11institutetext: Heidelberger Institut für Theoretische Studien, Schloss-
Wolfsbrunnenweg 35, 69118 Heidelberg, Germany 22institutetext: Astronomisches
Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg,
Mönchhofstr. 12-14, 69120 Heidelberg, Germany 33institutetext: University of
Oxford, St Edmund Hall, Oxford, OX1 4AR, United Kingdom

# Pre-supernova evolution and final fate of stellar mergers and accretors of
binary mass transfer

F.R.N. Schneider<EMAIL_ADDRESS>evolution and final
fate of stellar mergers and accretors of binary mass transferPre-supernova
evolution and final fate of stellar mergers and accretors of binary mass
transferPre-supernova evolution and final fate of stellar mergers and
accretors of binary mass transferPre-supernova evolution and final fate of
stellar mergers and accretors of binary mass transfer Ph. Podsiadlowski Pre-
supernova evolution and final fate of stellar mergers and accretors of binary
mass transferPre-supernova evolution and final fate of stellar mergers and
accretors of binary mass transferPre-supernova evolution and final fate of
stellar mergers and accretors of binary mass transferPre-supernova evolution
and final fate of stellar mergers and accretors of binary mass transfer E.
Laplace Pre-supernova evolution and final fate of stellar mergers and
accretors of binary mass transferPre-supernova evolution and final fate of
stellar mergers and accretors of binary mass transfer

(Received 1 September 2023 / Accepted 21 February 2024)

The majority of massive stars are expected to exchange mass or merge with a
companion during their lives. This immediately implies that most supernovae
(SNe) are from such post-mass-exchange objects. Here, we explore how mass
accretion and merging affect the pre-SN structures of stars and their final
fates. To this end, we model these complex processes by rapid mass accretion
onto stars of different evolutionary stages and follow their evolution up to
iron core collapse. We use the stellar evolution code Mesa and infer the
outcome of core-collapse using a neutrino-driven SN model. Our models cover
initial masses from $11$ to $70\,{\rm M}_{\odot}$ and the accreted mass ranges
from 10–200% of the initial mass. All models are non-rotating and for solar
metallicity. The rapid-accretion model offers a systematic way to approach the
landscape of mass accretion and stellar mergers. It is naturally limited in
scope and serves as a clean zeroth order baseline for these processes. We find
that mass accretion in particular onto post-main-sequence (post-MS) stars can
lead to a long-lived blue supergiant (BSG) phase during which stars burn
helium in their cores. In comparison to genuine single stars, post-MS
accretors have small core-to-total mass ratios, regardless of whether they end
their lives as BSGs or cool supergiants (CSGs), and they can have genuinely
different pre-SN core structures. As in single and binary-stripped stars, we
find black-hole (BH) formation for the same characteristic CO core masses
$M_{\mathrm{CO}}$ of $\approx 7\,{\rm M}_{\odot}$ and $\gtrsim 13\,{\rm
M}_{\odot}$. In models with the largest mass accretion, the BH-formation
landscape as a function of $M_{\mathrm{CO}}$ is shifted by about $0.5\,{\rm
M}_{\odot}$ to lower masses, i.e. such accretors are more difficult to
explode. We find a tight relation between our neutron-star (NS) masses and the
central entropy of the pre-SN models in all accretors and single stars,
suggesting a universal relation that is independent of the evolutionary
history of stars. Post-MS accretors explode both as BSGs and CSGs, and we show
how to understand their pre-SN locations in the Hertzsprung–Russell (HR)
diagram. Accretors exploding as CSGs can have much higher envelope masses than
single stars. Some BSGs that avoid the luminous-blue-variable (LBV) regime in
the HR diagram are predicted to collapse into BHs of up to $50\,{\rm
M}_{\odot}$ while others explode in supernovae and eject up to $40\,{\rm
M}_{\odot}$, greatly exceeding ejecta masses from single stars. Both the BH
and supernova ejecta masses increase to about $80\,{\rm M}_{\odot}$ in our
models when allowing for multiple mergers, e.g., in initial triple-star
systems, and they can be even higher at lower metallicities. Such high BH
masses may fall into the pair-instability-supernova mass gap and could help
explain binary BH mergers involving very massive BHs as observed in GW190521.
We further find that some of the BSG models explode as LBVs, which may lead to
interacting SNe and possibly even superluminous SNe.

###### Key Words.:

binaries: general – Stars: massive – Stars: evolution – Stars: black holes –
Stars: neutron – supernovae: general

##  1 Introduction

Most massive stars are born in binary, triple and other multiple-star systems
(e.g. Kobulnicky & Fryer 2007; Mason et al. 2009; Sana et al. 2012; Chini et
al. 2012; Sana et al. 2013; Kobulnicky et al. 2014; Sana et al. 2014; Moe & Di
Stefano 2017; Offner et al. 2023). For example, almost all O-type stars are
found in binary systems and the triple and higher-order multiple-star fraction
of these stars even exceeds 50% (e.g. Sana et al. 2014; Moe & Di Stefano 2017;
Offner et al. 2023). As a consequence, the majority of massive stars will
exchange mass with a companion at some point in their evolution, i.e. also
most supernovae (SNe) are from stars that evolved through past mass exchange
with a companion (e.g. Podsiadlowski et al. 1992; Sana et al. 2012; Moe & Di
Stefano 2017). In about 40–50% of the cases, mass exchange proceeds via stable
Roche-lobe overflow (RLOF), but it can also lead to stellar mergers in about
20–30% and common-envelope phases in 10–20% of cases111Some stars in multiple
systems never exchange mass with a companion and evolve as effectively single
stars. (see e.g. Sana et al. 2012; Schneider et al. 2015; Stegmann et al.
2022; Henneco et al. 2024).

During stable mass transfer and in common-envelope events, stars lose their
hydrogen-rich envelopes. These binary-stripped stars develop distinctly
different pre-SN core structures compared to single stars, affecting their
final fate (e.g. Timmes et al. 1996; Wellstein & Langer 1999; Brown et al.
2001; Podsiadlowski et al. 2004; Woosley 2019; Ertl et al. 2020; Schneider et
al. 2021; Laplace et al. 2021; Aguilera-Dena et al. 2023; Schneider et al.
2023). For example, they explode in SN Ib/c and the inherently different pre-
SN core structures result in neutron-star (NS) formation over a much larger
initial mass range than in genuine single stars; they also explode with on
average higher explosion energies, which results in larger nickel yields and
faster NS kicks (Schneider et al. 2021); the systematically higher nickel
yields are confirmed by observations (e.g. Anderson 2019; Meza & Anderson
2020). Moreover, their chemical yields differ from those of single stars
(Farmer et al. 2021, 2023), and they may produce BHs of universal masses of
about $9\,{\rm M}_{\odot}$ and $\gtrsim 16\,{\rm M}_{\odot}$ over a large
range of metallicities (Schneider et al. 2023). First hints for the existence
of such universal BH masses may have been found in the chirp and BH mass
distributions of binary BH mergers observed by gravitational-wave detectors
(e.g. Tiwari & Fairhurst 2021; The LIGO Scientific Collaboration et al. 2021b;
Tiwari 2022; Edelman et al. 2023).

Compared to the situation of binary-stripped stars, less is known about the
pre-SN core structures and the final fates of the accretors of mass transfer
and stellar mergers. It is estimated that about half of all Type II SNe are
from such accretors in binary systems, with mergers being the dominant
contribution (e.g. Zapartas et al. 2019). Main-sequence (MS) accretors likely
rejuvenate and adjust their structure to the new mass such that they continue
their evolution similar to that of single stars (e.g. Hellings 1983; Braun &
Langer 1995; Dray & Tout 2007; Schneider et al. 2016, 2019, 2020). In
contrast, post-MS accretors can become core-helium-burning blue supergiants
(BSGs) and explode as such stars (Hellings 1983, 1984; Podsiadlowski & Joss
1989; Podsiadlowski et al. 1990; Claeys et al. 2011; Vanbeveren et al. 2013;
Justham et al. 2014; Menon & Heger 2017; Menon et al. 2023). They may,
however, also evolve and explode as cool supergiants (CSGs)222In this paper,
we mainly distinguish between blue and cool supergiants. The latter includes
red and yellow supergiants, and we define cool supergiants to have effective
temperatures of $\log T_{\mathrm{eff}}/\mathrm{K}<3.9$., but with different
core masses than what is expected from single CSGs (e.g. Zapartas et al.
2021).

The well-studied SN 1987A was from a BSG that is likely the result of a merger
(see e.g. reviews by Podsiadlowski 1992, 2017b and some of the latest models
for SN 1987A by Menon & Heger 2017, Urushibata et al. 2018 and Menon et al.
2019). More massive BSGs from stellar mergers have further been suggested to
explode as luminous blue variables (LBVs) in interacting and possibly
superluminous SNe (Justham et al. 2014), and hydrogen-rich, pair-instability
supernovae (PISNe; Vigna-Gómez et al. 2019). Furthermore, very massive merged
stars with cores not massive enough for electron-positron pair production but
large envelope masses might form BHs with masses in the pair-instability mass
gap (e.g. Di Carlo et al. 2020; Renzo et al. 2020a; Costa et al. 2022; Ballone
et al. 2023).

In this paper, we systematically model the response of stars to rapid mass
accretion on a thermal timescale in an attempt to approximate the complex
physics occurring in accretors of stable RLOF in semi-detached binaries and
stellar mergers. We follow the evolution of these models until the onset of
core collapse to understand how such accretion events may modify the core
structures and hence affect the final fates and supernova explosions of these
stars. While our models are simplified and, e.g., do not take into account
rotation and the complex mixing in stellar mergers, they nevertheless allow us
to study the resulting structures of mass accretors in a uniform and
controlled way. In light of these approximations, we caution the reader when
using these models. To explore the outcome of core collapse, we use the pre-SN
stellar structures of our accretor models as input to the neutrino-driven SN
model of Müller et al. (2016). We describe how we model accretors in the
stellar evolution code Mesa (Paxton et al. 2011, 2013, 2015, 2018, 2019) and
explain the coupling to the SN code in Sect. 2. We present our results on the
pre-SN evolution and core structures of the accretors at core collapse in
Sect. 3. In Sect. 4, we show our findings on the final fates of accretors with
a focus on their explodability, compact-remnant and SN-ejecta masses, and pre-
SN locations in the Hertzsprung–Russell (HR) diagram. Further explosion
properties will be presented in a forthcoming publication (Schneider et al.
2024). We discuss our results concerning model simplifications and
uncertainties, and wider implications, e.g., on the maximum BH and SN-ejecta
masses from accretors at solar metallicity in Sect. 5. A summary of our main
findings is given in Sect. 6.

##  2 Methods

The stellar models are computed with revision 10398 of the Modules-for-
Experiments-in-Stellar-Astrophysics (Mesa) software package (Paxton et al.
2011, 2013, 2015, 2018, 2019). We use the same basic Mesa setup as in
Schneider et al. (2021) and briefly summarise it in Sect. 2.1. In Sect. 2.2,
we explain how we model binary-star accretion. For comparison of these
accretion models with genuine single stars, we employ the single-star models
of Schneider et al. (2021). In Sect. 2.3, we briefly describe how we couple
our models to a SN code to study their final fates and possible SN explosions.
An overview of all models and key properties discussed in this work are
provided in Table LABEL:tab:models. Mesa inlists and other settings required
to reproduce our models are published online at
https://zenodo.org/doi/10.5281/zenodo.10731998.

###  2.1 Stellar-model physics

All models are non-rotating and have the initial chemical composition of
Asplund et al. (2009), i.e. an initial hydrogen mass fraction of $X=0.7155$,
helium mass fraction of $Y=0.2703$ and metallicity of $Z=0.0142$. Convection
is described by mixing-length theory (Henyey et al. 1965) and we use a mixing-
length parameter of $\alpha_{\mathrm{mlt}}=1.8$. The Ledoux criterion is
applied and semi-convection is treated with an efficiency factor of
$\alpha_{\mathrm{sc}}=0.1$. We enable Mesa’s MLT++ scheme that boosts
convective energy transport in some stellar envelopes. Convective-boundary
mixing is only applied to convective cores during hydrogen and helium burning,
and is included via so-called “step overshooting” of $0.2$ pressure-scale
height. We use the stellar-wind prescription of Schneider et al. (2021) that
closely follows Mesa’s “Dutch” scheme. Cool-star winds follow Nieuwenhuijzen &
de Jager (1990), while hot-star winds apply the Vink et al. (2000, 2001)
rates. For Wolf–Rayet (WR) stars, we follow Nugis & Lamers (2000). The
metallicity scaling of the cool-star and WR winds are as in Mauron & Josselin
(2011) and Vink & de Koter (2005), respectively. No additional wind-mass loss
is applied to luminous blue variables (LBVs). However, we mark models that
spend more than $10^{4}\,\mathrm{yr}$ on the cool side of the diagonal S
Doradus instability strip in the Hertzsprung–Russell (HR) diagram (Smith et
al. 2004) and/or at effective temperatures
$T_{\mathrm{eff}}<12500\,\mathrm{K}$ while exceeding a luminosity of
$\log\,L/{\rm L}_{\odot}=5.5$ (i.e. connecting the hot-part of S Doradus
instability strip with the absence of red supergiants (RSGs) of $\log\,L/{\rm
L}_{\odot}\gtrsim 5.5$ at $T_{\mathrm{eff}}=12500\,\mathrm{K}$, Davies et al.
2018); such stars likely experience significant mass loss possibly leading to
the removal of the entire hydrogen-rich envelope. Nuclear burning is
implemented via the approx21_cr60_plus_co56.net reaction network. While this
network properly reflects energy production by nuclear burning, it cannot
follow accurately the electron fraction or proton-to-neutron ratio because of,
e.g., missing weak reactions. Reaction rates themselves are from the JINA
REACLIB database V2.2 (Cyburt et al. 2010) and, for example, the important
${}^{12}\mathrm{C}(\alpha,\gamma){}^{16}\mathrm{O}$ reaction rate is taken
from Xu et al. (2013).

###  2.2 Binary-star accretion models

####  2.2.1 Model assumptions

Mass transfer by stable RLOF in semi-detached binaries and stellar mergers are
events during which the masses of stars increase rapidly. Hence, to zeroth
order, these processes may be viewed as accretion events where one star,
called accretor from hereon, effectively gains mass. The accretion timescales
during these events cover the entire spectrum from dynamic-timescale accretion
in stellar mergers to thermal- and nuclear-timescale accretion in binary mass
transfer. In stable Case A RLOF in semi-detached binaries, most mass is often
accreted on a thermal timescale (e.g. Wellstein et al. 2001), but this can
vary from case to case. In our accretion models, we accrete a certain fraction
of the initial mass of a star, $f_{\mathrm{acc}}\equiv\Delta
M_{\mathrm{acc}}/M_{\mathrm{ini}}$, on the momentary thermal timescale of the
accretor, i.e. at a rate of

$\displaystyle\dot{M}=\frac{M}{\tau_{\mathrm{KH}}}=\frac{2RL}{GM},$ (1)

where $L$, $R$ and $M$ are the momentary luminosity, radius and mass of the
accretor, respectively, $\tau_{\mathrm{KH}}$ is the Kelvin-Helmholtz timescale
and $G$ is the gravitational constant. The chemical composition of the
accreted mass is always the same as that on the surface of the accretor.

