# Accreted or Not Accreted? The Fraction of Accreted Mass in Galaxies from
Simulations and Observations

Rhea-Silvia Remus1 and Duncan A. Forbes2
1Universitäts-Sternwarte München, Fakultät für Physik, LMU München,
Scheinerstr. 1, D-81679 München, Germany
2Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn VIC 3122, Australia E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In the two-phase scenario of galaxy formation, a galaxy’s stellar mass growth
is first dominated by in-situ star formation, and subsequently by accretion.
We analyse the radial distribution of the accreted stellar mass in $\sim$500
galaxies from the hydrodynamical cosmological simulation Magneticum.
Generally, we find good agreement with other simulations in that higher mass
galaxies have larger accreted fractions, but we predict higher accretion
fractions for low-mass galaxies. Based on the radial distribution of the
accreted and in-situ components, we define 6 galaxy classes, from completely
accretion dominated to completely in-situ dominated, and measure the
transition radii between in-situ and accretion-dominated regions for galaxies
that have such a transition. About 70% of our galaxies have one transition
radius. However, we also find about 10% of the galaxies to be accretion
dominated everywhere, and about 13% to have two transition radii, with the
centre and the outskirts both being accretion dominated. We show that these
classes are strongly correlated with the galaxy merger histories, especially
with the mergers’ cold gas fractions. We find high total in-situ (low
accretion) fractions to be associated with smaller, lower mass galaxies, lower
central dark matter fractions, and larger transition radii. Finally, we show
that the dips in observed surface brightness profiles seen in many early-type
galaxies do not correspond to the transition from in-situ to accretion-
dominated regions, and any inferred mass fractions are not indicative of the
true accreted mass. Instead, these dips contain information about the
galaxies’ dry minor merger assembly history.

###### keywords:

galaxies: structure – galaxies: evolution – galaxies: formation – methods:
numerical – methods: observational

††pubyear: 2021††pagerange: Accreted or Not Accreted? The Fraction of Accreted
Mass in Galaxies from Simulations and Observations–Accreted or Not Accreted?
The Fraction of Accreted Mass in Galaxies from Simulations and Observations

##  1 Introduction

In the two-phase scenario of galaxy formation (e.g., Oser et al., 2010;
Pillepich et al., 2014), galaxies undergo two main phases of growth: first the
in-situ, and subsequently the ex-situ (or accretion) growth phase. In the
former phase, stars are formed within the primary galaxy. In the latter phase,
mass growth occurs through accretion of satellite galaxies. A pioneering work
in this area was presented by Oser et al. (2010), using a set of zoom
simulations of massive galaxies, who found that the in-situ phase occurs
between redshifts 6 and 2, giving rise to a ‘galaxy core’ of size $\sim$2 kpc,
followed by the accretion dominated growth at lower redshifts. They also find
higher mass galaxies to have higher fractions of accreted mass, and that the
accreted mass is more often deposited in the galaxy outer halo regions. This
has been subsequently supported by parameter studies using binary merger
simulations of different mass ratios, demonstrating that larger mass ratios
for host and satellite galaxies are more likely to lead to a deposition of the
accreted stellar mass at larger radii, while small merger mass ratios usually
lead to full mixture of the accreted and in-situ formed stars (e.g., Hilz et
al., 2012; Karademir et al., 2019). However, the picture is not that clear, as
the orbital configurations of the mergers have been shown to influence the
radius of mass deposition for satellite galaxies especially for mergers of
larger mass ratios, with circular orbits leading to mass depositions at larger
radii than radial merger orbits (e.g., Amorisco, 2017; Karademir et al.,
2019).

This two-phase scenario has been further refined over the last decade with
increasingly sophisticated, full cosmological simulations. Those focusing on
predictions for the accreted stellar component of early-type galaxies include
the dark matter particle tagging approach of Cooper et al. (2013); Cooper et
al. (2015) and, more recently, hydrodynamical cosmological simulations like
Illustris (Pillepich et al., 2014; Rodriguez-Gomez et al., 2016), EAGLE
(Davison et al., 2020), and Illustris-TNG (Tacchella et al., 2019; Pulsoni et
al., 2020).

In particular, Cooper et al. (2013) modeled 1872 central galaxies in the mass
range $10.7<\log M_{*}<11.4$ (i.e., $11.5<\log M_{\mathrm{200}}<14.0$) using a
semi-analytic method to tag dark matter particles, not including a full
treatment of gas physics in the simulation itself. Being mainly massive
galaxies, the sample is dominated by early-type galaxies. They found that
accretion leads to a break, or change, in the slope of the stellar mass
surface density profile at the radius where the accreted material starts to
dominate over that formed in-situ. They fit Sérsic profiles to the in-situ and
accreted stars separately in surface density space, finding that the resulting
double Sérsic profiles provided a good overall fit. They showed that the
fraction of accreted material approaches 100% for the most massive early-type
galaxies, with more accretion dominated galaxies having shallower density
profiles with little, or no, obvious transition in the overall profile. This
study was further expanded for the very high mass end ($\log
M_{\mathrm{200}}\sim 14$) by Cooper et al. (2015). They found double Sérsic
profiles to be a good fit to the stellar mass surface density profiles, with
the inner component having Sérsic indices of $n\sim 4$ and the outer component
having $n\sim 1$ (similar to observational results for BCGs, e.g., Seigar et
al. (2007); Kluge et al. (2019)). However, these inner and outer components
were found to corresponded to the relaxed and unrelaxed accreted stars rather
than the in-situ and accreted stars.

Results from the fully hydrodynamical cosmological simulations all confirm the
idea of the two-phase scenario of galaxy formation, and the general trend for
higher mass galaxies to have larger amounts of accreted stars and shallower
radial stellar density profiles (Pillepich et al., 2014; Rodriguez-Gomez et
al., 2016; Tacchella et al., 2019; Davison et al., 2020). They also generally
agree that the transition radius between accretion-dominated and in-situ-
dominated stars is generally smaller for more massive galaxies. However, they
show very different results when it comes to the details of the accreted
versus in-situ components of galaxies, caused by the different subgrid models
describing the star formation and feedback processes (see Vogelsberger et al.,
2020, for a recent review).

For example, Rodriguez-Gomez et al. (2016) find for the old Illustris
simulations that all galaxies reveal a clear ‘transition’ radius for which the
in-situ and accreted components contribute equally (i.e., 50:50), while
Tacchella et al. (2019) reported for the new Illustris-TNG simulations that
their most massive galaxies can be dominated by accreted stars at all radii.
In addition, Tacchella et al. (2019) find much higher accreted mass fractions
at a given stellar mass for the new Illustris-TNG simulations than Rodriguez-
Gomez et al. (2016) found for the old Illustris simulatios. Using the EAGLE
simulation, Davison et al. (2020) found similar ex-situ fractions as a
function of stellar mass as Tacchella et al. (2019). They also confirmed
previous studies findings that the majority of accreted material is deposited
in the outer regions, while for the most massive galaxies the accreted mass
can dominate over in-situ material in the inner regions. However, the mass-
size relations found for the galaxies from these two simulations are
different, clearly showing that the details of galaxy formation still strongly
differ between the different simulations.

More recently, these studies were also broadened to study the radial kinematic
profiles of galaxies as possible tracers for the transition radii from in-situ
to accretion dominated parts of galaxies (e.g., Schulze et al. (2020) using
the Magneticum simulations and Pulsoni et al. (2020) using the Illustris-TNG
simulations). Both studies find that the shape of the kinematic profile (i.e.,
$v/\sigma$) does not, in general, trace the transition between in-situ and ex-
situ dominated galaxy regions. Schulze et al. (2020) showed that the kinematic
profiles only for a special subset of galaxies that only experienced very
small mergers since $z\sim 1$ can be used as tracer for the transition radius,
Pulsoni et al. (2020) split their galaxies into four classes of stellar mass
density profiles based on the variation of the in-situ and ex-situ fractions
with radius. We comment further on their findings in the main sections of this
work.

Observationally, the in-situ and accreted components of galaxies are much
harder to tackle. Deep imaging studies of massive early-type galaxies (ETGs)
found that around 3/4 of their galaxies reveal evidence for substructures in
their stellar halos (e.g., Schweizer & Seitzer (1992); Forbes & Thomson
(1992); Tal et al. (2009); Duc et al. (2015); Kluge et al. (2019)). Such
substructures, in the form of shells, plumes, envelopes etc, likely represent
the debris of accreted satellite galaxies. After stacking a large sample of
42,000 luminous red galaxies (with $\log M_{*}>11$), Tal & van Dokkum (2011)
found that within about $8R_{\mathrm{e}}$ their surface brightness profiles
could be well represented by a single Serisc profile, but beyond that an extra
component was required. A transition at similar radii have been reported in
the globular cluster systems of massive ETGs (e.g., Forbes & Remus (2018)).
D’Souza et al. (2014) fit 45,500 galaxies (avoiding edge-on disks) in several
stellar mass bins, finding good fits to a double Sérsic. There have also been
claims that the surface brightness profiles of elliptical galaxies are better
represented by three components (Huang et al., 2013), with radii of
$<1R_{\mathrm{e}}$, $\sim 2.5R_{\mathrm{e}}$, and $\sim 10R_{\mathrm{e}}$, all
with Sérsic values of $n\sim{}$1–2. Thus, a range of radii from much smaller
than $1R_{\mathrm{e}}$ to 8–10$R_{\mathrm{e}}$, have been reported in the
literature as key transition radii. Very few studies have quantified the
transition radius between different galaxy components and the mass associated
with each component. This has, however, been attempted by Spavone et al.
(2017) and Spavone et al. (2020), using deep imaging of ETGs from the VEGAS
survey. Fitting double Sérsic functions to the galaxy surface brightness
profiles and deriving transition radii from this they inferred outer halo mass
fractions, finding evidence for higher mass fractions in the outer component
for more massive galaxies.

