Black hole to breakout: 3D GRMHD simulations of collapsar jets reveal a wide range of transients - arXiv
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MNRAS 000, 1–14 (2021) Preprint 9 November 2021 Compiled using MNRAS LATEX style file v3.0
Black hole to breakout: 3D GRMHD simulations of collapsar jets reveal a
wide range of transients
Ore Gottlieb1? , Aretaios Lalakos1 , Omer Bromberg2 , Matthew Liska3 , Alexander Tchekhovskoy1
1 Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA), Physics & Astronomy, Northwestern University, Evanston, IL 60202, USA
2 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
3 Institute for Theory and Computation, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA; John Harvard Distinguished Science and ITC
Fellow
arXiv:2109.14619v2 [astro-ph.HE] 6 Nov 2021
9 November 2021
ABSTRACT
We present a suite of the first 3D GRMHD collapsar simulations, which extend from the self-consistent jet launching by an
accreting Kerr black hole (BH) to the breakout from the star. We identify three types of outflows, depending on the angular
momentum, l, of the collapsing material and the magnetic field, B, on the BH horizon: (i) subrelativistic outflow (low l and high
B), (ii) stationary accretion shock instability (SASI; high l and low B), (iii) relativistic jets (high l and high B). In the absence
of jets, free-fall of the stellar envelope provides a good estimate for the BH accretion rate. Jets can substantially suppress the
accretion rate, and their duration can be limited by the magnetization profile in the star. We find that progenitors with large
(steep) inner density power-law indices (& 2), face extreme challenges as gamma-ray burst (GRB) progenitors due to excessive
luminosity, global time evolution in the lightcurve throughout the burst and short breakout times, inconsistent with observations.
Our results suggest that the wide variety of observed explosion appearances (supernova/supernova+GRB/low-luminosity GRBs)
and the characteristics of the emitting relativistic outflows (luminosity and duration) can be naturally explained by the differences
in the progenitor structure. Our simulations reveal several important jet features: (i) strong magnetic dissipation inside the star,
resulting in weakly magnetized jets by breakout that may have significant photospheric emission and (ii) spontaneous emergence
of tilted accretion disk-jet flows, even in the absence of any tilt in the progenitor.
Key words: gamma-ray bursts — methods: numerical — stars: jets — stars: Wolf–Rayet
1 INTRODUCTION winds. Since it is anticipated that different layers in the star are cou-
pled to each other, at least to some extent, it is expected that the
The emergence of long duration gamma-ray bursts (GRBs) in active
angular momentum loss should result in a slowdown of the stel-
regions of star-forming galaxies (Bloom et al. 2002), links them to
lar core rotation. Furthermore, strong differential rotation generates
the death of massive stars (e.g., Galama et al. 1998; Modjaz et al.
magnetic torques, which strengthen the coupling and further decel-
2006), thus associating GRBs with core-collapse supernovae (CC-
erate the rotation of the Fe core (Heger et al. 2005). The anticipated
SNe). The collapsar model (MacFadyen & Woosley 1999) suggests
relatively low angular momentum of WR cores calls into question
that during the collapse of a massive star, the iron core collapses
their ability to form viable accretion disks (MacFadyen & Woosley
to form a Kerr black hole (BH). Subsequently, owing to fast rota-
1999, but see possible resolutions in e.g., Levan et al. 2016; Bavera
tion an accretion disk forms, allowing the extraction of energy from
et al. 2021). The difficulty in forming the accretion disk required for
the BH in the form of Poynting flux, which powers a relativistic jet
jet launching may also be a key issue in understanding why only
that punches through the star and generates the GRB signal after it
some CCSN of type Ic are associated with GRBs while others are
breaks out of the star. Wolf-Rayet (WR) stars have been marked as
not. Another possibility is that jets are launched in some (Modjaz
promising progenitors of GRBs thanks to their strong winds, which
et al. 2016) or even in all CCSN Ic progenitor stars, but many fail
can deplete their hydrogen shells, thereby mitigating the journey of
to break out and produce the observed GRB emission (e.g., Mazzali
the jet through the stellar envelope (Woosley 1993). Indeed, to date,
et al. 2008), e.g. due to the presence of an extended envelope (e.g.,
all SNe observed in coincidence with GRBs have been suggested to
Margutti et al. 2014; Nakar 2015). Indeed, previous works have sug-
be of type Ic, associated with WR stars (e.g., Cano et al. 2017), and
gested that a non-negligible fraction of all long GRB jets are choked
implying that depletion of the helium envelope may be required to
inside the stellar envelopes, from both observational considerations
allow GRBs to emerge.
(Bromberg et al. 2011; Sobacchi et al. 2017) and numerical perspec-
As the progenitor star sheds its outer envelope, it transports some
tive (Lazzati et al. 2012).
of its angular momentum to the interstellar medium (ISM) via strong
GRB jets are most likely powered electromagnetically by the rota-
tional energy of a BH through e.g. the Blandford-Znajek (BZ) mech-
anism (Blandford & Znajek 1977; Narayan et al. 1992; Kawanaka
? ore@northwestern.edu et al. 2013). Komissarov & Barkov (2009) suggested that within this
© 2021 The Authors2 O. Gottlieb et al.
framework a necessary condition for jet launching, in addition to Thus, an important condition necessitated by the progenitor is suffi-
sufficiently high angular momentum, is that the magnetic energy ciently high angular momentum, so that the infalling gas hits a cen-
density has to exceed the plasma energy density in the vicinity of trifugal barrier at a circulation radius rcirc > rISCO , where rISCO is the
the BH. They verified this argument using 2D relativistic magne- innermost stable circular orbit around the BH. Instability processes
tohydrodynamic (RMHD) simulations. Tchekhovskoy & Giannios in the disk such as magneto-rotational instability (MRI) can then
(2015) proposed that once the jet is launched, its power initially de- amplify the magnetic field, increase the coupling between the disk
pends on the available magnetic flux but is eventually limited by the layers and sustain angular momentum transport that is crucial for
accretion rate onto the BH, which by itself is governed by the free- accretion. Dynamo processes can then transform some of the pre-
fall time of stellar material on the BH. Once the accretion rate drops dominantly toroidal (pointing in the φ -direction) field formed by,
and the jet can no longer support a high Poynting power, the jet en- e.g., the disk shear or the MRI, into the poloidal field (pointing in
gine ceases to operate. However, this simple analytic model ignores the R- and z-directions; e.g., Mösta et al. 2015; Liska et al. 2020),
the complex processes that take place in the disk and can affect the which facilitates the extraction of rotational energy from the BH as
magnetic flux on the BH, as well as the jet-disk feedback and jet-star electromagnetic Poynting flux via the BZ process.
interplay. To account for these processes General relativistic magne- For the launching of a two-sided jet, the BZ power depends on
tohydrodynamic (GRMHD) simulations are required. Such simula- the BH spin (assumed to be aligned with the disk) and on the mag-
tions enable us to obtain a self-consistent model of the jet launching, netic flux threading the BH horizon at rh (e.g., Tchekhovskoy et al.
with which one can address fundamental questions such as what are 2011)1 :
the conditions that allow the jet to be launched and whether the en-
10−3 2 2
gine operates for a sufficiently long time to allow the jet to break out L≈ Φh Ωh f (Ωh ) ≈ 1051 M 2BH,5 B2h,15 a2−0.1 erg s−1 , (1)
from the progenitor star and produce the GRB. c
Another important question concerns the magnetohydrodynamic where Qx denotes the value of the quantity Q in units of 10x
evolution of the jet material after it is launched and in particular its times its c.g.s. units, except for Mx which is given in units of M .
