# The velocity distribution of outflows driven by choked jets in stellar
envelopes

Matteo Pais  Matteo Pais,1 Tsvi Piran,1 and Ehud Nakar2
1Racah Institute for Physics, The Hebrew University, Jerusalem 91904, Israel
2School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(November 2022)

###### Abstract

Many stripped envelope supernovae (SNe) present a signature of high-velocity
material responsible for broad absorption lines in the observed spectrum.
These include SNe that are associated with long gamma-ray bursts (LGRBs) and
low-luminosity GRBs (llGRBs), and SNe that are not associated with GRBs.
Recently it was suggested that this high velocity material originates from a
cocoon that is driven by a relativistic jet. In LGRBs this jet breaks out
successfully from the stellar envelope, while in llGRBs and SNe that are not
associated with GRBs the jet is choked. Here we use numerical simulations to
explore the velocity distribution of an outflow that is driven by a choked jet
and its dependence on the jet and progenitor properties. We find that in all
cases where the jet is not choked too deep within the star, the outflow
carries a roughly constant amount of energy per logarithmic scale of proper
velocity over a wide range of velocities, which depends mostly on the cocoon
volume at the time of its breakout. This is a universal property of jets
driven outflows, which does not exist in outflows of spherically symmetric
explosions or when the jets are choked very deep within the star. We therefore
conclude that jets that are choked (not too deep) provide a natural
explanation to the fast material seen in the early spectra of stripped
envelope SNe that are not associated with LGRBs and that properties of this
material could reveal information on the otherwise hidden jets.

###### keywords:

stars: jets – gamma-ray burst: general – supernovae: general – hydrodynamics

##  1 Introduction

Both long and short GRB jets have to cross a significant amount of matter (the
stellar atmosphere for long GRBs and the merger’s ejecta in short ones) before
producing the observed $\gamma$-rays. This understanding has lead to great
interest in jet propagation within surrounding matter and the question was
explored both analytically (e.g., Blandford & Rees, 1974; Begelman & Cioffi,
1989; Mészáros & Waxman, 2001; Matzner, 2003; Lazzati & Begelman, 2005;
Bromberg et al., 2011) and numerically (e.g., Marti et al., 1995; Martí et
al., 1997; Aloy et al., 2000; MacFadyen et al., 2001; Reynolds, 2001; Zhang et
al., 2004; Mizuta et al., 2006; Morsony et al., 2007; Wang et al., 2008;
Lazzati et al., 2009; Mizuta & Aloy, 2009; Morsony et al., 2010; Nagakura et
al., 2011; López-Cámara et al., 2013; Ito et al., 2015; López-Cámara et al.,
2016; Harrison et al., 2018). This, naturally, raises the possibility that
some jets are “choked” during their propagation and are unable to break out of
the surrounding dense medium. The observed temporal distributions of both long
(Bromberg et al., 2012) and short (Moharana & Piran, 2017) GRBs suggest that
this happens in both types of events and there are indications that this also
happens in some Supernovae (Piran et al., 2019).

In cases that the jet does not emerge we may still observed the signature of
the cocoon that forms. First, the breakout of the shock driven by the cocoon
produces a bright flash. For example, a cocoon breakout is most likely the
origin of low-luminosity GRBs (llGRBs) (Kulkarni et al., 1998; MacFadyen et
al., 2001; Tan et al., 2001; Campana et al., 2006; Wang et al., 2007; Waxman
et al., 2007; Katz et al., 2010; Nakar & Sari, 2012; Nakar, 2015). These type
of GRBs are rarely observed, however, when their low luminosity is taken into
account it was realized that they are more numerous than regular LGRBs
(Soderberg et al., 2006). Another signature arises from the fast cocoon
material that engulfs the star once the hot cocoon material breaks out and
spreads. Specifically, this material leads to very broad absorption lines that
are visible as long as it is optically thick (Piran et al., 2019). Such lines
have been observed in several SNe some accompanied by llGRBs (Galama et al.,
1998; Iwamoto et al., 1998; Modjaz et al., 2006; Mazzali et al., 2008; Bufano
et al., 2012; Xu et al., 2013; Ashall et al., 2019; Izzo et al., 2019) and
others without (Mazzali et al., 2000, 2002; Mazzali et al., 2009). Finally,
the cooling emission of the cocoon will also generate a potentially detectable
UV-optical transient on time scale of hours to days (Nakar & Piran, 2017).

The important signature that helps determining the origin of the broad
absorption lines is the energy-velocity distribution of the fast moving
material. Regular spherical explosion result in a very steep distribution with
roughly $\mathrm{d}E(v)/\mathrm{d}\ln v\propto v^{-5}$ (e.g., Nakar & Sari,
2010). However, when a jet is involved in the explosion this distribution is
expected to be much shallower with much more energy at high velocities.
Recently, Eisenberg et al. (2022) have shown that when the jet is successful
the cocoon generates a unique energy-velocity flat distribution with
$\mathrm{d}E/\mathrm{d}\ln\Gamma\beta\propto{\rm const.}$ over a wide range of
velocities from sub to mildly relativistic, where $\beta=v/c$ and $\Gamma$ is
the corresponding Lorentz factor. They also found that, when the jet is
choked, it leaves a unique signature of a flat energy-velocity distribution.
However, in the case of choked jets the flat distribution covers a range of
velocities that is narrower than that of outflows driven by successful jets.
Motivated by these results we examine here in detail the energy-velocity
distributions of different chocked jet, focusing on the relation between the
properties of the choked jet and the final energy-velocity distribution of the
outflow after it becomes homologous.

For our study we use a large set of 2D relativistic hydrodynamical
simulations. We consider explosions that are driven by choked jets in which we
vary the opening angle and the engine working time of the jet as well as the
structure of the progenitor. We follow the simulations until the entire
outflow becomes homologous and examine what is the relation between these
properties and the outflow energy-velocity distribution.

The paper is structured as follows. In Section 2 we describe the numerical
procedure adopted for the simulations. In Section 2.1 we describe the code
choice and the composite mesh structure adopted, while in Section 2.2 we
report in detail the setup for the stellar and interstellar environment and
the initial conditions for the relativistic jet. Numerical aspects are
discussed in two appendixes: a resolution study is described in Appendix A and
in Appendix B we explore the different use of a numerical smoothing function
for the stellar density profile. In Section 3 we explore the results of our
set of simulations. We summarize our findings and consider the implications to
observations in Section 4.

##  2 Methodology

###  2.1 Simulation Setup

Our simulations are performed using the open source massively parallel
multidimensional relativistic magneto-hydrodynamic code pluto (v4.3) (Mignone
et al., 2007). The code uses a finite-volume, shock-capturing scheme designed
to integrate a system of conservation laws where the flow quantities are
discretized on a logically rectangular computational grid enclosed by a
boundary. We use the special relativistic hydrodynamics module in 2D
cylindrical coordinates. We perform our calculations using a parabolic
reconstruction scheme combined with a third-order Runge-Kutta time stepping.
We also force the code to reconstruct the 4-velocity vectors at each time
step.

The 2D simulations enables us to reach high resolution with reasonable
computational resources. 3D simulations carried by Harrison et al. (2018)
suggest a similar generic evolution of the jet for the same parameters. The
propagation of the outer cocoon and the jet head are similar both in 2D and 3D
simulations and the differences for the velocity distribution of the system
are negligible - especially for choked jets - as shown in Eisenberg et al.
(2022). The main difference arises in the morphology of the jet head in 2D
simulations that is affected by a plug at the head front. This plug diverts
some of the jet elements sideways to dissipated their energy in oblique shocks
but it is irrelevant for the cocoon structure in which we are interested here.
Another difference is the stability of the boundary between the jet and the
inner cocoon (Gottlieb et al., 2021). However, all properties that are
important for the cocoon (like mixing at the head, mixing between inner and
outer cocoon and the propagation of the outer cocoon are all similar for 2D
and 3D.

