Stochastic Chemical Evolution of Galactic Subhalos and the Origin of r-Process Elements

Mergers of compact binaries (of a neutron star and another neutron star or a black hole, NSMs) are suggested to be the promising astrophysical site of the r-process. While the average coalescence timescale of NSMs appears to be>100 Myr, most of previous chemical evolution models indicate that the observed early appearance and large dispersion of [r/Fe] in Galactic halo stars at [Fe/H]<-2.5 favors shorter coalescence times of 1-10 Myr. We argue that this is not the case for the models assuming the formation of the Galactic halo from clustering of subhalos with different star formation histories as suggested by Ishimaru et al. (2015). We present a stochastic chemical evolution model of the subhalos, in which the site of the r-process is assumed to be mainly NSMs with a coalescence timescale of 100 Myr. In view of the scarcity of NSMs, their occurrence in each subhalo is computed with a Monte Carlo method. Our results show that the less massive subhalos evolve at lower metallicities and generate highly r-process-enhanced stars. An assembly of these subhalos leaves behind the large star-to-star scatters of [r/Fe] in the Galactic halo as observed. However, the observed scatters of [Sr/Ba] at low metallicities indicate the presence of an additional site that partially contributes to the enrichment of light neutron-capture elements such as Sr. The high enhancements of [r/Fe] at low metallicities found in our low-mass subhalo models also qualitatively reproduce the abundance signatures of the stars in the recently discovered ultra-faint dwarf galaxy Reticulum II. Therefore, our results suggest NSMs as the dominant sources of r-process elements in the Galactic halo.