We compute models for accretion fractions $f_{\mathrm{acc}}$ of 10%, 25%, 50%,
75%, 100%, 150% and 200%. The latter two fractions are not achievable by
stellar mergers in isolated binary evolution but can be achieved by stable
RLOF333The accretors in a semi-detached binary are initially less massive than
the donor and can possibly accrete more than their initial mass, e.g. in an
initially $10+3\,{\rm M}_{\odot}$ binary system.. In stellar mergers,
$f_{\mathrm{acc}}>1$ may be possible in triple systems and dense stellar
clusters (see Sect. 5.4 for further discussion). To better distinguish these
cases visually, we show the $f_{\mathrm{acc}}=150\%$ and 200% models in blue
colour in the rest of the paper. All other accretion fractions are possible in
both stable binary mass transfer and stellar mergers.

As with envelope stripping by RLOF (e.g. Schneider et al. 2021), the future
evolution of stars depends critically on the exact evolutionary moment when
the matter is accreted. In analogy to envelope stripping, we use the terms
Case A, B and C, but now applied to the evolutionary stage of the _accretor_.
Hence, Case A, Case B and Case C refer to stars that accrete mass when they
burn hydrogen in their cores, have finished core hydrogen burning but have not
started core helium burning yet, and have finished core helium burning,
respectively. We have defined the end of core helium burning as that point in
evolution when the central hydrogen mass fraction drops below $10^{-6}$. We
also distinguish between “early” and “late” Case A and B. In our case, early
and late Case A accretion refer to when the central hydrogen mass fraction of
the accretor drops below 0.35 and 0.05, respectively. In Case B accretors,
early means that the accretor has just left the main sequence while late is
when the accretor has developed a convective envelope and is shortly before
core helium ignition.

####  2.2.2 Model limitations

The simple accretion models cannot reproduce all the important physics of
binary mass transfer and stellar mergers, and only account for one such phase
while several mass-accretion and mass-loss episodes are possible in general
(see also Schneider et al. 2021). Nevertheless, applying such models allows us
to understand and study the resulting stellar structures in a uniform and
controlled way, and our models offer a zeroth-order approximation to these
complex physical processes. We mention some limitations of our approach here
and further discuss them in Sect. 5.

Our models are non-rotating while in particular accretors of binary mass
transfer are likely spun up to near critical surface rotation (see e.g. Langer
2012). Such rapid rotation can induce mixing of chemical elements and angular
momentum that we do not take into account. The mixing can enlarge the cores of
MS accretors and thereby affect the core structure at the pre-SN stage.
Moreover, rotation changes the mass-loss history of stars.

Multi-dimensional hydrodynamic simulations show that mixing also occurs in
stellar mergers (e.g. Lombardi et al. 2002; Ivanova et al. 2002; Glebbeek et
al. 2013; Schneider et al. 2019). It can enlarge the effective core sizes in
mergers of two MS stars increasing the effect of rejuvenation but can also
lead to smaller cores in mergers with post-MS stars. Moreover, nuclear-
processed core material of the two stars may be mixed in the outer layers of
the merger remnant, which we do not take into account. Consequently, our
models cannot reproduce the chemical surface enrichment observed in such
simulations (c.f. Lombardi et al. 2002; Ivanova 2002; Glebbeek et al. 2013).
Our accretion models also neglect strong magnetic fields that may be generated
in the merging process and that can affect the evolution of merged stars,
e.g., via internal angular-momentum transport and magnetic braking (Schneider
et al. 2019, 2020).

Lastly, our models cannot reproduce the surface chemical abundances expected
for products of binary mass transfer and stellar mergers. For example, they do
not show the nitrogen and helium enrichment from accreting nuclear-processed
material during the later accretion phases in stable binary mass transfer (see
e.g. review by Langer 2012). In some of our models, stellar winds have removed
parts of the stellar envelope such that helium-rich layers are visible at the
surface. In these cases, we then accrete helium-enriched material.

###  2.3 Semi-analytic supernova model

Pre-SN stellar structures are analysed with the semi-analytic, parametric-SN
code of Müller et al. (2016) once the Mesa models reach an iron-core infall
velocity of $950\,{\rm km\,s^{-1}}$. The code explores the explodability of
pre-SN stellar models within the framework of neutrino-driven SNe and takes
into account the entire interior structure of our models at core collapse. It
predicts SN properties such as explosion energy, synthesized nickel masses,
compact remnant masses and NS kick velocities. We apply the same calibrations
of this neutrino-driven-explosion model as in Schneider et al. (2021), i.e. an
average explosion energy of Type IIP SNe from single-star models of $0.69\pm
0.17\,{\rm B}$ and a NS kick-velocity distribution that is in agreement with
the Maxwellian with $\sigma=265\,{\rm km\,s^{-1}}$ of Hobbs et al. (2005). The
maximum, gravitational NS mass is chosen to be $2\,{\rm M}_{\odot}$.

We classify SN explosions as SN IIb if the mass of the hydrogen-rich envelope
is smaller than $1\,{\rm M}_{\odot}$ following Sravan et al. (2019). The mass
of the hydrogen-rich envelope is defined as the layer with a hydrogen mass
fraction $X_{\mathrm{H}}>0.01$ and a helium mass fraction
$X_{\mathrm{He}}>0.1$. If the hydrogen mass in the SN ejecta is smaller than
$0.1\,{\rm M}_{\odot}$ (Hachinger et al. 2012), we say that the SN is of Type
Ib/c. We do not distinguish between SN Ib and SN Ic in this work. In all other
cases with more hydrogen and larger hydrogen-rich envelopes, we define the SN
to be of Type II.

##  3 Pre-supernova evolution and structure

Rapid accretion of mass can change the stellar structure in such a way that
the accretors (merged stars) spend most of their core-helium-burning lifetime
and immediate pre-SN evolution as BSGs (e.g. Hellings 1983; Podsiadlowski &
Joss 1989; Podsiadlowski et al. 1990; Braun & Langer 1995; Claeys et al. 2011;
Vanbeveren et al. 2013; Justham et al. 2014; Menon & Heger 2017; Urushibata et
al. 2018). We describe the occurrence and structure of such BSGs in Sect. 3.1.
A star’s final fate is mostly determined already by the end of core helium
burning and depends on the then CO core mass $M_{\mathrm{CO}}^{\prime}$ and
central carbon mass fraction $X_{\mathrm{C}}$ (Chieffi & Limongi 2020; Patton
& Sukhbold 2020; Schneider et al. 2021). In our models, the CO core masses at
the end of core helium burning are almost identical to those at core collapse,
$M_{\mathrm{CO}}$. We show how core masses such as $M_{\mathrm{CO}}$ and the
important central carbon mass fraction $X_{\mathrm{C}}$ are affected by mass
accretion in Sect. 3.2 and Sect. 3.3, respectively, before presenting the
structures of our models at core collapse in Sect. 3.4.

###  3.1 Blue supergiants

In Fig. 1, we show an exemplary Hertzsprung–Russell (HR) diagram of initially
$13\,{\rm M}_{\odot}$ stars accreting mass shortly after finishing core
hydrogen burning (early Case B 444HR diagrams for Case A, late Case B and Case
C accretors can be found in Figs. 17 and 18, respectively.). We find three
qualitatively different outcomes of mass accretion: the accretor burns helium
in its core (i) as a CSG ($f_{\mathrm{acc}}=0.10$, $0.25$), (ii) as a BSG but
explodes as a CSG ($f_{\mathrm{acc}}=0.50$) and (iii) as a BSG and explodes as
a BSG ($f_{\mathrm{acc}}\geq 0.75$). Here, we define BSGs as supergiants with
effective temperatures of $\log\,T_{\mathrm{eff}}/\mathrm{K}>3.9$ while CSGs
are cooler than that. When burning helium in the core as a BSG, stars do so
for a nuclear timescale, i.e. for several $10^{5}\text{--}10^{6}\,\mathrm{yr}$
(see Figs. 19 and 20, and Table LABEL:tab:models), and we refer to them as
long-lived BSGs555Our single star models only spend a thermal timescale as
BSGs, which is 1–2 orders of magnitude shorter than these long-lived BSGs from
the accretor models.. The required amount of accreted mass to turn stars into
BSGs depends on the mass and evolutionary stage of the accretors. For example,
none of our early Case A accretors turn into BSGs and only one model does so
in late Case A accretion ($f_{\mathrm{acc}}=0.25$; Fig. 17). Most of the Case
A models fully rejuvenate and the accretors continue their evolution similarly
to single stars with a higher mass (see below for more details on the
rejuvenation). In late Case B accretion, the models with
$f_{\mathrm{acc}}=1.0$, $1.5$ and $2.0$ are BSGs during core helium burning
and also explode as BSGs, while this only occurs for $f_{\mathrm{acc}}=1.5$
and $2.0$ in the Case C accretors (Fig. 18). In general, we find that the more
evolved the stars are, the more mass needs to be accreted to turn them into
BSGs.

Figure 1: Hertzspung–Russell diagram of initially $13\,{\rm M}_{\odot}$ stars
accreting a fraction $f_{\mathrm{acc}}$ of their initial mass (color coded)
shortly after they finished core hydrogen burning (early Case B). The dotted
parts of the tracks show the accretion phase and the grey dashed evolutionary
track is for a single star. Plus symbols are added every $2\times
10^{5}\,\mathrm{yr}$ after finishing core hydrogen burning to indicate in
which region of the HR diagram stars burn helium in the core. The
observationally-inferred maximum luminosity of RSGs of $\log L/{\rm
L}_{\odot}=5.5$ (Davies et al. 2018) is shown by a dotted black line.
Explosion sites are indicated as well as the likely supernova type. The grey
bands show the hot and cool part of the S Doradus instability strip (Smith et
al. 2004). For $f_{\mathrm{acc}}\gtrsim 0.50$, the models burn helium as a BSG
and for $f_{\mathrm{acc}}\gtrsim 0.75$ they also explode as a BSG.

We have not considered the mixing of helium-rich material out of the core into
the envelope. It is well known that the dredge-up of helium favours the
occurrence of BSG structures (Barkat & Wheeler 1988; Saio et al. 1988).
Therefore, in the context of a stellar merger, the combination of the addition
of matter to the envelope and the dredge-up of part of the helium-rich core,
which increases the helium abundance in the evelope, but also further
decreases the core to total mass ratio, is particularly favourable for
producing BSGs (e.g. Hillebrandt & Meyer 1989; Podsiadlowski et al. 1990,
1992; Ivanova & Podsiadlowski 2002; Menon & Heger 2017; Podsiadlowski 2017a).
The more recent, systematic study of such mergers by Menon & Heger (2017) also
shows that the ultimate supernova explosions resulting from such mergers can
reproduce the properties of peculiar supernovae, such as SN 1987A, much better
than single BSG progenitors (Menon et al. 2019). Similarly, merger products
also better reproduce the observed population of BSGs (Bernini-Peron et al.
2023; Menon et al. 2023). Since we do not include these dredge-up effects,
this means that the accretion thresholds for BSG formation mentioned above are
upper limits when compared to actual stellar mergers. Interior mixing from
stellar mergers also affects Case A mergers, which is not accounted for in our
models either but, e.g., can further enhance complex morphologies in star
clusters above and around the MS turn-off region (see e.g. Wang et al. 2020).
In Case B and C accretors of binary mass transfer, such mixing is not expected
to occur, and the extra mixing and growth of the convective core in Case A
accretors is covered by our models. We show an overview of our accretor model
grid in Fig. 19 that illustrates and summarises which of our models evolve and
end their lives as CSGs and BSGs. For further, general considerations of
stellar structures that may spend time as a BSG, see e.g. Schootemeijer et al.
(2019), Farrell et al. (2022) and Sibony et al. (2023).

Figure 2: Kippenhahn diagrams of initially $13\,{\rm M}_{\odot}$ early Case B
systems accreting (a) 10% and (b) 100% of their initial mass. The moment of
mass accretion is indicated by an arrow. Energy generation by nuclear burning,
$\epsilon_{\mathrm{nuc}}$, and loss by neutrinos, $\epsilon_{\nu}$, are color-
coded. The right $y$-axes show the mass fractions of carbon, $X_{\mathrm{C}}$,
and oxygen, $X_{\mathrm{O}}$, in the center of the stars. Carbon starts being
produced when core helium burning begins. The grey hatched areas show
convective regions, purple hatchings are for convective overshooting, yellow
stands for thermohaline mixing and brown is for semi-convection. The blue
dotted lines show where the hydrogen mass fraction drops below 0.5, i.e. they
trace the helium core mass. Hydrogen shell burning in the
$f_{\mathrm{acc}}=0.1$ model is radiative and the star evolves into a CSG. In
the $f_{\mathrm{acc}}=1.0$ model, hydrogen shell burning proceeds convectively
and the star evolves as a BSG (c.f. Fig. 1).

The interior structures of the long-lived BSGs are characterised by hydrogen
shell burning driving a large convective region (Fig. 2). This occurs if the
accreted mass is so large that the temperature at the base of the hydrogen-
rich envelope becomes high enough to drive convection by large nuclear energy
generation from hydrogen shell burning (c.f. Sect. 4.1). For
$f_{\mathrm{acc}}=0.1$, the accretion modifies the temperature stratification
at the base of the hydrogen-rich envelope such that hydrogen shell burning is
pushed to lower mass coordinates but does not turn convective (Fig. 2a).
Contrarily, for $f_{\mathrm{acc}}=1.0$, shell burning becomes convective (Fig.
2b). The structure of such BSGs may be viewed as that of a (relatively)
unevolved MS star on top of a helium star. In this picture, the convective
hydrogen-burning shell corresponds to the core of the MS star. This analogous
picture also explains intuitively why the envelope of such BSGs remains
radiative and why the extent of the convective hydrogen-burning shell
decreases during core helium burning666Because of a continuously decreasing
electron-scattering opacity from converting hydrogen into helium, i.e. for the
same reason why the convective cores of MS stars recede during hydrogen
burning. (Fig. 2b).

To further illustrate the structural differences between stars evolving
through a long-lived BSG phase and those becoming CSGs, we compare in Fig. 3
the interior nuclear burning at the end of core oxygen burning of an initially
$20\,{\rm M}_{\odot}$ single star and two early Case B accretors with initial
masses of $20\,{\rm M}_{\odot}$ and $23\,{\rm M}_{\odot}$, and accretion
fractions of $f_{\mathrm{acc}}=0.1$ and $f_{\mathrm{acc}}=1.0$, respectively.
All models have similar CO core masses of $M_{\mathrm{CO}}=5.62\,{\rm
M}_{\odot}$, $5.64\,{\rm M}_{\odot}$ and $5.89\,{\rm M}_{\odot}$,
respectively, but evolve differently. In particular, the accretor model with
$f_{\mathrm{acc}}=1.0$ evolves as a long-lived BSG while the other two models
become CSGs. The largely different hydrogen-shell burning in the BSG
$f_{\mathrm{acc}}=1.0$ accretor is visible (c.f. Fig. 2) as well as markedly
different strengths, mass coordinates and extensions in mass of the nuclear
reaction rates due to carbon and neon burning (helium burning is also
different). Carbon and neon burning are particularly important for the final
core structures (see also Sect. 3.3). As is shown later, such differences can
distinguish between successful SN explosions or direct collapse into a BH.

