[1,2]Athira Menon

[1]Instituto de Astrofísica de Canarias, Avenida Vía Láctea s/n, 38205 La
Laguna, Tenerife, Spain

[2]Universidad de La Laguna, Departamento de Astrofísica, Avenida Astrofísico
Francisco Sánchez s/n, 38206 La Laguna, Tenerife, Spain

3]Argelander Institut für Astronomie,Auf dem Hügel 71, DE-53121, Bonn, Germany

4]Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr.
25/8, 6020 Innsbruck, Austria

5]OzGrav: Australian Research Council Centre of Excellence for Gravitational
Wave Discovery, Clayton, VIC 3800, Australia

6]School of Physics and Astronomy, Monash University, Clayton, Victoria 3800,
Australia

7]Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, DE-53121, Bonn,
Germany

8]Department of Physics & Astronomy, Louisiana State University, Baton Rouge,
LA, 70803, USA

9]Hearne Institute of Theoretical Physics, Louisiana State University, Baton
Rouge, LA, 70803, USA

# Evidence for stellar mergers of evolved massive binaries: blue supergiants
in the Large Magellanic Cloud

<EMAIL_ADDRESS>Andrea Ercolino  Miguel A. Urbaneja  Daniel J. Lennon
Artemio Herrero  Ryosuke Hirai  Norbert Langer  Abel Schootemeijer  Emmanouil
Chatzopoulos  Juhan Frank  Sagiv Shiber  *  *  [  [  [  [  [  [  [

###### Abstract

Blue supergiants are the brightest stars in their host galaxies and yet their
evolutionary status has been a long-standing problem in stellar astrophysics.
In this pioneering work, we present a large sample of 59 early B-type
supergiants in the Large Magellanic Cloud with newly derived stellar
parameters and identify the signatures of stars born from binary mergers among
them. We simulate novel 1D merger models of binaries consisting of supergiants
with hydrogen-free cores (primaries) and main-sequence companions
(secondaries) and consider the effects of interaction of the secondary with
the core of the primary. We follow the evolution of the new-born
16–40$\,\textrm{M}_{\odot}$ stars until core-carbon depletion, close to their
final pre-explosion structure. Unlike stars which are born alone, stars born
from such stellar mergers are blue throughout their core helium-burning phase
and reproduce the surface gravities and Hertzsprung-Russel diagram positions
of most of our sample. This indicates that the observed blue supergiants are
structurally similar to merger-born stars. Moreover, the large nitrogen-to-
carbon and oxygen ratios, and helium enhancements exhibited by at least half
our data sample are uniquely consistent with our model predictions, leading us
to conclude that a large fraction of blue supergiants are indeed products of
binary mergers.

###### keywords:

stars: massive, binaries: close, stars: supergiants, stars: fundamental
parameters, binaries: close, binaries: spectroscopic, supernovae: general

The evolution of massive stars
(M${}_{\textrm{initial}}>8$$\,\textrm{M}_{\odot}$) has a strong impact on
their host galaxies and are progenitors of supernovae, long gamma ray bursts
and gravitational-wave systems. They are thus among the science drivers of
large-scale surveys such as the upcoming Rubin-LSST which will observe
millions of optical transients and ongoing LIGO-VIRGO-KAGRA gravitational-wave
detectors, along with components of the future spectroscopic surveys such as
WEAVE, 4MOST and SDSS-V/APOGEE.

Due to their immense luminosities, blue supergiants (BSGs) are excellent
spectroscopic calibrators of extragalactic distances and interstellar
extinction (e.g., [1]), the galactic stellar mass-metallicity relation (e.g.,
[2]) and to trace galactic chemical evolution (e.g., [3]). However, their
evolution is poorly understood. For decades, the question of whether these
BSGs are core-hydrogen burning stars in an extended main sequence (MS), or
core-helium burning stars that are either evolving redwards in the
Hertzsprung-Russell diagram (HRD) or have looped back into the blue
(performing a ‘blue loop’) after a red supergiant (RSG) phase, has been
debated. The answer will have a profound impact on our understanding of
massive star and galaxy evolution. While clues as to their evolutionary status
is provided by their stellar parameters and surface composition, single-star
models have so far had limited success in explaining the overall observed
population [4, 5, 6, 7].

About $50-70$% of massive stars in our cosmic neighbourhood (the Milky Way and
Magellanic Clouds) are observed to live in binary systems with close stellar
companions [8, 9]. Binary evolution offers additional channels for the
creation of BSGs, for example as mass gainers in binary systems [10]. The low
incidence of BSGs as components of binaries [11, 12] however, argues against
this mechanism being a significant contributor to the population.

By contrast, stellar merger models have had great success in explaining both
the light curve of supernova SN 1987A which occurred in the Large Magellanic
Cloud (LMC) and the properties of its BSG progenitor [13, 14, 15], the B3 I
supergiant Sk-$69\,^{\circ}202$ [16]. Based on the ‘slow merger’ paradigm [17,
18], this model involves the merger of a core-He depleted red supergiant (RSG)
with a main sequence (MS) companion via common envelope (CE), to produce a
$20-24$$\,\textrm{M}_{\odot}$ core-C burning BSGs resembling
Sk-$69\,^{\circ}202$ that lives for $40-80$ kyr before exploding. The
explosions of such post-merger models have also been able to reproduce the
light curves of other Type-II peculiar SNe like SN 1987A [14].

Such ‘evolved’ binary mergers, where the primaries are post-MS giants, have
been evoked to explain luminous blue variables [19], B[e] supergiants [20],
supermassive stars like $\eta$ Carinae [21] and other supergiants observed
with circumstellar rings and bipolar nebulae similar to Sk-$69\,^{\circ}202$,
like Sher 25[22, 23].

Inspired by the success of binary mergers in explaining SN 1987A and its
progenitor, we build new binary merger models to confront a large sample of 59
B0 I – B3 I stars in the LMC, for which we derive new stellar parameters and
elemental abundances. We contrast our findings with single-star models and
argue that stellar mergers are likely to be the largest contributors to the
observed BSG population, and discuss some predictions of this scenario.

##  1 Observational data

The 59 B-type supergiants discussed here (the ‘BSG sample’) are drawn from
three sources; a ‘FEROS’ dataset (discussed by [24]), the Flames Survey of
Massive Stars data (FSMS, [25, 26, 27]) and the VLT-FLAMES Tarantula Survey
(VFTS, [11, 28]). For these sources we have derived stellar parameters, such
as their effective temperatures (log T${}_{\textrm{eff}}$), surface gravity
(log g), surface luminosity (log L/L⊙) and projected rotational velocity
(vsini), along with the surface helium (He), carbon (C), nitrogen (N) and
oxygen (O) abundances using the stellar atmosphere-wind code FASTWIND v10.6.4
[29]. Further details may be found under sections 5.1 and 5.2 , while results
for individual stars are listed in Table 6 under Extended Data. FSMS also
gives CNO abundances for MS O and B stars in the LMC [27], providing the LMC
abundance baseline reference point for the present work.

We express abundances as
$\epsilon\textrm{(X)}=\textrm{log}_{10}(N_{\textrm{X}}/N_{\textrm{H}})+12$,
where $N_{\textrm{X}}$ and $N_{\textrm{H}}$ are the number fractions of the
element and hydrogen respectively, except for helium, which is given as
$Y=N_{\textrm{He}}/N_{\textrm{H}}$. We obtain log (N/C) values from close to
the LMC baseline to as high as 1.8 dex, an enhancement factor of $\approx 60$.
BSGs classified as type BC by [30] have close to baseline nitrogen abundances,
consistent with the hypothesis that they are N-deficient relative to
‘morphologically normal’ supergiants. However, they are also oxygen rich
relative to the LMC baseline by $\approx 0.2$ to $-0.3$ dex. Two possible
explanations are: (i) some systematic abundance offsets between B-dwarfs (used
to define the baseline) and BSGs; (ii) the baseline metallicity in the LMC may
contain some scatter [31].

Noteworthy for the present paper is the enhancement of helium we find in a
third of the sources. This is also in good agreement with the recent work on
Galactic BSGs [32].

##  2 Evolutionary Models

To build our merger models, we first select precise binary models that are
flagged to enter CE when the primary star is a post-MS star and the secondary
is a MS star. At this point the binary evolution is terminated. The selected
binaries are thereafter merged via a detailed 1D process, and the post-merger
star is evolved until core C-depletion, close to its pre-supernova state.

###  2.1 Computational pipeline for binary mergers

For our binary models, we use the ‘LMC Bonn grid’ from [33] (see section 5.3
for the main physics assumptions). Their initial mass fractions are the LMC
mixture of [34]: $X_{\textrm{H}}=0.7391$, $X_{\textrm{He}}=0.2562$ and Z
(metallicity) = 0.0047 ($\approx\textrm{Z}_{\odot}/2$). Among these, ‘Case B’
binaries that have primaries with He-rich cores dominate the grid, while ‘Case
C’ binaries which have primaries with He-depleted cores form only 0.4% of the
grid. We assume all these CE-flagged models can merge (see section 5.4 for CE
termination conditions).

At the onset of the merger, the secondary star spirals-in towards the centre
of the primary due to viscous drag forces and may get tidally disrupted by the
H-free core of the primary, resulting in a stream of H-rich secondary material
to penetrate the core. In the meantime, the rest of the secondary dissolves in
the envelope and the merger is complete [35, 36]. To simulate our merger
models, we adapt the above scenario to 1D based on the method of [13], to the
stellar evolution code MESA v.21.1 [37, 38, 39, 40, 41].

We consider the effect of penetration for two limits: at its maximum depth
inside the H-free core ($M_{\textrm{r, max}}$) and just up to the boundary of
the H-free core ($M_{\textrm{r, H-free core}}$). In both cases, the secondary
is mixed uniformly down to the required depth. A first-order estimate of
$M_{\textrm{r, max}}$ can be made by assuming the secondary stream hits the
core ballistically. We define the core dredge-up factor as
$f_{\textrm{c}}=1-\left(\displaystyle\frac{M_{\textrm{r,\,penetration}}}{M_{\textrm{r,
H-free core}}}\right)$, where
$M_{\textrm{r,\,penetration}}=M_{\textrm{r,\,max}}$ or $M_{\textrm{r,\,H-free
core}}$ (see section 5.5 for the derivation).

###  2.2 An exemplary Case B merger model

Figure 1: Evolutionary tracks of a 32$\,\textrm{M}_{\odot}$ (primary) +
3$\,\textrm{M}_{\odot}$ (secondary) binary that merges when the primary is a
30$\,\textrm{M}_{\odot}$ star, with two core-penetration limits:
f${}_{\textrm{c,max}}$ (system 1, green line) and f${}_{\textrm{c}}=0$ (system
2, brown line). The direction of post-merger evolution is indicate by
$\triangle$ symbols, along with tick (|) marks separated by 0.1 Myrs during
the core He-burning phase (light blue shading). Also plotted are the primary
ZAMS position (grey $\bullet$), start (grey $\circ$) and end of merger (green
and brown $\circ$), core He-depletion (filled $\Diamond$) and core-C depletion
($\star$), close to their respective pre-explosion BSG (blue $\star$) and RSG
(red $\star$) position; the latter is connected by a red dotted line to its
core He-burning phase. The secondary star is barely evolved from ZAMS and is
not shown here.

In Fig. 1, we consider the merger of a binary model where both stars begin
their evolution at the Zero Age Main Sequence (ZAMS) with an initial period of
P${}_{\textrm{i}}=26.6$ d. The initial mass of the primary is
M${}_{\textrm{1,i}}=32$$\,\textrm{M}_{\odot}$ and of the secondary is
M${}_{\textrm{2,i}}=3$$\,\textrm{M}_{\odot}$. By nearly 6.2 Myr, the primary
star has lost $\approx 2$$\,\textrm{M}_{\odot}$ due to winds and just begun
core He-burning. Subsequently, it rapidly transfers mass at a rate of $\approx
10^{-1}$$\,\textrm{M}_{\odot}$/yr to its MS companion and undergoes unstable
Roche lobe overflow, at which point a CE is expected to ensue.

The onward merger evolution is modelled for the two limits of stream-core
interaction discussed in section 2.1: system 1 with $f_{\textrm{c,max}}=28$%,
and system 2 with $f_{\textrm{c}}=0$. The merger occurs over a duration of 300
years, after which the post-merger star has a total mass of
33$\,\textrm{M}_{\odot}$. As the secondary mass is accreted with the same
surface angular momentum as the primary, the immediate post-merger star in
both systems spins up to critical velocity. Rotationally-enhanced mass loss
quickly causes the star to shed $\approx 3$$\,\textrm{M}_{\odot}$ and spin
down to sub-critical equatorial velocity ($\approx 100$ km/s). As the
homogeneously mixed envelope of the post-merger star has no molecular weight
gradients, the star zooms towards the blue region of the HRD.