Those differences are not significant for our purposes.

We chose the equation of state of the fluid to be ideal and with a constant
relativistic polytropic index of $4/3$. This equation of state is applicable
for a relativistic gas (as in the jet) as well to a radiation dominated
Newtonian gas, such as the shocked stellar envelope.

To study the long term evolution of the jet and the cocoon from the star, we
use a large grid spanning for several orders of magnitude. This allows us to
track the evolution of the system for at least two minutes after the breakout.
At that time the entire stellar envelope is shocked by the cocoon and it
expands enough to become homologous. We use a grid of size $4736\times 4636$
cells, with the radial cylindrical coordinate111Throughout the paper $r$ is
used for the 2D cylindrical radius while $R$ stands for the 3D radius.
extending within the range $r=[0,350]\times 10^{10}\mathrm{cm}$ and the
vertical coordinate extending within the range $z=[0.1,360]\times
10^{10}\mathrm{cm}$.

We use a combination of a uniform and two non-uniform mesh grids in $r-z$
coordinates with a decreasing resolution from the inner region of the
simulation box to the outer boundaries. The grid mesh is uniform in the inner
part to maintain a high resolution of the jet injection and the formation of
the resulting high pressure cocoon. The uniform mesh has $1000\times 900$ grid
points extending in the ranges $r=[0,1]\times 10^{10}\mathrm{cm}$ and
$z=[0.1,1]\times 10^{10}\mathrm{cm}$ with a resolution along both coordinates
of $\Delta(r,z)_{\mathrm{unif.}}=10^{7}\mathrm{cm}$. Next to the uniform mesh
we placed a stretched mesh with $1278^{2}$ grid points extending along both
coordinates within the range $(r,z)=[1,6]\times 10^{10}\mathrm{cm}$ with a
stretching ratio of $\sim 1.0018$ The number of grid points for this mesh is
chosen such that its initial grid spacing is the same of to the adjacent
uniform mesh
$\Delta(r,z)_{\mathrm{s,init}}=\Delta(r,z)_{\mathrm{unif.}}=10^{7}\mathrm{cm}$
and its final grid spacing is
$\Delta(r,z)_{\mathrm{s,final}}=10^{8}\mathrm{cm}$. We cover the remaining
grid at larger distances with a logarithmic spaced mesh with $2458^{2}$ grid
points extending within the range $(r,z)=[6,360]\times 10^{10}\mathrm{cm}$.
The number of grid points is chosen such that the grid spacing of the mesh at
$(r,z)=6\times 10^{10}\mathrm{cm}$ coincides to the resolution of the
stretched mesh, such that
$\Delta(r,z)_{\mathrm{log,init}}=\Delta(r,z)_{\mathrm{s,final}}=10^{8}\mathrm{cm}$.
In this way we ensure a smooth increase of the resolution without jumps for
the entire simulation grid. A detailed resolution study for these simulations
is reported in Appendix A.

We inject the jet along the inner lower $z$ boundary, denoted $z_{0}$ (see
2.2.2). Otherwise, we impose a reflective boundary condition at this boundary
as it approximates the equatorial plane of the system. We impose axial-
symmetric conditions for the inner vertical boundary. Both outer boundaries
are set to outflow.

###  2.2 Initial conditions

####  2.2.1 The star

We approximate the stellar density profile as a continuous power law that
mimics the sharp decline of density in radius near the stellar edge222This
profile diverges at the origin but this region does not influence the jet
propagation and it is not included in our computational domain.

$\rho(R)=\begin{cases}\rho_{*}\left(\dfrac{R_{*}}{R}-1\right)^{2}+\rho_{0},&\mathrm{for}~{}R\leq
R_{*}\ ,\\\ \rho_{0},&\mathrm{for}~{}~{}R>R_{*}\ .\end{cases}$ (1)

Here we choose $\rho_{*}=100~{}\mathrm{g}~{}\mathrm{cm}^{-3}$ and
$R_{*}=3\times 10^{10}\mathrm{cm}$. The total integrated mass of the star is
$M_{*}=(9\pi/5)~{}M_{\odot}$ (see 2.2.3 for scaling of these parameters to
other values.)

For this density profile the local slope,
$\alpha\equiv\mathrm{d}\log\rho(R)/\mathrm{d}\log R={-2}/{(1-R/R_{*})}$. The
slope, $\alpha$, reaches the critical value of 3 for $R=R_{*}/3$. Beyond this
value a spherical blast wave accelerates and eventually looses causality. We
present our results for this specific density profile. However, in Sec. 3.4 we
show that the results for different stellar density profiles (both inner and
outer) are qualitatively similar.

Surrounding the star we have an external CSM density of $\rho_{0}=1.67\times
10^{-21}~{}\mathrm{g}~{}\mathrm{cm}^{-3}$. This exact value is unimportant as
it is added just to avoid a numerical vacuum. The interaction of the jet or
the cocoon outflow with this CSM is insignificant. To avoid numerical
artifacts arising from the sudden drop in density at the edge of the star we
smooth the density of the outer edge of the star with a power law:

$\rho_{\mathrm{smooth}}(R)=\rho_{\mathrm{s}}\left(\dfrac{R}{R_{\mathrm{s}}}+1\right)^{-8}\
,$ (2)

with $\rho_{\mathrm{s}}=0.05~{}\mathrm{g}~{}\mathrm{cm}^{-3}$ and a gradient
scale of $R_{\mathrm{s}}=5\times 10^{8}\mathrm{cm}$. We verified that this
arbitrary choice of the smoothing function does not affect our results (see
Appendix B). In order to avoid any initial random motion we set a uniform and
low ambient pressure of $P=3.5~{}\mathrm{keV}~{}\mathrm{cm}^{-3}$ within the
simulation grid.

####  2.2.2 The jet

We inject a collimated jet with a constant luminosity $L_{\mathrm{j}}$,
operating for $t_{\mathrm{e}}$ so that the total injected energy is
$E_{0}=L_{\mathrm{j}}\times t_{\mathrm{e}}=10^{51}\mathrm{erg}$. A uniform jet
is injected through a nozzle with a velocity in the $z$ direction with an
initial bulk Lorentz factor $\Gamma_{0,\mathrm{j}}$ a density
$\rho_{\mathrm{j}}$ and a specific enthalpy $h_{\mathrm{j}}\gg 1$. Being
relativistically hot the jet spreads quickly to form an initial opening angle
$\theta_{\mathrm{j}}\simeq 1/(1.4\Gamma_{0,\mathrm{j}})$ (see details on this
injection method at Mizuta & Ioka 2013 and Harrison et al. 2018).

The jet is numerically initialized by the injection of density, pressure, and
momentum along the $z-$direction through a nozzle parallel to the $z-$axis
with a radius $r_{\mathrm{j}}$ at an initial height $z=z_{0}$. The head cross
section is then $\Sigma_{\mathrm{j}}=\pi r_{\mathrm{j}}^{2}$. For an initial
opening angle $\theta_{\mathrm{j}}>0.1~{}\mathrm{rad}$, we set up
$r_{\mathrm{j}}=10^{8}~{}\mathrm{cm}$, allowing a sufficient mesh coverage
over the nozzle and we set the initial injection height at
$z_{0}=10^{9}~{}\mathrm{cm}$.

