# Impact of Dark Photon Emission on Massive Star Evolution and Pre-Supernova
Neutrino Signal

A. Sieverding A. Sieverding<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA  E. Rrapaj  Department of
Physics, University of California, Berkeley, CA 94720, USA  School of Physics
and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA  G. Guo
Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan  Y.-Z. Qian
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA

###### Abstract

We study the effects of additional cooling due to the emission of a dark
matter candidate particle, the dark photon, on the final phases of the
evolution of a $15\,M_{\odot}$ star and resulting modifications of the pre-
supernova neutrino signal. For a substantial portion of the dark photon
parameter space the extra cooling speeds up Si burning, which results in a
reduced number of neutrinos emitted during the last day before core collapse.
This reduction can be described by a systematic acceleration of the relevant
timescales and the results can be estimated semi-analytically in good
agreement with the numerical simulations. Outside the semi-analytic regime we
find more complicated effects. In a narrow parameter range, low-mass dark
photons lead to an increase of the number of emitted neutrinos because of
additional shell burning episodes that delay core collapse. Furthermore,
relatively strong couplings produce a thermonuclear runaway during O burning,
which could result in a complete disruption of the star but requires more
detailed simulations to determine the outcome. Our results show that pre-
supernova neutrino signals are a potential probe of the dark photon parameter
space.

##  1 Introduction

Dark matter, which represents more than 80% (Tanabashi et al., 2018) of the
matter density of the universe and whose nature remains one of the biggest
mysteries in physics, could be part of a dark sector which weakly interacts
with Standard Model (SM) particles. Such scenarios of dark sectors naturally
appear in many extensions of the SM where dark matter particles only interact
with the SM via a mediator. There is a rich experimental program searching for
signatures of such mediators (Essig et al., 2013; Alexander et al., 2016;
Aaboud et al., 2019). However, if the coupling to SM particles is too weak,
these particles could evade the searches and remain hidden. Astrophysical
probes can greatly extend the reach of the search for dark matter candidates,
trading the precision associated with the controlled environment of a
laboratory for the vast range of densities and temperatures of stars (Raffelt,
1996). In one of the simplest extensions of the SM, the dark sector interacts
with ordinary matter through the exchange of light vector bosons that couple
to SM conserved currents (Holdom, 1986; Rajpoot, 1989; Nelson & Tetradis,
1989; Batell et al., 2014a). Dark matter is charged under a local U(1)
symmetry in which the mediator couples to the SM electric charge $Q$, and is
described by the spin-one field $A^{D}_{\mu}$, called the dark photon, which
mixes kinetically with the standard photon $A_{\mu}$. Dark photons are
characterized by two independent parameters, their mass $m_{A}$ and the
reduced coupling strength with normal matter $\varepsilon$. Several general
dark matter searches have established experimental bounds on this parameter
space (Batell et al., 2014b; Essig et al., 2013).

Complementary constraints are provided by core collapse supernovae (SNe). For
instance, parameters that would result in a noticeable reduction of the
observed neutrino burst duration from SN 1987A can be excluded (Sung et al.,
2019; Chang et al., 2017; Rrapaj & Reddy, 2016; Raffelt, 1990; Choplin et al.,
2017). Additional constraints have been derived from observational signatures
of our sun and other low-mass stars (An et al., 2013). If beyond SM particles
can be produced at relatively low temperatures, they would affect stellar
evolution during He-burning and onward. This phase lasts millions of years and
is sampled broadly by observations that can be used to derive stringent
constraints (Raffelt & Dearborn, 1987). Dark matter particles that are more
massive, however, can only be produced at much higher temperatures and
therefore only affect the short, advanced stages of the evolution of more
massive stars.

This work is a continuation of Rrapaj et al. (2019) to study the potential
impact of the dark photon on the final stages of a $15\,M_{\odot}$ star.
During these stages temperatures in the star become high enough for dark
photons to be emitted from electron-positron pair annihilation while the
densities are still far below the regime where other processes, e.g.,
bremsstrahlung, become important. We therefore only include dark photons from
pair annihilation in this work and consider values of $m_{A}$ ranging from
$2m_{\mathrm{e}}$ to $10\,\mathrm{MeV}$, where $m_{\mathrm{e}}$ is the
electron mass. Due to energy-momentum conservation, dark photons with masses
below $2m_{\mathrm{e}}$ cannot be produced by electron-positron pair
annihilation and the temperatures needed for the emission of particles with
masses more than $10\,\mathrm{MeV}$ are not reached before core collapse. We
explore a wide range of coupling strengths $\varepsilon$, between $10^{-13}$
and $10^{-6}$, thereby probing a part of the parameter space unconstrained by
neutrino observations of SN 1987A (Chang et al., 2017; Hardy & Lasenby, 2017).

If the dark photon is lighter than all other particles in the dark sector it
decays back into electron-positron pairs, either still inside the star or
later in the interstellar medium. In the latter case, the $\gamma$-ray
signature of the subsequent annihilation of positrons can be used to derive
constraints (DeRocco et al., 2019). If the dark photons decay inside the star,
this may act as an additional heating mechanism leading to further constraints
from the observed explosion energies of SNe (Sung et al., 2019). In contrast
to these scenarios, we assume that the dark sector contains lighter particles
that the dark photon ultimately decays into, as also discussed in Rrapaj et
al. (2019). In this case, any energy carried away by the dark photon is leaked
into the dark sector, and the emission of the dark matter particles always
acts as a cooling mechanism without additional signatures. Under this
assumption the constraints discussed by DeRocco et al. (2019) and Sung et al.
(2019) do not apply.

As a messenger of the stellar interior, we look at the emission of pre-SN
neutrinos and find that, in a large part of our selected dark photon parameter
space, the extra cooling leads to a speed-up of the final burning stages and
systematically reduces the number of neutrinos emitted during the last day
before core collapse. We also show that this reduction can be estimated with
good accuracy just based on the baseline stellar model by adjusting the time-
integration of the neutrino luminosity. Our results suggest that this effect
may be used to constrain the dark photon parameter space if pre-SN neutrinos
are detected in the future. In a very small region of the parameter space we
also find that the effect of the extra cooling results in a slight increase of
the neutrino emission. In addition, for strong couplings unstable and
explosive O burning occurs. While the latter case may potentially also provide
constraints on the dark photon parameters, improved simulations are required
to better determine the final outcome and observational signatures.

Our paper is organized as follows. In §2 we describe the setup for our
calculations. In §3 we discuss the details of our fiducial model of the
evolution of a $15\,M_{\odot}$ star calculated without extra cooling. In §4 we
provide an overview of the relevant parameter space and discuss details of the
three types of effects that we find. In §5 we briefly outline the possibility
of deriving constraints from future observations of pre-SN neutrinos and
summarize our results.

