# Bright Supernova Precursors by Outbursts from Massive Stars with Compact
Object Companions

Daichi Tsuna TAPIR, Mailcode 350-17, California Institute of Technology,
Pasadena, CA 91125, USA  Research Center for the Early Universe (RESCEU),
School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo
113-0033, Japan  Tatsuya Matsumoto Department of Astronomy, Kyoto University,
Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan  Hakubi Center,
Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan
Department of Physics and Columbia Astrophysics Laboratory, Columbia
University, Pupin Hall, New York, NY 10027, USA  Samantha Chloe Wu TAPIR,
Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA
Jim Fuller TAPIR, Mailcode 350-17, California Institute of Technology,
Pasadena, CA 91125, USA

###### Abstract

A fraction of core-collapse supernovae (SNe) with signs of interaction with a
dense circumstellar matter are preceded by bright precursor emission. While
the precursors are likely caused by a mass ejection before core-collapse,
their mechanism to power energetic bursts, sometimes reaching
$10^{48}$–$10^{49}\ {\rm erg}$ that are larger than the binding energies of
red supergiant envelopes, is still under debate. Remarkably, such a huge
energy-deposition should result in an almost complete envelope ejection and
hence a strong sign of interaction, but the observed SNe with precursors show
in fact typical properties among the interacting SNe. More generally, the
observed luminosity of $10^{40-42}\,\rm erg\,s^{-1}$ is shown to be
challenging for a single SN progenitor. To resolve these tensions, we propose
a scenario where the progenitor is in a binary system with a compact object
(CO), and an outburst from the star leads to a super-Eddington accretion onto
the CO. We show that for sufficiently short separations, outbursts with
moderate initial kinetic energies of $10^{46}$–$10^{47}$ erg can be energized
by the accreting CO so that their radiative output can be consistent with the
observed precursors. We discuss the implications of our model in relation to
CO binaries detectable with Gaia and gravitational wave detectors.

Eruptive phenomena; Stellar mass-loss; Circumstellar matter; Core-collapse
supernovae

##  1 Introduction

Mass loss is an important process in massive stars that characterizes their
evolution and the appearances of their final supernova (SN) explosions (e.g.,
Smith, 2014). A wide variety of core-collapse explosions show signatures of
dense circumstellar matter (CSM) around the progenitor, indicating suddenly
enhanced mass-loss months to centuries prior to core-collapse. These span from
SNe with narrow lines classified as Type IIn/Ibn/Icn (Schlegel, 1990;
Pastorello et al., 2008; Gal-Yam et al., 2022), and possibly Type II-P/L SNe
that comprise about half of core-collapse SNe (Moriya et al., 2011; Khazov et
al., 2016; Morozova et al., 2017, 2018; Yaron et al., 2017; Förster et al.,
2018; Bruch et al., 2021, 2023).

For a fraction of these SNe, the dense CSM is linked to bright optical flares
observed from months to years before core-collapse, often called SN
precursors. Since the first finding of a precursor in a Type Ibn SN 2006jc two
years before the SN (Nakano et al., 2006; Pastorello et al., 2007), precursors
have been observed in many Type IIn SNe (e.g., Mauerhan et al., 2013; Fraser
et al., 2013; Ofek et al., 2013, 2014a; Margutti et al., 2014; Elias-Rosa et
al., 2016; Strotjohann et al., 2021; Fransson et al., 2022; Hiramatsu et al.,
2023) and a few SNe of other types including Type Ibn SN 2019uo (Strotjohann
et al., 2021) and Type II-P SN 2020tlf (Jacobson-Galán et al., 2022).
Understanding these precursors is important as clues to probing the mass-loss
mechanism that produces dense CSM in the final years of massive stars.

The observed precursors are bright with luminosities of $10^{40}$–$10^{42}$
erg s-1, which are orders of magnitude higher than the Eddington limit of
massive stars. While the super-Eddington energy injection may naturally
explain the generation of dense CSM (e.g., Matsumoto & Metzger, 2022a), the
mechanism that triggers such powerful energy injection is yet unclear. For the
wave-driven mass loss mechanism where specific predictions exist, the
luminosity of the (possible) outbursts months to years before core-collapse
are estimated to be only $\lesssim 10^{6}L_{\odot}$ (Shiode & Quataert 2014;
Fuller 2017; Fuller & Ro 2018; Leung et al. 2021; Wu & Fuller 2022a, see also
discussion in Strotjohann et al. 2021). Model-agnostic radiation
hydrodynamical simulations of partial envelope ejection triggered by energy
injection from the core, aimed to phenomenologically reproduce the CSM of
interacting SNe, find a similar upper limit in the precursor luminosity (e.g.,
Quataert et al., 2016; Kuriyama & Shigeyama, 2020, 2021; Tsang et al., 2022;
Tsuna et al., 2023b).

Figure 1: Schematic picture of our model (not to scale). The SN progenitor in
a close binary system with a CO undergoes a mass outburst that creates the
dense CSM. This generally results in super-critical accretion of a fraction of
the CSM onto the CO, which leads to outflows that boost the energy of the CSM
and power the precursor emission.

In this work, we suggest that the observed bright precursors can be reproduced
if the progenitor is in a binary system with a compact object (CO) companion,
i.e., a stellar-mass black hole (BH) or a neutron star (NS). Most of young
massive stars ($>70\%$) live as interacting binaries (Sana et al., 2012), and
a significant fraction of them can result in such star-CO systems. Binary
interaction is also considered to be the main formation channel for stripped-
envelope SNe of Type Ibc (e.g., Shigeyama et al., 1990; Podsiadlowski et al.,
1992; Eldridge et al., 2008; Smith et al., 2011), where the companions can be
COs if they were the more massive of the initial binary and underwent core-
collapse first.

The schematic picture of our model is shown in Figure 1. The erupted material
that encounters the CO’s Bondi sphere would be accreted, forming a disk around
it. For binaries close enough the accretion can become super-Eddington and
launch an energetic outflow, which can lead to additional energy injection
into the remaining CSM (e.g., Dexter & Kasen, 2013; Kimura et al., 2017;
Moriya et al., 2018a). This can boost the energy budget of the precursor
(e.g., Mcley & Soker, 2014; Danieli & Soker, 2019), drastically reducing the
required energy deposition for the partial eruption to values in line with
proposed theoretical models.

In Section 2 we first elaborate on the difficulty of reproducing the observed
bright precursors by eruptions of single massive stars. Then in Section 3 we
describe our detailed modeling of precursors from massive star-compact object
(CO) binary systems, and in Section 4 demonstrate that such systems can
generally explain the energetics of the observed precursor emission. In
Section 5 we discuss the formation channel of these binaries, as well as ways
to test the model with independent observations. We conclude in Section 6.

##  2 Precursors from Single Stars

In this section, we estimate the radiative output of a single mass eruption
event from an isolated (non-binary) star, and show its difficulty to reproduce
the observed precursor events.

The radiated energy in the observed precursors, of $10^{47}$–$10^{49}$ erg and
typically of the order of $10^{48}$ erg (see e.g., Table 4 of Strotjohann et
al. 2021 and Table 1 of Matsumoto & Metzger 2022a), requires that at least a
comparable energy is injected to the envelope. This disfavors an eruption from
a red supergiant (RSG) progenitor, which would result in (near-complete)
ejection of the hydrogen-rich envelope with its small binding energy of

$\displaystyle E_{\rm bind}$ $\displaystyle\sim$
$\displaystyle\frac{GM_{*}M_{\rm env}}{R_{*}}$ (1) $\displaystyle\sim$
$\displaystyle 4\times 10^{47}\ {\rm erg}\left(\frac{M_{*}}{10\\!\
M_{\odot}}\right)\\!\left(\frac{M_{\rm env}}{5\\!\
M_{\odot}}\right)\\!\left(\frac{R_{*}}{500\\!\ R_{\odot}}\right)^{\\!-1}.$

Here $G$ is the gravitational constant, and $M_{*}$, $M_{\rm env}$ and $R_{*}$
are respectively the total mass, envelope mass, and radius of the star. Such a
near-complete envelope ejection is unlikely for the observed precursors,
because (i) the mass of the CSM constrained from the rise time of the
subsequent SN is typically $\lesssim$ a few $M_{\odot}$ and (ii) Type IIn SNe
with precursors are not on the luminous end of the entire IIn population
(Strotjohann et al., 2021). For the former, the rise times for these SN
samples are estimated to be $t_{\rm rise}<25$ days (Strotjohann et al., 2021,
Table 4), and the timescale of radiative diffusion from the swept-up CSM being
shorter than this limits the mass of the CSM as (e.g., Moriya, 2015)

$\displaystyle M_{\rm CSM}$ $\displaystyle\lesssim$ $\displaystyle\frac{4\pi
ct_{\rm rise}(v_{\rm sh}t_{\rm rise})}{3\kappa}$ (2) $\displaystyle\approx$
$\displaystyle 3\ M_{\odot}\left(\frac{t_{\rm rise}}{25\ {\rm
day}}\right)^{2}\left(\frac{\kappa}{0.3\ {\rm cm^{2}\ g^{-1}}}\right)^{-1}\ ,$

where we assume a rather large SN shock velocity of $v_{\rm sh}=10^{4}$ km
s-1, $c$ is the speed of light, and $\kappa$ is the opacity assumed to be
constant in time and radius. For the latter, the peak magnitude of SNe with
precursors span from $-19$ to $-17$ mag (roughly correspoding to $3\times
10^{42}-10^{43}\,\rm erg\,s^{-1}$, Strotjohann et al., 2021; Matsumoto &
Metzger, 2022a), which agrees with the range seen in typical Type IIn SNe
(Kiewe et al., 2012; Ofek et al., 2014b; Nyholm et al., 2020). This indicates
that the mass of CSM for SNe with bright precursors should be comparable to
those without, and should not be biased towards higher masses.

For Type Ibn/Icn SNe from helium (He)/Wolf-Rayet (WR) stars, light curve
modelling of these SNe in general suggests a CSM mass of $\sim 0.01$–$1\
M_{\odot}$ (Maeda & Moriya, 2022; Dessart et al., 2022; Wu & Fuller, 2022b;
Takei et al., 2023), which may also favor a partial envelope ejection.

This energetics problem, on the difficulty to produce the observed precursor’s
energy without significantly modifying the progenitor’s structure, can be
avoided if the progenitor is more compact than RSGs, such as blue supergiants
(luminous blue variables) or He/WR stars. However, even in these cases, a
single episode of partial mass ejection struggles to reproduce the bright
luminosity of the precursors because of losses to adiabatic expansion, as we
show below.

Suppose that the precursor event ejects a CSM of mass $M_{\rm CSM}$ with a
velocity $v_{\rm CSM}=\eta v_{\rm esc}$ (here $\eta$ is a parameter) and

$\displaystyle v_{\rm esc}=\sqrt{\frac{2GM_{*}}{R_{*}}}\approx 90\ {\rm km\
s^{-1}}\left(\frac{M_{*}}{10\ M_{\odot}}\right)^{1/2}\left(\frac{R_{*}}{500\
R_{\odot}}\right)^{-1/2}$ (3)

is the surface escape velocity. Under the assumption that the energy is
sourced from near or inside the base of the envelope111There are scenarios for
injecting energy close to the surface, such as wave heating in the case of
stripped stars (e.g., Fuller & Ro, 2018; Leung et al., 2021). However, in this
scenario energy injection is continuous (i.e. longer than the dynamical time
near surface), and such steady-state energy injection driving a wind-like CSM
have difficulties to power emission highly exceeding the Eddington luminosity
(Quataert et al., 2016; Matsumoto & Metzger, 2022a) unless strong velocity
gradients develop internal shocks in the wind acceleration region. , for a
partial ejection of the envelope we expect that the bulk CSM velocity is
comparable to the surface escape velocity, i.e. $\eta\sim 1$ (Linial et al.,
2021).

Similar to SNe (Arnett, 1980), as the CSM expands and its optical depth drops,
photons in the CSM escape to power the precursor. Its timescale is governed by
the diffusion through the expanding CSM, which is

$\displaystyle t_{\rm prec,0}$ $\displaystyle\approx$
$\displaystyle\sqrt{\frac{3\kappa M_{\rm CSM}}{4\pi cv_{\rm CSM}}}$ (4)
$\displaystyle\approx$ $\displaystyle 80\ {\rm day}\left(\frac{\kappa}{0.3\
{\rm cm^{2}\ g^{-1}}}\right)^{1/2}$ $\displaystyle\times\left(\frac{M_{\rm
CSM}}{0.1\ M_{\odot}}\right)^{1/2}\left(\frac{v_{\rm CSM}}{100\ {\rm km\
s^{-1}}}\right)^{-1/2}.$

The internal and kinetic energy $E_{\rm kin}=M_{\rm CSM}v_{\rm CSM}^{2}/2$ are
comparable upon the eruption, but until $t=t_{\rm prec,0}$ the internal energy
is converted to kinetic energy by adiabatic expansion and reduced by a
factor222This factor is under the assumption of radiation-dominated gas. In
case gas pressure is important the energy budget of radiation decreases, only
strengthening the conclusion of this section. $\epsilon_{\rm
rad}\approx(R_{*}/v_{\rm CSM}t_{\rm prec,0})\leq 1$. The luminosity of the
precusor is then

$\displaystyle L_{\rm prec,0}\approx\frac{\epsilon_{\rm rad}E_{\rm
kin}}{t_{\rm prec,0}}\approx\eta^{2}\frac{4\pi
GM_{*}c}{3\kappa}=\frac{\eta^{2}}{3}L_{\rm Edd}$ (5)

where

$L_{\rm Edd}=1.7\times 10^{39}\ {\rm erg\ s^{-1}}\left(\frac{M_{*}}{10\
M_{\odot}}\right)\left(\frac{\kappa}{0.3\ {\rm cm^{2}\ g^{-1}}}\right)^{-1}$
(6)

is the Eddington luminosity.

