# The Influence of $\beta$-Decay Rates on r-Process Observables

Kelsey A. Lund Department of Physics, North Carolina State University,
Raleigh, NC 27695 USA  Theoretical Division, Los Alamos National Laboratory,
Los Alamos, NM 87544 USA  Joint Institute for Nuclear Astrophysics—Center for
the Evolution of the Elements, USA  J. Engel Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27599 USA  G.C.
McLaughlin Department of Physics, North Carolina State University, Raleigh, NC
27695 USA  Joint Institute for Nuclear Astrophysics—Center for the Evolution
of the Elements, USA  M.R. Mumpower Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87544 USA  Center for Theoretical Astrophysics, Los
Alamos National Laboratory, Los Alamos, NM 87544 USA  Joint Institute for
Nuclear Astrophysics—Center for the Evolution of the Elements, USA  E.M. Ney
Department of Physics and Astronomy, University of North Carolina, Chapel
Hill, NC 27599 USA  R. Surman University of Notre Dame, Notre Dame, IN 46556
USA  Joint Institute for Nuclear Astrophysics—Center for the Evolution of the
Elements, USA

###### Abstract

The rapid neutron capture process (r-process) is one of the main mechanisms
whereby elements heavier than iron are synthesized, and is entirely
responsible for the natural production of the actinides. Kilonova emissions
are modeled as being largely powered by the radioactive decay of species
synthesized via the _r_ -process. Given that the _r_ -process occurs far from
nuclear stability, unmeasured beta decay rates play an essential role in
setting the time scale for the _r_ -process. In an effort to better understand
the sensitivity of kilonova modeling to different theoretical global beta-
decay descriptions, we incorporate these into nucleosynthesis calculations. We
compare the results of these calculations and highlight differences in
kilonova nuclear energy generation and light curve predictions, as well as
final abundances and their implications for nuclear cosmochronometry. We
investigate scenarios where differences in beta decay rates are responsible
for increased nuclear heating on time scales of days that propagates into a
significantly increased average bolometric luminosity between 1-10 days post-
merger. We identify key nuclei, both measured and unmeasured, whose decay
rates are directly impact nuclear heating generation on timescales responsible
for light curve evolution. We also find that uncertainties in beta decay rates
significantly impact ages estimates from cosmochronometry.

r-process,beta decay,kilonova,neutron stars

##  1 Introduction

Since its emergence in the 1950s, one of the biggest questions in the field of
nuclear astrophysics remains the main production site of some of the heaviest
elements, synthesized via the rapid neutron capture process (_r_ -process)
(Burbidge et al., 1957; Cameron, 1957). It is currently hypothesized that this
process, thought to be responsible for roughly half the material heavier than
iron, as well as the only process for producing the actinides, occurs to some
extent in the neutron-rich outflows of neutron star mergers (NSM). In addition
to the many theoretical advances made in the last few decades, the recent
electromagnetic observations accompanying the gravitational wave event
GW170817 (Abbott et al., 2017a, b; Díaz et al., 2017) lend the most support to
the long-standing idea of a kilonova (KN) explosive transient powered by the
radioactive decay of freshly synthesized _r_ -process material (Lattimer &
Schramm, 1974, 1976; Li & Paczyński, 1998; Metzger et al., 2010; Roberts et
al., 2011; Barnes & Kasen, 2013; Grossman et al., 2014; Wollaeger et al.,
2018; Fontes et al., 2020).

The luminosity and morphology of the light curve associated with AT2017gfo
(Chornock et al., 2017; Cowperthwaite et al., 2017; Nicholl et al., 2017;
Perego et al., 2017) offer unique insight into the physics of these extreme
environments. The bright but rapidly decaying component of the light curve
observed at shorter wavelengths indicates at least some portion of ejecta
material with little to no lanthanide or actinide abundances (Metzger et al.,
2010; Roberts et al., 2011; Evans et al., 2017; Miller et al., 2019). On the
other hand, a dimmer "red" signal that dominates on timescales of days (when
the "blue" signal has faded away) indicates that at least some portion of the
ejected material is composed of high-opacity lanthanides and possibly
actinides (Barnes & Kasen, 2013; Tanaka & Hotokezaka, 2013; Kasen et al.,
2017). These combined observations suggest distinct nucleosynthesis sites
within the merger ejecta, each of which might be capable of producing a robust
_r_ -process pattern.

Despite the wealth of information provided by the data from GW170817, the
larger endeavour of modeling KN signals for the purpose of reliably
interpreting future signals remains subject to many unknown quantities and
large uncertainties. While it is generally accepted that NSMs are a site for
_r_ -process production, it remains unclear whether these sites alone are
capable of producing the entire observed _r_ -process pattern. Studies of
material ejected on dynamical timescales, either via tidal forces or
compression between the coalescing neutron stars, predict material that is
neutron-rich enough to produce out to the second and third _r_ -process peaks
(Rosswog et al., 1999; Goriely et al., 2011; Korobkin et al., 2012; Bauswein
et al., 2013; Wanajo et al., 2014; Sekiguchi et al., 2015; Radice et al.,
2018).

While there is general agreement on the robustness of _r_ -process production
in dynamically ejected channels, there is more uncertainty regarding the
extent to which _r_ -process production occurs in in late-time accretion disk
outflows driven by viscous heating. Some part of this uncertainty comes from
the central remnant scenario (Shibata et al., 2005; Agathos et al., 2020;
Nedora et al., 2021; Kashyap et al., 2022). When prompt collapse does not
occur, neutrino interactions are capable of driving the $\rm{Y_{e}}$ up enough
to stifle _r_ -process production. Neutrino oscillations can also play a large
part in determining the extent of this effect, as only electron neutrinos act
to reduce the neutron-richness of the ejecta (Malkus et al., 2012; Siegel &
Metzger, 2017; Li & Siegel, 2021).

Simulating _r_ -process nucleosynthesis is also subject to large uncertainties
due to its trajectory far from nuclear stability where many important
quantities remain unmeasured (Mendoza-Temis et al., 2015; Eichler et al.,
2015; Mumpower et al., 2016b; Nikas et al., 2020). Detailed calculations
incorporate nuclear heating contributions from multiple decay modes, and these
impact the energy released, the thermalization efficiency with which the decay
products deposit energy into the system, and the composition of material that
is ultimately synthesized (Beun et al., 2008; Barnes et al., 2016; Mumpower et
al., 2018; Even et al., 2019; Sprouse et al., 2020).

The incorporation of theoretical spontaneous fission and alpha decay rates
have been found to largely impact the uncertainty in the nuclear heating on
time scales relevant for the evolution of the light curve, in some cases
adding up to an order of magnitude to the uncertainty (Vassh et al. (2019);
Giuliani et al. (2020); Z21; B21). These combine with additional uncertainties
including the nuclear equation of state (Oechslin et al., 2007; Bauswein et
al., 2013; Hotokezaka et al., 2013; Sekiguchi et al., 2015; Lehner et al.,
2016), the nature of neutrino oscillations and neutrino transport in the
ejecta (Miller et al., 2019; Kullmann et al., 2021), as well as atomic line
energy calculations for high-opacity lanthanides and actinides.

