# Constraints on $R$-process Nucleosynthesis from 129I and 247Cm in the Early
Solar System

Projjwal Banerjee,1 Meng-Ru Wu,2,3 and Jeena S K1
1Department of Physics, Indian Institute of Technology Palakkad, Kerala
678558, India
2Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan
3Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, 10617,
Taiwan E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

GW170817 has confirmed binary neutron star mergers as one of the sites for
rapid neutron capture (r) process. However, there are large theoretical and
experimental uncertainties associated with the resulting nucleosynthesis
calculations and additional sites may be needed to explain all the existing
observations. In this regard, abundances of short-lived radioactive isotopes
(SLRIs) in the early solar system (ESS), that are synthesized exclusively by
r-process, can provide independent clues regarding the nature of r-process
events. In this work, we study the evolution of r-process SLRIs 129I and 247Cm
as well as the corresponding reference isotopes 127I and 235U at the Solar
location. We consider up to three different sources that have distinct
129I/247Cm production ratios corresponding to the varied r-process conditions
in different astrophysical scenarios. In contrast to the results found by Côté
et al. (2021), we find that 129I and 247Cm in the ESS do not come entirely
from a single major event but get contributions from at least two more minor
contributors. This has a dramatic effect on the evolution of the 129I/247Cm
ratio, such that the measured ESS value in meteorites may not correspond to
that of the “last” major r-process event. Interestingly, however, we find that
the 129I/247Cm ratio, in combination with the observed 129I/127I and
247Cm/235U ratio in the ESS, can still provide important constraints on the
properties of proposed r-process sources operating in the Milky Way.

###### keywords:

keyword1 – keyword2 – keyword3

††pubyear: 2015††pagerange: Constraints on $R$-process Nucleosynthesis from
129I and 247Cm in the Early Solar System–A

##  1 Introduction

The astrophysical site(s) for the synthesis of heavy elements via the rapid
neutron capture process (r-process) has been a long standing puzzle. The
electromagnetic counterpart following the detection of gravitational waves
from the first ever binary neutron star merger (BNSM) event GW170817 have
confirmed the presence of heavy elements in the merger ejecta and thus
confirming BNSMs as an r-process source (Abbott et al., 2017a, b; Kasen et
al., 2017; Cowperthwaite et al., 2017; Tanaka et al., 2017). Despite this
seminal discovery, very little is known about the details of r-process
nucleosynthesis including the possibility of additional sources that have
varying levels of neutron richness. The additional sources may be needed to
explain the evolution of r-process elements in the Galaxy as well as
neighbouring dwarf galaxies. In particular, the origin of r-process in the
early Galaxy as observed in very metal-poor stars (see e.g., a recent review
by Cowan et al. (2021) and the references therein) as well the evolution
r-process elements in the disk at late times may require additional sources
(Côté et al., 2019a). A number of sites have been proposed as candidates for
additional r-process sources that include NS – black hole (BH) mergers
(Lattimer & Schramm, 1974; Rosswog, 2005; Kyutoku et al., 2013; Foucart et
al., 2014), neutrino-driven winds from core-collapse supernovae (CCSNe)
(Woosley et al., 1994; Takahashi et al., 1994; Qian & Woosley, 1996; Arcones &
Thielemann, 2013), magneto-rotational supernovae (MRSNe) (Nishimura et al.,
2006; Winteler et al., 2012; Mösta et al., 2018), collapsars (Siegel et al.,
2019; Miller et al., 2020), accretion disk outflows during the common envelope
phase of NS–massive star system (Grichener & Soker, 2019), CCSNe triggered by
the hadron-quark phase transitions (Fischer et al., 2020), etc.

In this regard, measurement of the abundances of short-lived radioactive
isotopes (SLRIs) that are exclusively produced by r-process in the early solar
system (ESS), as determined from meteorites as well as Earth’s deep-sea
sediments, can be used to infer the properties of r-process events that
occurred in the Solar neighbourhood (Wallner et al., 2015; Hotokezaka et al.,
2015a; Bartos & Marka, 2019; Côté et al., 2021; Wallner et al., 2021). In a
recent study Côté et al. (2021) showed that the observed ratio of the
abundances of r-process SLRIs 129I and 247Cm present in the ESS can be used to
constrain the “last” r-process event that contributed to the solar abundance
before the formation of the SS. In particular, the authors showed that due to
the almost identical lifetimes of SLRIs 129I and 247Cm, the value of their
ratio is insensitive to the uncertainties associated with Galactic chemical
evolution and the observed ratio of $438\pm 184$ corresponds to the value
produced by the “last” r-process event. Additionally, using the fact that
129I/247Cm is extremely sensitive to the neutron richness, they concluded that
the observed value is consistent with medium neutron rich conditions that are
most likely associated with the disk ejecta following BNSMs (Fernández &
Metzger, 2013; Just et al., 2015; Fujibayashi et al., 2018; Miller et al.,
2019). Tidally-disrupted ejecta from NS–BH mergers or BNSMs (Freiburghaus et
al., 1999; Goriely et al., 2011; Korobkin et al., 2012) produce 129I/247Cm
ratios that are too low whereas those from the MRSNe result in values that are
too high to be compatible with the measurement111We note that recent BNSM
simulations that include the effect of weak interactions generally predict
less neutron-rich conditions in the early-time ejecta, particularly for the
component that originates from the collisional interface of the two NSs
(Wanajo et al., 2014; Radice et al., 2016; Bovard et al., 2017; Vincent et
al., 2020; George et al., 2020; Kullmann et al., 2021). However, the exact
electron fraction ($Y_{e}$) distribution of this component still vary
substantially in different works and require further improved treatment of
weak interactions in simulations..

The above conclusion reached by Côté et al. (2021) is based on the result that
for the typical values of frequency of various r-process sources, the
probability of more than one source contributing to 129I and 247Cm in the ESS
is negligible based on a stochastic chemical evolution model. In this paper we
use the turbulent gas diffusion formalism similar to Hotokezaka et al. (2015b)
to simulate the evolution of r-process isotopes at the birth location of the
Sun. We find that the “last” major r-process event only accounts for $\sim
50$–$75\%$ of the total 129I and 247Cm measured in the ESS and at least three
events are required to account for $\gtrsim 95\%$. The minor contributing
events can dramatically affect the 129I/247Cm ratio when there are more than
one r-process source with distinct production ratios for 129I/247Cm
(corresponding to different levels of neutron richness). Consequently, we find
that the measured meteoritic value may not correspond to the value due to a
single “last” r-process event. Surprisingly, we find that the 129I/247Cm ratio
can nevertheless provide important constraints on the neutron richness of
r-process when it is combined with the meteoritic measurement for the
129I/127I and 247Cm/235U ratio in the ESS.

##  2 Method

In order to calculate the abundance evolution of r-process elements, the
frequency of r-process events as well as their locations are required. The
frequency of an r-process source depends on the star formation history (SFH)
of the Milky Way. We use the omega+ code along with the parameters for the
best-fit Milky Way model of Côté et al. (2019b) to calculate the SFH and the
resulting CCSN rate and mass loss factor $f(t)$ (see Appendix A). The rate of
r-process events like BNSMs as well as the rate of formation of neutron star
binaries (NSBs) are derived by assuming a fraction $f_{r}$ of CCSNs result in
the formation of NSBs. We adopt a fiducial value of $f_{r}=8\times 10^{-4}$
that corresponds to a merger frequency $\nu_{0}\sim 10$ Myr-1 at the present
time.

We simulate multiple realisations of r-process events in the Milky Way. For
each realisation, the birth times for binaries, which lead to BNSM like
events, are generated from a probability distribution proportional to the NSB
birth rate calculated using omega+. The cylindrical radial coordinates for the
birth locations in the MW are generated according to a distribution $\propto
R\exp(-R/R_{d})$, where $R_{d}$ is the radial scale length for the surface SFR
density. In order to account for the inside-out formation of the MW, we adopt
time dependent $R_{d}$ from Schönrich & McMillan (2017). The corresponding
vertical heights are generated according to the distribution of the estimated
molecular and atomic gas density in the local solar neighbourhood by McKee et
al. (2015). The merger time $t_{\rm merge}$ for each NSB is sampled from a DTD
$\propto t^{-1}$ with a minimum and maximum delay time of 10 Myr and 10 Gyr,
respectively. Each NSB is assigned a kick velocity $\vec{v}_{\rm kick}$ at the
time of its birth. The magnitude of $\vec{v}_{\rm kick}$ is generated from a
distribution $\propto\exp(-v_{\rm kick}/v_{0})$ with a fiducial value of
$30\,{\rm km\,s}^{-1}$, whereas the direction of the kick is generated from a
uniform isotropic distribution. In order to find the final location of the NSB
at the time of its merger, we use galpy (Bovy, 2015) to trace the motion of
the NSB under the influence of the Galactic potential as described in Banerjee
et al. (2020). The birth time and location for MRSN or collapsar like events
are generated similar to the NSBs as described above. In this case, however,
there are no delays or offsets due to natal kicks.