We note that several late-type galaxies have been studied in order to measure
their outer halo light, e.g., the deep imaging of Merritt et al. (2016) using
the Dragonfly camera. This study revealed a large range in the fraction of
halo light in late-type galaxies beyond 5 disk scale lengths from $\sim 10\%$
to $<0.01\%$. However, it is not clear in how far this represents the accreted
component of the disk galaxies as this outer light could also come from
extended thick disk components.

In this work, we investigate the in-situ and accreted components using
galaxies from the hydrodynamical cosmological simulation Magneticum. We
compare their total and radial accretion properties to results from other
simulations as well as observations of massive early-type galaxies, especially
addressing the question in how far the transition radii from accreted to in-
situ components can be inferred from Sérsic fits to the radial surface density
profiles. In Sec. 2 we present the simulations and the details of our
classification of in-situ and accreted. Results from the simulations are
presented in Sec. 3. This includes the classification of the radial density
profiles into 6 classes (3.1), probing their assembly history (3.2), a
comparison with other simulations (3.3), and a study of the correlation of the
in-situ/accreted fractions with various galaxy properties (3.4) and the
transition radii (3.5). Sec. 4 provides a comparison between the 2D density
profiles from simulations with observations and the question of possible
recovery of the accreted fractions from the projected profiles. Finally we
present our summary and conclusions in Sec. 5.

##  2 The Magneticum Pathfinder Simulations

We use the Magneticum Pathfinder111www.magneticum.org simulations (Dolag et
al. 2021, in prep.), which are a set of cosmological hydrodynamical SPH-
simulations of several boxes with volumes ranging from
$(2688~{}\mathrm{Mpc}/h)^{3}$ to $(48~{}\mathrm{Mpc}/h)^{3}$ and different
resolutions, with the lowest having $m_{\mathrm{Gas}}=2.6\times
10^{9}M_{\odot}/h$ and the currently highest having
$m_{\mathrm{Gas}}=7.3\times 10^{6}M_{\odot}/h$. Each gas particle can spawn up
to four stellar particles during its lifetime, and as such the average mass of
a stellar particle is $1/4$th of the gas particle for each resolution. A WMAP7
$\Lambda$CDM cosmology (Komatsu et al., 2011) is adapted throughout all
simulations, with $\sigma_{8}=0.809$, $h=0.704$, $\Omega_{\Lambda}=0.728$,
$\Omega_{\mathrm{M}}=0.272$, $\Omega_{\mathrm{B}}=0.0451$, and an initial
slope for the power spectrum of $n_{\mathrm{s}}=0.963$.

All simulations are performed with a version of GADGET-3 that includes various
updates in the formulation of SPH (Dolag et al., 2004, 2005; Donnert et al.,
2013; Beck et al., 2016) as well as in the sub-grid physics, especially with
respect to the star formation and metal enrichment descriptions (Tornatore et
al., 2004, 2007; Wiersma et al., 2009) and the black hole feedback (Fabjan et
al., 2010; Hirschmann et al., 2014). For more details on the physics included
in the Magneticum Pathfinder simulations we refer the reader to Hirschmann et
al. (2014); Teklu et al. (2015) and Dolag et al. (2017). Structures are
identified using a modified version of SUBFIND (Springel et al., 2001; Dolag
et al., 2009).

As shown in previous works, the Magneticum Pathfinder simulations successfully
reproduce many observational results over a broad range of masses, from galaxy
clusters down to field galaxies. Most relevant for the work presented here,
they capture the evolution and properties of black holes (BHs) and AGN
(Hirschmann et al., 2014; Steinborn et al., 2015, 2016), and the angular
momentum, kinematic, and dynamical properties of galaxies at low and high
redshifts (Teklu et al., 2015; Remus et al., 2017; Schulze et al., 2018; Teklu
et al., 2018; van de Sande et al., 2019). Especially relevant for the work
presented in this paper, the Magneticum spheroidal and disk galaxies
successfully match the observed size-mass relation up to redshifts of $z=2$
(Remus et al., 2017; Schulze et al., 2018).

###  2.1 High resolution simulation

As we are focusing on the internal properties of galaxies in this work, we use
the currently largest volume of Magneticum with the highest resolution level
available. This box has a size of $(48~{}\mathrm{Mpc}/h)^{3}$. It initially
contains a total of $2\times 576^{3}$ (dark matter and gas) particles. The
mass resolution for the dark matter, gas, and stellar particles is
$m_{\mathrm{DM}}=3.6\times 10^{7}M_{\odot}/h$, $m_{\mathrm{Gas}}=7.3\times
10^{6}M_{\odot}/h$, and $m_{\mathrm{*}}\simeq 2\times 10^{6}M_{\odot}/h$,
respectively, with a softening of
$\epsilon_{\mathrm{DM}}=\epsilon_{\mathrm{Gas}}=1.4~{}\mathrm{kpc}/h$ for dark
matter and gas particles, and $\epsilon_{\mathrm{*}}=0.7~{}\mathrm{kpc}/h$ for
stellar particles.

We choose a lower stellar mass limit of $M_{\mathrm{*}}\geq 2\times
10^{10}M_{\odot}$ to ensure sufficient resolution for radial density profile
fits, and additionally limit the sample to central galaxies to ensure a proper
treatment of the in-situ/ex-situ classification. The highest mass galaxies in
this simulation are $M_{\mathrm{*}}\sim 10^{12}M_{\odot}$. With these
restrictions, we select 511 galaxies, which include 4 galaxies that are
brightest cluster galaxies (BCGs), and 43 galaxies which are brightest group
galaxies.

###  2.2 Definition of Accreted and In-situ Stars

We are interested in the accreted (ex-situ) and the in-situ components of the
galaxies. Therefore, we have to trace all stars that are part of a galaxy at
z=0 back to their formation redshift. If the star is born inside the main-
branch progenitor of the galaxy, it is considered to be formed “in-situ”. If
the star was born outside the virial radius of the main-branch progenitor of
the galaxy, and only later in its life accreted onto that galaxy, then it is
considered to be “accreted” independent of whether it was accreted smoothly or
as part of another galaxy. If the star particle is born inside the virial
radius of the main-branch progenitor but in the wake of a gas-rich merger, we
still consider the star to be formed “in-situ”, as otherwise all stars would
be accreted since ultimately all gas has been accreted onto the galaxy. These
stars have been handled differently in the literature, however, for the sake
of a clean classification with respect to the accreted fraction we use the
classification described above. This gives us the smallest possible fraction
of accreted stars, and the largest possible fraction of in-situ formed stars.

Figure 1: Examples for three of the six different in-situ/accreted profile
classes, from left to right: class A (extremely accretion dominated), class B
(accretion dominated), and class C (classic). The vertical black dashed lines
in the upper panels indicate the half mass radius. Upper panels: Radial
stellar density profiles for all stars (black lines), in-situ formed stars
(red lines) and accreted stars (blue lines). Middle panels: Relative mass
fractions of the in-situ (red) and accreted (blue) subcomponents. Bottom
panels: The assembly history of the stellar mass of the example galaxies.
Dashed red lines show major mergers (mass ratios of 1:1 to 3:1), green lines
show minor mergers (mass ratios of 3:1 to 10:1), and blue lines show mini
mergers (mass ratios below 10:1).

Figure 2: Same as Fig. 1 but for the other three profile classes, from left to
right: class D (double cross-over), class E (balanced), and class F (in-situ
dominated).

###  2.3 Galaxy Classification

The sample of 511 Magneticum galaxies includes all galaxy types. However, in
the second part of this study, we restrict our investigation to spheroidal (or
early-type) galaxies. They are selected using the $b$-value

$b=\log_{10}\left(\frac{j_{*}}{\mathrm{kpc}~{}\mathrm{km/s}}\right)-\frac{2}{3}\log_{10}\left(\frac{M_{*}}{\mathrm{M_{\odot}}}\right),$

which effectively gives a galaxies’ position in the $M_{*}$–$j_{*}$ plane as
discussed by Teklu et al. (2015). At $z=0$, galaxies with a $b$-value of
$b\leq-4.73$ are classified as spheroidals, while galaxies with $b\geq-4.35$
are classified as disks (Teklu et al., 2017). Galaxies with $b$-values in
between these limits have intermediate properties, that is they include S0
galaxies and disk galaxies with large bulges, but also a small number of
ongoing-merger and interacting galaxies. On this basis our sample includes 154
spheroidal, 105 disk, and 252 intermediate galaxies. This is the same
classification that has been used by Teklu et al. (2017), Schulze et al.
(2018), and Schulze et al. (2020).

##  3 Accreted and In-situ Formed Stars in Magneticum Galaxies

###  3.1 Radial Stellar Mass Density Profiles

We calculate the radial stellar mass density profiles for the Magneticum
galaxies using equal particle bins with at least 200 particles per bin, in
spherical shells around the galaxy center. For the in-situ and the accreted
components, the same radial bins are used as for the total profile to ensure a
direct radial comparability of the two components. To accommodate for the
softening, we exclude the inner 1.4 kpc, but we do not use an outer radial
limit.

Figure 3: Random 2D projected views of the example Magneticum galaxies for
each of the six profile classes. For each galaxy, a box with a length of 200
kpc centred on the galaxy is shown, with the left plot showing the intensity
map derived from all stars in the galaxy and the right plot showing the origin
of the stars colour coded according to the in-situ/accreted fraction (with
blue colours showing 100% in-situ fractions and red colours showing 100%
accreted fraction). From left to right, top to bottom, the shown galaxies are
from class A (upper left), class B (upper right), class C (central left),
class D (central right), class E (bottom left), and class F (bottom right). As
can be clearly seen, the amount of in-situ stars increases from the upper left
class (A) to the lower right class (F).

Inspecting the 3D radial stellar density profiles (in
$M_{\odot}/\mathrm{kpc}^{3}$) of the Magneticum galaxies, we find six
different classes based on their in-situ/accreted behaviour. Examples for each
class are shown in Fig. 1 and Fig. 2, and Fig. 3, and described in the
following:

  * •

Class A: Extremely accretion dominated profiles. For these galaxies, the
accreted stellar component is always dominant, even in the inner regions (see
left panels of Fig. 1 and upper left panels of Fig. 3). About 7% of all
galaxies show this kind of behaviour (see Tab. 1). Such galaxies have no clear
transition radius between in-situ and accreted stellar mass.

