy band and 0.7-3.0 keV temperature range (see the right panel
in Fig. 9 for our result in comparison with the literature). Lovisari et al.
(2021) presents an argument about the truly expected slope of this relation
based on the behaviour of the X-ray emissivity in the low-temperature regime.
They claim that instead of $L_{\rm X}\propto\sigma_{\rm v}^{4}$, the scaling
should follow $L_{\rm X}\propto\sigma_{\rm v}^{3+2\gamma}$, where $\gamma$ is
a constant determined based on the temperature range, energy band, and
metallicity (see Table 1 in Lovisari et al., 2021). Indeed, the most deviating
results are obtained for galaxy groups at lower X-ray luminosities, compared
to our sample. We detail the parameters of the $\sigma_{\rm v}-L_{\rm X}$
relation in Table 3. We notice almost no change in the scatter of both XMM and
ROSAT relations, which we attribute to the level of intrinsic scatter being
much larger than the scatter between XMM-Newton and RASS luminosities.
Although there is a noticeable difference in the normalization of the scaling
relation between RASS and XMM samples, once we retain only the high-quality
measurements of velocity dispersion by limiting the sample to that of at least
8 spectroscopic members, the difference disappears. This means that our XMM-
Newton subsample is representative of the 8 or more member AXES groups, while
the full catalogue is somewhat different. Moreover, we investigate the effect
of the number of spectroscopic members on the relation, and we find that
adding poor galaxy groups (${N_{\rm gal}}<8$) does not affect the quality of
the fit in terms of the statistical errors and the intrinsic scatter (on the
1-$\sigma$ level). Up to our level of precision, there is no clear evidence of
substructure (NG) deviations in the $\sigma_{\rm v}-L_{\rm X}$ relation as
shown in Table 3. Additionally, we matched the AXES-2MRS sample and the XMM-
Newton subsample with The Third Cambridge Catalog (3C) and its revised version
(3CR) within 5 arcmin to test whether including galaxy groups with an active
radio AGN changes the scaling relations. We removed a total of 6 systems from
the full AXES-2MRS sample and 1 system from the XMM subsample, and we did not
find any noticeable change in the scaling relations, concluding that ongoing
AGN activity is not immediately seen in the group properties. In the
subsequent analysis, we no longer split the sample based on the radio
properties.

Figure 13: Left panel: Temperature vs. X-ray luminosity for the XMM-Newton
subsample used in this work (black solid line) in comparison with the
literature. BC and NC indicate bias-corrected and non-corrected relations,
respectively. TL and CEL are total and core-excised $L_{\rm X}$, respectively.
TT and CET are total and core-excised $kT$, respectively. The type of the
sample used and the regression method are also indicated (see section 4.5 for
more details). Grey and black points are high and low concentration groups,
respectively, defined at $c_{200}=2$. Shaded region is the 1-$\sigma$
uncertainty. Right panel: Comparison of the slope of the $kT-L_{\rm X}$
relation with the literature. Slope labels are the same as relation labels
from the left panel. Only band-limited $L_{0.1-2.4}$ relations are considered.
Figure 14: MCMC fitting results with one and two-dimensional projections of
the posteriors of the $kT-L_{\rm X}$ scaling relation. Other details are the
same as Fig. 8.

Tracking the efforts done studying the $\sigma_{\rm v}-L_{\rm X}$ relation, we
find that Sohn et al. (2019) used a sample of 74 groups from the COSMOS survey
(Finoguenov et al., 2007; Gozaliasl et al., 2019) and reported a scaling of
the form $\sigma_{\rm v}\propto L_{\rm X}^{0.21\pm 0.03}$. Their sample
suffered from anomalously low velocity dispersion values so they reduced their
sample to 7 velocity dispersion bins and fitted them to get the above result.
Notably, they state that the intrinsic scatter of their relation came high,
however, they do not report a specific value. Lopes et al. (2009b) used a
(group + cluster) sample of 97 systems from the NoSOCS-SDSS survey (Lopes et
al., 2009a) and reported a result of the form $\sigma_{\rm v}\propto L_{\rm
X}^{0.27\pm 0.03}$ with an intrinsic scatter ($S_{\mathrm{ln}Y|X}$) of
$0.29\pm 0.03$. Osmond & Ponman (2004) was one of the earliest efforts in
constraining this relation on the group scale. They used ROSAT data for a
group sample of 60 systems and reported a result of the form $\sigma_{\rm
v}\propto L_{\rm X}^{0.43\pm 0.11}$. Next, Vajgel et al. (2014) used a small
sample of 14 groups from the X-Boötes survey (Murray et al., 2005) and
reported a relation of the form $\sigma_{\rm v}\propto L_{\rm X}^{0.46\pm
0.01}$ and intrinsic log-normal scatter of 0.3. Lastly, Connelly et al.
(2012), using a sample of 38 high redshift X-ray groups with luminosities
around $10^{42}$ ergs s-1 measured using deep XMM-Newton observations, derived
a scaling of the form $\sigma_{\rm v}\propto L_{\rm X}^{0.56\pm 0.18}$, with
$L_{\rm X}$ estimates similar to those obtained in the COSMOS field, so the
difference is mainly in the covered luminosity range. We report the least
intrinsic scatter in the $\sigma_{\rm v}-L_{\rm X}$ relation in the literature
so far with a value of $0.15\pm 0.07$. We note that the original relations
reported in Sohn et al. (2019); Osmond & Ponman (2004); Vajgel et al. (2014);
Connelly et al. (2012) were in the form $L_{\rm X}-\sigma_{\rm v}$, while we
inverted them for the slope comparison.

###  4.4 Velocity Dispersion – X-ray Temperature Relation

We study the $\sigma_{\rm v}-kT$ for the full XMM sample of 24 groups we have
(the effect of adding the dynamically complex 2MRS ID: 5089 system is explored
later in this section). The importance of these two observables arises as they
are independent baryonic tracers for the depth of the group potential. In the
absence of non-gravitational heating, one expects the scaling to go as
$\sigma_{\rm v}\propto T^{0.5}$. We report a consistent relation of the form
$\sigma_{v}\propto T^{0.42\pm 0.24}$. Detailed parameter estimations for the
$\sigma_{\rm v}-kT$ relation are summarised in Table 3. The role of the
velocity substructure in enhancing the scatter is marginally supported by the
data with the Gaussian (G) groups and the full sample having 40% lower
intrinsic scatter than the non-Gaussian (NG) groups. Fig.12 shows the one and
two-dimensional projections of the posteriors of the relation.

