# Observability of inflated companion stars after supernovae in massive
binaries

Misa Ogata,1, Ryosuke Hirai2,3,4, Kotaro Hijikawa5
1Advanced Research Institute for Science and Engineering, Waseda University,
3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
2OzGrav: The Australian Research Council Centre of Excellence for
Gravitational Wave Discovery, Clayton, VIC 3800, Australia
3School of Physics and Astronomy, Monash University, VIC 3800, Australia
4Department of Physics, University of Oxford, Keble Rd, Oxford, OX1 3RH,
United Kingdom
5Department of Astronomy, Graduate School of Science, The University of Tokyo,
Bunkyo-ku, Tokyo, Japan E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We carry out a systematic study of the response of companion stars in massive
binaries after being impacted by supernova ejecta. A total of 720 1D stellar
evolution calculations are performed to follow the inflation and contraction
of the star in response to the energy injection and how it depends on various
parameters. We find that the maximum luminosity achieved during the inflated
phase is only dependent on the stellar mass and we derive an analytic formula
to describe the relation. There is also a tight correlation between the
duration of expansion and the intersected energy. These correlations will be
useful to constrain pre-supernova binary parameters from future detections of
inflated companions. We also discuss the possible outcomes of the binary
system when the companion inflation is taken into account. Based on simple
binary population synthesis, we estimate that $\sim$1–3 % of stripped-envelope
supernovae may have observable inflated companions. Finally, we apply our
models to the observed companion of SN 2006jc and place strong constraints on
the possible pre-supernova binary parameters.

###### keywords:

binaries: close – supernovae: general – supernovae: individual: SN 2006jc –
stars: kinematics and dynamics

††pubyear: 2020††pagerange: Observability of inflated companion stars after
supernovae in massive binaries–C

##  1 Introduction

Core-collapse supernovae are explosions that occur at the end of the lives of
massive stars ($>8~{}\textnormal{M}_{\odot}$), and they leave dense compact
remnants such as neutron stars or black holes. The outermost regions of the
progenitors usually consist of mostly hydrogen and helium which can be
spectroscopically identified in supernova (SN) observations. However, a
substantial fraction of CCSNe have much weaker or no trace of hydrogen,
indicating a diversity in the structure of CCSN progenitors (Li et al., 2011;
Smith et al., 2011). Such hydrogen-deficient SNe are classified as type
Ib/Ic/IIb depending on the spectroscopic features at the peak of the light
curve, and are collectively called stripped-envelope SNe (see Modjaz et al.,
2019, and references therein for details of the SN classification). The
progenitors of these stripped-envelope SNe lose their envelope during their
evolution leading to SN, through mass-loss mechanisms such as stellar winds or
interactions with a binary companion.

In binary systems, various binary interactions could change their evolution
greatly. The specific form of interaction and their outcomes are sensitive to
their initial separation and mass ratio. For example, the main mass-loss
processes are Roche lobe overflow and common-envelope evolution. Through a
combination of these interactions, stars can be stripped of their hydrogen
envelopes (e.g. Podsiadlowski et al., 1992; Yoon et al., 2010; Yoon, 2015;
Eldridge et al., 2008; Eldridge et al., 2017; Zapartas et al., 2019; Sravan et
al., 2019) and some of them may even lose their helium layers prior to
explosion (e.g. Tauris et al., 2013). However because of the vast variety of
possible evolutionary paths, the understanding of the whole picture is
difficult.

To understand binary evolution, the feedback from observations is essential.
One example is the observation of the SN themselves, which could be used to
constrain the binary parameters at the point of SN explosion. Especially since
one of the main channels for producing stripped-envelope SN progenitors is
through binary interactions, they would be good targets. Pre-SN observations
of the _progenitor_ provide extra means to directly probe the properties of
the progenitor (Smartt, 2009; Van Dyk, 2017). For the case of stripped-
envelope SNe, only several progenitors have been detected: type IIb SNe SN
1993J (Aldering et al., 1994), SN 2011dh (Maund et al., 2011), SN 2013df (Van
Dyk et al., 2014), SN 2008ax (Folatelli et al., 2015), SN 2016gkg (Bersten et
al., 2018), and type Ib SNe iPTF13bvn (Cao et al., 2013), SN 2019yvr
(Kilpatrick et al., 2021). Based on the pre-SN information, various binary
evolution paths were advocated for individual cases (e.g. Bersten et al.,
2014; Eldridge et al., 2015; Hirai, 2017; Sravan et al., 2018). On the other
hand, post-SN observations are also extremely important in confirming whether
the progenitor had a binary companion or not. So far there are four stripped-
envelope SNe with _companion_ detections, SN 1993J (Maund et al., 2004), SN
2001ig (Ryder et al., 2018), SN 2006jc (Maund et al., 2016; Sun et al., 2019)
and SN 2011dh (Folatelli et al., 2014; Maund, 2019), confirming that some
stripped-envelope SN progenitors are indeed produced through the binary
channel. These companions are observed a few years after the explosion, when
the luminosity of the SN itself has faded sufficiently below that of the
companion star.

Moreover the observation of SN in binaries help our understanding not only on
the progenitors but also on the evolution after SN. After the first direct
detection of gravitational waves (Abbott et al., 2016), compact binaries such
as binary NSs or binary BHs have attracted wide attention. In particular, the
detection of a binary NS merger event was followed by observations of
counterparts across the entire electromagnetic spectrum, providing valuable
information for not only astrophysics, but also gravitational and nuclear
physics (Abbott et al., 2017a, b). Their evolutionary paths are not completely
understood, but it is clear that the binary should have experienced and
survived at least two SN events. Each SN has a finite probability of
disrupting the system due to the natal kick imparted to the new-born NS,
strongly influencing the formation rate of binary NS systems. SNe are also one
of the most observable phases during the evolution of a massive binary system.
Therefore, it is extremely important to understand the outcome of SNe in
binary systems and establish ways to extract information of the binary
properties from SN observations.

To study the nature of the binary companion after SNe, the effect of the
interaction between SN ejecta and the companion star (ejecta-companion
interaction; ECI) has been studied for a long time through analytic modelling
(Wheeler et al., 1975; Meng et al., 2007), numerical simulations (e.g. Fryxell
& Arnett, 1981; Marietta et al., 2000; Podsiadlowski, 2003; Pan et al., 2010,
2012; Liu et al., 2012; Shappee et al., 2013; Maeda et al., 2014; Zeng et al.,
2020) and even laboratory experiments (García-Senz et al., 2019). The
application to CCSNe are still much more limited (Hirai et al., 2014; Hirai &
Yamada, 2015; Liu et al., 2015; Rimoldi et al., 2016; Hirai et al., 2020).

Hirai et al. (2018) (hereafter H18) carried out a large set of 2D
hydrodynamical simulations of ECI with massive main-sequence (MS) companions
($10$–$20~{}\textnormal{M}_{\odot}$). They found that the classical analytical
model based on momentum conservation (Wheeler et al., 1975) fails to reproduce
their simulation results for the amount of unbound mass and imparted momentum.
They constructed an alternative analytical model that can explain the imparted
momentum along with the injected energy and how they are dependent on the
stellar structure and orbital parameters. Moreover, they evolved the MS
companion for 10,000 years after explosion, and found that the massive
companion can become very luminous and inflated for up to tens of years due to
the energy excess in the surface layers of the envelope.

One example of the companion observation of SNe is SN 2006jc. SN 2006jc is a
type Ibn SN which is one of the hydrogen-deficient SNe that have prominent
narrow lines of He I in their spectra (Foley et al., 2007). An outburst was
also observed in 2004 (Pastorello et al., 2007). The SN explosion was
discovered by K. Itagaki on 2006 Oct 9 (Nakano et al., 2006), and occurred in
the outskirt of UGC 4904. Maund et al. (2016) reported a late-time observation
in 2010, and they discovered a faint blue source at the SN position. Sun et
al. (2019) also report a late-time observation in 2017, and the magnitude in
V-band was not so different between observations in 2010 and 2017. The stable
optical brightness for over 7 years exclude other possibilities such as
circumstellar matter interaction or light echoes, and conclude that the source
is a stellar companion. Moreover the companion was inconsistent with being a
MS star and was instead located in the Hertzsprung gap that is known as an
extremely short phase of stellar evolution. Sun et al. (2019) interpreted that
the progenitor system is a binary whose mass ratio is close to unity and the
companion has just recently evolved off its MS phase.

In this paper, we investigate the ECI in CCSNe in greater detail. H18 only
followed the later evolution of these companions for a few models. They also
did not refer to the post-SN evolution of the “binary” in detail. In this
paper we aim to reveal the evolution of the systems after SN explosions using
their formulation for much wider parameter regions and calculate the
observability of such inflated companions. By studying the parameter
dependence of the companion inflation, it would be useful for constraining
pre-explosion binary parameters from follow-up observations of SNe.

We review the results of H18 in Section 2.1, and discuss possible outcomes of
the binary in the presence of expanding envelopes in Section 2.2. We will
outline our calculation method and parameters in Section 3. We will show
results of stellar modelling in Section 4. The classification of systems after
SN and the observable distance will also be presented. In Section 5, we will
present the rate that the expanding secondary is observed and the application
to SN 2006jc. Implications of future detections of inflated companions will
also be discussed. Finally we will summarize our work in Section 6. Detailed
calculation methods are shown in the Appendices A, B and C.

##  2 Evolution of Secondary Star with ECI

###  2.1 Post-ECI inflation of the companion

In this section we briefly review the model and results of H18. They
considered close binaries where the primary explodes as a CCSN with a massive
MS companion. In systems with close separations, the primary should have
experienced mass transfer prior to explosion and therefore should have lost
most of the hydrogen envelope, i.e. the primary explosion is a stripped-
envelope SN.