##  2 Calculations

We implement a tabulation of the dark photon emission rates from Rrapaj et al.
(2019) in the stellar evolution and hydrodynamics code KEPLER (Weaver et al.,
1978; Woosley & Heger, 2007) and calculate the evolution of a $15\,M_{\odot}$
star with an initial composition of solar metallicity (Lodders & Palme, 2009).
The mass loss prescription is based on Nieuwenhuijzen & de Jager (1990) and
the mixing length for convection is equal to the pressure scale height. Semi-
convection is treated as described in Woosley & Weaver (1988), limiting the
convective diffusion coefficient to $10\%$ of the thermal value (see also
Woosley et al. 2002). Overshoot and thermohaline mixing (Kippenhahn et al.,
1980) are also included. Neutrino energy loss and the resulting total neutrino
luminosity are based on Itoh et al. (1996).

We evolve the models until the onset of core collapse, which we define as the
point when the infall velocity exceeds $5000\,\mathrm{km}/\mathrm{s}$. This
limiting value is higher than the value of $1000\,\mathrm{km}/\mathrm{s}$ used
in previous studies (Woosley & Heger, 2007; Woosley et al., 2002) because the
additional dark matter cooling tends to accelerate contraction.

Figure 1: Track of core temperature vs. density for our fiducial stellar model
without extra cooling. The time until core collapse is indicated at several
points. In the region above the dotted contours for two sets of $m_{A}$ and
$\varepsilon$, the corresponding dark photon emission dominates neutrino loss.

In order to explore the dark photon parameter space we have calculated more
than 470 stellar models and to limit the computational cost we only use an
approximate 19-isotope nuclear reaction network to calculate the energy
generation rates. At a temperature of 3.5 GK the code switches to solving for
the composition in quasi-statistical equilibrium (QSE) (Hix et al., 1998). For
even higher temperatures and when O is depleted full nuclear statistical
equilibrium (NSE) is assumed. Furthermore, we limit the number of zones in our
models to 2000.

##  3 Fiducial model

Figure 1 shows the track of the fiducial stellar model without extra cooling
in terms of central temperature $T_{\mathrm{C}}$ and density
$\rho_{\mathrm{C}}$. This track is in good agreement with the results from
Woosley & Heger (2007) and also with those from different stellar evolution
codes, such as MESA (Paxton et al., 2015). For the parameters that we are
studying, dark photon production is not relevant for temperatures below $\sim
1$ GK. The dotted contours in Figure 1 indicate where energy loss due to dark
photon emission equals the neutrino loss, marking the boundary of the
temperature and density regime where dark photon loss would dominate. This
boundary depends on the values of $m_{A}$ and $\varepsilon$. For instance,
lighter dark photons with stronger coupling strengths start to dominate at
lower temperatures. The details of these contours were discussed by Rrapaj et
al. (2019). As illustrated for $m_{A}=2\,\mathrm{MeV}$ and
$\varepsilon=10^{-11}$ in Figure 1, dark photons may already become important
right after central C depletion, which occurs around 64 years before core
collapse. Therefore, for context, we provide a short description of the
evolution after C depletion.

Figure 2: Kippenhahn diagram of the fiducial stellar model for the last 60
years before collapse. Hatched areas denote convective and semi-convective
regions that are indicative of nuclear burning. Blue (red) background color
indicates regions where cooling (nuclear energy generation) dominates.

The convective burning phases over the last 60 years are indicated in Figure
2, where hatched areas indicate convection and the color code shows the net
energy generation or loss. After C core and shell burning, Ne burning ignites
at the center at $T_{\mathrm{C}}=1.35$ GK, around 1500 days before collapse
under slightly degenerate conditions with a value of the degeneracy parameter
$\eta=\mu_{\mathrm{e}}/k_{\mathrm{B}}T\approx 4$, where $\mu_{\mathrm{e}}$ is
the chemical potential of electrons, $T$ the temperature and $k_{\mathrm{B}}$
the Boltzmann constant. The partial electron degeneracy leads to an initially
rapid rise of the temperature, followed by core expansion at almost constant
temperature. Once Ne is depleted, neutrino cooling leads to a decrease in
temperature before the core continues to contract and eventually heats up
again. This leads to the loop in the $T_{\mathrm{C}}$-$\rho_{\mathrm{C}}$
diagram shown in Figure 1. Ne burning is visible as the small hatched peak in
Figure 2, which also shows that this episode is very short.

About 650 days before collapse, O burning ignites centrally at
$T_{\mathrm{C}}=1.7$ GK, which leads to a rapid rise of the central
temperature and subsequent expansion. In contrast to the Ne-burning loop, at
the end of O burning in the core the temperature does not decrease and the
core is stabilized by shell burning. Once the shell burning ceases, neutrino
loss reduces $T_{\mathrm{C}}$ while the core contracts. At higher densities,
neutrino loss is suppressed and the track resumes its upward climb, while a
second O-burning shell is ignited. Between core O depletion and Si ignition,
He and C shell burning are still active.

Figure 3: Stellar profiles of our fiducial 15 $M_{\odot}$ model at core
collapse. The top panel shows temperature and density and the bottom panel
mass fractions for the most important isotopes.

Convective Si burning ignites at the center around 3.5 days before collapse,
after the second O-burning shell is extinguished. Around 5.6 hours before
collapse, central Si burning finishes and leaves a hot $1.1M_{\odot}$ Fe core
behind. This initial core grows further by shell burning that ignites about
one hour before collapse and continues until the core mass exceeds its
effective Chandrasekhar limit and collapses. Figure 3 shows the temperature,
density, and mass fraction profiles of the most important isotopes of our
fiducial model at core collapse. The He core encompasses 4.26 $M_{\odot}$, the
C/O core 3.02 $M_{\odot}$, the O/Ne core up to 2.70 $M_{\odot}$, and the Fe
core encompasses 1.60 $M_{\odot}$ at a radius of $1336$ km.

Multi-dimensional hydrodynamics simulations of the last minutes before core
collapse (Yadav et al., 2020; Yoshida et al., 2019; Müller et al., 2016) have
shown deviations from spherically symmetric models using mixing length theory,
but we do not expect our results to be qualitatively affected by these
differences.

