# Comparison of Electron Capture Rates in the N=50 Region using 1D Simulations
of Core-collapse Supernovae

Zac Johnston Department of Physics and Astronomy, Michigan State University,
East Lansing, MI 48824, USA  Joint Institute for Nuclear Astrophysics – Center
for the Evolution of the Elements, Michigan State University, East Lansing, MI
48824, USA<EMAIL_ADDRESS>Sheldon Wasik Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA  Rachel Titus
Department of Physics and Astronomy, Michigan State University, East Lansing,
MI 48824, USA  National Superconducting Cyclotron Laboratory, Michigan State
University, East Lansing, MI 48824, USA  MacKenzie L. Warren Department of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Joint Institute for Nuclear Astrophysics – Center for the Evolution of the
Elements, Michigan State University, East Lansing, MI 48824, USA  Department
of Physics, North Carolina State University, Raleigh, NC 27695, USA  Evan P.
O’Connor The Oskar Klein Centre, Department of Astronomy, Stockholm
University, AlbaNova, SE-106 91 Stockholm, Sweden  Remco Zegers Department of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Joint Institute for Nuclear Astrophysics – Center for the Evolution of the
Elements, Michigan State University, East Lansing, MI 48824, USA  National
Superconducting Cyclotron Laboratory, Michigan State University, East Lansing,
MI 48824, USA  Sean M. Couch Department of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  Joint Institute for Nuclear
Astrophysics – Center for the Evolution of the Elements, Michigan State
University, East Lansing, MI 48824, USA  National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, MI 48824, USA  Department
of Computational Mathematics, Science, and Engineering, Michigan State
University, East Lansing, MI 48824, USA

(Received 18 February 2022; Revised 9 September 2022; Accepted 16 September
2022)

###### Abstract

Recent studies have highlighted the sensitivity of core-collapse supernovae
(CCSNe) models to electron-capture (EC) rates on neutron-rich nuclei near the
$N=50$ closed-shell region. In this work, we perform a large suite of one-
dimensional CCSN simulations for 200 stellar progenitors using recently
updated EC rates in this region. For comparison, we repeat the simulations
using two previous implementations of EC rates: a microphysical library with
parametrized $N=50$ rates (LMP), and an older independent-particle
approximation (IPA). We follow the simulations through shock revival up to
several seconds post-bounce, and show that the EC rates produce a consistent
imprint on CCSN properties, often surpassing the role of the progenitor
itself. Notable impacts include the timescale of core collapse, the electron
fraction and mass of the inner core at bounce, the accretion rate through the
shock, the success or failure of revival, and the properties of the central
compact remnant. We also compare the observable neutrino signal of the
neutronization burst in a DUNE-like detector, and find consistent impacts on
the counts and mean energies. Overall, the updated rates result in properties
that are intermediate between LMP and IPA, and yet slightly more favorable to
explosion than both.

††journal: ApJ††software:  FLASH666https://flash.rochester.edu/site/ (Fryxell
et al., 2000; Dubey et al., 2009), NSCL Weak-rate Library v1.2 (Sullivan,
2015), NuLib (O’Connor, 2015), SNOwGLoBES (Scholberg, 2012), GLoBES (Huber et
al., 2005), Matplotlib777https://matplotlib.org (Hunter, 2007),
NumPy888https://www.numpy.org (Harris et al., 2020),
SciPy999https://www.scipy.org (Virtanen et al., 2020), yt101010https://yt-
project.org (Turk et al., 2011), Pandas111111https://pandas.pydata.org (Pandas
development team et al., 2022), xarray121212https://xarray.pydata.org (Hoyer &
Hamman, 2017; Hoyer et al., 2020), Astropy131313https://www.astropy.org
(Astropy Collaboration et al., 2013, 2018),
flashbang141414https://github.com/zacjohnston/flashbang (Johnston, 2022a),
flash_snowglobes151515https://github.com/zacjohnston/flash_snowglobes
(Johnston, 2022b).

##  1 Introduction

Massive stars ($\gtrsim$8\text{\,}\mathrm{M_{\odot}}$$) are destined to
undergo iron core-collapse, either imploding entirely into a black hole (BH),
or violently ejecting their outer layers and leaving behind a proto-neutron
star (PNS) in a core-collapse supernova (CCSN; see reviews in Janka et al.,
2007; Müller, 2020). Of the myriad physical processes that contribute to these
stellar deaths, the capture of electrons onto protons via the weak interaction
plays a central role.

Electron-capture (EC) regulates the deleptonization of nuclear matter during
collapse, and thus helps to set the initial conditions of the shock at core
bounce (see reviews in Langanke & Martínez-Pinedo, 2003; Langanke et al.,
2021). The uncertainties on electron capture rates can span orders of
magnitude and produce larger variations in core-collapse properties than
changes to the nuclear equation of state (EOS) or stellar progenitor (Sullivan
et al., 2016; Pascal et al., 2020).

It is experimentally and computationally difficult to constrain EC rates under
astrophysical conditions, especially for the heavy, neutron-rich nuclei
relevant to core-collapse. For this reason, CCSN simulations typically rely on
parametrized approximations, particularly those of Bruenn (1985) and Langanke
et al. (2003). Recent decades, however, have seen the continued development of
tabulated rates for larger numbers of nuclei based on shell-model calculations
(e.g., Oda et al., 1994; Langanke & Martínez-Pinedo, 2000; Langanke et al.,
2003; Suzuki et al., 2016).

A systematic study by Sullivan et al. (2016) showed that core-collapse
simulations were most sensitive to changes in the electron capture rates of
neutron-rich nuclei near the $N=50$, $Z=28$ closed-shell region. The drawback
is that most of these rates relied on the parametrization of Langanke et al.
(2003), which is extrapolated from rates on nuclei near the valley of
stability. In a follow-up study focusing on 74 nuclei in the high-sensitivity
region, Titus et al. (2018) showed that these rates are likely overestimated
by up to two orders of magnitude, further emphasizing the need for updated
rates. Using new experimental measurements of the
$\mathrm{{}^{86}Kr}(t,\mathrm{{}^{3}He}+\gamma)$ charge-exchange reaction,
Titus et al. (2019) calculated microphysical rates on 78 nuclei in this
region, confirming that the parametrized rates are overestimated.

In the meantime, Raduta et al. (2017) improved upon the parametrization of
Langanke et al. (2003) for neutron-rich nuclei by accounting for temperature,
electron density, and odd-even effects. Comparing one-dimensional (1D) core-
collapse simulations using this improved parametrization, Pascal et al. (2020)
demonstrated the expected increase in core mass and electron fraction due to
an average decrease in EC rates. Pascal et al. (2020) also performed a rate
sensitivity study, independently verifying the findings of Sullivan et al.
(2016) and Titus et al. (2018) that EC rates in the $N=50$ region remain the
most crucial for CCSNe.

