# Jets from main sequence and white dwarf companions during common envelope
evolution

Yangyuxin Zou, Luke Chamandy, Jonathan Carroll-Nellenback, Eric G. Blackman
and Adam Frank
Department of Physics and Astronomy, University of Rochester, Rochester, NY
14620, USA
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

It has long been speculated that jet feedback from accretion onto the
companion during a common envelope (CE) event could affect the orbital
evolution and envelope unbinding process. We present global 3D hydrodynamical
simulations of CE evolution (CEE) that include a jet subgrid model and compare
them with an otherwise identical model without a jet. Our binary consists of a
$2M_{\odot}$ red giant branch primary and a $1M_{\odot}$ or $0.5M_{\odot}$
main sequence (MS) or white dwarf (WD) secondary companion modeled as a point
particle. We run the simulations for 10 orbits (40 days). Our jet model adds
mass at a constant rate $\dot{M}_{\mathrm{j}}$ of order the Eddington rate,
with maximum velocity $v_{\mathrm{j}}$ of order the escape speed, to two
spherical sectors with the jet axis perpendicular to the orbital plane. We
explore the influence of the jet on orbital evolution, envelope morphology and
envelope unbinding, and assess the dependence of the results on jet mass-loss
rate, launch speed, companion mass, opening angle, and accretion rate. In line
with our theoretical estimates, jets are choked around the time of first
periastron passage and remain choked thereafter. Subsequent to choking, but
not before, jets efficiently transfer energy to bound envelope material. This
leads to increases in unbound mass of up to $\sim 10\%$, as compared to the
simulations without jets. We also estimate the cumulative effects of jets over
a full CE phase, finding that jets launched by MS and WD companions are
unlikely to dominate envelope unbinding.

###### keywords:

binaries: close – stars: jets – hydrodynamics – stars: winds, out flows –
white dwarfs

††pubyear: 2021††pagerange: Jets from main sequence and white dwarf companions
during common envelope evolution–A

##  1 Introduction

The common envelope (CE) phase of binary stellar evolution (Paczynski, 1976;
Ivanova et al., 2013) is believed to play a key role in the lead-up to several
astrophysical phenomena, including certain types of planetary nebulae and
supernovae and mergers of neutron stars (NSs) or stellar mass black holes
(BHs). However, CE evolution (CEE) is still not very well understood, partly
owing to its intrinsic 3D nature, the vast range of spatial and temporal
scales involved, and the multitude of physical effects that could be
important. In CEE, the core of the primary star and the much smaller and more
compact secondary companion transfer their mutual orbital energy and angular
momentum to the envelope as they inspiral, reducing its binding energy.

Energy sources other than orbital may also be important for envelope
unbinding. One possibility is potential energy that is liberated as envelope
material accretes onto the secondary. Main sequence (MS) stars in post-CE
binary systems that are inflated in size (Jones et al., 2015) or chemically
enriched with carbon (Miszalski et al., 2013) might be signatures of accretion
during CEE. While CEE theoretical models differ in the treatment of accretion
– which generally occurs at subgrid scales – they concur that envelope mass
can flow toward the secondary at rates much higher than the photon Eddington
limit $\dot{M}_{\mathrm{Edd}}$ (Ricker & Taam 2008, Ricker & Taam 2012,
Blackman & Lucchini 2014, MacLeod et al. 2017, Chamandy et al. 2018; see the
latter reference for a derivation of $\dot{M}_{\mathrm{Edd}}$). This suggests
that some accretion at or in excess of the Eddington rate is plausible.

Part of the liberated accretion energy might be directed into bipolar jets,
which would in turn help to transport this energy to other parts of the
envelope (e.g. Soker, 2004). A jet could be launched from the centre of an
accretion disk and be accelerated and collimated by a hydromagnetic Blandford-
Payne magneto-centrifugal launch mechanism (Blandford & Payne, 1982) or a
magnetic tower (Lynden-Bell, 2003). The jet in turn removes mass, angular
momentum, and pressure from the accretion flow and could facilitate super-
Eddington accretion. Indeed, simulations have shown that accretion discs can
develop around the secondary, supporting the idea that jets could be launched
in some cases (Murguia-Berthier et al., 2017; Chamandy et al., 2018). The
secondary can also accrete material just before CEE, and thus potentially
exhibit a jet upon entering CEE (e.g. Chen et al. 2017).

Various theoretical studies have explored the role of jets launched from the
secondary star during CEE. Several have focused on NS or BH secondaries. In
these cases, photons could be trapped and advected by the accretion flow, but
accretion energy and pressure buildup can be released by weakly interacting
neutrinos. So-called hypercritical accretion, at rates $\gtrsim
10^{4}\dot{M}_{\mathrm{Edd}}$, may thus be common (Chevalier, 1993; Fryer et
al., 1996; Brown et al., 2000; Armitage & Livio, 2000; Chevalier, 2012; López-
Cámara et al., 2019; Cruz-Osorio & Rezzolla, 2020). These studies have
generally employed hydrodynamic simulations with “wind-tunnel” setups that are
local in simulating only a small region around the secondary. They are
generally appropriate only for modeling the earliest stages of CEE, before the
binary separation $a$ becomes comparable to the Bondi-Hoyle-Lyttleton
accretion radius (Chamandy et al., 2019b; Everson et al., 2020).

Shiber et al. (2019) presented the first global CE simulations to model jet
feedback and allow orbital evolution and compared these simulations to others
which did not include jets. These adaptive mesh refinement (AMR) simulations
involved a $0.88M_{\odot}$ red giant branch (RGB) star of radius $83R_{\odot}$
with a $0.39M_{\odot}$ core (represented by a point particle) along with a
$0.3M_{\odot}$ MS secondary (also represented by a point particle and with jet
properties chosen assuming it to be an MS star).

Comparing similar simulations with and without a jet, Shiber et al. (2019)
concluded that the presence of a jet increased the outward flux of material
through a sphere of radius $1\,\mathrm{au}$ or $2\,\mathrm{au}$ centred on the
origin of the simulation domain and extending out to the boundary. The
fraction of the material which had positive energy density flowing out of this
sphere was also determined to be higher when a jet was included. Furthermore,
they found that the orbits of the particles were quite different between jet
simulations and corresponding no-jet simulations, with jet runs settling into
a larger-separation orbit by the end of the simulation compared to no-jet
runs. They also found that jets can get choked as the secondary enters denser
envelope gas, but that they can eventually break out along the polar
directions.

Recently, Lopez-Camara et al. (2021) performed local (in space and time) CE
accretion/jet simulations that read in global simulation data from
simulation#5 (without jet) of Shiber et al. (2019). Their accretion and jet
subgrid models were more sophisticated and realistic than those of Shiber et
al. (2019) and they employed $16\times$ higher maximum resolution. Lopez-
Camara et al. (2021) found that jets are choked before the first periastron
passage, earlier than in Shiber et al. (2019). Moreover, whereas Shiber et al.
(2019) found that a jet can break out again after $\sim 3\,{\rm d}$, Lopez-
Camara et al. (2021) found that the jet had still not broken out $\sim
10\,{\rm d}$ after choking, when their simulations ended.

Aside from the 3D simulation studies mentioned above, there have been other
theoretical studies finding that jets can be important for envelope unbinding.
It has been suggested that CEE can be avoided if the system instead enters a
quasi-steady “grazing envelope evolution” (GEE) phase, where the jet of the
secondary continuously removes material from the outskirts of the envelope
(Soker, 2015). However, GEE has so far not been shown to take place in a 3D
hydrodynamical CE simulation that includes all relevant gravitational
interactions. Another possibility is that a jet launched by the secondary
could help to unbind what remains of the envelope during the late stages of
CEE (Soker, 2017).

The primary goal of the present work is to investigate to what degree a jet
launched from the secondary can affect envelope unbinding during CEE, and we
also study how jets affect orbital evolution and envelope morphology. The
organization of the paper is as follows. In Section 2, we describe the
numerical methods and model. Then in Section 3 we present some order-of-
magnitude analytical estimates that can assist in the interpretation of
numerical results and also allow for extrapolation to regions of the parameter
space inaccessible in the simulations. In Section 4, we present the results of
our simulations. We discuss the implications of our results for envelope
unbinding in Section 5, and compare our findings to the literature in Section
6. In Section 7 we summarize our key results and conclude.

##  2 Methods

Hydrodynamic simulations in this work were carried out using the AMR code
astrobear, (Cunningham et al., 2009; Carroll-Nellenback et al., 2013). The
simulation setups are very similar to those in our previous works (Chamandy et
al., 2018, 2019b) to which we now add a jet. We use an ideal gas equation of
state with adiabatic index $\gamma=5/3$. The primary is an RGB star of mass
$1.96M_{\odot}$ and radius $48.1R_{\odot}$, and its core is modeled using a
$0.367M_{\odot}$ point particle according to the method of Ohlmann et al.
(2017), along with an added iteration over particle mass (Chamandy et al.,
2018). For most of our runs the secondary is a point particle with initial
mass $0.978M_{\odot}$, but for two runs we use $0.489M_{\odot}$ instead. The
spline softening radius for the primary core and secondary particles is
$2.41R_{\odot}$ and the simulations employ $4$ levels of AMR with base
resolution $\delta_{0}=2.25R_{\odot}$ and highest resolution
$\delta_{4}=0.14R_{\odot}$. The density and pressure of the ambient medium
surrounding the star are $6.7\times 10^{-9}\,{\rm g\,cm^{-3}}$ and $1.0\times
10^{5}\,{\rm dyn\,cm^{-2}}$, and the cubic domain has side length
$1150R_{\odot}$. The simulation is conducted in a reference frame in which the
centre of mass remains approximately fixed. The two stars are initialized in a
circular orbit at $t=0$, with $a_{\mathrm{i}}=49R_{\odot}$, and the envelope
has no initial rotation.

