# The Compactness of Galaxy Groups in the Sloan Digital Sky Survey

Yun-Liang Zheng (郑云亮) Key Laboratory for Research in Galaxies and Cosmology,
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai, China, 200030  University of Chinese Academy of Sciences,
No.19A Yuquan Road, Beijing, China, 100049  Shi-Yin Shen (沈世银) Key Laboratory
for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory,
Chinese Academy of Sciences, 80 Nandan Road, Shanghai, China, 200030  Key Lab
for Astrophysics, Shanghai, China, 200034  Shuai Feng (冯帅) College of Physics,
Hebei Normal University, 20 South Erhuan Road, Shijiazhuang, 050024, China
Hebei Key Laboratory of Photophysics Research and Application, 050024
Shijiazhuang, China

(Received 2021.10.28; Revised 2021.12.13; Accepted 2021.12.14)

###### Abstract

We use an updated version of the halo-based galaxy group catalog of Yang et
al., and take the surface brightness of the galaxy group ($\mu_{\rm lim}$)
based on projected positions and luminosities of galaxy members as a
compactness proxy to divide groups into sub-systems with different
compactness. By comparing various properties, including galaxy conditional
luminosity function, stellar population, active galactic nuclei (AGN)
activity, and X-ray luminosity of the intra-cluster medium of carefully
controlled high (HC) and low compactness (LC) group samples, we find that the
group compactness plays an essential role in characterizing the detailed
physical properties of the group themselves and their group members,
especially for low mass groups with $M_{h}\lesssim 10^{13.5}h^{-1}M_{\odot}$.
We find that the low-mass HC groups have a systematically lower magnitude gap
$\Delta m_{12}$ and X-ray luminosity than their LC counterparts, indicating
that the HC groups are probably in the early stage of group merging. On the
other hand, a higher fraction of passive galaxies is found in the HC group,
which however is a result of systematically smaller halo-centric distance
distribution of their satellite population. After controlling of both $M_{h}$
and halo-centric distance, we do not find any differences for both the
quenching faction and AGN activity of the member galaxies between the HC and
LC groups. Therefore, we conclude that the halo quenching effect, which result
in the halo-centric dependence of galaxy population, is a faster process
compared to the dynamical relaxed time-scale of galaxy groups.

keyword1 – keyword2 – keyword3

††journal: ApJ

##  1 Introduction

In the paradigm of hierarchical structure formation, galaxies are believed
form via baryonic gas cooling and condensation within dark matter halos. It is
generally assumed that halo mass ($M_{h}$) is the most fundamental parameter
that determines the evolutionary path of galaxies inside (e.g., Weinmann et
al., 2006; Wetzel et al., 2012; Woo et al., 2013, 2015; Bluck et al., 2014,
2016; Wang et al., 2018; Davies et al., 2019). Based on this assumption,
numerous models for linking galaxies to dark matter halos have been developed,
such as the halo occupation distribution model (HOD, Jing et al., 1998;
Peacock & Smith, 2000; Zheng et al., 2005; Zu & Mandelbaum, 2016), abundance
matching model (HAM, Mo et al., 1999; Kravtsov et al., 2004; Vale & Ostriker,
2006; Guo et al., 2010; Simha et al., 2012; Reddick et al., 2013), the
conditional luminosity function (CLF, Yang et al., 2003, 2008; Cooray, 2006;
van den Bosch et al., 2007), and halo-based empirical models (Yang et al.,
2013; Lu et al., 2014; Moster et al., 2018). In these models, dark matter
halos have evolved hierarchically through repeated mergers. Galaxies that form
the main branch of a merger tree generally reside near the halo center (Lange
et al., 2018) and are referred to as central galaxies, while others that form
in the sub-branch and orbit the central galaxies are referred to as satellite
galaxies. These two galaxy populations have experienced different evolutionary
paths so that their properties are different statistically as a result (e.g.,
Weinmann et al., 2006; van den Bosch et al., 2008; Peng et al., 2010; Knobel
et al., 2015; Davies et al., 2019).

From an observational point of view, the physical properties of galaxies are
strongly correlated with their stellar mass ($M_{\star}$) and environment
(e.g., Dekel & Birnboim, 2006; Peng et al., 2010, 2012, 2014; Liu et al.,
2019; Zhang et al., 2021). Specific mechanisms have been proposed for mass
dependency, including supernovae and active galactic nuclei (AGN) feedback
(Croton et al., 2006; Fabian, 2012; Harrison, 2017), while the physical
mechanisms of the environmental effects on the dwarf/satellite galaxies
include tidal interaction (Farouki & Shapiro, 1981), ram-pressure stripping
(Gunn & Gott, 1972), and harassment (Moore et al., 1996), etc. In the
framework of halo model, the observed environmental effects of galaxies can be
interpreted as a dependence on their host halos (e.g., Liu et al., 2010;
Wetzel et al., 2012; Henriques et al., 2017; Wang et al., 2018; Davies et al.,
2019). Indeed, Haas et al. (2012) demonstrate that the environmental
parameters of galaxies defined in a traditional way (e.g., projected local
density, projected distance to the $i$th nearest galaxy, etc.) are in effect
measures of their host halo mass $M_{h}$. However, there are also
observational indications that galaxies exhibit more complex environmental
effects beyond the simple $M_{h}$ model. For example, More et al. (2011) and
Mandelbaum et al. (2016) revealed that the stellar-to-halo mass relation is
further correlated with the physical properties of central galaxies. From an
evolutionary point of view, dark matter halos form via mergers of smaller
halos. When dark matter halos merge, infalling galaxies do not necessarily
merge with central galaxies in a short timescale but keep orbiting and
interacting with the central galaxies (Stewart et al., 2009). Therefore, in
addition to $M_{h}$, the halo formation time (or the assembly history) has
long been used as the second parameter to profile the dark matter halos when
analyzing the simulation data (e.g., Matthee et al., 2017; Zehavi et al.,
2018; Cui et al., 2021).

In observation, the host $M_{h}$ of galaxy systems can be estimated from the
total stellar mass or richness of the group members (e.g., Yang et al., 2005,
2007, 2021; Robotham et al., 2011). However, it is not clear how to link the
observational properties of galaxy groups to their assembly history. Several
studies on fossil groups of galaxies (e.g., Dariush et al., 2010; Raouf et
al., 2014, 2018; Khosroshahi et al., 2017; Li & Cen, 2020) have suggested that
$\Delta m_{12}$, the magnitude gap between the first and second brightest
members, might characterize the assembly history of a galaxy group. However,
recent studies (e.g., Shen et al., 2014; Trevisan & Mamon, 2017) have shown
that the origin of large $\Delta m_{12}$ is quite complex and cannot be simply
accounted for by the halo formation time. On the other hand, like the size of
galaxies, the size (or compactness) of galaxy groups might be another
observationally accessible parameter encoding the assembly histories of halos.
Indeed, it has long been known that there is a special class of galaxy groups,
compact groups of galaxies (CGs), which is defined as a group system
containing a few galaxies separated by distance of an order of galaxy size
(e.g., Hickson, 1982; Barton et al., 1996; Díaz-Giménez & Zandivarez, 2015;
Sohn et al., 2015, 2016; Díaz-Giménez et al., 2018; Zheng & Shen, 2020).
Because of the observational facts that tidal interactions and mergers
frequently appear among CG members, (e.g., Barnes, 1989; Amram et al., 2004;
Coziol & Plauchu-Frayn, 2007; Torres-Flores et al., 2009, 2010, 2014), CGs are
believed to represent the sites for massive members that are about to merge.
Moreover, based on simulation, Farhang et al. (2017) suggested that CGs are
early-formed systems that depart from ordinary hierarchical assembly and
stressed that CGs appear to be a distinct halo environment. Inspired by the
ideas about the peculiarity of CGs, several comparative studies on the galaxy
properties of CGs and non-CGs have been performed. For instance, Coenda et al.
(2012) showed that CGs contain a larger fraction of red, early-type galaxies
in comparison to loose groups and fields at a given absolute magnitude, while
Cluver et al. (2020) found no significant difference in the scale relation
between quenched fraction and $M_{\star}$ in group systems with different
levels of compactness.

The connection between the observational selected CGs and halo-model-based
groups has been recently investigated by Zheng & Shen (2021). In that study,
we show that the observationally selected CGs are physically heterogeneous
systems and can be mainly separated into two categories, the isolated systems
and those embedded in rich groups or clusters. For these isolated CGs, we find
that they have lower dynamical masses compared to the non-compact ones at the
same group luminosity. However, we cannot distinguish whether this lower
dynamical mass state is a result of a merging process or the dynamical
evolution of post-mergers. In this paper, we expand our studies on CGs (Zheng
& Shen, 2020, 2021) and for the first time, take the compactness of galaxy
groups as a general parameter to explore its correlations with other physical
properties of galaxy groups and group members when $M_{h}$ is controlled.
Moreover, since the X-ray emission from the intragroup medium contains rich
information on the dynamical status of galaxy groups, for the first time, we
also compare the X-ray properties of galaxy groups with different compactness
using a statistical method in this study. Our goal is to figure out whether
the group compactness plays a role as a second observational parameter in
describing the properties and merger histories of galaxy groups.

This paper is organized as follows. We describe the halo-based group samples
in Section 2, and the measurements of relevant properties in section 3. We
present our results for the global and individual properties of groups with
different compactness in section 4 and 5, respectively. We provide a brief
discussion in Section 6 and summarize our conclusions in Section 7. Throughout
this paper, we assume a flat $\Lambda$CDM cosmology with parameters:
$\Omega_{m}=0.27$ and $H_{0}=100h$ km s-1 Mpc-1 with $h=0.7$.

##  2 Sample: groups of galaxies

In this work, we take use of the group catalog of Yang et al. (2007, 2012,
hereafter Y07), which has been derived from the latest version of New York
University Value-Added Galaxy Catalog (VAGC, Blanton et al., 2005) based on
the SDSS legacy survey (Abazajian et al., 2009) with a set of improved
reduction. Y07 has applied a halo-based group finder to assign each galaxy in
the SDSS-DR7 Main Galaxy Sample (MGS) within the redshift range of
$0.01<z<0.20$ and $r$-band magnitude brighter than $17.72$ mag to a unique
group. In Y07, three versions of a group catalog have been constructed based
on different redshift sources. In this work, we use sample III of the Y07
group catalog containing $\sim 640,000$ galaxies, where a small fraction of
the members ($\sim 37,000$) without spectroscopic redshifts have been assigned
with the redshifts of their nearest neighbors. Therefore, sample III of the
Y07 group catalog has $100\%$ completeness of member galaxies, while it also
contains contamination from background or foreground galaxies.