Figure 3: TULIPS diagram (Laplace 2022) of the nuclear energy generation rate
$\epsilon_{\mathrm{nuc}}$ at the end of core oxygen burning (central oxygen
mass fraction $<10^{-4}$). Shown are the interiors of a single star of
$M_{\mathrm{ini}}=20\,{\rm M}_{\odot}$ (bottom), an early Case B accretor
model of $M_{\mathrm{ini}}=20\,{\rm M}_{\odot}$ and $f_{\mathrm{acc}}=0.1$
(top left), and an early Case B accretor model of $M_{\mathrm{ini}}=23\,{\rm
M}_{\odot}$ and $f_{\mathrm{acc}}=1.0$ (top right). The three models have a
similar CO core masses of $M_{\mathrm{CO}}=5.62\,{\rm M}_{\odot}$, $5.64\,{\rm
M}_{\odot}$ and $5.89\,{\rm M}_{\odot}$, respectively. The
$f_{\mathrm{acc}}=0.1$ accretor model is a CSG while the
$f_{\mathrm{acc}}=1.0$ model is a BSG, which can be recognized by the strong
hydrogen-shell burning that drives a thick convection zone (not indicated here
for clarity, but c.f. Figs. 2 and 21).

Whether Case A accretors can become long-lived BSGs depends on whether they
adjust their interior structure to the new mass, i.e. whether they rejuvenate.
If an accreting MS star fully rejuvenates, the convective core will have the
same size as that of a single star with the same mass. This means that also
the helium core masses at the terminal-age main sequence (TAMS) will be the
same and thus also the average helium mass fraction $\left<Y\right>$. The
relative deviation of $\left<Y\right>$ at the TAMS between our Case A
accretors and genuine single stars shows that early Case A accretors (almost)
fully rejuvenate for accretion fractions larger than ${\approx}\,10\%$ (Fig.
4a). The same is true for late Case A accretors for $f_{\mathrm{acc}}\gtrsim
25\%$ (Fig. 4b). These findings and in particular that late Case A accretors
need to accrete more mass than early Case A accretors to fully rejuvenate
agree qualitatively with the results of Braun & Langer (1995). As explained by
Braun & Langer, semi-convection regulates whether a convective core can grow
and mix through chemical gradients upon mass accretion. The chemical gradients
are steeper at a later MS phase such that more accretion is needed to allow
semi-convection to mix through the gradient and the convective core to grow to
a value found in genuine single stars of the same mass. Compared to the models
of Braun & Langer (1995), semi-convection with $\alpha_{\mathrm{sc}}=0.1$ is
much more efficient in our models such that most of them tend to fully
rejuvenate. According to Schootemeijer et al. (2019), semi-convection may even
be more efficient than assumed here such that all Case A accretors would fully
rejuvenate.

Figure 4: Rejuvenation of early (panel a) and late (panel b) Case A accretors.
The rejuvenation is quantified as the difference of the average helium mass
fraction $\left<Y\right>$ of the accretors at the terminal-age main sequence
(TAMS) relative to that of single stars. The relative differences are plotted
as a function of mass at the TAMS, $M_{\mathrm{TAMS}}$. Negative differences
mean that the accretors have burnt less hydrogen into helium compared to
single stars, i.e. that they have less massive convective cores. This is a
sign that accretors did not (fully) adjust their structure to the new total
mass after accretion, i.e. that they did not (fully) rejuvenate.

In our models, not all Case A accretors fully rejuvenate (Fig. 4). However,
this does not automatically mean that they can also burn helium in their cores
as BSGs. To evolve into long-lived BSGs, they must also reach conditions to
drive convective hydrogen shell burning. The initially $13\,{\rm M}_{\odot}$
late Case A accretor with $f_{\mathrm{acc}}=0.25$ does not fully rejuvenate
(Fig. 4b) and has accreted sufficient mass such that it becomes a BSG during
core helium burning (Fig. 17b). Contrarily, the $f_{\mathrm{acc}}=0.10$ model
also does not fully rejuvenate but accretes too little mass to develop
convective hydrogen shell burning (Fig. 17b).

###  3.2 Helium core, carbon-oxygen core and final stellar masses

In the evolution of stars, the core mass plays an important role. Here, we
concentrate on the helium and carbon-oxygen (CO) core masses $M_{\mathrm{He}}$
and $M_{\mathrm{CO}}$, respectively, that are mostly set by the extent of
convection during core hydrogen and helium burning. Both core masses can vary
during the evolution (e.g. due to shell burning or dredge-ups) and we measure
them at the onset of core collapse in this work. We define $M_{\mathrm{He}}$
as the mass coordinate where the hydrogen mass fraction $X_{\mathrm{H}}$ drops
below $<10^{-5}$ while $M_{\mathrm{CO}}$ is defined by the helium mass
fraction $X_{\mathrm{He}}$ becoming smaller than $0.5$.

Figure 5: CO core masses $M_{\mathrm{CO}}$ at core collapse as a function of
initial mass $M_{\mathrm{ini}}$ for Case A (panel a), Case B (panel b) and
Case C (panel c) accretors. Colours show the accretion fraction
$f_{\mathrm{acc}}$ of the models, and single stars are shown in blue squares
for comparison in panel (a). Panels (b) and (c) show the difference in
$M_{\mathrm{CO}}$ relative to the corresponding single-star core masses. To
guide the eye, data points for $f_{\mathrm{acc}}=0.1$ (mostly CSGs) and $2.0$
(BSGs) are connected by dashed lines and highlighted.

In Fig. 5, we show $M_{\mathrm{CO}}$ as a function of initial mass
$M_{\mathrm{ini}}$ for single stars and Case A, B and C accretors. The trends
for the helium core masses are qualitatively similar to those of the CO core
masses and are therefore not shown in the main text but only in Fig. 22. Most
of our Case A accretors fully adjust their interior structures to the new mass
and then continue the evolution as if they were born with that high mass, i.e.
most of them fully rejuvenate (see Fig. 4 and Sect. 3.1). This is exemplified
by the increasing helium and CO core masses at constant initial mass and
increasing accretion fractions. The rejuvenation can even be seen more
directly in Fig. 6, where we plot the masses of stars at core collapse,
$M_{\mathrm{final}}$, as a function of $M_{\mathrm{CO}}$. Case A accretors
mostly follow the trend of single stars, i.e. they produce the same core mass
for a given total final mass, except for the models that do not fully
rejuvenate and have higher final masses (c.f. Fig. 4). This deviation from the
single-star final masses is larger in the late than the early Case A models.
In particular, some non-rejuvenating, late Case A accretors (partly) burn
helium in their cores as BSGs and not CSGs (Braun & Langer 1995). In turn,
they avoid the stronger CSG wind mass loss and retain higher final masses at
core collapse.

Figure 6: Similar to Fig. 5 but showing the final masses of stars as a
function of the CO core mass $M_{\mathrm{CO}}$ at core collapse. In all three
panels, single-star models are plotted as blue squares. We further indicate
models that experience SN fallback by grey squares while crosses mark those
models that should have experienced enhanced LBV-like mass loss, which is not
accounted for in our models (see Sect. 2.1). The black dotted line is a one-
to-one relation.

Some Case A accretor models with $f_{\mathrm{acc}}\gtrsim 0.75$ at
$M_{\mathrm{CO}}>20\,{\rm M}_{\odot}$ appear to be offset to higher
$M_{\mathrm{CO}}$ relative to the single stars (Figs. 5a and 6a). These models
are so massive after accretion that their strong winds erode their envelopes
and the final mass is the same as the CO core mass. At
$M_{\mathrm{CO}}>10\,{\rm M}_{\odot}$, winds have removed most of the
hydrogen-rich envelope, explaining the almost linear relation of single stars
and Case A accretors in Fig. 6a.

Case B and C accretors cannot rejuvenate as is visible by the relative
difference of their helium and CO core masses in comparison to single stars
(Figs. 5b and 22b). We find that they differ at most by an order of
$\mathcal{O}(10\%)$. This immediately implies that – when comparing to single
stars – such accretors roughly keep their core masses constant while mass is
mostly added to their envelopes. One may thus expect that, to zeroth order,
the cores of such accretors evolve similarly to core collapse as those of the
corresponding single stars with the same initial mass. This expectation is met
in accretors of binary mass transfer but not in fully realistic stellar
mergers. In the latter, parts of the core are mixed into the envelope of the
merged star, which reduces the core mass. This effect is not covered by our
models (see Sect. 2.2.2), i.e. the core masses of our Case B and C accretors
are upper limits in comparison to what is expected from fully realistic
stellar mergers.

In Case B accretors, the CSG and BSG sequences can be distinguished in Fig. 5b
and 22b: models with $f_{\mathrm{acc}}=0.1$ are generally CSGs while those
with $f_{\mathrm{acc}}=2.0$ always become BSGs (see Fig. 19 for a detailed
overview of which minimum accretion fractions are required in our models to
turn accretors into BSGs). Post-accretion CSGs hardly deviate from the core
masses of the single stars while the BSGs deviate by a few tens of percent in
$M_{\mathrm{He}}$ and $M_{\mathrm{CO}}$; they follow the indicated trend for
$f_{\mathrm{acc}}=2.0$. For $M_{\mathrm{ini}}\lesssim 15\,{\rm M}_{\odot}$,
the Case B accretors have helium and CO core masses at core collapse that are
larger than those of single stars while they are smaller for
$M_{\mathrm{ini}}\gtrsim 15\,{\rm M}_{\odot}$777Because of overshooting, the
models studied here show no drastic second dredge-up. Thus, the larger
$M_{\mathrm{He}}$ and $M_{\mathrm{CO}}$ of the accretors compared to single
stars are not related to a second dredge-up in the single stars which is
naturally absent in BSG accretors without a deep convective envelope (c.f.
Podsiadlowski et al. 2004).. For all initial masses, the convective cores of
accretors _at the beginning of core helium burning_ are less massive than
those of single stars and the location of the hydrogen-burning shell is at
lower mass coordinates (i.e. smaller $M_{\mathrm{He}}$ at this time in
evolution as can be seen in Fig. 2). Because of the hydrogen shell burning,
the helium core and with it the convective core grow in mass. In Case B
accretors, the convective core grows relatively more in less massive stars.
For example, for $M_{\mathrm{ini}}\gtrsim 30\,{\rm M}_{\odot}$, the convective
core masses remain almost constant during core helium burning (Fig. 21) while
they grow significantly for $M_{\mathrm{ini}}\lesssim 15\,{\rm M}_{\odot}$
(Fig. 2).

In Case C accretors, there is not enough remaining time before core collapse
for hydrogen and helium shell burning to add significantly to
$M_{\mathrm{He}}$ and $M_{\mathrm{CO}}$. This is indeed borne out by the
helium core masses being almost unchanged (they change by an order of
$\mathcal{O}(1\%)$) regardless of whether stars evolve as CSGs or BSGs (Fig.
22c). The picture is seemingly different for the CO core masses at
$M_{\mathrm{ini}}\gtrsim 20\,{\rm M}_{\odot}$ that can exceed the single-star
$M_{\mathrm{CO}}$ by an order of $\mathcal{O}(10\%)$. This increase is not
caused by helium shell burning but by a brief revival of core helium
burning888Note that $X_{\mathrm{He}}=10^{-6}$ at the point of mass accretion
and core helium burning has almost ceased but is still ongoing. because of the
mass accretion (see also Sec. 3.3). The nuclear burning drives a transient
convective core that mixes in fresh helium fuel, thereby leading to more
massive $M_{\mathrm{CO}}$. For $M_{\mathrm{ini}}\lesssim 20\,{\rm M}_{\odot}$,
we do not find such a strong revival of core helium burning and
$M_{\mathrm{CO}}$ does not grow as much.

The final masses of rejuvenated Case A accretors closely follow those of
single stars of the same CO core mass, which is not the case in non-
rejuvenating Case A and most Case B accretors (Fig. 6). In the latter cases,
stars retain more mass and we find final stellar masses of up to
$100\text{--}200\,{\rm M}_{\odot}$ in our models. So while $M_{\mathrm{He}}$
and $M_{\mathrm{CO}}$ change at most by some tens of percent in comparison to
single stars, the final core-to-stellar mass ratios can be reduced by up to an
order of magnitude: Case AB accretors have comparably small cores for their
total stellar mass.

As mentioned in Sect. 2, our models do not account for LBV-like mass loss.
Even disregarding all models that likely experience significant LBV mass loss
(marked by crosses in Fig. 6), Case B accretors have the largest final stellar
masses, followed by Case C and Case A accretors (and single stars). Case C
accretors generally have smaller final masses than Case B accretors for the
following reasons. Firstly, when accretion starts, Case C models have been
subject to strong wind mass loss on the RGB branch while Case B accretors can
avoid this entirely as BSGs and thus reach larger final masses. Secondly, even
if Case B accretors do not evolve through a long-lived BSG phase but also
become CSGs, they have smaller helium and CO core masses than Case C accretors
and single stars. As we will show below, the luminosity of CSGs scales with
their core mass such that Case C accretors and single stars have stronger wind
mass loss and smaller final masses because of their higher luminosities. We
will further discuss the final stellar masses in relation to BH and SN-ejecta
masses in Sect. 4.2.

###  3.3 Central carbon abundance after core helium burning

As explained above, the central carbon mass fraction $X_{\mathrm{C}}$ after
core helium burning is an important parameter that sets the initial conditions
for the later nuclear-burning stages and thereby the pre-SN core structure. It
is determined by helium burning that proceeds in two steps. First, the
triple-$\alpha$ reaction produces ${}^{12}\mathrm{C}$. Second, once
$\alpha$-particles become rarer and ${}^{12}\mathrm{C}$ nuclei have been
formed, the ${}^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ nuclear reaction
eventually becomes dominant and consumes ${}^{12}\mathrm{C}$ to its final
abundance $X_{\mathrm{C}}$ (see e.g. Langer 1989; Woosley et al. 1993; Brown
et al. 2001; Chieffi & Limongi 2020; Sukhbold & Adams 2020; Laplace et al.
2021, and Figs. 2 and 21). How much carbon is depleted depends on the central
density999The triple-$\alpha$ reaction is proportional to the third power of
the $\alpha$-particle number density while the
${}^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ reaction scales linearly
with this density. More massive stars have denser cores such that the latter
reaction becomes dominant earlier during helium burning than in lower-mass
stars, and more carbon is depleted and converted into oxygen., i.e. the (core)
mass of the star, and how much the convective core grows during helium
burning, which can mix in fresh $\alpha$-particles that are then captured onto
${}^{12}\mathrm{C}$ to produce ${}^{16}\mathrm{O}$.

