# The Three Hundred project: Galaxy groups do not survive cluster infall

Roan Haggar,1,2 Ulrike Kuchner,1 Meghan E. Gray,1 Frazer R. Pearce,1 Alexander
Knebe,3,4,5 Gustavo Yepes,3,4 Weiguang Cui6

1School and Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD,
UK 2Waterloo Centre for Astrophysics, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada 3Departamento de Física Teórica, Módulo 15 Universidad
Autónoma de Madrid, 28049 Madrid, Spain 4Centro de Investigación Avanzada en
Física Fundamental (CIAFF), Universidad Autónoma de Madrid, 28049 Madrid,
Spain 5International Centre for Radio Astronomy Research, The University of
Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009,
Australia 6Institute for Astronomy, University of Edinburgh, Royal
Observatory, Edinburgh EH9 3HJ, UK E-mail<EMAIL_ADDRESS>
(Accepted 2022 September 26. Received 2022 September 26; in original form 2022
March 29)

###### Abstract

Galaxy clusters grow by accreting galaxies as individual objects, or as
members of a galaxy group. These groups can strongly impact galaxy evolution,
stripping the gas from galaxies, and enhancing the rate of galaxy mergers.
However, it is not clear how the dynamics and structure of groups are affected
when they interact with a large cluster, or whether all group members
necessarily experience the same evolutionary processes. Using data from The
Three Hundred project, a suite of 324 hydrodynamical resimulations of large
galaxy clusters, we study the properties of 1340 groups passing through a
cluster. We find that half of group galaxies become gravitationally unbound
from the group by the first pericentre, typically just $0.5-1$ Gyr after
cluster entry. Most groups quickly mix with the cluster satellite population;
only $8\%$ of infalling group haloes later leave the cluster, although for
nearly half of these, all of their galaxies have become unbound, tidally
disrupted or merged into the central by this stage. The position of galaxies
in group-centric phase space is also important – only galaxies near the centre
of a group ($r\lesssim 0.7R_{200}$) remain bound once a group is inside a
cluster, and slow-moving galaxies in the group centre are likely to be tidally
disrupted, or merge with another galaxy. This work will help future
observational studies to constrain the environmental histories of group
galaxies. For instance, groups observed inside or nearby to clusters have
likely approached very recently, meaning that their galaxies will not have
experienced a cluster environment before.

###### keywords:

galaxies: clusters: general – galaxies: groups: general – galaxies: general –
methods: numerical

††pubyear: 2022††pagerange: The Three Hundred project: Galaxy groups do not
survive cluster infall–A

##  1 Introduction

Galaxy clusters grow by the gravitational accretion of smaller cosmic
structures. These accreted structures vary in size, from individual galaxies,
to galaxy groups containing tens or hundreds of member galaxies, to major
cluster-cluster mergers involving thousands of galaxies (Moore et al., 1999;
Frenk & White, 2012). Such hierarchical structure formation is one of the
cornerstones of the Lambda cold dark matter ($\Lambda$CDM) model of the
Universe (White & Rees, 1978; Navarro et al., 1996). The wide range in sizes
of their dark matter haloes, plus the existence of other structures such as
cosmological filaments and walls (Bond et al., 1996; Hahn et al., 2007),
results in a variety of cosmic environments in which galaxies can be found.

It is now well-established that the properties of galaxies strongly depend on
where they are located. An early study by Dressler (1980) revealed that
cluster environments contain mostly early-type galaxies, whereas galaxies in
field regions typically have late-type morphologies111This idea had been noted
previously in other works, such as Hubble (1936) and Zwicky (1937)..
Furthermore, cluster galaxies have quenched star formation rates (Balogh et
al., 1999; McNab et al., 2021) and lower gas fractions (Jaffé et al., 2015)
compared to field galaxies, across a large range of redshifts (Quadri et al.,
2012). Numerous mechanisms can explain this difference in gas content and star
formation rate, including slow quenching processes such as galaxy starvation
(Larson et al., 1980; Maier et al., 2016; Maier et al., 2019), and rapid
processes such as ram pressure stripping (Gunn & Gott, 1972; Abadi et al.,
1999; Zabel et al., 2019).

Although they represent the densest, most extreme galaxy environments,
clusters are not the only structures that can dramatically impact galaxy
evolution. Intermediate density environments can also play an important role:
for instance, galaxy groups have been shown to enhance the rate of galaxy
mergers, due to their combination of a high galaxy number density, and low
velocity dispersion222The relative velocities of merging galaxies are usually
$<500$ km s-1 (Lotz et al., 2008; An et al., 2019), but dark matter haloes
with masses greater than $10^{14}\ M_{\odot}$ typically have velocity
dispersions greater than $500$ km s-1 (McClintock et al., 2019; Wetzell et
al., 2021). Consequently, mergers are more likely in group-sized haloes, with
masses less than $10^{14}\ M_{\odot}$. (Jian et al., 2012). Mergers
drastically impact the evolution of galaxies, altering their morphology and
potentially triggering outflows and AGN feedback that can remove gas.

A consequence of this connection between galaxies and their environments is
that a galaxy’s evolution is not just impacted by the environment in which it
is currently found – it can also be affected by the environments through which
a galaxy has previously passed. In the context of clusters, ‘pre-processing’
describes the environmental mechanisms that act on a galaxy before it is
accreted by a cluster. For example, galaxies can enter a cluster through
cosmological filaments, which can quench star formation similarly to clusters,
albeit to a lesser degree (Kraljic et al., 2018; Laigle et al., 2018). This
results in degeneracy, as it is not immediately clear whether cluster galaxies
have been quenched by the cluster itself, or are quenched due to pre-
processing. However, it is clear that these filaments are an important factor
to consider: for instance, Kuchner et al. (2022) found that $45\%$ of cluster
galaxies are accreted via filaments.

As galaxies can also enter clusters as members of galaxy groups, these are
another contributor to pre-processing, although the exact degree of groups’
contribution is debated. Some simulations (McGee et al., 2009; Han et al.,
2018) and observations (Dressler et al., 2013) find that close to half of all
cluster members have been accreted as members of galaxy groups, while others
(Arthur et al., 2019) find a much lower fraction. There are multiple
explanations for this. For example, previous studies have shown that this
fraction depends on the stellar mass of the accreted galaxies (De Lucia et
al., 2012), and whether hydrodynamical or $N$-body simulations are used
(Haggar et al., 2021). Additionally, the definition of a ‘galaxy group’ is not
standardised, and different definitions can lead to different conclusions.
Various studies have identified group members as galaxies that lie within the
radius of a host group halo (Arthur et al., 2019; Donnari et al., 2021), that
satisfy a boundness criterion (Han et al., 2018; Choque-Challapa et al.,
2019), or by using a Friends-of-Friends algorithm (Benavides et al., 2020),
all of which can result in different selections of group members. Furthermore,
Berrier et al. (2009) found that, although $30\%$ of cluster members (with
dark matter halo masses greater than $10^{11.5}\ h^{-1}M_{\odot}$) are
accreted via group haloes, half of these ‘groups’ only contain two or three
galaxies. Clearly, the minimum (and maximum) size of what constitutes a group
is also an important consideration.

Both theoretical and observational studies have shown that the effects of a
group environment on the evolutionary processes in galaxies can be enhanced
even further when a group enters a cluster. Galaxy mergers (Benavides et al.,
2020) and gas removal (Pallero et al., 2019; Kleiner et al., 2021) are common
in infalling groups, and multiple studies have connected this galaxy evolution
to the external forces acting on a group, such as the effects of large-scale
structure and clusters. Vijayaraghavan & Ricker (2013) used cosmological
simulations to show that mergers, ram pressure stripping, and tidal truncation
of galaxy haloes are all enhanced further when their groups enter clusters,
for a variety of reasons – for example, the intra-group medium is shocked
during a group-cluster merger, increasing its density and thus increasing the
ram pressure stripping of the group members. Similar mechanisms have been
described in previous works, such as Mauduit & Mamon (2007), who showed that
near the centres of clusters lying in the core of the Shapley Supercluster,
galaxies have lower radio loudness than galaxies elsewhere. They attributed
this to the enhanced ram pressure stripping experienced by galaxies in shocked
regions of merging clusters. In a related observational study, Roberts &
Parker (2017) found that dynamically relaxed groups, which are typically
isolated and slowly-growing, contain a smaller fraction of star-forming
galaxies than unrelaxed groups. Again, this indicates that the processing of
galaxies in groups is dependent on the disturbance of these groups by the
larger environment in which they are located (see also Gouin et al., 2021).

All of this means that galaxies that have joined clusters as members of a
group have experienced different evolutionary processes to those that have
joined as individuals. Bahé et al. (2019) used the Hydrangea suite of
hydrodynamical simulations (Bahé et al., 2017) to study the survival fractions
of galaxies entering clusters – in their case, galaxies that do not ‘survive’
are no longer resolved in the simulations, meaning they have either merged
into a more massive galaxy (often a group central), or have been stripped
below the total mass limit of $5\times 10^{8}\ M_{\odot}$. Bahé et al. (2019)
showed that, after an infalling group enters a cluster, only $\sim 50\%$ of
its member galaxies survive to $z=0$. In contrast, they found that more than
$90\%$ of galaxies that have not experienced any pre-processing survive to
$z=0$. This survival fraction is higher than in some other studies, although
much of the prior work in this field has used $N$-body simulations (e.g. Gill
et al., 2004b), in which substructure can be more easily stripped (Smith et
al., 2016). The results of Bahé et al. (2019) show that group members are
particularly strongly influenced within clusters, and that they can be very
heavily disturbed during accretion onto a cluster.

Moreover, previous work has hinted that galaxy groups themselves can be
heavily disrupted when entering a cluster. Choque-Challapa et al. (2019) found
that, using dark matter-only simulations and a similar group definition as is
used in this work, over $90\%$ of group members become unbound after a group
enters a cluster, and that these galaxies quickly form part of the cluster
population of galaxies. Furthermore, González-Casado et al. (1994) showed that
tidal forces from clusters can rapidly increase the internal energy of
infalling groups, by up to a factor of 10 for the smallest groups. This can
allow these groups to be disrupted, although it should be noted that absorbing
more energy than the binding energy does not necessarily lead to the complete
disruption of groups (van den Bosch et al., 2018).

However, beyond this, there is little work that has examined in detail how the
dynamics of galaxy groups evolve when they are accreted by a cluster,
particularly with large numbers of clusters in hydrodynamical simulations.
While previous studies have looked at the overall disruption of groups that
enter a cluster and the subsequent ‘post-processing’ of their constituent
galaxies, we do not currently have a detailed understanding of the timescales
over which groups change, and how the evolutionary processes that galaxies
experience are affected by the group dynamics (Cohn, 2012; Bahé et al., 2019).

