# Brightest Cluster Galaxies and Intra-Cluster Light: Their Mass Distribution
in the Innermost Regions of Groups and Clusters

E. Contini1,2 and Q. Gu1,2 1School of Astronomy and Space Science, Nanjing
University, Nanjing 210093, China<EMAIL_ADDRESS><EMAIL_ADDRESS>2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing
University), Ministry of Education, China

###### Abstract

We improve the model presented in Contini & Gu (2020) that describes the
radial mass distribution of brightest cluster galaxies (BCGs) and the diffuse
component also known as intra-cluster light (ICL), by assuming that the global
BCG+ICL radial mass distribution follows the sum of three profiles: a Jaffe
and an exponential profiles for the bulge and disk of the BCG, respectively,
and a modified version of an NFW profile for the ICL. We take advantage of a
wide sample of BCG+ICL systems simulated with our state-of-art semi-analytic
model to: (a) investigate the reliability of our BCG+ICL distribution by
looking at several scaling relations between the BCG+ICL stellar mass within
different apertures and the total BCG+ICL/halo mass, at different redshift;
(b) make a prediction of the distance where the radial distribution
transitions from BCG to ICL dominated. We find that our model nicely
reproduces all the observed scaling relations investigated at the present time
with a compelling degree of precision, but slightly biased-low with respect to
observations at higher redshifts ($z\gtrsim 0.5$). The transition radius
predicted by our model is in good agreement with recent observational results,
and spans a range between $\sim 15$ kpc and $\sim 100$ kpc. It mostly depends
on the morphology of the BCG, whether it is bulge or disk dominated, on the
amount of ICL with respect to the bulge and/or disk, and on the dynamical
state of the group/cluster.

galaxies: evolution - galaxy: formation.

##  1 Introduction

Discovered for the first time by Zwicky (1937), the Intra-Cluster Light
(hereafter ICL) is a diffuse light in groups and clusters from stars that are
not gravitationally bound to any galaxy, and which can extend to several
hundreds kpc (Gonzalez et al. 2005; Zibetti et al. 2005; Krick & Bernstein
2007; Gonzalez et al. 2013; Giallongo et al. 2014; Zhang et al. 2019; Henden
et al. 2020 and references therein). In the recent past, many attempts, both
theoretical and observational (Puchwein et al. 2010; Rudick et al. 2011;
Presotto et al. 2014; Contini et al. 2014; Burke et al. 2015; DeMaio et al.
2015; Edwards et al. 2016; Tang et al. 2018; Montes & Trujillo 2019; Kluge et
al. 2020; Contini & Gu 2020; Spavone et al. 2020; Furnell et al. 2021, and
references therein), have been made in order to study the formation and
evolution of this peculiar component, and to highlight its importance in the
context of the growth of galaxy clusters and their dynamical state.

Most of the above quoted studies (and many others) agree on the fact that the
ICL forms mainly via tidal stripping of intermediate/massive galaxies and
mergers of galaxies orbiting around the potential well of the cluster (see
Contini et al. 2018, 2019 and references therein for a detailed discussion, or
Mihos et al. 2017; Montes 2019 for reviews). From the theoretical point of
view, it is almost trivial111For the sake of truth it is not very
straightforward to distinguish between merger and stripping channels in the
very moment two galaxies are starting to merge simply because there is not a
clear definition of when a merger starts. This isssue is particularly
important in simulations. to isolate the contribution of each channel of the
formation of the ICL. Observationally speaking, it is possible to look at
properties of the ICL such as metallicity and color and compare them with the
typical metallicity or color of galaxies (e.g., DeMaio et al. 2015; Morishita
et al. 2017; Iodice et al. 2017; Montes & Trujillo 2018; Iodice et al. 2020)
to understand what channel contributes the most.

Regardless of the processes that bring to the formation of the ICL, or the
main contributors to it, in the very last years a few authors (e.g., Montes &
Trujillo 2019; Kluge et al. 2020; Alonso Asensio et al. 2020; Contini & Gu
2020; Deason et al. 2021; Poliakov et al. 2021) have focused on the link
between the ICL and the dark matter distributions, to figure out whether or
not the ICL can be used as a tracer of the dark matter in clusters. By using a
sample of six clusters from the Hubble Frontier Fields (Lotz et al. 2017),
Montes & Trujillo (2019) compared the bi-dimensional distribution of the dark
matter with that of the ICL making use of the Modified Hausdorff distance
(MHD), which gives the idea of how far two distributions are from each other.
They found that the average distance between the ICL and the dark matter
within 140 kpc from the center is around 25 kpc, which is a very short
distance considering the size of typical galaxy clusters. Kluge et al. (2020)
investigated on the ICL-cluster alignment with a sample of around 50 local
clusters and focused on the ICL-cluster and brightest cluster galaxy
(hereafter BCG)-cluster alignments. They found that the ICL is better aligned
than the BCG with the host cluster in terms of both position and centering.
These results strongly suggest that the ICL follows the global dark matter
distribution and can be used as a tracer of it. Similarly, Alonso Asensio et
al. (2020) used the Cluster-EAGLE simulations (Barnes et al. 2017; Bahé et al.
2017) to test the results mentioned above. By using the same procedure used in
Montes et al., they found that the stellar mass distribution follows that of
the total, including dark matter, but their radial profiles differ
substantially.