magnetization upon exiting the star. Understanding this is important Φh = 4πrh2 |Bh | is the integrated magnetic flux on one hemisphere
for the identification of the energy dissipation mechanism at large of the BH horizon, |Bh | is the value of the radial contravariant
distances responsible for the prompt GRB emission. Bromberg & magnetic field on the horizon, MBH and a are the BH mass and
Tchekhovskoy (2016) performed 3D relativistic magnetohydrody- spin, Ωh = ac/(2rh ) is the angular velocity at the BH horizon, and
namic (RMHD) simulations of Poynting flux dominated jets formed
√ −1
f (Ωh ) ≈ 1 + 1.38x2 − 9.2x4 , where x ≡ 0.5a 1 + 1 − a2 .A
by the rotation of a magnetized compact object. They found that the
second necessary condition for a successful launching of a BZ jet
jet is subject to magnetic kink instability, which slows down the jet
is that the BZ jet power will be sufficient to overcome the accre-
head while it propagates through the stellar envelope. The instabil-
tion power of the infalling material along the jet path (Burrows et al.
ity is responsible for substantial magnetic energy dissipation, which
2007; Komissarov & Barkov 2009). Prior to the jet launching, mat-
takes place at the location where the jet runs into the stellar envelope
ter is free-falling on the BH quasi-spherically with a power on one
and pinches, forming a collimation nozzle: this reduces the jet mag-
hemisphere of
netization to σ ≡ B2 /(4πρc2 ) ∼ 1, where B is the comoving mag-
netic field and ρ is the comoving mass density. Their simulations Ṁc2 = 4πrg2 ρh βhr c3 ≈ 2 × 1051 M 2BH,5 βhr ρh,7 erg s−1 , (2)
however, did not include the effect of gravity on the stellar envelope,
considered neither GR effects on the jet launching nor the disk-jet where ρh is the mass density on the horizon at the time of the jet
connection. launching, and βhr is the dimensionless radial velocity of the infalling
In this paper we perform the first 3D GRMHD simulations that material at that time, which is expected to be close to unity. Com-
follow the relativistic jets launched self-consistently by the spin of paring the jet luminosity from Eq. 1 with the accretion power from
BHs at the center of collapsing WR-like stars, propagate ∼ 5 orders Eq. 2, we get a necessary condition for the strength of magnetic flux
of magnitude to the stellar surface and break out. In §2, we present on the horizon that allows a successful jet launching
analytic conditions for jet launching and estimate their duration. In p r
βh ρh,7
27
§3, we describe our numerical setup. Motivated by the simulation re- Φh & Φh,min ≈ 7 × 10 G cm2 , (3)
a−0.1
sults, we develop self-consistent estimates of the jet launching con-
ditions, luminosity and duration. We show that the analytic criteria corresponds to minimal magnetic field on the horizon Bh,min ≈
for jet launching near the BH event horizon are consistent with the 1.4 × 1015 G for the same normalization of parameters. Note that
simulations (§4), infer the accretion rate onto the BH (§5), and the since a and βhr are both of order unity, the minimal magnetic field
consequent jet power and work-time (§6). In §7, we discuss the jet is primarily dictated by the central density of the star, which in turn
and disk evolution including the magnetic dissipation in the jet in- depends on the stellar mass, radius and density profile.
side the star and the tilt of the disk and jet axis by infalling material.
In §8 we associate the breakout times with types of GRB progenitors
and comment on potential implications on the prompt emission. We 2.2 Jet engine work-time
summarize and discuss the consequences of our results in §9.
We assume that the accretion timescale of the stellar envelope is set
by the free-fall time, such that a shell consumed by the BH at a time
t is coming from an initial radius
2 ANALYTIC MODEL OVERVIEW r0 (t) = (2GM BH t 2 )1/3 , (4)
2.1 Jet launching conditions
A successful launch of a relativistic BZ jet requires an accretion disk 1 Note that here the normalization is different since we are using both sides
to form around the central BH (e.g., MacFadyen & Woosley 1999). of the jet.
MNRAS 000, 1–14 (2021)Collapsar simulations reveal a range of transients 3
where we neglect gravitational effects of the accreted shells on each
other, and the effect of the jet-cocoon outflows blocking some of
the infalling gas. In the absence of a jet, the gas free-falls onto the
10
accretion disk, whose size might be changing slowly with time. If 10
the shell originates from an initial density power-law profile ρ(r) =
ρ0 r−α , we can use Eq. 4 to estimate the mass accumulation rate on
the accretion disk, 10
5
8πρ0 α 2α
Ṁd = (2GM BH )1− 3 t 1− 3 , (5)
3
0
10
where we approximate the disk size to be constant in time. To model
the accretion rate onto the BH we assume that a fraction of the mass
that falls on the disk is lost to winds during the accretion process,
obtaining (e.g. Blandford & Begelman 1999; Tchekhovskoy & Gi- 10
6
10
7
10
8
10
9
10
10
annios 2015)
βl
rw Figure 1. The simple stellar density profile that we use (Eq. 8) is consistent
Ṁ = Ṁd , (6) with stellar evolution models. Shown here are model 16Ti (Woosley & Heger
rd
2006, blue), compared with our density profile modeling with α = δ = 2.5.
where rw is the characteristic radius from which the wind is ejected, Both models feature very similar density profiles over most of the star and
rd is the outer radius of the accretion disk and βl is a constant that hence would result in very similar mass accretion rates (except for factor of
controls the strength of the mass loss rate. a few at very early times).
The jet can operate as long as the accretion rate onto the BH is
high enough and efficiently converted to Poynting flux to sustain
the subsequent jet launching and follow its propagation through the
the necessary power for a jet launching. High efficiency is naturally
star until it breaks out from the stellar surface.
obtained when the disk is in a magnetically arrested (MAD) state
The initial mass profile in the progenitor star just prior to the col-
(e.g. Tchekhovskoy et al. 2011), where the magnetic pressure that
lapse is modeled as
supports the jet is of the order of the infalling mass ram pressure.
−α
We therefore first assume a MAD state with a accretion to jet energy r δ
r
conversion efficiency of ρ(r) = ρ0 1− , (8)
rh R?
1 L
Z
(Ttr − ρc2 )dAΩ = where ρ0 is normalized to obtain rRh ? ρ(r)d 3 r = M? . This type of
R
η= ∼1, (7)
Ṁc rh Ṁc2 progenitor fits stellar-evolution models outside of their core. For in-
where Ttr is the total energy flux in the radial direction, dAΩ is a stance, a commonly used progenitor model 16Ti (Woosley & Heger
differential area of a solid angle element and the integration is done 2006) which, like other stellar-evolution models, has a uniform den-
on the BH horizon. With this definition L is the available jet power sity core (r . 2 × 107 cm) followed by a steep power-law den-
that can preform work. Note that in our definition of η, all quantities sity. Outside the core, the model can be well fitted by Eq. 8 with
are instantaneous and η may temporarily deviate from unity, how- α = δ = 2.5 (see Fig. 1). We emphasize that the collapse of model
ever when averaged over & 10 ms, it is of order unity. In this case 16Ti is indistinguishable from the analytic model at times t & 10
the accretion rate reflects the jet luminosity and can be used to ob- ms, after the collapse of the flat density profile core, whose mass is
tain an upper limit on the work-time of the jet engine, teng,max . For negligible compared to the mass of the envelope.
