# Supernova Precursor Emission and the Origin of Pre-Explosion Stellar Mass-
Loss

Tatsuya Matsumoto  Department of Physics and Columbia Astrophysics Laboratory,
Columbia University, Pupin Hall, New York, NY 10027, USA  Brian D. Metzger
Department of Physics and Columbia Astrophysics Laboratory, Columbia
University, Pupin Hall, New York, NY 10027, USA  Center for Computational
Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA

###### Abstract

A growing number of core collapse supernovae (SNe) which show evidence for
interaction with dense circumstellar material (CSM) are accompanied by
“precursor” optical emission rising weeks to months prior to the explosion.
The precursor luminosities greatly exceed the Eddington limit of the
progenitor star, implying they are accompanied by substantial mass-loss. Here,
we present a semi-analytic model for SN precursor light curves which we apply
to constrain the properties and mechanisms of the pre-explosion mass-loss. We
explore two limiting mass-loss scenarios: (1) an “eruption” arising from shock
break-out following impulsive energy deposition below the stellar surface; (2)
a steady “wind” due to sustained heating of the progenitor envelope . The
eruption model, which resembles a scaled-down version of Type IIP SNe, can
explain the luminosities and timescales of well-sampled precursors, for ejecta
masses $\sim 0.1-1\,{M}_{\odot}$ and velocities $\sim 100-1000\,\rm
km\,s^{-1}$. By contrast, the steady-wind scenario cannot explain the highest
precursor luminosities $\gtrsim 10^{41}\,\rm erg\,s^{-1}$, under the
constraint that the total ejecta mass not exceed the entire progenitor mass
(though the less-luminous SN 2020tlf precursor can be explained by a mass-loss
rate $\sim 1\,{M}_{\odot}\,\rm yr^{-1}$). However, shock interaction between
the wind and pre-existing (earlier ejected) CSM may boost its radiative
efficiency and mitigate this constraint. In both eruption and wind scenarios
the precursor ejecta forms compact ($\lesssim 10^{15}$ cm) optically-thick CSM
at the time of core collapse; though only directly observable via rapid post-
explosion spectroscopy ($\lesssim$ few days before being overtaken by the SN
ejecta), this material can boost the SN luminosity via shock interaction.

XXX

##  1 Introduction

A fraction of massive stars undergo strongly enhanced mass-loss near the very
ends of their lives, forming a dense circumstellar medium (CSM) around
themselves (e.g., Smith 2014). When the stars explode as supernovae (SNe), the
CSM manifests through narrow emission lines in the optical spectra generated
by the as-yet-unshocked slowly expanding CSM, photoionized by the SN light
(e.g., SN Type IIn, Ibn or Icn, depending on whether the spectrum is hydrogen-
rich, hydrogen-poor but helium-rich, or both hydrogen- and helium-poor,
respectively; Schlegel 1990; Filippenko 1997; Foley et al. 2007; Pastorello et
al. 2008; Nyholm et al. 2020; Gal-Yam et al. 2022; Fraser et al. 2021). The
related technique of “flash spectroscopy” (e.g., Gal-Yam et al. 2014; Khazov
et al. 2016) demonstrates that the progenitor’s mass-loss rate is elevated
just before core collapse, among a significant fraction of even nominally CSM-
free SNe (e.g., Yaron et al. 2017; Bruch et al. 2021).

The light curves of some of the most luminous SNe are powered at least in part
by shock interaction between the SN ejecta and dense CSM released from the
progenitor star in the days to weeks to years prior to its terminal core
collapse (e.g., Smith & McCray 2007; Chevalier & Irwin 2011; Ginzburg &
Balberg 2012; Svirski et al. 2012; McDowell et al. 2018; Suzuki et al. 2019).
At typically larger radii and lower densities, CSM shock interaction can also
produce thermal and non-thermal X-ray and radio emission (e.g., Chandra et al.
2012; Margutti et al. 2014; Dwarkadas et al. 2016; Chakraborti et al. 2016;
Margutti et al. 2017; Chiba et al. 2020).

If the pre-SN stellar mass-loss is modeled as a steady wind, the inferred
mass-loss rates in Type IIn SNe, $\dot{M}\sim
10^{-4}-0.1M_{\odot}\mathrm{\,yr^{-1}}$ (Kiewe et al., 2012; Moriya et al.,
2014), are significantly larger than can be explained by line-driven winds
$\dot{M}\lesssim 10^{-5}M_{\odot}\mathrm{\,yr^{-1}}$ (Vink et al., 2001;
Smith, 2014). Instead, several alternative physical processes have been
proposed to give rise to enhanced mass-loss just prior to core collapse. Mass-
loss can occur due to intense heating of the stellar envelope by damping of
waves excited by vigorous convection in the stellar core (e.g., Quataert &
Shiode 2012; Shiode & Quataert 2014; Fuller 2017; Fuller & Ro 2018; Leung &
Fuller 2020; Wu & Fuller 2022). In this scenario, the timing of the mass-loss
should correspond to late stages of nuclear burning, which occurs on
timescales of days to weeks prior to explosion for silicon burning; to several
months for neon burning; to several years for oxygen burning (e.g., Woosley et
al. 2002). Another mechanism for generating pre-SN mass-loss invokes sudden
energy release deep inside the star due to instabilities associated with late-
stages of nuclear shell burning (e.g., Meakin & Arnett 2007; Smith & Arnett
2014; Fields & Couch 2021; Varma & Müller 2021; Yoshida et al. 2021). Mass-
loss due to interaction with a close binary companion may also play a role in
some events (e.g., Chevalier 2012; Sun et al. 2020). In very massive rotating
stars which experience efficient mixing, centrifugally-induced mass-loss may
accompany contraction during the star’s final burning stages (e.g., Aguilera-
Dena et al. 2018).

“Precursor” outbursts prior to the main SN explosion (e.g., Ofek et al. 2014)
offer a clue to distinguish these various mass-loss mechanisms. Several
precursor events accompanying interacting SNe have been observed in recent
years typically weeks to months prior to the main explosion, the most well-
studied of which accompanied SN 2009ip (Fraser et al., 2013; Mauerhan et al.,
2013; Pastorello et al., 2013; Prieto et al., 2013; Graham et al., 2014;
Margutti et al., 2014; Mauerhan et al., 2014; Levesque et al., 2014; Martin et
al., 2015; Graham et al., 2017; Reilly et al., 2017; Smith et al., 2022),
2010mc (Ofek et al., 2013), 2015bh (Elias-Rosa et al., 2016; Ofek et al.,
2016; Thöne et al., 2017; Jencson et al., 2022), 2016bhu (Pastorello et al.,
2018), LSQ13zm (Tartaglia et al., 2016), and 2020tlf (Jacobson-Galán et al.,
2022) as well as the implication of pre-SN activity in SN 2010bt (Elias-Rosa
et al., 2018), 2013gc (Reguitti et al., 2019), 2018cnf (Pastorello et al.,
2019), 2019zrk (Fransson et al., 2022), 2021foa (Reguitti et al., 2022), and
19 events compiled in Strotjohann et al. (2021). Efforts to monitor a large
number of massive stars in nearby galaxies to determine which explode as SNe
(Kochanek et al., 2008) and systematic analyses of interacting SNe (Ofek et
al., 2014; Bilinski et al., 2015; Strotjohann et al., 2021), rule out luminous
precursors accompanying all SNe (Kochanek et al., 2017; Johnson et al., 2018).
However, Strotjohann et al. (2021) found that $\simeq 25\,\%$ of Type IIn SNe
(themselves accounting for $\simeq 10\,\%$ of core collapse SNe; e.g., Perley
et al. 2020) exhibit precursors brighter than $-13$ mag (or luminosity
$\gtrsim 5\times 10^{40}\,\mathrm{erg\,s^{-1}}$) three months before the SN
explosion.

Figure 1 and Table 1 summarize the bolometric light curves and other
observable properties of a set of well-sampled SN precursors. Most of these
SNe are classified as Type II, so we focus in this work on precursors from
hydrogen-rich stars (though precursor outbursts have also been observed in
some hydrogen-poor SNe; e.g., Foley et al. 2007; Pastorello et al.
2008).111Some massive stars also generate bright eruptions$-$so-called “giant
eruptions” in luminous blue variable phase or “SN imposters” (e.g., Pastorello
et al. 2010; Pastorello & Fraser 2019), which are potentially related to
instabilities in massive star envelopes (e.g., Humphreys & Davidson 1994;
Owocki 2015; Jiang et al. 2018) and accompanied by mass-loss (e.g., Gal-Yam et
al., 2007; Gal-Yam & Leonard, 2009), but typically are not coincident with the
terminal core collapse event. Here we focus on SN precursors which precede the
terminal explosion by at most a few years and thus are likely to be causally
connected to late-stages of nuclear burning. The luminosities of the observed
precursors $\sim 10^{40}-10^{41}$ erg s-1 are typically well above the
Eddington limit for typical SN progenitor stars of mass $\simeq
10-20\,{M}_{\odot}$,222The progenitors of SN 2009ip and 2015bh may be more
massive $\gtrsim 35-50\,{M}_{\odot}$ (Smith et al., 2010; Foley et al., 2011;
Boian & Groh, 2018). indicating that this emission phase is accompanied by
substantial mass-loss, which should then contribute to CSM interaction with
the supernova light or ejecta in the post-explosion phase.

In this paper, we present a model for the light curves of SN precursors from
hydrogen-rich stars, which we apply to the observed precursor sample in order
to constrain the pre-SN mass-loss phase. With the exception of radiation
hydrodynamical simulations by Dessart et al. (2010), few previous theoretical
works have attempted to model the precursor emission phase. Instead, most
constraints on the mass and radial distribution of the CSM around SN
progenitors have come from the post-explosion interaction (e.g., Smith &
McCray 2007; Chevalier & Irwin 2011; Chatzopoulos et al. 2013; McDowell et al.
2018; Suzuki et al. 2019). Though our light curve model makes several
simplifying assumptions, these same features make it flexible and applicable
to a wide parameter space of potential pre-explosion mass-loss behavior. As we
shall discuss, we find that the bulk of the CSM generated during the observed
precursor outbursts will usually be located so close to the progenitor star at
the time of the explosion as to be quickly overtaken by the SN ejecta, making
this material challenging to directly observe by other means.

This paper is organized as follows. In Section 2 we introduce two idealized
models for the nature of the pre-SN mass-loss phase (dynamical “eruption”
versus steady “wind”) and describe our model for calculating the precursor
light curve from each case. In Section 3 we present the results of our light
curve calculations and interpret them analytically. We also assess the ability
of the eruption and wind models to explain the observed SN precursor events
and determine the ejecta parameters required in each case. We discuss the
resulting CSM from precursor events in Section 4 and finally summarize our
results and conclude in Section 5.

Figure 1: Sample of SN precursor light curves, showing the luminosity as a function of time relative to the onset of the SN (which we define by the luminosity rising abruptly by a factor of $\gtrsim 10$). Shaded regions show the characteristic range of timescales before core collapse corresponding to core neon and silicon burning in $13-20\,{M}_{\odot}$ stars (Woosley et al., 2002). References to the light curve data are given in Table 1. Table 1: Observed Properties of SN Precursor Emission Event | $L_{\rm pre}^{(a)}$ | $t_{\rm pre}^{(b)}$ | $E_{\rm pre}^{(c)}$ | $v_{\rm obs}$ | Ref.
---|---|---|---|---|---
| [erg/s] | [d] | [erg] | [$10^{3}$ km/s] |
SN 2009ip | $3.6\times 10^{41}$ | 44¶ |  $1.4\times 10^{48}$¶ | 0.8-1.4, 8-9, 14-15‡ | 1,2,3
SN 2010mc | $1.6\times 10^{41}$ | 31 | $4.4\times 10^{47}$ | 1-3 (6 d)† | 4
SN 2015bh | $9.3\times 10^{40}$ | 95¶ |  $7.6\times 10^{47}$¶ | 0.6-1, 0.9-1.5, 2.6-6‡ | 5
SN 2016bdu | $1.2\times 10^{41}$ | 97¶ |  $1.0\times 10^{48}$¶ | 0.4 ($\simeq 10$ d)† | 6
SN 2020tlf | $8.8\times 10^{39}$ | 127 | $9.7\times 10^{46}$ | 0.05-0.2 (10 d)† | 7
(a) Average luminosity of precursor emission, defined as $L_{\rm pre}=E_{\rm
pre}/t_{\rm pre}$.
(b) Duration from the first precursor detection to the time of the SN
explosion.
(c) Total radiated energy of the precursor emission.
¶Lower limit due to the lack of detection or flux upper limit.
‡Obtained via spectroscopy during the precursor emission itself. The H$\alpha$
line
profile is fitted by multiple velocity components.
†Obtained via flash-spectroscopy $t_{\rm flash}$ (shown in parentheses) after
the SN
explosion. Note that these velocities are sometimes too low to be associated
with
mass ejection responsible for the precursor emission because the latter would
have
been overtaken by the SN ejecta by the time of the flash-spectroscopy ($t_{\rm
flash}$).
Ref. 3: Mauerhan et al. (2013), 2: Pastorello et al. (2013),
3: Margutti et al. (2014), 4: Ofek et al. (2013), 5: Elias-Rosa et al. (2016),
6: Pastorello et al. (2018), 7: Jacobson-Galán et al. (2022).