The post-merger stars of the two systems have different structures: that of
system 1 has a reduced H-free core mass of M${}_{\textrm{H-free, core}}\approx
9$$\,\textrm{M}_{\odot}$, a thick H-burning shell and a
21$\,\textrm{M}_{\odot}$ H-rich envelope, while that of system 2 has an intact
H-free core mass (M${}_{\textrm{H-free, core}}\approx
12.5$$\,\textrm{M}_{\odot}$), a thinner H-burning shell and a smaller envelope
of 17.5$\,\textrm{M}_{\odot}$.

Once core He-burning begins in gusto (log
(L${}_{\textrm{He}}$/L${}_{\odot})\gtrapprox 4$), the post-merger star expands
and loops back to cooler temperatures. Nevertheless in both cases, they retain
their BSG nature throughout their core He-burning phase, which lasts for 0.45
Myr for system 1 (due to its smaller H-free core mass) and 0.38 Myr for system
2. As the H-burning shell of the post-merger star of system 1 is more luminous
than its He-burning core, it retains a hotter T${}_{\textrm{eff}}$, while the
reverse is true for system 2, causing the post-merger star to expand to cooler
temperatures.

The evolution after core He-burning depends on the luminosities of the
H-burning shell and the He-burning shell, formed during core contraction after
core He-burning. In system 2, the thin H-burning shell expands and gets
eventually extinguished. With just a single shell-burning source (the He-
burning shell), the post-merger star continues to expand to a
28$\,\textrm{M}_{\odot}$ RSG by the end of core C-burning, and subsequently
explode as a Type II-plateau SN. On the other hand in system 1, the dominance
of the H-burning shell persists. With two active shell-burning sources, the
star undergoes an overall contraction and returns to the blue region of the
HRD for core-C burning as a 30$\,\textrm{M}_{\odot}$ BSG, subsequently
exploding as Type II-peculiar SN.

The envelopes of these post-merger stars were mixed down to the H-burning
shell of their primaries, leading to surface abundances close to CNO
equilibrium. At the same time, carbon and oxygen are somewhat replenished due
to the mixing of the MS secondary (see Fig. 6 in section 5.5 for the
composition profiles). As we shall see, these processes lead to a range of N/C
and N/O ratios (and He abundances) which is not easily accessible by born-
alone stars.

###  2.3 The binary-merger models in this work

Due to the stellar-wind and mass-transfer prescriptions in the Bonn grid (see
section 5.4), the Case C models are restricted to
$10-11$$\,\textrm{M}_{\odot}$ core C-burning primaries, whose merger products
directly inflate to $11-15$$\,\textrm{M}_{\odot}$ RSGs. Hence, we are unable
to simulate the more massive Case C mergers required for the
20-24$\,\textrm{M}_{\odot}$ progenitor of SN 1987A [13].

In this study, we present the results of 47 Case B binary merger models with
M${}_{\textrm{1,i}}=10-40$$\,\textrm{M}_{\odot}$,
M${}_{\textrm{2,i}}=1-32$$\,\textrm{M}_{\odot}$ and P${}_{\textrm{i}}=16-3162$
days, which undergo core He-burning as $16-40$$\,\textrm{M}_{\odot}$ BSGs and
are evolved until core C-depletion (the full parameter list of the models are
in Table 6).

###  2.4 Born-alone stars

We consider born-alone stellar models of initial masses of
$16-40$$\,\textrm{M}_{\odot}$, whose evolutionary tracks overlap with our
observed sample on the HRD.

Some of our sample BSGs coincide with the HRD positions of O-type MS stars. In
the LMC, O-type MS stars are observed to have projected rotational velocities
that peak at vsini$\approx 80$ km/s and tail off at $\approx 200$ km/s [42].
We consider the evolution of single-star models with initial equatorial
velocities of v${}_{\textrm{i,rot}}\approx 220$ km/s from [34], corresponding
to the upper-limit contribution of rotating MS stars to our observed sample.

During core He-burning, single stars may perform blue loops as they return
from the RSG to the BSG phase. These can be further categorized in to
“classical" blue-loops, which typically occur in the low-luminosity regime
(log L/L${{}_{\odot}}<5.4$) and the high-luminosity blue-loops.

A combination of high semiconvection ($\alpha_{\textrm{sc}}$) and moderate
overshooting efficiency parameters ($\alpha_{\textrm{ov}}=0.33$) were found to
efficiently produce long-lived classical blue-loops in models of the Small
Magellanic Cloud ($\approx\textrm{Z}_{\odot}/5$ metallicity) [43]. We use the
most effective combination of mixing parameters from [43] corresponding to
$\alpha_{\textrm{sc}}=100$ and $\alpha_{\textrm{ov}}=0.33$ and build new LMC
models with the same initial composition as the binary models. Of these, only
the 16$\,\textrm{M}_{\odot}$ and 20$\,\textrm{M}_{\odot}$ models perform
classical blue-loops.

High-luminosity blue loops occur due to wind-driven mass loss in their
preceding RSG phase, causing stars to lose most of their H-rich envelopes
before becoming BSGs. We consider the GENEVA suite of LMC models [44] (built
with solar-scaled initial composition), among which the 32 and
40$\,\textrm{M}_{\odot}$ models perform high-luminosity blue loops that
overlap with our sample T${}_{\textrm{eff}}$ range.

##  3 Comparison between data and models

To evaluate the evolutionary status of our sample BSGs, we simultaneously
compare their surface abundances, HRD positions and surface gravities with the
models.

###  3.1 Surface abundances

Figure 2: Log number ratios of N/C and N/O of our considered binary-merger and
born-alone stellar models. The sample is marked according to the three groups
as discussed in the text, along with the nebular abundances of the BSG
progenitor of SN 1987A (dark orange $\triangle$, [45]) and He-rich stars
($Y\geq 0.12$). The grey dashed and solid lines depict the boundaries for the
ON and CN cycle respectively using the analytical approximations of [46].
Figure 3: Number fractions of C, N, O and He expressed in log units as
explained in section 1, for merger models, classical blue- loops and TAMS
stars, along with our sample groups (as plotted in Fig. 2). $Y$ abundance of
Sk-$69\,^{\circ}202$ is taken from [47]. Dotted lines in each panel represent
the initial abundances: $\epsilon\textrm{(C,i)}=7.75$,
$\epsilon\textrm{(N,i)}=6.90$, $\epsilon\textrm{(O,i)}=8.35$ and
$Y_{\textrm{i}}=0.087$.

Guided by the N/C and N/O number ratios of the models, we broadly split our
observed sample into three groups (within the error-bars of the sample), as
indicated in Fig. 2:

  1. 1.

Group 1 ($\textrm{log\,(N/C)}<-0.2$, $\textrm{log\,(N/O)}\lessapprox-1$):
containing 9 stars whose number ratios are explained by single-star models
with v${}_{\textrm{i,rot}}\lessapprox 220$ km/s.

  2. 2.

Group 2 ($-0.2\leq\textrm{log\,(N/C)}\lessapprox 0.35$,
$-1<\textrm{log\,(N/O)}\lessapprox-0.25$): is a mixed group of 23 stars whose
number ratios can be explained with classical blue-loop models or terminal-age
MS (TAMS) models with v${}_{\textrm{i,rot}}\geq 220$ km/s or even merger
models, as we shall in this section.

  3. 3.

Group 3 ($\textrm{log\,(N/C)}\gtrapprox 0.35$,
$\textrm{log\,(N/O)}\gtrapprox-0.25$): contains 25 stars whose number ratios
are explained by merger models, but not by the born-alone stellar models.

From Fig. 2, we see that born-alone stars may explain the N/C and N/O ratios
of the group 1 sample as MS stars, and also of the group 2 sample as
classical-blue loop stars or as TAMS stars with fast v${}_{\textrm{i,rot}}$.
The high-luminosity blue-loop models have extreme N/C and N/O ratios that are
far larger than the observed range, as these models have been stripped off
most of their H-rich envelopes. Thus these high-luminosity blue-loop models do
not explain the abundances of our sample.

The group 3 sample which contains stars with the highest N/C and N/O ratios
are not explained by blue-loop or MS stars, and are uniquely replicated by
merger-born stars. This is evident when examining individual elemental
abundances in Fig. 3: mergers result in significant nitrogen enhancements (1.1
to 1.7 dex) and carbon and oxygen depletions (up to $\approx 0.75$ and 0.25
dex, respectively), while classical blue loops or rotational mixing only lead
to moderate nitrogen enhancements ($\lessapprox 1$ dex) and minor carbon and
oxygen depletions relative to mergers.

It is worth noting that potential systematic offsets in the data (discussed in
section 1), may cause some stars with moderate log(N/C) ratios to lie outside
the ON equilibrium line in Fig. 2, due to their measured oxygen abundance
being higher than the baseline. This offset could lead to an increase in the
N/O ratio for some stars in group 2, aligning them with group 3, which is
exclusively explained by mergers.

About one-third of the sample displays helium enrichment by a factor of 1.6 to
2.5 times higher than baseline, which is not only consistent with the helium
abundances of merger models but also with Sk-$69\,^{\circ}202$ [47]. These He-
rich BSGs are predominantly found in group 3, with a few also in group 2. Such
helium enhancements are not produced by fast-rotating or classical blue-loop
born-alone stellar models. High-luminosity blue-loop models tend to exceed the
observed range of helium enhancements.

Some stars in the sample are substantially enriched in nitrogen ($\geq 1$ dex)
but only have baseline helium. This pattern is challenging to explain with any
evolutionary scenario.However these stars are predominantly cool
(T${}_{\textrm{eff}}\lesssim$ 20,000 K) and their He abundance depends
primarily on the He I lines, offering rather weak diagnostics for the He
abundance. The suspicion therefore arises that their He abundance and hence,
the number of He-rich BSGs in this sample, are underestimated. Considering
these uncertainties, we may infer that over half of our sample aligns with the
abundance predictions of our merge models.

###  3.2 HRD positions and surface gravities

(a) Core He-burning BSG phase of our merger models (light blue shading) and
their corresponding SN progenitors ($\star$ symbols) and their masses at core
C-depletion, as in Fig. 2.2. For BSG progenitors with log
T${}_{\textrm{eff}}\lessapprox 4.3$ and RSG progenitors (red, log
T${}_{\textrm{eff}}<4.0$), we connect their full HRD trajectory. Errors of MS
sample are about 0.1 dex for both log L/L⊙ and log T${}_{\textrm{eff}}$.

(b) Blue-loop phase (blue dotted lines) of the classical (yellow tracks) and
high-luminosity (light brown lines) categories, along with their mass in this
phase. Background grey evolutionary tracks are of single-star models with
v${}_{\textrm{i,rot}}\approx 220$ km/s from [34].

Figure 4: HRD showing the BSG phase of merger-born (top) and born-alone stars
(bottom) and our sample (as in Fig. 2), along with the position of
Sk-$69\,^{\circ}202$ [48]. Tick marks indicate intervals of 0.1 Myr in time in
the BSG phase. For reference, the ZAMS and TAMS curves of v${}_{\textrm{i,
rot}}\approx 220$ km/s single star models [34] are also plotted.  Figure 5:
Distribution of post-merger and blue-loop single-star models, along with the
observed sample in the bolometric luminosity (log L/L⊙) and spectroscopic
luminosity space (log $\mathcal{L}$/$\mathcal{L}_{\odot}$). Symbols are as
explained in Figs. 2 and 4.

We next consider the disposition of the above groups in the HRD in Fig. 4. Our
$16-40$$\,\textrm{M}_{\odot}$ post-merger stars are BSGs throughout their core
He-burning phase, spanning $1.6-0.6$ Myrs, and covering nearly the entire
observed HRD. The blue-loop models, on the other hand, only last for
$\lessapprox 50$% of their core He-burning lifetime and mostly fall short of
log T${}_{\textrm{eff}}\geq 4.3$ (K), where the majority of the sample lies.
Thus the high-luminosity sample (log L/L${}_{\odot}\geq 5.4$), where most of
the group 3 members also lie, is exclusively explained by the merger models.

In the lower-luminosity regime, covered by models under $\lessapprox
30$$\,\textrm{M}_{\odot}$, the classical blue-loops of single stars only
extend up to log T${}_{\textrm{eff}}\approx 4.1$ (K) and fail to overlap with
the observed BSGs. In contrast, merger models reproduce this low-luminosity
domain, which also includes the group 2 stars. Group 1 BSGs coincide with the
FSMS-MS sample on the HRD and in their abundances; we thus infer that these
BSGs may be MS stars.

Insight into the internal structure and the current mass of the observed BSGs
can be obtained from their surface gravities. Rather than comparing the
gravities directly, which depend sensitively on the stellar radius, we
consider their spectroscopic luminosity $\mathcal{L}=T_{\textrm{eff}}^{4}/g$
in Fig. 5, which is proportional to $L/M$. Because massive post-MS single
stars largely evolve at constant luminosity proportionally to their mass, the
HRD locations of their BSG models in Fig. 5, move further to the right the
more mass they lose. This is particularly useful to distinguish whether the
observed stars have massive H-rich envelopes or have lost their H-rich
envelopes.