In this paper, we present simulations of CCSNe through to shock
revival/failure for 200 stellar progenitors using the updated $N=50$ rates
from Titus et al. (2019). We run corresponding simulations using the baseline
rate library from Sullivan et al. (2016) with the improved approximation of
Raduta et al. (2017), and a third set using the independent-particle
approximation (IPA) of Bruenn (1985). By comparing the model sets, we
investigate the impact of updated EC rates on core collapse, shock revival,
and neutrino emission across a variety of progenitors.

The paper is structured as follows. In Section 2 we describe our methods,
including the EC rate tables (§ 2.1), the setup of the CCSN simulations (§
2.2), and the calculation of observable neutrino signals (§ 2.3). In Section
3, we compare the simulation results, with a detailed comparison of three
reference progenitors (§ 3.1), the impact across the full progenitor
population (§ 3.2), the compact remnants (§ 3.3), and the predicted neutrino
signal (§ 3.4). In Section 4 we interpret our results and compare them to
previous studies, and give concluding remarks in Section 5.

Figure 1: Post-bounce evolution of shock radius ($r_{\mathrm{sh}}$) and
neutrino heating efficiency in the gain region ($\eta_{\mathrm{heat}}$) for
three reference progenitors (§ 3.1). The 0.01$\times{}$ and 10$\times{}$
models consist of the LMP and LMP+N50 rates systematically scaled by factors
of $0.01$ and $10$ (§ 2.1). Note the different $\eta_{\mathrm{heat}}$ ranges.

##  2 Methods

To explore the impact of EC rates on CCSNe, we run multiple large sets of 1D
simulations using the FLASH code (Fryxell et al., 2000; Dubey et al., 2009).
For initial conditions, we use 200 stellar progenitor models from Sukhbold et
al. (2016) with zero-age main-sequence (ZAMS) masses between
$9120\text{\,}\mathrm{M_{\odot}}$ (§ 2.2).

For each progenitor, we run simulations using three different implementations
of EC rates, which include a independent-particle approximation, microphysical
calculations, and updated experimental rates (§ 2.1). For the
$122040\text{\,}\mathrm{M_{\odot}}$ progenitors, we also run simulations with
the microphysical rates scaled by factors of $0.01$ and $10$. In total, this
results in 612 supernova models evolved to between $15\text{\,}\mathrm{s}$
post-bounce.

###  2.1 EC Rates

The first of our three EC rate sets uses the IPA on a mean nucleus (Fuller et
al., 1982; Bruenn, 1985). These rates were used in the comparable FLASH
simulations of Couch et al. (2020) and Warren et al. (2020, though see § 2.2
for a note on differences). Crucially, IPA assumes that EC completely halts
for nuclei with $N\geq 40$ due to Pauli blocking, thus only permitting
captures on free protons at densities above
$\sim${10}^{10}\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$, where
neutron-rich nuclei dominate the composition (Langanke et al., 2003).

The second set, which we label LMP, uses a library of microphysical rates
compiled by the National Superconducting Cyclotron Laboratory (NSCL) Charge-
Exchange Group
111https://github.com/csullivan/weakrates,222https://groups.nscl.msu.edu/charge_exchange/weakrates.html
(Sullivan et al., 2016; Titus et al., 2018). This library includes rates from
Fuller et al. (1982), Oda et al. (1994), Langanke & Martínez-Pinedo (2000),
Langanke et al. (2003), Pruet et al. (2003) and Suzuki et al. (2016). For
nuclei that are not covered by the above calculations, the single-state
parametrization from Raduta et al. (2017, Model 3) is used, which extends the
approximation of Langanke et al. (2003) to account for temperature, electron
density, and odd-even effects. By unblocking EC on neutron-rich nuclei, these
modern calculations replace the assumption in IPA that deleptonization is
dominated at high densities by EC on free protons (Langanke et al., 2003).

The third set, which we label LMP+N50, is the same as LMP except for updated
microphysical rates in the region around $N=50$, $Z=28$, where core-collapse
is known to be highly sensitive (Sullivan et al., 2016; Titus et al., 2018;
Pascal et al., 2020). These new rates were calculated for 78 nuclei by Titus
et al. (2019) using a quasi-particle random-phase approximation (QRPA) using
constraints from $(t,\mathrm{{}^{3}He}+\gamma)$ charge-exchange experiments.
The LMP set largely relies on the Raduta et al. (2017) parametrization for
these rates, and are up to two orders of magnitude higher than the LMP+N50
set. This is due to the fact that in the LMP+N50 set Pauli-blocking effects
that reduce the rates play an important role as only transitions from the
ground state are considered (Titus et al., 2018, 2019).

Recently, Dzhioev et al. (2020) argued that Pauli unblocking at finite
temperature may actually reduce or eliminate this gap. Indeed, during the
latter stages of our present study, Giraud et al. (2022) reported new finite-
temperature calculations for the $N=50$ region, which resulted in rates around
an order of magnitude higher than LMP+N50 for
$T\lesssim$10\text{\,}\mathrm{G}\mathrm{K}$$ and $\rho
Y_{e}=${10}^{11}\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$. This
improvement brings the rates closer to the original LMP approximation, but
still lower by about a factor of 5. The calculations by Dzhioev et al. (2020)
and Giraud et al. (2022) both indicate that the temperature-dependent effects
significantly increase the EC rates compared to those estimated on the basis
of captures on the ground state only by Titus et al. (2019). Although a future
study will be required to determine the impact of the rates developed in
Giraud et al. (2022), we note that the LMP+N50 rates by Titus et al. (2019)
should be regarded as lower limits, and on the basis of Giraud et al. (2022),
the LMP rates provide a more realistic estimate but are likely still
overestimated.

For the $122040\text{\,}\mathrm{M_{\odot}}$ progenitors, in addition to the
three EC rate sets above, we also systematically scale the rates of LMP and
LMP+N50 following the approach used in Sullivan et al. (2016), whereby the
rates of all nuclei with atomic mass numbers of $A>4$ are scaled by factors of
$0.01$ and $10$ (hereafter labeled 0.01$\times{}$ and 10$\times{}$).