  * •

Class B: Accretion dominated profiles. For these galaxies, the fraction of in-
situ and accreted stars near the galaxy center is equal, but for all larger
radii the accreted fraction dominates (see middle panels of Fig. 1 and upper
right panels of Fig. 3). This is a rare class, with only about 2% of all
Magneticum galaxies in this class (see Tab. 1). It could also be interpreted
as an extreme case of class A, but we here study it as a separate class. The
transition radius for these galaxies is very small, and is not a real
transition in all cases as the in-situ component does not necessarily dominate
in the center but are sometimes simply equal amounts of in-situ and accreted.

  * •

Class C: Classic profiles. The inner regions of these galaxies are dominated
by in-situ formed stars, while in the outskirts the accreted stellar component
is dominant (see right panels of Fig. 1 and central left panels of Fig. 3).
This is by far the most common class of profiles, with 72% of all Magneticum
galaxies showing this behaviour (see Tab. 1). This is also the behaviour found
most commonly in previous work for example by Cooper et al. (2010); Rodriguez-
Gomez et al. (2016); Pulsoni et al. (2020). These galaxies have a clear
transition radius from in-situ to accretion dominated.

  * •

Class D: Double cross-over profiles. These galaxies have a large accreted
fraction dominating in the inner and the outer regions, with their
intermediate-radii regions dominated by in-situ formed stars (see left panels
of Fig. 2 and central right panels of Fig. 3). 13% of all Magneticum galaxies
fall in this category (see Tab. 1), making this the second most common profile
type. Given its nature, these profiles have two transition radii.

  * •

Class E: Balanced profiles. A small fraction ($\approx 4\%$, see Tab. 1) of
all Magneticum galaxies reveal profiles for which the in-situ and accreted
contributions are nearly equal over a large radial range (see middle panels of
Fig. 2 and bottom left panels of Fig. 3). For these profiles, we usually find
a transition radius at very large radii, however, even if the outer parts are
slightly dominated by accreted stars, the fraction of accreted stars usually
stays below 60%.

  * •

Class F: In-situ dominated profiles. Galaxies in this class have radial
density profiles that are always dominated by in-situ formed stars at all
radii, even at their outskirts (see right panels of Fig. 2 and bottom right
panels of Fig. 3). Only 2.7% of all Magneticum galaxies show this behaviour,
with the in-situ fraction always larger than the accreted fraction (Tab. 1).
As for class A galaxies, there is no transition radius for these galaxies.

Class | All | Spheroidals | Disks
---|---|---|---
|  N |  % |  N |  % |  N |  %
Class A |  36 |  7.1 |  15 |  9.7 |  1 |  0.95
Class B |  9 |  1.8 |  6 |  3.9 |  1 |  0.95
Class C |  367 |  71.8 |  100 |  64.9 |  78 |  74.3
Class D |  66 |  12.9 |  24 |  15.6 |  18 |  17.1
Class E |  19 |  3.7 |  7 |  4.6 |  2 |  1.9
Class F |  14 |  2.7 |  2 |  1.3 |  5 |  4.8
Table 1: Numbers and percentage for the six different profile classes for all
511 galaxies from the Magneticum simulation used in this work, and for those
galaxies that are classified as spheroidals (154 galaxies) or disks (105
galaxies).

Recently, a similar analysis of radial in-situ and accreted profiles has been
presented by Pulsoni et al. (2020) using the Illustris-TNG simulations.
Differently to our six classes of 3D mass density profiles, they reported four
classes based on 2D mass surface density profiles: Their class 1 (20% of the
sample) galaxies are in-situ dominated at all radii, equivalent to our class F
galaxies, albeit our class F only covers 2.7% of the galaxies. Class 2 is
their most common profile (57%), and is equivalent to our class C, albeit we
have more galaxies of class C (72%). This is also the kind of in-situ/accreted
profiles that have solely been reported for Illustris galaxies (Rodriguez-
Gomez et al., 2016). Their class 3 (15%) is closest to our double cross-over
profiles (class D), and with 13% of our galaxies being class D the numbers are
very similar between the two simulations. Their class 4 galaxies are accretion
dominated and their least common profile at 8%. In our work such profiles were
divided into class A (extremely accretion dominated) and B (accretion
dominated), representing a total of about 9% of galaxies, again in good
agreement with each others. We also classified $\sim 4\%$ of our galaxies to
be ‘balanced’ with similar in-situ and accreted components, a class that does
not appear in the classifications by Pulsoni et al. (2020). Bearing in mind
that the relative proportions of each profile class will depend on the range
of galaxy types and masses in each simulation, the main differences are that
we find more classic profiles (72% vs 57%) and fewer in-situ dominated
profiles (2.7% vs 20%), clearly highlighting the intrinsic differences between
the simulations.

###  3.2 Assembly History

One of the first steps to understand the origin of the different profile
classes is to test whether they correlate with the stellar mass of the galaxy.
In the left panel of Fig. 4 we show the normalised distribution of the
different profile classes as a function of stellar mass. We find a clear trend
that galaxies of classes E and F, where the in-situ component is 50% or
larger, are always galaxies with relatively low stellar masses, while galaxies
of classes A and B, which are everywhere accretion dominated, usually reside
at the high mass end. This mass trend is expected, as more massive galaxies
have generally accreted more mass than low mass galaxies. Similar trends were
seen by Pulsoni et al. (2020) in their study. Interestingly, the two most
common classes of galaxies, namely classes C and D, are most likely to be
found in the middle mass range of our galaxy sample, with the galaxies having
stellar masses of $10.5<\log(M_{*})<11$. This indicates that it is not the
frequency of mergers, but the type of merger that is crucial in establishing
the differences.

Figure 4: Profile class dependence on stellar mass and merger frequency. Left
panel: Histogram of the stellar mass distribution, colour coded by profile
class. Low mass galaxies are dominated by classes C, E and F, while classes A
and B are common for high mass galaxies. Right panels: Histogram showing the
frequency of major (red), minor (green), and mini (blue) mergers since $z=2$
for each class. Major mergers are least common for galaxies of class C, where
nearly 60% of the galaxies have had no major merger since z=2.  Figure 5: Mass
fraction added via major, minor, mini mergers and all mergers together since
$z=2$ for the six different profile classes. Classes A/B/C/D/E/F are given as
red/orange/blue/cyan/yellow/green filled circles, respectively. Top panel:
Fraction of stellar mass accreted through mergers of different types. Bottom
panel: Fraction of gas mass accreted through mergers of different types. Gas
mass includes hot and cold gas. For the most massive galaxies a significant
fraction of the gas is in a hot (and thus not star forming) phase.

To understand this in more detail, we study the assembly history of all
galaxies with respect to their main formation branch from $z=2$ to $z=0$. We
distinguish three different kind of mergers:

  * •

Major mergers with mass ratios of 1:1 to 3:1.

  * •

Minor mergers with mass ratios of 3:1 to 10:1.

  * •

Mini mergers with mass ratios below 10:1.

For the mini mergers, there is no lower mass limit in general, however, due to
the resolution limit of the simulation, the smallest mergers that we can
resolve here are on the order of 100:1 in mass ratio. Below this limit, we do
not call accretion events mergers, even for those few cases where this would
still be resolvable, for the sake of completeness in numbers.

Major mergers are known to have strong impacts on the main progenitor galaxy
at all radii, however, this is not in general the case for minor and mini
mergers. While minor mergers, especially in the mass range around 5:1, can
still strongly influence the mass distribution of the progenitor galaxies even
at their centres (e.g., Hilz et al., 2013; Karademir et al., 2019), mini
mergers mostly contribute to the outer halos of galaxies and only play a role
for the central evolution of the host galaxy for radial infall orbits or head-
on collisions (Karademir et al., 2019). Since we focus on radial ranges beyond
the half-mass radius, all types of mergers can play a role in establishing the
different profile classes.

Examples of the assembly history for the six classes are given in the lower
panels of Fig. 1 and Fig. 2, with the history always belonging to the galaxy
for which the radial density profiles are shown in the upper panels of the
same figures, and the intensity maps are shown in the according panels of Fig.
3. The redshifts of past merger events are marked as red/green/blue dashed
lines for major/minor/mini mergers, respectively. These six examples show that
all galaxies, but one, experience major mergers, with the galaxy that
experiences no major merger being of class C. In the case of the example
galaxies from classes A and B (the accretion dominated profile classes), both
show two major merger events since $z=2$. Interestingly, the galaxy from class
F, which is dominated by in-situ stars at all radii, experiences a major
merger at a redshift of about $z=1$.

More quantitatively, in the right panel of Fig. 4 we show for each profile
class histograms of the frequency of major (red dashed lines), minor (green
dash-dotted lines), and mini mergers (blue solid line) since redshift $z=2$
($\sim 10~{}\mathrm{Gyr}$ in look-back time). Furthermore, Fig. 5 shows the
amount of stellar mass relative to the present-day stellar mass that was
accreted through the mergers of different mass ratios and through all mergers
for each accretion class in the upper panel, while the lower panel shows the
fraction of gas relative to the present-day stellar mass accreted through
these mergers for the different accretion classes.

In general, about half of all Magneticum galaxies have experienced at least
one major merger since $z=2$. However, when considering individual profile
classes, we find large differences. Most strikingly, major mergers generally
play a significant role in the formation of galaxies from accretion classes A,
B, D, and E, while they only play a minor role in the evolution of galaxies
from accretion classes C and F. This reflects what could already be seen from
the example cases, but more statistically we find the following accretion
history patterns for our six accretion profile classes:

  * •

Class A: The accretion history of this “overmerged” class of galaxies is not
surprisingly completely dominated by merger events, with a total of nearly 70%
of the present-day stellar mass being accreted (see upper panel of Fig. 5).
Surprisingly, these mergers are not necessarily dry. Especially the major
mergers contribute an average of 25% of the total stellar mass in gas, but the
shear amount of accreted stellar mass is enough to dominate the final galaxy
at all radii. Additionally, most of these galaxies are rather massive, and as
such a larger amount of the gas will be present as a hot gas halo and not
participate in the star formation process. Galaxies of this profile class are
rather common for the spheroidals, but only one of these is found among the
disk galaxies222This is a very special case in which the accretion occurs
along a plane along the galaxies disk-plane, similar to what was discussed for
mini mergers by Karademir et al. (2019).