###  2.1 Jet model

We use a slightly modified version of the jet subgrid model of Federrath et
al. (2014), with a scheme that conserves mass, momentum and angular momentum
while minimizing discretization errors, as detailed in Appendix A. For the
duration of the simulations, gas is added to the grid at the rate
$\dot{M}_{\mathrm{j}}$, divided between two oppositely oriented spherical
sectors centred around the companion particle. The jet axis is constrained to
be perpendicular to the orbital plane. Each sector extends from the vicinity
of the companion out along the jet axis to the distance
$r_{\mathrm{j}}=\delta_{0}=16\delta_{4}=2.25R_{\odot}$, and from the axis to
the polar angle $\theta_{\mathrm{h}}$ (the jet half-opening angle). The jet
density and radial velocity fall off at higher polar angle $\theta$, and the
jet velocity is strongly peaked near the jet axis within
$\theta<\theta_{\mathrm{h}}/6$. Further, the jet mass is concentrated around
$r\sim r_{\mathrm{j}}/2$ (see Federrath et al. 2014 for details of the jet
geometry). In addition, the jet material is initialized with an extra velocity
component equal to the instantaneous orbital velocity of the secondary. Owing
to the Federrath et al. (2014) density and velocity profiles chosen for the
jet, the rate of kinetic energy supplied to the jet (excluding the smaller
variable contribution from the orbital motion) is equal to about
$\dot{M}_{\mathrm{j}}v_{\mathrm{j}}^{2}/40$, with $v_{\mathrm{j}}$ the peak
outward velocity of the jet, which occurs at $\theta=0$. The jet temperature
is chosen to be $T_{\mathrm{j}}=10^{4}\,\mathrm{K}$, but the results are
insensitive to temperature because the jet is highly supersonic. The jet is
initialized to be non-rotating.

The jet model we use (Federrath et al., 2014) is structured to include a high
momentum spine with small opening angle and a lower momentum “wide angle
wind”. This structured radial momentum distribution in the jet is in contrast
to simple constant density, constant velocity “top hat” jets. While simpler,
top hat jets are not as realistic since it is expected that collimated flows
are driven by magnetocentrifugal processes from the star/disc system. Both
analytic models (Shu et al., 2000) and numerical simulations (Banerjee &
Pudritz, 2006; Machida et al., 2008) of such systems show that the outflows
they produce always show momentum falling off sharply as one moves away from
the axis of the flow.

The initial jet speed at the stellar surface may be expressed as $Q$ times the
Keplerian value, i.e.
$v_{\mathrm{surf}}=432Q\,\mathrm{km\,s^{-1}}\,(M_{2}/M_{\odot})^{1/2}(R_{2}/R_{\odot})^{-1/2}$,
with $Q>1$ (e.g. Blackman & Lucchini, 2014). The value of the peak speed
$v_{\mathrm{j}}$, at the jet initialization radius of $r\sim
r_{\mathrm{j}}/2\approx 1R_{\odot}$, is then chosen by setting $Q$ to some
value between $2$ and $4$. For models for which the secondary point particle
represents a WD ($R_{2}\sim 0.01R_{\odot}$), we also take into account the
influence of gravity from the secondary when choosing $v_{\mathrm{j}}$.

Our jet prescription is not intended to capture the detailed mechanism of any
specific jet model such as magneto-centrifugal launch (MCL) models (Blandford
& Payne, 1982; Pelletier & Pudritz, 1992) or magnetic tower (MT) (Lynden-Bell,
2003; Huarte-Espinosa et al., 2013). We merely specify the kinematics and
opening angle. However, because we specify the kinematics of the jet to be
dominated by a velocity flow, our specification is most consistent
asymptotically with MCL type models rather than MT models because the latter
can remain magnetically dominated out to larger distances from the engine. MCL
models are asymptotic hydrodynamically dominated, and so our prescription is
essentially taking the asymptotic properties of the MCL as the input.

In our model the secondary launches a jet with constant $\dot{M}_{\mathrm{j}}$
and peak velocity $v_{\mathrm{j}}$ from time $t=0$. This effectively assumes
that the (subgrid) processes governing accretion and jet launching are in a
steady state that is unaffected by the changing environment of the secondary
over the timescale of the simulation. Aside from simplifying the numerical
implementation, this choice minimizes the number of variables, thus
facilitating interpretation of the simulation results.

For most of our runs, the secondary is not permitted to accrete from the
surrounding envelope. For some runs we allow it to accrete at roughly the
Bondi-Hoyle-Lyttleton rate, as computed by our accretion subgrid scheme, which
is modeled after Krumholz et al. (2004) (see also Chamandy et al. 2018).
Unlike Krumholz et al. (2004) our scheme allows the point particle secondary
to remove angular momentum from the gas it accretes. The specific angular
momentum of gas inside the accretion zones that is not accreted is conserved
during the accretion step, as in the scheme of Federrath et al. (2010). For
our WD run, we cap the accretion rate at $\approx 10^{4}\dot{M}_{\mathrm{j}}$.
As radiative feedback is not included, the accretion rates that obtain can be
considered upper limits.

Separate tracers are applied to the jet, envelope and ambient, which allows us
to distinguish these components in post-processing. Note that co-spatial jet
gas and envelope/ambient gas have different densities, but the same velocity
and temperature. Simulations are carried out for $40\,{\rm d}$, or about $10$
orbits in most of our models; this duration was chosen to optimize
computational resources.

Model | Type | $M_{2}(t=0)$ | $\dot{M}_{\mathrm{j}}$ | $\dot{M}_{\mathrm{a}}$ | $\dot{M}_{2}$ | $v_{\mathrm{j}}$ | $\theta_{\mathrm{h}}$
---|---|---|---|---|---|---|---
|  | [$0.978M_{\odot}$] | [$10^{-3}M_{\odot}\mathrm{\,yr^{-1}}$] |  |  | [$\\!\,\mathrm{km\,s^{-1}}$] | [∘]
J1 | MS | $1$ | $2$ | $0$ | $-\dot{M}_{\mathrm{j}}$ | $864$ | $15$
NJ1 | — | $1$ | $0$ | $0$ | $0$ | — | —
J2 | MS | $0.5$ | $20$ | C18 | $\dot{M}_{\mathrm{a}}-\dot{M}_{\mathrm{j}}$ | $864$ | $15$
NJ2 | — | $0.5$ | $0$ | $0$ | $0$ | — | —
J3 | MS | $1$ | $2$ | $0$ | $0$ | $864$ | $15$
J4 | MS | $1$ | $2$ | C18 | $\dot{M}_{\mathrm{a}}-\dot{M}_{\mathrm{j}}$ | $864$ | $15$
J5 | MS | $1$ | $20$ | C18 | $\dot{M}_{\mathrm{a}}-\dot{M}_{\mathrm{j}}$ | $864$ | $15$
J6 | MS | $1$ | $2$ | $0$ | $-\dot{M}_{\mathrm{j}}$ | $1728$ | $15$
J7 | MS | $1$ | $2$ | $0$ | $-\dot{M}_{\mathrm{j}}$ | $864$ | $30$
J8 | WD | $1$ | $0.02$ | C18 | $\dot{M}_{\mathrm{a}}-\dot{M}_{\mathrm{j}}$ | $8640$ | $15$
Table 1: Models labeled with ‘J’ refer to runs with a jet, and those labeled
with ‘NJ’ refer to runs without a jet. Jet parameters were selected based on
whether the secondary is modeled as a main sequence (MS) star, or a white
dwarf (WD). The quantity $v_{\mathrm{j}}$ is the peak jet velocity. For all
runs, $r_{\mathrm{soft}}=2.41R_{\odot}$,
$r_{\mathrm{j}}=\delta_{0}=2.25R_{\odot}$ and
$T_{\mathrm{j}}=10^{4}\,\mathrm{K}$. The Eddington accretion rate
$\dot{M}_{\mathrm{Edd}}\sim 2\times
10^{-3}(R_{2}/R_{\odot})~{}M_{\odot}\mathrm{\,yr^{-1}}$ (e.g. Chamandy et al.
2018, which is denoted as ‘C18’ in the table).

###  2.2 Runs

Table 1 describes the runs performed. J1 is the fiducial jet run and NJ1 is
the fiducial no-jet ($\dot{M}_{\mathrm{j}}=0$) run. In nature, we might expect
$\dot{M}_{\mathrm{j}}/\dot{M}_{\mathrm{a}}\sim 0.1$, where
$\dot{M}_{\mathrm{a}}$ is the accretion rate, and
$\dot{M}_{\mathrm{a}}/\dot{M}_{\mathrm{Edd}}\lesssim 10$ with the upper bound
$>1$ because the Eddington limit can be modestly exceeded in non-spherical
flows. For most of our runs, $\dot{M}_{\mathrm{j}}$ is taken to equal the
estimated Eddington rate for a $1R_{\odot}$ MS star,
$\dot{M}_{\mathrm{j}}=\dot{M}_{\mathrm{Edd}}\approx 2\times
10^{-3}M_{\odot}\mathrm{\,yr^{-1}}$ (e.g. Chamandy et al., 2018). In some
cases we adopt the more extreme value $\dot{M}_{\mathrm{j}}\approx
10\dot{M}_{\mathrm{Edd}}$.

For most of our models, the secondary loses mass at the rate
$\dot{M}_{2}=-\dot{M}_{\mathrm{j}}$ to conserve mass and momentum. The
secondary along with the gas inside the softening sphere loosely represent a
star+accretion disc system with small-scale (subgrid) processes enforcing a
fixed value of $\dot{M}_{\mathrm{j}}$. To test the sensitivity to this aspect
of the modeling, we perform Run J3, which is the same as J1 except that
$M_{2}$ is fixed. Furthermore, as mentioned above, for some of our models the
secondary point particle is permitted to accrete from its surroundings at very
high rates (up to $\sim 10^{3}\dot{M}_{\mathrm{Edd}}$ for our MS star runs and
$\sim 10^{4}\dot{M}_{\mathrm{Edd}}$ for our WD run). These accretion rates are
upper limits, and allow us to explore qualitatively the effects of accretion
by comparing to those runs without accretion. For example, Run J4 is like J1
except that it allows accretion.

To test the dependence of the results on the parameters, we have Runs J6 and
J7, which are like J1 except with $v_{\mathrm{j}}$ or $\theta_{\mathrm{h}}$
doubled. Run J5 is like J4 but with a 10 times higher $\dot{M}_{\mathrm{j}}$.
Run J2 (corresponding no-jet run NJ2) is like J5 except that the secondary
mass is halved. Finally, in Run J8 we adopt jet parameters that are
appropriate for a WD companion. J8 is like J4 but with a 10 times larger jet
speed and 100 times smaller jet mass-loss rate, resulting in a jet with
kinetic energy input rate comparable to that of J4 or J1.

