# Two Sites of r-Process Production Assessed on the Basis of the Age-tagged
Abundances of Solar Twins

Takuji Tsujimoto National Astronomical Observatory of Japan, Mitaka, Tokyo
181-8588, Japan

(Received xxx; Revised xxx; Accepted 2021 October 2)

###### Abstract

Solar twins, i.e., stars that are nearly identical to the Sun, including their
metallicities, in the solar vicinity show ages widely distributed from 0$-$10
Gyr. This fact matches the orbital history of solar twins in the new paradigm
of galactic dynamics, in which stars radially move on the disk when they
encounter transient spiral arms. This finding suggests that older twins were
born closer to the Galactic center and traveled a longer distance to reach
their present location, according to the hypothesis that chemical enrichment
occurs more quickly and that solar metallicity is attained on a shorter
timescale with a decreasing Galactocentric distance ($R_{\rm GC}$). We show
that abundance patterns covering a wide range of heavy elements for solar
twins sharing similar ages are identical and that their variation among
different age groups can be understood on the basis of the age-$R_{\rm GC}$
connection within the framework of Galactic chemical evolution. This study
identifies the Galactic bulge as the birthplace of the oldest solar twins.
Based on this scheme, we find that the relation between [$r$-process/Fe] and
$R_{\rm GC}$ for the inner Galactic region is incompatible with the hypothesis
of a sole site for $r$-process production, that is, neutron star mergers,
whose delay time distribution could be approximated by the power-law form
($\propto t^{n}$). Alternatively, this relation suggests the presence of two
distinct sites for $r$-process production: short-lived massive stars, ending
with specific core-collapse supernovae, and neutron star mergers that are
heavily inclined to emerge with longer delay times, as represented by
$n$=0$-$0.5.

Galactic archaeology (2178); Galactic bulge (2041); Galaxy chemical evolution
(580); Galaxy dynamics (591); Milky Way disk (1050); Solar abundances (1474)

##  1 Introduction

One of the remarkable advancements in the astrophysics field in recent years
is our understanding of the origin of $r$-process elements. After an intense
debate on this issue for more than half a century, we eventually identified
neutron star mergers (NSMs) as at least one site of $r$-process elements via
the discovery of gravitational waves from NSM GW170817 and the subsequent
detection of multiwavelength electromagnetic counterparts (e.g., Smartt et
al., 2017; Pian et al., 2017; Cowperthwaite et al., 2017). Our recent interest
has been directed toward the question of whether NSMs are the sole (major)
site of the $r$-process (e.g., Ji et al., 2019; Kobayashi et al., 2020;
Cavallo et al., 2021). This question arises from the following arguments.
Theoretically, some specific core-collapse supernovae (CCSNe), such as
magnetorotational SNe (e.g., Winteler et al., 2012; Nishimura et al., 2015) or
collapsars (e.g., Siegel et al., 2019), are capable of providing the physical
conditions necessary to realize the $r$-process. The property of these CCSN
candidates is a fast release of $r$-process products owing to a short lifetime
of massive stars, in stark contrast to the long delay time, typically a few
Gyrs, of NSMs (e.g., Dominik et al., 2012), which is exemplified by the
GW170817 case (6.5 Gyr; Im et al., 2017, 11.2 Gyr; Fong et al. 2017). In fact,
the presence of a population exhibiting such prompt $r$-process enrichment is
implied by the stellar record of the abundances of these products in very
metal-poor stars (e.g., Roederer, 2013) as well as in Galactic disk stars
(Côté et al., 2019; Siegel et al., 2019). On the other hand, the delay time
distribution (DTD) of NSMs, ranging from a few 10 Myrs to a Hubble time, is
not well constrained (Greggio et al., 2021, and references therein), allowing
the possibility of NSMs being the only source of $r$-process elements. In the
end, a full understanding of $r$-process sites is still lacking.