We consider a constant jet luminosity $L_{\mathrm{j}}$. This determines the
product $\rho_{\mathrm{j}}h_{\mathrm{j}}$ as:

$\rho_{\mathrm{j}}h_{\mathrm{j}}=\dfrac{L_{\mathrm{j}}}{\Sigma_{\mathrm{j}}\Gamma_{0,\mathrm{j}}^{2}c^{3}}\
.$ (3)

We choose $h_{\mathrm{j}}=100$. This choice of the enthalpy is arbitrary, as
long as $h_{\mathrm{j}}\gg 1$. The jet’s pressure is given by
$P_{\mathrm{j}}=(h_{\mathrm{j}}-1){\rho_{\mathrm{j}}c^{2}}/{4}$.

We explored the parameter space running simulations for different initial
values of $L_{\mathrm{j}}$ at steps of $2.5\times
10^{50}~{}\mathrm{erg}~{}\mathrm{s}^{-1}$ from $2.5\times
10^{50}~{}\mathrm{erg}~{}\mathrm{s}^{-1}$ to $2\times
10^{51}~{}\mathrm{erg}~{}\mathrm{s}^{-1}$ for a total of 9 different
luminosities. For each value of the luminosity, we run simulations for a set
of different opening angles
$\theta_{\mathrm{j}}=[0.05,0.1,0.2,0.4,0.6]~{}\mathrm{rad}$. As we keep the
total jet energy fixed these conditions translate to different engine working
times $t_{\mathrm{e}}=10^{51}\mathrm{erg}/L_{\mathrm{j}}$. For each of the 9
values of the luminosity we run 5 different values of the opening angle for a
total of 45 simulations.

####  2.2.3 Scaling relations

While we consider specific numerical values for the stellar and jet
parameters, our solutions can be scaled to other values. The equation of
motion of the forward shock speed $\beta_{\mathrm{h}}$, is regulated by
$\tilde{L}$ (Matzner, 2003; Bromberg et al., 2011):

$\tilde{L}\simeq\dfrac{L_{\mathrm{j}}}{\Sigma_{\mathrm{j}}\rho(R)c^{3}}\ ,$
(4)

with

$\beta_{\mathrm{h}}=\dfrac{1}{1+{\tilde{L}}^{-1/2}}\ .$ (5)

The stellar size $R_{*}$, is the scale length of t the system. The scalings
$\Sigma_{\mathrm{j}}\propto R_{*}^{2}$ and $\rho(R/R_{*})\propto\rho_{*}$ we
can express $\tilde{L}$ as

$\tilde{L}\propto\dfrac{E_{0}}{t_{\mathrm{e}}\rho_{*}R_{*}^{2}}\ .$ (6)

If we scale the stellar radius as $R_{*}=\lambda R_{*}^{\prime}$ we have to
scale the density and the jet luminosity accordingly in order to maintain
$\tilde{L}$ and $\beta_{\mathrm{h}}$ unchanged.

As we show later the location where the jet is choked (i.e. where the last
element launched by the jet reaches the head) with respect to the stellar
radius has also to be constant. The choking location $z_{\mathrm{ch}}$ is
roughly proportional to the engine time $t_{\mathrm{e}}$ (Nakar, 2015):

$z_{\mathrm{ch}}=\int_{0}^{t_{\mathrm{ch}}}\beta_{\mathrm{h}}c\mathrm{d}t\simeq\beta_{\mathrm{h}}ct_{\mathrm{ch}}=\dfrac{\beta_{\mathrm{h}}c}{1-\beta_{\mathrm{h}}}t_{\mathrm{e}}\
.$ (7)

where $t_{\mathrm{ch}}={t_{\mathrm{e}}}/{(1-\beta_{\mathrm{h}})}$ is the
choking time. If $\beta_{\mathrm{h}}$ is kept constant than any transformation
on $R_{*}$ will leave $z_{\mathrm{ch}}/R_{*}$ unchanged if
$t_{\mathrm{e}}=\lambda t_{\mathrm{e}}^{\prime}\propto R_{*}$. Thus, scaling
$E_{0}=\eta E_{0}^{\prime}$ and $R_{*}=\lambda R_{*}^{\prime}$ we require that
and $\rho_{*}=\eta\lambda^{-3}\rho_{*}^{\prime}$.

Because $M_{*}\propto\rho_{*}R_{*}^{3}$, when keeping this scaling of
$t_{\mathrm{e}}$ and $R_{*}$, we can rewrite Eq. 4 as $\tilde{L}\propto
E_{0}/M_{*}$, which means that since $\tilde{L}$ is kept constant the typical
velocity of the system $v_{0}=(2E_{0}/M_{*})^{1/2}$ is also conserved under
these transformations.

Turning to the jet parameters we recall that the only relevant quantities are
$L_{\mathrm{j}}$, $t_{\mathrm{e}}$ and $\theta_{\mathrm{j}}$. The first two
determine $E_{0}$ and the latter determined $\Gamma_{0,\mathrm{j}}$. The
luminosity, together with the stellar parameters determine the produce
$\rho_{\mathrm{j}}h_{\mathrm{j}}$ with the condition that $h_{\mathrm{j}}$
while arbitrary should be much larger than one.

In summary, given the physical scales $R_{*}$, $\rho_{*}$, $E_{0}$, $v_{0}$,
$t_{\mathrm{e}}$ and the scalings $R_{*}=\lambda R_{*}^{\prime}$, $E_{0}=\eta
E_{0}^{\prime}$, the parameters defining the physics of our system, i.e.
$\tilde{L}$, $z_{\mathrm{ch}}/R_{*}$, will not change if
$t_{\mathrm{e}}=\lambda t^{\prime}_{\mathrm{e}}$ and
$\rho_{*}=\eta\lambda^{-3}\rho^{\prime}_{*}$.