Since $\sim 13,000$ out of $\sim 37,000$ spectroscopically unmeasured galaxies
have achieved new redshifts from SDSS-DR16 (Ahumada et al., 2020), GAMA-DR2
(Liske et al., 2015), and LAMOST-DR7 (Luo et al., 2015), the contamination of
the group catalog can be significantly reduced (e.g., Shen et al., 2016; Feng
et al., 2019; Zheng & Shen, 2020). Here, we use these new redshift
measurements to update the sample III of the Y07 group catalog. We find $\sim
8500$ out of $\sim 13,000$ galaxies with new redshift measurements are
concordant with their nearest neighbors, thus, their group ID remains
unchanged. For the remaining $\sim 4500$ galaxies having updated redshifts but
inconsistent with their nearest neighbors, we reassign them to a particular
group following the criteria of Y07 and update the original halo properties
accordingly. For the remaining $\sim 24,000$ out of $\sim 37,000$ galaxies
without new redshift detection, we keep them unchanged.

Owing to our aim of investigating how group properties vary with group
compactness, which is defined based on at least three members, we only
consider $N\geq 3$ groups in this work. Moreover, we have also noticed that
bad photometry (e.g., one large extended galaxy deblending into various small
pieces or a fragment of a satellite trail identified as a chain of several
galaxies, McConnachie et al., 2009; Zheng & Shen, 2020) will contaminate the
following results because these problematic sources are more likely to be
identified as groups with higher compactness. Therefore, we have made visual
inspections of the images of the $N\geq 3$ groups and removed $\sim 300$ fake
detections, most of which belong to low $M_{h}$ groups.

Y07 has estimated the characteristic stellar mass ($M_{19.5}$) and luminosity
($L_{19.5}$) of each group by summing up the masses and luminosities of all
the members with ${}^{0.1}M_{r}-5\log h\leq-19.5$ mag, respectively. For the
distant groups ($z>0.09$) where the corresponding faint-end limit brighter
than ${}^{0.1}M_{r}-5\log h=-19.5$ mag, their $M_{19.5}$ and $L_{19.5}$ have
been corrected for the missing members. Y07 has also estimated two versions of
$M_{h}$ for each group based on their $M_{19.5}$ or $L_{19.5}$, respectively.
Unless stated otherwise, we adopt the values based on $M_{19.5}$ as
recommended in Y07 paper. For some groups, their $M_{h}$ have been updated due
to the slight revision of their group membership, especially for those groups
containing the galaxies with discordant new redshifts. There are also $\sim
700$ $N\geq 3$ groups without $M_{h}$ calculated because none of their members
is brighter than ${}^{0.1}M_{r}-5\log h\leq-19.5$ mag, thus we remove them in
the following analysis.

Finally, there are 22,468 $N\geq 3$ groups with $M_{h}$ assigned, which have
140,930 member galaxies and 129,622 of them ($\sim 92\%$) have spectroscopic
redshifts. The lower redshift completeness of the final sample compared to the
overall galaxy sample ($\sim 95\%$) is caused by the fact that almost all of
the galaxies with assigned redshifts are members of $N\geq 2$ systems.

In Y07, the dark matter halos are defined as having an over-density of 180, we
recompute the halo radius ($R_{180}$) and group velocity dispersion ($\sigma$)
given by the Equation (5) and (6) in Y07 paper due to the revision of $M_{h}$
estimate. Unless stated otherwise, the central galaxy of each group are
identified to be the most massive one.

Figure 1: The $\mu_{\rm lim}-$redshift diagrams for groups in different
$M_{h}$ bins. The bin number and mass range of $M_{h}$ are given in the lower-
left corner of each subplot. The solid step lines in each subplot represent
the $25$th and $75$th percentiles of the $\mu_{\rm lim}$ in each redshift bin
with a bin size of 0.01. The data points above the upper step lines are
regarded as HC groups (red circle), those lying in the middle are regarded as
IC groups (grey square), and those below the lower step lines are LC groups
(blue diamond).

###  2.1 Compactness of galaxy groups

In this work, we use the group surface brightness as a proxy for group
compactness. Following Hickson (1982), the surface brightness of a group
system is defined as:

$\mu_{\rm
lim}=-2.5\log{\left(\frac{\sum\limits_{i=1}^{N}10^{-0.4m_{i}}}{\pi\theta_{\rm
G}^{2}}\right)},$ (1)

where $\mu_{\rm lim}$ is the surface brightness averaged over the smallest
circle enclosing $N$ galaxy members brighter than the magnitude limit within
an angular radius of $\theta_{\rm G}$ in units of arcseconds, and $m_{i}$ is
the apparent magnitude of the $i$th brightest member.

To fairly define the compactness level of galaxy groups, we divide the groups
into 12 $M_{h}$ bins with equal sample size and then plot $\mu_{\rm lim}$
against redshift for each $M_{h}$ bin in figure 1. For each $M_{h}$ bin, we
further divide groups into 20 redshift bins with a bin width of 0.01, and then
calculate the $25^{\rm th}$ and $75^{\rm th}$ percentile values of $\mu_{\rm
lim}$ distribution in each redshift bin. The solid lines in each panel of
figure 1 show the separation lines of the $25^{\rm th}$ and $75^{\rm th}$
percentiles, respectively. We see that there is a systematical redshift
dependence of $\mu_{\rm lim}$ quartiles, where the higher redshift groups are
biased toward higher $\mu_{\rm lim}$ values. This redshift bias is likely
caused by the segregation effect of galaxies with different masses (see
section 6 for a detailed discussion).

Here, we are not interested in well-defining the compactness of groups of
galaxies with a quantitative $\mu_{\rm lim}$ criterion. On the contrary, our
aim is to make a fair comparison of galaxy groups with different compactness
while their other main physical features are well constrained. To do that, we
consider the groups of each $M_{h}$ bin that lie in the top quartile to be
high compactness (HC) groups, while those lying in the middle two quartiles
are considered as intermediate compactness (IC) groups and those in the bottom
quartile are considered to be low compactness (LC) groups. With this
configuration, our HC and LC groups have the same $M_{h}$ and redshift
distribution, and thus any differences in between these two samples would
mainly correlate with the compactness of galaxy groups. In figure 2, we show
the $M_{h}$ distribution of the Y07 groups with different levels of
compactness. As expected, these three different samples have nearly the same
$M_{h}$ distributions, which peaks around $\sim 10^{13.5}h^{-1}M_{\odot}$.

Moreover, for the $M_{h}\lesssim 10^{13.5}h^{-1}M_{\odot}$ groups, the
separation line of the top quartile is close to the criterion $\mu_{\rm
lim}\sim 26.0$ mag arcsec-2 that is used for defining the compact groups of
galaxies (Hickson, 1982). That is to say, there might be a good correspondence
between the classical compact groups and our low-mass HC groups. On the other
hand, for high-mass systems ($M_{h}\gtrsim 10^{13.5}h^{-1}M_{\odot}$), the
majority of the groups in the high compactness quartiles are actually not
‘compact’ enough ($\mu_{\rm lim}>26.0$ mag arcsec-2).

Figure 2: The $M_{h}$ distributions for the HC (red dashed), IC (grey solid),
and LC (blue filled) groups with a bin width of 0.1 dex.

###  2.2 X-ray emission of galaxy groups

We perform X-ray detections for Y07 groups in the ROSAT All Sky Survey (RASS)
image data, which mapped the sky in the soft X-ray band ($0.1-2.4$ keV) with
exposure time varying between 400 and 40000 seconds, depending on the
direction in the sky. The resolution of the RASS images is 45 arcsec.

The algorithm we use to estimate the X-ray luminosity is similar to those of
Shen et al. (2008) and Wang et al. (2014). As the first step, we determine the
X-ray center of each group. Optically, the position of the central galaxy is
usually taken to define the group center. Indeed, the X-ray peak position
might be offset from the central galaxy (e.g., Dai et al., 2007; Koester et
al., 2007; Shen et al., 2008; Wang et al., 2014). We locate the RASS fields
where the group reside, then apply a maximum-likelihood (ML) search algorithm
to generate the X-ray source catalog with detection likelihood $L>7$ in the
$0.5-2.0$ keV band (Voges et al., 1999). We match all the detected sources
within a tolerance of $0.1R_{180}$ from the central galaxies and regard the
closest one as the X-ray center. The other X-ray sources within $0.1R_{180}$
are regarded as part of the extended X-ray emission from this group, and those
that lie in $R>0.1R_{180}$ region are regarded as a contaminated source that
are not associated with the emission of intra-group medium. If none of the
X-ray sources is detected within $0.1R_{180}$, the central galaxy then is
regarded as the X-ray center, most of our samples ($\sim 90\%$) belong to such
cases.

To obtain an unbiased estimate of the typical X-ray luminosity of a subset of
galaxy groups, we stack each subsample with similar $M_{h}$ and group
compactness. The stacking method is described in Shen et al. (2008). Briefly,
we center each group image on the X-ray center. All the images are scaled to
the same size in units of $R_{180}$ and the contaminated sources located at
$R>0.1R_{180}$ are masked out. The X-ray background is estimated as the
average flux inside an annulus with inner radius of $R_{180}$ and outer radius
of $1.3R_{180}$. By subtracting the background, a one-dimensional count rate
profile as a function of radius is then obtained.

For a stack of $N$ groups, the source count of photons, $N_{\rm S}$, that
could be detected in RASS is given as

$N_{\rm
S}=\sum^{N}_{i=1}L_{\text{X},i}g\left(N_{\text{H},i},T_{i},z_{i}\right)t_{i},$
(2)

where $L_{\rm{X},i}$ is X-ray luminosity of the i$th$ group, $t_{i}$ is
average RASS effective exposure time of i$th$ group, and
$g\left(N_{\rm{H},i},T_{i},z_{i}\right)$ is a coefficient converting
bolometric X-ray luminosity to observed count rate as functions of redshift
$z_{i}$, average galactic hydrogen column density $N_{\rm{H},i}$ in the line
of sight and gas temperature within the i$th$ group. We assume that the gas
metal abundance is assumed to be $0.3$ of the solar value, whereas
$N_{\rm{H},i}$ is taken from the map of Dickey & Lockman (1990) and the X-ray
emission has a thermal spectrum with temperature $T$ obtained from the
empirical $T-\sigma$ scaling relation of White et al. (1997). However, the
slope of the $T-\sigma$ relation for small groups is not necessarily the same
as massive clusters, and the gas temperature of low $M_{h}$ groups is even not
well-constrained (Lovisari et al., 2021, and references therein). Moreover,
the conversion factor for hot gas with $T\lesssim 0.3$ keV (corresponding to
$M_{h}\lesssim 10^{13}h^{-1}M_{\odot}$) becomes very different for different
X-ray spectral models (e.g., power law, thermal bremsstrahlung, Raymond-Smith
plasma, and blackbody spectrum) over this range (Briel et al., 1996). However,
until there is more negative evidence, there is reason to believe that the
$N_{\rm{H}}$ and $T$ distributions for groups at fixed $M_{h}$ but with
different compactness are similar, i.e., they have similar
$g\left(N_{\rm{H},i},T_{i},z_{i}\right)$ distributions. Indeed, we have
checked that the difference between the resulting X-ray luminosity of the LC
groups and HC ones (we will show in figure 6) is caused by the difference
between their detected X-ray photon counts rather than the specific values of
the conversion factor $g\left(N_{\rm{H},i},T_{i},z_{i}\right)$.

