# Simulating the negative jet feedback mechanism in common envelope jets
supernovae

Aldana Grichener  Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Coral Cohen
Department of Education in Science and Technology, Technion, Haifa, 3200003,
Israel<EMAIL_ADDRESS>Noam Soker Department of Physics,
Technion, Haifa, 3200003, Israel<EMAIL_ADDRESS><EMAIL_ADDRESS>Guangdong Technion Israel Institute of
Technology, Shantou 515069, Guangdong Province, China

###### Abstract

We use the stellar evolution code MESA to study the negative jet feedback
mechanism in common envelope jets supernovae (CEJSNe) where a neutron star
(NS) launches jets in the envelope of a red supergiant (RSG), and find that
the feedback reduces the mass accretion rate to be $\chi_{\rm j}\simeq
0.04-0.3$ times the mass accretion rate without the operation of jets. We
mimic the effect of the jets on the RSG envelope by depositing the energy that
the jets carry into the envelope zones outside the NS orbit. The energy
deposition inflates the envelope, therefore reducing the density in the NS
vicinity, which in turn reduces the mass accretion rate in a negative feedback
cycle. In calculating the above values for the negative jet feedback
coefficient (the further reduction in the accretion rate) $\chi_{\rm j}$, we
adopt the canonical ratio of jets power to actual accretion power of $0.1$,
and the results of numerical simulations that show the actual mass accretion
rate to be a fraction of $0.1-0.5$ of the Bondi-Hoyle-Lyttleton mass accretion
rate.

stars: jets – stars: massive – stars: neutron – binaries: close – stars:
supergiants

##  1 Introduction

Common envelope jets supernovae (CEJSNe) are transient events in which a
neutron star (NS) or a black hole (BH) launch jets as they spiral in inside
the envelope of a red supergiant (RSG) star and later thorough its core (Soker
& Gilkis, 2018) . When inside the RSG, the compact object accretes mass from
its surroundings through an accretion disk. The accretion disk launches two
jets that propagate more or less perpendicular to the orbital plane in
opposite directions. The jets deposit their kinetic energy in the giant star
and eventually explode it in a CEJSN event. If the compact object does not
penetrate the core then the transient event is termed CEJSN impostor.

A CEJSN event resembles a core collapse supernova (CCSN) in many respects. The
explosion energy in both cases is $\approx 10^{50}-10^{52}~{}\rm{erg}$ and the
ejecta mass about equals the RSG envelope mass. However, the spiraling-in
process of the NS/BH in the RSG envelope ejects some mass with which the
ejecta collide later. Therefore, the light curve itself resembles those of
energetic CCSNe with circumstellar matter (CSM).

The scenario that leads to a CEJSN (see for instance Fig. 1 in Grichener &
Soker 2019a) begins with two massive main sequence stars, one of them heavier
than the other. The more massive star $M_{1}$ evolves into a RSG while its
companion $M_{2}$ is still on the main sequence. The remaining main sequence
star is massive enough to spin-up the envelope of the RSG to be synchronized
with the orbital motion such that the tidal force vanishes. This prevents
further spiraling-in, i.e. prevents a common envelope evolution (CEE) at this
stage. The RSG keeps evolving until it explodes as a CCSNe, leaving a NS/BH
behind. If the natal kick of the compact object (mainly the NS) at this stage
is not too high, and if enough mass remains bound to the binary system, the
CCSN explosion leaves a binary system of a NS/BH and a main sequence star. The
main sequence star eventually evolves into a second RSG that can engulf the
NS/BH initiating a CEE stage that might lead to a CEJSN event. The initially
more massive star should be of mass $M_{1}\gtrsim 8.5M_{\rm\odot}$ to form a
NS, while its companion should also be massive since the NS is required to
spiral-in into the envelope of $M_{2}$ as it becomes a RSG, and for that it
cannot force the RSG envelope to synchronization. We estimate the condition to
be that its initial mass is $M_{2}\gtrsim 6M_{\odot}$ as it can grow by mass
accretion from the primary before it evolves to become a RSG. However, to
prevent fine tuning with the other parameters, like initial semi-major axis
and eccentricity, we consider CEJSNe to results from more massive stars, i.e.,
initial masses of $M_{1}\gtrsim 15M_{\odot}$ and $M_{2}\gtrsim 10M_{\odot}$.