In our last paper (Contini & Gu 2020, hereafter CG20), we developed an ICL
profile to study its distribution in galaxy groups and clusters for a sample
of more than 300 BCGs, whose properties were obtained by means of our semi-
analytic model. We started from the idea, supported by the recent
observational results mentioned above, that the ICL can follow the dark matter
distribution. Hence, we assumed that the ICL follows an NFW (Navarro et al.
1997) profile and added a new parameter, that we called $\gamma$, which is
linked to the halo concentration by the relation $c_{ICL}=\gamma c_{DM}$. The
parameter $\gamma$ has been then set to reproduce the observed
$M^{*}_{100}-M_{500}$ relation, between the BCG-ICL stellar mass in the
innermost 100 kpc and the halo mass. By testing our model against several
observational and theoretical results, we found that $\gamma=3$ is required to
match the observed $M^{*}_{100}-M_{500}$ at $z=0$, while larger values of
$\gamma$ are needed at higher redshift, with also hints for a possible halo
mass dependence. We then concluded that an NFW like profile is a good
description of the ICL distribution in groups and clusters, making the ICL a
reliable tracer of the dark matter.

The idea of adopting an NFW profile to describe the distribution of the ICL is
not original. For example, Zibetti et al. (2005) found that the observed ICL
profile can be approximated by an NFW. They analysed the spatial distribution
and color of the ICL taking advantage of a sample counting almost 700 galaxy
clusters between $z\sim 0.2$ and $z\sim 0.3$ selected from the first release
of the Sloan Digital Sky Survey (SDSS-DR1, Abazajian et al. 2003), finding
that the radial distribution of the ICL is more centrally concentrated than
that of the galaxies, a result which well agrees with our assumption.
Recently, some evindence come from the study of Zhang et al. (2019). These
authors use around 300 galaxy clusters between $z\sim 0.2$ and $z\sim 0.3$
selected from the Dark Energy Survey (DES, McClintock et al. 2019) and focus
on the detection of the ICL. Among various interesting results, they find that
the ICL radial profile is self-similar and scales with the virial radius of
the clusters and that the ICL brightness is a good tracer of the cluster
radial mass distribution, in agreement with the studies quoted above. There is
evidence also from the theoretical side. Just to mention one, Pillepich et al.
(2018) studied the properties of the ICL as predicted by the IllustrisTNG
hydrodynamical simulation and found that the radial distribution of the ICL is
as shallow as that of the dark matter, meaning that the ICL can effectively
trace the overall cluster mass distribution.

The aim of this paper is to further improve the model developed in CG20, where
BCGs and ICL have been treated as separated components when we focused on the
$M^{*}_{100}-M_{500}$ relation, but BCGs have been assumed to be entirely
contained within 100 kpc, a fair assumption in many cases. Here, we treat a
BCG as a two-component system, bulge and disk, and model each of them in order
to get their mass within any given distance from the center. This approach
allows us to enter the very innermost regions of the clusters where BCGs are
placed, and compare our model predictions with a variety of observed
relations. The ICL is still assumed to follow a modified NFW profile as in
CG20, and we focus mainly at the present time given the fact that, in CG20, we
found the best match between our predictions and observed relation at $z=0$.
Once we show that our improved model matches a plethora of observed results,
we use it to investigate the differential mass distribution as a function of
clustercentric distance of the three component, bulge, disk and ICL, for
different case studies. We will show that, despite common assumptions
frequently used, the ICL can be the dominant component even at very low
distances from the cluster centre.

The paper is organized as follows. In Section 2 we briefly describe our semi-
analytic model and the profile used to describe any single component. In
Section 3 we present our analysis, which will be fully discussed in Section 4.
In Section 5 we sum up our main conclusions. Throughout this paper we use a
standard cosmology, namely: $\Omega_{\lambda}=0.76$, $\Omega_{m}=0.24$,
$\Omega_{b}=0.04$, $h=0.72$ , $n_{s}=0.96$ and $\sigma_{8}=0.8$. Stellar
masses are computed with the assumption of a Chabrier (2003) Initial Mass
Function (IMF), and all units are $h$ corrected.

##  2 Methods

We make use of the semi-analytic model developed in Contini et al. (2014), and
further improved in Contini et al. (2018, 2019), which contains a full
description of the formation and evolution of the ICL. For a detailed
description of the mechanisms that bring to the formation of the ICL, we refer
the reader to the papers quoted above, while below we give a short summary of
the main features of the semi-analytic model (hereafter SAM) in the context of
this work.

As most of nowadays SAMs, ours runs on the merger trees of N-body simulations.
Mergers trees contain all the information about the formation and growth of
dark matter haloes and subhaloes and constitute the input of our SAM, which,
according to the initial conditions (cosmological parameters) of the numerical
simulations, places galaxies in haloes and follow them along time by
considering the most important physical mechanisms in galaxy formation, i.e.
gas cooling, star formation, SN and AGN feedback, mergers and stellar
stripping.