Unlike $r$-process elements, the production sites of other heavy elements are
well understood. For instance, $\alpha$-elements such as O and Mg are
exclusively produced in CCSNe, while Fe-peak elements are produced in two
sites, i.e., CCSNe and type Ia SNe (SNe Ia), whose DTDs are given by $\propto
t_{\rm delay}^{-1}$ (Maoz et al., 2014). In addition, $s$-process elements
represented by Ba are dominantly produced in asymptotic giant branch (AGB)
stars in the late evolutionary stage of low- and intermediate-mass stars.
Owing to the different release timescales among the multiple production sites
for individual elements, the elemental abundance pattern from $\alpha$\- to
neutron-capture elements will change with the elapsed time, and its time
variation will depend on the star formation rate in situ. Here, suppose that
several detailed elemental abundance patterns that result from different star
formation rates, i.e., different speeds of chemical enrichment, are
observationally set up. Then, a comparison of the variation in $r$-process
abundances generated by a slow to high enrichment speed with the variation in
other well-understood elements would enable a precise assessment of the site
of $r$-process element production associated with their release timescale.

Remarkably, this supposed setup is realized thanks to two recent advancements
in the study of the dynamics and chemistry of Galactic stars. The improved
understanding of Galactic dynamics suggests that stars radially move on the
disk when they encounter transient spiral arms that are naturally generated
during the process of disk formation (Sellwood & Binney, 2002; Ros̆kar et al.,
2008; Grand et al., 2012; Baba et al., 2013). This so-called radial migration
of stars predicts that the birthplaces of a considerable number of disk stars
are different from their current positions. According to this prediction, the
stars in the solar vicinity represent the mixture of stars born at various
Galactocentric distances ($R_{\rm GC}$) over the disk. The impact of this
migration on the local chemistry is of significance since the stars born
outside the solar vicinity follow chemical enrichment tracks that differ from
the local tracks. This argument is based on our established knowledge that the
chemical evolution of the disk differs in accordance with $R_{\rm GC}$, which
is observationally evidenced by current radial abundance gradients showing
higher metallicity at smaller $R_{\rm GC}$ values. This negative abundance
gradient is theoretically understood in the context of galaxy formation, with
the inner region forming faster and becoming more metal-rich than the outer
region (the so-called inside-out scenario; Chiappini et al., 2001). In other
words, the rate at which the metallicity reaches a certain value (e.g., the
solar metallicity) increases as $R_{\rm GC}$ decreases.

Accordingly, for nearby stars sharing the same metallicity, we have two
arguments: (i) the ages could vary among these stars as an outcome of radial
migration, and (ii) older stars would have been born at smaller $R_{\rm GC}$
values. Therefore, the precise measurement of the age and elemental abundance
of these local stars, if obtained, could provide elemental features as a
function of $R_{\rm GC}$. At different $R_{\rm GC}$ values, chemical
enrichment proceeds at a different speed.

Locally, there exist stars that are nearly identical to the Sun, so-called
solar twins, which exhibit stellar atmospheric characteristics quite similar
to the solar values. Thanks to this close resemblance, the stellar ages of
solar twins can be precisely determined, with a typical uncertainty of 0.4
Gyr, together with highly accurate (an error $<$ 0.01 dex) chemical abundances
(Bedell et al., 2018; Spina et al., 2018). Here, we show that the precise age-
tagged abundances of solar twins offer new insight in the study of the
$r$-process.

##  2 Abundance patterns as a function of the solar twin’s age

We compare the chemical abundances of 28 elements, from carbon to dysprosium,
with respect to Fe for 79 solar twins (Bedell et al., 2018). A quick overview
indicates that stars with similar ages broadly share similar elemental
abundance patterns, which is anticipated on the basis of individual [X/Fe]
ratios as a function of the stellar age (Bedell et al., 2018). Then, we search
for the representative patterns for each age group to understand how the
patterns change with age via the following procedure. First, we separately
collect the patterns for several age bins and choose a typical pattern for
each group by eye. Then, we assign stars to each group by calculating
deviations from a chosen one and select those below a fixed value of
deviation. This procedure is repeated by altering age bins and/or the
selection of typical patterns until the best classification of age-tagged
abundance patterns is identified.