##  3 Results

Figure 1: A canonical jet ($\theta=0.2$ rad;
$L_{\mathrm{j}}=10^{51}\mathrm{erg}\ \mathrm{s}^{-1}$ and $t_{e}=1\mathrm{s}$)
launched in a test star with a density profile given by Eq. 1. The four panels
show, from left to right, the relativistic $\Gamma\beta$ factor, the density
$\rho$, the pressure $P$ and a scalar tracer of the ejected jet material (an
unitary value implies pure jet material with no mixing). All the quantities
but the $\Gamma\beta$ factor are normalized to the respective maximum values
in order to increase the color contrast. The scale for the velocity of our
system is dictated by $v_{0}$. With the above parameters
$\beta_{0}=v_{0}/c=0.014$ and the relativistic regime begins when
$\beta\Gamma/\beta_{0}\approx 50$. The different rows show the evolution of
the jet at $t=t_{\mathrm{e}}=1~{}\mathrm{s}$ (when the engine stops, first
row) - the jet is clearly seen here surrounded by a cocoon,
$t=1.3~{}\mathrm{s}$ (choking time, second row) - the jet has disappeared as
its tail reached the head, $t=14.2~{}\mathrm{s}$ (shortly after the breakout,
third row), and $t=16.0~{}\mathrm{s}$ (sideways spreading outside the star,
fourth row). To enhance the color contrast we change the normalization scale
of $\rho/\rho_{\mathrm{max}}$ and $P/P_{\mathrm{max}}$ in the third and fourth
row in order to capture the tenuous material and pressure spilling outside the
star.  Figure 2: 4-panel figure of the breakout of an unchoked jet at
$t=t_{\mathrm{e}}=$4 s with an opening angle of $\theta_{\mathrm{j}}=0.1$ rad
with a luminosity of $L_{\mathrm{j}}=2.5\times
10^{50}~{}\mathrm{erg}~{}\mathrm{s}^{-1}$. In this case the jet tail did not
catch up the jet head, resulting in an unchoked breakout and with a head
propagation operating as the jet engine were still active. We use the same
color scale of the fourth row of Fig. 1 for comparison. Figure 3: _Left_ : A
schematic figure of how the aspect ratio $\theta(t)$ is defined (Eq. 8)
superposed on the density map of the jet cocoon while it is still inside the
star. The dashed line represents the maximal width of the cocoon, while the
dashed-dotted line represents the maximal height. _Centre and right_ :
Evolution of the aspect ratio of the jet as a function of time (central panel)
and of the cocoon/jet head location $z_{\mathrm{h}}$ (right panel). The square
dots represent the choking time of the jet while the triangular dots represent
the breakout time of the cocoon. In both frames the thick dashed horizontal
line represents the condition for an isotropic blast wave
($r_{\mathrm{c}}=z_{\mathrm{h}}$), while the horizontal dotted line represent
the original opening angle of the launched jet ($0.2~{}\mathrm{rad}$ for the
figure). On the right-hand side the dashed-dotted vertical line represents the
star edge, located at $R_{*}=3\times 10^{10}~{}\mathrm{cm}$. Figure 4:
Classification of the energy-velocity distribution grouped according to their
different cocoon volume $V_{\mathrm{bo}}$ at $t=t_{\mathrm{bo}}$. Each curve
is marked by a different shade of color and a triplet of numbers indicating,
in order: the opening angle in radians, $t_{\mathrm{e}}$ in seconds, and then
$\sqrt{V_{*}/V_{\mathrm{bo}}}$. The dashed black line represents the isotropic
case. Figure 5: Correlation between $\sqrt{V_{*}/V_{\mathrm{bo}}}$ and the
cutoff of the energy-velocity distribution for the simulated set of jets for a
cutoff value of $0.25$ of the maximum of each differential energy distribution
in the plots of Fig. 4. The red-colored area represents 1-standard deviation
error of the red fit curve. From the fitting formula we see that the
distribution cutoff corresponds to 4 times the the square root of the volume
ratio at the breakout for values of $(\Gamma\beta)_{\mathrm{cut}}$ above 3.
The black dashed line represents the linear limit for high cutoffs. Figure 6:
Correlation between the cocoon volume at the breakout and the choking height
of the jet for different values of the initial opening angle
$\theta_{\mathrm{j}}$. The gray-shaded region represents the 1$\sigma$ error
for the fits.

###  3.1 The jet-cocoon system

We start analyzing our simulation set considering a jet with our _canonical_
parameters of 1-sided luminosity of
$L_{\mathrm{j}}=10^{51}\mathrm{erg}/\mathrm{s}$ and
$\theta_{\mathrm{j}}=0.2~{}\mathrm{rad}\simeq 10^{\circ}$.

While advancing through the stellar atmosphere the interaction of the
relativistic jet with the stellar material results in a forward-reverse shock
structure that is called the head of the jet (Blandford & Rees, 1974; Begelman
& Cioffi, 1989; Mészáros & Waxman, 2001; Matzner, 2003; Lazzati & Begelman,
2005; Bromberg et al., 2011). The jet head propagation velocity,
$\beta_{\mathrm{h}}$, is much lower than the jet velocity before it reaches
the head and for typical GRB jets it is Newtonian. The shock-heated jet and
stellar material that enters the head flow sideways because of the high head
pressure and form a pressurized cocoon which enshrines the jet. The contact
discontinuity between the material shocked in forward and the reverse shocks
divides the cocoon to inner and outer parts. The inner cocoon is composed of
tenuous jet material which has crossed the reverse shock while the outer
cocoon is composed of denser shocked stellar material. The cocoon exerts a
pressure on the jet such that, if sufficiently high, collimates it, thus
reducing its opening angle and consequently reducing the jet cross section
compared to the uncollimated jet.

Within our chosen stellar structure model the jet head moves at a constant
velocity at the inner region of the core where the local density slope
$\alpha=2$. If the jet reaches outer regions where $\alpha>2$ it starts
accelerating.

The jet is _choked_ if the engine stops while the jet is propagating within
the stellar envelope and the last jet element launched by the engine (_tail_ ,
hereafter) catches up with the jet head before the latter breaks out of the
star. In this case all the engine’s energy goes into the cocoon. Clearly, the
choking height (Eq. 7) satisfies $z_{\mathrm{ch}}<R_{*}$. Otherwise we define
the jet as unchoked or successful. Throughout the following analysis we focus
mostly on jets choked at various depths inside the star. For comparison we
show also the case of an unchoked jet breaking out of the star. For a detailed
study on the energy-velocity distribution of stellar explosions which are
driven by successful jets see Eisenberg et al. (2022).

We divide the evolution of the jet to three different phases: 1) the injection
phase and choking phase $t<t_{\mathrm{ch}}$, 2) the cocoon expansion phase
$t_{\mathrm{ch}}<t<t_{\mathrm{bo}}$ and 3) the cocoon breakout phase
$t>t_{\mathrm{bo}}$. The different phases of a choked jet are shown in Fig. 1.

####  3.1.1 Injection and Choking: $t\leq t_{\mathrm{ch}}$

The engine operates for $t_{\mathrm{e}}$ producing a jet. This is clearly seen
in the first row of Fig. 1 in both the rightmost panel showing $\beta\Gamma$
and the leftmost one showing that the tracer of the jet material is
concentrated mostly within a narrow cylinder along the symmetry axis $z$ with
radius $r\simeq r_{\mathrm{j}}$ (color dark red). This behaviour is typical of
the collimated regime (Bromberg et al., 2011).

After the jet engine stops the last jet material launched at the injection
nozzle propagates upwards. At $t_{\mathrm{e}}$ the jet head is still unaware
that the engine stopped and the jet head continues to propagate with
$\beta_{\mathrm{h}}$ (in this specific simulation $\beta_{\mathrm{h}}\simeq
0.2)$ after the central jet engine is switched off. However, as
$\beta_{\mathrm{h}}<1$ while the jet material moves at
$\beta_{\mathrm{t}}\simeq 1$ the jet tail catches up with the head. Only at
that time the information that the engine stopped reaches the head and the
reverse shock within the jet disappears. This is the time where the jet is
choked.

As the head propagates with a velocity of $\beta_{\mathrm{h}}c$ and the tail
propagates at $\beta_{\mathrm{t}}\simeq 1$, we can estimate the the time that
the jet tail will catch up with the head at $t_{\mathrm{ch}}$, as defined in
Eq. 7. Until $t<t_{\mathrm{ch}}$ the jet continues to drive the head forward
through the stellar atmosphere. The second row of Fig. 1 shows the system at
$t=t_{\mathrm{ch}}$, roughly 0.3 seconds after the end of the engine activity,
in this specific simulation. One can see that the very fast jet material
around the core disappeared. At this stage all the jet’s energy has been
dissipated and given to the surrounding cocoon.

####  3.1.2 Cocoon expansion: $t_{\mathrm{ch}}<t<t_{\mathrm{bo}}$

After the jet choking, the cocoon becomes less and less collimated and
proceeds spreading sideways while the forward shock decelerates when it is
deep within the envelope and accelerates as it reaches the steep density
gradient near the stellar edge (Irwin et al., 2019). During the propagation
the inner cocoon transfers energy to the freshly shocked material (via PdV
work).