Figure 7: Similar to Fig. 5 but showing the central carbon mass fraction at
the end of core helium burning $X_{\mathrm{C}}$ as a function of the CO core
mass $M_{\mathrm{CO}}$.

We show the carbon mass fraction $X_{\mathrm{C}}$ at the end of core helium
burning of our models as a function of the CO core mass in Fig. 7 (a similar
diagram as a function of the initial mass can be found in Fig. 23). As
discussed in Sect. 3.2, most Case A accretors fully rejuvenate and adjust
their interior structure to the new total mass such that they have the same
$X_{\mathrm{C}}$ as in single stars of the same $M_{\mathrm{CO}}$ (Fig. 7a).
We also find again that those Case A accretors, which spend most of their time
burning helium in their cores as BSGs, deviate from the single-star trend and
deplete relatively more ${}^{12}\mathrm{C}$. The reason is the same as in Case
B accretors discussed next.

In Case B accretors, we find that the CSG and BSG sequences differ
considerably in $X_{\mathrm{C}}$ (Fig. 7b). The post-accretion BSG models
roughly follow the $f_{\mathrm{acc}}=2.0$ models and end core helium burning
with less carbon than single stars while the CSG models follow the single-star
trends. We find that Case B accretors with $M_{\mathrm{CO}}\lesssim 15\,{\rm
M}_{\odot}$ ($M_{\mathrm{ini}}\lesssim 40\,{\rm M}_{\odot}$) can have up to
80% less carbon in their cores than single stars while this is ${<}\,10\%$ for
$M_{\mathrm{CO}}\gtrsim 15\,{\rm M}_{\odot}$.

The differences are due to different growths of the convective cores during
helium burning. At $M_{\mathrm{ini}}=13\,{\rm M}_{\odot}$ (Fig. 2), the
convective core driven by helium burning still grows considerably more in mass
in the $f_{\mathrm{acc}}=1.0$ model after reaching a maximum carbon mass
fraction of ${\approx}\,0.55$ than in the $f_{\mathrm{acc}}=0.1$ model. The
larger core growth during this phase mixes more $\alpha$-particles into the
convective core and the ${}^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$
nuclear reaction thus burns more ${}^{12}\mathrm{C}$ into ${}^{16}\mathrm{O}$.
The $f_{\mathrm{acc}}=1.0$ model then ends core helium burning with
$X_{\mathrm{C}}\approx 0.13$ while the $f_{\mathrm{acc}}=0.1$ model has a
larger abundance of $X_{\mathrm{C}}\approx 0.30$ (Fig. 2). For a higher
initial mass of $30\,{\rm M}_{\odot}$, the convective cores in the
$f_{\mathrm{acc}}=0.1$ and $f_{\mathrm{acc}}=1.0$ models grow similarly after
reaching the maximum core carbon mass fraction of ${\approx}\,0.55$ (Fig. 21).
Consequently, similar amounts of $\alpha$-particles are mixed into the
convectively helium-burning cores such that both models reach similar final
core carbon mass fractions of about $0.2$.

The carbon mass fractions $X_{\mathrm{C}}$ of the $f_{\mathrm{acc}}=2.0$ early
Case B accretors never reach the same values in single stars as a function of
the CO core mass (Fig. 7b) while they are the same as in single stars as a
function of initial mass for $M_{\mathrm{ini}}\,{\gtrsim}\,35\,{\rm
M}_{\odot}$ (Fig. 23b). At first sight, this may sound contradictory but it is
not. The accretor models follow a different
$M_{\mathrm{CO}}$–$M_{\mathrm{ini}}$ relation than single stars. For
$M_{\mathrm{ini}}>15\,{\rm M}_{\odot}$, they have ${\approx}\,20\%$ smaller CO
cores than single stars of the same initial mass (Fig. 5b). Generally,
$X_{\mathrm{C}}$ drops with increasing $M_{\mathrm{CO}}$ and
$M_{\mathrm{ini}}$ (Figs. 7 and 23). At the same initial mass, both single
stars and Case B accretors have the same $X_{\mathrm{C}}$. Translating into a
$X_{\mathrm{C}}$–$M_{\mathrm{CO}}$ relation, the $X_{\mathrm{C}}$ of the
accretor models are then compared to those of single stars of
${\approx}\,20\%$ smaller $M_{\mathrm{CO}}$, i.e. higher $X_{\mathrm{C}}$.

In Case C accretors, we find $X_{\mathrm{C}}$ values reduced by
${\approx}\,70\%$ for $7\lesssim M_{\mathrm{CO}}/{\rm M}_{\odot}\lesssim 25$
and $f_{\mathrm{acc}}\gtrsim 0.1$ (Fig. 7c). In these models, accretion
increases the star’s central temperature. Helium burning is revived and drives
a transient convective core that mixes in fresh helium fuel. As
${}^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ is the dominant reaction in
these models at this moment of evolution, significantly more carbon is
converted into oxygen compared to the equivalent single-star models. Once core
helium burning ceases entirely, $X_{\mathrm{C}}$ is reduced to 4–6%. In the
other models with different $M_{\mathrm{CO}}$, this mechanism does not operate
and $X_{\mathrm{C}}$ remains close to that found in the corresponding single-
star models.

As explained above, fully realistic Case B and C stellar mergers have smaller
cores than in our accretion models. The same mechanism of mixing fresh helium
into the helium-burning core will also operate in such stars. Hence, we expect
that also such stars show different $X_{\mathrm{C}}$ branches depending on
whether the merged star becomes a long-lived BSG or evolves into a CSG.

###  3.4 Pre-SN stellar structure

The different starting conditions for the later nuclear-burning stages, i.e.
$M_{\mathrm{CO}}$ and $X_{\mathrm{C}}$ (see Sects. 3.2 and 3.3), can lead to
inherently different core structures during the advanced burning stages and at
core collapse. We summarise the pre-SN stellar structures of our models by
their iron core mass, $M_{\mathrm{Fe}}$, the central specific entropy,
$s_{\mathrm{c}}$, and the so-called compactness parameter,

$\displaystyle\xi_{M}=\frac{M/{\rm M}_{\odot}}{R(M)/1000\,\mathrm{km}},$ (2)

which is the ratio of a mass coordinate $M$ and the radius at this location
$R(M)$ (O’Connor & Ott 2011). Usually, $M=2.5\,{\rm M}_{\odot}$, and
$\xi_{2.5}{\gtrsim}\,0.45$ indicates that stars are unable to explode in SNe
and form BHs (e.g. O’Connor & Ott 2011; Ugliano et al. 2012; Sukhbold &
Woosley 2014; Ertl et al. 2016; Sukhbold et al. 2016; Müller et al. 2016;
Chieffi & Limongi 2020; Patton & Sukhbold 2020; Schneider et al. 2021, 2023).
$M_{\mathrm{Fe}}$, $s_{\mathrm{c}}$ and $\xi_{\mathrm{2.5}}$ are closely
related and they provide similar insights into the pre-SN structure (see also
Fryer 2014; Schneider et al. 2021; Takahashi et al. 2023; Temaj et al. 2024).
For example, the effective Chandrasekhar mass that an iron core approaches
when it starts to collapse is a function of the electron fraction
$Y_{\mathrm{e}}$ and electronic entropy $s_{\mathrm{e,c}}$ (the average
electronic entropy is roughly
$\left<s_{\mathrm{e,c}}\right>\approx\left<s_{\mathrm{c}}\right>/3$ but
depends on each case; see e.g. Baron & Cooperstein 1990; Timmes et al. 1996),

$\displaystyle M_{\mathrm{Ch}}\approx 1.457\,{\rm
M}_{\odot}\,\left(\frac{Y_{\mathrm{e}}}{0.5}\right)^{2}\left[1+\left(\frac{s_{\mathrm{e,c}}}{\pi
Y_{\mathrm{e}}k_{\mathrm{B}}N_{\mathrm{A}}}\right)^{2}\right].$ (3)

This relation directly links $s_{\mathrm{c}}$ to $M_{\mathrm{Fe}}$ at core
collapse and $s_{\mathrm{c}}$ is further a measure of the mass-radius relation
of the iron core, i.e. it relates to $\xi_{\mathrm{2.5}}$ (Schneider et al.
2021). Also, the gravitational binding energy of the material exterior of the
iron core could have been used as a summary proxy of the structure of stars at
core collapse (e.g. Sukhbold & Woosley 2014; Temaj et al. 2024). All of these
quantities follow similar trends such as peaks and troughs at the same
$M_{\mathrm{ini}}$ and $M_{\mathrm{CO}}$.

Figure 8: Core compactness $\xi_{2.5}$ (panels a–c), iron core mass
$M_{\mathrm{Fe}}$ (panels d–f) and central specific entropy $s_{\mathrm{c}}$
(panels g–i) as a function of CO core mass $M_{\mathrm{CO}}$ at core collapse.
As in other figures, we distinguish between Case A, B and C accretors and
color-code the mass accretion fraction $f_{\mathrm{acc}}$. Accretion fractions
of 25%, 75% and 200% are connected by dashed lines to better visualize trends
and systematic shifts.

The evolution of massive stars beyond core helium burning is driven by thermal
neutrino losses and the delicate balance between these losses, energy gain
from nuclear burning, and energy release and consumption from changes in the
gravitational potential of stars. For example, the central carbon abundance
$X_{\mathrm{C}}$ at the end of core helium burning generally decreases with
increasing $M_{\mathrm{CO}}$ (Fig. 7). With fewer carbon atoms available for
carbon burning, there is a critical $M_{\mathrm{CO}}$ beyond which energy loss
from neutrinos overcomes energy generation from core carbon burning.
Consequently, the carbon burning proceeds no longer under convective but
radiative conditions, the evolution speeds up and the stellar cores at core-
collapse become more compact, reach higher $M_{\mathrm{Fe}}$ and have larger
entropy. With further increasing $M_{\mathrm{CO}}$, the next nuclear burning
stages ignite more quickly after the beginning of core carbon burning, become
more vigorous and reduce $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$
at core collapse. For yet higher $M_{\mathrm{CO}}$, also neon burning becomes
neutrino dominated, turns radiative and again leads to increased $\xi_{2.5}$,
$M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$ at core collapse (e.g. Patton &
Sukhbold 2020; Sukhbold & Adams 2020; Schneider et al. 2021, 2023; Laplace et
al. 2023).

This sequence of changes in the nuclear burning conditions in the deep
interior of stars leads to peaks and troughs in $\xi_{2.5}$, $M_{\mathrm{Fe}}$
and $s_{\mathrm{c}}$ at core collapse with $M_{\mathrm{CO}}$ (Fig. 8). In our
single-star models, a peak in $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and
$s_{\mathrm{c}}$ is found at $M_{\mathrm{CO}}\approx 7\,{\rm M}_{\odot}$ and
they increase and reach generally high values for $M_{\mathrm{CO}}\gtrsim
13\,{\rm M}_{\odot}$. We find an indistinguishable $\xi_{2.5}$,
$M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$ landscape with $M_{\mathrm{CO}}$
compared to single stars in our Case A accretors (Fig. 8a, d, g); even in the
non-rejuvenated Case A systems, there is no obviously deviating trend. We
attribute the latter to the fact that also the non-rejuvenated Case A
accretors show the same relation between $X_{\mathrm{C}}$ and
$M_{\mathrm{CO}}$ as the single stars (Fig. 7a).

In the Case B accretors, the $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and
$s_{\mathrm{c}}$ landscapes are qualitatively similar to those of single stars
but we find a systematic shift of the peaks and troughs to lower
$M_{\mathrm{CO}}$ (Fig. 8b, e, h). The magnitude of the shift depends on the
amount of accreted mass. For $f_{\mathrm{acc}}=2.0$, the compactness peak is
shifted by about $0.5\,{\rm M}_{\odot}$ to a lower $M_{\mathrm{CO}}$ of
${\approx}\,6.5\,{\rm M}_{\odot}$ and the second major increase of compactness
is shifted by about $1\,{\rm M}_{\odot}$ to
$M_{\mathrm{CO}}{\approx}\,12\,{\rm M}_{\odot}$. In these models, the carbon
abundance at the end of core helium burning is reduced (Fig. 7) such that core
carbon and core neon burning become neutrino dominated at slightly lower
$M_{\mathrm{CO}}$. This finding is analogous but in the opposite direction to
binary-stripped stars where the removal of the hydrogen-rich envelope results
in a larger $X_{\mathrm{C}}$ at the end of core helium burning and thus in a
systematic shift of the compactness landscape to higher $M_{\mathrm{CO}}$
(Woosley 2019; Schneider et al. 2021, 2023; Laplace et al. 2021). In the
example in Fig. 3, the accretor model with $f_{\mathrm{acc}}=1.0$ falls into
the shifted compactness peak for such models ($\xi=0.63$) while the single
star and $f_{\mathrm{acc}}=0.1$ accretor have rather low compactness
parameters of $0.24$ and $0.12$, respectively. This illustrates further that
accretor models can achieve inherently different core structures compared to
single stars despite similar CO core masses.

The relation of $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$ as a
function of $M_{\mathrm{CO}}$ in Case C accretors is qualitatively different
from those of the single stars and Case AB accretors at
$M_{\mathrm{CO}}\gtrsim 5\,{\rm M}_{\odot}$ (Fig. 8c, f, i). The much reduced
$X_{\mathrm{C}}$ in these models (Fig. 7c) results in a shortened core-carbon-
burning phase and generally larger $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and
$s_{\mathrm{c}}$ at $M_{\mathrm{CO}}\gtrsim 5\,{\rm M}_{\odot}$ compared to
the single-star models. The first compactness peak occurs at lower
$M_{\mathrm{CO}}$ and is broadened with respect to that of the single stars
and we find a dependence on the amount of accreted mass. The compactness peak
at $M_{\mathrm{CO}}\approx 7\,{\rm M}_{\odot}$ is broadened the most for
$f_{\mathrm{acc}}=0.25$ while it is broadened to a lesser extent for higher
accretion fractions. Because of this, the second major increase in compactness
is pushed to $M_{\mathrm{CO}}\approx 18\,{\rm M}_{\odot}$ in the
$f_{\mathrm{acc}}=0.25$ models and to $M_{\mathrm{CO}}\approx 14\,{\rm
M}_{\odot}$ for $f_{\mathrm{acc}}=2.0$. The $f_{\mathrm{acc}}=0.1$ models,
where we found no clear trend in the reduction of $X_{\mathrm{C}}$ (Fig. 7c),
also seem to show no clear trend in $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and
$s_{\mathrm{c}}$.

The mixing of core material into the envelope in fully realistic stellar
mergers may affect both the envelope and the pre-SN core structures. However,
the qualitative trends of $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$
with $M_{\mathrm{CO}}$ are expected to be similar. We expect fully realistic
mergers that end their lives as CSGs (BSGs) to follow similar relations of
$\xi_{2.5}$, $M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$ with $M_{\mathrm{CO}}$ as
is found in pre-SN CSGs (BSGs) from our accretors. However, the mapping of
$M_{\mathrm{CO}}$ to $M_{\mathrm{ini}}$ will differ and, as explained above,
which $f_{\mathrm{acc}}$ thresholds result in long-lived BSGs from Case B and
C stellar mergers and hence which mergers correspond to the most strongly
shifted $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$ curves.