INTRODUCTION
Most of the elements with atomic numbers greater than Z = 30-40 are produced through neutron-capture processes, and about half of such heavy elements originate from the rapid neutron-capture process (r-process). However, the astrophysical site of the r-process has not been specified, which remains a long-standing problem in nuclear astrophysics (see, e.g., Thielemann et al. 2017 for a recent review).
Stellar abundances of Galactic halo stars serve as the fossils of the early history of the Galaxy, providing us with important clues to the Galactic formation and early chemical evolution. Metal-poor stars, those with metallicities of [Fe/H] 1 −2.5, are thought to be among the oldest objects in the Galaxy, which presumably have been formed in the first few hundred Myr of its life. Spectroscopic observations of metalpoor halo stars show a large star-to-star scatter of about 2.5 dex in the abundances of Eu (as representative of r-process elements) with respect to Fe, [Eu/Fe] (e.g., Honda et al. 2004;François et al. 2007;Suda et al. 2008;Sneden et al. 2008). In particular, several metalpoor stars such as CS 22892-052 (Sneden et al. 2003) and CS 310812-001 (Siqueira Mello et al. 2013) show extremely high ratios of [Eu/Fe] = 1.6-1.7. These unique abundance signatures indicate that Eu does not share common astrophysical conditions with α or iron-group elements.
It is also well known that such r-process-enhanced stars, which account for about 10% of all metal-poor stars, exhibit fairly robust abundance distributions that agree with the solar system r-process pattern. On the one hand, the excellent agreement for the heavier side (Z > 50; Sneden et al. 2008) indicates the presence of the single robust "main" r-process 2 site. On the other hand, the less remarkable agreement for the lighter side (Z < 50;Siqueira Mello et al. 2014) as well as the bulk of (r-process-deficient) stars showing higher [Sr/Ba] than the solar r-process ratio (McWilliam 1998;Johnson & Bolte 2002) implies the presence of another "weak" r-process that produces only light neutroncapture elements . In fact, several metal-poor stars showing a descending trend of neutron-capture elements toward the heavier side have been identified (Honda et al. 2006(Honda et al. , 2007Aoki et al. 2017), which may reflect such a process.
Inhomogeneous chemical evolution models (Ishimaru & Wanajo 1999;Argast et al. 2000;Tsujimoto et al. 2000;Ishimaru et al. 2004;Cescutti 2008), which account for incomplete mixing of nucleosynthetic ejecta in the interstellar medium (ISM), have demonstrated that the observed dispersion of [r/Fe] ratios (where r indicates an r-process element) in Galactic halo stars can be explained if CCSNe from a limited initial stellar mass range are the sources of r-process elements. However, recent nucleosynthesis studies show difficulties in producing heavy r-process elements (Z 50) in the physical conditions relevant to CCSNe (Wanajo et al. 2011(Wanajo et al. , 2018Wanajo 2013;Bliss et al. 2018), which can be at best the sources of light neutron-capture elements made by a weak r-process. Effects of a strong magnetic field also have been discussed (Thompson 2003;Suzuki & Nagataki 2005;Winteler et al. 2012;Nishimura et al. 2015), although their roles on the r-process are under debate (Nishimura et al. 2017; Thompson & ud-Doula 2017;Mösta et al. 2017).
By contrast, recent nucleosynthesis calculations based on the hydrodynamical simulations of NSMs reasonably reproduce the solar r-process abundance curve (Wanajo et al. 2014;Goriely et al. 2015;Radice et al. 2016;Wu et al. 2016).
The discovery of an electromagnetic emission (kilonova; Li & Paczyński 1998;Metzger et al. 2010) associated with the gravitationalwave source GW170817 (Abbott et al. 2017) also supports NSMs as the site of the r-process in the universe. In fact, the inferred amount of the r-process material ejected from this event, about 0.03-0.05 M ⊙ (e.g., Pian et al. 2017), appears to be sufficient to account for the total mass of r-process elements in the Galaxy, provided that GW170817 is representative of NSM events.
However, binary population synthesis models (e.g., Dominik et al. 2012) as well as observations of binary neutron stars (e.g.,  suggest an average NSM coalescence timescale t NSM 100 Myr, which appears too long to allow for the observed appearance of, e.g., Eu at metallicities as low as [Fe/H] ∼ −2.5 on the basis of one-zone chemical evolution models (Argast et al. 2004). Such models suggest t NSM = 1 − 10 Myr to reproduce the observed early evolution of [r/Fe] (Argast et al. 2004;De Donder & Vanbeveren 2004;Matteucci et al. 2014;Tsujimoto & Shigeyama 2014;Cescutti et al. 2015;Wehmeyer et al. 2015). Previous chemical evolution models based on the hierarchical structure formation of the Galactic halo also favor short NSM coalescence timescales (Komiya et al. 2014;van de Voort et al. 2015). In addition, it is argued that the inferred low event rate (0.4 − 77.4 Myr −1 ; Dominik et al. 2012) of NSMs causes too large [r/Fe] dispersions at higher metallicity to be compatible with observations (Qian 2000;Argast et al. 2004).
Chemical evolution studies assuming multiple rprocess sites such as NSMs and magnetorotationally driven CCSNe attempt to explain the observed [r/Fe] evolution of the metal-poor stars (Wehmeyer et al. 2015;Shibagaki et al. 2016). However, the uniqueness of the abundance patterns in r-process-enhanced stars apparently disfavors multiple r-process sites with different abundance distributions (e.g., Sneden et al. 2008) considered in these studies. Prantzos (2006) suggested that the observed appearance of Eu at low metallicity as well as the large dispersion of [Eu/Fe] could be naturally explained if the Galactic halo was formed from merging subhalos with different star formation histories and if the production sites of Fe and r-process elements evolved on different timescales. Ishimaru et al. (2015, hereafter IWP15) have first explored this idea using a semianalytical model of merging subhalos, each of them evolving on a different timescale-depending on its massin a homogeneous way (i.e., the gas is assumed to be well mixed within each subhalo). According to IWP15, NSMs start occurring at the metallicity [Fe/H] −3 and contributing to the enrichment of Eu even with the coalescence timescale of 100 Myr if less massive subhalos evolve with lower star formation efficiency. Similar results can be found in recent semianalytic (Komiya & Shigeyama 2016) and chemodynamical (Shen et al. 2015;Hirai et al. 2015 studies (see also Côté et al. 2017 for a comparison of several chemical evolution studies mentioned above with their own model).
IWP15 also show that various star formation efficiencies make a difference in the numbers of cumulative NSMs occurring in subhalos. In fact, the number of cumulative average NSMs for the lightest subhalo with a stellar mass of 10 4 M ⊙ (similar to that of an ultra-faint dwarf galaxy, UFD) predicted in IWP15 is ∼ 0.1, implying that only one 1 of 10 small subhalos experiences an NSM event. They suggested that a single NSM occurring in the least massive systems would lead to a very high [r/Fe] of the inter stellar medium. The recently discovered UFD, Reticulum II (Ret II; Ji et al. 2016a;Roederer et al. 2016;Ji et al. 2016b;Ji & Frebel 2018), could be such an example as anticipated by IWP15, in which seven (out of nine) stars exhibited high [r/Fe] ratios comparable to those in the most r-process-enhanced Galactic halo stars.
In this paper, we extend the study of IWP15 to explain the presence of such r-process-enhanced stars and the scatter of [r/Fe] ratios in the Galactic halo as well as in Ret II. We also aim to examine the chemical evolution of Sr to test if our model is compatible with those of light neutron-capture elements. For this purpose, we construct a chemical evolution model, in which individual subhalos stochastically experience NSM events. In section 2, the concept and setup of our model are presented in detail. In section 3, we discuss the enrichment histories of Ba and Eu as representative of r-process elements. We also compare our results with the observations of a light neutron-capture element Sr (section 4) as well as of the r-process-enhanced stars in Ret II (section 5). Finally, we summarize and conclude our work in section 6.

CHEMICAL EVOLUTION MODEL
Based on the hierarchical structure formation scenario, our Galactic halo is formed from subhalos with various masses. Depending on their masses, the subhalos are expected to have different star formation histories, and the sum of the subhalos weighted by their mass function is assumed to become the Galactic halo. Within each subhalo, the occurrence of NSMs is treated stochastically (see Sec. 2.3).

