# Nuclear fission reaction simulations in compact stars

Alex Deibel<EMAIL_ADDRESS>Department of Astronomy, Indiana University,
Bloomington, IN 47405, USA  M. E. Caplan<EMAIL_ADDRESS>Department of
Physics, Illinois State University, Normal, IL 61790, USA  C. J. Horowitz
<EMAIL_ADDRESS>Center for Exploration of Energy and Matter and Department
of Physics, Indiana University, Bloomington, IN 47405, USA

###### Abstract

Type–Ia supernovae are powerful stellar explosions that provide important
distance indicators in cosmology. Recently, we proposed a new SN Ia mechanism
that involves a nuclear fission chain-reaction in an isolated white dwarf [PRL
126, 1311010]. Here we perform novel reaction network simulations of the
actinide-rich first solids in a cooling white dwarf. The network includes
neutron-capture and fission reactions on a range of U and Th isotopes with
various possible values for 235U enrichment. We find, for modest 235U
enrichments, neutron-capture on 238U and 232Th can breed additional fissile
nuclei so that a significant fraction of all U and Th nuclei may fission
during the chain-reaction. Finally, we compute the energy release from the
fission chain-reaction for various uranium enrichments; a novel result that is
a necessary input for thermal diffusion simulations of carbon ignition.

##  I Introduction

Type–Ia supernovae (SN Ia) are widely used distance indicators in cosmology
Abbott  _et al._ (2019); Howell (2011); Sullivan (2010), but significant
tension remains between the Hubble constant determined from SN Ia and the
value determined from other data Riess  _et al._ (2016, 2021); Di Valentino
_et al._ (2021). Despite the importance of SN Ia for cosmology, their
progenitor systems and explosion mechanism are still somewhat uncertain.

Traditionally, SN Ia are thought to involve the thermonuclear explosion of a
C/O white dwarf (WD) in a binary system. Here the companion is either a
conventional star (single-degenerate mechanism) or another WD (double-
degenerate mechanism) Wang and Han (2012); Hillebrandt  _et al._ (2013); Ruiz-
Lapuente (2014). Recently we proposed an additional SN Ia mechanism that may
occur in isolated WDs Horowitz and Caplan (2021a, b) wherein the cooling WD
core rapidly precipitates a fission-critical uranium crystal within $\lesssim
30\,\mathrm{s}$. If a fission chain-reaction proceeds in the crystal, it is
unknown, a priori, the fraction of uranium consumed or the resulting energy
release.

In the present paper we perform nuclear reaction network simulations of
fission chain-reactions in compact stars. Our goal is to determine the
fraction of fissile fuel consumed during a fission chain-reaction and the
resulting energy release and how this depends on the uranium enrichment
$f_{5}$ – the fraction of all uranium that is 235U. This is important input
for thermal diffusion simulations of carbon ignition in an isolated WD. We
present these simulations in a separate paper Horowitz (2022) where we find
that carbon ignition is likely, at high densities.

Our simulations are novel, apparently the first such calculations for a
compact star. To provide context, we briefly review fission chain-reactions in
conventional nuclear reactors and nuclear weapons in Sec. II. Our reaction
network formalism is described in Sec. III, results presented in Sec. IV and
we conclude in Sec. VI.

##  II Context

A fission reaction in a compact star is a unique hybrid between a nuclear
reactor and a nuclear weapon. Like a nuclear weapon the chain-reaction is
expected to proceed extremely rapidly. A WD is degenerate, however, and the
temperature can rise without a large increase in pressure. As a result, the
system does not rapidly disassemble as in a nuclear weapon. This allows time –
as in a nuclear reactor – for fertile isotopes such as 238U or 232Th to
capture neutrons and breed additional fissile material.

Many conventional nuclear reactors slow neutrons to (terrestrial) thermal
energies to take advantage of the large fission cross section of 235U at low
energies. In the WD core, however, the temperature is of order $\sim
1\,\mathrm{keV}$ and the 235U fission cross section is much smaller. Even if
plenty of light nuclei are present to moderate the neutrons, the neutron
energy will only be reduced to the $\sim 1\,\mathrm{keV}$ ambient temperature.
Therefore, unlike a terrestrial nuclear reactor, a stellar system can not take
advantage of the large low-energy 235U cross section.