##  4 Final fate and SN explosions

Applying the SN code of Müller et al. (2016) to our models at core-collapse
(Sect. 2), we find that a compactness threshold of $\xi_{2.5}\,{<}\,0.44$ for
successful SN explosions agrees with the model of Müller et al. (2016) in more
than 95% of cases. Similarly, the two-parameter explodability criterion of
Ertl et al. (2016), $\mu_{4}<k_{1}M_{4}\mu_{4}+k_{2}$ with constants
$k_{1}=0.200$ and $k_{2}=0.072$ (see Schneider et al. 2023), agrees in 89% of
cases. In the Ertl et al. criterion, $\mu_{4}=\mathrm{d}m/\mathrm{d}r$ is the
radial mass derivative in units of ${\rm M}_{\odot}/1000\,\mathrm{km}$ and
$M_{4}$ the mass in ${\rm M}_{\odot}$ at a specific entropy of $s=4$. In
particular, all explodability criteria predict BH formation in the compactness
peak at $M_{\mathrm{CO}}\,{\approx}\,7\,{\rm M}_{\odot}$ and generally also
for high compactness at $M_{\mathrm{CO}}\,{\gtrsim}\,13\,{\rm M}_{\odot}$. The
$\xi_{2.5}$-landscape found in this work is thus a good proxy for whether
stars may explode in neutrino-driven SNe101010Similarly, one can define
explodability criteria based on $M_{\mathrm{Fe}}$, $s_{\mathrm{c}}$ and the
gravitational binding energy outside the iron core. (see also the review by
Heger et al. 2023). This agrees with the expectation that stars with high
compactness, large iron core mass and high gravitational binding energy
exterior to the iron core are more prone to collapsing into BHs (see e.g.
Sukhbold & Woosley 2014; Heger et al. 2023). Note, however, that there are
core-collapse simulations that show shock revival in stars with high core
compactness (see e.g. Ott et al. 2018; Burrows et al. 2020; Chan et al. 2018,
2020; Kuroda et al. 2018; Couch et al. 2020; Powell & Müller 2020), but this
does not immediately imply a successful SN explosion (Heger et al. 2023).

We first present the pre-SN locations of our models in the HR diagram in Sect.
4.1 and then the resulting compact remnant and SN ejecta masses using the SN
model of Müller et al. (2016) in Sect. 4.2.

###  4.1 Pre-SN locations in the HR diagram

We show the pre-SN locations of all our models in the HR diagram in Fig. 9.
Because we do not apply LBV-like mass loss, our models at $\log\,L/{\rm
L}_{\odot}\,{\gtrsim}\,5.5$ do not lose large fractions of their envelopes and
do not become WR stars. Moreover, we do not model stars with initial masses
${<}\,13\,{\rm M}_{\odot}$, which explains the lowest pre-SN luminosity of
$\log\,L/{\rm L}_{\odot}\,{\approx}\,4.7$.

Figure 9: Pre-SN locations of the Case A, B and C accretors in the HR diagram.
The dotted line at $\log\,L/{\rm L}_{\odot}=5.5$ (Davies et al. 2018) is the
maximum luminosity of RSGs and the grey shaded regions indicate the hot and
cool parts of the S Doradus instability strip. The dashed line is the
Humphreys–Davidson limit (Humphreys & Davidson 1979). The dark-grey boxes
indicate the locations of the RSG N6946-BH that likely collapsed into a BH
without a bright SN (Gerke et al. 2015; Adams et al. 2017b, a; Basinger et al.
2021), and the BSG progenitor of SN 1987A (Woosley 1988). The grey
evolutionary tracks are single-star models with the masses given by the
labels.

The single-star models reach core collapse at the same sites in the HR diagram
as the Case A accretors, and are not shown for clarity (Fig. 9a). Both reach
core collapse along a well-defined branch in the HR diagram. Moreover, also
those Case B and C accretors that have no thick hydrogen-burning shell at
core-collapse follow the same branch. The cores of these pre-SN stars after
core helium burning evolve independently from the envelope because of a drop
in pressure by many orders of magnitude at the core–envelope boundary. The
photon luminosity is produced by nuclear burning which takes place in and near
the core, i.e. these stars follow a core-mass–luminosity relation (Fig. 10a).
The long-lived BSGs with their thick hydrogen-burning shell do not follow such
a relation as shown below. For the core mass, we take $M_{\mathrm{CO}}$ at
core collapse but using $M_{\mathrm{He}}$ gives a similar result. For
comparison, we show the fit function $\log\,L/{\rm
L}_{\odot}=4.372+1.268\log\,M_{\mathrm{CO}}/{\rm M}_{\odot}$ from Temaj et al.
(2024) in Fig. 10a. This function has been obtained by fitting the pre-SN
luminosities of single stars with different convective core overshooting
parameters. It also represents well the pre-SN luminosities of Case A, B and C
accretors that do not explode as BSGs. The luminosities of the CSG Case C and
B accretors fit well up to $M_{\mathrm{CO}}\,{\approx}\,10\,{\rm M}_{\odot}$
and for larger $M_{\mathrm{CO}}$ the slope of the relation steepens slightly
from $1.268$ to ${\approx}\,1.350$ (Fig. 10a).

Figure 10: Core-mass–luminosity relation (panel a) and the relationship
between the effective temperature and the average hydrogen mass fraction in
the envelope (panel b) of single stars and Case A, B and C accretors at core
collapse. In the core-mass–luminosity relation, only CSG Case B and C
accretors are shown as the BSG accretors with their thick hydrogen-burning
shell follow a different mass-luminosity relation (see main text and Fig. 11).
The black dashed curve in panel (a) is the fit of the core-mass–luminosity
relation from Temaj et al. (2024).

The effective temperature of CSGs (i.e. the Hayashi line) is set by opacity
and also the convective mixing-length efficiency, $\alpha_{\mathrm{MLT}}$.
With an increasing mass of stars, more and more of the hydrogen-rich envelope
is lost, and stars become hotter and appear as BSGs until they are eventually
hot helium stars. We find a close relationship between the effective
temperature of single stars and Case A, B and C accretors at core collapse and
the average hydrogen mass fraction in their envelope,
$\left<X_{\mathrm{env}}\right>_{\mathrm{cc}}$ at effective temperatures larger
than those of CSGs (i.e.
$\log\,T_{\mathrm{eff,cc}}/\mathrm{K}\,{\gtrsim}\,3.9$) and for
$\left<X_{\mathrm{env}}\right>_{\mathrm{cc}}\lesssim 0.5$ (Fig. 10b). As
expected, stars with lower hydrogen content in their envelopes have a hotter
effective temperature. Together with the core-mass–luminosity relation, this
explains the location of pre-SN single stars, the Case A accretors and some
Case B and C accretors in the HR diagram. The effective temperatures of BSGs
with a thick hydrogen-burning shell and stars on the Hayashi line do not
depend on $\left<X_{\mathrm{env}}\right>_{\mathrm{cc}}$.

Our models develop high compactness at $M_{\mathrm{CO}}\,{\approx}\,7\,{\rm
M}_{\odot}$ and $M_{\mathrm{CO}}\,{\gtrsim}\,13\,{\rm M}_{\odot}$ (Fig. 8a, d)
that results in BH formation (see also Sect. 4.2). Because of the core mass-
luminosity relation, this translates into luminosities of $\log\,L/{\rm
L}_{\odot}\,{\approx}\,5.4$ and $\log\,L/{\rm L}_{\odot}\,{\gtrsim}\,5.8$
where BHs are formed by stars that avoided the aforementioned BSG phase caused
by the thick convective hydrogen burning shell. To date, N6946-BH is the only
star known that may have collapsed into a BH without giving rise to a
SN111111After submission of this work, new JWST observations revealed an
infrared source at the location of N6946-BH that, e.g., could be the dust
enshrouded missing RSG or dust debris from a failed SN (Beasor et al. 2023;
Kochanek et al. 2024). Hence, only future observations will finally clarify
whether this RSG indeed collapsed into a BH or not. (Gerke et al. 2015; Adams
et al. 2017b, a; Basinger et al. 2021). Interestingly, its location in the HR
diagram seems to agree with where we predict BH formation from the compactness
peak (Fig. 9; see also Sukhbold & Adams 2020). Progenitors of SN IIP are
observed up to luminosities of $\log\,L/{\rm L}_{\odot}\,{\approx}\,5.1$
(Smartt 2015), but these luminosities remain uncertain (see e.g. Davies et al.
2018, Davies & Beasor 2020, Temaj et al. 2024 and also the vastly different
luminosities of $\log\,L/{\rm L}_{\odot}\,{=}\,4.74\pm 0.07$ and $\log\,L/{\rm
L}_{\odot}\,{=}\,5.1\pm 0.2$ inferred for the progenitor of SN 2023ixf by
Kilpatrick et al. 2023 and Jencson et al. 2023, respectively).

The long-lived BSG structures in some of the Case B and C accretors are
characterised by a thick convective shell driven by hydrogen burning and we
describe it in Sect. 3.1 as if a main sequence star with a convective core was
added on top of a helium star. The surface photon luminosity of these stars is
primarily given by the luminosity produced by hydrogen shell burning such that
they follow an envelope mass-luminosity relation and no longer a core-
mass–luminosity relation (Fig. 11a). The mass-luminosity exponent is larger at
low ($M_{\mathrm{env,cc}}\lesssim 30\,{\rm M}_{\odot}$) and smaller at high
envelope masses ($M_{\mathrm{env,cc}}\lesssim 80\,{\rm M}_{\odot}$) than the
value of $1.60$ found by fitting the entire mass range.

Figure 11: Envelope mass-luminosity relation (panel a) and H-envelope base
temperature-effective temperature relation (panel b) for Case B accretors.
Single stars are shown for comparison and models with an average hydrogen mass
fraction in their envelopes of
$\left<X_{\mathrm{env}}\right>_{\mathrm{cc}}<0.4$ are marked. These stars may
appear as BSGs at core collapse but their hot $T_{\mathrm{eff,cc}}$ is due to
the low hydrogen content in their envelopes and not due to a thick convective
hydrogen burning shell. Power-law fits are added and the dotted lines in panel
(b) indicate $\log\,T_{\mathrm{eff,cc}}/\mathrm{K}=3.9$ and the transition
base temperature at $\mu_{\mathrm{env,cc}}M_{\mathrm{env,cc}}/R_{*}\approx
0.02\,{\rm M}_{\odot}\,{\rm R}_{\odot}^{-1}$ where we find the BSG solutions.
Case B-e and B-l refer to early and late Case B accretors, respectively.

Convection during hydrogen burning occurs if the produced energy is so large
that it cannot be transported away by photons (Kippenhahn & Weigert 1994). In
general, energy generation by nuclear burning has a very strong temperature
sensitivity and this tends to increase for isotopes with higher nucleon
numbers121212In hydrogen burning via the CNO cycle, the energy generation rate
$\epsilon_{\mathrm{CNO}}$ scales with $T^{13\ldots 23}$ for $T=10\ldots
50\times 10^{6}\,\mathrm{K}$ (Kippenhahn & Weigert 1994).. Convective hydrogen
shell burning is thus expected to happen if the temperature at the base of the
envelope exceeds a certain threshold. From homology relations of main sequence
stars, one finds that the central temperature scales as
$T_{\mathrm{c}}\propto\mu M/R$, where $\mu$ is the mean molecular weight, $M$
the total mass of a star and $R$ the radius. Within this analogy, the
temperature at the base of the hydrogen-rich envelope then scales as
$T_{\mathrm{base}}\propto\mu_{\mathrm{env}}M_{\mathrm{env}}/R_{\mathrm{env}}$.
The helium core of such stars has a negligible radius ($\mathcal{O}(1\,{\rm
R}_{\odot})$) compared to the envelope ($\mathcal{O}(10\text{--}100\,{\rm
R}_{\odot})$) such that $R_{\mathrm{env}}$ can be approximated by the radius
of the entire star, $R_{*}$. Indeed, the pre-SN stars with thick convective
hydrogen shell burning follow a close relation in the
$T_{\mathrm{eff}}$–$T_{\mathrm{base}}$ diagram (Fig. 11b). Beyond the critical
value of

$\displaystyle
T_{\mathrm{base}}\propto\frac{\mu_{\mathrm{env,cc}}M_{\mathrm{env,cc}}}{R_{*}}\approx
0.02\,{\rm M}_{\odot}\,{\rm R}_{\odot}^{-1},$ (4)

stars have a hot enough $T_{\mathrm{base}}$ such that hydrogen shell burning
is convective and the stars reach core-collapse as BSGs with effective
temperatures exceeding $\log\,T_{\mathrm{eff,cc}}/\mathrm{K}\,{\approx}3.9$.

We also find high compactness in Case B and Case C accretors for
$M_{\mathrm{CO}}\,{\approx}\,7\,{\rm M}_{\odot}$ and
$M_{\mathrm{CO}}\,{\gtrsim}\,13\,{\rm M}_{\odot}$ that leads to BH formation
and no SN explosion (Figs. 8 and 12). In the HR diagram, BH formation occurs
close to the hot side of the S Doradus instability strip because of the
compactness peak at $M_{\mathrm{CO}}\,{\approx}\,7\,{\rm M}_{\odot}$ and at
cooler effective temperatures and larger luminosities because of high
compactness at $M_{\mathrm{CO}}\,{\gtrsim}\,13\,{\rm M}_{\odot}$ (Fig. 9b, c).
Hence, some of our accretor models may explode in SNe while appearing as LBVs
and they may have experienced significant mass loss before their SN. Other
models in a very similar location in the HR diagram may just collapse into BHs
but also these stars could have experienced significant mass loss before their
collapse. Accretor models that are too hot to enter the LBV regime in the HR
diagram are all predicted to explode in SNe.

While our models generally predict exploding BSGs in a region of the BSG
progenitor of SN 1987A, a Case C model is required to explain this particular
object (see e.g. Podsiadlowski 1992, 2017b; Menon & Heger 2017; Urushibata et
al. 2018). None of our current Case C accretors reproduce its location in the
HR diagram (Fig. 9c), but these models do not dredge up core material and mix
it throughout the envelope (see Sect. 2.2). Models including such mixing can
explain the BSG location and further observed characteristics (see discussion
in Sect. 5). Moreover, all our long-lived BSG accretors are likely
systematically biased to too-cold effective temperatures compared to what is
expected from fully realistic mergers because of this missing mixing of core
material into the envelope.

###  4.2 Compact-remnant and SN-ejecta masses

We show the gravitational NS and BH masses in Fig. 12. The NS masses appear
uncorrelated with the amount of accreted mass and the mass-accretion Cases A,
B and C. Generally, the NS masses are on average more massive in stars with
larger CO core masses, and the largest NS masses are found at CO core masses
close to regions of the highest compactness (Fig. 12d, e, f), where the models
are on the brink of not to result in a successful neutrino-driven explosion.