In this work we use the merger trees extracted from the same set of high-
resolution simulations used in CG20, whose details can be found in Contini et
al. (2012). Our sample accounts for 361 groups and clusters at $z=0$ with a
virial mass ($M_{200}$) that spans over two orders of magnitude, from
$10^{13}\,M_{\odot}/h$ and up to more than $10^{15}\,M_{\odot}/h$.

As in CG20, we model the formation of the ICL by making use of the model
”Tidal Radius” described in Contini et al. (2014) and further improved in
Contini et al. (2018). The prescription is applied to satellite galaxies
orbiting around central galaxies. The SAM derives a tidal radius $R_{t}$ for
each satellite galaxy, which is assumed to be a two-component system, bulge
and disk. If $R_{t}$ is contained within the bulge, the satellite is assumed
to be completely disrupted, while if $R_{t}$ is larger than the bulge but
smaller than the size of the disk, the mass in the shell between $R_{t}$ and
the radius of the disk is stripped. In both cases, all the stripped stellar
mass is added to the ICL component associated with the BCG. Another
prescription, which considers the stellar mass of satellites stripped during
mergers, is implemented and accounts for the so-called merger channel. At each
merger, a percentage of 20% of the stellar mass of the satellite is added to
the ICL component that is associated with the BCG, and the rest of the mass
ends up to the BCG stellar mass.

The details of the other minor assumptions concerning the description of the
mechanisms that brings to the formation of the ICL can be found in the papers
quoted above. Below, we describe how we model the mass distribution of BCGs,
bulge and disk, and the mass distribution of the ICL.

###  2.1 Bulge and Disk Profiles

We assume a BCG to be a two-component system: bulge and disk. The bulge is
assumed to follow a Jaffe profile (Jaffe 1983):

$\rho_{J}(r)=\frac{M_{J}r_{j}}{4\pi r^{2}(r_{j}+r)^{2}},$ (1)

where $M_{J}$ is the total mass of the bulge and $r_{J}$ the scale length. By
integrating, the mass contained within a sphere of radius $r$ is given by

$M_{J}(r)=\frac{M_{J}r_{j}}{r_{j}+r}.$ (2)

For the disk, we assume an exponential profile and its mass within a given
radius $r$ is given by:

$M_{D}(r)=M_{D,tot}\left[1-\left(1+\frac{r}{R_{D}}\right)e^{-r/R_{D}}\right],$
(3)

where $M_{D,tot}$ is the total stellar mass in the disk, and $R_{D}$ the scale
radius of the disk. By splitting the BCG in the two components, we are able to
drop the assumption made in CG20, that is the stellar mass of the BCG is
entirely contained in the innermost 100 kpc. With this approach, we can probe
distances much below 100 kpc.

###  2.2 ICL Profile

We model the mass profile of the ICL exactly as in CG20, by assuming a
modified version of the NFW profile. The reason behind our choice is fully
discussed in CG20 and in Section 1 of this paper. The NFW profile reads as
follows:

$\rho(r)=\frac{\rho_{0}}{\frac{r}{R_{s}}\left(1+\frac{r}{R_{s}}\right)^{2}}\,,$
(4)

where $\rho_{0}$ is the characteristic density of the halo at the time of its
collapse and $R_{s}$ is its scale radius. The concentration of a dark matter
halo is defined as

$c=\frac{R_{200}}{R_{s}}\,,$ (5)

where $R_{200}$ is the virial radius of the halo. The modification we made in
CG20 to the classic NFW profile described in Equation 4 is by introducing a
free parameter in Equation 5. We define the ICL concentration as

$c_{ICL}=\gamma\frac{R_{200}}{R_{s}}\,.$ (6)

In CG20 we argued that the parameter $\gamma$ could be a function of halo
properties such as the halo mass, and redshift. We set $\gamma$ by matching
the observed $M^{*}_{100}-M_{500}$ at different redshifts (not considering any
halo dependence) and we found that $\gamma=3$ at $z=0$ and higher values at
higher redshifts, which means that the ICL is more concentrated than the dark
matter halo in which it is contained. As mentioned in CG20, this is in good
agreement with recent theoretical works (e.g. Contini et al. 2018; Pillepich
et al. 2018) and observational studies (e.g. Montes & Trujillo 2019).

##  3 Results

In this section we test our model of BCG-ICL in the innermost regions of
clusters, below 100 kpc, by looking at the relation between $M_{X}$ (where $X$
is a given distance from the centre) and the halo/BCG mass, and their fraction
with respect to the total (Section 3.1). We show that our model compares well
with observational results, and we use the success of our model in reproducing
those observed quantities to investigate the mass distribution of each
component as a function of distance from the cluster centre for a few case
studies (Section 3.2).

Figure 1: Relation between the BCG-ICL stellar mass enclosed between 10-100
kpc and the halo mass $M_{500}$, for groups and clusters at different
redshifts as indicated in the legend. Dashed lines represent $\pm 3\sigma$
scatter around the average relations, marked with solid lines. The model
predictions are compared with the observed set of data by DeMaio et al.
(2020), with $\gamma=3$ at $z=0$ and $\gamma=5$ at higher redshifts. Our model
agrees well with the observed data in the local Universe, and fairly well at
higher redshifts, although our predictions are somewhat biased low with
respect to observations.