In this work, we use The Three Hundred project, a mass-complete sample of 324
galaxy clusters taken from a $1\ h^{-1}$ Gpc cosmological volume. These are
resimulated out to distances of several times the $R_{200}$ of the cluster,
where $R_{200}$ is the radius within which the mean density of a cluster is
equal to 200 times the critical density of the Universe. We use these
simulations to study the evolution of groups as they enter galaxy clusters,
and the processes that galaxies in these groups experience in their subsequent
passage through the cluster halo. Specifically, we look at how the phase space
of groups evolves: that is, how the positions and speeds of galaxies change,
relative to the group that they are bound to. We make comparisons between
groups before and after they pass through a cluster, to find the cumulative
effect that a cluster has on the dynamics and structure of galaxy groups.
Then, we look at the fates of group galaxies, categorising them based on the
processes they experience in the several Gyr after entering a cluster (such as
mergers and stripping), and how this depends on the structure of groups.
Finally, we discuss how this theoretical work can help observational studies.

The paper is structured as follows: In Section 2 we introduce the simulation
data that we use, and the methods we use to analyse groups. In Section 3 we
show how the internal dynamics of groups change as they pass through a
cluster, and in Section 4 we focus on the state of galaxies and groups after
passing through a cluster. Finally, we summarise our findings in Section 5.

##  2 Simulations & Numerical methods

Below we detail the methods and data used in this work. Much of this,
particularly Section 2.1 and Section 2.2, build on the analysis in our
previous work, Haggar et al. (2021), in which we compare the substructure of
galaxy groups and galaxy clusters in hydrodynamical and dark matter-only
simulations.

###  2.1 Simulation data

This work utilises data from The Three Hundred project, a suite of 324
hydrodynamical resimulations of large galaxy clusters. The simulations were
produced by extracting the 324 most massive clusters from the dark matter-only
MDPL2 MultiDark simulation (Klypin et al., 2016)333The MultiDark simulations
are publicly available from the cosmosim database, https://www.cosmosim.org.,
and resimulating each from its initial conditions with baryonic physics. This
was done by taking all dark matter particles within $15\ h^{-1}$ Mpc of the
cluster centre at $z=0$ (between $7-10R_{200}$ for the range of cluster masses
in the sample), tracing the particles back to their initial positions, and
then splitting each one into a dark matter and a gas particle, with masses set
by the baryonic matter fraction of the Universe. Lower-resolution particles
were used beyond $15\ h^{-1}$ Mpc to model any tidal effects of the
surrounding large-scale structure.

The MDPL2 simulation involves a box with sides of comoving length $1\ h^{-1}$
Gpc, simulated using Planck cosmology ($\Omega_{\rm{M}}=0.307$,
$\Omega_{\rm{B}}=0.048$, $\Omega_{\Lambda}=0.693$, $h=0.678$,
$\sigma_{8}=0.823$, $n_{\rm{s}}=0.96$) (Planck Collaboration et al., 2016).
The same box size and cosmology are used for each of the cluster simulations
in The Three Hundred, so that each cluster is embedded in a comoving box of
size $1\ h^{-1}$ Gpc, most of which is occupied by the low-resolution
particles described in the previous paragraph. Consequently, the lengths and
distances quoted throughout this work are also given in comoving coordinates.

The hydrodynamical resimulations were carried out using the GadgetX code.
GadgetX is a modified version of the Gadget3 code, which is itself an updated
version of the Gadget2 code, and uses a smoothed-particle hydrodynamics scheme
to fully evolve the gas component of the simulations (Springel, 2005; Beck et
al., 2016). The final dataset comprises of a mass-complete cluster sample from
$M_{200}=5\times 10^{14}\ h^{-1}M_{\odot}$ to $M_{200}=2.6\times 10^{15}\
h^{-1}M_{\odot}$, where $M_{200}$ is the mass contained within a sphere of
radius $R_{200}$. The dark matter and gas particles in the simulations have
masses of $m_{\rm{DM}}=1.27\times 10^{9}\ h^{-1}M_{\odot}$ and
$m_{\rm{gas}}=2.36\times 10^{8}\ h^{-1}M_{\odot}$ respectively. The
simulations also contain stellar particles of variable masses, typically with
$m_{\rm{star}}\sim 4\times 10^{7}\ h^{-1}M_{\odot}$, produced by the
stochastic star-formation model that is implemented by GadgetX (Tornatore et
al., 2007; Murante et al., 2010; Rasia et al., 2015). A Plummer equivalent
gravitational softening length of 6.5 $h^{-1}$ kpc is used for the dark matter
and gas particles, and 5 $h^{-1}$ kpc for the stellar particles. The Three
Hundred dataset is described in more extensive detail in Cui et al. (2018),
and has been used in numerous previous studies to examine galaxy groups
(Haggar et al., 2021), environment (Wang et al., 2018), cosmic filaments
(Kuchner et al., 2020; Rost et al., 2021; Kotecha et al., 2022), backsplash
galaxies (Haggar et al., 2020) and ram pressure stripping (Arthur et al.,
2019; Mostoghiu et al., 2021), among other areas. The full simulation suite
also includes simulations with different physics models, however in this work
we only use the GadgetX simulations.

####  2.1.1 Galaxy identification and tree-building

The data for each cluster in The Three Hundred consists of 129 snapshots saved
between $z=16.98$ and $z=0$, separated by approximately $0.3$ Gyr at low
redshift. To identify the haloes and subhaloes, each snapshot was processed
using the Amiga Halo Finder, ahf444http://popia.ft.uam.es/AHF (see Gill et al.
(2004a) and Knollmann & Knebe (2009) for further details). ahf operates by
identifying peaks in the matter density field, and returns the positions and
velocities of haloes and subhaloes, as well as their radii, their mass in gas,
stars and dark matter, and a host of other properties.

The halo merger trees were built using mergertree, a tree-builder that forms
part of the ahf package. For each halo in a given snapshot, this tree-builder
calculates a merit function with respect to all haloes in previous snapshots;
specifically, mergertree uses the merit function $M_{\rm{i}}$, as described in
Table B1 of Knebe et al. (2013). This merit function is then used to identify
a main progenitor, plus other progenitors, based on the number of particles
that they share with the halo of interest. The tree-builder has the ability to
skip snapshots, and thus is able to ‘patch’ over gaps in the merger tree, for
example when a subhalo is near to the centre of its host halo and so is not
easy to identify against the high background density (Onions et al., 2012). We
also place a limit on the change in mass permitted between successive
snapshots, such that no halo can more than double in dark matter mass. This
helps to prevent ‘mismatches’, caused by a subhalo located close to the centre
of a larger halo being detected as the main halo (as shown in Behroozi et al.,
2015). Additional information on ahf and mergertree can be found in Knebe et
al. (2011a) and Srisawat et al. (2013).

###  2.2 Sub-sample of clusters

Some of the clusters exhibit some minor problems in the trees constructed by
mergertree, which we describe below. However, thanks to the large dataset that
we are using, we can identify and remove these objects, and still be left with
a large sample of simulated clusters.

In some cases the merit function used by mergertree can incorrectly assign
links between haloes in different snapshots. This can lead to an apparent
‘jump’ in the position of a halo or subhalo (in box coordinates), as well as a
sudden change in its properties, due to one halo being incorrectly labelled as
the progenitor of another. These mismatching events are uncommon, typically
only affect a small number of snapshots, and are fairly inconsequential when
they affect individual galaxy haloes. However, the merger tree of the main
cluster halo can also be affected in this way, leading to a sudden change in
the position of the main halo. Such a change in position is particularly
problematic in this work, because it will result in many galaxies and groups
being erroneously tagged as members of a cluster.

These merger tree mismatches are especially common during a major merger
between two haloes. Behroozi et al. (2015) showed that various halo finders
experience this same problem, where two merging haloes of similar size can be
accidentally switched by a tree-builder, leading to the sizes and positions of
haloes appearing to change suddenly and dramatically. Many of the clusters in
The Three Hundred experience major mergers; a recent study, Contreras-Santos
et al. (2022), discusses cluster mergers in The Three Hundred simulations in
detail. In fact, we find that 59 of our 324 simulated clusters experience a
change in position of $>0.5R_{200}(z)$ between two snapshots after $z=1$. We
find that, given that the typical time elapsing between snapshots at this
redshift is $\sim 0.3$ Gyr, this distance is non-physical and so likely due to
these tree-builder issues.

In some cases, the tree-builder instead misses a link in the merger tree,
causing a branch of the merger tree to end prematurely and the history of the
halo before this link to be lost. For 17 clusters, the central cluster halo is
affected in this way, and the evolution of the cluster halo cannot be tracked
back further than $z=0.5$. We choose to also remove these clusters from our
analysis, in order to avoid affecting our results with clusters that do not
have complete, reliable merger trees. Nine of these clusters also experience
the halo mismatches described in the previous paragraph, resulting in a total
of 67 clusters that we choose to remove from our sample.

The remaining 257 clusters have $M_{200}$ masses (dark matter, gas and stars,
including subhaloes) ranging from $5\times 10^{14}\ h^{-1}M_{\odot}$ to
$2.6\times 10^{15}\ h^{-1}M_{\odot}$, with a median value of $8\times 10^{14}\
h^{-1}M_{\odot}$. Their radii ($R_{200}$) range from $1.3\ h^{-1}$ Mpc to
$2.3\ h^{-1}$ Mpc, with a median of $1.5\ h^{-1}$ Mpc.

###  2.3 Galaxy and group selection

In this work, we place lower limits on the total mass (including dark matter,
gas and stars) and the stellar mass of galaxies in the simulations, so that
all the haloes we keep from our halo finder represent real, physical galaxies.
We only examine galaxy haloes with a total mass of $M_{200}\geq 10^{10.5}\
h^{-1}M_{\odot}$, which corresponds to approximately $100$ particles in the
high-resolution regions containing the clusters. We also only use galaxies
with a stellar mass $M_{\rm{star}}\geq 10^{9.5}M_{\odot}$. We consider these
to be physical galaxies that have built up a substantial population of stars –
this cut is approximately equivalent to removing all galaxies with a
luminosity $L<10^{8}L_{\odot}$, whilst keeping all galaxies with
$L>10^{9}L_{\odot}$. This stellar mass cut also allows us to investigate a
similar population of galaxies to upcoming observational studies, such as the
WEAVE555https://www.ing.iac.es//confluence/display/WEAV Wide-Field Cluster
Survey, which will study cluster galaxies down to stellar masses of $\sim
10^{9}\ M_{\odot}$ (e.g. Kuchner et al., 2020). Finally, we remove all
galaxies from our simulations that contain more than $30\%$ of their mass in
stars. These objects are generally found extremely close to the centre of a
larger halo, and so have been heavily stripped (Knebe et al., 2020), leaving
remnants with high stellar mass fractions, whose properties (such as their
radii and masses) are not well-defined by our halo finder. These objects are
very rare, and make up only $1\%$ of all haloes within $5R_{200}$ of the
clusters, so we make the decision to remove these objects from our analysis.
By applying these three constraints to our simulations, we consider all
remaining objects to be realistic galaxies with a significant population of
stars at $z=0$.

