11institutetext:  Instituto de Física, Pontificia Universidad Católica de
Valparaíso, Casilla 4059, Valparaíso, Chile
11email<EMAIL_ADDRESS>22institutetext: Department of Astronomy,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai,
200240, People’s Republic of China 33institutetext: Key Laboratory for
Particle Astrophysics and Cosmology (MOE), Shanghai 200240, China
44institutetext: Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai 200240, China

#  The history and mass content of cluster galaxies in the EAGLE simulation

Cristóbal Sifón 11 Jiaxin Han 223344

(Received XXX; accepted XXX)

###### Abstract

Aims. We explore the mass content of galaxies residing in galaxy clusters at
$z=0$ in the EAGLE cosmological hydrodynamical simulation as well as the
galaxies’ mass build-up through cosmic time.

Methods. We use a galaxy catalogue generated with the hbt+ algorithm, which
identifies subhaloes consistently over time by tracking their dynamical
evolution throughout the simulation.

Results. The satellite subhalo-to-stellar mass relation (SHSMR) is well
described by a double power-law, becoming increasingly steeper with stellar
mass. At stellar masses $9<\log m_{\star}/\mathrm{M}_{\odot}<10$, satellites
have 20-25% the subhalo mass of central galaxies at fixed stellar mass. At
high stellar masses, $m_{\star}>2\times 10^{11}\,\mathrm{M}_{\odot}$, the
satellite SHSMR is consistent with that of centrals. The satellite SHSMR
decreases steeply for satellites closer to the cluster centre, even in
projection, broadly consistent with recent weak lensing measurements. The
scatter in the satellite SHSMR is larger than that of central galaxies at all
cluster masses and cluster-centric distances $R<R_{\mathrm{200m}}$. The SHSMR
scatter decreases with stellar mass by about 12% over an order of magnitude,
but this dependence can be explained by the mixing of infall times when
binning by stellar mass. By splitting satellites into direct and indirect
infallers (those that fell into their current host as a central galaxy or as
satellite of an infalling group, respectively) we clearly show the impact of
preprocessing separately on satellite galaxies’ dark and stellar mass. There
is significant dark matter preprocessing; the most recent infallers into
massive ($M_{\mathrm{200m}}\gtrsim 10^{14}\,\mathrm{M}_{\odot}$) clusters had
already lost up to 50% of their dark matter by the time of infall,
particularly if they fell in indirectly as satellites of another host. On the
contrary, on average satellite galaxies are still gaining stellar mass at the
time of infall and they do so for another 2 Gyr afterwards, although we see
evidence of a slowing growth for indirect infallers. How much and for how long
they continue to gain stellar mass depends primarily on the gas mass fraction
available at infall. Overall, pre- and post-processing each have similar
impacts on the satellite SHSMR. Finally, we provide a simple prescription to
infer the mean mass loss experienced by satellites as a function of cluster-
centric distance based on a comparison to central galaxies, convenient for
observational weak lensing measurements.

##  1 Introduction

Over time, galaxies evolve and co-evolve as they are pulled together by
gravity and merge into larger galaxies or form larger systems such as groups
and clusters of galaxies. Dynamical timescales in these objects are long, so a
large fraction of the galaxies that fall into them survive as satellite
galaxies. As this process unfolds, tidal interactions as well as hydrodynamic
processes transfer mass from the satellites to the host system, changing the
composition of a galaxy depending on its environment, and the dark matter
haloes hosting these galaxies become subhaloes bound to the parent structure
(e.g., Gunn & Gott, 1972; White & Rees, 1978; Springel et al., 2001). The
clearest observational evidence of this interaction is the rapid removal of
gas from infalling galaxies and subsequent shutdown of star formation, most
notably observed in fraganti in so-called “jellyfish” galaxies (Ebeling et
al., 2014; Jaffé et al., 2018). Thus blue, star-forming spiral galaxies are
transformed to “red and dead”, passively-evolving ellipticals. This
interaction gives rise to the well-known morphology-density relation (Dressler
et al., 1980) and a tight red sequence which is most prominent in massive
galaxy clusters (Gladders & Yee, 2000; Koester et al., 2007).

A similarly dramatic experience is expected for the dark matter halo of a
galaxy driven by tidal interactions. Dark matter is more loosely bound than
stars, which are heavily concentrated in the centres of galaxies, and is
therefore more susceptible to tidal stripping (e.g., Chang et al., 2013; Smith
et al., 2016). Thus, as a galaxy spends more time within a larger halo it
progressively loses preferentially more dark matter—its total-to-stellar mass
ratio decreases with time. Directly probing the dark matter content of
galaxies requires gravitational lensing or spectroscopic measurements of
stellar velocities within galaxies (i.e., rotation curves or velocity
dispersions). Such measurements have provided tentative evidence of mass
stripping using mass segregation as a proxy, i.e., taking distance from the
cluster centre as a proxy for time-since-infall, and evaluating the subhalo-
to-stellar mass ratio or, in some cases, the mean sizes of galaxies. Early
gravitational lensing analyses suggested that galaxies are significantly more
concentrated the closer they are to a cluster centre, which was taken as
evidence of mass loss through stripping (e.g., Limousin et al., 2005, 2007;
Natarajan et al., 2009), although it is possible that these results were at
least partly driven by the modeling choices (e.g., Mira et al., 2011). More
recently weak lensing analyses measuring the average total mass in subhaloes
have found that the total-to-stellar mass ratio decreases with cluster-centric
distance, as expected (Sifón et al., 2015, 2018; Niemiec et al., 2017; Li et
al., 2016; Kumar et al., 2022; Wang et al., 2023), but a quantitative
comparison between works is complicated by differing definitions of subhalo
mass and density profiles (Sifón et al., 2018).

The impact of tidal effects on the dark matter (sub)haloes has been assessed
to great extent in dark matter only simulations, which tend to agree that dark
matter subhaloes can lose as much as 90% of their mass over the timescale of a
few Gyrs after entering a massive halo (e.g. Rhee et al., 2017). A significant
fraction of subhaloes in $N$-body simulations are completely disrupted because
of these interactions, but most of this may be attributed to numerical noise
(Klypin et al., 1999a; Han et al., 2016; van den Bosch et al., 2018) or to the
difficulty in accurately separating subhaloes from parent haloes (Knebe et
al., 2011; Han et al., 2018), which complicates their interpretation.

In comparing to observations, $N$-body simulations provide a valuable first
approach but are limited when performing detailed comparisons, as highlighted
in several well-known discordances (e.g., Klypin et al., 1999b; Moore et al.,
1999; Boylan-Kolchin et al., 2011; Jauzac et al., 2016), which are largely
alleviated when gas heating by supernovae and active galactic nuclei are
incorporated into the simulations (e.g., Gao et al., 2004; Zolotov et al.,
2012; Brooks et al., 2013; Bahé, 2021). Therefore, simulations that
incorporate these hydrodynamical effects are required in order to inform our
interpretation—be it astrophysical or cosmological—of observations in more
detail. Such efforts are now possible thanks to high-resolution cosmological
hydrodynamical simulations which allow predictions of galaxy observables
instead of only dark matter haloes (e.g., Vogelsberger et al., 2014; Schaye et
al., 2015; Pillepich et al., 2018). Indeed, recent studies have made use of
hydrodynamical simulations to establish more realistic expectations for the
mass content of satellite galaxies (Engler et al., 2021; Niemiec et al., 2022;
O’Neil et al., 2023). Characterizing the relation between luminous and dark
matter in satellite galaxies is also useful to allow the population of dark
matter only simulations with galaxies with realistic properties and evolution
(e.g., Lacey et al., 2016; Jiang et al., 2021; Icaza-Lizaola et al., 2023).

In this paper we explore the mass content of satellite galaxies using an
improved galaxy catalogue from the largest hydrodynamical simulation within
the Evolution and Assembly of Galaxies and their Environments (EAGLE) project
(Crain et al., 2015; Schaye et al., 2015). The catalogue was produced with the
hbt+ algorithm (Han et al., 2012, 2018), which natively tracks subhaloes
consistently over time. We describe the subhalo-to-stellar mass relation at
$z=0$ and how it has evolved over time, paying particular attention to the
mass loss experienced by infalling galaxies and the build-up of the relation
between total and stellar mass through time. In this context, we discuss the

This paper is organized as follows. We describe the EAGLE simulation and the
galaxy and cluster samples used in Section 2. In Section 3, we look at the
present-day relation between total and stellar mass and compare to available
observations. In Section 4 we summarize the history of cluster galaxies and
their mass content. In Section 5 we discuss the implications for the subhalo-
to-stellar mass relation over time, and in Section 6 we discuss observational
proxies of mass loss. We summarize our results in Section 7.

Throughout, we use $\log$ to refer to base 10-logarithm. We use the notations
$\langle X\mid Y\rangle$ and $\sigma(X\mid Y)$ to represent the mean and
scatter in $X$ at fixed $Y$, respectively. Unless specified otherwise, masses
and other properties refer to the present time, $z=0$. Lookback times are in
units of Gigayears ago, expressed Gya.

##  2 Simulation data

We use the output of a simulation in the Evolution and Assembly of Galaxies
and their Environment (EAGLE) project (Crain et al., 2015; Schaye et al.,
2015). EAGLE is a suite of cosmological hydrodynamical simulations with
varying box sizes, resolutions, and baryonic feedback prescriptions including
supernova and AGN feedback. The simulation we use in this work is labelled
RefL0100N1504 and has a box size of $(100\,\mathrm{Mpc})^{3}$, with $1504^{3}$
particles and particle masses of $1.81\times 10^{6}\,\mathrm{M}_{\odot}$ and
$9.70\times 10^{6}\,\mathrm{M}_{\odot}$ for baryons and dark matter,
respectively. The simulation was run assuming a flat $\Lambda$CDM cosmology
with $\Omega_{\mathrm{m}}=0.307$, $\Omega_{\mathrm{b}}=0.04825$,
$\sigma_{8}=0.8288$, and $h=0.6777$.