In the lower-luminosity regime (log L/L${}_{\odot}<5.4$) corresponding to
blue-loop models with initial masses $\lessapprox 30\,$M⊙, we see that mass
loss is not significant as the positions of the observed stars agree well with
our merger models. The indicated high-luminosity single-star models, however,
only return to the BSG region after having lost most of their H-rich envelope
and do not match the observations. Notably, because of the strong mass loss,
these models also lost copious amounts of angular momentum, and fall short of
recovering the observed rotation rates of the BSGs (Table 6) by one order of
magnitude. Therefore in the high-luminosity regime also, the observed stars
agree with the positions of the merger models. Overall thus, we may conclude
that the observed BSGs are core He-burning stars with homogeneously mixed,
massive H-rich envelopes similar to our merger models.

Finally, a few of the observed high-luminosity BSGs with log
T${}_{\textrm{eff}}\lessapprox 4.3$ (K) are located on the HRD in the region
where Sk-$69\,^{\circ}202$ is also located. These stars may also be explained
with massive Case C merger models [13]. Indeed, their N/C and N/O ratios are
also consistent with the predictions from these models.

##  4 Conclusions

Our work shows how Case B binary-mergers lead to BSGs whose surface
parameters, core-envelope structures and final fates are distinctly different
from those born alone. We have found substantial evidence for imprints of such
binary mergers in our observational sample of nearly 60 BSGs in the LMC. The
HRD positions and surface gravities of most of the sample are successfully
reproduced by our merger models, indicating that early B-type supergiants
structurally resemble 16–40$\,\textrm{M}_{\odot}$ stars with homogeneous
massive H-rich envelopes that will sustain their BSG nature throughout their
core He-burning lives. Combined with the possibility of spin-up during the
merger process and contrasting with available single-star models, we conclude
that a large fraction of early B-type supergiants, and potentially other
classes of BSGs, are likely to be the results of evolved (Case B or C) stellar
mergers.

The helium enrichments of a third of the sample and the high N/C and N/O
ratios of over half the sample are uniquely explained by our merger models,
and are also consistent with the measured nuclear abundances of the BSG
progenitor of SN 1987A. While the fate of our merger-born stars with
$\textrm{M}<30$$\,\textrm{M}_{\odot}$ is to explode as Type II-peculiar SNe
(like SN 1987A), the more massive ones may also explode as Type II-plateau
SNe.

Our work thus identifies key imprints of massive-binary mergers that
distinguishes them from born-alone stars, and paves the way towards exploring
their contributions to the resolved stellar populations and core-collapse
supernovae of galaxies in the Local Group and beyond.

Acknowledgments AM, DL and AH are supported by the Spanish Government
Ministerio de Ciencia e Innovación through grants PID2021-122397NB-C21 and SEV
2015-0548. AM also acknowledges the Juan de la Cierva Incorporación (2022)
fellowship. AE is supported by the DFG through grant LA 587/22-1. AM also
thanks Alexander Heger for supporting the development of this work.

##  5 Methods

###  5.1 Spectroscopic Data

As noted, the spectroscopic data for the BSG sample is drawn from three
sources, designated FEROS, FSMS and VFTS in the main text. The FEROS dataset
is a high resolution ($R\sim 48\,000$) survey of B-type supergiants in the LMC
performed using the FEROS echelle spectrograph on the ESO 2.2m telescope, the
data covering the wavelength region 3500–9200 Å. The FSMS [25] and VFTS [28]
data are taken from two large programs that used the Flames multi-object
spectrograph on the ESO VLT to observe OB stars in the Magellanic Clouds, the
former being a high resolution survey ($R\sim 20\,000$) of OB stars in both
Clouds, covering regions 3950–4750 Å and 6300–6700Å, while latter was an
intermediate resolution ($R\sim 7\,000$) survey of OB stars in the Tarantula
Nebula of the LMC covering regions 3960–5070 Å and 6400–6820 Å. For these
latter two wide-ranging surveys we restrict the sample to apparently single
stars with spectral types between B0 and B3 inclusive, the range of validity
of the spectroscopic analysis, and luminosity class I. A small number of
variable and peculiar objects were excluded, as were a several hypergiants
with very high mass-loss rates. All the spectra considered here have signal-
to-noise ratios of approximately 200.

One should note that this work is focused on early to mid B-type supergiants,
hence the absence of late-B and A/F-type supergiants in the sample. In fact
there are very few BSGs in this cooler temperature range with published CNO
abundances, albeit [49] finds a mean under-abundance in [C/H] of only
$\sim$0.2 dex for a sample of 9 F-type supergiants in the LMC using an LTE
analysis.

###  5.2 Spectroscopic Analysis

The FSMS and VFTS data have been reported on extensively in the literature,
that included reports on the surface composition the B-type supergiant
populations [26, 27, 11]. Additionally, preliminary results for the FEROS
dataset are also available [24]. However these results are based on an
analysis using a TLUSTY [50] plane parallel NLTE model atmosphere grid [51]
that adopted a normal helium abundance, and did not take account of the impact
of the stellar wind on the spectrum. While the effect of neglecting the wind
may have only moderate impact on the derived CNO abundances [51], the helium
abundance provides important additional critical diagnostic value.
Furthermore, the VFTS results [11] focus solely on the abundance of nitrogen.
Therefore we have re-determined the stellar parameters and surface HeCNO
abundances using FASTWIND v10.6.4 [52, 29] that accounts explicitly for the
presence of a stellar wind. The approach was broadly similar in methodology to
that employed by [53, 54], with the exception that the C ii model atom of [55]
was adopted, and has been widely used in the determination of the surface
composition of O-type stars and B-type supergiants in Local Group Galaxies[2,
56, 53]. For our FSMS subsample of 13 objects we find good agreement with
published abundances [27], deriving mean CNO abundances of 7.69$\pm$0.11,
7.82$\pm$0.32 and 8.39$\pm$0.12 compared with 7.62$\pm 0.20$, 7.70$\pm$0.38
and 8.36$\pm$0.09. Similarly, for our sample VFTS sources we derive a mean
nitrogen abundance of 7.72$\pm$0.33, compared with a mean value of
7.70$\pm$0.38 for the same 14 sources [11]. Results are listed in Table 1,
along with projected rotational velocities (vsini) previously published for
the VFTS and FSMS subsamples. In addition we list vsini measurements for the
FEROS subsample, obtained using the same Fourier Transform technique [57].

###  5.3 Stellar physics assumptions

We summarise the main MESA input parameters used in all models presented in
this paper. Further details about their assumptions and justifications are
available in [33].

For convection, we use the Ledoux criterion and a mixing length parameter of
$\alpha_{\text{MLT}}=1.5$ and a semiconvection efficiency parameter of
$\alpha_{\text{sc}}=0.01$. The efficiencies of other mixing processes are
implemented as follows: thermohaline mixing with $\alpha_{\text{th}}=1$,
H-burning core overshooting with $\alpha_{\textrm{ov}}=0.335$ of the
convective boundary pressure scale height, rotational mixing efficiency of
$f_{\text{c}}=1/30$ and an efficiency parameter $f_{\mu}=0.1$ against
molecular weight gradients.

The adopted wind mass-loss recipes are based on the $T_{\textrm{eff}}$ being
hotter or cooler than the bi-stability jump temperature,
$T_{\textrm{jump}}\approx 25\,\textrm{kK}$ [58]. In the cold regime, the
maximum mass-loss rate between the [58] and [59] is used while in the ‘hot’
regime, the recipe depends on the surface hydrogen abundance ($X$): the [58]
recipe for $X>0.7$, the [60] recipe (reduced by a factor of 10) for $X<0.4$
and for $0.7>X>0.4$, a linearly interpolated rate between the two ‘hot’
recipes.

The nuclear network used is initially set to the basic net during the MS phase
and extended to include $\alpha$-capture chains for later stages (the
synthetic approx21 network).

###  5.4 The binary grid

The LMC Bonn grid is a series of binary star models computed with MESA
(Modules for experimental astrophysics) r8118 [37, 38, 39], in which both
stars are evolved simultaneously over an initial parameter space of primary
masses M${}_{\textrm{1,i}}=10,\ldots,40\,\textrm{M}_{\odot}$, mass ratio
q${}_{\textrm{i}}=\frac{\textrm{M}_{\textrm{2,i}}}{\textrm{M}_{\textrm{1,i}}}=0.10,\ldots,0.95$
and orbital period P${}_{\textrm{i}}=1.4,\ldots,3162.3\,\textrm{d}$ [33].
These models are computed using the ‘Roche-lobe’ binary mass-transfer scheme
and assumed to be tidally synchronized at the zero age main sequence (ZAMS).

Each binary model is initiated with both stars on the ZAMS and terminated
either at core carbon(C)-depletion or when a relevant criterion for CE is met,
such as: if the model reaches the maximum mass-loss rate that can be expelled
via radiation from a disk surrounding the accretor, or, when the mass-transfer
rate reaches $0.1\,\textrm{M}_{\odot}\textrm{yr}^{-1}$ (which is higher than
thermal-timescale mass transfer) or, at the first instance of mass transfer in
systems with q${}_{\textrm{i}}\leq\textrm{q}_{\textrm{crit}}=0.25$, which is
expected to be dynamically unstable.

Case B and Case C mass transfer is highly inefficient since a small amount
($\lesssim 10$%) of accreted mass is sufficient to spin up the secondary to
critical rotation. Hence most of the transferred mass is not accreted by the
secondary but is instead assumed to be lost from the system carrying the same
specific orbital angular momentum as the accretor.

The parameter space for late Case B and Case C mass transfer is affected by
several physics assumptions and their implementations. Our current findings
are for a rather inefficient semiconvection parameter of
$\alpha_{\textrm{sc}}=0.01$. The wind mass-loss rates may also be too high for
RSGs, which in turn affect their radius evolution and binary separations. The
‘Roche-lobe’ mass-transfer scheme used prevents the deep convective envelope
of RSGs from expanding during mass transfer and causes convergence failures in
some wide Case C systems of the grid. Including an extended atmosphere or
pulsations may also widen the Case C parameter space [61].

###  5.5 Our 1D merger method

In our 1D merger approximation, we take the exact structures of the post-MS
primary and MS secondary stars when the binary model is flagged to enter a CE
phase. We further assume that the MS star is always tidally disrupted by the
H-free core of the primary as it spirals in through the CE, before eventually
merging with it [36].

To simulate this in 1D, we accrete the averaged composition of the secondary
mass on the primary at a rate of 0.01$\,\textrm{M}_{\odot}$/yr and with the
same entropy and angular momentum as its surface, while simultaneously mixing
it up to a certain defined boundary inside the primary. By the end of the
merger, the resultant star has a chemically homogeneous envelope and a He core
mass determined by the penetration depth of the secondary. Although in reality
this is a 3D process, it is expected that the penetration and the subsequent
mixing occurs rapidly enough to justify our 1D approach.

We assume that the secondary star is always disrupted by the H-free core of
the primary during the merger in all our models. Although we do not consider
the angular momentum injected by the secondary as it spirals-in through the
CE, we do consider the envelope spin-up caused by the accretion of the
secondary.

For a first-order calculation for the maximum depth inside the He core that
the secondary stream can penetrate, M${}_{\textrm{r,max}}$, we assume that the
secondary stream ballistically hits the He core with a free-fall velocity
($v_{\textrm{st}}$) and ram pressure (P${}_{\textrm{st}}$) calculated as:

$\displaystyle
P_{\textrm{st}}=\frac{1}{2}\rho_{\textrm{st}}v_{\textrm{st}}^{2},\textrm{where}\,\,v_{\textrm{st}}=\sqrt{\cfrac{GM_{\textrm{H-free
core}}}{R_{\textrm{H-free core}}}}$ (1)

In the absence of shocks, we assume that the secondary star material conserves
its entropy when it is disrupted into a stream and approaches the core. In
this case, the equation of state can be written as $P\propto\rho^{\gamma}$,
where $\gamma$ is the adiabatic index. By assuming the stream is in lateral
pressure equilibrium with the surrounding gas, we can estimate the stream
density as:

$\displaystyle\rho_{\textrm{st}}$
$\displaystyle=\rho_{\textrm{2,central}}\Biggl{(}\frac{P_{\textrm{H-free
core}}}{P_{\textrm{2,central}}}\Biggl{)}^{\frac{1}{\gamma_{\textrm{H-free
core}}}}$ (2)

where, $P_{\textrm{2,central}}$ and $\rho_{\textrm{2,central}}$ are the
central pressure and density of the secondary star. We use these quantities
since the entropy at the centre of the secondary is the lowest and hence has
the highest chance to penetrate the He core. While the adiabatic index of the
stream may change as it changes pressure, we use $\gamma_{\textrm{H-free
core}}$ (its initial value as it enters the He core) for simplicity and
dimensional consistency. The location where $P_{\textrm{st}}$ equals the
pressure inside the He core corresponds close to the lowest mass-coordinate
that the stream can penetrate (M${}_{\textrm{r,max}}$). In reality, the stream
could be shocked or could ignite explosive nuclear burning while penetrating
the core, dramatically increasing its entropy before penetrating deep into the
core. For this reason, we expect that our formula for the penetration depth
corresponds to a conservative upper limit.

