# The Collapse and Three-Dimensional Explosion of Three-Dimensional Massive-
star Supernova Progenitor Models

David Vartanyan1, Matthew S. B. Coleman2, Adam Burrows2
1Department of Physics and Astronomy, University of California, Berkeley, CA
94720, USA
2Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University,
Princeton, NJ 08544, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The explosion outcome and diagnostics of core-collapse supernovae depend
sensitively on the nature of the stellar progenitor, but most studies to date
have focused exclusively on one-dimensional, spherically-symmetric massive
star progenitors. We present some of the first core-collapse supernovae
simulations of three-dimensional massive star supernovae progenitors, a 12.5-
and a 15-M⊙ model, evolved in three-dimensions from collapse to bounce through
explosion with the radiation-hydrodynamic code Fornax. We compare the results
using those starting from three-dimensional progenitors to three-dimensional
simulations of spherically-symmetric, one-dimensional progenitors of the same
mass. We find that the models evolved in three dimensions during the final
stages of massive star evolution are more prone to explosion. The turbulence
arising in these multi-dimensional initial models serve as seed turbulence
that promotes shock revival. Detection of gravitational waves and neutrinos
signals could reveal signatures of pre-bounce turbulence.

###### keywords:

stars - supernovae - general

††pubyear: 2021††pagerange: The Collapse and Three-Dimensional Explosion of
Three-Dimensional Massive-star Supernova Progenitor Models–References

##  1 Introduction

The successful explosion of a core-collapse supernova (CCSN) has posed a
theoretical challenge to the astrophysical community for over half a century
(Colgate & White, 1966; Janka, 2012; Burrows, 2013). The advent of a growing
number of detailed multi-dimensional simulations (Janka et al., 2016; Roberts
et al., 2016; O’Connor & Couch, 2018; Vartanyan et al., 2018, 2019a; Nagakura
et al., 2019a, c; Ott et al., 2018; Summa et al., 2018; Glas et al., 2019a;
Burrows et al., 2018; Burrows et al., 2019; Burrows et al., 2020; Kuroda et
al., 2020; Vartanyan et al., 2021), many of which produce robust explosions,
has recently transformed the simulation landscape (Burrows & Vartanyan, 2021)
with important implications for future gravitational wave and neutrino
observations (Fuller et al., 2015; Seadrow et al., 2018; Morozova et al.,
2018; Radice et al., 2019; Andresen et al., 2019; Arca Sedda et al., 2020;
Srivastava et al., 2019; Pajkos et al., 2019, 2020; Vartanyan et al., 2019b;
Vartanyan & Burrows, 2020; Kuroda et al., 2017; Nagakura et al., 2021a, b).

Despite these remarkable advances, most studies to date are fundamentally
limited in their use of spherically-symmetric progenitors (such as those
derived by MESA, e.g., Paxton et al. 2013, and KEPLER, e.g., Woosley & Heger
2007; Sukhbold et al. 2016; Sukhbold et al. 2018) as initial conditions. The
explosion outcome depends sensitively on the details of the progenitor
structure. An important next step for progress on CCSNe theory is the study of
the role of multi-dimensional progenitors pre-collapse.

The exploration of stellar evolution in multiple dimensions has progressed
through various phases over the last several decades. Early multi-dimensional
studies focused on the sun and solar-mass stars (Pinsonneault et al., 1989;
Sofia et al., 2012), as well as more massive stars (Endal & Sofia, 1978),
often including rotation. These studies, focused on stars no more massive than
10-M⊙, generally evolved the pre-collapse evolution during phases no further
than core carbon-ignition and focused on the validity of the mixing-length
approach in approximating multi-dimensional convection (Chan & Sofia, 1986).

Subsequent studies focused on asphericities in more massive progenitors pre-
collapse (e.g. Arnett et al. 1989; Fryxell et al. 1991; Arnett 1994; Bazan &
Arnett 1994; Asida & Arnett 2000; Meakin & Arnett 2007; Hirschi et al. 2017;
Jones et al. 2017) through two-, and later three-dimensional simulations of
oxygen and silicon shell burning. These models did not generally proceed to
core collapse, but rather focused again on the viability of mixing length
theory in modeling convective overshoot and entrainment to ultimately improve
hydrodynamic modeling in spherically-symmetric one-dimensional simulations
(e.g., Arnett et al. 2015). Massive stars are unique in that their dynamical
timescales are sufficiently short to require no more than a few convective
turnover timescales to generate saturated turbulence, making them well suited
for the current capabilities of three-dimensional simulations.

Some attention was given to the growth of turbulence on collapse (Goldreich et
al., 1997; Lai & Goldreich, 2000; Abdikamalov et al., 2021) and to the shock-
turbulence interaction (Abdikamalov et al., 2016; Radice et al., 2018). These
various explorations evolved in the last decade into efforts focused on the
role of perturbations as seeds for turbulence behind the shock (Couch & Ott,
2013, 2015; Müller & Janka, 2015; Burrows et al., 2018). Initial turbulence
seeds post-shock turbulence, thereby providing a greater stress behind the
shock, which is favorable for shock revival into explosion (Burrows et al.,
1995).