One particularly important data set for an _r_ -process nucleosynthesis
calculation is a description of the beta decay rates involved (Möller et al.,
2003; Caballero et al., 2014; Marketin et al., 2016; Shafer et al., 2016; Ney
et al., 2020; Robin et al., 2022; Kullmann et al., 2022). At early times, the
extent of r-process production is sensitive to the beta decay rates of the
nuclei involved, as these determine the relative abundances of connected
isotopic chains during $(n,\gamma)-(\gamma,n)$ equilibrium and compete
directly with neutron capture when equilibrium fails. At later times, and once
a population of high mass number species is synthesized, beta decay rates play
a further role in determining the time scale of beta decay chains which are
important for heating as well as for populating species which contribute
significantly to spontaneous fission and alpha decay heating. Additionally,
theoretical beta decay rates can compete with theoretical alpha decay and
spontaneous fission branching ratios which are crucial for determining the
shape and magnitude of the light curve Z21.

We aim to incorporate different global beta decay descriptions into
nucleosynthesis calculations and compare their impact with those of other
astrophysical and nuclear sources of uncertainty on nuclear energy generation,
light curve evolution, and predictions relevant to nuclear cosmochronometry.
In section 2, we describe the methods we use in generating and compiling
nuclear data as well as the computational methods we use for calculating
relevant quantities. In section 3, we show the results of our calculations of
nuclear energy generation. We also show the impact these uncertainties in
nuclear effective heating can have on a bolometric light curve. We conclude
the presentation of our results with section 4, which makes age estimates for
several _r_ -process-enhanced metal poor stars. Finally, we provide some
concluding remarks in section 5.

##  2 Method

We seek to quantify the leverage of beta decay rates on key aspects of
kilonova modeling when compared with other sources of nuclear and
astrophysical uncertainty. The evolution of the nucleosynthetic abundances
throughout the _r_ -process determines the energy output in the form of
nuclear heating. The thermalization profile of this released energy will
determine how it is transported away from the system, which in turn affects
the shape and magnitude of the observable light curve. The final abundance
that is produced in a given merger even can then be used in a stellar dating
technique if it is interpreted as being the sole source of a star’s _r_
-process material. In this section, we describe our data set and the methods
used in the calculation of these quantities.

###  2.1 Model Set and Nucleosynthesis

We use the Portable Routines for Integrated nucleoSynthesis Modeling (PRISM)
to perform nucleosynthesis calculations using a suite of prepared input files
describing astrophysical conditions and nuclear properties for a wide range of
nuclei. The extent of _r_ -process production is sensitive to the beta decay
rates of the nuclei involved, as these compete with neutron capture rates.
Currently, many methods exist to compute beta decay rates, but few are applied
to large sections of the chart of the nuclides. In order to investigate the
extent of the impact of different sets of beta decay rates in our
calculations, we construct separate beta decay and coupled beta decay/beta-
delayed fission reaction data sets consistent with three different beta decay
calculations.

The description contained in the work of Ney et al. (2020) (calculations using
these rates will hereafter be referred to as NES) uses the finite amplitude
method with Skyrme density functionals to compute the beta decay half-lives
for neutron rich species. The work of Marketin et al. (2016) (hereafter MKT)
uses a covariant density functional theory approach with Gogny interactions to
do the same. Möller et al. (2019) (hereafter MLR) uses a finite range droplet
model in a quasiparticle random phase approximation to obtain $\beta$-strength
functions for neutron-rich species. We use the three sets of beta decay rates
described and compute beta delayed neutron emission and beta-delayed fission
probabilities and daughter product distributions using Mumpower et al.
(2016a). We show the base-ten logarithm of the ratio of all three sets of beta
decay rates with respect to those of Möller et al. (2003) (MLR03) in Figure 1.

Figure 1: Logarithmic ratios of NES (left), MLR (center), and MKT (right) beta
decay rates with respect to MLR03 beta decay rates. The color bar is adjusted
to values of -2,2 to reveal trends more clearly. The area covered by the
Nubase 2016 data set (Audi et al., 2017) is removed.

In addition to beta decays, the nuclei involved in the _r_ -process are also
subject to other reactions and decays. The energy associated with these decays
is important for the nuclear heating, and plays a large part in determining
the shape and magnitude of the light curve. We calculate reaction and decay
rates, as well as Q-values, consistent with the eight nuclear models listed in
Table 1. We incorporate theoretical alpha decay rates obtained using a Viola-
Seaborg relation. We use neutron capture and neutron-induced fission rates
calculated using the statistical Hauser-Feshbach code, CoH (Kawano et al.,
2016).

For spontaneous fission rates, we use the barrier-height-dependent
prescription from Karpov et al. (2012) and Zagrebaev et al. (2011). We adopt
mass models with appropriate fission barrier height descriptions: ETFSI
(Aboussir et al., 1995) with ETFSI, TF (Myers & Świątecki, 1999) with TF,
HFB14 (Goriely et al., 2009) with HFB22 and HFB27, and FRLDM (Möller et al.,
2015) for all others. We consider two possible fission fragment distributions;
the first is a symmetric split, where the daughter products each equal one-
half of the parent nucleus, while the second is the double Gaussian
distribution described by Kodama & Takahashi (1975) (hereafter K&T). We make
an exception for the fission fragment distribution of the spontaneous fission
of 254Cf, where we use the more detailed calculation from Zhu et al. (2018).

Finally, where experimental or evaluated data is available, we overwrite
theory rates with data from Nubase (Audi et al., 2017), and calculate Q-values
using experimentally determined masses from AME2016 (Wang et al., 2017).

Table 1: Nuclear mass models (and associated references) used in nucleosynthesis calculations. Abbreviation | References
---|---
DZ33 | Duflo & Zuker (1995)
ETFSI |  Aboussir et al. (1995); Mamdouh et al. (2001),
FRDM2012 | Möller et al. (2016)
HFB22,HFB27 | Goriely et al. (2009, 2013)
SLY4 | Chabanat et al. (1998); Möller et al. (2015)
TF | Myers & Świątecki (1996, 1999)
UNEDF1 | Kortelainen et al. (2012); Möller et al. (2015)
WS3 | Liu et al. (2011); Möller et al. (2015)

For the thermodynamic evolution of the ejecta, we use a parameterized wind
model with an initial entropy per baryon of $s/k=40$ and an expansion
timescale of 20 ms. The network begins in nuclear statistical equilibrium with
initial seed nuclei determined using the SFHo equation of state (Steiner et
al., 2013). We use the initial electron fraction, $\rm{Y_{e}}$, as a proxy of
variation in the astrophysical conditions of the ejecta in order to compare
with uncertainties from the previously described variations from theoretical
nuclear models. We use single-$\rm{Y_{e}}$ trajectories with initial values of
0.02, 0.18, and 0.21 for the full suite of theoretical nuclear inputs. Based
on the work in Zhu et al. (2021), we consider that these can be taken to
represent varying degrees of contribution from fission to the total heating,
with $\rm{Y_{e,i}}=0.21$ yielding the smallest contribution.

We also consider a set of trajectories to more closely model an ejecta with
non-uniform composition. To do this, we perform nucleosynthesis calculations
using the FRDM2012 subset of nuclear inputs on single-$\rm{Y_{e}}$
trajectories ranging from 0.01 to 0.35 in increments of 0.01. We map these
onto an analytic probability distribution in order to sample a range of
distributions. We show these mappings in Figure 2. Previous work found that
solar-like abundances can be obtained by combining individual trajectories
with both high initial $\rm{Y_{e}}$ and well as low initial $\rm{Y_{e}}$ (Z21.
Our goal is not necessarily to obtain a solar-like final abundance pattern (we
direct the reader to, for example, Ristic et al. (2022), which addresses _r_
-process universality), since we do not know if kilonova events produce a
solar pattern, but rather to sample a variety of combined trajectories
containing material with a variety of neutron richness. This is consistent
with most modern kilonova models predict multiple ejecta components (see for
example Radice et al. (2018); Miller et al. (2019); Nedora et al. (2021);
Stewart et al. (2022)). We utilize these combinations of simulations for
effective heating, as well as light curve calculations out to 50 days, and use
the final (1 Gyr) abundances for cosmochronometry calculations (see section
2.4). We show these resulting abundance patterns in Figure 3.