Numerous studies of CEJSNe and CEJSN impostors were conducted over the past
few years (e.g., Soker & Gilkis 2018; Gilkis et al. 2019; Soker et al. 2019;
Grichener & Soker 2019a; López-Cámara et al. 2019; López-Cámara et al. 2020
Grichener & Soker 2021; Soker 2021). Other studies explore the properties of
NS/BH - core mergers, yet do not explicitly include jets (e.g., Fryer &
Woosley 1998; Chevalier 2012; Schrøder et al. 2020). The most generic property
of CEJSNe is non-monotonic light curves. This results from the early non-
spherical ejection of envelope gas that forms a relatively dense CSM and from
the operation of jets that might have variable intensity. One natural outcome
of the envelope mass ejection by jets is that the CEE efficiency parameter can
become larger than unity. Indeed, some scenarios of NS/BH spiraling-in inside
RSG envelopes require values of $\alpha_{\rm CE}>1$ (e.g. Fragos et al. 2019;
Broekgaarden & Berger 2021; Zevin et al. 2021; García et al. 2021). We note
that there is an ongoing debate on the accretion rate by the NS/BH relative to
the Bondi-Hoyle-Lyttleton mass accretion rate (see for instance Schreier et
al. 2021 for a discussion). More research is needed to better constrain all
parameters of CEJSNe.

Due to their wide diversity CEJSNe might be responsible for various
astrophysical phenomena from rare peculiar explosions such as iPTF14hls (Soker
& Gilkis, 2018) and AT2018cow (Soker et al., 2019), to more general processes
in high energy astrophysics, such as heavy r-process nucleosynthesis in the
early universe (Grichener & Soker 2019a; Grichener & Soker 2019b) and the PeV
neutrinos emission detected by IceCube in 2013 (Grichener & Soker, 2021). We
note that many of the scenarios for double NSs (e.g., Vigna-Gómez et al.
2018), double BHs (e.g., Mapelli 2020) and NS-BH binary (e.g., Hoang et al.
2020) formation and merger involve a phase where a NS/BH spirals in inside a
RSG envelope. Therefore, many of them must evolve through a CEJSN impostor
phase. This further stresses the increasing importance of CEJSNe and CEJSN
impostors and the ongoing research in the field.

In the present study we use approximate methods to estimate the effect of
envelope inflation on the accretion rate when a NS launches jets in the
envelope of a very massive RSG. The jets that a compact companion launches
inside the envelope of a giant star operate in a feedback cycle that has both
a positive and a negative component (see Soker 2016 for a review). The
positive feedback comes from the removal of energy and angular momentum from
the immediate vicinity of the accreting compact star. Since jets are likely to
remove high entropy gas at large velocities (about the escape speed from the
compact object and more), they reduce the pressure near the accreting object
and allow more gas to flow-in, increasing the mass accretion rate (e.g.,
Shiber et al. 2016; Chamandy et al. 2018). In the case of CEJSNe where the
accreting object is a NS, neutrino cooling carries most of the energy. This is
the positive feedback component of the cycle in the sense that a low accretion
rate does not allow for neutrino cooling to occur (Houck & Chevalier, 1991;
Chevalier, 1993, 2012).

The negative component of the feedback cycle comes from the energy and
momentum that the jets deposit to the common envelope matter (CEM) as the
compact star orbits inside the envelope (Soker 2016). This removes CEM from
the vicinity of the compact object by both direct local effects and by the
inflation of the entire envelope. The decrease in the local density reduces
the accretion rate, closing the negative feedback cycle.

In section 2 we describe our assumptions and approximated methods to mimic the
negative feedback cycle, and portray the numerical simulation of energy
injection into a spherical RSG envelope. In section 3 we describe our results.
We summarise our main conclusions in section 4.

##  2 Mimicking the negative feedback mechanism

We examine the approximate influence of the jet feedback mechanism on the mass
accretion rate and jets power by making several approximations and
assumptions, as we list below. We perform one-dimensional simulations of a RSG
to find the mass accretion rate onto a NS in a CEJSN event. However, to truly
explore the jet feedback mechanism (section 1) one needs to conduct full
three-dimensional hydrodynamical simulations that include jets in a CEE (e.g.,
Hillel et al. 2021).