that purpose, we calculate the time interval between the time 5% of We use a spherically symmetric, specific angular momentum pro-
the star was accreted and the time at which 95% of the star was ac- file of the stellar envelope, such that it depends only on the radial
creted. In practice, it is possible that η may drop well below unity at coordinate r̂ (except for model α1BcLz where the rotation is cylin-
t teng,max , thereby shutting off the relativistic jet at a time much drical, i.e. r → rsinθ ). The profile is chosen such that it is increasing
earlier than teng,max . The efficiency of the jet launching depends on until ∼ 70rg , and then it becomes constant (see e.g., Takiwaki &
the strength of the poloidal magnetic field threading the BH horizon Kotake 2011, for a similar profile):
and on its length scale. Before reaching the BH, the magnetic field is 2
r2
amplified in the accretion disk by MRI and dynamo processes. MRI ω0 r g
r < 70rg
produces turbulence in the disk, which can disrupt the shape of the l(r) = , (9)
magnetic field, reducing its typical length scale and lead to an early
2 2
ω0 (70 rg ) r > 70rg
engine shutoff, as we show in §6.1.
where the angular velocity ω0 is constant. Last, to set the magnetic
field we assume that the star has a distinct magnetic core at the end of
3 SETUP its life (i.e. it did not lose its ordered magnetic field by e.g. interlayer
mixing), carrying a uniform magnetic field, B0 , in the ẑ direction and
To test and calibrate the analytic assumptions in §2 we run a suite of adopt an initial dipole-like vector potential outside the core (here-
GRMHD simulations of self-consistent jet launching in collapsars, after core+dipole), which is fully contained within the star, r < R? ,
using the 3D GPU-accelerated code H - AMR (Liska et al. 2019). For
the simulated collapsars we take a WR-like star with a radius R? = sinθ
r2 R2?
4 × 1010 cm and mass M? ≈ 14 M . A Kerr BH is placed at the A = Aφ (r, θ ) = µ · max 3 − , 0 , (10)
r r + rc3 R3? + rc3
center of the star with a mass M BH = 4M (not included in M? )
and a dimensionless spin a = 0.8. The simulation begins with the where µ ≈ B0 rc3 is the magnetic moment of the uniformly magnetic
gravitational collapse of the star. We monitor the disk formation and core, and rc = 108 cm is the core radius.
MNRAS 000, 1–14 (2021)4 O. Gottlieb et al.
Bh
ω0
grid, thus the asymptotic Lorentz factor Γ∞ ≈ σmax is much smaller
Model α log Bh,min log Variation Outflow
ω0,min than the values inferred from GRB observations, Γ∞ & 100.
α0BcLc 0.0 0.0 0.0 canonical Weak jet We employ a spherical grid in Kerr-Schild coordinates, use local
adaptive time-step, and 2 levels of adaptive mesh refinement (AMR).
α0BshLc 0.0 0.0 0.0 B(r > rc ) ∝ r−2 Jet The refinement criterion is based on the entropy of the fluid in or-
der to properly resolve both the cocoon and the jet. In simulations
α0BwLc 0.0 -0.5 0.0 canonical SASI where an axisymmetric outflow is generated, we avoid its interac-
tion with the polar axis by directing the rotational axis along the x̂
α0BsLc 0.0 0.5 0.0 canonical Jet direction, such that the disk forms in the ŷ − ẑ plane. However, to
avoid confusion with the standard convention, in all figures we label
α0BcLs 0.0 0.0 -2.0 canonical Subrelativistic
the jet axis as the conventional ẑ direction. The cells are distributed
logarithmically in the radial direction, extending out to r = 2R? , and
α0BsLs 0.0 1.0 -2.0 canonical Subrelativistic
uniformly in the polar and azimuthal directions. The grid resolution
α0.5BcLc 0.5 0.0 0.0 canonical Jet at the higest AMR level is 1152 × 288 × 256 in r̂ − θ̂ − φ̂ directions,
respectively. We verify that this resolution is sufficient to resolve the
α0.5BsLc 0.5 0.5 0.0 canonical Jet fastest growing MRI wavelength by measuring the Q parameter, de-
fined as the ratio between the MRI wavelength and the proper length
α1BcLc 1.0 0.0 0.0 canonical Jet of a cell in the azimuthal direction. We find that the typical Q pa-
rameter is 102 − 103 , much larger than Q ∼ 10 which is required to
α1BcLz 1.0 0.0 0.0 l(r) = l(rsinθ ) Jet properly resolve the MRI (e.g., Hawley et al. 2011).
α1BcLcRs 1.0 0.0 0.0 R? = 109 cm Jet
α1BtLc 1.0 0.0 0.0 B(r > 500rg ) = 0 Jet 4 OUTFLOW
α1BwLc 1.0 -1.0 0.0 canonical SASI
4.1 GRB jet launching
One requirement for a successful jet launching is linked to the an-
α1BcLms 1.0 0.0 -1.0 canonical Subrelativistic gular momentum distribution within the star. Depending on whether
the rotational profile is such that rcirc is smaller or larger than rISCO ,
α1BcLs 1.0 0.0 -2.0 canonical Subrelativistic
we find two different regimes of jet launching. For our angular
α2BcLc 2.0 0.0 0.0 canonical Jet momentum profile the specific angular momentum is maximum at
ri ≥ 70rg . Therefore, to form an accretion disk, the material at this
α2.5BcLc 2.5 0.0 0.0 canonical Jet radius needs to possess a high enough angular momentum so that
once the gas reaches r ∼ rISCO (after a free-fall time of a couple of 10
α2.5BwLc 2.5 -1.0 0.0 canonical SASI ms), an accretion disk forms and the jet is launched. In general, for
a radius-dependent angular velocity, ωrot , an accretion disk forms if
α2.5BcLs 2.5 0.0 -2.0 canonical Subrelativistic there is a radius ri at which the specific angular momentum satisfies
ri2 ωrot (ri ) & rISCO c. It then follows that the minimal angular velocity
α2.5BwL0 2.5 -1.0 −∞ canonical None
at radius ri with which a jet can be launched is
Table 1. Details of the numerical models: Model name is composed of the r c
ωrot,min (ri ) ≈ 2 ISCO . (11)
index of α, the strength of the magnetic field on the horizon B (w: weak, c: ri ( ωrot,min )
canonical, t: truncated, s: strong, and sh: shallow), and angular momentum
L (s: slow, mildly slow: ms, c: canonical, 0: zero, and axisymmetric: z). the The formation of a disk is imperative for a stable jet launching as
inner density profile α, logarithmic of the ratio between magnetic flux on the it requires a significant magnetic amplification, which can only take
horizon Bh at the accretion disk formation time and the critical magnetic flux place in accretion disks. As we show later, our models suggest that if
Bh,min , logarithmic of the ratio between the circularization radius rcirc and the an accretion disk does not form, the accumulation of magnetic field
gravitational radius rg , variation of the model with respect to the canonical around the BH can only power a weak jet that operates briefly, since
model, and the outcome of the simulation (see §4). the initial poloidal field is not strong enough. Our simulations also
show that if an accretion disk does not form early on, the emerging
outflow of the weak jet may inhibit the later formation of a rela-
tivistic GRB jet. It also implies that a GRB jet should be launched
We vary α, B0 and ω0 in our simulations to examine their effect within a few tens of ms after the initial collapse, as was also found by
on the jet launching and propagation. We also carry out several sim- Mösta et al. (2014), where the precise time depends on the angular
ulations where we vary the magnetic field profile and R? to verify momentum, magnetic field and density profiles. Thus, hereafter for
our conclusions. The full list of models is given in Tab. 1. We re- our analytic analysis we assume that an accretion disk forms during
strict the value of B0 so that the initial σ everywhere in the star is the first . 0.05 s, and use our simulation parameters ri = 70rg and
below unity. We also keep δ = 3 and show analytically that its value ti ≈ 20 ms as the initial radius and the corresponding free-fall time
does not affect the results considerably. Similarly, we do not vary the of the matter that forms the disk.