In order to calculate the evolution of r-process elements at the Solar
location, we use the turbulent gas diffusion formalism by Hotokezaka et al.
(2015b). The number density of an isotope at a location $\vec{r}$ and time $t$
is given by

$n(\vec{r},t)=\sum_{t_{j}<t}\frac{Y_{r,j}e^{-f(t)\Delta t_{j}}}{K_{j}(\Delta
t_{j})}\exp\left[-\frac{\left|\vec{r}-\vec{r}_{j}\right|^{2}}{4D\Delta
t_{j}}-\frac{\Delta t_{j}}{\tau}\right],$ (1)

where $t_{j}$ is the occurrence time for the $j$th r-process event, $\Delta
t_{j}=t-t_{j}$, $Y_{r,j}$ is the number of atoms of the isotope produced by
the $j$th r-process event, $f(t)$ is the time-dependent loss factor due to
star formation and galactic outflows calculated from Milky Way model using
omega+, $\tau$ is the lifetime of the isotope. $D$ is the turbulent diffusion
coefficient, and $K_{j}(\Delta t_{j})$ is given by

$K_{j}(\Delta t_{j})=\mathrm{min}[(4\pi D\Delta t_{j})^{3/2},8\pi h_{z}D\Delta
t_{j}],$ (2)

where $h_{z}=0.3$ kpc is the vertical scale height.

The overall mixing of metals depend on the parameters $D$ and the frequency
$\nu$ of r-process events with typical mixing timescale $\tau_{\rm mix}$ given
by (Hotokezaka et al., 2015b)

$\tau_{\rm mix}\approx 200\left(\frac{\nu}{30~{}{\rm
Myr}^{-1}}\right)^{-2/5}\left(\frac{D}{0.1~{}{\rm
kpc^{2}~{}Gyr^{-1}}}\right)^{-3/5}~{}{\rm Myr}$ (3)

A recent study by Beniamini & Hotokezaka (2020) found that in order to satisfy
various independent observational constraints such as scatter of stable
r-process elements, highest observed r-process enrichment in the solar
neighbourhood, as well as constraints from radioactivity in the ESS, it
requires that $D\gtrsim 0.1~{}{\rm kpc^{2}~{}Gyr^{-1}}$ and $\nu\lesssim
40~{}{\rm Myr}^{-1}$ with typical mixing timescale of $\tau_{\rm mix}\approx
200$ Myr. This also means that $D$ and $\nu$ are related to each other by

$D\approx 0.3\left(\frac{\nu}{10~{}{\rm Myr}^{-1}}\right)^{-2/3}~{}{\rm
kpc^{2}~{}Gyr^{-1}}.$ (4)

Because $\tau_{\rm mix}$ is a fixed parameter, varying the value of $\nu$ is
always associated with a corresponding change in $D$. Thus, it is sufficient
to consider a single value of $\nu$. In our Milky Way model, the current rate
of $\nu_{0}=10~{}{\rm Myr^{-1}}$ corresponds to a rate of r-process events of
$\nu\approx 15~{}{\rm Myr^{-1}}$ at the time of SS formation of $t_{\odot}\sim
9.2$ Gyr. We adopt values of $D=0.1$ and $0.3~{}{\rm kpc^{2}~{}Myr^{-1}}$ that
covers values of $D$ slightly above and below the corresponding values given
by Eq. 4 for $\nu\approx 15~{}{\rm Myr^{-1}}$ and correspond to $\tau_{\rm
mix}\approx 140$– $260$ Myr.

Table 1: List of important parameters used in this work. Parameter | Definition | Values | Unit
---|---|---|---
$D$ | Turbulent diffusion coefficient | 0.1, 0.3 | ${\rm kpc^{2}~{}Myr^{-1}}$
$\nu$ | Total r-process frequency during SS formation | $\sim 15$ | ${\rm Myr}^{-1}$
$\nu_{LM}$ | Ratio of frequency of sources $S_{L}$ to $S_{M}$ | 1 | –
$\nu_{LH}$ | Ratio of frequency of sources $S_{L}$ to $S_{H}$ | 1 | –
$M_{LM}^{\rm ej}$ | Ratio of ejecta masses of sources $S_{L}$ to $S_{M}$ | 1/3,1,3 | –
$M_{LH}^{\rm ej}$ | Ratio of ejecta masses of sources $S_{L}$ to $S_{H}$ | 1/3,1,3 | –
$\lambda_{L}$ |  129I/247Cm production ratio for $S_{L}$ | 10 | –
$\lambda_{L}$ |  129I/247Cm production ratio for $S_{M}$ | 100 | –
$\lambda_{H}$ |  129I/247Cm production ratio for $S_{H}$ | 1000 | –

We compute the number abundance of r-process isotopes using the diffusion
prescription discussed above at the solar radius $R_{\odot}$ at the time when
the gas decoupled from the inter stellar medium (ISM) at $t=t_{\rm iso}$ to
form the SS. In this work, we consider three different types of r-process
sources $S_{L}$, $S_{M}$, and $S_{H}$ defined by distinct 129I/247Cm
production ratios of $\lambda_{L}$, $\lambda_{M}$, and $\lambda_{H}$. The
subscripts $L$, $M$, and $H$ refer to low, medium and high values,
respectively, for the 129I/247Cm production ratio corresponding to different
astrophysical sites with varying neutron richness. We adopt fiducial values of
$\lambda_{L}=10$, $\lambda_{M}=100$, and $\lambda_{H}=1000$. The adopted
values roughly correspond to values expected from neutron rich dynamical
ejecta during a BNSM or NS–BH merger ($\lambda_{L}$), moderately neutron rich
disk ejecta following BNSMs ($\lambda_{M}$), and a low neutron rich ejecta
from MRSN events ($\lambda_{H}$) from theoretical models reported in Côté et
al. (2021). For each r-process event the isotopic production ratio is taken to
be 129I/127I= 1.46, 247Cm/235U= 0.20, 244Pu/238U= 0.40, and 235U/238U= 1.05.
These values for actinide ratios are generally consistent with the predictions
from Mendoza-Temis et al. (2015) and Wu et al. (2016) which computed the
r-process nucleosynthesis in the BNSM dynamical ejecta and in the BH–accretion
disk outflows using different nuclear physics inputs. The value for 129I/127I
ratio corresponds to the solar r-process value adapted from Sneden et al.
(2008). We also account for the contribution of s-process to the ESS value of
127I by multiplying the a factor 1.06 which is consistent the solar abundance
decomposition by Sneden et al. (2008).

We consider three scenarios, where only two out of the three different types
of r-process sites contribute. Additionally, we consider the scenario where
all three types of sources contribute. The relevant parameters that impact the
evolution of isotopic ratios of SLRs produced by r-process are the frequency
$\nu_{i}$ of each r-process source, the corresponding production ratio
$\lambda_{i}$, the relative ratio of the ejected mass $M^{\rm ej}_{ij}=M^{\rm
ej}_{i}/M^{\rm ej}_{j}$ for sources $S_{i}$ and $S_{j}$, and the value of the
diffusion coefficient $D$. We list the definitions and the adopted values of
relevant model parameters in Table 1.

Figure 1: Evolution of 129I/127Cm at the solar location for models with two
equally frequent ($\nu_{LM}=1$) sources $S_{L}$ and $S_{M}$ with
$\lambda_{L}=10$ and $\lambda_{M}=100$ with $D=0.1$ kpc2 Gyr-1 for three
different values of $M^{\rm ej}_{LM}$.

##  3 Results

We first consider two r-process sources $S_{L}$ and $S_{M}$ that are BNSM
like. We assume that both of them are equally frequent, i.e.,
$\nu_{LM}=\nu_{L}/\nu_{M}=1$ with $\nu_{L}+\nu_{M}=\nu$, and have the same
kick velocity distribution. For their relative ejecta mass ratios, we consider
three different values of $M^{\rm ej}_{LM}=1/3,1,$ and $3$. $M^{\rm
ej}_{LM}=1$ corresponds to the scenario where both $S_{L}$ and $S_{M}$
contribute equally to the main r-process. The other two values of $M^{\rm
ej}_{LM}$ correspond to scenarios where one of the sources is the dominant
site for the main r-process. Figure 1 shows the evolution of 129I/247Cm ratio
at the solar location for $D=0.1$ kpc2 Gyr-1 with three different values of
$M^{\rm ej}_{LM}$ for one single representative realisation. As expected, the
value of the ratio varies between $\lambda_{L}$ and $\lambda_{M}$. However,
the ratio not only takes extremal values, but also assumes values of $\sim
30-70$ for substantial lengths of time. This occurs even when the ejecta mass
is higher for one of the sources i.e, $M^{\rm ej}_{LM}=1/3$ and $3$.