####  2.3.1 Group identification

Throughout this work, we identify galaxy groups by taking each galaxy,
assuming its halo to be the host halo of a galaxy group, and then determining
if any other galaxies in the same snapshot are associated with it. We identify
galaxies as being associated with a halo (and thus members of the group) using
the same approach as Han et al. (2018). They assume that a group’s dark matter
halo follows a spherically symmetric NFW density profile (Navarro et al.,
1996), truncated at $R_{200}$. Using this to calculate the gravitational
potential of the group halo, they identify group members as those that satisfy
the criterion given below:

$\frac{v^{2}}{2}+\Phi\left(r\right)<\Phi\left(2.5R_{200}^{\rm{grp}}\right)\,.$
(1)

Here, $v$ is the relative velocity of a galaxy with respect to its group host,
$\Phi(r)$ is the gravitational potential due to the group host at a distance
$r$ from its centre, and $R_{200}^{\rm{grp}}$ is the radius of the group host
halo. It is important to note that this is different to the radius of the host
cluster in each simulation, which is subsequently referred to by
$R_{200}^{\rm{clus}}$. Any galaxies that are less massive than their group
host and that satisfy Eq. 1 are taken to be bound members of this group.
Although we hereafter refer to these group members as being ‘bound’ to their
host group, it is important to note that this definition is not technically
equivalent to gravitational binding. Previous work (e.g. Behroozi et al.,
2013) has shown that halo particles can be gravitationally balanced against
the Hubble flow out to $\sim 4R_{200}$ from the halo centre. However, Eq. 1
places an artificial radial limit on groups, so that galaxies can only be
found as far as $2.5R_{200}^{\rm{grp}}$ from the centre of the group.

This outer limit is the same as was used by Han et al. (2018): their choice
was motivated by the work of Mamon et al. (2004), who showed that backsplash
galaxies are typically found out to approximately $2.5R_{200}$ from their host
halo, but rarely any further. By setting this as the outer limit of a group,
we include almost all galaxies that are on bound orbits around the group
(having passed through its central halo at least once), whilst excluding
galaxies that have not entered the group halo before. Furthermore, the
relative velocity term in Eq. 1 means that only slow-moving galaxies at large
distances are included as group members. Galaxies moving at greater velocities
are excluded from the group, as these are likely ‘fly-by’ galaxies or
‘renegade subhaloes’ (Knebe et al., 2011b), which happen to be passing near to
the group, but are not bound to it.

If a halo has four or more galaxies associated with it that each have a
smaller total mass (including dark matter, gas, and stars) than the halo, we
define this as a group, with the halo being the ‘group host’ halo. Throughout
this work we assume that the central group galaxy in each of these group host
haloes exists at the centre of the halo, which has been shown to be the case
in previous work. Lin & Mohr (2004) used X-ray observations of groups and
clusters with masses similar to those in this work ($10^{13.5}\ M_{\odot}<$$\
M_{200}<\ $$10^{15.3}\ M_{\odot}$) to show that in $75\%$ of these haloes, the
brightest galaxy is located within $0.06R_{200}$ of the halo centre; this
result is corroborated by both Hwang & Lee (2008) and Stott et al. (2012).

Very small groups with fewer than five members are common, but for the mass
constraints that we put in place in Section 2.3, a collection of $\gtrsim 5$
associated galaxies is typically required to define a group (see Tully, 2015,
for example). Additionally, we only study groups with 50 or fewer members, as
detailed in the following section. Again, we stress that this limit applies to
the number of group members that satisfy the mass constraints in Section 2.3,
as is the case throughout the rest of this work unless stated otherwise.

####  2.3.2 Infalling groups

The focus of this work is on the evolution of galaxy groups as they enter and
pass through a galaxy cluster. In order to study this, we identify a sample of
infalling galaxy groups at all redshifts, in the same way as our previous
study (Haggar et al., 2021), and other previous work (e.g. Choque-Challapa et
al., 2019). To do this, we identify all galaxies that have just fallen into
the cluster; these are galaxies that are within $R_{200}^{\rm{clus}}$ of the
cluster centre, having been outside of the cluster in the previous snapshot.
These objects are referred to as the ‘infalling’ galaxies. Note that we do not
distinguish by the time at which the galaxies entered the cluster – these
infall events can happen at any time over a cluster’s history.

We then examine each of these galaxies using the method described in Section
2.3.1, to determine whether each object is the host halo of a galaxy group
that has passed within the radius of the cluster. We keep groups with between
five and 50 members (including the host object) that each satisfy the mass
constraints given in Section 2.3. Groups of this richness are considered to be
small or intermediate sized groups (Tully, 2015), but are large enough to
provide an environment that can strongly impact galaxy evolution (Hester,
2006). Because of the upper limit of 50 members on the group size, major
cluster-cluster mergers are not included in this study.

Fig. 1 shows a schematic view of a galaxy group at the moment of infall, and
its subsequent passage through a cluster. Finally, we exclude any groups that
have passed through the cluster once previously, so that all of the groups in
our sample are entering a cluster for the first time. Groups on a second (or
subsequent) infall make up less than $1\%$ of the groups we identify, so we
assume that this will not strongly impact our results.

Figure 1: Schematic of a galaxy group halo (dark circle) passing within
$R_{200}^{\rm{clus}}$ of a cluster (light circle) for the first time. Red
crosses represent galaxies that are members of this group; note that these are
not limited to be within $R_{200}$ of the cluster or the group at infall, but
are just defined based on Eq. 1. The position, $r$, and velocity, $v$, of one
galaxy relative to its host group are also labelled. The subsequent path of
this group through the cluster is shown by the thick, grey line, and the black
squares on this line represent the moments of pericentre, apocentre, and
second infall of the group (marked $P$, $A$ and $I_{\rm{2}}$ respectively),
which are used extensively in Section 4 of this work.

Overall, we identify 1340 infalling groups across the 257 clusters that we use
in this work, with a median richness (number of galaxies) of $8^{+7}_{-3}$
members ($1\sigma$ spread). This indicates that, although we permit groups to
contain up to 50 members, groups of this richness are rare compared to the
large number of poorer groups – only $8\%$ of the groups contain more than 20
members. The average mass, $M_{200}^{\rm{grp}}$, of these groups at cluster
infall is $10^{13.5\pm 0.4}\ h^{-1}M_{\odot}$ (median and $1\sigma$ spread).
This means that the typical mass ratio between a group and cluster is roughly
$1:20$, although this varies across the range of group and cluster masses,
from approximately $1:5$ to $1:100$. Finally, these groups enter the cluster
over a wide range of redshifts, with a median value of
$z_{\rm{infall}}=0.4^{+0.6}_{-0.3}$.

####  2.3.3 Tidal radius of groups

Subhaloes passing through a larger halo can experience strong tidal stripping,
and group-sized haloes can often lose a large fraction of their mass due to
stripping from a cluster (Muldrew et al., 2011; Bahé et al., 2019). Similarly,
galaxies can be tidally stripped from these groups (González-Casado et al.,
1994; Choque-Challapa et al., 2019), although the extent of this stripping
varies between different studies. For instance, Vijayaraghavan et al. (2015)
found that the central regions of galaxy groups are largely unaffected by a
cluster potential, and are only disrupted by dynamical friction after several
Gyr.

The tidal radius of a group or dark matter halo is an effective way to predict
and explain tidal stripping. Generally, the tidal radius is defined as the
distance from a smaller object at which the self-gravity of that object is
less than the tidal force due to a larger object. However, the tidal radius is
not precisely defined, and different definitions exist for different scenarios
(see van den Bosch et al., 2018, for a detailed summary). Perhaps the simplest
example is the Roche limit, the tidal radius of a point mass that is being
tidally influenced by another point mass. More physically motivated scenarios
such as an extended subhalo within a larger extended halo (as is used in this
work) require more complex descriptions.

Calculating a tidal radius is complicated further by the fact that subhalo
properties are often poorly defined by a subhalo finder, and can be strongly
dependent on the distance of a subhalo from the group centre. Muldrew et al.
(2011) test the ability of ahf and another halo finder, subfind (Springel et
al., 2001), to recover subhalo properties. They find that ahf performs better
at identifying all the particles of a subhalo, and thus constrains the subhalo
mass more effectively. However, for subhaloes within $\sim 0.5R_{\rm{vir}}$
($\sim 0.7R_{200}$)666Muldrew et al. (2011) use the definition of virial
radius presented in Bryan & Norman (1998). For their cosmology, the mean
density of a halo within the virial radius is $101\rho_{\rm{crit}}$, where
$\rho_{\rm{crit}}$ is the critical density of the Universe. Hence, for the
clusters used in their work and ours, $R_{\rm{vir}}=R_{101}\approx
1.3R_{200}$, although it is important to note that this conversion depends on
the concentrations and density profiles of dark matter haloes., both halo
finders underestimate the number of particles in the subhalo. This makes it
challenging to predict the mass, and therefore the radius, of subhaloes in
these regions. Furthermore, in our work we wish to combine the data from
multiple galaxy groups (of different sizes) in multiple galaxy clusters (also
of different sizes). It is therefore convenient to have an expression for the
group tidal radius that is independent of the cluster or group size, and
solely depends on the separation between these two.

We define the tidal radius of an infalling subhalo by adapting the
descriptions in Klypin et al. (1999) and van den Bosch et al. (2018).
Specifically, they give the tidal radius in terms of a function, $f(d)$, whose
value is the minimum of two expressions:

$R_{\rm{t}}=d\left(\frac{M_{\rm{<d}}^{\rm{grp}}(R_{\rm{t}})}{M_{\rm{<d}}^{\rm{clus}}(d)}\frac{1}{2-f(d)}\right)^{\frac{1}{3}}\,,$
(2)

$f(d)={\rm{min}}\left[1,\
\left.\frac{{\rm{d(ln}}M_{\rm{<d}}^{\rm{clus}})}{{\rm{d(ln}}d)}\right|_{d}\right]\,.$
(3)

Here, $R_{\rm{t}}$ is the tidal radius of the group, $d$ is distance from a
group to the cluster centre, and $M$ are the radial enclosed mass profiles of
the group and the cluster. We assume the radial density of the dark matter
haloes follow an NFW profile (Navarro et al., 1996), given by

$\rho(d)=\frac{\rho_{0}}{x\left(1+x\right)^{2}}\,,$ (4)

$x=\frac{d}{R_{\rm{s}}}\,,$ (5)

where $\rho(d)$ is the radial density of the halo in terms of the distance to
its centre, $\rho_{0}$ is a characteristic density, and $R_{\rm{s}}$ is the
scale radius of the halo. We also define the quantity $x$ to make the
equations in this section more easily readable. The concentration of a halo,
$c$, is equal to the ratio between $R_{200}$ and $R_{\rm{s}}$:

$R_{\rm{s}}=\frac{R_{200}}{c}\,.$ (6)

Integrating the NFW profile, Eq. 4, gives the enclosed mass in a sphere of
radius $d$:

$M_{\rm{<d}}=4\pi\rho_{0}R_{\rm{s}}^{3}\left[{\rm{ln}}\left(1+\frac{d}{R_{\rm{s}}}\right)-\frac{d}{d+R_{\rm{s}}}\right]\,.$
(7)

This can then be used to rewrite Eq. 3. For a general NFW profile, $f(d)=1$ in
the region $d\lesssim 2.2R_{\rm{s}}$. However, $f(d)<1$ outside of this
region, and so must be calculated for each subhalo. Solving the derivative in
the expression of $f(d)$ gives:

$f(d)={\rm{min}}\left[1,\
\left(\frac{(x/(1+x))^{2}}{{\rm{ln}}(x+1)-x/(1+x)}\right)\right]\,,$ (8)

where $x$ is defined the same as in Eq. 5.