Terrestrial nuclear weapons, on the other hand, disassemble extremely rapidly
because of the large energy release. This necessitates using fast neutrons in
the chain-reaction because slow neutrons simply take too long to cause
additional fissions. By the time a slow neutron arrives to cause another
fission the fuel may have been blown apart. This reliance on fast neutrons
requires the use of highly enriched uranium or plutonium in a nuclear weapon.

The necessary uranium enrichment may be reduced if it is possible for the
chain-reaction to breed additional fissile nuclei. For example 238U in a
nuclear reactor can capture a neutron to become 239U that in turn beta decays
twice to produce 239Pu. Therefore, a fission chain-reaction in a star could
breed some of its nuclear fuel as the reaction progresses.

We now begin our study by discussing the composition of the first solids to
form as a WD cools. Next, we list the fission and neutron-capture reactions
that are included in our network simulations and present the results for
composition and energy release as a function of time. We end with a discussion
of sensitivity to uranium enrichment and to the 238U fission cross section. We
conclude that for modest enrichments a large fraction of the available fuel is
expected to fission producing a large energy release that could ignite carbon
burning.

##  III Formalism

Initial abundances: The composition of the first solids to form as material in
a WD just starts to crystallize has been studied using free energy models and
with molecular dynamics simulations Horowitz and Caplan (2021a, b). The
material is U and Th rich since these elements have the highest charge $Z$. In
addition some Pb is present because the solar system abundance of Pb is 100
times that of U. The initial abundance in nuclei per baryon
$Y_{i}=n_{i}/n_{b}$ are listed in Table 1. Here $n_{i}$ is the number density
of species $i$ and $n_{b}$ is the baryon density.

In addition to heavy nuclei, some C and O may be present in the first solids.
Elastic scattering from the light C and O nuclei can lower the energy of
fission neutrons. At this time the amount of C and O is uncertain and may be
zero. For simplicity in this first study, we assume there is no C and O
present. As a result there will be little moderation of the initial neutron
energies. The fission spectrum has a most probable energy near $\approx
1\,\mathrm{MeV}$. In this paper we simply assume all neutrons have an energy
of $1\,\mathrm{MeV}$ and evaluate all cross sections at this energy. This
assumption of mono-energetic neutrons greatly simplifies thermally averaged
reaction rates that are proportional to the cross section times the relative
velocity,

$\langle\sigma(E)v\rangle\approx\sigma({\rm 1\,MeV})v_{0}\,,$ (1)

with $v_{0}$ the velocity of a $1\,\mathrm{MeV}$ neutron. Furthermore, this
thermal average is independent of temperature.

Table 1:  Initial abundances $Y_{i}$, electron fraction $Y_{e}$, and baryon number density $n_{b}$. U | Th | Pb | $Y_{e}$ |  $n_{b}$ ($\mathrm{cm}^{-3}$)
---|---|---|---|---
$1.3\times 10^{-3}$ | $9.51\times 10^{-4}$ | $2.28\times 10^{-3}$ | 0.391 | $6.02\times 10^{31}$
Table 2:  Cross sections for neutron absorption $(n,\gamma)$ and fission reactions for 1 MeV neutrons on Th and U isotopes. Also listed is the average number of neutrons emitted per fission $\bar{\nu}$. Data from NND .  Isotope |  $\sigma_{n,\gamma}$ (b) |  $\sigma_{f}$ (b) | $\bar{\nu}$
---|---|---|---
232Th | 0.14 | 0.0013 | 2.18
233Th | 0.068 | 0.094 | 2.69
235U | 0.11 | 1.20 | 2.53
236U | 0.17 | 0.36 | 2.49
237U | 0.082 | 0.68 | 2.57
238U | 0.13 | 0.014 | 2.69
239U | 0.097 | 0.38 | 2.91
240U | 0.086 | 0.007 | 2.69
241U | 0.17 | 0.24 | 2.88