####  3.1.3 Brekout: $t>t_{\mathrm{bo}}$

After the breakout the cocoon material spreads both radially and tangentially
to engulfs the stellar surface, quickly shrouding the breakout point from most
observers (see the last row of Fig. 1). The star is blanketed by the ejecta in
a time equal to $t_{\mathrm{wrap}}\simeq\pi R_{*}/2v_{\mathrm{bo}}$, where
$v_{\mathrm{bo}}$ is the breakout velocity of the cocoon near the pole. The
shock driven by the cocoon also moves tangentially towards the equator at a
slower pace until the entire stellar envelope is shocked at
$t_{\mathrm{shock}}\simeq\pi R_{*}/(2v_{\mathrm{p}})$ where $v_{\mathrm{p}}$
is the pattern velocity at which the spilled material travels along the
stellar surface (Irwin et al., 2021). Shortly after reaching the equator the
shocked material propagates almost radially and outwards and it becomes
homologous once the outflow reaches $\sim 2R_{*}$.

####  3.1.4 Successful jets

Jets whose engine operates long enough break out from the stellar envelope
before the end of the activity of the central engine. These jets are not
choked and can preserve an ultra-relativistic velocity once they get out of
the star. We show an example of a successful jet in Fig. 2. Because the jet
broke out without being choked the cocoon structure inside the star is mostly
collimated along the vertical axis. From the first and fourth panel which show
$\Gamma\beta$ and the jet tracer respectively, it is evident how the innermost
region is still dominated by tenuous, highly relativistic jet material.
Comparing the last row of Fig. 1 and Fig. 2 we notice that for the same
normalized density and normalized pressure scale, the longer duration of the
unchoked jet results in expulsion of denser and faster stellar material with
respect to the choked jet.

###  3.2 The spreading angle and the cocoon volume

To describe quantitatively the geometry of the jet-cocoon during its
propagation within the stellar envelope we use the aspect ratio, defined as

$\theta(t)=\dfrac{\mathrm{max}(r_{\mathrm{c}}(t))}{z_{\mathrm{h}}(t)}\ ,$ (8)

where $r_{\mathrm{c}}$ is the cocoon cylindrical radius and $z_{\mathrm{h}}$
is the head position. For $\theta\ll 1$ the aspect ratio is a good
approximation of the cocoon spreading angle. The expanded cocoon at the moment
of the breakout is shown in the third row of Fig. 1. The steep density
transition results in an elongation and acceleration of the cocoon and the
ejection of low-density material from the star, which rapidly engulfs the
star’s external layers. We define the breakout angle $\theta_{\mathrm{bo}}$ as
the geometric opening angle measured at the breakout time $t=t_{\mathrm{bo}}$,
namely

$\theta_{\mathrm{bo}}=\theta(t_{\mathrm{bo}})=\dfrac{\mathrm{max}(r_{\mathrm{c}}(t_{\mathrm{bo}}))}{R_{*}}\
.$ (9)

The evolution of $\theta(t)$ for $\theta_{\mathrm{j}}=0.2~{}\mathrm{rad}$ and
several values of $t_{\mathrm{e}}$ is reported in Fig. 3. At first,
immediately after injection, the aspect ratio starts growing. The growth
continues until $z_{\mathrm{h}}$ is roughly twice the injection radius,
$z_{0}$, at which point the aspect ratio starts decreasing, approaching the
point where the cocoon opening angle is comparable to $\theta_{\mathrm{j}}$.
This evolution reflects the time it takes the pressure in the cocoon to build
up to the point that it starts collimating the jet effectively (see Harrison
et al. 2018 for details). The evolution of the aspect ratio changes
dramatically immediately as the jet is fully choked. Since there is no more
fresh jet material to drive the head its velocity drops sharply. At the same
time the cocoon pressure, and thus its sideways expansion, is not affected.
The result is that the aspect ratio grows continuously after
$t_{\mathrm{ch}}$. There is a short episode, just before and after the
breakout when the aspect ratio decreases, as the head accelerates near the
edge of the star and after the breakout. Soon after that the aspect ratio
starts increasing rapidly as some of the material that broke out of the star
spreads sideways at speed that is close to the speed of light. One clear
property that is seen in the figure is that jet that are choked more deeply
have longer time to expand before they breakout and therefore a deeper choking
results in a wider cocoon with a larger volume at the time of breakout. As we
show next this fact has important implications for the energy-velocity
distribution of the outflow.

The volume of the cocoon at breakout, $V_{\mathrm{bo}}$ is another parameter
that describes the properties of the jet-cocoon system. As the energy of a
choked jet is given to the cocoon, for a given energy, the cocoon mass (and
hence volume) at breakout, will corresponds to a typical expansion velocity of
the cocoon material. As the volume-averaged density in the shocked cocoon
material and the volume-averaged density of the star are roughly the same, we
can define a characteristic velocity at the breakout linked with the breakout
volume, namely:

$\beta_{\mathrm{bo}}\simeq\beta_{0}\sqrt{\dfrac{V_{*}}{V_{\mathrm{bo}}}}\ .$
(10)

###  3.3 The Energy-velocity distribution

Fig. 4 depicts the energy-velocity distribution of the entire set of
simulations for different values of the engine working time $t_{\mathrm{e}}$
and different initial opening angles $\theta_{\mathrm{j}}$ at
$t=120\mathrm{s}$ (when the outflow is homologous and kinetic energy
dominates). The $x$-axis is normalized by $\beta_{0}$ and the $y$-axis by
$E_{0}$. Each curve is differentiated by color and labeled by a triplet of
numbers describing, respectively, $\theta_{\mathrm{j}}$, $t_{\mathrm{e}}$, and
$\sqrt{V_{*}/V_{\mathrm{bo}}}$. We grouped the different curves according to
$\sqrt{V_{*}/V_{\mathrm{bo}}}$. For a comparison we superposed the energy-
velocity distribution of an isotropic spherically symmetric explosion (black-
dashed line) for all panels.

Fig. 4 shows, first, that in all cases the energy-velocity distribution
exhibits a roughly constant energy per logarithmic scale of $\Gamma\beta$ over
a range of velocities. The distribution rises quickly before this rough
plateau starts and decays sharply after it ends. The rough plateau always
starts at $\beta_{0}$ and its highest velocity is determined almost entirely
by $V_{\mathrm{bo}}$, with a weak dependence on the jet opening angle. To
estimate the highest velocity of the flat part of the distribution we define
$\beta_{\rm cut}$ as the velocity obtained when the energy-velocity
distribution drops to 1/4 of its maximum value. This arbitrary definition
provides a velocity that is slightly larger than the end of the plateau (e.g.,
in the spherical case $\beta_{\rm cut}=2\beta_{0}$). Fig 5 shows that there is
a strong positive correlation between $\beta_{\rm cut}$ and
$\sqrt{V_{*}/V_{\mathrm{bo}}}$. For small values of
$\sqrt{V_{*}/V_{\mathrm{bo}}}<3$, where typically $z_{\mathrm{ch}}\ll R_{*}$,
we see that $\beta_{\rm cut}\approx\beta_{\mathrm{bo}}$. However, for larger
values of $\sqrt{V_{*}/V_{\mathrm{bo}}}$ where the choking takes place not
very deep within the stellar envelope, $\beta_{\rm cut}>\beta_{\mathrm{bo}}$.
The origin of the material faster than $\beta_{\mathrm{bo}}$ in these cases is
the inner cocoon, which retain a significant fraction of its energy at the
time of the breakout and outer cocoon material that is close to the edge of
the star, where the forward shock is faster than $\beta_{\mathrm{bo}}$.