It was not until recently, however, that various groups have evolved in three-
dimensions the convective silicon- and oxygen-burning regions of a massive
star during the final minutes before core collapse (Couch & Ott, 2013; Couch
et al., 2015; Müller, 2016; Yadav et al., 2020; Müller et al., 2019; Yoshida
et al., 2019; Fields & Couch, 2020, 2021; Yoshida et al., 2021b), with several
models including rotation (Yoshida et al., 2021a; McNeill & Müller, 2021), and
with implications for both stellar evolution (McNeill & Müller, 2020; Varma &
Müller, 2021) and explosion outcome (Müller et al., 2017; Müller, 2019; Bollig
et al., 2021). However, detailed multi-dimensional core-collapse simulations
of multi-dimensional progenitors are still lacking. To that end, we here
employ our radiation-hydrodynamics code Fornax to follow the core collapse,
bounce, and explosion of two three-dimensional progenitors. We find that the
three-dimensional pre-collapse progenitors explode more readily than their
spherically-symmetric counterparts, emphasizing the role of preexisting
progenitor turbulence in the early growth of turbulence that promotes shock
revival to explosion.

The paper is organized as follows. In §2, we summarize the simulation setup
and introduce the 3D progenitors studied here. In §3, we discuss the
morphology of the stellar models and the results of our simulations, focusing
on the dependence of explosion outcome and diagnostics on the structure and
dimensionality of the progenitor models. In §4 we discuss the role of
progenitor structure on explosion and comment on both the emergent
gravitational waves and the presence of the lepton emission self-sustained
asymmetry (LESA). We summarize our conclusions in §5.

##  2 Numerical Methods and Computational Setup

Fornax (Skinner et al., 2019) is a multi-dimensional, multi-group radiation
hydrodynamics code constructed to study core-collapse supernovae. It features
an M1 solver (Vaytet et al., 2011) for neutrino transport with detailed
neutrino microphysics and a monopole approximation to general relativity
(Marek et al., 2006).

For all iterations here, the models are evolved out to 20,000 km with a
resolution of 678$\times$128$\times$256 in
($r$,$\mathrm{\theta}$,$\mathrm{\phi}$) using twelve fully-coupled energy
bins. We use the SFHo EOS of Steiner et al. (2013), consistent with most known
laboratory nuclear physics constraints (Tews et al., 2017). Additionally, we
note that Fornax treats three species of neutrinos: electron-neutrinos
($\nu_{e}$), electron anti-neutrinos ($\bar{\nu}_{e}$), and $\mu$,$\tau$
neutrinos and their antiparticles bundled together as heavy-neutrinos,
“$\nu_{\mu}$.” The setup here is otherwise identical to those discussed in
Vartanyan et al. (2019a). We introduce no perturbations of our own into the
models.

We study the 12.5-M⊙ progenitor presented in Müller et al. (2019) and the
15-M⊙ progenitor from Fields & Couch (2020) (labeled there as 3D32kmPert).
Müller et al. (2019) simulate convective burning in the active oxygen shell in
3D for the last few minutes prior to collapse following Müller et al. (2016).
Fields & Couch (2020) simulate the last $\sim$424 seconds before core
collapse, evolving the Si and O burning shells in 3D. To isolate the role of
multi-dimensional effects in the progenitor, we also study the outcomes of
otherwise equivalent, but 1D spherically-symmetric, 12.5-M⊙ and 15-M⊙ initial
models evolved with MESA (provided by B. Müller and C. Fields, respectively;
private correspondence). In total, we perform four 3D simulations,
encompassing two stellar progenitors each evolved until core collapse in both
1D, labeled subsequently, M12.5-1D3D and FC15-1D3D, and in 3D, labeled
M12.5-3D3D and FC15-3D3D, for the Müller et al. (2019) and Fields & Couch
(2020) models, respectively. The 1D3D models are mapped to 3D 10 ms after
core-bounce such that, after core-bounce, all models are evolved in 3D.

Many existing studies of multi-dimensional progenitors use simplified neutrino
physics and radiation transport (e.g., the ray-by-ray or fast-multigroup
approximations to neutrino radiation transport). Our work here has the
advantage of using detailed neutrino microphysics with the robust M1 closure
scheme. We show in Fig. 1 the angle-averaged density profiles of the four
models studied here. The FC15 models show only slight variations between the
1D and 3D progenitor models when density is angle-averaged. In contrast, the
M12.5 3D progenitor has a sharper density dropoff at the Si/O interface than
the 1D progenitor, which we find promotes explosion (see §3). In Fig. 2, we
illustrate the nature of the Si/O interface for the M12.5-3D3D progenitor for
a variety of viewing angles as a function of enclosed mass and radius. We see
a much smaller variation by angle in the pre-collapse FC15-3D3D model around
the Si/O interface. The interface itself is quite narrow $-$ roughly 30 km and
$\sim$0.02 M⊙ wide. With early infall speeds of $\sim$1000 km s-1 and
accretion rates of 1 M⊙ s-1, the corresponding time to accrete across the
interface is $\sim$10-20 ms.

The M12.5-3D3D and FC15-3D3D models differ significantly in the extent of
turbulence, with the former showing no evidence of turbulence below the sharp
Si/O interface. In Fig.3, we illustrate the initial velocity slices for the
M12.5-3D3D model (left) and the FC15-3D3D model (right). In both instances,
the 3D evolution around the oxygen-burning shells is evident. We note that
both models feature unique simulation artifacts: the M12.5 model shows yin-
yang grid stitching at large scales, and the FC15 model shows a large-scale
quadrupolar symmetry. In both cases, the artifacts lie outside one of the main
region of interest, the Si/O interface. The artifacts at large scales are
either not accreted at all, or are accreted so late that they have no
influence on the explosion evolution. The artifacts at small scales are
accreted so rapidly, becoming part of the PNS, that they also have no
noticeable impact on the explosion evolution.