We consider neutron-capture $(n,\gamma$) and neutron-induced fission reactions
on the U and Th isotopes. We use 1 MeV cross sections from the ENDF 2011 data
set available at the National Nuclear Data Center NND , see Table 2. Our
reaction network has 232Th, 233Th, and 234Th isotopes and U isotopes from 235U
to 242U. In addition we have neutrons and fission fragments. We do not
distinguish different possible fission fragments and simply assume that any
fission will produce two fragments. Our network has a total of 13 species
consisting of 3 Th isotopes, 8 U isotopes, $n$ and fission fragments.

There are simple equations for the change in abundance $dY_{i}/dt$ from a
given reaction Lippuner and Roberts (2017). For example the change in neutron
abundance $Y_{n}$ from the fission of nucleus $A$ of abundance $Y_{A}$ is

$\frac{dY_{n}}{dt}=(\bar{\nu}-1)n_{b}\sigma_{f}v_{0}Y_{n}Y_{A}\,.$ (2)

Here each fission produces an average number of neutrons $\bar{\nu}$ (see
Table 2, and one neutron was absorbed to cause the fission). The fission also
increases the abundance of fission fragments $Y_{\rm ff}$ where we include a
factor of two for the two fragments,

$\frac{dY_{\rm ff}}{dt}=2n_{b}\sigma_{f}v_{0}Y_{n}Y_{A}\,.$ (3)

Likewise neutron absorption on nucleus $A$ decreases its abundance and
increases the abundance of nucleus $A+1$,

$\frac{dY_{A+1}}{dt}=-\frac{dY_{A}}{dt}=n_{b}\sigma_{n,\gamma}v_{0}Y_{n}Y_{A}\,.$
(4)

We sum terms with these forms over all of the $(n,\gamma)$ and fission
reactions in Table 2.

Table 3:  Four cases of enrichment $f_{5}$, fission cross section of 238U, and neutron absorption cross section of fission fragments $\sigma_{\rm ff}$. Also listed are the initial $n$ abundance $Y_{n}$, the fission heating $S$, the fraction of U and Th that fission, and the final temperature $T_{f}$. Case | $f_{5}$ |  $\sigma_{f}(^{238}$U) | $\sigma_{\rm ff}$ | $Y_{n}$ | S | U | Th | $T_{f}$
---|---|---|---|---|---|---|---|---
|  | (b) | (b) |  | (MeV/A) | % | % |  $10^{9}$ K
A | 0.14 | 0.014 | 0 | $10^{-9}$ | 0.026 | 9.6 | 0.4 | 1.6
B | 0.14 | 0.04 | 0 | $10^{-6}$ | 0.358 | 95 | 58 | 5.9
C | 0.20 | 0.014 | 0 | $10^{-6}$ | 0.356 | 94 | 58 | 5.9
D | 0.14 | 0.04 | 0.01 | $10^{-6}$ | 0.325 | 89 | 49 | 5.6

The initial abundance of 235U is $f_{5}Y_{u}$ with $Y_{u}$ the uranium
abundance from Table 1 and $f_{5}$ the uranium enrichment. Likewise the 238U
abundance is $Y_{u}(1-f_{5})$. Finally for the initial neutron abundance one
can use any small seed value. The cases explored are listed in Table 3.

The energy released per fission is about $200\,\mathrm{MeV}$, or
$100\,\mathrm{MeV}$ per fission fragment. Therefore the heating rate in MeV
per baryon per time is

$\dot{S}=(100\,{\rm MeV})\,dY_{\rm ff}/dt,$ (5)

and the total energy released by a time $t_{\rm final}$ in MeV per baryon is

$S=(100\ {\rm MeV})\,Y_{\rm ff}(t_{\rm final})\ .$ (6)

The large fission energy release will raise the temperature of the system. We
assume the reaction proceeds at constant pressure. Previously we calculated
the heat capacity at constant pressure and obtained the final temperature
$T_{f}$ Horowitz and Caplan (2021b),

$T_{f}\approx\Bigl{(}\frac{8\epsilon_{F}S}{5\pi^{2}Y_{e}}\Bigr{)}^{1/2}\,.$
(7)

Here $\epsilon_{F}$ is the electron Fermi energy and $Y_{e}$ is the electron
fraction, see Table 1.