Table 2: Probability distribution of 129I/247Cm ratio in the ESS for criteria T0 (see text) for two equally frequent r-process sources $S_{L}$ and $S_{M}$ with $\lambda_{L}=10$, $\lambda_{M}=100$. Unit for $D$ is kpc2 Gyr-1. Model Parameters | Probability of ESS 129I/247Cm Ratio within an interval |  |
---|---|---|---
$(D,M^{\rm ej}_{LM})$ | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | 50–70 | 70–80 | 80–90 | 90–110
(0.3,1/3) | 0.41 | 0.11 | 0.07 | 0.04 | 0.04 | 0.04 | 0.05 | 0.08 | 0.18
(0.1,1/3) | 0.45 | 0.06 | 0.04 | 0.03 | 0.03 | 0.03 | 0.03 | 0.04 | 0.30
(0.3,1) | 0.56 | 0.09 | 0.06 | 0.05 | 0.04 | 0.03 | 0.03 | 0.04 | 0.09
(0.1,1) | 0.63 | 0.06 | 0.04 | 0.03 | 0.01 | 0.02 | 0.02 | 0.03 | 0.17
(0.3,3) | 0.75 | 0.07 | 0.04 | 0.03 | 0.02 | 0.02 | 0.02 | 0.02 | 0.03
(0.1,3) | 0.75 | 0.04 | 0.03 | 0.02 | 0.02 | 0.01 | 0.02 | 0.03 | 0.10

###  3.1 Results with Minimum Criteria

The formation of meteorites in the ESS is always preceded with some isolation
time. This is because there is always a interval $\Delta_{\rm iso}$ between
the time when the molecular cloud, from which the SS formed, decouples from
the ISM at $t_{\rm iso}$ and the formation of the SS at $t_{\odot}=t_{\rm
iso}+\Delta_{\rm iso}$. During this time, the gas from which the SS was formed
does not receive any contribution for r-process isotopes and the SLRs free
decay for a period of $\Delta_{\rm iso}$. Thus, in order for the isotope ratio
129I/127I and 247Cm/235U to be a viable candidate for the measured value at
ESS formation time, the SLR isotopic ratio resulting from chemical evolution
model has to be greater than the ESS meteoritic values. Because SLR 244Pu is
also produced exclusively by r-process, this also applies to the 244Pu/238U
ratio. We define this as the minimum criteria and refer to this criteria as T0
with the adopted observed ESS abundance of 129I/127I$=1.28\times 10^{-4}$
(Ott, 2016), 247Cm/235U$=5.6\times 10^{-5}$ (Tang et al., 2017), and
244Pu/238U$=7\times 10^{-3}$ (Hudson et al., 1989). We simulate 1000
realisations of r-process enrichment in the Milky Way that satisfy criteria
T0. This gives us a distribution of ESS value of the 129I/247Cm ratio that
ranges roughly from $\lambda_{L}$ to $\sim\lambda_{M}$. The distribution
allows us to directly calculate the probability of finding the ESS 129I/247Cm
ratio within a given interval between $\lambda_{L}$ and $\sim\lambda_{M}$.
Table 2 shows the results using $\lambda_{M}=100$ and $\lambda_{L}=10$ for
three different choices of $M^{\rm ej}_{LM}$, and two different values of $D$.
Interestingly, for all cases, the probability always peaks at values close to
$\lambda_{L}$. In all cases with $D=0.3$ kpc2 Gyr-1, there is a $\gtrsim
12$–$20\%$ probability of getting intermediate 129I/247Cm values of $30$–$70$
that is comparable to the probability of getting values close to
$\lambda_{M}$. When $D=0.1$ kpc2 Gyr-1, the probability for intermediate
values are somewhat lower but is still comparable to the probability for
getting values close to $\lambda_{M}$.

In order to understand the probability distribution of the 129I/247Cm ratio,
the history of enrichment of r-process SLRs at the solar location needs to be
analysed. In general, the abundance of r-process isotopes at the solar
location $R_{\odot}$ at $t_{\odot}$ receives contribution from all r-process
events that has occurred before $t_{\rm iso}$. However, for SLRs, only a
handful of events that has occurred at $t^{\prime}$ where $t_{\rm
iso}-t^{\prime}$ is within a few lifetimes of the SLRs at locations relatively
close to the Sun can contribute. For any isotope including SLRs, the
contribution from all the past events at a particular $(R_{\odot},t)$ can be
sorted in terms of the fraction contributed to the total amount. It is useful
to define the quantity $f^{\rm iso}_{\rm h1}$ as the highest fraction of the
ESS value contributed by a single event for a particular isotope. This is the
same as $f_{\rm last}$ defined by Beniamini & Hotokezaka (2020). We can
further define $f^{\rm iso}_{\rm h2}$ and $f^{\rm iso}_{\rm h3}$ as the
fractions contributed by the second and third highest contributors for a
particular isotope, respectively. Table 3 shows the values of $f_{\rm h1}$,
$f_{\rm h2}$, and $f_{\rm h3}$ for 129I corresponding to the models listed in
Table 2. As can be seen from the table, on average, the single major
contributor accounts for only $70$–$84\%$ and at least three events are
required to account for $\gtrsim 95\%$ of the total 129I in the ESS.

Table 3: The mean and median values of $f_{\rm h1}$, $f_{\rm h2}$, and $f_{\rm h3}$ along with the 95th percentile range for 129I for the models listed in Table 2. Unit for $D$ is kpc2 Gyr-1. $(D,M^{\rm ej}_{LM})$ |  | $f^{\rm{}^{129}I}_{\rm h1}$ | $f^{\rm{}^{129}I}_{\rm h2}$ | $f^{\rm{}^{129}I}_{\rm h3}$ |  $f^{\rm{}^{129}I}_{\rm h1+h2+h3}$
---|---|---|---|---|---
(0.3,1/3) | mean | 0.702 | 0.167 | 0.062 | 0.931
median | 0.714 | 0.158 | 0.039 | 0.961
95th percentile | 0.350–0.988 | 0.006–0.394 | 0.002–0.191 | 0.753–0.999
(0.1,1/3) | mean | 0.838 | 0.119 | 0.028 | 0.985
median | 0.908 | 0.062 | 0.006 | 0.997
95th percentile | 0.488–0.999 | 0.000–0.400 | 0.000–0.128 | 0.923–1.000
(0.3,1) | mean | 0.701 | 0.164 | 0.062 | 0.927
median | 0.705 | 0.155 | 0.044 | 0.959
95th percentile | 0.349–0.9991 | 0.005–0.374 | 0.001–0.189 | 0.746–0.999
(0.1,1) | mean | 0.823 | 0.127 | 0.032 | 0.982
median | 0.893 | 0.076 | 0.008 | 0.996
95th percentile | 0.497–0.999 | 0.000–0.398 | 0.000–0.142 | 0.913–1.000
(0.3,3) | mean | 0.719 | 0.158 | 0.058 | 0.936
median | 0.753 | 0.140 | 0.033 | 0.970
95th percentile | 0.348–0.990 | 0.005–0.384 | 0.001–0.190 | 0.772–0.999
(0.1,3) | mean | 0.826 | 0.124 | 0.031 | 0.982
median | 0.906 | 0.068 | 0.007 | 0.997
95th percentile | 0.459–0.999 | 0.000–0.395 | 0.000–0.148 | 0.905–1.000