For simplicity, we use the conversion factors given by the table 2, 3, and 5
of (Böhringer et al., 2004). For the groups with gas temperature below the
lowest threshold of each table, we take use of the values in that row. Based
on the assumption that the X-ray luminosities of the sources in a stack are
similar, the weighted average X-ray luminosity, $L_{\rm X,S}$, of the stack
can be given as

$L_{\rm X,S}=\cfrac{N_{\rm
S}}{\sum\limits^{N}_{i=1}g\left(N_{\text{H},i},T_{i},z_{i}\right)t_{i}},$ (3)

Following Wang et al. (2014), the X-ray extension radius is set as a fixed
value of $R_{\rm X}=0.5R_{180}$ and the source count rates are integrated into
the X-ray extension radius $R_{X}$. By changing this fixed value from $R_{\rm
X}=0.3R_{180}$ to $0.6R_{180}$, none of our main results is significantly
impacted. Finally, to estimate the flux that is missed outside $R_{\rm X}$, we
make a $\beta-$profile extension correction to make up the X-ray luminosity
missed in the range $R_{\rm X}\leq R\leq R_{180}$ (see Böhringer et al., 2000;
Shen et al., 2008; Wang et al., 2014) with $\beta=2/3$ and core radius
$R_{c}=0.14R_{180}$. If $\beta$ is set as a free parameter in profile fitting,
we find all the best fittings are in the range of $0.40-0.70$, but with large
uncertainties. Therefore, we use a fixed value of $\beta$ to make the profile
fitting and flux calibration.

##  3 Sample: physical properties of group member galaxies

###  3.1 Quenched and Star-forming Galaxies

The $M_{\star}$ and star formation rate (SFR) of individual galaxies are
directly retrieved from the MPA/JHU-DR7 release (Brinchmann et al., 2004;
Kauffmann et al., 2003) based on SDSS spectroscopy. There are $\sim 123,000$
out of ($140,930$) galaxies in our sample have been detected. These galaxies
without match in MPA/JHU catalog are mainly caused by the fiber collision
effect during the SDSS spectroscopy target. In other words, their redshifts is
obtained from other surveys rather than SDSS-DR7. Due to the need for
determining central galaxies, the $M_{\star}$ of these unmeasured galaxies are
obtained by the relation between stellar mass-light ratio and $(g-r)$ colors
of Bell et al. (2003). Following Li et al. (2006), the catastrophic $(g-r)$
color outside the $3\sigma$ ranges from the mean color-magnitude relations of
both blue and red sequences is replaced by the mean color of the red or blue
cloud, depending on their location on the diagram.

Based on the bimodality of galaxies distributed in the $\text{SFR}-M_{\star}$
diagram, galaxies can be divided into star-forming and quenched population. We
adopt the demarcation line suggested by Bluck et al. (2016), which is parallel
to but 1 dex below the star formation main sequence:

$\log{\text{SFR}}=0.73\log{M_{\star}}-1.46\log{h}-8.3,$ (4)

where the reduced Hubble constant $h$ is included here to convert the units of
$M_{\star}$ from $M_{\odot}$ used in Bluck et al. (2016) to $h^{-2}M_{\odot}$
used here. We have also examined the results based on the demarcation line
given by Woo et al. (2013), and find negligible changes.

Figure 3: Classification of spectral types for strong- and weak-line galaxies
in Y07 groups. Left Panel: The BPT diagnostic diagram for strong-line galaxies
with [O iii]/H$\beta$ versus [N ii]/H$\alpha$ emission-line ratios. The
demarcation lines are taken from Kewley et al. (2001, K01) and Kauffmann et
al. (2003, K03). Right Panel: The WHAN diagram for weak-line galaxies in Y07
groups. The SF/composite/AGN demarcation lines are transposed from K01 and
K03, respectively (Cid Fernandes et al., 2010), and the retired/non-retired
demarcation line is taken from Cid Fernandes et al. (2011).

###  3.2 AGN classification

####  3.2.1 Optical AGNs

The emission lines related to AGN classification are also taken from the
MPA/JHU catalog. The optical AGN are classified based on the BPT diagram
(Baldwin et al., 1981), which is traditionally used to separate type II AGN
from star-forming and composite galaxies. Here, we use the demarcation lines
proposed by Kewley et al. (2001) and Kauffmann et al. (2003) (the left panel
of figure 3). Following Pasquali et al. (2009), the fluxes of the four
emission lines are required to have a signal-to-noise ratio of $\rm S/N\geq
3$. This results in a sample of $\sim 50,000$ AGN out of $\sim 123,000$
strong-line galaxies. Besides, there are $\sim 28,000$ weak-line galaxies
whose H$\alpha$ and [N ii] lines are both detected with an $\rm S/N\geq 3$,
but either or both of H$\beta$ and [O iii] lines have a lower $\rm S/N$. Then,
following Cid Fernandes et al. (2010, 2011), we use the WHAN method, which
only requires [N ii]/H$\alpha$ and H$\alpha$ equivalent width ($W_{{\rm
H}\alpha}$) to select weak-line AGN. We show the classification diagram for
weak-line galaxies in the right panel of figure 3. Note that the emission line
fluxes have been corrected for intrinsic extinction based on the Balmer
decrement and a dust attenuation curve of the form $\lambda^{-0.7}$ (Charlot &
Fall, 2000) by assuming an intrinsic ${{\rm H}\alpha}/{{\rm H}\beta}=2.86$
(Osterbrock, 1989).

####  3.2.2 Radio-loud AGN

Radio-loud AGNs are obtained by matching our sample galaxies with the radio
galaxy catalog of Best & Heckman (2012), which was established by cross-
matching the MPA/JHU catalog with the NVSS (National Radio Astronomy
Observatory Very Large Array Sky Survey, Condon et al., 1998) and FIRST (Faint
Images of the Radio Sky at Twenty centimeters, Becker et al., 1995) surveys
following the method of Best et al. (2005). The catalog is down to a flux
density level of 5 mJy, which means that the catalog probes down to a radio
luminosity of $L_{1.4\rm GHz}=10^{23}{\rm W}\cdot{\rm Hz}^{-1}$ at $z=0.1$, we
thus only consider the galaxies with $z\leq 0.1$ (Wang et al., 2018) when
investigating the radio-loud AGN fraction.

Figure 4: The best-fit parameters ($M_{\rm cen}^{\star}$, $\sigma_{\rm
cen}^{\star}$, $M_{\rm sat}^{\star}$, and $\alpha_{\rm sat}^{\star}$) of CLFs
for the HC (red), IC (grey), and LC (blue) groups as a function of $M_{h}$.
The vertical error bars show the errors of the parameter in each $M_{h}$ bin.
The plot in the inset in the upper right corner demonstrates an example of a
CLF fitting for the groups in $M_{h}$ Bin $=6$ (shaded region) derived via
both a nonparametric stepwise (symbols with error bars) and parametric ML
estimator (solid lines), the fitting for each subsample is manually shifted
for clarity.

##  4 The global properties of galaxy groups: dependence on group compactness

In this section, we start with a detailed investigation of how the global
properties of a galaxy group varied with the group compactness.

###  4.1 CLFs

The dynamical state of a group system is directly related to the luminosity
distribution of its galaxy members (Martínez & Zandivarez, 2012). As the first
step in our investigation of how the group properties varied with group
compactness, we use the CLF to describe the luminosity distribution of
galaxies in groups with different levels of compactness at a given $M_{h}$.
Following Yang et al. (2008), the total CLF can be modeled as the sum of the
CLFs of central and satellite galaxies 111In this section, we take the
brightest galaxy in the group as the central galaxies instead of the most
massive one. Tests have shown that both of the definitions yield
indistinguishable results.:

$\Phi\left({}^{0.1}M_{r}|M_{h}\right)=\Phi_{\rm
cen}\left({}^{0.1}M_{r}|M_{h}\right)+\Phi_{\rm
sat}\left({}^{0.1}M_{r}|M_{h}\right),$ (5)

The CLF of central galaxies is assumed to be a lognormal form:

$\Phi_{\rm
cen}\left({}^{0.1}M_{r}|M_{h}\right)\propto\exp\left[-\frac{{\left({}^{0.1}M_{r}-M_{\rm
cen}^{\star}\right)}^{2}}{2{\sigma_{\rm cen}^{\star}}^{2}}\right],$ (6)

where $M_{\rm cen}^{\star}$ is the expected value for the absolute magnitude
of the central galaxy, and $\sigma_{\rm cen}^{\star}$ is the standard
deviation of $M_{\rm cen}^{\star}$. The CLF of satellite galaxies takes a
modified Schechter (1976) function:

$\begin{split}\Phi_{\rm
sat}\left({}^{0.1}M_{r}|M_{h}\right)\propto&10^{-0.4\left(\alpha_{\rm
sat}^{\star}+1\right)\left({}^{0.1}M_{r}-M_{\rm sat}^{\star}\right)}\\\
&\cdot\exp{\left[-10^{-0.8\left({}^{0.1}M_{r}-M_{\rm
sat}^{\star}\right)}\right]},\end{split}$ (7)

where $M_{\rm sat}^{\star}$ is the characteristic magnitude and $\alpha_{\rm
sat}^{\star}$ is the faint-end slope of the satellite galaxies. Here, we focus
on the shape of these CLFs only, and the parameters ($M_{\rm cen}^{\star}$,
$\sigma_{\rm cen}^{\star}$, $M_{\rm sat}^{\star}$, and $\alpha_{\rm
sat}^{\star}$) of CLFs are obtained using ML estimator (Tammann et al., 1979),
where the binned CLFs are obtained via nonparametric stepwise maximum
likelihood method (Efstathiou et al., 1988). The results for each parameter as
a function of $M_{h}$ are shown in figure 4, where we also plot an example of
the fitting for the subsamples in $M_{h}$ Bin $=5$ derived via both methods to
show the consistency of the results.