To further describe our results, we find it useful to define some averaging
procedures: for an arbitrary fluid/radiation variable $f$ we define the shell-
average and density-weighted shell-average as

$\displaystyle\left<f\right>_{\Omega}\left(t,r\right)$
$\displaystyle\equiv\int f\left(t,r,\theta,\phi\right)\;{\rm d}\Omega,$ (1)
and $\displaystyle\left<f\right>_{\rho}\left(t,r\right)$
$\displaystyle=\dfrac{\left<\rho
f\right>_{\Omega}}{\left<\rho\right>_{\Omega}},$ (2)

respectively.

##  3 Results

The four CCSNe models were evolved in 3D for roughly half a second to one
second post bounce to determine their early fates. In the early initial pre-
bounce phases we plot in Fig. 4 the behavior of the RMS tangential velocity
profiles versus mass coordinate for both 3D progenitor models, which show
nearly no evolution for the M12.5-3D3D model and significant evolution below
the Si/O interface for the FC15-3D3D model. In Fig. 5, we illustrate the
angle-averaged shock radii (left) and accretion rates at 100 km (right). All
models except for the M12.5-1D3D explode, with shocks reaching runaway growth
within $\sim$300 ms. We find that the 3D progenitor models explode
preferentially $-$ the FC15-3D3D model explodes $\sim$100 ms earlier than its
FC15-1D3D counterpart, whereas the M12.5-3D3D explodes, while the M12.5-1D3D
fails to explode.

In Fig. 6, we illustrate the angle-averaged neutrino luminosities and mean
energies for the four models studied here. The FC15 models have harder spectra
for all species until $\sim$400 ms, at which point the FC15-3D3D model
explodes, accretion is reversed, and the electron neutrino and anti-neutrino
energies decrease relative to those of the non-exploding model FC15-1D3D. The
FC15-1D3D explodes, and the mean energies respond $\sim$100 ms later.
Otherwise, the neutrino luminosity and energy of these two models behave
similarly. Note that the explosion barely affects the heavy ‘$\mu$’ neutrino
energies, which remain harder throughout, suggesting a deeper origin in the
proto-neutron star. We see similar behavior for the neutrino luminosities,
with explosion and accretion reversal truncating the electron-neutrino and
anti-neutrino luminosities, but leaving the ‘heavy’ neutrino luminosities
unaffected. The non-exploding M12.5-1D3D model has the highest electron-
neutrino and anti-neutrino energies at late times due to sustained accretion.
Also interesting is that model M12.5-1D3D shows markedly different neutrino
luminosities from the M12.5-3D3D model in the first 100 ms, even though the
latter has not yet exploded. This is due to progenitor profile differences
that manifest in a higher accretion rate for the 1D model than the 3D model.
Additionally, model M12.5-1D3D sustains a higher ‘heavy’ neutrino luminosity
than the M12.5-3D3D model for the entirety of the simulation, even before the
latter explodes.

In Fig. 7, we show late-time 2D slices of the entropy for the four models
studied here. All the models that do explode do so asymmetrically, with
multiple plumes in various directions. Model M12.5-3D3D maintains a dipolar
outflow at late times near the north-south axis, while models FC15-3D3D and
FC15-1D3D develop unipolar ejecta. We show 3D renderings of all four models at
early and late times in Figs. 8-11. The more massive models FC15-3D3D and
FC15-1D3D explode more asymmetrically than model M12.5-3D3D and with higher
entropies at late times. Some salient metrics for all the models are
summarized in Table 1. We note that the compactness parameter does not prove a
valuable metric of explodability for these models (see also e.g., Burrows et
al. 2020).111Model FC15-3D3D explodes preferentially to, but with a larger
compactness parameter than, model FC15-1D3D. Model M12.5-3D3D has a smaller
compactness parameter, but explodes, whereas model M12.5-1D3D does not.

Table 1: Explosion Properties

Model | Explosion? | Compactness Parameter | Gravitational Overburden [erg]
---|---|---|---
(M⊙) |  3D3D |  1D3D |  3D3D |  1D3D |  3D3D |  1D3D
M12.5 |  ✓✓ |  ✗ |  0.312 |  0.368 |  -2.80$\times$1049 |  -4.31$\times$1049
FC15 |  ✓✓ |  ✓ |  0.424 |  0.414 |  -4.66$\times$1049 |  -8.24$\times$1049

Table of our 3D simulation results: models with a checkmark explode, and
models with an ✗ do not explode. Models with two checkmarks explode first for
a given row, if the the counterpart explodes at all. We also show the
compactness parameter (at 1.75 M⊙) and gravitational binding energy, or an
overburden, exterior to our grids(in erg). The 1D3D models had larger
gravitational overburdens than the 3D3D models. Note that models with a
smaller compactness parameter are not necessarily more explodable, as model
FC15-3D3D has a higher compactness, but explodes earlier than model FC-1D3D.