The contributions from the second and third highest contributors have a
critical impact on the 129I/247Cm ratio. To illustrate this, let us consider
the case with $D=0.1$ kpc2 Gyr-1 and $M^{\rm ej}_{LM}=1$ where $\approx 99\%$
of the total 129I is produced by three events with mean values of $f_{\rm
h1}=0.84$, $f_{\rm h2}=0.12$, and $f_{\rm h3}=0.03$. In this case both $S_{L}$
and $S_{M}$ produce the same amount of 129I whereas the former produces a
factor 10 higher amount of 247Cm. If $N_{0}$ denotes the number of atoms of
129I produced by each source, then the corresponding yield of 247Cm is
$0.1N_{0}$ and $0.01N_{0}$ for $S_{L}$ and $S_{M}$, respectively. For
simplicity, let us assume that the three highest contributing events account
for all the 129I. In this case, for these three events $({\rm h1,h2,h3})$, the
8 different possible combinations are $(S_{L},S_{L},S_{L})$,
$(S_{L},S_{L},S_{M})$, $(S_{L},S_{M},S_{L})$, $(S_{L},S_{M},S_{M})$,
$(S_{M},S_{M},S_{M})$, $(S_{M},S_{L},S_{M})$, $(S_{M},S_{L},S_{L})$, and
$(S_{M},S_{M},S_{L})$, which all have equal occurrence probability of $12.5\%$
and produce equal amounts of 129I ($0.99N_{0}$ atoms). Summing over the
produced 247Cm atoms gives rise to values of 129I$/^{247}$Cm equal to 10.1,
10.4, 11.3, 11.7, 101.0, 48.3, 42.7, 79.4 for these 8 scenarios, respectively.
Clearly, the first four combinations ($50\%$ probability) where the dominant
129I contributor ${\rm h1}=S_{L}$, the total number of 247Cm atoms is
dominated by the $S_{L}$ source, leading to 129I/247Cm ratios close to
$\lambda_{L}=10$. In contrast, in the latter 4 combinations where ${\rm
h1}=S_{M}$, except for the case $(S_{M},S_{M},S_{M})$, the total 247Cm gets
substantial contribution from $S_{L}$ sources from ${\rm h2}$ and ${\rm h3}$
events as at least one them is of $S_{L}$. Thus, in the latter 4 cases,
129I/247Cm ratio is close to $\lambda_{M}$ only for $(S_{M},S_{M},S_{M})$ with
a probability of $12.5\%$. For the other three combinations, the 129I/247Cm
ratio takes intermediate values.

Table 4: Probability distribution of 129I/247Cm ratio in the ESS for concordance criteria T10 and T20 (see text) for the models listed in Table 2. Values with the minimal criteria T0 are the same as in Table 2. Unit for $D$ is kpc2 Gyr-1. Model |  | Probability of 129I/247Cm within an interval |  |
---|---|---|---|---
$(D,M^{\rm ej}_{LM})$ | Criteria | 10–20 | 20–30 | 30–40 | 40–50 | 50–60 | 60–70 | 70–80 | 80–90 | 90–110
\- (0.3,1/3) | T0 | 0.41 | 0.11 | 0.07 | 0.04 | 0.04 | 0.04 | 0.05 | 0.08 | 0.18
T10 | 0.00 | 0.10 | 0.56 | 0.32 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.00 | 0.24 | 0.35 | 0.28 | 0.13 | 0.00 | 0.00 | 0.00 | 0.00
(0.1,1/3) | T0 | 0.45 | 0.06 | 0.04 | 0.03 | 0.03 | 0.03 | 0.03 | 0.04 | 0.30
T10 | 0.00 | 0.10 | 0.59 | 0.29 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.00 | 0.22 | 0.34 | 0.27 | 0.15 | 0.02 | 0.00 | 0.00 | 0.00
(0.3,1) | T0 | 0.56 | 0.09 | 0.06 | 0.05 | 0.04 | 0.03 | 0.03 | 0.04 | 0.09
T10 | 0.32 | 0.68 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.46 | 0.47 | 0.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.1,1) | T0 | 0.63 | 0.06 | 0.04 | 0.03 | 0.01 | 0.02 | 0.02 | 0.03 | 0.17
T10 | 0.35 | 0.63 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.50 | 0.39 | 0.11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.3,3) | T0 | 0.75 | 0.07 | 0.04 | 0.03 | 0.02 | 0.02 | 0.02 | 0.02 | 0.03
T10 | 0.98 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.93 | 0.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.1,3) | T0 | 0.75 | 0.04 | 0.03 | 0.02 | 0.02 | 0.01 | 0.02 | 0.03 | 0.10
T10 | 0.98 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.95 | 0.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00

Figure 2: Probability distribution of 129I/127Cm ratio in the ESS for two
equally frequent r-process sources $S_{L}$ and $S_{M}$ with $\lambda_{L}=10$
and $\lambda_{M}=100$ for $M_{LM}^{\rm ej}=1$ (black), 3 (red), and 1/3 (blue)
and tolerances T0 (left vertical panel), T10 (middle vertical panel), and T20
(right vertical panel) corresponding to the values listed in Table 4. All
models have $D=0.1$ kpc2 Gyr-1.

The probabilities estimated from above simple illustration using the mean
values of $f_{\rm h1}$, $f_{\rm h2}$, and $f_{\rm h3}$ agrees qualitatively
with the values listed Table 2. The quantitative differences is due to the
fact that the mean value does not fully represent the distribution of $f_{\rm
h1}$, $f_{\rm h2}$, and $f_{\rm h3}$. This is particularly true for models
with lower value of $D=0.1$ kpc2 Gyr-1 which has distribution with a long
tail, as evident from the large 95th percentile ranges along with large
differences between the mean and median values. Nevertheless, the simple
analysis clearly illustrates the fact that 247Cm can receive substantial
contribution from the subdominant ${\rm h2}$ and ${\rm h3}$ events and
directly impacts the 129I/247Cm ratio. In particular, this shows why the
probability distribution peaks at values closer to $\lambda_{L}$ rather than
$\lambda_{M}$ and why it also takes intermediate values.

Table 5: The mean values of $f_{\rm h1}$, $f_{\rm h2}$, and $f_{\rm h3}$ for 129I for the models listed in Table 2. Values with the minimal criteria T0 are the same as in Table 3. Unit for $D$ is kpc2 Gyr-1. $(D,M_{LM}^{ej})$ | Criteria | $f^{\rm{}^{129}I}_{\rm h1}$ | $f^{\rm{}^{129}I}_{\rm h2}$ | $f^{\rm{}^{129}I}_{\rm h3}$ |  $f^{\rm{}^{129}I}_{\rm h1+h2+h3}$
---|---|---|---|---|---
(0.3,1/3) | T0 | 0.702 | 0.167 | 0.062 | 0.931
T10 | 0.693 | 0.196 | 0.071 | 0.960
T20 | 0.591 | 0.196 | 0.094 | 0.882
(0.1,1/3) | T0 | 0.838 | 0.119 | 0.028 | 0.985
T10 | 0.690 | 0.198 | 0.073 | 0.961
T20 | 0.691 | 0.198 | 0.070 | 0.960
(0.3,1) | T0 | 0.701 | 0.164 | 0.062 | 0.927
T10 | 0.458 | 0.292 | 0.116 | 0.865
T20 | 0.483 | 0.273 | 0.111 | 0.868
(0.1,1) | T0 | 0.823 | 0.127 | 0.032 | 0.982
T10 | 0.525 | 0.340 | 0.087 | 0.952
T20 | 0.561 | 0.311 | 0.085 | 0.957
(0.3,3) | T0 | 0.719 | 0.158 | 0.058 | 0.936
T10 | 0.639 | 0.249 | 0.072 | 0.959
T20 | 0.626 | 0.200 | 0.079 | 0.905
(0.1,3) | T0 | 0.826 | 0.124 | 0.031 | 0.982
T10 | 0.656 | 0.246 | 0.063 | 0.965
T20 | 0.768 | 0.168 | 0.040 | 0.977

###  3.2 Results with Concordant Decay Interval

So far, we have considered Milky Way realisations that satisfy the minimum
criteria T0, namely, that the ratios of 129I/127I, 247Cm/235U, and 244Pu/238U
are higher than the mean measured values in the ESS. This, however, does not
guarantee that the ratios are actually compatible with the measurements. As
mentioned before, after the star forming gas in the molecular or giant
molecular cloud decouples from the ISM, each SLR undergoes free decay for the
same length of isolation time $\Delta_{\rm iso}$ before the formation of
meteorites in the ESS (Wasserburg et al., 2006; Lugaro et al., 2018). The
value of $\Delta_{\rm iso}$ can be directly calculated for each SLR separately
from the relation

$\left(\frac{N_{R}}{N_{I}}\right)_{\rm
ESS}\simeq\left(\frac{N_{R}}{N_{I}}\right)_{\rm iso}e^{-\Delta_{\rm
iso}/\tau_{R}},$ (5)

where $(N_{R}/N_{I})_{\rm ESS}$ is the meteoritic ratio of the SLR $R$ with
respect to the stable or long-lived isotope $I$ and $(N_{R}/N_{I})_{\rm iso}$
is the corresponding ratio when the star forming gas decouples from the ISM at
$t_{\rm iso}$222The result in Eq. 5 holds exactly when the long-lived isotope
$I$ is stable or when the SLR $R$ does not decay to $I$.. In order for the
ratios to be compatible with the ESS measurements, the value of $\Delta_{\rm
iso}$ calculated for 129I/127I, 247Cm/235U, and 244Pu/238U need to match. This
criteria, however, is not meaningful as the probability for the three
different values of $\Delta_{\rm iso}$ to be exactly identical is zero. We can
however, define compatibility within some tolerance range about the mean
measured value of $(N_{R}/N_{I})_{\rm ESS}$. In this case, for each of the
three SLR to stable isotope ratios, we get a range of $\Delta_{\rm
iso}=(\Delta^{\rm min}_{\rm iso},\Delta^{\rm max}_{\rm iso})$. We consider a
realisation to be compatible (concordant) with the ESS measurements if the
range of the three $\Delta_{\rm iso}$ values overlap.