###  2.1 Mass accretion rate

We take the actual NS mass accretion rate to be a fraction $\xi$ of the Bondi-
Hoyle-Lyttleton mass accretion rate (Hoyle & Lyttleton 1939; Bondi & Hoyle
1944)

$\dot{M}_{\rm BHL}=\pi R^{2}_{\rm acc}v_{\rm rel}\rho,$ (1)

where $R_{\rm acc}$ is the accretion radius, $v_{\rm rel}=v_{\rm K}-v_{\rm
rot}$ is the NS-envelope relative velocity, $v_{\rm K}$ is the Keplerian
velocity of the NS inside the RSG, $v_{\rm rot}$ is the envelope rotation
velocity and $\rho$ is the envelope density at the location of the NS. We
neglect the rotation of the envelope such that $v_{\rm rel}\simeq v_{\rm K}$.
This is a plausible assumption since the NS cannot bring the envelope to
synchronization before it enters the envelope. Namely, already when the NS is
on the RSG surface, $a=R_{\rm RSG}$, we have the inequality $v_{\rm
rot}(R_{\rm RSG})<v_{\rm Kep}(R_{\rm RSG})$. We assume a solid body rotation
of the envelope $\Omega_{\rm env}={\rm constant}$, such that $v_{\rm
rot}=\Omega_{\rm env}r$. The Keplerian velocity varies as $\propto r^{-1/2}$.
At the location of the NS $r=a$ we find that $v_{\rm rot}(a)/v_{\rm
Kep}(a)\simeq(a/R_{\rm RSG})^{3/2}[v_{\rm rot}(R_{\rm RSG})/v_{\rm Kep}(R_{\rm
RSG})]<(a/R_{\rm RSG})^{3/2}$. Therefore, when the NS is deep inside the
envelope such that $a\lesssim 0.5R_{\rm RSG}$ we can neglect the rotation of
the envelope. Moreover, to the accuracy of our study the expression for
$R_{\rm acc}$ takes the form

$R_{\rm acc}\simeq\frac{2GM_{\rm NS}}{v^{2}_{\rm K}}.$ (2)

Three-dimensional hydrodynamical simulations over the last 4 decades (e.g.,
Livio et al. 1986; Ricker & Taam 2008; Chamandy et al. 2018) have shown that
the actual accretion rate onto a NS is $\dot{M}_{\rm acc}=\xi\dot{M}_{\rm
BHL}$ where in most cases $\xi\simeq 0.1-0.5$, yet smaller values are also
possible (e.g., MacLeod & Ramirez-Ruiz 2015a; MacLeod & Ramirez-Ruiz 2015b).
Chamandy et al. 2018 consider that the jets inflate low density cocoons that
in turn lead to a lower accretion rate by the mass-accreting body in a
negative feedback cycle. The other studies do not take into account the effect
of the jet feedback mechanism.

In the present study we consider the role of jets in reducing the accretion
rate by inflating the RSG envelope, hence reducing the density at the vicinity
of the NS by a factor of

$\chi_{\rm j}\equiv\frac{\rho}{\rho_{\rm 0}},$ (3)

where $\rho$ is the density at the location of the NS in the inflated
envelope, and $\rho_{0}$ is the density in the same location in the
unperturbed envelope. This implies that the actual accretion rate due to the
effect of the jets would be

$\dot{M}_{\rm acc,j}=\chi_{\rm j}\dot{M}_{\rm acc,0}=\chi_{\rm
j}\xi\dot{M}_{\rm BHL,0},$ (4)

where the subscript ‘0’ indicates values in the unperturbed envelope. The
correct radius to evaluate the densities in equation (3) is at the location of
the NS. However, in the spherical model that we use in this study, we cannot
follow the spiraling-in orbit of the NS and the propagation of the jest inside
the envelope since the location of the NS and the jets introduce high-non-
spherical flows that our spherical model cannot account for (for these non-
spherical effects see Hillel et al. 2021). Therefore, We will consider a
general evolution of several years during which the NS plunges deep into the
envelope, and examine the value of $\chi_{\rm j}$ at three different radii
deep inside the envelope.