Figure 2: Distribution of combined single-$\rm{Y_{e}}$ trajectories. The color
of each line indicates the average $\rm{Y_{e}}$ of the "combined trajectory".
A selection of combinations are shown as a guide to highlight the double-peak
structures within the distribution. This color scheme is used throughout this
work to refer to results for each combined trajectory, with the bluer
combinations having less neutron-rich (low $\rm{Y_{e}}$) material and the more
brown combinations having more neutron-rich material. Figure 3: Abundance
patterns for combined trajectories displayed in Figure 2. The coloring
convention follows that of Figure 2, with brown representing more neutron-rich
combinations, and brown more neutron-poor. Solar abundances (Asplund et al.,
2009) are plotted as black stars.

###  2.2 Nuclear Heating

The evolution of the nuclear heating is given by coupling the energy from the
radioactive decays with the efficiency with which their products thermalize.
We therefore take the outputs from the nucleosynthesis calculations described
and calculate the effective heating rate by combining the Q-value from the
radioactive decays with the thermalization efficiency. We follow the method in
Kasen & Barnes (2019) and Z21, with the total effective heating rate given by

$\dot{Q}(t)=\sum_{i}\dot{q}_{i}(t)f_{i}(M_{ej},v_{ej},t)M_{ej},$ (1)

where the sum is over all reactions and decays. The thermalization, $f_{i}$,
is calculated as in Kasen & Barnes (2019) and is dependent on the ejecta mass
and velocity. We use a total ejecta mass of $0.05M_{\odot}$ and an ejecta
velocity of $0.15c$.

###  2.3 Semi-Analytic Light Curve Model

We construct a semi-analytic light curve model following Z21 and Metzger
(2020). We include some of the relevant details of the calculation here for
convenience. We divide the ejecta into 100 layers with $0.1c<v<0.4c$, with a
velocity-dependent mass distribution and density profile given by

$\displaystyle M_{v}$ $\displaystyle=M_{ej}\left(\frac{v_{o}}{v}\right)^{3}$
(2) $\displaystyle\rho(v,t)$ $\displaystyle=\frac{3M_{v}}{4\pi v^{3}t^{3}}.$
(3)

The evolution of each shell is calculated independently using a forward-Euler
scheme from 0.01 days to 50 days post-merger, obeying

$\frac{dE_{v}}{dt}=\frac{M_{v}}{M_{ej}}\dot{Q}(t,v)-\frac{E_{v}}{t}-L_{v}.$
(4)

The last term,

$L_{v}=\frac{E_{v}}{t_{d,v}+t_{lc}},$ (5)

is the luminosity of a shell and is dependent on the diffusion timescale,
$t_{d,v}$, and the light-crossing time, $t_{lc}$ of that shell:

$\displaystyle t_{\rm d,v}$ $\displaystyle=\frac{M_{ext}\kappa}{4\pi
vtc},\>\text{ and}$ (6) $\displaystyle t_{\rm lc}$
$\displaystyle=\frac{vt}{c}.$ (7)

Here, $M_{ext}$ is the mass exterior to the shell with velocity v. $\kappa$ is
the opacity of the layer, which is calculated as a function of the
temperature, $T_{v}$,

$T_{v}=\left(\frac{E_{v}\rho(v,t)}{aM_{norm}}\right)^{\frac{1}{4}}\\\ $ (8)

of the layer, as:

$\kappa=\begin{cases}\kappa_{\rm
max}\left(\frac{T}{4000\;\rm{K}}\right)^{5.5},\>&\text{ $T<4000$ K }\\\
\kappa_{\rm max}\>&\text{ otherwise }\end{cases}$ (9)

The value of $\kappa_{\rm max}$ depends on the composition of the ejecta.
Specifically, the presence of lanthanides and actinides contribute large
opacities which are important for the "red" component of the light curve that
is relevant on timescales of days (and the focus of this work). Given the
temperature-dependent treatment we have selected (as opposed to a "grey"
opacity), we adopt a value of $\kappa_{\rm max}$ of $100\,\rm cm^{2}g^{-1}$
for all simulations, as this represents the scale of the maximum opacity from
more detailed calculations of low-$\rm{Y_{e}}$, lanthanide-bearing ejecta on
timescales of a few days (Kasen et al., 2013; Tanaka et al., 2020).

###  2.4 Nuclear Cosmochronometry

Table 2:  Names of select r-ii stars with their observed abundances. These stars are sorted by increasing actinide enhancement. Star Name | $\rm{log_{\epsilon}(Eu)}$ | $\rm{log_{\epsilon}(Th)}$ | $\rm{log_{\epsilon}(U)}$ | Reference(s)
---|---|---|---|---
CS22892-052 | -0.95 | -1.57 | -2.3 | Sneden et al. (2003)
HE1523-0901 | -0.62 | -1.2 | -2.06 | Frebel et al. (2007)
CS29497-004 | -0.66 | -1.16 | -2.20 | Hill, V. et al. (2017)
CS31082-001 | -0.72 | -0.98 | -1.92 | Siqueira Mello, C. et al. (2013)
|  |  |  | Hill, V. et al. (2002)
J2038-0023 | -0.75 | -1.24 | -2.14 | Placco et al. (2017)
J0954+5246 | -1.19 | -1.31 | -2.13 | Holmbeck et al. (2018b)

The material produced in the NSM can act as an enrichment source for a nearby
stellar environment, i.e. the final abundance of an _r_ -process-producing
event can be taken as the initial abundance of a star. These can then be
compared with spectral observations of stars, and a time can be obtained by
comparing the measured decay timescales and abundances of radioactive species.
To this end, we make a selection of six _r_ -process-enhanced
($[\rm{Eu}/\rm{Fe}]>+1.0$), metal-poor ($[\rm{Fe}/\rm{H}]<-2$) stars of
varying actinide richness, listed in Table 2.

From these observations, we use the measurements of europium (Z=63), thorium
(Z=90), and uranium (Z=92). We take the initial abundances to be those
produced in a NSM and remaining 1 Gyr post-merger. Running the calculations
out this long allows us to only rely on the theoretical abundances of the
long-lived isotopes of thorium (232Th) and uranium (238U),which have half-
lives of 14 Gyr and 4.47 Gyr, respectively. We compare these with the final
abundances of the two most stable isotopes of europium (151Eu and 153Eu).

If the NSM is taken to be the sole source of _r_ -process enrichment, and is
interpreted as occurring at t=0, then the observed spectra can be interpreted
as being taken at time t given by the following relations:

$\displaystyle t$ $\displaystyle=46.67\ \rm{Gyr}\
[log_{\epsilon}(Th/Eu)_{0}-log_{\epsilon}(Th/Eu)_{\rm{obs}}]$ (10)
$\displaystyle t$ $\displaystyle=14.84\ \rm{Gyr}\
[log_{\epsilon}(U/Eu)_{0}-log_{\epsilon}(U/Eu)_{\rm{obs}}]$ (11)
$\displaystyle t$ $\displaystyle=21.80\ \rm{Gyr}\
[log_{\epsilon}(U/Th)_{0}-log_{\epsilon}(U/Th)_{\rm{obs}}].$ (12)

These compare the initial abundances, $\rm{log_{\epsilon}(X/Y)_{0}}$, to
abundances, $\rm{log_{\epsilon}(X/Y)_{\rm{obs}}}$. This approach, while
useful, has a tendency to yield inconsistent results, especially when applied
to actinide-boost stars, which are overabundant in thorium and uranium
(Holmbeck et al. (2019) classifies these as having $\rm
log_{\epsilon}(Th/Dy)>-0.90$).