Using the hydrodynamical code HORMONE (Hirai et al., 2016), they carried out
2D hydrodynamical simulations of SN blast waves colliding with their
companions. As the ejecta hits the surface of the star, they form a forward
shock that sweeps through the whole star and escapes from the other side.
During this process, part of the kinetic energy of the ejecta is deposited
into the envelope and drives it out of thermal equilibrium. There is hardly
any mass stripped in this process, with only $\lesssim$1 % of the mass lost
even in the most extreme cases. They identified that most of the deposited
energy is located near the surface layers and the excess energy declines as it
goes deeper into the star. They also found that the total injected energy is
$\sim$8–10 % of the energy intersected by the secondary. This can be
understood as a product of three factors; 1) a factor
$(\gamma-1)/(\gamma+1)\sim 1/4$, where $\gamma$ is the adiabatic index, which
accounts for the energy thermalised at the bow shock that forms ahead of the
star, 2) a geometrical factor 1/2 accounting for the deflection of ejecta at
the stellar surface, 3) a factor $\sim$2/3 that accounts for the reduction in
cross section due to stellar compression during the ejecta passage. Since this
is a purely kinematic model, this should be applicable for any companion model
as long as the stripped mass is negligible.

They used the stellar evolution code MESA (Paxton et al., 2010; Paxton et al.,
2013, 2015, 2018, 2019) to follow the secondary evolution for 10,000 years
after ECI. They focused on a model in which the secondary mass is
$M_{2}=10~{}\textnormal{M}_{\odot}$, has a radius
$R_{2}=5~{}\textnormal{R}_{\odot}$, and the explosion model with an ejecta
mass $M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot}$ and explosion energy
$E_{\mathrm{expl}}=10^{51}~{}\mathrm{erg}$. The assumed metallicity was
$Z=0.02$.

In their hydrodynamical simulations, the excess energy is distributed
uniformly around the envelope surface down to a certain radius and then
decreases inverse proportionally to mass as it goes in. They created a fitting
formula for the excess energy distribution $\Delta\varepsilon$,

$\Delta\varepsilon(m)=\frac{E_{\mathrm{heat}}}{m_{h}}\frac{{\mathrm{min}}\left(1,m_{h}/m\right)}{1+\ln\left(M_{2}/m_{h}\right)},$
(1)

where $m$ is the mass coordinate from the surface. $E_{\mathrm{heat}}$ is the
total injected energy which is estimated from the energy intersected by the
secondary assuming the explosion is spherically symmetric,

$E_{\mathrm{heat}}=p\times E_{\mathrm{expl}}\times\tilde{\Omega},$ (2)

where $p\sim$8–10 % is the energy injection efficiency. $\tilde{\Omega}$ is
the fractional solid angle of the companion looking from the primary

$\tilde{\Omega}=\frac{\Omega}{4\pi}=\frac{1}{2}\left[1-\sqrt{1-\left(\frac{R_{2}}{a_{\mathrm{i}}}\right)^{2}}\right],$
(3)

where $a_{\mathrm{i}}$ is the pre-SN orbital separation. $m_{h}$ is the mass
coordinate from the surface marking the transition from flat to decreasing
energy excess, which can be estimated by

$m_{h}=\frac{\tilde{\Omega}M_{\mathrm{ej}}}{2}.$ (4)

Using these formulae, they inject energy into the companion star models and
evolved them for 10,000 years.

They obtained the time evolution of radius, and find that the secondary
greatly expands and stays in the inflated state for several to tens of years.
In some cases the maximum achieved radius far exceeds the initial orbital
separation, meaning that the remnant of the primary could be embedded in the
secondary envelope if the binary survives. This could give rise to various
interesting phenomena, which will be discussed in the following section.

###  2.2 Possible outcomes

Here we will discuss the possible outcomes of the system immediately after the
first SN. The post-SN state can be classified depending on the post-SN orbital
parameters and the degree of companion inflation due to ECI.

We identify 6 states;

  * -

state 1: the primary and secondary cores collide,

  * -

state 2: the binary survives and the primary orbit is fully embedded in the
inflated secondary envelope,

  * -

state 3: the binary survives and the primary orbit is partially embedded in
the inflated secondary envelope,

  * -

state 4: the binary survives and the primary orbit does not intersect with the
inflated secondary envelope at any point but could cause Roche lobe overflow,

  * -

state 5: the binary survives in an orbit too wide to interact with each other,

  * -

state 6: the binary is disrupted and evolve as two single stars.

State 1 occurs when the NS natal kick is directed such that the post-SN
periastron distance between the two stars is smaller than the original
secondary star radius. In this case the primary NS is expected to rapidly
inspiral within the secondary envelope and merge with each other. There could
be some explosive phenomena as the NS accretes material in the envelope,
leading to an unbinding of the outer layers and halting the spiral-in (Fryer
et al., 1996). Contrarily, the NS may spiral in all the way to the centre,
forming a Thorne-Żytkow object (TŻO; Thorne & Żytkow, 1977; Leonard et al.,
1994) which can have peculiar surface abundances such as enhanced lithium
(Podsiadlowski et al., 1995). The ultimate fate of TŻOs are not known, but
could lead to envelope collapse and BH formation or explosions (Podsiadlowski
et al., 1995; Fryer et al., 1996; Moriya, 2018).

States 2 and 3 are when the binary survives with an orbit tight enough to be
fully or partially engulfed by the inflated part of the secondary envelope. In
Fig. 1 we show an example of how the density structure inside the inflated
part of the envelope can be. Only a very small amount of mass is heated and
inflated, so the density in the region between the inflated and original
surfaces is extremely low. This density is not sufficient to impart a
meaningful drag force on the NS, therefore not being able to affect the
orbital evolution. However, there could still be efficient accretion onto the
NS that gives rise to some X-ray transients. It will only be observable if the
outflow from the accretion is able to make its way out of the optically thick
envelope. If the accretion feedback is significant, it may fully unbind the
inflated layers of the envelope, enabling high energy radiation to escape and
also suppresses the post-ECI inflation of the secondary. This means that even
if the effect of ECI on the companion is the same, their appearances could be
significantly different depending on whether the binary survived or not and
what the post-SN orbit is.

Figure 1: Density distributions for the model with
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=7~{}\textnormal{R}_{\odot},a_{\mathrm{i}}=40~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$. The purple curve is the density distribution of the inflated
secondary and the green curve is the original profile.

State 4 is a case where the inflated secondary envelope can exceed its post-SN
effective Roche lobe at periastron. It is possible that the secondary
experiences Roche lobe overflow to the new-born NS, creating a very brief
X-ray binary phase. The stellar inflation may again be suppressed due to the
mass transfer, although it is not clear how these inflated stars respond to
mass-loss and the mass transfer process in eccentric binaries is still poorly
understood. Another possible outcome of state 4 is the formation of planets.
Nakamura & Piran (1991) proposed that if part of the inflated envelope can be
transferred to the NS and form a proto-planetary disk, it could explain the
origin of pulsar planets.

State 5 is a trivial case where the binary survives but the post-ECI secondary
does not inflate enough to exceed its Roche lobe. In such cases nothing
particularly interesting would happen and would just continue its evolution as
a regular binary.

In state 6, the binary disrupts and continues its evolution as two separate
single stars. This case occurs when more than half of the total system mass is
lost in the SN explosion, or the NS natal kick is large and directed in
certain directions.

In this paper, we will explore on the observability of inflated companions
after the first SN in the system. We will also take into account the post-SN
binary properties as it may affect the maximum extent of the companion
inflation. By studying the relative occurrence rates of each state, it can
provide some insight into the observability of brief X-ray transients or
pulsar planet formation.

##  3 Numerical method

Based on the methodology established in H18, we extend the study of ECI and
secondary expansion to a much wider parameter space. We investigate the
parameter dependence of the expansion radius of the secondary envelope and the
inflated timescale. In this work we vary 5 model parameters, resulting in a
total of 720 models. The relevant parameters are the companion mass $M_{2}$,
the companion radius $R_{2}$, the binary separation $a_{\mathrm{i}}$, the
ejecta mass $M_{\mathrm{ej}}$ and the explosion energy $E_{\mathrm{expl}}$.
The combination of parameters used are summarized in Table 1. The lower end of
the secondary mass range is taken from 3 $\textnormal{M}_{\odot}$, below which
stars may not exist as companions to massive stars ($\gtrsim
8~{}\textnormal{M}_{\odot}$) because the star could not end its contraction
phase within the massive stars’ lifetime (Neugent et al., 2020). The largest
secondary mass we use is $20~{}\textnormal{M}_{\odot}$. Although it is
possible that higher mass companion stars exist for CCSNe, such systems are
less abundant so we ignore them in our study. We use the typical CCSN
explosion energy of $10^{51}~{}\mathrm{erg}$, but we also consider hypernova-
class energies of $10^{52}~{}\mathrm{erg}$ to see the energy dependence. The
minimum orbital separations are chosen such that the secondary is close to
Roche lobe filling, and the maximum separations are where the effect of ECI
becomes negligible.

For all our calculations we use the 1D stellar evolution code MESA (v10398).
The initial stellar model is evolved up to the specified radius $R_{2}$, to
account for the radius increase during the MS phase. For the
$M_{2}=3,5~{}\textnormal{M}_{\odot}$ models, we fix the radius to the zero-age
main-sequence value as we do not expect much radius evolution within the
lifetimes of CCSN progenitors. In real close binaries, the secondary should
have slightly different structures from that of single stars due to mass
transfer from the primary (e.g. Braun & Langer, 1995), but we ignore this
effect and use normal single MS models.