Because the evolution of the stellar core is mostly decoupled from the surface
during the final stages, neutrinos are unique messengers that may provide
detailed information about the processes and conditions in the core shortly
before collapse (e.g., Guo & Qian, 2016; Kato et al., 2020). Current and near-
term neutrino detectors are expected to be able to detect the neutrinos from a
nearby massive star only within the last day before collapse (e.g., Guo et
al., 2019; Kato et al., 2020). The top panel of Figure 4 shows the neutrino
luminosity during the last 10 days before core collapse for our fiducial
model. In general, the luminosity increases as the star contracts and heats
up, following the track in Figure 1. The two peaks visible in Figure 4 are
caused by the ignition of nuclear burning that leads to expansion and cooling,
temporarily delaying collapse. The peak at around 3.5 days before collapse
corresponds to Si ignition at the center and the peak at about one hour before
collapse corresponds to the ignition of Si shell burning.

Here we focus on the $\bar{\nu}_{\mathrm{e}}$ for two reasons. Firstly, future
scintillation detectors are likely to observe the pre-SN
$\bar{\nu}_{\mathrm{e}}$, mainly by inverse beta decay (IBD,
$\bar{\nu}_{\mathrm{e}}+\mathrm{p}\rightarrow\mathrm{e}^{+}+\mathrm{n}$).
Secondly, towards the end of the life of a massive star
$\bar{\nu}_{\mathrm{e}}$ are mostly produced by electron-positron pair
annihilation. In contrast to the emission of $\nu_{\mathrm{e}}$, this thermal
process is relatively insensitive to the details of the stellar composition
and does not depend on the uncertainties related to electron capture on
nuclei. We calculate the spectral neutrino flux from pair annihilation as in
Guo & Qian (2016), using temperature, density, and electron fraction profiles
as functions of time from our stellar models. The $\bar{\nu}_{\mathrm{e}}$
luminosity due to pair annihilation alone is shown in the top panel of Figure
4 in comparison to the total neutrino luminosity. At one day before collapse,
$\bar{\nu}_{\mathrm{e}}$ constitute almost 30% of the total luminosity. This
fraction decreases towards collapse, as $\nu_{\mathrm{e}}$ from electron
captures become increasingly important. At $10\,\mathrm{s}$ before collapse,
however, $\bar{\nu}_{\mathrm{e}}$ from pair annihilation still account for
$10\,\%$ of the total neutrino luminosity.

Figure 4: Top panel: Neutrino luminosities during the last 10 days before
collapse from our fiducial model, without considering flavor transformations,
for all neutrino species and processes as well as for only
$\bar{\nu}_{\mathrm{e}}$ from pair annihilation. Bottom panel: Expected IBD
event rates at JUNO for $\bar{\nu}_{\mathrm{e}}$ from pair annihilation
assuming a distance of $500\,\mathrm{pc}$ to the source. The expected event
rate depends on the neutrino mass ordering (normal or inverted hierarchy). The
backgrounds from geo-neutrinos and nearby reactors are indicated by the
horizontal gray lines.

Based on the $\bar{\nu}_{\mathrm{e}}$ fluxes from the stellar evolution
models, we calculate the expected pre-SN neutrino signal at the Jiangmen
Underground Neutrino Observatory (JUNO) following Guo et al. (2019). With the
spectral number luminosity $\phi_{\bar{\nu}_{\mathrm{e}}}(E_{\nu},t)$ of the
star, the expected energy-differential event rate at a distance $d$ is

$\frac{d^{2}N}{dE_{\nu}dt}=\frac{1}{4\pi
d^{2}}\epsilon_{\mathrm{eff}}N_{p}\sigma_{\rm
IBD}(E_{\nu})\phi_{\bar{\nu}_{\mathrm{e}}}(E_{\nu},t),$ (1)

where $\epsilon_{\mathrm{eff}}=0.73$ is the detector efficiency and
$N_{p}=1.45\times 10^{33}$ is the number of protons based on 20 kt of active
detector material with a $12\,\%$ proton fraction (An et al., 2016). The IBD
cross-section $\sigma_{\rm IBD}(E_{\nu})$ is calculated as in Guo & Qian
(2016). For the event rate we integrate Equation (1) over
$E_{\nu}=1.8$–$4\,\mathrm{MeV}$, which is the optimal energy window for
detection. Due to flavor transformations the detection rate depends on the
neutrino mass ordering and the expected number of events for the normal
hierarchy is about 3.4 times higher than for the inverted hierarchy (e.g., Guo
et al., 2019). Figure 4 shows the expected event rates from our fiducial model
assuming a distance of $500\,\mathrm{pc}$. We also show the main sources of
background, geo-neutrinos and $\bar{\nu}_{\mathrm{e}}$ from nearby reactors.
For the inverted hierarchy about 15 events are expected during the last day,
which are approximately the same as the background. For the normal hierarchy,
however, about 50 events are expected during the last day, which are a factor
of $\sim 3$ more than the background. Therefore, a pre-SN neutrino signal may
be detected above the background during the final day before core collapse,
especially in the case of the normal mass hierarchy.

##  4 Results

We have calculated stellar models for a grid of the dark photon parameters
$m_{A}$ and $\varepsilon$ laid out in Figure 5. For
$3\,\mathrm{MeV}<m_{A}<10\,\mathrm{MeV}$ we have looked at increments of
$0.5\,\mathrm{MeV}$ and for $2\,m_{\mathrm{e}}<m_{A}\leq 3\,\mathrm{MeV}$ we
have taken smaller steps of $0.1\,\mathrm{MeV}$. For $\varepsilon$ we looked
at logarithmic intervals of $0.5\,\mathrm{dex}$ spanning values from
$10^{-13}$ to $10^{-6}$. For reference, the region of the parameter space that
is already excluded by the detection of the neutrinos from SN 1987A (Chang et
al., 2017) is indicated as the top gray shaded area in Figure 5. Note also
that the region corresponding to $m_{A}<2\,m_{\mathrm{e}}$ is not studied
here. The different symbols in Figure 5 indicate the outcomes of the models
that we have calculated.

Figure 5: Overview of the results for a grid of the dark photon parameters
$m_{A}$ and $\varepsilon$. The gray shaded area at the top indicates the
parameter space already excluded by the detection of the neutrinos from SN
1987A (Chang et al., 2017). Note also that we do not consider values of
$m_{A}<2\,m_{\mathrm{e}}$. The green squares mark models that are not
noticeably affected by the dark photon emission. The blue circles mark models
that exhibit a significant reduction of pre-SN neutrino emission (see §4.1),
whereas the red crosses mark the three cases with noticeably increased
neutrino emission (see §4.2). For the red shaded area in the top left corner,
runaway nuclear burning can be expected from simple arguments (see §4.3) but
full exploration is beyond the scope of this paper. The red stars indicate
models that feature such explosive behavior.

We find three qualitatively different ways in which the extra cooling changes
the late phases of stellar evolution and the last-day pre-SN neutrino signal.
The green squares in Figure 5 mark models that exhibit only negligible
deviations from the fiducial model. In contrast, the blue circles mark cases
in which the pre-SN neutrino emission is reduced by more than $30\,\%$ due to
accelerated Si burning. Part of the parameter space with the blue circles is
currently unconstrained, suggesting that future observations of pre-SN
neutrinos may impose new constraints on the dark photon properties.