##  2 Precursor Emission Model

We calculate SN precursor light curves largely following the semi-analytical
model developed in Matsumoto & Metzger (2022); Metzger et al. (2021). The
general approach is as follows: the pre-SN mass-loss is divided into multiple
ejecta shells ordered based on their velocity, and for each shell we calculate
its thermal evolution and resulting emission by means of a one-zone model. The
total light curve is then obtained by summing the luminosity contributions
from each shell. For simplicity we assume the ejecta to be spherically
symmetric, even though in some mass-loss scenarios it could possess a non-
spherical (e.g., equatorial disk-like) geometry.

A key input to the model is the velocity distribution of the pre-SN ejecta. We
explore two physically-motivated scenarios for its form, which bracket
extremes along a continuum of possible behaviors.

  1. 1.

An “eruption,” in which mass-loss occurs in a single event on a timescale
comparable or less than the dynamical time at the ejection radius. Physically,
a temporally-concentrated injection of energy deep inside the star generates a
shock wave, which propagates radially outwards and accelerates the stellar
envelope, unbinding a portion of its mass (Dessart et al., 2010; Kuriyama &
Shigeyama, 2020; Linial et al., 2021; Ko et al., 2021). Although we remain
indifferent to the origin of the sudden energy injection, one possible
physical realization would be a dynamical instability associated with unstable
nuclear shell burning.

  2. 2.

A continuous “wind,” in which mass-loss occurs from the star at a roughly
constant rate, over timescales much longer than the dynamical time.
Physically, this is expected to occur when the stellar envelope is heated well
above the Eddington luminosity at a roughly constant rate (e.g., Smith &
Owocki 2006; Quataert et al. 2016). This scenario may be approximately
realized in wave-heating scenarios (e.g., Quataert & Shiode 2012; Fuller
2017).

The following subsections describe the eruption and wind scenarios
individually. Throughout this work we assume that the precursor emission
arises directly from a single mass-loss event or mass-loss phase prior to the
terminal explosion; however, we caution that some events may arise from more
complicated circumstances (e.g., multiple episodes of mass-loss giving rise to
collisions between ejecta shells; dim SNe followed by bright CSM interaction;
or binary star mergers) in particular for SN 2009ip (e.g., Pastorello et al.
2013; Soker & Kashi 2013; Smith et al. 2014).

###  2.1 Eruption Scenario

In the eruption scenario, energy is injected suddenly into the stellar
envelope, driving a shock wave towards its surface (e.g., Dessart et al.
2010). As a result, we assume the ejecta achieve a homologous density profile
$\rho(v=r/t)$, defined such that $M_{\rm
ej}(>v)=4\pi\int_{v}^{\infty}\rho(v)r^{2}dr$ is the ejecta mass above a given
velocity $v$.

Depending on the magnitude of the injected energy, the ejecta can be
accelerated to an arbitrarily high velocity as long as the latter exceeds the
escape speed at the radius $R_{0}$ matter is ejected (e.g., Linial et al.
2021),

$\displaystyle v_{\rm esc}$ $\displaystyle=\sqrt{\frac{2GM_{\star}}{R_{0}}}$
(1) $\displaystyle\simeq 200{\,\rm
km\,s^{-1}\,}\biggl{(}\frac{M_{\star}}{10\,{M}_{\odot}}\biggl{)}^{1/2}\biggl{(}\frac{R_{0}}{10^{2}\,{R}_{\odot}}\biggl{)}^{-1/2}\
,$ (2)

where $G$ is the gravitational constant and $M_{\star}$ is the progenitor
mass. Spectroscopic observations of SN precursors, in the few cases available,
indicate high ejecta speeds $\sim 10^{2}-10^{3}$ km s-1 (Table 1). This
suggests the deposited energy being comparable to the stellar binding energy
if released at radii $R_{0}\sim 1-100\,{R}_{\odot}$ deep below the surface
$R_{\star}\sim 10^{3}\,{R}_{\odot}$ of a RSG progenitor (or with energy
greatly exceeding the local binding energy if deposited closer to the
surface).

We now describe our method to calculate the emission from a single shell of
mass $M$ and velocity $v$ ejected at $t=0$ from radius $R_{0}$. The radial
distance of the shell from the progenitor’s center is given by $R=v\cdot
t+R_{0}$ and its volume is given by $V=4\pi R^{2}\Delta R$, where $\Delta R$
is the width of the shell. For a homologously expanding ejecta, the width is
given by $\Delta R=dv\cdot t+\Delta R_{0}$, where $dv$ is the velocity
difference from the next shell and $\Delta R_{0}$ is the initial width of the
shell. The initial internal energy of the shell is assumed to equal its
kinetic energy $E_{0}=Mv^{2}/2$.333In detail, for this initial condition, the
shell will accelerate after the ejection by $PdV$ work within several
dynamical times $\sim R_{0}/v$; however, since the increase of the velocity is
by at most a factor of $\sqrt{2}\simeq 1.4$, we neglect this acceleration
phase and assume the shells expand at constant velocity from $t=0$. The
internal energy is comprised of ideal gas, radiation, and ionization energy:

$\displaystyle E$ $\displaystyle=\frac{3}{2}(1+\bar{x})Nk_{\rm
B}T+aT^{4}V+\sum_{i}NA_{i}x_{i}\varepsilon_{i}\ ,$ (3)

where $\bar{x}=\sum_{i}A_{i}x_{i}$, $A_{i}$, $x_{i}$, and $\varepsilon_{i}$
are the mean ionization degree, abundance fraction, degree of ionization, and
ionization energy of species $i$, respectively. We calculate $\bar{x}$ and
$x_{i}$ by solving the Saha equation taking into account singly-ionized
hydrogen and helium. The other quantities are the total number of nuclei $N$,
the Boltzmann constant $k_{\rm B}$, temperature $T$, and radiation constant
$a$.

The thermal evolution of the ejecta is described by the first law of
thermodynamics:

$\displaystyle\frac{dE}{dt}=-(\gamma_{3}-1)\frac{E}{V}\frac{dV}{dt}-L\ ,$ (4)

where $\gamma_{3}$ and $L$ are the adiabatic index and radiated luminosity,
respectively. In the adiabatic loss term, we have used the fact the pressure
is given by $P=(\gamma_{3}-1)E/V$. The adiabatic index is calculated from the
density, temperature, and the ionization state of the shell as in Matsumoto &
Metzger (2022); Kasen & Ramirez-Ruiz (2010). The radiative loss term is
approximated by the photon diffusion luminosity

$\displaystyle L=\frac{E_{\rm rad}}{S(t)t_{\rm d}+t_{\rm lc}}\ ,$ (5)

where $E_{\rm rad}=aT^{4}V$ is the radiation’s internal energy, $t_{\rm d}$ is
the photon diffusion time, and $S(t)$ is a suppression factor discussed below.
The diffusion time is that over which photons escape radially through the
ejecta,

$\displaystyle t_{\rm
d}=\frac{R\tau}{c}\,\,\,\text{and}\,\,\,\tau=\int_{R}\kappa\rho dr\ ,$ (6)

where $c$ is the speed of light and $\kappa(\rho,T)$ is Rossland-mean opacity,
which we approximate using the analytic expression provided in Matsumoto &
Metzger (2022) for solar metallicity composition material. The dominant form
of opacity near peak light is electron scattering (under conditions of partial
or full ionization), though Kramer’s opacity can become relevant at high
densities and low temperatures. The suppression factor,

$\displaystyle S(t)\equiv\frac{e^{t_{\rm d}/t}-1}{t_{\rm d}/t}\ ,$ (7)

acts to reduce the radiative losses exponentially for $t\ll t_{\rm d}$.
Without this correction, the luminosities of all radiation-pressure-dominated
shells reach a value $L\propto R_{0}v/\kappa$ starting immediately after their
ejection and contribute almost equally (up to a factor of $v$) regardless of
their optical depth; this behavior is unphysical, however, because at
$t\lesssim t_{\rm d}$ most photons are still trapped within ejecta and only
the tiny fraction in the “diffusion tail” can escape (see also Piro & Nakar,
2013). The light crossing time $t_{\rm lc}=R/c$ in the denominator of Eq. (5)
limits the photon escape timescale from the system at late times.

The mass profile of the ejecta depends on the details of the ejection process.
As discussed further below, the process of shock-breakout motivates a power-
law profile (Nakar & Sari, 2010),

$\displaystyle M_{\rm ej}(>v)$ $\displaystyle=M_{\rm ej}(v/v_{\rm
ej})^{-\beta}\ ,\,\,v>v_{\rm ej}\ ,$ (8)

where $M_{\rm ej}$ is the total ejecta mass and $v_{\rm ej}$ is the minimum
velocity. Assuming the stellar envelope can be described as a polytrope of
index $n$, and that the energy injection occurs close to the stellar surface,
then we can set the initial radius $R_{0}\sim R_{\star}$ for all shells and
the process of shock break-out will impart a self-similar velocity profile of
the form $v\propto\rho^{-\mu}$ (Gandel’Man & Frank-Kamenetskii, 1956; Sakurai,
1960), where $\mu\simeq 0.22$ (for $n=3/2$) and $\mu=0.19$ (for $n=3$). Since
the density profile and external mass in this scenario obey $\rho\propto
x^{n}$ and $M(>r)\propto\rho xR_{\star}^{2}$, where $x=R_{\star}-r$ is the
depth measured from the surface, the power-law index entering Eq. (8) in this
scenario becomes $\beta=(n+1)/{\mu n}\simeq 7.6$ for $n=3/2$ and 7.0 for
$n=3$. The shock break-out solution also gives a relation between the velocity
and initial depth of each shell $v=v_{\rm ej}(x/x_{\rm ej})^{-\mu n}$, where
the depth of the innermost slowest shell ($v=v_{\rm ej}$) obeys $x_{\rm
ej}/R_{\star}\sim(M_{\rm ej}/M_{\star})^{\frac{1}{n+1}}$. The initial width of
each shell is likewise given by

$\displaystyle\Delta R_{0}(v)=\biggl{|}\frac{dx}{dv}\biggl{|}dv=\frac{x_{\rm
ej}}{\mu nv_{\rm ej}}\biggl{(}\frac{v}{v_{\rm ej}}\biggl{)}^{-\frac{\mu
n+1}{\mu n}}dv\ .$ (9)

Note that above prescription holds only for energy injected into layers of the
star close to the surface ($R_{0}\sim R_{\star}$ and $M_{\rm ej}\ll
M_{\star}$). Nevertheless, in what follows we shall apply $R_{0}=R_{\star}$
also to the eruptions with greater ejecta masses, seeded by energy deposition
at deeper layer $\ll R_{\star}$. Realistically, after the shock passage, each
mass shell is quickly imparted comparable internal and kinetic energies, and
hence the initial radius $R_{0}$ should be smaller than $R_{\star}$ and
different for each shell. While this complicated hydrodynamics should be
studied by numerical approaches with more specific stellar model to set more
accurate initial condition, we assume for simplicity that all shells are
ejected at $t=0$ from $R_{\star}$ with equal internal and kinetic energies as
adopted for analytical light curve modeling of SNe (Arnett, 1980, 1982; Popov,
1993). Physically, heating near the surface will occur as the ejecta from
deeper layers collides and shocks with material closer to the surface.

In summary, the eruption scenario is described by four main parameters: total
ejecta mass $M_{\rm ej}$, initial radius $R_{0}$ (which we canonically set to
$R_{0}=R_{\star}$, motivated by the shock-breakout prescription), minimum
ejecta speed $v_{\rm ej}$ (which must exceed $v_{\rm esc}$; Eq. 2), and the
time of the eruption prior to the SN explosion, $t_{\rm erupt}$. The power-law
index of mass profile $\beta$ is also a free parameter, though we shall take
$\beta=7.6$ as fiducial (corresponding to RSG progenitors), motivated by above
discussion. Although the self-similar shock break-out solution described above
holds only near the stellar surface, we find that the light curve properties
are not sensitive to the precise value of $\beta$ for otherwise fixed values
of $M_{\rm ej}$ and $v_{\rm ej}$ (see Matsumoto & Metzger 2022; their Fig. 6).
We assume initial shell widths given by Eq. (9); however, the calculated light
curve properties are also not sensitive to this choice (assuming a constant
width for all shells, $\Delta R_{0}=\rm const$, gives a similar result).