We consider both group and group + cluster samples when comparing our fit to
the literature, with the results on the slope summarised in Fig. 11. Xue & Wu
(2000) studied a low-redshift sample of 274 groups and clusters with $kT<$
10.1 keV, and found a relation of the form $\sigma_{\rm v}\propto T^{0.64\pm
0.08}$ using the ODR method with no characterization of intrinsic scatter. Our
data is more uniform because all the X-ray observations used were obtained
from the same instrument, and we have more control over the velocity
dispersion values as they were all computed using the same (gapper) estimator.
In contrast, they assembled their X-ray and spectroscopic samples from 20
different sources. Using a total of 19 groups + clusters with $kT<$ 5.5 keV
taken from the XMM Cluster Survey (Mehrtens et al., 2012), Wilson et al.
(2016) reported a steep relation of the form $\sigma_{\rm v}\propto T^{0.86\pm
0.16}$. Lastly, Farahi et al. (2018) used a 138 group + cluster sample
selected from the XXL survey. X-ray temperatures ($kT<$ 6.5 keV) were taken
from the XXL analysis of Adami et al. (2018), and spectroscopic data for
galaxies were obtained from 7 different literature sources. They report a
relation in the form $\sigma_{\rm v}\propto T^{0.63\pm 0.05}$. Despite the
formal difference in the slope values, as illustrated in Fig.11, all the fits
are broadly consistent with our data points. Therefore, obtaining a different
slope of the scaling relation might be a result of a different sampling of the
data. Exploring larger datasets and understanding the covariance of the
scatter with the group properties affecting the selection is needed to
establish a full picture.

###  4.5 X-ray Temperature – Luminosity Relation

The soft-band X-ray luminosity and temperature are two X-ray observables
obtained with minimal covariance. In our analysis, X-ray temperatures were
obtained from spectral modelling of the X-ray observations. In contrast, X-ray
luminosities were derived from the flux measurements in the 0.5–2 keV energy
band, which, at the redshifts of this study, are primarily sensitive to gas
density and metallicity. Accordingly, the $kT-L_{\rm X}$ relation is one of
the most studied relations in the literature. The gravitational self-similar
expectation is $L_{\rm bol}\propto T^{2}$, where $L_{\rm bol}$ is the
bolometric luminosity (e.g., Böhringer et al. (2012), and references therein).
However, the expected behaviour of the band-limited [0.1-2.4 keV] relation is
$L_{0.1-2.4}\propto T^{1.5+\gamma}$ with $\gamma=-0.04$ for a typical metal
abundance of $Z=0.3\,Z_{\odot}$ and a temperature range of 0.4–3.0
keV(Lovisari et al., 2021). Thus, the slope of the $kT-L_{\rm X}$ relation is
expected to be $\sim$ 0.67. We report a relation of the form $kT\propto L_{\rm
X}^{0.3{\pm}0.08}$ for the XMM-Newton subsample representing AXES-2MRS. We
show our relation, together with that of Kettula et al. (2015); Lovisari et
al. (2015); Markevitch (1998) in Fig. 13. To ensure an in-depth comparison, we
list several properties of the literature relations in the left panel of the
figure. In particular, we specify whether the relation is based on a group,
cluster, or group + cluster sample. We also indicate whether it employs a
selection bias correction (BC) or not (NC), and specify whether the central
regions were excluded in estimating $L_{\rm X}$ (core excised, CEL) or the
total $L_{\rm X}$ was used (TL). Furthermore, total (TT) vs. core-excised
(CET) temperature is clarified together with the regression method. These
acronyms are summarised in Table. 1. To correct for the sampling bias of our
XMM-Newton subsample, we provide a detection vector to linmix to take into
account the distribution of source over the luminosity, where we set
temperatures for the systems without XMM-Newton temperature measurements to
its upper limit of 20 keV. We do not find differences in the results obtained
without a detection vector, which illustrates that the sampling bias is
negligible.

Table 1: Acronyms used in the $kT-L_{\rm{X}}$ relation Acronym | Definition
---|---
BC | Selection-bias corrected
NC | Selection-bias non-corrected
CEL | Core-excised X-ray luminosity
TL | Total X-ray luminosity
CET | Core-excised X-ray temperature
TT | Total X-ray temperature

The literature scaling relations shown in Fig. 13 are detailed as follows:
firstly, Lovisari et al. (2015) compiled a group sample by applying a flux
limit of $F_{0.1-2.4}=5\times 10^{-12}$ erg s-1cm-2 and two redshift cuts
($0.01<z<0.035$) to the northern ROSAT all-sky galaxy cluster survey
(Böhringer et al., 2000, NORAS) and the ROSAT-ESO flux-limited X-ray galaxy
cluster survey (Böhringer et al., 2004, REFLEX) catalogues. The resulting
23-group sample span a similar temperature ($0.85<kT<2.8$ keV) and band-
limited luminosity range ($0.4<L_{\rm X}<5.3\times 10^{43}$ erg s-1) as our
XMM-subsample, and uses total luminosities and core-excised temperatures.
Moreover, multiple regression methods were used, and in Fig. 13 we show the
relation using two of them (ODR and BCES Y—X) as well as the BC and NC
relations. On the other hand, Kettula et al. (2015) produced the $L_{\rm
X}-kT$ relation based on a compiled sample of 12 low-mass clusters from the
XMM-CFHTLS survey (Mirkazemi et al., 2015), 10 low-mass systems from the
COSMOS field (Kettula et al., 2013), and 48 high-mass clusters from the
Canadian Cluster Comparison Project (Hoekstra et al., 2015; Mahdavi et al.,
2013, CCCP). The combined sample has a temperature and luminosity range of
$1-12$ keV, and $10^{43}-10^{45}$ erg s-1, respectively. In Fig. 13, we show
their relation with the combined group + cluster sample and that with only the
low-mass COSMOS systems since the latter shows greater similarity in the
parameter ranges with our study. Nevertheless, we note that they exclusively
used CEL and CET. As a high-mass reference, Markevitch (1998) used a low-
redshift ($0.04<z<0.09$) 35-cluster sample selected from RASS that excludes
systems with $kT\leq 3.5$ keV and has a mean TL of $\sim 3.86\times 10^{44}$
erg s-1. We show their relation using CEL and CET as well as that using TL and
TT. In the right panel of Fig. 13, we separately compare the slope of the
relation with the mentioned works.