Figure 1: Elemental abundance patterns of solar twins for the elements in the
range 6 $\leq$$Z$$\leq$ 66 with respect to Fe for three groups classified by
their ages. The mean ages are indicated. The observed abundances of solar
twins are from Bedell et al. (2018). Here, we remove the element Pr ($Z$=59)
from our study since all stars exhibit a large deviation from the solar Pr/Fe
ratio. For reference, offsets of 0.05 dex from the solar pattern (i.e.,
[X/Fe]=0) are shown by the dotted lines. In the top panel, the individual
elemental abundance ratios measured thus far for bulge stars with
[Fe/H]$\approx$0 are represented by the light-green stripes. The observed
abundance data for Galactic bulge stars are from Schultheis et al. (2017, for
O, Mg, Ca, Mn, and Ni), Duong et al. (2019, for Na, Al, Sc, Cr, Co, Cu, Zr,
La, Ce, and Nd), Gonzalez et al. (2011, for Si), Lucertini et al. (2021, for
S), Bensby et al. (2017, for Ti, Zn, Y, and Ba), Lomaeva et al. (2019, for V),
and Forsberg et al. (2019, for Eu). Regarding [C/Fe], we assign the observed
range exhibited by three stars at [Fe/H]$\sim$0.2$-$0.4 (the ratios could
theoretically be higher at solar metallicity; Romano et al., 2020).

Finally, we obtain five groups and show the elemental abundance patterns for
the three of them according to age, i.e., $\langle{\rm age}\rangle$=8.7, 5.9,
and 0.7 Gyr, in Figure 1. We clearly see the difference among the three
patterns. Note that the other two groups with $\langle{\rm age}\rangle$=7.3
and 3.4 Gyr exhibit the intermediate patterns among the three in Figure 1.
Before discussing Galactic chemical evolution based on these results, we
elaborate on two findings.

###  2.1 Bulge migrators and the Sun’s birthplace

The first finding is that the abundance pattern of the oldest group is quite
similar to that of Galactic bulge stars with [Fe/H]$\approx$0\. In the top
panel of Figure 1, the elemental abundances of the Galactic bulge around
[Fe/H]=0 measured by several authors are superimposed over those of solar
twins with $\langle{\rm age}\rangle$=8.7 Gyr. This good agreement between the
two abundance patterns suggests that the oldest solar twins would migrate from
inside of the bulge to the solar vicinity. The dynamic process of escaping
from the Galactic bulge potential could be induced by perturbation of the bar
(Raboud et al., 1998, see also Di Matteo et al. 2014). It should be noted that
the fraction of local solar twins belonging to the oldest group is
approximately 10% (8/79) and thus is not negligible. This implies that
migrators from the bulge would be an important population among nearby stars,
particularly for super metal-rich stars.

Second, we find that the solar abundance pattern is identical to those of
solar twins which range in age from 4.5 to 7.9 Gyr with $\langle{\rm
age}\rangle$=5.9 Gyr and thus are typically older than the Sun (4.56 Gyr).
This revealed association of the Sun with older solar twins suggests a common
birthplace, specifically the region closer to the Galactic center. In this
common region, interstellar matter was enriched to a mean metallicity [Fe/H]
$\approx$ 0 approximately 6 Gyr ago, not 4.6 Gyr ago, with some variation in
[Fe/H] over time. Considering the age of Galactic thin disk of $\sim$8$-$9 Gyr
(e.g., Fantin et al., 2019), [Fe/H] reached the solar value within a short
timescale of $\sim$2$-$3 Gyr at the Sun’s birthplace. The implied metal-rich
environment at the Sun’s birthplace and formation time is supported by
measured silicon isotopic ratios, i.e., 29Si/28Si and 30Si/28Si, in presolar
silicon carbide grains (Clayton, 1997; Tsujimoto et al., 2020).