For a typical $10\ M_{\odot}$ star, the fact that $\eta$ cannot be much
greater than unity limits the luminosity of the eruption to the order of
$10^{39}\ {\rm erg\ s^{-1}}$, which is much dimmer than the observed
precursors with luminosities of $10^{40}$–$10^{42}$ erg s-1. This scaling in
the luminosity is also roughly seen in more detailed light curve modelling.
For instance, radiation hydrodynamical simulations of partial envelope
ejections find a luminosity of $10^{38}$–$10^{39}$ erg s-1 for the resulting
transient (Fuller, 2017; Kuriyama & Shigeyama, 2020; Tsuna et al., 2023b). A
semi-analytical model by Matsumoto & Metzger (2022a) finds that for a RSG of
$M_{*}=10M_{\odot},\,R_{*}=10^{3}R_{\odot}$, a precursor luminosity of $L_{\rm
prec}\approx 10^{41}$ erg s-1 requires an eruption with CSM velocity of
$\approx 400$–$700$ km s-1 for $M_{\rm CSM}=0.1$–$10\ M_{\odot}$ (their Figure
4), which corresponds to $\eta\sim 10$.

We can consider two ways to break the limit on the luminosity (equation 5),
which is similar to the models proposed for super-luminous SNe (e.g., Moriya
et al., 2018b; Gal-Yam, 2019; Nicholl, 2021). The first possibility is to
invoke a pre-existing CSM surrounding the progenitor at the time of the
eruption. The collision of the two CSM shells can (re-)convert the kinetic
energy into internal energy, effectively raising $\epsilon_{\rm rad}$. This
however still requires an energy budget for $E_{\rm kin}$ of at least the
radiated energy in the precursors, of $10^{47}$–$10^{49}$ erg (Strotjohann et
al., 2021; Jacobson-Galán et al., 2022), which again is disfavored for RSG
eruptions. Even for more compact progenitors, the mechanism that can realize
this large energy deposition is unclear. Recent models of wave heating find
that energy injections of $\sim\\!10^{48}$ ergs are realized in a limited set
of progenitors only during the end stages of nuclear burning ($\lesssim 0.1$
yr before core-collapse; Wu & Fuller 2021; Leung et al. 2021), while the
precursors are typically seen several months to years before the SN.

The second possibility we explore in the following is a binary with a CO,
which acts as a “central engine” to power the light curve. For binaries with
sufficiently close separations, the gravitational trapping of a fraction of
the CSM by the CO can lead to super-Eddington accretion. As we show below,
this accretion can generate outflows that carry energy greatly exceeding the
original outburst energy, enhancing the energy budget in equation 5. Thus this
scenario is an energetically favorable solution compared to the scenario
invoking CSM interaction.

##  3 Precursors from Binaries with Compact Object Companions

We consider a circular binary system with separation $a_{\rm bin}$, composed
of a CO of mass $M_{\bullet}$ and a massive star of mass $M_{*}$ that
undergoes a partial envelope ejection. We restrict the separation to larger
than that for Roche-lobe overflow to occur, i.e. $a_{\rm bin}\geq 3R_{*}$
(Eggleton, 1983) for mass ratios of $M_{*}/M_{\bullet}\gtrsim 0.3$ of interest
in this work, which results in non-interacting binaries until the pre-SN
outburst.

We estimate the properties of the energy injection from the CO, and calculate
the time-dependent emission from the CSM. The formulations of this model are
similar to Kimura et al. (2017), but we newly construct a one-zone light curve
model including time-dependent effects of the accretion and ionization of the
CSM. The model parameters we employ are summarized in Table 1. The details of
each parameter will be described in the following sections.

Parameters | RSG-BH(NS) | HeHighMass-BH(NS) | HeLowMass-BH(NS)
---|---|---|---
CO mass ($M_{\bullet}$) |  |  $10M_{\odot}$ ($1.4M_{\odot}$) |
Star mass ($M_{*}$) | $10M_{\odot}$ | $5M_{\odot}$ | $3M_{\odot}$
Stellar radii ($R_{*}$) | $500R_{\odot}$ | $5R_{\odot}$ | $50R_{\odot}$
Opacity ($\kappa$) |  $0.3$ cm2 g-1 |  $0.1$ cm2 g-1 |  $0.1$ cm2 g-1
Ionization temperature ($T_{I}$) |  $6000$ K |  $10000$ K |  $10000$ K
Initial CSM velocity in units of $v_{\rm esc}$ ($\xi$) |  | 2 |
CSM mass ($M_{\rm CSM}$) | 0.1–3$M_{\odot}$ | 0.01–1$M_{\odot}$ | 0.01–1$M_{\odot}$
Binary separation ($a_{\rm bin}$) |  | (3–30)$R_{*}$ |
Table 1: Parameters adopted in our precursor model. Values shown in only one
column are common to all three systems.

###  3.1 Accretion Disk Wind from the CO Companion

We assume that the CSM is expanding homologously and has a uniform density
profile for simplicity. Hereafter, we define the velocity as that in the rest
frame of the star. The CSM has a velocity ranging from $v_{\rm min}\equiv\chi
v_{\rm esc}$ to $v_{\rm max}\equiv\xi v_{\rm esc}$, where the minimal velocity
is set by the condition that the CSM reaches the CO without falling back to
the progenitor:

$\displaystyle\chi=\sqrt{1-R_{*}/a_{\rm bin}}\ .$ (7)

This definition of $\chi$ is approximate as we ignore the gravity from the CO,
but for a uniform density profile varying $\chi$ would lead to only slight
changes in the density for a fixed CSM mass. We treat $\xi\gtrsim 1$ as a free
parameter not much larger than unity, based on the arguments in Section 2.

Given the velocity range, the (uniform) CSM density at time $t$ from the
eruption is given by

$\displaystyle\rho_{\rm CSM}$ $\displaystyle=$ $\displaystyle\frac{3M_{\rm
CSM}}{4\pi(v_{\rm esc}t)^{3}(\xi^{3}-\chi^{3})}$ (9) $\displaystyle\sim$
$\displaystyle\frac{7\times 10^{-11}}{\xi^{3}-\chi^{3}}\ {\rm g\
cm^{-3}\,}\left(\frac{M_{\rm CSM}}{0.1\ M_{\odot}}\right)$
$\displaystyle\times\left(\frac{v_{\rm esc}}{100\ {\rm km\
s^{-1}}}\right)^{-3}\left(\frac{t}{100\ {\rm day}}\right)^{-3}.$

The CSM starts to be accreted when it reaches the CO at at time

$\displaystyle t_{\bullet}$ $\displaystyle\equiv$ $\displaystyle\frac{a_{\rm
bin}}{v_{\rm max}}$ (10) $\displaystyle\sim$
$\displaystyle\frac{200}{\xi}{\,\rm day\,}\left(\frac{a_{\rm
bin}}{5R_{*}}\right)\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)\left(\frac{v_{\rm
esc}}{100\,\rm km\,s^{-1}}\right)^{-1}\ .$

The gravitational pull of the CO traps the CSM (which is moving approximately
homologously at a velocity of $a_{\rm bin}/t$) within the CO’s Bondi radius

$\displaystyle r_{\rm B}=\frac{GM_{\bullet}}{V^{2}}\approx 1\times 10^{13}\
{\rm cm}\left(\frac{M_{\bullet}}{10\ M_{\odot}}\right)\left(\frac{V}{100\,\rm
km\,s^{-1}}\right)^{-2}$ (11)

where we define

$\displaystyle V$ $\displaystyle\equiv$ $\displaystyle\sqrt{(a_{\rm
bin}/t)^{2}+v_{\rm orb}^{2}}\ ,$ (12) $\displaystyle v_{\rm orb}$
$\displaystyle=$ $\displaystyle\sqrt{{G(M_{*}+M_{\bullet})}/{a_{\rm bin}}}$
(13) $\displaystyle\approx$ $\displaystyle 40\ {\rm km\
s^{-1}}\left(\frac{M_{*}+M_{\bullet}}{20\ M_{\odot}}\right)^{1/2}$
$\displaystyle\times\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{-1/2}\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{-1/2}\
.$

Equation (13) is the orbital velocity of the CO with respect to the star. Note
that when the CSM reaches the CO at $a_{\rm bin}$, the internal energy in the
CSM is much smaller than the kinetic energy, and hence the contribution from
the sound speed in the CSM to $V$ can be neglected.

As the homologously expanding CSM contains velocity asymmetry of
$\sim\\!r_{\rm B}/t$ due to a velocity difference over the Bondi radius, it
carries a specific angular momentum of $j\sim r_{\rm B}(r_{\rm B}/t)$. This
results in a formation of an accretion disk around the CO with outer radius
(Shapiro & Lightman, 1976)

$\displaystyle r_{\rm disk}$ $\displaystyle\approx$
$\displaystyle\frac{j^{2}}{GM_{\bullet}}$ (14) $\displaystyle\sim$
$\displaystyle 8\times 10^{10}\xi^{2}{\,\rm
cm\,}\left(\frac{M_{\bullet}}{10\,\mathrm{M}_{\odot}}\right)^{3}\left(\frac{V}{100{\,\rm
km\,s^{-1}}}\right)^{-8}$ $\displaystyle\times\left(\frac{v_{\rm
esc}}{100\,\rm km\,s^{-1}}\right)^{2}\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{-2}\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{-2}\left(\frac{t}{t_{\bullet}}\right)^{-2}\
.$

Hence an accretion disk can form outside the CO if $r_{\rm disk}>r_{\rm in}$,
where we adopt $r_{\rm in}=6GM_{\bullet}/c^{2}\sim 9\times 10^{6}\ {\rm cm}\
(M_{\bullet}/10M_{\odot})$. The accretion rate for $t_{\bullet}<t<a_{\rm
bin}/v_{\rm min}$ is given by the Bondi-Hoyle-Lyttleton rate as (Hoyle &
Lyttleton, 1939; Bondi & Hoyle, 1944; Bondi, 1952)

$\displaystyle\dot{M}_{\rm acc}$ $\displaystyle\approx$ $\displaystyle 4\pi
r_{\rm B}^{2}\rho_{\rm CSM}V$ $\displaystyle\sim$
$\displaystyle\frac{10^{4}\xi^{3}\dot{M}_{\rm
Edd}}{\xi^{3}-\chi^{3}}\left(\frac{M_{\bullet}}{10\
\mathrm{M}_{\odot}}\right)\left(\frac{M_{\rm
CSM}}{0.1\,\mathrm{M}_{\odot}}\right)$ $\displaystyle\times\left(\frac{V}{100\
{\rm km\ s^{-1}}}\right)^{-3}\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{-3}\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{-3}\left(\frac{t}{t_{\bullet}}\right)^{-3}\
,$

where we assumed $r_{\rm B}\ll a_{\rm bin}$. Here the Eddington accretion rate
is defined by $\dot{M}_{\rm Edd}\equiv\epsilon_{\rm sd}^{-1}L_{\rm
Edd}/c^{2}\sim 2\times 10^{19}\ {\rm g\ s^{-1}}(\epsilon_{\rm
sd}/0.1)^{-1}(M_{\bullet}/10M_{\odot})(\kappa/0.3\ {\rm cm^{2}\
g^{-1}})^{-1}$, where $\epsilon_{\rm sd}\sim 0.1$ is the radiative efficiency
of a standard disk.