We use the subhalo catalogue produced with the upgraded Hierarchical Bound
Tracing (hbt+, Han et al., 2012, 2018) post-processing of EAGLE, in which
subhaloes are found consistently across time by tracing their evolution as
well as their instantaneous properties. This makes hbt+ both more robust in
identifying subhalos and more stable in building their evolutionary histories.
In particular, hbt+ is robust against background obscuration (Han et al.,
2012) that prevents most subhalo finders from identifying subhaloes near the
center the host (Knebe et al., 2011; Muldrew et al., 2011). By construction,
it is also immune to the flip-flop problem when linking subhaloes across time
(Behroozi et al., 2015). These advantages make hbt+ an ideal choice for this
work.

###  2.1 Mass definitions and sample

Throughout this work we speak of ‘subhaloes’ as all dark matter haloes that
inhabit a larger halo, including central subhaloes (the subhaloes associated
with central galaxies); and of ‘galaxies’ as the subset of subhaloes which
contain at least one star particle. In practice, we limit our study to
galaxies with stellar masses $m_{\star}\geq 10^{9}\,\mathrm{M}_{\odot}$. All
such subhaloes have at least 1,000 particles in total, of which at least 500
stellar particles. Total masses of both subhaloes and galaxies are defined as
the total mass of all bound particles (i.e., the sum of all dark matter, gas,
star, and black hole particles, defined in hbt+ as Mbound), and stellar masses
are similarly defined as the total mass in all bound star particles according
to hbt+. Central and satellite subhaloes are hosted within a ‘host halo’,
whose total mass is defined as an overdensity mass $M_{\mathrm{200m}}$ with
respect to the mean density of the Universe at the corresponding redshift,
contributed by the masses of all the particles within the corresponding
radius, $R_{\rm 200m}$.

Although it is customary in analyses of numerical simulations to express
masses in units of $h^{-1}\mathrm{M}_{\odot}$, throughout this work we take
into account the value $h=0.6777$ assumed in the simulations and express
masses in units of $\mathrm{M}_{\odot}$ alone.

We select all satellite galaxies with stellar masses
$m_{\star}>10^{9}\,\mathrm{M}_{\odot}$ in galaxy clusters with
$M_{\mathrm{200m}}>10^{13}\,\mathrm{M}_{\odot}$ at $z=0$, as calculated by
hbt+. There are 216 clusters above this mass limit in EAGLE, of which 68 have
$M_{\mathrm{200m}}>5\times 10^{13}\,\mathrm{M}_{\odot}$ and 12 have
$M_{\mathrm{200m}}>10^{14}\mathrm{M}_{\odot}$; the most massive cluster in
EAGLE has a mass $M_{\mathrm{200m}}=6.1\times 10^{14}\,\mathrm{M}_{\odot}$.
These 216 clusters host a total of 3,740 satellite galaxies above our stellar
mass limit of $10^{9}\,\mathrm{M}_{\odot}$, of which 1,494 reside in
$M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$ clusters.

##  3 Present-day mass content of cluster galaxies in EAGLE

Figure 1: Subhalo-to-stellar mass relation (SHSMR) for satellite galaxies in
galaxy clusters with masses $M_{\mathrm{200m}}>10^{13}\,\mathrm{M}_{\odot}$ in
the EAGLE simulation. The coloured background shows the occupation of the
$m_{\mathrm{sub}}-m_{\star}$ plane, while the thick blue line shows the best-
fitting Equation 1, which has been fit only over the range in which the blue
line is solid. The dashed magenta line shows the corresponding fit to central
galaxies while black dots show the centrals in the
$M_{\mathrm{200m}}>10^{13}\,\mathrm{M}_{\odot}$ clusters included in this
work. The dashed black line black line shows the satellite SHSMR in
IllustrisTNG (Niemiec et al., 2022) and large circles with errorbars show the
weak lensing measurements of Sifón et al. (2018). The inset shows the
logarithmic scatter around the mean relation, which can be approximated by a
lognormal distribution with standard deviation $\sigma=0.41$.

We begin by studying the relation between satellite galaxies’ subhalo and
stellar masses at $z=0$. In Sections 4 and 5 we then study the evolution of
satellites and their mass content over cosmic time, with the aim of
understanding the physical origin of the present-day SHSMR.

Figure 2: Dependence of the satellite subhalo-to-stellar mass relation on host
cluster mass, $M_{\mathrm{200m}}$, as described by the colour bar, where jumps
in colour mark $M_{\mathrm{200m}}$ bin edges. Shaded regions are standard
errors on the means. The black dashed line with circles shows the mean SHSMR
of central galaxies.

Figure 3: Total-to-stellar mass ratio (i.e., SHSMR) as a function of cluster-
centric distance normalized by cluster size, $R_{\mathrm{200m}}$, binned by
stellar mass. Left: 3-dimensional distances. The straight gray line shows a
linear dependence with cluster-centric distance, for reference. Right:
distances projected along the $z$ direction. Weak lensing measurements from
Sifón et al. (2018), Kumar et al. (2022), and Wang et al. (2023) are shown
with black circles, green squares, and blue triangles, respectively. Numbers
in parentheses correspond to the mean stellar mass in each study, in units of
$10^{10}\,\mathrm{M}_{\odot}$.

###  3.1 Subhalo-to-stellar mass relation

Figure 1 shows the joint distribution of subhalo and stellar masses for
satellite galaxies in massive clusters in EAGLE, as well as the subhalo-to-
stellar mass relation (SHSMR), i.e., $\langle m_{\mathrm{sub}}\mid
m_{\star}\rangle$, for central and satellite galaxies. Following Niemiec et
al. (2022) we fit the satellite and central SHSMRs separately using double
power-laws of the form

$\langle m_{\mathrm{sub}}\mid
m_{\star}\rangle=2N\left[\left(\frac{m_{\star}}{m_{1}}\right)^{-\beta}+\left(\frac{m_{\star}}{m_{1}}\right)^{\gamma}\right]m_{\star}.$
(1)

Table 1: Best-fit subhalo-to-stellar mass relation parameters from Equation 1 for central and satellite galaxies. Parameter | Centrals | Satellites
---|---|---
$\log m_{1}/\mathrm{M}_{\odot}$ |  $10.54$$\,\pm\,$ | $0.05$ |  $10.56$$\,\pm\,$ | $0.09$
$\beta$ |  $0.46$$\,\pm\,$ | $0.02$ |  $0.36$$\,\pm\,$ | $0.04$
$\gamma$ |  $0.57$$\,\pm\,$ | $0.03$ |  $1.11$$\,\pm\,$ | $0.16$
$N$ |  $12.55$$\,\pm\,$ | $0.11$ |  $3.39$$\,\pm\,$ | $0.19$

Best-fit parameters for centrals and satellites are listed in Table 1. The
ratio of satellite-to-central SHSMR increases from $0.19\pm 0.04$ at
$m_{\star}=10^{9}\,\mathrm{M}_{\odot}$ to consistent with unity at
$m_{\star}>2\times 10^{11}\,\mathrm{M}_{\odot}$. As shown in Figure 1, over
the range $9<\log m_{\star}/\mathrm{M}_{\odot}\lesssim 11$ the EAGLE SHSMR
agrees with the SHSMR of satellite cluster galaxies in the IllustrisTNG
simulation derived by Niemiec et al. (2022). At larger stellar masses the
EAGLE SHSMR tends to lie above that from IllustrisTNG. This could be largely
explained by the different subhalo finders used. The most massive subhaloes
usually have their outer part well mixed with their host haloes, making it
difficult to separate the two using only a single snapshot. In IllustrisTNG,
subhaloes were identified using the (instantaneous) configuration-space
subhalo finder subfind, which has difficulty determining the masses of
subhaloes deep into their host haloes (e.g. Knebe et al., 2011; Onions et al.,
2012; Han et al., 2018). As hbt+ uses temporal information when identifying
subhalos, it is able to recover a much more complete list of member particles
for these massive subhaloes. Most notably, the SHSMR for satellites at the
highest mass end approaches that for central galaxies according to hbt+. This
is consistent with the discussions in Han et al. (2018) that the most massive
subhaloes experience weaker tidal stripping and stronger dynamical friction,
so that the surviving massive ones are not much different from centrals.
Nevertheless, to draw conclusions that are not sensitive to this halo-finding
systematic, in the remainder of this work we study satellite galaxies over the
range $9<\log m_{\star}/\mathrm{M}_{\odot}<11$. As we discuss next, this is
also the range over which there are observational measurements to compare
with.

We also compare the EAGLE SHSMR with galaxy-galaxy lensing measurements of
satellites galaxies in massive ($M_{\mathrm{200m}}\gtrsim
10^{14}\,\mathrm{M}_{\odot}$) clusters (Sifón et al., 2018), which is
consistent with measurements at lower cluster masses (e.g., Dvornik et al.,
2020), in Figure 1. The mass range probed by Sifón et al. (2018) stops right
where EAGLE predicts a turn in slope so that the weak lensing measurements are
not able to assess the double power law nature of the SHSMR, but over the
range probed the EAGLE SHSMR agrees with the observational measurements.
Figure 2 shows that the SHSMR111In order to reduce the dynamical range, we
represent the SHSMR as $\langle m_{\mathrm{sub}}/m_{\star}\mid
m_{\star}\rangle$ hereafter. depends weakly on cluster mass. At low stellar
masses, $m_{\mathrm{sub}}\lesssim 10^{10}\,\mathrm{M}_{\odot}$, there is a
constant $\sim$10% difference in $m_{\mathrm{sub}}/m_{\star}$ over an order of
magnitude in cluster mass. At higher stellar masses. $10^{10}\lesssim
m_{\star}/\mathrm{M}_{\odot}\lesssim 10^{11}$, galaxies have approximately
twice as much total mass at fixed stellar mass in $M_{\mathrm{200m}}\sim
10^{13}\,\mathrm{M}_{\odot}$ than in $M_{\mathrm{200m}}\sim
10^{14}\,\mathrm{M}_{\odot}$ clusters, although this trend seems to stall for
larger cluster masses. That this difference manifests itself only for more
massive satellites can be generally ascribed to their shorter dynamical
timescales. The trend with cluster mass, on the other hand, can be due to two
effects, noting that—as discussed later—dark matter is more susceptible to
mass loss than stellar mass. Firstly, tidal stripping is expected to be
stronger in more massive clusters. Secondly, galaxies in more massive clusters
are expected to have been more strongly preprocessed, such that they enter
massive clusters already with a reduced dark matter fraction (Bahé et al.,
2019). We return to this in Section 5.