Figure 12: Black-hole and neutron-star masses of Case A (panels a, d), Case B
(panels b, e) and Case C (panels d, f) accretors as a function of the final CO
core mass $M_{\mathrm{CO}}$. The bottom row is a zoom into the NS-mass regime.
The NS and BH masses of the single-star models are shown for comparison and
accretor models with $f_{\mathrm{acc}}\,{=}\,0.25$, $0.75$ and $2.00$ are
connected by lines to guide the eye. Systems likely experiencing SN fallback
are marked by grey boxes and models that should have experienced LBV-like mass
loss are indicated by crosses (c.f. Sect. 2.1).

As shown by Temaj et al. (2024) for single stars with different degrees of
convective boundary mixing, the NS masses follow a tight relation with the
central specific entropy $s_{\mathrm{c}}$ of the SN progenitors. We find a
similar relation in our models and the exponential best fit of Temaj et al.,

$\displaystyle M_{\mathrm{NS,grav}}/{\rm
M}_{\odot}=0.048\,\exp\left(2.76\,s_{\mathrm{c}}/N_{\mathrm{A}}\,k_{\mathrm{B}}\right)+0.880,$
(5)

also fits our data well despite the different evolutionary histories of their
and our models (Fig. 13). In particular, the neutron star mass–entropy
relation is the same in our Case A, B and C accretors. Both studies apply the
same SN code of Müller et al. (2016), where shock reversal and hence SN
explosions are triggered outside the iron core near a mass coordinate of
$M_{4}$, i.e. within the oxygen shell (c.f. figure A.1 in Schneider et al.
2021). Hence, the resulting NS masses are directly proportional to $M_{4}$ and
thus $M_{\mathrm{Fe}}$. Because $M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$ are
closely related (see Eq. 3 and Fig. 8), it is not surprising that we find a
strong correlation between $M_{\mathrm{NS,grav}}$ and $s_{\mathrm{c}}$, but it
could not be foreseen that this relation is so tight and does not carry
information on the evolutionary history of the stars (Fig. 13). It thus seems
that $s_{\mathrm{c}}$ is a good indicator of that mass coordinate at which
neutrino-driven SNe within the model of Müller et al. (2016) are launched.
Whether such a connection is more general and can also be made in other
neutrino-driven SN models remains unclear and warrants further investigation.

Figure 13: Gravitational neutron star mass $M_{\mathrm{NS,grav}}$ as a
function of central specific entropy $s_{\mathrm{c}}$ of the progenitor stars
for Case A (panel a), Case B (panel b) and Case C (panel c) accretion. The
black dashed line is the exponential fit to the neutron star masses of single
stars computed for different convective boundary mixing found by Temaj et al.
(2024). Figure 14: Same as Figs. 6 and 12 but showing the SN ejecta mass
$M_{\mathrm{ej}}$. Models marked by crosses (i.e. also those with the highest
SN ejecta mass) are expected to have experienced LBV-like mass loss such that
these ejecta masses are upper limits.

The BH masses (Fig. 12a, b, c) are closely linked to the final stellar masses
(Fig. 6), i.e. to the mass loss history of stars. Because of rejuvenation,
most Case A accretors evolve as single stars of higher initial masses and thus
collapse to BHs of similar mass for the same $M_{\mathrm{CO}}$. Only the non-
rejuvenating accretors have smaller cores compared to their total mass and
lose less mass, thus forming more massive BHs (Fig. 12a). In Case B and C
accretors and regardless of whether stars explode as red or blue supergiants,
we find that all BH masses are larger than those of single stars with the same
$M_{\mathrm{CO}}$ because of the larger final stellar masses (Fig. 12b, c; see
also Sect. 3.2). The compactness peak at $M_{\mathrm{CO}}\,{\approx}\,7\,{\rm
M}_{\odot}$ results in BH formation and this island of BH formation is shifted
by about $0.5\,{\rm M}_{\odot}$ to lower $M_{\mathrm{CO}}$ in Case B models
that accreted enough mass to evolve through a long-lived BSG phase (Fig. 12b;
c.f. Sect. 3.4). The generally higher compactness at core-collapse in Case C
accretors is reflected by BH formation over a larger $M_{\mathrm{CO}}$ range
compared to Case B accretors (e.g. at $M_{\mathrm{CO}}\,{\approx}\,10\,{\rm
M}_{\odot}$, some Case C accretors collapse into BHs while this is not the
case in the Case B models). Neglecting enhanced mass loss in massive CSGs and
LBVs, BH masses are in the range of $12.5\text{--}50\,{\rm M}_{\odot}$,
$15\text{--}190\,{\rm M}_{\odot}$ and $15\text{--}150\,{\rm M}_{\odot}$ in our
Case A accretors/single stars, Case B accretors and Case C accretors,
respectively. The minimum BH masses (${\approx}\,12.5\,{\rm M}_{\odot}$) are
for stars in the compactness peak.

The SN-ejecta masses (Fig. 14) are similar to those of BHs (Fig. 12): if a
star does not collapse into a BH, it explodes and ejects a mass that we assume
is given by the final total stellar mass minus the (baryonic) mass of the
formed NS. Stars in our models explode and eject significant mass for
$M_{\mathrm{CO}}\,{\lesssim}\,13\,{\rm M}_{\odot}$ and outside the compactness
peak at $M_{\mathrm{CO}}\,{\approx}\,7\,{\rm M}_{\odot}$ (Fig. 14). Fully
rejuvenating Case A accretors eject the same mass as single stars of the same
$M_{\mathrm{CO}}$ while the non-rejuvenating models retain higher final mass
and eject correspondingly more mass in their SN explosions (Fig. 14a). The
ejecta masses of Case B and C accretors scale directly with the amount of
accreted mass and greatly exceed those of single stars (Fig. 14b, c). While
single stars eject $8.6\text{--}14.0\,{\rm M}_{\odot}$, the ejecta masses
reach values of up to $30\,{\rm M}_{\odot}$, $100\,{\rm M}_{\odot}$ and
$70\,{\rm M}_{\odot}$ in our (non-rejuvenated) Case A, B and C accretors,
respectively.

As with the final masses (Fig. 6), some BH and SN ejecta masses (Figs. 12 and
14, respectively) are rather upper limits because of unaccounted for mass
loss. For example, no RSG has been observed today at $\log\,L/{\rm
L}_{\odot}\,{\gtrsim}\,5.5$ (Davies et al. 2018) and this might be because of
enhanced mass loss in such luminous RSGs that erode the hydrogen-rich envelope
and turn stars into hotter, e.g. WR, stars. Similarly, eruptive mass loss,
e.g., because of envelope instabilities in LBVs may result in the loss of the
entire hydrogen-rich envelope. Typical averaged LBV mass-loss rates at
$\log\,L/{\rm L}_{\odot}\,{\gtrsim}\,6.0$ are $10^{-4}\,{\rm
M}_{\odot}\,\mathrm{yr}^{-1}$ (Smith 2014), suggesting that such LBVs would
lose $1\,{\rm M}_{\odot}$ within $10^{4}\,\mathrm{yr}$. We identify all models
that would be subject to such LBV-like mass loss for longer than this time,
i.e. those models that likely lose more than $1\,{\rm M}_{\odot}$ before core
collapse. For example, none of our initially $13\,{\rm M}_{\odot}$ early Case
B models are expected to be subject to such enhanced mass loss (e.g. Fig. 1)
while some of the initially $30\,{\rm M}_{\odot}$ models are (e.g. Fig. 15).

Figure 15: Same as Fig. 1 but for initially $30\,{\rm M}_{\odot}$ stars. The
black dashed line is the Humphreys–Davidson limit (Humphreys & Davidson 1979).
Cross symbols indicate models that may experience enhanced, LBV-like mass loss
and may thus rather explode at hotter effective temperatures as stripped-
envelope SNe of Type Ib/c. Plus symbols are separated by $\Delta
t\,{=}\,5\times 10^{4}\,\mathrm{yr}$.

The initially $30\,{\rm M}_{\odot}$ early Case B accretors burn helium in
their cores as BSGs for $f_{\mathrm{acc}}\,{\gtrsim}\,0.50$ and otherwise as
CSGs (Fig. 15). Only the $f_{\mathrm{acc}}\,{=}\,1.0$ and $2.0$ models enter
the LBV instability regime for less than $10^{4}\,\mathrm{yr}$ and are thus
probably not much affected by LBV-like mass loss. The other models may
encounter enhanced mass loss as an LBV star just before the SN explosion.
Models with $f_{\mathrm{acc}}\,{=}\,0.50$ and $0.75$ explode in the S Dor
instability strip and may be LBV SN progenitors. The models evolving into CSGs
(including the initially $30\,{\rm M}_{\odot}$ single star) are expected to
lose significant mass that we do not account for in our models. They may even
lose their entire hydrogen-rich envelope and then rather explode as a
stripped-envelope supernova (SN Ib/c).

Figure 16: Maximum BH and SN ejecta mass of single stars and accretors with
$f_{\mathrm{acc}}\,{\leq}\,1.0$ and $f_{\mathrm{acc}}\,{\leq}\,2.0$ as a
function of the maximum time spent in the LBV-region of the HR diagram. The
grey-shaded region indicates models for which we expect significant LBV-like
mass loss such that the maximum BH and SN ejecta masses should be considered
upper limits.

To better understand the possible maximum BH and SN ejecta masses of our
models, we show them as a function of the maximum time spent in the LBV region
of the HR diagram in Fig. 16. Models indicated by the grey-shaded areas are
the same as those marked by cross symbols throughout this paper, i.e. for
which considerable mass loss is expected. Generally, the maximum BH masses are
larger than the SN ejecta masses and, if a single-star model evolves into the
LBV region in the HR diagram, it spends most of its core-helium burning in
these locations (i.e. ${>}10^{5}\,\mathrm{yr}$). Considering
$10^{4}\,\mathrm{yr}$ as the limiting time a star can spend as an LBV in the
HR diagram before it loses considerable fractions of its mass, we find maximum
BH (SN ejecta) masses of about $13$, $50$ and $77\,{\rm M}_{\odot}$ ($12$,
$37$ and $84\,{\rm M}_{\odot}$) for our single stars and accretors with
$f_{\mathrm{acc}}\,{\leq}\,1.0$ (e.g. mergers in isolated binary stars) and
$f_{\mathrm{acc}}\,{\leq}\,2.0$ (e.g. allowing also for repeated mergers in
triples), respectively. These values reduce to $13$, $36$ and $36\,{\rm
M}_{\odot}$ ($11$, $32$ and $42\,{\rm M}_{\odot}$) for models that avoid the
LBV-region in the HR diagram altogether – they can thus be considered lower
limits. Neglecting LBV-like mass loss, we find maximum BH (SN ejecta) masses
of $47$, $112$ and $190\,{\rm M}_{\odot}$ ($14$, $62$ and $98\,{\rm
M}_{\odot}$), respectively.

The exact BH and ejecta masses depend on a multitude of modelling
uncertainties and the limitations of our models to fully represent mass
gainers of binary mass transfer and stellar mergers (see discussion in Sect.
5.1). However, regardless of the uncertainties and limitations, accretors of
binary mass transfer and stellar mergers lead to a much larger range of BH and
SN ejecta masses that single stars can only reproduce at much lower
metallicities (c.f. Sect. 5.3).

##  5 Discussion

###  5.1 Main model uncertainties and limitations

Our effective merger and binary mass-transfer prescription of accreting
certain fractions $f_{\mathrm{acc}}$ of the initial mass of stars over the
accretor’s thermal timescale (see Sect. 2.2) is simplified and cannot fully
catch the complex physics of these phases (see Sect. 2.2.2). In the following,
we discuss how our assumptions of the chemical composition of the accreted
mass, the absence of mixing of parts of the helium core in mergers and the
dynamic mass loss in mergers affect our results. We also touch upon the role
of wind mass loss, rotation of our accretors, possible magnetic fields
produced in stellar mergers, and multiple and extended mass exchange episodes
in binary stars. Moreover, we discuss to which extent the findings from our
solar-metallicity models can be generalised to accretors of other
metallicities. The role of semi-convection is discussed already in Sect. 3.1.
As discussed in the following and mentioned previously, our assumptions for
the merger models are such that one should expect to find a higher fraction of
post-merger BSGs than what is predicted by our models (i.e. BSG solutions
occur at lower $f_{\mathrm{acc}}$ than predicted by our current models).

####  5.1.1 Chemical composition of accreted mass

We assume that the accreted mass has a chemical composition of that of the
surface of the accretor, i.e. almost the primordial chemical composition of
our models. However, the accreted material is certainly enriched in helium in
stellar mergers and possibly also during binary mass-transfer events. In Case
B and C accretors, a larger mean molecular weight in the accreted material
will increase the temperature at the base of the hydrogen-rich envelope,
$T_{\mathrm{base}}\propto\mu_{\mathrm{env}}M_{\mathrm{env}}/R_{*}$; this might
also occur in Case A accretors that do not fully rejuvenate. Hence, we expect
to find BSGs for smaller $f_{\mathrm{acc}}$ compared to our current models
such that more models evolve as BSGs and less as CSGs (c.f. Eq. 4). The BSGs
themselves will then have a higher luminosity and cooler effective temperature
because the luminosities and effective temperatures of the BSGs mostly depend
on the hydrogen-rich envelopes which evolve analogously to main-sequence stars
(Sects. 3.1 and 4.1). In RSGs, the larger mean molecular weight of accreted
mass decreases the opacity and thereby increases the effective temperature
according to the new location of the Hayashi line in the HR diagram (Hayashi &
Hoshi 1961). If enough helium-rich material is accreted, such stars may no
longer look like RSGs but more like yellow super-/hypergiants, and show a
clear surface helium and nitrogen enrichment.

####  5.1.2 Core mixing in stellar mergers and SN 1987A

In all stellar mergers (i.e. Case A, B and C), parts of the helium-enriched
core of the more evolved star can be dredged up and replenished by hydrogen-
rich material (see e.g. Podsiadlowski 1992; Ivanova et al. 2002; Ivanova &
Podsiadlowski 2002; Podsiadlowski et al. 2007; Glebbeek et al. 2013; Menon &
Heger 2017; Schneider et al. 2019). This mechanism may operate during the
dynamic merger phase but also during the later thermal relaxation of the
merger product (see also Schneider et al. 2020). In Case A mergers, the mixing
can prolong the core hydrogen burning phase and thereby extend the MS turn-off
to cooler temperatures and larger luminosities. In Case B and C mergers, the
partial mixing of the helium core out into the envelope has two main
consequences: (i) the helium core becomes effectively less massive and the
envelope correspondingly more massive, and (ii) the dredged-up helium results
in a higher mean molecular weight of the envelope. The effect of the larger
mean molecular weight in the envelope is already discussed above, and the
smaller helium core mass/larger envelope mass effectively has similar
consequences as an increased mean molecular weight. Because of a larger
$M_{\mathrm{env}}$,
$T_{\mathrm{base}}\propto\mu_{\mathrm{env}}M_{\mathrm{env}}/R_{*}$ is hotter
and stars more easily become BSGs. Moreover, a larger $M_{\mathrm{env}}$ also
increases the star’s luminosity and effective temperature (see Sect. 4.1).
This will also affect the structures of stars at core collapse and their
explodability, and can lead to vastly different outcomes of core collapse
because of the non-monotonic relation of various interior properties (e.g.
compactness and central entropy) with the stellar core mass.

