# On the response of massive main sequence stars to mass accretion and outflow
at high rates

Ealeal Bear  Noam Soker  0000-0003-0375-8987 Department of Physics, Technion,
Haifa, 3200003, Israel<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

With a one-dimensional stellar evolution model, we find that massive main-
sequence stars can accrete mass at very high mass accretion rates without
expanding much if they lose a significant fraction of this mass from their
outer layers simultaneously with mass accretion. We assume the accretion
process is via an accretion disk that launches powerful jets from its inner
zones. These jets remove the outer high-entropy layers of the mass-accreting
star. This process operates in a negative feedback cycle, as the jets remove
more envelope mass when the star expands. With the one-dimensional model, we
mimic the mass removal by jets by alternative mass addition and mass removal
phases. For the simulated models of $30M_{\odot}$ and $60M_{\odot}$, the star
does not expand much if we remove more than about half of the added mass in
not-too-short episodes. This holds even if we deposit the energy the jets do
not carry into the envelope. As the star does not expand much, its
gravitational potential well stays deep, and the jets are energetic. These
results are relevant to bright transient events of binary systems powered by
accretion and the launching of jets, e.g., intermediate luminosity optical
transients, including some luminous red novae, the grazing envelope evolution,
and the 1837-1856 Great Eruption of Eta Carinae.

###### keywords:

Stars: jets; stars: massive; stars: mass-loss; Common envelope evolution; Red
supergiant stars

##  1 Introduction

Intermediate luminosity optical transients (ILOTs; Berger et al. 2009; Kashi &
Soker 2016a; Muthukrishna et al. 2019) are a heterogeneous group of transients
in the visible band with typical peak luminosities between those of classical
novae and those of supernovae (e.g., Mould et al. 1990; Bond et al. 2003; Rau
et al. 2007; Ofek et al. 2008; Mason et al. 2010; Kasliwal 2011; Tylenda et
al. 2013; Kasliwal et al. 2012; Kaminski et al. 2018; Boian & Groh 2019; Cai
et al. 2019; Jencson et al. 2019; Kashi et al. 2019; Pastorello et al. 2019;
Blagorodnova et al. 2020; Banerjee et al. 2020; Howitt et al. 2020; Jones
2020; Kamiński et al. 2020, 2021; Klencki et al. 2021; Stritzinger et al.
2020a, b; Blagorodnova et al. 2021; Mobeen et al. 2021; Pastorello et al.
2021, 2023; Addison et al. 2022; Cai et al. 2022a; Wadhwa et al. 2022;
Kamiński et al. 2023; Karambelkar et al. 2023; Mobeen et al. 2024; Kaminski
2024). There is no consensus on the naming of this heterogeneous transient
group. We use the term ILOTs for all transients powered by gravitational
energy, whether triggered by a merger process or a mass transfer. Other
researchers (e.g., Jencson et al. 2019; Cai et al. 2022b) use other terms,
like gap transients, luminous red novae (LRNe), red novae, or intermediate
luminosity red transients. There is also no consensus on the sub-classes of
this group of transients (e.g., Kashi & Soker 2016a versus Pastorello et al.
2019 and Pastorello & Fraser 2019). We use the term LRNe for ILOTs that are
powered by a complete merger that leaves one stellar remnant (Kashi & Soker,
2016a), and for a common envelope evolution (CEE) as during the bright phase
there is only one photosphere. We also include ILOTs from the grazing envelope
evolution of low-mass stars (to distinguish from eruptions of luminous blue
variables) under LRNe.

CEE LRNe might be powered even by a sub-stellar companion, i.e., a planet or a
brown dwarf (e.g., Retter & Marom 2003; Metzger, Giannios, & Spiegel 2012;
Yamazaki, Hayasaki, & Loeb 2017; Kashi et al. 2019; Gurevich, Bear, & Soker
2022; De et al. 2023; O’Connor et al. 2023), although most events are thought
to be powered by a stellar companion (e.g., Tylenda et al. 2011; Ivanova et
al. 2013; Nandez et al. 2014; Kamiński et al. 2015; Pejcha et al. 2016a, b;
Soker 2016a; Blagorodnova et al. 2017; MacLeod et al. 2017, 2018; Segev et al.
2019; Howitt et al. 2020; MacLeod & Loeb 2020; Qian et al. 2020; Schrøder et
al. 2020; Blagorodnova et al. 2021; Addison et al. 2022; Zhu et al. 2023;
Tylenda et al. 2024). In CEE LRNe, the primary energy sources might be the
dynamical interaction of the companion inside the envelope of the engulfing
larger star, the recombination of the ejected common envelope (e.g., Matsumoto
& Metzger 2022), or the accretion energy onto the more compact companion. The
dynamical interaction is the gravitational energy of the companion that
spirals in and ejects the envelope. Radiation can result from the collision of
ejected envelope gas with itself in and near the equatorial plane (e.g.,
Pejcha et al. 2016a, b, 2017; Metzger & Pejcha 2017; Hubová, & Pejcha 2019),
or from the heating of the envelope by the spiraling-in companion.

We adopt the view that the most efficient energy source to power a bright ILOT
is the accretion energy of gas onto one of the two stars of a binary system
followed by the launching of jets (e.g., Soker 2020). The bipolar ejecta of
spatially-resolved ILOTs (e.g., Kaminski 2024) suggest that most, or even all,
bright ILOTs are powered by jets (e.g., Soker 2023, 2024). The companion
accretes mass via an accretion disk and launches jets. When the companion is a
neutron star or a black hole the event can mimic a core-collapse supernova
(e.g., Soker & Gilkis 2018; Gilkis et al. 2019; Grichener & Soker 2019;
Yalinewich & Matzner 2019; Schreier et al. 2021), and the event is termed
common envelope jet supernova rather than an LRN. In a CEE LRN (including an
LRN in a grazing envelope evolution; Soker 2016a), the companion launches jets
that collide with the common envelope or with the circumstellar material
(CSM), a process that transfers kinetic energy to thermal energy and radiation
(e.g., Soker 2020; Soker & Kaplan 2021).