Such accretion should lead to an advection-dominated accretion flow (ADAF,
e.g., Ichimaru, 1977; Narayan & Yi, 1994), where we expect mass outflows of
high velocities (e.g., Ohsuga et al., 2005; Jiang et al., 2014; Sadowski et
al., 2014; Kitaki et al., 2021). Here we use the ADAF model by Blandford &
Begelman (1999) to estimate the luminosity of the disk wind333We call this
energy injection as “wind” in the rest of the paper, but we note that other
forms of energy injection, such as bright X-ray emission or jets, may be
realized (e.g., Narayan et al., 2017; Soker, 2022). that will energize the
CSM. In this model most of the accreted material is blown away by the wind,
and only a small fraction reaches the compact object. The accretion rate is
assumed to be a power-law in radius (from the CO) as $\dot{M}(r)\propto r^{p}$
for a range $r_{\rm in}<r<r_{\rm disk}$, with $r_{\rm disk}$ defined in
equation (14). We adopt $p=0.5$, which is consistent with the range inferred
from simulations and observations of radiatively inefficient accretion flows
($0.3<p<0.8$; Yuan & Narayan 2014). Assuming that the wind velocity at each
radius is the local escape velocity, the mass-averaged velocity of the wind is
obtained from integration over radius as (e.g., Metzger, 2012; Tsuna &
Kawanaka, 2019; Fuller & Lu, 2022)

$\displaystyle\langle v_{\rm wind}\rangle$ $\displaystyle\approx$
$\displaystyle\sqrt{\frac{p}{1-p}\frac{GM_{\bullet}}{r_{\rm
in}}\left[\left(\frac{r_{\rm in}}{r_{\rm disk}}\right)^{p}-\left(\frac{r_{\rm
in}}{r_{\rm disk}}\right)\right]}$ $\displaystyle\overset{r_{\rm in}\ll r_{\rm
disk}}{\sim}$ $\displaystyle\sqrt{\frac{p}{1-p}\frac{GM_{\bullet}}{r_{\rm
in}}\left(\frac{r_{\rm in}}{r_{\rm disk}}\right)^{p}}$
$\displaystyle\overset{p=0.5}{\sim}$ $\displaystyle
0.04\xi^{-0.5}\,c\,\left(\frac{M_{\bullet}}{10\,~{}\mathrm{M}_{\odot}}\right)^{-1/2}\left(\frac{V}{100{\,\rm
km\,s^{-1}}}\right)^{2}$ $\displaystyle\times\left(\frac{v_{\rm
esc}}{100{\,\rm km\,s^{-1}}}\right)^{-1/2}\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{1/2}$
$\displaystyle\times\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{1/2}\left(\frac{t}{t_{\bullet}}\right)^{1/2}\
.$

The approximation $r_{\rm in}\ll r_{\rm disk}$ is used only in this section to
demonstrate the approximate scaling. The wind kinetic energy luminosity is
given as

$\displaystyle L_{\rm wind}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2}\dot{M}_{\rm acc}\langle v_{\rm wind}\rangle^{2}$
(17) $\displaystyle\overset{\begin{subarray}{c}r_{\rm in}\ll r_{\rm disk},\\\
p=0.5\end{subarray}}{\sim}$ $\displaystyle\frac{2\times
10^{41}\xi^{2}}{\xi^{3}-\chi^{3}}{\,\rm erg\
s^{-1}\,}\left(\frac{M_{\bullet}}{10\
\mathrm{M}_{\odot}}\right)\left(\frac{M_{\rm
CSM}}{0.1\,\mathrm{M}_{\odot}}\right)$ $\displaystyle\times\left(\frac{V}{100\
{\rm km\ s^{-1}}}\right)\left(\frac{v_{\rm esc}}{100{\,\rm
km\,s^{-1}}}\right)^{-1}$ $\displaystyle\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{-2}\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{-2}\left(\frac{t}{t_{\bullet}}\right)^{-2}\
.$

Since the wind luminosity steeply declines with time as $Vt^{-2}\propto
t^{-3}$ at early times, the total available kinetic energy of the wind to
power the precursor is

$\displaystyle E_{\rm wind}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2}t_{\bullet}L_{{\rm wind},t=t_{\bullet}}$
$\displaystyle\overset{p=0.5}{\sim}$
$\displaystyle\frac{10^{48}\xi}{\xi^{3}-\chi^{3}}{\,\rm
erg\,}\left(\frac{M_{\bullet}}{10\
\mathrm{M}_{\odot}}\right)\left(\frac{M_{\rm
CSM}}{0.1\,\mathrm{M}_{\odot}}\right)\left(\frac{V}{100\ {\rm km\
s^{-1}}}\right)$ $\displaystyle\times\left(\frac{v_{\rm esc}}{100{\,\rm
km\,s^{-1}}}\right)^{-2}\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{-1}\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{-1}\
.$

A luminous precursor can be powered if this energy is injected to the CSM at a
timescale comparable to the diffusion timescale, analogous to the central
engine models proposed for super-luminous SNe (e.g., Kasen & Bildsten, 2010;
Woosley, 2010).

###  3.2 Transients from Wind-CSM Interaction

The disk wind is injected to the rest of the CSM, and we assume that the
kinetic energy of the injected wind quickly thermalizes in the CSM (for the
validity of this assumption see Kimura et al., 2017). We calculate the
dynamics of the CSM and the resulting light curves using the following one-
zone model.

####  3.2.1 Governing Equations

We assume that the opacity in the CSM is uniform and constant, which is
justified for the characteristic density of the CSM (e.g., see Figure 3 in
Matsumoto & Metzger, 2022b) unless significant amounts of dust are formed
within the CSM. We can then solve the evolution by the following set of
equations (e.g., Section 2.2 of Metzger et al., 2015)

$\displaystyle\frac{dE_{\rm int}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{E_{\rm int}}{R_{\rm CSM}}v_{\rm CSM}+L_{\rm inj}-L_{\rm
rad}$ (19) $\displaystyle\frac{dR_{\rm CSM}}{dt}$ $\displaystyle=$
$\displaystyle v_{\rm CSM}$ (20) $\displaystyle\frac{dv_{\rm CSM}}{dt}$
$\displaystyle=$ $\displaystyle\frac{5}{3}\frac{E_{\rm int}}{M_{\rm CSM}R_{\rm
CSM}}$ (21) $\displaystyle L_{\rm rad}$ $\displaystyle=$
$\displaystyle\frac{E_{\rm int}}{t_{\rm diff}}$ (22)

where $E_{\rm int}$ is the internal energy, $L_{\rm rad}$ is the radiative
luminosity, $t_{\rm diff}\approx 3\kappa M_{\rm CSM}/(4\pi cR_{\rm CSM})$ is
the diffusion timescale in the CSM at time $t$, and $L_{\rm inj}$ is the rate
of energy injection due to the disk wind (with $\langle v_{\rm wind}\rangle$
in equation 3.1)

$\displaystyle L_{\rm inj}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{@{}l@{\quad}l}L_{\rm
wind}&\left(t_{\bullet}<t<a_{\rm bin}/v_{\rm min}\ {\rm and}\ r_{\rm
disk}>r_{\rm in}\right)\\\ 0&({\rm otherwise})\end{array}\right.$ (25)

Equation (19) is the first law of thermodynamics with PdV work in the first
term, and equation (21) accounts for the increase of the kinetic energy
$E_{\rm kin}=3M_{\rm CSM}v_{\rm CSM}^{2}/10$ due to PdV work444The prefactor
$3/10$ is for a homologously expanding, uniform ejecta (Arnett, 1982), but we
note that the corresponding equation in the original paper contains a typo
(3/5 should be 5/3; see also Wheeler et al. 2015).. As the wind carries
negligible mass compared to the rest of the CSM, we can approximate $M_{\rm
CSM}$ as a constant.

####  3.2.2 Accounting for recombination

Initially the CSM is ionized and the adoption of uniform electron scattering
opacity for $\kappa$ is justified. However, as the CSM expands and its
photospheric temperature

$\displaystyle T_{\rm ph}=(L_{\rm rad}/4\pi\sigma R_{\rm CSM}^{2})^{1/4}\ ,$
(26)

where $\sigma$ is the Stefan-Boltzmann constant, becomes smaller than the
ionization temperature $T_{\rm I}$, recombination of the CSM sets in.
Afterwards, a sharp ionization front develops and recedes within the CSM, with
its radius given as $R_{\rm I}=(L_{\rm rad}/4\pi\sigma T_{\rm I}^{4})^{1/2}$.
In order to calculate the light curve in this regime, the location of the
ionization front should be tracked (Popov 1993; Dexter & Kasen 2013; Matsumoto
et al. 2016, Matsumoto et al. in prep).

The radius of the ionization front can be parameterized by the dimensionless
variable $x_{i}\equiv R_{\rm I}/R_{\rm CSM}(\leq 1)$, which is 1 when the CSM
is fully ionized. Since the outside of $R_{\rm I}$ is transparent, Equation
(19) is then modified to consider the change of internal energy in the ionized
region of volume $V_{i}=4\pi(x_{i}R_{\rm CSM})^{3}/3$, as

$\displaystyle\frac{1}{\epsilon_{\rm int}}\frac{\partial\epsilon_{\rm
int}}{\partial t}+\frac{4}{x_{i}R_{\rm CSM}}\frac{\partial(x_{i}R_{\rm
CSM})}{\partial t}=\frac{L_{\rm inj}-L_{\rm rad}}{\epsilon_{\rm int}V_{i}},$
(27)

where $\epsilon_{\rm int}=E_{\rm int}/V_{i}$ is the internal energy density.
Assuming that the ionized CSM is still optically thick, both the Thomson
optical depth and light-crossing time in the ionization region are $\propto
x_{i}$, so the diffusion time where photons escape is modified to
$x_{i}^{2}t_{\rm diff}$. The radiative luminosity is thus modified by a factor
$x_{i}^{-2}$ as

$\displaystyle L_{\rm rad}=\frac{\epsilon_{\rm int}V_{i}}{x_{i}^{2}t_{\rm
diff}}=4\pi(x_{i}R_{\rm CSM})^{2}\left(\frac{\epsilon_{\rm int}R_{\rm
CSM}}{3x_{i}t_{\rm diff}}\right),$ (28)

which is equated with the luminosity from the ionization front $L_{\rm
rad}=4\pi(x_{i}R_{\rm CSM})^{2}\sigma T_{I}^{4}$ to obtain

$\displaystyle\epsilon_{\rm int}=\frac{3x_{i}t_{\rm diff}}{R_{\rm CSM}}\sigma
T_{I}^{4}.$ (29)

Substituting this to equation (27), we obtain a differential equation for
$x_{i}$ as

$\displaystyle\frac{dx_{i}}{dt}=-\frac{2x_{i}}{5}\frac{v_{\rm CSM}}{R_{\rm
CSM}}-\frac{1}{5x_{i}t_{\rm diff}}\left[1-\frac{L_{\rm inj}}{4\pi(x_{i}R_{\rm
CSM})^{2}\sigma T_{I}^{4}}\right].$ (30)

Note that if the acceleration is negligible and the CSM expands freely with
the relation $R_{\rm CSM}=v_{\rm CSM}t$, we recover the evolution of $x_{i}$
in Dexter & Kasen (2013) (their equation A12; note that their $t_{\rm d}$
differs from our $t_{\rm diff}$ as $t_{\rm diff}=t_{\rm d}^{2}/t$). However
the energy injection from the accreting CO can accelerate the CSM via PdV
work. The PdV work in equation (21) now only comes from the ionized region,
but under the one-zone approximation we assume that this pushes the entire CSM
with an acceleration given as

$\displaystyle\frac{dv_{\rm CSM}}{dt}$ $\displaystyle=$
$\displaystyle\frac{5}{3}\frac{1}{M_{\rm CSM}v_{\rm
CSM}}\times\frac{\epsilon_{\rm int}}{3}\frac{dV_{i}}{dt}$ (31)
$\displaystyle=$ $\displaystyle\frac{4\pi(x_{i}R_{\rm CSM})^{2}\epsilon_{\rm
int}}{3M_{\rm CSM}v_{\rm CSM}}\frac{d(x_{i}R_{\rm CSM})}{dt}.$

By solving equations (20) and (28) - (31), we can obtain $R_{\rm CSM},L_{\rm
rad},\epsilon_{\rm int}$, $x_{i}$ and $v_{\rm CSM}$ as a function of $t$ after
the recombination starts.

####  3.2.3 Methods of our light curve model with recombination

Generally, the energy injection raises the precursor luminosity and sustains
the ionization of the CSM. One important difference between our situation and
existing modeling (e.g., Dexter & Kasen 2013 and Matsumoto et al. in prep.) is
that the energy injection is delayed, happening only after the CSM reaches the
CO. For the CSM from compact progenitors in tight binaries, this is not a
significant effect because the CSM reaches the CO early and energy injection
is done well before recombination sets in. In this case the formulation is the
same as the modeling of normal type II-P SNe (Popov, 1993; Kasen & Woosley,
2009).

However, for RSG progenitors the CSM generally recombines before the energy
injection. The former timescale is estimated by solving equation (30) for
$L_{\rm inj}=0$ (e.g., Matsumoto et al., 2016):

$\displaystyle t_{\rm rec}$ $\displaystyle\approx$ $\displaystyle
100\xi^{-5/14}{\,\rm day\,}\left(\frac{M_{\rm
CSM}}{0.1\,\mathrm{M}_{\odot}}\right)^{5/14}$ (32)
$\displaystyle\times\left(\frac{v_{\rm esc}}{100{\,\rm
km\,s^{-1}}}\right)^{-5/14}\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{1/7}\
,$

which could be shorter than the latter one $t_{\bullet}\approx
200\xi^{-1}\,\rm days$ (see equation 10). In this case, the recombined CSM is
re-ionized by the delayed energy injection and then recombines again at late
times. In our one-zone prescription, the sudden energy injection at
$t=t_{\bullet}$ to an almost neutral ($x_{i}\ll 1$) CSM results in a prompt
ionization, due to the last term $\propto L_{\rm inj}/x_{i}^{3}$ in equation
(30). While this instantaneous ionization essentially comes from our one-zone
assumption, we believe that sudden re-ionization probably does occur in
nature. This is because the ionization would result from shock heating
following the wind-CSM collision, and this occurs over the wind-crossing
timescale of the CSM $\sim R_{\rm CSM}/\langle v_{\rm wind}\rangle$ which is
much faster than the dynamical timescale $\sim R_{\rm CSM}/v_{\rm CSM}$.