Table 6: Probability distribution of 129I/247Cm ratio in the ESS for concordance criteria T0, T10 and T20 (see text) for two equally frequent r-process sources: one MRSN like $S_{H}$ source with $\lambda_{H}=1000$ and one BNSM like $S_{L}$ source with $\lambda_{L}=10$. Model |  | Probability of 129I/247Cm within an interval
---|---|---
$(D,M^{\rm ej}_{LH})$ | Criteria | 10-20 | 20-30 | 30-50 | 50-70 | 70-80 | 80-90 | 90-110 | 110-130 | 130-150 | $>150$
(0.1,1) | T0 | 0.63 | 0.06 | 0.05 | 0.03 | 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | 0.17
T10 | 0.04 | 0.69 | 0.27 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.16 | 0.49 | 0.34 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.3,1) | T0 | 0.53 | 0.09 | 0.08 | 0.04 | 0.02 | 0.01 | 0.02 | 0.01 | 0.02 | 0.17
T10 | 0.03 | 0.84 | 0.13 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.21 | 0.50 | 0.29 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.1,1/3) | T0 | 0.47 | 0.05 | 0.06 | 0.03 | 0.02 | 0.01 | 0.02 | 0.01 | 0.01 | 0.32
T10 | 0.00 | 0.00 | 0.25 | 0.60 | 0.10 | 0.04 | 0.01 | 0.00 | 0.00 | 0.00
T20 | 0.00 | 0.00 | 0.33 | 0.42 | 0.12 | 0.07 | 0.05 | 0.00 | 0.00 | 0.00
(0.3,1/3) | T0 | 0.37 | 0.08 | 0.09 | 0.06 | 0.02 | 0.01 | 0.03 | 0.02 | 0.01 | 0.31
T10 | 0.00 | 0.10 | 0.88 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.00 | 0.00 | 0.45 | 0.41 | 0.10 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00
(0.1,3) | T0 | 0.76 | 0.04 | 0.05 | 0.02 | 0.01 | 0.00 | 0.01 | 0.01 | 0.00 | 0.10
T10 | 0.98 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.88 | 0.12 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.3,3) | T0 | 0.70 | 0.07 | 0.06 | 0.03 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.09
T10 | 0.96 | 0.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.87 | 0.13 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00

We adopt tolerance levels of $10\%$ (criteria T10) and $20\%$ (criteria T20)
about the mean measured meteoritic value in the ESS for each of the three
ratios. Table 4 shows the resulting probabilities for the ESS 129I/247Cm ratio
in different intervals between $\sim\lambda_{L}$ to $\lambda_{M}$. When
concordance is imposed with the adopted tolerances, the probability
distributions are dramatically different compared to case where only the
minimum criteria (T0) is imposed (see Fig. 2). For both concordance criteria,
the probability distribution for the 129I/247Cm ratio is $\lesssim
6\lambda_{L}$ for all models. In fact, for $M^{\rm ej}_{LM}=1$ and $3$, the
129I/247Cm ratio is limited to values $\leq 4\lambda_{L}$. Only the models
with $M^{\rm ej}_{LM}=1/3$ have non-negligible probability of up to 17% for
129I/247Cm ratio to be larger than $>5\lambda_{L}$. Thus, remarkably, even in
the case where $\lambda_{M}$ is the major contributor to r-process, i.e.,
$M^{\rm ej}_{LM}=1/3$, the probability for the 129I/247Cm ratio to be close to
$\lambda_{M}$ is zero. As we can see from Table 4, in all models with T10 and
T20 criteria, the probability is strongly peaked at values either close to
$\lambda_{L}$ or at intermediate values of $\sim 3-5\lambda_{L}$. Models with
$M^{\rm ej}_{LM}=3$ i.e., $S_{L}$ as the dominant r-process source, where the
ratio is almost entirely limited to values close to $\lambda_{L}$, are the
only ones where the measured 129I/247Cm ratio corresponds to the true
production ratio of one of the sources with almost 100% probability. In all
other cases, the ESS value of 129I/247Cm can only provide constraints on the
range spanned by $\lambda_{L}$ and $\lambda_{M}$ but cannot directly constrain
their exact values.

Figure 3: Probability distribution of 129I/127Cm ratio in the ESS for two
equally frequent r-process sources $S_{L}$ and $S_{H}$ with $\lambda_{L}=10$
and $\lambda_{H}=1000$ for $M_{LM}^{\rm ej}=1$ (black), 3 (red), and 1/3
(blue) and tolerances T0 (left vertical panel), T10 (middle vertical panel),
and T20 (right vertical panel) corresponding to the values listed in Table 6.
All models have $D=0.1$ kpc2 Gyr-1.

The dramatic change in the probability distribution when we apply the T10 or
T20 criteria is primarily due to the requirement of concordance for 129I/127I
and 247Cm/235U ratios. Because 129I and 247Cm have almost identical lifetimes
and, 127I and 235U are stable or relatively long-lived, the relative ratio of
129I/127I to 247Cm/235U is essentially unaffected by the free decay during the
typical isolation time $\Delta_{\rm iso}\lesssim 50$ Myr. Therefore, the
criteria for concordance requires the 129I/127I and 247Cm/235U ratios to
overlap within the tolerance level at $t_{\rm iso}$. In the scenario where
$\nu_{L}=\nu_{M}$, both type of sources contribute equally to 127I whereas
235U is mostly from the $S_{L}$ source as it produces ten times more 235U per
event relative to $S_{M}$. In addition, if we consider the simplified
approximation where the SLRs 129I and 247Cm mostly come from one major event,
the contribution of this event to the total amount of 127I and 235U
accumulated from all past events is negligible. Thus, the 129I/127I ratio
after the last major event is the same for both $S_{M}$ and $S_{L}$ as both
sources produce identical amounts of 129I. However, the 247Cm/235U ratio is a
factor of $\sim 10$ lower if the last major event is $S_{M}$ compared to
$S_{L}$ as the former produces 10 times less 247Cm. With the adopted
production factors, only the scenario where the last major event is $S_{L}$
can result in 129I/127I ratio comparable to that of 247Cm/235U at
$(R_{\odot},t_{\odot})$. If the last major event is $S_{M}$, the 247Cm/235U is
$\sim$ an order of magnitude lower, which substantially lowers the chances of
concordance. This leads to a probability distribution of 129I/247Cm ratio that
is mostly limited to low to intermediate values for criteria T10 and T20.

The criteria for concordance also affects the fraction contributed from the
last three highest contributing events. Table 5 shows the mean values of
$f_{\rm h1},~{}f_{\rm h2},$ and $f_{\rm h3}$ for 129I with the concordance
criteria T10 and T20 along with the the minimum criteria T0. Overall, compared
to T0, for T10 and T20, the fraction contributed from the highest contributing
event h1 decreases while the fraction contributed from h2 and h3 increases.
With the concordance criteria, the mean values $f_{\rm h1},~{}f_{\rm h2},$ and
$f_{\rm h3}$ ranges form $\sim 0.53$–0.76, 0.17–0.34, and 0.04–0.12 compared
to $0.70$–0.84, 0.12–0.17, and 0.03–0.06, respectively, for the minimum
criteria.