Luminous blues variables are also a group of ILOTs. The most famous is the
Great Eruption of Eta Carinae, which ejected a bipolar nebula, the Homunculus.
The bipolar morphology of the Homunculus testifies that jets powered the Great
Eruption of Eta Carinae (e.g., Soker 2001; Kashi & Soker 2010). The mass of
the companion that accreted the gas during the great eruption and launched the
jets is $M_{2}\simeq 30-80M_{\odot}$ (Kashi & Soker, 2016b). We will examine
the response of such a companion to mass accretion accompanied by jet
launching.

Another group of ILOTs includes pre-explosion outbursts, where a massive star
experiences an outburst years to days before a core-collapse supernova
explosion. The energy source of these outbursts can be accretion onto a
companion (e.g., Mcley & Soker 2014; Danieli & Soker 2019; Tsuna et al. 2024;
see Soker 2022 for a review). The compact companion accretes mass and launches
jets that, when colliding with the CSM, power the pre-explosion outburst. We
here consider cases where the compact object is a main sequence companion with
a mass of $M_{2}\ga 1.5M_{\odot}$ and therefore has a radiative envelope.

In this study, we examine one aspect of ILOTs that are powered by the
accretion process onto massive main sequence stars, i.e., those that have a
radiative envelope. We further consider that the mass-accreting main sequence
star launches jets. We will not study the entire accretion via an accretion
disk that launches jets but rather mimic the process with the numerical
spherical code mesa (section 2). We present our results in section 3. In
section 4 we summarize our results, compare them to a study by Schürmann &
Langer (2024), and discuss their implications to ILOTs and CEE with massive
main sequence companions.

##  2 Method

We used version 23.05.1 of the stellar evolution code Modules for Experiments
in Stellar Astrophysics (mesa; Paxton et al. 2011, 2013, 2015, 2018, 2019) in
its single star mode. We evolve a stellar model with an initial zero-age main
sequence (ZAMS) mass of $M=30M_{\odot}$ and $M=60M_{\odot}$ and metallicity of
$z=0.019$. We start mass accretion and energy deposition on the main sequence
at the age of $t_{\rm MS}=1.3\times 10^{6}{~{}\rm yr}$ in all simulations of
$M=30M_{\odot}$ and $t_{\rm MS}=0.9\times 10^{6}{~{}\rm yr}$ in all
simulations of $M=60M_{\odot}$. We base our simulations on the example of 20M
pre ms to core collapse for both $M=30M_{\odot}$ and $M=60M_{\odot}$; all
other parameters remain as in the default of mesa.111 The default capabilities
of mesa-single relay on the MESA EOS that is a blend of the OPAL (Rogers &
Nayfonov, 2002), SCVH (Saumon et al., 1995), FreeEOS (Irwin, 2004), HELM
(Timmes & Swesty, 2000), PC (Potekhin & Chabrier, 2010), and Skye (Jermyn et
al., 2021) EOSes. Radiative opacities are primarily from OPAL (Iglesias &
Rogers, 1993, 1996), with low-temperature data from Ferguson et al. (2005) and
the high-temperature, Compton-scattering dominated regime by Poutanen (2017).
Electron conduction opacities are from Cassisi et al. (2007) and Blouin et al.
(2020). Nuclear reaction rates are from JINA REACLIB (Cyburt et al., 2010),
NACRE (Angulo et al., 1999) and additional tabulated weak reaction rates
Fuller et al. 1985; Oda et al. 1994; Langanke & Martínez-Pinedo 2000.
Screening is included via the prescription of Chugunov et al. (2007). Thermal
neutrino loss rates are from Itoh et al. 1996.

We mimic the process by which the jets remove envelope mass from the mass-
accreting star using small alternating pulses of accretion and mass removal;
each mass-addition part has a duration of $\Delta t_{\rm p}$. The code mesa
removes and adds mass with the same properties, like entropy, as in the
outermost shell of the stellar model. Our pulse has two parts. In the first
part, when we accrete mass, we also deposit energy. In the second part of the
pulse, we remove a mass, either $\eta_{\rm MR}=\frac{2}{3}$ or $\eta_{\rm
MR}=\frac{1}{3}$ of the mass accreted in the first part of the pulse. We
deposit energy to the envelope’s outer $0.1$ (by radius). The amount of added
energy is $\eta_{\rm acc}=0.25$ of the accreted energy. We found (see section
3), that if we remove a small fraction which for many simulations is
$\eta_{\rm MR}=\frac{1}{3}$ of the accreted mass, the star inflates to a very
large radii, that violent the assumption of the model, like hydrostatic
equilibrium. In all simulations, we start the pulses once the star is in the
main sequence; the exact time at the main sequence when we start the
simulations has little influence on the results.

##  3 Results

We aim to present a process by which massive main sequence stars can accrete
mass at high rates without expanding much. We focus, therefore, on presenting
the evolution of the stellar radius following a high mass accretion rate under
different conditions. We emphasize that the process we are mimicking is of
accretion from an accretion disk and a mass loss by jets launched from the
inner zone of the accretion disk and its boundary with the stellar surface.
The jets remove more material from the stellar outskirts. The inflow-outflow
occurs simultaneously. However, we alternate mass accretion and mass loss
because of the limitations of the one-dimensional stellar model. We refer to
each mass addition phase as a pulse of duration $\Delta t_{\rm p}$. During the
mass addition pulse, we also inject energy into the envelope. After this
pulse, we remove a mass. In most cases, the duration of mass removal is equal
to that of mass addition (the pulse duration). In two simulations, the mass
removal time is longer.

To avoid complicated graphs, the first three figures present only the mass
addition and energy deposition pulses but not the mass removal phases. This
causes a discontinuity in the lines from one pulse to the next. Namely, after
the mass removal phase, the star contracts, and its mass decreases; therefore,
the following line segment (next pulse) starts below and to the left of the
endpoint of the line segment of the previous pulse.