For the assumed mass profile, the diffusion time is

$\displaystyle t_{\rm d}$
$\displaystyle\simeq\frac{\beta}{\beta+2}\frac{\kappa M(>v)}{4\pi cR}$
$\displaystyle\underset{\beta=7.6}{\simeq}{240\,\rm
d\,}\left(\frac{M(>v)}{{M}_{\odot}}\right)\left(\frac{R}{10^{3}R_{\odot}}\right)^{-1}\
,$ (10)

where the second equality assumes a constant electron scattering opacity
$\kappa=\kappa_{\rm es}\simeq 0.32\,\rm cm^{2}\,g^{-1}$ (hereafter adopted to
other analytic estimates unless otherwise specified). The total ejecta kinetic
energy is given by

$\displaystyle E_{\rm kin}$ $\displaystyle=\int_{v_{\rm
ej}}\frac{dM}{dv}\frac{v^{2}}{2}dv\simeq\frac{\beta}{2(\beta-2)}M_{\rm
ej}v_{\rm ej}^{2}$ $\displaystyle\underset{\beta=7.6}{\simeq}$ $\displaystyle
5.2\times 10^{47}{\,\rm erg\,}\biggl{(}\frac{M_{\rm
ej}}{{M}_{\odot}}\biggl{)}\biggl{(}\frac{v_{\rm ej}}{v_{\rm
esc}}\biggl{)}^{2}\biggl{(}\frac{M_{\star}}{10\,{M}_{\odot}}\biggl{)}\biggl{(}\frac{R_{0}}{10^{2}\,{R}_{\odot}}\biggl{)}^{-1}\
.$ (11)

Based on the physical requirement that the total energy radiated by SN
precursors $E_{\rm pre}\sim 10^{47}-10^{48}$ erg (Table 1) not exceed $E_{\rm
kin}$, we note that the minimum velocity must obey $v_{\rm ej}\gtrsim
340\,\mathrm{km\,s^{-1}}\,(E_{\rm pre}/10^{48}{\,\rm erg\,})^{1/2}(M_{\rm
ej}/{M}_{\odot})^{-1/2}$.

###  2.2 Wind Scenario

In the steady wind scenario, the precursor emission is generated by a mass
outflow from the star with an assumed constant mass-loss rate $\dot{M}$ and
velocity $v_{\rm w}$. We begin our calculation at the sonic radius $R_{\rm
s}$, where the thermal energy flux of the wind is comparable to its kinetic
power,

$\displaystyle\dot{E}_{\rm w}$ $\displaystyle=\frac{1}{2}\dot{M}v_{\rm
w}^{2}\simeq 1.2\times 10^{40}\,\mathrm{erg\,s^{-1}}$
$\displaystyle\times\biggl{(}\frac{\dot{M}}{M_{\odot}\mathrm{\,yr^{-1}}}\biggl{)}\biggl{(}\frac{v_{\rm
w}}{v_{\rm
esc}}\biggl{)}^{2}\biggl{(}\frac{M_{\star}}{10\,{M}_{\odot}}\biggl{)}\biggl{(}\frac{R_{\rm
s}}{10^{2}\,{R}_{\odot}}\biggl{)}^{-1}\ ,$ (12)

where we have scaled the wind velocity to the escape speed $v_{\rm esc}$ at
the sonic radius (Eq. 2, but replacing $R_{0}$ with $R_{\rm s}$).

To treat the problem within the same framework as the eruption case, the wind
is divided into shells of equal mass $M=\dot{M}\Delta t$, radius $R=v_{\rm
w}(t-t_{0})$, width $\Delta R=v_{\rm w}\Delta t$, and volume $V=4\pi
R^{2}\Delta R$, where $t_{0}$ is the time a given shell is released relative
to the start of the wind ($t=0$) and $\Delta t$ is the fixed time interval
separating the ejection of successive shells. The procedure to calculate the
light curve from each shell follows that of the eruption scenario, except that
the suppression factor (Eq. 7) is now given by

$\displaystyle S(t)=\frac{\left[e^{t_{\rm d}/(t-t_{0})}-1\right]}{(t-t_{0})}\
.$ (13)

While the diffusion timescale is calculated by Eq. (6), its estimate is given
by

$\displaystyle t_{\rm d}$ $\displaystyle\simeq\frac{\kappa\dot{M}}{4\pi
cv_{\rm w}}\simeq 32{\,\rm d\,}\biggl{(}\frac{\dot{M}}{{M}_{\odot}\,\rm
yr^{-1}}\biggl{)}\times$ $\displaystyle\biggl{(}\frac{v_{\rm w}}{v_{\rm
esc}}\biggl{)}^{-1}\biggl{(}\frac{M_{\star}}{10\,{M}_{\odot}}\biggl{)}^{-1/2}\biggl{(}\frac{R_{\rm
s}}{10^{2}\,{R}_{\odot}}\biggl{)}^{1/2}\ ,$ (14)

which follows by taking the density as $\rho=\dot{M}/4\pi r^{2}v_{\rm w}$ (for
a wind density profile $\rho\propto r^{-2}$, the diffusion time receives
roughly equal contributions from all decades in radius; e.g., Chevalier &
Irwin 2011).

In summary, the wind scenario is described by three parameters: wind mass-loss
rate $\dot{M}$, wind velocity $v_{\rm w}$ (equivalently, sonic point $R_{\rm
s}$ for a given progenitor mass $M_{\star}$), and the duration of the wind
prior to the SN explosion $t_{\rm w}$. Rather than define the latter as the
entire duration of mass-loss prior to the explosion, $t_{\rm active}$, we
define $t_{\rm w}$ as the more limited duration over which the luminosity has
reached its roughly constant value, i.e. $t_{\rm w}=t_{\rm active}-t_{\rm
rise},$ where $t_{\rm rise}$ is the light curve rise-time (see below).

What range of wind properties are expected physically? Quataert et al. (2016)
show that energy injection at some radius $R_{\rm in}$ below the stellar
surface at a super-Eddington rate $\dot{E}\gg L_{\rm Edd}$ drives a continuum
radiation pressure-driven outflow. For sufficiently high values of
$\dot{E}\gg\dot{E}_{\rm Q}$, most of the injected power is eventually
converted into the kinetic energy of the wind, i.e. $\dot{E}_{\rm
w}\simeq\dot{E}$, where the threshold power is given by

$\displaystyle\dot{E}_{\rm Q}$ $\displaystyle\equiv f^{5/2}\frac{M_{\rm
env}}{M_{\star}}\frac{v_{\rm esc}^{5}}{G}\simeq 2\times
10^{40}\,\mathrm{erg\,s^{-1}}$
$\displaystyle\times\left(\frac{f}{0.3}\right)^{5/2}\left(\frac{M_{\rm
env}}{10^{-2}M_{\star}}\right)\left(\frac{M_{\star}}{10\,{M}_{\odot}}\right)^{5/2}\left(\frac{R_{\rm
in}}{10^{2}\,{R}_{\odot}}\right)^{-5/2}.$ (15)

Here, $M_{\rm env}\lesssim 10^{-3}-10^{-2}\,M_{\star}$ is the mass of the
stellar envelope above the energy injection point up to the sonic radius and
$f\sim 0.1-1$ is a dimensionless parameter that depends on the envelope
structure (Quataert et al. 2016; their Fig. 11). Thus, we typically have
$\dot{E}_{\rm Q}\sim 10^{39}-10^{41}{\,\rm erg\,s^{-1}}\sim 1-100\,L_{\rm
Edd},$ where the Eddington luminosity

$\displaystyle L_{\rm Edd}$ $\displaystyle=\frac{4\pi GM_{\star}c}{\kappa_{\rm
es}}\simeq 1.6\times
10^{39}{\,\mathrm{erg\,s^{-1}}\,}\left(\frac{M_{\star}}{10\,{M}_{\odot}}\right).\
$ (16)

In the limit $\dot{E}\gg\dot{E}_{\rm Q}$, Quataert et al. (2016) show that the
terminal velocity and mass-loss rate of the wind are related according to
(their Eqs. 20, 26):

$\displaystyle v_{\rm w}$ $\displaystyle\simeq
1.7\left(\frac{M_{\star}}{M_{\rm env}}G\dot{E}\right)^{1/5}\simeq 430\,{\rm
km\,s^{-1}}$ $\displaystyle\times\left(\frac{M_{\rm
env}}{10^{-3}\,M_{\star}}\right)^{-1/5}\left(\frac{M_{\star}}{10\,{M}_{\odot}}\right)^{1/5}\left(\frac{\dot{E}}{100L_{\rm
Edd}}\right)^{1/5}\ ,$ (17) $\displaystyle\dot{M}$ $\displaystyle\simeq
0.15\,\frac{v_{\rm w}^{3}}{G}\frac{M_{\rm env}}{M_{\star}}$
$\displaystyle\simeq 0.96\,M_{\odot}{\rm\,yr^{-1}}\,\left(\frac{M_{\rm
env}}{10^{-3}\,M_{\star}}\right)\left(\frac{v_{\rm w}}{300\,{\rm
km\,s^{-1}}}\right)^{3}.$ (18)

The lower range of ejecta velocities inferred from SN precursor observations,
$v_{\rm w}\sim 100-500$ km s-1 (Table 1), can be obtained for $\dot{E}\sim
1-100\,L_{\rm Edd}\gtrsim\dot{E}_{\rm Q}$ and $M_{\rm env}\sim
10^{-4}-10^{-2}\,M_{\star}$, corresponding to range of $\dot{M}\sim
1-100\,M_{\odot}$ yr-1. On the other hand, the higher end of the observed
precursor velocity $\gtrsim 10^{3}\,\mathrm{km\,s^{-1}}$ would require both a
small envelope mass $M_{\rm env}\lesssim 10^{-5}\,M_{\star}$ and large
injection luminosity $\dot{E}\gtrsim 10^{3}\,L_{\rm Edd}$.

Although $v_{\rm w}$ represents the final velocity of the wind, Quataert et
al. (2016) found that the initial expansion rate of the overlying envelope
material, after the onset of heating, is significantly slower than $v_{\rm
w}$. This rising velocity inevitably leads to shock interaction between the
wind and the overlying envelope material. Additional heating from such shocks
may boost the transient luminosity compared to the constant-velocity wind
modeled here (see Sec. 3.2.1).

##  3 Results

We now summarize the light curve results, separately in the eruption and wind
scenarios.

###  3.1 Eruption Scenario

Figure 2 shows example light curves in the eruption scenario, calculated for:
$M_{\rm ej}=1\,{M}_{\odot}$, $M_{\star}=10\,{M}_{\odot}$,
$R_{0}=R_{\star}=10^{3}\,{R}_{\odot}$, and $v_{\rm
ej}=1000\,\mathrm{km\,s^{-1}}$. For these parameters, the ejecta is radiation-
pressure-dominated and the light curve is powered by the escape of the initial
internal energy. Hydrogen begins to recombine once the temperature decreases
to a value $T_{\rm ion}\sim 10^{4}\,\rm K$, and the resulting sharp drop in
opacity creates a well-defined recombination front. As the front recedes
through the ejecta shell, this shapes a characteristic plateau in the light
curve in the same way as in Type IIP SNe (Arnett, 1980; Popov, 1993; Kasen &
Woosley, 2009). The top panel shows the effective temperature, $T_{\rm
eff}=\left[L/(4\pi\sigma_{\rm SB}R_{\rm ph}^{2})\right]^{1/4}$, where
$\sigma_{\rm SB}$ is the Stephan-Boltzmann constant and the photosphere radius
$R_{\rm ph}$ is defined by $\tau=\int_{R_{\rm ph}}\kappa\rho dr=2/3$.

Just after the eruption, the effective temperature is high and the peak of the
assumed blackbody spectrum is in the UV range. We approximate the optical-
wavelength light curve (black curve) by subtracting the UV luminosity (defined
as radiation with frequency $\nu>10^{15}\,\rm Hz$) from the bolometric
luminosity (gray curve). Colored solid lines denote the single-shell light
curves for different velocities ($v=1000\,\mathrm{km\,s^{-1}}$ to
$1750\,\mathrm{km\,s^{-1}}$, corresponding to colors from purple to red). As a
result of the suppression factor (Eq. 7), low velocity shells do not
contribute significantly to the total luminosity at early times, until their
diffusion times become shorter than the expansion times.