Fig. 13 indicates that all the previous studies of the $T-L_{\rm X}$ relation
in groups are consistent with our data and differences in the slope can be
attributed to the selection effects or the fitting method used. All these
slopes are shallower than the cluster slope of Markevitch (1998), indicating
that the groups are hotter than expected from the extrapolation of the cluster
relation, which can be attributed to the effect of feedback, to higher
concentration, or an earlier formation epoch. The effect of feedback on the
scaling relations is largely removed when using total mass measurements,
instead of temperature, while the effect of the difference in the
concentration would remain in those studies. This motivates us to consider the
scaling relations using the total mass and to measure the concentration for
our sample.

Table 2: Fitting formulas of the scaling relations.

Relation | Fitting formula
---|---
$L_{\rm XMM}-L_{\rm RASS}$ |  $\ln(L_{\rm XMM}/10^{43}$ erg s${}^{-1})=A+B\ln(L_{\rm RASS}/10^{43}$ erg s${}^{-1})+S_{\mathrm{ln}Y|X}$
$\sigma_{\rm v}-L_{\rm X}$ |  $\ln(\sigma_{\rm v}$ km s${}^{-1})=A+B\ln(L_{\rm X}/10^{43}$ erg s${}^{-1})+S_{\mathrm{ln}Y|X}$
$\sigma_{\rm v}-kT$ |  $\ln(\sigma_{\rm v}$ km s${}^{-1})=A+B\ln(kT/1.5$ keV) + $S_{\mathrm{ln}Y|X}$
$kT-L_{\rm X}$ |  $\ln(kT$ keV${}^{-1})=A+B\ln(L_{\rm X}/10^{43}$ erg s${}^{-1})+S_{\mathrm{ln}Y|X}$
$\sigma_{\rm v}-M_{10000}$ |  $\ln(\sigma_{\rm v}$ km s${}^{-1})=A+B\ln(M_{10000}/10^{13}M_{\odot})+S_{\mathrm{ln}Y|X}$
$c_{200}-L_{\rm X}$ |  $\ln(c_{200})$ = $A$ \+ $B$ $\ln(L_{\rm X}/8\times 10^{42}$††footnotemark: † erg s-1) + $S_{\mathrm{ln}Y|X}$
444$A$, $B$, and $S_{\mathrm{ln}Y|X}$ are the normalization, slope, and
intrinsic scatter of the scaling relations, respectively (see Section 4 for
more details). $$\dagger$$$$\dagger$$footnotetext: $8\times 10^{42}$ erg s-1
is the median LX of the concentration sample and is found to decrease the
slope uncertainty by a factor of 2.

Table 3: Scaling relations.

Relation | Sample | linmix | ODR
---|---|---|---
intercept | slope | scatter | intercept | slope
|  | $A$ | $B$ | $S_{\mathrm{ln}Y|X}$ | $A$ | $B$
$L_{\rm XMM}-L_{\rm RASS}$ | Full XMM (23) | -0.18 $\pm$ 0.1 | 0.83 $\pm$ 0.19 | 0.43 $\pm$ 0.08 | -0.31 $\pm$ 0.12 | 1.56 $\pm$ 0.31
$\sigma_{\rm v}-L_{\rm X}$ | AXES-2MRS ($N_{\rm gal}>4$) (252) | 5.99 $\pm$ 0.02 | 0.19 $\pm$ 0.01 | 0.14 $\pm$ 0.03 | 6.01 $\pm$ 0.02 | 0.19 $\pm$ 0.01
| AXES-2MRS ($N_{\rm gal}>7$) (134) | 6.03 $\pm$ 0.03 | 0.17 $\pm$ 0.02 | 0.17 $\pm$ 0.04 | 6.1 $\pm$ 0.03 | 0.17 $\pm$ 0.01
| AXES-2MRS ($N_{\rm gal}>$ 7, G) (102) | 6.07 $\pm$ 0.03 | 0.17 $\pm$ 0.02 | 0.16 $\pm$ 0.04 | 6.1 $\pm$ 0.03 | 0.17 $\pm$ 0.01
| AXES-2MRS ($N_{\rm gal}>$ 7, NG) (32) | 6.06 $\pm$ 0.06 | 0.17 $\pm$ 0.05 | 0.23 $\pm$ 0.08 | 6.09 $\pm$ 0.05 | 0.16 $\pm$ 0.04
| Full XMM (23) | 6.17 $\pm$ 0.07 | 0.21 $\pm$ 0.11 | 0.15 $\pm$ 0.07 | 6.19 $\pm$ 0.06 | 0.22 $\pm$ 0.09
| XMM (G) (14) | 6.14 $\pm$ 0.1 | 0.23 $\pm$ 0.18 | 0.14 $\pm$ 0.09 | 6.14 $\pm$ 0.05 | 0.23 $\pm$ 0.09
| XMM (NG) | 6.17 $\pm$ 0.2 | 0.17 $\pm$ 0.27 | 0.41 $\pm$ 0.24 | 6.25 $\pm$ 0.12 | 0.23 $\pm$ 0.16
$\sigma_{\rm v}-kT$ | Full XMM (24) | 6.11 $\pm$ 0.07 | 0.43 $\pm$ 0.24 | 0.15 $\pm$ 0.07 | 6.12 $\pm$ 0.05 | 0.44 $\pm$ 0.18
| XMM (G) (15) | 6.09 $\pm$ 0.08 | 0.29 $\pm$ 0.41 | 0.16 $\pm$ 0.09 | 6.1 $\pm$ 0.05 | 0.28 $\pm$ 0.2
| XMM (NG) (9) | 6.1 $\pm$ 0.18 | 0.48 $\pm$ 0.54 | 0.39 $\pm$ 0.23 | 6.16 $\pm$ 0.11 | 0.52 $\pm$ 0.31
$kT-L_{\rm X}$ | Full XMM (23) | 0.51 $\pm$ 0.05 | 0.3 $\pm$ 0.08 | 0.21 $\pm$ 0.04 | 0.4 $\pm$ 0.04 | 0.36 $\pm$ 0.07
| XMM (G) (14) | 0.49 $\pm$ 0.07 | 0.22 $\pm$ 0.12 | 0.2 $\pm$ 0.06 | 0.42 $\pm$ 0.04 | 0.16 $\pm$ 0.08
| XMM (NG) (9) | 0.53 $\pm$ 0.14 | 0.35 $\pm$ 0.19 | 0.35 $\pm$ 0.15 | 0.43 $\pm$ 0.07 | 0.47 $\pm$ 0.12
$\sigma_{\rm v}-M_{10000}$ | Full mass sample (30) | 6.1 $\pm$ 0.06 | 0.13 $\pm$ 0.05 | 0.12 $\pm$ 0.06 | 6.15 $\pm$ 0.04 | 0.12 $\pm$ 0.04
| XMM (G) (15) | 6.13 $\pm$ 0.09 | 0.22 $\pm$ 0.18 | 0.14 $\pm$ 0.08 | 6.13 $\pm$ 0.05 | 0.13 $\pm$ 0.06
| XMM (NG) (9) | 6.23 $\pm$ 0.18 | 0.31 $\pm$ 0.26 | 0.34 $\pm$ 0.22 | 6.31 $\pm$ 0.11 | 0.34 $\pm$ 0.13
$c_{200}-L_{\rm X}$ | Full $c_{200}$ sample (24) | –0.12 $\pm$ 0.96 | 0.15 $\pm$ 5.37 | 0.76 $\pm$ 0.31 | 9.51 $\pm$ 17.75 | –1.51 $\pm$ 3.55