Figure 1 shows the evolution of the stellar radius of a main sequence star
with $M_{\rm ZAMS}=60M_{\odot}$ for different pulse durations as indicated in
the inset and caption. The total number of pulses is different between the
simulations (see caption), as we stop the simulations after we identify the
stellar behavior, namely if it rapidly expands if it reaches a more or less
constant radius or only a very slow expansion when we add more mass. In the
four cases of Figure 1 we add mass and energy at rates of $\dot{M}_{\rm
add}=0.03M_{\odot}{~{}\rm yr}^{-1}$ and $\dot{E}_{\rm add}=6.32\times
10^{39}{~{}\rm erg}{~{}\rm s}^{-1}$, respectively, and remove mass at a rate
of $\dot{M}_{\rm rem}=-0.02M_{\odot}{~{}\rm yr}^{-1}$. The net average mass
accretion rate and energy power are $\dot{M}_{\rm acc}=0.005M_{\odot}{~{}\rm
yr}^{-1}$ and $E_{\rm acc}=3.16\times 10^{39}{~{}\rm erg}{~{}\rm s}^{-1}$,
respectively. The power of the accretion is a fraction of $\eta_{\rm
acc}=0.25$ of the gravitational energy that the accreted mass releases (see
second 2).

Figure 1: Stellar radius (on a log scale) vs. stellar mass of a main sequence
star with $M_{\rm ZAMS}=60M_{\odot}$, and for four different mass addition
pulse durations, as follows. We present only the evolution during mass
addition (pulses) but not during mass removal; this causes discontinuity in
each line. Black line (total of 24 pulses) is for a pulse duration of $\Delta
t_{\rm p}=2{~{}\rm yr}$, blue line (48 pulses) is for $\Delta t_{\rm
p}=1{~{}\rm yr}$, cyan line (24 pulses) represents pulses of $\Delta t_{\rm
p}=0.5{~{}\rm yr}$, and the red line (72 pulses) represents pulses of $\Delta
t_{\rm p}=0.1{~{}\rm yr}$. For all pulses, the mass accretion rate when we add
mass is $\dot{M}_{\rm add}=0.03M_{\odot}{~{}\rm yr}^{-1}$. The mass removal
rate when we remove mass is $\dot{M}_{\rm rem}=-0.02M_{\odot}{~{}\rm
yr}^{-1}$, i.e., $\eta_{\rm MR}=|\dot{M}_{\rm rem}|/\dot{M}_{\rm add}=0.67$,
the mass addition and removal times are equal, and the net mass accretion rate
is $\dot{M}_{\rm acc}=0.01M_{\odot}{~{}\rm yr}^{-1}$. We add energy to the
outer $10\%$, of the radius of the stellar envelope when we add mass at a
power of $\dot{E}_{\rm add}=6.32\times 10^{39}{~{}\rm erg}{~{}\rm s}^{-1}$,
which is a fraction of $\eta_{\rm acc}=0.25$ of the gravitational energy that
the accreted mass releases (see second 2). The net energy deposition power (as
we add energy only during half the cycle) is $E_{\rm acc}=3.16\times
10^{39}{~{}\rm erg}{~{}\rm s}^{-1}$.

In three cases of Figure 1, the star does not expand much; in one case, it
rapidly expands. The four simulations of Figure 1 have $\eta_{\rm
MR}=|\dot{M}_{\rm rem}|/\dot{M}_{\rm add}=0.67$. We found that when $\eta_{\rm
MR}=0.33$, i.e., when we remove less mass after a mass addition pulse, the
star rapidly expands in all cases, namely, for all pulse durations as in
Figure 1. This shows that to prevent rapid stellar expansion, the star must
lose its outer high-entropy layers. According to our assumption, the jets from
the accretion disk carry high-entropy gas, remove energy, and remove high-
entropy gas from the envelope outskirts. We mimic this process.

Figure 1 shows that when the pulse duration is short, the star rapidly expands
(red line for $\Delta t_{\rm p}=0.1{~{}\rm yr}$). We present more simulated
cases in Figure 2 to investigate this phenomenon. The thin-red line presents
the same case as the thin-red line in Figure 1.

Figure 2: Similar to Figure 1 but for more cases. The thin-red line (72
pulses) is the same as the thin-red line in Figure 1, i.e., mass addition
pulse lasts for $\Delta t_{\rm p}=0.1{~{}\rm yr}$ and the mass removal rate is
$\eta_{\rm MR}=0.67$ of the mass addition rate and last for the same length of
$0.1{~{}\rm yr}$. The magenta line represents the same $\eta_{\rm MR}=0.67$
but for a pulse duration of $\Delta t_{\rm p}=0.15{~{}\rm yr}$ (72 pulses).
The green line (24 pulses) is for a variation of the parameters where we let
the star evolve on the red track. i.e., $\Delta t_{\rm p}=0.1{~{}\rm yr}$ when
$\eta_{\rm MR}=0.67$ (for 55 pulses) and when the stellar radius is
$16R_{\odot}$ we switch to $\Delta t_{\rm p}=0.5{~{}\rm yr}$ at $\eta_{\rm
MR}=0.67$ (for 24 additional pulses). The blue line represents simulation with
$\Delta t_{\rm p}=0.1{~{}\rm yr}$ but with a larger mass removal rate such
that $\eta_{\rm MR}=0.8$. In this case, the star does not expand (336 pulses).
The black line (168 pulses) represents a simulation with $\Delta t_{\rm
p}=0.1{~{}\rm yr}$ and mass addition rate of $\dot{M}_{\rm
add}=0.03M_{\odot}{~{}\rm yr}^{-1}$, but the mass removal rate is
$\dot{M}_{\rm rem}=-0.004M_{\odot}{~{}\rm yr}^{-1}$ and it lasts for $5\Delta
t_{\rm p}=0.5{~{}\rm yr}$; this implies an effective value of $\eta_{\rm
MR}=0.67$. In this case, the net average mass accretion rate is $\dot{M}_{\rm
acc}=0.00167M_{\odot}{~{}\rm yr}^{-1}$; the star does not expand. For all
simulations in this figure, the power at which we add energy to the envelope
during the mass addition phase (the pulse) is $\dot{E}_{\rm add}=6.32\times
10^{39}{~{}\rm erg}{~{}\rm s}^{-1}$ which is equal to $\eta_{\rm acc}=0.25$
accreted energy and is inserted in the outer $10\%$ of the star (by radius;
see text).