###  2.2 The power of the jets

The energy that the two jets, one on each side of the equatorial plane,
deposit to the RSG envelope is

$\dot{E}_{\rm 2j}=\eta\frac{GM_{\rm NS}\dot{M}_{\rm acc,j}}{R_{\rm
NS}}=\zeta\frac{GM_{\rm NS}\dot{M}_{\rm BHL,0}}{R_{\rm NS}},$ (5)

where $M_{\rm NS}=1.4M_{\rm\odot}$ and $R_{\rm NS}=12~{}\rm{km}$ are the mass
and the radius of the NS used in this study, respectively, and

$\zeta\equiv\eta\chi_{\rm j}\xi.$ (6)

We take the efficiency parameter to be $\eta=0.1$ (e.g., Schrøder et al.
2020), and perform several simulations in which we inject jets energies with
different values of $\zeta$ to the RSG envelope (subsection 2.3). In each
simulation we fix the value of $\zeta$, and we determine the value of
$\chi_{\rm j}$ by equation (3) at three orbital separations after energy
deposition to the envelope. We then compute the value of $\xi$ from
$\xi={\zeta}/{\eta\chi_{\rm j}}$ (section 3).

###  2.3 The Numerical scheme and model

We use the stellar evolution code MESA (e.g., Paxton et al. 2010; Paxton et
al. 2013; Paxton et al. 2015; Paxton et al. 2018; Paxton et al. 2019) to
obtain a non-rotating spherical massive RSG model with zero age main sequence
(ZAMS) mass of $M_{\rm ZAMS}=60M_{\rm\odot}$ and a metalicity of $Z=0.02$. We
let the star evolve until it reaches a radius of $R_{\rm
RSG}=800R_{\rm\odot}$. At this stage, the stellar mass is $M_{\rm
RSG}=59.8M_{\rm\odot}$ due to mass loss by stellar winds. We assume that the
RSG swallows a NS of mass $M_{\rm NS}=1.4M_{\rm\odot}$ at this point. The NS
launches jets inside the envelope of the RSG, driving it to expand due to
deposition of the kinetic energy of the jets to the envelope as they collide
with the envelope gas. We assume that the NS spirals-in throughout the
envelope in a constant rate until the end of our simulations. Hence, we inject
the jets energy into the envelope in spherical shells and with a constant
power per unit mass over a time of $t=0$ to $t=\tau_{\rm j}$.

At each time step $\Delta t$ we inject the energy of the jets $\Delta E_{\rm
2j}=\dot{E}_{\rm 2j}\Delta t$ into a thick spherical shell bounded from inside
by the sphere $r_{\rm in}=a(t)$, i.e., the location of the NS, and from
outside by the sphere $r_{\rm out}=700R_{\rm\odot}$, i.e., a sphere somewhat
below the surface of the RSG before the onset of the CEE. Due to envelope
expansion the density decreases with time, and we calculate at each time step
the mass $M_{\rm TS}$ in a thick shell into which we deposit the energy of the
jets. We assume that the jets propagate to large distances, and therefore
distribute the energy in a thick spherical shell from the location of the NS
at each time step to the above value of $r_{\rm out}$. Into each spherical
numerical shell of mass $\Delta M_{\rm shell}$ we inject during this time step
an energy of $\Delta E_{\rm shell}=(\Delta M_{\rm shell}/M_{\rm TS})\Delta
E_{\rm 2j}$. Typically there are hundreds up to about a thousand numerical
shells into which we deposit the energy.

We perform two sets of simulations. In the first set, we inject energy to the
envelope using equation (5), until the NS reaches an orbital separation of
$a=100R_{\rm\odot}$. We assume that the time it takes the NS to spiral in to
that radius is $\tau_{\rm j}=5~{}\rm{yr}$, which is $5.4$ times the Keplerian
orbital period on the surface of the RSG. Hence, we inject the energy from the
location of the NS to $r_{\rm out}=700R_{\rm\odot}$ during this time. We
conduct ten different simulations for ten equally spaced values of $\zeta$ in
the range $\zeta=5\times 10^{-4}-5\times 10^{-3}$. In the second set of
simulations we let the NS dive deeper, and inject the energy until the NS
reaches an orbital separation of $a=20R_{\rm\odot}$. As the spiral-in slows
down at deeper layers of the envelope (e.g., Passy et al. 2012; Ricker & Taam
2012; Ivanova et al. 2013; Sand et al. 2020; Glanz & Perets 2021), we take the
energy injection time to be $\tau_{\rm j}=10~{}\rm{yr}$ in this case. We
conduct ten simulations with the same values of $\zeta$ as in the previous
set.