In order to quantify the impact of new features of the light curve model
introduced in this paper, a dashed line show the calculation performed with
the original model of Matsumoto & Metzger (2022), which does not include the
early-time suppression effect (Eq. 7) and used a local estimate for the radial
optical depth rather than a full integral taken over the external shells. The
new calculation light curve exhibits a shallower decay, but its overall
luminosity and duration, do not change appreciably from the Matsumoto &
Metzger (2022) model.

Figure 2: Example SN precursor light curve calculation in the eruptive mass-
loss scenario. A solid black curve shows the optical light curve (bottom) and
evolution of effective temperature (top) for a fiducial case with ejecta mass
$M_{\rm ej}={M}_{\odot}$, initial radius
$R_{0}=R_{\star}=10^{3}\,{R}_{\odot}$, and minimum ejecta velocity $v_{\rm
ej}=1000\,\mathrm{km\,s^{-1}}$ for an assumed stellar mass
$M_{\star}=10\,{M}_{\odot}$. The optical luminosity is calculated by excluding
radiation of frequency $>10^{15}$ Hz from the bolometric light curve (gray
curve). Colored solid lines show the luminosity contributed by single shells
with velocity $v=1000\,\mathrm{km\,s^{-1}}$ to $1750\,\mathrm{km\,s^{-1}}$
(separated by $250\,\mathrm{km\,s^{-1}}$) and shell width of
$dv=2\,\mathrm{km\,s^{-1}}$. Dashed curves show an otherwise identical
calculation calculated following the original model of Matsumoto & Metzger
(2022), which neglects the early-time suppression factor (Eq. 7) and uses a
local estimate for the radial optical depth.

Due to the close similarity of the physical conditions in the precursor ejecta
to that of Type IIP SNe, the characteristic luminosity and duration of the
light curve can be estimated using the analytic expressions from Popov (1993):

$\displaystyle L_{\rm Popov}$ $\displaystyle=2.0\times 10^{41}{\,\rm
erg\,s^{-1}\,}\left(\frac{M_{\rm ej}}{{M}_{\odot}}\right)^{1/3}$
$\displaystyle\times\left(\frac{R_{0}}{10^{3}\,{R}_{\odot}}\right)^{2/3}\left(\frac{v_{\rm
ej}}{10^{3}\,\rm km\,s^{-1}}\right)^{5/3}\ ,$ (19) $\displaystyle t_{\rm
Popov}$ $\displaystyle=74{\,\rm d\,}\left(\frac{M_{\rm
ej}}{{M}_{\odot}}\right)^{1/3}\left(\frac{R_{0}}{10^{3}\,{R}_{\odot}}\right)^{1/6}\left(\frac{v_{\rm
ej}}{10^{3}\,\rm km\,s^{-1}}\right)^{-1/3}\ ,$ (20)

where we have approximated the mean ejecta velocity as $v_{\rm ej}$ and have
adopted a normalization calibrated from the radiative transfer simulations of
Sukhbold et al. (2016) (see also Blagorodnova et al. 2021).

Figure 3 shows light curves in the eruption scenario, calculated for different
variations of the ejecta parameters relative to the fiducial case in Fig. 2.
The resulting changes in the plateau emission properties roughly agree with
the analytic scaling relations in Eqs. (19), (20). For example, increasing the
ejecta mass increases the duration and luminosity of the plateau, while
increasing the minimum velocity increases the plateau luminosity but shortens
its duration.

For comparison in Fig. 3 we show the light curves of observed SN precursors
(Fig. 1). In order to roughly match the observations, we set the time of the
eruption prior to core collapse to $t_{\rm erupt}=140$, $100$, and $50$ days
for $v_{\rm ej}=200\,\mathrm{km\,s^{-1}}$ (black and yellow),
$500\,\mathrm{km\,s^{-1}}$ (blue), and $1000\,\mathrm{km\,s^{-1}}$ (red),
respectively. Before exploring the detailed model parameters required to fit
the observations, we first note that$-$broadly speaking$-$the luminosities
predicted in the outburst model, $L_{\rm pre}\sim 10^{41}\,\rm erg\,s^{-1}$,
match those of observed precursors. In detail, our outburst model predicts
light curves which decay monotonically following the initial peak, in conflict
with observed precursor emission (in particular, SN 2015bh and 2016bdu), which
instead rises for a few months leading up to the SN. However, this discrepancy
may at least in part arise from some of the simplifying assumptions of our
model, such as the velocity profile and initial shell radii; indeed, the
radiation hydrodynamic simulations of precursor outbursts by Dessart et al.
(2010) predict rising light curves.444Adopting the same ejecta parameters as
Dessart et al. (2010), our calculation predicts a plateau luminosity and
duration that agree with their numerical results to within a factor of a few.

Figure 3: Same as in Fig. 2 but showing light curves in the eruption scenario
for different ejecta parameters, and compared with observed SN precursors
(Fig. 1). Blue, black, and red curves show models with a common ejecta mass
$M_{\rm ej}={M}_{\odot}$ but different ejecta velocities $v_{\rm ej}=200$,
500, and $1000\,\mathrm{km\,s^{-1}}$, respectively. A yellow curve shows a
model with $v_{\rm ej}=200\,\mathrm{km\,s^{-1}}$ but larger $M_{\rm
ej}=3\,{M}_{\odot}$. The stellar mass $M_{\star}=10\,{M}_{\odot}$ and radius
$R_{0}=10^{3}\,{R}_{\odot}$ are fixed for all models. The time of the eruption
prior to the SN is set to $t_{\rm erupt}$ = 140 days (except for high-velocity
models shown with blue and red curve, for which $t_{\rm erupt}=100$ and $50$
d) to roughly match the onset of the observed precursor emission. The
dependence of the predicted plateau luminosity and duration on the ejecta
properties broadly agree with the Popov analytic scalings (Eqs. 19, 20).

Figure 4 shows an estimate of the plateau luminosity $L_{\rm pl}$ and duration
$t_{\rm pl}$, calculated from a large grid of models in the space of ejecta
mass $M_{\rm ej}$ and ejecta velocity $v_{\rm ej}$ for different assumptions
above initial (and progenitor) radii $R_{0}=R_{\star}=10$, $10^{2}$, and
$10^{3}\,{R}_{\odot}$. Here, we define the duration $t_{\rm pl}$ as that over
which 90% of the total energy is radiated, $0.9E_{\rm pl}=\int^{t_{\rm
pl}}Ldt$, where $E_{\rm pl}=\int Ldt$. The average luminosity is then defined
as $L_{\rm pl}=E_{\rm pl}/t_{\rm pl}$. Note that the model-predicted plateau
duration $t_{\rm pl}$ only represents an upper limit on the observed precursor
duration if the SN explosion occurs before the end of the eruption emission
phase. However, due to the relatively flat shape of the predicted light curve,
the model-predicted value of $L_{\rm pl}$ will still be comparable to the
time-averaged observed luminosity $L_{\rm pl}$, even when the eruption
emission is prematurely terminated by the SN.

The distributions of plateau luminosity and duration roughly follow those
predicted by Eqs. (19) and (20), namely $M_{\rm ej}\propto v_{\rm ej}^{-5}$
and $M_{\rm ej}\propto v_{\rm ej}$, for fixed values of $L_{\rm Popov}$ and
$t_{\rm Popov}$, respectively. For lower ejecta velocities, comparable to the
RSG surface escape speed $\sim 10-100$ km s-1, gas pressure instead of
radiation pressure dominates in the ejecta and other assumptions of the Popov
analytic estimates (e.g., the neglect of hydrogen recombination energy) are
violated (Matsumoto & Metzger, 2022). On the other hand, for ejecta masses
$\sim{M}_{\odot}$, velocities $\sim 10^{3}\,\rm km\,s^{-1}$, and radii close
to the surface of the RSG progenitors ($R_{0}\sim R_{\star}\sim
10^{3}\,{R}_{\odot}$), the predicted transient properties are consistent the
observed precursors. The high required ejecta velocity exceeding the surface
escape speed by factors $\gtrsim 10$ may point to energy deposition deep
within the star $\sim 10\,{R}_{\odot}$. To reproduce the observed precursors
from more compact progenitor stars such as blue supergiants, would require
more massive ejecta $\gtrsim 3\,{M}_{\odot}$.

Figure 4: Contours of the characteristic optical precursor luminosity $L_{\rm
pl}$ (after subtracting the UV luminosity; left column) and duration $t_{\rm
pl}$ (right column) associated with eruptive mass-loss events in the space of
ejecta velocity and mass, for different initial radii $R_{0}=10^{3}$,
$10^{2}$, and $10\,{R}_{\odot}$ (from top to bottom). Black shaded regions are
excluded because the ejecta velocity is smaller than the surface escape speed
for a star of mass $M_{\star}=10\,{M}_{\odot}$ and radius $R_{\star}\leq
R_{0}$. To the right of the white contours radiation-pressure exceeds gas-
pressure at the initial radius.

###  3.2 Wind Scenario

Figure 5 shows examples of the light curve and effective temperature evolution
in the wind mass-loss scenario, for a fixed progenitor mass
$M_{\star}=10\,{M}_{\odot}$. A thick black curve depicts the chosen fiducial
model, with wind mass-loss rate $\dot{M}=M_{\odot}\mathrm{\,yr^{-1}}$ and
speed $v_{\rm w}=200\,\mathrm{km\,s^{-1}}$ (corresponding to $R_{\rm
s}=100\,{R}_{\odot}$, Eq. 2). As described in Sec. 2.2, the total light curve
is obtained by adding those from successive single shells (an example of which
is shown by a thin black curve in Fig. 5). Since the total luminosity
approaches a steady-state on the timescale roughly set by the characteristic
duration of the single shell emission, we denote the latter $t_{\rm rise}$ and
calculate its value in the same way as the eruption model at the stationary
state. The rise-time is roughly equal to that over which photons diffuse out
from the shell, $t_{\rm d}$ (Eq. 14, or Eq. A5 for high values of $\dot{M}$;
see below).

The single-shell light curve exhibits a gradual rise followed by sharp peak,
powered by the escape of energy that accompanies a rapid decrease in the
photon diffusion timescale at hydrogen recombination. For the fiducial model,
radiation pressure dominates (marginally) over gas pressure during the
recombination phase, rendering the contribution of recombination energy to the
radiated energy small. The luminosity reached during the steady plateau phase
is roughly given by the product of the single shell’s characteristic
luminosity calculated in the same way as in the eruption model, by the number
of shells ejected during the rise phase $\simeq t_{\rm rise}/\Delta t$. We
remark that the rise-time $t_{\rm rise}(\sim t_{\rm d})$ is independent of the
choice of shell thickness $v\Delta t$ because it does not enter in Eq. (14).

Figure 5: Light curves of SN precursor emission in the wind mass-loss scenario
for progenitor mass $M_{\star}=10\,{M}_{\odot}$. A thick black curve shows the
fiducial case with wind mass-loss rate $\dot{M}=1{M}_{\odot}\,\rm yr^{-1}$ and
wind velocity $v_{\rm w}=200\,\mathrm{km\,s^{-1}}$ ($R_{\rm
s}=10^{2}\,{R}_{\odot}$). The wind duration is set to $t_{\rm w}=130$ d to
match the SN2020tlf precursor duration (shown for comparison red points, along
with the other precursors labeled as in Fig. 1). A thin black curve shows the
light curve from a single shell of thickness $\Delta t=4\,\rm d$, whose
duration$-$being similar for all shells$-$controls the rise-time of the total
light curve. Colored lines show the light curve that results by increasing the
wind mass-loss rate or speed relative to the fiducial model. A black dashed
curve shows the effect of tripling the “sonic” radius at which the wind
thermal energy is initialized, to crudely mimic the effects of re-heating of
the ejecta by internal shocks (Sec. 3.2.1).

Colored curves in Fig. 5 show additional light curve calculations varying the
wind mass-loss rates and velocities from the fiducial model. Because the rise-
time is roughly set by the diffusion timescale $t_{\rm d}\propto\dot{M}/v_{\rm
w}$ (Eq. 14), the steady-state luminosity is achieved faster for higher wind
velocities. Similarly, winds with higher mass-loss rates take longer to reach
steady-state. As discussed further below, this rules out large $\dot{M}$-winds
from generating the observed precursors because they are not able to generate
quasi-steady light-curves like those observed (Fig. 1) before the star would
have lost most of its mass.