555Column 1 gives the scaling relation (the exact fitting formulas are shown
in Table. 2). Column 2 provides the sample and the number of systems used in
the relation (in parenthesis). Columns 3 to 7 show the parameters of the
scaling relations using the Bayesian regression package linmix, and the
orthogonal distance regression (ODR), respectively. See Section 4 and Table 2
for details.

###  4.6 Velocity Dispersion – Mass Relation

Figure 15:  Velocity dispersion vs. mass within a sphere of overdensity
$10^{4}$ relative to the critical density of the Universe. Blue dashed and red
dotted lines represent the scaling relation of our full mass sample (see
Section 4.6 and the Gaussian subsample, respectively. Black and green points
are our XMM-Newton subsample and AXES-2MRS published groups, respectively.
Figure 16: MCMC fitting results with one and two-dimensional projections of
the posteriors of the $\sigma_{\rm v}-M_{10000}$ scaling relation. Other
details are the same as Fig. 8.

As most of our mass measurements correspond to a high overdensity, finding a
correspondence to our measurement of velocity dispersion, which traces virial
mass, is sensitive to halo concentration. In this section, we consider the
scatter in the estimate of the central mass this effect introduces and include
an effect of complexity in the velocity dispersion profile into consideration.
The obtained relation for our mass sample is shown in Fig.15 while the details
of the fitting parameters and the one and two-dimensional posteriors are
presented in Table 3 and Fig. 16, respectively. While we have concentrated on
the XMM-Newton analysis on galaxy group scales, an analysis of AXES-2MRS
clusters can be readily found in the literature (e.g. Schellenberger &
Reiprich, 2017). To improve on the mass coverage, we included previously
published results from the HIFLUGCS cluster sample (Schellenberger & Reiprich,
2017), in overlap with our velocity dispersion measurements. We report a
nearly flat relation for the full mass sample of $\sigma_{\rm v}\propto
M_{10000}^{0.13\pm 0.05}$. The deviations of the slope from the $M^{1/3}$
expectation can be explained either if the contribution of the non-thermal
pressure to the mass estimate at the overdensities of 10000 is high in galaxy
groups, or if our sample is dominated by the low concentration systems, which
we explore below. Splitting the systems, based on the Gaussianity of velocity
dispersion, does not lead to any detectable changes in the parameters of this
scaling relation.

###  4.7 Concentration – X-ray Luminosity Relation

Figure 17: Halo concentration vs. X-ray luminosity for the $c_{200}$ sample
used in this work. Figure 18: MCMC fitting results with one and two-
dimensional projections of the posteriors of the $c_{200}-L_{\rm X}$ scaling
relation. Other details are the same as Fig. 8.

Owing to the hypersensitivity of the concentration calculation to variations
in the velocity dispersion, coupled with the presence of relatively large
errors in our velocity dispersion measurements, approximately on the order of
20-30%, the determination of the concentration parameter poses a significant
challenge. As a result, we were only able to confidently constrain the
concentration of 22 systems within our XMM-Newton subsample (see Table 6).
Additionally, due to the intricate nature of this analysis, we could extend
our concentration determinations to only two systems from previously published
datasets (Group IDs: 6041 and 6093). The result of the $c_{200}-L_{\rm X}$
scaling relation is shown in Figs. 17 and 18 and the details of the parameters
are shown in Table 3. We used $K=4$ Gaussian components in fitting this
relation with linmix instead of 3 (see Section 4) as it was found to increase
the significance level of the parameters by a factor of 2. Remarkably, we
measured a low halo concentration across the majority of systems within our
XMM subsample, with values below 1. In performing the linear regression
analysis, as an input on concentration we have described the probability
distribution of concentrations in the expected range between 1 and 10 using a
log-normal distribution, as most points have asymmetric profiles of the
probability distribution function, and of relevance is the correct estimate of
this probability in the range of expected solution. Our result on the slope of
the scaling of $-0.12\pm 0.96$ broadly agrees with the expectations based on
the numerical modelling of the $c_{\rm 200}-M$ relation on the scales of
groups and clusters of galaxies.