##  3 Theoretical estimates

###  3.1 Jet launching

From $t=0$, material is added to the jet spherical sectors at the rate
$\dot{M}_{\mathrm{j}}$, with vertical speed of order $v_{\mathrm{j}}$. This
added material does not propagate out of the jet sectors immediately owing to
the inertia of the ambient gas. The material in the jet sectors is accelerated
to the initialization velocity after a time $t_{v}$, when sufficient momentum
has been imparted. We crudely estimate this time neglecting the variation in
the density and vertical component of the velocity across the jet. At $t=0$,
when the pressure and gravity of the ambient gas are negligible, we have

$\frac{\dot{M}_{\mathrm{j}}v_{\mathrm{j}}}{2}\sim\rho
V\frac{v_{\mathrm{j}}}{t_{v}},$ (1)

where the factor of $2$ accounts for the two jet spherical sectors, $V=2\uppi
r_{\mathrm{j}}^{3}\left(1-\cos\theta_{\mathrm{h}}\right)/3$ is the volume of
each sector, and $\rho=\rho_{\mathrm{amb}}$. Solving for $t_{v}$ we obtain

$t_{v}\sim\frac{4\uppi\rho
r_{\mathrm{j}}^{3}(1-\cos\theta_{\mathrm{h}})}{3\dot{M}_{\mathrm{j}}},$ (2)

which evaluates to $t_{v}\sim 30\,{\rm s}$ for our fiducial model. At this
time the material in the spherical sectors moves at a speed of order
$v_{\mathrm{j}}$ and still has a density comparable to $\rho_{\mathrm{amb}}$.

Note that $\rho_{\mathrm{amb}}$ is lower than the anticipated final density
$\rho_{\mathrm{j}}\sim\dot{M}_{\mathrm{j}}/(2Av_{\mathrm{j}})\sim 5\times
10^{-7}\,{\rm g\,cm^{-3}}$, where
$A\sim\uppi(r_{\mathrm{j}}/2)^{2}\tan^{2}\theta_{\mathrm{h}}$ is the jet
cross-sectional area and the numerical estimate is for our fiducial model. The
jet is truly launched when material is advected out of the spherical sectors,
which happens after a time $t_{\mathrm{adv}}\sim
r_{\mathrm{j}}/v_{\mathrm{j}}\sim 2\times 10^{3}\,{\rm s}$ for our fiducial
model. This is also approximately equal to the time it takes for the density
in the spherical sectors to reach $\rho_{\mathrm{j}}$, given by $t_{\rho}\sim
2V\rho_{\mathrm{j}}/\dot{M}_{\mathrm{j}}\sim V/Av_{\mathrm{j}}\sim
r_{\mathrm{j}}/v_{\mathrm{j}}$.

###  3.2 Jet choking

As the secondary plunges deeper into the envelope, the ram pressure force of
the jet on the surrounding gas remains roughly constant but the gravitational
force due to the secondary increases because of the higher density. As the jet
velocity and mass flux density in our model are strongly peaked within
$\theta<\theta_{\mathrm{h}}/6$ (Federrath et al., 2014), we focus on the
choking of this central part of the jet. The jet is choked when the net force
reduces to zero,

$F_{\mathrm{net}}\sim\frac{(\dot{M}_{\mathrm{j}}/5)v_{\mathrm{j}}}{2}-\frac{GM_{2}\rho_{\mathrm{c}}(V/36)}{(r_{\mathrm{j}}/2)^{2}}=0,$
(3)

where $\rho_{\mathrm{c}}$ is the critical density just outside the jet launch
region. Here $\dot{M}_{\mathrm{j}}$ is divided by $5$ since we are considering
only the fraction of the mass flux associated with the central part of the
jet, $\int_{0}^{\theta_{\mathrm{h}}/6}\rho(r,\theta)v(\theta)\sin\theta
d\theta\Big{/}\int_{0}^{\theta_{\mathrm{h}}}\rho(r,\theta)v(\theta)\sin\theta
d\theta\approx 0.2$, and $V$ is divided by $36$ since the central part
occupies the portion
$[1-\cos(\theta_{\mathrm{h}}/6)]/(1-\cos\theta_{\mathrm{h}})\approx 1/36$ of
the total volume. Solving for $\rho_{\mathrm{c}}$, we obtain

$\rho_{\mathrm{c}}\sim\frac{0.4\dot{M}_{\mathrm{j}}v_{\mathrm{j}}}{GM_{2}r_{\mathrm{j}}(1-\cos\theta_{\mathrm{h}})},$
(4)

which evaluates to $\rho_{\mathrm{c}}\approx 7\times 10^{-6}\,{\rm
g\,cm^{-3}}$ for our fiducial parameter values. In the initial envelope, this
density occurs at a radius of about $37R_{\odot}$, so we estimate that the jet
could choke at this separation, $a_{\mathrm{c}}\sim 37R_{\odot}$. In an
identical simulation for which a jet is not launched, this separation is
reached after $t_{\mathrm{c}}\sim 9\,{\rm d}$ (Chamandy et al., 2018, or Model
NJ1). We do not expect the small mass loss rate of the jet to have a large
effect on the orbit at this time (this is confirmed by our numerical results,
presented below). Therefore, we might expect the jet to choke at about this
time.

Additionally, we can use equation (2) with $\rho=\rho_{\mathrm{c}}$ to find
the time it would take to accelerate envelope material in the jet launch
region up to ${\color[rgb]{1,0,0}\sim}v_{\mathrm{j}}$, now assuming that the
jet ram pressure force still dominates over gravity. Since
$\rho_{\mathrm{c}}/\rho_{\mathrm{amb}}\sim 10^{3}$ we obtain
$t_{\mathrm{v,q}}\sim 3\times 10^{4}\,{\rm s}$ or about $0.3\,{\rm d}$. This
can be compared with the time it takes for new envelope material to enter the
jet launch region

$t_{\mathrm{orb}}\sim\frac{2r_{\mathrm{j}}\tan\theta_{\mathrm{h}}}{v_{\mathrm{orb}}},$
(5)

where $v_{\mathrm{orb}}$ is the orbital velocity at $t=9\,{\rm d}$. Computing
the tangential component of the relative velocity of the secondary with
respect to the primary core particle from the simulation, we obtain
$v_{\mathrm{orb}}\approx 100\,\mathrm{km\,s^{-1}}$. This gives
$t_{\mathrm{orb}}\sim 0.1\,{\rm d}$, which is $\lesssim t_{\mathrm{v,q}}$, so
the jet has become unable to clear away envelope material before new material
takes its place. In reality the secondary alters the local conditions in the
envelope, so these estimates are only rough.

We conclude that jet choking is expected to commence by the first periastron
passage. This conclusion applies to our other runs as well.

###  3.3 Role of the jet in unbinding the envelope

####  3.3.1 Overall contribution during the CE phase

Even if the jet does not contribute significantly to envelope unbinding over
the $10$-orbit timescale of our simulations, it may be significant over the
longer full CE phase, whose termination is presumably marked by stabilization
of the orbit and complete ejection of the envelope.

The timescale for a jet from the secondary to supply energy comparable to the
binding energy of the envelope is $E_{\mathrm{b}}/\dot{E}_{\mathrm{j}}$. Here
$E_{\mathrm{b}}=1.9\times 10^{47}\,\mathrm{erg}$ is the magnitude of the
binding energy of the envelope, including contributions from thermal energy
and potential energy terms involving the envelope self-gravity and the
gravitational interaction between the primary core point particle and
envelope.

The jet is expected to be choked early on (Section 3.2). Consider a more
extreme case than those simulated where the jet injects energy into the
envelope at a constant rate of
$\dot{E}_{\mathrm{j}}\sim\tfrac{1}{2}\dot{M}_{\mathrm{j}}v_{\mathrm{j}}^{2}$
thereafter. For the $\dot{M}_{\mathrm{j}}$ and $v_{\mathrm{j}}$ used in Runs
J1 or J8, this gives $\dot{E}_{\mathrm{j}}\sim 5\times
10^{38}\,\mathrm{erg\,s^{-1}}$, which implies
$E_{\mathrm{b}}/\dot{E}_{\mathrm{j}}\sim 13\,{\rm yr}$, whereas for $10$ times
higher $\dot{M}_{\mathrm{j}}$, we obtain
$E_{\mathrm{b}}/\dot{E}_{\mathrm{j}}\sim 1.3\,{\rm yr}$.

This is the time it would take for a very powerful, continuously active jet to
unbind the envelope – acting alone without any other sources of energy – if
jet energy could be transferred with maximal efficiency
$\alpha_{\mathrm{j}}\equiv\dot{E}_{\mathrm{b}}/\dot{E}_{\mathrm{j}}=1$, where
$\dot{E}_{\mathrm{b}}$ is the rate at which energy is transferred to (bound)
envelope material. In nature and simulations, $\alpha_{\mathrm{j}}<1$ because
a portion of the jet energy is transferred to already unbound (former)
envelope gas and a portion could directly leave the envelope.111In nature,
radiative cooling of envelope gas would also reduce $\alpha_{\mathrm{j}}$ (but
our simulations assume an adiabatic ideal gas, so that inefficiency is not
accounted for). The time for the jet to unbind the envelope assuming a
constant $\alpha_{\mathrm{j}}$ is

$t_{\mathrm{j}}=\frac{E_{\mathrm{b}}}{\alpha_{\mathrm{j}}\dot{E}_{\mathrm{j}}}.$
(6)

If $\alpha_{\mathrm{j}}=0.25$, for example, then we obtain
$t_{\mathrm{j}}\approx 50\,{\rm yr}$ if
$\dot{M}_{\mathrm{j}}\sim\dot{M}_{\mathrm{Edd}}$. (In our simulations the jet
adds kinetic energy at a rate about $20$ times smaller due to the jet profile
chosen, so $t_{\mathrm{j}}$ would be predicted to be $20$ times longer if the
full CE phase could be simulated.)

Crudely extrapolating the mass unbinding seen in simulations of an almost
identical system to the one considered here (but without accretion or jets)
leads to estimated unbinding times in the range $t_{\mathrm{CE}}\sim
1$-$10\,{\rm yr}$, where the lower value includes the recombination and
thermal energies in determining which gas is unbound, and the higher value
does not (Prust & Chang, 2019). Extrapolations of CE simulations involving
asymptotic giant branch (AGB) primaries suggest it might take of order
$t_{\mathrm{CE}}\sim 10\,{\rm yr}$ to unbind the envelope (Chamandy et al.,
2020; Sand et al., 2020).

Taken as a whole, the above estimate suggests that jets from MS and WD
secondaries could contribute significantly to unbinding for sustained jet
mass-loss rates equal to or exceeding the Eddington value, but could not
dominate envelope unbinding.