For energy injection into a neutral CSM ($x_{i}\ll 1$), the ionization will
proceed as in equation (30), and saturates when either $x_{i}$ reaches 1 (full
ionization) or $dx_{i}/dt=0$. In the latter case, the corresponding degree of
ionization is obtained by solving for $x_{i}$ in equation (30):

$\displaystyle x^{2}_{\rm sat}$ $\displaystyle=$ $\displaystyle-\frac{R_{\rm
CSM}}{4v_{\rm CSM}t_{\rm diff}}$ (33) $\displaystyle+\sqrt{\left(\frac{R_{\rm
CSM}}{4v_{\rm CSM}t_{\rm diff}}\right)^{2}+\frac{R_{\rm CSM}}{2v_{\rm
CSM}t_{\rm diff}}\frac{L_{\rm inj}}{4\pi R_{\rm CSM}^{2}\sigma T_{I}^{4}}}\ .$

Then, the location of the ionization front at saturation is estimated as
$x_{i,\bullet}={\rm min}(1,x_{\rm sat})$. For a reliable calculation of the
light curve, we need to smoothly evolve the ionization from the $x_{i}$ just
before energy injection to $x_{i,\bullet}$, and also accurately evolve the
equations that control the energy budget ($E_{\rm int},E_{\rm kin}$) of the
precursor.

For every time $t$, we evolve the first set of equations (19) – (22) when the
CSM is fully ionized ($x_{i}=1$), and when $x_{i}<1$ ($R_{\rm I}<R_{\rm CSM}$)
we evolve the second set of equations (20), (28) – (31) that also follow the
evolution of $x_{i}$. We control the timestep of the integration by the
equations shown in Appendix A. If the CSM becomes fully ionized during the
energy injection, we switch back to solving the equations for $x_{i}=1$, with
$E_{\rm int}=\epsilon_{\rm int}(4\pi R_{\rm CSM}^{3}/3)$ at that time.

For the initial conditions at $t=0$, we set $R_{\rm CSM,0}=R_{*}$, and assume
the kinetic and internal energy are equally shared with total energy $3M_{\rm
CSM}(\xi v_{\rm esc})^{2}/10$, i.e. $E_{\rm kin}=E_{\rm int}=3M_{\rm CSM}(\xi
v_{\rm esc})^{2}/20$. We set a floor value for $x_{i}$ of $10^{-5}$, as it
increasingly becomes small as the CSM recombines. We verified that enhancing
the floor value to $10^{-4}$ and $10^{-3}$ results in almost identical light
curves.

There are eight parameters characterizing our binary precursor model,
$M_{\bullet}$, $M_{*}$, $R_{*}$, $\kappa$, $T_{I}$, $\xi$, $M_{\rm CSM}$, and
$a_{\rm bin}$. For simplicity, in this work we fix $\xi=2$ as in Kimura et al.
(2017), and consider two values $M_{\bullet}=\\{10,1.4\\}\ M_{\odot}$
representing a BH and NS companion respectively. We discuss the effect of
changing $\xi$ in Appendix B, where we find it to not greatly affect the light
curves. For the SN progenitor, we consider a RSG and two He star progenitors,
with values ($M_{*},R_{*},\kappa,T_{I}$) in Table 1. We consider a high-mass
He star with a compact radius that is expected to be a typical progenitor of
stripped-envelope SNe, and a low-mass He star with mass around $3M_{\odot}$
that expands to a radius of around $\sim\\!100R_{\odot}$ before core-collapse
(e.g., Woosley, 2019; Ertl et al., 2020; Laplace et al., 2020; Wu & Fuller,
2022b). The values of $\kappa$ and $T_{I}$ for the two types of progenitors
are motivated from singly-ionized hydrogen and helium respectively (e.g.,
Kleiser & Kasen, 2014). We consider a range of values for the remaining free
parameters $M_{\rm CSM}$ and $a_{\rm bin}$, which should greatly vary for each
SN progenitor. For $M_{\rm CSM}$ we adopt $0.1M_{\odot}\leq M_{\rm CSM}\leq
3M_{\odot}$ for RSGs and $0.01M_{\odot}\leq M_{\rm CSM}\leq 1M_{\odot}$ for He
stars (see Section 2).

##  4 Results

###  4.1 Example Light Curves

Figure 2: Light curves and CSM velocities for a few parameter sets of ($a_{\rm
bin}$, $M_{\rm CSM}$), for the case of a $10M_{\odot}$ BH companion (other
parameters are in Table 1). Dotted lines show the wind (injection) luminosity
$L_{\rm wind}$ for each parameter, and dashed lines show the injection-free
cases, i.e., a single star progenitor. The points show the light curves of
eight well-observed SN precursor samples: SN 2019zrk (Fransson et al., 2022),
2006jc (Pastorello et al., 2007), 2019uo (Strotjohann et al., 2021), and
others from Matsumoto & Metzger 2022a. For SN 2006jc and 2019uo, the
precursors are observed $\approx 2\,\rm yr$ and $\approx 300\,\rm day$ before
the SNe, respectively. For the data points, the times of SN explosions are set
so that the points are roughly aligned with our light curves. The post-peak
part of the SN light curve are in small empty circles, to distinguish with the
precursor emission (large filled circles) we aim to reproduce. The horizontal
dash-dotted lines show our luminosity limit for the single star case (equation
5). The right panels show the observed CSM velocities of these events by
Strotjohann et al. (2021) (SN 2019zrk and SN 2019uo); Foley et al. (2007) (SN
2006jc); Matsumoto & Metzger (2022a) and references therein (others). Note
that in the center panel the blue line is overlapped with the gray line; while
larger $M_{\rm CSM}$ gives larger $L_{\rm wind}$, the increased inertia
results in an acceleration independent of $M_{\rm CSM}$.  Figure 3: Same as
Figure 2, but for the case of a $1.4M_{\odot}$ NS companion.

Figure 2 depicts the evolution of luminosities and the CSM velocities for
different parameter sets and progenitors, in the case of a BH companion of
$M_{\bullet}=10\,\mathrm{M}_{\odot}$. The left panels show the light curves
(solid curves) and corresponding wind (injection) luminosity $L_{\rm wind}$
(dotted curves). For RSG-BH binaries (top panel) and our fiducial parameters
of $M_{\rm CSM}=0.1\,\mathrm{M}_{\odot}$ and $a_{\rm bin}=5\,R_{*}(\approx
4\times 10^{14}\,\rm cm)$, the CSM completely recombines $\approx 50\,\rm
days$ after the ejection. The CSM starts to accrete onto the CO at
$t_{\bullet}\approx 100\,\rm days$, which triggers a powerful disk wind and
energy injection until $\approx 250\,\rm days$. Since the diffusion timescale
is almost comparable to the energy injection time $t_{\bullet}$, the resulting
light curve traces the injected $L_{\rm wind}$. Therefore, the light curve
shows two distinct components: an early injection-free part and a later part
due to energy injection. It should be noted that the former corresponds to the
precursor emission from a single star, and its luminosity $\sim\\!10^{39}\,\rm
erg\,s^{-1}$ is consistent with our luminosity limit in equation (5). As shown
in the middle panel, once the energy injection starts the CSM is accelerated
until recombination sets in.

Inspecting the dependence of the light curve on the parameters $a_{\rm bin}$
and $M_{\rm CSM}$, a more massive CSM (gray curve) increases the diffusion
time (equation 4) and wind luminosity (equation 17). Since $t_{\bullet}$ is
independent of the CSM mass, the longer diffusion time makes (i) the light
curve broader and deviate from $L_{\rm wind}$, and (ii) delays the
recombination which results in higher CSM velocity. A larger binary separation
(orange curve) simply delays the energy injection, which leads to smaller
$L_{\rm wind}$ and makes the light curve almost identical to the evolution of
$L_{\rm wind}$. Therefore, in this regime the duration and luminosity of the
precursor are given by

$\displaystyle t_{\rm prec,1}$ $\displaystyle\approx\frac{a_{\rm bin}}{v_{\rm
min}}-\frac{a_{\rm bin}}{v_{\rm max}}$ $\displaystyle\sim 200{\,\rm
day\,}\left(\frac{1}{\chi}-\frac{1}{\xi}\right)\left(\frac{v_{\rm
esc}}{100\,\rm km\,s^{-1}}\right)^{-1}$ $\displaystyle\times\left(\frac{a_{\rm
bin}}{5R_{*}}\right)\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)\ ,$
(34) $\displaystyle L_{\rm prec,1}$ $\displaystyle\approx L_{\rm wind}$
$\displaystyle\sim 2\times 10^{41}{\,\rm erg\
s^{-1}\,}\left(\frac{\xi^{3}}{\xi^{3}-\chi^{3}}\right)\left(\frac{M_{\bullet}}{10\
\mathrm{M}_{\odot}}\right)$ $\displaystyle\times\left(\frac{M_{\rm
CSM}}{0.1\,\mathrm{M}_{\odot}}\right)\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{-2}\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{-2}\
,$ (35)

where we used $V=\xi v_{\rm esc}$ and $t=t_{\bullet}$ to estimtate the
luminosity.

For the compact He star models, energy injection generally happens before
recombination in the CSM completes. In particular for the high-mass He star
models (middle panels) the injection timescale of $t_{\bullet}\approx
0.1-1\,\rm day$ is shorter than the diffusion timescale of $t_{\rm
prec,0}\approx 10\,\rm day$. Therefore the evolution is practically the same
as the CSM ejected with an initial radius of $a_{\rm bin}$ and energy of
$E_{\rm wind}$. Before the recombination, the luminosity is given by

$\displaystyle L$ $\displaystyle\approx 1\times 10^{41}{\,\rm
erg\,s^{-1}\,}\left(\frac{\xi^{2}}{\xi^{3}-\chi^{3}}\right)$
$\displaystyle\times\left(\frac{M_{\bullet}}{10\,\mathrm{M}_{\odot}}\right)\left(\frac{v_{\rm
esc}}{1000\,\rm km\,s^{-1}}\right)^{-1}\ ,$ (36)

where we used equation (5) but replacing $E_{\rm kin}$, $R_{*}$, and $v_{\rm
esc}$ with $E_{\rm wind}$, $a_{\rm bin}$, and $v_{\rm acc}\equiv\sqrt{10E_{\rm
wind}/3M_{\rm CSM}}$ (the CSM velocity after the acceleration), respectively.
Note that the normalization of the escape velocity is different from the case
of RSG progenitors. Interestingly the luminosity is independent of the CSM
mass and binary separation, as roughly seen in our results. By using this
luminosity, the onset time of the recombination is estimated by

$\displaystyle t_{\rm i}$ $\displaystyle\equiv\left(\frac{L}{4\pi v_{\rm
acc}^{2}\sigma T_{\rm I}^{4}}\right)^{1/2}\approx 6{\,\rm
day\,}\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{1/2}\left(\frac{R_{*}}{5\,\mathrm{R}_{\odot}}\right)^{1/2}\
.$ (37)

During the recombination, the light curve is described by the Popov model
(Popov, 1993), which gives the duration and luminosity (Matsumoto et al. 2016
and Matsumoto et al. in prep)

$\displaystyle t_{\rm prec,2}$ $\displaystyle\approx 7^{5/14}t_{\rm
i}^{2/7}t_{\rm prec,0}^{5/7}$ $\displaystyle\sim 20{\,\rm
day\,}\left(\frac{\xi^{2}}{\xi^{3}-\chi^{3}}\right)^{-\frac{5}{28}}\left(\frac{M_{\bullet}}{10\,\mathrm{M}_{\odot}}\right)^{-\frac{5}{28}}\left(\frac{M_{\rm
CSM}}{0.1\,\mathrm{M}_{\odot}}\right)^{\frac{5}{14}}$
$\displaystyle\times\left(\frac{v_{\rm esc}}{10^{3}\,\rm
km\,s^{-1}}\right)^{\frac{5}{28}}\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{\frac{9}{28}}\left(\frac{R_{*}}{5\,\mathrm{R}_{\odot}}\right)^{\frac{9}{28}}\
,$ (38) $\displaystyle L_{\rm prec,2}$
$\displaystyle\approx\frac{7^{3/7}56}{55}\pi\sigma T_{\rm I}^{4}v_{\rm
acc}^{2}t_{\rm i}^{8/7}t_{\rm prec,0}^{6/7}$ $\displaystyle\sim 1\times
10^{41}{\,\rm
erg\,s^{-1}\,}\left(\frac{\xi^{2}}{\xi^{3}-\chi^{3}}\right)^{\frac{11}{14}}\left(\frac{M_{\bullet}}{10\,\mathrm{M}_{\odot}}\right)^{\frac{11}{14}}$
$\displaystyle\times\left(\frac{M_{\rm
CSM}}{0.1\,\mathrm{M}_{\odot}}\right)^{\frac{3}{7}}\left(\frac{v_{\rm
esc}}{10^{3}\,\rm km\,s^{-1}}\right)^{-\frac{11}{14}}$
$\displaystyle\times\left(\frac{a_{\rm
bin}}{5R_{*}}\right)^{-\frac{3}{14}}\left(\frac{R_{*}}{5\,\mathrm{R}_{\odot}}\right)^{-\frac{3}{14}}\
.$ (39)

Here, again we replaced $v_{\rm CSM}$ in $t_{\rm prec,0}$ with $v_{\rm acc}$.
In particular, the duration weakly depends on the parameters.