We report a logarithmic intrinsic scatter on the concentration of $S(\ln
c_{200})=0.76\pm 0.31$. Within errors, our mean value is slightly higher than
the results on the $c_{\rm 200}-M$ relation from cosmological simulations of
$S(\ln c_{200})\approx 0.33$ (Wechsler et al., 2002; Neto et al., 2007; Macciò
et al., 2007, 2008; Bhattacharya et al., 2013; Child et al., 2018).
Observational efforts on the $S(\ln c_{200})$ parameter, despite being limited
and mostly inferred from the $c_{\rm 200}-M$ relation, show a fair agreement
with the simulations (e.g., Wojtak & Łokas, 2010; Amodeo et al., 2016; Umetsu
et al., 2020). In particular, Wojtak & Łokas (2010) assembled a low-redshift
($z<0.1$) sample of 41 relaxed and rich ($N_{\rm gal}\geq 70$) clusters from
the NASA/IPAC Extragalactic Database (NED) and The WIde field Nearby Galaxy
cluster Survey (Cava et al., 2009, WINGS). The median mass of the sample is
$5\times 10^{14}M_{\odot}$ and the median concentration is 7. They report an
intrinsic scatter of $S(\ln c_{200})=0.35$. On the other hand, Amodeo et al.
(2016) used a high-redshift ($z\geq 0.4$) sample of 47 clusters collected from
the _Chandra_ public archive and represents the high-mass end of the cluster
population with a median mass of $1.3\times 10^{15}M_{\odot}$. They report a
mean log-normal scatter of $S(\ln c_{200})=0.32$. Finally, Umetsu et al.
(2020) reported a low upper limit on $S(\ln c_{200})$ of 0.24 using the X-ray
selected XXL cluster sample (Adami et al., 2018). They claim that this low
scatter can be attributed to a selection bias related to the dynamical state
of the clusters or a result of overestimation of the $c_{200}$ measurement
errors.

It is important to note that Amodeo et al. (2016) has shown that the intrinsic
scatter decreases with increasing mass. In fact, the weak but systematic anti-
correlation of mass and intrinsic scatter is also supported by the simulation
work of Neto et al. (2007). Given the low masses of our XMM-Newton subsample,
this effect could potentially explain the relatively high mean value of the
scatter. Additionally, a larger sample with higher luminosity and
concentration ranges can improve the significance of the scatter measurement.

##  5 Conclusions and Summary

In this work, we presented a new catalogue of AXES-2MRS galaxy groups, which
has a selection based on the baryonic content at $M_{500}$ and we examined its
properties. We have significantly enhanced the representation of the under-
explored low-redshift low-luminosity galaxy groups. In addition, our sample is
prevalent in low-mass systems ($<$ 10${}^{14}M_{\odot}$, see Table 6), which
enhances the completeness of the galaxy group catalogues, potentially
addressing the longstanding problem of missing faint low-mass systems.

The main parameter in common between various subsamples studied here is
velocity dispersion. We find the main sample to exhibit a comparable scaling
relation between the X-ray luminosity and the velocity dispersion, and in
particular, to exhibit a similar scatter. The value of the scatter is high,
which is in agreement with the conclusion of Damsted et al. (2023) that the
scatter at $z<0.15$ is very large. We can see that using our measurements of
intra-group medium temperatures does not resolve this problem, and the scatter
is still large. Our use of masses, as opposed to temperature, reduces the
scatter, which indicates that feedback effects contribute significantly to the
scatter. Our reported $\sigma_{\rm v}-kT$ relation is also marginally flatter
than the self-similar expectation, which also points to either the importance
of non-gravitational heating or the effect of halo concentration. Our analysis
of the $c_{200}-L_{\rm X}$ relation reveals a large intrinsic scatter that we
deem representative of galaxy groups. Thus, we conclude that both feedback and
halo concentration are at the root of the large scatter of properties of X-ray
groups. We believe that a combination of large scatter and group selection can
explain differences in the mean scaling relations for galaxy groups, as all
published relations pass through some of the points presented in this study.
The main question that remains open is what is the right balance of groups to
be included in the samples of X-ray properties.

We note a similarity in the slope and normalization of the scaling relation
between the velocity dispersion and X-ray luminosity between AXES-2MRS galaxy
group sample and the distant COSMOS sample, which is obtained using a similar
X-ray detection technique. Besides, the XMM-Newton-observed subsample is
comparable to the full AXES-2MRS catalogue in this relation and can be safely
held as representative of the full AXES-2MRS sample.

Using linmix with intrinsic scatter, the fitting parameters come out
consistent with our results using orthogonal distance regression. The reported
uncertainties of linmix are however larger by a factor of 1.5-2. This might
help resolve previously found tensions in the fit parameters obtained without
the intrinsic scatter. As we have demonstrated the scatter of the scaling
relations is a meaningful parameter that allows one to access the physics of
galaxy groups.

As discussed in the paper, the differences in the literature results on the
scaling relations might be associated with the selection of the sample. Our
study has employed the largest angular scales ever considered in X-ray source
identification. Our results on the scaling relations are within the difference
between the literature results, which limits the scope of contribution from
the ”known unknowns”. The number of AXES sources combined with the spatial
scales employed in searching for the X-ray emission approaches the source
confusion limit. While further detailed studies of AXES sources will be
beneficial, deeper surveys will have to use smaller spatial scales in
searching for X-ray emission, leading to different selection effects. With the
advances in hydrodynamical simulations, modelling of the X-ray emission from
the outskirts of groups and clusters of galaxies becomes reliable, which makes
AXES the most suitable sample for comparison to simulations.

###### Acknowledgements.

ET acknowledges the Estonian Research Council grant PRG1006 and the Centre of
Excellence ‘Foundations of the Universe’ (TK202) funded by the Ministry of
Education and Research.

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##  Appendix A Details of the XMM-Newton Observations

Table 4 lists the optical parameters of our XMM-Newton subsample. The
positions (RA and Dec) correspond to the X-ray peak emission centres taken as
a centroid of a wavelet reconstruction of the 0.5–2 keV image on scales 0.5-4
arcmin. Also shown are the median group redshifts, the optical line-of-sight
velocity dispersions determined using the gapper method, and the number of
member galaxies.

Table 4: XMM-Newton subsample of AXES-2MRS groups.