Also using Eq. 7, we can produce an expression for $M_{200}$, as
$M_{\rm{<d}}(d=R_{200})=M_{200}$. Substituting this into Eq. 7 gives

$\begin{split}M_{\rm{<d}}=M_{200}&\left[{\rm{ln}}\left(1+x\right)-\frac{x}{1+x}\right]\\\
&\hskip
11.0pt\times\left[{\rm{ln}}\left(1+c\right)-\frac{c}{1+c}\right]^{-1}\,.\end{split}$
(9)

This expression can then be substituted into the equation for tidal radius,
Eq. 2, for the cluster enclosed mass, $M_{\rm{<d}}^{\rm{clus}}(d)$, and for
the mass enclosed within the tidal radius of a group,
$M_{\rm{<d}}^{\rm{grp}}(R_{\rm{t}})$. This gives the expression for tidal
radius below,

$\begin{split}\frac{R_{\rm{t}}}{R_{200}^{\rm{grp}}}=\frac{d}{R_{200}^{\rm{clus}}}\left(\frac{1}{2-f(d)}\right)^{\frac{1}{3}}\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\
\times\left(\frac{\left[{\rm{ln}}\left(1+\frac{C_{\rm{c}}d}{R_{200}^{\rm{clus}}}\right)+\left(1+\frac{C_{\rm{c}}d}{R_{200}^{\rm{clus}}}\right)^{-1}-1\right]}{\left[{\rm{ln}}\left(1+C_{\rm{c}}\right)-\frac{C_{\rm{c}}}{1+C_{\rm{c}}}\right]}\right)^{-\frac{1}{3}}\\\
\times\left(\frac{\left[{\rm{ln}}\left(1+\frac{C_{\rm{g}}R_{\rm{t}}}{R_{200}^{\rm{grp}}}\right)+\left(1+\frac{C_{\rm{g}}R_{\rm{t}}}{R_{200}^{\rm{grp}}}\right)^{-1}-1\right]}{\left[{\rm{ln}}\left(1+C_{\rm{g}}\right)-\frac{C_{\rm{g}}}{1+C_{\rm{g}}}\right]}\right)^{\frac{1}{3}}\,,\end{split}$
(10)

where $C_{\rm{c}}$ and $C_{\rm{g}}$ are the concentrations of the cluster and
group haloes, respectively, and $f(d)$ is given by Eq. 8. Finally, we take the
halo concentrations to be constant for all of the clusters, and all of the
groups. Specifically, we set the value of $C_{\rm{c}}$ equal to the median
value for our clusters, $C_{\rm{c}}=3.9$, and $C_{\rm{g}}$ equal to the median
value for our groups, $C_{\rm{g}}=4.4$. Approximating these concentrations as
constant has a small effect because Eq. 10 is not strongly dependent on them.
For a group at a distance $d=0.2R_{200}^{\rm{clus}}$ from the cluster centre,
the value of $R_{\rm{t}}$ varies from its median value by $20\%$ across the
full range of cluster concentrations (from $C_{\rm{c}}=2.3$ to
$C_{\rm{c}}=7.7$). At greater distances from the cluster centre, this
variation is even smaller. Similarly, the $1\sigma$ deviation777There are a
small number of groups with highly concentrated haloes, so we use the
$1\sigma$ spread in $C_{\rm{g}}$ to avoid skewing our data. in $C_{\rm{g}}$,
between $2.6$ and $6.9$, leads to a variation in $R_{\rm{t}}$ of less than
$18\%$. This variation is used as the uncertainty in the tidal radii that we
calculate in Section 3.2.1.

By making these assumptions, we are able to reach an expression for the tidal
radius of a group in units of $R_{200}^{\rm{grp}}$ that depends only on the
distance from the group to the cluster centre. As $R_{200}^{\rm{grp}}$ of a
group can change over the course of infall, the tidal radius could be scaled
by this changing group radius. However, we instead choose to scale the tidal
radius by $R_{200}^{\rm{grp}}$ at the moment of cluster infall, to allow us to
stack groups and study their evolution more clearly – this is explained in
further detail in Section 3.2.

Eq. 10 is an ideal form of the tidal radius for our analysis, as it allows us
to calculate the average tidal radius for all groups in a radial bin across
many clusters. This form of the tidal radius may also be useful in future
studies, both observational and theoretical, that wish to stack substructure
on multiple different size scales.

##  3 Phase space evolution

Much of the work in this paper revolves around studying the phase space of
galaxies within galaxy groups, as the groups enter and pass through a cluster,
and how the distribution of galaxies within this phase space changes over
time. This analysis follows the same basic process as in Haggar et al. (2021);
the phase space consists of the radial distance of a galaxy from its host
group halo in terms of the group halo radius, $R_{200}^{\rm{grp}}$, and the
galaxy’s velocity relative to the group halo, in units of $v_{\rm{cir}}$, the
circular orbital velocity at $r=R_{200}^{\rm{grp}}$. It is important to stress
that this work involves looking at the phase space of galaxies relative to
their host group, not the cluster (as has been done by numerous previous
studies, such as Jaffé et al., 2015; Arthur et al., 2019, for example). This
method can provide detailed information, by showing both the spatial and
velocity distribution of galaxies in groups, and telling us how the speed and
acceleration of galaxies differ in different regions of the group.

Fig. 2 shows the average distribution of galaxies in phase space, for an
infalling group – a group that has just passed within $R_{200}^{\rm{clus}}$ of
the cluster centre for the first time, as shown in Fig. 1. Similarly to our
previous work (Haggar et al., 2021), we produce a smoothed distribution of
galaxies using a 2D kernel density estimation (KDE) with an optimised
bandwidth. In the remainder of this section, we examine how this phase space
changes as a group passes through a cluster.

Figure 2: Distribution of galaxies in group phase space, for groups at the
moment of infall into the host cluster. Data for all groups from all 257
clusters that are used in this analysis are shown, stacked together. Lighter
colours represent regions of phase space with more galaxies – the maximum
value is at $r=0.65R_{200}^{\rm{grp}}$, $v=1.15v_{\rm{cir}}$, representing the
region of this phase space in which group members are most likely to be found.
The red line represents the boundness criterion for galaxies Eq. 1; galaxies
above this line are not considered group members, and so are excluded from
this figure. Contours are at densities of [0.2, 0.4, 0.6, 1, 2, 4, 6]
$(R_{200}^{\rm{grp}}v_{\rm{cir}})^{-1}$. The data in this figure, and
subsequent phase space diagrams in this work, are smoothed using a 2D kernel
density estimation (KDE) with an optimised bandwidth, typically $\sim 0.2$
virial units.

The 1340 infalling groups that we identify represent an average of 5.2
accreted groups per cluster – this might appear to be a small number, but it
is important to note that this is not the entire accreted group population, as
this only accounts for intermediate size groups (with between five and 50
members). Although they use different mass limits to this work, Berrier et al.
(2009) demonstrate that about half of galaxy groups contain only two or three
members, and such groups are not included in our analysis. If we do include
these poor groups, we find that approximately $14\%$ of $z=0$ cluster galaxies
in our simulations were accreted as members of a group, comparable to the
results from other studies presented in Section 1 (see also Haggar et al.,
2021).

###  3.1 Groups beyond the cluster outskirts

Before studying groups passing through clusters, we first study how this phase
space changes in groups that are not under the influence of a cluster, and are
located far from the cluster centre (greater than $3R_{200}^{\rm{clus}}$ from
the cluster). This can then be used as a control, showing how the distribution
of galaxies changes for a group evolving secularly, as an (approximately)
isolated system. Fig. 3 shows the direction and rate at which galaxies in
groups move around this phase space, for groups between $3R_{200}^{\rm{clus}}$
and $10R_{200}^{\rm{clus}}$ from the centre of a cluster, between redshifts of
$z=0.1$ and $z=0$. We assume that these groups are isolated, as they are
sufficiently far from a cluster that they are not subject to its strongest
effects. We did not study groups at greater cluster distances because the
resolution of the simulations decreases outside of this distance. This figure
includes only galaxies that were bound to a group (i.e. that lay below the
thick red line) at $z=0.1$, which we then follow until $z=0$ (about 1.3 Gyr).

Figure 3: Motion of group members in phase space of host group, for groups
located beyond the influence of a cluster. The thick red line shows boundness
criterion, providing an approximate measure of galaxies that have become
unbound from their group. Colours of arrows represent the rate at which
galaxies are moving in this phase space, with darker arrows indicating that
galaxies are moving at a greater rate in this phase space. This plot shows
stacked data for 2769 groups, located between $3R_{200}^{\rm{clus}}$ and
$10R_{200}^{\rm{clus}}$ from the centre of a cluster, between $z=0.1$ and
$z=0$. This shows how galaxies move in the phase space of groups when the
group is not affected by the external environment. All galaxies lie below the
bounded line at $z=0.1$, but some move above the red line and become unbound,
although many remain bound to the group. The motion of the bound galaxies
follows a characteristic pattern, rather than being in random directions.

Throughout this section, in order to study how the speeds and positions of
group galaxies change, we examine the changes of these properties for bound
group members, relative to $R_{200}^{\rm{grp}}$ and $v_{\rm{cir}}$ of their
host group measured at a previous time. In Fig. 3 and the following figures in
Section 3.2 we show how the phase space of groups changes over time. In these
plots, the direction of arrows shows the average direction that galaxies in
this region are moving in phase space, and darker arrows mean that the
galaxies are moving across the phase space more quickly. For example, a galaxy
going from $[1.0R_{200}^{\rm{clus}},0.5v_{\rm{cir}}]$ to
$[2.0R_{200}^{\rm{clus}},1.5v_{\rm{cir}}]$ in 2 Gyr would be represented by an
arrow located at $[1.5R_{200}^{\rm{clus}},1.0v_{\rm{cir}}]$, pointing at a
$45^{\circ}$ angle to the top-right, with a colour of $\sim 0.71$. Note that
the horizontal and vertical axes in Fig. 3 are dimensionless, as they have
been normalised to prior values of $R_{200}^{\rm{grp}}$ and
$v_{\textrm{cir}}$, and so we describe the distance moved across this phase
space in a given time with the term ‘virial units per Gyr’.