###  3.2 Subhalo mass segregation

We now turn to the spatial trends we may expect to see in observational data.
Han et al. (2016) studied the spatial distribution of subhaloes in dark
matter-only simulations, finding that they closely follow the dark matter
particles. This implies that dynamical friction does not play an important
role, at least beyond $R>0.02R_{\mathrm{200m}}$. They showed that differences
in the distribution of subhaloes and dark matter particles in the literature
are due not to dynamical friction but to the selection of subhaloes based on
present-day subhalo mass instead of mass at accretion (see also van den Bosch
et al., 2016). Our selection is based on present-day _stellar_ mass, which is
expected to be less affected by these selection biases, and any such biases
will also be present in observations, for which we aim to provide a reference
point.

We show the present-day mass segregation of EAGLE galaxies, expressed as the
ratio $m_{\mathrm{sub}}/m_{\star}$ as a function of cluster-centric distance
at fixed stellar mass, in the left panel of Figure 3. Over the range
$0.05<R/R_{\mathrm{200m}}<1$, this ratio is well described by a power law,
$m_{\mathrm{sub}}/m_{\star}\propto(R/R_{\mathrm{200m}})^{0.7}$. This is
significantly shallower than the relation
$m_{\mathrm{sub}}/m_{\mathrm{sub}}^{\mathrm{max}}\propto(R/R_{\mathrm{200m}})^{1.3}$
for dark matter subhaloes (where $m_{\mathrm{sub}}^{\mathrm{max}}$ is the
maximum mass the subhalo has had throughout cosmic time) found by Han et al.
(2016), although it is close to the slope of 0.6 in the latter relation found
by van den Bosch et al. (2016) in a number of $N$-body simulations. However,
in doing this comparison the strong dependence on sample selection pointed out
by van den Bosch et al. (2016) should be kept in mind. We further discuss the
connection between stellar mass and maximal subhalo mass in Section 4. As
discussed in Section 5.3, at smaller radii the flattening of the relation is
influenced by EAGLE’s mass resolution, whereby only the most massive systems
are present in the simulation.

Recently, large optical surveys with high image quality have allowed
observational measurements of the dependence of the subhalo-to-stellar mass
ratio with (projected) cluster-centric distance, i.e., subhalo mass
segregation, using weak gravitational lensing. Although a detailed comparison
between these results can be affected by modelling and analysis differences,
overall these results do suggest mild segregation (Sifón et al., 2015, 2018;
Li et al., 2016; Niemiec et al., 2017; Kumar et al., 2022; Wang et al., 2023).
In order to allow a comparison with observational work, we also show the
dependence with projected distance, which we refer to as $R_{xy}$,222Here and
throughout we use projected distances measured along the $x-y$ plane. We have
verified that using the $x-z$ or $y-z$ planes produces statistically identical
results, as expected. in the right panel of Figure 3. The decrease in
$m_{\mathrm{sub}}/m_{\star}$ is still present but is reduced due to projection
effects—to roughly a factor 2 instead of 10 between $R_{xy}=R_{\mathrm{200m}}$
and $R_{xy}=0.05R_{\mathrm{200m}}$ at fixed stellar mass. The segregation
observed in EAGLE is consistent with the results of both Sifón et al. (2018)
and Kumar et al. (2022), although the measurements of Kumar et al. (2022) tend
to lie above the EAGLE relations, while Wang et al. (2023) found stronger
segregation than seen in EAGLE. As discussed by Sifón et al. (2018), this
could be due to differing definitions of subhalo mass as well as stellar mass.
An additional complication in the comparison to observations is that our
analysis is entirely free of interlopers, i.e., galaxies mistakenly identified
as cluster members. Because of the steepness of the dependence with cluster-
centric distance and because central galaxies have significantly higher
subhalo-to-stellar mass ratios than satellites, this contamination will tend
to increase the inferred mean mass. We leave a detailed assessment of
interlopers for future work.

The dependence of $\langle m_{\mathrm{sub}}/m_{\star}\mid
R/R_{\mathrm{200m}}\rangle$ on stellar mass means that observationally one
must ensure galaxies at different cluster-centric radii have comparable
stellar mass distributions. In particular, if the mean stellar mass increases
with radius by roughly half a dex or more then we would not expect to observe
any segregation. However this is not the case for any of the recent weak
lensing measurements, where the change is at most a factor 2, i.e., less than
the bin size in Figure 3. We would therefore expect the slopes seen in
observations to be comparable with those in the right panel of Figure 3.
Although there is considerable point-to-point variation, there is overall
agreement between observations and simulations.

The strength of segregation, on the other hand, does not depend on stellar
mass (i.e., all curves in Figure 3 have the same slope). This is expected if
the trend is due mostly to tidal stripping, since stellar mass is more
centrally-concentrated than dark matter and is therefore the component least
affected by external tidal forces (Chang et al., 2013; Smith et al., 2016).

Figure 4: Dependence of the logarithmic scatter in the present-day SHSMR on
host halo mass. The dashed black line with circles shows the scatter of
central galaxies, for comparison.

###  3.3 The scatter in the relation between total and stellar masses in
cluster galaxies

Figure 5: Dependence of intrinsic scatter in the SHSMR on three-dimensional
cluster-centric distances, for satellite galaxies of different stellar masses.

Another important aspect of the relation between total and stellar mass is the
scatter in the relation, as it can tell us about the variety of phenomena that
affect said relation. For instance, Niemiec et al. (2022) found that the
scatter is mostly correlated with distance to the centre and the compactness
of galaxies (in addition to properties specifying the history of the subhalo,
which we explore in Section 5), but it is unrelated to the properties of the
host cluster.

As shown in Figure 4, the scatter in the present-day satellite SHSMR tends to
decrease slightly with stellar mass, and this behaviour shows no significant
dependence on cluster mass. Considering all subhaloes, the scatter is $0.43\pm
0.01$ dex at $m_{\star}=10^{9}\,\mathrm{M}_{\odot}$, while at
$m_{\star}=10^{11}\,\mathrm{M}_{\odot}$ the scatter is $0.36\pm 0.04$ dex,
i.e., a decrease of $(17\pm 11)$% over two orders of magnitude in stellar
mass. This is statistically consistent with Niemiec et al. (2022) who find no
dependence of SHSMR scatter with stellar mass. The scatter in the satellite
SHSMR is larger than the central SHSMR, which also decreases slowly with
stellar mass, from 0.20 dex at $m_{\star}=10^{9}\,\mathrm{M}_{\odot}$ to
roughly 0.15 dex at $m_{\star}=10^{11}\,\mathrm{M}_{\odot}$.

On the other hand, there is a slight but noticeable dependence of the SHSMR
scatter on cluster-centric distance (Figure 5), with a scatter approaching
that of centrals at the closest separations, $R\lesssim 0.05R_{\mathrm{200m}}$
(70 kpc for a $M_{\mathrm{200m}}=10^{14}\,\mathrm{M}_{\odot}$ cluster at
$z=0$). The scatter increases slowly outward from $0.20\pm 0.03$ dex at
$R<0.02R_{\mathrm{200m}}$ to $0.43\pm 0.02$ dex at $R>R_{\mathrm{200m}}$, when
averaging over all stellar masses. As with $\langle
m_{\mathrm{sub}}/m_{\star}\rangle$, this reduction in scatter may be due in
part to the disruption of subhaloes, be it artificial or physical, whereby
only the most massive end of the subhalo population survives very close to the
cluster centre.

Figure 5 shows the dependence of scatter on three-dimensional radius. There is
no such trend in projection—there is even a hint of an inverted trend, with
intrinsic scatter decreasing up to 20% from the innermost regions to just
inside $R_{\mathrm{200m}}$, due to the mixing of satellites at different
physical distances. Beyond $R_{xy}=R_{\mathrm{200m}}$, the intrinsic scatter
increases again to $\approx 0.40$ dex for all stellar masses.

Figure 6: Correlation between the different lookback times listed in Table 2,
in Gigayears ago (Gya). _Diagonal panels_ show kernel density estimates of
each lookback time distribution, using a Gaussian kernel with
$\sigma=0.1\,\mathrm{Gyr}$, and the median marked by a blue line. Orange and
blue curves and dashed vertical lines show distributions and medians for
satellites in $M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$ and
$10^{13}\,\mathrm{M}_{\odot}<M_{\mathrm{200m}}<5\times
10^{13}\,\mathrm{M}_{\odot}$. Each off-diagonal panel in the _lower triangle_
shows the corresponding two-dimensional occupation; the color scale runs from
white (zero) to a color given by the Pearson’s $r$ correlation between the two
times, shown in the corner of each panel, and coded in the bottom colour bar.
The brightness scaling is consistent across panels, with the darkest cells
containing 2.5% of the population or more. Each panel in the _upper triangle_
shows the logarithm of the mean subhalo-to-stellar mass ratio at $z=0$ in each
cell, as coded in the upper colour bar. Numbers in the corners of each panel
also correspond to the correlation coefficients.