As an aside, we remark that for winds with low velocities and/or
high-$\dot{M}$, our calculation overestimates the photosphere radius and hence
underestimates (by less than a factor of $\lesssim$ 2) the effective
temperature of the emission. As a result of the shallow wind density profile
$\rho\propto r^{-2}$, material ahead of the recombination front (which quickly
cools to $\lesssim 10^{3}\,\rm K$) can contribute significantly to the optical
depth through molecular absorption. However, at such low temperatures, our
analytic opacity law (which adopts a constant value $\kappa_{\rm mol}=0.1Z\sim
10^{-3}$ cm2 g-1) overestimates the true molecular opacity. Unless significant
dust formation occurs to boost the opacity further (Freedman et al., 2008;
Pejcha et al., 2016), the true photosphere location should therefore coincide
with the recombination front, at somewhat smaller radii than our model
predicts.

The top panels of Fig. 6 show the rise-time $t_{\rm rise}$ (upper right panel)
and steady-state luminosity $L_{\rm pl}$ (upper left panel) as a function of
the wind velocity $v_{\rm w}$ and mass-loss rate $\dot{M}$, for fixed
progenitor mass $M_{\star}=10\,{M}_{\odot}$. Regions of the phase space for
which the wind kinetic energy is below the Eddington luminosity of the star
are blacked out, based on the expectation that the star would respond to sub-
Eddington energy deposition by driving convection or expanding, rather than
give rise to significant mass-loss (e.g., Quataert et al. 2016). Likewise,
regions for which the total mass-loss in wind ejecta experienced over the
light curve rise-time, exceeds the total stellar mass (i.e., $M_{\rm
ej}\simeq\dot{M}t_{\rm rise}<M_{\star}$) are shaded gray. White shaded regions
denote the mass-loss rate $\dot{M}(v_{\rm w})$ predicted by Quataert et al.
(2016) (Eq. 18) for a range of heated envelope masses $M_{\rm env}\sim
10^{-3}-10^{-1}M_{\star}$.

Figure 6: Top: Steady-state $L_{\rm pl}$ luminosity (left panel) and rise-time
$t_{\rm rise}$ (right panel) of SN precursor emission from wind-like mass-
ejection as a function of the wind mass-loss rate $\dot{M}$ and velocity
$v_{\rm w}$ (equivalently, sonic radius for an assumed progenitor star mass
$M_{\star}=10\,{M}_{\odot}$; top horizontal axis). Black and gray shaded
regions are excluded based on the wind kinetic luminosity being sub-Eddington
and the total ejecta mass during the rise-time exceeding the progenitor star
mass (Eq. A8). The other solid black lines correspond to $M_{\rm
ej}=0.1M_{\star}$ and $0.01M_{\star}$. Thin shaded regions represent the mass-
loss rate $\dot{M}(v_{\rm w})$ predicted by Quataert et al. (2016) (Eq. 18),
for a range of envelope masses $M_{\rm env}/M_{\star}=0.001-0.1$. A black
dashed line shows the critical mass-loss rate $\dot{M}_{\rm ion}$ (Eq. A7 for
$T_{\rm ion}=10^{4}$ K) above which the drop in opacity from hydrogen
recombination determines the transient rise-time. Radiation pressure exceeds
gas pressure at the sonic point to the right of the white line, i.e. across
most of the parameter space. Bottom: Same as top panels, but showing the
ratios of steady-state $L_{\rm pl}$ luminosity to the wind kinetic power,
$\dot{E}_{\rm w}$ (Eq. A2; left panel), and to the maximum recombination
luminosity, $L_{\rm rec}$ (Eq. 21; right panel).

For characteristic outflow velocities $\sim 100-3000$ km s-1 of most SN
precursors (Table 1), explaining the observed luminosities $L_{\rm pre}\sim
10^{40}-10^{41}$ erg s-1 is barely possible, requiring very large $\dot{M}\sim
1-10\,M_{\odot}\mathrm{\,yr^{-1}}$; however, in this region of
$\\{\dot{M},v_{\rm w}\\}$ parameter space, the predicted rise-times exceed
several months, in tension with observed precursor plateau durations (Fig. 1)
and pushing up against the $M_{\rm ej}<M_{\star}$ limit. These constraints
impose severe limits on the wind scenario (see further discussion below).

The bottom left panel of Fig. 6 shows the ratio of steady-state transient
luminosity to the wind kinetic power $L_{\rm pl}/\dot{E}_{\rm w}$, i.e., a
measure of the wind’s radiative efficiency. The efficiency decreases as the
wind power $\dot{E}_{\rm w}\propto\dot{M}v_{\rm w}^{2}$ increases, roughly
according to $L_{\rm pl}/\dot{E}_{\rm w}\propto(\dot{E}_{\rm w}/L_{\rm
Edd})^{-2/3}$ (see Eq. A3). Larger values of $\dot{M}$ push the diffusion
radius ($\sim v_{\rm w}t_{\rm d}$) outwards, while larger $v_{\rm w}$ shrink
the sonic radius, both effects of which result in greater adiabatic losses
prior to where radiation can escape from the wind.

The bottom right panel of Fig. 6 shows the ratio of $L_{\rm pl}$ to maximum
recombination luminosity,

$\displaystyle L_{\rm rec}=\frac{\varepsilon_{\rm H}X\dot{M}}{m_{\rm p}}\simeq
6.1\times
10^{38}{\,\mathrm{erg\,s^{-1}}\,}\biggl{(}\frac{\dot{M}}{M_{\odot}\mathrm{\,yr^{-1}}}\biggl{)}\
,$ (21)

where $\varepsilon_{\rm H}=13.6\,\rm eV$ is the recombination energy per
hydrogen atom of mass $m_{\rm p}$ and $X=0.74$ is the hydrogen mass fraction
of the ejecta at solar metallicity. Unlike in the case of most stellar merger
transients (Metzger et al., 2021; Matsumoto & Metzger, 2022), the radiated
luminosity generally exceeds the recombination luminosity insofar as radiation
pressure dominates over gas pressure (as satisfied to the right of the solid
white line in Fig. 6).

In Appendix A we derive analytic estimates of the steady-state luminosity as a
function of the wind properties. There are two regimes to consider, depending
on whether the ejecta is still ionized or has begun to recombine by the radius
of emission (as dictated by whether $\dot{M}$ is below or above, respectively,
a critical value $\dot{M}_{\rm ion}$; Eq. A7). For $\dot{M}<\dot{M}_{\rm ion}$
we find $L\propto\dot{M}^{1/3}v_{\rm w}^{2/3}$ (Eq. A3; see also Quataert et
al. 2016; Shen et al. 2016), while for $\dot{M}>\dot{M}_{\rm ion}$ we find
$L\propto\dot{M}^{3/4}v_{\rm w}^{3/4}$ (Eq. A6). These dependencies roughly
agree with our findings in Figs. 5, 6.

For $\dot{M}<\dot{M}_{\rm ion},$ we invert Eqs. (14), (A3) to obtain the wind
mass-loss rate and velocity in terms of the observed precursor luminosity and
rise-time555Constraining the rise-time is challenging in practice because
current observations do not probe the earliest stages of the precursor
emission (Fig. 1); furthermore, the luminosity evolves slowly even before
reaching steady-state, making it hard to cleanly distinguish the rising
phase.:

$\displaystyle v_{\rm w}$ $\displaystyle\simeq
940{\,\mathrm{km\,s^{-1}}\,}\left(\frac{M_{\star}}{10\,{M}_{\odot}}\right)^{-2/3}\left(\frac{t_{\rm
rise}}{10{\,\rm d}}\right)^{-1/3}\left(\frac{L_{\rm pl}}{10^{40}{\,\rm
erg\,s^{-1}}}\right)\ ,$ (22) $\displaystyle\dot{M}$ $\displaystyle\simeq
1.5{\,M_{\odot}\mathrm{\,yr^{-1}}\,}\left(\frac{M_{\star}}{10\,{M}_{\odot}}\right)^{-2/3}\left(\frac{t_{\rm
rise}}{10{\,\rm d}}\right)^{2/3}\left(\frac{L_{\rm pl}}{10^{40}{\,\rm
erg\,s^{-1}}}\right)\ .$ (23)

These expressions reveal that very large mass-loss rates $\dot{M}\gtrsim
1-10\,M_{\odot}\mathrm{\,yr^{-1}}$ are required to explain the SN precursor
luminosities, $L_{\rm pre}\sim 10^{40}-10^{41}$ erg s-1. In fact, a
conservative upper limit can be placed on the steady-state plateau luminosity
in the wind-scenario under the condition that total wind ejecta mass $M_{\rm
ej}$ not exceed the progenitor star mass. This translates into an upper limit
on the wind mass-loss rate: $M_{\rm ej}=t_{\rm active}\dot{M}<M_{\star}$,
where $t_{\rm active}$ is the duration of mass-loss episode (from the wind
launch to the SN explosion). This timescale may be reasonably approximated
$t_{\rm active}\sim{\rm max}(t_{\rm rise},\,t_{\rm pre})$ as the longer of the
(theoretical) light curve rise-time, $t_{\rm rise}$, and the observed
precursor plateau duration, $t_{\rm pre}$. When the rise-time is the longer of
the two, the maximal $\dot{M}$ and corresponding maximum steady-state
luminosity $L_{\rm max}$ are derived in Appendix A (Eqs. A9, A11). When the
observed plateau duration is the longer of the two timescales, the maximal
luminosity follows by substituting $\dot{M}=\dot{M}_{\rm max}=M_{\star}/t_{\rm
pre}$ into Eqs. (A3), (A6).

This theoretical maximum luminosity is shown with a solid black line in Fig. 7
as a function of $v_{\rm w}$ for $M_{\star}=10\,{M}_{\odot}$ and a plateau
duration $t_{\rm pre}=100\,\rm d$ close to those observed (Fig. 1). For lower
wind speeds $v_{\rm w}\lesssim 1000\,\mathrm{km\,s^{-1}}$, the light curve
rise-time exceeds the observed plateau duration $t_{\rm rise}>t_{\rm pre}$,
placing the tighter limit on the maximal luminosity. For comparison, colored
symbols show the time-averaged luminosities and spectroscopically-measured
outflow velocities for individual SN precursors (Table 1). The precursor
luminosities typically exceed the theoretical limit by at least a factor of a
few, thus challenging the wind scenario.

In precursor events for which flash-spectroscopy observations were conducted
soon after the SN explosion, the measured CSM velocities from these
observations ($v_{\rm obs}$ in Table 1) are in two cases (SN 2016bdu and SN
2020tlf) so low the SN ejecta would have swept up any precursor-ejected
material before the observations were taken (hence, the CSM velocities
measured must arise from an earlier mass-loss episode prior to the precursor
rise). This hypothetical requirement on the measurable CSM velocity can be
written:

$\displaystyle v_{\rm obs}\gtrsim v_{\rm engulf}\equiv v_{\rm SN}\frac{t_{\rm
flash}}{t_{\rm pre}}$ $\displaystyle\simeq
1000\,\mathrm{km\,s^{-1}}\,\left(\frac{v_{\rm
SN}}{10^{4}{\,\mathrm{km\,s^{-1}}}}\right)\left(\frac{t_{\rm flash}}{10{\,\rm
d}}\right)\left(\frac{t_{\rm pre}}{100{\,\rm d}}\right)^{-1},$ (24)

where $v_{\rm SN}$ and $t_{\rm flash}$ are the SN ejecta velocity and time of
the flash-spectroscopy observation measured since the SN explosion,
respectively.666Note that here we define the critical velocity not by the
entire wind active period $t_{\rm active}$ but rather the observed duration
$t_{\rm pre}$; this is justified because emission from the maximal CSM
velocity material typically rises quickly $t_{\rm rise}\lesssim t_{\rm pre}$.
As a corollary, CSM with velocities $v_{\rm obs}>v_{\rm engulf}$ could in
principle (if present) be observed in systems with flash spectroscopy
observations, thus enabling an upper limit on $v_{\rm w}<v_{\rm engulf}$ based
on the lack of detection. This maximal observable velocity $v_{\rm engulf}$
are shown as left-facing arrows in Fig. 7 for $v_{\rm
SN}=10^{4}\,\mathrm{km\,s^{-1}}$.