###  2.4 Neglecting the orbital energy

To reveal the role of the jets we neglect the effect of the orbital energy
that the core of the RSG star and the NS release as they spiral-in towards one
another in a CEE, $E_{\rm orb}(r)\simeq GM_{\rm RSG}(r)M_{\rm NS}/2r$. We use
a massive RSG model such that $E_{\rm orb}(r)$ in the orbital separations
range where we follow the spiraling-in NS is not only smaller than the energy
that the jets deposit in the RSG envelope, $E_{\rm 2j}(r)$, but it is also
smaller than the binding energy of the envelope in this volume. This assures
us that the orbital energy does not change much the envelope, which is
inflated mainly by the energy that the two jets deposit as they collide with
the envelope gas.

The energy that the jets deposit to the RSG envelope until the NS reaches the
radii $r=20R_{\rm\odot}$ and $r=100R_{\odot}$ are (for our different models)
$E_{\rm 2j}(20R_{\rm\odot})=5.5\times 10^{50}-5.5\times 10^{51}~{}\rm{erg}$
and $E_{\rm 2j}(100R_{\rm\odot})=5.7\times 10^{49}-5.7\times
10^{50}~{}\rm{erg}$, respectively. Although we take the duration of the
spiraling-in to $r=20R_{\odot}$ to be twice as long as that to
$r=100R_{\odot}$, the extra spiraling-in takes place in a much denser envelope
that results in a much larger mass accretion rate and hence in jets with much
higher powers. Therefore $E_{\rm 2j}(20R_{\rm\odot})$ is larger than $E_{\rm
2j}(100R_{\rm\odot})$ by more than a factor of two.

At these radii, we find that the orbital energy of the unperturbed system is
$E_{\rm orb}(20R_{\rm\odot})=8\times 10^{48}~{}\rm{erg}$ and $E_{\rm
orb}(100R_{\rm\odot})=1.6\times 10^{48}~{}\rm{erg}$. We see indeed that
$E_{\rm 2j}\gg E_{\rm orb}$ in our simulations. Moreover, the binding energies
of the unperturbed envelope down to a radius of $r=20R_{\rm\odot}$ and
$r=100R_{\rm\odot}$ are $E_{\rm bind}(20R_{\rm\odot})=2.8\times
10^{49}~{}\rm{erg}$ and $E_{\rm bind}(100R_{\rm\odot})=4.1\times
10^{48}~{}\rm{erg}$, respectively. We note that in both cases $E_{\rm
bind}(r)>E_{\rm orb}(r)$. Therefore, we can safely neglect the orbital energy
in the simulations.

###  2.5 Neglecting dynamical effects

We can neglect the spin-up of the envelope by the spiralling-in NS. The moment
of inertia of the unperturbed spherical RSG envelope is $I_{\rm env,0}=5\times
10^{5}M_{\odot}R^{2}_{\odot}$. The moment of inertia of the NS on the surface
of the RSG is larger, $I_{\rm orb,0}=M_{\rm NS}R^{2}_{\rm RSG}=9\times
10^{5}M_{\odot}R^{2}_{\odot}$. However, after envelope expansion to a radius
of $R_{\rm RSG}\simeq 4.1\times 10^{3}-9.6\times 10^{3}R_{\odot}$ depending on
the value of $\zeta$, we find a gross typical value of $I_{\rm env}\simeq
10^{8}M_{\odot}R^{2}_{\odot}\gg I_{\rm orb,0}$. We therefore expect, assuming
a solid body rotation, the angular velocities of the envelope in regions where
the NS spirals-in to be much lower than the Keplerian velocities at the
respective orbital locations of the NS.

We also neglect the dynamical response time of the envelope. Namely, we will
use the stellar evolutionary model under the assumption of an hydrostatic
equilibrium of the inflated RSG envelope. This is not an accurate assumption
for the very outer zones of the inflated envelope. We inject the jets energy
within 5 or 10 years. The dynamical time at the stellar radius $R_{\rm RSG}$
is $\tau_{\rm RSG,d}=(G\bar{\rho})^{-1/2}=5(R_{\rm
RSG}/10^{4}R_{\odot})^{3/2}~{}\rm{yr}$ for an average density $\bar{\rho}$ of
the RSG of mass $M_{\rm RSG}=59.8M_{\odot}$. In inner zones the dynamical
times are much shorter and therefore the RSG has time to hydrostatically
arrange itself during the spiraling-in phase. Namely, besides the very outer
regions, the model is stable. As we are interested in the response of the
envelope at radii of $r<1000R_{\odot}$ (see section 3), the effect of not
including the dynamical response time of the envelope is not substantial. In
any case, we expect that the outer parts of the inflated RSG envelope will be
lost in a strong wind during the CEE.