  * •

Class B: Galaxies from this accretion class have a similar accretion history
to galaxies of class A, with more than 60% of their stellar mass being
accreted. The main difference here is that the mergers were all gas poor,
resulting in the lowest amount of gas accreted since $z=2$ in the whole sample
(see lower panel of Fig. 5).

  * •

Class C: The “classic” profile shows a significantly different behaviour from
all the others, namely 2/3 of the galaxies in this class have never
experienced a major merger since $z=2$ (see right panel of Fig. 4). Even those
class C galaxies with major mergers only get about 10% of their mass from this
pathway, implying that the major mergers happen rather early for this class
(see upper panel of Fig. 5). In general, galaxies of this class only accrete
about 30% of their stellar mass through mergers, which is the lowest fraction
found for all classes. Furthermore, the mass accreted through the major and
minor mergers is approximately equal, and the relative contribution of the
mini mergers is rather large for galaxies of this group. This clearly shows
the importance of minor and mini mergers in the assembly history of class C
galaxies. In addition, we find that the mergers that a galaxy of class C
experiences, are usually rather dry, and contribute only about 20% of the
total stellar mass in gas (see lower panel of Fig. 5). This also explains the
dominance of the accreted material at large radii and the dominance of the in-
situ components in the center, as the minor and mini merger, especially when
dry, often do not reach the central parts of the host galaxy at all but rather
deposit their mass at large radii (Purcell et al., 2007; Amorisco, 2017;
Karademir et al., 2019).

  * •

Class D: Major mergers are important for half of the galaxies in this profile
class, but those mergers are relatively dry (gas-poor). The host galaxy, on
the other hand, is relatively wet (gas-rich) at the time of merging, and
through the merger the gas is moved outwards into a ring-like structure, where
the star formation occurs. Our simulation does not have the resolution to
confirm this, but this may be one possible way to form an (old) bulge inside a
gas-rich galaxy. In addition, these galaxies are equally common for both disks
and spheroidals (seen Tab. 1).

  * •

Class E: The mass accretion history of galaxies from this class is dominated
by a single merger which is either a major merger (60%) or a massive minor
merger (see right panels of Fig. 4). These mergers were gas-rich, causing a
starburst after accretion, which effectively leads to this special profile
case where the in-situ and accreted radial fractions are identical over a
broad radial range. As known from classical binary merger simulations
Hernquist & Barnes (e.g., 1991), these mergers usually result in a spheroidal
galaxy as long as the merger is not in-plane or has a very high gas fraction
(Springel & Hernquist, 2005). This is reflected in the low fraction of disk
galaxies in this class (only 1.9%), while for the spheroidals they account for
4.6%, as seen in Tab. 1.

  * •

Class F: Galaxies of class F show a similar behaviour to galaxies of class C,
as only half of them experience a major merger and only about 40% of their
stellar mass is accreted (see right panel of Fig. 4 and upper panel of Fig.
5). The major difference is the amount of gas accreted through the merger
events independent of the merger mass ratio: We find that all mergers deliver
significantly more gas than for the galaxies of class C, with a total of about
80% of the stellar mass being accreted in gas mass (see lower panel of Fig.
5), which is the highest frequency found for the different profile classes.
Since all these galaxies have stellar masses well below $10^{11}M_{\odot}$,
they do not host a large hot gas halo, and thus most of that gas is cold and
contributes to star formation, resulting in an overall in-situ dominated
radial density profile. This clearly shows that the origin of these overall-
in-situ dominated profiles is gas-rich accretion, and as such it is surprising
that two of these galaxies are actually spheroidals.

###  3.3 Accreted Mass Fractions and Galaxy Mass

Figure 6: Ex-situ (accreted) fractions for Magneticum galaxies in comparison
to other simulations. Upper left panel: Mean ex-situ fraction versus critical
halo mass $M_{\mathrm{200c}}$ for Magneticum (solid blue line), with the
$1\sigma$ scatter shown in light blue. For comparison, mean values are also
shown for Illustris-TNG100 (Pillepich et al. (2018), dash-dot-dot-dotted pink
line and shaded area), and the particle tagging models from Cooper et al.
(2013) (dash-dotted lines). Lower left panel: Same as upper panel but showing
the individual values for the Magneticum galaxies, with the colours marking
the different profile classes as indicated in the legend. The solid line (here
black instead of blue for better visibility) and blue shaded area mark the
mean and $1\sigma$ scatter for this distribution, as in the upper panel. Upper
right panel: Ex-situ fraction versus stellar mass $M_{*}$. The blue solid line
and shaded area show the mean and the $1\sigma$ scatter for the Magneticum
galaxies, as in the left panel. For comparison, we include the relations for
four other fully cosmological simulations: Illustris (Rodriguez-Gomez et al.
(2016), dashed black line and gray shade), Eagle (Davison et al. (2020),
yellow dash-dotted line and yellow shade), Illustris-TNG (Tacchella et al.
(2019), pink dash-dot-dot-dotted line and shade), and Horizon-AGN (Dubois et
al. (2016), green dashed lines, light green without AGN, dark green with AGN).
Additionally, we include the mean values from two zoom-simulations that cover
a range of stellar masses: the SPH-based GADGET from Oser et al. (2010) as
x-circle symbols, and the AMR-based ENZO simulations from Lackner et al.
(2012) as +-circle symbols. The mean values in three mass bins from the semi-
analytic modeling by Lee & Yi (2013) are shown as black half-circles. Finally,
we also include the mean values for the zoom-simulations by Hirschmann et al.
(2015) (square for non-feedback, diamond for the feedback-runs, joined by a
vertical line). These data points were extracted from Rodriguez-Gomez et al.
(2016). We include these simulations as they nicely demonstrate the impact of
the stellar feedback on the simulations. Lower right panel: Same as the upper
right panel, but for the individual Magenticum galaxies with the colours
marking the different profile classes as in the lower left panel. Again, the
Magneticum mean is shown as solid black line and the blue shaded area marks
the $1\sigma$ scatter for this distribution, as in the upper panel. Magneticum
galaxies tend to have higher accretion fractions at low masses compared to
other simulations.

Overall, there is broad agreement between simulations and models of different
kinds that the fraction of accreted stars is correlated with the stellar (and
halo) mass of galaxies. However, the different simulations vary strongly in
the actual (average) values found for the accreted (or ex-situ) fractions at a
given mass. This can clearly be seen in Fig. 6 for halo mass (upper left
panel) and for stellar mass (upper right panel). Here, we compile data from
the literature for the five large hydrodynamical simulations:

  * •

Magneticum from this study, shown as the blue solid line and shaded area in
both panels.

  * •

Illustris-TNG, shown as the pink dash-dot-dot-dotted line and shaded area,
from Pillepich et al. (2018) for the halo mass comparison (left panel) and
from Tacchella et al. (2019) for the stellar mass comparison (right panel).

  * •

EAGLE, shown as dash-dotted yellow line and shaded area, from Davison et al.
(2020), only for the stellar mass comparison (right panel).

  * •

Horizon-AGN, shown as green dashed lines, from Dubois et al. (2016), only for
the stellar mass comparison (right panel) for the runs with (dark green) and
without (light green) AGN.

  * •

Illustris, shown as black dotted line and gray shaded area, from Rodriguez-
Gomez et al. (2016), only for the stellar mass comparison (right panel).

As can be seen immediately, Magneticum galaxies have larger accreted fractions
at the low mass end compared to the other simulations, while at the high mass
end the accreted fractions for all the simulations converge to around 70-80%
for galaxies of stellar masses above $3\times 10^{11}M_{\odot}$ (or halo
masses above $1\times 10^{13}M_{\odot}$).

There are different reasons for these simulations to show such different
behaviour, and it is up to now still unclear which ones are closest to
reality. One commonly discussed reason for the different results with respect
to the amount of accreted and in-situ components in galaxies is the stellar
and/or AGN feedback, which is modeled slightly different in each simulation.
That both processes have strong effects on the resulting accretion fractions
has been reported by previous studies: Hirschmann et al. (2015) used a subset
of the galaxies presented by Oser et al. (2010) and simulated them with and
without strong stellar feedback (see open black diamond and square in the
upper right panel of Fig. 6, respectively). They found that, while the stellar
mass of the galaxies did not change much, the accreted fraction dropped
significantly, on average from about 50% to 20%, when the stellar feedback was
switched on. This is due to the fact that the feedback from the stars heats up
the gas inside the galaxies and thus suppresses star formation, especially for
the lower mass galaxies, leading to lower stellar masses for the galaxies in
general and thus smaller amounts of accreted stars. However, the merger events
still lead to starbursts and subsequent star formation, but this is then in-
situ star formation. An opposite effect was reported by Dubois et al. (2016,
see green dashed lines in Fig. 6) and Dubois et al. (2013) for the AGN
feedback. Here, the simulation runs without AGN feedback (light green) result
in lower accreted fractions than the simulation with AGN feedback (dark
green).

As all five fully hydrodynamical simulations shown here include both types of
feedback, albeit in different implementations, it is not possible to know
which of the processes is the main driver of the differences between the
simulations. Note, for example, that the values found for Magneticum are very
similar to the average values found by Oser et al. (2010) in their zoom-
simulations (x-circles), but the latter has no AGN feedback and the galaxies
found in that simulation also differ strongly from those in Magneticum in
other parameters (see e.g., Remus et al., 2017). For Magneticum, we know from
Teklu et al. (2017) that our stellar feedback is slightly too weak for the low
mass end and our AGN feedback is slightly too strong at the high mass end when
looking at the baryon convergence efficiency, which is most likely also the
reason for the difference at the low mass end between Magneticum and EAGLE and
Illustris-TNG.