Taken together, we conclude that only SN 2020tlf can be explained by the wind
scenario for a reasonable ejecta mass $\lesssim 0.1\,M_{\star}$, and in this
case only if the wind velocity (inaccessible to flash spectroscopy) were to
obey $600\,\mathrm{km\,s^{-1}}\lesssim v_{\rm ej}\lesssim
1000\,\mathrm{km\,s^{-1}}$. Interestingly, these parameters are consistent
with the wind solution of Quataert et al. (2016) for the envelope mass of
$M_{\rm env}\sim 10^{-4}-10^{-5}\,M_{\star}$. Given the implied small sonic
radius of $R_{\rm s}\sim 10\,{R}_{\odot}$, we can estimate the mass included
between the energy injection and sonic radii by $M_{\rm env}\sim\dot{M}R_{\rm
s}/v_{\rm w}\sim 10^{-5}\,{M}_{\odot}\sim 10^{-4}\,M_{\star}$, consistent with
the required value of $M_{\rm env}.$

Figure 7: The steady wind scenario is challenged to explain high luminosities
of SN precursor emission for reasonable ejecta masses. Black lines shows the
maximum steady-state luminosity in the wind scenario as a function of wind
velocity so as not to overproduce the total wind ejecta mass $M_{\rm ej}$
(Eqs. A9, A11); specifically, the gray shaded region is excluded by the
requirement $M_{\rm ej}>M_{\star}=10\,{M}_{\odot}$ (see text for details).
Colored circles represents observed SN precursor luminosities and velocities
(Table 1). Left-directed triangles show an estimate of the wind velocity
($v_{\rm engulf}$; Eq. 24) invisible to flash spectroscopy observations (too
slow wind material would be overtaken by the SN shock prior to the
observations).

####  3.2.1 Shock Interaction

A potential mechanism to boost the luminosity of the wind is through shock
interaction. By reheating the wind ejecta well above the sonic surface, shocks
reduce the adiabatic losses experienced before the flow reaches the diffusion
radius where the remaining thermal energy is radiated (Eq. A1). One source of
shock interaction are internal collisions within the wind itself, due to time
variability in the wind speed.

Consider two shells with the same mass $M$ but different velocities. The
slower shell with velocity $v$ is ejected $\delta t$ before the faster one
with velocity $(1+\delta_{v})v$. The latter one catches up with former one in
a time $\delta t/\delta_{v}$ and they merger to form a single shell. The
collision radius is estimated by $R_{\rm col}\sim(2+1/\delta_{v})R_{\rm s}$,
where we have assumed both shells originate from the same radius $\sim R_{\rm
s}$ separated by a time $\delta t\sim R_{\rm s}/v$ roughly equal to the
dynamical time at the launching surface. From momentum and energy
conservation, the final velocity of the merged-shell is $(1+\delta_{v}/2)v$
and the energy dissipated by the collision is $M(v\delta_{v})^{2}/2$.

The velocity difference $\delta_{v}$ has two competing effects on the radiated
luminosity of the shocked wind. While smaller values of $\delta_{v}$ increase
the collision radius and thus reduce the adiabatic losses, such collisions
also dissipate less energy than for larger $\delta_{v}$. Adiabatic losses
exterior to $R_{\rm col}$ reduce the observed luminosity by an amount
$L\propto R_{\rm col}^{-2/3}\propto(2+1/\delta_{v})^{-2/3}$ (Eq. A2), while
the boost in thermal energy $\propto(\delta_{v})^{2}$; cases with the largest,
order-unity fluctuations $\delta_{v}\sim 1$ therefore give the biggest
luminosity boost.

To demonstrate the potential effects of shock heating, we recalculate the
fiducial wind scenario light curve for the (most luminous) case $\delta_{v}=1$
by artificially increasing the “sonic” radius $R_{\rm s}$, at which the wind
thermal energy is initialized in our calculation, by a factor of three
compared to the fiducial case. As shown in Fig. 5, the radiated luminosity
roughly doubles compared to the case without shock interaction. Our simplified
treatment of the effects of shock heating results in a smooth plateau-shaped
light curve similar to the case without shocks. In reality, however, the light
curve should exhibit variability due to the stochastic nature of the heating,
though this could be smoothed out by photon diffusion effects. Most SN
precursor light curves are not perfectly smooth (Fig. 1), with the exception
of SN 2020tlf (for which shock heating is anyways not strictly required in the
wind scenario; Fig. 7).

Another source of shock interaction is between the wind and pre-existing CSM
ejected earlier from the star, or, equivalently, if the wind velocity rises in
time. In fact, Quataert et al. (2016) found that shock interaction inevitably
happens before the wind settles into a steady state. After the onset of energy
injection below the stellar surface, the stellar envelope is initially only
weakly accelerated to form a slowly-expanding atmosphere. Continuous heating
decreases the density at the heating site, accelerating the material there to
higher velocity (Eq. 17). The high-speed wind then collides with the slowly-
expanding atmosphere, generating shocks which can dominate the radiated
luminosity, at least initially. Although we defer a detailed study to future
work, shock interaction due to an accelerating wind provide another way to
increase the wind’s radiative efficiency, particularly at early times.

##  4 Precursor-Generated CSM

The mass ejected from the progenitor star during the precursor phase becomes a
source of CSM for shock interaction with the SN ejecta following the
explosion. The application of our precursor light curve models to individual
events enables us to constrain their ejecta masses and velocities, which we
can then check for consistency with the CSM properties inferred by modeling
interaction signatures during the SN phase.

###  4.1 Eruption scenario

We have found (Sec. 3.1) that the eruption scenario can account for the
luminosities and timescales of SN precursor emission for physically reasonable
ejecta properties. The ejecta mass and initial radius in this scenario are
obtained by inverting Eqs. (19), (20) in terms of the precursor luminosity
$L_{\rm pl}$ and duration $t_{\rm pl}$:

$\displaystyle M_{\rm ej}$ $\displaystyle\simeq
6.7\,{M}_{\odot}\left(\frac{v_{\rm
ej}}{10^{3}\,\mathrm{km\,s^{-1}}}\right)^{3}\left(\frac{t_{\rm pl}}{100\,\rm
d}\right)^{4}\left(\frac{L_{\rm pl}}{10^{41}{\,\rm erg\,s^{-1}}}\right)^{-1}\
,$ (25) $\displaystyle R_{0}$ $\displaystyle\simeq
140\,{R}_{\odot}\left(\frac{v_{\rm
ej}}{10^{3}\,\mathrm{km\,s^{-1}}}\right)^{-4}\left(\frac{t_{\rm pl}}{100\,\rm
d}\right)^{-2}\left(\frac{L_{\rm pl}}{10^{41}{\,\rm erg\,s^{-1}}}\right)^{2}\
.$ (26)

Unfortunately, these estimates depend sensitively on the ejecta speed $v_{\rm
ej}$, which is challenging to measure and comes with large uncertainties.
Furthermore, the observed precursor duration $t_{\rm pre}$ provides only a
lower limit on the total duration ($t_{\rm pl}$) which enters these formulae
if the SN prematurely terminates the precursor emission.

Table 2 provides the ejecta mass, kinetic energy ($E_{\rm kin}=M_{\rm
ej}v_{\rm obs}^{2}/2$), and progenitor radius for our precursor sample (Table
1) as obtained from the inverted Popov formulae (Eqs. 25, 26). Again, the
observed precursor duration is only a lower limit on the total plateau
duration and hence we obtain only a lower (upper) limit on the ejecta mass
(progenitor radius). All values are calculated assuming lowest
spectroscopically observed velocity range. However, we again caution that our
estimates are highly dependent on the adopted ejecta speed; a factor of two
difference in the velocity changes the results by an order of magnitude or
more.

Table 2: CSM Properties implied by Eruption Model Event | $M_{\rm ej}^{(a)}$ | $E_{\rm kin}^{(a)}$ | $R_{0}^{(b)}$
---|---|---|---
| [${M}_{\odot}$] | [$10^{48}$erg] | [$10^{3}{R}_{\odot}$]
SN 2009ip | $0.036-0.19$ | $0.23-3.8$ | $2.4-23$
SN 2010mc | $0.039-1.0$ | $0.39-94$ | $0.046-3.7$
SN 2015bh | $1.3-5.9$ | $4.6-59$ | $0.13-1.0$
SN 2016bdu | $0.32$ | $0.51$ | $8.4$
SN 2020tlf | $0.025-1.6$ | $0.00062-0.63$ | $0.42-110$
Note. All quantities calculated by applying the observed
precursor properties (Table 1) to Eqs. (25), (26), assuming
the lowest spectroscopically observed velocity (which gives
the most physical ejecta mass; Fig. 8).
(a) Lower limit, based on the requirement that $t_{\rm pl}>t_{\rm pre}$.
(b) Upper limit, based on the requirement that $t_{\rm pl}>t_{\rm pre}$.

At the time of the SN explosion, the characteristic outer radius and density
of the precursor ejecta (hereafter CSM) are thus given, respectively, by

$\displaystyle R_{\rm CSM}$ $\displaystyle\sim v_{\rm ej}t_{\rm pre}\simeq
8.6\times 10^{14}{\,\rm cm\,}\left(\frac{v_{\rm ej}}{10^{3}\,\rm
km\,s^{-1}}\right)\left(\frac{t_{\rm pre}}{100\,\rm d}\right)\ ,$ (27)
$\displaystyle\rho_{\rm CSM}$ $\displaystyle\sim\frac{M_{\rm
ej}}{\frac{4\pi}{3}R_{\rm CSM}^{3}}\gtrsim 5.0\times 10^{-12}{\,\rm
g\,cm^{-3}\,}$ $\displaystyle\times\left(\frac{t_{\rm pre}}{100\,\rm
d}\right)\left(\frac{L_{\rm pre}}{10^{41}{\,\rm erg\,s^{-1}}}\right)^{-1}\ ,$
(28)

where the second equality in Eq. (28) makes use of Eq. (25). The upper limit
on $\rho_{\rm CSM}$ results because $t_{\rm pl}>t_{\rm pre}$ and is notably
independent of the (poorly constrained) ejecta velocity. Also note the perhaps
counter-intuitive result that less luminous precursors generate more massive,
denser CSM.

The left panel of Fig. 8 shows the estimated CSM density $\rho_{\rm CSM}$ and
radius $R_{\rm CSM}$ in the eruption scenario for the observed precursors
(Fig. 1, Table 1). As discussed in Sec. 3.1 and shown in Fig. 3, the required
CSM masses are typically in the range $\sim 0.1-1\,{M}_{\odot}$. Since the
light curve properties depend only weakly on the ejecta velocity profile
$M_{\rm ej}\propto v^{-\beta}$ (Eq. 8), the slope of the resulting CSM radial
density profile $\rho\propto r^{-\beta-3}$ (at $r>R_{\rm CSM}$ or equivalently
$v>v_{\rm ej}$) is not well-constrained by the precursor observations.
However, as long as $\beta\gtrsim 0$, the CSM mass will be concentrated in the
$\sim v_{\rm ej}$ shell on the radial scale $\sim R_{\rm CSM}$.

Several implications follow from the implied CSM properties. The inferred CSM
is sufficiently massive and compact as to be optically thick ($\rho\gtrsim
10^{-14}{\,\rm g\,cm^{-3}\,}(R_{\rm CSM}/10^{14}{\,\rm cm})^{-1}$), thus
implying that (1) the SN shock break-out will occur from the precursor-
generated CSM shell instead of the original stellar surface; (2) flash
spectroscopy cannot readily probe the mean velocity of the precursor ejecta
(though it may probe its high-velocity tail).

Shock heating of the CSM shell by the supernova ejecta will dissipate the
kinetic energy of the latter and boost the luminosity of the supernova
relative to the CSM-free case. The luminosity of this shock interaction can be
crudely estimated as777This scaling differs from that for interacting SN with
wind-like CSM because of the steeper density profile.

$\displaystyle L_{\rm sh}\sim\frac{\frac{1}{2}\varepsilon M_{\rm ej}v_{\rm
SN}^{2}}{t_{\rm diff}}\frac{R_{\rm CSM}}{v_{\rm SN}t_{\rm diff}}$
$\displaystyle\simeq 6\times 10^{42}{\,\rm
erg\,s^{-1}}\left(\frac{\varepsilon}{0.1}\right)\left(\frac{R_{\rm
CSM}}{10^{14}\,\rm cm}\right)\left(\frac{v_{\rm SN}}{10^{4}\,\rm
km\,s^{-1}}\right)^{2}\ ,$ (29)

where $t_{\rm diff}=\sqrt{\frac{\kappa M_{\rm ej}}{4\pi cv_{\rm SN}}}$ is the
characteristic photon diffusion time through the shocked CSM, and
$\varepsilon$ is the radiative efficiency. Interestingly, the luminosity
depends only on the CSM radius. For CSM radii $R_{\rm CSM}\simeq 3\times
10^{14}\,\rm cm$ implied by the precursor emission (Fig. 4) the predicted
luminosities $L_{\rm sh}\sim 10^{43}\,\rm erg\,s^{-1}$ broadly agree with
those of the SNe following the precursor events (Fig. 1). However, for SN
2020tlf with $R_{\rm CSM}\sim 10^{14}\,\rm cm$, the shock-boosted SN
luminosity would be too low and another CSM component, for example optically-
thick CSM extending to large radii $\sim 10^{15}\,\rm cm$, is required to
power the early SN light curve via CSM interaction.