###  2.2 Galactic chemical evolution across the disk

Figure 2: Top panel: The correlation of the [$\alpha$/Fe] ratio for five
groups of solar twins classified by mean stellar age relative to the elapsed
time until the formation of individual twin groups, defined as
$t_{\rm[Fe/H]=0}$, which is estimated from $\Delta t$ between the age of the
disk (= 9 Gyr) and each mean age. The observed ratios are indicated by the
filled circles for two elements (O and Mg), while the theoretical correlation
of [Mg/Fe] predicted by the models for Galactic chemical evolution is shown by
the red solid curve. This decreasing trend is interpreted as an outcome of an
increasing contribution of Fe from SNe Ia into interstellar matter with
increasing $t_{\rm[Fe/H]=0}$. Middle panel: The same as in the top panel but
for [$s$-process/Fe] ($s$-process = Y and Ba). The predicted curve is for
[Ba/Fe]. The increasing trend is attributed to a gradual accumulation of
$s$-process elements in interstellar matter released from AGB stars in
accordance with a longer elapsed time. Bottom panel: The correlation for
[$r$-process/Fe]. The [$r$-process/Fe] ratio decreases with increasing
$t_{\rm[Fe/H]=0}$ and upturns at approximately $t_{\rm[Fe/H]=0}$= 4 Gyr. These
two discrete trends can be understood in the context of the properties of the
$r$-process production site.

Next, we argue how the variation in elemental abundance patterns in accordance
with the mean age of solar twins can be understood within the framework of
Galactic chemical evolution. A key hypothesis is that the abundance pattern of
the older twin group is an end result of faster chemical enrichment at smaller
$R_{\rm GC}$. By taking a close look at the transitions of [$\alpha$/Fe] and
[$s$-process/Fe] with age, we show that a fundamental change in patterns is
explained based on this hypothesis. First, we convert stellar ages to elapsed
time, $t_{\rm[Fe/H]=0}$, for chemical enrichment up to [Fe/H]=0 from the start
of star formation, assuming a disk age of 9 Gyr. Note that the oldest group
with $\langle{\rm age}\rangle$=8.7 Gyr is considered to originate in the
bulge, which is older than the disk, and that $t_{\rm[Fe/H]=0}$ for this group
should, in practice, be larger than 0.3 Gyr. The top panel of Figure 2 shows
the [$\alpha$/Fe] ratio as a function of $t_{\rm[Fe/H]=0}$ for five groups
classified by mean stellar age (filled circles). We see a decreasing trend of
[$\alpha$/Fe] with increasing $t_{\rm[Fe/H]=0}$. This is naturally interpreted
to be an outcome of the larger amount of Fe released from SNe Ia into
interstellar matter owing to more SNe Ia with longer delay times. In addition,
the case for [$s$-process/Fe] is shown in the middle panel, which conversely
exhibits an increasing trend. This feature is also reasonable if we consider a
contribution of $s$-process elements from low- and intermediate-mass AGB stars
which overwhelms the increase in Fe ejected from delayed SNe Ia.

To validate the above arguments, we perform calculations of the Galactic
chemical evolution. Models for the chemical evolution of the disk within the
solar orbit (i.e., $R_{\rm GC}$ $\;\buildrel<\over{\sim}\;$8 kpc) and of the
bulge are designed to realize more efficient chemical enrichment as $R_{\rm
GC}$ decreases, leading to a faster increase in metallicity and higher
metallicity. We model each one for different $R_{\rm GC}$ by changing the
timescales of star formation ($\tau_{\rm SF}$) and the supply of gas from the
halo ($\tau_{\rm in}$). Here, star formation is assumed to be proportional to
1/$\tau_{\rm SF}$, and $\tau_{\rm in}$ is introduced in exponential form. The
values of ($\tau_{\rm SF}$, $\tau_{\rm in}$) in units of Gyr are assumed to
vary from (0.25, 0.3) for the bulge to (3.3, 5) for the solar vicinity. For
the DTD of SNe Ia, we assume DTD $\propto t_{\rm delay}^{-1}$, with a range of
$0.1\leq t_{\rm delay}\leq 10$ Gyr (Maoz et al., 2014). We adopt the AGB yield
for Ba which is empirically deduced from the abundances of Ba stars (Tsujimoto
& Bekki, 2012). The validity of this approach is guaranteed by agreement with
the theoretical stellar yields (Cseh et al., 2018, see also Prantzos et al.
2018 and Kobayashi et al. 2020). Then, we calculate the evolutions of [Mg/Fe]
and [Ba/Fe] and depict the results as red solid curves in the top and middle
panels. Our results agree with the correlation of [Mg/Fe] and [Ba/Fe] with
$t_{\rm[Fe/H]=0}$, that is, the mean age for solar twins (indicated at the
top). This agreement confirms that the age-tagged abundances of solar twins
are understood within the scheme of Galactic chemical evolution that combines
the inside-out scenario with the galactic dynamics of radial migration.