Firstly, we note from Figure 2 that even for $\Delta t_{\rm p}=0.15{~{}\rm
yr}$, keeping other parameters as in the cases of Figure 1, the star rapidly
expands (the thick magenta line). However, if we remove more mass during the
mass removal phase, the star does not expand much, as the blue line shows for
the case where the mass removal rate is $\eta_{\rm MR}=0.8$ of the mass
accretion rate; namely, the new average mass accretion rate is $\dot{M}_{\rm
acc}=0.003M_{\odot}{~{}\rm yr}^{-1}$. This simulation was run for 336 pulses.
The green line in Figure 2 shows that once the star rapidly expands, it is
difficult to halt this expansion. The green line represents a simulation that
starts like the thin-red line, with $\Delta t_{\rm p}=0.1{~{}\rm yr}$ and
$\eta_{\rm MR}=0.67$. Then, when the star has expanded as a result of mass
accretion to a radius of $16R_{\odot}$ we switch to $\Delta t_{\rm
p}=0.5{~{}\rm yr}$ keeping $\eta_{\rm MR}=0.67$. The star continues to expand,
although at a somewhat slower rate. We recall that when we start with $\Delta
t_{\rm p}=0.5{~{}\rm yr}$ the star does not expand (cyan line in figure 1).

In another simulation, we kept the ratio of total removed mass to total added
mass but removed the mass at a lower rate for a longer time: The duration of
the accretion pulses and the mass addition rate are still $\Delta t_{\rm
p}=0.1{~{}\rm yr}$ and $\dot{M}_{\rm add}=0.03M_{\odot}{~{}\rm yr}^{-1}$,
respectively, but the mass removal rate is $\dot{M}_{\rm
rem}=-0.004M_{\odot}{~{}\rm yr}^{-1}$ during a time of $5\Delta t_{\rm
p}=0.5{~{}\rm yr}$ (for 168 pulses); namely, effectively we have $\eta_{\rm
MR}=0.67$. The total cycle time is $0.1+0.5=0.6{~{}\rm yr}$, with a net
addition of $0.01M_{\odot}$. The net average mass accretion rate is
$\dot{M}_{\rm acc}=0.00167M_{\odot}{~{}\rm yr}^{-1}$.

We also examine the response of a main sequence stellar model of $M_{\rm
ZAMS}=30M_{\odot}$. We present the results of some cases in Figure 3. The mass
addition rate during the mass addition phases (the pulses) is $\dot{M}_{\rm
add}=0.015M_{\odot}{~{}\rm yr}^{-1}$, half of that for the $M_{\rm
ZAMS}=60M_{\odot}$ simulations. Figure 3 shows that when we remove enough mass
in the mass removal phase of each cycle, namely, $\eta_{\rm MR}\ga 2/3$ (we
did not scan the parameter space as we explain in section 4) the star does not
expand much. The black and magenta lines represent such cases. The red line
represents the case with $\Delta t_{\rm p}=0.5{~{}\rm yr}$ and a low value of
$\eta_{\rm MR}=1/3$; expansion occurs later after more mass is accreted. For
$\Delta t_{\rm p}=1{~{}\rm yr}$ and $\eta_{\rm MR}=1/3$ that the blue line
represents, expansion occurs earlier since insufficient mass is being removed.

Figure 3: Stellar radius (on a log scale) vs. stellar mass. Similar to 1 but
for an initial mass of $M_{\rm ZAMS}=30M_{\odot}$. This image represents
different accretion pulses. Black line (36 pulses) represents $\Delta
t_{p}=1{~{}\rm yr}$ and mass removal at $\eta_{\rm MR}=\frac{2}{3}$. Blue line
(18 pulses) represents $\Delta t_{\rm p}=1{~{}\rm yr}$ and mass removal at
$\eta_{\rm MR}=\frac{1}{3}$. Magenta line (36 pulses) represents $\Delta
t_{\rm p}=0.5{~{}\rm yr}$ and $\eta_{\rm MR}=\frac{2}{3}$. Red line (36
pulses) represents $\Delta t_{\rm p}=0.5{~{}\rm yr}$ for mass removal rate of
$\eta_{\rm MR}=\frac{1}{3}$. The accretion rate for all lines is
$0.015M_{\odot}{~{}\rm yr}^{-1}$, which is calibrated by half of
$0.03M_{\odot}{~{}\rm yr}^{-1}$ which was the accretion rate of $M_{\rm
ZAMS}=60M_{\odot}$. Energy is deposited along with the mass at a power of
$\dot{E}_{\rm add}=2.34\times 10^{39}$, which is $\eta_{\rm acc}=0.25$ of the
accretion energy; the energy is deposited in the outer $10\%$ of the star (by
radius).

In Figure 4, we present entropy and density profiles at several points as we
add more mass to the $M_{\rm ZAMS}=60M_{\odot}$ stellar model. The insets give
the pulse number of each profile. The first row shows the entropy profile
given by MESA as a function of the mass coordinate in the outer region where
we add mass, while the second row gives the entropy as a function of radius.
The bottom row gives the density as a function of radius. Figure 5 presents
similar profiles for two simulations of $M_{\rm ZAMS}=30M_{\odot}$.