Though constraints on the CSM properties from the literature of our precursor
sample are limited, we comment briefly on those available. For SN 2009ip,
Margutti et al. (2014); Moriya (2015) interpret the SN emission as arising due
to shock breakout emission from a dense shell produced by the precursor
outburst; they estimate a precursor mass of $\sim 0.1\,{M}_{\odot}$ and radius
$\sim 10^{15}\,\rm cm$ at explosion, by equating the SN rise-time to the
photon diffusion time through the dense shell. Our result is essentially
consistent with theirs for this event, for an assumed ejecta velocity equal to
the lowest measured spectroscopically, $v_{\rm ej}\simeq
800-1400\,\mathrm{km\,s^{-1}}$. For SN2020tlf, Chugai & Utrobin (2022) model
the precursor emission as arising from an ejecta mass of $M_{\rm ej}\sim
0.1\,{M}_{\odot}$, also consistent with our result. They find that subsequent
SN emission is not effected by this compact CSM, but that an additional wind-
like CSM extending up to $\sim 10^{15}\,\rm cm$ is required (we return to this
event within the context of the wind-scenario in the next section).

Figure 8: CSM density profile resulting from eruption (left) and wind (right)
mass-loss models. The gray lines in left (right) panel show corresponding
ejecta mass (mass-loss rate) of $M_{\rm ej}(\dot{M})=0.1$, 1, and 10
${M}_{\odot}$ (${M}_{\odot}\,\rm yr^{-1}$). For the same event, different
points are calculated by different velocity obtained by observations (see
Table 1).

###  4.2 Wind scenario

Detecting the rising phase of the light curve in the wind scenario is
typically challenging (see Fig. 5). We therefore assume that the emission has
already reached its stationary state by the time of the observations.
Inverting Eq. (A3), we derive the required wind mass-loss rate in terms of
observed precursor properties:

$\displaystyle\dot{M}$ $\displaystyle\simeq
1.5{\,M_{\odot}\mathrm{\,yr^{-1}}\,}\left(\frac{M_{\star}}{10\,{M}_{\odot}}\right)^{-2}$
$\displaystyle\times\left(\frac{v_{\rm w}}{10^{3}{\,\rm
km\,s^{-1}}}\right)^{-2}\left(\frac{L_{\rm pre}}{10^{40}{\,\rm
erg\,s^{-1}}}\right)^{3}\ .$ (30)

Insofar as the outer radius of the wind ejecta at the time of the SN explosion
is given by $R_{\rm CSM}\sim v_{\rm w}t_{\rm active}\gtrsim v_{\rm w}t_{\rm
pre}$, Eq. (27) with $v_{\rm ej}$ replaced by $v_{\rm w}$ gives a lower limit
on $R_{\rm CSM}$ while the rise-time is typically shorter than $t_{\rm pre}$
for relevant wind velocities. This in turn implies an upper limit on the CSM
density at $R_{\rm CSM}$,

$\displaystyle\rho_{\rm CSM}$ $\displaystyle=\frac{\dot{M}}{4\pi R_{\rm
CSM}^{2}v_{\rm w}}$ (31) $\displaystyle\lesssim 1.0\times 10^{-13}{\,\rm
g\,cm^{-3}\,}\left(\frac{M_{\star}}{10\,{M}_{\odot}}\right)^{-2}$
$\displaystyle\times\left(\frac{v_{\rm w}}{10^{3}{\,\rm
km\,s^{-1}}}\right)^{-5}\left(\frac{L_{\rm pre}}{10^{40}{\,\rm
erg\,s^{-1}}}\right)^{3}\left(\frac{t_{\rm pre}}{100\,\rm d}\right)^{-2}\ .$
(32)

which depends sensitively on the wind velocity. Assuming the wind mass-loss
rate is roughly constant prior to the SN, the CSM density at $r<R_{\rm CSM}$
will follow a wind-like $\rho\propto r^{-2}$ profile.

The right panel of Fig. 8 shows the estimated CSM density of observed
precursors in the wind mass-loss scenario. As discussed in Sec. 3.2, the
required CSM density, or equivalently mass-loss rate $\dot{M}$, is enormous,
exceeding that allowed by the total mass budget of the star in most events.

Only in SN 2020tlf can the observed precursor be reasonably accommodated, for
the maximally-allowed velocity $v_{\rm w}\sim v_{\rm engulf}\simeq
10^{3}\,\mathrm{km\,s^{-1}}$ (and we note that $\dot{M}\simeq const$ mass-loss
may indeed be suggested for this event by the flat shape of the precursor
light curve). The implied radial extent of the CSM $\sim 10^{15}\,\rm cm$ in
the wind scenario is consistent with an upper limit based on X-ray non-
detections (Jacobson-Galán et al., 2022) and light-curve modeling (Chugai &
Utrobin, 2022) (though we note that the density required in our scenario is
$3-10$ times greater than found by these authors). Given uncertainties on
parameters, such as the progenitor mass and wind velocity, and potential for
internal-shock-heating (Sec. 3.2.1), this level of discrepancy may
nevertheless be tolerable. We note that the CSM radius obtained for SN 2020tlf
in the eruption scenario (Table 2) is also consistent with the modeling of
Chugai & Utrobin (2022).

As in the eruption case, the inferred CSM density (wind mass-loss rate) should
not be so large that it overproduces the SN luminosity via CSM shock
interaction. For freely expanding SN ejecta into a wind-like density profile,
the SN luminosity can be estimated as (e.g., Chevalier & Irwin 2011, but here
ignoring deceleration of the shocked CSM shell):

$\displaystyle L_{\rm sh}$ $\displaystyle\sim\frac{\frac{1}{2}\varepsilon
M_{\rm swpt}v_{\rm SN}^{2}}{t_{\rm diff}}$ (33) $\displaystyle\simeq 3\times
10^{43}{\,\rm
erg\,s^{-1}}\left(\frac{\varepsilon}{0.1}\right)\left(\frac{\dot{M}}{{M}_{\odot}\,\rm
yr^{-1}}\right)$ $\displaystyle\times\left(\frac{v_{\rm SN}}{10^{4}\,\rm
km\,s^{-1}}\right)^{3}\left(\frac{v_{\rm w}}{10^{3}{\,\rm
km\,s^{-1}}}\right)^{-1}\ ,$ (34)

where $M_{\rm swpt}\sim\rho R_{\rm diff}^{3}$ is the CSM wind mass swept-up by
SN ejecta on the wind diffusion timescale $t_{\rm
diff}=\frac{\kappa\dot{M}}{4\pi cv_{\rm w}}$ (Eq. 6), and we assumed the
radius $R_{\rm diff}\sim v_{\rm SN}t_{\rm diff}\sim 6\times 10^{14}{\,\rm
cm\,}(\dot{M}/M_{\odot}\mathrm{\,yr^{-1}})(v_{\rm SN}/10^{4}{\,\rm
km\,s^{-1}})(v_{\rm w}/10^{3}\,\rm km\,s^{-1})^{-1}$ is smaller than the outer
edge of the CSM, $R_{\rm CSM}$.

The estimate for $L_{\rm sh}$ in the case of SN 2020tlf is roughly consistent
with its SN luminosity, while we find $L_{\rm sh}\gg L_{\rm SN}$ given the
much higher values $\dot{M}$ required to explain other precursors. This again
favors the eruption scenario over the steady-wind scenario as the origin of
most SN precursors, at least neglecting if shock-heating within the wind (Sec.
3.2.1).

##  5 Summary

We have modeled the precursor optical emission detected from a growing sample
of core collapse SNe using an extension of the semi-analytical light curve
model described in Matsumoto & Metzger (2022). The observed precursors can be
regarded as extreme cases of the mass-loss events increasingly inferred to
occur just before the terminal collapse, whose direct emission can provide
clues to the nature of the mass-loss mechanisms at work and more generally
about the final stages of massive star evolution. We develop light curve
models in the context of two scenarios (‘eruption’ and ‘wind’) for the nature
of the pre-SN mass-loss phase.

In the eruption model, energy is deposited by near the base of stellar
envelope on a timescale shorter than the dynamical time. While our model is
indifferent to the mechanism responsible for this energy injection, it could
in principle be caused by a violent outburst associated with late stages of
nuclear shell burning. This sudden energy deposition creates a shock wave,
which propagates outwards towards the surface of the star, ejecting a portion
of its envelope. For the eruption scenario, our findings are summarized as
follows:

  * •

The eruption light curve is characterized by a recombination-driven plateau
similar to that of Type IIP SNe (Fig. 3). The luminosity and duration of the
plateau are reasonably estimated by the so-called “Popov formulae” (Popov,
1993) which relates these observables to the ejecta mass and progenitor radius
(Eqs. 19, 20). However, because the entirety of precursor light curve is
generally not observable (due to interruption by the SN explosion), we can
obtain only lower limits on the ejecta mass $M_{\rm ej}$ and kinetic energy
$E_{\rm kin}$ for individual events. Adopting an ejecta velocity $v_{\rm
obs}\sim 100-1000\,\rm km\,s^{-1}$ consistent with those inferred from
spectroscopy, the observed precursor luminosities are reproduced for $M_{\rm
ej}\gtrsim 0.1-1\,{M}_{\odot}$ and $E_{\rm kin}\gtrsim 10^{48}-10^{50}\,\rm
erg$ (Table 2). We caution that the latter estimate depends sensitively on the
ejecta velocity, which is not available in all events and could in principle
be confused with CSM from other, earlier phases of mass-loss.

  * •

A wide range of precursor luminosities can be produced in this model, $L_{\rm
pre}\sim 10^{39}-10^{42}\,\rm erg\,s^{-1}$ (Fig. 4, see also Dessart et al.
2010). The luminosity mainly depends on the progenitor radius and ejecta
velocity. Bright precursors of $L_{\rm pre}\gtrsim 10^{41}\,\rm erg\,s^{-1}$
require large progenitor radii $R_{\star}\gtrsim 10^{2}\,{R}_{\odot}$ such as
RSGs. For these progenitors, super-Eddington luminosities $L_{\rm pre}\gtrsim
L_{\rm Edd}\sim 10^{39}\,\rm erg\,s^{-1}\,(M_{\star}/10\,{M}_{\odot})$ are
achieved only for the ejecta speeds larger than the escape speed of the
stellar surface, supporting the source of energy deposition occurring deep
inside the stellar envelope.

The second scenario we considered is one of steady wind-like mass-loss, which
results from a continuous energy deposition at the base of stellar envelop
(mainly motivated by the wave-heating scenario; e.g., Quataert & Shiode 2012;
Fuller 2017). For the wind scenario, our findings are summarized as follows:

  * •

The light curve undergoes a gradual rise, before settling into a stationary
state. The rise-time is essentially given by the diffusion timescale of the
wind, and the steady-state luminosity as $L\sim L_{\rm
Edd}(\frac{1}{2}\dot{M}v_{\rm w}^{2}/L_{\rm Edd})^{1/3}$ (Eq. A3, A6; see also
Quataert et al. 2016; Shen et al. 2016). For typical values of the observed
precursor ejecta velocity, very large mass-loss rates $\dot{M}\gtrsim
10\,{M}_{\odot}\,\rm yr^{-1}$ are required to achieve the observed precursor
luminosities $L_{\rm pre}\sim 10^{41}\,\rm erg\,s^{-1}$ (Fig. 6). Such large
mass-loss rates would also increase the light curve rise-time which (in
conjunction with the large required $\dot{M}$) would exceed the mass budget of
the star. This constraint can be formalized by defining a theoretical maximum
precursor luminosity for a given wind velocity (Fig. 7; Appendix A).

Among the five well-observed precursors in our sample, only SN 2020tlf with
its relatively low luminosity, can be explained by the wind model, for a mass-
loss rate of $\dot{M}\simeq 1\,{M}_{\odot}\rm\,yr^{-1}$ and velocity of
$v_{\rm w}\simeq 10^{3}\,\rm km\,s^{-1}$. The corresponding sonic radius is
again deep inside the star, $R_{\rm s}\sim 10\,{R}_{\odot}\ll
10^{2}-10^{3}\,{R}_{\odot}$, requiring energy deposition at the base of
stellar envelope.