The positions and velocities of the galaxies in Fig. 3 are scaled relative to
$R_{200}^{\rm{grp}}$ and $v_{\rm{cir}}$ of each group at $z=0.1$. Some regions
of phase space do not contain any arrows because of a lack of data, indicating
that almost no galaxies were found in this region across all the groups – for
example, there are no galaxies in the top-right of Fig. 3, because they were
all below the red line just a short time previously. Some galaxies are still
found above the line, because they have become unbound between $z=0.1$ and
$z=0$.

The phase space of these groups is not in equilibrium, and bound galaxies in
the centres of these groups appear to be moving downwards on this plot (i.e.
losing speed but remaining a similar distance from the group centre). This
indicates that energy is being dissipated during their orbits. Dynamical
friction is strongest in the group centres, and so this is likely responsible
for the loss of energy during these orbits. Fig. 2 in Arthur et al. (2019)
shows analogous behaviour to this for the phase space of a galaxy cluster:
subhaloes move horizontally in phase space when approaching the centre of
their host halo, then move sharply downwards when they are near to the halo
centre, resulting in the apparent ‘spiral’ motion of galaxies in Fig. 3.

This trend could also potentially be explained by the destruction of some
inner galaxies by mergers before they have time to leave the group centre.
However, we find that this is not the case, as the majority ($82\%$) of
galaxies within $0.5R_{200}^{\rm{grp}}$ of the group centre survive to $z=0$
without merging into another halo or being heavily stripped (see Section 4.3
for further discussion of the fates of group members). Furthermore, if we
remove these galaxies from our analysis, there is a negligible change in the
trends in Fig. 3.

###  3.2 Groups passing through clusters

To study groups falling into clusters, the phase space diagrams that we
present are instead scaled relative to $R_{200}^{\rm{grp}}$ and $v_{\rm{cir}}$
of each group at $z_{\rm{infall}}$, the moment of cluster infall. We scale the
positions and speeds of galaxies by these values, even in subsequent snapshots
after $z_{\rm{infall}}$. This approach is not perfect, because the radius and
circular velocity of a host group halo changes as the group approaches the
centre of a cluster, likely due to tidal stripping. Despite this, we choose to
measure these properties only at the moment of infall because, in the central
regions of a large halo, the mass and radius of a subhalo are not well-
defined; due to the high background density in the centre of the cluster halo,
it can be challenging for a halo finder to identify the overdensity of a
subhalo. Consequently, near the centre of a cluster, the mass and radius of a
group ($M^{\rm{grp}}_{200}$ and $R^{\rm{grp}}_{200}$) are not reliable
(Muldrew et al., 2011). Scaling by the values of $R_{200}^{\rm{grp}}$ and
$v_{\rm{cir}}$ at $z_{\rm{infall}}$ allows us to visualise how the absolute
values of the distance and speed of galaxies relative to their groups are
changing. This means that galaxies lying below the line of boundness after
infall are not strictly bound to the group, but the approach still provides a
good approximation.

In this section we consider groups that are entering the cluster for the first
time, and so have not previously experienced a cluster potential. We also only
include groups at times between their first infall, and their first apocentric
passage after entering the cluster (the turnaround in their cluster orbit). It
is important to note that this is not necessarily the true ‘first apocentre’
of an orbit, as haloes are not accreted onto clusters in perfectly radial
orbits. Instead, they have some tangential component to their velocity,
meaning that some haloes will pass an apocentre before their entry to the
cluster (Ghigna et al., 1998; Tollet et al., 2017). However, as the focus of
this paper is on the evolution of groups after their cluster infall, we will
hereafter refer to the first apocentric passage post-infall as the ‘first
apocentre’. Finally, we do not separate groups by redshift – for example, some
of these groups have passed their first pericentre by $z=0$, but some have not
and so are absent from the post-pericentre analysis.

Fig. 4 shows how the phase space of these groups changes as they pass through
a cluster. We find that the behaviour of groups as they enter and pass through
a cluster can be approximately split into two main phases, with the group
dynamics changing suddenly as a group makes its closest approach to the
cluster centre, as shown by the two panels in this figure. The left panel of
Fig. 4 shows groups on their infall, moving from the cluster outskirts towards
their first pericentric passage, near to the cluster centre. Generally, the
galaxies in these groups move upwards on this plot, showing an increase in
their velocity relative to their host group. This data is for galaxies that
are bound to groups at the moment of infall ($z_{\rm{infall}}$), but some of
these move above the red line and so become unbound from their host group.
Similarly to in Fig. 3, the direction of arrows shows the average direction
that galaxies are moving in phase space, for galaxies in this region of phase
space. It is important to note that the arrows in the left panel (‘pre-
pericentre’) are pointing vertically upwards, with a very small horizontal
component. This shows that, although these galaxies have a large change in
speed, their distance to the group centre does not change very much; galaxies
within $R_{200}^{\rm{grp}}$ remain within $R_{200}^{\rm{grp}}$.

Figure 4: Same as Fig. 3, but showing motion of group members in phase space
of host group for two epochs, before and after pericentre. Data shown are for
galaxies that are bound to groups at the moment of infall, for groups on their
first passage through the cluster. The red line shows boundness criterion at
the moment of infall, and so provides an approximate measure of galaxies that
have become unbound from their group. Left panel shows data for groups before
reaching their first pericentric passage of the cluster, moving between
$R_{200}^{\rm{clus}}$ and cluster centre, and the right panel shows groups
moving between the cluster centre and $R_{200}^{\rm{clus}}$, which have passed
their pericentre and are now receding from the cluster, moving towards their
first apocentric passage. For pre-pericentre groups, the bulk motion of the
galaxies is upwards, representing an increase in their group-centric speed,
but little change in the spatial separation of galaxies from their host group.
In contrast, for groups that have passed pericentre and are now receding from
the cluster, group members are moving approximately horizontally in phase
space, increasing their distance from the group to which they were previously
bound.

This behaviour is different for groups that have passed the pericentre of
their orbit, shown in the right panel of Fig. 4. This panel shows data for
groups at snapshots when they have passed pericentre, but have not yet reached
their first apocentre, and so are receding from the cluster centre. Groups are
also only included here at stages of their orbit when they are between the
cluster centre and $R_{200}^{\rm{clus}}$, to allow us to compare the two
panels in Fig. 4. Instead of increasing their velocity, most galaxies in these
post-pericentre, receding groups keep a fairly constant relative velocity, and
instead move horizontally on this plot, becoming spatially separated from the
centre of their host group. This behaviour is stronger for galaxies that have
become unbound from the group, moving above the boundness line – these move to
greater distances from the group centre, often with an accompanying slight
increase in relative speed. Galaxies that are still bound to a group instead
experience a drop in their relative speed, as well as an increase in
separation from the group centre.

In summary, Fig. 4 shows that there are two phases of evolution for a galaxy
group passing through a large cluster. First, galaxies are given a kinetic
energy kick, increasing their speed relative to their host group. This rapid
boost in kinetic energy is manifested after the group passes pericentre, which
typically occurs $\sim 0.5$ Gyr after entering the cluster, by being converted
into potential energy as the galaxies recede from the group centre.

####  3.2.1 Tidal effects and dynamical friction

In Fig. LABEL:fig:phasespace_arrows we break down the results from Section 3.2
into individual steps, separating the infalling groups into bins based on
their cluster-centric distance, both before and after passing pericentre. This
gives a much more detailed view of how this phase space changes over the
average course of a group through a cluster. We note that each panel does not
represent an identical sample of groups, as most groups will not have a
snapshot in all of these radial bins, and so this data represents the
evolution of all groups that are found in this radial range. If we instead
select only groups that have passed through each of these bins, there is only
a minimal impact on our results, but large amounts of noise are introduced due
to the small number of groups.

In each panel, the tidal radius (based on the approximations detailed in
Section 2.3.3) for a group in the centre of this bin is also marked, in units
of the group radius at infall. The closer a group is to the cluster centre,
the stronger the effect of the cluster will be, and this is demonstrated by
the decrease and subsequent increase of the tidal radius as groups pass
through the cluster. Interestingly, across the eight panels, the tidal radius
appears to mark a transition, such that the group dynamics evolve differently
within the tidal radius, compared to beyond the tidal radius. Outside the
tidal radius, galaxies first experience a kinetic kick and then recede from
the group centre, as detailed in the previous section. However, inside the
tidal radius, galaxies generally behave in a way similar to that seen in the
centres of isolated groups in Fig. 3 – they mostly move downwards on these
plots, showing a decrease in speed.

Physically, this distinction indicates how the dynamics in some regions of the
group are dominated by the group itself, whilst others are dominated by the
effects of the cluster. As described in Section 3.1, galaxies in isolated
groups experience dynamical friction due to the group’s halo (Vijayaraghavan
et al., 2015). This is particularly strong in the dense central regions of the
group, where dynamical friction will cause galaxies to slow down and spiral
inwards, dominating over the effect of the cluster. However, beyond the tidal
radius, tidal effects from the cluster dominate this dynamical friction,
meaning that the movement of galaxies in phase space is dictated by the
cluster, not the group. The change in the tidal radius means that the two
phases of group evolution are clearer in the outskirts of a group, as the
dynamics of these regions are dominated by the cluster for much of the group’s
journey. Conversely, galaxies in the group centres ($r<0.5R_{200}^{\rm{grp}}$)
decrease in velocity at almost all times, as they are almost always within the
tidal radius. The only exception to this is in the very deepest parts of the
cluster (such as in panel (d) in Fig. LABEL:fig:phasespace_arrows). Dekel et
al. (2003) showed that at the very centre of a dark matter halo, tidal forces
can become fully compressive – this could explain why all group galaxies
change their orbits around their host groups, with their speeds increasing and
their distances either remaining the same or decreasing.

Finally, as the change from an increase in $v$ to an increase in $r$ is
dependent on the tidal radius, this switch in behaviour is not instantaneous
as it might appear to be in Fig. 4. Once a group reaches a distance of
approximately $0.3R_{200}^{\rm{clus}}$ beyond pericentre (panel (f) in Fig.
LABEL:fig:phasespace_arrows), the motion of galaxies away from the group
begins in the centre, and then spreads throughout the group as it once again
dominates over the cluster. Eventually, for groups that are long past
pericentre (panel (h) of Fig. LABEL:fig:phasespace_arrows), all galaxies are
either decreasing in relative speed, or their speed is staying the same. All
the galaxies remaining in groups at this stage are also moving away from the
group centre, towards the bottom-right of the phase space, which is
characteristic of galaxies approaching the apocentre of a bound orbit around a
group.

To help visualise this behaviour, Appendix A shows how group galaxies move
around phase space for a single example group as it passes through a cluster,
clearly showing the two main phases of group evolution.