####  3.3.2 Expected extra unbinding in simulations with jets

We can try to predict the relative importance of jets to envelope unbinding by
comparing the energy input from jets with that from orbital decay during the
simulation. In simulations of the binary system considered here without
accretion or a jet, only $\sim 10\%$ of the envelope is unbound in the first
$40\,{\rm d}$ (Ohlmann et al. 2016; Chamandy et al. 2019a; Prust & Chang 2019
and see Section 4.3).

We can roughly estimate the orbital energy at first periastron passage for a
companion mass of $0.978M_{\odot}$ ($t\approx 12.8\,{\rm d}$) as
$-Gm_{2}M_{\mathrm{1,int}}/2a\approx-1.06\times 10^{47}\,\mathrm{erg}$, where
$a\approx 14R_{\odot}$ is the separation and $M_{\mathrm{1,int}}\approx
0.8M_{\odot}$ is the mass of the unperturbed primary inside $r=a$. Subtracting
this from the initial orbital energy
$-Gm_{2}M_{1}/2a_{\mathrm{i}}\approx-7.4\times 10^{46}\,\mathrm{erg}$ gives
$\sim 3\times 10^{46}\,\mathrm{erg}$ of orbital energy released up to the
first periastron passage. By comparison, the jet would inject only $3\times
10^{44}\,\mathrm{erg}$ of kinetic energy in our most extreme model (Run J5)
during this time.222To obtain this estimate we are ignoring the kinetic energy
due to the orbital motion of the jet, which is somewhat smaller than that due
to the outward motion of jet material. At the end of the simulation without
accretion or jet at $t=40\,{\rm d}$, $a\approx 7R_{\odot}$ and
$M_{\mathrm{1,int}}\approx 0.5M_{\odot}$. From $t=12.8\,{\rm d}$ to $40\,{\rm
d}$, the orbital energy liberated is $\sim 6\times 10^{46}\,\mathrm{erg}$,
whereas the jet in J5 injects about $6\times 10^{44}\,\mathrm{erg}$ of kinetic
energy. Therefore, the rate of kinetic energy supplied by the jet amounts to
only $\sim 1\%$ of the change in orbital energy for Run J5, and $\sim 0.1\%$
for Run J1 or J8.

However, the efficiency with which released energy is used to unbind envelope
mass may change when a jet is present, and in Section 4.3 we will see that the
impact of jets on envelope unbinding in the simulations is, in fact, more
substantial than the above estimate suggests.

##  4 Simulation Results

###  4.1 Jet evolution

We first verified from the simulations that the jet is activated as discussed
in Section 3.1. The timescales for the central part of the jet to accelerate
to $v_{\mathrm{j}}$ and for the jet material to be advected out of the
spherical sectors are roughly as predicted.

####  4.1.1 Jet choking

Figure 1: Diagram of the coordinate system used in this paper. The origin of
this primed system is located at the point particle representing the secondary
($P2$) and the axes rotate with the binary system. The $z^{\prime}$ axis
aligns with the vertical axis of the simulation lab frame $z$. The
$y^{\prime}$ axis is in the orbital plane of the binary and on a line joining
the primary core ($P1$) and the secondary particle. The $x^{\prime}$ axis is
also in the orbital plane and orthogonal to the $y^{\prime}$ axis. Later, we
show snapshots taking slices that are orthogonal to the orbital plane, and
either as viewed from the position of the primary (the $x^{\prime}-z^{\prime}$
plane) or sliced through both particles (the $y^{\prime}-z^{\prime}$ plane).

Figure 2: Snapshots showing the density of jet gas (i.e. jet tracer) in
$\,{\rm g\,cm^{-3}}$ at various times throughout the simulation model J1. The
secondary is located at the centre with its softening radius shown by a cyan
circle. The jet is initially vertical and extended but becomes deformed and
eventually choked inside dense envelope gas. In the top two rows, slices are
orthogonal to the orbital plane and to the line connecting the primary core
and the secondary particles (the $x^{\prime}-z^{\prime}$ plane), shown as
would be viewed from the position of the primary core particle (the azimuthal
component of the secondary’s orbital motion is toward the left). The bottom
two rows show the same times, but now sliced through both particles,
perpendicular to the orbital plane (the $y^{\prime}-z^{\prime}$ plane), with
the primary core particle situated left-of-centre. Its softening sphere is
shown by a magenta circle.

Figure 3: Evolution of the inter-particle separation in the orbital plane.
Top: Runs with initial companion mass $M_{2}(0)=0.978$ and without subgrid
accretion. Inset shows the full time evolution of the no-jet model NJ1, with
snapshot times of Fig. 2 indicated by short blue lines. While the orbits of
all runs are almost identical up to the first periastron passage, all jet runs
without subgrid accretion eventually develop a shorter orbital period compared
to NJ1. Bottom: Runs with $M_{2}(0)=0.978$ that include subgrid accretion,
except for NJ1, which is repeated from the top panel for ease of comparison.
All jet runs with subgrid accretion eventually develop a longer orbital period
than NJ1.  Figure 4: Snapshots of total gas density (including jet, envelope
and ambient gas) comparing models J1 and NJ1. Times are the same as in the
first four snapshots of Fig. 2. Velocity vectors are relative to the motion of
the secondary. The left two columns are sliced through the secondary
perpendicular to both the orbital plane and the line joining the particles,
(the $x^{\prime}-z^{\prime}$ plane). The left-most column shows our fiducial
jet run J1, and the second column shows NJ1, which is the same except without
a jet. By the last snapshot, at approximately the first periastron passage
(bottom row), the differences between the jet and no-jet slices are very
small. The right two columns show the slice through both particles and
perpendicular to the orbital plane, with NJ1 in the right-most column and J1
to its left (the $y^{\prime}-z^{\prime}$ plane). The softening sphere of the
RGB core is shown as a magenta circle. Note the bow shocks below the secondary
at $t=10.42\,{\rm d}$ (third row, third panel from the left), which are
produced by the jet as it chokes.  Figure 5: Slices at the final time
$t=40.0\,{\rm d}$, taken in the same view as the first two rows in Fig. 2,
show, from top to bottom rows, (i) tracer density, (ii) gas density with
velocity vectors in the rest frame of the secondary, (iii) tracer density
(zoomed in) and (iv) gas density (zoomed in with velocity vectors in the rest
frame of the secondary). Left column: Run J1 (fiducial, no subgrid accretion).
Middle column: Run J4 (As J1 but with subgrid accretion turned on). Right
column: Run J5 (As J4 but with jet mass-loss rate $\dot{M}_{\mathrm{j}}$
increased by a factor of $10$). In the runs with subgrid accretion, jet gas
has extended above and below the orbital plane (first and third rows). This is
caused by entrainment of thermalized jet gas by envelope gas flowing through
the funnel-shaped channel formed by the (highly super-Eddington) accretion,
after the jet has choked. Note that jet gas remains subdominant in mass (and
energy) to co-spatial envelope gas in J4
($\dot{M}_{\mathrm{j}}=\dot{M}_{\mathrm{Edd}}$), which implies that the jet
has not broken out. In J5 ($\dot{M}_{\mathrm{j}}=10\dot{M}_{\mathrm{Edd}}$)
the jet gas has comparable density and energy to co-spatial envelope gas –
since jet gas does not significantly displace envelope gas, we do not consider
this to be jet breakout.

Fig. 2 shows the density of only that part of the gas which was injected by
our jet subgrid model (the jet tracer gas), for our fiducial run J1. The top
two rows show slices through the secondary perpendicular to the orbital plane
and to the line joining the two particles, with the direction of the primary
core particle out of the page. The secondary is at the centre of the frame
with softening radius labeled by a circle, and the component of its motion
parallel to the slice is toward the left. The bottom two rows show slices
through both particles and perpendicular to the orbital plane, with the
primary core situated to the left of the secondary and its softening sphere
shown with a magenta circle. The evolution of the orbital separation between
the particles $a(t)$ is shown in the top panel of Fig. 3, with Run J1
represented by a solid green line.

By $t=1.1\,{\rm d}$, the jet has propagated out to $\sim\pm 60R_{\odot}$ from
the secondary. By $t=5.8\,{\rm d}$, as the secondary plunges into the
envelope, the jet begins to lose coherence and connectivity. Then, by
$t=10.4\,{\rm d}$, the jet has acquired a “<” morphology (with
$\phi$-component of its motion toward the left; see the third panel in the top
row). To explain this, consider that the vertical component of the jet
velocity has reduced owing to the resistance provided by envelope material in
and around the jet launch region. Further, jet material is being entrained, to
some extent, by envelope gas which moves at a speed of order
$v_{\mathrm{orb}}$ with respect to the secondary. As the two speeds become
comparable, the jet acquires a significant angle with respect to the vertical,
and this angle increases with time.

By $t=12.7\,{\rm d}$, around the time of the first periastron passage, the
angle of the “<” is smaller and the jet is less extended and narrower due to
its confinement by surrounding gas. Evidence of Kelvin-Helmholtz instability –
caused by the relative motion between the jet and envelope material – is
visible. The jet then oscillates between “<” and “>” morphologies as the
secondary dives into dense material and re-emerges during the second
periastron passage at $t\approx 17.5\,{\rm d}$. There is quite a bit of
symmetry between the upper and lower portions of the jet, but at $t=10.4\,{\rm
d}$ significant asymmetry is present and the top portion is more deformed than
the bottom. Similar asymmetry when the jet begins to be choked is seen in
other runs as well, e.g. the angles made with the vertical by the top and
bottom portions of the jet can be quite different. By the end of the
simulation, the jet material is confined to a quasi-spherical region. Inside
of this region, the structure of the jet is still somewhat bi-polar, as seen
in the right panel in the second row of Fig. 2. The jet has been completely
choked by the surrounding envelope gas. The energy it continues to eject is
rapidly thermalized. After the first periastron passage, jet material added to
the grid remains bound to the secondary and has Mach number $\mathcal{M}<1$.
Other runs show an overall similar choking of the jet.

To further study jet choking and how the jet affects envelope gas, we plot in
Fig. 4 snapshots of the total gas density (including jet, envelope and ambient
gas) overlaid with velocity vectors in the rest frame of the secondary, for
runs with and without a jet. Both cuts of Fig. 2 (perpendicular to the line
joining the particles and through the particles) are shown, with Run J1 in the
first and third columns and Run NJ1 in the second and fourth columns. Times
are the same as the first four snapshots of Fig. 2, and the last snapshot
approximately coincides with the first periastron passage.