We explore the impact of changing the description of beta decay rates on the
final abundance pattern where relevant for cosmochronometry calculations using
the stars listed in Table 2. The abundance patterns we use to perform these
calculations are constructed from the subset of individual trajectories using
the FRDM2012 mass model, with initial electron fractions ranging from 0.01 to
0.35, as described in section 2.1.

##  3 Effective Heating and Light Curve

We demonstrate the influence of beta decay rates on the effective heating and
the light curve by first considering single trajectory models which have a
single value of the initial electron fraction. We then turn to multi-
trajectory models which account for ejecta which has a weighted range of
initial neutron richness, as illustrated in Figure 2. In all cases, we take
the total heating to be the summed contribution of the effective heating (as
described in Section 2.2) from beta decay, spontaneous fission, and alpha
decay reactions.

###  3.1 Single Trajectory

We begin with three different single trajectory models which are chosen to
access different physics. The $\rm{Y_{e}}$$=0.02$ case is chosen to probe very
neutron rich ejecta that experiences fission cycling, where the daughter
products of the first nuclei which fission capture enough neutrons to make it
back to very heavy nuclei that will fission again. The $\rm{Y_{e}}$$=0.18$
case is chosen because a significant number of nuclei fission, but there is
limited cycling, since at this $\rm{Y_{e}}$, the number of neutrons is not
enough to allow nuclei to fission twice. Finally, the $\rm{Y_{e}}$$=0.21$ case
was chosen because material with this neutron richness makes a full r-process
but does not have enough neutrons for much fission to occur.

The first row of Figure 4 shows the range of total heating curves resulting
from these single-$\rm{Y_{e}}$ trajectories of 0.02 (left), 0.18 (center), and
0.21 (right). The width of any one shaded band comes from the use of different
mass models, and corresponding fission barriers, with each band color
corresponding to one set of beta decay rates from Figure 1. Overlap in the
bands for different $\rm{Y_{e}}$ cases appear as a darker region on the plot.
An immediately noticeable trend is that the two lower $\rm{Y_{e}}$ cases have
a wider spread in the prediction of total heating than the highest
$\rm{Y_{e}}$ simulation does. In these lower $\rm{Y_{e}}$ cases, the NES
simulations (darkest blue region) tend to show the highest total heating
rates, i.e. they provide an upper limit for the total heating. Conversely, the
MKT simulations (light blue region) tend to show less total heating and
therefore provide a lower limit for the same.

To explore the reason for these effects, in the next three rows of the figure,
we plot the contribution to the effective heating that stems directly from
beta decaying nuclei (second row), fissioning nuclei (third row) and alpha-
decaying nuclei (bottom row). In all cases, the upper limit of the total
heating shown in the top row is reproduced here as a faint dashed line for
comparison. The narrow width of the beta decay heating bands, as well as their
overlap, indicates that uncertainties in total effective heating, seen in the
top row, cannot directly be attributed to differences in the beta decay
heating. The third and fourth rows of Figure 4 indicate that, in fact, the
largest variation in total heating instead comes from differences in the
contribution of spontaneous fission and alpha decay heating. Looking at the
right-most column, the $\rm{Y_{e}}$$=0.21$ case, we see that the total heating
is dominated by beta decay, with alpha decay and spontaneous fission making up
a relatively small portion of the total heating. As a consequence, the total
heating for this case (top right) exhibits the least variation with different
beta decay rates. At about 1 day, this variation spans only about a factor of
2. However, looking at the left-most column ($\rm{Y_{e}}$$=0.02$), we see that
fission can substantially affect both the total heating and the uncertainty in
the total heating. The choice of beta decay rates is indirectly but strongly
influencing the total heating.

For some low $\rm{Y_{e}}$ simulations, the contribution of fission is sub-
dominant, but for others, fission is the majority contribution, leading to the
substantial spread in the results. Alpha decay plays a similar role in the
middle $\rm{Y_{e}}$ cases (second column), where it largely controls the width
of the band. The NES simulations contribute the upper limit in alpha decay in
the middle $\rm{Y_{e}}$ simulations as well as the upper limit to the heating
at a day. Finally we note that the lower limits on the total heating, which
are very similar between the three sets of beta decay rates, occur in
simulations with minimal alpha decay and fission, and are determined primarily
by the beta decay rates.

Figure 4:  Range of effective heating rate predictions for all nuclear mass
models from Table 1 when a single $\rm{Y_{e}}$ trajectory is considered.
Looking at the columns from left to right, the initial electron fraction
increases from 0.02 to 0.18 to 0.21. The width of each band corresponds to the
range of heating predictions for the NES (dark blue), MLR (pink), or MKT
(light blue) simulations with the different mass models and fission yields for
which we calculate heating. For comparison, the maximum total heating is shown
for each set of simulations as a series of dotted lines.

The shape and peak luminosity of the late-time ("red") light curve are
expected to be substantially influenced by the evolution of the nuclear
heating (Z21, B21). Thus, as described in Section 2.3, we compute the
evolution of the light curve on a timescale of days, out to 40 days post-
merger. Analogous to Figure 4, we show the ranges of these results in Figure
5. We see that the beta decay rates which produce the upper limit in the
overall heating produce a corresponding upper limit in the light curve.
Similarly, the range of uncertainty follows the pattern of heating bands with
the largest variations coming from the lowest $\rm{Y_{e}}$ cases. It is
interesting to note that in the $\rm{Y_{e}}$$=0.21$ scenarios, the full range
of variation is largely captured by the MLR rates.

Figure 5:  Uncertainty range of analytic light curve models for all
theoretical nuclear models (from Table 1, when a single $\rm{Y_{e}}$
trajectory is considered, with the calculation carried out to 40 days post-
merger. The initial electron fraction increases from 0.02 to 0.18 to 0.21 from
top to bottom. The width of each band corresponds to the range of luminosity
predictions when the NES (dark blue), MLR (pink), or MKT (light blue) beta
decay rates are used, and include both 50/50 and KT fission yields.

While Figures 4 and 5 show the broad uncertainty in heating and light curve
evolution that can be obtained by changing the beta decay rates, they do not
show the specific simulations that are sensitive to these changes. The
predictions of heating, and therefore of the light curve, for some mass models
are relatively insensitive to the beta decay rates. On the other hand, the
predictions from other mass models show substantial sensitivity to the beta
decay rates. To illustrate this point, we have separated the total heating
rate results by mass model in Table 3.