Figure 1:  Abundance per baryon $Y_{i}$ versus time for Case A in panel (a), B
in (b) and C in (c). Panel (d) shows abundance per baryon of neutrons (black
curves) and fission fragments (red curves) vs time. Results are shown for Case
B without neutron capture on fission fragments (dotted) and for Case D with
capture (solid), see Table 3.

##  IV Results

###  IV.1 Abundance evolution

We now present results for three cases of enrichment and 238U fission cross
section as listed in Table 3. In all cases the chain-reaction proceeds
rapidly, in less than $10^{-11}\,\mathrm{s}$, as shown in Fig. 1. This is
because of the high density of the system, the large neutron cross sections,
and the high velocity of $1\,\mathrm{MeV}$ neutrons. In general, the reaction
proceeds in two stages. In the first stage neutrons from 235U fission
transform or breed some 238U and 232Th nuclei into more easily fissionable
239U and 233Th. In the second stage, or breeder reaction, most of the original
U and a significant fraction of the Th fission.

In Case A we use $f_{5}=0.14$ and the unmodified $1\,\mathrm{MeV}$ fission
cross section for 238U of $0.014\,\mathrm{b}$. One needs an enrichment of at
least $f_{5}\approx 0.12$ for the system to be critical. If $f_{5}$ is smaller
than that no chain-reaction will take place. If $f_{5}$ is only slightly
larger than 0.12 we expect the chain-reaction to burn a small amount of 235U
until the system becomes sub-critical and the chain reaction stops. This will
only release a small amount of fission heating.

In Fig. 1 (a) we show results for $Y_{i}$ versus time for Case A. The system
fissions $\approx 9.6,\%$ of the total U. This includes over half of the
original 235U and only a small amount of 238U. Only a small amount of Th
fissions near $\approx 0.4\,\%$. We see that the neutron abundance rises
exponentially with time until enough 235U has been burned so that the system
is no longer critical. After that $Y_{n}$ decreases as the remaining neutrons
are captured.

Figure 2:  Heating rate per baryon $\dot{S}(t)$ versus time for the four cases
in Table 3.  Figure 3:  Total fission heating per baryon $S(t)$ versus time
for the four cases in Table 3.

Results are very sensitive to the small 238U fission cross section. This is
$0.014\,\mathrm{b}$ at $1\,\mathrm{MeV}$ but rises rapidly at higher energies.
A detailed calculation averaging the energy-dependent cross section over the
neutron fission spectrum may give a larger value. Alternatively, as the
temperature of the medium rises, 238U nuclei will occupy a range of excited
states and these may have higher fission cross sections for $1\,\mathrm{MeV}$
neutrons. For example, Zhu and Pei calculate that the spontaneous fission half
life of 240Pu decreases by 12 orders of magnitude as the temperature is
increased from $0\,\mathrm{MeV}$ to $0.1\,\mathrm{MeV}$ Zhu and Pei (2016). In
Case B we use $\sigma_{f}=0.04\,\mathrm{b}$ instead of $0.014\,\mathrm{b}$ for
$\sigma_{f}(^{238}$U). The results in Fig. 1 (b) show dramatic differences
from Case A. The reaction now proceeds in two stages. First mostly 235U
fissions. This releases enough neutrons so the $n$ capture converts both 232Th
and 238U into odd A nuclei with significant fission cross sections. In the
second or breeder reaction stage these nuclei fission. As a result fully 95%
of the U and 58% of the Th fissions. Note that in a terrestrial nuclear
reactor there is time for 239U to decay to 239Pu. Here there is not enough
time and 239U instead can be used directly as a fuel.