BH spin and mass between simulations as they can only change by a A second requirement is related to the strength of the magnetic
factor of . 2 implying a change of up to half an order of magnitude field. Consider a shell at a radius r0 having width ∆r0 . r0 and mass
in the jet luminosity (Eq. 1). All GRMHD codes need to use den- m = 4πρ(r0 )r02 ∆r0 free-falling onto the BH. Its density on the hori-
sity floors to maintain the stability of the numerical scheme. In this zon is ρh ≈ ρ(r0 )(r0 /rh )2 ∆r0 /∆rh , where ∆rh is the shell width on
spirit, we limit the magnetization to σmax = 25 everywhere in the the horizon. Free-fall dictates that r ∝ t 2/3 , implying that the shell
MNRAS 000, 1–14 (2021)Collapsar simulations reveal a range of transients 5
width on the horizon increases quasi-linearly as ∆rh /∆r0 ∝ r0 /rh . 55
10
It follows that the shell density on the horizon also increases quasi-
linearly as ρh ≈ ρ(r0 )r0 /rh . Since the jet is launched at t = ti , it
encounters a central density ρh ≈ 100ρ(ri ). Plugging this into Eq. 3,
we find that at the time of the jet launching, the minimal magnetic 10
52
flux on the horizon which allows a successful launching is
q
r
βh,−0.2 ρ5 (100rg )
Φh,min ≡ 6 × 1027 Mx , (12) 10
49
a−0.1
where the calibration βhr ≈ 10−0.2 βh,−0.2
r is set by our simulations
which show, as expected, that the infalling material is reaching the
46
BH at radial velocities close to the speed of light. Equivalently, we 10
0 0.5 1 1.5 2 2.5
can use Eqs. 1 and 2 to associate the density with a minimal BZ jet
luminosity,
erg Figure 2. Allowed stellar inner density power-law indices α based on the jet
Lmin,0 = 1.6 × 1051 M 2BH,5 βh,−0.2
r
ρ5 (100rg ) . (13) luminosity. Jets in progenitors with α & 2 can only be launched if their lumi-
s
nosity (blue, Eq. 13) is & 1052 erg s−1 (above black dashed line), in tension
Since M BH,5 cannot be much smaller than unity and GRB observa- with observations of long GRBs (e.g., Shahmoradi & Nemiroff 2015). Also
tions suggest that L . 1052 erg s−1 (e.g., Shahmoradi & Nemiroff shown is the maximum jet luminosity (red), based on 100% jet energy pro-
2015), it follows from Eq. 13 that GRB progenitors cannot have duction efficiency (jet power = accretion power). Luminosities are shown for
ρ(100rg ) & 106 g cm−3 . Due to the magnetic and density profiles, R? = 4 × 1010 cm (thick lines) and R? = 4 × 1011 cm (thin lines). The lines
our magnetization profile peaks at ∼ 100rg . are plotted for δ = 3, with shaded areas mark the range of L between δ = 1
and δ = 5 (see Eq. 8).
Our estimates also apply in the case where the angular momen-
tum has a cylindrical profile, such as in model α1BcLz. Although a
cylindrical angular momentum creates a low density funnel on the
polar axis, it is only important at r rg since the density that the rcirc & rISCO rcirc < rISCO
jet encounters upon launching is set by the infalling material onto
the disk, on timescales shorter than the time it takes to form a funnel Φh & Φh,min Relativistic jet Subrelativistic outflow
through rotation.
Φh . Φh,min SASI Gravitational collapse
One can use the maximum central density that allows a jet launch-
ing to constrain the inner density profile of GRB progenitors. As- Table 2. The four possible outcomes of outflow from the BH horizon, de-
suming for simplicity our canonical M? = 14 M , we find the den- pending on the specific angular momentum of the star which determines if
sity at 100rg for each power-law index α, and plot in Fig. 2 the min- an accretion disk forms, and magnetic field in the star which dictates whether
imal jet luminosity upon launching, Lmin,0 (blue) as a function of α. the jet power is sufficient to overcome the infalling accretion power.
The shaded area represents the variation in Lmin,0 between values of
δ = 1 and δ = 5. For comparison we also plot in red the maximum
luminosity obtained if 100% of the accretion power at the time of the relation between ω0 , Bh and their critical values ω0,min , Φh,min
the jet launching, is channeled to the jet (η = 1). The actual power obtained from Eqs. 11 and 12, respectively. The outflow types are
of a successful GRB jet will lie between those two lines. Since the listed in Tab. 2 (see also Tab. 1), and illustrated in Fig. 3 showing
density is linear in M? , which cannot be much smaller than 14 M , a snapshots of logarithmic density maps of three cases in which an
change in the stellar mass does not affect this result to a large extent. outflow emerges:
On the other hand, R? may vary by an order of magnitude2 and its i) SASI ( rcirc & rISCO and Φh . Φh,min ): a rapid rotation with
effect on the density is superlinear. Thus, we plot the critical lumi- ω0 & 10 s−1 results in the formation of an accretion disk around the
nosity both for R? = 4 × 1010 cm (thick lines) and R? = 4 × 1011 BH. When the magnetic field on the horizon is weak, the Poynting
cm (thin lines). Fig. 2 demonstrates that only moderate density pro- power is insufficient to overcome the ram pressure of the infalling
file slopes at the stellar interior (α . 1.5) are allowed as they lead material and no jet is launched. Wind from the hot accretion disk
to sufficiently low central densities, which support the formation of collides with the falling material forming a quasi spherical shock
jets with observed luminosities of L . 1052 erg s−1 . Jets in progeni- around the disk. The continual flow of wind energizes the shock and
tors with α > 2 are ruled out as GRB candidates due to their exces- drives hydrodynamic instabilities, which lead to a behavior similar
sive power. One interesting example is 16Ti (where α = 2.5) which to that seen in accretion shocks around CCSNe, known as station-
maintains ρ(100rg ) ≈ 109 g cm−3 , and thus can only launch a jet ary accretion shock instability (SASI; Blondin et al. 2003; Blondin
with a luminosity that is much higher than all detected long GRBs. & Mezzacappa 2006). We stress that the formation of the spheri-
cal shock wave in CCSNe is different, as it is driven by the stellar
core-bounce and stalled at ∼ 107 cm due to neutrino losses and the
4.2 Types of outflows dissociation of heavy nuclei such as iron into nucleons (see e.g.,
Burrows et al. 1995; Mezzacappa et al. 1998). In the absence of
Following the discussion above, our simulation results can be di-
neutrino scheme and nuclear physics in our simulations, the shock
vided into four types of outflows from the BH horizon, depending on
is not halted at any point, and thus only resembles the early evo-
lution of SASI. Nevertheless, both types of shocks show a similar
2 Note that our definition for R is set by the density profile in Eq. 8, and
?
behaviour and lead to symmetry breaking of the explosion. We find
does not include the case of a progenitor that is embedded in an extended that in all simulations in which an accretion disk forms, the spiral
envelope. mode (m = 1) of the accretion shock grows by nearly axisymmet-
MNRAS 000, 1–14 (2021)6 O. Gottlieb et al.