For easier comparison, we list the average ratio of total heating for NES:MLR
and MKT:MLR. We also list the ratio of luminosity in parentheses, averaged
over two different time periods: 1-10 days and 10-100 days. For both the
average heating as well as the average luminosity, we highlight those
instances where using a different set of beta decay rates results in a change
of 50% or more in bold. In the following discussion, we refer to these models
with the format nuclearmodel.ye.

|  | 1-10 Days | 10-100 Days
---|---|---|---
Nuclear Model | $\rm{Y_{e}}$ | NES | MKT | NES | MKT
SLY4 | 0.02 |  1.616 (1.338) | 0.856 (0.917) | 1.583 | 0.761
0.18 | 1.797 (1.642) | 1.015 (1.217) | 2.662 | 0.937
0.21 | 1.174 (1.112) | 1.469 (1.374) | 1.078 | 1.118
UNEDF1 | 0.02 | 1.166 (1.231) | 0.823 (0.888) | 1.182 | 1.009
0.18 |  1.517 (1.404) | 0.557 (0.657) | 1.027 | 0.454
0.21 | 0.878 (0.904) | 1.167 (1.16) | 0.801 | 1.125
DZ33 | 0.02 |  1.674 (1.459) | 0.678 (0.82) | 2.224 | 1.089
0.18 |  1.631 (1.416) |  0.432 (0.549) | 0.938 | 0.849
0.21 |  0.416 (0.504) | 0.656 (0.709) | 0.384 | 1.376
ETFSI | 0.02 | 1.114 (1.042) | 0.553 (0.683) | 1.779 | 1.432
0.18 | 1.24 (1.179) | 0.333 (0.499) | 1.799 | 0.841
0.21 | 0.923 (0.914) | 1.241 (1.263) | 1.12 | 1.249
FRDM2012 | 0.02 | 1.27 (1.089) | 0.548 (0.752) | 1.438 | 0.633
0.18 | 1.377 (1.261) |  0.453 (0.556) | 0.785 | 0.333
0.21 | 0.538 (0.589) | 0.642 (0.692) | 0.318 | 0.822
HFB22 | 0.02 | 6.591 (2.806) | 1.215 (0.529) | 4.528 | 1.501
0.18 | 1.069 (0.993) |  0.412 (0.551) | 0.437 | 0.333
0.21 | 0.895 (0.902) | 1.177 (1.2) | 0.949 | 3.147
HFB27 | 0.02 | 2.002 (1.74) | 0.26 (0.348) | 2.085 | 0.399
0.18 | 1.177 (1.083) | 0.302 (0.462) | 0.697 | 0.299
0.21 | 0.962 (0.952) | 1.264 (1.27) | 1.011 | 3.158
TF | 0.02 | 1.25 (1.068) | 0.406 (0.761) | 1.171 | 0.406
0.18 | 1.078 (1.11) | 0.586 (0.786) | 1.818 | 0.364
0.21 | 0.642 (0.719) | 0.54 (0.634) | 0.584 | 0.282
WS3 | 0.02 |  1.549 (1.277) | 0.729 (0.856) | 1.918 | 0.976
0.18 | 1.293 (1.253) | 0.578 (0.664) | 0.709 | 0.828
0.21 | 0.886 (0.893) | 1.389 (1.33) | 1.131 | 4.588
Table 3: Average of the ratios of total effective heating using NES or MKT to
total effective heating using MLR beta decay rates, over different time
periods. In the 1-10 days column, the average ratio of bolometric luminosity
is listed in parentheses. Bold-faced values indicate an average change of $\pm
50\%$ or more.

The heating at the later timescale of 10-100 days is in many instances
dominated by the spontaneous fission of the long-lived 254Cf, which is a
marker for actinide production. However, some models, such as the HFB models,
facilitate the contribution from additional spontaneous fission heaters during
this time. On the other hand, models with lower fission barrier heights tend
to suppress these extra possible contributions to the fission heating, leaving
only that of 254Cf (Z21, B21).

During the earlier time period of 1 - 10 days, the difference in total heating
stems largely from competition between spontaneous fission, alpha decay, and
beta decay heating. This highlights the sensitivity of some mass models to
both $\rm{Y_{e}}$ and beta decay rates. For example, in the case of sly4.18,
we find that the NES simulations show approximately 60% of the total heating
coming from alpha decay by 5 days. Meanwhile, the corresponding MLR simulation
is dominated by beta decay with a contribution from alpha decay that only
rises to about 40% by 7 days. Similarly, in the case of etfsi.18, the MLR
simulations show more than 80% of the total heating coming from alpha decay as
early as 2 days. When the MKT rates are used, there is still a significant
contribution from alpha decay, but only up to a maximum of about 64% around 6
days.

Figure 6: Ratio of beta decay rates used in NES:MLR (top) or MKT:MLR (bottom)
calculations in the region feeding into important alpha decay heaters
(outlined in black boxes) identified in, for example, sly4.18. We show here
only the rates subject to theoretical models; those covered by the Nubase2016
data set are shown in grey.

In both these cases, it was the alpha decays of 212Po, 214Po, or 216Po that
were among the top most significant contributors to the total heating. These
lie in a region where nuclei undergo alpha decay on very short timescales and
with a relatively large Q-value. Because of the very short timescales on which
these decays occur, it is actually the populations of 224Ra and 222Rn (with
half-lives of 3.6 and 3.8 days, respectively) that determine the overall
contribution of the decays of their daughter polonium isotopes. The alpha
decay feeders into, for example, 224Ra and 222Rn decay on timescales that are
too slow (with half-lives of 1.9 years and 1600 years,respectively) to be
directly responsible for differences in heating on a timescale of a few days.
Thus we conclude that the main source of differences lies in the unmeasured
beta decay rates feeding into 224Ra and 222Rn, as well as directly into
212-216Po (highlighted in figure 6); these are critical for determining the
amount of material that is available for alpha decay heating, thereby
determining the dominant source of total heating. We find that it is the
cumulative effect of slight differences in the beta decay rates in the large
feeder region, rather than any one specific feeder nucleus. The use of overall
slower rates (NES) feeding into this alpha decay region resulted in a large
enough heating contribution from alpha decay to dominate significantly over
the beta decay heating that determined the total heating (and light curve) at
earlier times.

Spontaneous fission reactions occurring on time scales of days have the
largest potential to make a significant difference in the overall heating, as
well as the light curve, due to the large Q-values involved as well as the
high thermalization efficiency of the reaction products. We find that,
especially in the cases using the HFB theoretical nuclear models, spontaneous
fission heating has the potential to dominate the total heating as early as 1
day post-merger. The enhanced heating seen in hfb22.02, hfb27.02, and
hfb27.18, for example can be attributed largely to differences in the
predicted spontaneous fission heating rates. However, there is still a great
deal of variety.

By one day post-merger, hfb22.02 shows approximately $77\%$ of the total
heating as coming from spontaneous fission heating, compared to only $33\%$ at
the same time in the corresponding MLR simulation. By three days, the
spontaneous fission heating in the MLR simulation loses out to beta decay
heating, while the NEs simulation shows it continuing to dominate the total
heating out past ten days. There are two mechanisms largely responsible for
this behavior. One involves directly competing theoretical branching ratios
for potential fission heaters. The colored regions of figure 7 indicate where
this occurs, and show the theoretical branching ratios for alpha decay,
spontaneous fission, and beta decay. We found that the isotopes 272No and
271Lr appeared to consistently be responsible for a large part of the total
heating in NES simulations. The significantly slower beta decay rates
predicted in these cases allowed for the spontaneous fission mechanism to
compete with beta decay. Contrarily, the beta decay rates predicted in MLR and
MKT are fast enough to yield a beta decay branching ratio of almost 100%.

The isotopes of Rutherfordium (Z=104) appeared to contribute to different
degrees in the NES and MLR calculations, yet were not among the top heaters in
the MKT calculation. In NES calculations, the heavier isotopes (N=168,169,172)
appeared to contribute the most to the total heating at early times. For
example, the spontaneous fission of 273Rf alone was responsible for
approximately 28% of the total heating at on day post-merger. On the other
hand, it was the "lighter" isotopes (N=166,167) that contributed the most to
the total heating in MLR calculations, and did so for a more extended period
of time in both MLR and NES calculations.

We attribute this to the second mechanism responsible for differences in the
role of fission in our calculations: differences in feeder decay chains that
build up different abundances available for decay. One very obvious example of
this is that of 271Db. This isotope only appears in the MLR calculation, as
this is the only one in which decay into it is allowed. Furthermore, in the
MLR calculation in which it appears, the heating from its decay via
spontaneous fission competes with or even exceeds that of alpha decay, despite
a very small fission branching.