Figure 6: Composition of the primary star at the onset of merger (left) and of
the post-merger star at the end of the merger (right), corresponding to system
1 in section 2.2.

##  6 Extended data

Stellar parameters and abundances from FASTWIND analysis. YHe is the helium to hydrogen ratio by number, Vt is the adopted microturbulent velocity, and spectral types are taken from [30] and references in section 3.2. Also listed are the measured projected rotational velocities, vsini, for the FEROS sample and, for convenience, the values published for the VFTS and FSMS samples [11, 27]. |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---
Name | Spectral type | vsini | logL/L⊙ | Teff | log$g$ | YHe | Vt | [C/H] | [N/H] | [O/H]
|  | (km s-1) |  | (K) |  |  | (km s-1) |  |  |
FEROS
Sk-66 1 | B2Ia | 51 | 5.67$\pm$0.03 | $19862^{+479}_{-502}$ | $2.40^{+0.08}_{-0.09}$ | $0.08^{+0.02}_{-0.00}$ | $15.5^{+2.4}_{-1.8}$ | $7.46^{+0.12}_{-0.15}$ | $8.10^{+0.09}_{-0.11}$ | $8.37^{+0.10}_{-0.08}$
Sk-66 5 | B2.5Ia | 9 | 5.77$\pm$0.04 | $17224^{+294}_{-297}$ | $2.09^{+0.06}_{-0.07}$ | $0.08^{+0.01}_{-0.00}$ | $15.1^{+2.3}_{-1.8}$ | $7.62^{+0.13}_{-0.11}$ | $8.16^{+0.09}_{-0.15}$ | $8.65^{+0.08}_{-0.11}$
Sk-66 35 | BC1Ia | 23 | 5.74$\pm$0.03 | $22214^{+686}_{-939}$ | $2.67^{+0.11}_{-0.13}$ | $0.13^{+0.03}_{-0.03}$ | $10.3^{+2.8}_{-2.2}$ | $7.98^{+0.18}_{-0.15}$ | $7.42^{+0.18}_{-0.20}$ | $8.61^{+0.13}_{-0.09}$
Sk-66 106 | B1.5Ia | 49 | 5.66$\pm$0.03 | $21730^{+718}_{-758}$ | $2.65^{+0.12}_{-0.11}$ | $0.08^{+0.03}_{-0.00}$ | $20.4^{+4.0}_{-1.9}$ | $7.26^{+0.20}_{-0.14}$ | $8.03^{+0.09}_{-0.10}$ | $8.29^{+0.09}_{-0.08}$
Sk-66 118 | B2Iab | 8 | 5.54$\pm$0.03 | $20133^{+591}_{-677}$ | $2.47^{+0.11}_{-0.11}$ | $0.18^{+0.04}_{-0.03}$ | $16.5^{+2.5}_{-2.6}$ | $7.61^{+0.18}_{-0.15}$ | $8.26^{+0.13}_{-0.09}$ | $8.33^{+0.15}_{-0.09}$
Sk-67 14 | B1.5Ia | 44 | 5.72$\pm$0.03 | $21245^{+1003}_{-808}$ | $2.59^{+0.14}_{-0.12}$ | $0.13^{+0.03}_{-0.05}$ | $21.2^{+3.1}_{-3.6}$ | $7.66^{+0.12}_{-0.20}$ | $8.22^{+0.11}_{-0.08}$ | $8.19^{+0.11}_{-0.10}$
Sk-67 28 | B0.7Ia | 43 | 5.50$\pm$0.03 | $24645^{+509}_{-553}$ | $2.73^{+0.08}_{-0.11}$ | $0.14^{+0.03}_{-0.04}$ | $19.1^{+5.5}_{-3.8}$ | $7.38^{+0.13}_{-0.28}$ | $8.00^{+0.13}_{-0.16}$ | $8.41^{+0.10}_{-0.11}$
Sk-67 78 | B3Ia | 9 | 5.49$\pm$0.03 | $16638^{+435}_{-260}$ | $2.10^{+0.07}_{-0.07}$ | $0.08^{+0.00}_{-0.00}$ | $15.0^{+1.9}_{-2.0}$ | $7.73^{+0.09}_{-0.12}$ | $8.01^{+0.17}_{-0.11}$ | $8.34^{+0.14}_{-0.07}$
Sk-67 90 | B1.5Ia | 53 | 5.81$\pm$0.03 | $21446^{+476}_{-651}$ | $2.51^{+0.08}_{-0.09}$ | $0.12^{+0.03}_{-0.03}$ | $20.4^{+1.9}_{-2.6}$ | $7.67^{+0.12}_{-0.13}$ | $8.14^{+0.10}_{-0.07}$ | $8.22^{+0.08}_{-0.08}$
Sk-67 112 | B0.5Ia | 49 | 5.62$\pm$0.03 | $25573^{+718}_{-415}$ | $2.82^{+0.11}_{-0.09}$ | $0.13^{+0.02}_{-0.04}$ | $27.5^{+2.0}_{-4.4}$ | $7.40^{+0.20}_{-0.20}$ | $7.84^{+0.16}_{-0.18}$ | $8.22^{+0.11}_{-0.12}$
Sk-67 150 | B1.5Ia | 28 | 5.50$\pm$0.04 | $23246^{+521}_{-716}$ | $2.66^{+0.10}_{-0.11}$ | $0.18^{+0.04}_{-0.03}$ | $23.3^{+3.0}_{-4.0}$ | $7.42^{+0.17}_{-0.22}$ | $8.23^{+0.10}_{-0.09}$ | $8.24^{+0.10}_{-0.07}$
Sk-67 169 | B1Ia | 26 | 5.51$\pm$0.03 | $23798^{+566}_{-552}$ | $2.73^{+0.10}_{-0.09}$ | $0.10^{+0.03}_{-0.02}$ | $19.1^{+2.8}_{-2.7}$ | $7.83^{+0.15}_{-0.10}$ | $7.78^{+0.10}_{-0.19}$ | $8.35^{+0.10}_{-0.09}$
SK-67 172 | B2.5Iab | 15 | 5.47$\pm$0.03 | $19331^{+443}_{-449}$ | $2.53^{+0.08}_{-0.09}$ | $0.08^{+0.02}_{-0.00}$ | $16.2^{+1.6}_{-1.9}$ | $7.67^{+0.11}_{-0.08}$ | $7.95^{+0.10}_{-0.10}$ | $8.33^{+0.09}_{-0.09}$
Sk-67 173 | B0Ia | 53 | 5.70$\pm$0.03 | $27480^{+510}_{-670}$ | $2.95^{+0.09}_{-0.10}$ | $0.09^{+0.02}_{-0.02}$ | $25.0^{+3.7}_{-3.0}$ | $7.35^{+0.18}_{-0.20}$ | $7.69^{+0.18}_{-0.20}$ | $8.53^{+0.13}_{-0.17}$
Sk-67 206 | BN0.5Ia | 59 | 5.61$\pm$0.02 | $25067^{+609}_{-438}$ | $2.73^{+0.11}_{-0.08}$ | $0.19^{+0.03}_{-0.04}$ | $15.1^{+4.0}_{-2.4}$ | $7.48^{+0.22}_{-0.28}$ | $8.07^{+0.16}_{-0.14}$ | $8.35^{+0.09}_{-0.15}$
Sk-67 256 | BC1Ia | 47 | 5.66$\pm$0.05 | $21993^{+545}_{-603}$ | $2.59^{+0.10}_{-0.10}$ | $0.10^{+0.03}_{-0.02}$ | $14.8^{+2.5}_{-2.5}$ | $7.98^{+0.12}_{-0.12}$ | $7.37^{+0.20}_{-0.16}$ | $8.58^{+0.11}_{-0.07}$
Sk-68 26 | BC2Ia | 48 | 5.71$\pm$0.02 | $17765^{+328}_{-381}$ | $2.14^{+0.07}_{-0.08}$ | $0.08^{+0.03}_{-0.00}$ | $13.0^{+2.9}_{-2.3}$ | $7.80^{+0.10}_{-0.14}$ | $7.23^{+0.26}_{-0.17}$ | $8.49^{+0.10}_{-0.08}$
Sk-68 41 | B0.5Ia | 67 | 5.66$\pm$0.06 | $25158^{+405}_{-540}$ | $2.76^{+0.08}_{-0.09}$ | $0.11^{+0.02}_{-0.03}$ | $22.4^{+3.9}_{-4.0}$ | $7.35^{+0.21}_{-0.18}$ | $7.84^{+0.12}_{-0.18}$ | $8.59^{+0.12}_{-0.09}$
Sk-68 45 | B0Ia | 60 | 5.65$\pm$0.03 | $27545^{+644}_{-475}$ | $2.90^{+0.09}_{-0.08}$ | $0.08^{+0.02}_{-0.00}$ | $25.1^{+3.5}_{-2.9}$ | $7.76^{+0.19}_{-0.13}$ | $7.43^{+0.19}_{-0.36}$ | $8.57^{+0.12}_{-0.15}$
Sk-68 92 | B1.5Ia | 42 | 5.62$\pm$0.02 | $21238^{+765}_{-765}$ | $2.62^{+0.11}_{-0.13}$ | $0.08^{+0.02}_{-0.00}$ | $22.8^{+2.9}_{-3.0}$ | $7.52^{+0.12}_{-0.21}$ | $8.00^{+0.08}_{-0.10}$ | $8.37^{+0.09}_{-0.09}$
Sk-68 171 | B0.7Ia | 53 | 5.62$\pm$0.03 | $24147^{+695}_{-442}$ | $2.73^{+0.11}_{-0.10}$ | $0.15^{+0.04}_{-0.03}$ | $28.4^{+1.6}_{-5.5}$ | $7.60^{+0.20}_{-0.20}$ | $8.02^{+0.09}_{-0.19}$ | $8.57^{+0.09}_{-0.10}$
Sk-69 43 | B1Ia | 42 | 5.58$\pm$0.04 | $23419^{+706}_{-382}$ | $2.74^{+0.12}_{-0.07}$ | $0.10^{+0.01}_{-0.03}$ | $24.9^{+2.2}_{-3.3}$ | $7.80^{+0.08}_{-0.14}$ | $8.02^{+0.08}_{-0.10}$ | $8.36^{+0.09}_{-0.05}$
Sk-69 89 | B2.5Ia | 9 | 5.62$\pm$0.03 | $17959^{+445}_{-337}$ | $2.23^{+0.08}_{-0.08}$ | $0.09^{+0.04}_{-0.02}$ | $20.0^{+2.2}_{-2.5}$ | $7.72^{+0.12}_{-0.12}$ | $8.41^{+0.11}_{-0.09}$ | $8.31^{+0.09}_{-0.10}$
Sk-69 214 | B0.7Ia | 41 | 5.68$\pm$0.02 | $24507^{+533}_{-697}$ | $2.96^{+0.10}_{-0.15}$ | $0.12^{+0.02}_{-0.03}$ | $23.3^{+3.2}_{-3.0}$ | $7.82^{+0.15}_{-0.19}$ | $7.92^{+0.13}_{-0.11}$ | $8.24^{+0.07}_{-0.10}$
Sk-69 237 | B1IaNstr | 49 | 5.68$\pm$0.02 | $24568^{+486}_{-657}$ | $2.90^{+0.10}_{-0.10}$ | $0.10^{+0.01}_{-0.03}$ | $21.6^{+2.4}_{-2.5}$ | $7.48^{+0.21}_{-0.17}$ | $8.08^{+0.09}_{-0.10}$ | $8.14^{+0.06}_{-0.10}$
Sk-69 270 | B2.5Ia | 10 | 5.72$\pm$0.03 | $16858^{+257}_{-334}$ | $2.06^{+0.06}_{-0.08}$ | $0.08^{+0.02}_{-0.00}$ | $14.0^{+1.7}_{-2.6}$ | $7.64^{+0.11}_{-0.16}$ | $8.32^{+0.14}_{-0.11}$ | $8.60^{+0.09}_{-0.11}$
Sk-69 274 | B2.5Ia | 8 | 5.74$\pm$0.03 | $18104^{+326}_{-404}$ | $2.22^{+0.08}_{-0.07}$ | $0.08^{+0.04}_{-0.01}$ | $16.3^{+1.4}_{-2.9}$ | $7.83^{+0.13}_{-0.10}$ | $8.41^{+0.09}_{-0.14}$ | $8.35^{+0.11}_{-0.08}$