8.2 8.2
7.8 7.8
7.5 7.5
7.1 7.1
6.8 6.8
6.4 6.4
6.1 6.1
5.7 5.7
5.4 5.4
5.0 5.0
6.1
5.6
5.2
4.7
4.3
3.8
3.4
2.9
2.5
2.0
4.6
3.7
2.8
1.8
0.9
0.0
-0.9
-1.8
-2.8
-3.7
Figure 3. A variety of outflows from different progenitors, based on Tab. 2. Displayed logarithmic mass density maps of: a) SASI: high specific angular
momentum to form an accretion disk, and weak magnetic flux which is insufficient to launch a jet. Shown is an edge-on view (a1) and head-on view (a2), taken
from model α1BwLc, 0.3 s after the collapse; b) subrelativistic outflow: specific angular momentum is too low to form a disk and magnetic field is strong to
extract energy from the BH. Shown is an edge-on view from model α1BcLs, 1.2 s after the collapse; c) jet: high specific angular momentum and strong magnetic
field support jet launching. Shown is an edge-on view when the forward shock reaches ∼ 2R? from model α1BcLz, 12 s after the collapse. The central engine
shut itself off before the jet breakout and thus the breakout is of the shocked jet material. In panels (b,c) the metric is tilted such that in all panels the accretion
disk lies on the x̂ − ŷ plane. Videos are available at http://www.oregottlieb.com/collapsar.html.
MNRAS 000, 1–14 (2021)Collapsar simulations reveal a range of transients 7
ric accretion (Fig. 3a1,a2), as was found in 3D simulations of CC- (a)
SNe (e.g., Blondin & Mezzacappa 2007; Fernández 2010). When 10-2
the magnetic field on the horizon is too small to power a jet, the
SASI-like structure grows unperturbed, as in a case of a SN that is
not accompanied by a GRB jet. We point out that the growth of ac-
cretion shock increases the central density, shuttering any hope for a
10-4
delayed jet launching.
ii) Subrelativistic outflow ( rcirc . rISCO and Φh & Φh,min ): the
small circularization radius places the centrifugal barrier at a radius
smaller than rISCO and an accretion disk does not form. However, the
infalling magnetic flux can initially thread the spinning black hole 10-1 10-2 10-1 100 (b)
10 1
and extract some BH spin energy to briefly power a weak jet. In the
absence of a disk that can amplify the magnetic field, the flux on
the BH escapes via reconnection on the equatorial plane and weak-
10-2
ens until the jet launching ceases. Such an outflow might have been
found by Burrows et al. (2007) who described it as a pre-cursor jet.
The outflow structure at later times depends on the density profile.
A quasi-spherical subrelativistic outflow emerges if the jet deceler- 10-3
ates in a rather flat density profile (Fig. 3b), thus its breakout may
resemble a weak SN explosion that is continuously powered by the
10-2 10-1 100 (c)1
10
inefficient energy extraction from the BH by a weak field. If the core
density profile is steep such that the outflow accelerates, it may keep
101
its elongated shape in what may be the source of low-luminosity,
soft GRBs via e.g. sub-relativistic shock breakout emission, which
may yield a quasi-isotropic signal. 100
iii) Relativistic jet ( rcirc & rISCO and Φh & Φh,min ): the first cri-
terion indicates that the stellar envelope collapses until reaching the 10-1
centrifugal barrier to form an accretion disk, which drives a BZ jet
launching. The second criterion implies that the jet power is suffi-
cient to overcome the ram pressure of the central density and a rel- 10-2 10-1 100 (d)
10 1
ativistic jet is successfully launched from the horizon. The jet prop-
agates in the stellar envelope and forms a hot weakly-magnetized
cocoon which collimates the jet (Fig. 3c). Since in our simulations
both the jet and the accretion shock are driven by the instabilities in 101
the disk, they are built over the same timescale such that the growth
of the cocoon disrupts the otherwise spiral structure of the accre-
tion shock. We discuss the jet propagation in the envelope in §7,
and leave a detailed analysis of the jet-SASI interplay (Nagakura &
Yamada 2008) for a future work.
100
10-2 10-1 100 101
Figure 4. Analytic model for free-fall mass accretion rate (black) provides
an upper limit to Ṁ from our simulations (colored lines). The power-laws are
in agreement in the absence of the jet, Ṁ ∝ t (3−2α)/3 , whereas when a jet is
present, Ṁ ∝ t (2−2α)/3 . (a)-(c): our canonical progenitor is considered with
a variety of density profiles, α = 0, 1, 2.5. (d): the accretion rate for a small
5 ACCRETION RATE
star, R? = 109 cm, showing that the time at which the accretion rate drops is
We show that the accretion rate onto the BH can be estimated by a in agreement between the analytic and numerical results.
spherical free-falling rate of the stellar envelope with a few caveats:
i) we neglect the unbound mass that could be ejected if there is an
explosion prior to the onset of the simulation. ii) our gravitational where r0 (t) is the original radius from which matter is reaching the
potential is fixed and does not change with the accretion of mass disk3 at time t.
onto the BH. iii) the presence of the cocoon may deviate the accre- Fig. 4 shows the accretion rate onto the BH, Ṁ, as found in the
tion from a spherical free-fall. Our simulations are carried out for simulations (colored lines), for a variety of α, B0 and ω0 , compared
progenitors with M? = 14 M and δ = 3. We emphasize that the ac-
with the analytic expression for the mass flow rate on the disk in
cretion rate temporal profile is solely dictated by the value of α, as
M? and δ affect only the normalization through the density constant Eq. 14 (black lines). In the top three panels, which show the results
ρ0 . We begin with calculating the free-falling mass flow rate on the for progenitors with R? = 4 × 1010 cm, the analytic accretion rate
disk using Eqs. 4 and 8
8πρ0 r0 (t) 3 At t . 10 ms there is no disk to stall the matter from reaching the BH and
Ṁd (t, α) = 3
−r(t)5−α + 3R? r(t)4−α − 3R2? r(t)3−α + R3? r(t)2−α ,
3R? t at t 10 ms, r0 is much larger than the size of the disk, so the free-falling
(14) time to the BH and the disk is similar.
MNRAS 000, 1–14 (2021)8 O. Gottlieb et al.
evolves as a power-law throughout the entire simulation. In simula- time4 . The true luminosity depends on the conversion efficiency of
tions where there is no jet (subrelativistic outflow/SASI in red and accreted mass flow to jet power, η as L = η Lmax .
yellow lines) the measured accretion rate is in a remarkable agree-
ment with the analytic prediction of free-fall time, both in the slope
and the normalization which differs by up to a factor of . 3. This 6.1 Engine activity
factor can be explained by a combination of an aspherical accretion To evaluate the maximum possible engine activity time, teng,max , we
in the case of a subrelativistic outflow whose cocoon blocks some follow the method presented in §2.2, which connects this time with
of infalling matter from reaching the disk, and winds which lead to the time where 90% of the stellar material is accreted. Since our sim-
a loss of some of the accreting mass onto the accretion disk before ulations do not last through the entire stellar collapse, we estimate
reaching the BH as discussed in §2.2. If the factor is attributed to this time analytically by integrating Eq. 14 over the entire accretion
winds from the disk, an agreement between the analytic accretion time. As stated before, teng,max is an upper limit for the jet activity
rate given in Eq. 6 and the simulated ones can be obtained by taking time, and thus that the accretion onto the BH can be approximated
the size of the accretion disk rd ≈ 100rg from the simulation, and as Ṁ = Ṁd and that the conversion efficiency to jet power is η = 1.