Table 1: Neutrino detection channels used in the SNOwGLoBES analysis for a DUNE-like liquid argon detector Channel | Reaction | Flavor | $(\%)$
---|---|---|---
$\nu_{e}\mathrm{CC}$ | $\nu_{e}+\mathrm{{}^{40}Ar}\rightarrow e^{-}+\mathrm{{}^{40}K}$ | $\nu_{e}$ | $7080\text{\,}$
$\bar{\nu}_{e}\mathrm{CC}$ | $\bar{\nu}_{e}+\mathrm{{}^{40}Ar}\rightarrow e^{+}+\mathrm{{}^{40}Cl}$ | $\bar{\nu}_{e}$ | $\lesssim 1$
NC | $\nu+\mathrm{{}^{40}Ar}\rightarrow\nu+\mathrm{{}^{40}Ar}$ |  $\nu_{e}$, $\bar{\nu}_{e}$, $\nu_{x}$ | $1020\text{\,}$
ES | $\nu+e^{-}\rightarrow\nu+e^{-}$ |  $\nu_{e}$, $\bar{\nu}_{e}$, $\nu_{x}$ | $\sim 7$

###  2.2 Numerical Methods

To simulate the collapse and explosion of massive stars, we use the FLASH
hydrodynamics code (Fryxell et al., 2000; Dubey et al., 2009) with the
Supernova Turbulence in Reduced-dimensionality (STIR) framework (Couch et al.,
2020; Warren et al., 2020).

STIR enhances the explodability of 1D CCSN models by using time-dependent
mixing-length theory (MLT) to approximate convective turbulence in 1D. We use
a mixing length parameter of $\alpha_{\Lambda}=1.25$, chosen to reproduce the
convective velocities of 3D simulations, which is multiplied by the pressure
scale height to obtain the mixing length (for details, see Couch et al.,
2020).

We use a recently implemented hydrodynamics solver (Couch et al., 2020), which
uses a fifth-order finite-volume weighted essentially non-oscillatory (WENO)
spatial discretization, and a method-of-lines Runge-Kutta time integration.

For neutrino transport, we use the “M1” scheme, an explicit two-moment method
with an analytic closure (described in O’Connor & Couch, 2018), with three
neutrino species ($\nu_{e}$, $\bar{\nu}_{e}$, and
$\nu_{x}{}=\\{\nu_{\mu},\,\nu_{\tau},\,\bar{\nu}_{\mu},\,\bar{\nu}_{\tau}\\}$)
and 18 logarithmically spaced energy groups between
$1300\text{\,}\mathrm{MeV}$.

We generate neutrino opacity tables using the open source neutrino interaction
library NuLib333https://github.com/evanoconnor/nulib (O’Connor, 2015). The
interaction rates largely follow Bruenn (1985) and Burrows et al. (2006), with
corrections for weak magnetism and nucleon recoil from Horowitz (2002).
Separate tables are calculated using the neutrino emissivities derived from
each EC rate set described in Section 2.1. We note that our tables do not
include the many-body effects and virial corrections to neutrino-nucleon
scattering from Horowitz et al. (2017). These corrections aid explodability by
enhancing neutrino heating in the gain region (O’Connor et al., 2017), and
thus our simulations result in fewer explosions than the corresponding models
in Couch et al. (2020) and Warren et al. (2020). Nevertheless, this does not
impede our goal of a comparison study between the EC rates.

We use the SFHo EOS from Steiner et al. (2013), and assume nuclear statistical
equilibrium (NSE) abundances everywhere in the domain. For the IPA EC rates,
the average nucleus from the NSE distribution is used. Self-gravity is
included using an approximate general-relativistic effective potential (Marek
et al., 2006; O’Connor & Couch, 2018).

For initial conditions, we use 200 stellar progenitor models from Sukhbold et
al. (2016), the same set used with FLASH+STIR in Couch et al. (2020) and
Warren et al. (2020). These progenitors are spherically symmetric, solar-
metallicity, nonrotating, and nonmagnetic, with ZAMS masses ranging from
$9120\text{\,}\mathrm{M_{\odot}}$. The set spans core compactness values of
$0\lesssim\xi_{2.5}\lesssim 0.54$ (as defined in O’Connor & Ott, 2011) and
iron core masses of $1.29\lesssim
M_{\mathrm{Fe}}\lesssim$1.84\text{\,}\mathrm{M_{\odot}}$$.

The simulation domain extends from the center of the star to
$r=$15\,000\text{\,}\mathrm{km}$$. The domain is divided into 15 adaptive mesh
refinement blocks, each containing 16 zones. We allow up to nine levels of
mesh refinement, resulting in a zone resolution of $62.5\text{\,}\mathrm{km}$
at the coarsest level and $0.244\text{\,}\mathrm{km}$ at the finest level. The
adaptive mesh refinement results in a total of roughly $1000$ zones.

###  2.3 Neutrino Observables

Following the approach used in Warren et al. (2020), we calculate simulated
observations of the neutrino burst at core bounce using
SNOwGLoBES444https://github.com/SNOwGLoBES/snowglobes (Scholberg, 2012), which
uses the GLoBES555www.mpi-hd.mpg.de/personalhomes/globes (Huber et al., 2005)
framework to predict event rates for a given detector material.

As input for SNOwGLoBES, we calculate from our simulations the neutrino flux
at Earth assuming a CCSN distance of
$10\text{\,}\mathrm{k}\mathrm{p}\mathrm{c}$ and a pinched neutrino spectrum
with a Fermi-Dirac parametrization (Keil et al., 2003). We include adiabatic
neutrino flavor conversions from Mikheyev–Smirnov–Wolfenstein (MSW) matter
effects (Dighe & Smirnov, 2000). For each model, we apply three separate cases
of flavor mixing: no flavor mixing, normal neutrino mass ordering, and
inverted mass ordering (Appendix A).

We calculate detection events for a $40\text{\,}\mathrm{k}\mathrm{t}$ liquid
argon detector, representing the under-construction Deep Underground Neutrino
Experiment (DUNE; Abi et al., 2021) capable of detecting large numbers of
$\nu_{e}$ from nearby CCSNe (Kato et al., 2017). Table 1 summarizes the
different interaction channels: charged-current (CC) reactions on
$\mathrm{{}^{40}Ar}$ by $\nu_{e}$ and $\bar{\nu}_{e}$; neutral-current (NC)
reactions on $\mathrm{{}^{40}Ar}$ for all flavors; and electron scattering
(ES) for all flavors. The $\nu_{e}\mathrm{CC}$ reaction channel accounts for
approximately $7080\text{\,}\mathrm{\char 37\relax}$ of the total counts in
our models.

We capture the neutronization burst by integrating events over
$100\text{\,}\mathrm{m}\mathrm{s}$ centered on the bounce, using
$5\text{\,}\mathrm{m}\mathrm{s}$ time bins and $0.2\text{\,}\mathrm{MeV}$
energy bins. For each model and flavor-mixing case, we thus obtain the total
neutrino counts and the mean detected neutrino energy, ${\langle E\rangle}$,
summed over all detection channels.