Sk-69 228 | BC1.5Ia | 66 | 5.55$\pm$0.03 | $21020^{+811}_{-674}$ | $2.51^{+0.12}_{-0.11}$ | $0.08^{+0.03}_{-0.00}$ | $23.0^{+3.4}_{-3.1}$ | $7.91^{+0.16}_{-0.16}$ | $7.15^{+0.18}_{-0.16}$ | $8.25^{+0.10}_{-0.13}$
Sk-70 78 | B0.7Ia | 40 | 5.83$\pm$0.02 | $23483^{+632}_{-645}$ | $2.73^{+0.11}_{-0.12}$ | $0.08^{+0.03}_{-0.00}$ | $24.1^{+3.4}_{-3.1}$ | $7.58^{+0.18}_{-0.20}$ | $7.81^{+0.11}_{-0.16}$ | $8.39^{+0.09}_{-0.07}$
SK-70 111 | B0.5Ia | 10 | 5.80$\pm$0.05 | $26014^{+481}_{-476}$ | $2.93^{+0.09}_{-0.08}$ | $0.14^{+0.03}_{-0.02}$ | $25.7^{+1.9}_{-2.7}$ | $7.30^{+0.20}_{-0.13}$ | $7.95^{+0.12}_{-0.09}$ | $8.15^{+0.10}_{-0.07}$
Sk-70 120 | B1Ia | 36 | 5.69$\pm$0.02 | $22693^{+472}_{-730}$ | $2.58^{+0.09}_{-0.10}$ | $0.08^{+0.03}_{-0.00}$ | $21.7^{+3.0}_{-2.9}$ | $7.64^{+0.11}_{-0.18}$ | $7.81^{+0.09}_{-0.15}$ | $8.36^{+0.08}_{-0.08}$
Sk-71 42 | B2Ia | 43 | 5.83$\pm$0.02 | $19522^{+486}_{-534}$ | $2.41^{+0.09}_{-0.10}$ | $0.09^{+0.04}_{-0.01}$ | $18.9^{+2.2}_{-2.4}$ | $7.54^{+0.10}_{-0.19}$ | $8.34^{+0.12}_{-0.08}$ | $8.32^{+0.09}_{-0.10}$
VFTS
VFTS003 | B1Ia+ | 48 | 5.91$\pm$0.05 | $22546^{+513}_{-436}$ | $2.64^{+0.09}_{-0.07}$ | $0.08^{+0.03}_{-0.00}$ | $20.5^{+2.1}_{-2.8}$ | $7.15^{+0.31}_{-0.07}$ | $8.01^{+0.11}_{-0.13}$ | $8.29^{+0.09}_{-0.08}$
VFTS028 | B0.7IaNwk | 50 | 5.73$\pm$0.10 | $24495^{+584}_{-332}$ | $2.67^{+0.09}_{-0.07}$ | $0.08^{+0.03}_{-0.00}$ | $17.4^{+2.7}_{-2.2}$ | $7.42^{+0.19}_{-0.23}$ | $7.54^{+0.16}_{-0.27}$ | $8.47^{+0.12}_{-0.09}$
VFTS069 | B0.7Ib-Ia | 46 | 5.61$\pm$0.09 | $25072^{+325}_{-559}$ | $2.83^{+0.08}_{-0.10}$ | $0.13^{+0.05}_{-0.03}$ | $18.8^{+3.0}_{-4.4}$ | $7.28^{+0.22}_{-0.16}$ | $7.96^{+0.21}_{-0.15}$ | $8.35^{+0.10}_{-0.10}$
VFTS082 | B0.5Ib-Iab | 49 | 5.28$\pm$0.12 | $26713^{+596}_{-402}$ | $3.02^{+0.11}_{-0.07}$ | $0.11^{+0.05}_{-0.02}$ | $15.9^{+3.3}_{-4.1}$ | $7.44^{+0.17}_{-0.27}$ | $8.08^{+0.20}_{-0.19}$ | $8.32^{+0.08}_{-0.11}$
VFTS232 | B3Ia | 42 | 5.01$\pm$0.06 | $17277^{+326}_{-278}$ | $2.48^{+0.07}_{-0.05}$ | $0.08^{+0.01}_{-0.00}$ | $5.0^{+1.9}_{-0.0}$ | $7.28^{+0.10}_{-0.09}$ | $7.66^{+0.09}_{-0.10}$ | $8.20^{+0.10}_{-0.10}$
VFTS302 | B1.5Ib | 34 | 4.90$\pm$0.07 | $23063^{+517}_{-709}$ | $3.14^{+0.02}_{-0.16}$ | $0.08^{+0.04}_{-0.00}$ | $10.2^{+1.6}_{-2.9}$ | $7.54^{+0.15}_{-0.15}$ | $7.87^{+0.21}_{-0.13}$ | $8.42^{+0.09}_{-0.06}$
VFTS315 | B1Ib | 31 | 4.73$\pm$0.08 | $24899^{+485}_{-772}$ | $3.28^{+0.03}_{-0.09}$ | $0.10^{+0.02}_{-0.02}$ | $11.4^{+1.6}_{-2.2}$ | $7.41^{+0.20}_{-0.16}$ | $7.49^{+0.14}_{-0.25}$ | $8.58^{+0.05}_{-0.09}$
VFTS431 | B1.5IaNstr | 41 | 5.80$\pm$0.10 | $20824^{+499}_{-611}$ | $2.43^{+0.10}_{-0.10}$ | $0.15^{+0.03}_{-0.04}$ | $14.8^{+1.4}_{-2.8}$ | $7.53^{+0.15}_{-0.18}$ | $8.31^{+0.14}_{-0.11}$ | $8.53^{+0.08}_{-0.12}$
VFTS533 | B1.5Ia+pNwk | 57 | 5.92$\pm$0.07 | $19210^{+579}_{-429}$ | $2.31^{+0.12}_{-0.08}$ | $0.09^{+0.03}_{-0.01}$ | $15.0^{+2.6}_{-1.7}$ | $7.40^{+0.17}_{-0.21}$ | $7.65^{+0.15}_{-0.20}$ | $8.66^{+0.10}_{-0.10}$
VFTS590 | B0.7Iab | 60 | 5.85$\pm$0.07 | $25318^{+397}_{-370}$ | $2.72^{+0.07}_{-0.07}$ | $0.12^{+0.05}_{-0.03}$ | $14.6^{+5.1}_{-2.1}$ | $7.23^{+0.23}_{-0.13}$ | $8.16^{+0.17}_{-0.17}$ | $8.15^{+0.11}_{-0.12}$
VFTS696 | B0.7Ib-abNwk | 53 | 5.61$\pm$0.08 | $24628^{+578}_{-414}$ | $2.76^{+0.09}_{-0.08}$ | $0.08^{+0.03}_{-0.00}$ | $15.1^{+3.2}_{-2.1}$ | $7.36^{+0.21}_{-0.20}$ | $7.49^{+0.15}_{-0.24}$ | $8.51^{+0.09}_{-0.10}$
VFTS732 | B1.5IapNwk | 45 | 5.61$\pm$0.08 | $22404^{+492}_{-059}$ | $2.69^{+0.10}_{-0.15}$ | $0.08^{+0.04}_{-0.00}$ | $13.5^{+2.1}_{-2.4}$ | $7.45^{+0.20}_{-0.22}$ | $7.39^{+0.15}_{-0.24}$ | $8.46^{+0.09}_{-0.09}$
VFTS831 | B5Ia | 41 | 5.25$\pm$0.06 | $16220^{+215}_{-122}$ | $2.17^{+0.05}_{-0.05}$ | $0.08^{+0.03}_{-0.00}$ | $8.5^{+1.1}_{-1.3}$ | $7.45^{+0.09}_{-0.08}$ | $7.78^{+0.23}_{-0.35}$ | $8.45^{+0.12}_{-0.10}$
VFTS867 | B1IbNwk | 32 | 4.99$\pm$0.07 | $26274^{+248}_{-471}$ | $3.37^{+0.01}_{-0.09}$ | $0.09^{+0.03}_{-0.01}$ | $13.2^{+2.1}_{-1.4}$ | $7.49^{+0.13}_{-0.25}$ | $7.27^{+0.17}_{-0.17}$ | $8.54^{+0.07}_{-0.08}$
FSMS
N11-001 | B2Ia | 52 | 5.81$\pm$0.03 | $18594^{+527}_{-882}$ | $2.27^{+0.09}_{-0.13}$ | $0.08^{+0.04}_{-0.00}$ | $15.5^{+2.7}_{-2.3}$ | $7.53^{+0.17}_{-0.15}$ | $8.26^{+0.13}_{-0.14}$ | $8.29^{+0.16}_{-0.13}$
N11-002 | B3Ia | 53 | 5.45$\pm$0.03 | $17035^{+568}_{-343}$ | $2.21^{+0.09}_{-0.09}$ | $0.08^{+0.04}_{-0.00}$ | $12.4^{+2.3}_{-2.2}$ | $7.82^{+0.15}_{-0.14}$ | $8.14^{+0.19}_{-0.17}$ | $8.26^{+0.18}_{-0.16}$
N11-008 | B0.5Ia | 43 | 5.39$\pm$0.03 | $25540^{+682}_{-545}$ | $2.87^{+0.11}_{-0.11}$ | $0.09^{+0.05}_{-0.02}$ | $16.8^{+5.9}_{-3.5}$ | $7.52^{+0.19}_{-0.16}$ | $7.75^{+0.16}_{-0.18}$ | $8.31^{+0.10}_{-0.11}$
N11-015 | B0.7Ib | 58 | 5.30$\pm$0.03 | $26082^{+665}_{-619}$ | $3.03^{+0.13}_{-0.09}$ | $0.15^{+0.06}_{-0.03}$ | $9.6^{+2.8}_{-2.4}$ | $7.72^{+0.19}_{-0.17}$ | $7.52^{+0.21}_{-0.35}$ | $8.56^{+0.14}_{-0.10}$
N11-016 | B1Ib | 54 | 5.19$\pm$0.03 | $22832^{+876}_{-671}$ | $2.74^{+0.12}_{-0.10}$ | $0.08^{+0.05}_{-0.01}$ | $13.7^{+3.4}_{-2.0}$ | $7.67^{+0.17}_{-0.12}$ | $7.91^{+0.16}_{-0.14}$ | $8.37^{+0.11}_{-0.12}$
N11-024 | B1Ib | 30 | 5.05$\pm$0.03 | $23671^{+719}_{-986}$ | $2.91^{+0.13}_{-0.14}$ | $0.13^{+0.03}_{-0.04}$ | $11.0^{+3.2}_{-2.8}$ | $7.65^{+0.16}_{-0.15}$ | $8.02^{+0.16}_{-0.16}$ | $8.32^{+0.12}_{-0.11}$
N11-036 | B0.5Ib | 30 | 5.06$\pm$0.16 | $25118^{+777}_{-690}$ | $3.10^{+0.14}_{-0.11}$ | $0.08^{+0.05}_{-0.01}$ | $11.1^{+2.6}_{-3.1}$ | $7.59^{+0.12}_{-0.17}$ | $7.85^{+0.16}_{-0.15}$ | $8.32^{+0.14}_{-0.09}$
N11-054 | B1Ib | 45 | 5.14$\pm$0.14 | $25600^{+877}_{-811}$ | $3.15^{+0.12}_{-0.14}$ | $0.08^{+0.04}_{-0.00}$ | $8.2^{+2.4}_{-2.6}$ | $7.80^{+0.12}_{-0.17}$ | $7.10^{+0.20}_{-0.20}$ | $8.55^{+0.08}_{-0.13}$
NGC2004-010 | B2.5Iab | 30 | 5.16$\pm$0.03 | $18038^{+490}_{-537}$ | $2.45^{+0.09}_{-0.11}$ | $0.08^{+0.05}_{-0.01}$ | $12.5^{+2.6}_{-2.6}$ | $7.63^{+0.11}_{-0.14}$ | $8.18^{+0.13}_{-0.22}$ | $8.19^{+0.19}_{-0.15}$
NGC2004-011 | B1.5Ia | 40 | 5.23$\pm$0.03 | $22531^{+982}_{-785}$ | $2.73^{+0.14}_{-0.12}$ | $0.12^{+0.05}_{-0.03}$ | $13.8^{+3.1}_{-3.2}$ | $7.67^{+0.14}_{-0.15}$ | $7.83^{+0.19}_{-0.15}$ | $8.38^{+0.15}_{-0.12}$
NGC2004-012 | B1.5Iab | 40 | 4.99$\pm$0.05 | $23801^{+845}_{-855}$ | $2.99^{+0.14}_{-0.13}$ | $0.12^{+0.03}_{-0.05}$ | $9.8^{+2.6}_{-3.1}$ | $7.75^{+0.15}_{-0.17}$ | $7.89^{+0.19}_{-0.14}$ | $8.42^{+0.11}_{-0.12}$