Only three models with small dark photon masses, indicated by the red crosses
in Figure 5, exhibit a slight increase in the pre-SN neutrino emission. As we
will show in §4.2, this result is caused by an additional shell burning
episode that delays core collapse. For sufficiently small dark photon masses
and intermediate coupling strengths, the dark photon emission predominantly
originates not from the stellar center, but from the region of the final the
O/Ne shell.

For the parameters in the red shaded region in the upper left corner of Figure
5, the nuclear timescale exceeds the hydrodynamic timescale during O burning
and thermonuclear runaway is to be expected, as we will explain in §4.3. We
did not perform full calculations to cover this whole region due to the
complications associated with this explosive scenario. While the final fate of
the corresponding models requires further investigation beyond the scope of
this paper, we can speculate that all these cases may even lead to
thermonuclear SNe and the disruption of the whole star.

In general, energy loss determines the timescales of the advanced burning
stages in massive star evolution (Woosley & Heger, 2007). The balance between
energy generation due to nuclear burning and energy loss determines the
temperature of hydrostatic burning and thus the rates at which nuclear
reactions occur. Furthermore, energy loss determines how fast the stellar core
can contract because part of the resulting gain in gravitational binding
energy needs to be radiated away. Thus, emission of dark matter particles that
increases the energy loss rate generally leads to a speed-up of stellar
processes. The consequences of this acceleration, however, depend on where the
speed-up occurs and on whether the star can adjust sufficiently fast. Below we
give more detailed explanations for the various effects that we observe.

###  4.1 Reduced neutrino emission

As long as the stellar path to core collapse is guided by energy loss, an
additional cooling agent, such as dark photon emission, leads to a speed-up of
the evolution. Once the core becomes hot enough, the bulk of the energy loss
is taken over by the dark matter particles instead of the neutrinos, and the
accelerated evolution leaves less time for neutrino emission.

Figure 6: Kippenhahn diagram for the model with the parameters
$m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-12}$. The onset of convective Si
burning in the core, indicated by the hatched region starting about 0.9 day
before collapse, is much closer to core collapse than in the fiducial model
without extra cooling shown in Figure 2.

As an example, Figure 6 shows the convective burning phases for a calculation
with the parameters $m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-12}$.
Comparison with Figure 2 shows that the stellar structure remains largely the
same as in the fiducial model, but the evolution timescales change greatly.
The period between the onset of convective Si burning and collapse is only
around 0.9 day in Figure 6, compared to 3.5 days in the fiducial model (see
Figure 2). That between the end of convective Si burning at the center and
collapse is less than 3 hours, compared to 8 hours in the fiducial model. The
radial extent of the convective regions, on the other hand, remains almost
identical. The slight reduction of the extent of the convective zone due to
the more efficient energy loss, along with the shorter time for shell burning
due to the faster evolution of the core, results in a slightly smaller final
Fe core mass of $1.58\,M_{\odot}$, compared with $1.60\,M_{\odot}$ for the
fiducial model. With the remnant mass estimated by the point where the entropy
decreases below $4\,k_{\mathrm{B}}/$baryon (Woosley et al., 2002), dark photon
emission slightly reduces the potential neutron star mass from
$1.78\,M_{\odot}$ to $1.76\,M_{\odot}$.

For the above values of $m_{A}$ and $\varepsilon$, the overall changes to the
stellar structure relative to the fiducial model are minor. In contrast, the
integrated number of neutrinos emitted during the last day is already
decreased by a factor of two. This result shows that the main effect of the
additional cooling is a speed-up of Si burning and core contraction that
scales with the energy loss rate. This speed-up of the evolutionary clock
reduces the integrated neutrino emission with dark photons also contributing
to the energy loss. The effects on the $\bar{\nu}_{\mathrm{e}}$ luminosity
from pair annihilation and the associated signal are illustrated in Figure 7
for models with $m_{A}=2\,\mathrm{MeV}$ and a range of $\varepsilon$ values.
In addition to a suppression of the luminosity, some features of the signal
are shifted closer to the time of collapse. This shift is clearly visible for
the peak associated with Si shell burning, which appears at about one hour
before collapse for the fiducial model (see §3), but at only
$30\,\mathrm{minutes}$ and $12\,\mathrm{minutes}$ before collapse for
$\varepsilon=10^{-12}$ and $10^{-11.5}$, respectively. Note that for
$\varepsilon=10^{-11}$, the evolution of the $\bar{\nu}_{\mathrm{e}}$
luminosity becomes less smooth due to complications discussed in §4.3.

Figure 7: Evolution of $\bar{\nu}_{\mathrm{e}}$ luminosity from pair
annihilation and associated event rate as in Figure 4, but for models
including dark photons with $m_{A}=2\,\mathrm{MeV}$. The event rate is
calculated assuming the normal neutrino mass hierarchy and a distance of
$500\,\mathrm{pc}$. With increasing values of $\varepsilon$ the luminosity is
suppressed more and the peak associated with Si shell burning moves closer to
the time of collapse, indicating the acceleration of the evolution.

For all dark photon masses covered in our study, we find the same type of
reduction of the neutrino emission, and the necessary coupling strength to
achieve the same reduction increases with mass. In the following we show that
this reduction can be estimated semi-analytically, just based on the fiducial
stellar model. Without dark photon cooling, the timescales during the final
stages of evolution are determined mostly by the neutrino loss. We expect
$dt\propto L_{\nu}(t)^{-1}$, where $L_{\nu}(t)$ is the total neutrino
luminosity from the whole star at time $t$. With the additional dark photon
cooling, we expect

$dt^{\prime}=\frac{L_{\nu}(t)}{L_{\nu}(t)+L_{A}(t)}dt,$ (2)

where $L_{A}(t)$ is the dark photon luminosity from the whole star. This
change affects the results of any time-integrated quantity in two ways.
Firstly, the time measure itself is changed. Secondly, the effective
integration limits are changed. The time of core collapse, $t_{\mathrm{CC}}$,
marks the final time with a well-defined physical condition. In order to
accumulate a fixed time interval $\Delta t$, e.g., one day, up to
$t_{\mathrm{CC}}$, the starting point needs to be modified such that

$\Delta
t=\int\limits_{t_{0}}^{t_{\mathrm{CC}}}dt=\int\limits_{t_{0}^{\prime}}^{t_{\mathrm{CC}}}dt^{\prime}.$
(3)

With $dt^{\prime}<dt$ due to dark photon cooling, we have
$t_{0}^{\prime}<t_{0}$, so $\Delta t$ samples earlier stages of the evolution.