Figure 4: Entropy (upper two rows) and density (lower row) profiles along
several evolutionary points as we add mass, for two simulations with $M_{\rm
ZAMS}=60M_{\odot}$ and $\Delta t_{\rm p}=0.5{~{}\rm yr}$. Entropy is as given
by mesa, which is the entropy per gram in units of $N_{\rm A}k_{\rm B}$ (the
Avogadro number times Boltzmann constant). The three left panels are for the
simulation with large removal mass $\eta_{\rm MR}=2/3$, and the three right
panels (with dotted lines) are for a simulation with low removal mass
$\eta_{\rm MR}=1/3$. The color of the lines is according to the pulse number
given in each column’s inset.  Figure 5: Similar to Figure 4, but for $M_{\rm
ZAMS}=30M_{\odot}$ and $\Delta t_{\rm p}=0.5{~{}\rm yr}$. The left column is
for $\eta_{\rm MR}=2/3$, and the right column is for $\eta_{\rm MR}=1/3$.

The density profiles in the bottom row of Figures 4 and 5 show, as expected,
the expansion of the envelope. In the case of large mass removal (left
column), the star does not expand much.

The entropy profiles, shown in the upper row as a function of mass, are
important to our discussion. They show that as we add mass, a sharp entropy
rise develops at the edge of the envelope, i.e., a sharp entropy rise in a
very thin mass layer. If we remove this high-entropy thin mass layer, the star
contracts. This explains why mass removal substantially reduces, or even
prevents, the star’s expansion. A star with a radiative envelope can grow in
mass without expending much if the high-entropy outer layers are removed
alongside the mass accretion. These outer layers have less mass than the added
mass, so the net average mass accretion is positive.

We turn to discuss the mass removal by jets and their operation in negative
feedback, which determines the parameters to prevent expansion.

##  4 Discussion and Summary

We mimic a process where a very massive main sequence star accretes mass via
an accretion disk, and this accretion disk launches energetic jets from its
inner zones attached to the star. Namely, the jets carry most of the
gravitational energy that the accreted mass releases. Moreover, we assume that
the jets remove the outer layers of the mass-accreting star. This occurs
because the star expands as it accretes mass at a high rate to radii larger
than the inner radius of the accretion disk. The jets that the accretion disk
launches from its inner zone collide with the rarefied outer layers of the
swelled star and remove mass from them. We alternately added and removed mass
from the stellar models to mimic this process. The numerical code mesa
(section 2) adds and removes mass from the outer layer. The accreted and
removed mass has the same properties, like entropy, as the outermost layer.
After a mass accretion episode (the pulse), the star expands, and the
outermost layer has higher entropy than the former outermost payer (Figures 4
and 5). In the next phase, we remove mass from the outer stellar layer; this
layer has high entropy, so the star shrinks.

We simulated the accretion process onto non-rotating spherically symmetric
main sequence stellar models of $30M_{\odot}$ and $60M_{\odot}$. We found
(section 3) that for not too short alternating inflow-outflow cycles and for
an outflow that carries more than about half the accreted mass, the mass loss
substantially reduces the rate of the expansion of the mass-accreting star
(Figures 1 \- 3). We did not explore a large parameter space for two reasons.
Firstly, the simulations take time, and we aim to show that substantial mass
removal, as might occur when the accretion disk launches energetic jets,
allows for a high mass accretion rate without much expansion. Secondly, mass
removal by jets operates in a negative feedback cycle. If the star expands,
the jets immediately remove more mass, so the star contracts. If the star
contracts, the jets are less efficient in removing envelop mass, and the
accretion leads to expansion. The feedback mechanism operates in a way that
prevents much expansion (unless the mass accretion rate is much too high).

Schürmann & Langer (2024) use mesa to thoroughly study main sequence stars
accreting at constant rates. The parameter space they cover with their
simulations is much larger than ours. They simulate only pure accretion, i.e.,
accretion without mass loss. The main difference from our study is that they
did not include mass loss during the mass-accretion process. This is a crucial
difference in the accretion process and in the results. They also did not
inject energy alongside the accretion process, while we did.

We compare here the results for $M=30M_{\odot}$, as Schürmann & Langer (2024)
do not simulate a star of $M=60M_{\odot}$. Schürmann & Langer (2024) find that
the general trend is that when the mass accretion timescale is shorter than
the thermal timescale, the star expands. For the $M=30M_{\odot}$ stellar
model, Schürmann & Langer (2024) find that at an accretion rate of
$\dot{M}_{\rm acc}=3\times 10^{-5}M_{\odot}{~{}\rm yr}^{-1}$ the star expands
by $30\%$ as its mass grows to $34M_{\odot}$. For accretion rates of
$\dot{M}_{\rm acc}\ga 10^{-4}M_{\odot}{~{}\rm yr}^{-1}$ their stellar model
expands unstably. We find, as an example, that when we add mass at a rate of
$\dot{M}_{\rm acc}=0.015M_{\odot}{~{}\rm yr}^{-1}$ in the mass addition part
of the cycle and remove $2/3$ of this mass in the mass removal part, namely, a
net accretion rate of $\dot{M}_{\rm acc,n}=0.0025M_{\odot}{~{}\rm yr}^{-1}$
the stellar model does not expand much (black lines in Figure 3). This holds
even as we inject extra energy into the envelope with the parameter of
$\eta_{\rm acc}=0.25$, i.e., a fraction of $0.25$ of the accretion energy is
deposited to the envelope (the rest is carried by jets). This further
emphasizes the result that mass removal, even when we inject energy that acts
to expand the star, prevents a large expansion.