Figure 2: Radial matter profiles at core bounce versus enclosed mass for the
$20\text{\,}\mathrm{M_{\odot}}$ progenitor (left), and all 200 progenitors
between $9120\text{\,}\mathrm{M_{\odot}}$ (right). From top to bottom:
electron fraction ($Y_{e}$); specific entropy ($S$); density ($\rho$); and
radial velocity ($v_{r}$). Differences between the EC rate are typically
larger than differences between the progenitors. For the
$20\text{\,}\mathrm{M_{\odot}}$ progenitor, the IPA and 0.01$\times{}$ models
fail to explode, whereas the LMP, LMP+N50, and 10$\times{}$ models
successfully explode (Fig. 1).  Figure 3: Lepton fraction ($Y_{l}$) versus
density ($\rho$) at core bounce for the $20\text{\,}\mathrm{M_{\odot}}$
progenitor. Of the baseline EC rate sets, IPA has the weakest deleptonization
at densities
$\gtrsim$1\text{\times}{10}^{12}\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$,
but the strongest at lower densities.

##  3 Results

Our collection of 612 simulations can be sorted into three groups based on the
explosion outcomes for each progenitor. Firstly, there are those models that,
for a given progenitor, fail to explode for all three EC rate sets. Secondly,
there are those with mixed explosion outcomes between the rate sets. And
thirdly, there are those that successfully explode for all three sets.

Of the 200 progenitors, 126 fail for all three rate sets, 29 have mixed
explosion outcomes, and 44 explode for all sets. The
$10.25\text{\,}\mathrm{M_{\odot}}$ progenitor simulations experience numerical
crashes mid-shock revival, and are excluded from discussions of explosion
outcome. Of the mixed-outcome group, the IPA models always fail, whereas LMP
and LMP+N50 both explode in 26 cases, and LMP+N50 is the only explosion for
the remaining three cases ($2227.433\text{\,}\mathrm{M_{\odot}}$). In summary,
there are 44 successful explosions for IPA, 70 for LMP, and 73 for LMP+N50.

The data presented here, and the codes used to analyze it, are publicly
available (Appendix B).

###  3.1 Reference Progenitors

We here present detailed simulation comparisons for the
$122040\text{\,}\mathrm{M_{\odot}}$ progenitors. These progenitors are
representative of the three groups of explosion outcomes, respectively: all EC
rate sets fail to explode; mixed outcomes; and all successfully explode.

The evolution of the shock radius, $r_{\mathrm{sh}}$, and the neutrino heating
efficiency, $\eta_{\mathrm{heat}}$, are shown in Figure 1. Here,
$\eta_{\mathrm{heat}}$ is the fraction of the total $\nu_{e}$ and
$\bar{\nu}_{e}$ luminosity absorbed in the gain region, which we estimate
following O’Connor & Ott (2011). All $12\text{\,}\mathrm{M_{\odot}}$ models
fail to explode and all $40\text{\,}\mathrm{M_{\odot}}$ models successfully
explode. For the $20\text{\,}\mathrm{M_{\odot}}$ models, the LMP, LMP+N50, and
both 10$\times{}$ models explode, whereas the IPA and both 0.01$\times{}$
models fail.

A consistent hierarchy of $r_{\mathrm{sh}}$ evolution is seen across the
reference progenitors. Overall, the IPA models reach the smallest
$r_{\mathrm{sh}}$, experience the earliest shock recession
($1220\text{\,}\mathrm{M_{\odot}}$), and the latest shock revival
($40\text{\,}\mathrm{M_{\odot}}$), followed closely by the 0.01$\times{}$
models. The LMP and LMP+N50 models reach larger $r_{\mathrm{sh}}$ before
recession ($12\text{\,}\mathrm{M_{\odot}}$) and undergo earlier shock revival
($2040\text{\,}\mathrm{M_{\odot}}$). Finally, the 10$\times{}$ models reach
the largest $r_{\mathrm{sh}}$ before recession
($12\text{\,}\mathrm{M_{\odot}}$) and the earliest shock revivals
($2040\text{\,}\mathrm{M_{\odot}}$).

For the $20\text{\,}\mathrm{M_{\odot}}$ progenitor, LMP+50 appears to require
a smaller heating efficiency for shock revival than LMP, suggesting more
favorable explosion conditions. The 10$\times{}$ models experience a surge in
$\eta_{\mathrm{heat}}$ around $250\text{\,}\mathrm{m}\mathrm{s}$, which
appears to contribute to an early shock runaway. In contrast, IPA does not
reach sufficient $\eta_{\mathrm{heat}}$ before its shock contracts, shrinking
the available gain region for neutrino interactions.

The matter profiles at core bounce are shown in Figure 2 for the
$20\text{\,}\mathrm{M_{\odot}}$ progenitor (left) and the full set of 200
progenitors (right). We define bounce as the moment when the peak entropy in
the core reaches $3\,k_{\mathrm{B}}\,\mathrm{baryon^{-1}}$. We also define the
inner core mass at bounce, $M_{\mathrm{core}}$, as the mass enclosed within
this point (also known as the homologous core mass).

As with $r_{\mathrm{sh}}$, the models maintain a consistent hierarchy, even
across the entire population of progenitors. The IPA models have the largest
inner core mass and electron fraction, entropy, density, and infall velocity.
This trend is followed, in order, by the 0.01$\times{}$, LMP+N50, LMP, and
finally 10$\times{}$ models. This ordering is reversed for the $Y_{e}$ outside
the shock, where IPA is the lowest and 10$\times{}$ the largest.

In Figure 3, the lepton fraction is shown versus density at bounce for the
$20\text{\,}\mathrm{M_{\odot}}$ progenitor. Above the neutrino-trapping
densities of
$\sim$1\text{\times}{10}^{12}\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$,
the 0.01$\times{}$ models have the largest $Y_{l}$, followed by IPA, LMP+N50,
LMP, and 10$\times{}$.

Figure 4: Core bounce properties versus progenitor iron core mass,
$M_{\mathrm{Fe}}$, for all simulations (§ 3.2). From top to bottom: electron
fraction at bounce in the $M=$0.1\text{\,}\mathrm{M_{\odot}}$$ mass shell
($Y_{e}$); inner core mass at bounce ($M_{\mathrm{core}}$); gravitational
potential at the shock at bounce ($V_{b}$); and convergence time of the shock
with the $\nu_{e}$ sphere, relative to bounce ($t_{\nu_{e}}-t_{b}{}$). The
0.01$\times{}$ and 10$\times{}$ rate-scaled models are marked by downward and
upward-pointing triangles, respectively (appearing from left to right:
$122040\text{\,}\mathrm{M_{\odot}}$). In most cases, the differences between
EC rates are larger than the dependence on stellar progenitor.