When recombination sets in (seen as the small kinks in the light curves), the
light curve temporarily rises and then steeply drops. This is due to the rapid
recession of the ionization front, which quickly reduces the diffusion time
and initially enhances the luminosity. We do not expect that this will always
happen in reality, as the detailed evolution of the ionization front will
depend on the density profile of the CSM.

A sample of six well-observed precursors of SN II (taken from Matsumoto &
Metzger, 2022a; Fransson et al., 2022) and two precursors of SN Ibn
(Pastorello et al., 2007; Strotjohann et al., 2021) are plotted as colored
dots. For the case of a BH companion, the range of luminosity observed in
these precursors ($\sim\\!\\!10^{40}$–$10^{42}$ erg s-1) is well reproduced by
the range in binary separation, given a CSM mass typical for these SNe. In
particular, we find that qualitative properties of these precursor light
curves could be dictated by two timescales, the beginning of energy injection
$t_{\bullet}$ and diffusion timescale $t_{\rm prec,0}$ (see below for a more
quantitative discussion). While the smooth light curve of SN 2020tlf may
indicate $t_{\rm prec,0}\lesssim t_{\bullet}$, the other precursors have
longer diffusion timescales. However, we note that the detailed morphology of
the light curve can depend on the CSM profile, especially when the diffusion
timescale in the CSM is shorter than the wind duration. Thus we do not attempt
to make a detailed fit of the light curves here, and leave this to a future
numerical study that considers a realistic density/velocity profile of the
ejected CSM.

Figure 3 depicts the results for the NS companion case, where the same
calculations are done but for a reduced $M_{\bullet}=1.4~{}M_{\odot}$. While
the qualitative properties are the same as the BH case, the precursor is
dimmer and CSM acceleration is weaker due to the smaller Bondi accretion rate.
In this case, the low-mass He star model can reproduce the observed luminosity
of SN Ibn precursors.

###  4.2 Parameter Exploration of Light Curves

We next conduct a parameter study to understand the expected range of light
curve properties by the model parameters ($M_{\rm CSM}$, $a_{\rm bin}$). We
logarithmically sample the range of $M_{\rm CSM}$ and $a_{\rm bin}$ in Table 1
by ten points, resulting in a total of 100 parameter sets for each progenitor
model. We focus on the light curve from $t=t_{\bullet}$, and define the
precursor duration as the timescale over which $10$% to $90$% of the total
radiated energy is radiated away. In other words, we solve for $t_{\rm
10},t_{\rm 90}$ from

$\int_{t_{\rm 10}}^{\infty}L_{\rm rad}dt=\int_{t_{\bullet}}^{t_{\rm 90}}L_{\rm
rad}dt=0.9\int_{t_{\bullet}}^{\infty}L_{\rm rad}dt,$ (40)

and then define the duration and luminosity of the precursor respectively as

$\displaystyle t_{\rm prec}=t_{\rm 90}-t_{\rm 10},\ L_{\rm
prec}=\frac{1}{t_{\rm 90}-t_{\rm 10}}\int_{t_{\rm 10}}^{t_{\rm 90}}L_{\rm
rad}dt.$ (41)

We show the dependence of these light curve properties on ($M_{\rm CSM}$,
$a_{\rm bin}$) in Figure 4. For the RSG-BH models, a wide range of luminosity
($10^{39}$–$10^{42}$ erg s-1) and timescale (month to years) are found with
varying $M_{\rm CSM}$ and $a_{\rm bin}$. For a fixed CSM mass, increasing
$a_{\rm bin}$ always makes the precursors dimmer and longer. This is because
the energy injection is delayed, and the more tenuous CSM gives a lower Bondi
accretion rate.

On the other hand, increasing $M_{\rm CSM}$ basically results in brighter and
longer precursors, except for large $a_{\rm bin}$ where the duration becomes
nearly independent of $M_{\rm CSM}$. This behavior is understood by comparing
the diffusion and energy injection timescales. As we discussed in the previous
section, for smaller $a_{\rm bin}$, the energy injection happens before the
radiative diffusion plays a role, and the light curve is practically the same
as that described by Popov (1993): $t_{\rm prec}\propto M_{\rm
CSM}^{5/14}a_{\rm bin}^{9/28}$ (equation 38) and $L_{\rm prec}\propto M_{\rm
CSM}^{3/7}a_{\rm bin}^{-3/14}$ (equation 39). In the opposite limit, the
precursor basically traces the wind luminosity and hence we have $t_{\rm
prec}\propto a_{\rm bin}$ (equation 34) independent of $M_{\rm CSM}$, and
$L_{\rm prec}\propto M_{\rm CSM}a_{\rm bin}^{-2}$ (equation 35). The boundary
between these two limits can be obtained by comparing $t_{\rm prec,0}$ and
$a_{\rm bin}/\xi v_{\rm esc}$:

$\displaystyle\frac{a_{\rm bin}}{R_{*}}\left(\frac{M_{\rm
CSM}}{0.1\,\mathrm{M}_{\odot}}\right)^{-1/2}\sim
2\xi^{1/2}\left(\frac{R_{*}}{500\,\mathrm{R}_{\odot}}\right)^{-1}\left(\frac{v_{\rm
esc}}{100{\,\rm km\,s^{-1}}}\right)^{1/2}\ .$ (42)

Combining this relation with equations (38) and (39), the boundary on which
$t_{\rm prec,0}=t_{\bullet}$ holds, is given by

$\displaystyle L_{\rm prec}$ $\displaystyle\sim 2\times 10^{41}{\,\rm
erg\,s^{-1}\,}\left(\frac{t_{\rm prec}}{100\,\rm
day}\right)^{18/29}\left[\frac{\xi^{81/116}}{(\xi^{3}-\chi^{3})^{26/29}}\right]$
$\displaystyle\times\left(\frac{M_{\bullet}}{10\,\mathrm{M}_{\odot}}\right)^{{26}/{29}}\left(\frac{v_{\rm
esc}}{100\,\rm km\,s^{-1}}\right)^{-{32}/{29}}\left(\frac{T_{\rm I}}{6000\,\rm
K}\right)^{60/29}\ ,$ (43)

where we show the dependence on the recombination temperature explicitly. We
plot this boundary in Figure 4, which roughly captures the transition of two
cases.

For the case of a NS companion, the resulting duration and luminosity are
basically the same as the BH case, but the luminosity is reduced by an order
of magnitude (note $L_{\rm prec,1}\propto M_{\bullet}$, and $L_{\rm
prec,2}\propto M_{\bullet}^{11/14}$). At large $a_{\rm bin}$ where the
radiated luminosity traces the wind luminosity, the duration only weakly
increases as $a_{\rm bin}$ increases. This is because the accretion disk
formation is delayed for a NS companion until the CSM velocity at the Bondi
radius, $V\approx a_{\rm bin}/t$, becomes small enough so that $r_{\rm
disk}>r_{\rm in}$ is satisfied.

The He star models generally predict shorter durations of weeks to months,
with the high mass models generally resulting in dimmer and faster light
curves when the parameters are fixed. This is because for a fixed $a_{\rm
bin}/R_{*}$, the high-mass models with smallest $R_{*}$ have the shortest
accretion time scales, and lose most of the injected internal energy into
adiabatic expansion before it can be radiated.

In Figure 4 we also show a more detailed comparison with the observed Type IIn
and Ibn precursors in the luminosity-duration phase space. We note that the
observed duration gives only a lower limit on the possible intrinsic duration,
because when the SN explosion occurs, precursors are outshined by the SN
radiation. While the RSG-BH models reproduce the range of precursor
luminosities, the RSG-NS models can reproduce only the dimmer precursors of
$L_{\rm prec}\lesssim 10^{41}$ erg s-1. The parameters of the precursors for
the He star models are found to cluster in a narrower range of
$10^{40}$–$10^{42}$ erg s-1, with timescales from weeks to months. These
roughly overlap with the two precursors observed for type Ibn SNe, with SN
2006jc (SN 2019uo) better reproduced by the HeHighMass-BH (HeLowMass-NS)
model.

Figure 4: Light curve properties of our precursor models (dots), compared with
the observed well-sampled precursors in Figure 2 (red stars) and selected
well-sampled events of Strotjohann et al. (2021) with 5 or more unbinned
detections in multiple bands (green stars). The SN IIn precursors have been
shining until core collapse, so their durations are lower limits. The dashed
orange line is the binding energy (equation 1) for the RSG model, and the
horizontal dash-dotted lines are the luminosity limit for single stars
(equation 5). The solid blue line in the left panel represents the boundary on
which $t_{\rm prec,0}=t_{\bullet}$ holds (equation 43). Below (above) the
line, the light curve is described by the wind luminosity (Popov model; Popov
1993).

###  4.3 Velocity of Accelerated CSM

Figure 5 shows the final velocity of the precursor ejecta as a function of
$(M_{\rm CSM},a_{\rm bin})$. For the RSG-BH models, the final velocities
generally agree with CSM velocities observed in SN IIn, from 100 km s-1 to
$\sim\\!10^{3}$ km s-1 (Kiewe et al., 2012; Taddia et al., 2013). It is
noteworthy that this is reproduced even for RSG progenitors which are
generally expected to have slow CSM velocities.

However, a direct comparison to the observed CSM velocities on an event-by-
event basis is not straightforward. For instance, broad emission lines of
$1000-3000$ km s-1 are observed in SN 2010mc, but broad wings are not solely
due to fast CSM, and can also be created by Thomson scattering in the CSM if
the CSM is optically thick (e.g., Chugai, 2001; Huang & Chevalier, 2018; Ishii
et al., 2023). Furthermore, for the CSM velocities measured after explosion
(2010mc, 2016bdu, 2019zrk, 2020tlf), they have additional contributions from
radiative acceleration by the bright SN emission (Chevalier & Irwin, 2011;
Kochanek, 2019; Tsuna et al., 2023a).

For SN 2009ip, multiple velocity features are observed during the outburst,
from $800$–$1400$ km s-1 typical for CSM in SN IIn, but also with a broad
$\sim\\!10^{4}$ km s-1 absorption feature (Pastorello et al., 2013; Margutti
et al., 2014). Similar coexistence of slow components ($600$–$1000$ km s-1)
and fast emission line wings ($2600$–$6000$ km s-1) are observed in SN 2015bh
during the outburst phase (Elias-Rosa et al., 2016). While the slow component
can be explained by our model, the fast part cannot. It could be explained if
we are directly observing the accretion disk wind as well as the CSM (equation
3.1; see also Tsebrenko & Soker, 2013). Multi-dimensional simulations would
help understand the observability of such fast outflows.

For SN Ibn, the observed CSM velocities are typically of the order of
$\sim\\!10^{3}$ km s-1. For compact He stars this can be easily achieved, as
the CSM can be close to this velocity even without energy injection. On the
other hand, for the low-mass extended He star models, significant energy
injection is needed to accelerate the CSM to the observed values. This would
require either a small orbital separation or a BH companion, but the latter
may be in tension with observations for some SN Ibn, as it predicts bright
precursors (Figure 4). Radiative acceleration may also bring the CSM velocity
of the low-mass He star models closer to the observed values. Due to its
slower velocity, the precursor CSM for these models is likely to be more
confined at the time of SN, and will receive significant acceleration by the
subsequent SN radiation (e.g., Figure 6 of Tsuna et al. 2023a).

While the HeHighMass-BH and HeLowMass-NS models predict similar precursor
luminosities that are consistent with the observed ones, the final velocities
of the CSM are quite different. As shown in Figure 5, the former would lead to
CSM velocities of a few $1000$ km s-1, while the latter would have velocities
less than 1000 km s-1 for the most of the parameter space. These are
consistent with the measured velocities of SN 2006jc (2000–3000 km s-1; Foley
et al. 2007) and SN 2019uo (550–1000 km s-1; Strotjohann et al. 2021)
respectively. This implies that the precursors of SN Ibn may be a mixture of
two binary populations, low-mass (a few $M_{\odot}$) He stars with NS
companions and more massive He stars with BH companions.

---
Figure 5: Final velocity of the precursor CSM as a function of $a_{\rm bin}$
and $M_{\rm CSM}$. The HeHighMass-NS model has a negligibly small acceleration
for the entire parameter space (see also Figure 3), while other models predict
larger acceleration for smaller $a_{\rm bin}$ and larger $M_{\rm CSM}$.

##  5 Discussion

###  5.1 Possible Formation Channels and Rates

So far we have discussed a phenomenological model of a mass ejection in close
binaries, and demonstrated that they can reproduce the energetics of observed
precursors. Here we outline the binary channels that produce these systems,
and roughly estimate their rates.

Our channel requires a massive star in a binary with a compact object, at a
separation of $\sim\\!$ 3–30$~{}R_{*}$ at the time of core-collapse.
Population synthesis predicts that the outcome of massive binary systems are
nearly equally split among stellar merger, envelope stripping and non-
interacting binaries (Sana et al., 2012). For the latter two channels, the
binary can become unbound following the core-collapse of the first object, due
to the natal kick of the CO. While the fraction is very sensitive to the
magnitude of natal kicks, population synthesis calculations find a binary
survival fraction of 14${}^{+20}_{-10}$%, with comparable amounts of NS and BH
remnants (Renzo et al., 2019, Figure 4). The surviving binaries with NS
companions are weighted towards close separations and massive secondaries, due
to NSs typically having larger kicks and (SN) ejecta masses than BHs.