###  3.1 Model-Observation Comparisons

In Figure 1 we show the predicted $M_{10-100}-M_{500}$ relation, i.e. the BCG-
ICL stellar mass between 10-100 kpc and the halo mass at different redshifts
(different panels), compared with the observed relation by DeMaio et al.
(2020) at similar redshift. As explained above, the values of $\gamma$ used
are $\gamma=3$ at $z=0$ and $\gamma=5$ at higher redshifts. This figure is
similar to Figure 2 in CG20, with the exception that the first 10 kpc are not
considered. As in CG20, we find a reasonable agreement with the observed
relations, in particular at the present time. Nevertheless, it appears clear
that our model tends to underpredict, although for $\sim 0.3$ dex, the
observed relation at $z\geq 0.5$. The degree of the agreement at $z=0$,
instead, is definitely excellent as it is in Figure 2 of CG20 where we account
for also the first 10 kpc.

Figure 2: Ratio between the BCG-ICL stellar mass enclosed within 10 kpc and
that within 100 kpc as a function of the halo mass $M_{500}$, for groups and
clusters at different redshifts as indicated in the legend. Dashed lines
represent $\pm 1\sigma$ scatter around the average relations, marked with
solid lines. As in Fig 1, the model predictions are compared with the observed
set of data by DeMaio et al. (2020), with $\gamma=3$ at $z=0$ and $\gamma=5$
at higher redshifts. Similarly to Fig 1, our model predictions compare fairly
well with observations, particularly at the present time. Regardless of the
redshift, a weak trend is visible: less massive haloes contain more BCG-ICL
mass in the innermost 10 kpc than more massive ones.

In order to investigate whether our model overpredicts the amount of stellar
mass in the first 10 kpc, in Figure 2 we plot the ratio between the stellar
mass within 10 kpc and that within 100 kpc, as a function of the halo mass and
at different redshifts (different panels). Once again, our predictions (color
lines) are compared with the observed data by De Maio et al. (black symbols).
The agreement looks better than that found in Figure 1, at least at $z\leq
0.5$, although it is clear that our predictions are somewhat higher than the
observed data. Considering the fact that the predicted mass within 100 kpc is
either similar or lower than observed, it suggests that our model predicts a
slightly higher amount of stellar mass in the first 10 kpc. Moreover,
regardless of the redshift, less massive haloes have higher $M_{10}$ than more
massive ones. This results from the fact that the parameter $\gamma$ is fixed
with halo mass, but less massive haloes are intrinsecally more concentrated
than more massive ones. Hence, on average, the percentage of ICL within the
first 10 kpc is higher for less massive haloes.

Figure 3: BCG-ICL stellar mass within different apertures (30, 50 and 70 kpc)
as a function of the halo mass $M_{500}$, in the local universe. Dashed lines
represent the 16-th and 84-th percentiles, while solid lines represent the
median of the distributions. Our model predictions (black lines) are compared
with observed data (red stars) by Kravtsov et al. (2018). Our model is able to
predict the right amount of BCG-ICL mass within different apertures in a wide
range of halo mass.

To give more credit to our model, we now investigate on the BCG-ICL mass
within different apertures, and on their ratio over the total, as a function
of halo and BCG mass. Since observations are so far better reproduced by our
model at $z=0$, we will focus only at the present time. In Figure 3 we plot
the BCG-ICL mass within 30, 50 and 70 kpc as a function of halo mass, as
predicted by the model (black lines), and as observed (red symbols) by
Kravtsov et al. (2018). Our model can predict the right amount of BCG-ICL mass
within different apertures for a wide range of halo mass. Similarly, Figure 4
shows the same information, but as a function the BCG-ICL mass. In order to be
consistent with the set of observed data, we derived the amount of BCG-ICL
mass within a given radius $R_{out}$ that is a function of halo mass (see
Kravtsov et al. 2018 for further details). Once again, our model is in very
good agreement with the observed data.

Figure 4: BCG-ICL stellar mass within different apertures (30, 50 and 70 kpc)
as a function of the amount of BCG-ICL mass within a given radius (see text
for further details), at $z=0$. Dashed lines represent the 16-th and 84-th
percentiles, while solid lines represent the median of the distributions. Our
model predictions (black lines) are in good agreement with the observed data
(red stars) by Kravtsov et al. (2018). Figure 5: BCG-ICL stellar mass within
different apertures (30, 50 and 70 kpc) normalized to the amount of BCG-ICL
mass within a given radius $R_{out}$ (see text for further details), as a
function of the halo mass $M_{500}$, in the local universe. Dashed lines
represent the 16-th and 84-th percentiles, while solid lines represent the
median of the distributions. As in Fig 3, our model predictions (black lines)
are compared with observed data (red stars) by Kravtsov et al. (2018) and show
a good agreement. Similarly to Fig 2, in less massive haloes the BCG-ICL mass
is more concentrated in the innermost regions than more massive haloes. The
trend here is clearer because the normalization accounts for a higher amount
of BCG-ICL stellar mass.