##  4 Discussion

Here, we investigate the preferential explosion of the 3D progenitors relative
to their 1D progenitor counterparts. In the left panel of Fig. 12, we
illustrate the net explosion energy (in Bethes, or 1051 ergs), including the
gravitational overburden of material exterior to our grid, as a function of
time after bounce for all models. As suggested by the explosion morphology of
Fig. 7, and the discussion in §3, models with unipolar ejecta maintain higher
explosion energies than those with dipolar ejecta (such as M12.5-3D3D, see
also Burrows & Vartanyan 2021), perhaps due to the preferential neutrino
heating into one hemisphere. Note that, within the first half second to second
$-$ the temporal extent of our simulations $-$ only model FC15-3D3D reaches
positive explosion energies. This is not concerning. Rather, simulations
require multiple seconds to reach their final asymptotic explosion energies
(Müller et al., 2017; Burrows et al., 2020; Burrows & Vartanyan, 2021; Bollig
et al., 2021). Indeed, the energy growth rates for the M12.5-3D and FC15-3D3D
models within the first half second are $\sim$0.15 and $\sim$0.3 Bethes per
second, both larger than their 1D3D counterparts by $\sim$0.1 Bethes per
second. The evolution of the M12.5-3D3D progenitor shows a similar diagnostic
binding energy within the first $\sim$500 ms post bounce (the extent of our
simulation) as Müller et al. (2019) (their Fig. 3). Similarly, in both
studies, the shock radius reaches 1000 km approximately 500 ms after core-
bounce (theirs is delayed by $\sim$60 ms as it stalls at $\sim$200 km), with
shock revival corresponding to the accretion of the Si/O interface.
Furthermore, we find that the 3D progenitors typically have smaller
gravitational binding energies exterior to our grid. In the right panel of
Fig. 12, we illustrate the baryonic proto-neutron star (PNS) masses and radii,
with the models reaching final baryonic PNS masses between 1.6 and 1.7 M⊙.

In Fig. 13 we illustrate, for each of the four models studied here, the
maximum and minimum shock radius, the accretion rate, and the location of the
Si/O (or Fe/Si, in models FC15-3D3D and FC15-1D3D) interface as a function of
time after bounce (in seconds).222Historically, this transition has been
associated with a silicon/oxygen boundary (see also Fryer 1999; Ott et al.
2018). However, we found that the density drop corresponding to a
compositional interface, especially if fragmented, could also correspond to
iron/silicon in the FC15-3D3D and FC15-1D3D. We refer to the Si/O interface,
and more broadly, the silicon-group and oxygen-group compositional boundary,
interchangeably (see also Vartanyan et al. 2021). Accretion of a sharp
interface across a stalled shock leads to a sharp drop in ram pressure, which
may rejuvenate the stalled shock into a successful explosion (Fryer, 1999;
Vartanyan et al., 2018; Ott et al., 2018). Indeed, this is the case for all
models except for the FC15-3D3D, which begins to explode before accreting even
the Fe/Si, let alone the Si/O, interface. In both FC15 models, shock revival
precedes accretion of the Si/O interface, whose subsequent accretion further
accelerates shock expansion. Even for the non-exploding model M12.5-1D3D,
accretion of the Si/O interface coincides with a small bump in shock radii,
albeit insufficient for explosion. The tangential kinetic energy reacts to the
accretion of the Si/O interface in a similar manner (see Fig. 14).

In Fig. 15, we depict Mollweide projections of the normalized density
variations (top), accretion rate variations (middle), and shock surface
variations (bottom) for model M12.5-3D3D as a case study of the role of
progenitor profile on explosion outcome. We illustrate the density variations
at the Si/O interface located at $\sim$350 km at 93 ms post bounce, roughly 10
ms before explosion, the accretion rate (at 200 km), and the shock surface
just interior. Regions with high density have a higher accretion rate that
suppresses shock expansion. The Si/O interface regulates the accretion rate
and modulates the geometric development of the shock 150 km interior.
Ultimately, the shock expands in the regions of lower density determined by
the morphology of the matter exterior, here encompassing the Si/O interface.
Notice how even small pockets of higher/lower density manifest as regions of
increased/decreased accretion rate and decreased/increased shock radius
expansion, respectively. These correlations continue to be imprinted on the
shock radius and ram pressure to at least 100 ms post bounce (compare Fig. 15
to the upper-left panel of Fig. 16). From here on out the shock takes the path
of least resistance, signified by the anti-correlation of ram pressure and
shock radius shown in Fig. 16 for the two M12.5 models and in Fig. 17 for the
two FC15 models.

In Fig. 18, we look at the gravitational wave data for our models. In the left
panel we plot the strains for both $+$ and $\times$ polarizations. Unlike in
the 1D3D case, the 3D3D model evinces nonzero strain before the onset of inner
PNS convection $\sim$30 ms post bounce, and even pre-bounce. The nonzero
strain reveals early quadrupolar asymmetries in the 3D3D model present before
bounce. In the right panel, we show the gravitational wave energies for the
three models. Again, models developed in 3D pre-bounce show an earlier onset
of turbulence.

###  4.1 Effects of Turbulence

As we have alluded to earlier in this paper, the disparity in the turbulent
nature of the 1D-to-3D and 3D-to-3D models is significant. Turbulence is
inherently a 3D phenomena, and the 1D-to-3D have zero initial turbulence,
which subsequently plays a significant role in their evolution. While the
importance of turbulence in aiding the successful explosion of a CCSN is not
new (see also Nagakura et al. 2019b on the effects of resolution on
turbulence), here we show how the initial, physically-consistent, turbulence
of the 3D-to-3D models aids their resulting explosions by directly comparing
them to their 1D-to-3D counterparts.

The shell burning occurring just above the Si/O interface during the final
stages of stellar evolution sets the stage for the convective instability to
generate the initial turbulence found in the 3D models we obtained from Müller
et al. (2019) and Fields & Couch (2020). Due to the intrinsic connection
between the Si/O interface and turbulence, the accretion of the Si/O interface
through the shock can significantly modify the evolution of the explosion (or
lack thereof). We find that when the Si/O interface is accreted before
explosion, the turbulence between the mass coordinate of the initial Si/O
interface and the shock radius is significantly enhanced (see Figs. 19 and
20). For the same amount of input energy, turbulent stresses provide more
support than thermal pressure, allowing this increase in turbulence to foster
the explosion (Burrows et al., 1995; Burrows et al., 2018).