Another possible reason for the different accretion fractions found in the
different simulations could be the use of AMR versus SPH codes. Those
simulations, both fully cosmological and zoom-in, that were performed with AMR
codes (namely Horizon-AGN (Dubois et al., 2016) and the simulations by Lackner
et al. (2012)) show significantly lower accreted fractions at all mass ranges
than those simulations that were performed with SPH codes (namely Magneticum,
EAGLE, and the simulations by Oser et al. (2010) and Hirschmann et al.
(2014)), independent of the included feedback. This is especially interesting
given that the studies by Lackner et al. (2012) and Oser et al. (2010) both do
not include the AGN feedback, but show the strongest differences. It is well
know that all codes have their shortcomings, and many improvements have been
implemented in recent years, but to understand how much influence the choice
of the code has on the accreted fractions, however, would require a detailed
comparison study, which has not been done so far.

Note that, for the five large cosmological simulations shown here, the
definitions of in-situ and accreted largely agree in that we only count those
stars as accreted that were already born at infall, and count those stars that
were born from gas that was accreted through a merger but only formed after
the merger event from this gas as in-situ (see also Rodriguez-Gomez et al.,
2016; Tacchella et al., 2019). This means that, effectively, our accreted
fractions are lower limits, and the values could only get higher for more
elaborate definitions of in-situ and accreted, but thus this definition is not
responsible for the difference found between these simulation samples.

We also include the accreted fractions with mass obtained from two semi-
analytic models (SAMs): Cooper et al. (2013) used a particle-tagging method on
top of the SAM presented by Guo et al. (2011) based on the Millenium II
simulation (Boylan-Kolchin et al., 2009), and the resulting accretion
fractions for different halo masses are shown as dash-dotted lines in the left
panel of Fig. 6, split according to the morphology as disks (gray line) or
spheroidals (black line). The results found for their sample of spheroidal
galaxies is close to the average values found for the Magneticum galaxies in
this work, albeit our sample includes both disks and spheroidals. When we
separate our disks and spheroidals, we find a general trend for the disks to
have lower accreted fractions than spheroidals of the same mass in agreement
with Cooper et al. (2013), but the trend is much less pronounced and the
scatter is large. Lee & Yi (2013) use a SAM built on their own Gadget-2 based
dark matter only simulation, and provide accreted fractions for a range of
stellar masses (see half-circles in the upper panel of Fig. 6). Their values
lie in between the five different fully cosmological simulations, and are
especially at the low-mass end not in agreement with our Magneticum results.

So far, we have discussed the mean values of the accreted fractions with
stellar and halo mass for the Magneticum simulation in comparison to other
simulations, now we want to take a closer look at the distribution of the
individual galaxies with regard to their accretion classes A–F. They are shown
in the lower two panels of Fig. 6 in comparison to the mean value lines shown
in black. As can be seen immediately, there are strong differences between the
galaxies of the different accretion classes: The galaxies from the over-merged
classes A and B all show high accretion fractions, well above 60%, with no
real trend with mass visible. On the other hand, the in-situ dominated
galaxies of class F all have, as expected, low accretion fractions below the
mean Magneticum values, and their spread in mass is too small to see any trend
with mass for both stellar and halo mass. Similarly, galaxies of the major
merger class E also show no trend in mass, and the overall accretion fractions
are around 50%. For the other two classes, C and D, we find a clear
correlation of the accreted fraction with both stellar and halo mass, with a
tendency for the galaxies of class D to be slightly above the mean Magneticum
accretion values per mass, and for class C to be slightly below on average.
Interestingly, class C also includes the lowest accretion fractions at all
mass bins, even lower than the galaxies of the in-situ dominated class F,
clearly demonstrating that the accretion fractions can be really low if most
of the accretion is provided by dry minor and mini mergers in the outskirts of
a galaxy, while the centre is left undisturbed.

Figure 7: Left panel: Stellar mass-size relation for the Magneticum galaxies,
colour coded according to their profile class (see legend). The 3D half-mass
radius is shown against total stellar mass. For comparison, the observed mass-
size relation from the GAMA survey by Lange et al. (2015) is shown for ETGs
(dashed line) and LTGs (solid line). Note that the observations measure half-
light radii instead of half-mass radii. Right panel: In-situ fraction versus
3D half-mass radius for the Magneticum galaxies, colour coded as in the left
panel. Only class C reveals a clear correlation between in-situ fraction and
galaxy size.

###  3.4 Accreted Mass Fractions and Global Galaxy Properties

Next we examine trends between the 3D half-mass radius and dark matter
fraction with total stellar mass and in-situ fraction. It has been shown
already by Remus et al. (2017) and Schulze et al. (2018) that the Magneticum
galaxies successfully reproduce the observed stellar mass-size relations
(e.g., GAMA, by Lange et al., 2015), but here we now take a closer look at the
different profile classes in this relation (left panel of Fig. 7). The
overmerged galaxies of classes A and B show only small scatter close to the
observed relation over the whole mass range, but due to the fact that they are
the by far most common class at high stellar masses, they are also most common
among the large galaxies. The classical profile galaxies of class C, however,
show a significantly different behaviour from the galaxies of the other
classes, in that they are the clearly dominant class for the small galaxies at
all stellar mass ranges. Albeit the scatter is large for this class and there
are also very large galaxies among them, they clearly dominate the small size
end especially at the lower stellar mass end. This reflects our previous
findings that these galaxies are dominated by compact star formation in their
centers and only little accretion mostly to the outskirts, resulting in a
rather compact central part and consequently a smaller half-mass radius. On
the other hand, we see a very different behaviour for those galaxies of
classes D, E, and F, all of which have large sizes for their stellar masses,
clearly dominating the region of the mass-size relation that is usually
occupied by disk galaxies. This is in good agreement with the fact that all of
them have large amount of cold gas accreted through their formation history
since z=2, resulting in in-situ star formation in disks and thus larger half-
mass radii (even if in case of class D galaxies the large central accreted
component will prevent its classification as a disk given its massive bulge-
like nature).

As stellar mass $M_{*}$ and 3D half-mass radius $r_{1/2}$ of a galaxy are
correlated, it is not surprising that we also find a correlation between the
in-situ fraction of a galaxy and its half-mass radius (right panel of Fig. 7).
It can best be seen for the galaxies of accretion class C, as they cover the
largest range of both half-mass radii and in-situ fractions, with a clear
tendency for smaller galaxies to have larger in-situ fractions and large
galaxies to have small in-situ fractions. A similar behaviour is found for
galaxies of classes D and E, albeit class E only covers such a small range of
in-situ fractions that no clear correlation between size and in-situ fraction
can be inferred from these galaxies alone. In general, the in-situ fraction
decreases with increasing halfmass radius, i.e., accretion leads to a growth
in the scaled size (e.g., Oser et al., 2012), which indicates that most of the
accreted stars are deposited at large radii (see also Amorisco, 2017; Lagos et
al., 2018; Karademir et al., 2019; Davison et al., 2020).

Figure 8: Dark matter fraction within the half-mass radius trends. Left panel:
Dark matter fraction $f_{\mathrm{DM}}$ versus stellar mass $M_{*}$, with
colours as in the right panel. Observations for LTGs from the SPARCS survey
(Tortora et al., 2019) are included as a lilac solid line and shaded area, and
observations for ETGs from the SPIDER survey (Tortora et al., 2014) are
included as an aqua solid line and shaded area. Right panel: Dark matter
fraction $f_{\mathrm{DM}}$ versus in-situ fraction $f_{\mathrm{in-situ}}$ for
the Magneticum galaxies, with colours indicating the different accretion
classes.

For galaxies of the overmerged classes A and B we find a similarly large range
of half-mass radii, but only a small range of in-situ fractions around 20%,
and they reveal no correlation at all between size and in-situ fraction. This
well reflects the known fact that a major merger results in a much more
compact galaxy than a series of minor mergers that bring in the same total
mass as the major merger but deposit their masses at different radii (Naab et
al., 2009; Hilz et al., 2012). So while all galaxies of these two classes had
plenty of mergers, we find the differences in the individual merger mass
ratios mirrored in the size distribution. Galaxies of the in-situ dominated
class F show the strongest deviation from the correlation between size and in-
situ fraction: while all the in-situ fractions are rather high, the sizes are
generally larger than those of the class C galaxies of similar in-situ
fraction, in agreement with our previous finding that class F galaxies are
more similar to disks than the average class C galaxy.

Finally, we investigate if the fraction of dark matter within the half-mass
radius, $f_{\mathrm{DM}}$, is correlated with the in-situ fraction and stellar
mass. As can be seen in Fig. 8, there is a broad tendency for galaxies with
smaller in-situ fractions to have larger central dark matter fractions,
indicating that (massive) accretion events lead to larger fractions of dark
matter in the center by either enhancing the relative amount of dark matter in
the center or dispersing the baryonic matter. This tendency can be seen for
galaxies of all accretion classes but those of class F, the in-situ dominated
class. Galaxies of that class show much higher central dark matter fractions
than galaxies of class C with similar in-situ fractions. This is in good
agreement with our previous conclusion that class F galaxies closely resemble
the typical behaviour of disk galaxies, since observations show that LTGs
have, at the same stellar mass, larger central dark matter fractions than ETGs
(Tortora et al., 2019, but also e.g., Courteau & Dutton (2015); Genzel et al.
(2020), albeit they use velocity dispersions instead of stellar masses and
different definitions of central radius). This can also be seen in the left
panel of Fig. 8 where we included the observational results for LTGs and ETGs
from Tortora et al. (2019) and Tortora et al. (2014), respectively. As can
immediately be seen, most of the class D and F galaxies clearly resemble the
properties of the LTGs, while the clear observed correlation between
$f_{\mathrm{DM}}$ and $M_{*}$ for ETGs is most strongly populated by galaxies
of the classical accretion profile class C, in good agreement with the idea
that dry merging lowers the central dark matter fractions while wet merging
and smooth gas accretion lead to larger central dark matter fractions.
However, the details of the interactions between the baryons and the dark
matter in the centers of galaxies and the influence of gas and feedback on
this interaction are currently under debate and are beyond the scope of this
work.