In models for the progenitor of SN 1987A (see e.g. Hillebrandt & Meyer 1989;
Podsiadlowski et al. 1990; Podsiadlowski 1992; Menon & Heger 2017;
Podsiadlowski 2017a), the dredge-up of helium-rich material from the core of
the more evolved star is required to explain observations for two main
reasons. First, the triple-ring nebula is enriched in helium and nitrogen,
i.e. by-products of hydrogen burning (e.g. Arnett et al. 1989; Lundqvist &
Fransson 1996). Second, the expansion timescale of the rings surrounding SN
1987A is only about $20,000\,\mathrm{yr}$ (e.g. Burrows et al. 1995), i.e. the
proposed merger must have occurred after core-helium burning as a Case C
system. The mixing of helium-rich material into the envelope of the merged
star helps the progenitor models to become a BSG (see the discussion in Sect.
3.1). Indeed, we do not find BSGs from Case C accretors with
$f_{\mathrm{acc}}\leq 1.0$ that can explain the position of SN 1987A’s
progenitor in the HR diagram (see e.g. Fig. 9). Most recently, this has again
been demonstrated by the merger models of Menon & Heger (2017) and Urushibata
et al. (2018) that show a good match to the BSG progenitor of SN 1987A. Such
models can then also reproduce the light curve of SN 1987A, along with other
peculiar Type II SNe (Menon et al. 2019). Moreover, Case B mergers with
similar physics can also explain several properties of observed BSGs (Menon et
al. 2023).

####  5.1.3 Mass loss via binary mass transfer and mergers

We do not take into account mass loss from non-conservative binary mass
transfer or dynamic stellar mergers. In this regard, our accretion fractions
are “effective” fractions of how much mass is truly accreted by stars. Hence,
mass loss is only relevant when linking our different accretion scenarios to
the initial binary-star configurations. Binary mass transfer can be fully
conservative for Case A systems but also almost entirely non-conservative when
the donor star is more evolved (e.g. Podsiadlowski et al. 1992; Pols 1994;
Wellstein et al. 2001; de Mink et al. 2007; Henneco et al. 2024). In stellar
mergers, Glebbeek et al. (2013) find that up to ten percent of the total
binary mass is dynamically ejected and the ejection fraction depends on the
binary mass ratio $q=M_{2}/M_{1}$,
$f_{\mathrm{ej}}\,{\approx}\,0.3q/(1+q)^{2}$ ($M_{1}>M_{2}$). The authors
conduct head-on collisions where the ejection fraction depends strongly on the
exact collision setup (e.g. collision energy and impact parameter, see also
Freitag & Benz 2005) and similar mergers in binary systems on Keplerian orbits
find lower ejection fractions of less than one percent (Schneider et al.
2019). In general, the mass loss fraction depends on the evolutionary state of
the merging stars, and mergers of more evolved stars lead to higher mass loss
because their binding energies are lower. While mergers of two main-sequence
stars may eject less than one percent of their total mass (e.g. Schneider et
al. 2019), mergers of giant stars may proceed more like traditional common-
envelope phases and could lead to full envelope ejection, i.e. ejection
fractions of 50% and higher (e.g. Moreno et al. 2022). The triple-ring nebula
of SN 1987A is an example of left-over debris from a stellar merger (see e.g.
Morris & Podsiadlowski 2007, 2009).

####  5.1.4 Mass loss via winds and LBV-like outflows

Stellar wind mass-loss rates are still substantially uncertain (easily by a
factor of 2–3 depending on the stellar type, see e.g. Smith 2014). In Case A
accretors, wind mass loss directly influences the core evolution and stars
adjust their structure to the modified total mass. In Case C accretors, the
core and envelope are already decoupled such that mass loss mainly reduces the
envelope mass and leaves the further core evolution untouched. This is not
true for Case B accretors where a reduced envelope mass can still affect the
further core evolution. With a lower envelope mass (i.e. higher mass loss),
BSG formation becomes more difficult as discussed in Sects. 5.1.1 and 5.1.2
and some initial BSGs might evolve to the red giant branch when hydrogen shell
burning turns radiative.

In our hydrogen-rich accretors, WR wind mass loss is negligible, and the wind
mass loss applied to hot ($T_{\mathrm{eff}}\,{\geq}\,11,000\,\mathrm{K}$) and
cool ($T_{\mathrm{eff}}\,{\leq}\,10,000\,\mathrm{K}$) stars mostly affects our
results. The applied cool wind mass loss is generally higher than that applied
to our hot stars such that models that remain hot for most of their evolution
lose considerably less mass than models that evolve into CSGs. Hence, models
that end as CSGs tend to be more affected by uncertainties in the applied wind
mass-loss rates than those that end as BSGs. Smith (2014) suggest that the
mass-loss rates as applied in our models are an overestimate. If true, our
models would lose less mass and could result in higher SN ejecta masses and
more massive BHs.

In massive stars, winds are mostly line-driven and therefore scale with
metallicity. This is likely not true for LBV-like winds and eruptions. These
have been suggested to be connected to helium opacity in the outer envelopes
of massive stars (Jiang et al. 2018; Grassitelli et al. 2021) and such a
mechanism would operate at all metallicities (albeit less severely at lower
metallicity with generally lower opacity and luminosity-to-mass ratios). This
seems to be confirmed observationally: Davies et al. (2018) find that their
re-evaluated Humphreys–Davidson limit for cool supergiants is the same in the
Milky Way, LMC and SMC, and LBVs have been found in low-metallicity galaxies
(e.g. SMC; Humphreys & Davidson 1994; Weis & Bomans 2020). So while line-
driven winds of hot and cool stars become less strong at lower metallicity,
LBV-like mass loss might be always present. This is particularly relevant for
our massive accretors that evolve through or close to the S Dor instability
strip and that may lose a significant fraction of their hydrogen-rich
envelopes.

Our models employ Mesa’s MLT++ scheme. Switching this off, stars would develop
more strongly inflated envelopes and would more easily evolve into the LBV
region in the HR diagram. This could potentially lead to enhanced mass loss.

####  5.1.5 Rotation

Accretors of binary mass transfer can be easily spun up to critical rotation
if tides are not efficient enough to slow them down (see e.g. de Mink et al.
2013). Merger products may also show unusual rotation velocities.
Traditionally, it is assumed that such stars would rotate critically because
of the large amount of angular momentum in the pre-merger binary orbit.
However, Schneider et al. (2019, 2020) have shown that merged stars may
actually spin slowly. They attribute this to the efficient coupling of the
merged star to a toroidal accretion structure right after the merger that
could transport away angular momentum and to an unusual moment of inertia of
the merged star right after the coalescence that does not allow the merged
star to gain high spin angular momentum as a solid body rotator. The merger of
two stars gives birth to a new, more massive star and this situation might be
comparable to the star-formation process. Also, most young massive stars do
not seem to rotate unusually rapidly (e.g. Dufton et al. 2013; Ramírez-Agudelo
et al. 2013, 2015) and this may well apply to merger products. Moreover,
merged stars could lose angular momentum in a variety of ways, e.g., through
magnetic braking and mass ejection episodes (see below and also Schneider et
al. 2020). In lower-mass blue straggler stars originating from collisions of
stars, slow rotation seems to be favored over rapid rotation (e.g. Leonard &
Livio 1995; Sills et al. 2005; Ferraro et al. 2023), further supporting our
hypothesis that merged massive stars may actually not rotate rapidly.

While most of the above is true for MS stars and hence Case A accretors, less
is known for Case B and C accretors. The larger binary orbits of such systems
imply an overall larger orbital angular momentum which could enhance the
likelihood of obtaining more rapidly rotating Case B and C accretors than
those from Case A accretion. However, as shown by Urushibata et al. (2018),
most of the available angular momentum of their merger models must be lost by
some mechanism such that the merged star can become massive enough to explain
the BSG progenitor of SN 1987A. This would also speak in favour of a mechanism
that can efficiently remove angular momentum from Case B and C mergers.

Our models are for non-rotating stars. While this assumption may be adequate
for some of our accretors as described above, some will certainly rotate
rapidly. Rotation could lead to enhanced mass loss, internal mixing, reduced
ejecta masses and spin up of the remnants. As discussed in Sect. 5.1.4, the
enhanced mass loss makes BSG formation more difficult. The additional mixing
of chemical elements has the opposite effect and enhances the likelihood of
BSG formation because it likely mixes helium-enriched material out into the
envelope.

####  5.1.6 Magnetic fields

Observationally, strong, large-scale surface magnetic fields are observed in
about 5–10% of massive stars (e.g. Donati & Landstreet 2009; Wade et al. 2014;
Ferrario et al. 2015; Fossati et al. 2015; Schöller et al. 2017; Grunhut et
al. 2017). It has been shown that such magnetic fields can be produced in the
merger of two stars (Schneider et al. 2019). In stars accreting mass by binary
mass transfer, strong, large-scale magnetic fields are probably not being
produced (Langer 2014; Schneider et al. 2016) as exemplified by the absence of
detected magnetic fields in Be stars of which at least some are expected to be
post-mass-transfer accretors (e.g. Wade et al. 2014; Neiner et al. 2015).
Hence, only the accretor models representing stellar mergers may lack physics
ingredients to account for the effects of magnetic fields.

The magnetic fields in merger products can influence the further evolution in
various ways (see e.g. Meynet et al. 2011; Potter et al. 2012; Petermann et
al. 2015; Petit et al. 2017; Quentin & Tout 2018; Keszthelyi et al. 2019;
Schneider et al. 2020). Most importantly, magnetic fields may effectively
reduce wind mass loss by confinement, lead to enhanced angular momentum
transport in the stellar interior, inhibit convective boundary mixing, and
magnetic braking can spin stars down. The reduced mass loss and, in
particular, the inhibition of convection can make it more likely that merged
stars evolve as BSGs (Petermann et al. 2015). Given that our accretor models
are non-rotating, they may be viewed as models with extremely efficient
magnetic braking.

####  5.1.7 Multiple and extended binary mass-transfer phases

In the evolution of binary stars, there can be multiple phases of mass
exchange with a companion before a star reaches core collapse. Each phase can
leave its characteristic fingerprint on the final stellar structure and may
influence the outcome of core collapse. For example, a star that accreted mass
in the first mass-transfer episode may be stripped off its envelope in a
second mass-transfer phase. For example, Renzo et al. (2023) modelled a
rejuvenated former accretor that evolves into a supergiant with a reduced
envelope binding energy because of the past mass accretion. The supergiant
then engulfs a companion and initiates a common-envelope phase whose outcome
may now be affected by the previous mass accretion. Similarly, previous mass
exchange in a binary can affect the structures of pre-merger stars (see e.g.
Menon et al. 2023). Such additional mass-exchange phases are not covered by
our models. They may affect the accretor models representing mass gainers of
binary mass transfer and stellar mergers (the pre-merger stars may have
already evolved through a phase of binary mass transfer).

As detailed in Sect. 2.2, our models accrete mass on a thermal timescale but
this timescale can be much longer, e.g. on a nuclear timescale in case of
accretion from an MS donor star (classical Case A binary systems, _not_ to be
confused with a Case A accretor). In such systems, we expect continued
rejuvenation. The evolution of such systems can differ from our models, most
notably by the chemical composition and specific angular momentum of the
accreted matter. Also in the case of stellar mergers, there can be a preceding
mass-transfer episode that modifies the structures of the pre-merger stars and
then gives rise to a different merger outcome. Such effects are not accounted
for in our models.

###  5.2 Metallicity

Our models are for the solar metallicity of Asplund et al. (2009), but to
which degree can our results be generalised to other metallicities $Z$? While
the quantitative results will differ for models of another metallicity, the
qualitative picture should remain the same. For example, the general landscape
of $\xi_{2.5}$, $M_{\mathrm{Fe}}$ and $s_{\mathrm{c}}$ with $M_{\mathrm{CO}}$
in binary-stripped stars is robust for a large range of metallicities
(Schneider et al. 2023) and we expect the same outcome for accretors of binary
mass transfer and stellar mergers. Similarly, long-lived BSGs may form from
merging at various metallicities. These stars will always lose the least mass
in stellar winds, hence they will produce the most massive BHs and the largest
SN ejecta masses for a given $Z$. As such they could naturally explain some
high-mass BHs in the observed population of binary BH mergers (The LIGO
Scientific Collaboration et al. 2021a, b). Furthermore, the accretor models
may always have larger final masses than single stars of the same $Z$ and
larger envelope-to-core-mass ratios.

The metallicity enters stellar evolution mainly through the opacity of the
stellar matter and the strength of line-driven stellar winds. At lower
metallicities, stars are more compact and have weaker winds. This leads to the
general picture that wind mass loss is most important for the evolution of
single stars at high $Z$ while rotation and rotationally-induced mixing become
more important at low $Z$ where less angular momentum is lost through winds.
As discussed in Sect. 5.1.4, weaker winds at lower $Z$ increase the likelihood
of the formation of long-lived BSGs (and vice versa at higher $Z$ with
stronger winds). Because of the increased importance of rotation at low $Z$,
our accretor models for mass gainers of binary mass transfer become less and
less applicable because they are for non-rotating stars. This may not apply to
stellar mergers as discussed in Sect. 5.1.5.

###  5.3 Maximum BH and SN ejecta mass

The maximum BH and SN ejecta masses are both closely connected to the maximum
final mass of our models and thus to wind mass loss (Sects. 3.2 and 4.2). Wind
mass-loss rates of CSGs and WR stars are stronger than those of stars on the
main sequence (e.g. Smith 2014). Hence, stars that undergo core helium burning
as BSGs can avoid the strong CSG and WR winds, and, if they are also not
subject to LBV-like mass loss, they will retain the highest mass. A natural
way to obtain such stars is by stellar mergers as shown here131313It is also
possible to tune the physics of stars to obtain core-helium-burning BSGs from
single stars as explored in detail when trying to understand the BSG
progenitor of SN 1987A (see e.g. the review in section 3 of Podsiadlowski
1992). Vink et al. (2021) show that this is also possible for stars with
initial masses of $90\text{--}100\,{\rm M}_{\odot}$. In many cases, however,
the choice of physics might be in tension with other observations of stars
(e.g. unusually little or no convective boundary mixing). (see also Justham et
al. 2014).