Figure 5: Fractional and absolute differences of LMP+N50 and IPA models
relative to LMP, versus progenitor iron core mass

(§ 3.2). The quantities compared, from top to bottom: time to bounce from
start of simulation ($t_{b}$); accretion rate through
$r=$500\text{\,}\mathrm{k}\mathrm{m}$$ at bounce ($\dot{M}_{b}$); and maximum
shock radius reached for failed explosion models
($\mathrm{max}(r_{\mathrm{sh}})$). Note the grid resolution of the simulation
limits the precision of $r_{\mathrm{sh}}$ here to
$\approx$0.5\text{\,}\mathrm{k}\mathrm{m}$$.

Figure 6: Absolute difference of compact remnant properties relative to LMP (§
3.3), versus progenitor iron core mass. Top: proto-neutron star mass at the
end of the simulation ($M_{\mathrm{PNS}}$) for exploding models. Bottom: time
from bounce to BH formation ($t_{\mathrm{BH}}$).

###  3.2 Population Comparisons

A selection of properties at core bounce for all 200 progenitors are plotted
versus the progenitor iron core mass, $M_{\mathrm{Fe}}$, in Figure 4. These
quantities in particular demonstrate large differences between the EC rates
compared to differences between the progenitors.

The electron fraction of the inner core at bounce, $Y_{e}$, is taken at the
$M=$0.1\text{\,}\mathrm{M_{\odot}}$$ enclosed mass coordinate (see also Fig.
2). The IPA models have the largest $Y_{e}$ (i.e., weakest deleptonization),
followed by LMP+N50 and LMP.

The extent of deleptonization translates directly into the inner core mass at
bounce, $M_{\mathrm{core}}$. The IPA rates produce systematically larger
$M_{\mathrm{core}}$ than LMP by around $0.08\text{\,}\mathrm{M_{\odot}}$
($\approx$15\text{\,}\mathrm{\char 37\relax}$$), whereas LMP+N50 are around
$0.03\text{\,}\mathrm{M_{\odot}}$ ($\approx$5\text{\,}\mathrm{\char
37\relax}$$) larger.

The density profile at bounce (Fig. 2) determines the gravitational potential
at the shock, $V_{b}$. Following previous trends, IPA results in a potential
around $15\text{\,}\mathrm{\char 37\relax}$ deeper than LMP, compared to
LMP+N50, which is consistently $\approx$5\text{\,}\mathrm{\char 37\relax}$$
deeper.

Also shown is $t_{\nu_{e}}$, the time when the shock crosses the $\nu_{e}$
sphere immediately following the bounce. IPA reaches $t_{\nu_{e}}$
consistently $\approx$1.5\text{\,}\mathrm{m}\mathrm{s}$$ earlier than LMP,
while LMP+N50 is $\approx$0.5\text{\,}\mathrm{m}\mathrm{s}$$ earlier. This
relatively small but persistent difference impacts the neutrino signal of the
deleptonization burst (§ 3.4).

Quantities that have much larger variation between progenitors than between
the EC rates are illustrated in Figure 5. For clarity, we emphasize the
changes due to EC rates by plotting the fractional or absolute difference
relative to LMP for each progenitor. For example, the difference of a given
quantity $X$ corresponds to $\Delta X=X-X_{\mathrm{LMP}}$. For quantities
where the absolute value depends somewhat arbitrarily on the initial
conditions, we instead compare the fractional difference, i.e., $\Delta
X/X=(X-X_{\mathrm{LMP}})/X_{\mathrm{LMP}}$.

The time of core bounce from the start of the simulation, $t_{b}$, illustrates
the speed of collapse from a common starting point. IPA reaches core bounce
between $2.58.5\text{\,}\mathrm{\char 37\relax}$ earlier than LMP
(corresponding to approximately $1017\text{\,}\mathrm{m}\mathrm{s}$), whereas
LMP+N50 reaches bounce $\lesssim$1.5\text{\,}\mathrm{\char 37\relax}$$ later
than LMP (corresponding to $\lesssim$2\text{\,}\mathrm{m}\mathrm{s}$$). In
both cases, the difference is the smallest for larger progenitor iron core
masses.

The mass accretion rate through $r=$500\text{\,}\mathrm{k}\mathrm{m}$$ at
bounce, $\dot{M}_{b}$, further illustrates the strength of collapse. Overall,
IPA reaches accretion rates around
$0.1\text{\,}\mathrm{M_{\odot}}\,\mathrm{s}^{-1}$ larger than LMP, whereas
LMP+N50 is $\approx$0.1\text{\,}\mathrm{M_{\odot}}\,\mathrm{s}^{-1}$$ smaller.

The maximum shock radius reached for failed explosions,
$\mathrm{max}(r_{\mathrm{sh}})$, further supports the differences in shock
evolution seen in the reference progenitors (Fig. 1). IPA reaches the smallest
$r_{\mathrm{sh}}$, generally around $520\text{\,}\mathrm{k}\mathrm{m}$ smaller
than LMP, whereas LMP+N50 is around $05\text{\,}\mathrm{k}\mathrm{m}$ larger.

###  3.3 Compact Remnants

The compact remnant properties are compared in Figure 6 as the absolute
difference relative to LMP, as used in Section 3.2.

The PNS mass at the end of the simulation, $M_{\mathrm{PNS}}$, is compared for
exploding models. Here, we define $M_{\mathrm{PNS}}$ as the baryonic mass
contained in the region above a density of
$1\text{\times}{10}^{12}\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$.

IPA tends to produce a heavier PNS, with progenitors of
$M_{\mathrm{Fe}}\gtrsim$1.45\text{\,}\mathrm{M_{\odot}}$$ around
$0.020.1\text{\,}\mathrm{M_{\odot}}$ larger than LMP. On the other hand,
LMP+N50 tends to produce a similar or slightly lighter PNS, at most up to
$0.010.02\text{\,}\mathrm{M_{\odot}}$ lighter than LMP. All of the exploding
models for $M_{\mathrm{Fe}}\lesssim$1.45\text{\,}\mathrm{M_{\odot}}$$ are
within $0.02\text{\,}\mathrm{M_{\odot}}$ of LMP.

The post-bounce time to BH formation, $t_{\mathrm{BH}}$, is compared for the
subset of models that reach PNS collapse. Not all failed-explosion models
reach PNS collapse within the time simulated (between $15\text{\,}\mathrm{s}$
post-bounce). For the handful of models that do allow comparison with LMP, IPA
reaches BH formation around $2080\text{\,}\mathrm{m}\mathrm{s}$ earlier,
whereas LMP+N50 is only around $27\text{\,}\mathrm{m}\mathrm{s}$ earlier.

Assuming the entire star is accreted, there would be no difference in the
final BH mass between the EC rates.

Figure 7: Neutrino emission at $r=$500\text{\,}\mathrm{k}\mathrm{m}$$ for the
$20\text{\,}\mathrm{M_{\odot}}$ progenitor. Top row: neutrino luminosity,
$L_{\nu}$. Bottom row: mean neutrino energy, ${\langle E_{\nu}\rangle}$. Left:
electron-neutrino ($\nu_{e}$) emission. Right: electron antineutrino
($\bar{\nu}_{e}$) and heavy-lepton neutrino ($\nu_{x}$) emission. Note the
different time and $L_{\nu}$ ranges.