##  4 A brief history of satellite galaxies

Having established the statistical properties of cluster satellites at
present, we begin this section by reviewing the history of the satellite
galaxies they host, which provides some context for the results of the
following subsections (and the rest of the paper) where we explore how the
mass content of satellite galaxies changes over cosmic time and what has
driven these changes.

###  4.1 Milestones in the life of a satellite galaxy

We identify the following key moments333For each of these moments we use $i$
to refer to the snapshot number (increasing with cosmic time), $t$ to refer to
the time, and $z$ to the redshift, depending on the context. in the life of a
satellite galaxy: after being formed, we label $i_{\mathrm{cent}}$ the last
snapshot in the simulation when a galaxy was labelled as central and
$i_{\mathrm{sat}}$ the first snapshot it was labelled a satellite of any host,
independent of the host’s mass. We also label the first time a galaxy is
identified as a satellite of its current host as $i_{\mathrm{infall}}$, and
the first snapshot after which it never leaves its current host again as
$i_{\mathrm{acc}}$. By definition, then,
$z_{\mathrm{sat}}\geq\\{z_{\mathrm{cent}},z_{\mathrm{infall}}\\}\geq
z_{\mathrm{acc}}$. Additionally, we register the snapshots where each subhalo
reached maximal total mass ($i_{\mathrm{sub,max}}$), stellar mass
($i_{\mathrm{\star,max}}$), and gas mass ($i_{\mathrm{gas,max}}$). Throughout,
we use “infall” to refer to infall to the $z=0$ host specifically. The times
$t_{\mathrm{sat}}$ and $t_{\mathrm{infall}}$ are sometimes referred to as
first and last infall, and are analogous to $t_{\mathrm{branch}}$ and
$t_{\mathrm{main}}$ in Bahé et al. (2019).444Bahé et al. (2019) defined
$t_{\mathrm{branch}}$ and $t_{\mathrm{main}}$ as the middle of the relevant
snapshot interval, whereas we used the specified snapshot time. This
introduces typically a $\approx 30$ Myr difference between those authors’ and
our definitions at $z<1$ and differences much smaller at earlier times; these
are insignificant for our purposes. $t_{\mathrm{infall}}$ is also similar to
the residence time defined by O’Neil et al. (2023), except they defined it as
the time since the subhalo first crossed $R_{\mathrm{200m}}$, while we define
it based on whether it is gravitationally bound as calculated by hbt+.

Table 2:  Statistics for present-day satellite galaxies’ characteristic lookback times in Gigayears ago (Gya). Brackets show redshifts corresponding to the quoted lookback times. Times are, from top to bottom: sat: first time as a satellite (see text for caveats), cent: last time as a central, infall: first time labelled as a satellite of current host, acc: definitively labelled as a satellite of current host. We also list the times of maximum stellar, subhalo (i.e., total), and gas masses.  Event | Lookback times (Gya)
---|---
Median | 50% range | 90% range
sat | 9.4 [1.46] | 6.0 – | 12.7 | 1.8 – | 13.2
cent | 5.3 [0.52] | 2.8 – | 7.9 | 0.4 – | 9.6
infall | 5.1 [0.48] | 2.7 – | 7.5 | 0.7 – | 9.0
acc | 4.0 [0.35] | 1.8 – | 6.7 | 0.2 – | 8.5
$m_{\mathrm{\star,max}}$ | 1.5 [0.11] | $\,\,\,<4.8$ | $\,\,\,<7.0$
$m_{\mathrm{sub,max}}$ | 6.1 [0.63] | 3.3 – | 8.4 | 0.4 – | 9.7
$m_{\mathrm{gas,max}}$ | 8.2 [1.07] | 6.2 – | 9.7 | 2.1 – | 10.7

We summarize the median and 50% and 90% ranges for all characteristic times in
Table 2. For instance, the time elapsed between being first labelled a
satellite of its current host, $t_{\mathrm{infall}}$, and definitively so,
$t_{\mathrm{acc}}$, is roughly 1 Gyr. This means we are excluding from our
sample galaxies that have entered a halo but have temporarily left it until
they re-enter after apocenter, so-called backsplash galaxies (Ludlow et al.,
2008; Diemer, 2021; Fong & Han, 2021). We make no attempt to include these
galaxies in our sample.

We take a closer look at these characteristic times in Figure 6, which shows
joint histograms in the bottom-left corner plot and the logarithm of the mean
subhalo-to-stellar mass ratio at $z=0$ in the upper-right corner, as a
function of pairs of lookback times. A few interesting features can be noted.
For instance, most panels in the bottom-left triangle show that most galaxies
occupy the $y\approx x$ zone—that is, most of these events happen within a few
hundred Myr of other events labelled here: a present-day satellite typically
spent little or no time as satellites of other host ($t_{\mathrm{acc}}\approx
t_{\mathrm{infall}}\approx t_{\mathrm{cent}}\approx t_{\mathrm{sat}}$). The
time of maximal subhalo mass is also highly correlated with these events, most
notably with the last time a satellite was labelled a central. This is
expected since, no matter the details, after this time a galaxy is or has been
under the influence of a larger host and can therefore be subject to mass loss
processes.

There are three notable exceptions to this strong correspondence between
events. The first is the time of maximal gas mass, which is
$t_{\mathrm{gas,max}}^{\mathrm{lookback}}\sim 6-10\,\mathrm{Gya}$ mostly
independent of other events. Like $t_{\mathrm{sub,max}}$,
$t_{\mathrm{gas,max}}$ is most (though still quite poorly) correlated with
$t_{\mathrm{cent}}$. The most important relation between maximal gas mass and
other characteristic times is that $t_{\mathrm{gas,max}}$ is the first event
to take place for the vast majority of galaxies. Given that stellar mass
continues to grow well past this time, the reason for the early
$t_{\mathrm{gas,max}}$ is not so much gas stripping but the termination of the
gas _supply_. While some level of quenching is observed in low-mass host
(e.g., Brown et al., 2017; Zinger et al., 2018), this is consistent with the
finding in some simulations (including EAGLE) that it requires a massive host
to fully shut down star formation (e.g., Pallero et al., 2022; Hough et al.,
2023).

The second exception is the time of maximal stellar mass, which is by and
large the last event to take place. When this happens can be divided into two
populations: most galaxies attained their maximal stellar mass almost 3 Gyr
after their maximal subhalo mass (although this difference increases at later
times); and a second population—15% of satellites—have maximal amounts of
stellar mass today, i.e., $t_{\mathrm{\star,max}}^{\mathrm{lookback}}=0.0$
Gya. This population is naturally interpreted as consisting of satellites that
have not yet reached their maximal stellar mass, and as shown in Figure 6 can
have fallen into their present host as much as 6 Gya. The median infall time
of these satellites is 2.3 Gya, compared to 6.2 Gya for the rest of the
population; similarly, they were last classified as centrals 2.5 Gya compared
to 7.0 Gya for the rest of the population. These “maximal stellar mass”
satellites are four times as massive (in total subhalo mass) and reside in
clusters 44% as massive (median values) as the rest of the population. This
results from a combination of these satellites being preferentially more
massive while also having spent less time under the influence of a massive
host. As seen in the diagonals of Figure 6, the time of maximal stellar mass
displays the largest difference between satellites in low- and high-mass
clusters today: the fraction of satellites with maximal stellar masses at
$z=0$ is 10% in high-mass ($M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$)
clusters and 42% in low-mass ($M_{\mathrm{200m}}<5\times
10^{13}\,\mathrm{M}_{\odot}$) clusters, and the median time of maximal stellar
mass occurs 4.5 Gyr earlier for satellites currently residing in massive
clusters compared to those in low-mass clusters. These trends are
qualitatively consistent with trends for quenched fractions found by e.g.,
Park et al. (2023).

The third exception concerns $t_{\mathrm{sat}}$: 906 satellites—a full 32% of
the present-day satellite population—became a satelllite more than 12 Gya
(i.e., at most 1.3 Gyr after their formation in the simulation). Unlike the
case for the satellites currently at their maximal stellar mass time, these
“early satellite” population does not appear any different from the rest of
the satellite population: even though their $t_{\mathrm{sat}}$ happened 5 Gyr
earlier than the general population, no other characteristic time follows a
significantly different distribution from the rest of the population, with the
mean characteristic times being different by more than 0.1 Gyr, _including
$t_{\mathrm{cent}}$_. Similarly, both their total and stellar masses are
statistically identical to those galaxies that became satellites for the first
time later on, both at the present day and at all milestones; they reside in
hosts with the same median mass today; and occur in the same proportion in
low- and high-mass clusters. All these early satellites were _born_ as
centrals, and while for many of them it appears the assignment as a satellite
may be an algorithmic glitch—after only one or two snapshots they return to
being centrals for a long time—this is not true for all of them. We therefore
choose to leave $t_{\mathrm{sat}}$ as is for these galaxies. However, the
peculiar distribution must be kept in mind in any discussions involving
$t_{\mathrm{sat}}$.

There is also a notable spike in $t_{\mathrm{acc}}$ roughly 1.1 Gya (partially
smoothed out by the kernel in Figure 6), where 12% of the entire satellite
population definitively became members of their current host within a single
snapshot. This spike is dominated by two major merger events: one where 115
subhaloes definitively became satellites of HostHaloId 1, currently hosting
239 satellites and with $M_{\mathrm{200m}}=2.1\times
10^{14}\,\mathrm{M}_{\odot}$, and another one where 88 subhaloes definitively
became satellites of HostHaloId 5, currently hosting 103 subhaloes and with
$M_{\mathrm{200m}}=1.9\times 10^{14}\,\mathrm{M}_{\odot}$.555All of these
numbers refer to satellites with $m_{\star}>10^{9}\,\mathrm{M}_{\odot}$ only.
In fact, the accretion (or final infall) times in massive clusters follow a
much more punctuated distribution than the overall population, meaning major
mergers are more relevant overall, although this is influenced by small number
statistics; as already mentioned, there are only 12 clusters with
$M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$.