Alternatively, even if the 238U cross section is only $0.014\,\mathrm{b}$, one
can obtain a breeder reaction stage by modestly increasing $f_{5}$. Case C has
$\sigma_{f}=0.014$ b but uses an enrichment of $f_{5}=0.20$ instead of 0.14.
The results in Fig. 1 panel (c) show two well-separated reaction stages and
the fission of a large fraction of available nuclei similar to Case B.

We have assumed that the fission fragments are essentially inert. While there
may not be time for beta-decay, the fission fragments could capture neutrons.
Many fission fragments have $(n,\gamma)$ cross sections for $1\,\mathrm{MeV}$
neutrons of order $\sim 0.01\,\mathrm{b}$ NND . Therefore to explore this we
simply assign all fission fragments a $0.01\,\mathrm{b}$ capture cross section
for Case D, see Table 3. This case is otherwise identical to Case B. Note that
we are not keeping track of the identity of each fission fragment so when a
fragment captures a neutron its identity does not change. Therefore, neutron-
capture on fission fragments simply acts as a neutron sink.

Figure 1 panel (d) shows that capture on fission fragments somewhat reduces
the abundance of neutrons. Indeed Case B has a finite abundance of neutrons
remaining. This artificial result reflects the limitations of the reaction
network. In Case D all neutrons are eventually captured. The reduction in
neutrons slows the production of fission fragments somewhat. However by the
time the reaction is over, the total number of fission fragments and therefore
the total fission energy released is only slightly smaller in Case D with
capture than originally in Case B.

Figure 4:  Final temperature as a function of $f_{5}$ and
$\sigma_{f}(^{238}$U) using constant $\sigma_{\rm ff}$ = 0 (left) and
$0.01\,\mathrm{b}$ (center). Right panel shows the final temperature as a
function of $f_{5}$ and $\sigma_{\rm ff}$ at constant $\sigma_{f}(^{238}$U) =
$0.014\,\mathrm{b}$. We compute 368 networks in (a) and (b) and 483 networks
in (c); the jagged regions are an artifact of the finite resolution of the
grid. Scenarios A, B, C, and D from Table 3 are indicated in the panels. The
Red region approximately reaches carbon ignition Timmes and Woosley (1992).

Figure 5:  Percentage of Th that fissions (top) and total fraction of U that
fissions (bottom) for the same simulations as in Fig. 4.

###  IV.2 Heat release and final temperature

The heating rate for the different scenarios is shown in Fig. 2 and the total
heating is plotted in Fig. 3 and listed in Table 3. The final temperature is
plotted in Fig. 4 for a range of $f_{5}$ values. There is a minimum value of
$f_{5}$ for the system to be critical. Below this value there is almost no
fission heating and $T_{f}$ is small. Next there is a modest range of $f_{5}$
values where significant 235U fissions but little 238U or Th fissions. Here
$T_{f}$ is between $\approx 1\textrm{--}2\times 10^{9}\,\mathrm{K}$. Finally
there is a sharp transition when there are enough neutrons for an essentially
complete breeder reaction stage that fissions most of the U and 58% of the Th.

Note that the breeder reaction stage leads to a large total energy release of
$\approx 0.36\,\mathrm{MeV}/\mathrm{nucleon}$ as listed in Table 3; this leads
to $T_{f}\approx 6\times 10^{9}\,\mathrm{K}$. As the 238U fission cross
section increases, the necessary $f_{5}$ for the breeder reaction decreases.
In general, the breeder reaction uses all available fuel resulting in a nearly
complete burn.

According to Fig. 6 of Timmes and Woosley Timmes and Woosley (1992) a final
temperature of $T_{f}\approx 6\times 10^{9}\,\mathrm{K}$ for a
$5\,\mathrm{mg}$ mass may be hot enough to ignite carbon burning. However,
hydrodynamical simulations should be performed to explicitly verify that the
energy release $S$ heats the system enough to start carbon burning, which we
reserve for future work. We note that the presence of Pb in Table 1
significantly increases the heat capacity without increasing the fission
energy released. If the amount of Pb were less (or absent) the system would
reach higher temperatures.