Figure 8: Neutrino burst signal in a DUNE-like liquid argon detector for all
200 progenitors, versus progenitor iron core mass. Top row: total neutrino
counts. Bottom row: mean detected neutrino energy. From left to right are the
adiabatic flavor mixing implementations: no flavor mixing, normal mass
ordering, and inverted mass ordering. The signal is integrated over
$100\text{\,}\mathrm{m}\mathrm{s}$ centered on the bounce (§ 2.3), across all
detection channels (Table 1), and assuming a distance of
$10\text{\,}\mathrm{k}\mathrm{p}\mathrm{c}$. The error bars show typical
$1\sigma$ uncertainties due to Poisson counting statistics alone. Note the
different $y$-axis ranges.

###  3.4 Neutrino Signal

The neutrino emission at $r=$500\text{\,}\mathrm{k}\mathrm{m}$$ is shown in
Figure 8 for the $20\text{\,}\mathrm{M_{\odot}}$ progenitor. The three
baseline sets reach similar peak electron-neutrino luminosities of
$L_{\nu_{e}}\approx$5\text{\times}{10}^{53}\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}$$.
The 0.01$\times{}$ luminosities peak roughly $10\text{\,}\mathrm{\char
37\relax}$ higher, whereas the 10$\times{}$ peak roughly
$10\text{\,}\mathrm{\char 37\relax}$ lower. Additionally, the IPA and
0.01$\times{}$ models peak slightly earlier than the LMP, LMP+N50, and
10$\times{}$ models, which have $\nu_{e}$ emission spread out to later times.
The mean neutrino energies, ${\langle E_{\nu}\rangle}$, follow a similar
pattern. The $\bar{\nu}_{e}$ and $\nu_{x}$ emission is largely reversed, with
IPA and 0.01$\times{}$ generally having the largest luminosities and mean
energies, followed by LMP+N50, LMP, and 10$\times{}$.

The predicted neutrino burst signal in a DUNE-like detector is shown for all
612 models across 200 progenitors in Figure 8. For all three MSW flavor-mixing
cases, there are consistent differences in the detected neutrinos between the
EC rate sets.

Overall, LMP and LMP+N50 produce similar total neutrino counts and mean
detected energies, ${\langle E\rangle}$, with larger differences for IPA,
particularly in ${\langle E\rangle}$. When no flavor mixing is assumed, IPA
results in the lowest counts and ${\langle E\rangle}$. The effect is reversed
when flavor mixing is included for both normal and inverted neutrino mass
ordering. Overall, the inclusion of flavor mixing results in fewer counts and
larger ${\langle E\rangle}$.

##  4 Discussion

The choice of EC rates has a clear impact on our simulations of CCSNe. Perhaps
the starkest difference in outcome is whether the models undergo successful
shock revival and explosion. This variation in outcome can largely be traced
to the pre-bounce core-collapse phase, where the EC rates control
deleptonization and set the conditions at the core bounce for the subsequent
shock evolution. The EC rates also influence the formation of the compact
remnant and the observable neutrino signals.

###  4.1 Collapse and Bounce

Due to the lack in IPA of forbidden transitions and thermal unblocking
effects, the rates are limited to EC on free protons when the average nucleus
has a neutron number of $N\geq 40$ (Hix et al., 2003). Because protons are
less abundant than neutron-rich nuclei at high densities, the total number of
ECs are suppressed, resulting in a larger $Y_{e}$ in the inner core at bounce
compared to LMP and LMP+N50, which do allow EC for $N\geq 40$. On the other
hand, ECs are actually enhanced in IPA at low densities below the $N=40$
threshold because the rates are overestimated compared to the LMP-based tables
(Lentz et al., 2012).

The combined effect is a larger $Y_{e}$ and $Y_{l}$ for IPA in the inner core
region
($\rho\gtrsim$1\text{\times}{10}^{12}\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$)
and lower values in the outer core compared to LMP and LMP+N50 (Fig. 2 and 3).
This enhanced deleptonization of matter passing through lower densities
accelerates the collapse to core bounce, leading to stronger accretion rates,
larger infall velocities, larger inner core mass, deeper gravitational
potential, and higher densities (Fig. 4 and 5).

These bounce profile differences between the IPA and LMP-based rates are well-
established in the literature, noted by Langanke et al. (2003) and Hix et al.
(2003), and reproduced in subsequent studies (e.g., Lentz et al., 2012;
Sullivan et al., 2016; Richers et al., 2017; Pascal et al., 2020).

###  4.2 Shock Evolution and Explosion Outcome

Initially, owing to the larger inner core mass, the IPA shock has more kinetic
energy and less overlying material to pass through than LMP and LMP+N50.
Competing with these favorable conditions, however, are the higher densities
(and thus deeper gravitational potential), faster infall of the overlying
material (Fig. 2 and 4), and stronger neutrino cooling. Despite rapid
expansion at early times in IPA, the shock is soon overwhelmed by accretion
and neutrino cooling, leading to earlier stalling at smaller radii (Fig. 5).
Compounded by the smaller gain region now available for neutrino heating, the
result is an earlier shock recession or a delayed explosion (Fig. 1). By
contrast, LMP+N50 tends to have more favorable conditions for a successful
explosion, with the slowest collapse to bounce and smallest accretion rates
(Fig. 5).

The impact of the EC rates on shock evolution is borne out by the incidence of
successful explosions. Of the 73 progenitors with at least one explosion among
the EC rate sets, LMP+N50 explodes in all 73 and LMP explodes in 70. In
contrast, IPA explodes in only 43 of these cases, and for no progenitor is IPA
the sole explosion.

###  4.3 Compact Remnants

The EC rates also impact the formation of the compact remnant (§ 3.3). In
successful explosions, the PNS mass, $M_{\mathrm{PNS}}$, is effectively
determined by the total mass accreted through the shock before shock revival
unbinds the remaining material. The two deciding factors are thus the
accretion rate and the elapsed time before shock runaway. IPA typically
experiences both higher accretion rates (Fig. 5) and later shock revival (Fig.
1), resulting in the largest $M_{\mathrm{PNS}}$ (Fig. 6). The converse effects
on LMP+N50 result in smaller $M_{\mathrm{PNS}}$.