The above prescription allows us to estimate the energy emitted in
$\bar{\nu}_{\mathrm{e}}$ from pair annihilation during the last day before
collapse as

$E_{\mathrm{1day}}^{\prime}=\int\limits_{t_{0}^{\prime}}^{t_{\mathrm{CC}}}L_{\bar{\nu}_{\mathrm{e}},\mathrm{pair}}(t)\left(\frac{L_{\nu}(t)}{L_{\nu}(t)+L_{A}(t)}\right)dt,$
(4)

where $L_{\bar{\nu}_{\mathrm{e}},\mathrm{pair}}(t)$ is the
$\bar{\nu}_{\mathrm{e}}$ luminosity due to pair annihilation from the whole
star. Note that $L_{\nu}(t)$ and $L_{A}(t)$ are based on the temperature and
density profiles of the fiducial model without dark photon emission. This
approximation is valid as long as the effects of extra cooling on stellar
structure are negligible. In this case, the change of the
$\bar{\nu}_{\mathrm{e}}$ emission can be estimated semi-analytically without
computing new stellar models with the additional energy loss.

Figure 8: Energy carried away by $\bar{\nu}_{\mathrm{e}}$ over the last day
before collapse relative to the fiducial model. Semi-analytic estimates based
on Eq. (4) are displayed as solid curves and compared to the results from full
simulations (symbols connected with dotted line segments). The gray horizontal
bar indicates no change from the fiducial model.

Figure 8 displays the energy emitted in $\bar{\nu}_{\mathrm{e}}$ from pair
annihilation over the last day before core collapse, normalized to the
fiducial model, as a function of the dark photon coupling strength for a range
of $m_{A}$. The symbols show the results from full stellar evolution
calculations and demonstrate that increasing dark photon emission leads to a
reduction of the neutrino emission. The solid curves show the estimates based
on Eq. (4), which successfully describe the onset of the effects of dark
photons as well as the overall trend for many of the calculated models,
especially when the reduction of the neutrino emission is $\lesssim 50\%$.
Deviations from the actual calculations are expected when the dark photon
cooling starts to change the temperature and density profiles for sufficiently
large values of $\varepsilon$. For a large part of the dark photon parameter
space, however, the main effect is the relatively straightforward reduction of
the neutrino emission without significant changes to the stellar structure as
discussed above. Those cases are indicated by the blue circles in Figure 5 and
cover an unconstrained region of the parameter space.

Figure 8 also shows large deviations of the estimates from the actual
calculations for $m_{A}=1.2$ MeV. As we will explain in §4.2, these deviations
are due to changes of the shell-burning evolution.

###  4.2 Increased neutrino emission

Figure 9: Kippenhahn diagram for the model with the parameters
$m_{A}=1.2\,\mathrm{MeV}$ and $\varepsilon=10^{-12}$. At 0.25 days before
collapse, an additional convective O burning shell ignites between the
enclosed mass coordinates of $1.6\,M_{\odot}$ and $2\,M_{\odot}$, which delays
core collapse and allows more time for neutrino emission.

As discussed in §4.1, the reduction of the pre-SN neutrino emission results
almost entirely from the modification of the timescales of core burning and
contraction. Below we show that the deviations from this explanation for low
$m_{A}$ (see Figure 8) are due to the appearance of shell burning phases with
a stabilizing effect, which leads to the cases of the increased neutrino
emission marked by the red crosses in Figure 5.

For the dark photon parameters under our consideration, only three models with
$m_{A}=2\,m_{\mathrm{e}}$, $1.1\,\mathrm{MeV}$, and $1.2\,\mathrm{MeV}$, all
with $\varepsilon=10^{-12}$, have an increase of $\lesssim 50\%$ in the pre-SN
neutrino emission relative to the fiducial model. The comparison of the
neutrino emission with the fiducial model is shown for the case of
$m_{A}=1.2\,\mathrm{MeV}$ in Figure 8. The Kippenhahn diagram for this case is
shown in Figure 9, which reveals an additional episode of convective O shell
burning between an enclosed mass of $1.6\,M_{\odot}$ and $2\,M_{\odot}$. This
episode starts at $0.25\,\mathrm{days}$ before collapse, following the end of
core Si burning, but before the onset of Si shell burning. Such an episode
occurs neither in the fiducial model nor in the model with the reduced
neutrino emission shown in Figure 6. Heating by this additional O-shell
burning phase relieves the pressure on the Si-core and delays the ignition of
Si-shell burning and hence core collapse. This delay gives the star more time,
thereby increasing the total number of pre-SN neutrinos emitted.

For $m_{A}=1.2\,\mathrm{MeV}$, the additional O-shell burning phase also
occurs for larger values of the dark photon coupling, but it cannot compensate
for the effects of the accelerated burning and core contraction. Consequently,
the integrated neutrino emission for such cases is still reduced (see Figure
8), although not by as much as estimated from Eq. (4), which does not take
into account the additional O-shell burning. We can only find a net increase
of the pre-SN neutrino emission for a very narrow range of $\varepsilon$,
which is strong enough to cause an additional burning episode, but weak enough
to limit the acceleration of the evolution.

In order to understand the apparent association of the additional burning
episodes with the lightest dark photons in our study, we look at the time-
integrated energy loss as a function of the enclosed mass $M_{r}$ inside
radius $r$,

$E_{\mathrm{loss},x}(M_{r})=\int\limits_{t_{0}}^{t_{CC}}\dot{q}_{x}(M_{r},t)dt,$
(5)

where $x$ is either $\nu$ for the neutrino with the energy loss rate
$\dot{q}_{\nu}(M_{r},t)$ or $A$ for the dark photon with the energy loss rate
$\dot{q}_{A}(M_{r},t)$, and $t_{0}$ is the time of C ignition when the energy
loss due to neutrinos and dark photons becomes relevant. Figure 10 shows the
ratio
$R_{A/\nu}(M_{r})=E_{\mathrm{loss,A}}(M_{r})/E_{\mathrm{loss},\nu}(M_{r})$
relative to the value for the central zone $R_{A/\nu}(M_{r,0})$ based on the
fiducial model. Due to the scaling of $R_{A/\nu}(M_{r})$ with $\varepsilon$,
the quantity $R_{A/\nu}(M_{r})/R_{A/\nu}(M_{r,0})$ is independent of
$\varepsilon$. It is, however, quite sensitive to $m_{A}$.

Figure 10: Ratio of dark photon to neutrino energy loss rates
$R_{A/\nu}(M_{r})$ (normalized to the central zone) as a function of the
enclosed mass for the fiducial stellar model. The background colors indicate
the shells with different compositions shown in Figure 3. For
$m_{A}<2\,\mathrm{MeV}$, the strongest dark photon emission originates not
from the Fe core, but from the Si and O/Ne shells due to the suppression at
high densities.