We therefore expect that roughly 1 – 10% of core-collapse SNe can arise in the
presence of a CO companion. This fraction is divided into hydrogen-rich
progenitors (our RSG-CO models in most cases) and stripped-envelope
progenitors (our He star-CO models), which is further complicated by various
binary interaction processes (and their uncertainties) preceding the second
core-collapse. While we cannot predict the relative fractions and their
expected binary separations without in-depth population synthesis
calculations, we expect that for most surviving binaries the pre-SN secondary
will reside close to the CO, since it is likely to expand to a similar size to
that of the primary during its post-main sequence evolution. The separation is
further favored to be close for stripped progenitors, which are usually
stripped by the CO companion. Another population synthesis estimate finds that
the number of stripped-envelope SNe with CO companions is $\sim\\!1$%
(Zapartas et al., 2017), which is roughly consistent with this assumption.

As discussed below, the precursor rate is expected to be $\sim\\!1\%$ of core-
collapse SNe for bright IIn precursors, and $\sim\\!0.1\%$ for Ibn precursors.
Hence, our scenario can plausibly reproduce the observed precursors if pre-SN
outbursts or enhanced pre-SN mass loss is relatively common, with a fraction
of at least $\gtrsim 10$%. For RSG progenitors the common presence of confined
dense CSM in Type II SNe may support this (Morozova et al., 2018; Förster et
al., 2018; Bruch et al., 2021, 2023), although there are alternative
possibilities to explain the dense CSM (Dessart et al. 2017; Kochanek 2019;
Soker 2021; see also Davies et al. 2022).

We note that for SN 2006jc, a point source was detected several years after
the SN, whose spectral energy distribution is most consistent with a companion
star of the SN progenitor (Maund et al., 2016; Sun et al., 2020). As the
inferred stellar mass for the surviving companion ($\lesssim 12\ M_{\odot}$)
rules out a luminous blue variable, these works conclude that it is indeed the
progenitor that made the precursor outburst, and not the companion. If our
scenario for the precursor is correct, this may imply that the progenitor of
SN 2006jc was in a hierarchical triple system, with the surviving companion
star as the outer tertiary.

###  5.2 Implications for Other Observations

For the RSG-BH progenitors, we may find the progenitors of these systems in
Gaia when they are in their main-sequence (MS) phase. Assuming a circular
orbit, the orbital period is

$\displaystyle P_{\rm orb}$ $\displaystyle\approx$ $\displaystyle
2\pi\sqrt{\frac{a_{\rm bin}^{3}}{G(M_{\rm BH}+M_{*})}}$ (44)
$\displaystyle\approx$ $\displaystyle 3.9\ {\rm yr}\left(\frac{a_{\rm
bin}}{10^{14}\ {\rm cm}}\right)^{3/2}\left(\frac{M_{\rm
BH}+M_{*}}{20M_{\odot}}\right)^{-1/2}.$

Since these progenitors will be non-interacting before the precursor, we may
expect that the orbital period has not greatly changed from the first BH
formation. The required orbital separations of $a\lesssim$ a few $\times
10^{14}$ cm for powering bright precursors of $L_{\rm prec}\gtrsim 10^{41}$
erg s-1 overlap with the binary periods up to which Gaia can detect their
astrometric motions, of $\lesssim\\!10$ years (e.g., Shikauchi et al., 2022;
El-Badry et al., 2023a). Theoretical calculations also predict that such
detached OB-BH binaries with long periods are more common than interacting
ones seen as X-ray binaries (e.g., Langer et al., 2020; Janssens et al.,
2022).

We estimate the number of such systems in the Milky Way, assuming that they
explain the observed precursors of Type IIn SNe. Adopting a bright precursor
fraction in SN IIn of $f_{\rm prec}=5$–$69$% (Strotjohann et al., 2021)555We
note that this fraction is for precursors brighter than -13 mag. If we assume
RSG-NS systems are responsible for precursors dimmer than -14 mag ($\approx
10^{41}$ erg; Section 4.2) and instead put the threshold at -14 mag, this
fraction decreases by a factor of a few (Figure 9 of Strotjohann et al.
2021)., SN IIn fraction among core-collapse SNe of $f_{\rm IIn}=5\%$ (Cold &
Hjorth, 2023), MW core-collapse SN rate of $R_{\rm SN,MW}\approx 1.6$ per
century (Rozwadowska et al., 2021), and assumed lifetime of the MS star
$t_{\rm MS}\sim 5\ {\rm Myr}$ after the companion becomes a BH, we expect

$f_{\rm prec}f_{\rm IIn}R_{\rm SN,MW}t_{\rm MS}\sim 200-2800$ (45)

such MS-BH binaries in the Milky Way. The value of $t_{\rm MS}$ is assumed to
be roughly half of the typical lifetime of a BH progenitor $\sim 10$ Myr
(e.g., Woosley et al., 2002), but should greatly vary for each binary system.

Population synthesis models predict that there are $\sim 10^{4}$–$10^{5}$
detached BH binaries in the Milky Way that have period within $10$ years
accessible by Gaia, with a good fraction of those with longer periods of years
(Shikauchi et al. 2023; see also Chawla et al. 2022; Rodriguez et al. 2023).
The mass of the luminous star is typically $\sim\\!1\ M_{\odot}$, with massive
ones of $\gtrsim 8\ M_{\odot}$ comprising typically only $\sim(1$–few)%
depending on binary population synthesis models. Thus the number of RSG-BH
binary systems are estimated to be of the order of 100 – 1000, which roughly
agrees with that estimated from the SN IIn precursor rate although both still
contain large uncertainties. Ongoing observations by Gaia can lead to a better
constraint on our scenario, as long-period binary systems with BHs are
starting to be discovered (El-Badry et al., 2023a, b; Tanikawa et al., 2023;
Chakrabarti et al., 2023).

For precursors from He stars that are progenitors of SNe Ibn, the event rates
are less certain as we have only a few detections. For the case of HeHighMass-
BH models that may explain the SN Ibn precursors, we may compare the precursor
event rates to the NS-BH merger rate. For these binaries with separations of
our interest, these two can be linked because the merger timescale by
gravitational wave (GW) emission (Peters, 1964)

$\displaystyle t_{\rm GW}$ $\displaystyle=$ $\displaystyle\frac{5a_{\rm
bin}^{4}c^{5}}{256G^{3}M_{\rm NS}M_{\rm BH}(M_{\rm NS}+M_{\rm BH})}$ (46)
$\displaystyle\approx$ $\displaystyle 9.4\ {\rm Gyr}\left(\frac{a_{\rm
bin}}{10R_{\odot}}\right)^{4}$ $\displaystyle\times\left(\frac{M_{\rm
NS}}{1.4\ M_{\odot}}\right)^{-1}\left(\frac{M_{\rm BH}}{10\
M_{\odot}}\right)^{-1}\left(\frac{M_{\rm NS}+M_{\rm BH}}{11.4\
M_{\odot}}\right)^{-1}$

is less than a Hubble time for separations of $a_{\rm bin}\lesssim
10~{}R_{\odot}(M_{\rm BH}/10M_{\odot})^{1/2}$, and for compact stellar radii
$R_{*}\lesssim$ (a few) $R_{\odot}$ expected in massive He stars (e.g., Yoon
et al., 2010; Woosley, 2019; Laplace et al., 2020). We infer the bright
precursor fraction of Type Ibn SNe to be $\sim\\!10\%$ since one precursor was
detected out of 11 targets in Strotjohann et al. (2021), while they obtain an
upper limit of $<31\%$ for precursors brighter than -13 mag. This translates
to an event rate of $\sim\\!100\ {\rm Gpc^{-3}}\ {\rm yr^{-1}}$ or an upper
limit of $\lesssim 300\ {\rm Gpc^{-3}}\ {\rm yr^{-1}}$, assuming a SN Ibn
fraction of $1\%$ among all core-collapse SNe (Pastorello et al., 2008; Maeda
& Moriya, 2022). This event rate is similar to the NS-BH merger rate of
$130^{+112}_{-69}\ {\rm Gpc^{-3}}\ {\rm yr^{-1}}$, inferred from the GW
detections of two NS-BH mergers by the LIGO-Virgo-KAGRA collaboration (Abbott
et al., 2021).

The Ibn precursor rate is uncertain, as it could be increased relative to the
CO merger rate because it can operate in binaries with periods too long to
merge via GWs. However, it could be suppressed relative to the CO merger rate
if pre-SN outbursts are uncommon. Although we do not claim that these kinds of
binaries are the sole channel for NS-BH mergers, this approximate agreement
may motivate further investigations of the connection between these peculiar
optical transients and formation of binary compact objects.

Finally, there is a handful of known high-mass X-ray binaries hosting BHs
(Remillard & McClintock, 2006), and recently a few detached binaries with a
massive O-type star and a putative BH companion have been identified
(HD130298, Mahy et al. 2022; VFTS 243, Shenar et al. 2022). These binaries
have rather tight orbits with periods of $\lesssim 10$ days, and they may lead
to He star-BH binaries if the O-stars lose their hydrogen-rich envelopes
through future binary interactions.

###  5.3 Possible Caveats and Future Avenues of our Model

We have restricted our model to the case of CSM created from outbursts, which
has the advantage of both the CSM mass and velocity being well-defined.
However, there are other mechanisms to lift material to larger radii, such as
a steady-state wind (e.g., Yoon & Cantiello, 2010; Moriya, 2014; Quataert et
al., 2016), or envelope inflation through energy injection and/or intense
shell burning (e.g., Soker, 2013; Smith & Arnett, 2014; Fuller & Ro, 2018;
Ouchi & Maeda, 2019; Wu & Fuller, 2022a, b) resulting in rapid mass transfer.
This more gradual mass expulsion from the star can result in a steady-state
energy injection from the CO, and lead to precursors more long-lasting than
what we have simulated here. It would be useful future work to extend our
precursor model to such cases.

Two major multi-dimensional effects are not captured in our one-zone light
curve modeling. First, as energy injection starts soon after the fastest end
of the CSM reaches the CO, the CSM is initially heated not from the center but
from the outside. This may affect the initial evolution of the CSM, while the
effect may be smaller at late times. Second, we do not consider the detailed
angular dependence of the energy injection due to disk wind. In our work, the
angle subtended by the wind is somewhat idealized like Kimura et al. (2017),
i.e. wide enough so that it can heat the entire CSM, but not too wide such
that feedback would prevent the CSM from accreting onto the CO. These effects
are important caveats that should be investigated in multi-dimensional
hydrodynamical simulations.

Related to these multi-dimensional effects, if a powerful disk wind or a
relativistic jet hypothetically launched from the CO emerges from the CSM, it
may produce radio/X-ray precursors of SN explosions. Detecting such signals
would be a smoking gun signature of our model. We plan to study their
detectability in future work.

##  6 Conclusion

We studied the emission from pre-SN outbursts from massive stars with a CO
companion, aiming to reproduce the observed bright precursors of interacting
SNe. We model the super-Eddington accretion of (a fraction of) the ejected CSM
onto the CO, and the resulting energy injection into the CSM by the disk wind
launched from the CO. We considered SN progenitors consisting of a RSG, a
high-mass compact He star, and a low-mass extended He star (with parameters in
Table 1), each with a 10 $M_{\odot}$ BH or a 1.4 $M_{\odot}$ NS companion.

We constructed a one-zone light curve model including time-dependent energy
injection, emission and acceleration of the CSM. We find that binary systems
with separations of 3–30 times the stellar radii can generally reproduce both
the luminosity and duration of the observed precursors, as well as the
velocity of the CSM inferred from spectroscopy.

For the RSG model motivated for precursors of hydrogen-rich (Type II) SNe, a
BH companion leads to precursors with broad ranges of luminosity
$10^{39}$–$10^{42}$ erg s-1 and durations of months to years. We find that
this can successfully explain the observed precursors, for a realistic range
of CSM masses in Type II SNe corresponding to outburst energies only of the
order of $10^{46}$–$10^{47}$ ergs. The case of NS companions can explain
dimmer precursors of $\lesssim 10^{41}$ erg s-1 which are observed in a
fraction of Type IIn SNe (Strotjohann et al., 2021) but not the brightest ones
approaching $\sim\\!10^{42}$ erg s-1.

For the He star models motivated for Type Ibn SNe, the luminosity and duration
of the precursors also agree with the two observed precursors of SN 2006jc and
2019uo. From the precursor luminosity, timescale, and the CSM velocity
observed in the SN phase, we conclude that the progenitor of SN 2006jc is
likely a compact (Wolf Rayet-like) He star with a BH companion, whereas that
of SN 2019uo is likely an inflated He star with a NS companion. This may
indicate a diverse progenitor channel for Type Ibn SNe, in addition to the
claim that a fraction of these SNe may not be from massive stars (Hosseinzadeh
et al., 2019).

We find that the predicted event rates of our model roughly agree with the
observed precursor rates, if enhanced mass-loss is relatively common months to
years before core-collapse. However, we urge a more in-depth population
synthesis study for rate predictions, as future observations will likely
narrow down the uncertainty in the observed rate. Moreover we point out a
possible link between the systems powering the precursors and CO binary
systems observable by Gaia and gravitational-wave detectors, which can also be
investigated by such population synthesis studies.