Figure 7: Theoretical branching ratios for a selection of key nuclei that have
the potential to be important spontaneous fission heaters, as identified in,
for example, hfb22.02. The grey-filled isotopes have experimentally determined
values; the colored isotopes remain unmeasured. Light green, teal, and dark
green represent the branchings expected for alpha decay, spontaneous fission,
and beta decay, respectively. In the bottom right panel, we show the chemical
names for highlighted isotopes for ease of identification.

This difference in populations also affects the extent to which fission
heaters with measured rates are able to contribute to the total heating. These
are also highlighted in figure 7, and include 254Cf, 256Cf, and 259Fm. In the
MKT calculation, the population of 259Fm is blocked via beta decay. The
population of its alpha decay feeder, 263No, is also blocked via beta decay,
resulting in 259Fm not being able to contribute significantly to the heating
in MKT calculations. However, in NES calculations, enough material is able to
decay into 259Fm such that its contribution to the spontaneous fission heating
is significant; alone it is responsible for roughly $38\%$ of the total
heating at 2 days post-merger.

While these calculations represent the results obtained by using a
single-$\rm{Y_{e}}$ trajectory, they highlight the influence that the choice
of mass model can have when combined with $\rm{Y_{e}}$. The use of certain
mass models with a given $\rm{Y_{e}}$ unlocked a wide variety of potential
heaters that impacted the evolution of the light curve. We emphasize that
experimental data for several of these unmeasured isotopes would prove highly
valuable in constraining this uncertainty.

###  3.2 Combined Trajectories

Figure 8: Range of effective heating rate predictions of FRDM2012 mass model
when a composite-$\rm{Y_{e}}$ trajectory is considered. From left to right,
the beta decay description used is NES, MLR, MKT. The color scheme corresponds
to that shown in Figure 2. As in Figure 4, we replicate the total heating
curves shown in the top row in the bottom three rows as light grey lines for
easier comparison. Figure 9: Uncertainty range of analytic light curve models
for FRDM2012 nuclear model with multiple-$\rm{Y_{e}}$ composition. From left
to right, the beta decay description used is NES, MLR, MKT. The color scheme
corresponds to that shown in Figure 9, with the most striking blue
representing the behavior of the most neutron-poor material and the darkest
brown that of the most neutron-rich material.

Since it is expected that element synthesis will occur in outflows with a
range of electron fractions, we now turn to the variation in outcomes that is
produced by employing different sets of beta decay rates in scenarios with
multiple trajectories but using a single mass model. Accordingly, Figure 8
shows the nuclear heating obtained from the linear combinations of individual
$\rm{Y_{e}}$ trajectories (combined trajectories) using only the FRDM2012 mass
model, as described in section 2.1. The coloring of the individual lines
corresponds to those in Figure 2, i.e. more brown representing a combination
weighted toward low-$\rm{Y_{e}}$ and bluer representing one weighted toward
high $\rm{Y_{e}}$. The left, center, and right columns represent NES, MLR, and
MKT simulations, respectively. Note that in all cases we have a substantial
fraction of high $\rm{Y_{e}}$ ($\rm{Y_{e}}$$>0.2)$ material that has little to
no fission.

As in Figure 4, the top row of Figure 8 shows the total heating for each of
the combined trajectories. Similarly, the second, third, and fourth rows show
the individual contributions from beta decay, spontaneous fission, and alpha
decay heating, respectively and the total heating is shown as faint grey lines
for comparison. We find the heating out to at least one day is dominated by
beta decay for all three beta sets of beta decay rates. Furthermore, the beta
decay heating is roughly independent of the combined trajectory, as can be see
from a comparison of panels in the second row.

As mentioned, each of the composite trajectories has a significant amount of
high-$\rm{Y_{e}}$ material and this can "dilute" the heating contributions
from spontaneous fission and alpha decay. Indeed, the spontaneous fission
heating does not appear to significantly dominate the shape of the total
heating until tens of days, for any of the combined trajectories. We find the
most potential for early-time (order days) contribution to the total heating
from spontaneous fission in the low-$\rm{Y_{e}}$ weighted NES simulations.
Similarly, the effect of the alpha decay heating is diluted enough that for no
combined trajectory does it ever dominate the total heating. However it is
apparent from Figure 8 that the description of the total heating is not
complete without accounting for both the spontaneous fission as well as the
alpha decay heating.

In Section 3.1, we saw the largest contribution from spontaneous fission in
the lowest $\rm{Y_{e}}$ case. We see this same behavior in the third row of
Figure 8. We see the combined trajectories with the largest proportions of the
lowest-$\rm{Y_{e}}$ material showing the most spontaneous fission heating. As
a larger proportion of high-$\rm{Y_{e}}$ material is included, we begin to see
this dilution effect, and the amount of spontaneous fission heating decreases
accordingly. Section 3.1 also showed the largest amount of alpha decay heating
in the "semi-neutron rich" ($\rm{Y_{e}}$ of 0.18) case. This behavior is
reflected in the fourth row of Figure 8, and is most apparent for the NES
simulations. There is an increase in the contribution of alpha decay heating
as more neutron-rich material is included until a point where the material
becomes too neutron-rich and material is more efficiently deposited into the
higher-Z fissioning region, thus contributing less to the alpha decay heating.

In the case of both spontaneous fission and alpha decay, the point in time at
which all these effects occur depends on the beta decay rates. For example,
the number of days after the merger at which beta decay no longer closely
approximates the total heating occurs sooner in NES simulations than MLR,
which in turn occurs sooner than in MKT simulations. We find that the
contribution of alpha decay heating has the most potential to be significant
in NES simulations that are more heavily weighted toward the middle of our
$\rm{Y_{e}}$ range. There is also a more significant difference between the
potential for significant alpha decay heating for mid-$\rm{Y_{e}}$ combined
trajectories compared to those weighted toward low $\rm{Y_{e}}$ for NES
simulations than both MLR and MKT simulations. In these latter cases, there is
more similarity between low- and mid-$\rm{Y_{e}}$ weighted combined
trajectories. This is consistent with the result obtained in Section 3.1,
where we found the largest contribution to alpha decay heating to be in the
$\rm{Y_{e}}$$=0.18$ case in NES simulations.

We find that the use of a different set of beta decay rates changes the time
scales on which these differences appear, and the extent to which they affect
the total heating. However, when these differences are propagated through the
light curve calculations, we find that the differences are more subtle. We
show the light curves resulting from the combined trajectory effective heating
results in Figure 9, following the same coloring convention as the heating.
The left, middle and right panel show the light curves for NES, MLR and MKT
simulations, respectively. In all three cases, we find that the overall shape
of the light curves are consistent for all three sets of calculations, e.g.
there are no plateaus or bumps present in some but not others.

The most apparent difference in the light curves lies in the behavior after
approximately 4 days. The NES and MLR simulations show similar behavior with
the NES simulations yielding a higher peak luminosity for mid to
high-$\rm{Y_{e}}$ weighted combined trajectories. In addition to a smaller
peak magnitude, the low-$\rm{Y_{e}}$ weighted combinations show an earlier
peak than the more neutron rich trajectories. Furthermore, these tend to decay
more quickly. The MKT calculations yield slightly different results, with the
lowest-$\rm{Y_{e}}$ material yielding the largest peak magnitude. This is
consistent with the total heating behavior observed in the MKT simulations,
shown in the top row of Figure 8. We attribute this to the dominant heating
mechanisms in MKT calculations being beta decay for a longer period of time,
which is largest in the higher-$\rm{Y_{e}}$ weighted trajectories. Thus the
alpha decay heating contribution, which is largest for low-$\rm{Y_{e}}$
dominated trajectories, is unable to compete until later in time. This point
is reflected in the MKT light curves, in which there is a flip, and the
low-$\rm{Y_{e}}$ trajectories have the highest luminosity.