using rw ≈ 10rg , and βl ≈ 0.5, consistent with the values found in In Fig. 5 we show teng,max as a function of α for our canonical pro-
other numerical studies (e.g., McKinney et al. 2012). genitor of R? = 4 × 1010 cm, denoted as teng,max,10.6 . The obtained
When a jet is present (blue lines), the cocoon expands fast (in values are of the order of teng,max ' 110 ± 30 s for 0 ≤ α ≤ 2.5. Eq. 4
contrast to the subrelativistic outflow) and suppresses some fraction suggests a scaling relation of engine activity time with stellar radius
of the free-falling gas into the disk. The main gas component that
ultimately powers the accretion flows along the lower sides of the 1.5
expanding cocoons of the two jets towards the equator, where it hits R?
teng,max (α, R? ) = teng,max,10.6 (α), (15)
a strong accretion shock and is stalled. Thus, it takes longer to reach 4 × 1010 cm
the disk, depending on the shape of the cocoon and on the location implying that stars with R? ≈ 4 × 1010 cm could be more consistent
6−α
of the accretion shock at the equator, which evolves as t 5−α (Gottlieb with typical durations of GRB jets, whereas stars with R? & 1011 cm
& Nakar 2021). However, the accretion rate is not only linked to the might be more consistent with ultra-long GRBs.
cocoon width. First, while part of the shocked gas in the cocoon be- As we pointed in §2.2, efficient conversion of accretion energy to
comes unbound, some of the cocoon gas does fall back onto the disk. jet power strongly depends on the conditions of the magnetic fields
Second, interaction between the cocoon and the counter cocoon on at the vicinity of the BH and may result in teng teng,max . In partic-
the equator drives shocks which render the pressure in the cocoon ular the curvature radius of the poloidal field component needs to be
nonuniform. Therefore, we do not provide an analytic solution to larger than a few tens rg in order to efficiently convert the BH rota-
the influence? of the cocoon on the accretion rate, but study the ac- tional energy to Pointing flux (e.g. Chashkina et al. 2021). The ini-
cretion rate in this case from the numerical simulations. Our simula- tial field in our simulations consists of an inner core with a uniform
tions show that as long as the accretion originates in the inner den- vertical field at r . 108 cm surrounded by an outer dipole field. To
sity power-law segment, the accretion rate scales as Ṁ ∼ t (2−2α)/3 control the magnetization we modify the density normalization ρ0
(slower than the analytic estimate of Ṁ ∝ t (3−2α)/3 ). and power-law index, α under the constraint that σ < 1 everywhere
Finally, our simulations with R? = 4 × 1010 cm do not last long inside the star. The resultant magnetization profile holds σ ∝ rα in-
enough to reach the expected drop in the accretion rate at t & 100 side the core and σ ∝ rα−6 outside of it. Accretion of such a large
s, which takes place when the outer layers of the envelope are ac- scale field builds enough magnetic flux on the BH horizon to trigger
creted. Thus, in order to examine the change in the accretion rate a short jet launching episode at the onset of the simulation. How-
at late times, we also simulate a smaller progenitor with R? = 109 ever, this flux does not survive as it leaks out over dynamical time
cm, while keeping M? = 14 M (model α1BcLcRs ). In this progen- scales, due to magnetic reconnections that take place on the equato-
itor, as shown in Fig. 4d, the drop in the accretion rate should take rial plane. The continuous powering of the jet requires a buildup of
place at t ≈ 0.5 s according to Eq. 14, enabling us to explore the the flux by magnetic field coming from the accretion disk, where the
final phase of the accretion. We first note that unlike our canonical length scale of this field is set by various instability processes in the
sized progenitors, here the sharp decline in the stellar edge affects disk, such as MRI, which can generate turbulent motions and disrupt
the density profile already at early times, and thus the initial accre- the initial field configuration.
tion is roughly, but not fully, consistent with the arguments above In order for turbulence to grow in the disk, the fastest growing
for jets. One can see that the accretion rate has a clear break in its MRI mode in the z direction must be of the order of (or smaller
power-law behavior at a time that is consistent with the analytic ex- than) the disk scale height H = cs / ωrot , where cs is the local sound
pectation and it starts falling as t −2 (similar to what was found in the speed and ωrot is the local angular velocity. In ideal MHD this
simulations of Christie et al. 2019). We conclude that the accretion mode is λMRI = 2πvA,z / ωrot (e.g. Masada & Sano 2008), where
duration is also consistent with the analytic estimate. vA,z = B2z /4πρ is the Alfven velocity in the z direction. We can thus
−1/2
define the ratio λ̃ = λMRI /H ∝ β p , and use the condition λ̃ . 1
as an approximate threshold for a successful growth of small scale
6 JET WORK-TIME & LUMINOSITY turbulence in the disk that can disrupt the ordered magnetic field
structure and facilitate reconnection that destroy the overall flux that
Using the accretion rate behavior, one can infer two fundamental
goes into the BH. Fig. 6 displays the radial profile of λ̃ , weighted by
jet quantities: the central engine work-time and the jet luminosity.
the mass density at each point and averaged over all angles, which
The jet launching duration, teng , is governed by the accretion rate,
picks out the disk region out of the infalling material. In the case
magnetization and by the magnetic field profile of the infalling ma-
terial. The maximum luminosity is simply the total accreted power,
Lmax ≡ Ṁc2 , as shown in Fig. 2 (red line) at the time of the jet initial 4 Since this is an upper limit estimate, we ignore the numerical correction of
launching, where Ṁ is calculated under the assumption of free-fall a factor of a few between Ṁ and the free-fall approximation.
MNRAS 000, 1–14 (2021)Collapsar simulations reveal a range of transients 9
on the magnetic field normalization, we find that the typical jet du-
140
ration in such cases is . 1 s. Moderate values of α ∼ 1 can support
130 jet launching for teng & 1 s, but do not seem to reconcile with the
typical GRB durations of teng > 10 s (see e.g. yellow curve in Fig.
120 7 where the luminosity drops after ∼ 2 s). Longer duration can be
obtained with stronger initial magnetic fields, however in this case
110
the jet luminosity might be too high. Progenitors with high values
100 of α & 2 seem to be the ones compatible with engine activity of
teng ≈ teng,max , but are ruled out due to excessive luminosity and
90 temporal evolution of the lightcurve (§6.2).
80
One possibility of why teng is shorter than the observed long GRB
durations is differences in the magnetic profile. For example a mag-
70 netic field that drops radially slower from the core, such as the one
0 0.5 1 1.5 2 2.5
used in model α0BshLc, results in a shallower magnetization pro-
file, σ ∝ rα−5 . This can prolong the duration of jet engine activity,
Figure 5. An upper limit to the central engine activity time, determined by thereby bringing progenitors with α = 1 into agreement with typical
the free-fall time of the stellar upper envelope, for R? = 4 × 1010 cm. GRB durations as well. Alternatively, if the disk viscosity is high,
like in the case of efficient neutrino cooling, λMRI may increase and
lead to larger scale turbulence and a longer jet launching duration
(Masada et al. 2007). A variety of density and magnetic field pro-
files among GRB progenitors may play a major part in setting the
1BcLc wide distribution of GRB durations, from bursts that last less than
0BcLc a second to hundreds of seconds. However, we point out that while
101 stronger magnetic fields and steeper density profiles lead to more
luminous jets and longer central engine activity times, observations
point at an anti-correlation relationship, L ∝ tγ−1.52 (Dainotti et al.