At first, the jet is prominent in the snapshots and drills through the
envelope by displacing envelope gas. By the first periastron passage, however,
the J1 and NJ1 snapshots are almost indistinguishable. At $t=10.4\,{\rm d}$,
shocks caused by the marginally choked jet are visible on either side of the
orbital plane in the slice through both particles. The shocks are much more
visible below the orbital plane in the figure, and it is apparent from the
velocity vectors that the lower half of the jet is choked a bit later than the
top half.

In tandem, these various results show that the jet is choked around the time
of first periastron passage, in broad agreement with the theoretical estimates
presented in Section 3.2.

####  4.1.2 Role of accretion and lack of jet breakout

To understand how accretion can affect the results, we compare Runs J1 and J4,
which are identical except that J4 also allows accretion as in Model B of
Chamandy et al. (2018). Accretion rates after the first periastron passage are
in the range $\sim(0.2$–$2)M_{\odot}\mathrm{\,yr^{-1}}$ (Chamandy et al.,
2018). This is equal to $\sim(100$–$1000)\dot{M}_{\mathrm{Edd}}$ for an MS
star and $\sim(10^{4}$–$10^{5})\dot{M}_{\mathrm{Edd}}$ for a WD. The Eddington
value always assumes spherically symmetric flow. While it can be exceeded by a
factor of $\sim 10$ in axially symmetric flows, we consider the rates quoted
above to be upper limits. We thus study how maximally strong accretion affects
CEE that includes a jet, keeping in mind that smaller accretion rates might be
expected to result in the same effects, only weaker.

In Fig. 5 we present various snapshots of density at the end of the simulation
at $t=40.0\,{\rm d}$ for fiducial run J1 (left column), subgrid accretion run
J4 (middle) and $10$-fold higher jet mass, subgrid accretion run J5 (right).
From top to bottom, we have the density of jet material alone (as in Fig. 2),
total gas density (as in Fig. 4), density of jet material (zoomed in) and
total gas density (zoomed in). To facilitate direct comparison between the
panels, the same colour table is used throughout.

Up until $t\approx 30\,{\rm d}$, the morphologies of the jets in J1 and J4 are
quite similar. At about this time, the blob of jet material in J4 starts to
expand upward. Just before the simulation ends, the jet tracer gas suddenly
reverses direction and begins to extend downward. Jet gas remaining from the
first episode is visible above the secondary, and the gas involved in the
ongoing episode is visible below the secondary in Fig. 5. The jet gas in J4 is
able to expand in the polar directions because of the channel that forms when
the accretion subgrid model is active (Chamandy et al., 2018). The polar
regions become relatively depleted of envelope gas, providing a channel
through which jet gas can flow. Comparing the density of jet material with the
total gas density we see that the density is dominated, as in J1, by envelope
material even where the jet material is located. As the temperature and
velocity are the same for co-spatial jet gas and envelope gas in our
simulation, the envelope gas dominates energetically as well. We thus conclude
that this behaviour is caused by entrainment of thermalized jet material by
envelope material flowing through the double-funnel-shaped partially evacuated
region inside the torus.

While some of the material entering the subgrid accretion sphere of radius
$4\delta_{4}\approx 0.56R_{\odot}$ accretes, much of the gas flowing toward
the torus centre instead passes through to the other side. The same behaviour
is seen in simulations with accretion but no jet, and even in those
simulations, the flow switches directions seemingly at random.333 This
behaviour was observed in the runs with subgrid accretion of Chamandy et al.
(2018), but is being reported here for the first time. It is not clear how
physical this particular flow pattern is, and greater fidelity in this aspect
would require more sophisticated modeling of the (unresolved) secondary (e.g.
Prust, 2020). Because the jet is completely dominated by envelope gas, we do
not consider this to be a true breakout of the jet. On the other hand, if
$\dot{M}_{\mathrm{j}}$ is increased by a factor of $10$, as in Run J5, the jet
density is comparable to the density of envelope gas in the polar regions, as
seen in the right column of Fig. 5. Even in this extreme case, however, the
jet does not truly break out because it does not significantly displace
envelope material, and jet gas barely contributes to the total gas density.
Other runs show less or no entrainment of jet gas by envelope gas along the
polar directions. Therefore, true jet breakout never occurs in our runs.
Moreover, the high accretion rates which are found to promote entrainment of
jet material represent an upper limit, and it remains to be seen whether this
behaviour would still occur if more realistic accretion rates for an MS or WD
($\lesssim 10\dot{M}_{\mathrm{Edd}}$) were obtained or assumed.

On the other hand, if the jet continues to inject energy at the same rate up
until the envelope ejects, then at some point before that the jet would
naturally break out. With this in mind, we can conclude from the lack of jet
breakout in our simulations that jet breakout could only happen well after the
$10$-orbit mark, and much later still for more realistic accretion rates at
least $1$–$2$ orders of magnitude smaller than those obtained. The later the
jet breaks out, the longer it stays coupled to the envelope, enabling
efficient energy transfer that assists unbinding. Therefore, longer
simulations which can help to constrain the time of jet breakout would be
valuable.

###  4.2 Orbital evolution

The evolution of the orbital separation $a(t)$ for all of the runs involving a
$M_{2}(0)=0.978M_{\odot}$ companion is shown in Fig. 3. We plot the runs
without subgrid accretion in the top panel and those with it in the bottom
panel, with NJ1 repeated in both panels for ease of comparison.

In the top panel, we see that all runs including a jet but not subgrid
accretion end up with slightly smaller orbital periods than the run without a
jet. Our fiducial run, J1, is more than half a period out of phase with NJ1 by
$t=40\,{\rm d}$, and the mean separation is correspondingly smaller. However,
the majority of this difference is apparently caused by the mass loss
$\dot{M}_{2}=-\dot{M}_{\mathrm{j}}$, since the orbit in Run J3, which is
identical to Run J1 but with $\dot{M}_{2}=0$, is closer to that of NJ1 than to
that of J1. It is known that making the companion mass smaller results (for
equal initial separation) in an initially longer period, followed by a shorter
period, and that the smaller the companion mass, the later the transition
happens (Passy et al., 2012; Chamandy et al., 2019b), and this likely explains
the difference between J1 and J3.

While the dynamical friction drag on the secondary would be slightly enhanced
by jet material that remains near to it, it is not clear whether this explains
why J3 ends up with a slightly smaller period than NJ1. Doubling the jet
speed, in going from J1 to J6, leads to a slight increase in the period,
implying weaker drag. This may be because less of the jet material remains
bound. Run J7, which is like J1 except that the jet opening angle is doubled,
also has a larger orbital period compared to J1, perhaps because the wider jet
is better at preventing envelope material from accumulating near the
secondary.

We now turn to the runs with accretion, shown in the bottom panel of Fig. 3.
All runs with accretion lengthen the orbital period compared to those without
accretion. This is consistent with the known result (mentioned above) that
more massive secondaries result in longer periods (eventually). But another
cause might be that accretion clears away material near the secondary,
reducing drag. Increasing $\dot{M}_{\mathrm{j}}$ by an order of magnitude from
J4 to J5 causes the period to first be longer, and eventually to be shorter,
with the transition occurring at $t\approx 35\,{\rm d}$. This seems consistent
with the companion accreting somewhat less mass after $t\approx 22\,{\rm d}$
in J5 than in J4, as seen in Fig. 6, though the reason that a larger
$\dot{M}_{\mathrm{j}}$ leads to a dip in the accreted mass is not immediately
clear.

The WD run J8 has a separation curve with period only marginally longer than
NJ1; the shorter period compared to J4 or J5 could be due to the smaller
accretion rate. Finally, we turn to runs J2 and NJ2, which are similar to J5
and NJ1, respectively, but with companion mass only half as large. These runs
are plotted in Fig. 7. J2 has a somewhat longer orbital period than NJ2 by the
end of the simulation, likely due to rapid accretion by the companion.

###  4.3 Envelope unbinding

Figure 6:  Change in mass of the secondary $\Delta M_{2}(t)=M_{2}(t)-M_{2}(0)$
for the runs with subgrid accretion turned on. The accretion model is the same
as that used in Chamandy et al. (2018), except that the accretion rate in Run
J8 has been capped at a constant value. In addition to mass gain due to
accretion, $\Delta M_{2}$ depends on a small constant rate of mass loss to
supply the jet (see Tab. 1). The right vertical axis shows $\Delta M_{2}$
divided by the initial envelope mass. Figure 7: Evolution of the inter-
particle separation in the orbital plane for Runs J2 and NJ2, which have
initial companion mass $M_{2}(0)=0.489M_{\odot}$, i.e. half of that used in
the other runs.  Figure 8:  Top: Mass of all unbound gas (envelope, jet,
ambient) relative to the initial value, for Run J1 (fiducial jet model) and
Run NJ1 (fiducial model without jet). Unbound mass of the ambient gas alone is
also shown. Note that the unbound mass expressed as a percentage of the
initial envelope mass is shown on the right vertical axis. Middle: Difference
in unbound mass between simulations J1 and NJ1, shown for all unbound gas
(solid) and ambient gas (dotted). The mass of jet material is relatively small
and is shown as a dotted line. Bottom: Difference in the unbound mass between
Runs J1 and NJ1, including envelope and jet material but excluding ambient
material. At $t=40.0\,{\rm d}$, J1 has unbound an extra $\sim
10^{-3}M_{\odot}$ of envelope material, i.e. an extra $\sim 1\%$ compared to
the total unbound envelope mass of $\sim 0.11M_{\odot}$
($=0.15M_{\odot}-0.04M_{\odot}$ from the top panel).  Figure 9:  Same as
bottom panel of Fig. 8, now for all of the runs. Top: Runs without subgrid
accretion. Middle: Runs with subgrid accretion involving a $\sim 1M_{\odot}$
secondary. Bottom: Run J1, involving a $\sim 1/Msun$ secondary, as compared to
its no-jet counterpart. In each plot, dotted lines represent the accumulated
jet mass $\dot{M}_{\mathrm{j}}t$ (see Tab. 1).