To each stellar model, we artificially inject energy following Eq. (1)–(4),
over a timescale of $\sim$1 yr. For the lower mass models
($M_{2}=3,5~{}\textnormal{M}_{\odot}$), we choose a slightly longer injection
time of $\sim$3–10 yr to avoid numerical difficulties due to the longer
thermal timescale of the star. All injection timescales are set to be
significantly shorter than the thermal timescale, and we confirmed that the
timescale does not have a huge effect on the evolution as long as this
condition is satisfied. The true energy injection timescale is much shorter
($\sim$few hr, corresponding to the shock crossing time), and therefore we
define the time since the end of the artificial heating phase as the time
since SN. We use a fixed energy injection efficiency $p=0.08$111Strictly
speaking, this is dependent on the stellar structure and orbital separation,
where $p$ increases as the orbital separation increases (H18). However, the
variation is small ($p\sim$0.08–0.1) so we fix the value for simplicity.. A
fixed metallicity $Z=0.02$ is used, however we run a few additional models to
explore the metallicity dependence. We use the method adopted in Henyey et al.
(1965) with the mixing length parameter set to $\alpha_{\textsc{mlt}}=2.0$ for
convection, and the so-called MLT++ feature is switched off (Section 7.2 in
Paxton et al., 2013). We apply the “Dutch wind” scheme (see Glebbeek et al.,
2009) throughout, although the effect of winds is negligible for our
applications. Simple grey atmosphere boundary conditions are applied to the
outer boundary. The MESA inlist and run_star_extras.f files for the
calculations are made publicly available in Zenodo
222https://zenodo.org/record/4624586##.YHH6Cq_7Q2w. The models are run for
10,000 yr, which encases the whole expansion and contraction phase after
energy injection.

$M_{2}[\textnormal{M}_{\odot}]$ | $R_{2}[\textnormal{R}_{\odot}]$ | $a_{\mathrm{i}}[\textnormal{R}_{\odot}]$ | $M_{\mathrm{ej}}[\textnormal{M}_{\odot}]$ | $E_{\mathrm{expl}}[\mathrm{erg}]$
---|---|---|---|---
3 | 2 | 10, 20, 30, 40, 60 | 2, 3, 5 | $10^{51},10^{52}$
5 | 2.7 | 10, 20, 30, 40, 60 | 2, 3, 5 | $10^{51},10^{52}$
10 | 6, 7, 8, 9, 10 | 20, 30, 40, 60 | 2, 3, 5, 7.1, 10 | $10^{51},10^{52}$
15 | 6, 7, 8, 9, 10 | 20, 30, 40, 60 | 2, 3, 5, 7.1, 10 | $10^{51},10^{52}$
20 | 6, 7, 8, 9, 10 | 20, 30, 40, 60 | 2, 3, 5, 7.1, 10 | $10^{51},10^{52}$
Table 1: The parameter space explored in our study. The stellar radius for
$M_{2}=3,5~{}\textnormal{M}_{\odot}$ are fixed to zero-age main-sequence
values. The total number of models are 720.

We also consider the binary properties immediately after the SN. It is
determined by the amount of mass loss from the system, the NS natal kick and
the impact velocity which is defined as the velocity given to the companion
due to the momentum imparted by the ejecta. We assume initially circular
orbits for all models. This is a reasonable assumption in the relevant
parameter regions because the orbit should have been tidally circularised at
close separations. We analytically calculate the post-SN orbital parameters by
assigning a kick velocity and direction. In Fig. 2 we show the coordinate
system we use for the orbit calculations. We set the origin at the centre of
the primary star (exploding star), with the x-axis in the direction of the
initial orbital velocity $V_{i}$ and the y-axis directed from the companion to
the primary. The z-axis is antiparallel to the orbital angular momentum. We
also define angles $\theta$ and $\phi$ as shown in the lower panel of Fig. 2.
The impact velocity always points in the negative y-direction whereas the
natal kick direction is arbitrary. Details of how we calculate the post-SN
orbital separation $a_{\mathrm{f}}$, eccentricity $e$ and Roche-lobe radius
$R_{\mathrm{rl}}$ are described in Appendix A.

Figure 2: Schematic diagrams of the coordinate system used for the orbit
calculations. (Upper panel) the red large circle expresses the secondary MS
star and the blue small circle is the primary. The three arrows represent the
orbital velocity (orange), the impact velocity (green) and the kick velocity
(light green). The dotted circle expresses the circular orbit before
explosion. (Lower panel) the definition of angles for the kick velocity. The
x- and y-axis is the same as the upper panel.

The outcome of the SN can be classified by comparing the orbital parameters
and the inflated envelope radius,

  * -

state 1: $e<1\wedge a_{\mathrm{f}}(1-e)<R_{2}$,

  * -

state 2: $e<1\wedge a_{\mathrm{f}}(1+e)<R_{\mathrm{2,max}}$,

  * -

state 3: $e<1\wedge a_{\mathrm{f}}(1-e)<R_{\mathrm{2,max}}$,

  * -

state 4: $e<1\wedge R_{\mathrm{rl,f}}<R_{\mathrm{2,max}}$,

  * -

state 5: $e<1\wedge R_{\mathrm{rl,f}}>R_{\mathrm{2,max}}$,

  * -

state 6: $e>1$,

where $R_{\mathrm{rl,f}},R_{\mathrm{2,max}}$ are the post-SN effective Roche-
lobe radius at periastron and the maximum radius of the secondary during the
inflated phase.

Our methodology is based on various assumptions. First, we have assumed that
the explosion is spherically symmetric. In reality, CCSNe are known to be
rather aspherical, and therefore the energy intersected by the secondary can
vary depending on the degree of asphericity. However, this uncertainty will be
masked by the wide range in possible explosion energies, so we do not take the
asphericity into account. Another assumption is that the ejecta velocity is
much faster than the orbital velocity. This only breaks down in extremely
tight orbits such as for ultra-stripped SNe (see e.g. Tauris et al., 2013). In
such situations, the mass loss can no longer be treated as an impulsive event
and hence the orbital response will behave differently (Fryer et al., 2015).
We do not deal with ultra-stripped SNe in this paper, so the impulsive
approximation is always valid. We also assume that there is no mass stripping
by the ECI. This is a good approximation for MS companions where the unbound
masses are $\lesssim$ 1 % (Liu et al., 2015; Rimoldi et al., 2016; Hirai et
al., 2018). This is invalid for more evolved companions such as Hertzsprung
gap stars or red supergiants where there can be significant mass loss (Hirai
et al., 2014; Hirai et al., 2020), but these cases are rare and we ignore such
systems in this paper.

##  4 Results

###  4.1 Time evolution of stellar properties

In the upper panel of Fig. 3 we show the time evolution of radius for the
$M_{2}=3~{}\textnormal{M}_{\odot}$ and $15~{}\textnormal{M}_{\odot}$
($M_{\mathrm{ej}}=2~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}~{}\mathrm{erg}$)
models. The initial radius for the $M_{2}=15~{}\textnormal{M}_{\odot}$ models
are $R_{2}=6~{}\textnormal{R}_{\odot}$, and the separations are varied between
$20~{}\textnormal{R}_{\odot}$ and $60~{}\textnormal{R}_{\odot}$. Most curves
have qualitatively the same shape, where the radius stays roughly constant and
then abruptly shrinks back to its original radius after $\sim$1–20 yr. The
star is still quite overluminous for $\sim$100–10,000 yr even after the radius
has retracted. However, the temperature is much hotter once the star has
contracted, so the drop in luminosity at longer wavelengths should be more
prominent than the shape in this figure.

For a fixed stellar mass, it is clear that the expansion radius and timescale
increase with decreasing orbital separation. Especially, the
$a_{\mathrm{i}}<20~{}\textnormal{R}_{\odot}$ models expand largely for more
than 10 years, which may be long enough to be observable as the SN itself
fades away. For fixed orbital separations, the expansion timescales are
shorter for larger secondary mass. This is because the luminosity of the
inflated stars are larger and therefore takes less time to radiate away the
excess energy. Although we only show the time evolution for select systems,
the overall shape and parameter dependences are similar for all other models.

Figure 3: Time evolution of the secondary radius (upper panel) and luminosity
(lower panel) after energy injection for the models with
$M_{2}=3~{}\textnormal{M}_{\odot},R_{2}=2~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=2~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$ (solid curves), and
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=6~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=2~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$ (dashed curves). Different colours denote different orbital
separations $a_{i}=10$ (only for $M_{2}=$3 $\textnormal{M}_{\odot}$), 20, 30,
40, 60 $\textnormal{R}_{\odot}$. The grey solid lines mark the original
stellar radii $R_{2}$ and the original stellar luminosities. The time is
measured from the end of the artificial heating.

In Fig. 4 we show the evolution on the HR-diagram for the models
$(M_{2},R_{2})=(3,2),(10,6),(15,6)$,
$M_{\mathrm{ej}}=2~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}~{}\mathrm{erg}$.
As the envelope expands, the secondary jumps to the cooler and more luminous
region in the HR-diagram and stays there for several years. Some models
inflate close to the Hayashi limit, but do not fully reach it. After that it
rapidly returns to the original state as the envelope contracts. The maximum
radius is larger for shorter separations, therefore the closer models reach
lower temperatures. On the other hand, the maximum luminosity is almost
independent of any parameters other than initial mass.

Figure 4: HR-diagram after explosion for the models
$(M_{2},R_{2})=(3,2),(10,6),(15,6),M_{\mathrm{ej}}=2~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$. Grey solid curves are the normal evolutionary tracks of
$3,5,10,15,20~{}\textnormal{M}_{\odot}$ stars. Grey dashed lines are equal
radius lines of $5,10,20,40,80,160,320,640~{}\textnormal{R}_{\odot}$. The
squares, circle and triangles are time stamps at $1,5,10,20,$ and $50$ years
after SN explosion.