Group ID | RA | Dec | $z_{\rm{med}}$ |  $\sigma$v | $N_{\rm{gal}}$
---|---|---|---|---|---
2MRS | (J2000) | (J2000) |  | (km s-1) |
361 | 16.853 | 32.399 | 0.016 | 370 $\pm$ 83 | 18
505 | 21.445 | –1.395 | 0.0171 | 438 $\pm$ 76 | 29
827 | 35.780 | 42.986 | 0.0197 | 510 $\pm$ 117 | 17
859 | 36.401 | 36.961 | 0.0353 | 480 $\pm$ 167 | 8
1571 | 61.642 | 30.379 | 0.0179 | 370 $\pm$ 103 | 12
1830 | 74.732 | –0.484 | 0.0144 | 320 $\pm$ 85 | 13
2009 | 86.370 | –25.936 | 0.0388 | 925 $\pm$ 201 | 19
2161 | 96.162 | –37.337 | 0.0329 | 569 $\pm$ 151 | 13
2533 | 315.436 | –13.311 | 0.0278 | 409 $\pm$ 125 | 10
2541 | 117.844 | 50.202 | 0.0229 | 521 $\pm$ 170 | 9
2657 | 125.154 | 21.072 | 0.017 | 373 $\pm$ 104 | 12
3551 | 164.543 | 1.612 | 0.0405 | 339 $\pm$ 110 | 9
3718 | 170.612 | 24.296 | 0.027 | 478 $\pm$ 166 | 8
4050 | 182.018 | 25.239 | 0.023 | 319 $\pm$ 98 | 10
4808 | 202.351 | 11.765 | 0.0239 | 347 $\pm$ 101 | 11
5089 | 210.908 | –33.983 | 0.0139 | 238 $\pm$ 66 | 12
5841 | 244.338 | 34.903 | 0.0303 | 313 $\pm$ 96 | 10
5914 | 247.417 | 40.826 | 0.0318 | 591 $\pm$ 111 | 25
6015 | 254.498 | 27.858 | 0.0345 | 309 $\pm$ 108 | 8
6116 | 260.202 | –1.039 | 0.0286 | 538 $\pm$ 137 | 14
6407 | 281.827 | –63.332 | 0.015 | 471 $\pm$ 83 | 28
6666 | 304.458 | –70.819 | 0.0131 | 420 $\pm$ 137 | 9
6916 | 316.840 | –25.459 | 0.0359 | 577 $\pm$ 201 | 8
7427 | 348.942 | –2.389 | 0.0234 | 473 $\pm$ 145 | 19
7727 | 181.04 | 20.293 | 0.0248 | 445 $\pm$ 94 | 20
666 RA and Dec are coordinates of the peak X-ray emission centres.

In Table 5, we provide a detailed summary of the observations for our XMM
subsample. The table includes Group ID (XMM-Newton group identifier), OBS-ID
(XMM-Newton observation identifier), DATE-OBS (date of the observation),
Clean-EXP (clean exposure time), kT (X-ray gas temperature), and $L_{\rm X}$
(0.1-2.4 keV X-ray luminosity). It also includes aspec, bspec, and $\theta$
which are the semi-major axis, semi-minor axis, and position angle of the
elliptical extraction regions used in the spectral analysis, respectively (see
the green regions in Fig. 19).

Table 6 complements Table 5 with more X-ray properties obtained from the XMM-
Newton observations. In particular, it contains the details of the surface
brightness profile fitting using a single $\beta$-model and mass estimates.
The columns are $\beta$ (slope of the surface brightness profile), RkT (the
outer radius of the initial mass estimate), $\Delta_{kT}$ (the overdensity of
the measurement at RkT), M${}_{\Delta_{R_{kT}}}$ (the mass estimate at the
initial overdensity $\Delta_{kT}$), RC (the core radius of the $\beta$ model),
DS (the distance scale), M10000 (mass estimate at the overdensity covered by
the data), and c200 (halo concentration).

Table 5: XMM observations and basic properties of AXES-2MRS.