####  4.3.1 Definition

We designate gas as “unbound” if
$\mathcal{E}_{\mathrm{kin,gas}}+\mathcal{E}_{\mathrm{int,gas}}+\mathcal{E}_{\mathrm{pot,gas-
gas}}+2\mathcal{E}_{\mathrm{pot,gas-1}}+2\mathcal{E}_{\mathrm{pot,gas-2}}>0$,
where the terms are respectively the kinetic energy density of bulk motions,
the internal energy density, the potential energy density due to gas self-
gravity, twice the potential energy density due to the gravitational
interaction between gas and the RGB core particle, and twice the potential
energy density due to the gravitational interaction between gas and the
secondary (see Chamandy et al. 2020 for details). We choose to include the
factors of two in the last two terms because for the gas and cores to
ultimately unbind from one another, the total potential energy must be
balanced, including the half nominally contained in the core particles (e.g.
§2.1 of Binney & Tremaine, 2008). No diagnostic predicts with certainty
whether a given gas parcel will eventually be ejected, so any such choice is
somewhat arbitrary, and the choice used here is fairly conservative (for a
discussion see Ivanova et al. 2013).

####  4.3.2 Extra unbinding caused by jets

The difference in the mass of unbound gas with respect to the simulation
initial condition $\Delta M_{\mathrm{ub}}(t)\equiv
M_{\mathrm{ub}}(t)-M_{\mathrm{ub}}(0)$ is plotted in Fig. 8 for Runs J1 and
NJ1. Here $M_{\mathrm{ub}}(0)$ is the $\sim 98\%$ of ambient gas mass that is
already unbound at $t=0$. Mass flux through the boundaries of the grid is
fully accounted for in the plots. The top panel shows $\Delta M_{\mathrm{ub}}$
for all the gas, as well as for ambient gas alone. Note that all of the
ambient gas that is bound at $t=0$ (found closest to the binary) is rendered
unbound by $t\approx 16\,{\rm d}$ in each simulation.

About $12\%$ of the envelope mass is unbound by the first periastron passage
(obtained by subtracting the unbound ambient gas mass from the unbound gas
mass). Thereafter, the unbound mass plateaus and slowly decreases. This
decrease is inconsequential for the analysis below because we are interested
in differences in the unbound mass between the runs.444The decrease may be
caused by energy transfer from marginally unbound gas causing it to become
marginally bound, subtle effects stemming from our chosen definition for
“unbound” or energy transfer to the ambient medium. We have explored other
definitions (factors of $2$ omitted or thermal energy excluded) and still find
a decrease with time. This is consistent with the results of Prust & Chang
(2019) (with initial conditions almost identical to our own), who find a
decrease until about $150\,{\rm d}$ using a definition of “unbound” that
excludes our factors of $2$ and thermal energy. However, such a decrease is
not seen in Ohlmann (2016), who employs similar initial conditions.

The middle panel shows the difference in $\Delta M_{\mathrm{ub}}$ between J1
and NJ1, i.e. the excess unbound mass caused by the jet. Finally, the bottom
panel subtracts the ambient mass and therefore shows the excess unbound
envelope mass in J1 compared to NJ1.555To be precise, this quantity also
includes the unbound jet mass. However, the total jet mass (cyan dotted line
in Fig. 8) is relatively small, implying that its contribution to the unbound
mass is fairly negligible. Note that the right vertical axis shows the unbound
mass as a percentage of the initial envelope mass $M_{\mathrm{e}}(0)\approx
1.6M_{\odot}$.

We can measure the relative contribution of the jet to envelope unbinding by
dividing the excess mass of unbound envelope gas in the jet run by the mass of
all unbound envelope gas. At the end of our simulations at $t=40.0\,{\rm d}$,
this amounts to $\sim 10^{-3}M_{\odot}/(0.15M_{\odot}-0.04M_{\odot})\sim 1\%$,
where we have used the top panel and subtracted the unbound ambient mass from
the unbound gas mass to obtain the unbound envelope mass. Therefore, the jet
causes an additional $\sim 1\%$ of envelope mass to be unbound by the end of
Run J1, as compared to Run NJ1. Prior to the end of the simulation, the excess
unbound mass in the jet simulation can be a few times higher, but between the
first and second periastron passages it is actually negative (i.e. less mass
is unbound in J1 than NJ1 at that time). Taking the simulation as a whole,
including a jet leads to a small increase in unbound envelope mass.

In Fig. 9, we plot the relative difference in the unbound mass compared to the
corresponding no-jet run (as in the bottom panel of Fig. 8) for all of the jet
runs. The top (middle) panel includes runs without (with) subgrid accretion,
while the bottom panel shows Run J2 (smaller secondary mass) relative to Run
NJ2.

In all jet runs, the peak unbound mass, which occurs near the first periastron
passage, is higher as compared to the corresponding no-jet run. Jets are
generally seen to enhance the unbound envelope mass by a few percent. The
excess unbound mass caused by the jet peaks at some point after $t=26\,{\rm
d}$ for runs involving a $\sim 1M_{\odot}$ secondary, but sometimes decreasing
fairly sharply after peaking. There are two cases where the excess unbound
mass caused by the jet is significantly higher. In Run J2, the peak unbound
envelope mass, which occurs at $t\approx 20\,{\rm d}$, near the first
periastron passage, is about $8\%$ of the initial envelope mass. This is about
$20\%$ higher than in Run NJ2. However, by the end of the simulation the extra
unbinding in J2 compared to NJ2 is marginal. This is probably because the
final orbit of J2 is much wider than that of NJ2 owing to accretion (Section
4.2), resulting in less liberation of orbital energy. In Run J8, our WD run,
almost an additional $1\%$ of the envelope mass, or $\sim 10\%$ of the unbound
envelope mass, is unbound by the end of the simulation, as compared to NJ1.

In summary, we generally find that simulations with jets (or jets+accretion)
lead to between about $\sim 1\%$ and $\sim 10\%$ of extra unbinding compared
to identical simulations without jets (or jets+accretion).

####  4.3.3 Parameter space exploration

Here we compare the various jet runs to better understand the effects of the
model parameters. Run J3 is the only run for which the jet mass is not
subtracted from that of the secondary, and is otherwise identical to Run J1.
We see from the top panel of Fig. 9 that J3 unbinds about twice as much extra
envelope mass (compared to NJ1) as does J1. This must be caused by the larger
mass of the secondary. Note that a dip to negative values of the excess
unbound envelope mass immediately following the first periastron passage at
$t\approx 13\,{\rm d}$ is seen in all runs except J3. This tells us that this
brief period of reduction in the unbound envelope mass as compared with the
corresponding no-jet run is at least partly caused by the reduction of the
secondary mass as it feeds the jet (in Runs J2, J4, J5 and J8, accretion
eventually produces a net increase in $M_{2}$).

Run J6 is like J1 except with twice larger jet speed $v_{\mathrm{j}}$. This
results in a $4$-fold higher rate of kinetic energy release into the jet.
Consistent with this higher jet power, the excess unbound envelope mass is a
few times larger than in J1, but, for reasons not yet understood, decreases at
the end of the simulation to be comparable to J1.

Run J7 is like J1 except with a twice larger jet opening angle. The unbinding
curve is similar to that of J1, but shows slightly more unbound mass at the
first periastron passage. This may be due to a larger surface of interaction
between the jet and envelope material as the secondary plunges in.

On the whole, turning on subgrid accretion does not have a large effect on the
unbound mass. This can be seen by comparing the surplus unbound mass in J1
(Fig. 9 top panel, green solid) with that in J4 (bottom panel green solid).
Therefore, differences in unbound mass between accretion+jet runs and no-
accretion/no-jet runs can primarily be attributed to the jets.

The wider orbits of the runs with subgrid accretion imply a smaller amount of
orbital energy transfer to the envelope. This results in lower unbound mass at
late times, and likely explains the dip to negative values of excess unbound
mass at the end of Runs J4 and J5. The jet in J5 is about $10$ times as
powerful compared to that in J4 due to its $10$ times larger
$\dot{M}_{\mathrm{j}}$. The peak value of excess unbound envelope mass in J5
is more than twice that in J4. The tendency for a greater jet power to lead to
more unbound material is compensated somewhat by the lower secondary mass
$M_{2}(t)$ in J5 (due to feeding the jet at a higher rate and to a smaller
accretion rate compared to J4).

Run J2 is the run that produces the highest relative increase in the unbound
mass at around the first periastron passage (peak of the unbound envelope mass
curve) at $t\approx 20\,{\rm d}$. The jet power is generally larger as
compared to the rate of orbital energy release in J2 compared to the other jet
runs. In addition, envelope material near the jet has less binding energy due
to the smaller secondary mass. Finally, the smaller secondary mass implies
less potential energy added by the jet. However, by the end of the simulation,
the wider orbit (due to accretion) negates any gain in unbound mass due to the
jet. With a smaller, more realistic accretion rate, extra unbinding at a level
of $\sim 20\%$ could perhaps be sustained for much longer.

The WD run J8 is the only case that exhibits a $\sim 10\%$ increase in the
unbound mass compared to NJ1 by the end of the simulation. Here
$\dot{M}_{\mathrm{j}}$ is $100$ times smaller than in J1, so a comparison with
J3 is more apt. The peak excess unbound mass is about twice as high in J8 as
compared to J3. The kinetic energy of the jet is the same in both runs due to
the $10$ times larger jet speed in J8. Once the jet is choked, this kinetic
energy transfers to heat in the envelope. However, while the kinetic energy
input is similar, J8 injects $100$ times less negative potential energy. This
might explain the greater unbinding in J8 as compared to J3. Since J8 is the
only run for which the net energy injected by the jet, including potential
energy, is positive, we consider this jet model to be the most realistic
(though the potential energy injected by the jet can be thought of as sourced
from the core particle).

##  5 Overall impact of jets on envelope unbinding

In Section 3.3.2 we estimated the ratio of the kinetic energy supplied by the
jet to the orbital energy released during inspiral and found this ratio to be
equal to about $(0.1$–$1)\%$. Sure enough, including a jet in our simulations
did not produce a drastic difference in the unbound envelope mass. However, it
produced an increase in the unbound mass of $\sim(1$–$10)\%$. Run J8, which
simulates a WD companion, is particularly noteworthy because it unbinds an
extra $\sim 10\%$ of envelope mass compared to Run NJ1, even though the
estimated energy ratio is $\sim 0.1\%$. Thus, we conclude that jets may cause
a fractional increase in unbound mass higher than their fractional energy
input. This is likely because the jet energy is being distributed efficiently
to unbind already marginally bound gas, or prevent marginally unbound gas from
becoming bound again.