To demonstrate why the maximum luminosity is roughly constant for a given
stellar mass, we show the luminosity distribution near the surface of the star
during the inflated phase in Fig. 5. The luminosity distribution increases
monotonically in the star because of the way the excess energy is injected
(Eq. (1)). It then flattens out beyond some radius very close to the surface
($M_{2}-m\lesssim 0.002~{}\textnormal{M}_{\odot}$) above which most of the
envelope is convective and the energy is efficiently transported. The bottom
of this convective region corresponds to where the luminosity intersects with
the local Eddington luminosity. At this specific layer, the local Eddington
luminosity is determined by the Fe opacity bump (slightly off the peak) which
is located at temperature regions of $\log(T/\mathrm{K})\sim$5.2–5.4. We find
a similar structure for all of our models, indicating that the maximum
luminosity after ECI is always determined by the Eddington luminosity
evaluated by the Fe opacity. Because the Fe opacity is sensitive to the
temperature and density, the actual opacity value at this layer is different
for different mass models but within a reasonably small range. For each
stellar mass, the average value of the opacity at the bottom of the convective
layer is $\kappa\sim$1.25, 1.00, 1.03, 0.85, 0.76 cm2 g-1 for the
$M_{2}=3,5,10,15,20~{}\textnormal{M}_{\odot}$ models respectively. The average
opacities decrease linearly as the secondary mass increases, and can be fit
with the function

$\kappa_{\mathrm{fit}}=1.24~{}\mathrm{cm}^{2}\mathrm{g}^{-1}\left(1-0.02\frac{M_{2}}{\textnormal{M}_{\odot}}\right).$
(5)

This is consistent with the fact that at a layer with a given temperature, the
density decreases as the mass of the star increases so the opacity decreases
too. Using this fitting formula, we can analytically estimate the maximum
post-ECI luminosity as

$L_{\mathrm{max,ana}}=\frac{4\pi GM_{2}c}{\kappa_{\mathrm{fit}}},$ (6)

where $G$ is the gravitational constant and $c$ is the speed of light. We
compare this against the simulated results for $L_{\mathrm{max}}$ in Fig. 6.
The analytical estimates are in good agreement with the simulated results.

Figure 5: Luminosity distribution in mass coordinate near the surface for the
model with
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=9~{}\textnormal{R}_{\odot},a_{\mathrm{i}}=60~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$ at the point of maximum expansion. The purple and green curves
are the stellar luminosity and the Eddington luminosity respectively. The blue
shaded region corresponds to the convective region. Figure 6: Maximum
luminosity as a function of secondary mass for our 720 models. The black curve
is the Eddington luminosity computed with Eq. (6).

###  4.2 Systematic study

In our simulations, the response of the companion after ECI is almost like a
phase transition between the inflated and original states (Fig. 3), and
therefore can be characterised by just three variables: the maximum
luminosity, maximum radius and duration of expansion. We have already shown
that the maximum luminosity is only dependent on the stellar mass. In this
section we will study the maximum radius and the duration of expansion and how
they depend on the other input parameters.

The relation between the injected energy as defined in Eq. (2) and the maximum
radius is presented in Fig. 7. There is a relatively strong correlation
between the injected energy and maximum radius especially at the higher energy
end. Models that fall off the relation are the models with smaller initial
radius, and the expansion radius become small with the same injected energy
due to the higher specific binding energy of the envelope. We empirically fit
a curve to the maximum radii with the function

$\log_{10}\frac{R_{\mathrm{max}}}{\textnormal{R}_{\odot}}=-\frac{1}{14}\left(\log_{10}\frac{E_{\mathrm{heat}}}{0.08\times
10^{51}~{}\mathrm{erg}}\right)^{2}+3.1,$ (7)

which is shown as the black curve.

Figure 7: Relation between the energy intersected by the companion and the
maximum inflated radius. Open circles are simulated results and the black
curve is an empirical fit based on Eq. (7). Colours of the circles denote the
secondary mass.

In Fig. 8, we display the duration of expansion as a function of the injected
energy. The duration of expansion is defined as the time from the energy
injection up to when the radius shrinks to half of the maximum radius. There
is a remarkably tight correlation that can be expressed as power-laws and have
different coefficients according to the secondary mass. More massive
secondaries have shorter inflated timescales even if the injected energy is
the same. Solid lines represent analytical fits to the correlation with the
function

$\tau_{\mathrm{inf}}=\alpha\frac{E_{\mathrm{heat}}}{L_{\mathrm{max,ana}}}$ (8)

where $L_{\mathrm{max,ana}}$ is computed from Eq. (6) and $\alpha$ is a
fitting coefficient which we set to $\alpha=0.181$. Although in principle
$\alpha$ can be mass-dependent, we find that the analytic curves fit the
simulated results fairly well with a universal value. This implies that the
inflated state of the star is only maintained until $\sim$18.1 % of the
injected energy is radiated away. We have compared this timescale with the
thermal timescale of the heated part of the envelope ($m_{h}$ from the
surface), but found no clear correlation, implying that $m_{h}$ does not
directly determine the depth of the inflated part of the envelope. The
dependence on other parameters such as ejecta mass or initial stellar radius
seem to be much weaker.

Figure 8: Relation between the energy intersected by the companion and the
duration of expansion. Open circles are simulated results and solid lines are
analytical fits based on Eq. (8). Colours of the circles and lines denote the
secondary mass.

It should be noted that the maximum radius and luminosity can be rather
sensitive to the choices made for our calculations. Because the maximum
luminosity is restricted by the Eddington luminosity determined around the
iron bump, it is quite sensitive to the metallicity of the star. We compared
the evolution with different metallicities in Fig. 9 and found that the
maximum luminosity can increase by $\sim 60\%$ when the metallicity is reduced
by two orders of magnitude. The inflated radius increases and the duration of
expansion decreases correspondingly. We will postpone a more thorough
investigation of metallicity effects to future work.

The maximum radius can particularly be dependent on the treatment of the outer
boundary condition. As the large amount of energy excess makes its way towards
the surface, it could develop dynamical instabilities (e.g. Shaviv, 1999),
which cannot be correctly accounted for in our MESA simulations. Such effects
can possibly drive super-Eddington winds (Owocki et al., 2004) or dynamically
eject the outer layers. It is also known that dynamical treatments of the
outer boundary can change the stellar radius considerably compared to the
usual hydrostatic treatments (e.g. Poniatowski et al., 2021). Properly
accounting for these effects will require a full hydrodynamical treatment in
multi-dimensions, which is out of the scope of this paper. However, we expect
the maximum luminosity and inflated timescales are much less sensitive to the
numerical choices and the qualitative behaviour of the maximum radii should
still hold, i.e. larger energy injection should lead to larger inflation.

Figure 9: Time evolution of the secondary radius (upper) and luminosity
(lower) after energy injection with different metallicities. Model parameters
are
$M_{2}=10~{}\textnormal{M}_{\odot},R_{2}=6~{}\textnormal{R}_{\odot},a_{i}=20~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=2~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}~{}\mathrm{erg}$.
Different colours correspond to metallicities of $Z=0.014,0.0014,0.00014$.

###  4.3 The fate of the system after ECI

To investigate the post-SN state of the system, we computed the post-explosion
orbital parameters. The orbital separation and the eccentricity depend on the
magnitude and direction of the natal kick imparted to the new-born NS. We show
in Fig. 10 the states of the systems after explosion for a given kick velocity
for the model with
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=6~{}\textnormal{R}_{\odot},a_{\mathrm{i}}=60~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}~{}\mathrm{erg},V_{\mathrm{kick}}=300~{}\mathrm{km~{}s^{-1}}$.
The inflated secondary reached a radius of
$R_{\mathrm{max}}=86~{}\textnormal{R}_{\odot}$ in this model. It illustrates
how the six outcomes classified in section 2.2 appear depending on the kick
direction. In the case that the kick velocity is pointed opposite to the
orbital motion ($\theta\sim 180^{\circ}$), the relative velocity between the
primary and the secondary decreases and the two stars are destined to collide
(State 1). At slightly more moderate angles
($150^{\circ}\lesssim\theta\lesssim 120^{\circ}$), the stars can avoid
colliding and instead the full orbit is embedded in the inflated part of the
secondary envelope depending on $\phi$ (state 2). In the region
$100^{\circ}\lesssim\theta\lesssim 120^{\circ}$, the orbit widens after the SN
and becomes partially embedded in the inflated secondary envelope (state 3).
These cases are potentially interesting as they may cause some brief X-ray
transients as they plunge in and out of the inflated envelope. As the kick
angle decreases and becomes closer to being aligned with the orbit
($\theta\sim 0^{\circ}$), the relative velocity increases and they will simply
become unbound and evolve as two single stars.

Figure 10: Example system state map after explosion, colour-coded according to
the kick direction. The binary model parameters are
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=7~{}\textnormal{R}_{\odot},a_{\mathrm{i}}=40~{}\textnormal{R}_{\odot}$
and the kick velocity is fixed to
$v_{\mathrm{kick}}=300~{}\mathrm{km~{}s^{-1}}$. The two angles $\theta,\phi$
for the kick direction are defined in Fig. 2, and the states are defined in
Section 2.2: (1) primary and secondary collide, (2) binary survives and the
primary orbit is fully embedded in the secondary envelope, (3) binary survives
and the primary orbit is partially embedded in the secondary envelope, (4)
binary survives and secondary envelope exceeds its Roche lobe at periastron,
(5) binary survives in a wide orbit where there is no immediate interaction
and (6) binary is disrupted.