###  IV.3 Cross section and enrichment sensitivity

To explore sensitivity to input parameters we have performed large numbers of
reaction network simulations. The frames in Fig. 4 were prepared by computing
a grid of networks to find the final temperature as a function of $f_{5}$,
$\sigma_{f}(^{238}\mathrm{U})$, and $\sigma_{\rm ff}$. In Fig. 4a and 4b we
compare the final temperature when excluding neutron-captures on the fission
fragments (i.e., $\sigma_{\rm ff}=0.0\,\mathrm{b}$) and assuming a modest
$\sigma_{\rm ff}=0.01\,\mathrm{b}$ as an average neutron-capture cross
section, respectively. The grid is computed between $f_{5}=0.08$ and
$f_{5}=0.30$ at a resolution of $\Delta f_{5}=0.01$, and between
$\sigma_{f}(^{238}\mathrm{U})=0.00\,\mathrm{b}$ and $0.075\,\mathrm{b}$ with a
resolution of $\Delta\sigma_{f}(^{238}\mathrm{U})=0.005\,\mathrm{b}$, for a
total of 368 network calculations. The cases A, B, C, and D from Table 3 are
marked in Fig. 4.

In Fig. 4a, where no neutron-captures occur on the fission fragments, there is
a sharp transition between the incomplete burn or ‘fizzle’ behavior at low
$f_{5}$ and the complete burn at high $f_{5}$ where the final temperature is
$T_{f}\gtrsim 5\times 10^{9}\,\mathrm{K}$. For enrichments $f_{5}\lesssim
0.25$ the transition between incomplete/complete burns depends on
$\sigma_{f}(^{238}\mathrm{U})$, requiring larger
$\sigma_{f}(^{238}\mathrm{U})$ for a complete burn at lower enrichment
$f_{5}$. For enrichments $f_{5}\gtrsim 0.25$ a nonzero
$\sigma_{f}(^{238}\mathrm{U})$ always results in a complete burn. In Fig. 4b
the fission fragments act as a neutron sink with a cross section of
$\sigma_{\rm ff}=0.01\,\mathrm{b}$. In this case, the burning transition
shifts to higher $f_{5}$. However, at nonzero $\sigma_{f}(^{238}\mathrm{U})$
we still find a robust burn of the ${}^{238}\mathrm{U}$ and Th.

In Fig. 4c we explore the sensitivity to the fission fragments as a neutron
sink assuming constant $\sigma_{f}(^{238}\mathrm{U})$=0.014 b. We use a
resolution of $\Delta\sigma_{\rm ff}=0.002$ b; the resolution in $f_{5}$ is
the same as above, for a total of 483 networks. For low values of $\sigma_{\rm
ff}$ (i.e. $\sigma_{\rm ff}\lesssim\sigma_{f}(^{238}\mathrm{U})$), we find
that complete burns are still readily achieved for $f_{5}\gtrsim 0.20$. It is
only at $\sigma_{\rm ff}\gtrsim\sigma_{f}(^{238}\mathrm{U})$ that the fission
fragments begin to ‘outcompete’ the ${}^{238}\mathrm{U}$ for neutrons and
quench the burning. Thus, even with some degree of ‘poisoning’ due to neutron
captures onto fission fragments, or other impurities, the system may still
undergo a complete burn so long as $\sigma_{f}(^{238}\mathrm{U})$ (and
$f_{5}$) is sufficiently high.