We also note that $t_{\mathrm{infall}}$ shows a higher correlation with both
the times of maximal subhalo and stellar mass than $t_{\mathrm{acc}}$, which
is why—coupled with the very high correlation of $r=0.86$ between
$t_{\mathrm{infall}}$ and $t_{\mathrm{acc}}$—we choose to focus the later
discussion on $t_{\mathrm{infall}}$ and ignore $t_{\mathrm{acc}}$ from now on.
In words, once a galaxy has been identified as a satellite of its current host
it starts to become influenced by it; whether at some later time hbt+ deems it
to have left its host for some time (which, as discussed by Diemer, 2021, is
probably a true feature of membership to a cluster by most definitions) is of
secondary importance regarding the galaxy’s mass content.

The upper-right corner of Figure 6 shows the mean present-day stellar-to-
subhalo mass ratio depending on pairs of characteristic times.666This should
not be confused with the mean characteristic times of satellites with a given
$m_{\mathrm{sub}}/m_{\star}$. The largest gradients in mean present-day
stellar-to-subhalo mass ratio are seen in panels involving time of maximal
subhalo mass, which is closely related to $t_{\mathrm{cent}}$ (but the former
can be expected to depend less on subhalo finder details). We might expect a
similar dependence on $t_{\mathrm{sat}}$ but that is not as clear due to the
prevalence of early-$t_{\mathrm{sat}}$ satellites already discussed. It is
also apparent that recent infallers have higher mean subhalo-to-stellar mass
ratios, something we discuss at length next.

In Figure 6, we also show the marginalized distributions of satellites in the
most massive ($M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$) and the least
massive ($10^{13}\,\mathrm{M}_{\odot}<M_{\mathrm{200m}}<3\times
10^{13}\,\mathrm{M}_{\odot}$) clusters in our sample. There are 12 and 142
such clusters and 1,494 and 1,059 satellites in each respective sample. As
already discussed, only the time of maximal stellar mass shows a significant
dependence on host cluster mass.

###  4.2 Preprocessing stars and dark matter

Figure 7: Preprocessing stars and dark matter. Ratio of infall-to-maximum dark
matter (left) and stellar (right) masses for satellite galaxies, as a function
of infall time. Colours indicate bins in present-day host halo mass,
$M_{\mathrm{200m}}$. Solid and dashed lines show means of galaxies that fell
in as centrals (“direct infallers”) and as satellites of an infalling group
(“indirect infallers”), respectively. Note the different vertical scales.

Figure 8: Postprocessing stars and dark matter. Left: ratio of present-day-to-
infall dark matter mass as a function of time since infall, colour-coded by
total subhalo mass at infall. Right: ratio of present-day-to-infall stellar
mass as a function of time since infall, colour-coded by infall gas-to-stellar
mass ratio. In each case, the quantity shown in the colour bar is the one that
maximizes differences in behaviour. Line styles are as in Figure 7.

The concept of preprocessing is usually applied to the quenching of star
formation prior to infall onto the current host. It is generally accepted that
preprocessing plays a central role in shaping cluster galaxy populations, with
evidence found both in observations (Haines et al., 2015; Olave-Rojas et al.,
2018) and simulations (Bahé et al., 2019; Pallero et al., 2022). It is also
known that the domain of influence of a dark matter halo is much larger than
the virial radius (Fong & Han, 2021; Gao et al., 2023; Zhou & Han, 2023). Here
we discuss whether and to what extent galaxies gain or lose mass separately in
the form of stars or dark matter, independent of the mechanisms, before
falling into their current host.

As with star formation quenching, the mass growth of a galaxy can be
significantly slowed down or even stopped well before it entered its current
host—in some cases, any host at all. As summarized in Table 2, maximal subhalo
mass occured on average 1 Gyr before infall; as shown in Figure 6 in some
cases maximal subhalo mass can occur up to 8 Gyr before infall, and in a few
rare cases it even happens before a galaxy is ever labelled a satellite of any
host. The left panel of Figure 7 shows this dark matter preprocessing more
directly: on average, satellites in massive
($M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$) clusters fell into their
current host with 70–80% their maximal dark matter mass, and as low as 60% at
$z=0$. Even satellites currently residing in low-mass clusters
($10^{13}<M_{\mathrm{200m}}/\mathrm{M}_{\odot}<10^{14}$) fell in to the
present host with roughly 90% their maximal dark matter mass.777These overall
numbers can be extracted visually from Figure 7 by combining the solid and
dashed lines accounting for uncertainties, since both sets are mutually
exclusive. The curves have been omitted from the figure to avoid overcrowding.
In fact, the level of preprocessing depends strongly on whether a present-day
satellite fell in as a central or as a satellite in a smaller host. We use the
terms “direct infall” and “indirect infall” to refer to present-day satellites
that fell into their current host as central or satellite galaxies,
respectively. In the latter case, the subhalo became a sub-subhalo888Sub-
subhaloes are identified in hbt+ as having Depth $>1$., at least temporarily.
At any given time, between 30 and 40% of infallers in EAGLE do so as
satellites of an infalling group, i.e., through indirect infall. Galaxies that
fell in directly (i.e., as centrals) did so with almost their maximal dark
matter masses. The exception is directs infallers falling into the most
massive clusters, which fell in having lost almost 10% of their dark matter
mass on average _while being centrals_. Indeed, we may expect a small level of
dark matter loss even for central galaxies, because tidal stripping can act
well before a central halo reaches the virial radius of its neighbour
(Behroozi et al., 2014), or due e.g. to relaxation following a major merger or
tidal losses from close encounters (Lee et al., 2018). Indirect infallers, in
contrast, lost as much as 40–50% of their dark matter to preprocessing, on
average. These trends are also consistent with results in numerical
simulations (e.g., Joshi et al., 2017). As expected, the amount of dark matter
preprocessing increases with cosmic time for galaxies falling into their
present hosts indirectly as satellites of another host. The ratio of dark
matter at infall to maximal dark matter is constant with time for direct
infallers in the most massive clusters, but it increases with cosmic time for
clusters with $M_{\mathrm{200m}}\lesssim 10^{14}\,\mathrm{M}_{\odot}$, as
neighboring gravitational potentials get stronger.999We note that some
galaxies do reach maximal subhalo mass _after_ infall. However, this is not
the dominant behaviour for direct nor indirect infallers at any given time. On
average, therefore, it is appropriate to interpret the lower infall subhalo
mass as preprocessing rather than as late growth.

Stellar mass shows the opposite trend. As shown in Figure 6, it is quite rare
for a galaxy to reach maximal stellar mass before infall; only 12% of present-
day satellites had $z_{\mathrm{\star,max}}>z_{\mathrm{infall}}$. In fact, for
galaxies with $t_{\mathrm{infall}}^{\mathrm{lookback}}\gtrsim 3$ Gya maximal
stellar mass typically occurs 2 Gyr _after_ infall to the present host. More
recent infallers cover a wide range of $t_{\mathrm{\star,max}}$, with the
latter tending to happen earlier, simply because $t_{\mathrm{infall}}$ happens
late. As with dark matter mass, both the maximal stellar mass and the trend
with infall time depend on both (present-day) cluster mass and on whether the
galaxy fell in directly or indirectly, although the latter dependence is seen
only for massive ($M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot})$ clusters.
In particular, galaxies that entered their current host, say, 8 Gya, did so on
average with 60–75% their maximal stellar mass depending on cluster mass. (We
have verified that the dependence on cluster mass is more significant than
that on subhalo mass.) As already discussed, maximal stellar mass by and large
happens after infall; this means satellites in lower-mass clusters grew almost
twice as much stars after infall than their counterparts in higher-mass
clusters. However, galaxies that fell into
$M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$ clusters indirectly gained no
more (on average) than 20% additional mass afterwards. The fact that this
maximal gain has remained constant over the last 7–8 Gyr (i.e., since $z\simeq
1$) for indirect infallers only is another way of seeing star formation
quenching by preprocessing. In contrast, the post-infall stellar mass gain
continues to increase even to the present day for indirect infallers into low-
mass clusters.

###  4.3 Stars and dark matter after infall

Figure 8 shows what happens after Figure 7: upon infall, a galaxy steadily
loses dark matter, taking an average 5 Gyr for a galaxy to lose half the dark
matter it had at infall. The left panel of Figure 8 shows that the dark matter
loss _rate_ since infall, at fixed infall time, depends slightly on infall
mass. Among recent infallers ($t_{\mathrm{infall}}<2$ Gya), less massive
subhaloes have lost a fraction of dark matter approximately twice as high as
the more massive subhaloes. However for the very early infallers
($t_{\mathrm{infall}}\sim 10$ Gya) the more massive subhaloes have lost higher
dark matter fractions. Although the trend over time does not depend on cluster
mass, satellites in lower-mass clusters always retain roughly ten percentage
points more of their infall dark matter (not shown). For instance, satellites
that fell 10 Gya to a cluster with present-day mass $M_{\mathrm{200m}}\sim
2\times 10^{13}\,\mathrm{M}_{\odot}$ retain 15–20% of their infall dark
matter, while sateliltes that fell 10 Gya to a cluster with present-day
$M_{\mathrm{200m}}\sim 2\times 10^{14}\,\mathrm{M}_{\odot}$ retain 5-10%.
Similarly, satellites that fell in to their present host 2 Gya retain, on
average, 85% (75%) their infall dark matter mass if they reside in a
$M_{\mathrm{200m}}\sim 2\times 10^{13}\,\mathrm{M}_{\odot}$
($M_{\mathrm{200m}}\sim 2\times 10^{14}\,\mathrm{M}_{\odot}$) cluster. Whether
a subhalo fell in directly or indirectly is only apparent in terms of dark
matter loss within the first 2–3 Gyr after infall; the dark matter fraction
lost since infall does not depend on the status at infall for earlier
infallers. This timescale can be interpreted as the timescale it takes the new
host to take over the small-scale gravitational influence over the satellite
galaxy, and may in fact be associated with the dismembering of the infalling
host (Haggar et al., 2023).