NGC2004-021 | B1.5Ib | 40 | 4.83$\pm$0.06 | $22795^{+960}_{-724}$ | $2.97^{+0.12}_{-0.13}$ | $0.10^{+0.04}_{-0.03}$ | $9.3^{+2.8}_{-2.9}$ | $7.91^{+0.16}_{-0.15}$ | $7.34^{+0.20}_{-0.20}$ | $8.56^{+0.11}_{-0.12}$
NGC2004-022 | B1.5Ib | 30 | 4.77$\pm$0.03 | $23928^{+928}_{-890}$ | $3.19^{+0.14}_{-0.14}$ | $0.08^{+0.04}_{-0.00}$ | $8.4^{+2.7}_{-1.8}$ | $7.65^{+0.13}_{-0.18}$ | $7.81^{+0.15}_{-0.17}$ | $8.48^{+0.11}_{-0.12}$
Parameters of the Case B merger models: M${}_{\textrm{1,i}}$, M${}_{\textrm{2,i}}$ and P${}_{\textrm{i}}$ are the initial parameters of the binary system. M${}_{\textrm{1,merger}}$ is the mass of the primary at the time of the merger. BSG parameters are averaged during the BSG phase of the models, and are as reported in Figs. 2 and 3. M${}_{\textrm{C-dep}}$ is the mass at core C-depletion, close to its expected value at explosion and SN prog. indicates the expected structure of the star at explosion: ‘BSG’ for progenitors with log T$\textrm{eff}\geq 4.0$ and ‘RSG’ for progenitors which have log T$\textrm{eff}<4.0$. Where convergence was possible until core C-depletion, we have evolved a given binary system at both limits of core penetration. |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
M${}_{\textrm{1,i}}$ | M${}_{\textrm{2,i}}$ | P${}_{\textrm{i}}$ | f${}_{\textrm{c}}$ | M${}_{\textrm{1,merger}}$ | M${}_{\textrm{BSG}}$ | M${}_{\textrm{H-free core, BSG}}$ | t${}_{\textrm{BSG}}$ | log g${}_{\textrm{BSG}}$ | log $\left(\displaystyle\frac{L_{\textrm{BSG}}}{L_{\odot}}\right)$ | log T${}_{\textrm{eff,BSG}}$ | $\epsilon_{\textrm{C,BSG}}$ | $\epsilon_{\textrm{N,BSG}}$ | $\epsilon_{\textrm{O,BSG}}$ | $\epsilon_{\textrm{He,BSG}}$ | M${}_{\textrm{C-dep}}$ | SN prog.
($\,\textrm{M}_{\odot}$) | ($\,\textrm{M}_{\odot}$) | (d) | (%) | ($\,\textrm{M}_{\odot}$) | ($\,\textrm{M}_{\odot}$) | ($\,\textrm{M}_{\odot}$) | (Myr) | (cm/s2) |  | (K) |  |  |  |  | ($\,\textrm{M}_{\odot}$) |
10.0 | 1.0 | 24 | 0.0 | 10 | 10 | 2.2 | 2.0 | 2.2 | 4.4 | 4.0 | 7.6 | 7.9 | 8.3 | 0.14 | 10 | RSG
11.2 | 9.0 | 1413 | 0.06 | 8 | 17 | 2.7 | 1.4 | 3.4 | 5.0 | 4.4 | 7.4 | 8.2 | 8.2 | 0.17 | 17 | BSG
11.2 | 9.0 | 1413 | 0.0 | 8 | 17 | 2.8 | 1.3 | 3.5 | 5.0 | 4.5 | 7.4 | 8.2 | 8.2 | 0.17 | 17 | BSG
14.1 | 5.7 | 1884 | 0.1 | 11 | 17 | 3.6 | 1.0 | 3.3 | 5.0 | 4.4 | 7.4 | 8.2 | 8.2 | 0.15 | 17 | BSG
14.1 | 12.0 | 1995 | 0.07 | 10 | 22 | 3.6 | 1.0 | 3.2 | 5.3 | 4.4 | 7.4 | 8.3 | 8.2 | 0.2 | 22 | BSG
14.1 | 5.7 | 1884 | 0.0 | 11 | 17 | 4.0 | 0.9 | 3.3 | 4.9 | 4.4 | 7.4 | 8.1 | 8.2 | 0.14 | 17 | BSG
14.1 | 12.0 | 1995 | 0.0 | 10 | 22 | 4.0 | 0.9 | 3.1 | 5.3 | 4.4 | 7.4 | 8.3 | 8.2 | 0.2 | 22 | BSG
15.8 | 1.1 | 562 | 0.25 | 16 | 17 | 3.5 | 1.0 | 3.2 | 5.0 | 4.4 | 7.6 | 8.2 | 8.3 | 0.18 | 17 | BSG
15.8 | 1.9 | 596 | 0.2 | 16 | 18 | 3.7 | 1.0 | 3.1 | 5.1 | 4.4 | 7.6 | 8.1 | 8.3 | 0.17 | 18 | BSG
15.8 | 2.7 | 668 | 0.17 | 16 | 18 | 3.9 | 0.9 | 3.4 | 5.1 | 4.4 | 7.5 | 8.1 | 8.3 | 0.16 | 18 | BSG
15.8 | 10.3 | 2371 | 0.08 | 13 | 24 | 4.4 | 0.8 | 3.3 | 5.3 | 4.4 | 7.4 | 8.2 | 8.2 | 0.16 | 24 | BSG
15.8 | 1.6 | 47 | 0.0 | 16 | 16 | 3.4 | 1.0 | 2.6 | 5.0 | 4.2 | 7.6 | 8.1 | 8.3 | 0.18 | 16 | BSG
15.8 | 1.9 | 596 | 0.12 | 16 | 17 | 4.7 | 0.7 | 3.0 | 5.0 | 4.3 | 7.5 | 8.0 | 8.3 | 0.14 | 17 | BSG
15.8 | 2.7 | 668 | 0.0 | 16 | 18 | 3.3 | 1.1 | 3.6 | 5.1 | 4.5 | 7.6 | 8.1 | 8.3 | 0.18 | 18 | BSG
15.8 | 10.3 | 2371 | 0.0 | 13 | 24 | 4.8 | 0.7 | 3.2 | 5.3 | 4.4 | 7.4 | 8.2 | 8.2 | 0.15 | 24 | BSG
15.8 | 1.1 | 562 | 0.0 | 16 | 17 | 4.7 | 0.7 | 2.9 | 5.0 | 4.3 | 7.6 | 8.0 | 8.3 | 0.15 | 16 | BSG
15.8 | 1.1 | 1585 | 0.0 | 16 | 17 | 4.7 | 0.7 | 2.9 | 5.0 | 4.3 | 7.6 | 8.0 | 8.3 | 0.14 | 16 | BSG
15.8 | 1.6 | 47 | 0.0 | 16 | 16 | 4.6 | 0.8 | 2.3 | 5.0 | 4.2 | 7.6 | 8.0 | 8.3 | 0.14 | 16 | BSG
17.8 | 8.9 | 2661 | 0.09 | 16 | 25 | 5.2 | 0.7 | 3.2 | 5.3 | 4.4 | 7.4 | 8.2 | 8.2 | 0.15 | 25 | BSG
17.8 | 8.9 | 2661 | 0.0 | 16 | 25 | 5.7 | 0.6 | 3.1 | 5.3 | 4.4 | 7.4 | 8.1 | 8.2 | 0.14 | 25 | BSG
17.8 | 1.2 | 631 | 0.0 | 18 | 18 | 5.6 | 0.6 | 2.9 | 5.1 | 4.3 | 7.6 | 8.0 | 8.3 | 0.15 | 18 | BSG
17.8 | 1.2 | 1995 | 0.0 | 18 | 18 | 5.7 | 0.6 | 2.8 | 5.1 | 4.3 | 7.6 | 8.0 | 8.3 | 0.14 | 18 | BSG
20.0 | 11.0 | 2985 | 0.09 | 18 | 29 | 6.0 | 0.6 | 3.1 | 5.5 | 4.4 | 7.4 | 8.2 | 8.2 | 0.16 | 29 | BSG
20.0 | 11.0 | 2985 | 0.0 | 18 | 29 | 6.7 | 0.6 | 3.0 | 5.5 | 4.4 | 7.4 | 8.2 | 8.2 | 0.15 | 29 | BSG
20.0 | 15.0 | 3162 | 0.08 | 18 | 33 | 6.1 | 0.6 | 3.2 | 5.6 | 4.5 | 7.4 | 8.2 | 8.2 | 0.19 | 33 | BSG
20.0 | 15.0 | 3162 | 0.0 | 18 | 33 | 6.7 | 0.5 | 3.1 | 5.6 | 4.4 | 7.4 | 8.2 | 8.2 | 0.18 | 33 | BSG
20.0 | 1.0 | 708 | 0.0 | 20 | 20 | 6.6 | 0.6 | 2.7 | 5.2 | 4.3 | 7.6 | 8.1 | 8.3 | 0.15 | 20 | BSG
20.0 | 1.0 | 2113 | 0.0 | 20 | 20 | 6.7 | 0.6 | 2.6 | 5.2 | 4.3 | 7.6 | 8.1 | 8.3 | 0.15 | 20 | BSG
20.0 | 1.4 | 750 | 0.0 | 20 | 20 | 6.6 | 0.6 | 2.8 | 5.2 | 4.3 | 7.6 | 8.1 | 8.3 | 0.15 | 20 | BSG
20.0 | 1.4 | 2239 | 0.0 | 20 | 21 | 6.7 | 0.6 | 2.7 | 5.2 | 4.3 | 7.6 | 8.1 | 8.3 | 0.15 | 20 | BSG
20.0 | 2.0 | 16 | 0.0 | 20 | 20 | 6.5 | 0.6 | 2.5 | 5.2 | 4.2 | 7.6 | 7.9 | 8.3 | 0.14 | 20 | BSG
25.1 | 1.3 | 1000 | 0.0 | 24 | 25 | 9.2 | 0.5 | 2.2 | 5.4 | 4.2 | 7.6 | 8.1 | 8.3 | 0.16 | 25 | BSG
25.1 | 1.3 | 2661 | 0.0 | 24 | 25 | 9.3 | 0.5 | 2.4 | 5.5 | 4.3 | 7.5 | 8.1 | 8.3 | 0.16 | 24 | BSG
31.6 | 3.2 | 27 | 0.25 | 30 | 30 | 8.9 | 0.5 | 2.7 | 5.6 | 4.4 | 7.6 | 8.2 | 8.2 | 0.23 | 30 | BSG
31.6 | 13.3 | 28 | 0.28 | 23 | 29 | 10.8 | 0.4 | 2.5 | 5.6 | 4.3 | 7.2 | 8.4 | 8.1 | 0.21 | 29 | BSG
31.6 | 13.3 | 28 | 0.0 | 23 | 28 | 12.6 | 0.4 | 2.4 | 5.6 | 4.3 | 7.2 | 8.3 | 8.1 | 0.17 | 27 | RSG
31.6 | 7.9 | 126 | 0.0 | 30 | 32 | 12.6 | 0.4 | 2.2 | 5.6 | 4.2 | 7.4 | 8.1 | 8.2 | 0.15 | 31 | RSG
31.6 | 3.2 | 27 | 0.0 | 30 | 30 | 12.4 | 0.4 | 2.4 | 5.6 | 4.3 | 7.5 | 8.1 | 8.2 | 0.16 | 28 | RSG
35.5 | 17.7 | 668 | 0.12 | 19 | 36 | 13.1 | 0.4 | 2.4 | 5.8 | 4.3 | 7.0 | 8.4 | 8.1 | 0.25 | 35 | BSG
35.5 | 17.7 | 668 | 0.0 | 19 | 35 | 14.7 | 0.4 | 2.1 | 5.8 | 4.2 | 7.0 | 8.4 | 8.1 | 0.21 | 34 | BSG
39.8 | 20.7 | 631 | 0.14 | 22 | 40 | 15.1 | 0.4 | 2.1 | 5.9 | 4.3 | 7.0 | 8.5 | 8.1 | 0.27 | 40 | BSG
39.8 | 31.8 | 376 | 0.0 | 21 | 50 | 16.8 | 0.3 | 2.3 | 6.1 | 4.3 | 7.2 | 8.6 | 8.1 | 0.38 | 46 | BSG
39.8 | 4.0 | 30 | 0.0 | 37 | 36 | 16.8 | 0.3 | 2.4 | 5.8 | 4.3 | 7.4 | 8.2 | 8.2 | 0.18 | 33 | RSG
39.8 | 10.0 | 38 | 0.0 | 37 | 36 | 17.0 | 0.3 | 2.3 | 5.8 | 4.3 | 7.3 | 8.2 | 8.2 | 0.16 | 33 | RSG
39.8 | 20.7 | 631 | 0.0 | 22 | 41 | 17.0 | 0.3 | 2.3 | 5.9 | 4.3 | 7.0 | 8.4 | 8.1 | 0.23 | 39 | RSG