##  4 Applications for Nuclear Cosmochronometry

Figure 10: Maximum uncertainty from the use of different beta decay rates for
the stars listed in Table 2, with each row corresponding to a different star.
The left column shows ages computed using the 50/50 fission yields, while the
right column contains those obtained using the K&T fission yields. As
throughout this work, NES calculations are shown in dark blue, while MLR and
MKT calculations are shown in pink and light blue, respectively. The regions
are filled in to show the range of values obtained using the three different
chronometers, as described in Equations 10-12. Dashed horizontal lines
indicate a value of 13.7 GYr; solid horizontal lines indicate a value of 0
GYr.

Using the same combined trajectories described in Section 2.1 (which all use
the FRDM2012 mass model), we calculate the ages 111We use ”age” as a concise
way to refer to the time since the enrichment event- not necessarily the time
since the formation of the star itself. of several stars, as described in
Section 2.4. We show the range of results of these calculations in Figure 10.
We emphasize that the choice of mass model will quantitatively influence the
results shown in Figure 10 and the analysis here is presented only for FRDM
masses. The left and right columns of Figure 10 use the 50/50 and K&T fission
yields, respectively. To obtain these ranges, we use the quoted observational
values in Table 2 for $\rm log_{\epsilon}(X/Y)_{obs}$ and the theoretical
abundances from each combined trajectory for $\rm log_{\epsilon}(X/Y)_{0}$.
For the purposes of Fig. 10, we have not included observational uncertainties.
We will do so later in this section.

Global beta decay rate predictions are a critical component of
cosmochronomotery studies. For example, the use of theoretically calculated
lanthanide and actinide abundances can yield age estimates that show a
substantial spread depending on the choice of beta decay rates (Holmbeck et
al., 2018a). We show this effect using our calculations in Fig. 10. We show
the maximum uncertainty in age for the six stars in table 2 from NES (dark
blue), MLR (pink), and MKT (light blue) calculations.

Use of the rare earth - actinide chronometer pairs with NES or MLR simulations
tends to produce ages estimates that are high relative to the actinide-only
chronometer. This effect is most pronounced in actinide deficient stars, as
can be seen from top row of Figure 10. We attribute this to an overproduction
of actinides relative to rare earths, as shown in Fig. 11, where we have
plotted the composite, i.e. for the combined trajectories, abundances of
europium (red), thorium (teal), and uranium (pink). The overproduction is
largest for the 50/50 fission yields. This effect is ameliorated with the use
of the K&T fission yields since this model spreads out the fission daughter
nuclei over a larger range of mass number. In all cases, for a more actinide
deficient star, the amount of time that is necessary for the overproduced
actinide content to decay to match observed abundances is larger. We see this
reflected in Figure 10, where the stars are sorted in order of ratio of
increasing actinide to rare earth abundance.

In contrast, the use of MKT beta decay rates in some astrophysical conditions
yields theoretical initial production (Th/Eu) values that are lower than those
observed in some stars. This results in lower ages when Equations 10 and/or 11
are applied, as compared with the actinide-only chronometer. As can be seen in
Figure 11, MKT tends to produce simultaneously less actinides and more rare
earths than do other theoretical formulations of beta decay rates. This effect
is most pronounced with the actinide boost star, J0594, as well as stars at
the high end of actinide normal, e.g CS31802. Again, the effect is mitigated
slightly when using the more diverse fission daughter product distribution
(K&T), which increases somewhat the predicted europium yield, as well as the
yields between $57<Z<63$. Indeed, europium production in K&T simulations shows
less sensitivity to beta decay rates than simulations with the 50/50 fission
daughter product distribution. The lower actinide population in MKT
calculations is consistent with previous calculations (Z21,B21), where the
relatively fast MKT beta decay rates above the $\rm{N=126}$ shell closure
(which can be seen in figure 1) were found to inhibit the buildup of a
significant actinide population.

Figure 11: Composite values of initial production abundances of europium
(red), thorium (teal), and uranium (pink), separated out by beta decay rates
used. NES, MLR, and MKT simulations are shown in the top, middle, and bottom
rows, respectively. The left column shows results obtained using the 50/50
fission yields, and the right column shows those obtained using the K&T
fission yields. The horizontal axis shows average $\rm{Y_{e}}$ (with
increasing neutron richness) of the combined trajectories.

###  4.1 Actinide Constraint

While the actinide:lanthanide ratios yield a large spread of results, thorium
and uranium are generally produced concomitantly, resulting in smaller
uncertainties. In Figure 12 we focus on this uncertainty, obtained solely from
using variation in the theoretical values of $\rm{log_{\epsilon}(U/Th)}$ in
our models. Each colored bar corresponds to the range of results we obtain
using NES (dark blue), MLR (pink), and MKT (light blue) simulations, along
with the 50/50 (plotted on the left) and K&T (plotted on the right) fission
yields. We find from this figure that there is more star-to-star variation,
than variation from the use of different beta decay rates, fission
descriptions, or combined trajectory sets.

We also find here that similar actinide abundances are produced in NES and MLR
simulations, while MKT shows consistently different behavior. The NES and MLR
simulations produced comparable amounts of thorium as well as uranium, with
NES simulations never exceeding a factor 0.8-1.3 times the corresponding MLR
simulation abundance. The result of this is largely overlapping age estimates
stemming from the actinide chronometers, as seen in Figure 12.

In comparison, the MKT simulations yielded roughly only one third the actinide
abundances compared to MLR. However, the difference between thorium and
uranium production within MKT simulations was small, as can be seen in the
bottom panel of Figure 11. The overall effect of this translates into larger
age estimates for MKT simulations, as shown by the light blue error bars in
Figure 12 being consistently centered at larger values than the pink or dark
blue. We point out, though, that the ratio is not necessarily smaller because
either uranium or thorium specifically is less effectively produced. Rather
both are inefficiently produced yielding overall smaller abundances.

Figure 12: Uncertainty in age estimates due to uncertain nuclear physics and
$\rm{Y_{e}}$ for various stellar observations. The ages in this figure are
calculated solely using the U:Th abundance. A horizontal dotted line indicates
the value of 13.7 GYr.

###  4.2 Chronometric Agreement

Figure 13: Full range of predicted age values when the full range of
observational error bars plus the full uncertainty from the composite
trajectories are used. For the values plotted in this figure, the age
predictions made by all three chronometer ratios agree to within 1% of their
average.

An important assumption in Equations 10-12 is that of a single enrichment
event, i.e. that the lanthanide and actinides observed in the r-process
enhanced star stem from the same event which occurred at time $\rm{t_{0}}$.
Beyond this, the chronometry equations make no other assumptions and are
derived from the nuclear decay equation. Hence if the r-process elements in a
given star come from a single event and if the abundances from this event have
been correctly predicted, then the three chronometers should provide the same
age estimates within observational uncertainty.