The right panel of Figure 8, on the other hand, shows that stellar mass
continued to grow significantly after infall. This could already be seen from
Figure 6; Figure 8 shows that this is true specifically for satellites that
fell in to their present host with $m_{\mathrm{gas}}^{\mathrm{infall}}\gtrsim
m_{\star}^{\mathrm{infall}}$, independent of when they fell in. It is quite
interesting that very gas-rich galaxies, with
$m_{\mathrm{gas}}^{\mathrm{infall}}\gg m_{\star}^{\mathrm{infall}}$, gained a
_larger_ stellar mass fraction if they fell in indirectly than if they fell in
directly, independently of the time of infall. For galaxies to have less
stellar mass today than they had at infall requires either a long time since
infall, $t_{\mathrm{infall}}\gtrsim 10$ Gya, or a low availability of gas at
infall, $m_{\mathrm{gas}}^{\mathrm{infall}}\ll m_{\star}^{\mathrm{infall}}$.

Figure 9: Evolution in the subhalo mass-stellar mass plane within two
randomly-selected clusters in EAGLE, one in the low-mass end and the other in
the high-mass end of our sample (HostHaloId 16 and 1 at $z=0$ in the hbt+
catalog, respectively). From left to right, we show the motion in the
$m_{\mathrm{sub}}-m_{\star}$ plane of 38 and 50 subhaloes within each cluster,
respectively, selected randomly to span the full stellar mass range explored
in this work but including the 6 most massive subhaloes (including the central
subhalo, whose track ends on a white star), and only selecting satellites with
$t_{\mathrm{sat}}<12$ Gya (see Section 4.1). Note that subhaloes with present-
day stellar masses $m_{\star}<10^{9}\,\mathrm{M}_{\odot}$ are shown here for
illustration but are not part of the sample analyzed in this work. All three
panels show the same subhaloes per host. As labelled in the top left, each
panel shows the timespan between the birth of a subhalo and the first time
they were labelled a satellite of any host (left), since then until they fell
into their present host for the first time (middle), and from first infall
until the present day (right), except for the track of the central subhalo
which is reproduced from birth to the present day in all panels. Each
subhalo’s history is colour-coded by age of the Universe. The thin solid and
dash-dotted black lines show the present-day satellite and central SHSMR,
respectively, for reference. Gray dashed diagonal lines show constant subhalo-
to-stellar mass ratios, increasing by factors of ten from bottom to top, as
labelled.  Figure 10: Pre- and post-processing of the satellite SHSMR. From
top to bottom we show the SHSMR at $z=0$, at the time of infall to the present
host, and at the time each galaxy was labelled a satellite for the first time.
From left to right we show increasing times since infall, as labelled. Black
solid lines show the SHSMR for galaxies that fell in to their present hosts as
centrals while coloured lines show galaxies that fell in as satellites of
another host, colour-coded by time elapsed between first being labelled a
satellite and falling into the present host, for satellites residing in
$M_{\mathrm{200m}}>1e14\,\mathrm{M}_{\odot}$ only. Black dashed lines with
circles show the SHSMR of central galaxies at the median infall time of each
sample. Shaded regions show normal uncertainties on the mean. Note the
different vertical scales in each row.

##  5 Evolving the satellite HSMR

Having discussed the history of satellites and their mass content at length,
we now turn to the build-up of the SHSMR and its relation to the central SHSMR
today.

###  5.1 Evolution of the mean relation

Figure 9 illustrates the evolution of the mass content of subhaloes in detail
for two randomly-selected clusters, one on the low-mass end and another on the
high-mass end of our sample, in terms of the milestones described in Section
4.1. The most striking difference between satellites in either cluster is
_when_ subhaloes suffer most of the mass loss: in the more massive cluster
much of the mass loss happens prior to infall when satellites resided in
another host (i.e., in the middle panel), while in the lower-mass cluster most
of the mass loss suffered by subhaloes happens within their current host
(i.e., in the rightmost panel). Indeed, some subhaloes in the massive host
lost a significant fraction of their total mass before being satellites of any
host at all, and many spent a significant amount of time as satellites before
falling into their current host: they fell into the massive cluster indirectly
as satellites. On the contrary, most galaxies in the low-mass cluster do not
have a long trail in the middle panel, i.e., $t_{\mathrm{sat}}\approx
t_{\mathrm{infall}}$. Many galaxies in the low-mass cluster also kept growing
their stellar mass for a significant time after infall, while this is rarely
the case in the massive cluster.

Once mass loss begins it happens quickly; some of the latest-falling galaxies
already lost an order of magnitude in total mass within the last 2 Gyr. This
may be due to a number of effects. For one, the host cluster is continuously
growing over time, which means the potential encountered by late infallers is
much deeper—the gravitational effects significantly stronger—than that
encountered by early infallers (halo growth can be traced approximately by the
growth of the central subhalo depicted in Figure 9). However, stellar mass
usually continues to grow well after total mass starts to decrease, in
particular for early infallers. As discussed in Section 4.1, galaxies that
fell early on may have still had a significant gas reservoir, while late
infallers had already lost their gas supply while satellites of previous hosts
and were not able to grow their stellar mass as much. The trends of Figure 7
are manifested in the fact that most tracks are already “moving” down in the
bottom middle panel—decreasing $m_{\mathrm{sub}}$ before infall to the massive
cluster—and that most tracks in the right-most panels have some rightward
movement (i.e., increasing $m_{\star}$ after infall), although less so in the
massive cluster (bottom panel). The same behaviour has been described in the
IllustrisTNG simulation by Engler et al. (2021).

The impact of the combination above in the SHSMR over time is shown in Figure
10. Anticipating the discussion, we note two differences with respect to the
rest of the paper, which also apply to Figure 11. First, we only show the
SHSMR of satellites in massive clusters,
$M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$, to avoid cluttering. For
lower-mass hosts we see a reduction in the impact of preprocessing, which for
$M_{\mathrm{200m}}<5\times 10^{13}\,\mathrm{M}_{\odot}$ is approximately half
that discussed below hosts, as shown in Figure 15. Second, we include all
satellites regardless of their present mass in each panel, i.e., we include
satellites that have gone below $m_{\star}=10^{9}\,\mathrm{M}_{\odot}$ at
$z=0$ but have large enough masses at infall or when first becoming a
satellite, as appropriate for each panel. This latter change makes the
discussion less sensitive to artificial disruption (e.g. Han et al., 2016;
Bahé et al., 2019).101010We also exclude galaxies with $t_{\mathrm{sat}}>12$
Gya, which reduces the statistical power of these results but, as discussed in
Section 4.1, has no impact on our conclusions. The top row shows that the
present-day SHSMR depends on preprocessing time but only for galaxies
infalling in the last 8 Gyr. It appears as though it takes at least 2 Gyr for
preprocessing to become apparent in the SHSMR at late times
($t_{\mathrm{infall}}<4$ Gya). At any rate, the dependence of the SHSMR on
time since infall is at least comparable to the dependence on processing time.
This is consistent, for instance, with recent results on galaxy quenching in
IllustrisTNG (Park et al., 2023) and as inferred in observations of clusters
at $z\sim 1$ (Werner et al., 2022). The shorter extent of the dark purple line
in the top-left panel means, roughly speaking, that there are no massive
satellites ($m_{\star}>10^{10}\,\mathrm{M}_{\odot}$) which became satellites
for the first time in the last 4 Gyr and then fell into their current host
within 1 Gyr.

The middle row shows the preprocessing of the SHSMR directly as the SHSMR at
the time of infall, binned in an identical manner as discussed above. Galaxies
with longer preprocessing times fall into their current hosts with
significantly lower SHSMRs, and the dependence with time since infall is
mostly removed, as expected. Here it is clear that 1 Gyr is enough
preprocessing time to significantly shift the SHSMR of galaxies falling into
massive clusters (at least until roughly 4 Gya).

Finally, the bottom panels of Figure 10 show the SHSMR at the time present-day
satellites first became satellites of any host, i.e., $t_{\mathrm{sat}}$. The
left panel shows, surprisingly, that satellite galaxies have a lower SHSMR
than central galaxies _even when they first become satellites_. This shows
that will-be satellite galaxies are influenced by the larger-scale environment
differently than galaxies that will remain centrals up to the present day.
There is a hint that this effect may be stronger for galaxies with _shorter_
preprocessing times, however this is not statistically significant. The lower
SHSMR of will-be satellites is even apparent at the time of maximal subhalo
mass (not shown) instead of $t_{\mathrm{sat}}$, showing this is not an
artifact of the labelling of a galaxy as satellite by hbt+. In words, dark
matter preprocessing is more dramatic than suggested by the discussion in
Section 4.2: before removing mass it halts growth. Galaxies which will become
satellites in the future have already began being preprocessed by the time
they reach maximal mass. This has important implications for the estimation of
mass loss observationally, which we discuss in Section 6.

As mentioned, Figure 15 shows the pre- and post-processing of the SHSMR for
satellites currently residing in low-mass clusters. The levels of
preprocessing are smaller, as may be expected, but the trends are otherwise
equivalent. It is noteworthy that the SHSMR at $t_{\mathrm{sat}}$ is also
lower than that of centrals at the same times.

Figure 11: Impact of pre- and post-processing on the satellite SHSMR scatter
over time. Samples, binning, and styles are as in Figure 10.