###  3.5 Accreted Mass Fractions and Transition Radii

Figure 9: Transition radius trends for all profile classes as indicated in the
label. Classes A and F have no transition radii and are therefore not shown.
Class D galaxies (cyan diamonds) have two transition radii, so the outer ones
are shown as filled diamonds and the inner ones are shown as open diamonds.
Upper left panel: Stellar mass $M_{*}$ versus 3D transition radius
$r_{\mathrm{trans}}$ in kpc. Upper right panel: In-situ fraction
$f_{\mathrm{in-situ}}$ versus transition radius $r_{\mathrm{trans}}$ in kpc.
Lower left panel: Stellar mass $M_{*}$ versus 3D transition radius
$r_{\mathrm{trans}}$ normalised by 3D half-mass radius $r_{\mathrm{1/2}}$.
Lower right panel: In-situ fraction $f_{\mathrm{in-situ}}$ versus normalised
transition radius. The horizontal line in both lower panels represents a
normalised transition radius of 1 half-mass radius, i.e.,
$r_{\mathrm{trans}}/r_{\mathrm{1/2}}=1$. Dashed black lines and shaded areas
show the results from the Illustris simulation (Rodriguez-Gomez et al., 2016),
and the dash-dot-dot-dotted pink line and shaded area are the results found
for Illustris-TNG (Pulsoni et al., 2020). The horizontal dotted line marks
where the transition radius equals the half-mass radius.

As discussed before, for most galaxies there exists a radius at which the
contribution from in-situ and accreted stars is 50% each, that is at which the
dominance of the two components switches. We call this radius the transition
radius $r_{\mathrm{trans}}$. For our classic profile (class C), this is the
radius where the dominant stellar component switches from in-situ in the
center to accreted in the outskirts, and thus separates the inner, self-made
part of the galaxy from the outer, dry-merger dominated part.

Previous works by Cooper et al. (2013) and Rodriguez-Gomez et al. (2016)
already reported this radius to be smaller for larger stellar masses and
smaller in-situ fractions, and we can confirm these general trends for our
class C galaxies as shown in Fig. 9. However, we do not find a tight
correlation between the transition radius and stellar mass, and only a weak
correlation is seen between transition radius and in-situ fraction (see upper
panels of Fig. 9), with a large scatter. Only when moving to normalised
transition radius (i.e., the transition radius divided by the half-mass
radius), the trends become more clear: we even see a clear positive
correlation between normalised transition radius and in-situ fraction, very
similar to the correlation found by Rodriguez-Gomez et al. (2016) but slightly
less steep (see lower panels of Fig. 9). We also find a clear negative trend
between the in-situ fractions and the stellar mass, with galaxies that have
accreted a lot of material (i.e., high mass galaxies) tending to have
normalised transition radii of $r_{\mathrm{trans}}/r_{\mathrm{1/2}}\leq 1$.
However, this trend is more an upper limit for the in-situ fractions at a
given mass, as we also find low-mass galaxies with normalised transition radii
$r_{\mathrm{trans}}/r_{\mathrm{1/2}}\leq 1$, but basically no high-mass
galaxies with $r_{\mathrm{trans}}/r_{\mathrm{1/2}}>1$. The trend found for the
Magneticum galaxies is weaker than what has been found by Rodriguez-Gomez et
al. (2016), and much weaker than the trend reported for the Illustris-TNG
galaxies by Pulsoni et al. (2020). This reflects the result found already in
Fig. 6, namely that the galaxies in Magneticum have, on average, accreted more
dry stellar mass through mergers than galaxies in galaxies from Illustris or
Illustris-TNG.

So far, we only discussed those galaxies of profile class C as most previous
works only discussed this profile class with no mention of other profile
classes. For classes A and F, we cannot provide a transition radius as these
galaxies are always dominated by accreted or in-situ stars, respectively, but
for the profile classes B, D, and E such transition radii exist: Galaxies of
class B usually have a very small transition radius of only about
$2~{}\mathrm{kpc}$, close to the limits of our spatial resolution (and hence
may be somewhat smaller than indicated). We do not find any trend, positive or
negative, with stellar mass, and only a weak positive correlation between in-
situ fraction and normalised transition radius. This is not surprising as this
class if very close to being overmerged like class A, and thus we do not
expect the transition radius to have any relevant meaning.

In the case of class D, where the center and the outskirts are dominated by
accreted stars but the middle radial range is dominated by in-situ stars, even
two transition radii exist. For the outer transition radii of class D (filled
cyan symbols in Fig. 9) and class E we find the trend with in-situ fraction to
be very similar, but generally steeper than the correlation seen for class C.
This is even clearer for the normalized transition radii again. For both
classes we also find a tighter anti-correlation between normalised transition
radius and stellar mass, albeit the scatter is still large.

The inner transition radii for class D galaxies (open cyan diamonds in Fig. 9)
are usually comparably small and often well below the half-mass radius.
Generally, the inner transition radii of class D behave significantly
different from all other transition radii, as there is no trend at all for the
stellar mass neither with the transition radius nor with the normalised
transition radius, and there is actually a negative trend with in-situ
fraction, clearly showing that the larger the fraction of stars formed in-
situ, the smaller the accreted core in the centre, indicating that more stars
are formed in-situ if the mass accreted onto the centre was small compared to
the gas disk of the progenitor galaxy.

As these transition radii are very indicative of the accretion history of the
galaxies and may provide a method to estimate the in-situ fraction of a
galaxy, it would be very instructive to be able to measure this transition
radius observationally. Therefore, in the next section of this paper we
address the question of whether it is possible to measure the radius of the
transition from in-situ to accretion dominance from the observed surface
brightness profiles of galaxies, as suggested by Cooper et al. (2013) and
Rodriguez-Gomez et al. (2016).

##  4 Accreted Fractions from Sérsic Fits for ETGs

Figure 10: Upper panels: Example of a class C mass density profile in 3D and
2D projections. Top left panel: 3D total stellar mass density profile (black
curve) with the in-situ and accreted components in red and blue, respectively.
Top right panels: Projected 2D stellar mass density profiles from three
different projections: The projected total stellar profile is shown as black
curve, the in-situ and accreted components are shown as solid red and blue
lines, respectively. The dashed lines show the inner (red) and the outer
(blue) fits from the double Sérsic fits, and the single Sérsic fits (green) to
the total projected stellar density profiles. In the upper right of each panel
we list the transition radius in 3D and the crossing radii in 2D, in units of
kpc. In this example, the single Sérsic fit is never a good fit to any of the
projections. The double Sérsic fits describe the total profile very well in
all projections, and are also a good approximation to the in-situ and accreted
profiles in all cases. However, the crossing radii $R_{\mathrm{cross}}$ (i.e.,
the radius where the two Sérsic profiles cross) vary on the order of 1 kpc
between the three projections, and are in all three cases only about half as
large as the real 3D transition radius $r_{\mathrm{trans}}$. Lower panels:
Same as upper panels but for a class A profile. Class A galaxies are extremely
accretion dominated and have no transition radius between in-situ and accreted
components in their 3D or 2D stellar density profiles, as can be seen from the
solid red and blue curves in all four panels. However, a single Sérsic fit is
not a good fit to any of the projections and the double Sérsic fit is clearly
needed in all three projections to describe the total total stellar profiles.
Thus, we obtain crossing radii $R_{\mathrm{cross}}$ from these double Sérsic
components, that vary again between the three projections, but in no case are
they representative of the true in-situ or accreted components.

Motivated by previous simulation results from Cooper et al. (2013) and
Rodriguez-Gomez et al. (2016), there have been observational attempts to
measure the in-situ and accreted fractions of galaxies using a double Sérsic
fit (Sérsic, 1963) to the observed surface brightness profiles of galaxies.
This assumes that the inner Sérsic fit describes the in-situ component of the
galaxy, and the outer Sérsic fit describes the accreted component. In some
cases, a third fit to the very outskirts of a galaxy was carried out, under
the assumption that the third component describes the stellar halo of the
galaxy and not the galaxy itself (e.g., Spavone et al., 2017), but we will not
investigate this approach here. Instead, we investigate in this section from
our simulated galaxies if the double-Sérsic approach really supplies a good
measure for the in-situ and accreted components of galaxies. As the
observational work has so far focused mostly on early-type galaxies (ETGs), we
restrict our sample to Magneticum ETGs only.

Figure 11: Left panel: Crossing radius $R_{\mathrm{cross}}$ obtained from
double Sérsic fits to the 2D projected mass density profiles (random
projection) versus the 3D transition radius $r_{\mathrm{trans}}$ between in-
situ and accreted components, for all profile classes (colours as indicated in
the right panel) with well defined transition radii. Right panel: Fraction of
integrated mass from the outer Sérsic function fit to the 2D mass density
profiles versus the true accreted mass fraction. In both panels, the dashed
line shows a 1:1 relation. Error bars indicate the maximum and minimum values
obtained for edge-on and face-on projections.

###  4.1 Sérsic Fits to Projected Surface Density Profiles of Simulated ETGs

To compare the simulated mass density profiles with observed surface
brightness profiles, we need to create 2D projections of the simulated
galaxies. As this projection is rather arbitrary, we choose a random
projection along with the face-on and edge-on projections to test for each
galaxy in our sample. In all cases, we find that the profile class of our
galaxies does not change, and the in-situ to accreted relations stay the same
under all projections. Thus, for all galaxies that have a transition radius
($r_{\mathrm{trans}}$) in 3D, we also find a radius in the 2D projections
which indicates a transition from in-situ to accretion dominance.

To follow the observational approach, we fit the projected surface density
profiles with both single and double Sérsic fits. In most cases, a double
Sérsic fit is a better fit to the projected surface density profiles,
independent of the projection. In those cases where a single Sérsic fit is
sufficient, this is true for all tested projections. This is rather promising
as this clearly indicates that, if a double Sérsic fit is needed to describe
the observed surface brightness profile, then it is independent of the viewing
angle and reflects the underlying 3D density distribution. For the double
Sérsic fits to the 2D surface density profiles, we define the crossing radius,
$R_{\mathrm{cross}}$, as the radius where the inner and outer Sérsic profiles
cross each others.