For central galaxies, we see that there is a strong dependence of CLF shape on
$M_{h}$ values. However, for a given $M_{h}$, there is only a weak dependence
of CLF parameters on group compactness. For high $M_{h}$ bins ($M_{h}\gtrsim
10^{13}h^{-1}M_{\odot}$), the three categories of galaxy groups with different
compactness have almost identical $M_{\rm cen}^{\star}$ values. However, for
low $M_{h}$ bins ($M_{h}\lesssim 10^{13}h^{-1}M_{\odot}$), LC groups tend to
have slightly but systematically brighter $M_{\rm cen}^{\star}$ than the
others. The scatter of central galaxies $\sigma_{\rm cen}^{\star}$ are
consistent with each other inside uncertainties for all $M_{h}$ bins.

For the CLFs of satellite galaxies, we see not only a strong dependence on
$M_{h}$, but also an obvious dependence on group compactness. The $M_{\rm
sat}^{\star}$ of the LC groups is systematically fainter, while their faint-
end slope, $\alpha_{\rm sat}^{\star}$, is more negative and hence contains
more faint galaxy members than the HC groups. These two results are not
independent of each other because there is a degeneracy between $M_{\rm
sat}^{\star}$ and $\alpha_{\rm sat}^{\star}$ in modified Schechter function
fitting when the total luminosity is fixed. Besides, at a fixed total
luminosity, groups with more galaxy members (so that more negative
$\alpha_{\rm sat}^{\star}$) might also be biased toward having higher
$\mu_{\rm lim}$ values (because of the larger radius of the smallest enclosed
circle). To check to what extent the difference of richness between the LC and
HC groups at a given $M_{h}$ can explain the observed difference of $\mu_{\rm
lim}$ values, we perform a simple test. More specifically, for a group with
$N$ members at a given $M_{h}$, we take the median $\theta_{G}$ of the
observed samples as satisfying the conditions as expected at the radius
$R_{\rm m}(N,M_{h})$. By assuming the total luminosity of each group at a
fixed $M_{h}$ are the same, we thus can estimate the relative difference in
$\mu_{\rm lim}$ distributions for the mock groups with the richness
distribution assigned from HC and LC groups, respectively. Indeed, the
disparity of the richness between the LC and HC groups at a given $M_{h}$ can
only explain at most $\sim 20\%$ of their $\mu_{\rm lim}$ difference.

Figure 5: The median $\Delta m_{12}$ as a function of $M_{h}$ for the HC
(red), IC (grey), and LC (blue) groups. The vertical error bars show the
errors of the median $\Delta m_{12}$ in each $M_{h}$ bin. Note that the HC,
IC, and LC groups in each $M_{h}$ bin have been controlled to have similar
redshift and richness distributions.

###  4.2 Luminosity gap: $\Delta m_{12}$

The CLF results presented in the last section imply that the HC groups
(especially for those with $M_{h}<10^{13.5}h^{-1}M_{\odot}$) are biased to
have fainter central galaxies and brighter satellites. Therefore, the natural
conclusion is that the HC groups would be biased to have lower magnitude gap
$\Delta m_{12}$. In this subsection, we investigate in detail how $\Delta
m_{12}$ is correlated with the group compactness.

Based on the principle of order statistics, Shen et al. (2014) have shown that
$\Delta m_{12}$ is mainly determined by group richness, where groups with more
members would be naturally biased to have smaller $\Delta m_{12}$. On the
other hand, for groups with a given $M_{h}$, $\Delta m_{12}$ has long been
conjectured to reflect their dynamical states (e.g., More, 2012; Paranjape &
Sheth, 2012; Tal et al., 2012). This argument is based on the fact that the
dynamic friction drives massive major pairs to merge more quickly than other
mergers within the same halo. The massive satellites would be coalesced by the
central galaxies earlier, which drives $\Delta m_{12}$ to increase with time.
However, this simple scenario might not be enough to account for the observed
$\Delta m_{1,2}$ of galaxy groups since all galaxy systems are in hierarchical
merging and a recent merging event of group systems may significantly change
$\Delta m_{12}$ (Trevisan & Mamon, 2017).

To perform a fair comparison of $\Delta m_{12}$ between the groups with
different compactness, we do not compare their $\Delta m_{12}$ distribution in
a straightforward way. We match the LC (IC) groups to the HC groups at the
same $M_{h}$ bin against the confounding covariates, $X=\\{N,z\\}$. We search
for the counterparts within a distance of 1 scaled by a tolerance of
$\\{\Delta N,\Delta z\\}=\\{0,0.01\\}$ in the multi-dimensional space formed
by $X$. We select the closest one (two, for IC groups) of them not selected
yet. If all of the satisfied targets have been selected before, we pick the
closest one again. If none of the targets satisfy the criterion, this HC group
will be discarded.

Figure 5 shows the median $\Delta m_{12}$ for the HC, IC, and LC groups in
each $M_{h}$ bin. Here we see that $\Delta m_{12}$ decreases with increasing
$M_{h}$, mostly due to the richness being generally higher for massive groups.
On the other hand, the LC groups exhibit systematically larger $\Delta m_{12}$
than the HC groups. This result confirms the conclusion we have obtained from
the CLFs in the last section, which is also consistent with the fact that
classical compact groups are generally comprised of several relatively more
luminous/massive galaxies close in proximity to one another.

This result also raises a question on the dynamical status of the HC groups.
Previous studies have suggested that galaxy groups show similar dynamical
features (velocity dispersion, mean harmonic separation, and dynamical mass)
in the late collapse and dynamical friction phases, although the time interval
between these two phases is up to several gigayears (Mamon, 1993, 2007; Zheng
& Shen, 2021). Hence, there are two possibilities for these observed HC
groups, one is that the HC groups are dynamical young systems in the late
forming stage, and the other is that they are dynamical old systems in late
merging stage. Obviously, the systematically smaller $\Delta m_{12}$ for the
HC groups is much more likely to favor the former scenario. To further verify
this hypothesis, we investigate the X-ray properties of groups with different
compactness in the next section.

Figure 6: The stacked X-ray luminosity as a function of $M_{h}$ for the HC
(red) and LC (blue) groups. The vertical error bars show the errors
transported from the uncertainties of the count rate and the horizontal error
bars indicate the median deviation of the
$\log{\left[M_{h}/(h^{-1}M_{\odot})\right]}$ distribution in each bin. The
solid and dashed-dotted lines represent the results obtained by Wang et al.
(2014) and Mantz et al. (2010), respectively. The plot in the inset in the
upper left corner presents an example of surface brightness profiles (symbols
with error bars) with $\beta-$model fittings (lines) for stacks of HC (red)
and LC (blue) groups in $M_{h}$ Bin $=5-6$, corresponding to the shaded region
in the main plot.

###  4.3 X-ray Luminosities of Galaxy Groups

In this section, we investigate how the average X-ray luminosity of galaxy
groups depend on the group compactness. To increase the statistical
significance of our results, we rebin each pair of the adjacent $M_{h}$ bins
in the following. We stack the groups of similar $M_{h}$ and compactness
independently. In figure 6, we present the average X-ray luminosity given by
equation 3 as a function of $M_{h}$ for the stacks of each subsample and show
an example surface brightness profiles for the stacks of the HC and LC groups
in $M_{h}$ Bin $=5-6$. As one can see, the $S/N$ of the surface brightness
profile is sufficiently good and can be well represented by the $\beta-$model.

Broadly, the X-ray luminosity of galaxy groups has a good correlation with
$M_{h}$, in agreement with the results of Mantz et al. (2010) and Wang et al.
(2014). Note that the X-ray luminosity and halo mass are defined in a
different way in Mantz et al. (2010), we transform both of the quantities
using the method following Wang et al. (2014). For the groups with
$M_{h}\lesssim 10^{13.5}h^{-1}M_{\odot}$, the stacked X-ray luminosity of the
LC groups is significantly brighter than the HC groups at a given $M_{h}$.

X-ray emission of clusters/groups of galaxies has long been considered to
reflect the dynamical status of group systems. X-ray selected groups generally
have lower late-type fractions than typical optically-selected ones and are
known to be biased toward relaxed systems, particularly in the low-mass regime
(e.g., Osmond & Ponman, 2004; Eckert et al., 2011; Wen & Han, 2013).
Therefore, the systematically lower $L_{\rm X}$ for the HC groups, again,
implies that they are potentially in a dynamical young status. To find further
evidence of the correlation between the compactness of groups and their
dynamical status, we investigate the star formation properties of individual
galaxies living in groups with different compactness in the following section.

Besides the dynamical status, the X-ray emission of the intragroup medium is
also known to be correlated with the feedback process of the radio AGN of the
central galaxy (e.g., Shen et al., 2008). We will present a detailed study of
the AGN phenomena in groups of galaxies with different compactness in section
5.2.

Figure 7: The quenched galaxy fraction for (a). overall, (b). central, and
(c). satellite galaxies of the HCS (red), LCS (green), and LCRS (blue). Note
that the central galaxies of LCS and LCRS are exactly the same; thus, we do
not plot the results for the LCS in panel (b). In each panel, the errors in
each bin represent the 68 percent confidence limits estimated using 1000
bootstrap resamplings.

##  5 The properties of group member galaxies: dependence on group compactness

###  5.1 Stellar Population of Galaxies

We start by investigating the quenched fraction, which is defined as the
fraction of galaxies that are quenched according to the aforementioned
criterion of the HC and LC groups. Since our aim is to perform a fair
comparison between the galaxies within the HC and LC groups at a given
$M_{h}$, the bias due to different CLFs for these compared samples should be
mitigated. The control samples for the HC and LC group galaxies are derived
following the same method as described in section 4.2. Briefly, we match the
central and satellite galaxies of the LC groups to those in the HC groups
against the confounding covariates, $X_{\rm cen}(\text{or }X_{\rm
sat})=\\{\log{M_{\star}},z\\}$ with a tolerance of
$\\{\Delta\log{M_{\star}},\Delta z\\}=\\{0.2,0.01\\}$, respectively. The
control samples derived in this way are referred to as control samples in the
HC groups (HCS) and LC groups (LCS), respectively.