Group ID | OBS-ID | DATE-OBS | Clean-EXP | $k\,T$ | $L_{\rm X}$ | $a_{\rm spec}$ | $b_{\rm spec}$ | $\theta$
---|---|---|---|---|---|---|---|---
2MRS | XMM-Newton | (UTC) | (ks) | (keV) | (1043 erg s-1) | (arcmin) | (arcmin) | (deg)
361MM$M$Groups showing merging behaviour. | 0551720101 | 2008-07-01 | 24.1 | 1.898 $\pm$ 0.119 | 0.716 $\pm$ 0.004 | 4.5 | 3.5 | 82.9
505OSOS$OS$Over-split groups. Refer to section 2.3 for more details. | 0743700201 | 2015-01-09 | 69.4 | 1.417 $\pm$ 0.077 | 0.344 $\pm$ 0.004 | 2.7 | 1.4 | 33.9
827 | 0002970201 | 2002-02-05 | 13.5 | 2.159 $\pm$ 0.121 | 1.027 $\pm$ 0.009 | 4.7 | 3.9 | 277.7
859 | 0863880401 | 2020-07-21 | 15.1 | 1.708 $\pm$ 0.110 | 0.769 $\pm$ 0.02 | 2.8 | 2.2 | 278.4
1571 | 0883620101 | 2021-09-13 | 7.2 | 1.490 $\pm$ 0.251 | 0.355 $\pm$ 0.011 | 2.9 | 1.9 | 7.9
1830 | 0673180301 | 2012-02-24 | 3.1 | 1.170 $\pm$ 0.005 | 0.573 $\pm$ 0.01 | 3.2 | 2.6 | 309.7
2009 | 0302030101 | 2006-02-17 | 27.2 | 3.132 $\pm$ 0.156 | 1.956 $\pm$ 0.014 | 4.8 | 3.6 | 341.7
2161 | 0800761301 | 2017-10-11 | 14.1 | 1.651 $\pm$ 0.045 | 1.284 $\pm$ 0.013 | 3.0 | 2.5 | 344.4
2533 | 0864052501 | 2021-04-23 | 7.2 | 1.455 $\pm$ 0.046 | 1.241 $\pm$ 0.016 | 4.9 | 3.6 | 272.4
2541OSOS$OS$Over-split groups. Refer to section 2.3 for more details. | 0800761001 | 2018-04-19 | 8.4 | 1.477 $\pm$ 0.039 | 1.115 $\pm$ 0.018 | 2.9 | 2.1 | 29.5
2657 | 0108860501 | 2001-10-15 | 15.8 | 1.516 $\pm$ 0.054 | 0.343 $\pm$ 0.005 | 3.3 | 3.0 | 4.3
3551 | 0601930101 | 2009-05-26 | 18.1 | 2.090 $\pm$ 0.115 | 1.218 $\pm$ 0.01 | 4.6 | 3.1 | 334.2
3718 | 0112270301 | 2001-12-02 | 6.5 | 1.563 $\pm$ 0.057 | 1.156 $\pm$ 0.013 | 3.7 | 3.1 | 57.1
4050 | 0151400201 | 2003-05-26 | 8.2 | 1.361 $\pm$ 0.179 | 0.373 $\pm$ 0.008 | 3.3 | 2.1 | 291.6
4808 | 0041180801 | 2001-12-30 | 13.7 | 1.255 $\pm$ 0.073 | 0.56 $\pm$ 0.008 | 3.5 | 1.6 | 283.7
5089MM$M$Groups showing merging behaviour. | 0741930101 | 2014-07-25 | 90.5 | 2.271 $\pm$ 0.060 | 0.86 $\pm$ 0.004 | 4.8 | 3.4 | 313.0
5841 | 0800761701 | 2018-01-16 | 7.1 | 1.687 $\pm$ 0.179 | 0.742 $\pm$ 0.016 | 2.7 | 1.8 | 282.0
5914 | 0203710201 | 2004-09-07 | 3.5 | 1.175 $\pm$ 0.037 | 0.856 $\pm$ 0.02 | 3.0 | 2.9 | 0.0
6015 | 0654800201 | 2010-08-26 | 38.4 | 1.877 $\pm$ 0.097 | 2.14 $\pm$ 0.014 | 4.7 | 3.8 | 291.7
6116 | 0400930101 | 2006-08-25 | 23.1 | 2.265 $\pm$ 0.210 | – | 6.8 | 5.2 | 321.3
6407MM$M$Groups showing merging behaviour. | 0405550401 | 2006-09-07 | 17.1 | 1.168 $\pm$ 0.041 | 0.513 $\pm$ 0.008 | 2.8 | 1.9 | 23.9
6666OSOS$OS$Over-split groups. Refer to section 2.3 for more details. | 0022340101 | 2002-03-31 | 8.8 | 1.094 $\pm$ 0.051 | 0.475 $\pm$ 0.009 | 2.8 | 1.7 | 40.4
6916 | 0741581601 | 2014-10-22 | 5.1 | 2.213 $\pm$ 0.308 | 1.719 $\pm$ 0.025 | 4.7 | 5.2 | 77.0
7427 | 0501110101 | 2007-11-22 | 24.5 | 1.635 $\pm$ 0.047 | 1.631 $\pm$ 0.009 | 4.6 | 3.7 | 41.0
7727OSOS$OS$Over-split groups. Refer to section 2.3 for more details. | 0112270601 | 2003-01-02 | 2.43 | 1.055 $\pm$ 0.119 | 0.385 $\pm$ 0.013 | 2.6 | 2.3 | 275.9

777

Table 6: Surface brightness profiles, mass estimates, and halo concentrations
of our XMM subsample.