###  5.2 Evolution of scatter in the SHSMR

We now turn to the evolution of the scatter in the satellite SHSMR, which we
explore in Figure 11. The first interesting fact from these figures is that
when controlling for satellite history the scatter in the satellite SHSMR is
significantly reduced. This is already apparent in Figure 10, in that
$t_{\mathrm{infall}}$ clearly separates occupation in the
$m_{\mathrm{sub}}-m_{\star}$ plane. The scatter in the present-day SHSMR
decreases slightly for more recent infallers, and except for those galaxies
with the longest preprocessing times it is independent of preprocessing time
itself. This suggests, as does the discussion below, that it is the time spent
as a satellite, $t_{\mathrm{sat}}$, the time that dominates the present-day
SHSMR scatter. Whether the time spent as a satellite is spent within a less
massive or more massive host is of secondary importance in these terms.

As the bottom panels show, when becoming satellites the SHSMR shows as little
scatter as that of centrals. The middle panels, in turn, isolate the effect of
preprocessing, which increases the scatter by up to 0.2 dex, with a timescale
that depends on infall time (or, equivalently for this purpose, on
$t_{\mathrm{sat}}$). That is, early infallers suffer preprocessing more
quickly than late infallers, which seem to require several Gyr for the scatter
to increase. This increase in scatter is, in the end, equally large as for
recent infallers. As anticipated this suggests that $t_{\mathrm{sat}}$ is more
important than preprocessing time in establishing both the mean and scatter of
the SHSMR at infall.

As in the preceding section, we note that Figure 11 only shows satellites
currently residing in $M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$. Figure
16 shows the equivalent for satellites in $M_{\mathrm{200m}}<5\times
10^{13}\,\mathrm{M}_{\odot}$ clusters. The only difference may be that the
scatter in the SHSMR increases more quickly due to preprocessing for
satellites currently residing in low-mass clusters, but the trends are
otherwise consistent.

###  5.3 The evolution of segregation

Finally, we look at how the segregation described in Section 3.2 builds up
over time. Figure 12 clearly shows the trend in Figure 3 evolving since
infall, with galaxies both getting closer to the centre and losing
preferentially more dark matter over time. Galaxies that fell most
recently–within the last 2 Gyr or so—show little segregation within
$R_{\mathrm{200m}}$, and cannot be found closer in than $0.1R_{\mathrm{200m}}$
as they have yet to reach the centre of the cluster. Except for the innermost
radii, 10 Gyr after infall the SHSMR depends on cluster-centric distance as
$\propto(R/R_{\mathrm{200m}})^{0.70}$. The flattening at
$R<0.1R_{\mathrm{200m}}$ is due to our selection in present-day stellar mass,
which means for heavily stripped subhaloes we only retain those with high
$m_{\mathrm{sub}}/m_{\star}$. We confirm that selecting by infall mass instead
produces relations consistent with power laws in the entire
$R<R_{\mathrm{200m}}$ range, but the slopes at $R\gtrsim 0.1R_{\mathrm{200m}}$
do not change significantly. Using $t_{\mathrm{sat}}$ instead of
$t_{\mathrm{infall}}$ (not shown) produces trends that match each other more
closely than Figure 12, i.e., segregation does not depend as strongly on
$t_{\mathrm{sat}}$ as it does on $t_{\mathrm{infall}}$. This is expected since
preprocessing happens by definition while a galaxy is in the outskirts, or
even entirely outside, of its (eventual) host cluster, well before galaxies
have sunk into the cluster centre.

Recent infallers show different subhalo-to-stellar mass ratios at fixed
cluster-centric distance but this is no longer the case for
$t_{\mathrm{infall}}\gtrsim 7$ Gya. Of such early infallers, indirect
infallers have considerably lower ratios than direct infallers, due to the
effect of the infalling host. Interestingly, the observation in Figure 3 that
satellites have a lower $m_{\mathrm{sub}}/m_{\star}$ beyond
$R_{\mathrm{200m}}$ is due to indirect infallers alone, while direct infallers
have a constant $m_{\mathrm{sub}}/m_{\star}$ from $R_{\mathrm{200m}}$ outward.
The galaxies that make it beyond $R_{\mathrm{200m}}$ after infall are those
with the most elliptical orbits and therefore shortest pericentres, which
suffer the most mass loss (e.g., Bakels et al., 2021; Smith et al., 2022).
However, Haggar et al. (2023) have shown that galaxies in infalling groups are
largely shielded from the host cluster if they are within the tidal radius of
the infalling group, at least for some time. Combined, these effects explain
why only indirect infallers show a decrease in $m_{\mathrm{sub}}/m_{\star}$
when located outside $R_{\mathrm{200m}}$ of the host cluster.

We also note that the fact that earlier infallers tend to be located closer to
the cluster centre was discussed by van den Bosch et al. (2016), who showed
that the strong segregation of
$m_{\mathrm{sub}}/m_{\mathrm{sub}}^{\mathrm{infall}}$ of dark matter subhaloes
is due to the inside-out growth of haloes, i.e., that earlier infallers joined
the host halo closer to the centre than late infallers do.

Figure 12: Mass segregation build-up: subhalo-to-stellar mass ratio as a
function of present-day 3-dimensional distance to the cluster centre,
normalized by $R_{\mathrm{200m}}$ and split according to time since infall, as
shown in the colourbar. Solid and dashed lines show direct and indirect
infallers, respectively. The grey line shows a linear relation, for reference.

##  6 Quantifying mass loss in observations

Figure 13: Bias incurred in the estimation of mass loss by assuming it can be
calculated as the ratio of satellite-to-central subhalo mass at fixed stellar
mass (bottom). We calculate this bias as the ratio between the present-day-to-
maximal subhalo mass (i.e., mass loss itself; top), and
$\tau_{\mathrm{strip}}=1-m_{\mathrm{sat}}/m_{\mathrm{central}}$ (i.e., the
observational proxy; middle). All curves are colour-coded by projected
cluster-centric distance normalized by $R_{\mathrm{200m}}$, and shaded regions
show corresponding normal uncertainties. Dashed lines correspond to using the
outermost bin in the denominator instead; see Section 6 for details. Black
lines show quantities for all satellites together.

It is common among observational studies to attempt to quantify the mass loss
in satellite galaxies by comparing their subhalo-to-stellar mass ratio with
that of a sample of central galaxies with matching properties, usually stellar
mass (e.g., Kumar et al., 2022). Under the assumption (justified, as we have
seen) that stellar mass is much more stable than dark matter to mass loss,
differences in the SHSMR between the two samples are taken as indication of
mass loss. For instance, Wang et al. (2023) defined a stripping fraction as
$\tau_{\mathrm{strip}}=1-m_{\mathrm{sub}}/m_{\mathrm{central}}$, i.e.,
assuming $m_{\mathrm{central}}=m_{\mathrm{sub}}^{\mathrm{max}}$ at fixed
stellar mass, using the central SHSMR from Shan et al. (2017) to obtain
$m_{\mathrm{central}}$. Alternatively, Niemiec et al. (2017) defined
$\tau_{\mathrm{strip}}=1-m_{\mathrm{sub}}^{\mathrm{inner}}/m_{\mathrm{sub}}^{\mathrm{outer}}$,
where $m_{\mathrm{sub}}^{\mathrm{inner}}$ and
$m_{\mathrm{sub}}^{\mathrm{outer}}$ correspond to the mean masses of subhaloes
in the inner and outer regions of clusters, respectively.

As we have seen, stellar mass can increase quite significantly after infall.
This means that the timescales for dark matter mass and stellar mass build-up
in satellite galaxies are different, with maximal stellar mass attained
typically 4 Gyr after maximal dark matter mass. Therefore, the points of
comparison for dark matter loss and stellar mass loss are different. On the
other hand, stellar mass generally contributes a small fraction of the total
mass, even for heavily stripped subhaloes, so the stellar mass increase makes
little difference for the overall amount of mass lost due to stripping.

With the aim of providing an anchor for observational results such as the
lensing measurements discussed in Section 1, here we evaluate how does the
total, true mass loss compare to $\tau_{\mathrm{strip}}$. We calculate this
mass loss bias as

$\mathrm{bias}=\frac{\tau_{\mathrm{strip}}}{1-m_{\mathrm{sub}}/m_{\mathrm{sub}}^{\mathrm{max}}}-1=\frac{1-m_{\mathrm{sub}}/m_{\mathrm{central}}}{1-m_{\mathrm{sub}}/m_{\mathrm{sub}}^{\mathrm{max}}}-1$
(2)

where $m_{\mathrm{sub}}$ and $m_{\mathrm{central}}$ correspond to the subhalo
masses of satellite and central galaxies, respectively (including baryons and
dark matter). Figure 13 shows that this bias is approximately independent of
stellar mass but strongly dependent on projected cluster-centric radius,
increasing from 6% for the innermost bin shown to 51% in the outermost
bin.111111These values correpsond to inverse-variance weighted averages of
data points with $m_{\star}<5\times 10^{10}\,\mathrm{M}_{\odot}$, beyond which
uncertainties dominate. In particular, note how the bias for
$R_{01}>R_{\mathrm{200m}}$ diverges at higher stellar masses, because mass
loss itself approaches zero. We also calculate $\tau_{\mathrm{strip}}$
similarly to Niemiec et al. (2017), by taking the ratios between mean subhalo
masses in each of the three innermost radial bins and the outermost one,
covering $0.8<R_{01}/R_{\mathrm{200m}}\lesssim 2$; these lead to the dashed
lines. Because such a definition essentially ignores preprocessing, it leads
to an underestimation of mass loss. Niemiec et al. (2017) used for the
denominator a radial bin similar to the third radial bin in Figure 13, namely
$0.55<R/(h^{-1}\,\mathrm{Mpc})<1.0$; because EAGLE galaxies in that radial bin
have already lost 40–50% of their total mass already, this would lead to a
larger underestimation of mass loss. We list the mean mass loss bias, Equation
2, in Table 3, when averaging each sample over the whole stellar mass range.
Although the bias is significant it is well behaved in that it does not depend
on stellar mass; it can therefore be treated as a correction to
$\tau_{\mathrm{strip}}$ that depends only on the radial distribution of the
satellites under consideration.