In the era of the Rubin Observatory (Ivezić et al., 2019), the number of SN
precursors we can detect is expected to dramatically increase, and we may also
be able to probe deeper into the dimmer end of the precursor population (e.g.,
Tsuna et al., 2023b; Strotjohann et al., 2024). We expect that our framework
would be helpful to characterize the origin and diversity of such
precursors666We make our source code for calculating the light curves public
(upon acceptance of this paper), accessible from:
https://github.com/DTsuna/binary_precursor..

We thank Xiaoshan Huang and Kareem El-Badry for stimulating discussions. D. T.
thanks Morgan Macleod for discussions that led to inspiration of this model.
D. T. is supported by the Sherman Fairchild Postdoctoral Fellowship at the
California Institute of Technology. T. M. acknowledges supports from JSPS
Overseas Research Fellowship and the Hakubi project at Kyoto University. S. C.
W. is supported by the National Science Foundation Graduate Research
Fellowship under Grant No. DGE-1745301. J. F. is grateful for support from the
NSF through grant AST-2205974.

##  Appendix A Timestep Control for the Precursor Modeling

When solving the governing equations for modelling the precursor described in
Section 3.2.3, we carefully control the timestep of the light curve
calculation so that the parameters in the governing equations do not suddenly
evolve in a single step. Specifically, when $x_{i}=1$ the timestep is set by
the dynamical and diffusion timescales as

$\Delta t_{x=1}=10^{-2}{\rm min}\left(\frac{R_{\rm CSM}}{v_{\rm CSM}},t_{\rm
diff}\right).$ (A1)

When the wind is turned on with $x_{i}<1$, we impose another timestep limit
using equation (30) as

$\displaystyle\Delta t_{x<1}={\rm min}\left[\Delta t_{x=1},10^{-2}t_{\rm
diff}(5x_{i}x_{i,\bullet})\left|\frac{L_{\rm inj}}{4\pi x_{i}^{2}r^{2}\sigma
T_{I}^{4}}-1\right|^{-1}\right].$ (A2)

This makes $x_{i}$ evolve smoothly towards $x_{i,\bullet}$ (defined after
equation 33), even when the wind is turned on when $x_{i}\ll 1$.

##  Appendix B Dependence on the Velocity Parameter $\xi$

Here we check the dependence of our results on $\xi(\gtrsim 1)$, the velocity
of the ejected CSM normalized by the surface escape velocity of the
progenitor, by conducting the same calculation as in the main text that
assumed $\xi=2$ but for a reduced value of $\xi=1.4$. For a given time and
fixed $M_{\rm CSM}$, the effect of reducing $\xi$ is to make the CSM less
extended and denser. Thus we expect the energy injection by the disk wind to
start later, and be shorter duration with higher power.

Figure 6 shows the dependence of the precursor light curves and final CSM
velocities on $\xi$. We focused on the case for the BH companion, and adopted
the same parameter sets as in Figure 2. As expected, for a fixed ($a_{\rm
bin},M_{\rm CSM}$) we see that for $\xi=1.4$ the precursor light curve rises
at a later time than $\xi=2$, have a slightly brighter peak luminosity, and
have a slightly faster final CSM velocity. However these quantities are very
similar for the two cases of $\xi$, with differences typically within a few
10%.

We note that the precursor duration can become quite different for extended
stars with large $a_{\rm bin}$, as shown for the case of $a_{\rm bin}=15R_{*}$
for the top-left panel. This is because the precursor duration in this
parameter region (typically 100s of days) is governed by the duration of the
disk wind (equation 34), rather than the diffusion timescale of the CSM.
However since no precursors are observed for much longer than $100$ days, we
expect this finding to not affect the discussion on the comparison of our
model with the observed precursors in Section 4.

Figure 6: Dependence on the precursor light curves and final CSM velocities on
the initial velocity parameter $\xi$. The solid lines are the case for
$\xi=2$, and the dashed line are for $\xi=1.4$.