We can also track the turbulence directly by examining the turbulent power
spectra of the tangential velocity, $C_{\ell}^{2}\left(v_{\rm tan}\right)$,
defined as

$\displaystyle C_{\ell m}\left(f\right)$ $\displaystyle\equiv\int f\,Y_{\ell
m}\,{\rm d}\Omega$ (3a) $\displaystyle C_{\ell}^{2}\left(f\right)$
$\displaystyle\equiv\sum_{m=-\ell}^{\ell}\left|C_{\ell
m}\left(f\right)\right|^{2},$ (3b)

where $Y_{\ell m}$ are the spherical harmonics. In Fig. 21, we follow the
evolution of this power spectrum near the Si/O interface for all 4 models. The
models with 3D progenitors exhibit turbulence before bounce, whereas the other
models do not develop it until significantly after bounce (over 50 ms).
Furthermore, the M12.5-3D3D initial progenitor model clearly shows disparate
power spectra above and below the Si/O interface, with material below the
interface having extremely flat spectra. Seed perturbations enable a more
rapid growth of turbulence in the models with 3D progenitors, enabling them to
explode more easily.

We also followed the evolution of RMS $v_{\rm tan}$ for several mass
coordinates (the same ones shown in Fig. 21) surrounding and including the
Si/O interface in Fig. 22. For model M12.5-3D3D, the layers at and above the
Si/O interface show significant enhancement of the RMS of $v_{\rm tan}$ after
passing through the shock, compared to the layers below the Si/O interface
which do not experience significant initial turbulence, signifying the
potential importance of the seed turbulence present in the 3D progenitors.
Further solidifying this point is the fact that the M12.5-3D3D model explodes,
while the corresponding 1D-to-3D model fails to do so.

###  4.2 LESA

We comment here briefly on the development of the LESA, the lepton-emission
self-sustained asymmetry, first identified in Tamborra et al. (2014) and seen
in recent 3D studies (O’Connor & Couch 2018; Glas et al. 2019b; Vartanyan et
al. 2019b; Bollig et al. 2021, but see also Kuroda et al. 2020). In Fig. 23,
we illustrate the monopole and dipole components of the lepton asymmetry as a
function of time after bounce at 500 km. All models except for M12.5-3D3D
illustrate the development of the LESA to varying degrees except for
M12.5-3D3D. This low-mass model explodes relatively earlier, reversing
accretion, which may be necessary to power the LESA mechanism. Furthermore,
models FC15-3D3D and FC15-1D3D, which ran for an additional $\sim$250 ms
beyond the M12.5 models, show stronger evidence of LESA in terms of the lepton
asymmetry dipole magnitude than model M12.5-1D3D. This suggests that there may
be a stronger LESA at later times ($\sim$800 ms post bounce), as the lepton
asymmetry monopole magnitude continues to fall.

Figure 1: The angle-averaged initial density (in g cm-3) profiles as a
function of enclosed mass (in M⊙) for the four models studied here. Figure 2:
The pre-collapse Si/O interface density (in g cm-3) as a function of radius
for the stellar interior (in km) for the M12.5 model as a function of various
viewing angles in $\theta$, $\phi$. Note that the interface location is
isotropic (independent of angle) at its inner boundary, i.e. the Si/O
interface is at the same radius for all angles for this 3D progenitor. The
interface has a width of $\sim$30 km, or $\sim$0.02 M⊙. The density drops by a
factor of $\sim$3 across the interface, and varies by $\sim$50% with viewing
angle at the outer boundary.

Figure 3: The initial velocity profiles along the xz-plane for the M12.5 3D
model (left) and the FC15 3D model (right). Mean radial speeds are $\sim$-1000
km s-1 as expected at infall, with turbulent velocities $\sim$500 km s-1. We
identify a weak axial artifact and the remnants of the quadrupolar
perturbations from the Cartesian grid in the FC15 model. The M12.5 model,
which uses a yin-yang grid, does not manifest axial artifacts but shows signs
of the yin-yang grid stitching at large scales. Figure 4: RMS $v_{\rm tan}$
($\sqrt{\left<v_{\rm tan}^{2}\right>_{\rho}}$) profiles at several times
before bounce as indicated in the legends (in seconds). The vertical dotted
black line is the initial mass coordinate of the Si/O interface. For the
Müller et al. 3D-to-3D model (left), we can clearly see the Si/O interface in
$v_{\rm tan}$, while there is no such discernible feature in the FC15-3D3D
model (right). The evolution of $v_{\rm tan}$ before bounce in the M12.5-3D3D
model is minimal, while for the FC15-3D3D model $v_{\rm tan}$ changes
significantly.

Figure 5: Left: Mean shock radius (in km) as a function of time after bounce
(in seconds) for simulations of the 12.5- and 15-M⊙ models studied here for
both the 1D MESA model mapped to 3D, and the 3D progenitors evolved entirely
in 3D. For both models, the fully 3D progenitor is more explodable. For the
12.5-M⊙ set, only the fully 3D model (blue) explodes. The 1D MESA progenitor,
whether mapped to 3D or 2D, both 10 ms after bounce, fails to explode.
Explosion time of the 3D progenitor just after 100 ms corresponds with the
accretion of the Si/O interface. Right: Accretion rate (in M⊙ s-1) as a
function of time after bounce (in seconds).