For masses above $2\,\mathrm{MeV}$ the emission of dark photons dominates that
of neutrinos mostly in the final Fe core because the temperature is generally
higher at smaller radii. For masses below $2\,\mathrm{MeV}$, however, the dark
photon emission dominates further outside the core. This different behavior
for the lower values of $m_{A}$ can be understood as follows. Because the
number of $e^{\pm}$ pairs is reduced in degenerate conditions, dark photon
production by pair annihilation is suppressed at high densities in the core
(see Rrapaj et al. 2019 for details). This suppression does not occur in the
region at lower densities outside the core, but the temperature of this region
can only facilitate significant emission of the lower-mass dark photons.
Therefore, shell burning phases are accelerated and additional burning
episodes with stabilizing effects as described above tend to occur for
$m_{A}<2\,\mathrm{MeV}$. For heavier dark photons, their production requires
energetic $e^{\pm}$ pairs that can only be provided by the final core burning
stages. Therefore, the assumption that the effect of dark photon cooling is
limited to the core burning regions is justified for $m_{A}>2\,\mathrm{MeV}$,
but the evolution of the outer regions is noticeably affected for smaller dark
photon masses.

The stabilizing effect of additional shell burning phases due to the
accelerated evolution outside the core cannot be easily captured by a semi-
analytical prescription like the one presented in §4.1. Detailed stellar
evolution calculations are essential to the discovery and understanding of
this effect.

###  4.3 Runaway O Burning

Figure 11: Tracks of central temperature (upper panels) and degeneracy
parameter $\eta$ (bottom panels) vs. central density for models with
$m_{A}=2\,\mathrm{MeV}$ and different values of $\varepsilon$. For reference,
the fiducial model is shown in gray in all the panels.

The stellar models with those dark photon parameters indicated by the red
stars in Figure 5 exhibit a thermonuclear runaway during O burning. This
result may be understood as follows. With the dark photon cooling, the
accelerated burning timescales get shorter than the convective and eventually
also the hydrodynamic timescales. In addition, the increased electron
degeneracy further prevents the star from adjusting its structure in response
to the nuclear energy release. Consequently, the central temperature rises
rapidly to $\sim 5$ GK and O is exhausted in a convective region covering less
than the innermost $0.1M_{\odot}$ of material. Afterwards, O burning
propagates outward in a thin shell as a deflagration front. Figure 11 shows
the evolutionary tracks of $T_{\mathrm{C}}$ and $\rho_{\mathrm{C}}$ for
$m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-12}$, $10^{-11}$, $10^{-10.5}$,
and $10^{-10}$, respectively, in comparison with the fiducial model. The
degeneracy parameter $\eta$ is also shown in the bottom panels of this figure.
The onset of the runaway can be seen clearly for the largest two values of
$\varepsilon$: $T_{\mathrm{C}}$ almost vertically increases to $\sim
5\,\mathrm{GK}$ when we stopped the calculations due to the required very
small time steps.

Compared to the fiducial model, the tracks for $\varepsilon=10^{-12}$ in
Figure 11 do not show much deviation, but the density tends to be higher for
the same temperature, thereby increasing the degeneracy. While the track of
$T_{\mathrm{C}}$ vs. $\rho_{\mathrm{C}}$ for the fiducial model already
exhibits a loop, which arises from burning under degenerate conditions (see
also Figure 1), the track for $\varepsilon=10^{-11}$ shows two such loops and
several backward kinks resulting from shell flashes, signifying ignition of
shell burning under degenerate conditions. For $\varepsilon=10^{-10.5}$, O
burning becomes unstable and the central temperature explosively increases up
to $\sim 5$ GK. It can be seen from Figure 11 that in this case the core
expands somewhat initially, but the decrease of density fails to stop the
runaway heating. As $T_{\mathrm{C}}$ increases rapidly, convection also
quickly becomes inefficient to distribute the energy released from the nuclear
reactions. The detailed evolution in this case is discussed in Appendix A. We
find the same type of behavior as in this case for even larger values of
$\varepsilon$. For example, for $\varepsilon=10^{-10}$, the density response
to the rise in temperature due to O burning is barely visible in Figure 11,
indicating a further acceleration of the runaway process.

While the runaway burning described above results from the interplay of
several factors, including the competition of burning timescales with
convective and hydrodynamic timescales, as well as partial electron
degeneracy, below we present a simple argument for the association of a
thermonuclear runaway with large values of $\varepsilon$ and provide a
conservative estimate for the dark photon luminosity (per unit mass) at which
hydrostatic burning is no longer possible.

In configurations of stable nuclear burning, the energy release from nuclear
reactions is compensated by energy losses, leading to a stable consumption of
the nuclear fuel. For advanced burning phases without dark photons, the
temperature for hydrostatic burning can be estimated by equating the neutrino
energy loss and nuclear energy release (Woosley et al., 2002). Assuming

$\rho=10^{6}\left(T/\rm{GK}\right)^{3}\ \rm{g/cm}^{3}$ (6)

for O burning, the intersection of the O-burning energy generation and the
neutrino loss rate gives the typical O-burning temperature of $\approx 1.8$ GK
(see blue curves in Figure 12).

The additional energy loss due to dark photons changes the temperature for
hydrostatic O burning. Due to the very steep temperature dependence of the
nuclear reaction rate, a small change of temperature is sufficient to
compensate for a large increase of the energy loss. As shown in Figure 12, a
factor of $\approx 100$ increase of the energy loss for
$m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-11}$ only slightly increases the
nominal O-burning temperature to $\approx 2\,\mathrm{GK}$. Nevertheless, this
temperature increase is responsible for the speed-up of the evolution
discussed in §4.1. Of course, the burning temperature must increase
substantially for strong dark photon couplings. For example, it reaches
$\approx 3\,\mathrm{GK}$ for $\varepsilon=10^{-9}$ (see Figure 12).