In Fig. 5 we show the total percentage of Th and U that fissions in the grid
of networks computed in Fig. 4. It is clear that barely any Th (top) burns in
networks that fizzle at low $f_{5}$ (Fig. 5a), and that the conditions
required for a complete burn Th burning have a sharp transition. Neutron
captures on the fission fragments can suppress the total Th fraction that
fission (Figs. 5b and 5c), and the sharp turn-on is shifted to slightly
greater $f_{5}$. When we compare to the U fraction that burns (bottom) in the
networks that fizzle, we see that most of the 235U burns, but barely more than
$f_{5}$, so the heating is largely due to a 235U burning which then stalls
before igniting the breeder stage. We conclude that the final temperatures
observed in Fig. 4 can be explained by a steady increase in 235U burning with
increasing $f_{5}$ in the fizzling regime, with complete burns achieved after
a very sharp turn-on which burns the Th and 238U in a breeder stage.

##  V Discussion

In this section we discuss limitations in our reaction network and then carbon
ignition.

###  V.1 Limitations of reaction network

We now explore possible limitations in our reaction network. First we have
only included $(n,\gamma)$ and fission reactions. These reactions proceed very
rapidly on a timescale of $\tau\sim 10^{-15}\,\mathrm{s}$ and neglecting beta-
decay and $(\gamma,n)$ should be a good first approximation. We have neglected
$(n,2n)$ reactions because these should be unimportant except at high neutron
energies above $1\,\mathrm{MeV}$. In future work we will examine temperature-
dependent cross sections and any new reaction pathways that may result.

Our reaction network assumes $1\,\mathrm{MeV}$ neutrons. This is a reasonable
first approximation to the fission spectrum as long as the amount of light
nuclei such as C or O is small. If light nuclei are present, then nuclear
recoil following elastic scattering will reduce the neutron energies. In
future work we will explore sensitivity of the reaction network to the neutron
spectrum.

Our reaction network is somewhat incomplete and does not include reactions for
very neutron-rich Th or U isotopes. For U we include reactions on isotopes up
to 241U. The omission of reactions on 242U or heavier isotopes is not expected
to be important because most of the U fissions and only a tiny fraction
captures enough neutrons to reach 242U.

For Th we only include reactions on 232Th and 233Th. For Cases B, C and D a
significant fraction of the original Th is converted to 234Th (which is stable
in our network). Including reactions on heavier Th isotopes could lead to the
fission of more Th. In particular, the neutron-induced fission of 235Th could
be a significant addition to our reaction network. There may not be data for
this neutron-rich isotope, however, and further progress using our reaction
network likely will require theoretical rates for very neutron-rich isotopes.

We assumed a constant baryon density in Sec. III Eqs. 2,3, and 4. In reality,
the system will expand slightly because of the large fission energy release.
However this decrease in density is only about 25% because the electrons are
degenerate Horowitz and Caplan (2021b). This will slightly slow down the rate
of all neutron reactions and therefore the chain reaction will take slightly
longer to complete.

We have neglected cooling from heat conduction and neutrino emission. This
should be a good approximation during the fission reaction because the
reaction rates are so high. The fission heating rate in Fig. 2 is consistent
with the estimated rate in Ref. Horowitz and Caplan (2021b). The rate of
cooling from heat conduction via the large thermal conductivity of the
degenerate electrons is estimated in Ref. Horowitz and Caplan (2021b) to be
two to three orders of magnitude lower than the heating rate in Fig. 2.
Therefore heat conduction is unimportant during the fission reaction.

###  V.2 Carbon ignition and explosive yield

We now consider carbon ignition. After the fission chain reaction, the system
may be so hot that self-propagating thermonuclear carbon burning is initiated.
Such a scenario is unstudied and future hydrodynamical simulations will be
needed. In the meantime, however, we can examine the fission-scenario through
analogy with terrestrial nuclear weapons.

First for context, we discuss ignition of hydrogen isotopes in a terrestrial
nuclear weapon. The Classical Super was the original idea to use heat from an
atomic bomb to start fusion in a deuterium, tritium mixture. It is said that
losses from thermal radiation will help quench thermonuclear fusion in the
Classical Super. Instead, radiation implosion can be used to compress the
hydrogen fuel first. This compression increases the energy density from
hydrogen fusion without increasing the radiation losses. In a WD, the carbon
fuel is already at a very high density. Therefore radiation losses are likely
unimportant and our system may avoid this problem with the Classical Super.
Because of the high initial density there may be no need for radiation
implosion to compress the system further.