In Figure 5 we plot the ratio between the BCG-ICL within the same apertures
and the total BCG-ICL mass within $R_{out}$ as a function of halo mass (a
similar plot can be obtained as a function of the BCG-ICL mass with similar
information). Our model predictions (black lines) compare well with the
observed data (red symbols) for a wider range of halo mass with respect to the
similar plots in Figure 2. Overall, the model is able to capture the trend
shown by the observed data, and as already seen in Figure 2, with less massive
haloes having a larger concentration of mass in the innermost regions than
more massive haloes. In this case, being the normalization higher, the trend
is even more visible. However, contrary to Figure 2, here the model seems to
be somewhat biased low rather than high. We will come back on this in Section
4.

###  3.2 The Internal BCG-ICL Mass Distribution

As shown in the previous subsection, our model is able to predict several
observed scaling relations between the BCG-ICL mass within different apertures
and halo/BCG-ICL total mass, in particular at $z=0$. We now make use of our
model to probe the mass distributions as a function of distance from the
cluster centre of the three main components (bulge, disk and ICL) separately,
that is the main purpose of this study.

Figure 6: Differential stellar mass of bulge (black line), disk (red line),
ICL (green line) and their sum (purple line) as a function of clustercentric
distance for four BCG-ICL systems with different mass, as indicated in the
legend of each panel.  Figure 7: Continued: zoom-in of the innermost kpc of
each panel. Regardless of the BCG-ICL mass, our model suggests that the ICL
can start to dominate the total mass in the very innermost regions, even
within 15-20 kpc.

Figure 8: Same as Fig 6 for galaxies with similar BCG-ICL stellar mass, but
different disk/ICL mass ratio (0.25 in the upper panels and 0.5 in the bottom
panels).

Figure 9: Continued: zoom-in of the innermost kpc of each panel. When BCGs
have a more prominent disk compared to the amount of ICL associated to them
(bottom panels), the ICL starts to dominate the total mass at farther
distances ($>70$ kpc for c2 and $>100$ for d2).

Figure 10: Same as Fig 8 for galaxies with similar BCG-ICL stellar mass, but
different bulge/total mass ratio (interpreted as bulge/(bulge+disk)) as
indicated in the legend of each panel.

Figure 11: Continued: zoom-in of the innermost kpc of each panel. When BCGs
are more “disky” (upper panels), the ICL starts to dominate the total mass at
farther distances ($>60$ kpc).

The first four panels of Figure 6 show the diffferential stellar mass of bulge
(black line), disk (red line), ICL (green line) and their sum (purple line) as
a function of clustercentric distance for four BCG-ICL systems with different
mass, ranging from $\log(BCG+ICL)\sim 11.0$ (panel a1), to $\log(BCG+ICL)\sim
13.3$. These case studies have been chosen almost randomly from our sample,
with the only condition of being in the lowest quartile of the mass range
(a1), some percentiles before and after the median (b1 and c1), and in the
highest quartile (d1). No other condition regarding the mass in the bulge or
disk has been chosen, neither about the morpholgy of the BCGs, or halo mass.
For these “random” case studies, it is quite evident that the total BCG+ICL
stellar mass does not give a real indication of which component (and from
where) starts to dominate the total distribution. This is more clear in the
last four panels (a2, b2, c2 and d2) of the same figure, which are a zoom-in
of the previous four of the innermost regions. In all cases the bulge, as
expected, drops very fast, while disk and ICL dominate the total distributions
at different clustercentric distances, with the only exception given by d2 for
which the mass in the disk is small compared to that in the bulge. Very
interestingly, the ICL starts to dominate the total distribution at very small
radii, even at 15-20 kpc.

We now move the attention to other four case studies: two pairs of BCG-ICL
systems with similar mass and disk/ICL mass ratio. Their differential mass for
each component as a function of distance is shown in Figure 8, four a pair
with $\log(BCG+ICL)\sim 12.1$ and disk/ICL mass $\sim 0.25$ (panels a1 and
b1), and a pair with $\log(BCG+ICL)\sim 11.9$ and disk/ICL mass $\sim 0.5$
(panels c1 and d1). Again, the four BCGs have been chosen in a random manner
among those in our sample, with the only conditions of having same total mass
and disk/ICL mass ratio for members of each pair, and similar total mass but
different disk/ICL mass ratio for pair-pair members. For these case studies,
the picture looks quite different with respect to the previous ones. Indeed,
all case studies have a discreet disk/ICL mass ratio, that translates in a
more relevant importance of the disk compared to the ICL. From panel a1 to d1
(similar mass but increasing disk/ICL mass ratio) we can see that the disk is
increasingly important, and this makes the ICL dominate the total distribution
at larger radii than seen before. This trend is more visible in the zoom-in
shown in the last four panels of the same figure. As expected, the bulge drops
even faster because these are BCGs with a prominent disk, and the importance
of the disk with respect to the ICL makes all the difference. In the two cases
where the disk is a quarter of the ICL (a2 and b2), the latter starts to
dominate at radii $\sim 50-60$ kpc, but when the disk is half of the ICL, the
distribution of the latter meets the total one at farther distances, $>70$ kpc
in the case c2 and $>100$ kpc in case d2.