## Data availability

Observational sample and model parameters can be made available on request.
Please contact AM for the same.

## Code availability

The code used to calculate the merger models can be shared on reasonable
request. Please contact AM for the same.

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7-3 | Mean | 0.45 | 0.09 | 0.23 | 0.36 | 0.90∗∗ | 0.29 | 0.15 | 0.30 | 0.13 | 0.98∗∗
| T-stat | [1.30] | [0.28] | [0.56] | [0.82] | [2.34] | [0.77] | [0.37] | [0.65] | [0.25] | [2.08]
12-3 | Mean | 0.38 | 0.25 | 1.03∗ | 0.47 | 1.25∗∗∗ | 0.22 | 0.16 | 1.06∗ | 0.62 | 1.52∗∗∗
| T-stat | [1.05] | [0.67] | [1.69] | [0.93] | [2.97] | [0.51] | [0.39] | [1.70] | [1.08] | [3.00]
12-7 | Mean | 0.74∗∗ | 0.91∗∗ | 0.83∗ | 0.78 | 0.08 | 0.20 | 0.89∗∗ | 0.69 | 0.97∗ | 0.04
| T-stat | [2.03] | [2.38] | [1.87] | [1.54] | [0.17] | [0.50] | [2.14] | [1.47] | [1.72] | [0.08]

Table 34: Returns of long-short decile portfolios, formed on customer momentum
across different quintiles of relative customer size

This table reports the returns of long-short decile portfolios, formed on the
returns of customers’ portfolio at a given lag. Type represents a lag, which
is defined as (Beginning of return window as a number of months before the
current month)-(End of return window as a number of months before the current
month). This table reports the results separately for 5 quintiles of relative
customer size, defined as mean customer size/supplier size. So for each month
I divide the firms into 5 quintiles on relative customer size and then perform
sort on past customers’ returns within each size quintile. Reported results
are the returns of long-short decile portfolios, formed on the returns of
customers’ portfolio at a given lag. For example, 0.451 in the column ”Q1”
means that the return of long-short decile portfolio (largest decile by past
month customers’ returns minus the smallest decile) is equal to 0.451 within
the subsample of supplier-customer pairs with low (first quintile) relative
customer size.

Weights | Equal | Value |
---|---|---|---
Type | Statistic | Q1 | Q2 | Q3 | Q4 | Q5 | Q1 | Q2 | Q3 | Q4 | Q5
1-1 | Mean | 0.45 | 1.34∗∗∗ | 0.62 | 0.96∗∗ | 1.61∗∗∗ | 0.51 | 1.06∗∗ | 0.55 | 1.38∗∗∗ | 1.12∗∗
| T-stat | [1.19] | [2.93] | [1.33] | [2.08] | [3.32] | [1.11] | [2.08] | [1.00] | [2.66] | [2.12]
2-1 | Mean | 0.57 | 0.82∗ | 1.36∗∗ | 1.15∗∗∗ | 0.82 | 0.59 | 1.71∗∗∗ | 1.08∗ | 1.05∗∗ | 1.03∗
| T-stat | [1.52] | [1.83] | [2.31] | [2.58] | [1.61] | [1.27] | [3.38] | [1.82] | [2.14] | [1.79]
3-1 | Mean | 0.71 | 0.26 | 0.72 | 1.08∗∗ | 1.46∗∗∗ | 1.03∗∗ | 0.82∗ | 0.91 | 1.28∗∗ | 1.90∗∗∗
| T-stat | [1.59] | [0.61] | [1.17] | [2.20] | [2.75] | [2.04] | [1.68] | [1.34] | [2.22] | [3.10]
4-1 | Mean | 1.12∗∗ | 0.78∗ | 0.66 | 0.59 | 1.36∗∗ | 1.30∗∗∗ | 0.62 | 0.58 | 0.23 | 1.70∗∗∗
| T-stat | [2.34] | [1.76] | [1.25] | [1.13] | [2.49] | [2.60] | [1.22] | [1.08] | [0.39] | [2.66]
7-1 | Mean | 0.86∗ | 0.35 | 0.69 | 0.26 | 1.72∗∗∗ | 0.61 | 0.37 | 0.42 | 0.01 | 2.08∗∗∗
| T-stat | [1.74] | [0.73] | [1.26] | [0.48] | [3.12] | [1.21] | [0.69] | [0.71] | [0.01] | [3.44]
12-1 | Mean | 1.38∗∗ | 0.90∗ | 2.10∗ | 0.89 | 1.11∗ | 1.21∗∗ | 1.24∗∗ | 2.10∗ | 0.68 | 1.37∗
| T-stat | [2.56] | [1.75] | [1.93] | [1.19] | [1.73] | [1.97] | [2.42] | [1.88] | [0.88] | [1.90]
2-2 | Mean | 0.26 | -0.13 | 0.73 | 0.70 | 0.00 | -0.15 | 0.76 | 1.01 | 0.54 | 0.00
| T-stat | [0.68] | [-0.30] | [1.18] | [1.43] | [-0.01] | [-0.33] | [1.56] | [1.50] | [0.97] | [-0.01]
4-2 | Mean | 0.94∗∗ | 0.37 | -0.10 | 0.72 | 0.54 | 0.86∗ | -0.12 | 0.15 | -0.05 | 1.01∗
| T-stat | [2.03] | [0.80] | [-0.20] | [1.32] | [1.05] | [1.76] | [-0.23] | [0.27] | [-0.08] | [1.69]
7-2 | Mean | 0.85∗ | -0.25 | 0.95 | 0.16 | 1.68∗∗∗ | 0.68 | -0.11 | 0.89 | -0.04 | 2.08∗∗∗
| T-stat | [1.65] | [-0.47] | [1.34] | [0.27] | [2.70] | [1.25] | [-0.18] | [1.19] | [-0.06] | [3.02]
12-2 | Mean | 0.79 | 0.23 | 0.62 | 1.13 | 1.66∗∗ | 0.66 | 0.79 | 0.52 | 1.18 | 1.48∗∗
| T-stat | [1.41] | [0.41] | [1.03] | [1.54] | [2.56] | [1.04] | [1.40] | [0.86] | [1.53] | [2.08]
3-3 | Mean | 0.35 | -0.28 | -0.16 | 0.63 | 0.53 | 0.25 | 0.14 | -0.10 | 0.27 | 0.94
| T-stat | [0.76] | [-0.65] | [-0.33] | [1.21] | [1.00] | [0.56] | [0.29] | [-0.18] | [0.44] | [1.53]
7-3 | Mean | 0.40 | -0.57 | 0.79 | -0.29 | 0.69 | 0.70 | -0.54 | 0.66 | -0.42 | 1.09∗
| T-stat | [0.78] | [-1.09] | [1.14] | [-0.48] | [1.22] | [1.33] | [-0.97] | [0.91] | [-0.64] | [1.72]
12-3 | Mean | 0.60 | 0.09 | 0.01 | 0.92 | 0.72 | 0.32 | 0.29 | 0.02 | 1.00 | 0.60
| T-stat | [1.18] | [0.17] | [0.01] | [1.23] | [1.20] | [0.56] | [0.52] | [0.02] | [1.29] | [0.86]
12-7 | Mean | 0.73 | 0.78 | 0.50 | 1.65∗∗ | 0.25 | 0.60 | 0.66 | 0.41 | 1.93∗∗ | 0.38
| T-stat | [1.43] | [1.38] | [0.75] | [2.05] | [0.38] | [1.04] | [1.13] | [0.60] | [2.21] | [0.56]

Table 35: Fama-MacBeth regressions with interaction term

This table reports the results of Fama-MacBeth regression of returns on the
characteristics, including the interaction term between past customers’
returns and relative customers’ size. Cmom1-1 is returns of customers’
portfolio during the last month. Mom1-1 and Mom12-2 are past returns of stock
at lags 1-1 and 12-2. OP is operating profitability. CAR3 is abnormal return
during 3-day window, centered around earnings announcement. Standardized
unexpected earnings (SUE) are defined as the most recent year-to-year change
in earnings per share (EPS), divided by the standard deviation of the changes
in earnings during the recent eight announcements. Cmom1-1_cme is the
interaction term between returns of customers’ portfolio in the last month and
customer’s size. Cmom1-1_rme is the interaction term between returns of
customers’ portfolio in the last month and customer’s relative size. Relative
size of customer is defined as customer’s size/supplier’s size.

| Dependent variable:
---|---
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Mom12-2 |  |  |  |  |  | 0.002 | -0.004 | -0.004 | -0.004
|  |  |  |  |  | [0.85] | [-1.53] | [-1.45] | [-1.40]
Mom1-1 |  |  |  |  |  |  -0.04∗∗∗ |  -0.06∗∗∗ |  -0.06∗∗∗ |  -0.06∗∗∗
|  |  |  |  |  | [-5.13] | [-6.83] | [-6.99] | [-7.21]
Cmom |  0.05∗∗∗ |  0.05∗∗∗ |  0.04∗∗∗ | -0.07 | -0.04 |  0.05∗∗∗ |  0.05∗∗∗ | -0.01 | -0.01
| [3.39] | [3.33] | [3.28] | [-0.96] | [-1.15] | [3.60] | [3.78] | [-0.20] | [-0.38]
Cmom*cME |  |  |  |  0.01∗ |  |  |  | 0.01 |
|  |  |  | [1.71] |  |  |  | [0.96] |
Cmom*rME |  |  |  |  |  0.05∗∗ |  |  |  |  0.04∗
|  |  |  |  | [2.39] |  |  |  | [1.82]
log(ME) |  |  -0.16∗∗∗ |  -0.17∗∗∗ |  -0.19∗∗∗ |  -0.22∗∗∗ |  -0.19∗∗∗ |  -0.18∗∗∗ |  -0.19∗∗∗ |  -0.23∗∗∗
|  | [-2.82] | [-3.28] | [-3.50] | [-3.76] | [-3.64] | [-3.33] | [-3.54] | [-4.03]
log(B/M) |  | 0.08 | 0.12 | 0.15 | 0.12 | 0.15 | 0.20 |  0.26∗ | 0.20
|  | [0.61] | [0.79] | [0.96] | [0.82] | [1.03] | [1.34] | [1.67] | [1.33]
OP |  |  | 0.48 | 0.52 | 0.42 | 0.16 | 0.58 | 0.67 | 0.58
|  |  | [1.03] | [1.10] | [0.89] | [0.35] | [1.22] | [1.39] | [1.20]
CAR3 |  |  |  2.76∗∗ |  2.90∗∗ |  2.84∗∗ |  |  3.55∗∗∗ |  3.63∗∗∗ |  3.79∗∗∗
|  |  | [2.39] | [2.46] | [2.46] |  | [2.91] | [2.90] | [3.08]
SUE |  |  |  0.87∗∗∗ |  0.87∗∗∗ |  0.88∗∗∗ |  |  0.91∗∗∗ |  0.91∗∗∗ |  0.92∗∗∗
|  |  | [12.69] | [12.78] | [12.68] |  | [12.84] | [12.89] | [12.79]
Constant |  1.40∗∗∗ |  2.30∗∗∗ |  2.17∗∗∗ |  2.24∗∗∗ |  2.41∗∗∗ |  2.30∗∗∗ |  2.23∗∗∗ |  2.28∗∗∗ |  2.51∗∗∗
| [5.31] | [5.48] | [5.16] | [5.30] | [5.33] | [5.58] | [5.39] | [5.49] | [5.62]
Observations | 47,488 | 47,488 | 47,488 | 47,488 | 47,488 | 47,488 | 47,488 | 47,488 | 47,488
R2 | 0.002 | 0.003 | 0.007 | 0.007 | 0.008 | 0.003 | 0.008 | 0.008 | 0.009
Adjusted R2 | 0.002 | 0.003 | 0.007 | 0.007 | 0.008 | 0.003 | 0.008 | 0.008 | 0.008

Table 36: Returns of portfolios, formed on customers’ SUE

This table reports the returns of quintile and decile portfolios, formed by
sorts on SUE of customers’ portfolio. L/S corresponds to long-short portfolio,
defined as a difference in returns of tenth and first decile portfolios
(D10-D1) or fifth and first quintile portfolio (Q5-Q1). Tenth portfolio (D10)
is the portfolio of the firms with the largest customers’ returns. Standard
errors are calculated using Newey-West adjustment.