Our results are most important in vigorous binary interactions, where mass
transfer at high rates leads the mass-accreting star to launch jets, e.g., the
grazing envelope evolution. Such likely has been the case in the 1837-1856
Great Eruption of Eta Carinae (e.g., Soker 2001; Akashi & Kashi 2020). The
companion mass in the jet-powered model of the Great Eruption is in the range
of $M_{2}(\eta~{}Car)\simeq 30-80M_{\odot}$, and more likely in the upper part
of this range. Near periastron passages during the Great Eruption, the system
experienced a grazing envelope evolution. The secondary star accreted $M_{\rm
acc}\simeq 4M_{\odot}$ during these 20 years (Kashi & Soker, 2010). Our
results show that such a secondary can accrete mass at a high rate without
expanding much if jets indeed remove mass from the outer high-entropy layers
of the envelope. In that case, the gravitational potential well of the
accreting star stays deep, and the jets are powerful. Our results allow for
the high mass-accretion rate required by the jet-powered binary model of the
Great Eruption. 222The triple-star models of the Great Eruption (Portegies
Zwart & van den Heuvel, 2016; Hirai et al., 2021) suffer from difficulties,
and we consider them unlikely. ($i$) The Lesser Eruption in 1890-1895
(Humphreys, Davidson, & Smith 1999) required another merging star according to
the triple-star scenarios. Namely, a system of initial four stars. ($ii$) The
Homunculus and the present binary system share an equatorial plane (e.g.,
Madura et al. 2012). This requires a coplanar triple-stellar system; merger
scenarios require an unstable triple system that will likely not be coplanar.
($iii$) The scenario by Hirai et al. (2021) predicts the presence of a dense
gas in the equatorial plane; this is not observed in the Homunculus. Applying
a triple-star or a quadruple-star model to the two nineteenth-century
eruptions of Eta Carinae is unnecessary and problematic.  If this also holds
for lower-mass stars (under study), our results allow high-mass-accretion
rates in other types of ILOTs; some recent studies argue that only jets can
power luminous ILOTs (e.g., Soker 2024).

## Acknowledgments

This research was supported by a grant from the Israel Science Foundation
(769/20).