It appears clear that in disk-like BCGs the ICL is less likely to dominate
from the innermost regions, while for bulge-like BCGs, given the fact that the
bulge drops faster than the disk, it is more likely. We address this point in
Figure 10, where we show other two pairs of case studies with similar mass and
bulge/BCG mass ratio and plot again the differential mass of each component as
a function of distance. The first two cases (a1 and b1) are two disk-like BCGs
with B/T$<0.4$ and $\log(BCG+ICL)\sim 12.0$, and the second two cases (c1 and
d1) are two bulge-like BCGs with B/T$>0.7$ and $\log(BCG+ICL)\sim 13.0$. For
these four cases the conditions were medium size disky BCGs and two bulge-like
BCGs on the last quartile of the mass range. The reasons behind our choices is
that we want to see the difference bewteen BCGs of different morphology and,
at the same time, whether or not in a bulge-like BCG the ICL can dominate the
total distribution in a similar way as in a smaller disky BCG. In the first
two cases the disk is the component that ”competes” with the ICL and the bulge
does not contribute much, while in the last two cases the bulge makes the
difference and an almost absent disk does not contribute. The main features of
these plots can be seen in the zoom-in provided in the last four panels of the
same figure. Similar to previous cases where the disk is important with
respect to the ICL, the latter component starts to dominate the total
distribution at large distances (over 100 kpc), but the interesting aspect is
that in massive bulge-like BCGs the ICL can be important only at distances
beyond 60-70 kpc.

With all the previous case studies of different BCG-ICL systems we have shown
that the ICL can be the most important component, and so not neglected, even
at very small clustercentric distances such as 10-20 kpc. Moreover, the
morpholgy of the BCGs, the importance of the mass in the disk with respect to
that of the ICL as well as the relative importance of the total BCG mass with
respect to that of the bulge, are parameters that weigh on the distance at
which the ICL begins to be the only component at play, and the range of
distances can be as wide as $\sim 100$ kpc. In the next section we will
further discuss all our results, but mainly those of the second part of our
analysis in light of the common assumptions that are generally made when
distinguishing between BCG and ICL.

##  4 Discussion

One of the most serious problems in studying the ICL come directly from its
own definition (e.g., Rudick et al. 2011; Cui et al. 2014; Jiménez-Teja &
Dupke 2016; Montes & Trujillo 2018; Zhang et al. 2019). Indeed, different
authors use different definitions and this happens in both theoretical and
observational sides, which makes comparisons among different studies quite
complicated. Measuring the ICL is already a hard task, and attempts to
separate it from the BCG complicate the job even more, simply because the two
components are overlapped in the region dominated by the BCG and the
transition between them is supposed to be smooth. Theoretical speaking, it is
quite easy to define the ICL because in both semi-analytics and
hydrosimulations it is possible to take advantage of information not allowed
to observers (see, for example, Rudick et al. 2011 for a detailed review of
the most common methods).

On the other hand, in observations, the picture is rather different. Among the
years, mainly two are the methods that have been constantly used in
observational studies. The easiest way is to apply some cut in surface
brightness after removing all the contribution from other sources, such as
satellite galaxies and sky. All the light fainter than the cut is defined as
ICL, and the rest is given by the BCG (e.g Burke et al. 2015; Montes &
Trujillo 2018; Spavone et al. 2020; Iodice et al. 2020 and references
therein). As mentioned above, clearly this method might severely underestimate
the contribution of the ICL in the innermost regions where it overlaps with
the BCG, and thus, might overestimate that of the BCG. This approach is
similar, methodologically speaking, to assume a given distance from the center
where the light profile drops and follows a plateau, and assume that from
there on the ICL is the only component. The other common approach is to use
functional forms as we have done in CG20 (and in this paper), that allows to
distribute the light of BCGs and ICL (e.g., Kravtsov et al. 2018; Zhang et al.
2019; Kluge et al. 2020; Montes et al. 2021). To quote a few them, Montes et
al. (2021) use GALFIT (Peng et al. 2002) to fit a double (one for the BCG and
one for the ICL) Sersic (Sersic 1968) profile. Similarly to Kravtsov et al.
(2018), Zhang et al. (2019) used a combination of three Sersic profiles: one
for the inner core, one for the bulge and one for the ICL. Either with a
double or triple Sersic profile, the different components dominate a different
distance from the centre. We will come back on this point below.

###  4.1 Is Our Model Reliable?

In Section 3.1 we presented the analysis done in order to set our model, by
comparing our predictions with observational data by DeMaio et al. (2020) up
to redshift $z\sim 1.5$, and by Kravtsov et al. (2018) at the present time.
Overall we find a compelling agreement with observations, but a key point it
is worth discussing deeper. When we plot the $M_{10-100}-M_{500}$ relation at
different redshifts, we find a gap of about $\sim 0.3$ dex between our
predictions and the observed ones at redshift $z\gtrsim 0.5$. A similar gap
has been found in CG20 when we included the innermost 10 kpc. The question
that rises naturally is: Is this gap real or explainable by invoking intrisic
differences between the model and the observation? For real we mean that, as a
matter of fact, it implies an evolution in mass since redshift $z\sim 0.5$ of
a factor $\sim 2$ that it is not seen in the observed data. In CG20 we have
addressed this point. As a possible cause for the gap we have argued that,
while our predictions are exactly at that particular redshifts, $z\sim 0.5$
and present time, the observational data by DeMaio et al. span a wider range
of redshift with median at $z\sim 0.5$. Another possible reason is that our
BCG+ICL mass are actually lower than observed at that redshift, which could be
due to the difficulty in dectecting the stellar mass at higher redshift given
by all the masking done for separating the source from the sky. It might also
be that our model, at least in part, underestimates the BCG+ICL mass at
$z\gtrsim 0.5$.