In the case of failed explosions, the conditions at core bounce influence the
evolution and eventual collapse of the PNS. The larger core $Y_{e}$ and
$M_{\mathrm{core}}$ in IPA results in a stronger initial shock, producing a
larger PNS radius and stronger $\nu_{x}$ radiation (Fig. 8). The subsequent
cooling of the PNS results in IPA reaching collapse approximately
$2080\text{\,}\mathrm{m}\mathrm{s}$ earlier than LMP (Fig. 6). LMP+N50 has
only marginally more efficient PNS cooling than LMP, and collapses at most a
few milliseconds earlier. A change in the BH formation time would alter the
shutoff of multimessenger signals from neutrinos and gravitational waves.

###  4.4 Neutrino Emission

The effects of EC on deleptonization and shock formation also manifest in the
neutrino emission around bounce (§ 3.4). For IPA, a larger $M_{\mathrm{core}}$
at bounce and stronger initial shock, combined with smaller neutrino spheres
due to lowered opacities, leads to a faster convergence of the shock with the
$\nu_{e}$ sphere, producing an earlier peak in $L_{\nu_{e}}$ (Fig. 8). The
mean time between core bounce and the shock reaching the $\nu_{e}$ sphere was
$1.8\pm$0.1\text{\,}\mathrm{m}\mathrm{s}$$ for IPA,
$3.3\pm$0.1\text{\,}\mathrm{m}\mathrm{s}$$ for LMP, and
$2.8\pm$0.1\text{\,}\mathrm{m}\mathrm{s}$$ for LMP+N50 ($1\sigma$ standard
deviations; Fig. 4). These differences in $L_{\nu_{e}}$ have been noted in
previous EC rate studies (e.g., Hix et al., 2003; Lentz et al., 2012; Sullivan
et al., 2016; Pascal et al., 2020).

For LMP and LMP+N50, the extended emission of $\nu_{e}$ at larger
$L_{\nu_{e}}$ and ${\langle E_{\nu_{e}}\rangle}$ results in higher detected
neutrino counts and energies in a DUNE-like liquid argon detector when no
flavor mixing is assumed (Fig. 8).

When MSW flavor mixing is included with normal neutrino mass ordering,
approximately $98\text{\,}\mathrm{\char 37\relax}$ of the emitted $\nu_{e}$
are converted to $\nu_{x}$, and vice versa. This reduces the number of
$\nu_{e}$ available for detection in the dominant $\nu_{e}\mathrm{CC}$ channel
(Table 1), resulting in fewer total counts. There is also a shift to larger
${\langle E\rangle}$ because most of the $\nu_{e}$ that are now detected
originate as high-energy $\nu_{x}$. Because IPA has larger emitted
$L_{\nu_{x}}$ and ${\langle E_{\nu_{x}}\rangle}$, it now has the highest
counts and ${\langle E\rangle}$.

The inverted mass ordering case is somewhat intermediate, with only
$70\text{\,}\mathrm{\char 37\relax}$ of the emitted $\nu_{e}$ converted to
$\nu_{x}$, and vice versa. These favorable survival probabilities result in
overall counts and ${\langle E\rangle}$ that are between the previous two
cases. This appears to roughly coincide with the crossover point, where all
three EC rates produce very similar counts.

The large error bars from Poisson statistics alone suggest that these
differences would be difficult to detect under these narrow assumptions,
especially for ${\langle E\rangle}$, even if the progenitor was known.
Additionally, there are degeneracies between the EC rate and the impacts of
the progenitor star, neutrino mass hierarchy, and potentially the nuclear EOS
(not investigated here). Reducing the uncertainties via larger count numbers
could be achieved by combining measurements from multiple neutrino detectors,
or if the supernova occurred closer than the assumed
$10\text{\,}\mathrm{k}\mathrm{p}\mathrm{c}$. The degenerate signals might be
broken by incorporating into the analysis: additional parts of the neutrino
lightcurve (e.g., Segerlund et al., 2021); additional detectors sensitive to
other neutrino flavors (e.g., water Cherenkov detectors); or, given sufficient
counts, the full neutrino spectrum instead of an average energy.

##  5 Conclusion

We have produced a suite of 612 one-dimensional CCSN simulations, using three
sets of EC rates and 200 stellar progenitors between
$9120\text{\,}\mathrm{M_{\odot}}$.

The three EC rate sets were (§ 2.1): an IPA (Bruenn, 1985); a microphysical
library with parametrized rates in the high-sensitivity $N=50$ region (LMP;
Sullivan et al., 2016; Titus et al., 2018); and the same library with updated
$N=50$ rates (LMP+N50; Titus et al., 2019).

Of the 200 progenitors, there were 43 successful explosions for the IPA set,
70 for LMP, and 73 for LMP+N50. In general, the IPA models reached smaller
shock radii and exploded later than their LMP and LMP+N50 counterparts (§
3.1). Of the latter two, LMP+N50 appeared marginally more favorable to
explosion, with larger shock radii and earlier shock runaway than LMP.

At core bounce, IPA typically had the largest inner core mass, electron
fraction, density, accretion rate, infall velocity, and gravitational
potential (Fig. 2, 4, and 5). The next largest values were generally LMP+N50
followed by LMP, although LMP+N50 changed places with LMP for the collapse
time and accretion rate (Fig. 5). The standard ordering was also reversed for
$Y_{e}$ in the outer core, where IPA had the lowest values and LMP the highest
(Fig. 3).

For exploding progenitors, IPA produced a PNS mass around
$0.020.1\text{\,}\mathrm{M_{\odot}}$ larger than LMP due to higher accretion
rates and delayed shock revival, and LMP+N50 was typically
$\lesssim$0.02\text{\,}\mathrm{M_{\odot}}$$ smaller (Fig. 6). For failed
explosions, enhanced PNS cooling in IPA resulted in a collapse to BH roughly
$2080\text{\,}\mathrm{m}\mathrm{s}$ earlier than LMP, whereas LMP+N50 was at
most a few milliseconds earlier.

Without flavor mixing, the extended $\nu_{e}$ emission of LMP and LMP+N50
following bounce (Fig. 8) resulted in higher detected counts and mean energies
than IPA in a DUNE-like liquid argon detector (Fig. 8). Conversely, when
adiabatic flavor mixing is included, the enhanced $\nu_{x}$ emission in IPA is
converted to $\nu_{e}$, resulting in higher counts and energies than LMP and
LMP+N50. Given only $\sim 10^{2}$ counts, however, these differences were
typically smaller than the estimated uncertainties.

All of these results largely stem from the total rate of electron captures at
different densities during collapse (§ 4.1). For IPA, the total EC rate is
overestimated at lower densities, but subsequently underestimated at higher
densities due to Pauli blocking on a mean nucleus of $N\geq 40$. The LMP-based
rates unblock EC for neutron-rich nuclei, and so deleptonization proceeds
further than IPA during collapse. The updated $N=50$ rates in LMP+N50 are
lower than the parametrized rates in LMP, producing an intermediate case
between LMP and IPA (Fig. 4).