###  2.6 Spherical symmetry and limited evolution time

Three-dimensional hydrodynamical simulations have shown over more than three
decades that CEE interaction is highly non-spherical even before we consider
the effects of jets (e.g., Rasio & Livio 1996; Livio & Soker 1988; Ricker &
Taam 2008; Passy et al. 2012; Nandez et al. 2014; Ohlmann et al. 2016; Iaconi
et al. 2017; MacLeod et al. 2018; Glanz & Perets 2021). The jets add another
prominent non-spherical component to this asymmetry (e.g., Moreno Méndez et
al. 2017; Shiber & Soker 2018; López-Cámara et al. 2019; Shiber et al. 2019;
López-Cámara et al. 2020; Schreier et al. 2021). With the numerical scheme we
use, we are forced to assume a spherically symmetric energy deposition and
envelope response.

The non-spherical effects might be very significant at the final CEE phases
(e.g., Soker 1992; Reichardt et al. 2019; García-Segura et al. 2020; Zou et
al. 2020) and might include the formation of a circumbinary thick disk (e.g.,
Kashi & Soker 2011; Chen & Podsiadlowski 2017). For that reason, and for
neglecting the orbital energy (section 2.4) that increases to non-negligible
values at small orbital separations, we do not continue the evolution to late
phases when the orbital separation decreases below about $a=20R_{\odot}$.

##  3 Results: The effect of jets

We determine the envelope structure at $t=\tau_{\rm j}$, i.e., at the end of
jet-energy deposition in each one of our simulations. We present the results
for $\zeta=2.5\times 10^{-3}$ for both $\tau_{\rm j}=5~{}\rm{yr}$, where the
NS spirals in down to $a=100R_{\rm\odot}$ (Fig. 1, green curve; Fig. 2, red
curve), and $\tau_{\rm j}=10~{}\rm{yr}$, where the NS dives deeper to
$a=20R_{\rm\odot}$ (Fig. 1, purple curve; Fig. 3, red curve). For each case we
also present the stellar density profile at these times for a case where we do
not inject energy to the star (blue curve).

Figure 1: Mass profile of the red supergiant (RSG) for $\zeta=2.5\times
10^{-3}$ in the first set of our simulations after $5~{}\rm{yr}$ when the
orbital separation is $a=100R_{\rm\odot}$ (green curve), and in the second set
of our simulations after $10~{}\rm{yr}$ when the orbital separation is
$a=20R_{\rm\odot}$ (purple curve).  Figure 2: Density profile of the red
supergiant (RSG) in the first set of our simulations after $5~{}\rm{yr}$ when
the orbital separation is $a=100R_{\rm\odot}$, for $\zeta=2.5\times 10^{-3}$
(red curve) and for $\zeta=0$ (no energy injection; blue curve).  Figure 3:
Same as Fig. 2 for our second set of simulations after $10~{}\rm{yr}$ of
energy injection, when the orbital separation is $a=20R_{\rm\odot}$.

Figs. 2 and 3 show the extended RSG envelope that the jets inflate. In the
case where we mimic the spiraling-in of the NS down to $100R_{\rm\odot}$ (Fig.
2), the jets inflate the envelope of the giant star to a radius of $R_{\rm
RSG}=6.4\times 10^{3}R_{\rm\odot}$. When the NS spirals-in deeper, down to
$20R_{\rm\odot}$ (Fig. 3), the envelope expands to an even higher radius of
$R_{\rm RSG}=9\times 10^{3}R_{\rm\odot}$.

Our aim is to determine the values of the feedback coefficient $\chi_{\rm j}$
under the assumption that the jet feedback mechanism operates in our scenario.
In each of our simulations we inject energy to the envelope with a fixed value
of $\zeta$ (equation 6). We use the densities depicted in Figs. 2 and 3 to
find the values of $\chi_{\rm j}$ according to equation (3). Since there is a
large uncertainty regarding the propagation of the jets inside the RSG, we
compute $\chi_{\rm j}$ at three radii at the end of each simulation (5 or 10
years) according to equation (3). We limit the calculations to radii deep
within the extended envelope, where dynamical effects can be neglected since
the RSG envelope in these regions has time to rearrange itself in a
hydrostatic equilibrium within a time much shorter than our simulation time
(subsection 2.5).