## References

  * Addison et al. (2022) Addison H., Blagorodnova N., Groot P. J., Erasmus N., Jones D., Mogawana O., 2022, MNRAS, 517, 1884.
  * Akashi & Kashi (2020) Akashi M., Kashi A., 2020, MNRAS, 494, 3186.
  * Angulo et al. (1999) Angulo, C., Arnould, M., Rayet, M., et al. 1999, Nucl. Phys. A, 656, 3.
  * Banerjee et al. (2020) Banerjee, D. P. K., Geballe, T. R., Evans, A., Shahbandeh, M., Woodward, C. E., Gehrz, R. D., Eyres, S. P. S., Starrfield, S. & Zijlstra, A. 2020, ApJ, 904, L23.
  * Berger et al. (2009) Berger, E., Soderberg, A. M., Chevalier, R. A., et al. 2009, ApJ, 699, 1850
  * Blagorodnova et al. (2020) Blagorodnova, N., Karambelkar, V., Adams, S. M., et al. 2020, MNRAS, 496, 5503. doi:10.1093/mnras/staa1872
  * Blagorodnova et al. (2021) Blagorodnova N., Klencki J., Pejcha O., Vreeswijk P. M., Bond H. E., Burdge K. B., De K., et al., 2021, A&A, 653, A134. doi:10.1051/0004-6361/202140525
  * Blagorodnova et al. (2017) Blagorodnova, N., Kotak, R., Polshaw, J., et al. 2017, ApJ, 834, 107
  * Blouin et al. (2020) Blouin, S., Shaffer, N. R., Saumon, D., & Starrett, C. E. 2020, ApJ, 899, 46.
  * Boian & Groh (2019) Boian, I., & Groh, J. H. 2019, A&A, 621, A109.
  * Bond et al. (2003) Bond, H. E., Henden, A., Levay, Z. G., et al. 2003, Nature, 422, 405
  * Cai et al. (2019) Cai Y.-Z., Pastorello A., Fraser M., Prentice S. J., Reynolds T. M., Cappellaro E., Benetti S., et al., 2019, A&A, 632, L6.
  * Cai et al. (2022a) Cai Y.-Z., Pastorello A., Fraser M., Wang X.-F., Filippenko A. V., Reguitti A., Patra K. C., et al., 2022a, A&A, 667, A4.
  * Cai et al. (2022b) Cai Y., Reguitti A., Valerin G., Wang X., 2022b, Univ, 8, 493.
  * Cassisi et al. (2007) Cassisi, S., Potekhin, A. Y., Pietrinferni, A., Catelan, M., & Salaris, M. 2007, ApJ, 661, 1094.
  * Chugunov et al. (2007) Chugunov, A. I., Dewitt, H. E., & Yakovlev, D. G. 2007, Phys. Rev. D, 76, 025028.
  * Cyburt et al. (2010) Cyburt, R. H., Amthor, A. M., Ferguson, R., et al. 2010, ApJS, 189, 240.
  * Danieli & Soker (2019) Danieli B., Soker N., 2019, MNRAS, 482, 2277.
  * De et al. (2023) De K., MacLeod M., Karambelkar V., Jencson J. E., Chakrabarty D., Conroy C., Dekany R., et al., 2023, Natur, 617, 55.
  * Ferguson et al. (2005) Ferguson, J. W., Alexander, D. R., Allard, F., Barman, T., Bodnarik, J. G., Hauschildt, P. H., Heffner-Wong, A., & Tamanai, A. 2005, ApJ, 623, 585.
  * Fuller et al. (1985) Fuller, G. M., Fowler, W. A., & Newman, M. J. 1985, ApJ, 293, 1.
  * Gilkis et al. (2019) Gilkis, A., Soker, N., & Kashi, A. 2019, MNRAS, 482, 4233
  * Grichener & Soker (2019) Grichener A., Soker N., 2019, ApJ, 878, 24.
  * Gurevich, Bear, & Soker (2022) Gurevich O., Bear E., Soker N., 2022, MNRAS, 511, 1330.
  * Hirai et al. (2021) Hirai R., Podsiadlowski P., Owocki S. P., Schneider F. R. N., Smith N., 2021, MNRAS, 503, 4276.
  * Howitt et al. (2020) Howitt, G., Stevenson, S., Vigna-Gómez, A., et al. 2020, MNRAS, 492, 3229
  * Hubová, & Pejcha (2019) Hubová, D., & Pejcha, O. 2019, MNRAS, 489, 891
  * Humphreys, Davidson, & Smith (1999) Humphreys R. M., Davidson K., Smith N., 1999, PASP, 111, 1124.
  * Iglesias & Rogers (1993) Iglesias, C. A. & Rogers, F. J. 1993, ApJ, 412, 752.
  * Iglesias & Rogers (1996) Iglesias, C. A. & Rogers, F. J. 1996, ApJ, 464, 943.
  * Irwin (2004) Irwin, Alan. W. (2004). The FreeEOS Code for Calculating the Equation of State for Stellar Interiors (cit. on p. xxi).
  * Itoh et al. (1996) Itoh, N., Hayashi, H., Nishikawa, A., Kohyama, Y. 1996, ApJS, 102, 411.
  * Ivanova et al. (2013) Ivanova, N., Justham, S., Avendano Nandez, J. L., & Lombardi, J. C. 2013, Science, 339, 433
  * Jencson et al. (2019) Jencson, J. E., Kasliwal, M. M., Adams, S. M., et al. 2019, ApJ, 886, 40
  * Jermyn et al. (2021) Jermyn, A. S., Schwab, J., Bauer, E., Timmes F. X., & Potekhin A. Y. 2021, ApJ, 913, 72.
  * Jones (2020) Jones, D. 2020, arXiv e-prints, arXiv:2001.03337
  * Kaminski (2024) Kaminski T., 2024, arXiv, arXiv:2401.03919.
  * Kamiński et al. (2015) Kamiński, T., Mason, E., Tylenda, R., & Schmidt, M. R. 2015, A&A, 580, A34
  * Kamiński et al. (2020) Kamiński, T., Menten, K. M., Tylenda, R., et al. 2020, A&A, 644, A59. doi:10.1051/0004-6361/202038648
  * Kamiński et al. (2023) Kamiński T., Schmidt M., Hajduk M., Kiljan A., Izviekova I., Frankowski A., 2023, A&A, 672, A196.
  * Kamiński et al. (2021) Kamiński, T., Steffen, W., Bujarrabal, V., et al. 2021, A&A, 646, A1. doi:10.1051/0004-6361/202039634
  * Kaminski et al. (2018) Kaminski, T., Steffen, W., Tylenda, R., Young, K. H., Patel, N. A., & Menten, K. M. 2018, A&A, 617, A129
  * Karambelkar et al. (2023) Karambelkar V. R., Kasliwal M. M., Blagorodnova N., Sollerman J., Aloisi R., Anand S. G., Andreoni I., et al., 2023, ApJ, 948, 137.
  * Kashi et al. (2019) Kashi, A., Michaelis, A. M., & Feigin, L. 2019, Galaxies, 8, 2.
  * Kashi & Soker (2010) Kashi A., Soker N., 2010, ApJ, 723, 602.
  * Kashi & Soker (2016a) Kashi A., Soker N., 2016, RAA, 16, 99. doi:10.1088/1674-4527/16/6/099
  * Kashi & Soker (2016b) Kashi A., Soker N., 2016, ApJ, 825, 105.
  * Kasliwal (2011) Kasliwal, M. M. 2011, Bulletin of the Astronomical Society of India, 39, 375
  * Kasliwal et al. (2012) Kasliwal, M. M., Kulkarni, S. R., Gal-Yam, A., et al. 2012, ApJ, 755, 161
  * Klencki et al. (2021) Klencki, J., Nelemans, G., Istrate, A. G., & Chruslinska, M., 2021, A&A, 645, A54. doi:10.1051/0004-6361/202038707
  * Langanke & Martínez-Pinedo (2000) Langanke, K. & Martínez-Pinedo, G. 2000, Nucl. Phys. A, 673, 481.
  * MacLeod & Loeb (2020) MacLeod, M. & Loeb, A. 2020, ApJ, 895, 29. doi:10.3847/1538-4357/ab89b6