Mean radius ($R_{\mathrm{200m}}^{-1}$) | 0.035 | 0.1 | 0.4 | 1.5
---|---|---|---|---
Mass loss bias | 0.06 | 0.13 | 0.32 | 0.51
Table 3: Mass loss bias, Equation 2, for each bin in projected cluster-centric
distance in Figure 13, averaged over all stellar masses.

##  7 Conclusions

We study the mass content of satellite galaxies in galaxy clusters with masses
$M_{\mathrm{200m}}>10^{13}\,\mathrm{M}_{\odot}$ using an improved subhalo
catalogue produced with the hbt+ algorithm (Han et al., 2018) applied to the
100 Mpc-box EAGLE cosmological hydrodynamical simulation, both at present and
as a function of time. Consistent with previous work, we find satellites have
a significantly reduced subhalo-to-stellar mass ratio compared to present-day
central galaxies (Figure 1) and roughly 3 times higher intrinsic scatter in
the subhalo-to-stellar mass relation at fixed stellar mass (Figure 4.) We find
a strong dependence of the subhalo-to-stellar mass ratio on cluter-centric
distance, independent of stellar mass, which is consistent with recent weak
lensing measurements (Figure 3).

We also investigate the histories of present-day cluster galaxies. On average,
satellite galaxies have spent roughly 5 Gyr within their current host. 30-40%
of surviving satellites fell in to their present host indirectly as satellites
of less massive systems, and this has a significant impact on their mass
content: these indirect infallers had already lost 50–70% of their dark matter
by infall, while direct infallers, i.e., those which fell into their current
host as central galaxies, had lost $<10$%. Stellar mass, in contrast,
continues to grow for roughly 2 Gyr after infall, and most satellites still
have higher stellar masses today than they did at infall (Figures 7 and 8).

In fact, present-day satellites have lower SHSMR than centrals even at the
time of maximal subhalo mass several Gya (Figure 10, lower panels), suggesting
that while they were centrals they already experienced harsher environments
than those galaxies which are still centrals today. Similarly, even galaxies
in low-mass clusters (with
$10^{13}<M_{\mathrm{200m}}/\mathrm{M}_{\odot}<5\times 10^{13}$) show
significant signs of preprocessing, with the SHSMR _at infall_ of indirect
infallers being as low as 50% that of direct infallers into the same low-mass
clusters at the same times (Figure 15). Overall, the effects of
preprocessing—mass lost as a satellite of a previous host—and
postprocessing—mass lost within the $z=0$ host—are equally important in
shaping the SHSMR of present-day satellites.

The scatter in the satellite SHSMR depends most strongly on when a galaxy
first became a satellite, increasing continuously while galaxies are
satellites independently of stellar mass (Figure 11). The stellar mass
dependence of intrinsic scatter in the present-day SHSMR is therefore due to
the mixing of galaxies which have been satellites for different time spans.

Finally, we quantify the degree to which mass loss can be estimated by
comparing with the central SHSMR as done in weak lensing analyses. While this
comparison provides a biased estimate, the bias is well behaved, with a steep
dependence on cluster-centric radius but independent of stellar mass (Figure
13), suggesting it provides a good observational mass loss proxy.

## Acknowledgements

CS acknowledges support from the Agencia Nacional de Investigación y
Desarrollo (ANID) through FONDECYT Iniciación grant no. 11191125 and through
BASAL project FB210003. JH is supported by NSFC (11973032, 11890691), 111
project (No. B20019), the science research grants from the China Manned Space
Project (No. CMS-CSST-2021-A03) and the Yangyang Development Fund.

This work used the DiRAC@Durham facility managed by the Institute for
Computational Cosmology on behalf of the STFC DiRAC HPC Facility
(www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC
capital grants ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University
and STFC operations grant ST/R000832/1. DiRAC is part of the National
e-Infrastructure.

This work made use of Astropy: a community-developed core Python package and
an ecosystem of tools and resources for astronomy (Robitaille et al., 2013;
Collaboration et al., 2018, 2022, http://www.astropy.org). This work has
benefited from open-source software including Matplotlib (Hunter, 2007,
https://matplotlib.org/) NumPy (Harris et al., 2020, https://numpy.org/),
SciPy (Virtanen et al., 2020, https://scipy.org/) Parts of the results in this
work make use of the colormaps in the CMasher package (van der Velden, 2020,
https://cmasher.readthedocs.io/index.html).

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##  Appendix A Orphan galaxies and disrupted subhaloes

Dark matter simulations produce large populations of fully disrupted
subhaloes: subhaloes whose mass has fallen below the resolution of the
simulation. Consequently, semi-analytic models introduce the concept of
“orphan” galaxies: fully disrupted dark matter subhaloes whose stellar content
remains bound (e.g., Édouard Tollet et al. 2017; Han et al. 2016). However it
is now quite clear that a majority, if not all, of the disruption is numerical
rather than physical in origin (van den Bosch & Ogiya 2018; van den Bosch et
al. 2018). Baryons do not seem to alleviate this: Bahé et al. (2019) have
shown that subhaloes are disrupted essentially just as often in the Hydrangea
simulations (a set of zoom-in simulations run with the same hydrodynamics
model as EAGLE), and if a dark matter subhalo is disrupted so is the galaxy.
In other words, orphan galaxies essentially do not exist in the Hydrangea
simulation. Hydrangea includes clusters with
$M_{\mathrm{200m}}>10^{14}\,\mathrm{M}_{\odot}$; by exploring these issues in
EAGLE we therefore provide an extension of those results to lower cluster mass
here.

We show the fraction of disrupted satellites in our cluster sample in Figure
14, as a function of maximal subhalo mass. Here, we define a subhalo as
disrupted if it has a total of 20 bound particles or less, considering all
species. The disrupted fraction increases noticeably with decreasing maximal
mass but even subhaloes with maximal masses $>10^{12}\,\mathrm{M}_{\odot}$
have a nonnegligible probability of being disrupted. Overall, the fraction of
disrupted subhaloes is consistent with the fraction typical of dark matter
simulations (Han et al. 2016), despite a lower fraction at the very high mass
end which could be dominated by recent infallers.

Consistently with Bahé et al. (2019), we find little evidence for orphan
galaxies. There is only one galaxy in our sample of satellites with less than
20 dark matter particles (it has 18) and only seven with less than 100 dark
matter particles ($m_{\mathrm{DM}}<10^{9}\,\mathrm{M}_{\odot}$). Even at lower
stellar masses there are hardly any galaxies deficient in dark matter, with
only 18 galaxies (out of 28,937) with $m_{\star}>10^{7}\,\mathrm{M}_{\odot}$
and less than 20 dark matter particles ($m_{\mathrm{DM}}<2\times
10^{8}\,\mathrm{M}_{\odot}$). Only four satellite galaxy (residing in
$M_{\mathrm{200m}}>10^{13}\,\mathrm{M}_{\odot}$ clusters) have lost _all_ of
their dark matter, retaining $\leq 1$ dark matter particles. The most massive
of these four has a stellar mass $m_{\star}=5\times
10^{7}\,\mathrm{M}_{\odot}$ and therefore none of them are included in the
analyses in this paper. These results are also consistent with Jing et al.
(2019) who found only $\sim 3\%$ of satellites in EAGLE clusters have a dark
matter fraction $<50\%$.

Combined, these two results imply that, as already discussed by Bahé et al.
(2019), baryons do little to stop subhalo disruption in simulations. As it is
likely that a large fraction of the observed disruption is numerical, there is
a population of galaxies in observations which is entirely absent in
simulations, both in dark and stellar matter. Contrary to the assumption that
numerical disruption in $N$-body simulations should result in a population of
orphan galaxies, to a large extent it appears the dark matter subhaloes should
survive too. We do not know whether the properties of those subhaloes are
statistically identical to those of surviving subhalo, so this is a
significant source of systematic uncertainty in analyses of the satellite
populations in numerical simulations. While a full exploration of these issues
is beyond the scope of this work, we end by noting that there appears to be no
significant difference in the fraction of disrupted subhaloes as found by
subfind (Bahé et al. 2019), rockstar (van den Bosch & Ogiya 2018), or hbt+ as
in this work, so disruption seems to be a feature of the simulations and not
the subhalo finders.

Figure 14:  Top panel: Unevolved subhalo mass function for present-day
satellites in $M_{\mathrm{200m}}>10^{13}\,\mathrm{M}_{\odot}$ clusters. We
show all subhaloes (blue) and disrupted subhaloes, defined as subhaloes which
at present have less than 20 particles in total. Bottom panel: fraction of
disrupted subhaloes as a function of maximal subhalo mass. The black line
shows the differential fraction and the grey line shows the cumulative
fraction of disrupted subhaloes above a certain mass. The dashed line marks a
fraction of 0.5 for reference. All curves have been smoothed with Gaussian
kernel density estimates with $\sigma=0.1$ dex in mass. Note that the lower
mass limit approaches the numerical resolution of the simulation.

##  Appendix B Preprocessing and postprocessing of satellites in low-mass
clusters

Figure 15: Impact of pre- and post-processing on the satellite SHSMR over
time, for satellites currently residing in clusters with
$M_{\mathrm{200m}}<5\times 10^{13}\,\mathrm{M}_{\odot}$. Binning and styles
are as in Figure 10. Figure 16: Impact of pre- and post-processing on the
satellite SHSMR scatter over time, for satellites currently residing in
clusters with $M_{\mathrm{200m}}<5\times 10^{13}\,\mathrm{M}_{\odot}$.
Samples, binning, and styles are as in Figure 15.

Here we show the equivalent of Figures 10 and 11 but for satellites hosted by
low-mass clusters, $10^{13}<M_{\mathrm{200m}}/\mathrm{M}_{\odot}<5\times
10^{13}$. Trends are qualitatively consistent with Figures 10 and 11 but the
fractional amount of mass lost is lower in these satellites at all stages. See
Sections 5.1 and 5.2 for details.