Table 7: Probability distribution of 129I/247Cm ratio in the ESS for concordance criteria T0, T10 and T20 (see text) for three equally frequent r-process sources: one MRSN like $S_{H}$ source with $\lambda_{H}=1000$, one BNSM like $S_{M}$ source with $\lambda_{M}=100$, and one BNSM like $S_{L}$ source with $\lambda_{L}=10$.  Model |  | Probability of 129I/247Cm within an interval
---|---|---
$(D,M^{\rm ej}_{LH},M^{\rm ej}_{LM})$ | Criteria | 10-20 | 20-30 | 30-50 | 50-70 | 70-80 | 80-90 | 90-110 | 110-130 | 130-150 | $>150$
(0.1,1,1) | T0 | 0.44 | 0.05 | 0.06 | 0.03 | 0.02 | 0.04 | 0.18 | 0.02 | 0.02 | 0.14
T10 | 0.00 | 0.19 | 0.76 | 0.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.02 | 0.25 | 0.61 | 0.12 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.3,1,1) | T0 | 0.40 | 0.10 | 0.11 | 0.06 | 0.03 | 0.04 | 0.10 | 0.02 | 0.01 | 0.13
T10 | 0.00 | 0.16 | 0.84 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.00 | 0.32 | 0.61 | 0.07 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.1,3,3) | T0 | 0.69 | 0.04 | 0.05 | 0.03 | 0.01 | 0.02 | 0.11 | 0.01 | 0.01 | 0.05
T10 | 0.87 | 0.13 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.86 | 0.14 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.3,3,3) | T0 | 0.6 | 0.09 | 0.08 | 0.05 | 0.03 | 0.03 | 0.05 | 0.01 | 0.00 | 0.05
T10 | 0.91 | 0.09 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.84 | 0.16 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
(0.1,1,1/3) | T0 | 0.31 | 0.05 | 0.04 | 0.05 | 0.03 | 0.02 | 0.22 | 0.03 | 0.01 | 0.23
T10 | 0.00 | 0.00 | 0.01 | 0.25 | 0.18 | 0.16 | 0.40 | 0.01 | 0.00 | 0.00
T20 | 0.00 | 0.00 | 0.03 | 0.13 | 0.09 | 0.08 | 0.61 | 0.05 | 0.01 | 0.00
(0.3,1,1/3) | T0 | 0.21 | 0.08 | 0.09 | 0.07 | 0.03 | 0.04 | 0.17 | 0.04 | 0.02 | 0.26
T10 | 0.00 | 0.00 | 0.00 | 0.50 | 0.31 | 0.16 | 0.04 | 0.00 | 0.00 | 0.00
T20 | 0.00 | 0.00 | 0.06 | 0.25 | 0.14 | 0.15 | 0.38 | 0.02 | 0.00 | 0.00
(0.1,1/3,1) | T0 | 0.36 | 0.06 | 0.06 | 0.03 | 0.02 | 0.02 | 0.16 | 0.02 | 0.01 | 0.25
T10 | 0.00 | 0.00 | 0.09 | 0.46 | 0.15 | 0.13 | 0.17 | 0.01 | 0.00 | 0.00
T20 | 0.00 | 0.00 | 0.11 | 0.20 | 0.12 | 0.12 | 0.42 | 0.04 | 0.00 | 0.00
(0.3,1/3,1) | T0 | 0.28 | 0.09 | 0.10 | 0.06 | 0.03 | 0.03 | 0.09 | 0.02 | 0.03 | 0.27
T10 | 0.00 | 0.00 | 0.12 | 0.75 | 0.11 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00
T20 | 0.00 | 0.00 | 0.21 | 0.39 | 0.19 | 0.12 | 0.09 | 0.00 | 0.00 | 0.00

###  3.3 Results with Two Sources of Type $S_{L}$ and $S_{H}$

We also explored the scenario with two sources $S_{L}$ and $S_{H}$, where the
former is a BNSM like source and the latter is a MRSN like source with
$\lambda_{L}=10$ and $\lambda_{H}=1000$. As before, we assume that both
sources are equally frequent and consider three different relative ejecta mass
ratios of $M_{LH}^{\rm ej}=1/3,1,$ and 3. In the case of minimum criteria T0,
the probability distribution has a peak at values close to $\lambda_{L}$ with
a very long tail that extends up to $\lambda_{H}$. The reason is similar to
the scenario discussed before with $S_{L}$ and $S_{M}$. Even if the main
contributing h1 event is of $S_{H}$ type, a minor contribution from $S_{L}$
either as h2 or h3 produces a significant amount of 247Cm and decreases the
value of the 129I/247Cm ratio and brings it closer to $\lambda_{L}$. When the
criteria for concordance i.e, T10 or T20 is imposed, the probability
distribution for the 129I/247Cm again changes drastically (see Fig. 3). In
this case, the probability distribution is primarily limited to values
$\lesssim 7\lambda_{L}$ and is negligible for values $\gtrsim\lambda_{M}$ in
all cases. When $M_{LH}^{ej}=3$, the distribution strongly peaks at
$\sim\lambda_{L}$, whereas for $M_{LH}^{ej}=1$, the peak of the probability is
at $\sim 2-5\lambda_{L}$. Even when $S_{H}$ is the dominant source with
$M_{LH}^{ej}=1$, the peak of the probability distribution is at $\sim
5-7\lambda_{L}$ with zero probability for values $>11\lambda_{L}$.

The reason for the dramatic change in the probability distribution when T10
and T20 is imposed is again similar to the scenario discussed before with
$S_{L}$ and $S_{M}$. The allowed values for the 129I/247Cm ratio is governed
by the requirement of concordance for 129I/127I and 247Cm/235U. As before,
both sources contribute equally to the total 127I but almost all of the 235U
comes from $S_{L}$. Thus, if the last few events are dominated by $S_{H}$, the
247Cm/235U ratio is factor $\sim 100$ lower than 129I/127I which makes
concordance impossible.

###  3.4 Results with Two Sources of Type $S_{M}$ and $S_{H}$

Because we are dealing with isotopic ratios, the results for the evolution of
129I/247Cm ratio for a fixed value of $\lambda_{i}/\lambda_{j}$ and
$\nu_{i}/\nu_{j}$ can be used to get the corresponding results for
$\kappa\lambda_{i}$ and $\kappa\lambda_{j}$ by simply multiplying the results
for 129I/247Cm the corresponding factor $\kappa$. Thus, the results for the
case of two types of sources $S_{L}$ and $S_{M}$ with $\lambda_{L}=10$ and
$\lambda_{M}=100$ can be used to get the results for the case with two sources
$S_{M}$ and $S_{H}$ with $\lambda_{M}=100$ and $\lambda_{H}=1000$ with
$\nu_{MH}=1$ by scaling up the probability distribution of 129I/247Cm ratio by
a factor of 10. In this case, with the criteria T10 and T20, the probability
distribution is mostly limited to values of $\lesssim 5\lambda_{M}$ and
strongly peaks at values of $\sim 300$–500.

Figure 4: Probability distribution of 129I/127Cm ratio in the ESS for three
equally frequent r-process sources $S_{L}$, $S_{M}$, and $S_{H}$ with
$\lambda_{L}=10$, $\lambda_{M}=100$, and $\lambda_{H}=1000$, respectively, for
$(M_{LM}^{\rm ej},M_{LH}^{\rm ej})=(1,1)$ (black), $(3,3)$ (red), $(1/3,1)$
(blue), and $(1,1/3)$ (green) and tolerances T0 (left vertical panel), T10
(middle vertical panel), and T20 (right vertical panel) corresponding to the
values listed in Table 7. All models have $D=0.1$ kpc2 Gyr-1.

###  3.5 Result with Three Sources

Finally, we consider the scenario with three different sources $S_{L}$,
$S_{M}$, and $S_{H}$ with distinct values for 129I/247Cm ratio covering two
orders of magnitude ranging from low $\lambda_{L}=10$, to medium
$\lambda_{M}=100$, to high $\lambda_{H}=1000$. The frequency for all sources
are taken to be equal. In this case, there are two parameters for the ratio of
mass of r-process ejecta. They are denoted by $M_{LH}^{\rm ej}$ and
$M_{LM}^{\rm ej}$ corresponding to the ejecta mass ratio for $S_{L}$ relative
to $S_{H}$ and $S_{L}$ relative to $S_{M}$, respectively. We simulate four
different scenarios. The first one is where all sources contribute equally to
the main r-process, i.e, $M_{LH}^{\rm ej}=M_{LM}^{\rm ej}=1$. The other three
scenarios have one of the sources as the dominant main r-process source, which
has three times more ejecta mass than the rest of the two sources. As before,
for each scenario, we consider two different values of $D$. The probability
distribution of the 129I/247Cm ratio for the three different criteria T0, T10,
and T20 are listed in Table 7 along with the corresponding figure for $D=0.1$
kpc2 Gyr-1 in Fig. 4.