  * MacLeod et al. (2017) MacLeod, M., Macias, P., Ramirez-Ruiz, E., Grindlay, J., Batta, A., & Montes, G. 2017, ApJ, 835, 282
  * MacLeod et al. (2018) MacLeod, M., Ostriker, E. C., & Stone, J. M. 2018, ApJ, 868, 136.
  * Madura et al. (2012) Madura T. I., Gull T. R., Owocki S. P., Groh J. H., Okazaki A. T., Russell C. M. P., 2012, MNRAS, 420, 2064.
  * Mason et al. (2010) Mason, E., Diaz, M., Williams, R. E., Preston, G., & Bensby, T. 2010, A&A, 516, A108
  * Mcley & Soker (2014) Mcley L., Soker N., 2014, MNRAS, 445, 2492.
  * Matsumoto & Metzger (2022) Matsumoto T., Metzger B. D., 2022, ApJ, 938, 5.
  * Metzger, Giannios, & Spiegel (2012) Metzger B. D., Giannios D., Spiegel D. S., 2012, MNRAS, 425, 2778.
  * Metzger & Pejcha (2017) Metzger, B. D., & Pejcha, O. 2017, MNRAS, 471, 3200
  * Mobeen et al. (2021) Mobeen M. Z., Kamiński T., Matter A., Wittkowski M., Paladini C., 2021, A&A, 655, A100.
  * Mould et al. (1990) Mould, J., Cohen, J., Graham, J. R., et al. 1990, ApJ, 353, L35
  * Muthukrishna et al. (2019) Muthukrishna, D., Narayan, G., Mandel, K. S., Biswas, R., & Hložek, R. 2019, PASP, 131, 118002
  * Nandez et al. (2014) Nandez, J. L. A., Ivanova, N., & Lombardi, J. C., Jr. 2014, ApJ, 786, 39
  * O’Connor et al. (2023) O’Connor C. E., Bildsten L., Cantiello M., Lai D., 2023, ApJ, 950, 128. doi:10.3847/1538-4357/acd2d4
  * Oda et al. (1994) Oda, T., Hino, M., Muto, K., Takahara M., & Sato K. 1994, Atomic Data and Nuclear Data Tables, 56, 231.
  * Ofek et al. (2008) Ofek, E. O., Kulkarni, S. R., Rau, A., et al. 2008, ApJ, 674, 447
  * Pastorello & Fraser (2019) Pastorello, A., & Fraser, M. 2019, Nature Astronomy, 3, 676
  * Pastorello et al. (2019) Pastorello, A., Mason, E., Taubenberger, S., et al. 2019, A&A, 630, A75
  * Pastorello et al. (2021) Pastorello A., Valerin G., Fraser M., Elias-Rosa N., Valenti S., Reguitti A., Mazzali P. A., et al., 2021, A&A, 647, A93.
  * Pastorello et al. (2023) Pastorello A., Valerin G., Fraser M., Reguitti A., Elias-Rosa N., Filippenko A. V., Rojas-Bravo C., et al., 2023, A&A, 671, A158.
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15
  * Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34
  * Paxton et al. (2019) Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10,
  * Pejcha et al. (2016a) Pejcha, O., Metzger, B. D., & Tomida, K. 2016a, MNRAS, 455, 4351
  * Pejcha et al. (2016b) Pejcha, O., Metzger, B. D., & Tomida, K. 2016b, MNRAS, 461, 2527
  * Pejcha et al. (2017) Pejcha O., Metzger B. D., Tyles J. G., Tomida K., 2017, ApJ, 850, 59.
  * Portegies Zwart & van den Heuvel (2016) Portegies Zwart S. F., van den Heuvel E. P. J., 2016, MNRAS, 456, 3401.
  * Potekhin & Chabrier (2010) Potekhin, A. Y. & Chabrier, G. 2010, Contributions to Plasma Physics, 50, 82.
  * Poutanen (2017) Poutanen, J. 2017, ApJ, 835, 119.
  * Qian et al. (2020) Qian S.-B., Zhu L.-Y., Liu L., Zhang X.-D., Shi X.-D., He J.-J., Zhang J., 2020, RAA, 20, 163.
  * Rau et al. (2007) Rau, A., Kulkarni, S. R., Ofek, E. O., & Yan, L. 2007, ApJ, 659, 1536
  * Retter & Marom (2003) Retter, A., & Marom, A. 2003, MNRAS, 345, L25
  * Rogers & Nayfonov (2002) Rogers, F. J. & Nayfonov, A. 2002, ApJ, 576, 1064.
  * Saumon et al. (1995) Saumon, D., Chabrier, G., & van Horn, H. M. 1995, ApJS, 99, 713.
  * Schrøder et al. (2020) Schrøder, S. L., MacLeod, M., Loeb, A., Vigna-Gómez, A., & Mandel, I. 2020, arXiv:1906.04189
  * Schreier et al. (2021) Schreier, R., Hillel, S., Shiber, S., & Soker, N. 2021, MNRAS.
  * Schürmann & Langer (2024) Schürmann C., Langer N., 2024, arXiv:2404.08615.
  * Segev et al. (2019) Segev, R., Sabach, E., & Soker, N. 2019, ApJ, 884, 58
  * Soker (2001) Soker N., 2001, MNRAS, 325, 584.
  * Soker (2016a) Soker N., 2016a, NewA, 47, 16.
  * Soker (2020) Soker N., 2020, ApJ, 893, 20.
  * Soker (2022) Soker, N., 2022, Research in Astronomy and Astrophysics, 22, 122003.
  * Soker (2023) Soker N., 2023, OJAp, 6, 32. doi:10.21105/astro.2306.07702
  * Soker (2024) Soker N., 2024, arXiv:2404.19617.
  * Soker & Gilkis (2018) Soker, N., & Gilkis, A. 2018, MNRAS, 475, 1198
  * Soker & Kaplan (2021) Soker N., Kaplan N., 2021, RAA, 21, 090.
  * Stritzinger et al. (2020a) Stritzinger M. D., Taddia F., Fraser M., Tauris T. M., Contreras C., Drybye S., Galbany L., et al., 2020a, A&A, 639, A104.
  * Stritzinger et al. (2020b) Stritzinger M. D., Taddia F., Fraser M., Tauris T. M., Suntzeff N. B., Contreras C., Drybye S., et al., 2020b, A&A, 639, A103.
  * Timmes & Swesty (2000) Timmes, F. X. & Swesty, F. D. 2000, ApJS, 126, 501.
  * Tsuna et al. (2024) Tsuna, D., Matsumoto, T., Wu, S. C., Fuller J., 2024, ApJ, 966, 30. doi:10.3847/1538-4357/ad3637
  * Tylenda et al. (2011) Tylenda, R., Hajduk, M., Kamiński, T., et al. 2011, A&A, 528, A114
  * Tylenda et al. (2024) Tylenda, R., Kamiński, T., & Smolec, R. 2024, A&A, 685, A49. doi:10.1051/0004-6361/202244896
  * Tylenda et al. (2013) Tylenda, R., Kamiński, T., Udalski, A., et al. 2013, A&A, 555, A16
  * Wadhwa et al. (2022) Wadhwa S. S., De Horta A., Filipović M. D., Tothill N. F. H., Arbutina B., Petrović J., Djurašević G., 2022, RAA, 22, 105009.
  * Yalinewich & Matzner (2019) Yalinewich, A., & Matzner, C. D. 2019, MNRAS, 490, 312
  * Yamazaki, Hayasaki, & Loeb (2017) Yamazaki R., Hayasaki K., Loeb A., 2017, MNRAS, 466, 1421.
  * Mobeen et al. (2024) Mobeen, M. Z., Kamiński, T., Matter, A., Wittkowski M., Monnier J. D., Kraus S., Le Bouquin J.-B., et al., 2024, A&A, 686, A260. doi:10.1051/0004-6361/202347322
  * Zhu et al. (2023) Zhu C.-H., Lü G.-L., Lu X.-Z., He J., 2023, RAA, 23, 025021.