The upper row of Fig. 10 shows an example of a class C profile galaxy with its
well defined transition radius in 3D (here
$r_{\mathrm{trans}}=10.52~{}\mathrm{kpc}$, left panel). A transition is also
seen between the in-situ and accretion dominated regions of the galaxy in all
three 2D projections (upper right panels), however, the values of the
transition radii in the 2D projections are all smaller than the 3D transition
radius $r_{\mathrm{trans}}$. We find that this is not simply a matter of
unlucky projections but is rather a common feature of class C profiles (which
make up the majority of profiles). This disconnect between the transition
radii seen in 3D and the 2D profiles also occurs in all other classes with
well-defined transition radii, namely class B, D, and E.

While the transition radii are already disconnected from 3D to 2D, the matter
is even worse if we use the double Sérsic fits to describe the underlying in-
situ and accreted components: In a few cases like the one shown in the upper
panels of Fig. 10, the two Sérsic components are a good approximation of the
in-situ and accreted components, and the resulting crossing radius between the
two Sérsic components, $R_{\mathrm{cross}}$, is a good approximation to the 2D
transition radius. However, for most galaxies this is not the case. One
example of a galaxy that demonstrates the issue nicely is shown in the lower
panels of Fig. 10: This galaxy is of class A, i.e., is accretion dominated at
all radii and has no transition radius from in-situ to accretion dominated,
neither in 3D (left lower panel) nor in projection (three panels on the lower
right). However, the stellar 2D surface density profiles in this example,
under all projections, clearly require a double Sérsic fit, thus providing a
crossing radius $R_{\mathrm{cross}}$. The two resulting Sérsic components in
this case do not describe the underlying in-situ and accreted components. They
instead mark the radius where accretion due to massive mergers transitions
into accretion from small mergers and mini mergers that never reach the centre
of the galaxy.

To further quantify this issue, the left panel of Fig. 11 shows the
differences between the crossing radii of the double Sérsic fits
$R_{\mathrm{cross}}$ for the random projection (with error bars marking the
values for the edge-on and face-on projections), and the true 3D transition
radius $r_{\mathrm{trans}}$ between the in-situ and accreted components for
all galaxies where such a transition radius is well defined (classes B, C, D,
and E). The plot is largely a scatter diagram with little, or no,
correspondence between the two measured radii, independent of the profile
class. This is also true for the projected 2D transition radii, but as the
behaviour is nearly identical we do not show this plot here.

For galaxies of classes D and E, the 2D crossing radii are much smaller than
the real transition radii, while for galaxies of class B we find the opposite
trend (with two of them having such large crossing radii $R_{\mathrm{cross}}$
that they are well above the plotted radius range). Galaxies of class C show
both kind of behaviour, with $R_{\mathrm{cross}}$ both lower and higher than
the true $r_{\mathrm{trans}}$. Independent of the profile class, we conclude
that it is a lucky coincidence if the crossing radius $R_{\mathrm{cross}}$ of
the double Sérsic fits is a good approximation to the transition radius
$r_{\mathrm{trans}}$. In summary, the transition from an inner to an outer
Sérsic fit to an observed surface brightness profile bears little, or no,
connection to the true transition between the in-situ and accreted components
in a galaxy.

We also measure the integrated mass within the outer Sérsic component and
compare it to the true accreted mass fraction. This is shown in the right
panel of Fig. 11 for all galaxies where a double Sérsic fit was a good fit,
even those of profile classes A and E that are dominated by accreted or in-
situ stars at all radii, respectively. This plot reveals a large scatter with
a very weak trend about the unity line, indicating that an outer (inner)
Sérsic component fit to a surface brightness profile is a poor guide to the
true accreted (in-situ) mass fraction.

In summary, we find that fitting a double Sérsic profile to the 2D surface
density profile of an ETG will not reveal the true radius for the transition
from in-situ to accretion dominated material. This suggests that the dips seen
in observed surface brightness profiles can not, in general, be taken as a
signature of a division between in-situ and accreted components of the galaxy.
They may instead be more indicative of a transition from stars being formed
in-situ plus stars accreted by major mergers, to the component of stars mostly
accreted through minor or mini mergers. Furthermore, the integrated mass
associated with the inner and outer Sérsic functions provide a poor guide to
the true in-situ and accreted mass fractions of a galaxy, respectively. We
conclude that the (two) components visible in the (projected) density profiles
do not reflect the in-situ and accreted components in general, but rather mark
the transition from the inner part of the galaxy which can be dominated by in-
situ stars but also be dominated by a massive merger event, and the outer part
of the galaxy which is dominated by small minor or mini mergers that get
disrupted in the outskirts of the galaxy and never interact with its
center333Note that in the very inner parts of galaxies additional components
can be visible in the (projected) density profiles, caused for example by bars
and bulges, but we cannot include these structures in our analysis as the
resolution of the Magneticum galaxies is not high enough to resolve these
inner structures of the galaxies.. This is similar to the dynamical split of
the ICL and the BCG in galaxy clusters, and might be a way to distinguish
outer stellar halos of galaxies from the galaxies themselves instead.

###  4.2 Observational Comparison Samples

Some of the deepest imaging of nearby ETGs available comes from the VEGAS
survey (Capaccioli et al., 2015). The survey probes surface brightness
profiles out to $\sim 10R_{\mathrm{e}}$ and down to surface brightness levels
of $\sim 29~{}\mathrm{mag/arcsec}^{2}$ in the g-band. The survey is still
ongoing, however, results on the radial surface brightness profiles have been
published for several massive galaxies in group/cluster environments by
Spavone et al. (2017) and Spavone et al. (2020). Spavone et al. (2017) fit two
or three Sérsic profiles to 6 ETGs, with the Sérsic parameters $n$ constrained
to a narrow range. More recently, Spavone et al. (2020) fit 19 ETGs in the
Fornax cluster with either two or three Sérsic components. Here we focus on
the two component fit, for which $n$ was a free parameter. The two component
fits have a single intermediate radius, and the accreted mass fractions are
calculated from the second (outer) component. These are referred to as the
“relaxed” components following Cooper et al. (2015). Hence the approach to
provide unconstrained fits is more comparable to our approach, we focus on the
study by Spavone et al. (2020) instead of Spavone et al. (2017).

Another very deep imaging study has been carried out by Kluge et al. (2019),
who fit double Sérsic profiles to extremely low surface brightness profiles
targeting especially BCGs. Both single and double Sérsic fits were obtained,
as well as accreted fractions from the double Sérsic fits.

We also compare to the double Sérsic fits of $\sim 45,500$ galaxies, observed
at a mean redshift $z\sim 0.08$ and stacked in mass bins by D’Souza et al.
(2014). Taking mean values from their figure 13 for ETG-like galaxies, we note
that their data covers a similar stellar mass range to our modelled galaxies
and that they find effective radii of the inner and outer components to be
around 3 and 8 kpc, respectively. D’Souza et al. (2014) also provided such
measurements for their stacked LTG sample, however, we will focuss on their
results regarding the ETGs here.

###  4.3 Accreted Fractions from Double Sérsic Fits

Figure 12: Mass fraction of accreted stars versus stellar mass (same as the
upper right panel of Fig. 6) but with observations included as black symbols:
Data points from the VEGAS survey Spavone et al. (2020) as open diamonds, with
additional data points from the literature as presented by Spavone et al.
(2020) as open triangles. Data points for BCGs from Kluge et al. (2019) are
shown as filled black diamonds. The values for the stacked SDSS galaxies by
D’Souza et al. (2014) are shown as open squares, with the upper line the one
obtained for the ETG-like galaxies where a double Sérsic fit is a good
description, and the lower line for the LTG-like galaxies where a third Sérsic
fit is required. The blue solid line and shaded region indicate the Magneticum
average values from this work, and all other coloured lines and shaded areas
indicate other cosmological simulations as described in Fig. 6.

In Fig. 12 we reproduce the top right panel of Fig. 6, showing the ex-situ
accretion fraction versus stellar mass. Again, the lines and shading show the
results of various models, colors as indicated in Fig. 6, with the blue solid
line showing the average fraction of accreted mass for Magneticum galaxies.
This time, we included in Fig. 12 observational data from the literature.
These observations do not directly measure the fraction of accreted mass but
rather fit double Sérsic profiles to the surface brightness profiles of early-
type galaxies, as described in Sec. 4.2, inferring the mass fraction from the
outer Sérsic component.

Figure 13: Sérsic index $n$ versus stellar mass for single Sérsic fits to the
2D projected mass density profiles of all Magneticum galaxies. The dashed
black line show the results from simulations by Hopkins et al. (2009), with
the shaded area marking the scatter inferred from their simulations. The solid
black line is the mean relation from observations by (Graham, 2013) converted
into stellar mass assuming $M/L_{\mathrm{B}}=10$. Left panel: All Magneticum
galaxies are coloured according to their profile classes as indicated in the
legend. Right panel: Only the ETGs from the Magneticum galaxy sample are
shown. In addition, open black diamonds show observations of ETGs from
Kormendy et al. (2009), and the solid black diamonds show BCGs from the
observations by Kluge et al. (2019).

While all simulations suggest a general trend of increasing accretion fraction
for higher mass galaxies, there is no clear trend for the observational proxy
(the outer component mass) to vary with stellar mass for the Fornax galaxy
sample by Spavone et al. (2020) or the BCG sample by Kluge et al. (2019). Only
the stacked sample provided by D’Souza et al. (2014) shows a decrease in
accreted fraction with mass for their ETG sample. In fact, the different
observations differ strongly from each others and do not show a clear picture
of the accreted fraction estimated through the outer Sérsic fit component
being correlated with the total stellar mass of a galaxy. This further
supports our results from Sec. 4.1, that dips in the observed surface
brightness profile do not, in general, correspond to the true transition from
in-situ to accreted dominated material. An alternative approach to estimating
accretion fraction may come from star formation histories (Boecker et al.,
2020) or 2D chemo-dynamical analysis (Poci et al., 2019), and needs to be
investigated in future studies.