According to DeMaio et al., it is possible to reconcile the theoretical
prediction of our model (since it predicts a rapid evolution of the ICL at
late times, Contini et al. 2014) with their data if most of the ICL growth
actually occurs at $r>100$ kpc, which is a very reasonable explanation.
However, here we try to investigate the innermost tens kpc by assuming a mass
profile for every component of a BCG+ICL system, and a direct comparison with
their data shows that, although minor, our model does show some evolution
within 100 kpc from $z\sim 0.5$ to the present time. The current state of
affairs does not allow us to answer to the question above, because the gap is
small for making final statements. Nevertheless, while we suggest the need to
have more data at $z\sim 0.5$ in order to get a more complete picture and a
fairer comparison, we surely stress that the profiles we assumed might not
completely describe the internal BCG+ICL mass distribution at any redshift,
even though they do at the present time.

DeMaio et al. correctly explain that the BCG+ICL gains stellar mass in the
range 10-100 kpc more rapidly than within 10 kpc by invoking the “inside-out”
scenario (van Dokkum et al. 2010; Patel et al. 2013; Bai et al. 2014; van der
Burg et al. 2015; Zhang et al. 2016, 2019 and others) for the growth of the
stellar mass distribution. Despite the issue discussed above, our model agrees
well with this scenario. Indeed, albeit our predictions are slightly biased
low with respect to the observed data, the trend is reproduced, i.e., as a
cluster grows in halo mass, it gains more stellar mass from the outer regions
(see Figure 2).

We then conclude that our model is reliable at least at the present time,
while it needs more testing at higher redshift when more data are available.
Hence, a model with a Jaffe profile for the bulge, an exponential one for the
disk, and an NFW distribution of the ICL mass can be a good description of the
whole BCG+ICL system. In the light of this, we now discuss the main point of
this paper, that is, at what distance from the centre the ICL starts to
dominate the mass distribution.

###  4.2 Where Does the ICL Dominate the Mass Distribution?

In Section 3.2 we have investigated the mass distribution of several case
studies of BCG+ICL systems by looking at the distribution of each component.
We have shown that the distance from which the ICL starts to dominate the
total distribution strongly depends on the BCG properties, such as its
morphology and whether or not it has an extended disk. The radius at which the
ICL dominates the total distribution (for the sake of simplicity let’s call it
$R_{transition}$ hereafter) can be very short, around 10-20 kpc, but it might
reach even over 100 kpc, depending on the particular case. The total BCG+ICL
stellar mass does not seem to play an important role (Figure 6), while the
relative contribution of the disk with respect to the ICL (Figure 8) and that
of the bulge with respect to the total BCG mass (Figure 10) do.

The morphology, bulge/disk-like BCG, has a major role in determining from
where the ICL becomes the dominant component. It is plausible and somewhat
expected that BCG+ICL systems where the BCG has an extended disk had larger
$R_{transition}$, and bulge-like BCGs, depending on the mass of their bulge,
more or less shorter $R_{transition}$. Having or not an extendend disk depends
mostly on the merger history of the particular BCG, mainly whether or not it
has experienced a recent major merger, which totally destroys the disk.
However, the disk can further form again if there is enough cold gas and time.
Mergers, on the other hand, contribute also on the production of the ICL,
i.e., the more the number of mergers experienced, the more the mass that will
end up in the ICL component. We can distinguish among three different case
scenarios that might explain the BCG+ICL distribution for BCGs (a) bulge
dominated (the most common), (b) spheroidals with similar bulge and disk
components, and (c) disk-like BCGs (rare objects).

In the case (a), such BCGs either have had a troubled merger history ceasing
the star formation before a new disk could grow again (a1), or they had a very
recent major merger and no time available to grow a new disk (a2). In the case
(a1) the amount of ICL is expected to be large, given the fact that mergers do
contribute to the ICL and it is very likely that stellar stripping would play
an important role when satellite galaxies are merging. In the case (a2), most
depends whether the ICL formed through stellar stripping at late times. Case
(b) represents a sort of middle ground between (a) and (c). These BCGs might
have had mergers in the past (and so formed ICL), but no recent major merger
so that the disk could grow again. For disk-like BCGs, i.e. case (c), it is
very likely that they did not have any major merger in the recent past (or at
all), and they evolved mainly through smooth accretion/minor mergers and via
star formation until gas has been available. However, such BCGs are very rare
and constitute a class of BCGs for which their ICL should not be expected to
be the relevant component.