MRI /H >
2015), where tγ is the duration of the prompt emission.
100
<
6.2 Luminosity
We find that during the time of a stable jet launching, the jet lu-
minosity satisfies L ≈ Ṁc2 , implying that the efficiency remains at
10 1 η ∼ 1 (top panel of Fig. 7)5 , consistent with a MAD state. However,
106 107
r [cm] if the jet operates on the timescale of t . 0.1 s, it does not man-
age to sustain the high-efficiency η ∼ 1 state for a long time. This
Figure 6. Accretion disk magnetization determines its ability to retain the can happen either when β p is too high so large scale magnetic field
large-scale vertical magnetic flux. Model α1BcLc (blue), which starts with a structure is absent, or in the case of the weak subrelativistic outflow
shallower magnetization profile, ends up with a higher magnetization of the scenario (see §4). The bottom panel of Fig. 7 depicts the jet lumi-
disk and the vertical MRI wavelength exceeding the disk thickness. This pre- nosity for a variety of progenitors. To avoid the effects of numerical
vents the magnetic field from developing small-scale features and diffusing floors contaminating the measurements, the luminosity is calculated
outwards, resulting in a stable jet for ∼ 3 s before the magnetization in the slightly above the horizon at r = 2rg . We find that the luminosity is
disk drops. In contrast, when the magnetization profile is steeper, such as in approximately independent over the wide range of radii, 2rg − 20rg ,
model α0BcLc (orange), low magnetization matter is accreted early on, lead- as expected, since L is a conserved quantity and is expected to be a
ing to λMRI < H and the growth of small scale turbulence in the disk, which
constant in a steady state. Since L ∝ Ṁ, the jet luminosity features
disrupt the large scale profile of the magnetic filed. Shown are the radial dis-
a global temporal evolution in time, in addition to a short timescale
tributions of < λ̃ >, the ratio of the wavelength of the fastest growing MRI
mode to the disk scale height, weighted by the local density and averaged variability due to variations in the accretion power. The absence of
over the angular direction. The distribution is shown 0.2 s after the collapse observed temporal evolution in GRB lightcurves6 (McBreen et al.
in two models. Videos of the evolution in time of the disk in both cases are 2002b, but see McBreen et al. 2002a for changing lightcurves as
available at http://www.oregottlieb.com/collapsar.html. L ∝ t ±1 ) places a further constraint on the allowed values of α.
Jets in stars with α & 2 (purple curve in Fig. 7) are in tension
with observations due to both their very high luminosity needed to
of α = 0 (orange line), λ̃ drops below unity after t ≈ 0.1 s which overcome the high central density ram pressure, and a monotonically
corresponds to a free-fall time of matter that is located initially at a decreasing accretion rate that drops over orders of magnitude during
radius twice the size of the magnetized core. The initial magnetiza- the typical duration of long GRBs. A temporal variation may also
tion in this case scales like r−6 outside the core implying that the arise in progenitors with α ≈ 0, where the accretion mass rate, and
magnetization of the accumulating disk matter drops like t −2 . The subsequently the jet luminosity, increases with time, as can be seen
fast drop in magnetization leads to a hotter disk and to the reduction
of λ̃ below 1. In the case of α = 1 (blue line), the magnetization
profile is shallower and as a result λ̃ remains larger than one for a 5 Note that since the efficiency in Fig. 7 is measured at 2rg , the efficiency is
longer time, leading to a longer lasting jet. somewhat lower than unity.
We generally find that when α . 0.5 the magnetization profile is 6 If the radiation efficiency is of order unity, as suggested by observations,
steep and the average β p in the disk rapidly increases. Depending then the GRB light curve behavior reflects the jet power.
MNRAS 000, 1–14 (2021)10 O. Gottlieb et al.
100
7.5
7.0
10-2
6.5
6.0
5.5
10-1 100 101
1054
5.0
1053 4.5
4.0
1052
3.5
1051
3.0
50
10
1049 Figure 8. Deflection of infalling gas by the jet breaks the symmetry and re-
sults in a tilted accretion disk and jet axis. Shown is the logarithmic mass
density map from model α1BcLz, 2.3 s after the initial collapse. The en-
1048
tire tilt process can be seen in a movie in http://www.oregottlieb/
collapsar.html.
1047
1046 and stellar material to form a weakly-magnetized cocoon that colli-
10-1 100 101
mates the jet. The jet-cocoon-star interplay ultimately regulates the
jet evolution in the star. While this is not the main focus of this pa-
per, we report of two major features that are found in our simulations
Figure 7. Different density profiles lead to different efficiencies (top) and lu-
as the first self-consistent 3D GRMHD simulations of the collapsar
minosities (bottom), in the models that we specifically designed to be slightly
model. A detailed analysis of the two will be presented in a follow-
above the critical magnetic flux so they can launch jets. The quantities are
measured at r = 2rg as a function of time in a variety of progenitors: three up work.
with Φh = Φh,min : α = 0 (blue), α = 1 (yellow) and α = 2.5 (purple); and
one progenitor with Φh ≈ 3 Φh,min and α = 0 (red). We find that over time the
fastest growing mode of the MRI, λMRI , drops relative to the disk thickness,
7.1 Tilt of the disk
H, resulting in the magnetic flux diffusing outwards and jet efficiency drop-
ping (see §6.1). The jet power in progenitors with α ∼ 0 or α > 2 exhibits The high pressure that grows in accretion disks leads to release of
temporal evolution while η ∼ 1. winds from the equatorial plane towards the polar axis, which facil-
itate the jet collimation at its base. As the jet propagates farther in
during the first ∼ 0.5 s in model α0BcLc (blue curve). When the jet the star, its collimation becomes supported by the pressurized back-
is stronger (model α0BsLc, red curve) the suppression of accretion flowing material of the cocoon. Our simulations show that the heavy
by the cocoon may level the accretion rate such that it is nearly con- parts of the cocoon, which are close enough to the BH fail to be-
stant. Stars with α ∼ 1 are found to be the most consistent with ob- come unbound and fall towards the BH. When such relatively heavy
servations, maintaining both L ≈ 1051 erg s−1 and a rather constant material bumps into the jet, it is deflected sideways and falls onto
luminosity as long as the engine is highly active (yellow curve) as the accretion disk. If enough angular momentum is carried by such
indicated by observations. Once the global magnetic field structure blobs, or if this process reoccurs several times in the same direction,
around the BH is disrupted, the efficiency drops and the jet gradually it tilts the disk by virtue of altering its angular momentum, and sub-
shuts off (§6.1). This effect occurs mostly in systems with α < 1 and sequently tilts the jet launching direction as well. The relaunching of
is manifested as a break in the luminosity and η curves. The break the jet on an alternative path may considerably prolong its breakout
is present at t . 1 s when α = 0, and at t & 1 s when α & 1. The from the star and even result in a failed jet.
efficiency remains high for a longer time in systems with higher α Fig. 8 depicts a zoom-in logarithmic density map of the BH vicin-
as demonstrated in Fig. 7 for the cases of α = 2.5 (purple curve) and ity. It is shown that the disk and the jet are tilted by ∼ 40◦ (note that a
α = 1 (yellow curve). non-tilted disk lies on the x̂ − ŷ plane), and in some simulations may
reach up to ∼ 60◦ tilt. If the change in the jet launching direction
is substantial, and the time cycle over which it changes is compa-
rable to the GRB duration, the jet head may leave traces of its tilt
7 JET EVOLUTION
even after breaking out from the star. Such a process could have pro-
After the jet is launched, it interacts with the infalling dense stellar found implications on the expected emission from GRB jets, such as
envelope. The interaction of the jet head with the star shocks the jet a periodicity in the lightcurve over the precession timescales.