## References

  * Abbott et al. (2021) Abbott, R., Abbott, T. D., Abraham, S., et al. 2021, ApJ, 915, L5, doi: 10.3847/2041-8213/ac082e
  * Arnett (1980) Arnett, W. D. 1980, ApJ, 237, 541, doi: 10.1086/157898
  * Arnett (1982) —. 1982, ApJ, 253, 785, doi: 10.1086/159681
  * Blandford & Begelman (1999) Blandford, R. D., & Begelman, M. C. 1999, MNRAS, 303, L1, doi: 10.1046/j.1365-8711.1999.02358.x
  * Bondi (1952) Bondi, H. 1952, MNRAS, 112, 195, doi: 10.1093/mnras/112.2.195
  * Bondi & Hoyle (1944) Bondi, H., & Hoyle, F. 1944, MNRAS, 104, 273, doi: 10.1093/mnras/104.5.273
  * Bruch et al. (2021) Bruch, R. J., Gal-Yam, A., Schulze, S., et al. 2021, ApJ, 912, 46, doi: 10.3847/1538-4357/abef05
  * Bruch et al. (2023) Bruch, R. J., Gal-Yam, A., Yaron, O., et al. 2023, ApJ, 952, 119, doi: 10.3847/1538-4357/acd8be
  * Chakrabarti et al. (2023) Chakrabarti, S., Simon, J. D., Craig, P. A., et al. 2023, AJ, 166, 6, doi: 10.3847/1538-3881/accf21
  * Chawla et al. (2022) Chawla, C., Chatterjee, S., Breivik, K., et al. 2022, ApJ, 931, 107, doi: 10.3847/1538-4357/ac60a5
  * Chevalier & Irwin (2011) Chevalier, R. A., & Irwin, C. M. 2011, ApJ, 729, L6, doi: 10.1088/2041-8205/729/1/L6
  * Chugai (2001) Chugai, N. N. 2001, MNRAS, 326, 1448, doi: 10.1111/j.1365-2966.2001.04717.x
  * Cold & Hjorth (2023) Cold, C., & Hjorth, J. 2023, A&A, 670, A48, doi: 10.1051/0004-6361/202244867
  * Danieli & Soker (2019) Danieli, B., & Soker, N. 2019, MNRAS, 482, 2277, doi: 10.1093/mnras/sty2892
  * Davies et al. (2022) Davies, B., Plez, B., & Petrault, M. 2022, MNRAS, 517, 1483, doi: 10.1093/mnras/stac2427
  * Dessart et al. (2017) Dessart, L., Hillier, D. J., & Audit, E. 2017, A&A, 605, A83, doi: 10.1051/0004-6361/201730942
  * Dessart et al. (2022) Dessart, L., John Hillier, D., & Kuncarayakti, H. 2022, A&A, 658, A130, doi: 10.1051/0004-6361/202142436
  * Dexter & Kasen (2013) Dexter, J., & Kasen, D. 2013, ApJ, 772, 30, doi: 10.1088/0004-637X/772/1/30
  * Eggleton (1983) Eggleton, P. P. 1983, ApJ, 268, 368, doi: 10.1086/160960
  * El-Badry et al. (2023a) El-Badry, K., Rix, H.-W., Cendes, Y., et al. 2023a, MNRAS, 521, 4323, doi: 10.1093/mnras/stad799
  * El-Badry et al. (2023b) El-Badry, K., Rix, H.-W., Quataert, E., et al. 2023b, MNRAS, 518, 1057, doi: 10.1093/mnras/stac3140
  * Eldridge et al. (2008) Eldridge, J. J., Izzard, R. G., & Tout, C. A. 2008, MNRAS, 384, 1109, doi: 10.1111/j.1365-2966.2007.12738.x
  * Elias-Rosa et al. (2016) Elias-Rosa, N., Pastorello, A., Benetti, S., et al. 2016, MNRAS, 463, 3894, doi: 10.1093/mnras/stw2253
  * Ertl et al. (2020) Ertl, T., Woosley, S. E., Sukhbold, T., & Janka, H. T. 2020, ApJ, 890, 51, doi: 10.3847/1538-4357/ab6458
  * Foley et al. (2007) Foley, R. J., Smith, N., Ganeshalingam, M., et al. 2007, ApJ, 657, L105, doi: 10.1086/513145
  * Förster et al. (2018) Förster, F., Moriya, T. J., Maureira, J. C., et al. 2018, Nature Astronomy, 2, 808, doi: 10.1038/s41550-018-0563-4
  * Fransson et al. (2022) Fransson, C., Sollerman, J., Strotjohann, N. L., et al. 2022, A&A, 666, A79, doi: 10.1051/0004-6361/202243452
  * Fraser et al. (2013) Fraser, M., Magee, M., Kotak, R., et al. 2013, ApJ, 779, L8, doi: 10.1088/2041-8205/779/1/L8
  * Fuller (2017) Fuller, J. 2017, MNRAS, 470, 1642, doi: 10.1093/mnras/stx1314
  * Fuller & Lu (2022) Fuller, J., & Lu, W. 2022, MNRAS, 511, 3951, doi: 10.1093/mnras/stac317
  * Fuller & Ro (2018) Fuller, J., & Ro, S. 2018, MNRAS, 476, 1853, doi: 10.1093/mnras/sty369
  * Gal-Yam (2019) Gal-Yam, A. 2019, ARA&A, 57, 305, doi: 10.1146/annurev-astro-081817-051819
  * Gal-Yam et al. (2022) Gal-Yam, A., Bruch, R., Schulze, S., et al. 2022, Nature, 601, 201, doi: 10.1038/s41586-021-04155-1
  * Hiramatsu et al. (2023) Hiramatsu, D., Matsumoto, T., Berger, E., et al. 2023, arXiv e-prints, arXiv:2305.11168, doi: 10.48550/arXiv.2305.11168
  * Hosseinzadeh et al. (2019) Hosseinzadeh, G., McCully, C., Zabludoff, A. I., et al. 2019, ApJ, 871, L9, doi: 10.3847/2041-8213/aafc61
  * Hoyle & Lyttleton (1939) Hoyle, F., & Lyttleton, R. A. 1939, Proceedings of the Cambridge Philosophical Society, 35, 405, doi: 10.1017/S0305004100021150
  * Huang & Chevalier (2018) Huang, C., & Chevalier, R. A. 2018, MNRAS, 475, 1261, doi: 10.1093/mnras/stx3163
  * Ichimaru (1977) Ichimaru, S. 1977, ApJ, 214, 840, doi: 10.1086/155314
  * Ishii et al. (2023) Ishii, A. T., Takei, Y., Tsuna, D., Shigeyama, T., & Takahashi, K. 2023, arXiv e-prints, arXiv:2309.05344, doi: 10.48550/arXiv.2309.05344
  * Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111, doi: 10.3847/1538-4357/ab042c
  * Jacobson-Galán et al. (2022) Jacobson-Galán, W. V., Dessart, L., Jones, D. O., et al. 2022, ApJ, 924, 15, doi: 10.3847/1538-4357/ac3f3a
  * Janssens et al. (2022) Janssens, S., Shenar, T., Sana, H., et al. 2022, A&A, 658, A129, doi: 10.1051/0004-6361/202141866
  * Jiang et al. (2014) Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2014, ApJ, 796, 106, doi: 10.1088/0004-637X/796/2/106
  * Kasen & Bildsten (2010) Kasen, D., & Bildsten, L. 2010, ApJ, 717, 245, doi: 10.1088/0004-637X/717/1/245
  * Kasen & Woosley (2009) Kasen, D., & Woosley, S. E. 2009, ApJ, 703, 2205, doi: 10.1088/0004-637X/703/2/2205
  * Khazov et al. (2016) Khazov, D., Yaron, O., Gal-Yam, A., et al. 2016, ApJ, 818, 3, doi: 10.3847/0004-637X/818/1/3
  * Kiewe et al. (2012) Kiewe, M., Gal-Yam, A., Arcavi, I., et al. 2012, ApJ, 744, 10, doi: 10.1088/0004-637X/744/1/10
  * Kimura et al. (2017) Kimura, S. S., Murase, K., & Mészáros, P. 2017, ApJ, 851, 53, doi: 10.3847/1538-4357/aa988b
  * Kitaki et al. (2021) Kitaki, T., Mineshige, S., Ohsuga, K., & Kawashima, T. 2021, PASJ, 73, 450, doi: 10.1093/pasj/psab011
  * Kleiser & Kasen (2014) Kleiser, I. K. W., & Kasen, D. 2014, MNRAS, 438, 318, doi: 10.1093/mnras/stt2191
  * Kochanek (2019) Kochanek, C. S. 2019, MNRAS, 483, 3762, doi: 10.1093/mnras/sty3363
  * Kuriyama & Shigeyama (2020) Kuriyama, N., & Shigeyama, T. 2020, A&A, 635, A127, doi: 10.1051/0004-6361/201937226
  * Kuriyama & Shigeyama (2021) —. 2021, A&A, 646, A118, doi: 10.1051/0004-6361/202038637
  * Langer et al. (2020) Langer, N., Schürmann, C., Stoll, K., et al. 2020, A&A, 638, A39, doi: 10.1051/0004-6361/201937375
  * Laplace et al. (2020) Laplace, E., Götberg, Y., de Mink, S. E., Justham, S., & Farmer, R. 2020, A&A, 637, A6, doi: 10.1051/0004-6361/201937300
  * Leung et al. (2021) Leung, S.-C., Wu, S., & Fuller, J. 2021, ApJ, 923, 41, doi: 10.3847/1538-4357/ac2c63
  * Linial et al. (2021) Linial, I., Fuller, J., & Sari, R. 2021, MNRAS, 501, 4266, doi: 10.1093/mnras/staa3969
  * Maeda & Moriya (2022) Maeda, K., & Moriya, T. J. 2022, ApJ, 927, 25, doi: 10.3847/1538-4357/ac4672
  * Mahy et al. (2022) Mahy, L., Sana, H., Shenar, T., et al. 2022, A&A, 664, A159, doi: 10.1051/0004-6361/202243147
  * Margutti et al. (2014) Margutti, R., Milisavljevic, D., Soderberg, A. M., et al. 2014, ApJ, 780, 21, doi: 10.1088/0004-637X/780/1/21
  * Matsumoto & Metzger (2022a) Matsumoto, T., & Metzger, B. D. 2022a, ApJ, 936, 114, doi: 10.3847/1538-4357/ac892c
  * Matsumoto & Metzger (2022b) —. 2022b, ApJ, 938, 5, doi: 10.3847/1538-4357/ac6269
  * Matsumoto et al. (2016) Matsumoto, T., Nakauchi, D., Ioka, K., & Nakamura, T. 2016, ApJ, 823, 83, doi: 10.3847/0004-637X/823/2/83
  * Mauerhan et al. (2013) Mauerhan, J. C., Smith, N., Filippenko, A. V., et al. 2013, MNRAS, 430, 1801, doi: 10.1093/mnras/stt009
  * Maund et al. (2016) Maund, J. R., Pastorello, A., Mattila, S., Itagaki, K., & Boles, T. 2016, ApJ, 833, 128, doi: 10.3847/1538-4357/833/2/128
  * Mcley & Soker (2014) Mcley, L., & Soker, N. 2014, MNRAS, 445, 2492, doi: 10.1093/mnras/stu1952
  * Metzger (2012) Metzger, B. D. 2012, MNRAS, 419, 827, doi: 10.1111/j.1365-2966.2011.19747.x
  * Metzger et al. (2015) Metzger, B. D., Margalit, B., Kasen, D., & Quataert, E. 2015, MNRAS, 454, 3311, doi: 10.1093/mnras/stv2224
  * Moriya et al. (2011) Moriya, T., Tominaga, N., Blinnikov, S. I., Baklanov, P. V., & Sorokina, E. I. 2011, MNRAS, 415, 199, doi: 10.1111/j.1365-2966.2011.18689.x
  * Moriya (2014) Moriya, T. J. 2014, A&A, 564, A83, doi: 10.1051/0004-6361/201322992
  * Moriya (2015) —. 2015, ApJ, 803, L26, doi: 10.1088/2041-8205/803/2/L26
  * Moriya et al. (2018a) Moriya, T. J., Nicholl, M., & Guillochon, J. 2018a, ApJ, 867, 113, doi: 10.3847/1538-4357/aae53d
  * Moriya et al. (2018b) Moriya, T. J., Sorokina, E. I., & Chevalier, R. A. 2018b, Space Sci. Rev., 214, 59, doi: 10.1007/s11214-018-0493-6
  * Morozova et al. (2017) Morozova, V., Piro, A. L., & Valenti, S. 2017, ApJ, 838, 28, doi: 10.3847/1538-4357/aa6251
  * Morozova et al. (2018) —. 2018, ApJ, 858, 15, doi: 10.3847/1538-4357/aab9a6
  * Nakano et al. (2006) Nakano, S., Itagaki, K., Puckett, T., & Gorelli, R. 2006, Central Bureau Electronic Telegrams, 666, 1
  * Narayan et al. (2017) Narayan, R., SaÌ§dowski, A., & Soria, R. 2017, MNRAS, 469, 2997, doi: 10.1093/mnras/stx1027
  * Narayan & Yi (1994) Narayan, R., & Yi, I. 1994, ApJ, 428, L13, doi: 10.1086/187381
  * Nicholl (2021) Nicholl, M. 2021, Astronomy and Geophysics, 62, 5.34, doi: 10.1093/astrogeo/atab092
  * Nyholm et al. (2020) Nyholm, A., Sollerman, J., Tartaglia, L., et al. 2020, A&A, 637, A73, doi: 10.1051/0004-6361/201936097
  * Ofek et al. (2013) Ofek, E. O., Sullivan, M., Cenko, S. B., et al. 2013, Nature, 494, 65, doi: 10.1038/nature11877
  * Ofek et al. (2014a) Ofek, E. O., Sullivan, M., Shaviv, N. J., et al. 2014a, ApJ, 789, 104, doi: 10.1088/0004-637X/789/2/104
  * Ofek et al. (2014b) Ofek, E. O., Arcavi, I., Tal, D., et al. 2014b, ApJ, 788, 154, doi: 10.1088/0004-637X/788/2/154
  * Ohsuga et al. (2005) Ohsuga, K., Mori, M., Nakamoto, T., & Mineshige, S. 2005, ApJ, 628, 368, doi: 10.1086/430728
  * Ouchi & Maeda (2019) Ouchi, R., & Maeda, K. 2019, ApJ, 877, 92, doi: 10.3847/1538-4357/ab1a37
  * Pastorello et al. (2007) Pastorello, A., Smartt, S. J., Mattila, S., et al. 2007, Nature, 447, 829, doi: 10.1038/nature05825
  * Pastorello et al. (2008) Pastorello, A., Mattila, S., Zampieri, L., et al. 2008, MNRAS, 389, 113, doi: 10.1111/j.1365-2966.2008.13602.x
  * Pastorello et al. (2013) Pastorello, A., Cappellaro, E., Inserra, C., et al. 2013, ApJ, 767, 1, doi: 10.1088/0004-637X/767/1/1
  * Peters (1964) Peters, P. C. 1964, Physical Review, 136, 1224, doi: 10.1103/PhysRev.136.B1224
  * Podsiadlowski et al. (1992) Podsiadlowski, P., Joss, P. C., & Hsu, J. J. L. 1992, ApJ, 391, 246, doi: 10.1086/171341
  * Popov (1993) Popov, D. V. 1993, ApJ, 414, 712, doi: 10.1086/173117
  * Quataert et al. (2016) Quataert, E., Fernández, R., Kasen, D., Klion, H., & Paxton, B. 2016, MNRAS, 458, 1214, doi: 10.1093/mnras/stw365
  * Remillard & McClintock (2006) Remillard, R. A., & McClintock, J. E. 2006, ARA&A, 44, 49, doi: 10.1146/annurev.astro.44.051905.092532
  * Renzo et al. (2019) Renzo, M., Zapartas, E., de Mink, S. E., et al. 2019, A&A, 624, A66, doi: 10.1051/0004-6361/201833297
  * Rodriguez et al. (2023) Rodriguez, A. C., Cendes, Y., El-Badry, K., & Berger, E. 2023, arXiv e-prints, arXiv:2311.05685, doi: 10.48550/arXiv.2311.05685
  * Rozwadowska et al. (2021) Rozwadowska, K., Vissani, F., & Cappellaro, E. 2021, New A, 83, 101498, doi: 10.1016/j.newast.2020.101498
  * Sadowski et al. (2014) Sadowski, A., Narayan, R., McKinney, J. C., & Tchekhovskoy, A. 2014, MNRAS, 439, 503, doi: 10.1093/mnras/stt2479
  * Sana et al. (2012) Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, 337, 444, doi: 10.1126/science.1223344
  * Schlegel (1990) Schlegel, E. M. 1990, MNRAS, 244, 269
  * Shapiro & Lightman (1976) Shapiro, S. L., & Lightman, A. P. 1976, ApJ, 204, 555, doi: 10.1086/154203
  * Shenar et al. (2022) Shenar, T., Sana, H., Mahy, L., et al. 2022, Nature Astronomy, 6, 1085, doi: 10.1038/s41550-022-01730-y
  * Shigeyama et al. (1990) Shigeyama, T., Nomoto, K., Tsujimoto, T., & Hashimoto, M.-A. 1990, ApJ, 361, L23, doi: 10.1086/185818
  * Shikauchi et al. (2022) Shikauchi, M., Tanikawa, A., & Kawanaka, N. 2022, ApJ, 928, 13, doi: 10.3847/1538-4357/ac5329
  * Shikauchi et al. (2023) Shikauchi, M., Tsuna, D., Tanikawa, A., & Kawanaka, N. 2023, ApJ, 953, 52, doi: 10.3847/1538-4357/acd752
  * Shiode & Quataert (2014) Shiode, J. H., & Quataert, E. 2014, ApJ, 780, 96, doi: 10.1088/0004-637X/780/1/96
  * Smith (2014) Smith, N. 2014, ARA&A, 52, 487, doi: 10.1146/annurev-astro-081913-040025
  * Smith & Arnett (2014) Smith, N., & Arnett, W. D. 2014, ApJ, 785, 82, doi: 10.1088/0004-637X/785/2/82
  * Smith et al. (2011) Smith, N., Li, W., Filippenko, A. V., & Chornock, R. 2011, MNRAS, 412, 1522, doi: 10.1111/j.1365-2966.2011.17229.x
  * Soker (2013) Soker, N. 2013, arXiv e-prints, arXiv:1302.5037, doi: 10.48550/arXiv.1302.5037
  * Soker (2021) —. 2021, ApJ, 906, 1, doi: 10.3847/1538-4357/abca8f
  * Soker (2022) —. 2022, Research in Astronomy and Astrophysics, 22, 122003, doi: 10.1088/1674-4527/ac9782
  * Strotjohann et al. (2021) Strotjohann, N. L., Ofek, E. O., Gal-Yam, A., et al. 2021, ApJ, 907, 99, doi: 10.3847/1538-4357/abd032
  * Strotjohann et al. (2024) —. 2024, ApJ, 960, 72, doi: 10.3847/1538-4357/ad06b6
  * Sun et al. (2020) Sun, N.-C., Maund, J. R., Hirai, R., Crowther, P. A., & Podsiadlowski, P. 2020, MNRAS, 491, 6000, doi: 10.1093/mnras/stz3431
  * Taddia et al. (2013) Taddia, F., Stritzinger, M. D., Sollerman, J., et al. 2013, A&A, 555, A10, doi: 10.1051/0004-6361/201321180
  * Takei et al. (2023) Takei, Y., Tsuna, D., Ko, T., & Shigeyama, T. 2023, arXiv e-prints, arXiv:2308.10785, doi: 10.48550/arXiv.2308.10785
  * Tanikawa et al. (2023) Tanikawa, A., Hattori, K., Kawanaka, N., et al. 2023, ApJ, 946, 79, doi: 10.3847/1538-4357/acbf36
  * Tsang et al. (2022) Tsang, B. T. H., Kasen, D., & Bildsten, L. 2022, ApJ, 936, 28, doi: 10.3847/1538-4357/ac83bc
  * Tsebrenko & Soker (2013) Tsebrenko, D., & Soker, N. 2013, ApJ, 777, L35, doi: 10.1088/2041-8205/777/2/L35
  * Tsuna & Kawanaka (2019) Tsuna, D., & Kawanaka, N. 2019, MNRAS, 488, 2099, doi: 10.1093/mnras/stz1809
  * Tsuna et al. (2023a) Tsuna, D., Murase, K., & Moriya, T. J. 2023a, ApJ, 952, 115, doi: 10.3847/1538-4357/acdb71
  * Tsuna et al. (2023b) Tsuna, D., Takei, Y., & Shigeyama, T. 2023b, ApJ, 945, 104, doi: 10.3847/1538-4357/acbbc6
  * Wheeler et al. (2015) Wheeler, J. C., Johnson, V., & Clocchiatti, A. 2015, MNRAS, 450, 1295, doi: 10.1093/mnras/stv650
  * Woosley (2010) Woosley, S. E. 2010, ApJ, 719, L204, doi: 10.1088/2041-8205/719/2/L204
  * Woosley (2019) —. 2019, ApJ, 878, 49, doi: 10.3847/1538-4357/ab1b41
  * Woosley et al. (2002) Woosley, S. E., Heger, A., & Weaver, T. A. 2002, Reviews of Modern Physics, 74, 1015, doi: 10.1103/RevModPhys.74.1015
  * Wu & Fuller (2021) Wu, S., & Fuller, J. 2021, ApJ, 906, 3, doi: 10.3847/1538-4357/abc87c
  * Wu & Fuller (2022a) Wu, S. C., & Fuller, J. 2022a, ApJ, 930, 119, doi: 10.3847/1538-4357/ac660c
  * Wu & Fuller (2022b) —. 2022b, ApJ, 940, L27, doi: 10.3847/2041-8213/ac9b3d
  * Yaron et al. (2017) Yaron, O., Perley, D. A., Gal-Yam, A., et al. 2017, Nature Physics, 13, 510, doi: 10.1038/nphys4025
  * Yoon & Cantiello (2010) Yoon, S.-C., & Cantiello, M. 2010, ApJ, 717, L62, doi: 10.1088/2041-8205/717/1/L62
  * Yoon et al. (2010) Yoon, S. C., Woosley, S. E., & Langer, N. 2010, ApJ, 725, 940, doi: 10.1088/0004-637X/725/1/940
  * Yuan & Narayan (2014) Yuan, F., & Narayan, R. 2014, ARA&A, 52, 529, doi: 10.1146/annurev-astro-082812-141003
  * Zapartas et al. (2017) Zapartas, E., de Mink, S. E., Van Dyk, S. D., et al. 2017, ApJ, 842, 125, doi: 10.3847/1538-4357/aa7467