Group ID | $\beta$ | $R_{kT}$ |  $R_{C}$ | DS | $M_{\Delta_{R_{kT}}}$ | $\Delta_{R_{kT}}$ | $M_{10000}$ | $c_{200}$
---|---|---|---|---|---|---|---|---
2MRS |  | (kpc) |  (′′) | (kpc/′′) | (1013 M⊙) |  | (1013 M⊙) |
361 | 0.337 $\pm$ 0.042 | 83.7 | 0.001 | 0.346 | 0.65 $\pm$ 0.12 | 19161 | 0.72 $\pm$ 0.19 | 1.38 $\pm$ 0.99
505 | 0.363 $\pm$ 0.01 | 47.2 | 0.0391 | 0.366 | 0.3 $\pm$ 0.07 | 48373 | 0.37 $\pm$ 0.12 | 0.395 $\pm$ 0.205
827 | 0.337 $\pm$ 0.008 | 130.1 | 0.001 | 0.502 | 1.16 $\pm$ 0.09 | 8988 | 1.14 $\pm$ 0.13 | 0.67 $\pm$ 0.42
859 | 0.354 $\pm$ 0.008 | 107.9 | 0.001 | 0.714 | 0.8 $\pm$ 0.08 | 10710 | 0.8 $\pm$ 0.12 | 0.925 $\pm$ 0.745
1571 | 0.313 $\pm$ 0.017 | 54.7 | 0.00002 | 0.372 | 0.31 $\pm$ 0.1 | 32663 | 0.37 $\pm$ 0.17 | 0.83 $\pm$ 0.63
1830 | 0.363 $\pm$ 0.001 | 51.8 | 0.039 | 0.296 | 0.27 $\pm$ 0.05 | 33594 | 0.32 $\pm$ 0.08 | 0.98 $\pm$ 0.7
2009 | 0.374 $\pm$ 0.099 | 202.4 | 0.003 | 0.795 | 2.89 $\pm$ 0.37 | 5866 | 2.66 $\pm$ 0.47 | 0.265 $\pm$ 0.165
2161 | 0.390 $\pm$ 0.102 | 107.5 | 0.003 | 0.649 | 0.85 $\pm$ 0.2 | 11512 | 0.86 $\pm$ 0.29 | 0.46 $\pm$ 0.33
2533 | 0.334 $\pm$ 0.019 | 149.1 | 0.001 | 0.578 | 0.89 $\pm$ 0.06 | 4537 | 0.78 $\pm$ 0.08 | 1.185 $\pm$ 0.965
2541 | 0.484 $\pm$ 0.009 | 69.1 | 0.019 | 0.455 | 0.6 $\pm$ 0.08 | 31206 | 0.71 $\pm$ 0.13 | 0.625 $\pm$ 0.455
2657 | 0.371 $\pm$ 0.006 | 62.1 | 0.003 | 0.328 | 0.43 $\pm$ 0.07 | 30624 | 0.5 $\pm$ 0.12 | 1.03 $\pm$ 0.75
3551 | 0.301 $\pm$ 0.014 | 183.1 | 0.0002 | 0.778 | 1.41 $\pm$ 0.09 | 3867 | 1.21 $\pm$ 0.11 | 50.05 $\pm$ 49.95
3718 | 0.334 $\pm$ 0.01 | 106.7 | 0.016 | 0.521 | 0.68 $\pm$ 0.06 | 9538 | 0.68 $\pm$ 0.08 | 0.72 $\pm$ 0.57
4050 | 0.331 $\pm$ 0.011 | 75.5 | 0.001 | 0.455 | 0.42 $\pm$ 0.08 | 16473 | 0.45 $\pm$ 0.12 | 2.055 $\pm$ 1.755
4808 | 0.435 $\pm$ 0.137 | 76.4 | 0.01 | 0.468 | 0.51 $\pm$ 0.2 | 19484 | 0.56 $\pm$ 0.31 | 2.475 $\pm$ 2.225
5089* | 0.268 $\pm$ 0.004 | 68.4 | 0.001 | 0.274 | 0.51 $\pm$ 0.07 | 27368 | 0.58 $\pm$ 0.14 | –
5841 | 0.386 $\pm$ 0.01 | 84.5 | 0.001 | 0.614 | 0.67 $\pm$ 0.11 | 18874 | 0.74 $\pm$ 0.17 | 12.91 $\pm$ 12.39
5914 | 0.457 $\pm$ 0.169 | 112.2 | 0.264 | 0.634 | 0.74 $\pm$ 0.22 | 8824 | 0.72 $\pm$ 0.31 | 0.275 $\pm$ 0.175
6015 | 0.307 $\pm$ 0.001 | 175.7 | 0.006 | 0.685 | 1.24 $\pm$ 0.07 | 3872 | 1.07 $\pm$ 0.08 | –
6116† | – | – | – | – | – | – | – | –
6407 | 0.319 $\pm$ 0.044 | 43.9 | 0.0004 | 0.306 | 0.2 $\pm$ 0.07 | 40579 | 0.24 $\pm$ 0.12 | 0.24 $\pm$ 0.14
6666 | 0.527 $\pm$ 0.006 | 42.7 | 0.02 | 0.307 | 0.3 $\pm$ 0.07 | 66548 | 0.38 $\pm$ 0.17 | 0.715 $\pm$ 0.535
6916 | 0.346 $\pm$ 0.031 | 222.4 | 0.001 | 0.748 | 2.08 $\pm$ 0.16 | 3187 | 1.73 $\pm$ 0.19 | 0.67 $\pm$ 1.0
7427 | 0.418 $\pm$ 0.066 | 127.2 | 0.007 | 0.508 | 1.06 $\pm$ 0.14 | 8774 | 1.04 $\pm$ 0.2 | 1.16 $\pm$ 0.93
7727 | 0.349 $\pm$ 0.072 | 70.4 | 0.022 | 0.478 | 0.32 $\pm$ 0.1 | 15462 | 0.34 $\pm$ 0.15 | 0.335 $\pm$ 0.215

888 $\textdagger$$\textdagger$footnotetext: Has a large offset relative to
XMM’s FOV. $*$$*$footnotetext: Shows a significant merging behaviour. (see
section 2.3).

##  Appendix B XMM-Newton images of AXES-2MRS groups

The X-ray images for the XMM-subsample used in this work are shown in Fig. 19.
The size of the spectral extraction regions (green dashed ellipses) are listed
in Table 5.

Figure 19: 25 XMM-Newton X-ray pointings for the full sample used in this
work. Green dashed ellipses are spectral extraction regions. Red dashed
ellipses are background extraction regions. Small white circles represent
member galaxies in each group, each denoted with the respective group ID. The
color bar scale is counts/s/cm2/arcmin2. The original colour bar values were
re-scaled in SAOImage DS9 from counts/s/pixel for the XMM-Newton.

Figure 20: Continued.

##  Appendix C AXES-2MRS Group Catalogue

In Table7 we describe the X-ray properties of the Full AXES-2MRS Group
catalogue. Source flux and luminosities are based on RASS data. The optical
properties of the groups are calculated only for groups with at least 5 clean
members. The redshifts are reported in the CMB frame, using the catalogues of
Tempel et al. (2018). The catalogue is released on CDS.

Table 7: Descriptions of the columns of the AXE-2MRS catalogue. The full VAC is available online at CDS. Column | Unit | Description | Example
---|---|---|---
GROUP_ID (1) |  | 2MRS group identification number from Tempel et al. (2018) | 2150
AXES_ID (2) |  | Extended X-ray source ID in the AXES catalog | 93280903
RA (3) | deg | X-ray detection right ascension (J2000) | 95.56809
DEC (4) | deg | X-ray detection declination (J2000) | –64.67731
NMEM (5) |  | Number of spectroscopic members in 2MRS group catalogue | 23
NMEM_CLEAN (6) |  | Number of spectroscopic members after the cleaning | 23
ZSPEC (7) |  | 2MRS group redshift | 0.0281
ZSPEC_CLEAN (8) |  | Group redshift, assigned using median value of clean members | 0.0281
CLUVDISP_GAP (9) | km s-1 | Gapper estimate of the cluster velocity dispersion | 582.223
GAUSSIANITY (10) |  | Gaussianity, based on the substructure analysis | G
LX0124 (11) | ergs s-1 | Luminosity in the (0.1-2.4) keV band of the cluster, aperture $R_{\text{500c}}$ | $2.3\times 10^{43}$
ELX (12) | ergs s-1 | Uncertainty on LX0124 | $7.59\times 10^{41}$
FLUX052 (13) | ergs s-1 cm-2 | Galaxy cluster X-ray flux in the 0.5-2.0 keV band | $7.25\times 10^{-12}$
EFLUX052 (14) | ergs s-1 cm-2 | Uncertainty on FLUX052 | $2.39\times 10^{-13}$
R_E (15) | arcmin | Apparent radial extent of X-ray emission | 16.8