Panel A: Returns of equally-weighted quintile portfolios, sorted on SUE of
customers
---
statistic |  Q1 |  Q2 |  Q3 |  Q4 |  Q5 |  LS
Mean |  1.02∗∗∗ |  1.05∗∗∗ |  1.17∗∗∗ |  1.06∗∗∗ |  1.29∗∗∗ |  0.27
T-stat |  [3.34] |  [3.55] |  [4.02] |  [3.58] |  [4.25] |  [1.43]

Panel B: Returns of value-weighted quintile portfolios, sorted on SUE of
customers
---
statistic |  Q1 |  Q2 |  Q3 |  Q4 |  Q5 |  LS
Mean |  0.47 |  0.77∗∗ |  0.63∗∗ |  0.40 |  0.82∗∗ |  0.36
T-stat |  [1.52] |  [2.44] |  [2.09] |  [1.28] |  [2.56] |  [1.30]

Panel C: Returns of equally-weighted decile portfolios, sorted on SUE of
customers
---
statistic |  D1 |  D2 |  D3 |  D4 |  D5 |  D6 |  D7 |  D8 |  D9 |  D10 |  LS
Mean |  0.89∗∗ |  1.18∗∗∗ |  0.92∗∗∗ |  1.15∗∗∗ |  1.17∗∗∗ |  1.15∗∗∗ |  0.84∗∗∗ |  1.26∗∗∗ |  1.22∗∗∗ |  1.42∗∗∗ |  0.54∗∗
T-stat |  [2.57] |  [3.65] |  [2.87] |  [3.62] |  [3.74] |  [3.71] |  [2.63] |  [3.99] |  [3.81] |  [4.33] |  [2.02]

Panel D: Returns of value-weighted decile portfolios, sorted on SUE of
customers
---
statistic |  D1 |  D2 |  D3 |  D4 |  D5 |  D6 |  D7 |  D8 |  D9 |  D10 |  LS
Mean |  0.48 |  0.60∗ |  0.52 |  1.04∗∗∗ |  0.89∗∗∗ |  0.54∗ |  0.59∗ |  0.46 |  0.82∗∗ |  0.94∗∗∗ |  0.45
T-stat |  [1.37] |  [1.79] |  [1.41] |  [2.87] |  [2.63] |  [1.67] |  [1.72] |  [1.34] |  [2.42] |  [2.60] |  [1.31]

Table 37: Returns of portfolios, formed on customers’ CAR3

This table reports the returns of quintile and decile portfolios, formed by
sorts on CAR3 of customers’ portfolio. L/S corresponds to long-short
portfolio, defined as a difference in returns of tenth and first decile
portfolios (D10-D1) or fifth and first quintile portfolio (Q5-Q1). Tenth
portfolio (D10) is the portfolio of the firms with the largest customers’
returns. Standard errors are calculated using Newey-West adjustment.

Panel A: Returns of equally-weighted quintile portfolios, sorted on CAR3 of
customers
---
statistic |  Q1 |  Q2 |  Q3 |  Q4 |  Q5 |  LS
Mean |  1.15∗∗∗ |  0.92∗∗∗ |  1.19∗∗∗ |  1.14∗∗∗ |  1.21∗∗∗ |  0.06
T-stat |  [3.63] |  [3.10] |  [4.32] |  [3.98] |  [3.81] |  [0.32]

Panel B: Returns of value-weighted quintile portfolios, sorted on CAR3 of
customers
---
statistic |  Q1 |  Q2 |  Q3 |  Q4 |  Q5 |  LS
Mean |  0.76∗∗ |  0.60∗∗ |  0.52∗ |  0.55∗ |  0.66∗ |  -0.10
T-stat |  [2.31] |  [2.02] |  [1.88] |  [1.66] |  [1.94] |  [-0.36]

Panel C: Returns of equally-weighted decile portfolios, sorted on CAR3 of
customers
---
statistic |  D1 |  D2 |  D3 |  D4 |  D5 |  D6 |  D7 |  D8 |  D9 |  D10 |  LS
Mean |  1.22∗∗∗ |  1.10∗∗∗ |  1.13∗∗∗ |  0.74∗∗ |  1.14∗∗∗ |  1.23∗∗∗ |  0.97∗∗∗ |  1.28∗∗∗ |  1.31∗∗∗ |  1.07∗∗∗ |  -0.14
T-stat |  [3.45] |  [3.31] |  [3.51] |  [2.36] |  [3.81] |  [4.16] |  [3.34] |  [3.98] |  [3.77] |  [3.18] |  [-0.54]

Panel D: Returns of value-weighted decile portfolios, sorted on CAR3 of
customers
---
statistic |  D1 |  D2 |  D3 |  D4 |  D5 |  D6 |  D7 |  D8 |  D9 |  D10 |  LS
Mean |  0.84∗∗ |  0.87∗∗ |  0.81∗∗ |  0.61∗ |  0.43 |  0.79∗∗∗ |  0.68∗∗ |  0.66∗ |  0.71∗ |  0.67∗ |  -0.17
T-stat |  [2.17] |  [2.53] |  [2.27] |  [1.83] |  [1.32] |  [2.71] |  [2.00] |  [1.78] |  [1.89] |  [1.84] |  [-0.50]

Table 38: Daily returns of long-short portfolios, formed on customers momentum

This table reports the daily returns of long-short decile portfolios, formed
by sorts on customer momentum. Type corresponds to the width of the window (in
trading days) over which customer momentum is calculated. Columns correspond
to the length of the window over which the returns of the supplier is
calculated. For example, the intersection of row 10-1 and column ”15days”
contains the return of long-short decile portfolio from sorts on customer
momentum, where customer momentum is calculated over the past 10 days and the
return of the supplier is cumulative return over the period [t, t+15].
Standard errors are calculated using Newey-West adjustment.

Panel A: Equal weights |
---|---
type |  statistic |  1day |  5days |  10days |  15days |  20days |  30days
1-1 |  Mean |  0.17∗∗∗ |  0.28∗∗∗ |  0.38∗∗∗ |  0.49∗∗∗ |  0.53∗∗∗ |  0.58∗∗∗
|  T-stat |  [14.25] |  [11.09] |  [10.50] |  [10.89] |  [10.20] |  [8.83]
5-1 |  Mean |  0.12∗∗∗ |  0.32∗∗∗ |  0.54∗∗∗ |  0.71∗∗∗ |  0.79∗∗∗ |  0.92∗∗∗
|  T-stat |  [9.85] |  [12.59] |  [15.06] |  [16.04] |  [15.04] |  [14.13]
10-1 |  Mean |  0.12∗∗∗ |  0.37∗∗∗ |  0.66∗∗∗ |  0.85∗∗∗ |  0.96∗∗∗ |  1.13∗∗∗
|  T-stat |  [10.45] |  [15.06] |  [18.47] |  [18.90] |  [18.15] |  [17.34]
20-1 |  Mean |  0.11∗∗∗ |  0.40∗∗∗ |  0.68∗∗∗ |  0.88∗∗∗ |  0.98∗∗∗ |  1.21∗∗∗
|  T-stat |  [9.49] |  [15.33] |  [18.36] |  [19.30] |  [18.82] |  [18.61]
40-1 |  Mean |  0.09∗∗∗ |  0.36∗∗∗ |  0.64∗∗∗ |  0.87∗∗∗ |  1.06∗∗∗ |  1.36∗∗∗
|  T-stat |  [7.54] |  [13.79] |  [17.29] |  [18.61] |  [19.49] |  [20.25]

Panel B: Value weights |
---|---
type |  statistic |  1day |  5days |  10days |  15days |  20days |  30days
1-1 |  Mean |  0.12∗∗∗ |  0.17∗∗∗ |  0.22∗∗∗ |  0.29∗∗∗ |  0.35∗∗∗ |  0.36∗∗∗
|  T-stat |  [7.91] |  [4.88] |  [4.61] |  [5.01] |  [5.28] |  [4.45]
5-1 |  Mean |  0.08∗∗∗ |  0.15∗∗∗ |  0.29∗∗∗ |  0.42∗∗∗ |  0.48∗∗∗ |  0.66∗∗∗
|  T-stat |  [5.27] |  [4.40] |  [5.99] |  [7.40] |  [7.35] |  [8.27]
10-1 |  Mean |  0.08∗∗∗ |  0.20∗∗∗ |  0.41∗∗∗ |  0.56∗∗∗ |  0.58∗∗∗ |  0.76∗∗∗
|  T-stat |  [5.04] |  [5.85] |  [8.49] |  [9.69] |  [8.97] |  [9.64]
20-1 |  Mean |  0.10∗∗∗ |  0.28∗∗∗ |  0.46∗∗∗ |  0.61∗∗∗ |  0.73∗∗∗ |  0.94∗∗∗
|  T-stat |  [6.18] |  [7.87] |  [9.64] |  [10.90] |  [11.39] |  [11.98]
40-1 |  Mean |  0.08∗∗∗ |  0.31∗∗∗ |  0.52∗∗∗ |  0.65∗∗∗ |  0.70∗∗∗ |  0.86∗∗∗
|  T-stat |  [5.19] |  [8.73] |  [10.86] |  [11.41] |  [10.66] |  [10.36]

Table 39: Daily returns of long-short quintile portfolios, double-sorted on
investor attention and customer momentum

This table reports the daily returns of long-short quintile portfolios, formed
by sorts on investor attention to customer-supplier link. Attention to
customer-supplier link is measured as Normalized Abnormal Trading Volume (NAV)
of suppliers during the day of earnings announcement of the customer. Type
corresponds to the width of the window (in trading days) over which customer
momentum is calculated. Columns correspond to the length of the window over
which the returns of the supplier is calculated. For example, the intersection
of row 10-1 and column ”15days” contains the return of long-short decile
portfolio (within the first quintile of investor‘s attention) from sorts on
customer momentum, where customer momentum is calculated over the past 10 days
and the return of the supplier is cumulative return over the period [t, t+15],
minus the the return of long-short decile portfolio (within the fifth quintile
of investor‘s attention) from sorts on customer momentum. In other words, this
cell contains the returns of long-short portfolio, conditionally double-sorted
on investor attention and customer momentum. Standard errors are calculated
using Newey-West adjustment.

Panel A: Equal weights |
---|---
type |  statistic |  1day |  5days |  10days |  15days |  20days |  30days
1-1 |  Mean |  0.00 |  -0.15 |  -0.04 |  -0.04 |  0.06 |  0.45
|  T-stat |  [0.85] |  [-1.25] |  [-0.24] |  [-0.21] |  [0.25] |  [1.24]
5-1 |  Mean |  -0.00 |  -0.21∗ |  -0.32∗∗ |  -0.22 |  0.03 |  0.58∗
|  T-stat |  [-0.99] |  [-1.77] |  [-1.97] |  [-1.09] |  [0.11] |  [1.66]
10-1 |  Mean |  0.00 |  -0.00 |  -0.21 |  0.17 |  0.49∗∗ |  0.75∗∗
|  T-stat |  [-0.20] |  [-0.00] |  [-1.31] |  [0.88] |  [2.09] |  [2.16]
20-1 |  Mean |  0.00 |  -0.00 |  0.24 |  0.55∗∗∗ |  0.63∗∗∗ |  1.58∗∗∗
|  T-stat |  [-0.61] |  [-0.04] |  [1.61] |  [2.96] |  [2.90] |  [4.47]
40-1 |  Mean |  0.00 |  0.41∗∗∗ |  0.72∗∗∗ |  1.28∗∗∗ |  1.21∗∗∗ |  1.50∗∗∗
|  T-stat |  [0.93] |  [3.40] |  [4.42] |  [6.68] |  [5.23] |  [4.07]

Panel B: Value weights |
---|---
type |  statistic |  1day |  5days |  10days |  15days |  20days |  30days
1-1 |  Mean |  -0.00 |  -0.16 |  -0.19 |  -0.22 |  -0.18 |  -0.19
|  T-stat |  [-0.46] |  [-1.48] |  [-1.21] |  [-1.13] |  [-0.81] |  [-0.71]
5-1 |  Mean |  -0.00∗ |  -0.24∗∗ |  -0.28∗ |  -0.11 |  0.16 |  -0.26
|  T-stat |  [-1.83] |  [-2.06] |  [-1.76] |  [-0.59] |  [0.70] |  [-0.93]
10-1 |  Mean |  0.00 |  0.03 |  -0.04 |  0.03 |  -0.09 |  -0.30
|  T-stat |  [-0.79] |  [0.25] |  [-0.28] |  [0.15] |  [-0.42] |  [-1.14]
20-1 |  Mean |  0.00 |  0.00 |  0.18 |  0.17 |  0.20 |  0.06
|  T-stat |  [-0.13] |  [0.00] |  [1.20] |  [0.92] |  [0.90] |  [0.24]
40-1 |  Mean |  0.00 |  0.36∗∗∗ |  1.03∗∗∗ |  1.17∗∗∗ |  1.60∗∗∗ |  1.48∗∗∗
|  T-stat |  [0.86] |  [3.21] |  [6.77] |  [6.39] |  [7.42] |  [5.61]