Complications arise if the timescale $\tau_{\mathrm{n}}$ of nuclear burning
becomes shorter than the hydrodynamic timescale on which stellar structure can
adjust to the temperature change. We take the latter timescale to be the free-
fall timescale

$\tau_{\mathrm{ff}}=\frac{1}{\sqrt{G\rho}},$ (7)

where the density $\rho$ can be estimated from the temperature assuming Eq.
(6). The timescale of nuclear burning can be estimated as

$\tau_{\mathrm{n}}=XQ/\dot{q}_{\mathrm{nuc}},$ (8)

where $Q=5\times 10^{17}$ erg/g (Woosley et al., 2002) is the effective energy
release, $X=0.7$ is the mass fraction of the 16O fuel, and
$\dot{q}_{\mathrm{nuc}}$ is the specific energy generation rate per unit mass.
For O burning, $\tau_{\rm{n}}=\tau_{\rm{ff}}$ occurs at a critical specific
energy generation rate of

$\dot{q}_{\mathrm{nuc,crit}}\approx 9\times 10^{16}(T/{\rm GK})^{3/2}\
\rm{erg/g/s}.$ (9)

The contour for $\tau_{\rm{n}}=\tau_{\rm{ff}}$ is shown in Figure 12. It
intersects the curve for the energy release due to O burning at $T\approx 3$
GK and $\dot{q}_{\rm{nuc,crit}}\approx 4.6\times 10^{17}\,\rm{erg/g/s}$. At
this temperature and the corresponding density from Eq. (6), if the specific
dark photon energy loss rate exceeds $\dot{q}_{\rm{nuc,crit}}$, i.e.,

$\dot{q}_{A}(T=3\,\rm{GK})>4.6\times 10^{17}\ \rm{erg/g/s},$ (10)

a thermonuclear runaway is expected. As shown in Figure 12, this result occurs
for $m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-9}$.

Figure 12: Comparison of the specific energy generation rate for O burning
with the loss rates for neutrinos and for dark photons with
$m_{A}=2\,\mathrm{MeV}$. Equating the energy generation and loss rates gives
the nominal nuclear burning temperatures. For the red shaded area at the top,
the nuclear burning timescale $\tau_{\rm n}$ is shorter than the free-fall
timescale $\tau_{\mathrm{ff}}$, and a thermonuclear runaway is expected. See
text for detail.

The red shaded region in Figure 5 indicates the general parameter space of
$m_{A}$ and $\varepsilon$ for which the condition in Eq. (10) is fulfilled.
While the boundary of the region roughly reflects the appearance of the red
stars indicating runaway O burning, the stellar models already exhibit such
burning for much weaker couplings for $m_{A}\lesssim 2$ MeV. This result may
be explained by the complications due to electron degeneracy, which tends to
prevent the star from reacting to nuclear energy release, and due to the
competition of nuclear burning with convection. Both these complications are
ignored in the simple picture assumed for deriving Eq. (10). At O ignition,
our fiducial model already exhibits some degree of degeneracy with $\eta\sim
2$ (see Figure 11). When dark photon cooling is included, the temperature of
stable burning increases, which requires the stellar core to contract to a
higher density. Because degeneracy rises more steeply with density than the
temperature, values up to $\eta\sim 8$ is reached in the core at O-burning
temperatures when O burning becomes unstable. As the adjustment of the core is
slowed down, the O burning luminosity exceeds the maximum luminosity that can
be transported by convection (see Appendix A), which also favors a local
runaway.

The boundary of the red shaded region in Figure 5 reflects the dependence of
the specific dark photon energy loss rate on $m_{A}$ and $\varepsilon$. It
also tends to predict too small threshold values of $\varepsilon$ for runaway
burning for higher values of $m_{A}$. Specifically, for $(m_{A}/{\rm
MeV},\varepsilon)=(4,10^{-7})$, $(4.5,10^{-6.5})$, $(4.5,10^{-6})$, and
$(5,10^{-6})$ that are above the boundary, although O burning leads to a rapid
rise in temperature, the dark photon energy loss increases fast enough to
avoid a thermonuclear runaway.

##  5 Discussion and conclusions

We have studied the impact of extra cooling due to beyond SM particles on the
evolution of a $15\,M_{\odot}$ star. Specifically, we have implemented the
dark photon emission from $e^{\pm}$ pair annihilation in the stellar evolution
code KEPLER, assuming that these particles decay into other dark sector
components, thereby representing an additional mechanism of energy loss for
the star. We have considered dark photon masses of $m_{A}=2\,m_{\mathrm{e}}$
to 10 MeV and couplings of $\varepsilon=10^{-13}$ to $10^{-6}$, and found that
the dark photon can affect the O and Si burning phases. There are three types
of potentially observable effects, which are summarized in Figure 5.

Figure 13: Constraints on dark photon parameters. The light gray region would
be excluded if more than 17 pre-SN neutrinos were detected from a $15\
M_{\odot}$ star at a distance of $500\,\mathrm{pc}$ (the case of
$m_{A}<2\,m_{\mathrm{e}}$ is not considered). The dark gray regions have been
excluded by big bang nucleosynthesis (BBN, Fradette et al. 2014) and the
detection of the neutrinos from SN 1987A (Sung et al., 2019).

For broad ranges of $m_{A}$ and $\varepsilon$, the extra cooling allows the
star to contract faster and increases the temperature at which nuclear burning
proceeds in equilibrium with energy losses. These effects speed up the burning
processes and reduce the number of neutrinos emitted during the last day
before core collapse. We have developed a semi-analytical approach that
describes the reduction of the pre-SN neutrino emission for the relevant
ranges of $m_{A}$ and $\varepsilon$ (see §4.1).

We further find that, due to the density dependence of the emission process,
dark photons with $m_{A}<2\,\mathrm{MeV}$ are also produced outside the core,
which causes additional shell burning episodes, thereby increasing the pre-SN
neutrino emission for a very narrow range of parameters (see §4.2).

Because extra cooling increases the nominal nuclear burning temperatures,
sufficiently strong dark photon couplings are expected to produce a
thermonuclear runaway (see §4.3). We have found many cases of runaway O
burning in our models and speculate that thermonuclear SNe may be the outcome
in these cases (see Appendix A). Finding the actual outcome and possible
observables in these cases, however, requires more refined simulations that
can properly follow the propagation of narrow nuclear burning fronts. Were
dark photon cooling to cause complete disruption of a star, the relevant
parameter space may be constrained by the observed inventory of neutron stars
and stellar black holes, which requires most massive stars to undergo core-
collapse SNe.

The first type of effect, i.e., the suppression of the pre-SN neutrino
emission by dark photon cooling, may be constrained by the detection of such
neutrinos. In the most conservative approach, this suppression means that part
of the parameter space is only consistent with a non-detection of pre-SN
neutrinos. Therefore, any positive detection of a pre-SN neutrino signal from
a well-understood progenitor excludes this part of the dark photon parameter
space. For illustration, we take our $15\,M_{\odot}$ model to explode as a
core-collapse SN at a distance of $500\,\mathrm{pc}$. Assuming the normal
neutrino mass hierarchy, we can find the values of ($m_{A},\varepsilon$) that
reduce the pre-SN neutrino emission sufficiently to give an expected number of
events less than the background count of about $17$ events, in contrast to
about 50 events without dark photon cooling. The corresponding parameter space
is bounded by the red curve in Figure 13. The detection a number of events
significantly above the background level would exclude this part of the
parameter space, if the progenitor is similar to the model we have studied
here. Note that a small fraction of this parameter space may lead to a
thermonuclear runaway, and therefore, could be constrained by the observation
of a core-collapse rather than thermonuclear SN.