The overall results are roughly an average of the results derived for the
scenario with two sources ($S_{L}$, $S_{M}$) and ($S_{L}$, $S_{H}$). When the
minimum criteria T0 is applied, the probability distribution for the
129I/247Cm ratio has the highest peak close to $\lambda_{L}$ with values
ranging from $\sim 0.2$–$0.7$, and a relatively smaller peak at
$\sim\lambda_{M}$ with values ranging from $\sim 0.09$–$0.22$ along with
probability for intermediate values between $\lambda_{L}$ and $\lambda_{M}$
ranging from $\sim 0.09$-$0.17$. The probability distribution drops sharply
for values $\gtrsim\lambda_{M}$ and has a long and relative flat tail that
extends to values up to $\sim\lambda_{H}$. When the criteria for concordance
are imposed, the probability distribution changes drastically similar to the
cases with two sources. Firstly, the long tail of the probability distribution
above $\gtrsim\lambda_{M}$ vanishes completely. As with the models with two
sources, this change in the probability distribution is due to the requirement
of concordance for 129I/127I and 247Cm/235U such that if the last major
contributors are events of the $S_{H}$ type, concordance is impossible.

There is, however, a significant difference in the probability distribution
for 3 sources when compared to results from 2 sources. In the case of 2
sources, for all models, the probability distribution peaks at values
$\sim\lambda_{L}=10$ or intermediate values of $\sim 50$ with negligible or
very low probability for values close to $\lambda_{M}$. In contrast, for 3
sources, the probability distribution has either peaks or relatively high
values close to $\lambda_{M}$ for models where $S_{M}$ or $S_{H}$ are the
dominant r-process source. However, in all such models, there is a substantial
probability ranging from 0.16–0.60 for the ratio to have intermediate values
between 30–70. In the model with equal ejecta mass for all sources, the
probability distribution peaks at intermediate values. Only for the model
where $S_{L}$ is the dominant r-process source, the entire probability
distribution is limited close to $\lambda_{L}$ with a negligible probability
for values $\gtrsim 30$.

###  3.6 Isolation Time and Time of “Last" Event

It is interesting to consider the typical isolation time $\Delta_{\rm iso}$ in
our models. In addition, we can also consider the time interval $\delta_{\rm
h1}$ between the highest 129I contributing event and the beginning of
isolation at $t_{\rm iso}$. Among the three highest contributors, h1, h2, and
h3, h1 is not necessarily the last contributing event. Thus we additionally
define $\delta_{\rm last}$ as the time interval between the most recent event
among h1, h2, and h3 and $t_{\rm iso}$. The effective isolation time
corresponding to h1 and the last event can be defined as $\Delta_{\rm
iso}^{\rm h1}=\Delta_{\rm iso}+\delta_{\rm h1}$ and $\Delta_{\rm iso}^{\rm
last}=\Delta_{\rm iso}+\delta_{\rm last}$. We find that the distribution of
$\Delta_{\rm iso}$, $\Delta_{\rm iso}^{\rm h1}$, and $\Delta_{\rm iso}^{\rm
last}$ are similar for the four different scenarios consider in this work. For
the purpose of illustration, the distribution of $\Delta_{\rm iso}$,
$\Delta_{\rm iso}^{\rm h1}$, and $\Delta_{\rm iso}^{\rm last}$ are shown in
Fig. 5 for the scenario with two sources $S_{L}$ and $S_{M}$ for models with
$D=0.1$ kpc2 Gyr-1 and criteria T20. The distribution of $\Delta_{\rm iso}$
peaks at values of $\sim\lesssim 50$ Myr which is consistent with the typical
lifetimes associated with molecular and giant molecular clouds (Hartmann et
al., 2001; Murray, 2011). The 68th percentile range for $\Delta_{\rm iso}^{\rm
h1}$ ranges form $\sim 100$–140 Myr whereas the corresponding range for
$\Delta_{\rm iso}^{\rm last}$ is $\sim 80$–115 Myr. The central values are
broadly consistent with recent calculations by Côté et al. (2019b) where the
total isolation time for 129I was found to be $\sim 85$–116 Myr.
Interestingly, in our case, the probability for the effective isolation time
to be $\lesssim 50$ Myr is low with typical values $\lesssim 5\%$ but not
zero.

Figure 5: Probability distribution of $\Delta_{\rm iso}$, $\Delta_{\rm
iso}^{\rm h1}$, and $\Delta_{\rm iso}^{\rm last}$ for two equally frequent
r-process sources $S_{L}$ and $S_{M}$ with $\lambda_{L}=10$ and
$\lambda_{M}=100$ for $M_{LM}^{\rm ej}=1$ (black), 3 (red), and 1/3 (blue).
All models have $D=0.1$ kpc2 Gyr-1 and tolerance T20.

##  4 Discussion & Conclusions

In this work we explored the evolution of the SLRs produced by r-process at
the solar location due to two or three different r-process sources that have
distinct 129I/247Cm production ratios. In contrast to the conclusion reached
in Côté et al. (2021), we find that in general, the observed ESS ratio for
129I/247Cm does not correspond to a single “last” event. Although there is a
major contributing last event, it accounts for only $\sim 50$–75% of all the
129I in the ESS and at least two more minor contributing events are required
to account for $\gtrsim 95\%$ of the observed 129I. This has a large impact on
the probability distribution for the 129I/247Cm ratio in the ESS that depends
on the particular choice of parameters such as the ratio of ejecta masses, and
the relative frequency of the sources.

One of the reasons for the difference in our conclusion and the one reached by
Côté et al. (2021) is related to the prescription for modelling the evolution
of r-process elements. The turbulent gas diffusion prescription used in this
work is very different from the one used in Côté et al. (2021) where a
stochastic one-zone model was used. Although both prescriptions are able to
model the stochasticity of occurrence time of r-process events, the diffusion
prescription can also capture the stochasticity associated with the spatial
location, distance of the event from the solar location as well as the
corresponding dilution. The other important reason is the application of the
criteria for concordance used in our work, which involves using the ESS ratio
of 129I/127I and 247Cm/235U that is different from what was considered in Côté
et al. (2021). As mentioned earlier, the concordance imposes additional
constraints that changes the ESS 129I/247Cm ratio substantially..

Although we find that the ESS 129I/247Cm ratio measured in meteorites cannot
be used to directly constrain the “last” r-process event, the concordance
criteria used in our work still allows us to put interesting constraints on
the nature of r-process sources when combined with theoretical nucleosynthetic
calculations for various astrophysical sources. Below, we take the results of
r-process nucleosynthesis calculations for the various astrophysical scenarios
and nuclear physics inputs by Côté et al. (2021) as examples for illustration
and discussion. The astrophysical models considered in Côté et al. (2021) are,
i) dynamical ejecta from NS-NS and NS-BH mergers that are the most neutron
rich and have the lowest values of 129I/247Cm, ii) three different NS-NS
merger disk ejecta numbered 1, 2 and 3 with varying levels of neutron richness
resulting in higher values of 129I/247Cm compared to the dynamical ejecta, and
iii) MRSN ejecta that have the highest values of 129I/247Cm. The values of
129I/247Cm in all models are sensitive to the nuclear physics inputs where
three different nuclear reaction rates and three different fission fragment
distributions were considered for each model. The ranges of the 129I/247Cm
ratio in these models are roughly between 10–100 for the neutron-rich
dynamical ejecta, 250–1000 for disk ejecta 1, 50–250 for disk ejecta 2, 25–150
for disk ejecta 3, and 1000-6000 for MRSN ejecta. We associate the dynamical
ejecta from NS-NS and NS-BH mergers as the $S_{L}$ source, one of the three
different disk ejecta as the $S_{M}$ source, and the MRSN ejecta as the
$S_{H}$ source.

Considering two equally frequent r-process sources $S_{L}$ and $S_{M}$ with
$\lambda_{M}/\lambda_{L}\approx 10$, we can draw the following conclusions
when the concordance criteria (T10 or T20) is imposed;

  * •

Because the probability distribution can have a maximum value of
$\sim\lambda_{M}$, the possibility of NS-NS merger disk ejecta 3 as $S_{M}$ is
ruled out. This is simply due to the fact that for all NS-NS merger disk
ejecta 3 models, $\lambda\lesssim 150$ making it incompatible with the
observed value of $438\pm 184$.

  * •

The $S_{L}$ source as the NS-NS or NS-BH dynamical ejecta being the dominant
r-process source (i.e, $M_{LM}^{\rm ej}=3$) is highly disfavoured for all
models. This is because in this case the values for the 129I/247Cm ratio is
limited to $<3\lambda_{L}$ whereas the 129I/247Cm production ratio $\lesssim
90$ for all dynamical ejecta models considered by (Côté et al., 2021)333Other
than the TF(D3C*) model which is an outlier.. This is inconsistent with the
measured value of $438\pm 184$.

  * •

If the $S_{M}$ source is the dominant contributor to the r-process (i.e,
$M_{LM}^{\rm ej}=1/3$), NS-NS merger disk ejecta 1 is favoured while NS-NS
merger disk ejecta 2 is marginally consistent with the observed values.