MNRAS 000, 1–14 (2021)Collapsar simulations reveal a range of transients 11
7.2 Magnetic dissipation We stress that the evolution of the jet above the collimation nozzle
could be affected by the limit in our simulations of Γ∞ ≤ 25. Under
When a Poynting flux dominated jet is collimated, current driven
this limit, even moderate mixing with cocoon material that pene-
instabilities, most notably kink instability, grow in the jet and dissi-
trates the jet, may reduce the asymptotic Lorentz factor to Γ∞ . 3.
pate magnetic energy (e.g., Levinson & Begelman 2013). Bromberg
Had the jet been launched with Γ∞ > 100, as inferred from observa-
& Tchekhovskoy (2016) studied the propagation of such jets in stel-
tions, it is possible that it would have maintained a higher asymptotic
lar envelopes using 3D RMHD simulations. The jets were launched
Lorentz factor after the mixing.
by the rotation of a perfectly conducting sphere carrying a monopole
magnetic field into a medium with a power-law density profile. They
found that when the jets are collimated, they form narrow nozzles 8.1 Breakout
inside which local kink modes grow and dissipate the jet magnetic
energy to a level of σ ' 1. Above the nozzle, energy can continue In §6.2 we showed that progenitors with steep density profiles α & 2
to dissipate via stochastic turbulent motions, though at a lower rate are in tension with observations, since the jets that are able to break
(Bromberg et al. 2019). Such dissipation was not seen in the RMHD out of them are too powerful and show temporal evolution that is
simulations of Bromberg & Tchekhovskoy (2016), possibly since absent in typical GRB lightcurves. Our simulations also show that
the simulation box was too small and the jets were not sufficiently the combination of a very strong jet and a steep density profile leads
evolved. If magnetic dissipation is sufficient to reduce the magneti- to a fast (t ∼ 0.1 s) acceleration of the jet head to mildly-relativistic
zation to a level of σ 10−2 , the jet continues to evolve as a hydro- velocities. Once the head reaches vh ≈ c, after ∼ 0.5 s, it becomes
dynamic jet (Levinson & Begelman 2013; Gottlieb et al. 2020). In detached from the engine, namely even if the engine stops at that
this regime hydrodynamic instabilities growing on the jet boundary point, the information will fail to reach the head before it breaks
will cause strong mixing between jet and cocoon material and may out of the star. In a population of stars with a distribution of engine
disrupt the jet (Gottlieb et al. 2021a). In the intermediate regime of activity times, the inferred breakout time from observations will be
10−2 . σ . 1 jets are stable to both current-driven and boundary of the order of this critical time (e.g. Bromberg et al. 2012), implying
instabilities (Gottlieb et al. 2020). It is therefore important to care- that tb ∼ 1 s tb,obs . This supports our previous claim that stars with
fully analyze the jet magnetization above the nozzle and distinguish α & 2 cannot be typical progenitors of long GRBs. Note that the
between the various cases. results above were obtained for Φh ≈ Φh,min , and jets with weaker
Our GRMHD simulations are different from those of Bromberg initial magnetic field would fail to produce GRBs.
& Tchekhovskoy (2016) in two important properties. i) we do not In density profiles with α < 2, the jet head typically propagates
assume a magnetic field configuration on the horizon but let it ac- in subrelativistic velocities inside the stellar envelope. At distances
cumulate through accretion. ii) we account for gravity, thus the jet r & 109 cm, the jet material close to the head is weakly-magnetized
propagates into a medium that is free-falling to the center. We find (σ 1), and the free-fall time of the stellar material ahead of the
that qualitatively our results are in agreement with Bromberg & jet is larger than the jet propagation time. We can use the scal-
Tchekhovskoy (2016), as the jet plasma exhibits substantial mag- ing relations obtained for the propagation of hydrodynamic jets in
1
netic dissipation after it becomes collimated and passes through stationary envelopes, vh ∝ (t α−2 L) 5−α , (e.g. Bromberg et al. 2011;
the nozzle. The simulations show that the dissipation continues Harrison et al. 2018) to predict the head velocity at large distances
also above the nozzle reducing the magnetic energy to a level of from the center. We showed that the jet luminosity roughly follows
10−2 . σ . 10−1 , as shown in Fig. 9 (and was also recently found L ∝ t (2−2α)/3 , thus
by Gottlieb & Globus 2021). The reason for the stronger dissipa- α−4
vh ∝ t 3(5−α) , (16)
tion may lie in the fact that the jets propagate to a much larger dis-
tance than previously investigated, thus allowing more time for the i.e. all non-relativistic heads decelerate over time (and not only when
stochastic processes to reduce the magnetic field. Another possibil- α < 2 as in the case of a constant luminosity), with an absolute ve-
ity is insufficient resolution, especially at the nozzle region that leads locity that depends on Bh . Using this expression and the jet velocity
to a high numerical diffusivity and an excess of magnetic energy in our α . 0.5 simulations at early times, we deduce the jet velocity
dissipation. During the late stages of the jet evolution the nozzles at later times7 . In the case of Φh ≈ 10 Φh,min , integration over the
are resolved by ∼ 6 cells in the lateral direction at their narrow- jet velocity yields tb of a few dozen seconds. This result is roughly
est point. While a similar behavior is detected in a convergence test consistent with observations, but requires a different magnetic pro-
where the resolution is doubled, a detailed study of this dissipation file than the core+dipole used in our simulations, that can sustain
is needed and will be preformed in a separate work. The question re- teng 1 s (§6.1). In the progenitor with α ≈ 1 and Φh ≈ Φh,min
mains whether the properties of the weakly-magnetized jet above the (models α1BcLc, α1BcLz) the jet propagates at vh ≈ 0.15 c. Sub-
nozzle (e.g., Lorentz factor and energy) are compatible with those stituting α = 1 in Eq. 16 yields vh ∝ t −1/4 , which results in an
inferred from observations. estimated breakout time of tb ≈ tb,obs ≈ 10 s, consistent with the
breakout time found in our simulations. In the simulation the weak-
ening of the jet after a couple of seconds implies that teng < 10 s
(see yellow curve in Fig. 7), hence in these cases the breakout is of
8 IMPLICATIONS TO BREAKOUT & EMISSION the shocked jet material rather than the jet itself, and may resemble
a low-luminosity GRB, similar to the case when the disk is absent.
The jets ultimately break out of the star to power the luminous GRB
However, a slight modification of Bh or of the magnetic field profile
emission. Based on the jet propagation and properties at the end of
would enable the jet to work for a few dozen seconds, as suggested
the simulations, we derive the conditions needed for the jet to break
by observations.
out within a typical breakout time of tb,obs ∼ 10 s inferred from ob-
Our simulations focus on progenitors with R? = 4 × 1010 cm, and
servations (Bromberg et al. 2012), and discuss the implications for
the γ-ray signal. In this analysis we ignore potential effects of the tilt
on the jet breakout and emission, which are left for a future work. 7 In these simulations tb 10 s, so they do not last until jet breakout.
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