Figure 6: Neutrino luminosity (in 1051 erg s-1, left) and neutrino average
energy (in MeV, right) as a function of time after bounce (in seconds). We
include an inset for the electron-neutrino luminosity in the top left panel
with a different scale showing the early breakout peak.

Figure 7: Top: The entropy profiles along the xz-plane of the M12.5-3D3D
(left) and M12.5-1D3D (right) models at the last time step of our simulations.
The latter fails to explode (note the different scales). The former explodes
asymmetrically. Bottom: Same as above, but for the FC15-3D3D (left) and the
FC15-1D3D (right) models. Both models here explode. Note that the more massive
FC15 progenitors explosions are focused entirely in one hemisphere, whereas
the less massive M12.5-3D3D model explodes with plumes in both hemispheres.

Figure 8: Entropy visualization at various times for model M12.5-3D3D. The
cyan veil indicates the shock surface. The length scale changes with time.

Figure 9: Same as above, but for model M12.5-1D3D, with times chosen analogous
to the prior model to indicate the model’s time evolution. This model fails to
explode. Note the smaller scale turbulence in the leftmost figure compared
with model M12.5-3D3D, indicating the significance of pre-bounce perturbations
in driving large-scale turbulence.

Figure 10: Same as above, but for model FC15-3D3D.

Figure 11: Same as above, but for model FC15-1D3D. Note that the plume size is
again slight smaller at similar times post bounce compared to FC15-3D3D

Figure 12: Net explosion energy (in Bethes) as a function of time after bounce
(in s) for the models studied here. The energy calculated accounts for the
gravitational overburden exterior to our grid. At the end of our simulations,
the explosion energy is just breaking even, unbinding the star. Note that the
net explosion energy has a growth rate of $\sim$0.3 Bethes per second for the
FC15-M⊙ 3D progenitor, and $\sim$0.15 Bethes per second for the 12.5-M⊙ 3D
progenitor. The explosion energy growth rate for the 3D model is higher than
for the 1D MESA progenitor by $\sim$0.1 Bethes per second, 0.5 seconds after
bounce for the 15-M⊙ progenitor. Right: Baryonic PNS mass (in M⊙, right side)
and PNS radius, defined where the density is 1011 g cm-3 (in km, left side) as
a function of time after bounce (in seconds).

Figure 13: For each of the four models (top: M12.5, bottom: FC15, left: 3D3D,
right: 1D3D), we show maximum and minimum shock radii, the position of the
infalling Si/O (or Fe/Si for the FC15 models) interface (in km), and the
accretion rate (in M⊙ s-1) as a function of time after bounce (in s) for the
first 250 ms post bounce. Accretion of the sharp density interface onto the
stalled shock results in a drop in the accretion ram pressure exterior to it,
which oftimes leads to the rejuvenation of the shock wave. This is the case
for the M12.5 models: the 3D model accretes a sharper, deeper Si/O interface
earlier and revives $\sim$50 ms earlier than the MESA model, which ultimately
fails to explode. However, even the non-exploding M12.5-1D3D model experiences
a modest shock revival shortly after $\sim$150 ms, coincident with accretion
of the Si/O interface. For the FC15-3D3D model, however, the model has already
begun to explode, and the shock surface reaches the Fe/Si interface before the
latter accretes onto it. For the FC15-1D3D model, Fe/Si interface accretion
does correspond with shock revival. For both FC15 models, subsequent accretion
of the Si/O interface (not shown) occurs after shock revival but still
contributes to an accelerating shock front. See §4 for more details. Figure
14: Integrated tangential kinetic energy (for $\rho<10^{11}$ g cm-3) as a
function of time for the four models. This is used as a proxy to measure the
importance of turbulence in the models. The faded vertical lines show when the
Si/O interface mean radial location is equal to the mean shock radius for each
model (with colors and line styles denoted in the legend). In both cases, the
1D-to-3D model has substantially less turbulence. This difference is more
prominent shortly after bounce (see inset) which significantly affects the
production of gravitational waves at these early times (see Fig 18).
Additionally, the turbulence around the Si/O interface can be seen in the
3D-to-3D models before bounce.

Figure 15: Mollweide projection for model M12.5-3D3D of the density (top) at
the Si/O interface at 93 ms postbounce, accretion rate at 200 km (middle), and
shock radius (bottom), the latter two at 98 ms. Note how the morphology of the
density profile around the Si/O interface sets the stage for the geometry of
the accretion and the emergent shock surface (which is strongly anti-
correlated with the accretion rate, as seen here). This is all the more
interesting because of the large separation of the Si/O interface (at
$\sim$350 km) and the shock surface (at $\sim$ 200 km, concurrently). Strong
shock expansion occurs in the direction of lowest accretion ram pressure,
which aligns with regions of lower density.

M12.5-3D3D

M12.5-1D3D

Figure 16: Time series (top to bottom) of ram pressure variations at a fixed
radius with overlain contours of shock radius deviations for the two Müller et
al. models. For the 3D-to-3D model (left) the radius at which the ram pressure
is plotted is 260 km, while for the 1D-to-3D model (right) the radius is
chosen to be 220 km. These radii correspond to the mean shock radius at 200 ms
after bounce for each model. For both models the shock radius and ram pressure
are anti-correlated, indicating that the shock takes the path of least
resistance. Plotting accretion rate instead of ram pressure gives
qualitatively similar results.