In figure 7, we first plot the quenched fraction as a function of $M_{h}$ for
the overall, central, and satellite population in three different panels,
where the green solid and red dashed-dotted lines represent LCS and HCS,
respectively. As a global trend, the quenched fraction increases
systematically with $M_{h}$ for each subsample. Since SDSS MGS is a flux-
limited sample, the higher $M_{h}$ systems are biased toward higher redshift
than the lower $M_{h}$ ones. Therefore, galaxies within higher $M_{h}$ groups
are on average more massive, so they have a higher quenched fraction (known as
mass quenching). On the other hand, at a given $M_{\star}$, galaxies in higher
$M_{h}$ groups are also known to have a higher quenched fraction (known as
halo quenching). Therefore, the systematically higher quenched fraction of
galaxies in higher $M_{h}$ bins we show in figure 7 is a combined result of
both mass and halo quenching.

Besides the least massive $M_{h}$ bin (we will discuss it later), we find that
the relative quenched fraction for all galaxies in the LCS is systematically
lower than that in the HCS, which mainly results from their systematically
lower quenched fraction for satellite galaxies (the lower panel of figure 7).
This result is in agreement with earlier findings that the galaxies in
classical compact groups are on average redder than their counterparts living
in noncompact groups (e.g., Coenda et al., 2012, 2015). On the other hand,
recent studies (e.g., van den Bosch et al., 2008; Wetzel et al., 2012;
Kauffmann et al., 2013; Wang et al., 2018) have found a dependency of galaxy
quenching on halo-centric distance, where galaxies in central regions tend to
be more quenched than their counterparts in the outskirts. In our work,
galaxies in the LCS and HSC have been controlled to have the same $M_{h}$ so
that have the same $R_{180}$. Therefore, the satellite galaxies of the HCS, by
definition, naturally have systematically smaller $R/R_{180}$ distributions
than those of the LCS. That is to say, the systematically higher fraction of
quenched galaxies in the HCS that we have found might be a result of their
smaller average halo-centric distance.

To test this hypothesis, we construct another control sample of LC group
galaxies, the central population of which remains unchanged, by matching the
satellite population to those in the HCS with the confounding covariates,
$X_{\rm sat}=\\{\log{M_{\star}},z,R/R_{180}\\}$, scaled by a tolerance of
$\\{\Delta\log{M_{\star}},\Delta z,\Delta R/R_{180}\\}=\\{0.2,0.01,0.025\\}$.
The control sample of LC groups derived in this way is referred to as the
R-controlled sample in LC groups (LCRS). We plot the quenched fraction of
galaxies for the LCRS as the blue dotted lines in each panel of figure 7.

We see that, for galaxies in the LC groups, when $R/R_{180}$ is further
controlled, the quenched fraction of satellite galaxies has systematically
increases. The global quenched fraction of galaxies in the LCRS is now in
excellent agreement with the galaxies in the HCS. This result implies that the
halo quenching effect might be a faster process. In section 4, from the global
properties of groups (e.g., magnitude gap, X-ray luminosity), we have argued
that the HC groups are likely to be in the early stage of group merging (or
halo formation). Here, we see that although the HC groups are more likely to
be recently formed, their members already show significant halo-centric
dependence, i.e., galaxies in central regions of the halo are more likely to
be quenched than in the outskirts.

On the other hand, although galaxies in the HCS and LCRS have been controlled
to have the same $R/R_{180}$ distributions, galaxies in the HCS actually still
have systematically higher local density than those in the LCRS. The higher
local density means that they are more likely to show a galaxy-galaxy
interaction effect. Such an effect is manifested in the lowest $M_{h}$ bin,
where the local galaxy density (or $\mu_{\rm lim}$) of the HCS peaks (see also
figure 1). For the least massive $M_{h}$ bin where the halo quenching
mechanism might not be very significant, we see a local galaxy-galaxy
interaction effect, the galaxies in the HCS have systematically lower
quenching fraction (because of enhanced star formation) than the LCS. Besides
this bin, we see that the central galaxies in the HCS show a marginal increase
in their proportion of quenched galaxies, this result might be affected by the
magnitude measurement bias remaining in the pipeline, which we will discuss
this in more detail in section 6.3.

Figure 8: Upper Panel: The optical AGN fraction for central (hexagon) and
satellite (diamond) galaxies of the HCS (red) and LCRS (blue), respectively.
Lower Panel: The radio-loud AGN fraction for all galaxies of the HCS and LCRS.
In each subplot, the errors in each bin represent the 68 percent confidence
limits estimated using 1000 bootstrap resamplings.

###  5.2 AGN Properties

In the last section, we have shown that galaxies in the HCS have established a
halo-centric relation as galaxies in the LCS. Here, we further explore whether
the AGN phenomena of galaxies have been influenced by the formation phase of
galaxy groups. To better control the variables, we focus on the HCS and LCRS
only.

We show the fraction of optical AGN in central and satellite populations for
the HCS and LCRS in the upper two panels of figure 8, where the left panel
shows the fraction as a function of halo mass $M_{h}$, while the right panel
shows that as a function of their stellar mass $M_{\star}$. We see that the
global trend of optical AGN phenomena shows a complicated dependence on
$M_{h}$ and $M_{\star}$, which has been studied in detail by many previous
works (e.g., Pasquali et al., 2009; Wang et al., 2018; Zhang et al., 2021). We
give a brief overview here. First, the optical AGN phenomena peak around
$M_{\star}\sim 10^{10.5-11}h^{-2}M_{\odot}$, and decreases with increasing
$M_{h}$ at $M_{h}\gtrsim 10^{12}h^{-1}M_{\odot}$. The central population has a
higher fraction of optical AGNs than satellites at a given $M_{\star}$, which
is due to the fact that the former live in the groups with lower $M_{h}$. On
the other hand, the central galaxies show a higher average optical AGN
fraction than satellites at a given $M_{h}$ for $M_{h}\lesssim
10^{14}h^{-1}M_{\odot}$ because the former contains more $M_{\star}\sim
10^{10.5-11}h^{-2}M_{\odot}$ galaxies. In addition to these known overall
trends, for subsamples of the HCS and LCRS, we do not see any systematic
differences between the HCS and LCRS.

For radio-AGN phenomena, owing to the fact that radio-loud AGN are not very
abundant, and Wang et al. (2018) showed that there is no difference in radio
properties between central and satellite galaxies at given $M_{h}$ and
$M_{\star}$, we do not separate the galaxies into central and satellite
populations here. In the lower panels of figure 8, we show the radio-loud AGN
fraction for all galaxies in the HCS and LCRS as a function of $M_{\star}$
(left) and $M_{h}$ (right), respectively. We also do not see any significant
differences between the galaxies in HCS and LCRS. However, it is worth
mentioning that the radio-loud AGN fraction of small satellites in the HCS
sample seems systematically higher than that of their counterparts in the
LCRS. Such a difference is interesting but suspicious since we know that the
radio-loud AGN phenomena is linked with super-massive black holes which
generally live in massive galaxies. This suspiciously higher radio-loud
fraction at the low $M_{\star}$ end might come from the contamination during
the identification of radio-loud AGN. An FR-II galaxy have two radio lobes
separated out to a distance of $\sim 100$ kpc, which is very likely to
contaminate its neighbors if this radio galaxy is located in a higher density
region like the HC groups (Best et al., 2005). A detailed study of the origin
of this systematical difference is beyond the scope of this paper and deserves
to be carried out in the future.

In summary, we find no systematic differences in AGN phenomena between the
galaxies in the HCS and LCRS when their halo-centric effect has also been
controlled. This result is consistent with an earlier finding that there is no
dependence on optical nuclear activity within the local environment of
galaxies (e.g., Argudo-Fernández et al., 2016).

Figure 9: Redshift completeness for galaxies living in the HC (red dashed), IC
(grey solid), and LC (blue dotted) groups as a function of $M_{h}$. The insets
represent the ${}^{0.1}M_{r}$ distributions for the HC group galaxies with
(filled) or without (open) redshift measurement in each $M_{h}$ bin.

##  6 Discussion

So far, we have used a carefully controlled sample and quantitative
statistical methods to investigate how properties of a galaxy group and its
galaxy members depend on its structure parameter: group compactness. However,
there are still some biases remaining in the data analysis pipeline, which
could potentially affect our results. In this section, we further discuss on
these issues and present the robustness of our results.

###  6.1 Redshift Incompleteness

As mentioned in section 2.1, redshift incompleteness are more frequently
occurred in the HC groups, figure 9 presents the redshift completeness as a
function of $M_{h}$ for the HC, IC, and LC groups. A trend of increasing
redshift incompleteness with decreasing $M_{h}$ is clearly seen for the HC
groups, while there is no such trend for the IC and LC groups. This systematic
trend is a result of increasing compactness limit for the selection of HC
groups in low $M_{h}$ bins (figure 1), where the fiber collision effect is
easier to be trigger.

In our study, we have assigned redshifts from their nearest neighbor for these
galaxies without any spectroscopic redshifts. That is to say, there is a
larger fraction of galaxies in the HC groups being contaminated by foreground
or background galaxies, which biases the $M_{h}$ estimation of HC groups to
higher values. This bias introduces a question on our results in section 4.3,
where we have shown that the HC groups have systematically fainter X-ray
luminosities than LC groups at lower $M_{h}$ bins. In section 2, we show that
$\sim 65\%$ of the spectroscopy of unmeasured galaxies are genuine members of
their assigned groups. Therefore, we do not expect that this redshift
incompleteness bias on $M_{h}$ estimation will be significant. For the lowest
$M_{h}$ bin with the most serious incompleteness problem, the redshift
completeness of the HC groups is $\sim 78\%$, which results in an over-
estimation of $M_{h}$ for the HC groups at the level of $\sim 0.03$ dex, much
smaller than the bin width ($\sim 0.63$ dex). Such a small difference in
$M_{h}$ obviously can not account for the difference in $L_{\rm X}$ between
the low-mass HC and LC groups ($\sim 0.3$ dex) we have obtained in section 2.

Besides the global bias, as shown by the subplots in the inset of figure 9,
the redshift incompleteness is not uniform, but clearly biased toward fainter
(or satellite) galaxies. For instance, the redshift completeness of the
central galaxies and satellites of HC groups in the lowest $M_{h}$ bin are
$\sim 87\%$ and $\sim 72\%$, respectively.222This less significant bias might
stem from the fact that the brighter galaxies more easily obtain spectroscopic
reshifts from the LAMOST spectral survey easier (Shen et al., 2016). This non-
uniform bias in magnitude might have an impact on the CLF and magnitude gap
studies in section 4.

To further validate our results presented in section 4 compared with the
redshift incompleteness, we use a smaller but cleaner subsample for the
robustness test. To do that, we remove the groups with member completeness
$\leq 80\%$ to ensure that low richness ($3\leq N\leq 5$) groups have $100\%$
completeness and the bias on $M_{h}$ estimation for the high richness groups
($N\geq 6$) are negligible 333Most of the groups with $M_{h}\lesssim
10^{13.5}h^{-1}M_{\odot}$ have richness $3\leq N\leq 5$.. After removing $\sim
5,500$ groups not satisfying with this condition, we repeat all the studies in
section 4 and find consistent results inside statistical errors.