###  4.4 Indices from Single and Double Sérsic Fits

As a final test, we compare the Sérsic indices $n$ from the single and double
Sérsic fits to our projected simulated galaxies with observations to ensure
that the simulated and observed galaxy samples really are comparable in their
radial properties. Fig. 13 shows the Sérsic indices $n$ from single Sérsic
fits to the 2D projected mass density profiles of the Magneticum galaxies, for
all galaxies in the left panel and ETGs-only in the right panel. We also
include the observed relation for galaxies using eq. 2.7 from Graham (2013)
and simply assuming $M/L_{\mathrm{B}}=10$ for all galaxies as solid black
line. We find a trend of increasing Sérsic index $n$ for higher mass galaxies
that matches the observed mean trend well, clearly showing that the overall
mass distribution of the simulated galaxies is in good agreement with
observations. Galaxies of the different profile classes are well spread in
Sérsic index for a given stellar mass, with no clear trends apart from the
fact that the largest Sérsic indices are clearly found in class C galaxies. We
also include the range of Sérsic indices from the empirical model by Hopkins
et al. (2009), which cover a similar mass range than our simulated galaxies.
The model predicts on average Sérsic indices that are, at a given stellar
mass, somewhat larger than the Magneticum and the observed galaxies of Graham
(2013), especially at the high mass end, but the scatter range is very
similar, and the overall trend of larger Sŕsic indices with larger stellar
mass is in good agreement.

When limiting our galaxy sample to ETGs-only (right panel of Fig. 13), we find
the Sérsic indices to be slightly larger on average. We additionally include
the observational data for individual ETGs from Kormendy et al. (2009) and
Kluge et al. (2019), further showing that the simulations also provide good
descriptions of the stellar mass distributions of ETGs especially. There is a
slight tendency for galaxies with stellar masses above $\log(M_{*})>11.5$ to
have larger Sérsic indices in the observed samples than in our simulated
sample, however, at this mass range our simulated sample is statistically not
representative anymore, especially since there are only 4 BCGs in our sample
while the observations by Kluge et al. (2019) are BCGs only444For a more
detailed comparison of the BCG properties from a larger Magneticum simulation
volume with observations, see Remus et al., in prep..

Figure 14: Sérsic index $n$ versus stellar mass for double Sérsic fits to the
2D projected mass density profiles of the Magneticum galaxies, with colours as
indicated in the legend. Observations from the VEGAS survey (Spavone et al.,
2020) are shown as open black diamonds, and BCGs from Kluge et al. (2019) as
solid black diamonds. The black lines mark the values obtained by D’Souza et
al. (2014) from fits to over $45,500$ galaxies. Left panel: Inner Sérsic fits.
Right panel: Outer Sérsic fits.

For double Sérsic fits, we here focus only on ETGs as the observational
comparison samples only include ETGs. As shown in Fig. 14, we find the inner
profiles of our simulated ETGs to have Sérsic indices of $n\sim 2.5$ on
average (left panel), and the outer profiles to have Sérsic indices of
$n\sim{}$0.5–1, on average (right panel). We find little variation of these
indices with stellar mass, if all there is a slight average trend for the
inner Sérsic indices of more massive galaxies to be slightly larger. The
agreement with the 9 ETGs from the VEGAS survey (Spavone et al., 2020), with
masses $>10^{10}M_{\odot}$, is reasonable, with a slight tendency for the
obersved outer Sérsic indices to be larger than for the simulated ETGs.
Compared to the observations of BCGs by Kluge et al. (2019), we find that the
agreement with their outer Sérsic indices is rather good, while their inner
Sérsic are on average larger than those of our simulated galaxies. However,
this could also be due to the fact that most of the Magneticum ETGs in the
mass range comparable to the sample by Kluge et al. (2019) are not BCGs.

Additionally, we also show in Fig. 14 the inner and outer Sérsic indices from
the stacked observations by D’Souza et al. (2014), using their double Sérsic
fits (solid lines). As can clearly be seen they differ rather strongly from
the simulated galaxies, but also the other observations, with their inner
slopes generally smaller and their outer slopes much larger in comparison. We
note that D’Souza et al. (2014) also fit triple Sérsic profiles to their
highest mass galaxies. In this case, their outer Sérsic fits have $n\sim 1.5$,
which is much closer to our simulation values and the other observations,
indicating that their inner slopes are actually really “inner” slopes which we
do not fit in this work to avoid resolution issues. This clearly highlights
the importance of clear definitions regarding the fitted regions of galaxies
when performing comparisons.

Overall, we find reasonable agreement between the Sérsic indices $n$ predicted
by our simulated galaxies and the observed Sérsic indices, especially of ETGs,
for both single and double Sérsic fit indices, clearly showing that the
simulated galaxies used in this work capture the observed matter
distributions. This clearly shows that our result, that double Sérsic fits do
not describe the in-situ and accreted components of galaxies, is applicable to
observations. We rather suggest that the double Sérsic fit (excluding the
central inner bulge areas) describes the relaxed inner and the unrelaxed outer
stellar (halo) components of galaxies and could therefore be used to
distinguish the outer stellar halo from a galaxy.

##  5 Summary and Conclusions

Using the Magneticum simulation we have studied a sample of 511 model galaxies
in the log mass range $10.3<M_{*}<12$. These simulated galaxies reproduce well
the observed galaxy size-mass relation. We also find that the fraction of
accreted material, as a function of total halo and stellar mass, reveals
similar trends to those found by previous simulations. One notable difference
is that our simulations predict somewhat higher accretion fractions in the
lower mass galaxies.

We examined the stellar mass density profiles of our sample, split in accreted
and in-situ components, and classified them into 6 classes depending on the
profile type: The most common class reveals, at intermediate radii, a
transition radius where the accreted and the in-situ component are equal, with
the in-situ component being dominant in the centre and the accreted component
dominating the outskirts. This class is comparable to the profiles found in
previous studies. However, for about 30% of our galaxies we find different
profiles: Some are accretion dominated at all radii, even in the centre;
another group of galaxies is in-situ dominated at all radii; and most
interestingly, we find one class of galaxies that have even two transition
radii at which the in-situ and accreted material are equal, with an accretion
dominated core, an in-situ dominated shell around it, and an accretion
dominated outskirt. This is actually the second most common class of galaxies
in our sample.

We show that these profile classes correlate with galaxy mass, and that the
type of mergers they undergo help to shape their profiles. We especially show
that the amount of gas that is involved in these mergers is more important in
shaping these profiles than the actual merger, and that the most common class
is to about 70% not dominated by major mergers but rather smaller merger
events. Their outer regions are largely built up by dry minor and mini
mergers, clearly showing the importance of minor and especially mini mergers
in shaping the outer stellar halos of galaxies.

We find that galaxies with high in-situ fractions (low accretion fractions)
tend to be lower mass galaxies with smaller halfmass radii, and we see a weak
trend for high in-situ fraction galaxies to have lower central dark matter
fractions, with the exception of the overall in-situ dominated galaxies that
have clearly larger central dark matter fractions at a given stellar mass than
galaxies of the most common class, typical for what is found in disk galaxies.
We measure the radius between the in-situ and accretion-dominated regions for
those galaxies that reveal a clear transition, which are the majority of our
sample. This transition radius is found to weakly be inversely correlated with
stellar mass, but strongly correlated with the in-situ fraction for our most
common class of galaxies. However, galaxies of the other classes that have one
or even two transition radii, do not follow the same relations.

In order to compare our 3D profiles with observations, we projected the
stellar mass density in different angles for each galaxy to be able to
directly compare with observed surface brightness profiles. We find that the
transition radius from in-situ to accretion dominated profiles seen in many 3D
profiles also occurs in all projections, but always at different radii in the
2D profiles, usually at smaller radii. None of our galaxies changes its in-
situ-profile class during projections. We also find that, similar to
observations, our projected stellar mass surface density profiles usually
require a double Sérsic fit to be described accurately, with Sérsic indices
for both components similar to the range of observed Sérsic indices. However,
we clearly see that these two Sérsic components usually do not describe the
underlying in-situ and accreted components, but are rather disjunct from
those. Only in very few cases the crossing radius of the two Sérsic components
coincides with the transition radius of the galaxy. Even worse, we also
clearly see that most galaxies that are dominated by accreted stars at all
radii, still require a double Sérsic fit to describe the stellar surface
density profiles, thus having a crossing radius but no transition radius.

In other words, we clearly conclude that the dip seen in 2D profiles does not
correspond to the true transition radius between in-situ and accretion
dominated regions. Similarly, any mass inferred from these double-Sérsci fits
will not trace the true in-situ or accreted mass of a galaxy. Thus, fits to
the dips seen in some observed surface brightness profiles of early-type
galaxies are not a true measure of a galaxy’s accretion material. However,
they do hold some information about the assembly history of that galaxy, as we
find indications that these dips are more likely an indication for the
transition from the inner (in-situ and massive merger dominated) core of a
galaxy to its stellar halo, mostly accreted through minor and mini mergers,
similar to the ICL component around BCGs. To confirm this, a more detailed
study including also the radial kinematics of a galaxy in addition to its
density component is needed in the future, to disentangle the formation
pathways of galaxies from observational tracers.

## Acknowledgements

We thank Klaus Dolag and Felix Schulze for very useful discussions. We also
thank Thomas Davison, Enrica Iodice, and Marilena Spavone for their helpful
comments. We also acknowledge funding from the DAAD PPP Germany-Australia
Exchange Program. The Magneticum Pathfinder simulations were partially
performed at the Leibniz-Rechenzentrum with CPU time assigned to the Project
“pr86re”, supported by the DFG Cluster of Excellence “Origin and Structure of
the Universe”. We are especially grateful for the support by M. Petkova
through the Computational Center for Particle and Astrophysics (C2PAP).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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