It is interesting to compare the explosive yields in our system to those of
nuclear weapons. Previously we had estimated the initial mass of the uranium
rich crystal to be about $5\,\mathrm{mg}$ Horowitz and Caplan (2021a, b). In
Cases B, C, and D we find a significant fraction of the U and Th in this 5 mg
mass fissions. This will release energy equivalent to about $50\,\mathrm{kg}$
of TNT. The yield is much lower than the approximately $15\,\mathrm{kiloton}$
yield of the first atomic bombs because the $5\,\mathrm{mg}$ mass of our
system is much less than the multi kg core masses of conventional fission
weapons. Note that the efficiency of our system may be much higher, with
nearly all of the U and Th fissioning, compared to the few % efficiency of an
atomic bomb. Nevertheless, because the mass and critical mass are so much
smaller, our fission yield is almost a million times smaller. Therefore, if
carbon burning is not initiated, the fission chain reaction may have very
little effect on the star.

The situation is dramatically different if carbon burning is initiated. The
thermonuclear burning of a significant fraction of C in a WD will release
energy comparable to a SN Ia. This corresponds to an explosive yield of almost
$10^{29}\,\mathrm{megatons}$ (MT)! Thus, we propose using a $50\,\mathrm{kg}$
yield fission primary to ignite a $10^{29}\,\mathrm{MT}$ fusion secondary.

The temperature required for carbon ignition depends on the ignition scenario
including the system size, density and ignition timescale. Our system has a
density near $10^{8}\,\mathrm{g\ cm^{-3}}$ and a total mass of order
$5\,\mathrm{mg}$. The original work of Timmes and Woosley may be the most
directly relevant previous calculation of ignition temperature for our
conditions Timmes and Woosley (1992). They consider ignition in a C/O liquid
where they instantaneously replace a small mass of C/O by carbon burning ashes
and assume the temperature of this ash has been raised to $T_{i}$. They find
an ignition temperature of $T_{i}\approx 5\times 10^{9}\,\mathrm{K}$. If the
final temperature $T_{f}>T_{i}$ a flame may propagate in the surrounding C/O
liquid. If $T_{f}<T_{i}$ the system will cool via heat conduction without
initiating carbon burning. This ignition temperature of $5\times
10^{9}\,\mathrm{K}$ should be verified with future hydrodynamic simulations.

##  VI Conclusions

In this paper we have performed novel reaction network simulations of fission
chain-reactions in a cooling WD. The first solids to form when material in a
WD just starts to crystallize are expected to be U and Th rich because of
their high charges. These solids may support a fission chain-reaction if the
uranium enrichment $f_{5}$ is high enough $>0.12$.

We find that the reaction proceeds very quickly (within $\approx 10^{-11}$ s)
because the density is high and the neutron cross sections are large. In
general, the reaction proceeds in two stages. In the first stage neutrons from
235U fission transform or breed some 238U and 232Th nuclei into more easily
fissionable 239U and 233Th. In the second, or breeder reaction, stage most of
the original U and a significant fraction of the Th fission. These reaction
stages release $\approx 0.36\,\mathrm{MeV/nucleon}$ and raise the final
temperature to $T_{f}\approx 6\times 10^{9}\,\mathrm{K}$. This is important
input for our thermal diffusion simulations where we find that carbon ignition
is likely at high densities Horowitz (2022).

Acknowledgements: We thank Ezra Booker, Constantine Deliyannis, Erika
Holmbeck, Wendell Misch, Matthew Mumpower, Witek Nazarewicz, Catherine
Pilachowski, Tomasz Plewa, and Rebecca Surman for helpful discussions. The
work of CJH was performed in part at the Aspen Center for Physics, which is
supported by National Science Foundation grant PHY-1607611. This research was
supported in part by the US Department of Energy Office of Science Office of
Nuclear Physics grants DE-FG02-87ER40365 and DE-SC0018083 (NUCLEI SCIDAC).

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