##  3 Galactic $r$-process enrichment

Figure 3: Evolutionary tracks of [Eu/Fe] predicted by models with different
slopes ($n$=0.5, 0, -0.5, and -1.5) of the DTD for NSMs compared with the
observed data of nearby thin disk stars. The former three models include the
contribution of $r$-process elements from rare CCSNe while NSMs are the sole
site of the $r$-process in the model with $n$=-1.5. In each model, four tracks
calculated at different chemical enrichment speeds (green, light blue, blue,
and red, in order from fastest to slowest) are shown.

In this context, the [$r$-process/Fe]-$t_{\rm[Fe/H]=0}$ correlation must be
highlighted in terms of its direct association with two properties of
$r$-process elements, i.e., their production sites and release timescales,
both of which remain enigmatic. Here, we show the correlation of
[$r$-process/Fe] with $t_{\rm[Fe/H]=0}$ for two elements, Eu and Dy111The Th
abundance is measured for nearly 70% of solar twins (Botelho et al., 2019).
However, because of a large scatter in [Th/Fe] for each group arising from a
large measurement error ($>$ 0.1 dex), we exclude Th from our discussion., in
the bottom panel of Figure 2. Note that the percentages of each element
produced by the $r$-process in terms of the solar abundance are 95% and 85%
(Prantzos et al., 2020). Intriguingly, a peculiar feature$-$a decreasing trend
followed by an upturn$-$is identified. These two discrete trends imply the
mixture of two $r$-process sites with different, i.e., short and long, delay
times for the product release, which can be associated with specific CCSNe and
NSMs, respectively.

Based on this consideration, we calculate the chemical evolution of Eu/Fe by
using the models constructed for that of Mg/Fe and Ba/Fe, which are described
in section 2.2. The key requirement for the models is to reproduce an upturn
of [Eu/Fe] for $t_{\rm[Fe/H]=0}$ $\;\buildrel>\over{\sim}\;$4 Gyr; this
corresponding feature is largely determined by the DTD for NSMs.
Conventionally, the DTD form for NSMs is parameterized by a power law
($\propto t_{\rm delay}^{n}$), with the power index being obtained from the
DTD for short gamma-ray bursts (SGRBs). Although the index $n$ is poorly
constrained, mainly owing to the difficulty in characterizing the SGRB host
galaxy with a limited number of samples, studies to date are in rough
agreement that $n$$\;\buildrel>\over{\sim}\;$$-1$ (e.g., Zheng & Ramirez-Ruiz,
2007; Fong & Berger, 2013; Paterson et al., 2020). In this study, we consider
three cases of the DTD, i.e., $n$= -0.5, 0, and 0.5, with longer delay times
chosen to reproduce the upturn feature, with a range of 0.03 $\leq t_{\rm
delay}\leq 10$ Gyr. Together with these DTDs, we assume that NSMs occur at a
rate of one per 300 CCSNe and that the ejected mass of Eu from each event is
$2\times 10^{-5}$$M_{\odot}$ (Tsujimoto et al., 2020). In addition to NSMs, we
assign specific CCSNe to a separate production site of $r$-process elements.
For the production rate and yield, we adopt the results of magnetorotational
SNe deduced from chemical evolution (Tsujimoto & Nishimura, 2015) and
nucleosynthesis calculations (Nishimura et al., 2015); Eu ejection is assumed
to occur at a rate of one per 150 CCSNe, with an ejected Eu mass of $1.2\times
10^{-5}$$M_{\odot}$ from each SN. In addition to these hybrid models involving
two $r$-process sites, we consider the case of a single $r$-process site of
NSMs. For this model, we adopt a DTD with a steep slope of $-1.5$, which is
constrained by the Galactic chemical evolution (Hotokezaka et al., 2018; Côté
et al., 2019).