More extreme MS or WD jets with up to two orders of magnitude larger powers
are perhaps possible. We estimated in Section 3.3.1 that such powerful jets
might alone unbind the envelope within $\sim 10\,{\rm yr}$, comparable to the
CE timescale estimated based on extrapolation from simulations that did not
include jets. Therefore, it seems plausible that maximally powerful MS or WD
jets ($1$–$2$ orders of magnitude more powerful than those simulated) could
cause an $\mathcal{O}(1)$ increase in the rate of envelope mass unbinding, and
additional simulations are needed to investigate this possibility (though even
if such powerful jets could exist, it is doubtful whether they could be
sustained for the duration of the CE phase).

We find that the excess unbound mass caused by including the jet spikes around
the time of first periastron passage. This is also around the time that the
jet becomes choked and the overall unbound envelope mass peaks (in simulations
with or without a jet). However, the excess unbound mass generally increases
to its maximum values some time after this (Fig. 9). From the beginning of the
simulation up until about the first periastron passage, the jet interacts only
weakly with the envelope for two reasons. First, the tidal stream that
emanates from the envelope and wraps around the secondary is concentrated at
small polar angles relative to the secondary, where the jet is not present or
weak (Fig. 4; this effect is enhanced by our choice of a strongly peaked
angular jet velocity profile). Second, the jet drills through the low density
material of the outer envelope very easily, i.e., without transferring much of
its energy. We can estimate the fraction of jet energy that does work on the
surrounding gas to be of order $\sim t_{v}/t_{\mathrm{orb}}$, where $t_{v}$ is
given by equation (2) and $t_{\mathrm{orb}}$ by equation (5) (c.f. Soker,
2016). This gives an efficiency of about $(1$–$2)\%$, assuming
$a=(47$–$47.5)R_{\odot}$, corresponding to $t\approx(4.5$–$5)\,{\rm d}$ and
$v_{\mathrm{orb}}\approx 100\,\mathrm{km\,s^{-1}}$. However, some of this
energy would be transferred to already unbound gas, so the actual efficiency
would be $\lesssim 1\%$. On the other hand, once the jet chokes (and the
energy it releases is thermalized locally), virtually all of its energy is
transferred to bound envelope gas surrounding the secondary. Therefore, the
ability of the jet to unbind efficiently envelope material relies on the jet
remaining choked. Once it breaks out (which would happen eventually if it did
not turn off first), its effect on further envelope unbinding would likely be
marginal.

##  6 Comparison with previous studies

###  6.1 “Grazing envelope evolution”

“Grazing envelope evolution” (GEE; Soker 2015) is a proposed scenario where
the jet unbinds envelope material and thus prevents it from accumulating
around the secondary and causing dynamical friction drag.666An apparent
contradiction entailed by this scenario is that the secondary would need to be
accreting envelope gas in order to power the jet. This potential problem could
perhaps be circumvented if the accretion disc acquires enough mass before jet
activation to sustain accretion after it, but more work is needed.  Assuming
that the jet can remain powered at a steady rate (as in our simulations),
could the CE phase be prevented by “grazing”?

GEE is not seen in our simulations nor in those of Shiber et al. (2019), which
are, to our knowledge, the only global simulations to include all the relevant
gravitational forces as well as jets. To test the viability of the GEE
scenario, our simulations would ideally begin with a larger initial
separation. However, Shiber et al. (2019) did use a large initial separation
for some of their runs, and still did not report seeing GEE. If GEE were
common, one might expect a difference in the orbital evolution between jet and
no-jet runs at very early times. But the separation curves of our jet and no-
jet runs are extremely similar up to the first periastron passage (Figs. 3 and
7).

One reason we do not see evidence of GEE could be that the efficiency with
which the jet unbinds material before it chokes is very small, as discussed in
Section 5. However, a smaller mass companion, larger initial separation, or
jet that is more powerful, wider or inclined relative to the vertical may
enhance the likelihood of a GEE-like scenario unfolding.

###  6.2 Previous CE jet simulations

The global simulations by Shiber et al. (2019) had a similar setup to ours,
and were run for a similar number of orbits. As in the present work, they
found that jets choke as they becomes surrounded by dense envelope gas. Just
prior to choking they develop a “<” morphology (in the plane perpendicular to
the orbital plane and line joining the particles) similarly to what we found.
Unlike us, however, they found that the choked jet eventually breaks out. On
the other hand, Lopez-Camara et al. (2021) found that even the strongest jets
they simulate choke before the first periastron passage and do not break out
for the remainder of the simulation. These results agree qualitatively with
our own (though the parameter values of the systems simulated are very
different). However, it should be noted that all of the Lopez-Camara et al.
(2021) simulations end before the first pronounced periastron passage, which
occurs just prior to $70\,{\rm d}$ (see simulation#5 of Shiber et al. 2019).

The jet model of Shiber et al. (2019) removes material from their conical jet
initialization regions at rates similar to the highly super-Eddington
accretion rates ($\gtrsim 100\dot{M}_{\mathrm{Edd}}$) that result from our
accretion subgrid model. In our simulations, jet breakout does not occur but
jet gas does expand in the polar directions in runs that allow accretion,
owing to the partially evacuated funnel-shaped regions that form on either
side of the orbital plane and entrainment of jet gas by envelope gas flowing
through this region. Therefore, it seems likely that the jet breakout seen in
simulations of Shiber et al. (2019) is enabled by their mass removal
prescription, which (as they explain) is somewhat akin to an accretion subgrid
model. As in our simulations which allow accretion, they find that the orbital
period eventually becomes longer when jets are included.

When jet gas re-emerges from the dense quasi-spherical concentration of gas
around the secondary in our simulations with accretion, the jet material
remains energetically subdominant compared to the envelope gas in its
immediate surroundings, and thus the jet does not truly break out (Section
4.1.2). This is not the case in Shiber et al. (2019), where the low-density
jet is able to displace envelope material (i.e. break out) in at least some
cases.

While accretion in our simulations happens within $4\delta_{4}\approx
0.56R_{\odot}$ from the secondary, Shiber et al. (2019) continuously expunge
all envelope material within the jet cones, which extend out to $7R_{\odot}$
or $14R_{\odot}$, depending on the simulation. Furthermore, they pressurize
their jet by giving it thermal energy density comparable to that of the
envelope gas removed. As thermal energy of the envelope gas around the
secondary likely almost balances gravitational potential energy (Chamandy et
al., 2018), this choice likely further facilitates jet breakout. Buoyancy of
the dilute jet gas could also help the jet to break out. For these reasons,
the jet breakout observed in Shiber et al. (2019) may not be realistic.

Comparing similar simulations with and without a jet, Shiber et al. (2019)
concluded that the presence of a jet increased the outward flux of material
through a sphere of radius $1\,\mathrm{au}$ or $2\,\mathrm{au}$ centred on the
origin of the simulation domain and extending out to the boundary. The
fraction of the material which had positive energy density flowing out of this
sphere was also determined to be higher when a jet was included (but note the
different simulation durations of the jet and no-jet runs). Based on this they
claim that jets unbind roughly three times as much envelope mass as identical
simulations without jets. Firstly, this result may depend sensitively on the
ability of the jet to break out, which, as argued above, may rely on
questionable subgrid prescriptions. Secondly, they did not state whether the
fraction of the total gas mass that acquired a positive energy density – a
more direct measure of unbinding – was different for jet and corresponding no-
jet runs. As such, they did not actually determine whether the jets caused
more material to become unbound or only changed the spatial distributions of
bound and unbound material.

###  6.3 Neutron star and black hole secondaries

NSs and BHs can undergo so-called hypercritical accretion, with an accretion
rate several orders of magnitude above the Eddington rate. A NS jet with
$\dot{M}_{\mathrm{j}}=0.032M_{\odot}\mathrm{\,yr^{-1}}$ (about
$10^{6}\dot{M}_{\mathrm{Edd}}$ if $R_{2}=11.1\,\mathrm{km}$) and jet velocity
$v_{\mathrm{j}}=0.1c$ (at a distance of $r_{\mathrm{j}}/2\approx 1.1R_{\odot}$
from the NS) would have a power of $\sim 9\times
10^{43}\,\mathrm{erg\,s^{-1}}$, which is several orders of magnitude more
powerful than the jets in our simulations. This would also greatly exceed the
mean rate of orbital energy release of $\sim 3\times
10^{40}\,\mathrm{erg\,s^{-1}}$ in the simulations presented. Using equation
(4) with the fiducial values $r_{\mathrm{j}}=2.25R_{\odot}$,
$\theta_{\mathrm{h}}=15^{\circ}$ and $M_{2}(0)=0.978M_{\odot}$, we find that
the jet would choke when the envelope density near the secondary is $\sim
3.7\times 10^{-3}\,{\rm g\,cm^{-3}}$, which occurs at a radius $\sim
0.8R_{\odot}$ in the initial envelope profile. This is comparable to the final
separation predicted using the $\alpha_{\mathrm{CE}}$ energy formalism
(Chamandy et al., 2019a), so the jet may or may not choke during the CE phase.
If not, then our results suggest that this may reduce its capacity to unbind
the envelope.

To take another example, the same jet but now with
$\dot{M}_{\mathrm{j}}=10^{4}\dot{M}_{\mathrm{Edd}}$ instead of
$10^{6}\dot{M}_{\mathrm{Edd}}$ is predicted to choke at a density of $\sim
3.7\times 10^{-5}\,{\rm g\,cm^{-3}}$, which occurs at $26R_{\odot}$ from the
centre in the initial RGB profile. In this case, the jet would likely choke at
around the first periastron passage. Subsequently, if it continued to supply
energy at the same rate, its power would exceed the rate of orbital energy
release by a factor of a few, and the jet would thus likely dominate the
envelope unbinding. However, it might not remain choked and quickly break out,
reducing the efficiency of energy transfer to the envelope. In any case, these
rough examples suggest there may be a region of the CE parameter space
involving NS and BH secondaries where jets dominate envelope unbinding (e.g.
Hillel et al., 2021). Further studies involving global CE simulations with NS
or BH jets are needed.