The value of $V_{*}/V_{\mathrm{bo}}$ is expected to depend on the jet opening
angle and the choking depth. The jet opening angle determines the aspect ratio
of the cocoon as long as the head that is pushed ahead by the jet is feeding
the cocoon ($\theta\approx\theta_{\mathrm{j}}$), while the choking depth
determines by how much this aspect ratio increases until the breakout. Fig. 6
depicts the correlation between $V_{\mathrm{bo}}$ and $z_{\mathrm{ch}}$ for
different values of the initial $\theta_{\mathrm{j}}$. As expected,
$V_{\mathrm{bo}}$ is a function of $z_{\mathrm{ch}}$ and
$\theta_{\mathrm{j}}$. A deeper choking height and a wider jet correspond to a
larger cocoon volume upon breakout.

Figure 7: A sketch of the division of the stellar volume for the analysis of
the distribution of the stellar material. The cocoon shape taken at
$t=t_{\mathrm{bo}}$ is overlaid. We associate a scalar tracer to each of the
four sectors: I) internal-axis, II) external-axis, III) internal-equatorial,
and IV) external-equatorial.  Figure 8: Maps of the four tracers I, II, III
and IV (from left to right) associated with the four sectors of the stellar
material (see Fig. 7). The maps show the distribution of the stellar material
at $t=60~{}\mathrm{s}$ resulting from a jet with canonical parameters (see Sec
3.1).

####  3.3.1 The origin of ejecta with different final velocities

Figure 9: The energy-velocity distribution of the four different matter
tracers associated with the sectors depicted in Fig. 8. Matter from region II
(external along the axis) dominates the highest velocity region. Matter from
region III (internal equatorial) dominates the low velocity regime. Matter
from region I (internal along the axis) dominates the intermediate region and
the plateau. Matter from IV (external equatorial) is always subdominant.

To understand the origin of the various components of the outflow, we tracked
the distribution of the ejected material using four different scalar tracers
associated with four distinct regions of the star. The division to the
different regions is determined at the time of the breakout and it is shown in
Fig. 7. The tracers follow the mass in each of this region (at the time of
breakout): I) internal-axis, II) external-axis, III) internal-equatorial, IV)
external-equatorial.

Fig. 8 shows the distribution of the stellar material from each of the regions
at $t=60~{}\mathrm{s}$, roughly $46~{}\mathrm{s}$ after the breakout. Fig. 9
shows the energy-velocity distributions of the four sectors.

We see that the quasi-spherical outflow that leads the ejecta is made only of
tenuous material coming from the on-axis, external layers of the progenitor
directly above the expanding jet cocoon (region II). This component contains
only around $2\%$ of the total stellar mass but it contains 11% of the total
ejecta energy. Evidently, the fastest ejecta is dominated by this sector. The
material associated with the stellar core part that is along the axis (region
I; first panel in Fig. 8) is much more concentrated than that of the external-
axis region (II) but much more extended than the two equatorial sectors. It
contains $30\%$ of the total stellar mass and 46% of the outflow energy. This
section dominates the energy distribution over a wide range of velocities.
Almost all the rest of the mass and the energy are contained in the internal-
equatorial sector (III) which carries $60\%$ of the mass and 32% of the
energy. It dominates the energy at low velocities $\lesssim\beta_{0}$.
Finally, the outer-equatorial section carries $5\%$ of the ejecta mass and 11%
of its energy. All its material is moving at intermediate velocities and it is
subdominant at all velocities.

###  3.4 The effect of the stellar density profile

Figure 10: _Left_ : Energy-velocity distributions of jet simulations with canonical parameters for four different stellar density profiles. The blue line ($\rho\propto R^{-2}(R_{*}-R)^{2}$) represents the profile used in most of the previous simulations. _Right_ : A comparison of the energy distributions of the cases $\alpha=2,n=3$ and $\alpha=2.5,n=3$ with those arising from from two jets choked in a star with a canonical density profile at the same heights $z_{\mathrm{ch}}/R_{*}$, respectively. Jets |  $t_{\mathrm{e}}$ [s] |  $\theta_{\mathrm{j}}$ [rad] | $\rho(r)$ | $z_{\mathrm{ch}}/R_{*}$ |  $t_{\mathrm{bo}}$ [s]
---|---|---|---|---|---
Canonical | 1 | 0.2 | $\propto R^{-2}(R_{*}-R)^{2}$ | 0.21 | 13.9
$\alpha 2$_$n2.5$ | 1 | 0.2 | $\propto R^{-2}(R_{*}-R)^{2.5}$ | 0.25 | 11.2
$\alpha 2$_$n3$ | 1 | 0.2 | $\propto R^{-2}(R_{*}-R)^{3}$ | 0.25 | 9.5
$\alpha 2.5$_$n2$ | 1 | 0.2 | $\propto R^{-2.5}(R_{*}-R)^{2}$ | 0.28 | 6.8
$\alpha 2.5$_$n3$ | 1 | 0.2 | $\propto R^{-2.5}(R_{*}-R)^{3}$ | 0.37 | 4.0
Canonical_t1.33 | 1.33 | 0.2 | $\propto R^{-2}(R_{*}-R)^{2}$ | 0.25 | 11.5
Canonical_t2 | 2 | 0.2 | $\propto R^{-2}(R_{*}-R)^{2}$ | 0.37 | 8.3
Table 1: Properties of the jets injected in different density profiles. The
table lists the engine working time $t_{\mathrm{e}}$, the initial opening
angle $\theta_{\mathrm{j}}$, the density profile $\rho(r)$ used in the run,
the choking height relative to the star radius $z_{\mathrm{ch}}/R_{*}$, and
the breakout time $t_{\mathrm{bo}}$.

To study the effect of different stellar density profiles we consider stellar
density profiles that can be written as:

$\rho(R)=\rho_{*}\left(\dfrac{R_{*}}{R}\right)^{\alpha}\left(1-\dfrac{R}{R_{*}}\right)^{n}\
,$ (11)

where $n$ is the outer slope at the edge, and $\alpha$ is the inner slope,
with $\alpha<3$. The density profile described by Eq. 1, which is used through
the rest of the paper, is roughly equivalent to the case of $\alpha=2,~{}n=2$
and it will be referred as the _canonical profile_ hereafter. The profiles
that we consider are listed in Table. 1. For each profile we run a simulation
with our canonical jet parameters, $\theta_{\mathrm{j}}=0.2~{}\mathrm{rad}$,
$L_{\mathrm{j}}=10^{51}~{}\mathrm{erg}~{}\mathrm{s}^{-1}$, and we inject the
jets from the same initial height ($z_{0}=10^{9}~{}\mathrm{cm}$).

Fig. 10 shows a comparison of the energy-velocity distributions from different
stellar profiles. First, it shows that the distributions are all flat over a
range of velocities, implying that them main feature of the outflow from an
explosion driven by a choked jet is independent of exact stellar profile (a
similar result was found by Eisenberg et al. 2022 in the case of explosions
that are driven by successful jets). When looking in more detail, the right-
hand side shows two pairs of simulation. Each pair shows the results of
different stellar profiles with similar $\theta_{\mathrm{j}}$ and
$z_{\mathrm{ch}}$ (which dominates $V_{\mathrm{bo}}$). The distributions found
in the two simulations of each pair are very similar, implying that when the
cocoon properties are similar the stellar profile has a minor effect on the
outflow energy-velocity distribution.