The pattern in Fig. 10 can drastically change depending on the magnitude of
the kick velocity. In Fig.11, we show the probability of each outcome as a
function of the kick velocity. For each given kick velocity, the probilities
are computed by integrating the area in Fig. 10 assuming the kick direction is
isotropically distributed. We focus on the same binary model with
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=7~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}~{}\mathrm{erg}$
and the three panels display results for separations
$a_{\mathrm{i}}=20,40,60~{}\textnormal{R}_{\odot}$. The maximum inflated radii
are $R_{\mathrm{max}}=346,86,40~{}\textnormal{R}_{\odot}$ for the
$a_{\mathrm{i}}=20,40,60~{}\textnormal{R}_{\odot}$ models respectively. In all
panels the fraction of binary disruption (state 6) rapidly increases with the
kick velocity. In cases where the system keeps its binarity, the exact
configuration is strongly dependent on both the kick velocity and initial
orbital separation. The primary is more frequently embedded in the companion
envelope (state 2) in the models with closer separation. This is because the
secondary expansion is greater for smaller separation as seen in Fig. 3 and
satisfies $R_{\mathrm{max}}\geq a_{\mathrm{i}}$. Due to the fact that the
post-SN periastron distance cannot exceed the original separation regardless
of the kick velocity, part of the orbit will always intersect with the
envelope so states 4 and 5 cannot appear when state 2 is a possible outcome.
At larger separations, the companion inflation does not exceed the original
separation ($R_{\mathrm{max}}<a_{\mathrm{i}}$), so state 2 cannot appear.
Instead, we obtain systems in state 4, which may be interesting in terms of
pulsar planet formation.

Figure 11: System state rates as a function of kick velocity for the binary
model
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=7~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$. Separations are
$a_{\mathrm{i}}=20,40,60~{}\textnormal{R}_{\odot}$ from top to bottom panels
respectively. The states are defined in Section 2.2. The corresponding colours
are the same as in Fig. 10.

##  5 Discussion

###  5.1 How will the inflated companions be observed?

So far, we have shown the state of the system after explosion by combining the
results of the inflation calculation with the post-SN orbital parameters.
Based on these results we will also show how such systems may be observed by
considering the position on the HR diagram.

In the upper panel of Fig. 12, we show the position of the secondary on the HR
diagram in its maximally inflated phase. This corresponds to the end points of
the tracks in Fig. 4 and is where the star spends most of its time before
rapidly contracting back. In these figures we display models with
$M_{2}=15~{}\textnormal{M}_{\odot},a_{\mathrm{i}}=60~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}~{}\mathrm{erg}$,
with different initial secondary radii (corresponds to different ages at which
the primary SN occurs). It is clear that the positions of the secondary
affected by ECI are extremely far from the regular evolutionary track of a
$15~{}\textnormal{M}_{\odot}$ star and also even brighter than a
$20~{}\textnormal{M}_{\odot}$ star. In the lower panel, we show the same
figure as Fig. 12, but when the secondary expansion is suppressed by the
presence of the NS. Here we assume that the NS accretes or blows away any
matter that expands beyond the periastron distance of the post-SN orbit, while
keeping the same high luminosity due to the continuous supply of excess energy
diffusing out from the interior. This is a somewhat conservative assumption,
since there could be Roche lobe overflow and the star could be constrained by
its Roche lobe size instead. However, it is also possible that the NS just
plunges through the envelope without significantly disturbing it. The
periastron distance depends on the direction and magnitude of the kick. Here
we consider the smallest possible periastron distance for a given kick
velocity $V_{\mathrm{kick}}=100$ km s-1. Compared to Fig. 12 the larger
initial radius models ($R_{2}=$8–10$~{}\textnormal{R}_{\odot}$) are prohibited
to expand beyond the periastron distance, therefore the temperature and
luminosity of these models become indistinguishable.

(a) the envelope is not stripped

(b) the envelope is stripped

Figure 12: HR diagram for a model
$M_{2}=15~{}\textnormal{M}_{\odot},a_{\mathrm{i}}=60~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}~{}\mathrm{erg}$.
The grey curves are single star evolution tracks for
$10~{}\textnormal{M}_{\odot},15~{}\textnormal{M}_{\odot}$ and
$20~{}\textnormal{M}_{\odot}$ stars, and crosses represent the position of the
expanded secondary due to ECI at the maximum radius. In the upper panel, the
secondary envelope is not stripped by the primary, on the other hand, in the
lower panel the envelope is partially stripped by the primary NS.

We now focus on the observability of such inflated companions with various
telescopes. Assuming the inflated stars emit as a black body, we estimate how
far these systems could be observed with the Hubble Space Telescope (HST) and
the James Webb Space Telescope (JWST). We consider three filters,
`HST/WFC3_UVIS1.F300X` for UV, `JWST/NIRCam.F070W` for visible and
`JWST/NIRCam.F115W` for near-IR.

We show the observable distance of such inflated companions with each
wavelength in Fig. 13. The observable distance is defined as the maximum
distance where the apparent magnitude of the star is lower than a limiting
magnitude, which we set to $m_{\mathrm{lim}}=25$ mag. The thin lines show the
observable distance when there is no suppression of expansion. The thick
curves are the observable distances for stars with inflation suppressed at the
periastron distance of the orbit. Because the periastron distance depends on
the kick direction, we show the results for the most severely suppressed
cases. The true observable distance can be anywhere between the thin and thick
curves depending on the kick direction. How we calculate the distances are
explained in Appendix B. This model is for
$M_{2}=5~{}\textnormal{M}_{\odot},R_{2}=2.7~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=5~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\mathrm{erg}$.
The line colours and types indicate the different observation wavelengths and
initial separations. When the orbital separation is sufficiently close, we may
be able to see inflated $5~{}\textnormal{M}_{\odot}$ companions up to
$\lesssim 30$ Mpc in the IR, or $\lesssim 20$ Mpc in the visible band. This
will obviously be much higher for higher mass companions because the maximum
luminosity is proportional to mass. Basically, any suppression of expansion
will lower the detectable distance within the observational wavelengths we
show here, except for some regions of the UV where suppression leads to
brightening. It should also be noted that for the wider separation models, the
inflated companion may not be observed even if it is located at sufficiently
close distances ($\lesssim$5 Mpc). This is because the weaker energy models
have shorter inflated timescales and therefore may shrink back while the SN is
still bright.

Figure 13: The observable distance for the model
$M_{2}=5~{}\textnormal{M}_{\odot},R_{2}=2.7~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=5~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$. The blue, green and red curves denote the wavelength used to
observe, UV, visible and near IR respectively. Different line types show
results for different orbital separations
$a_{\mathrm{i}}=30,40,60~{}\textnormal{R}_{\odot}$. Thick curves assume that
the inflation can be suppressed by the primary, while the thin curves assume
no suppression.

###  5.2 Detection probability

Whether we can observe the inflated companions to CCSNe depend on both the
maximum luminosity during the inflated phase, which is determined solely by
the secondary mass, and the distance to the system. If we consider expansion
suppression by the primary, it will also depend on the orbital separation. The
combination of secondary masses and orbital separation is strongly dependent
on the binary evolution leading to the SN. We perform rapid binary population
synthesis calculations in order to obtain the distribution of secondary masses
and orbital separations and estimate the detection probability of inflated
secondaries.

We use the widely used BSE code (Hurley et al., 2002), to evolve 1,400,000
binaries based on simplified binary evolution prescriptions. Two sets of
models are computed with solar metallicity $Z=0.0134$ and sub-solar
metallicity $Z=0.00134$. The initial primary masses are drawn from the
Salpeter initial mass function (Salpeter, 1955), and the secondary mass is
drawn from a flat distribution for the mass ratio $q=M_{2}/M_{1}$. The initial
orbital separation is assumed to follow a log-flat distribution (Abt, 1983),
and the initial eccentricities are set to zero for simplicity. The initial
primary mass range is chosen to be $7<M_{1}/\textnormal{M}_{\odot}<300$, and
the initial mass ratio range is $0.08/M_{1}<q<1$ (this corresponds to an
initial secondary mass range of $0.08<M_{2}/\textnormal{M}_{\odot}<M_{1}$).
The range of the initial separation is
$a_{\mathrm{min}}/\textnormal{R}_{\odot}<a/\textnormal{R}_{\odot}<10^{6}$,
where $a_{\mathrm{min}}$ is the separation at which the primary fills its
Roche lobe. All models are evolved up to the point of the first SN in the
system. $\sim$400,000 of the binaries end up merging before exploding, so only
$\sim$1,000,000 binaries experience the first SN with a companion.

The binary physics adopted in our population synthesis calculations are as
follows: When a star fills its Roche lobe, the stellar material is transferred
to its companion. The mass transfer stability is determined by comparing
$\zeta_{\mathrm{L}}=\mathrm{d}\log R_{\mathrm{L,don}}/\mathrm{d}\log
M_{\mathrm{don}}$ with $\zeta_{\mathrm{ad}}=(\mathrm{d}\log
R_{\mathrm{don}}/\mathrm{d}\log M_{\mathrm{don}})_{\mathrm{ad}}$, where
$M_{\mathrm{don}}$, $R_{\mathrm{don}}$ and $R_{\mathrm{L,don}}$ are the mass,
radius and Roche lobe radius of the donor star, respectively.
$\zeta_{\mathrm{ad}}$ depends on the stellar type, and in this study we use
the following values: $-1+2M_{\mathrm{don}}/3M_{\mathrm{env,don}}$, 2.59,
6.85, 1.95 and 5.79 for the giant star with the convective envelope (Hjellming
& Webbink, 1987), main sequence, giant star with the radiative envelope
(Hjellming, 1989), helium main sequence and helium giant star (Ivanova et al.,
2002; Belczynski et al., 2008), where $M_{\mathrm{env,don}}$ is the envelope
mass of the donor giant star. If $\zeta_{\mathrm{L}}>\zeta_{\mathrm{ad}}$, the
mass transfer is dynamically unstable, and the binary system enters a common-
envelope phase. We use the $\alpha\lambda$ formalism for common-envelope
evolution (Webbink, 1984), and set the common-envelope efficiency parameter as
$\alpha_{\mathrm{CE}}=1$. Prescriptions in Xu & Li (2010a, b) are used for the
envelope binding energy parameter $\lambda_{\mathrm{CE}}$. We assume that the
binary system coalesces after the common-envelope phase if the radius of one
of the stars exceeds its Roche lobe radius (Hurley et al., 2002). If
$\zeta_{\mathrm{L}}<\zeta_{\mathrm{ad}}$, the donor mass is stably transferred
to its companion. We calculate the mass transfer rate using the Eq. (58) in
Hurley et al. (2002). In this study, we assume that the mass transfer is non-
conservative and only half of the transferred mass can be accreted onto the
companion. The non-accreted mass is assumed to be isotropically re-emitted,
ejected with the specific angular momentum of the accretor. If an SN explosion
occurs, we calculate the remnant mass from the carbon-oxygen core mass,
adopting the ‘rapid’ SN prescription in Fryer et al. (2012).