##  4 Groups after cluster infall

The results in Section 3 show how the dynamics of galaxy groups change as they
pass through a cluster. In this section, we discuss the differences in the
properties of a group before and after it passes through a cluster, in order
to understand how distinguishable are these two classes of groups.

###  4.1 Orbits of galaxy groups

The data used in Section 3 is for groups on their first passage through a
cluster, but not all of these groups will follow the same path. Just as some
galaxies that are accreted by a cluster can become ‘backsplash galaxies’
(Balogh et al., 2000; Gill et al., 2005; Haggar et al., 2020), some groups
will pass through the cluster and exit $R_{200}^{\rm{clus}}$ again, becoming
‘backsplash groups’ that can enter the cluster for a second time. Others will
‘stick’ to the cluster, remaining bound and not leaving $R_{200}^{\rm{clus}}$
once they have entered.

We find that, across the 257 clusters used in this work, most groups ($92\%$)
that fall into a cluster do not leave it again. By $z=0$, only 42% of the
groups that enter a cluster have reached their first turnaround (apocentre) in
their cluster orbit, while $58\%$ do not reach this stage. These groups do not
reach apocentre for multiple reasons; either they have merged with the cluster
halo before reaching apocentre rather than remain a substructure of the
cluster, they have been heavily stripped by the cluster and so fall below the
resolution limit before reaching apocentre, or they have simply not had time
to reach apocentre by $z=0$. Of the groups that do reach the apocentre of
their orbit, 20% have left the cluster after entering $R_{200}^{\rm{clus}}$,
while 80% reach apocentre within $R_{200}^{\rm{clus}}$ of the cluster centre,
and so remain ‘stuck’ to the cluster potential. We hereafter refer to these as
‘backsplash groups’ and ‘sticky groups’, respectively. Overall, just 8% of all
infalling groups go on to leave the cluster again. Finally, 81% of the
backsplash groups in our sample later experience a second cluster infall, and
19% are still outside of the cluster at $z=0$.

The paths that groups can take through a cluster can be described in terms of
the distance from a group to the cluster centre at pericentre and apocentre.
Interestingly, we find that the distance of a group halo from the cluster
centre at pericentre is very consistent, regardless of the group’s later
behaviour. Groups that do not reach apocentre have a median pericentric
distance of $0.36^{+0.14}_{-0.09}R_{200}^{\rm{clus}}$ from the cluster centre,
which is very similar to the pericentre of groups that do later reach
apocentre: backsplash groups and sticky groups have median pericentric
distances of $0.38\pm 0.13R_{200}^{\rm{clus}}$ and
$0.36^{+0.11}_{-0.08}R_{200}^{\rm{clus}}$, respectively. This justifies our
decision to normalise the figures throughout this paper by the group radius at
infall, as most groups pass well within $0.7R_{200}^{\rm{clus}}$, where
Muldrew et al. (2011) showed that subhalo sizes cannot be reliably measured.

This shows that most groups take a similar trajectory into clusters, passing
by the cluster centre at a similar distance. However, the subsequent orbits of
these groups can vary dramatically, with groups reaching a wide range of
apocentric distances, and some not being tracked to reach their apocentre at
all. By definition, the post-infall apocentric cluster distances of backsplash
groups and sticky groups are very different. Backsplash groups have a median
apocentric distance of $1.16^{+0.29}_{-0.12}R_{200}^{\rm{clus}}$, and sticky
groups of $0.63^{+0.22}_{-0.16}R_{200}^{\rm{clus}}$, which correspond to
median orbital eccentricities of $0.53\pm 0.12$ and $0.25^{+0.19}_{-0.15}$,
respectively.

###  4.2 Removal of galaxies from groups

The sample of backsplash groups that exit a cluster and then re-enter allow us
to directly compare how a single passage through a cluster permanently affects
the properties of a group. Comparing the same sample of groups at the moment
of first infall and second infall means that the groups are in approximately
the same configuration (at a distance of $\sim{R}_{200}^{\rm{clus}}$, falling
towards the cluster).

Overall, we find that groups on a second infall contain far fewer galaxies,
when compared to groups infalling for the first time. On their first infall,
the median number of galaxies in these groups was $6^{+3}_{-1}$ (note that
this is slightly smaller than the value of $8^{+7}_{-3}$ quoted in Section 3,
which includes groups that do not exit and re-enter the cluster). By their
second infall, the median richness of these same groups is $2\pm{1}$ members.
In fact, 46% of the groups that are infalling for the second time contain only
one member. Physically, these objects are not actually groups at all: a
‘group’ with one member instead represents a single galaxy that has no other
galaxies bound to it, having previously been the central galaxy in a group.
This shows that, in a single passage through a cluster, almost all galaxies
become unbound from groups. Often this process completely disrupts a group,
resulting in no galaxies remaining bound together.

Similarly, the dark matter haloes of these groups are heavily stripped during
their passage through the cluster. At first infall, the median radius,
$R_{200}^{\rm{grp}}$, of a group was $0.51^{+0.15}_{-0.10}\ h^{-1}$ Mpc. By
their second infall, these same groups had a median radius of
$0.32^{+0.10}_{-0.07}\ h^{-1}$ Mpc. Similarly, the median mass888The infall
mass of groups that later have a second infall is slightly smaller than the
average mass of all groups, $10^{13.5}\ h^{-1}M_{\odot}$ (Section 2.3.2).
However, as we discuss in Section 4.3, we still consider these to be a
representative sample of all infalling groups. of these groups,
$M_{200}^{\rm{grp}}$, decreases by a factor of three in this time, from
$10^{13.2}\ h^{-1}M_{\odot}$ to $10^{12.7}\ h^{-1}M_{\odot}$, consistent with
the decrease in the number of galaxies. This is comparable to the results from
other previous studies which have found that dark matter subhaloes are heavily
stripped; Muldrew et al. (2011) found that a halo passing through the centre
of a cluster has approximately half of its mass stripped away, and Taylor &
Babul (2004) used semi-analytic models to show that subhaloes on orbits
similar to our groups lose $>40\%$ of their mass with each pericentric passage
of a cluster. Some studies find even more extreme evidence of this removal of
dark matter: Smith et al. (2016) used hydrodynamical simulations to show that
a cluster halo can strip away $\sim 80\%$ of the dark matter in galaxy-sized
subhaloes.

###  4.3 The fates of group galaxies

We can investigate the removal of galaxies from groups further, by comparing
these groups at different stages of their infall and journey through a
cluster. As shown in Section 3, the speed of galaxies relative to their host
group increases before they have reached pericentre of their cluster orbit,
and their group-centric distance increases post-pericentre. Therefore,
although groups become spatially separated after pericentre, it is not clear
when the galaxies become unbound from these groups.

For backsplash groups that also have a second infall, their member galaxies
are removed from their host group very quickly. Of those galaxies that are
bound to a group at first cluster infall (i.e. that satisfy Eq. 1),
$60^{+20}_{-35}\%$ are no longer bound to the group by the first pericentre,
$76^{+24}_{-6}\%$ are removed by apocentre, and $89^{+11}_{-29}\%$ by the
second infall into the cluster (median and $1\sigma$ spread for backsplash
groups). These numbers are almost identical for backsplash groups that do not
have a second infall.

For groups that reach apocentre but do not leave the cluster (‘sticky
groups’), $75\pm{25}\%$ of previously bound galaxies are no longer group
members at pericentre, and $73^{+27}_{-33}\%$ at apocentre. Although it
appears that the number of unbound galaxies drops slightly between pericentre
and apocentre, this can actually be explained by the fact that the radius (and
thus mass) of a subhalo are artificially suppressed in the centre of a large
halo, making more galaxies appear to be unbound.

However, although these galaxies are no longer members of the group, this is
not necessarily because they have become gravitationally unbound from their
host group. In this section, we analyse the final fates of these galaxies
after their group enters a cluster. To do this, we separate the galaxies’
states into five categories:

  * •

Bound: galaxy is still bound to its host group, according to Eq. 1.

  * •

Unbound: galaxy does not satisfy Eq. 1, and so is no longer bound to its host
group.

  * •

Disrupted: no descendent of a group member has been found by the halo finder,
typically because its dark matter halo has been heavily stripped.

  * •

Merged with group: galaxy has been absorbed by the halo of its host group,
effectively merging with the Brightest Group Galaxy.

  * •

Other merger: merging with another, larger object (e.g. merging with a more
massive satellite galaxy). Alternatively, galaxy may be absorbed by the
cluster halo, effectively merging with the Brightest Cluster Galaxy.

The ‘disrupted’ galaxies in our sample represent a class of objects that have
physical similarities. However, because of the nature of the simulations and
tree-builder that we use in this work, the branches of their merger trees are
cut off prematurely, meaning that they appear to have no descendent halo in
the simulations and so their final fate cannot be determined. Before their
branches end, the dark matter masses of these galaxies are changing rapidly –
in their final ten snapshots before they are removed from the merger tree,
$76\%$ of these galaxies experience at least one drop of $>30\%$ in their halo
mass between two snapshots ($\sim 0.3$ Gyr), and $37\%$ have a measured drop
of $>40\%$. However, mergertree does not allow for the dark matter mass of an
object to change by more than a factor of two between snapshots (see Section
2.1.1 for an explanation of this). Consequently, if a galaxy’s dark matter
halo mass drops by $>50\%$ between snapshots, this change will not be
recorded, no descendent for the halo will be added to the catalogue, and this
branch in the merger tree will end. Despite this heavy stripping of dark
matter, very few of the disrupted galaxies violate the mass limits that are
imposed in Section 2.1.1; if we remove these mass limits, the median final
mass of these galaxies before their merger tree ends is
${\rm{log}}_{10}(M_{200}/h^{-1}M_{\odot})=11.3^{+0.7}_{-0.6}$, with a median
stellar mass of
${\rm{log}}_{10}(M_{\rm{star}}/h^{-1}M_{\odot})=10.4^{+0.5}_{-0.4}$, and a
ratio between these of $0.14^{+0.12}_{-0.08}$. Consequently, few of these
galaxies are removed from the merger trees due to violating these imposed mass
limits.

Fig. 5 shows the status of group member galaxies as their host group passes
through a cluster. This data is averaged across all groups that become
backsplash groups and then have a second cluster infall, meaning that we have
data for their entire passage through a cluster. Overlaid as solid, dashed,
dot-dashed and dotted lines are the boundaries between the coloured regions
when all groups are included. For example, this indicates the states of
galaxies at pericentre for all groups that reach their first pericentre,
regardless of what subsequently happens to the group. Similarly, the apocentre
data shows the fates of all galaxies in groups at apocentre, whether or not
this apocentre is outside of the cluster. This data closely follows the data
for groups that have a second infall, showing that these second infallers are
representative of the entire group sample. We therefore only discuss these
groups that later have a second infall, allowing us to make comparisons of the
same sample of groups at different stages of their orbit.