Pre-SN neutrinos have not been detected yet. Our results, however, indicate
that their observation in the future may offer a unique probe of not only the
dark photon but also other dark matter particles that are efficiently produced
in the temperature regime of $\sim 1$–$10\,\mathrm{GK}$ during stellar
evolution.

This work was supported in part by the US Department of Energy [DE-
FG02-87ER40328 (UM)]. Calculations were carried out at the Minnesota
Supercomputing Institute. ER was supported by the NSF (PHY-1630782) and the
Heising-Simons Foundation (2017-228). GG acknowledges support from the
Academia Sinica by Grant No. AS-CDA-109-M11. We thank Alexander Heger for
providing access to the KEPLER code.

##  Appendix A A model with Runaway O Burning

Figure 14: Snapshot profiles of the stellar core for the model with
$m_{A}=2\,\mathrm{MeV}$ and $\varepsilon=10^{-10.5}$. Time progresses from
left to right and convective regions are shown as red shaded areas. The top
row shows the temperature and density profiles, the second row shows the
nuclear energy release and combined neutrino and dark photon loss, the third
row shows the mass fractions of the key isotopes, and the bottom row shows the
velocity profile, where positive (outward) velocities are displayed as solid
curves and negative (infall) velocities as dashed curves. Panel (a) shows the
beginning of O burning when the energy release from nuclear reactions just
exceeds the combined neutrino and dark photon loss. Panel (b) shows a snapshot
275 s after panel (a), when the energy generation at the center starts to
decrease as O is exhausted. The high luminosity cannot be distributed
efficiently by convection and only the region of the inner 300 km finishes O
burning. Panel (c) is 82 s after panel (b) and shows that the burning is
compressed into a narrow burning front, where the temperature jumps from
$3\,\mathrm{GK}$ to $5\,\mathrm{GK}$.

In order to understand the phenomenon of runaway O burning, we investigate the
model for $m_{A}=2$ MeV and $\varepsilon=10^{-10.5}$ in more detail. Figure 14
shows a sequence of snapshots from the evolution of this model. The top row
shows the temperature and density profiles, the second row from the top shows
the nuclear energy release and combined energy loss due to neutrinos and dark
photons, the third row shows the mass fractions of the major isotopes to
illustrate the progress of nuclear burning, and the bottom row shows the
velocity profile, where positive (outward) velocities are shown as solid
curves and negative (infall) velocities are represented by dashed curves. The
red shaded regions are convective zones. Time runs from panel (a) on the left
to panel (c) on the right, spanning $357\,\mathrm{s}$. Panel (a) shows the
onset of O burning, which is confined to a convective region reaching up to
1000 km and encompassing about 0.1 $M_{\odot}$ of material. At $T\approx
2\,\mathrm{GK}$ and $\rho\approx 8\times 10^{7}\,\mathrm{g/cm}^{3}$, the
energy release from nuclear reactions exceeds the combined neutrino and dark
photon loss. In response to the excess heat, positive velocities develop,
indicating the onset of core expansion. Panel (b), however, shows that O
burning proceeds much faster than the adjustment of stellar structure. The
fuel in the center is almost depleted and the temperature has risen to
$4.7\,\mathrm{GK}$ while the density has only been reduced by about $40\,\%$
compared to panel (a).

At a temperature above $4\,\mathrm{GK}$, the luminosity due to O burning
exceeds the maximum value that can be transported by convection (Woosley &
Heger, 2015)

$L_{\mathrm{max}}\approx 4\pi r^{2}\rho v_{\mathrm{conv}}fC_{\mathrm{P}}T,$
(A1)

where $v_{\mathrm{conv}}$ is the convective velocity, $C_{\mathrm{P}}T$ is the
thermal energy content, and $f\ll 1$ indicates the efficiency of convection to
remove the internal energy. With $f=0.1$ we find
$L_{\mathrm{max}}<10^{50}\mathrm{erg/s}$, corresponding to an energy
generation rate of $\dot{q}_{\mathrm{max}}\approx 5\times
10^{17}\,\mathrm{erg/g/s}$ for the conditions of O burning assuming an initial
core mass of $0.1\,M_{\odot}$. The nuclear energy release in panel (b) exceeds
this luminosity already by four orders of magnitude and convection cannot
distribute the heat and the fuel throughout the convective region on the
burning timescale. Therefore, O is only depleted in the innermost
$300\,\mathrm{km}$ despite that the convective zone reaches out to
$2000\,\mathrm{km}$.

Without efficient convection, the very rapid burning gets confined into a
narrow burning front at the bottom of the nominally convective layer, as shown
in panel (c) of Figure 14. The change of composition and continuing energy
generation also show that Si burning immediately follows because of the high
temperature of the central region. In our model, the burning region consists
only of a few zones and the fuel in one zone is consumed before the zone above
it ignites. The propagation of the burning depends critically on the heat
transport and would need to be described by a model for flame propagation. For
a very narrow burning front, multi-dimensional effects are important for the
propagation of such a deflagration Fryxell & Woosley (1982) and the outcome is
highly sensitive to turbulence and mixing at the boundary of the burning front
(Jones et al., 2013, 2016). The bottom row of Figure 14 also shows that
significant outward velocities develop but remain subsonic. The above
situation cannot be adequately simulated by our model due to the lack of
resolution and the employed small reaction network and mixing-length treatment
of convection. Therefore, our results on the final outcome are speculative.

We have followed the calculation of our model further and find that the shell
burning steepens into a supersonic shock once it reaches the steep density
gradient at the upper edge of the convective zone around the radius of 2000
km, where Ne burning also provides additional energy. The outward velocity
reaches several 1000 km/s, which takes the pressure off the core, eventually
facilitating a rapid expansion and cooling. The $T_{\mathrm{C}}$ drops below
0.1 GK and $\rho_{\mathrm{C}}$ below $10^{5}$ g/cm3. When the shock reaches
the H envelope we find that the kinetic energy of $3.6\times 10^{50}$ erg
exceeds the gravitational binding energy of $1.4\times 10^{50}$ erg for the
star. The shock may thus disrupt the whole star or at least unbind a
significant fraction of the H envelope and lead in either case to an optical
transient. Even if the core eventually collapses, the runaway O burning would
delay the collapse by many hours up to days. Here we caution again that our
model is probably inadequate to describe this situation accurately and a
better treatment is needed to predict potentially observable signatures.

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