Finally, during the study of the properties of group member galaxies in
section 5, we only use the galaxies with spectroscopic measurements in SDSS-
DR7 and make comparisons using carefully selected control samples. Therefore,
we do not expect any biases to be produced by the redshift incompleteness
here.

Figure 10: An example of the median and scatter of $\mu_{\rm lim}$ (grey) and
$\mu_{3}$ (magenta) as a function of redshift for Y07 groups in the $M_{h}$
Bin $=6$.

###  6.2 Alternative Compactness Definition: $\mu_{N}$

Historically, Hickson (1982) originally used a fixed value of $\mu_{\rm
lim}=26.0$ mag arcsec-2 to define the classical compact groups. However, as
mentioned in section 2.1, the $\mu_{\rm lim}$ distribution shows a clear
dependence on redshift. This redshift bias might be due to the fact that
galaxies within groups are not randomly distributed. The brighter galaxy
members suffer the dynamical friction effect more significantly and are on
average located in more central regions (Saro et al., 2013). The faint members
in the outskirts are more likely to be out of the detection limit in the flux
limit sample, which makes the $\mu_{\rm lim}$ measurement be biased toward
lower values for higher redshift groups. Therefore, using a fixed value of
$\mu_{\rm lim}$ as the compact group threshold will result in an in-
homogeneous sample at different redshifts (Zheng & Shen, 2020). One option
that might alleviate this redshift bias is adopting an alternative compactness
definition, $\mu_{N}$, based on the brightest $N$ galaxy members.

Here, we use $\mu_{3}$, the surface brightness as given by Equation (1) but
calculated using the three brightest galaxies only, to test our hypothesis. We
show an example of the $\mu_{3}-$redshift relation for groups in $M_{h}$
Bin$=6$ in figure 10. Clearly, the redshift dependence is significantly
reduced, and the quartile values show little change with redshift. On the
other hand, using fewer galaxy members to calculate $\mu_{N}$ leads to a
larger random dispersion on group compactness. As can be seen in figure 10,
the scatter of $\mu_{3}$ is about 2 times larger than $\mu_{\rm lim}$ at low
redshift. At high redshift, because of the decreasing of the observable
members, $\mu_{3}$ becomes to be equivalent to $\mu_{\rm lim}$. We have
tested, using $\mu_{3}$ instead of $\mu_{\rm lim}$ to define the HC and LC
groups, and all of our conclusions remain unchanged but with much lower
significance.

###  6.3 Magnitude Measurement of Galaxies in the High Density Region

The magnitude measurement of galaxies in the high density region is known to
be easily biased to high values, i.e., underestimation of the flux because of
the difficulties in the subtraction of background (e.g, Lauer et al., 2007;
Von Der Linden et al., 2007).

For our HC groups, the surface brightness is as high as $\mu_{\rm lim}\sim 26$
mag arcsec-2, where the magnitude measurement bias very likely happens. If so,
the stellar masses of galaxy members of the HC groups, especially for the
central population, will be underestimated. Such an underestimation effect
might account for the systematically higher (although not significant)
quenching fraction of central galaxies within higher $M_{h}$ HC groups. On the
other hand, because of the underestimation of $M_{\star}$, $M_{h}$ will also
be underestimated. Such an underestimation effect on $M_{h}$ might compensate
for the overestimation effect discussed in section 6.1.

##  7 Conclusion

In this study, we have performed a systematic and statistical analysis of the
compactness of galaxy groups based on an $N\geq 3$ group sample drawn from the
halo-based group catalog of Yang et al. (2007) as well as the extra GAMA and
LAMOST redshift measurements. Based on the group surface brightness $\mu_{\rm
lim}$, the total luminosity averaged over the smallest circle enclosing all
the members, akin to the compactness criterion for classical compact groups
introduced by Hickson (1982), we divide the samples into the HC and LC groups
based on the aggregate relation of $\mu_{\rm lim}$ versus redshift in each
$M_{h}$ bin. We investigate various aspects of the individual and global
properties such as CLF, $\Delta m_{12}$, X-ray luminosity of ICM, quenched
fraction and AGN fraction. We reach the following conclusions and predictions:

  1. 1.

For massive halos ($M_{h}\gtrsim 10^{13.5}h^{-1}M_{\odot}$), there is no
significant dependence on group compactness. Also, at the same redshift,
massive groups are generally less compact (higher $\mu_{\rm lim}$) than small
groups in observation. This echoes the fact that the classical compact groups
are generally small galaxy systems (e.g., McConnachie et al., 2009; Zheng &
Shen, 2020).

  2. 2.

The HC groups are generally comprised of several members with comparable
$M_{\star}$ (or luminosity), while the LC groups generally contain a more
dominant central galaxy. This is relevant to the studies of fossil groups,
such as the studies by Farhang et al. (2017) and Raouf et al. (2019), which
have suggested that the fossil groups with a larger magnitude gap $\Delta
m_{12}$ tend to be older and the spatial distribution of their galaxy members
are less crowded compared to the non-relaxed groups. Combined with the fact
that the average X-ray luminosity of the LC groups is higher than the HC ones
in the range of $M_{h}\lesssim 10^{13.5}h^{-1}M_{\odot}$, we argue that those
small HC groups, on average, are in the early stages of group merging.

  3. 3.

The HC groups contain more passive galaxies than the LC groups with similar
$M_{h}$ at a given $M_{\star}$, which is a result of their different halo-
centric distance distributions for a satellite population. When controlled for
halo-centric distance further, this difference tends to disappear. We conclude
that the halo quenching effect in galaxy members, which makes the dependence
of passive fraction on halo-centric distance, is a faster process compared to
the dynamical relaxing time-scale for the groups in which they are hosted.

  4. 4.

The fraction of galaxies that host an optical or radio-loud AGN depends weakly
on group compactness, indicating the local environmental effect triggered by
group evolution does not correlate with the AGN activity of galaxies.

In summary, our results imply a relationship between the observed group
compactness and halo formation history. However, this relationship is not
consistent with the inferences obtained from simulations of pure dark matter
halo, where the halos with higher concentration are believed to be formed
earlier (e.g., Zhao et al., 2009; Ragagnin et al., 2019). To further clarify
the possible inconsistency between our observations and numerical simulations,
we need a joint effort for both. On the one hand, by expanding to a larger and
deeper galaxy sample (e.g., DESI, Dark Energy Survey Collaboration et al.,
2016), we would expand the definition of group compactness to include more
fainter group members. On the other hand, cosmological simulations including
both dark matter and baryonic components (e.g., IllustrisTNG, Pillepich et
al., 2018) are helpful for resolving the formation histories of groups with
different group compactness and exploring the relationship between halo
concentration and group compactness.

## Acknowledgements

We acknowledge the anonymous referee for helpful comments and detailed
suggestions. This work is supported by National Key R&D Program of China
No.2019YFA0405501 and the National Natural Science Foundation of China (grant
No. 12073059, 12103017).

This work has made use of data products from the Sloan Digital Sky Survey
(SDSS, http://www.sdss.org), the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope (LAMOST, http://www.lamost.org), the GAMA survey
(http://www.gama-survey.org. We are thankful for their tremendous efforts on
the surveying work.

The Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic
Telescope LAMOST) is a National Major Scientific Project built by the Chinese
Academy of Sciences. Funding for the project has been provided by the National
Development and Reform Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences.