Figure 5: Status of galaxies that were bound to groups at cluster infall, as
their host group passes through a cluster and begins its second infall. These
data are averaged across all groups that become backsplash groups, and
experience a second infall. All galaxies are bound at first infall, by
definition. Areas representing galaxies that have become bound, unbound,
disrupted, or merged with the group halo are labelled. The small, black region
represents other mergers, which is unlabelled for clarity. Solid/dashed/dot-
dashed/dotted lines show the boundaries between these regions for all groups
that reach this stage of their orbit, regardless of whether they go on to
reach apocentre or have a second infall.

As stated above, only approximately $40\%$ of group members are still members
of the group at the pericentric passage of the cluster centre (note that here
we use the mean behaviour of each group, as opposed to the median used earlier
on in this section, and so the quantities differ slightly). However, of the
$61\%$ of galaxies that are no longer group members at pericentre, only $45\%$
have become unbound from their host group, while $16\%$ have experienced one
of the other fates described above. As these groups exit the cluster and re-
enter, the number of galaxies becoming unbound increases slightly (to $53\%$),
but the number of galaxies leaving the group for another reason doubles, to
$32\%$, showing that these other processes are more important after a group’s
initial infall.

It is also important to note that these four stages in the group orbit –
infall, pericentre, apocentre and second infall – are not equally spaced in
time. For the groups shown in Fig. 5 (backsplash groups with a second infall),
pericentre, apocentre and the second infall occur an average of $0.5\pm 0.2$
Gyr, $2.6^{+0.4}_{-0.7}$ Gyr, and $3.5^{+1.2}_{-1.0}$ Gyr after the first
infall, respectively. Consequently, not only do most of the unbound galaxies
leave the group between infall and pericentre, but this process takes place in
just $\sim 0.5$ Gyr, compared to the $\sim 2$ Gyr between pericentre and
apocentre. We note that these timescales are redshift dependent: the time for
a galaxy entering a cluster to reach pericentre at $z=0$ can typically range
from $1-2$ Gyr (see Fig. B1 in Tollet et al., 2017, for further details), but
the time taken decreases at higher redshifts. Our method consequently returns
an average infall-to-pericentre time of less than $1$ Gyr, because we stack
data from groups at numerous different redshifts (for some additional
discussion of cluster crossing times, see Contreras-Santos et al., 2022).

Fig. 5 represents all group members at infall, however Fig.
LABEL:fig:phasespace_arrows shows that galaxies in different regions of the
group phase space will experience different processes, and so the likelihood
of each outcome is not the same for all galaxies in a group. Accordingly, we
also find that the evolution and fates of group galaxies is strongly dependent
on their position within the phase space of their host group. Fig. 6 and Fig.
7 show the evolution of members of groups that pass through and re-enter a
cluster, in the bottom-left999This selection specifically examines slow-moving
galaxies near the group centre, as fast-moving galaxies near the group centre
exhibit different behaviour. We elaborate on this in Section 4.3.1.
($r<0.5R_{200}^{\rm{grp}}$, $v<0.5v_{\rm{cir}}$) and bottom-right
($r>0.8R_{200}^{\rm{grp}}$) of the phase space shown in Fig. 2. These
represent the slow-moving galaxies deep within the group’s potential well, and
loosely-bound galaxies in the group outskirts, respectively.

Figure 6: Same as Fig. 5, but for slow-moving galaxies in the centre of groups
($r<0.5R_{200}^{\rm{grp}}$, $v<0.5v_{\rm{cir}}$). Again, the small ‘other
mergers’ region is unlabelled for clarity. Galaxies in this region are much
more likely to become heavily disrupted and have an incomplete merger tree,
although a substantial fraction merge with the group halo, mostly between
pericentre and apocentre. Figure 7: Same as Fig. 5, but for galaxies in the
outskirts of groups ($r>0.8R_{200}^{\rm{grp}}$). The ‘group mergers’ and
‘other mergers’ regions are unlabelled for clarity. Galaxies in the outskirts
of the groups are highly likely to become unbound from their host group, which
usually happens between infall and pericentric passage.

Clearly, galaxies in the central (Fig. 6) and outer (Fig. 7) regions of a
group have vastly different evolutionary histories. Slow-moving galaxies in
the centres of groups almost never become unbound from the group – instead,
the majority of them are disrupted by the time the group re-enters the
cluster, although a sizeable fraction ($17\%$) of them merge with the group
halo. Dynamical friction likely plays a role in this, by causing these
galaxies to spiral in towards the group centre, making them likely to merge
with their host group’s halo. This is in contrast to the outskirts of the
groups, where the vast majority of group members become unbound from the group
almost immediately after the group enters the cluster, and only a small
fraction are heavily disrupted. In both cases, the black lines on the figures
show that galaxies in other infalling groups experience similar evolution,
although slightly more galaxies are disrupted in groups that become backsplash
groups.

####  4.3.1 Galaxy fates across group phase space

Finally, we can take a more general approach to Section 4.3 by looking at the
phase space of the infalling groups, to determine the typical fates of
galaxies at the second cluster infall, as a function of their initial position
in this phase space. Fig. 8 shows how common different outcomes are for group
members, as a function of their relative position and speed at cluster infall;
this is in effect a generalisation of Fig. 6 and Fig. 7. For example, in the
bottom-left region of the phase space, there is a high density of ‘disrupted’
galaxies, showing that galaxies here during infall later became disrupted, in
agreement with Fig. 6.

Figure 8: Of the galaxies that are bound to a group at its first infall, each
panel shows the fraction of these in each state at the moment of second
infall. Phase space is defined by the position/speeds of the galaxies at the
first infall. Top-left panel shows the fraction of galaxies that remain bound
to the group. Top-right panel shows the fraction that become unbound from the
group. Bottom-left shows the fraction that are ‘disrupted’. Bottom-right shows
the fraction that merge with the group halo. Lighter colours represent regions
of the phase space with a greater number of galaxies. White regions either
represent the ‘unbound’ region, or regions where the number of galaxies is
very low.

The top-left and top-right panels of Fig. 8 show a substantial decrease in the
number of galaxies that remain bound to a group outside of $r\sim
0.7R_{200}^{\rm{grp}}$ from the group centre. This indicates that, for almost
all groups, virtually all galaxies outside of this radius are removed.
Similarly to in Section 3.2, the tidal stripping of groups can explain this
sharp cut. According to Eq. 10, a tidal radius of $0.7R_{200}^{\rm{grp}}$
corresponds to a group that is approximately $0.7R_{200}^{\rm{clus}}$ from the
cluster centre. This distance is the maximum typical pericentric distance that
we find for groups in our sample – almost all groups ($95\%$) have a
pericentric passage of $r\leq 0.7R_{200}^{\rm{clus}}$. Consequently, almost
all groups will have had a tidal radius of
$R_{\textrm{t}}=0.7R_{200}^{\rm{grp}}$ at some point in their orbit, but not
all groups will have experienced a tidal radius less than this. This explains
why some galaxies remain in the groups within $0.7R_{200}^{\rm{grp}}$, but
none remain beyond this distance.

Generally, only galaxies near to the group centre with high velocities remain
as bound group members. These are on longer, eccentric orbits – galaxies with
lower velocities spend more time nearer the group centre, and so are more
likely to merge with the group, or to be disrupted. Furthermore, the bottom-
left and bottom-right panels show that the disrupted galaxies inhabit
different parts of phase space, compared to those that later merge with the
group halo. Disrupted galaxies have large amounts of their dark matter
stripped in a short period of time: for two-thirds of these galaxies, in the
snapshot immediately after they are ‘disrupted’, more than $50\%$ of their
dark matter particles appear either in the halo of their host group or (less
often) their host cluster. This disruption by a larger halo is similar to how
galaxy harassment can occur in clusters (Moore et al., 1996). However, the
galaxies in the centre of these disrupted haloes do not immediately become
associated with the group halo – if this were the case, these objects would be
tagged as merging with the group halo, which they are not. This implies that a
tidal disruption is occurring, in which large amounts of material are removed
from the galaxy, forming a substructure in the group such as a tidal stream.
This substructure will most likely merge with the group halo at some later
time (Moore et al., 1998), effectively making this process a merger with the
group halo, but over a longer time period.

Disruption is more likely for galaxies in the centres of groups that are slow-
moving at the moment of infall, while galaxies with greater speeds are
somewhat more likely to merge with the group halo. One explanation for this
lies in the left panel of Fig. 4, showing pre-pericentre groups. Before a
group reaches pericentre, galaxies in the group centre with high speeds move
downwards in phase space, indicating that their speed is decreasing due to
dynamical friction, and they are slowly spiralling into the group centre where
they merge. Slow-moving galaxies are instead moving upwards on this plot,
indicating that they are experiencing strong, accelerating forces that can
disrupt their structure. Additionally, high-speed galaxies are on radial,
eccentric orbits, meaning that they pass the group centre infrequently. In
contrast, a low speed and low group-centric distance indicates that a galaxy
is on a small, circular orbit, and so will be able to make multiple orbits of
the group in a short period of time, providing more opportunities for heavy
stripping by the central group galaxy.

To more clearly show the differences between these classes of galaxies (those
that are bound to the group, unbound, disrupted, or have merged with the group
at second infall), we combine the four panels from Fig. 8 into a single
figure, Fig. 9. The contours in this show, for galaxies in each class, where
in phase space they were located at the moment of infall. It is important to
note that each contour is located at half of the maximum value for that class,
and so they are not scaled in the same way as some galaxy fates are more
common than others. Consequently, this plot does not show what outcome is most
likely for galaxies in each part of phase space. For example, many more
galaxies become unbound than remain bound to a group, so those in the top-left
of this diagram are far more likely to become unbound than remain bound – this
is more apparent when we compare the top two panels of Fig. 8.

Instead, Fig. 9 allows us to take a single class of galaxies (say, those that
are disrupted by second infall), and see from where they originated (in this
case, the low-velocity, inner regions of the group). Some regions are the
source of multiple classes of galaxies, while some are the source of only one.
For instance, group members that are later either bound or unbound can
originate from the the low-$r$, high-$v$ region of phase space, but only
unbound galaxies originate from the high-$r$, low-$v$ region. These results
from Fig. 8 and Fig. 9 broadly agree with the findings of Choque-Challapa et
al. (2019), who use dark matter-only simulations to study the fates of
galaxies in infalling groups as a function of their position in group phase
space. Among other results, they show that outside of $r\sim
0.8R_{200}^{\rm{grp}}$ there is a sharp increase in the fraction of members
becoming unbound from their host group, and that galaxies lying near to the
boundness line are more likely to be stripped from their groups.

Figure 9: Overlay of four panels from Fig. 8, to aid with comparison of
different phase space distributions, for galaxies bound to groups at cluster
infall. One highlighted region is shown for each of the four galaxy fates at
the time of second infall. These show the region from which each class of
galaxy most commonly originated – that is, where were they previously found at
the moment of first infall. Contour surrounding each region is placed at a
value equal to half of the maximum, from each panel in Fig. 8. Grey regions
either represent areas of phase space that contain few galaxies, or are not an
important producer of any of these four classes of galaxies. From this, it is
clear that galaxies in different regions of phase space later experience
different environments, and different evolutionary processes.