  * •

The $S_{L}$ source as NS-NS or NS-BH dynamical ejecta as an equal contributor
to the r-process (i.e., $M_{LM}^{\rm ej}=1$) requires the $S_{M}$ source to be
NS-NS merger disk ejecta 1. This is because when $S_{L}$ is an equal
contributor, the probability distribution for the 129I/247Cm ratio is limited
to values of $\lesssim\lambda_{M}/2$. Among all the disk models, this can only
be satisfied by the NS-NS merger disk ejecta 1 where the 129I/247Cm production
ratio can reach values of up to $\sim 900$ such that the probability
distribution for the 129I/247Cm ratio is non-negligible at the observed value
of $438\pm 184$.

  * •

Overall, the most favoured scenario is the NS-NS merger disk ejecta 1 as the
dominant $S_{M}$ source which gives the highest probability for the 129I/247Cm
ratio to be in the range consistent with the observed value of $438\pm 184$.

For the case with two equally frequent r-process sources $S_{M}$ and $S_{H}$
with $\lambda_{H}/\lambda_{M}\approx 10$ and MRSN ejecta the $S_{H}$ source,
all the three NS-NS merger disk ejecta are possible as they result in
probability distribution of 129I/247Cm that is compatible with the observed
value of $438\pm 184$.

On the other hand, with MRSN ejecta the $S_{H}$ source and NS-NS or NS-BH
dynamical ejecta as the $S_{L}$ source with $\lambda_{H}/\lambda_{L}\approx
100$ we conclude that;

  * •

NS-NS or NS-BH dynamical ejecta ($S_{L}$) as the dominant r-process source is
highly disfavoured. Again, in this case the 129I/247Cm ratio is limited to
$\lesssim 3\lambda_{L}$ whereas 129I/247Cm production ratio $\lesssim 90$ for
all dynamical ejecta models considered by (Côté et al., 2021). This makes it
incompatible with the observed value of $438\pm 184$.

  * •

NS-NS or NS-BH dynamical ejecta ($S_{L}$) as either equal or subdominant
contributor to total r-process is consistent with the observed value.

Finally, with three equally frequent r-process sources with NS-NS or NS-BH
dynamical ejecta as $S_{L}$, NS-NS merger disk ejecta as $S_{M}$ and MRSN
ejecta as $S_{H}$, the conclusions we draw are;

  * •

The probability distribution for the 129I/247Cm ratio is limited to values
between $\lambda_{L}$ to $\sim\lambda_{M}$. Thus, the ESS value of $438\pm
184$ disfavours NS-NS merger disk ejecta 3 and and is marginally consistent
with NS-NS merger disk ejecta 2 as the $S_{M}$ source whereas NS-NS merger
disk ejecta 1 is favoured.

  * •

Similar to the scenario with two sources, the $S_{L}$ source as the NS-NS or
NS-BH dynamical ejecta being the dominant r-process source (i.e $M_{LH}^{\rm
ej},M_{LM}^{\rm ej}=3$) is highly disfavoured for all models.

  * •

Overall, the most favoured scenario involves NS-NS merger disk ejecta 1 as the
$S_{M}$ source, with either the $S_{H}$ (MRSN ejecta) or the disk ejecta are
the dominant r-process source (i.e, $M_{LH}^{\rm ej},M_{LM}^{\rm ej}=1/3,1$ or
$M_{LH}^{\rm ej},M_{LM}^{\rm ej}=1,1/3$).

##  5 Summary and Outlook

We studied the evolution of r-process isotopes including SLRs at the birth
location of the sun and the prospect of using the 129I/247Cm ratio to
constrain the r-process sources. We find that the measured meteoritic value of
the 129I/247Cm ratio does not correspond to a single “last” r-process event
when there are multiple sources with distinct 129I/247Cm production ratios.
Instead, we find that the 129I/247Cm ratio can be used to put important
constraints on r-process sources when the ESS data of 129I/127I and 247Cm/235U
is taken into account. In particular, based on the nucleosynthesis calculation
by Côté et al. (2021) for various astrophysical sites, we find that models of
NS-NS or NS-BH dynamical ejecta that are neutron rich and have low 129I/247Cm
ratio cannot be the dominant source of r-process. This statement holds both in
the case of where there are just two sources as well as when all three sources
are considered. Interestingly, this is consistent with current detailed BNSM
merger simulations which predict that merger disk ejecta to be a more dominant
source of r-process than the dynamical ejecta (Shibata & Hotokezaka, 2019;
Metzger, 2020). If there is a MRSN like source that has a high value for the
production ratio of 129I/127I and is as frequent as BNSMs, then it can mix
with the dynamical ejecta to produce values that are compatible with
observations without the need for disk ejecta. However, from theoretical
expectations, this is unlikely as substantial amount of r-process is expected
from NS-NS merger disk ejecta. In a realistic scenario where the disk ejecta
is as frequent as the dynamical ejecta, NS-NS merger disk ejecta which has
medium values of 129I/247Cm production ratio is highly favoured when there are
no MRSN like source that is similarly frequent. However, if there is a MRSN
like source, any of the current merger disk models calculated in Côté et al.
(2021) are in fact possible.

Our analysis is based on simplifying assumptions such as equal frequency for
all sources and fixed values of $\lambda$ for the three types of r-process
sources and limited values of ejecta masses. In principle, future studies can
explore a larger parameter space to identify allowed regions that are
consistent with the ESS data. However, for some realistic scenarios which have
parameters that are not directly covered in this work, the results presented
here can be easily extrapolated. For example, in the scenario with all the
three sources, if the frequency of MRSN like event is lower, then it would
effectively reduce to the scenario with two sources of type $S_{L}$ and
$S_{M}$ and the conclusion drawn for such a scenario can be applied in this
case. Similarly, if there are two different $S_{M}$ sources that have similar
values of $\lambda$, then they can effectively be treated as a single $S_{M}$
source and the results for two sources with $S_{L}$ and $S_{M}$ can be applied
in this case. In future, with better simulations of astrophysical sites with
improved nuclear physics inputs and accurate nucleosynthetic models, the ESS
ratio of 129I/247Cm along with 129I/127I and 247Cm/235U could be used to
provide strong constraint that could help identify the main source of
r-process and even rule out certain astrophysical sites.

## Acknowledgements

M.-R. W. acknowledges supports from the Ministry of Science and Technology,
Taiwan under Grant No. 110-2112-M-001 -050, the Academia Sinica under Project
No. AS-CDA-109-M11, and the Physics Division, National Center for Theoretical
Sciences, Taiwan.

## Data Availability

Data is available upon request.

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##  Appendix A Milky Way Model

The model for the Milky Way is adapted from the best fit model from Côté et
al. (2019b). The star formation rate is assumed to be directly proportional to
the gas mass with a star formation efficiency $f_{\star}=2.6\times
10^{-10}~{}{\rm yr}^{-1}$. The two-infall prescription of Chiappini et al.
(1997) is adopted for the gas inflow rate given by

$\dot{M}_{\rm
inflow}=A_{1}\exp\left(\frac{-t}{\tau_{1}}\right)+A_{2}\exp\left(\frac{t_{\rm
max}-t}{\tau_{2}}\right),$ (6)

where $A_{1}=46\,M_{\odot}~{}{\rm yr}^{-1}$, $A_{2}=5.9\,M_{\odot}~{}{\rm
yr}^{-1}$, $\tau_{1}=0.68$ Gyr, $\tau_{2}=7$ Gyr, and $t_{\rm max}=1$ Gyr. An
initial time offset of 300 Myr is adopted to account for the lack of star
formation in the very early Galaxy. The outflow rate is assumed to be
proportional to the star formation rate with a proportionality constant
$\eta=0.52$. The nucleosynthetic yields for low- and intermediate-mass stars
are taken from Karakas (2010). The massive star yields are taken from
Kobayashi et al. (2006) with half the stars in the range of
$20$–$40~{}M_{\odot}$ assumed to explode as hypernovae. The number of SNe Ia
per stellar mass of star formation is taken to be $10^{-3}$ with a DTD
$\propto t^{-1}$ and a minimum delay time of $\sim 40$ Myr corresponding to
the lifetime of a $8\,M_{\odot}$ star. A fraction $f_{r}=8\times 10^{-4}$ of
massive stars are assumed to form $r$-process events which corresponds to a
current Galactic rate at $t_{\rm gal}=13.7$ Gyr of $\nu_{0}\sim 10\,{\rm
Myr}^{-1}$ and $\nu\sim 15\,{\rm Myr}^{-1}$ at $t_{\odot}=9.2$ Gyr.