Given these three possible scenarios for the formation and evolution of
BCG+ICL systems, $R_{transition}$ spans a wide range. It is the shortest for
systems where the BCG is a bulge-like galaxy in the low end of the mass rank,
and the highest where the BCG has a prominent disk. Overall, considering our
case studies, we find that 15 kpc $\lesssim R_{transition}\lesssim 100$ kpc,
but can be even larger than 100 kpc in very particular cases. Let’s discuss
our results in the context of the most recent observational evidence.

Recently, Montes et al. (2021) focused on a deep study of the ICL of the Abell
85 galaxy cluster by using archival data taken with Hyper Suprime-Cam on the
Subaru Telescope. They measured the BCG+ICL surface brightness profile out to
around 215 kpc in the g and i bands, and noticed that, at $\sim 75$ kpc, both
the surface brightness and color profiles become shallower. The interpretation
of this result is that the ICL starts to dominate at that radius. Not only, in
agreement with the prediction of our model (Contini et al. 2014, 2019) the
color profile at radius beyond 100 kpc is consistent with that of massive
satellite galaxies, suggesting that the stripping of these objects is
responsible for the build-up of the ICL. Their $R_{transition}\sim 70$ kpc is
also in good agreement with $R_{transition}\sim 80$ kpc by Zibetti et al.
(2005), who found a similar flattening of the profile at that radius. It must
be noted that the value of $R_{transition}$ by Zibetti et al. is the result of
a stacking of about 700 galaxy clusters taken by the SDSS. This value of the
transition radius at which the ICL becomes the dominant component agrees
fairly well with works (e.g., Gonzalez et al. 2005; Seigar et al. 2007; Iodice
et al. 2016; Gonzalez et al. 2021) that showed that BCG+ICL systems have such
a break in the profile in the range 60-80 kpc.

Similar numbers have been found by Zhang et al. (2019) from the analysis of
about 300 galaxy clusters at redshift $0.2<z<0.3$. These authors described the
BCG+ICL radial profile by using a triple Sersic model and found a dominant
core in the innermost 10 kpc, a bulge between 30 and 100 kpc and the ICL that
dominates the distribution beyond 200 kpc, but already outside 100 kpc the
profile transitions from the BCG to the ICL. They suggested that this
transition is a potential cut between the two components, in good agreement
with Pillepich et al. 2018 who noticed that simulated haloes more massive than
$5\cdot 10^{14}\,M_{\odot}$ have more ICL mass outside 100 kpc than within.

All the values of the transition radius, i.e. the radius at which the ICL
starts to become the dominant component, reported above are comparable with
the range predicted by our model. We stress again that, a wide range of
possible values of $R_{transition}$ is preferable to a narrow one for the
reasons discussed above. It strongly depends on the morphology of the BCG, of
the amount of ICL with respect to the other components and, not as last, to
the dymanical state of clusters, which reflects the merging history of the
BCG+ICL system and so its evolution with time.

##  5 Conclusions

Following the analysis done in our recent paper (Contini & Gu 2020), we have
further improved our model by assuming that the radial mass distribution of a
BCG+ICL system can be described by the sum of three profiles: a Jaffe profile
for the bulge, an exponential disk and a modified NFW for the ICL. We have
taken advantage of a wide sample of simulated galaxy groups and clusters from
our state-of-art semi-analytic model to study the innermost $\sim 100$ kpc of
BCG+ICL systems. Unlike the model used in our previous study, the new one
offers the opportunity to go below 100 kpc since bulge and disk are treated
separately. The main goals of this study were to confirm that our model can
reproduce several observed relations between the stellar mass at a given
radius and the total BCG+ICL/halo mass, at different redshift and, give an
idea of where the transition between the BCG and the ICL happens, i.e. when
the ICL component starts to dominate the total distribution. In the light of
the analysis done and the overall discussion, we conclude the following:

  * •

the radial mass distribution of a BCG+ICL system can be described by the sum
of three profile: a Jaffe profile for the bulge, an exponential disk and a
NFW-like profile for the ICL. The model can reproduce several observed scaling
relation between the BCG+ICL stellar mass within a given aperture and the
BCG+ICL/halo mass, at different redshift with an overall precision below 0.05
dex at $z=0$ and 0.3 dex at $z\gtrsim 0.5$ along the wide stellar mass/halo
mass ranges probed.

  * •

In good agreement with recent observational studies, our model predicts that
$R_{transition}$, i.e. the radius at which the radial distribution of BCG+ICL
systems, transitions from the BCG to the ICL, ranges from $\sim 15$ kpc to
$\sim 100$ kpc. The transition radius, however, depends on BCG properties such
as its morphology, bulge/disk-like galaxy, on the amount of ICL with respect
to the other components, and to the dymanical state of the group/cluster.

In a future study we aim to better set the model at higher redshifts by
investigating the parameter $\gamma$, that links the concentration of the ICL
to that of the halo, and its relation between halo mass and redshift. In order
to do that, we need more observational data at $z\gtrsim 0.5$ where our model
seems to show some bias with respect to the data used in this study.

## Acknowledgements

The authors thank the anonymous referee for his/her constructive comments
which helped to improve the manuscript. This work is supported by the National
Key Research and Development Program of China (No. 2017YFA0402703), and by the
National Natural Science Foundation of China (Key Project No. 11733002).

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