Because ECI will only be important in close binaries, we focus on the
hydrogen-deficient SNe. In population synthesis codes, it is not possible to
distinguish between type Ib and IIb SNe due to the way mass transfer is
treated. We simply define the models with no hydrogen envelope as stripped-
envelope SNe. This leaves us with $\sim$150,000 (180,000 for the sub-solar
metallicity model) binaries out of the initial 1,400,000. Moreover the number
of the post-common-envelope systems is $\sim$33,000 (67,000 for sub-solar
metallicity) in binaries that experience stripped-envelope SNe. Most of them
leave close binaries that are relevant for ECI, so as a result 77.9% (68.8%
for sub-solar metallicity) of the close ($a_{i}<60~{}\textnormal{R}_{\odot}$)
binaries including the stripped-envelope SN progenitors are the post-common-
envelope binaries.

For each binary in our population synthesis models, we use the secondary mass,
radius and orbital separation to calculate how much the star will inflate,
based on our ECI simulations333We ignore the effect of metallicity on post-ECI
inflation. Instead we apply our solar metallicity ECI results even to the sub-
solar metallicity population synthesis models.. We also use the CO core mass
to estimate the kick velocity (see Appendix C). By assuming the kick is
randomly directed isotropically, we calculate the probabilities of each state
which the system can end up in. We then add up all the probabilities to obtain
the rate of occurrence of each state over the whole stripped-envelope SN
population.

The results are summarised in Table 2. The most frequent case is state 5 (two
stars are bound, but nothing occurs) and the probability is about 97 %. The
separation is large in these models and therefore there is little expansion
due to ECI. The second most frequent case is state 6 (binary disruption) with
1-2%. The third most frequent case is state 2 (primary NS engulfed in the
secondary envelope) with $\sim$0.8%. State 2 and 3 is probably the most
interesting among other states. There could be accretion onto the NS that
cause brief X-ray transients, and/or it could suppress the expansion of the
envelope. Our rate estimate implies that possibly $\sim$0.8 % of stripped-
envelope SNe may be accompanied by brief X-ray transients if the environment
is sufficiently optically thin. On the other hand, state 4 seems quite rare in
both metallicities. This implies that pulsar planet formation through this
channel may be extremely rare.

| state 1 [%] | state 2 [%] | state 3 [%] | state 4 [%] | state 5 [%] | state 6 [%]
---|---|---|---|---|---|---
Solar | 0.27 | 0.82 | 0.064 | 2.6$\times 10^{-4}$ | 97.2 | 1.62
sub-Solar | 0.27 | 0.75 | 0.072 | 6.7$\times 10^{-4}$ | 96.8 | 2.10
Table 2: The rate that each states appear after hydrogen deficient SNe for two
metallicities ($Z=1.0~{}Z_{\odot},0.1~{}Z_{\odot}$). The different states are
defined in section 2.2.

We estimate the expected probability at which we will be able to detect an
inflated companion per stripped-envelope SN at five different wavelengths.
Again, our limiting magnitude is set to $m_{\mathrm{lim}}=25$ mag. We
calculate the detectability for five filters, `HST/WFC3_UVIS1.F300X` for the U
band, `HST/WFC3_UVIS1.F410M` for the B band, `HST/WFC3_UVIS1.J547M` for the V
band, `JWST/NIRCam.F070W` for the R band and `JWST/NIRCam.F115W` for the I
band. The detailed calculation methods are described in Appendix C.

The detection probabilities are summarized in Table 3. The inflated
secondaries could be observed best in the U band and the detection probability
is $\sim 1.0~{}\%$. This means that out of all stripped-envelope SNe in the
local Universe ($\leq$20 Mpc), there is $\sim$1 % chance of detecting an
inflated companion. The detection probabilities are higher for sub-solar
metallicity models. This is likely because the stellar winds are stronger in
higher metallicity, leading to wider separations and therefore less inflation
after ECI. 91.6 % (89.5 % for sub-solar metallicity) of the detectable systems
are post-common-envelope binaries in the suppressed case. The detectability is
$\sim$3 times higher in all bands when we lift the assumption of expansion
suppression. Since we do not know whether the inflation can actually be
inhibited by the presence of the NS, the true detectability can be anywhere in
between.

We also note that the detectability will likely be higher if we constrain our
targets to type Ib SNe. As we have mentioned before, the different types of
stripped-envelope SNe cannot be distinguished in rapid binary population
synthesis codes. Type Ib SNe are more severely stripped than type IIb SNe, so
the typical binary separation can be expected to be closer for type Ib
progenitor systems and therefore have more inflated companions. Type Ib SNe
comprise roughly half of the hydrogen-deficient SNe (e.g. Shivvers et al.,
2017), so the detection rates will roughly be twice higher. Also, we have only
derived rates for a particular set of assumptions in the population synthesis
calculations. For example if we choose a smaller value for the common-envelope
efficiency parameter, the typical post-common-envelope separations will be
smaller and could drastically affect our rate estimates. Conversely, we may be
able to use observed rates of inflated companions to constrain the binary
evolution physics. We will leave this exploration to future work.

Population model | U [%] | B [%] | V [%] | R [%] | I [%]
---|---|---|---|---|---
$1.0~{}Z_{\odot}$, suppressed | 0.90 | 0.27 | 0.39 | 0.03 | 0.0
$1.0~{}Z_{\odot}$, no suppression | 2.73 | 2.60 | 1.13 | 0.05 | 0.0
$0.1~{}Z_{\odot}$, suppressed | 1.08 | 0.31 | 0.44 | 0.06 | 0.0
$0.1~{}Z_{\odot}$, no suppression | 3.11 | 2.95 | 1.36 | 0.11 | 0.0
Table 3: Observable probabilities of inflated companions in each band. Results
are shown for two types metallicities ($Z=1.0~{}Z_{\odot},0.1~{}Z_{\odot}$),
and with or without expansion suppression.

###  5.3 Implications on binary evolution

With the relations presented in Section 4, it is now possible to derive the
complete set of pre-SN binary parameters from detections of inflated
companions. First, we can use Eq. (6) to calculate the secondary mass from the
luminosity $L$. By manipulating Eq. (6), the secondary mass can be expressed
as

$\frac{M_{2}}{\textnormal{M}_{\odot}}=\left(0.02+\frac{4\pi
Gc\textnormal{M}_{\odot}}{\kappa_{0}L}\right)^{-1},$ (9)

where $\kappa_{0}=1.24~{}\mathrm{cm}^{2}~{}\mathrm{g}^{-1}$. Note that this is
only valid if the observed source is in the inflated state. Second, we can use
Eq. (8), to compute the injected energy using the inflated timescale

$E_{\mathrm{heat}}=\frac{\tau_{\mathrm{inf}}L}{\alpha}.$ (10)

Then from $E_{\mathrm{heat}}=pE_{\mathrm{expl}}\tilde{\Omega}\sim
pE_{\mathrm{expl}}(R_{2}/a_{\mathrm{i}})^{2}/4$, we obtain

$\frac{R_{2}}{a_{\mathrm{i}}}\sim\sqrt{\frac{4\tau_{\mathrm{inf}}L}{\alpha
pE_{\mathrm{expl}}}},$ (11)

where $p\sim 0.08-0.10$ and $\alpha\sim 0.18$. Assuming that the explosion
energy is constrained from other diagnostics such as the SN light curve, these
relations allow us to explicitly compute the pre-SN binary parameters.
Obtaining the required information such as the companion luminosity $L$ and
the inflated timescale $\tau_{\mathrm{inf}}$ requires multi-band photometry
over multiple epochs. Single band photometry will only allow us to constrain
the luminosity very loosely. We can also only place lower limits on the
duration of expansion unless the contraction back from the inflated state is
observed.

There is also the possibility that the surface of the companion is polluted
with heavy elements captured from the slower parts of the SN ejecta. The
amount of pollution is usually very small ($\lesssim
10^{-3}~{}\textnormal{M}_{\odot}$; Liu et al., 2015; Hirai et al., 2018), and
it is also strongly dependent on the orbital separation. The accreted elements
will be quickly mixed through the surface convective layers of the inflated
envelope. It is not clear how observable these chemical features would be, but
it may provide an additional constraint on the orbital separation.

Extracting the full set of pre-SN binary parameters for an observed system may
have extremely important implications for binary evolution theory. The systems
that have sufficiently inflated companions are the systems with close orbital
separations, which mostly have experienced common-envelope evolution. Despite
its great importance in stellar astrophysics, common-envelope phases are one
of the least understood phases in binary evolution (see Ivanova et al., 2013,
for a review on common-envelope evolution). Of particular importance is the
orbital separation after common-envelope phases, which determines how the
binary evolves thereafter. Because the pre-SN orbital separations are closely
related to the post-common-envelope separation, the detections of inflated
companions may provide the best constraints on the outcome of common-envelope
phases to date. This will give insight into the long sought common-envelope
efficiency parameter ($\alpha_{\textsc{ce}}$), which has so far been difficult
to calibrate from observations especially in the massive star regime. For the
systems that only experienced stable mass transfer, the pre-SN orbital
separation would be determined by how conservative the mass transfer is, and
the mode of angular momentum loss. Observations of inflated companions can
help probe these effects too.