Figure 4: The theoretical correlation of [Eu/Fe] predicted by Galactic
chemical evolution models with $t_{\rm[Fe/H]=0}$ for the two models (dashed
curves) compared with that for five groups of solar twins classified by the
mean stellar age (crosses).

We first show four evolutionary paths of [Eu/Fe] against [Fe/H] calculated by
individual models in Figure 3. Each path is a result of different speeds of
chemical enrichment, corresponding to $t_{\rm[Fe/H]=0}$ = 0.3, 1.7, 3.1, and
8.3 Gyr. These model results are compared with the observed data of nearby
thin disk stars. This comparison demonstrates that the four [Eu/Fe] curves in
all models broadly overlap the domain of the observed data, indicating that
all models pass the first test, i.e., agreement with the renewed view of
Galaxy formation, in which stars in the solar vicinity represent a mixture of
different stars born at various $R_{\rm GC}$. On the other hand, a close look
at the trajectory of [Eu/Fe] at a late stage clarifies a difference among the
four models; the models with $n$=0.5 and 0 predict a switch from a decreasing
[Eu/Fe] trend to a slightly increasing one for the cases (red and blue curves)
in which a relatively slow chemical enrichment proceeds, while the other two
models predict a continious decreasing trend of [Eu/Fe] for all cases. Indeed,
the former two model results are compatible with the correlation of [Eu/Fe]
with $t_{\rm[Fe/H]=0}$ shown in the bottom panel of Figure 2.

This result is clearly demonstrated in Figure 4, which shows the sequence of
[Eu/Fe] values at [Fe/H]=0 as a function of the elapsed time reaching [Fe/H]=0
for the two models with $n$=0.5 and -1.5. We conclude that the joint
$r$-process production of specific CCSNe and NSMs with a DTD comprising longer
delay times ($n$$\;\buildrel>\over{\sim}\;$0) explains the variation in
$r$-process abundance among solar twins classified by their ages. Our obtained
results agree with those of a previous study, which suggested $n$=0.6 as the
most likely value within the allowable range of $-0.5<n<2.6$ (Nakar et al.,
2006).

##  4 Summary

The abundance patterns of solar twins are utilized to infer Galactic chemical
evolution across the disk under the hypothesis that (i) their widely spanning
ages are an outcome of radial migration from the inner region, including the
bulge, and (ii) older solar twins were born closer to the Galactic center,
where chemical enrichment occurs more quickly. This paradigm explains the
variation in solar twin abundance patterns relative to their ages and provides
new insight into the features of $r$-process production sites. We find that
two distinct $r$-process sites should exist to accelerate $r$-process
enrichment in both early and late phases of Galactic evolution.

A population of $r$-process progenitors with long delay times for ejection is
identified as NSMs whose power-law DTDs have a slope of
$n$$\;\buildrel>\over{\sim}\;$0\. On the other hand, an $r$-process site with
short delay times is likely to be associated with specific and rare CCSNe;
however, this suggests the alternative possibility of the presence of a
population ending promptly with NSMs (e.g., Beniamini & Piran, 2019; Wanajo et
al., 2021). The CCSNe or NSMs for the short-delayed $r$-process site could be
validated by future works, including an exploration of high-redshift SGRBs.

This work was performed in part at Aspen Center for Physics, which is
supported by National Science Foundation grant PHY-1607611. The author thanks
J. Baba for fruitful discussions on the stellar dynamics in relation to the
Galactic bar. This work was supported by JSPS KAKENHI Grant Numbers 18H01258
and 19H05811.

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