Our models are for solar-like metallicity. Even when assuming that the entire
hydrogen-rich envelope is lost when stars enter the LBV region in the HR
diagram, we find BHs and SN ejecta masses of almost $40\,{\rm M}_{\odot}$ and
$35\,{\rm M}_{\odot}$, respectively, from stellar mergers in binaries (i.e.
$f_{\mathrm{acc}}\,{\leq}\,1.0$). These numbers go up to about $50\,{\rm
M}_{\odot}$ and $40\,{\rm M}_{\odot}$, respectively, when assuming that LBV-
like mass loss can be neglected for stars spending less than
$10^{4}\,\mathrm{yr}$ in the LBV region of the HR diagram. At lower
metallicities, the final and thus also the BH and SN ejecta masses of all our
models would be even larger.

Already at solar metallicity and for rather conservative assumptions on LBV-
like mass loss, we predict BH masses that may populate the so-called PISN BH
mass gap at about $45\text{--}130\,{\rm M}_{\odot}$ (Woosley et al. 2002;
Woosley 2017; Farmer et al. 2019, 2020; Marchant & Moriya 2020; Renzo et al.
2020b; Farag et al. 2022). Avoiding the pair instability in merger models that
become BSGs is possible because such stars have relatively small cores and
large envelope masses. For example, an initially $27\,{\rm M}_{\odot}$ star
turns into a BSG upon accreting with $f_{\mathrm{acc}}\,{=}\,1.0$ as an early
Case B system and has a CO core mass of $M_{\mathrm{CO}}\,{\approx}\,6.5\,{\rm
M}_{\odot}$ at core collapse. This $M_{\mathrm{CO}}$ places the star in the
compactness peak and a BH of $M_{\mathrm{BH,grav}}\,{\approx}\,50\,{\rm
M}_{\odot}$ forms in our models (it spends some $9000\,\mathrm{yr}$ as an LBV
in the HR diagram). Such a core mass is far from the ${\gtrsim}\,35\,{\rm
M}_{\odot}$ required to trigger a PISN. In the event of multiple/repeated
mergers (e.g. $f_{\mathrm{acc}}\,{=}\,2.0$, see Sect. 5.4), the final BH mass
even reaches $80\,{\rm M}_{\odot}$ in our models (in this case, the model
evolves for ${\approx}\,4500\,\mathrm{yr}$ as an LBV in the HR diagram).

With the observation of the binary BH merger GW190521 with component masses of
${\approx}\,85\,{\rm M}_{\odot}$ and ${\approx}\,66\,{\rm M}_{\odot}$ (Abbott
et al. 2020a, b), two BHs in the PISN BH mass gap have been observed. This
raises the question of how such massive BHs can form. Unless we neglect LBV-
like mass loss in our models and/or consider mergers with
$f_{\mathrm{acc}}\,{>}\,1.0$, we do not predict the existence of such massive
BHs at solar-like metallicity. A lower metallicity and/or mergers with
$f_{\mathrm{acc}}\,{>}\,1.0$ are required. This is generally consistent with
other studies involving the merger of massive stars to obtain BSGs that avoid
significant mass loss and retain a large final mass to produce BHs in the PISN
mass gap (e.g. Di Carlo et al. 2019; Renzo et al. 2020a; Costa et al. 2022;
Ballone et al. 2023). As noted by Renzo et al. (2020a), however, the maximum
possible BH mass depends crucially on how strong the LBV-like wind mass loss
is and whether the merged stars can retain their large hydrogen mass. As shown
here, some BH-forming, merged BSGs may be able to avoid the LBV regime, but
their total mass is limited by the fact that these stars have to have CO core
masses which fall into the compactness peak where we anticipate BH formation.
Hence, even at lower metallicity, it seems that multiple/repeated mergers are
required to obtain BHs of ${\gtrsim}\,66\text{--}85\,{\rm M}_{\odot}$ when
assuming that LBV-like mass loss can erode most of the hydrogen-rich envelope
of stars.

The maximum SN ejecta mass in our models is closely related to the maximum BH
mass if we assume that models lose almost their entire envelope by LBV-like
mass loss. In this case, the most massive BH forms from a Case B accretor with
a core mass falling into the compactness peak while the SN with the largest
ejecta mass is a model that has a core just a little bit less massive such
that it does not reach high core compactness and explodes in a SN.
Consequently, we predict that the maximum SN ejecta mass is a little bit less
massive than the most massive BH (c.f. Fig. 16). Our accretors with high SN
ejecta mass explode as Type II SNe. Observationally, Hamuy (2003) report SN
ejecta masses in the range $14\text{--}56\,{\rm M}_{\odot}$ for SN IIP, and SN
ejecta masses of $10\text{--}60\,{\rm M}_{\odot}$ are reported for long-rising
SN 1987A-like Type II SNe (Taddia et al. 2016; Pumo et al. 2023). In our
models at solar metallicity, single stars have SN ejecta masses of
${\lesssim}\,15\,{\rm M}_{\odot}$ and can thus not account for the observed
large SN ejecta masses (see also Zapartas et al. 2021). Contrarily, the
accretor models can have larger final stellar masses and thus better match
these observations.

###  5.4 Formation channels for $f_{\mathrm{acc}}\,{>}\,1$ mergers

In the following, we denote the primary star in a binary system as the
initially more massive and thus more evolved star at the beginning of the
first phase of binary mass exchange (stable mass transfer or stellar merger).
In a merger, it is expected that its core forms the core of the merger remnant
because it is denser and less buoyant141414In almost equal-mass merger,
however, the core of the secondary often forms the core of the merger product
(see e.g. Glebbeek et al. 2013; Schneider et al. 2019).. In this picture, the
less massive secondary star accretes onto the more massive primary star and
$f_{\mathrm{acc}}\,{\leq}\,1$ in our models of standard stellar mergers in
isolated binary stars.

In binary mass transfer, $f_{\mathrm{acc}}\,{>}\,1$ is possible, because now
the initially less massive secondary accretes mass from the initially more
massive primary. In most cases, the accretor will be a main sequence star (
accretor). Case B and Case C accretors, i.e. accreting post-main sequence
stars, are only possible in binaries with initial mass ratios close to one
because the post-main sequence lifetime is short compared to that of the main
sequence. Hence, Case B and Case C accretors are also only possible up to
$f_{\mathrm{acc}}\,{\approx}\,1$ in the first mass transfer phase of binary
stars.

Triples and other multiples are common among massive stars (${>}\,50\%$ of all
systems, see e.g. Sana et al. 2014; Moe & Di Stefano 2017; Offner et al.
2023). In triples and higher order multiples, mergers with
$f_{\mathrm{acc}}\,{>}\,1$ are possible. For example, the initially most
massive star could merge with a close binary system whose individual masses do
not exceed the mass of the primary but its total mass does, possibly leading
to Case A, B and C mergers with $f_{\mathrm{acc}}\,{>}\,1.0$. Similarly, an
initially hierarchical triple/multiple system can be rendered unstable if,
e.g., mass transfer in a close binary leads to orbital widening (see e.g.
merger model in an initial triple for the formation of $\eta$ Car, Hirai et
al. 2021). Furthermore, Justham et al. (2014) estimate that 10% of early Case
B mergers might later accrete additional mass from an outer tertiary
companion, effectively leading to accretors with $f_{\mathrm{acc}}\,{>}\,1.0$.

In star clusters and other dense stellar systems, repeated mergers of stars
are possible on a short timescale (e.g. Portegies Zwart & McMillan 2002;
Portegies Zwart et al. 2004; Mapelli 2016; Gieles et al. 2018; Di Carlo et al.
2019, 2020). This mechanism can easily explain effective accretion fractions
of $f_{\mathrm{acc}}\,{>}\,1.0$ and even $f_{\mathrm{acc}}\,{>}\,2.0$ that are
not covered here. Such mergers have in fact been invoked to explain how BHs
may populate the PISN mass gap (e.g. Di Carlo et al. 2019, 2020) and they can
also form in sequential mergers in triples and higher-order multiples (Vigna-
Gómez et al. 2021).

##  6 Summary and conclusions

Envelope stripping because of binary mass exchange is known to affect the
structures of stars at core collapse and generally allows stars to explode in
supernovae over a larger initial mass range than what is expected from single
stars (Timmes et al. 1996; Wellstein & Langer 1999; Brown et al. 2001;
Podsiadlowski et al. 2004; Woosley 2019; Ertl et al. 2020; Schneider et al.
2021; Laplace et al. 2021; Schneider et al. 2023). In this work, we have
explored how mass _accretion_ as such in stable Roche-lobe overflow or in a
stellar merger affects the further evolution of stars and the stellar
structures just prior to core collapse. To this end, we assume that stars
accrete a fraction $f_{\mathrm{acc}}$ of their initial mass on a thermal
timescale at various points in their evolution. Our main findings can be
summarised as follows.

  * •

Beyond certain accretion thresholds and regardless of when stars accrete mass
in their evolution, they can evolve through a long-lived BSG phase during
which they burn helium in their cores. Some also reach core collapse as BSGs
and thereby avoid the CSG phase altogether. The BSG phase is related to a
thick, convective, hydrogen-burning shell in stars with a sufficiently large
envelope mass compared to their core mass such that hydrogen-shell burning
becomes convective.

  * •

The accretion thresholds depend on when stars accrete mass in their evolution,
the initial mass, and also the amount of mixing of core material into the
envelope (the latter is only relevant for stellar mergers and has not been
considered here). Most of our Case A accretors fully rejuvenate such that they
evolve just as single stars of higher mass. At low accretion fractions,
however, for which stars do not fully rejuvenate, we also find long-lived BSG
phases. Generally, the more evolved the post-main-sequence accretors are, the
more mass they need to accrete to become long-lived BSGs.

  * •

At core collapse, all (non-rejuvenated) accretors have a core size within
10–20% of that of single stars of the original initial mass but a possibly
much higher envelope mass. Because of the larger envelope mass, accretors end
core helium burning with a lower central carbon abundance than genuine single
stars. These fewer carbon atoms change the burning episodes beyond core-helium
burning and generally lead to pre-SN structures that are more difficult to
explode than those of single stars with the same final core mass. This is the
opposite trend of what is found in binary-stripped stars.

  * •

The final masses of non-rejuvenated Case A, and all Case B/C accretors are
larger than those of single stars, with those of Case B accretors being the
largest. In particular, accretors that evolve through a long-lived BSG phase
can avoid the higher wind mass-loss rates as CSGs and thereby retain the most
mass.

  * •

We find that the central entropy is a particularly useful summary quantity of
the pre-SN structures because it directly links to the effective Chandrasekhar
mass at which core collapse occurs and contains information on the mass-radius
relation of the inner core region. For these reasons, we prefer this quantity
over the often-used compactness parameter.

  * •

The trend of central entropy (and compactness) at core collapse with the CO
core mass $M_{\mathrm{CO}}$ is qualitatively similar to that found in single
and binary-stripped stars: there are two main characteristic $M_{\mathrm{CO}}$
of about $7\,{\rm M}_{\odot}$ and $13\,{\rm M}_{\odot}$ at which we find high
central entropy/compactness. These core masses are linked to central carbon
and neon burning becoming radiative as a consequence of a low carbon abundance
after core-helium burning. While the overall trend is shifted to larger
$M_{\mathrm{CO}}$ in binary-stripped stars, it is shifted to lower
$M_{\mathrm{CO}}$ in (non-rejuvenated) accretors. Stars that evolve through a
long-lived BSG phase show the largest shift towards lower $M_{\mathrm{CO}}$ of
about $0.5\,{\rm M}_{\odot}$.

  * •

When analysing our pre-SN interior structures with the neutrino-driven SN
model of Müller et al. (2016), we find that the explodability closely follows
the trend of central entropy with core mass. We find BH formation for the high
central entropy models with $M_{\mathrm{CO}}$ of $\approx 7\,{\rm M}_{\odot}$
and $\gtrsim 13\,{\rm M}_{\odot}$, successful SN explosions with NS formation
for $M_{\mathrm{CO}}\lesssim 7\,{\rm M}_{\odot}$, and partial fallback SNe
with likely BH formation in some models with $M_{\mathrm{CO}}\approx
7\text{--}13\,{\rm M}_{\odot}$.

  * •

The positions in the HR diagram of accretors that do not reach core collapse
in the long-lived BSG phase, are in the same region as those of single stars.
Because of the island of BH formation at $M_{\mathrm{CO}}\approx 7\,{\rm
M}_{\odot}$, accretors ending their lives as CSGs collapse into BHs at
luminosities of $\log L/{\rm L}_{\odot}\approx 5.4$, the same luminosity as in
BH-forming CSGs from genuine single stars. Accretors with
$M_{\mathrm{CO}}\approx 7\,{\rm M}_{\odot}$ that end their lives as BSGs and
collapse into BHs, populate a region along the hot side of the S Dor
instability strip.

  * •

Some of the accretors evolving as BSGs explode inside the S Dor instability
strip, i.e. our models predict that some LBVs may be SN progenitors. If the
supernova explodes into recently ejected material from an LBV eruption,
interacting and possibly superluminous SNe seem possible.

  * •

In all of our accretor models, we find a tight relation between their central
entropy at core collapse and the resulting NS mass. A similar relation has
already been found in single stars with varying degrees of convective boundary
mixing (Temaj et al. 2024) and the very same relation also fits our models.

  * •

Because accretors evolving and ending their lives as BSGs have high final
stellar masses, they can also result in large BH and SN ejecta masses.
Accounting only for models that likely do not experience strong LBV-like mass
loss and restricting $f_{\mathrm{acc}}$ to values of $\leq 1$ (i.e. values
achievable by stellar mergers in binary stars), we find maximum BH and SN
ejecta masses of up to $50\,{\rm M}_{\odot}$ and $40\,{\rm M}_{\odot}$,
respectively. The ejecta masses greatly exceed those from genuine single
stars. Allowing for repeated/multiple mergers (i.e. $f_{\mathrm{acc}}>1$),
both masses increase to about $80\,{\rm M}_{\odot}$. This could potentially
result in BH masses from stars at solar metallicity that fall into the PISN
mass gap.

We have shown here that accretor models approximating mass gainers in binary
mass transfer and products of stellar mergers reach core collapse with
interior structures that can be profoundly different from single stars. While
our models cannot fully account for the complex physics of binary mass
transfer and stellar mergers, they serve as a zeroth-order baseline. To obtain
a more complete picture of stellar mergers and specifically the mixing in such
events, further detailed multi-dimensional, hydrodynamic simulations and a
systematic exploration of the relevant parameter space are warranted. This
will be explored in upcoming work. We have focussed on the core structures at
core collapse, the explodability of models, and their final BH and SN ejecta
masses. In a forthcoming publication (Schneider et al. 2024), we will present
further explosion properties and implications for the zoo of supernova
transients.

###### Acknowledgements.

We thank the reviewer for their careful reading of our manuscript and helpful
and constructive comments. The authors acknowledge support by the Klaus
Tschira Foundation. This work has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant agreement No. 945806) and is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC 2181/1-390900948 (the Heidelberg STRUCTURES Excellence
Cluster). This research made use of NumPy (Oliphant 2006), SciPy (Virtanen et
al. 2020), Matplotlib (Hunter 2007) and Jupyter Notebooks (Kluyver et al.
2016).

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