###  5.4 Application to SN 2006jc

There are still only a limited number of SNe that have detections of companion
stars. It is also crucial to obtain multi-epoch observations of the companion
in order to constrain the inflated timescale, which can be used to estimate
the pre-SN binary parameters. One case that satisfies these requirements is SN
2006jc (Foley et al., 2007; Pastorello et al., 2007; Maund et al., 2016; Sun
et al., 2019).

The SN site was examined in 2010 and 2017, $\sim$3 and 10 years after the
explosion, both showing fluxes consistent with each other. From multi-band
photometry of the post-SN source, the temperature and luminosity of the
companion to SN 2006jc were constrained and is consistent with being a
Hertzsprung gap star of mass $M_{2}\sim$10–15 $\textnormal{M}_{\odot}$ (Fig.
14, Maund et al., 2016; Sun et al., 2019). Although it is feasible, it is also
a rare case because the Hertzsprung gap phases of massive stars are very
short-lived. Sun et al. (2019) investigated whether this could instead be a
low-mass star inflated due to ECI, and estimated the binary parameters based
on this scenario. However, there was one issue with the scenario. For models
that can inflate for more than 10 years, the maximum radius is also quite
large and tend to lie closer to the red supergiant regime, while the observed
source was bluer.

Here we will bring in the additional assumption that the stellar expansion
could have been inhibited by the presence of the NS. This would allow the
inflated star to appear hotter, while having the high luminosity and long
inflated timescales. Based on this scenario, we can already constrain the pre-
SN binary parameters using the analytical formulae we derived above. The
luminosity of the source is $\log(L/\textnormal{L}_{\odot})\sim 4.5$. From Eq.
9, the secondary mass should be $\sim 3~{}\textnormal{M}_{\odot}$. Because the
companion is inflated for $\gtrsim$10 yr, Eq. (10) gives
$E_{\mathrm{int}}\gtrsim 3\times 10^{48}$ erg. The estimated temperature of
the companion corresponds to a star with a radius $\sim
40~{}\textnormal{R}_{\odot}$. Assuming that the radius is constrained by the
periastron distance of the post-SN orbit, the pre-SN separation should be
$a_{\mathrm{i}}\gtrsim 40~{}\textnormal{R}_{\odot}$. This leaves us with only
the explosion energy as a free parameter.

The contours in Fig. 14 show the HR diagram constraints of the SN 2006jc
companion at two epochs. We overplot three post-ECI models run with parameters
summarized in Table 4. By assuming a suppressed expansion due to the presence
of the primary NS, all models lie within the observational constraints. The
expanded timescale is from a few to tens of years depending on the explosion
energy. From the follow-up observations, SN 2006jc has a flat light curve for
at least ten years. Therefore if the companion of SN 2006jc is indeed a post-
ECI star where its expansion is suppressed by the primary NS, the model with
relatively high explosion energy (model 3, $E_{\mathrm{expl}}\gtrsim
10^{52}\mathrm{erg}$) is compatible with the observations. This is consistent
with some theoretical estimates from the SN light curve (Tominaga et al.,
2008).

Figure 14: Evolutionary tracks of post-ECI models with $M_{2}=3~{}\textnormal{M}_{\odot},R_{2}=2~{}\textnormal{R}_{\odot},a_{\mathrm{i}}=40~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=2~{}\textnormal{M}_{\odot}$ on the HR diagram. The various cross symbols mark the locations if the envelope expansion is suppressed by the primary. The three models have different explosion energies, $E_{\mathrm{expl}}=1,5,10\times 10^{51}\ \mathrm{erg}$ respectively. The three solid curves are the evolutionary tracks in the suppressed model, and the dashed curves are the evolutionary tracks without the suppression. The square, circle and triangles are time stamps at $5,10,20$ and $30$ years after SN explosion, and these are also for the non-suppressed model. The contours show the probability of SN 2006jc companion’s position from 2010 and 2017 observations (Sun et al., 2019). | $M_{2}[\textnormal{M}_{\odot}]$ | $R_{2}[\textnormal{R}_{\odot}]$ | $a_{\mathrm{i}}[\textnormal{R}_{\odot}]$ | $M_{\mathrm{ej}}[\textnormal{M}_{\odot}]$ | $E_{\mathrm{expl}}[\mathrm{erg}]$ | timescale [yr]
---|---|---|---|---|---|---
model 1 | 3 | 2.0 | 40 | 2 | $10^{51}$ | 1.5
model 2 | 3 | 2.0 | 40 | 2 | $5\times 10^{51}$ | 7.0
model 3 | 3 | 2.0 | 40 | 2 | $10^{52}$ | 13.0
Sun et al. | 4 | 2.5 | 57 | 2 | $10^{52}$ | -
Table 4: Simulation parameters for the models plotted in Fig. 14.

Within this scenario, we expect the companion to contract back to its original
radius within a few years. For example in model 3, the temperature and
luminosity drop out of the contours $\sim$15 years after the SN (red
triangle). Therefore, it will be vital to carry out an additional observation
of the SN site some time soon to further constrain the model.

In Sun et al. (2019), the assumed binary parameters were
$a_{\mathrm{i}}=57~{}\textnormal{R}_{\odot},M_{2}=4~{}\textnormal{M}_{\odot},R_{2}=2.5~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=5~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{52}\mathrm{erg}$.
Compared with their results, the system of SN 2006jc could be a closer binary,
if the stellar expansion can indeed be inhibited by the primary.

##  6 Summary

We systematically study the outcome of SNe in binary systems whose companions
are MS stars. We carry out 720 1D stellar evolution calculations of companion
stars for 10,000 years after being impacted by SN ejecta. This is done using
the 1D stellar evlution code MESA by injecting energy into the star based on
the formulation of H18. We explore a wide parameter space, varying the
secondary mass, secondary radius (or age), orbital separation, ejecta mass and
explosion energy.

The companion star is driven out of thermal equilibrium and immediately
expands its envelope for several years before rapidly contracting back to its
original size. Most of the inflated companions exist in the Hertzsprung gap of
the HR diagram, and are much more luminous than its regular luminosity. The
maximum luminosities are determined by the Eddington luminosity computed with
Fe opacity. We derive an empirical formula for the relevant opacity as a
function of the secondary mass, which can be used to relate the post-SN
luminosity and mass of the companion. The duration of expansion is tightly
correlated with the injected energy. If we can observe inflated companions in
the future, the luminosity and inflated timescales can be used to determine
the full set of pre-SN binary parameters. Such detections may have extremely
important implications for understanding binary evolution physics, especially
on common-envelope evolution.

We discuss the various different states the binary can take after the SN. We
identify 6 different states (written in section 2.2) which are determined by
the kick velocity and direction. For example, part of the inflated envelope
may be transferred to the new-born NS, possibly leading to the formation of
pulsar planets. As another possible outcome, the NS may directly plunge into
the inflated part of the envelope, which could lead to short X-ray transients
and/or the accretion feedback may blow away the outer parts of the envelope.
This may be particularly interesting as it will affect the appearance of the
secondary star, making it smaller in size and therefore hotter in temperature.

We also performed simple binary population synthesis calculations to estimate
the observability of companions. We estimate that 1–3 % of hydrogen deficient
SNe within <20 Mpc may have observable inflated companions which will be most
detectable in the U band.

Finally, we apply our model to the observed companion of SN 2006jc which is a
type Ibn SN. Using our analytic formulae, we estimate the binary parameters to
be
$M_{2}=3~{}\textnormal{M}_{\odot},R_{2}=2~{}\textnormal{R}_{\odot},a_{\mathrm{i}}=40~{}\textnormal{R}_{\odot}$,
and the explosion energy should be relatively large ($E_{\mathrm{expl}}\sim
10^{52}$ erg). We apply a new assumption that the inflation could be
suppressed due to the presence of the NS. Compared with Sun et al. (2019), the
progenitor of SN 2006jc could be a closer binary. The inflated timescale can
strongly vary depending on the explosion energy. Therefore, if future
observations reveal that the companion has faded, it will provide additional
strong constraints on the explosion energy.

## Acknowledgements

The authors thank the anonymous referee for constructive comments that
improved the manuscript. The authors thank Ning-Chen Sun for providing data of
the HR diagram constraints for SN 2006jc. MO thanks Keiichi Maeda and the
COMPAS team for fruitful discussions during the development of the project.

## Data Availability

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

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##  Appendix A Post-SN orbital variables

The orbital variables after SN are computed based on an approximation that the
explosion is impulsive. First, we introduce velocities related to orbital
motion. As we assume an initially circular orbit, the orbital velocity just
before the explosion $V_{\mathrm{i}}$ is

$V_{\mathrm{i}}=\left(\frac{G(M_{1}+M_{2})}{a_{\mathrm{i}}}\right)^{1/2},$
(12)

where $G$ is the gravitational constant and $M_{1}$ the primary mass,
$a_{\mathrm{i}}$ is the initial orbital separation. We set
$M_{1}=M_{\mathrm{ej}}+m_{c}$, where $m_{c}$ is the primary core mass and
equal to the remnant NS mass, and we use $m_{c}=1.4~{}\textnormal{M}_{\odot}$
for all of our models. We also consider the natal kick imparted to the remnant
of the exploding star which is expressed as

$\displaystyle V_{\mathrm{kick,x}}$
$\displaystyle=V_{\mathrm{kick}}\sin\theta\cos\phi,$ (13) $\displaystyle
V_{\mathrm{kick,y}}$ $\displaystyle=V_{\mathrm{kick}}\cos\theta,$ (14)
$\displaystyle V_{\mathrm{kick,z}}$
$\displaystyle=V_{\mathrm{kick}}\sin\theta\sin\phi,$ (15)

where $V_{\mathrm{kick}}$ is the magnitude of the kick. The coordinate angles
are defined in Fig. 2. The impact velocity of the secondary is also taken into
account and estimated analytically based on the formula (H18)

$V_{\mathrm{im}}=\eta\left(2E_{\mathrm{expl}}M_{\mathrm{ej}}\right)^{1/2}\frac{\tilde{\Omega}}{M_{2}},$
(16)

where $\eta$ and $\tilde{\Omega}$ are the efficiency of momentum transfer
($\sim 1/3$) and the fractional solid angle to the companion ($=\Omega/4\pi$),
and the expression in the brackets is a rough estimate of the total momentum
of the ejecta.