## References

  * Abazajian et al. (2009) Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS, 182, 543. doi:10.1088/0067-0049/182/2/543
  * Ahumada et al. (2020) Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3. doi:10.3847/1538-4365/ab929e
  * Amram et al. (2004) Amram, P., Mendes de Oliveira, C., Plana, H., et al. 2004, ApJ, 612, L5. doi:10.1086/424482
  * Argudo-Fernández et al. (2016) Argudo-Fernández, M., Shen, S., Sabater, J., et al. 2016, A&A, 592, A30. doi:10.1051/0004-6361/201628232
  * Baldwin et al. (1981) Baldwin, J. A., Phillips, M. M., & Terlevich, R. 1981, PASP, 93, 5. doi:10.1086/130766
  * Barnes (1989) Barnes, J. E. 1989, Nature, 338, 123. doi:10.1038/338123a0
  * Barton et al. (1996) Barton, E., Geller, M., Ramella, M., Marzke, R. O., & da Costa, L. N. 1996, AJ, 112, 871
  * Becker et al. (1995) Becker, R. H., White, R. L., & Helfand, D. J. 1995, ApJ, 450, 559. doi:10.1086/176166
  * Bell et al. (2003) Bell, E. F., McIntosh, D. H., Katz, N., et al. 2003, ApJS, 149, 289. doi:10.1086/378847
  * Best et al. (2005) Best, P. N., Kauffmann, G., Heckman, T. M., et al. 2005, MNRAS, 362, 25. doi:10.1111/j.1365-2966.2005.09192.x
  * Best & Heckman (2012) Best, P. N. & Heckman, T. M. 2012, MNRAS, 421, 1569. doi:10.1111/j.1365-2966.2012.20414.x
  * Blanton et al. (2005) Blanton, M. R., Schlegel, D. J., Strauss, M. A., et al. 2005, AJ, 129, 2562
  * Bluck et al. (2014) Bluck, A. F. L., Mendel, J. T., Ellison, S. L., et al. 2014, MNRAS, 441, 599. doi:10.1093/mnras/stu594
  * Bluck et al. (2016) Bluck, A. F. L., Mendel, J. T., Ellison, S. L., et al. 2016, MNRAS, 462, 2559. doi:10.1093/mnras/stw1665
  * Briel et al. (1996) Briel, U. G., Aschenbach, B., Hasinger, G., et al. 1996, The ROSAT Users’ Handbook. A guide to the Astronomical X-Ray satellite ROSAT. Intended to provide ROSAT users with an overview of all technical aspects of ROSAT. With special emphasis on the calibration of the various instruments. Garching bei Muenchen, 1996. 212 pages.
  * Brinchmann et al. (2004) Brinchmann, J., Charlot, S., White, S. D. M., et al. 2004, MNRAS, 351, 1151. doi:10.1111/j.1365-2966.2004.07881.x
  * Böhringer et al. (2000) Böhringer, H., Voges, W., Huchra, J. P., et al. 2000, ApJS, 129, 435. doi:10.1086/313427
  * Böhringer et al. (2004) Böhringer, H., Schuecker, P., Guzzo, L., et al. 2004, A&A, 425, 367. doi:10.1051/0004-6361:20034484
  * Charlot & Fall (2000) Charlot, S. & Fall, S. M. 2000, ApJ, 539, 718. doi:10.1086/309250
  * Cid Fernandes et al. (2010) Cid Fernandes, R., Stasińska, G., Schlickmann, M. S., et al. 2010, MNRAS, 403, 1036. doi:10.1111/j.1365-2966.2009.16185.x
  * Cid Fernandes et al. (2011) Cid Fernandes, R., Stasińska, G., Mateus, A., et al. 2011, MNRAS, 413, 1687. doi:10.1111/j.1365-2966.2011.18244.x
  * Cluver et al. (2020) Cluver, M. E., Jarrett, T. H., Taylor, E. N., et al. 2020, arXiv:2006.07535
  * Coenda et al. (2012) Coenda, V., Muriel, H., & Martínez, H. J. 2012, A&A, 543, A119
  * Coenda et al. (2015) Coenda, V., Muriel, H., & Martínez, H. J. 2015, A&A, 573, A96. doi:10.1051/0004-6361/201424870
  * Condon et al. (1998) Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693. doi:10.1086/300337
  * Cooray (2006) Cooray, A. 2006, MNRAS, 365, 842. doi:10.1111/j.1365-2966.2005.09747.x
  * Coziol & Plauchu-Frayn (2007) Coziol, R. & Plauchu-Frayn, I. 2007, AJ, 133, 2630. doi:10.1086/513514
  * Croton et al. (2006) Croton, D. J., Springel, V., White, S. D. M., et al. 2006, MNRAS, 365, 11. doi:10.1111/j.1365-2966.2005.09675.x
  * Cui et al. (2021) Cui, W., Davé, R., Peacock, J. A., et al. 2021, Nature Astronomy. doi:10.1038/s41550-021-01404-1
  * Dai et al. (2007) Dai, X., Kochanek, C. S., & Morgan, N. D. 2007, ApJ, 658, 917. doi:10.1086/509651
  * Dariush et al. (2010) Dariush, A. A., Raychaudhury, S., Ponman, T. J., et al. 2010, MNRAS, 405, 1873. doi:10.1111/j.1365-2966.2010.16569.x
  * Dark Energy Survey Collaboration et al. (2016) Dark Energy Survey Collaboration, Abbott, T., Abdalla, F. B., et al. 2016, MNRAS, 460, 1270. doi:10.1093/mnras/stw641
  * Davies et al. (2019) Davies, L. J. M., Robotham, A. S. G., Lagos, C. del P., et al. 2019, MNRAS, 483, 5444. doi:10.1093/mnras/sty3393
  * Dekel & Birnboim (2006) Dekel, A. & Birnboim, Y. 2006, MNRAS, 368, 2. doi:10.1111/j.1365-2966.2006.10145.x
  * Díaz-Giménez & Zandivarez (2015) Díaz-Giménez, E. & Zandivarez, A. 2015, A&A, 578, A61. doi:10.1051/0004-6361/201425267
  * Díaz-Giménez et al. (2018) Díaz-Giménez, E., Zandivarez, A., & Taverna, A. 2018, A&A, 618, A157. doi:10.1051/0004-6361/201833329
  * Dickey & Lockman (1990) Dickey, J. M. & Lockman, F. J. 1990, ARA&A, 28, 215. doi:10.1146/annurev.aa.28.090190.001243
  * Eckert et al. (2011) Eckert, D., Molendi, S., & Paltani, S. 2011, A&A, 526, A79. doi:10.1051/0004-6361/201015856
  * Efstathiou et al. (1988) Efstathiou, G., Ellis, R. S., & Peterson, B. A. 1988, MNRAS, 232, 431
  * Fabian (2012) Fabian, A. C. 2012, ARA&A, 50, 455. doi:10.1146/annurev-astro-081811-125521
  * Farhang et al. (2017) Farhang, A., Khosroshahi, H. G., Mamon, G. A., et al. 2017, ApJ, 840, 58. doi:10.3847/1538-4357/aa6b00
  * Farouki & Shapiro (1981) Farouki, R., & Shapiro, S. L. 1981, ApJ, 243, 32
  * Feng et al. (2019) Feng, S., Shen, S.-Y., Yuan, F.-T., et al. 2019, ApJ, 880, 114
  * Gunn & Gott (1972) Gunn, J. E., & Gott, J. R. 1972, ApJ, 176, 1
  * Guo et al. (2010) Guo, Q., White, S., Li, C., et al. 2010, MNRAS, 404, 1111. doi:10.1111/j.1365-2966.2010.16341.x
  * Haas et al. (2012) Haas, M. R., Schaye, J., & Jeeson-Daniel, A. 2012, MNRAS, 419, 2133. doi:10.1111/j.1365-2966.2011.19863.x
  * Harrison (2017) Harrison, C. M. 2017, Nature Astronomy, 1, 0165. doi:10.1038/s41550-017-0165
  * Henriques et al. (2017) Henriques, B. M. B., White, S. D. M., Thomas, P. A., et al. 2017, MNRAS, 469, 2626. doi:10.1093/mnras/stx1010
  * Hickson (1982) Hickson, P. 1982, ApJ, 255, 382
  * Jiang et al. (2014) Jiang, C. Y., Jing, Y. P., & Han, J. 2014, ApJ, 790, 7. doi:10.1088/0004-637X/790/1/7
  * Jing et al. (1998) Jing, Y. P., Mo, H. J., & Börner, G. 1998, ApJ, 494, 1. doi:10.1086/305209
  * Kauffmann et al. (2003) Kauffmann, G., Heckman, T. M., White, S. D. M., et al. 2003, MNRAS, 341, 33. doi:10.1046/j.1365-8711.2003.06291.x
  * Kauffmann et al. (2003) Kauffmann, G., Heckman, T. M., Tremonti, C., et al. 2003, MNRAS, 346, 1055. doi:10.1111/j.1365-2966.2003.07154.x
  * Kauffmann et al. (2004) Kauffmann, G., White, S. D. M., Heckman, T. M., et al. 2004, MNRAS, 353, 713. doi:10.1111/j.1365-2966.2004.08117.x
  * Kauffmann et al. (2013) Kauffmann, G., Li, C., Zhang, W., et al. 2013, MNRAS, 430, 1447. doi:10.1093/mnras/stt007
  * Kewley et al. (2001) Kewley, L. J., Dopita, M. A., Sutherland, R. S., et al. 2001, ApJ, 556, 121. doi:10.1086/321545
  * Khosroshahi et al. (2017) Khosroshahi, H. G., Raouf, M., Miraghaei, H., et al. 2017, ApJ, 842, 81. doi:10.3847/1538-4357/aa7048
  * Knobel et al. (2015) Knobel, C., Lilly, S. J., Woo, J., et al. 2015, ApJ, 800, 24. doi:10.1088/0004-637X/800/1/24
  * Koester et al. (2007) Koester, B. P., McKay, T. A., Annis, J., et al. 2007, ApJ, 660, 239. doi:10.1086/509599
  * Kravtsov et al. (2004) Kravtsov, A. V., Berlind, A. A., Wechsler, R. H., et al. 2004, ApJ, 609, 35. doi:10.1086/420959
  * Lange et al. (2018) Lange, J. U., van den Bosch, F. C., Hearin, A., et al. 2018, MNRAS, 473, 2830. doi:10.1093/mnras/stx2434
  * Lauer et al. (2007) Lauer, T. R., Faber, S. M., Richstone, D., et al. 2007, ApJ, 662, 808. doi:10.1086/518223
  * Li et al. (2006) Li, C., Kauffmann, G., Jing, Y. P., et al. 2006, MNRAS, 368, 21. doi:10.1111/j.1365-2966.2006.10066.x
  * Li et al. (2020) Li, P., Wang, H., Mo, H. J., et al. 2020, ApJ, 902, 75. doi:10.3847/1538-4357/abb66c
  * Li & Cen (2020) Li, Z. & Cen, R. 2020, ApJ, 898, 39. doi:10.3847/1538-4357/ab9811
  * Liske et al. (2015) Liske, J., Baldry, I. K., Driver, S. P., et al. 2015, MNRAS, 452, 2087
  * Liu et al. (2010) Liu, L., Yang, X., Mo, H. J., et al. 2010, ApJ, 712, 734. doi:10.1088/0004-637X/712/1/734
  * Liu et al. (2019) Liu, C., Hao, L., Wang, H., et al. 2019, ApJ, 878, 69. doi:10.3847/1538-4357/ab1ea0
  * Lovisari et al. (2021) Lovisari, L., Ettori, S., Gaspari, M., et al. 2021, Universe, 7, 139. doi:10.3390/universe7050139
  * Lu et al. (2014) Lu, Z., Mo, H. J., Lu, Y., et al. 2014, MNRAS, 439, 1294. doi:10.1093/mnras/stu016
  * Luo et al. (2015) Luo, A.-L., Zhao, Y.-H., Zhao, G., et al. 2015, Research in Astronomy and Astrophysics, 15, 1095