In Figs. 4 and 5 we show the values of $\chi_{\rm j}$ computed at three
different radii inside the RSG as we indicate in the inset. At these radii we
calculate the density ratio of the perturbed to unperturbed envelope and find
the values of $\chi_{\rm j}$ as function of $\eta\xi=\zeta/\chi_{\rm j}$. We
find that the range of $\chi_{\rm j}$ that corresponds to the values of $\xi$
from previous hydrodynamical simulations is not much different when computing
$\chi_{\rm j}$ in the three different radii above.

Figure 4: The negative jet feedback coefficient $\chi_{\rm j}$ computed at
$r=200R_{\rm\odot}$ (blue dots), at $r=300R_{\rm\odot}$ (red dots) and at
$r=400R_{\rm\odot}$ (orange dots) for the first set of simulations
($a=100R_{\rm\odot}$, $\tau_{\rm j}=5~{}\rm{yr}$). As $\zeta=\eta\chi_{\rm
j}\xi$ each value of $\chi_{\rm j}$ corresponds to a different value of
$\eta\xi$.  Figure 5: Same as Fig. 4 for our second set of simulations
($a=20R_{\rm\odot}$, $\tau_{\rm j}=10~{}\rm{yr}$).

We note that although at the end of the first set of simulations the NS is at
$a=100R_{\odot}$ and at the end of the second set of simulations it is at
$a=20R_{\odot}$, we calculate the values of $\chi_{\rm j}$ (equation 3) at
somewhat larger radii of $r=200$, $300$ and $400R_{\odot}$. The reason for
this is that we inject the energy in spherical shells, and in our spherical
stellar model we cannot take into account the local effect of the jets, which
are larger near the NS location, including to zones inner to the NS orbit. Our
spherical modeling is limited in its accuracy, and gives only the order of
magnitude of the coefficient of the jet feedback mechanism $\chi_{\rm j}$.

As in most cases $\xi\simeq 0.1-0.5$ (e.g., Livio et al. 1986; Ricker & Taam
2008; Chamandy et al. 2018), we can conclude from Figs. 4 and 5 that the
effect of the jets in reducing the NS mass accretion rate in a CEE, i.e., the
jets negative feedback mechanism coefficient that is compatible with the above
values of $\xi$ is $\chi_{\rm j}\approx 0.04-0.3$.

##  4 Summary and discussion

We studied the effect of the negative jet feedback mechanism on the mass
accretion rate by a NS that spirals-in inside the envelope of a RSG and
launches jets, i.e., a CEJSN (or imposter) event. We performed one-dimensional
simulations of a massive RSG star using the stellar evolution code MESA
(section 2). We mimicked the effect of jets that a NS launches inside the RSG
envelope by depositing energy to the envelope outside the NS location with a
power according to equation (5). We performed two sets of simulations where we
followed the NS spiraling-in down to orbital separations of either
$a=100R_{\rm\odot}$ or $a=20R_{\rm\odot}$. For each set we performed
simulations with different values $\zeta$ that is the ratio of jets power to
the BHL accretion power from the unperturbed envelope (defined in equations 5
and 6).

As we demonstrate for two cases with $\zeta=2.5\times 10^{-3}$ in Figs. 2 and
3, in all our simulations the jets substantially inflate the envelope. We take
the negative jet feedback mechanism efficiency $\chi_{\rm j}$ to be the
density ratio between the perturbed and unperturbed envelope (equation 3). Due
to the limitation of the spherical model that we used to mimic the highly non-
spherical effects of the jets on the RSG envelope we take the density ratio
somewhat outside the NS orbital radius at the end of the simulations. Since
the exact radius is highly uncertain, we check the value of $\chi_{\rm j}$ at
three radii as we indicate in Figs 4 and 5.

Performing a a simple spherically symmetric mimicking of the energy of jets
launched by a NS inside a RSG envelope in a CEJSN (or impostor) event, we
could determine the reduction factor in the jets power due to the negative jet
feedback mechanism. The main conclusion from this study is that under our
assumptions, for values of $\eta\simeq 0.1$ as observations of jets indicate,
and for $\xi\simeq 0.1-0.5$ as numerical simulations indicate, the jet
feedback mechanism reduced the mass accretion rate onto a NS in the envelope
of a RSG by a factor $\chi_{\rm j}\simeq 0.04-0.3$ (Figs. 4 and 5).

## Acknowledgments

We thank an anonymous referee for very detailed comments that helped in
improving our paper. This research was supported by a grant from the Israel
Science Foundation (769/20).

Data availability

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

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