FC15-3D3D

FC15-1D3D

Figure 17: Same as Fig. 16, but for the two Fields-Couch models. For the
3D-to-3D model (left) the radius at which the ram pressure is plotted is 300
km, while for the 1D-to-3D model the radius is chosen to be 250 km. As before,
these radii correspond to the mean shock radius at 200 ms after bounce for
each model.

Figure 18: We illustrate the gravitational wave strain (in cm, left) and the
gravitational wave energy (right) as a function of time after bounce (in s)
for the models studied here. Note that the 1D MESA FC15 model (black dashed)
shows a delayed growth of its gravitational wave energy compared with the
fully 3D models, which show early growth $\sim$30 ms post bounce. This is a
manifestation of the earlier onset of turbulence in the fully 3D model than in
the spherically-symmetric MESA progenitor.

Figure 19: Space-time diagram of tangential velocity, $\left<v_{\rm
tan}\right>_{\Omega}(t,r)=\left<\sqrt{v_{\theta}^{2}+v_{\phi}^{2}}\right>_{\Omega}$
(a proxy for turbulence; see also Eqn. 1) for simulations with the 3D
progenitor (top) and 1D progenitor (bottom): Müller et al. models (left) and
Fields-Couch (right). The dashed gray line corresponds to the mass coordinate
of the Si/O interface of the initial progenitor. The shock radius ($R_{\rm
sh}$) is roughly where $v_{\rm tan}$ falls below $\sim 2\times 10^{3}$ km/s
and is noticeable in all the models. In all four models, $R_{\rm sh}$ is
greater than 200 km by 100 ms post bounce. For the 1D-to-3D Müller et al.
model, $R_{\rm sh}$ can be seen to reverse direction at $\sim 300$ms post
bounce, indicating a failed explosion, while all other models successfully
explode. One of the important features of this figure is the low-velocity
(purple) region present in the 1D-to-3D models beneath the shock up to $\sim
0.2$ s after bounce. This feature is absent in the 3D-to-3D models, signifying
the impact of the seed turbulence. This deficit in turbulence (compared to the
3D-to-3D case) is larger in the Müller et al. models and is still significant
at late time. This may explain why the 1D-to-3D Müller et al. model fails to
explode here.

Figure 20: Mass-time diagram of specific tangential kinetic energy
$\frac{1}{2}\left<v_{\rm tan}^{2}\right>_{\rho}(t,r)$ (a proxy for turbulence)
for simulations with the 3D progenitor (top) and 1D progenitor (bottom):
Müller et al. models (left) and Couch (right). The horizontal dashed gray line
corresponds to the mass coordinate of the Si/O interface of the initial
progenitor. All four models have a region of reduced turbulence below the
shock and above the region of PNS convection (which extends up to $\sim
1\,M_{\sun}$ shortly after bounce). For both the Müller et al. and Fields-
Couch cases, it is clear that this deficit in turbulence is much more
significant in the 1D-to-3D models compared to the 3D-to-3D model, as we point
out in Fig. 19 by noting the purple region just below the shock in 1D-to-3D
models at early times. This extra turbulence in the 3D-to-3D models likely
helps foster the explosion in these cases, explaining why these models explode
more easily compared to their 1D-to-3D counterparts.  Figure 21: Power spectra
of the tangential velocity, $C_{\ell}^{2}\left(v_{\rm tan}\right)$ (see Eqn.
3a), as a function of model (top to bottom), time (left to right) and mass bin
(color). For each model we choose seven adjacent mass bins of width $\delta
m=0.025\,M_{\sun}$ with the middle mass bin (gray) centered on the mass
coordinate of the initial Si/O interface, making blue lines below this
interface and yellow/red lines above it. The numeric labels in the legends are
the center of the mass bin in solar masses; these bins are held fixed for each
row/model. The dotted black line is $\propto\ell^{-5/3}$, indicating
Kolmogorov-like turbulence. There are 4 main features of this figure: _1)_ In
the 1D-to-3D models, the initial power spectra are flat for all shown mass
bins, demonstrating a lack of turbulence. With the exception of the blue lines
in the M12.5-1D3D model, this continues to be the case out to at least 50 ms
post bounce. _2)_ Conversely, the 3D-to-3D models have at least some bins that
turn down at large $\ell$ (small angular scales) to a power-law slope
resembling $-5/3$ indicating the existence of turbulence. For the Müller et
al. case, we see that the mass bins below the Si/O interface (blue lines) lack
turbulence while at and above this interface turbulence is present. The
Fields-Couch 3D-to-3D model exhibits turbulence initially at all shown mass
bins (both above and below the Si/O interface). _3)_ At later times, it
appears that turbulence exists in all shown mass bins for all models. _4)_ As
time progresses some mass bins dramatically lose power at all scales,
indicating that they have been accreted onto the PNS. Figure 22: RMS $v_{\rm
tan}$ ($\sqrt{\left<v_{\rm tan}^{2}\right>_{\rho}}$) as a function of time,
using the same mass bins as Fig. 21. As before, the gray line corresponds to
the Si/O interface. Here, the vertical dotted black line can be thought of as
when the Si/O interface is accreted; this line corresponds to the time when
the mean shock radius is equal to the radius of the mass coordinate
corresponding to the initial Si/O interface. For both 3D-to-3D models, we can
see that $v_{\rm tan}>0$ initially, while the 1D-to-3D models do not develop
noticeable tangential velocities until the shock has interacted with a given
mass bin. We also note that for the Müller et al. 3D-to-3D model the initial
region with $v_{\rm tan}>0$ does not extend below the Si/O interface, while
for the analogous Fields-Couch 3D-to-3D model it does. This appears to
significantly affect the evolution of turbulence as the shock migrates through
the model, where the mass bins at and above the Si/O interface in the Müller
et al. 3D-to-3D model receive a significant amplification of $v_{\rm tan}$.
Alternatively, in the Fields-Couch 3D-to-3D model the shock has already become
an explosion and is propagating outwards by the time it meets the Si/O
interface, providing more of a gentle “nudge” to those mass bins. This
illustrates that the Si/O interface plays a more important role in the
evolution of the Müller et al. model.  Figure 23: The monopole and dipole
components of the LESA (at 500 km) as a function of time after bounce (in
seconds). Note the presence of a strong dipole $\sim$30 ms after bounce in the
3D3D models absent in the 1D3D models. All models except M12.5-3D3D develop a
strong LESA within the first $\sim$500 ms post bounce except model M12.5-3D3D.
Note the strong dipole component shortly post bounce in the 3D3D models absent
in the 1D3D models.