On the left-hand side of Fig. 10 we compare the energy-velocity distributions
of jets with the exact same parameters (including $t_{\mathrm{e}}$) but
different envelope density profiles. It shows that the stellar profile affects
the velocity of the head (as was found previously by Bromberg et al. 2011;
Harrison et al. 2018) and therefore jets with the same properties are choked
at different heights when propagating in different density profiles. Since the
energy-velocity profile depends strongly on $z_{\mathrm{ch}}$ two jets with
the same properties that propagate at different stellar profiles will result
in outflows with different energy-velocity distributions, as shown on the
right-hand panel of this figure.

###  3.5 The energy velocity distribution at different viewing angles

Figure 11: Energy-velocity distributions for six different viewing angles
(chosen so that the corresponding wedges have the same volumes). _Top_ : Two
jets with the same choking height $(V_{*}/V_{\mathrm{bo}})^{1/2}=1.95$ but
different initial opening angles. _Bottom_ : Two jets choked at
$(V_{*}/V_{\mathrm{bo}})^{1/2}\sim 4$ (bottom panels) and with the same
opening angles of the jets on the top row, respectively. The black dashed line
represents the isotropic energy-velocity distribution for $1/6$ of the total
volume. The profiles are taken at $t=120~{}\mathrm{s}$.

Since jet driven explosions are aspherical, one expects that the outflow will
not be isotropic. Fig. 11 depicts the energy-velocity distribution of four
simulations. For each simulation we show the distributions at six different
sections, where each section is the sum of the ejecta within a range of polar
direction. To see the dependence on the initial conditions we show simulations
with two different jet opening angles ($0.2$ and $0.6$ rad) and two different
values of breakout volume $\sqrt{V_{*}/V_{\mathrm{bo}}}$. As expected, the
outflow is aspherical. A common, also expected, property of all four
simulations is that the maximal velocity of the outflow is around the jet axis
at lower polar angles. This result was found also for jet-driven explosions of
successful jets (Eisenberg et al., 2022). In the two simulations with the
small value of $\sqrt{V_{*}/V_{\mathrm{bo}}}$ (i.e., low $z_{\mathrm{ch}}$
and/or wide $\theta_{\mathrm{j}}$; top panels) the energy-velocity
distribution of the equatorial outflow ($\theta\gtrsim 60^{\circ}$) is similar
to that of a spherical explosion, with a typical velocity $\beta_{0}$. The
faster outflow is confined to lower angles. In the two simulations with the
large value of $\sqrt{V_{*}/V_{\mathrm{bo}}}$ (i.e., high $z_{\mathrm{ch}}$
and narrow $\theta_{\mathrm{j}}$; bottom panels) a large range of velocities
was seen in all directions, but still faster velocities are observed closer to
the jet axis.

Figure 12: Density profiles at different viewing angles of two jets with a
similar breakout volume $(V_{*}/V_{\mathrm{bo}})^{1/2}=1.95$ (top row) but
different initial opening angle compared to the density profile of jets choked
at $(V_{*}/V_{\mathrm{bo}})^{1/2}\sim 4$ (bottom row) with the same opening
angles, respectively. The solid thick black line in each panel represent a
power-law fit of $\rho(r)\propto v^{-5}$, corresponding to a flat distribution
of energy per logarithmic bin of the velocity. The profiles are taken at
$t=120~{}\mathrm{s}$.

For clarity we present in Fig. 12 the radial density distribution profiles,
$\rho(R)$, for different viewing angles of four different simulations. These
are the same simulations and the same divisions to angular sections as in Fig.
11. This presentation is often used in studies of SN ejecta and at sub-
relativistic velocity
$\rho(R)\propto\beta^{-5}\frac{\mathrm{d}E}{\mathrm{d}\log\beta}$.

##  4 Conclusions and implications to observations

We carried relativistic hydrodynamical simulations in 2D cylindrical
coordinates of stellar explosions driven by jets, focusing on configurations
of choked relativistic jets and exploring how those can lead to different
realizations of the velocity distribution of the outflow in its homologous
expansion phase. We followed the evolution of a relativistic jet from the
injection deep inside the star to the point where it is choked and then
continued to follow the cocoon as it emerges from the envelope and ultimately
unbinds it up to the point that the outflow becomes homologous. We scrutinized
the various stages of the jet inside the star and analyzed what happens during
the choking process and the adiabatic cocoon expansion. While the results are
given for a specific set of parameters we provided scaling relation for the
physical parameters of the jet and the star in order to facilitate a
dimensionless treatment of the problem. We stress that the scaling laws
presented in Sec. 2.2.3 don’t involve gravity and they are valid as long as
the gravitational binding energy of the star is subdominant with respect to
the total energy of the jet. A conditions that holds for the powerful jets
that have been observed in some SNe (Piran et al., 2019).

We summarize our findings as follows:

  * •

All jet driven explosions in which the jet is not choked too deep within the
star generate an outflow with a unique feature: a significant range of
velocities over which the outflow carries a roughly constant amount of energy
per logarithmic scale of the proper velocity ($\Gamma\beta$). This is a
universal property of jet driven explosions. The main difference between
different setups is the range of velocities over which the energy is constant.

  * •

The plateau of the energy-velocity distribution starts in all cases at
$v_{0}=\sqrt{E_{0}/M_{*}}$. The maximal velocity of the plateau depends mostly
on the cocoon volume upon breakout and the corresponding velocity is
$\beta_{\mathrm{bo}}=\beta_{0}\sqrt{V_{*}/V_{\mathrm{bo}}}$. For
$\sqrt{V_{*}/V_{\mathrm{bo}}}<3$ the maximal velocity is comparable to
$\beta_{\mathrm{bo}}$, while for larger values of
$\sqrt{V_{*}/V_{\mathrm{bo}}}$ the maximal velocity is larger than
$\beta_{\mathrm{bo}}$ and it can become mildly relativistic.

  * •

The volume of the cocoon upon breakout, $V_{\mathrm{bo}}$, depends on the
choking height, $z_{\mathrm{ch}}$, and on opening angle of the jet upon
launching. A higher $z_{\mathrm{ch}}$ and narrower opening angle leads to a
smaller $V_{\mathrm{bo}}$ and thus to an outflow that extends to higher
velocities.

  * •

The outflow from an explosion driven by a choked jet is not isotropic. In
general, the material along the poles (that is along the jet direction) is
faster while the material along the equator is slower.

A spherical explosions accelerate only a negligible fraction of the stellar
mass to very high velocities. Indeed, hydrodynamic simulations of core-
collapse supernova explosions have proved to be rather aspherical without
necessarily harboring a jet (e.g., Burrows et al., 2006; Janka et al., 2007;
Janka, 2012; Janka et al., 2016, for a review). However, the differential
mass-velocity distribution in those kind of simulations (Wongwathanarat et
al., 2015) does not show a significant fraction of the ejected material with
sufficient velocity to mimic the high-velocity tail seen in some SNe and
produced by the jets considered here.

We have shown here that the situation is drastically different when there is a
jet that breaks the symmetry. Such a jet can deposit a significant amount of
energy at high velocity matter, even in case that the jet is choked within the
envelope. This excess in high velocity outflow (compared to a spherical
explosion) is certainly expected when the entire stellar explosion is driven
by a jet, but it is also expected if the jet is accompanied by a simultaneous
more spherical explosion (see e.g., Eisenberg et al. 2022). If sufficiently
optically thick such a high velocity material that surrounds a SN would
produce a very broad absorption lines (with typical width corresponding to
0.1-0.2c) in the observed spectrum. It will be observed in the early spectra
but will disappear later when this outer envelope that is rapidly expanding
becomes optically thin.