It is important to emphasize the limitations of our study. While our STIR
framework (Couch et al., 2020) approximates the effects of turbulence in 1D,
there is ultimately no substitute for multidimensional simulations. Only high-
fidelity 3D simulations can hope to fully capture the interplay between fluid
instabilities, magnetohydrodynamics, and neutrino transport in CCSNe (e.g.,
Hanke et al., 2013; Lentz et al., 2015; O’Connor & Couch, 2018; Summa et al.,
2018; Müller et al., 2019). Our progenitors from Sukhbold et al. (2016) were
1D, solar-metallicity, nonrotating, and nonmagnetic, and do not represent the
full variety of stellar populations. The progenitor models also used
microphysical EC rates, and so the sudden transition to approximate rates in
our IPA models is somewhat artificial. Although we accounted for adiabatic
flavor mixing when calculating neutrino signals, there remains copious
uncertainty around the effects of flavor oscillations, which were not included
in our simulations.

We note that our study only considered the SFHo nuclear EOS, as previous works
have demonstrated that the EOS dependence of the collapse and early post-
bounce phase ($\lesssim$100\text{\,}\mathrm{m}\mathrm{s}$$) is dwarfed by the
impact of EC rate uncertainties (Sullivan et al., 2016). We posit that the
qualitative differences seen here between the EC rates beyond
$\approx$100\text{\,}\mathrm{m}\mathrm{s}$$ post-bounce are unlikely to be
dramatically altered by the EOS. Nevertheless, the choice of EOS can alter the
nuclear abundances that EC acts upon (Nagakura et al., 2019), and it would be
valuable to test our assumption in a future study.

Finally, the updated rates in LMP+N50 do not include temperature dependence
effects (Dzhioev et al., 2020). Very recently, Giraud et al. (2022) reported
new finite-temperature calculations for the $N=50$ region, with rates around
an order of magnitude higher than LMP+N50 and about a factor of 5 below the
LMP rates. Their simulations suggest that CCSN properties would be
intermediate between the LMP and LMP+N50 models presented here, which are
already in relatively good agreement compared to the commonly used IPA rates.
Work is underway to incorporate these new rates into NuLib so that they can be
freely used in future CCSN simulations.

EC plays a central role in deleptonization, shock formation, and neutrino
production during core-collapse. Our study has explored the effects of updated
EC rates in the high-sensitivity $N=50$ region, including a detailed
comparison between microphysical rates and a simple IPA. By producing
simulations across 200 progenitors, we have shown there are clear, systematic
impacts of EC rates on the core structure, shock dynamics, and neutrino
signals throughout the CCSN mechanism.

This work was supported in part by Michigan State University through
computational resources provided by the Institute for Cyber-Enabled Research.
SMC is supported by the U.S. Department of Energy, Office of Science, Office
of Nuclear Physics, under Award Numbers DE-SC0015904 and DE-SC0017955. RT and
RZ are supported by the US National Science Foundation under Grants
PHY-1913554 (Windows on the Universe: Nuclear Astrophysics at the NSCL),
PHY-1430152 (JINA Center for the Evolution of the Elements), and PHY-1927130
(AccelNet-WOU: International Research Network for Nuclear Astrophysics
[IReNA]). EOC is supported by the Swedish Research Council (Project No.
2020-00452).

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##  Appendix A Neutrino Flavor Mixing

When calculating detectable neutrino counts with SNOwGLoBES (§ 2.3), we
account for adiabatic neutrino flavor conversions due to MSW matter effects
(Dighe & Smirnov, 2000). Following similar approaches in Nagakura (2021) and
Segerlund et al. (2021), we calculate the neutrino flux at Earth, $F_{i}$, for
each neutrino flavor $i$, using

$\displaystyle F_{e}$ $\displaystyle=pF_{e}^{0}+(1-p)F_{x}^{0},$ (A1)
$\displaystyle\bar{F}_{e}$
$\displaystyle=\bar{p}\bar{F}_{e}^{0}+(1-\bar{p})\bar{F}_{x}^{0},$ (A2)
$\displaystyle F_{x}$
$\displaystyle=\frac{1}{2}(1-p)F_{e}^{0}+\frac{1}{2}(1+p)F_{x}^{0},$ (A3)
$\displaystyle\bar{F}_{x}$
$\displaystyle=\frac{1}{2}(1-\bar{p})\bar{F}_{e}^{0}+\frac{1}{2}(1+\bar{p})\bar{F}_{x}^{0},$
(A4)

where $p$ and $\bar{p}$ are the survival probabilities, and $F^{0}_{i}$ are
the emitted fluxes for each flavor $i$. Under normal neutrino mass ordering,
the survival probabilities are given by

$\displaystyle p$ $\displaystyle=\sin^{2}\theta_{13}\approx 0.02,$ (A5)
$\displaystyle\bar{p}$
$\displaystyle=\cos^{2}\theta_{12}\cos^{2}\theta_{13}\approx 0.69,$ (A6)

and under inverted mass ordering by

$\displaystyle p$ $\displaystyle=\sin^{2}\theta_{12}\cos^{2}\theta_{13}\approx
0.29$ (A7) $\displaystyle\bar{p}$ $\displaystyle=\sin^{2}\theta_{13}\approx
0.02,$ (A8)

where we use mixing parameters of $\sin^{2}\theta_{12}=0.297$ and
$\sin^{2}\theta_{13}=0.0215$ (Capozzi et al., 2017). The no-mixing case is
equivalent to $p=\bar{p}=1$. Note that our simulations use the combined heavy-
lepton species $\nu_{x}=\\{\nu_{\mu}$, $\nu_{\tau}$, $\bar{\nu}_{\mu}$,
$\bar{\nu}_{\tau}\\}$, and thus we assume

$F_{x}^{0}=\bar{F}_{x}^{0}\propto\frac{1}{4}L_{\nu_{x}},$ (A9)

where $L_{\nu_{x}}$ is the heavy-lepton neutrino luminosity from the
simulation (Fig. 8). The separated heavy-flavor inputs to SNOwGLoBES are then

$\displaystyle F_{\mu}=F_{\tau}$ $\displaystyle=F_{x},$ (A10)
$\displaystyle\bar{F}_{\mu}=\bar{F}_{\tau}$ $\displaystyle=\bar{F}_{x}.$ (A11)

The code used for these calculations is publicly available as a Python
package, flash_snowglobes (Johnston, 2022b).

##  Appendix B Data Availability

The data for all simulations presented in this work are publicly available in
a Mendeley Data repository (Johnston et al., 2022). The dataset includes
summarized simulation results, radial profiles at core bounce, time-dependent
quantities (e.g., shock radius and neutrino luminosities), and time-binned
SNOwGLoBES neutrino counts and energies.

Much of the model analysis and plotting were performed using our publicly
available Python package flashbang (Johnston, 2022a).