####  4.3.2 Observational analogues

This work focuses on simulations of groups and clusters, but these simulations
can be used to inform and interpret future observational studies.

It might appear that this preferential removal of outer group members could
lead to the formation of very dense, centrally-concentrated groups such as
Hickson Compact Groups (Hickson, 1982) in and around clusters. As Fig.
LABEL:fig:phasespace_arrows shows though, the galaxies that remain bound to a
group do not remain in the same region of phase space. Instead, the remaining
galaxies are redistributed throughout the group until they follow a similar
distribution to that shown in Fig. 2. These group remnants are no more
centrally concentrated than the infalling groups.

Most importantly, Fig. 5 and the top-left panel of Fig. 8 show that, of the
galaxies that are bound to a group when it approaches a cluster, almost none
are still bound to a group after just a single crossing of the cluster ($\sim
2$ Gyr later). Typically, only a very small number of galaxies remain in a
group, and so the remnant ‘groups’ are usually either single galaxies, or
galaxy binaries. Groups with five or more members are extremely unlikely to
have previously experienced a cluster environment; across all of our
simulations, less than $1\%$ of such groups entering a cluster after $z=0.1$
have previously passed through a cluster. Because of this, galaxy groups
nearby to a cluster (i.e. in the cluster outskirts, just outside of
$R_{200}^{\rm{clus}}$) are very unlikely to contain backsplash galaxies, which
typically make up about $50\%$ of the galaxies surrounding a cluster (Gill et
al., 2005; Haggar et al., 2020). Instead, these groups represent a population
of galaxies that are unprocessed by their host clusters, but have experienced
group effects in their past. This may also partly explain why unrelaxed
clusters, which contain more substructure and hence more galaxy groups, are
typically surrounded by fewer backsplash galaxies than relaxed clusters
(Haggar et al., 2020).

The fact that galaxy groups observed nearby to clusters are very likely to be
on their first approach to the cluster has important implications for
observational studies of galaxy evolution and environmental pre-processing.
Additionally, we can infer greater detail about the histories of the galaxies
in these groups. For example, cluster galaxies that are currently not in
groups are unlikely to have previously experienced the dense, central regions
of a group, as galaxies in group centres are much more likely to remain in
their groups. Similarly, galaxies associated with groups inside clusters have
almost certainly previously passed through the group centre, even if they now
reside in the group outskirts. This means that they will have experienced the
most extreme environmental impacts of the group. Hester (2006) showed that, in
groups of a similar size to those used in this work
($M_{200}^{\rm{grp}}=10^{13}M_{\odot}$), a disk galaxy with a dark matter mass
of $10^{11}M_{\odot}$ at $r=0.75R_{200}^{\rm{grp}}$ will have $\sim 20\%$ of
its disk gas removed, but if this galaxy passes within
$r=0.25R_{200}^{\rm{grp}}$ of the group centre, it can lose approximately
$90\%$ of its gas. They attribute this to the stronger ram pressure stripping
that takes place in group centres.

##  5 Conclusions

In this work we use hydrodynamical simulations to study the evolution of
intermediate sized galaxy groups (five to 50 members with stellar masses
$M_{\rm{star}}\geq 10^{9.5}M_{\odot}$) in the vicinity of large galaxy
clusters, and specifically from the time after the groups pass within
$R_{200}$ of the cluster. We begin by studying the positions and speeds of
galaxies relative to their host group in order to characterise how this ‘phase
space’ of the group changes over time, before studying the fates of group
members after the passage of their group through a cluster. Our findings are
summarised below.

  * •

On entering a cluster, galaxy groups typically pass within
$0.6R_{200}^{\rm{clus}}$ of the cluster centre. Most of these groups remain
permanently bound to the cluster, although a small fraction ($\sim 10\%$)
reach an apocentric distance outside of the cluster’s radius,
$R_{200}^{\rm{clus}}$.

  * •

The dynamics of these groups change in two phases. First, the member galaxies
increase their speeds relative to the group centre, often becoming
gravitationally unbound. Then, after the group passes the pericentre of its
cluster orbit (which typically occurs after $\sim 0.5$ Gyr in the cluster),
the distances of galaxies from the group centre increases.

  * •

The majority of galaxies bound to a group at its first cluster infall are no
longer in the group after a full passage through the cluster. Many of these
galaxies become either unbound from the group, heavily stripped, or merge with
the Brightest Group Galaxy, and the fate of a galaxy depends strongly on its
location within the group at the infall time.

  * •

Consequently, the overwhelming majority ($>99\%$) of groups that enter a
cluster are doing so for the first time in their histories. In observations,
groups that are seen just outside of a cluster are very unlikely to have
previously experienced a cluster environment.

Although the composition and structure of simulated galaxy groups is dependent
on the physical models that are used, the results from this work still allow
us to make conclusions about groups that can be applied to observational work.
Groups that are observed nearby to clusters are almost certainly recent
infallers, particularly groups with low velocity dispersions, as galaxies in
groups become gravitationally unbound almost immediately after entering a
cluster. Furthermore, any galaxies that are observed in a group that is inside
a cluster will have previously passed through the group centre, and so will be
severely stripped by the tidal forces and ram pressure of their host group.

In addition to the approach taken in this paper, which draws conclusions on
galaxy groups that can be applied observationally, work remains to be done on
the dynamics of these groups. In future work we plan to study the dynamics of
these groups in greater detail. For example, the binding energy-angular
momentum phase space, and the orbital parameters of galaxies, can tell us
about the anisotropy of group members’ orbits (Wojtak et al., 2008; Lotz et
al., 2019, for example), which can in turn be used to describe how virialised
is a group. In our future work we will study the time evolution of these
dynamical parameters.

The work in this paper further strengthens the case that galaxy groups provide
a unique way to study galaxy evolution, and particularly pre-processing. As
they are almost all first-infallers, groups in the outskirts of clusters will
have experienced no cluster processing, and will have a very low contamination
by backsplash galaxies. Consequently, pre-processing effects will dominate
over any effects from the cluster in these structures, and so studying these
objects in more detail will allow us to further disentangle the environmental
effects of clusters, and the effects of other cosmic environments. Processes
such as gas removal and morphological changes in these galaxies will
exclusively have occurred pre-infall in groups or cosmic filaments, and so
ultimately the properties of galaxies in groups will help inform us of the
role that environment plays in driving galaxy evolution.

## Acknowledgements

We thank the referee, Gary Mamon, for his helpful and thorough comments, which
have helped to improve the quality of this paper.

This work has been made possible by The Three Hundred
collaboration101010https://www.the300-project.org. This work has received
financial support from the European Union’s Horizon 2020 Research and
Innovation programme under the Marie Skłodowskaw-Curie grant agreement number
734374, i.e. the LACEGAL
project111111https://cordis.europa.eu/project/rcn/207630_en.html. The Three
Hundred simulations used in this paper have been performed in the MareNostrum
Supercomputer at the Barcelona Supercomputing Center, thanks to CPU time
granted by the Red Española de Supercomputación. For the purpose of open
access, the author has applied a creative commons attribution (CC BY) to any
author accepted manuscript version arising.

RH acknowledges support from STFC through a studentship. He also thanks Andrew
Benson for useful suggestions relating to the tidal radius of groups, and Liza
Sazonova for extremely helpful discussions relating to merger trees. AK is
supported by the Ministerio de Ciencia, Innovación y Universidades
(MICIU/FEDER) under research grant PGC2018-094975-C21.

This work makes use of the SciPy (Virtanen et al., 2020), NumPy (van der Walt
et al., 2011), Matplotlib (Hunter, 2007), SymPy (Meurer et al., 2017) and
pandas (McKinney, 2010) packages for Python121212https://www.python.org.

The authors contributed to this paper in the following ways: RH, UK, MEG and
FRP formed the core team. RH analysed the data, produced the plots and wrote
the paper along with UK, MEG and FRP. AK produced the halo catalogues and
merger trees. GY supplied The Three Hundred simulation data. WC assisted with
interpreting the time evolution of group members and group properties,
particularly in Section 4.3. All authors had the opportunity to comment on the
paper.

## Data availability

The data underlying this work has been provided by The Three Hundred
collaboration. The data may be shared on reasonable request to the
corresponding author, with the permission of the collaboration.

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##  Appendix A Evolution of a single galaxy group in phase space

Fig. 10 shows an example of one galaxy group entering, passing through, and
then re-entering a cluster, as a demonstration of the process discussed
throughout this work. The right column shows the dark matter halo of the
cluster, represented by the grey circle, and the paths of the galaxies in a
group as the group passes through the cluster. The system is rotated so that
the path of the group is in the plane of the page.

In the left column, the changing position of each group member in phase space
is shown (i.e. its changing position and speed relative to the host group).
Each line in phase space represents the path taken by one galaxy through phase
space, as the galaxy has followed the path through the cluster shown in the
right column. Six timesteps are shown altogether, from top to bottom. The two
phases of dynamical evolution shown in Fig. 4 and Fig.
LABEL:fig:phasespace_arrows can be seen in the evolving phase space of this
group, as the galaxies move upwards in phase space as the group approaches its
pericentric passage, and then from left to right after this.

For clarity, schematics are shown in the bottom-right of each panel, similarly
to those used in Fig. LABEL:fig:phasespace_arrows, to show where the group is
along its orbit through the cluster. From top to bottom, the six timesteps
show the group immediately after its first infall, one snapshot after
pericentric passage, shortly before exiting the cluster, shortly after exiting
the cluster, at apocentre, and immediately after its second cluster infall.

The right panels show the overall behaviour of groups that we find throughout
this work – a relatively compact group of galaxies remains coherent for a
short period after entering a cluster, but then becomes heavily disrupted, and
the galaxies are separated from each other to large distances (shown in the
final panel).

Figure 10: Phase space evolution of one group as it enters a cluster, passes
through, leaves the cluster and then re-enters, showing the initial increase
in $v$, and subsequent increase in $r$, of the member galaxies. Right panels
show the cluster halo (large grey circle), and a 2D projection of the paths
taken through it by the group halo (thick black line) and the galaxies bound
to this group at infall (thin coloured lines). Left panels show the
corresponding paths taken by these galaxies in the phase space diagram used in
Section 3, from infall to the current snapshot. For clarity, the positions of
galaxies at infall and at the present time are represented by dots and crosses
respectively. Six snapshots are shown altogether, with time increasing from
top to bottom. Schematics in the bottom-right of the left panels show where
the group is on its cluster orbit – for instance, the fifth timestep shows the
group at apocentre. The first and last panels are separated by 10 snapshots,
covering $\sim 3.2$ Gyr between $z=0.25$ and $z=0$ (continued on next page).
Figure 10: (continued from previous page) Evolution in phase space of an
example group.