The separation and eccentricity of the orbit after SN explosion
$a_{\mathrm{f}},e$ is computed with the following formula (Postnov &
Yungelson, 2014)

$\displaystyle a_{\mathrm{f}}$
$\displaystyle=a_{\mathrm{i}}\left[2-X\frac{(V_{\mathrm{kick,x}}+V_{\mathrm{i}})^{2}+(V_{\mathrm{kick,y}}+V_{\mathrm{im}})^{2}+V_{\mathrm{kick,z}}^{2}}{V_{\mathrm{i}}^{2}}\right]^{-1},$
(17) $\displaystyle e$
$\displaystyle=\left(1-X\frac{a_{\mathrm{i}}}{a_{\mathrm{f}}}\frac{(V_{\mathrm{kick,y}}+V_{\mathrm{im}})^{2}+V_{\mathrm{kick,z}}^{2}}{V_{\mathrm{i}}^{2}}\right)^{1/2},$
(18)

where $X=(M_{1}+M_{2})/(m_{c}+M_{2})$ is the mass ratio of the total masses
before and after explosion.

When $e>1$, the binary is disrupted. If the binary survives, the orbital
period $P_{\mathrm{orb}}$ is

$P_{\mathrm{orb}}=2\pi\left(\frac{a_{\mathrm{f}}^{3}}{G(m_{c}+M_{2})}\right)^{1/2}.$
(19)

The Roche lobe radius (Eggleton, 1983) of the secondary is

$R_{\mathrm{rl}}=\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})}a_{\mathrm{i}},$
(20)

where $q=M_{2}/M_{1}$ is the mass ratio. We use $R_{\mathrm{rl}}$ to evaluate
whether Roche-lobe overflow commences or not.

##  Appendix B Observable distance

To calculate the observable distance of inflated secondary, we approximate the
emission from the secondary as a black-body. The radiation intensity
$B(T,\lambda)$ is

$B(T,\lambda)=\frac{2hc^{2}}{\lambda^{5}}\left[\exp\left(\cfrac{hc}{kT\lambda}\right)-1\right]^{-1},$
(21)

where $h,c,k,\lambda$ and $T$ are the Planck constant, the speed of light, the
Boltzmann constant, emission wavelength and temperature respectively. Using
the radiation intensity we obtain the specific flux
$F_{\lambda}(10\mathrm{pc})$ for wavelength $\lambda$ at $10\mathrm{pc}$ away
in order to seek the absolute magnitude

$F_{\lambda}(10\mathrm{pc})=\pi
B(T,\lambda)\left(\frac{R}{10\mathrm{pc}}\right)^{2},$ (22)

where $R$ is the stellar radius. Then the absolute magnitude at a given
wavelength $\lambda$ follows from

$M_{\mathrm{abs}}=2.5\log_{10}\left[\frac{F_{\mathrm{zero\
point}}}{F_{\lambda}(10\mathrm{pc})}\right],$ (23)

where $F_{\mathrm{zero\ point}}$ is the zero point flux for each filter of the
telescopes. We then compute the maximum observable distance for a given
limiting magnitude from

$d=10^{\left(m_{\mathrm{lim}}-M_{\mathrm{abs}}+5-A(\lambda)\right)/5},$ (24)

where $m_{\mathrm{lim}},A(\lambda)$ are limiting magnitude and interstellar
extinction. We set the limiting magnitude to 25 mag for all wavelengths. We
use laws in Cardelli et al. (1989) for the interstellar extinction that
depends on the wavelength. As typical extinction values for SN Ibc, we use
$E(B-V)=0.4$ and $R_{\mathrm{V}}=3.1$ according to Drout et al. (2011).

##  Appendix C Rate calculation

To calculate the observable rate, we make full use of our binary population
synthesis data and our post-ECI inflation simulation results. First, we assign
a maximum expansion radius ($R_{\mathrm{2,max}}$) and maximum luminosity
($L_{\mathrm{max}}$) to each binary in the population synthesis data set by
interpolating between our MESA inflated star models with the closest binary
parameters $M_{2},R_{2},a$. We also randomly assign a distance to each binary
assuming they are uniformly distributed within 20 Mpc from the observer.

For a given binary model and observation wavelength $\lambda$, we can
calculate the minimum required absolute flux $F_{\lambda}$(10pc) such that the
apparent magnitude is brighter than the limiting magnitude
$m_{\mathrm{lim}}=25$ mag, using Eqs. (23)–(24). Then we plug this into Eq.
(22) and solve for $T$, which gives the minimum temperature required for the
star to have a specific flux above the detection limit ($\equiv
T_{\mathrm{min}}$). Finally, by assuming the star is emitting as a black body,
the minimum observable radius of the star is given by

$R_{\mathrm{min,obs}}=\left(\frac{L_{\mathrm{max}}}{4\pi\sigma
T_{\mathrm{min}}^{4}}\right)^{1/2}.$ (25)

Whenever the maximum expansion radius of the star exceeds the minimum
observable radius ($R_{\mathrm{2,max}}\geq R_{\mathrm{min,obs}}$), the star is
“detectable”. For the case where we do not take into account any expansion
suppression, we simply count the number of detectable systems and divide by
the total number of stripped-envelope SNe to obtain the detection rate.

For the case where we consider expansion suppression by the primary, we also
take into account the effect of the NS kick on the orbit. For the kick
magnitudes, we use the fitting formula presented in Vigna-Gómez et al. (2018),
that relates the velocity to the CO core mass $m_{\mathrm{CO}}$

$V_{\mathrm{kick}}=\left\\{\begin{array}[]{cc}35+1000(m_{\mathrm{CO}}-1.372),&1.372\leq
m_{\mathrm{CO}}<1.49\\\ 90+650(m_{\mathrm{CO}}-1.49),&1.49\leq
m_{\mathrm{CO}}<1.65\\\ 100+175(m_{\mathrm{CO}}-1.65),&1.65\leq
m_{\mathrm{CO}}<2.4\\\ 200+550(m_{\mathrm{CO}}-2.4),&2.4\leq
m_{\mathrm{CO}}<3.2\\\ 80+120(m_{\mathrm{CO}}-3.2),&3.2\leq
m_{\mathrm{CO}}<3.6\\\ 350+50(m_{\mathrm{CO}}-4.05),&4.05\leq
m_{\mathrm{CO}}<4.6\\\ 275-300(m_{\mathrm{CO}}-5.7),&5.7\leq
m_{\mathrm{CO}}<6.0\end{array}\right.$ (26)

This gives velocities ranging from $0$ to $700~{}\mathrm{km~{}s^{-1}}$.
Whenever $m_{\mathrm{CO}}$ falls outside the mass range, we assume a BH is
formed and therefore there was no strong SN. With the given kick velocity, we
calculate a cumulative probability distribution of the periastron distance
assuming the kick direction is uniformly distributed over all solid angles.
This corresponds to the probability of the post-SN radius of the star within
our scenario. In Fig. 15 we show some sample cumulative probability
distributions of the radius for a model
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=8~{}\textnormal{R}_{\odot},a_{\mathrm{i}}=40~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$. Kick velocities are $V_{\mathrm{kick}}=100,200$ and
$300~{}\mathrm{km~{}s}^{-1}$. The maximum radius is about
$R_{\mathrm{2,max}}\sim 220~{}\textnormal{R}_{\odot}$ in this model. The
distributions of 200 and 300 $\mathrm{km~{}s}^{-1}$ do not start from 0
because we set the radius to zero whenever the system ends in state 1 (stellar
collision).

Figure 15: Cumulative probability distributions of the post-ECI radius for the
model
$M_{2}=15~{}\textnormal{M}_{\odot},R_{2}=8~{}\textnormal{R}_{\odot},a_{\mathrm{i}}=40~{}\textnormal{R}_{\odot},M_{\mathrm{ej}}=7.1~{}\textnormal{M}_{\odot},E_{\mathrm{expl}}=10^{51}\
\mathrm{erg}$, assuming that the radius expansion is limited to the periastron
distance. Different colours denote different kick velocities
$V_{\mathrm{kick}}=100,200$ and $300~{}\mathrm{km~{}s}^{-1}$.

From the cumulative distribution function of radius, we can compute the
probability of the star to have a radius exceeding $R_{\mathrm{min,obs}}$,
which we denote as $P_{\mathrm{obs},i}$. This means the system $i$ has a
probability $P_{\mathrm{obs},i}$ of being observable. Then the detection
probability for the whole population is computed from

$P_{\mathrm{obs}}=\frac{1}{N}\sum_{i}P_{\mathrm{obs,i}},$ (27)

where $N$ is the total number of stripped-envelope SNe.

  *[SN]: supernova
  *[CCSNe]: Core-collapse supernova
  *[CCSN]: Core-collapse supernova
  *[SNe]: supernova
  *[NSs]: neutron star
  *[BHs]: black hole
  *[NS]: neutron star
  *[MS]: main-sequence
  *[BH]: black hole