##  7 Conclusions

We simulated a common envelope phase involving a $2M_{\odot}$, $48R_{\odot}$
RGB primary and a $1M_{\odot}$ or $0.5M_{\odot}$ secondary that continuously
launches a jet with approximately constant power. The jet was included using a
subgrid model that adds high-velocity gas to two spherical sectors on either
side of the orbital plane (see Section 2.1, Appendix A and Federrath et al.
2014). We explored jet opening angles of
$\theta_{\mathrm{h}}=(15$–$30)^{\circ}$ but the jet velocity was strongly
peaked inside $\theta_{\mathrm{h}}/6$. Our jets injected kinetic energy equal
to $\sim(0.025-0.25)\dot{M}_{\mathrm{Edd}}v_{\mathrm{j}}^{2}$, with
$\dot{M}_{\mathrm{Edd}}$ the Eddington accretion rate for an MS star or WD and
$v_{\mathrm{j}}$ the peak ($\theta=0$) jet speed, chosen to be of order
(slightly larger than) the escape speed. The results of our jet runs were
compared with identical runs without a jet, and we performed several runs to
explore the parameter space for MS and WD secondaries. Some of our runs
included subgrid accretion onto the secondary at rates that we consider to be
upper limits (Krumholz et al., 2004; Chamandy et al., 2018), but the
contribution of the jet to envelope unbinding was found not to be very
sensitive to whether or not subgrid accretion was turned on. The duration of
our simulations was about $10$ orbits. Our main results can be summarized as
follows:

  * •

Jets in the simulations get choked at around the time of the first periastron
passage, and remain choked for the duration of each simulation, in broad
agreement with rough analytic estimates and recent “wind-tunnel” type
simulations which used input from global simulations (Lopez-Camara et al.,
2021);

  * •

After choking, jets can enhance envelope unbinding by depositing energy
locally, whereas before choking, jets tend to interact relatively weakly with
envelope gas, hardly affecting envelope unbinding;

  * •

Jets can enhance the unbound envelope mass in the simulations by up to about
$10\%$, as for our simulation involving a WD companion (compare the curve
representing J8 in Fig. 9 with the top panel of Fig. 8);

  * •

The extra fractional unbound mass caused by jets during a simulation can far
exceed the fractional increase in the energy added, likely because the jets
can energize and unbind marginally bound gas or prevent marginally unbound gas
from rebinding;

  * •

Based on analytic estimates and rough extrapolation of simulation results
there remains a possibility that over the course of the full CE phase, MS or
WD jets might, in the most optimistic cases, contribute to envelope unbinding
at a level comparable to that resulting from orbital inspiral alone (Sections
3.3.1 and 5). But this would likely require extremely powerful jets that
remain choked, yet strongly active, over much of the CE phase;

  * •

Qualitative disagreement between our results and those of Shiber et al.
(2019), who find that jets can break out after choking and greatly enhance
envelope unbinding, may be attributable to two sources: (i) their jet subgrid
model artificially promotes jet breakout by continually removing all envelope
gas in the jet launch region and replacing it with jet material with pressure
equal to that of the material it replaces, and (ii) the measure they used for
envelope unbinding does not take into account all of the envelope gas;

  * •

We see no evidence whatsoever for a GEE phase (Soker, 2015), but we cannot
exclude the possibility of this type of scenario occurring for cases with
higher jet speeds, lower mass companions, larger initial separations, initial
rotation of the primary, larger jet opening angles, and jets which have axes
that are inclined relative to the orbital axis (e.g. Schreier et al., 2019);

  * •

We estimate that for CEE involving an NS or BH companion launching a jet, the
jet could play a more important, possibly dominant role in envelope unbinding
because these jets could be much more powerful.

Future studies should involve longer simulations at higher resolution, tie the
accretion rate to the jet mass-loss rate (perhaps keeping the fraction of
accreted mass that goes into the jet fixed) and explore maximally powerful MS
and WD companion jets. Similar simulations with NS or BH companions would also
be interesting. However, such cases are challenging owing to the small
timesteps necessitated by simulating high jet speeds (to say nothing of
modeling relativistic effects).

## Acknowledgements

This work used the computational and visualization resources in the Center for
Integrated Research Computing (CIRC) at the University of Rochester. The
authors acknowledge the Texas Advanced Computing Center (TACC) at The
University of Texas at Austin for providing HPC resources that have
contributed to the research results reported within this paper. These were
provided through allocation TG-AST120060 from the Extreme Science and
Engineering Discovery Environment (XSEDE) (Towns et al., 2014), which is
supported by National Science Foundation grant number ACI-1548562, and through
Frontera Pathways allocation AST20034, Financial support for this project was
provided by the Department of Energy grants DE-SC0020432 and DE-SC0020434, the
National Science Foundation grants AST-1813298 and PHY-2020249 (CMAP), and the
National Aeronautics and Space Administration grant 80NSSC20K0622. YZ
acknowledges financial support from University of Rochester Frank J. Horton
Graduate Research Fellowship.

## Data Availability

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

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##  Appendix A Details of the jet subgrid model

When injecting outflow mass, momentum, and angular momentum, it is desired to
accurately conserve these quantities which would otherwise not be conserved
due to the numerical discretization employed in subgrid models. Here we solve
for corrected values for mass and momentum injection that are close to the
desired values while constrained by the conservation laws. $\vec{R}$ and
$\vec{V}$ are the particles position and velocity that remain unchanged
following the injection of some amount of mass $M$, radial momentum $P_{r}$,
and angular momentum $\vec{J}$. We will also use primed quantities for those
in the particles frame $\vec{r}^{\prime}\equiv\vec{r}-\vec{R}$ and
$\vec{v}^{\prime}\equiv\vec{v}-\vec{V}$

  * •

Mass conservation

$\qquad\displaystyle\sum m_{i}=M;$ (7)

  * •

Conserving center of mass

$\qquad\displaystyle\sum m_{i}\vec{r}_{i}=M\vec{R}\rightarrow\sum
m_{i}\vec{r}^{\prime}_{i}=0;$ (8)

  * •

Momentum conservation

$\qquad\displaystyle\sum m_{i}\vec{v}_{i}=M\vec{V}\rightarrow\sum
m_{i}\vec{v}_{i}^{\prime}=0;$ (9)

  * •

Angular momentum conservation (with an additional source of angular momentum
from the particles spin $\vec{J}$)

$\displaystyle\qquad\displaystyle\sum
m_{i}\vec{r_{i}}\times\vec{v_{i}}=M\vec{R}\times\vec{V}+\vec{J}\rightarrow\sum
m_{i}\vec{r}_{i}^{\prime}\times\vec{v}_{i}^{\prime}=\vec{J}$ (10)

Note this follows from 7, 8, & 9.

  * •

Desired amount of radial momentum in particle’s frame

$\qquad\displaystyle\sum_{i}m_{i}\vec{v}_{i}^{\prime}\cdot\frac{\vec{r}_{i}^{\prime}}{|\vec{r}_{i}^{\prime}|}=P_{r}\rightarrow\sum_{i}m_{i}\vec{\xi}_{i}\cdot\vec{v}_{i}=P_{r}$
(11)

where
$\vec{\xi}_{i}\equiv\frac{\vec{r}_{i}^{\prime}}{|\vec{r}_{i}^{\prime}|}$.

Now writing $\vec{r}_{i}^{\prime}\equiv[x_{i},y_{i},z_{i}]$ and
$\vec{v}_{i}^{\prime}\equiv[u_{i},v_{i},w_{i}]$ the first two constraints for
total mass 7 & the center of mass 8 only depend on the mass and can be
expressed as

$\left[\begin{array}[]{rrrrr}1&1&1&...&1\\\ x_{1}&x_{2}&x_{3}&...&x_{n}\\\
y_{1}&y_{2}&y_{3}&...&y_{n}\\\ z_{1}&z_{2}&z_{3}&...&z_{n}\\\
\end{array}\right]\left[\begin{array}[]{r}m_{1}\\\ m_{2}\\\ m_{3}\\\ ...\\\
m_{n}\\\ \end{array}\right]=\left[\begin{array}[]{r}M\\\ 0\\\ 0\\\ 0\\\
\end{array}\right]$

This system is in general under-determined since the kernel will have more
points than constraints. However, it can be solved using a least squares
approach, where we find the solution for $m_{i}$ that is as close as possible
to our kernel function (i.e. injection profile) for mass at those points
$\mathcal{M}(\vec{r}_{i}^{\prime})$

If we consider the above matrix equation as $Ax=b$ and represent the target
kernel solution vector $d=\mathcal{M}(\vec{r}_{i}^{\prime})$, then the
solution $x$ closest to the target $d$ that satisfies the constraints is given
by

$x=d-A^{T}(AA^{T})^{-1}(Ad-b)$ (12)

Note this involves inverting the 4x4 matrix $(AA^{T})$. Once we have solved
for the masses, we can then solve for the velocities using the other 3
constraints (conservation of momentum 9, angular momentum 10, and desired
scalar momentum injection 11 treating $m_{i}$ as knowns and $u_{i}$, $v_{i}$,
and $w_{i}$ as unknowns. This gives us the matrix equation

$\left[\begin{array}[]{rrrrrrrrr}m_{1}&...&m_{n}&0&...&0&0&...&0\\\
0&...&0&m_{1}&...&m_{n}&0&...&0\\\ 0&...&0&0&...&0&m_{1}&...&m_{n}\\\
0&...&0&-m_{1}z_{1}&...&-m_{n}z_{n}&m_{1}y_{1}&...&m_{n}y_{n}\\\
m_{1}z_{1}&...&m_{n}z_{n}&0&...&0&-m_{1}x_{1}&...&-m_{n}x_{n}\\\
-m_{1}y_{1}&...&-m_{n}y_{n}&m_{1}x_{1}&...&m_{n}x_{n}&0&...&0\\\
m_{1}\xi_{1,1}&...&m_{n}\xi_{n,1}&m_{1}\xi_{1,2}&...&m_{n}\xi_{n,2}&m_{1}\xi_{1,3}&...&m_{n}\xi_{n,3}\\\
\end{array}\right]$ $\left[\begin{array}[]{r}u_{1}\\\ ...\\\ u_{n}\\\ v_{1}\\\
...\\\ v_{n}\\\ w_{1}\\\ ...\\\ w_{n}\\\
\end{array}\right]=\left[\begin{array}[]{r}0\\\ 0\\\ 0\\\ J_{x}\\\ J_{y}\\\
J_{z}\\\ P_{r}\\\ \end{array}\right]$

And again we can construct a target vector $d$ of velocities using our
velocity kernel $\vec{\mathcal{V}}(\vec{r}_{i}^{\prime})$ and solve for the
velocities $x$ that satisfy the constraint while being as close as possible to
the kernel.

Note that it is important that our various kernels for mass, radial momentum,
and angular momentum be somewhat close to satisfying the constraints to begin
with. The closer the solution is to satisfying the constraints, the smaller
the resulting deviation from the target profile. This is particularly
important for the mass. For instance, using an unnormalized mass kernel can
lead to negative masses when solving the constraints.