Lines that show an excess of high velocity material have been observed in
several SNe (Galama et al., 1998; Iwamoto et al., 1998; Mazzali et al., 2000,
2002; Modjaz et al., 2006; Mazzali et al., 2008; Bufano et al., 2012; Xu et
al., 2013; Ashall et al., 2019; Izzo et al., 2019). Our result show that, as
suggested by Piran et al. (2019), a chocked jet can lead to that high velocity
material. However, we have found that some conditions are needed to observe
the corresponding broad absorption lines. First, the jet must be chocked at
sufficiently large distance at the stellar atmosphere. The signature of jets
that are chocked too deep will not be so significant. Second, as there is less
fast moving material in directions far from the jet direction, the fast moving
matter will become optically thin earlier in these directions. As the broad
absorption line will fade faster, this implies that observers at such viewing
angles are less likely to observe the broad absorption line signature. These
last two facts imply that we may not observer broad emission line in all SNe
that harbour relativistic jets.

The excess in fast material was observed in various types of stripped envelope
SNe. This include SNe that are associated with long GRBs, SNE that are
associated with llGRBs and SNe that are not associated with GRBs at all. Long
GRBs must contain successful relativistic jets. llGRBs contain jets which may
very well be choked (Kulkarni et al., 1998; MacFadyen et al., 2001; Tan et
al., 2001; Campana et al., 2006; Wang et al., 2007; Waxman et al., 2007; Katz
et al., 2010; Nakar & Sari, 2012; Nakar, 2015). We do not know if SNe that are
not associated with GRBs harbour jets, but if they are then these jets must be
choked ones. A previous study by Eisenberg et al. (2022) have shown that
successful jets can generate the energy-velocity distribution which is
observed in SNe that are associated with long GRBs. Our finding here show that
choked jets can explain the energy-velocity distribution seen in SNe that are
associated with llGRBs and in SNe that are not associated with any type of
GRBs. This provides further support for the interpretation of the
“disappearing” early very broad absorption lines in some SNe as arising from
choked jets. These findings also show that such lines may not be detected in
all SNe that harbor choked jets. Further exploration of this model, including
estimates of the observed spectra and the fraction of events in which these
lines will be observed will be carried out in future work.

## Acknowledgments

We kindly thank Christopher Irwin for the stimulating discussions and
suggestions. We also thank our anonymous referee for a constructive and
helpful report. This work is supported by the ERC grants TReX (TP and MP) and
JetNS and an ISF grant 1995/21 (EN).

## Data Availability

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

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##  Appendix A Resolution Check

Figure 13: Results of the resolution test showing the average head forward
shock velocity as a function of the head position $z$ for the simulation with
$\theta_{\mathrm{j}}=0.1~{}\mathrm{rad}$ and
$L_{\mathrm{j}}=10^{51}\mathrm{erg}~{}\mathrm{s}^{-1}$ at different
resolutions. The resolutions are: $620\times 595$ (blue), $1240\times 1190$
(orange), $1860\times 1785$ (green), $2480\times 2380$ (red),and D)
$3720\times 3570$ (purple). Convergence is excellent for $z/R_{*}>0.5$. As the
numerical initial conditions of the jets depend on the resolution, convergence
is less clear for $z/R_{*}<0.5$ as in this regime the jet has to settle down
and “forget” its initial conditions (see text). Figure 14: The energy-velocity
distributions for simulations with $\theta_{\mathrm{j}}=0.1~{}\mathrm{rad}$
and $L_{\mathrm{j}}=10^{51}\mathrm{erg}~{}\mathrm{s}^{-1}$ with different
resolutions. For each curve we report the different choking height relative to
the stellar radius. The resolutions are the same as reported for the curves in
Fig. 13.

We tested the convergence of our jet evolution in 2D varying the grid
resolution of our simulation box for the case
$\theta_{\mathrm{j}}=0.1~{}\mathrm{rad}$ and
$L_{\mathrm{j}}=10^{51}~{}\mathrm{erg}/\mathrm{s}$. The engine powering the
jet in each simulation stops after 1 second and thus we expect that the jet is
choked inside the stellar atmosphere at around the same height and the cocoon
expands slowly before accelerating again close to the stellar edge.

We tested our setup for different resolutions: 1) $620\times 595$, 2)
$1240\times 1190$, 3) $1860\times 1785$, 4) $2480\times 2380$, and 5)
$3720\times 3570$ (purple). Resolution 4) is what we use for all the
simulations in this paper. For these different resolutions we investigated the
head velocity convergence. Figure 13 presents the head velocity as a function
of time for the five resolutions previously discussed.

We see how the lower resolution runs tend to propagate slightly slower than
the high resolution runs (red and purple dots) before the jet engine
terminates its activity. Low resolution runs also tend to converge slower to a
constant velocity in the internal part of the the stellar atmosphere after the
jet engine switches off and the choking happens. This results in a different
choking height position $z_{\mathrm{ch}}$ for the low resolution runs with
respect to the high resolution cases (red and purple). At around $R\simeq
R_{*}/2$ the head velocity stabilizes and reaches a stable value of
$0.06~{}\mathrm{c}$ for all the resolutions taken into account. The
convergence to this velocity is excellent, however it takes some time for the
jet to reach this velocity and “forget” about its initial conditions, which
depends on the resolution. After the forward shock reaches $R_{\ast}/2$ the
head accelerates as it nears the edge of the star, eventually resulting in a
shock breakout at $z_{\mathrm{ch}}/R_{*}=1$. After the shock breakout we
notice that the forward shock propagation for the low resolution runs is much
slower with respect to the high resolution simulations, a clear sign that the
numerical resolution is insufficient to treat the problem properly.

In Fig. 14 we plot the energy-velocity distribution for the five different
resolutions we analyzed for Fig. 13. We can clearly see how the distribution
converges to a similar shape (red and purple curves at the highest
resolutions) at the resolution increases. The two highest resolutions differ
very little qualitatively and the jets are choked almost at the same
$z_{\mathrm{ch}}$. This demonstrates that our results are sufficiently
converged for the runs used in the paper.

##  Appendix B Different smoothing function

Figure 15: Comparison of the energy-velocity distribution of two simulations
of a jet with canonical parameters with two different smoothing functions. The
blue line uses the smoothing function of Eq. 2, while the orange line uses the
smoothing of Eq. 12.

The smoothing function for the stellar density profile (Eq. 2) prevents the
computation to break down at the moment of the shock breakout because of the
sudden drop in density of $\sim$20 orders of magnitude at the edge of the star
in a relatively small distance. The functional form of the smoothing function
however is either arbitrary, so we tested whether the result is affected by
the particular choice of Eq. 2. We run a simulation for canonical jet
parameters with the sharper smoothing function

$\rho_{\mathrm{smooth,2}}(R)=\rho_{\mathrm{s}}\left(\dfrac{R}{R_{\mathrm{s}}}+1\right)^{-10}\
,$ (12)

with $\rho_{\mathrm{s}}=0.05~{}\mathrm{g}~{}\mathrm{cm}^{-3}$ and
$R_{\mathrm{s}}=5\times 10^{8}~{}\mathrm{cm}$. In Fig. 15 we report the
comparison of the energy-velocity distribution of this new run with the old
one which uses the smoothing of Eq. 2. We can immediately see that the the two
curves almost overlap and show insignificant differences which do not affect
the final result. This is also due to the fact that the external additional
mass given by these smoothing functions is of the order of $10^{-6}~{}M_{*}$,
which is completely negligible and do not alter the physical scale of the
system.