##  5 Conclusions

Only in the last five years have three-dimensional simulations fruitfully
modeled the final stages of the stellar evolution of massive stars up to core
collapse. We presented here a total of four 3D core-collapse supernovae
simulations encompassing two stellar models, a 12.5- and a 15-M⊙ progenitor,
each evolved to core collapse in spherically-symmetric 1D and in 3D. We found
that models evolved in 3D to core collapse, which better encapsulates the
turbulent convective environments in the later stages of stellar evolution,
preferentially exploded.

All of our models exploded, save for the model M12.5-1D3D. We found, most
notably for model M12.5-3D3D, that the density profile around the Si/O
interface exterior to the shock surface tracks the infalling accretion rate
prior to shock revival, and this in turn determines in part the geometry of
shock expansion. In other words, the morphology of the Si/O interface, and the
turbulence around it, sets the early explosion morphology. The turbulence
present in the 3D initial progenitor models, around and interior to the Si/O
interface, serve as seed turbulence which amplifies on collapse. This
facilitates the onset of robust turbulence behind the shock, thereby enhancing
shock expansion and promoting explosion. We found that, for model FC15-3D3D,
shock revival precedes accretion of the Si/O interface, in contrast to model
M12.5-3D3D, where accretion of the Si/O interface precipitates shock revival.

Although neither of the exploding models reach large positive explosion
energies within the first $\sim$600 ms simulated here, the 3D3D models
manifest significant explosion energy growth rates of several tenths of a
Bethe per second. We reiterate that multi-second post-bounce 3D simulations of
core-collapse supernovae are essential to witness the late-time behavior of
exploding massive stars, including their asymptotic explosion energy.

In addition, we provided corresponding neutrino and gravitational wave data,
which encode information about the explosion timing and geometry. We identify
the presence of a LESA in all models, but it is weakest in model M12.5-3D3D,
whose early explosion and low-mass may reverse accretion possibly responsible
for maintaining the LESA. We also see that, generally, the LESA manifests more
strongly at later times, after $\sim$500 ms, as the lepton-number asymmetry
monopole declines and the dipole begins to dominate. In particular, we also
note that the 3D3D models have an earlier gravitational wave signal at, and
even before, core bounce. The latter effect is a consequence of turbulence in
3D pre-collapse progenitors, with possible observational ramifications.

## Acknowledgements

The authors acknowledge Bernhard Müller, Carl Fields, and Sean Couch for
providing us with the progenitor models studied here and for valuable
correspondence throughout. We are also grateful to Chris White, David Radice,
and Hiroki Nagakura for helpful discussions. We acknowledge support from the
U.S. Department of Energy Office of Science and the Office of Advanced
Scientific Computing Research via the Scientific Discovery through Advanced
Computing (SciDAC4) program and Grant DE-SC0018297 (subaward 00009650) and
support from the U.S. NSF under Grants AST-1714267 and PHY-1804048 (the latter
via the Max-Planck/Princeton Center (MPPC) for Plasma Physics). A generous
award of computer time was provided by the INCITE program. That research used
resources of the Argonne Leadership Computing Facility, which is a DOE Office
of Science User Facility supported under Contract DE-AC02-06CH11357. We are
also grateful for our computational resources through the Texas Advanced
Computing Center (TACC) at The University of Texas at Austin via Frontera
Large-Scale Community Partnerships under grant SC0018297 as well as the
Leadership Resource Allocation under grant number 1804048. In addition, this
overall research project was part of the Blue Waters sustained-petascale
computing project, which is supported by the National Science Foundation
(awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters
was a joint effort of the University of Illinois at Urbana-Champaign and its
National Center for Supercomputing Applications. This general project was also
part of the “Three-Dimensional Simulations of Core-Collapse Supernovae” PRAC
allocation support by the National Science Foundation (under award
#OAC-1809073). Moreover, we acknowledge access under the local award #TG-
AST170045 to the resource Stampede2 in the Extreme Science and Engineering
Discovery Environment (XSEDE), which is supported by National Science
Foundation grant number ACI-1548562. Finally, the authors employed
computational resources provided by the TIGRESS high performance computer
center at Princeton University, which is jointly supported by the Princeton
Institute for Computational Science and Engineering (PICSciE) and the
Princeton University Office of Information Technology, and acknowledge our
continuing allocation at the National Energy Research Scientific Computing
Center (NERSC), which is supported by the Office of Science of the US
Department of Energy (DOE) under contract DE-AC03-76SF00098.

## Data Availability

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

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