With this in mind, we begin with our model set which includes all the
combinations of trajectories shown in Figure 2, each computed for all three
different beta decay rates. We then select only those models for which each of
the chronometers yield the same age within the quoted observational error
bars, terming this "chronometric agreement" (or simply "agreement"). We show
the results of this procedure in Figure 13. We see that for many stars no
chronometric agreement exists for our selection of combined trajectories with
the MLR and NES rates, consistent with the results of Fig. 10. However we
caution that all our simulations for this analysis were performed with the
FRDM mass model, and this conclusion may change when a wider variety of
theoretical predictions for off-stability masses are considered. Furthermore,
we have used only the combined trajectories from Fig 2. Other combinations,
particularly those weighted towards even higher values of $\rm{Y_{e}}$ (or
simply more heavily weighted toward the higher-$\rm{Y_{e}}$ range of our
selection) could produce agreement for these stars.

Figure 14: Combined trajectories (as described in Section 2, represented on
the vertical axis by $<\rm{Y_{e}}$) which yield chronometric agreement for
each of the stars in our sample. In this figure we show only those values
obtained with NES (dark blue) and MKT (light blue) calculations, but note that
there was overlap in the case of NES and MLR for CS31082 and J0954 when the
K&T fission yields were used.

On the other hand, for all stars there are some simulations with the MKT rates
that produce chronometric agreement, again due to their more limited
production of actinides as compared to the rare earths. For each star the
range of ages that are in agreement is larger in Fig. 13 than in Fig. 12. This
is because we have taken into account observational uncertainty in the former.

It is interesting to see what sort of distribution of electron fraction is
needed to produce the chronometric agreement. We show instances in which this
occurs in figure 14 for NES (dark blue) and MKT (light blue) calculations. As
is consistent with Figure 13, we find that some or all of the MKT calculations
yield chronometric agreement when observational error bars are taken into
account. In the case of the most actinide poor star, CS22892, all MKT
calculations agree when the K&T fission yields are used. This changes when the
50/50 yields are used, and only those combined trajectories weighted towards
high $\rm{Y_{e}}$ yield agreeing results. Contrarily, for the stars with the
largest actinide enhancement (CS31082 and J0954), MKT calculations show
agreement with those trajectories weighted towards mid and/or low
$\rm{Y_{e}}$.

For CS31082 and J0954, there is overlap in the NES and MLR calculations that
agree when the K&T fission yields are used. This is consistent with previous
work that showed agreement with K&T fission yields for the J0954 using the MLR
and FRDM combination as long as the average $Y_{e}$ was sufficiently high, and
the actinides where "diluted" (Holmbeck et al., 2018a). However, if the
fission daughter product distribution is taken to be 50/50, none of the MLR
trajectories we have considered yield agreement, as do only a selection of NES
calculations with trajectories weighted towards high $\rm{Y_{e}}$. This is in
stark contrast to the aforementioned behavior of the MKT calculations, which
favor low $\rm{Y_{e}}$ weighted trajectories. This contrast is consistent with
the abundance patterns resulting from these calculations, as seen in Figure
11. NES calculations more effectively reproduce a large actinide population,
to the point of overproducing actinides. Thus only a small amount of low
$\rm{Y_{e}}$ material is sufficient to produce a large actinide abundance.

##  5 Conclusion

We performed a targeted study to investigate the impact of global decay rates
on key aspects of _r_ -process nucleosynthesis and kilonova modeling. We
combined three sets of beta decay rates with nine different mass models and
two fission daughter product distributions for our nucleosynthesis
calculations. Furthermore, we considered three single-$\rm{Y_{e}}$
trajectories for the full suite of nuclear inputs in order to probe the role
of fission heating in our calculations. We also considered several ensembles
of trajectories for a subset of nuclear inputs. We compared the abundances
obtained 1 Gyr post-merger for these ensembles to astronomical observations
for six _r_ -process-enhanced metal-poor stars.

For the single trajectory cases, we found a substantial difference in the
predicted total heating from different mass models. The magnitude of this
difference was sensitive to the value of $\rm{Y_{e}}$, and almost entirely due
to differences in the predicted alpha decay and spontaneous fission heating.
This was especially the case in trajectories with initial fraction at or below
0.18, as this was where a significant amount of fission or alpha decay could
occur. We provided a closer investigation of some instances where the change
in beta decay rate translated to an increase of $50\%$ or more in the
bolometric luminosity averaged over one to ten days. We found these increased
luminosities could be attributed to both unmeasured nuclei that feed into
known nuclei (such was the case for the population of, for example, 224Ra), as
well as unmeasured nuclei directly responsible for heating (as was the case,
for example, for several isotopes of Rf).

We found the behavior of the single trajectory calculations was reflected in
the combined trajectories using the FRDM2012 mass model. In these cases, the
competition between alpha decay and spontaneous fission with beta decay was
not as large as in the single-trajectory cases, as all our combined
trajectories had a substantial amount of material with electron fractions
above 0.18. However, even in these circumstances, the description of the
heating past one day is still incomplete without the contribution from fission
and alpha decay. Furthermore, the point in time at which alpha decay and
fission begin to influence the overall magnitude of the heating differs, with
NES contributions becoming relevant prior to one day, MLR at approximately one
day, and MKT predicting a significant contribution closer to ten days.

Finally, we used our calculated abundances of the longest-lived isotopes of
europium, thorium, and uranium to perform cosmochronometry calculations for a
sample of six _r_ -process-enhanced metal-poor stars. We found a larger
uncertainty when we used actinide to europium ratios, as opposed to
uranium:thorium ratios. This is to be expected given the larger separation
between europium and the actinides in the nuclear chart. However, despite the
large uncertainty, we were able to draw interesting conclusions.

One is that the use of different beta decay rates predicted disparities in the
age estimated even from the actinide abundances alone. While there was
significant overlap between the predictions resulting from NES and MLR
actinide abundances, these differed from the MKT abundances, hinting at the
extent to which these different beta decay rates hinder or facilitate actinide
production. Secondly, we were able to use the lanthanide abundances, together
with the observational uncertainties to place a constraint on the age
predictions, in the context of our model. We showed that chronometric
agreement depends on the beta decay rates.

We look forward to additional experimental efforts to measure beta decay
properties (Gade & Sherrill, 2016; Aprahamian et al., 2018; Tain et al., 2018;
Horowitz et al., 2019; Savard et al., 2020; Allmond et al., 2020; Wu et al.,
2020; Schatz et al., 2022), which will greatly help to reduce this source of
uncertainty in the predictions of kilonova light curves and of abundance
predictions. We also look forward to new theoretical predictions of the
thermodynamic conditions in merging neutron stars, of fission yields and
daughter products, and of neutron capture and alpha decay rates, all of which
have an important role to play.

##  6 Acknowledgements

This work was partially supported by the Fission in r-Process Elements (FIRE)
topical collaboration in nuclear theory, funded by the U.S. DOE, contract No.
DE-AC5207NA27344. This work was partially supported by the Office of Defense
Nuclear Nonproliferation Research & Development (DNN R&D), National Nuclear
Security Administration, US Department of Energy. This work was also possible
due to support by the U.S. DOE through Los Alamos National Laboratory,
operated by Triad National Security, LLC, for the National Nuclear Security
Administration of the U.S. DOE. K.L. acknowledges support from the Seaborg
Institute for funding under LDRD project 20210527CR, as well as from the
Center for Nonlinear Studies. J.E. acknowledges support from the Nuclear
Computational Low Energy Initiative (NUCLEI) SciDAC-4 project under U.S.
Department of Energy Grant No. DE-SC0018223 and by the U.S. We acknowledge
support from the NSF (N3AS PFC) grant No. PHY-2020275, as well as from U.S.
DOE contract Nos. DE-FG0202ER41216 and DE-FG0295ER40934. This research was
supported in part by the National Science Foundation under Grant No.
PHY-1430152 (JINA Center for the Evolution of the Elements). This paper is
approved for unlimited release, assigned LA-UR 22-28160.

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