  * Mamon (1993) Mamon, G. 1993, N-body Problems and Gravitational Dynamics, 188
  * Mamon (2007) Mamon, G. A. 2007, Groups of Galaxies in the Nearby Universe, 203
  * Mandelbaum et al. (2016) Mandelbaum, R., Wang, W., Zu, Y., et al. 2016, MNRAS, 457, 3200. doi:10.1093/mnras/stw188
  * Mantz et al. (2010) Mantz, A., Allen, S. W., Ebeling, H., et al. 2010, MNRAS, 406, 1773. doi:10.1111/j.1365-2966.2010.16993.x
  * Martínez & Zandivarez (2012) Martínez, H. J. & Zandivarez, A. 2012, MNRAS, 419, L24. doi:10.1111/j.1745-3933.2011.01170.x
  * Matthee et al. (2017) Matthee, J., Schaye, J., Crain, R. A., et al. 2017, MNRAS, 465, 2381. doi:10.1093/mnras/stw2884
  * McConnachie et al. (2009) McConnachie, A. W., Patton, D. R., Ellison, S. L., & Simard, L. 2009, MNRAS, 395, 255
  * Mendel et al. (2011) Mendel, J. T., Ellison, S. L., Simard, L., Patton, D. R., & McConnachie, A. W. 2011, MNRAS, 418, 1409
  * Mo et al. (1999) Mo, H. J., Mao, S., & White, S. D. M. 1999, MNRAS, 304, 175. doi:10.1046/j.1365-8711.1999.02289.x
  * Moore et al. (1996) Moore, B., Katz, N., Lake, G., Dressler, A., & Oemler, A. 1996, Nature, 379, 613
  * More et al. (2011) More, S., van den Bosch, F. C., Cacciato, M., et al. 2011, MNRAS, 410, 210. doi:10.1111/j.1365-2966.2010.17436.x
  * More (2012) More, S. 2012, ApJ, 761, 127. doi:10.1088/0004-637X/761/2/127
  * Moster et al. (2018) Moster, B. P., Naab, T., & White, S. D. M. 2018, MNRAS, 477, 1822. doi:10.1093/mnras/sty655
  * Osmond & Ponman (2004) Osmond, J. P. F. & Ponman, T. J. 2004, MNRAS, 350, 1511. doi:10.1111/j.1365-2966.2004.07742.x
  * Osterbrock (1989) Osterbrock, D. E. 1989, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei, by Donald E. Osterbrock. Published by University Science Books, ISBN 0-935702-22-9, 408pp, 1989.
  * Paranjape & Sheth (2012) Paranjape, A. & Sheth, R. K. 2012, MNRAS, 423, 1845. doi:10.1111/j.1365-2966.2012.21008.x
  * Pasquali et al. (2009) Pasquali, A., van den Bosch, F. C., Mo, H. J., et al. 2009, MNRAS, 394, 38. doi:10.1111/j.1365-2966.2008.14233.x
  * Peacock & Smith (2000) Peacock, J. A. & Smith, R. E. 2000, MNRAS, 318, 1144. doi:10.1046/j.1365-8711.2000.03779.x
  * Peng et al. (2010) Peng, Y.-. jie ., Lilly, S. J., Kovač, K., et al. 2010, ApJ, 721, 193. doi:10.1088/0004-637X/721/1/193
  * Peng et al. (2012) Peng, Y.-. jie ., Lilly, S. J., Renzini, A., et al. 2012, ApJ, 757, 4. doi:10.1088/0004-637X/757/1/4
  * Peng et al. (2014) Peng, Y.-. jie ., Lilly, S. J., Renzini, A., et al. 2014, ApJ, 790, 95. doi:10.1088/0004-637X/790/2/95
  * Pillepich et al. (2018) Pillepich, A., Springel, V., Nelson, D., et al. 2018, MNRAS, 473, 4077. doi:10.1093/mnras/stx2656
  * Ragagnin et al. (2019) Ragagnin, A., Dolag, K., Moscardini, L., et al. 2019, MNRAS, 486, 4001. doi:10.1093/mnras/stz1103
  * Raouf et al. (2014) Raouf, M., Khosroshahi, H. G., Ponman, T. J., et al. 2014, MNRAS, 442, 1578. doi:10.1093/mnras/stu963
  * Raouf et al. (2018) Raouf, M., Khosroshahi, H. G., Mamon, G. A., et al. 2018, ApJ, 863, 40. doi:10.3847/1538-4357/aace57
  * Raouf et al. (2019) Raouf, M., Smith, R., Khosroshahi, H. G., et al. 2019, ApJ, 887, 264. doi:10.3847/1538-4357/ab5581
  * Reddick et al. (2013) Reddick, R. M., Wechsler, R. H., Tinker, J. L., et al. 2013, ApJ, 771, 30. doi:10.1088/0004-637X/771/1/30
  * Robotham et al. (2011) Robotham, A. S. G., Norberg, P., Driver, S. P., et al. 2011, MNRAS, 416, 2640. doi:10.1111/j.1365-2966.2011.19217.x
  * Saro et al. (2013) Saro, A., Mohr, J. J., Bazin, G., et al. 2013, ApJ, 772, 47. doi:10.1088/0004-637X/772/1/47
  * Schechter (1976) Schechter, P. 1976, ApJ, 203, 297
  * Shen et al. (2008) Shen, S., Kauffmann, G., von der Linden, A., et al. 2008, MNRAS, 389, 1074. doi:10.1111/j.1365-2966.2008.13647.x
  * Shen et al. (2014) Shen, S., Yang, X., Mo, H., et al. 2014, ApJ, 782, 23. doi:10.1088/0004-637X/782/1/23
  * Shen et al. (2016) Shen, S.-Y., Argudo-Fernández, M., Chen, L., et al. 2016, Research in Astronomy and Astrophysics, 16, 43
  * Simha et al. (2012) Simha, V., Weinberg, D. H., Davé, R., et al. 2012, MNRAS, 423, 3458. doi:10.1111/j.1365-2966.2012.21142.x
  * Sohn et al. (2015) Sohn, J., Hwang, H. S., Geller, M. J., et al. 2015, Journal of Korean Astronomical Society, 48, 381
  * Sohn et al. (2016) Sohn, J., Geller, M. J., Hwang, H. S., Zahid, H. J., & Lee, M. G. 2016, ApJS, 225, 23
  * Stewart et al. (2009) Stewart, K. R., Bullock, J. S., Barton, E. J., et al. 2009, ApJ, 702, 1005. doi:10.1088/0004-637X/702/2/1005
  * Tal et al. (2012) Tal, T., Wake, D. A., van Dokkum, P. G., et al. 2012, ApJ, 746, 138. doi:10.1088/0004-637X/746/2/138
  * Tammann et al. (1979) Tammann, G. A., Yahil, A., & Sandage, A. 1979, ApJ, 234, 775
  * Torres-Flores et al. (2009) Torres-Flores, S., Mendes de Oliveira, C., de Mello, D. F., et al. 2009, A&A, 507, 723. doi:10.1051/0004-6361/200911878
  * Torres-Flores et al. (2010) Torres-Flores, S., Mendes de Oliveira, C., Amram, P., et al. 2010, A&A, 521, A59. doi:10.1051/0004-6361/200913912
  * Torres-Flores et al. (2014) Torres-Flores, S., Amram, P., Mendes de Oliveira, C., et al. 2014, MNRAS, 442, 2188. doi:10.1093/mnras/stu1002
  * Trevisan & Mamon (2017) Trevisan, M. & Mamon, G. A. 2017, MNRAS, 471, 2022. doi:10.1093/mnras/stx1656
  * Vale & Ostriker (2006) Vale, A. & Ostriker, J. P. 2006, MNRAS, 371, 1173. doi:10.1111/j.1365-2966.2006.10605.x
  * van den Bosch et al. (2007) van den Bosch, F. C., Yang, X., Mo, H. J., et al. 2007, MNRAS, 376, 841. doi:10.1111/j.1365-2966.2007.11493.x
  * van den Bosch et al. (2008) van den Bosch, F. C., Pasquali, A., Yang, X., et al. 2008, arXiv:0805.0002
  * Voges et al. (1999) Voges, W., Aschenbach, B., Boller, T., et al. 1999, A&A, 349, 389
  * Von Der Linden et al. (2007) Von Der Linden, A., Best, P. N., Kauffmann, G., et al. 2007, MNRAS, 379, 867. doi:10.1111/j.1365-2966.2007.11940.x
  * Wang et al. (2014) Wang, L., Yang, X., Shen, S., et al. 2014, MNRAS, 439, 611. doi:10.1093/mnras/stt2481
  * Wang et al. (2018) Wang, E., Wang, H., Mo, H., et al. 2018, ApJ, 860, 102. doi:10.3847/1538-4357/aac4a5
  * Weinmann et al. (2006) Weinmann, S. M., van den Bosch, F. C., Yang, X., et al. 2006, MNRAS, 366, 2. doi:10.1111/j.1365-2966.2005.09865.x
  * Wen & Han (2013) Wen, Z. L. & Han, J. L. 2013, MNRAS, 436, 275. doi:10.1093/mnras/stt1581
  * Wetzel et al. (2012) Wetzel, A. R., Tinker, J. L., & Conroy, C. 2012, MNRAS, 424, 232. doi:10.1111/j.1365-2966.2012.21188.x
  * White et al. (1997) White, D. A., Jones, C., & Forman, W. 1997, MNRAS, 292, 419. doi:10.1093/mnras/292.2.419
  * Woo et al. (2013) Woo, J., Dekel, A., Faber, S. M., et al. 2013, MNRAS, 428, 3306. doi:10.1093/mnras/sts274
  * Woo et al. (2015) Woo, J., Dekel, A., Faber, S. M., et al. 2015, MNRAS, 448, 237. doi:10.1093/mnras/stu2755
  * Yang et al. (2003) Yang, X., Mo, H. J., & van den Bosch, F. C. 2003, MNRAS, 339, 1057. doi:10.1046/j.1365-8711.2003.06254.x
  * Yang et al. (2005) Yang, X., Mo, H. J., van den Bosch, F. C., et al. 2005, MNRAS, 356, 1293
  * Yang et al. (2007) Yang, X., Mo, H. J., van den Bosch, F. C., et al. 2007, ApJ, 671, 153
  * Yang et al. (2008) Yang, X., Mo, H. J., & van den Bosch, F. C. 2008, ApJ, 676, 248. doi:10.1086/528954
  * Yang et al. (2012) Yang, X., Mo, H. J., van den Bosch, F. C., et al. 2012, ApJ, 752, 41
  * Yang et al. (2013) Yang, X., Mo, H. J., van den Bosch, F. C., et al. 2013, ApJ, 770, 115. doi:10.1088/0004-637X/770/2/115
  * Yang et al. (2021) Yang, X., Xu, H., He, M., et al. 2021, ApJ, 909, 143. doi:10.3847/1538-4357/abddb2
  * Zehavi et al. (2018) Zehavi, I., Contreras, S., Padilla, N., et al. 2018, ApJ, 853, 84. doi:10.3847/1538-4357/aaa54a
  * Zhang et al. (2021) Zhang, C., Peng, Y., Ho, L. C., et al. 2021, ApJ, 911, 57. doi:10.3847/1538-4357/abd723
  * Zhang et al. (2021) Zhang, Z., Wang, H., Luo, W., et al. 2021, A&A, 650, A155. doi:10.1051/0004-6361/202040150
  * Zhao et al. (2009) Zhao, D. H., Jing, Y. P., Mo, H. J., et al. 2009, ApJ, 707, 354. doi:10.1088/0004-637X/707/1/354
  * Zheng & Shen (2020) Zheng, Y.-L., & Shen, S.-Y. 2020, ApJS, 246, 12
  * Zheng & Shen (2021) Zheng, Y.-L. & Shen, S.-Y. 2021, ApJ, 911, 105. doi:10.3847/1538-4357/abeaa2
  * Zheng et al. (2005) Zheng, Z., Berlind, A. A., Weinberg, D. H., et al. 2005, ApJ, 633, 791. doi:10.1086/466510
  * Zu & Mandelbaum (2016) Zu, Y. & Mandelbaum, R. 2016, MNRAS, 457, 4360. doi:10.1093/mnras/stw221