  * •

The luminosity of the wind can be boosted due to internal shocks, for instance
due to a time-variable wind speed, by a modest factor $\lesssim 2$. However,
the relatively smooth observed shape of the precursor light curves may
disfavor shock interaction.

While we do not study them in detail, external shocks between the precursor
ejecta and pre-existing CSM may also take place and contribute to the
luminosity (see also Quataert et al., 2016; Strotjohann et al., 2021;
Jacobson-Galán et al., 2022). Neglecting any losses of efficiency, we estimate
the shock luminosity $L_{\rm sh}\sim\frac{1}{2}\left(\frac{\dot{M}_{\rm
CSM}\dot{M}}{\dot{M}_{\rm CSM}+\dot{M}}\right)v_{\rm w}^{2}$, where
$\dot{M}_{\rm CSM}$ is the mass-loss rate of pre-existing CSM. This luminosity
can exceed the non-interacting wind luminosity (Eq. A3) for a sufficiently
large CSM mass-loss rate $\dot{m}_{\rm CSM}>\dot{m}/(\dot{m}^{2/3}-1)$, where
we have defined $\dot{m}\equiv\dot{M}/(2L_{\rm Edd}/v_{\rm w}^{2})$. Given
this potential promise, future work should study the external shock scenario
in greater detail.

  * •

Absent external shock interaction, the fact that observed SN precursors
violated the maximal luminosity in the wind scenario may favors eruptive
events (e.g., driven by unstable nuclear shell burning) over a more continuous
heating source (e.g., due to wave heating).

In both eruption and wind models, the precursor ejecta forms an optically-
thick compact CSM shell surrounding the progenitor star that will be present
at the time of collapse and could boost the luminosity of the supernova light
curve substantially through shock interaction. Our main finding is:

  * •

The radial extent of the CSM is typically so small $R_{\rm CSM}\lesssim
10^{15}\,\rm cm$ (see Eq. 27) that it is engulfed by SN ejecta less than ten
days after SN explosion. Therefore, flash spectroscopy carried out a few days
after SN may miss the signature of the precursor ejecta and probe only the CSM
released before the precursor mass-loss episode (see also Strotjohann et al.,
2021). In the eruption model, the density profile of the CSM is steeper than
the steady wind profile ($\rho\propto r^{-2}$), forming a dense CSM core.

We thank Avishay Gal-Yam and Eliot Quataert for helpful comments on an early
draft of the text. This work is supported in part by JSPS Overseas Research
Fellowships (T.M.). B.D.M. acknowledges support from the National Science
Foundation (grant number AST-2009255).

##  Appendix A Analytic Estimates of Precursor Emission in the Wind Mass-Loss
Scenario

The rise-time of the precursor light curve (equivalently, the duration of the
single shell light curve; Fig. 5) is determined by the diffusion timescale
through the shell, $t_{\rm d}$ (Eq. 14). Radiation thus escapes the wind
ejecta from the characteristic radius

$\displaystyle R_{\rm d}\simeq v_{\rm w}t_{\rm d}=\frac{\kappa\dot{M}}{4\pi
c}\simeq 5.4\times 10^{13}{\,\rm
cm\,}\biggl{(}\frac{\dot{M}}{M_{\odot}\mathrm{\,yr^{-1}}}\biggl{)}\ .$ (A1)

In this appendix, we normalize $\kappa$ to the fully-ionized electron
scattering opacity, $\kappa_{\rm es}\simeq 0.32\,\rm cm^{2}\,g^{-1}$. Because
photons are trapped, the wind material expands adiabatically between the sonic
and diffusion radii ($R_{\rm s}<r<R_{\rm d}$), such that
$T\propto\rho^{1/3}\propto r^{-2/3}$ under radiation-pressure-dominated
condition and $\rho\propto r^{-2}$ for a steady wind. The thermal and kinetic
luminosities of the wind are comparable at $R_{\rm s}$; thus, the photon
(advection) luminosity $L\simeq 4\pi r^{2}v_{\rm w}aT^{4}\propto
T^{4}r^{2}\propto r^{-2/3}$ evolves as

$\displaystyle L$ $\displaystyle\simeq\dot{E}_{\rm w}\biggl{(}\frac{r}{R_{\rm
s}}\biggl{)}^{-2/3}\ ,$ (A2)

up to the diffusion becomes important. Here $\dot{E}_{\rm w}$ is the wind
kinetic power (Eq. A2).

There are two cases to consider, depending on whether the wind material
remains fully ionized or has begun to recombine below $\sim R_{\rm d}$. When
$\dot{M}$ is sufficiently low (the precise threshold, $\dot{M}_{\rm ion}$, to
be defined below), the ejecta is still fully ionized ($T\gtrsim T_{\rm
ion}\simeq 10^{4}$ K) when photons start to diffuse our of the wind. Inserting
Eq. (A1) into Eq. (A2), we find

$\displaystyle L=L_{\rm Edd}\biggl{(}\frac{\dot{E}_{\rm w}}{L_{\rm
Edd}}\biggl{)}^{1/3}\simeq 3.2\times
10^{39}{\,\mathrm{erg\,s^{-1}}\,}M_{\star,1}^{2/3}\biggl{(}\frac{\dot{M}}{M_{\odot}\mathrm{\,yr^{-1}}}\biggl{)}^{1/3}\biggl{(}\frac{v_{\rm
w}}{200\,\mathrm{km\,s^{-1}}}\biggl{)}^{2/3}\ ,\dot{M}<\dot{M}_{\rm ion},$
(A3)

where $M_{\star,1}=M_{\star}/(10M_{\odot}).$ This expression agrees with
previous works (e.g., Quataert et al., 2016; Shen et al., 2016).

On the other hand, when $\dot{M}$ is sufficiently large, hydrogen
recombination begins below $R_{\rm d}$, which triggers photon diffusion by
reducing the opacity $\kappa\ll\kappa_{\rm es}$ and effectively shrinking the
diffusion radius. Recombination occurs at the temperature $T=T_{\rm
ion}=10^{4}T_{\rm ion,4}$ K, as is achieved at the radius

$\displaystyle R_{\rm ion}=R_{\rm s}\biggl{(}\frac{T_{\rm s}}{T_{\rm
ion}}\biggl{)}^{3/2}=\biggl{(}\frac{\dot{M}^{3}(2GM_{\star})^{2}}{(8\pi
aT_{\rm ion}^{4})^{3}v_{\rm w}}\biggl{)}^{1/8}\simeq 2.5\times 10^{14}{\,\rm
cm\,}M_{\star,1}^{1/4}T_{\rm
ion,4}^{-3/2}\biggl{(}\frac{\dot{M}}{M_{\odot}\mathrm{\,yr^{-1}}}\biggl{)}^{3/8}\biggl{(}\frac{v_{\rm
w}}{200\,\mathrm{km\,s^{-1}}}\biggl{)}^{-1/8}\ ,$ (A4)

on the timescale

$\displaystyle t_{\rm ion}=\frac{R_{\rm ion}}{v_{\rm w}}\simeq 140{\,\rm
d\,}M_{\star,1}^{1/4}T_{\rm
ion,4}^{-3/2}\biggl{(}\frac{\dot{M}}{M_{\odot}\mathrm{\,yr^{-1}}}\biggl{)}^{3/8}\biggl{(}\frac{v_{\rm
w}}{200\,\mathrm{km\,s^{-1}}}\biggl{)}^{-9/8}\ ,$ (A5)

where

$\displaystyle T_{\rm s}=\biggl{(}\frac{\dot{M}v_{\rm w}}{8\pi aR_{\rm
s}^{2}}\biggl{)}^{1/4}=\biggl{(}\frac{\dot{M}v_{\rm w}^{5}}{8\pi
a(2GM_{\star})^{2}}\biggl{)}^{1/4}\simeq 1.1\times 10^{5}{\,\rm
K\,}M_{\star,1}^{-1/2}\biggl{(}\frac{\dot{M}}{M_{\odot}\mathrm{\,yr^{-1}}}\biggl{)}^{1/4}\biggl{(}\frac{v_{\rm
w}}{200\,\mathrm{km\,s^{-1}}}\biggl{)}^{5/4}\ $

is the wind temperature at the sonic radius. The wind luminosity in the
high-$\dot{M}$ case (Eq. A2) is then given by

$\displaystyle L$ $\displaystyle=\dot{E}_{\rm w}\left(\frac{R_{\rm
ion}}{R_{\rm s}}\right)^{-2/3}=L_{\rm Edd}\biggl{(}\frac{\dot{E}_{\rm
w}}{L_{\rm Edd}}\biggl{)}^{1/3}\biggl{(}\frac{\dot{M}}{\dot{M}_{\rm
ion}}\biggl{)}^{5/12}=\biggl{(}\frac{\pi aT_{\rm
ion}^{4}(2GM_{\star})^{2}\dot{M}^{3}v_{\rm w}^{3}}{2}\biggl{)}^{1/4}\ ,$
$\displaystyle\simeq 1.2\times 10^{39}{\,\rm erg\,}M_{\star,1}^{1/2}T_{\rm
ion,4}\biggl{(}\frac{\dot{M}}{M_{\odot}\mathrm{\,yr^{-1}}}\biggl{)}^{3/4}\biggl{(}\frac{v_{\rm
w}}{200\,\mathrm{km\,s^{-1}}}\biggl{)}^{3/4}\,\,\,,\dot{M}>\dot{M}_{\rm ion}.$
(A6)

The critical mass-loss rate above which recombination defines the transient
luminosity can be estimated by equating Eqs. (A3) and (A6),

$\displaystyle\dot{M}_{\rm
ion}=\biggl{(}\frac{2^{7}\pi^{5}(2GM_{\star})^{2}c^{8}}{a^{3}\kappa^{8}T_{\rm
ion}^{12}v_{\rm w}}\biggl{)}^{1/5}\simeq
11{\,M_{\odot}\mathrm{\,yr^{-1}}\,}M_{\star,1}^{2/5}T_{\rm
ion,4}^{-12/5}\biggl{(}\frac{v_{\rm
w}}{200\mathrm{km\,s^{-1}}}\biggl{)}^{-1/5}\ .$ (A7)

For precursor emission which exhibits a steady-state plateau-like light curve,
the mass-loss must be active for a timescale $\gtrsim t_{\rm d}$ ($t_{\rm
ion}$). For a given allowed ejecta mass $M_{\rm ej}$, this implies a maximum
on the wind mass-loss rate can be obtained. For the low-$\dot{M}$ case,
demanding $t_{\rm d}\dot{M}<M_{\rm ej}$ gives

$\displaystyle\dot{M}_{\rm max}=\left(\frac{4\pi cv_{\rm w}M_{\rm
ej}}{\kappa}\right)^{1/2}\simeq
11{\,M_{\odot}\mathrm{\,yr^{-1}}\,}M_{\star,1}^{1/2}\left(\frac{M_{\rm
ej}}{M_{\star}}\right)^{1/2}\left(\frac{v_{\rm
w}}{200\mathrm{km\,s^{-1}}}\right)^{1/2}\ ,$ (A8)

and corresponding maximal steady-state luminosity is given by

$\displaystyle L_{\rm max}=7.0\times
10^{39}{\mathrm{erg\,s^{-1}}}M_{\star,1}^{5/6}\left(\frac{M_{\rm
ej}}{M_{\star}}\right)^{1/6}\left(\frac{v_{\rm
w}}{200\mathrm{km\,s^{-1}}}\right)^{5/6}\ .$ (A9)

On the other hand, for the low $\dot{M}$ case, the maximal mass-loss rate and
luminosity are given by

$\displaystyle\dot{M}_{\rm max}$ $\displaystyle=\left(\frac{(8\pi aT_{\rm
ion}^{4})^{3}v_{\rm w}^{9}M_{\rm
ej}^{8}}{(2GM_{\star})^{2}}\right)^{1/11}\simeq
11{\,M_{\odot}\mathrm{\,yr^{-1}}\,}M_{\star,1}^{6/11}T_{\rm
ion,4}^{12/11}\left(\frac{M_{\rm
ej}}{M_{\star}}\right)^{8/11}\left(\frac{v_{\rm
w}}{200\mathrm{km\,s^{-1}}}\right)^{9/11}\ ,$ (A10) $\displaystyle L_{\rm
max}$ $\displaystyle=6.8\times
10^{39}{\mathrm{erg\,s^{-1}}}M_{\star,1}^{10/11}T_{\rm
ion,4}^{20/11}\left(\frac{M_{\rm
ej}}{M_{\star}}\right)^{6/11}\left(\frac{v_{\rm
w}}{200\mathrm{km\,s^{-1}}}\right)^{15/11}\ .$ (A11)

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