Chemical Evolution of Different Mass Subhalos
The star formation rate (SFR) of a subhalo, ψ(t), is assumed to be proportional to its interstellar medium (ISM) mass, M ISM (t), where k SF is a time-independent star formation efficiency. All subhalos are expected to suffer gas outflow because of their weak gravitational binding (as a result of, e.g., gas heating, ram pressure, and tidal stripping) in which the interstellar material is efficiently mixed (i.e., spatially homogeneous). The gas outflow rate (OFR), ϕ(t), is assumed to be proportional to the SFR, where η is a time-independent coefficient. We introduce a time-independent outflow efficiency, k OF ≡ η k SF ; i.e., For the stellar initial mass function (IMF), we adopt Kroupa's IMF (Kroupa 2002) within the mass range of 0.05-1 M ⊙ and the slope of −2.7 is taken for the range of 1-60 M ⊙ (see also Kubryk et al. 2015). Massive stars of 10-60 M ⊙ are the progenitors of CCSNe in this study.
Although it is nearly impossible to directly observe ancient subhalos nowadays, they might be very close in nature to the local dwarf spheroidal galaxies that we observe now. The observed mass-metallicity relation of those galaxies by Kirby et al. (2013) shows a clear correlation between the stellar masses and the mean metallicities of the local dwarf galaxies, regardless of their morphologies. Therefore, we assume that subhalos also have the same correlation. This mass-metallicity relation scales as [Fe/H] ∝ log M ⋆ 0.3 within the mass range 10 3 − 10 8 M ⊙ , where [Fe/H] and M ⋆ are the mean metallicity and the stellar mass of a dwarf galaxy, respectively. Here, we assume that the mean metallicity is equal to the peak of a given metallicity distribution. This leads to η ∝ M ⋆ −0.3 (Prantzos 2008; IWP15). Thus, we adopt where M sub is the stellar mass of a subhalo and η 8 corresponds to the value for a subhalo with the mass 10 8 M ⊙ , which is one of the most massive subhalos assumed in our model. The values η 8 and 10 8 M ⊙ are chosen so that the subhalos with the final stellar mass 10 8 M ⊙ have a metallicity reaching [Fe/H] ∼ −1 at the final time t = 2 Gyr, which is realized with k SF = 0.20 Gyr −1 and k OF = 1.0 Gyr −1 ; i.e., η 8 = 5.0. Since Galactic halo stars show no trace of type Ia supernovae, we do not include their contribution to Fe yields in our model. For each value of η, we consider two extreme cases in order to determine k SF and k OF (as in Table 1 in IWP15). For case 1, k OF is assumed to be constant at k OF = 1.0 Gyr −1 while k SF ∝ M sub +0.3 . On the other hand, for case 2, k SF is assumed to be constant, k SF = 0.20 Gyr −1 , while k OF ∝ M sub −0.3 . The SFR ψ(M sub , t) is then obtained as a function of time and the stellar mass of a subhalo.
Element yields of CCSN are taken from Nomoto et al. (2013) with a linear interpolation between 10 and 40 M ⊙ and those at 40 M ⊙ in the range 40-60 M ⊙ . We do not consider the enrichment from the s-process, because the s-component of the heavy elements stems mainly from the AGB phase of low-mass stars and is not expected to contribute to metal-poor stars. Since we fo-cus mainly on metal-poor stars, we also treat Ba (an s-process-dominant element in the solar system) as an r-process element, a concept that was first suggested by Truran (1981); see, however, Prantzos et al. (2018) for the case of rotating massive stars, which may have a nonnegligible contribution to s-element production even at low metallicities.
The ISM is assumed to be well mixed; thus, enrichment by CCSNe is calculated in a continuous way by a simple chemical evolution model with gas outflow. Enrichment of r-process elements is computed in a stochastic way only when NSMs occur.

Subhalo Mass Distribution
Cosmological simulations based on the hierarchical structure formation scenario predict the dark matter function of subhalos, which is proportional to the inverse square of the subhalo dark mass (Diemand et al. 2007), i.e., proportional to the inverse square of the baryonic mass. Taking into account the effective yield from the mass-metallicity relation, which suggests outflow, the final stellar mass is smaller than the initial baryonic mass.
As a result, the subhalo stellar mass function becomes We consider the subhalos with stellar masses ranging from 10 4 M ⊙ to 2 × 10 8 M ⊙ . The lowest subhalo mass that we consider is comparable to the smallest local dwarf galaxies (Kirby et al. 2013) 4 . The highest mass is set to half of the mass of the Galactic halo, M halo = 4 × 10 8 M ⊙ (Bell et al. 2008). The subhalo stellar mass function is normalized as follows: By definition, the total number of subhalos within the mass range between M 1 and M 2 is given by The third column of Table 1 shows the numbers of subhalos between M according to Equation (5). Note that N sub is the same for both cas 2.3. Enrichment by NSMs 4 Note that the observed lowest stellar mass of the local dwarf galaxies from Kirby et al. (2013) is of the order of 10 3 M ⊙ (Segue 2); however, we set the lowest mass of subhalos be that of intermediate-mass ultra-faint dwarfs, 10 4 M ⊙ , because the massmetallicity relation scales well in the range 10 3.5 < M * /M ⊙ < 10 9 (Figure 9 in Kirby et al. 2013  The suggested typical coalescence timescale of NSM in the literature is 100 Myr (e.g., Belczynski et al. 2008;Dominik et al. 2012). Although these studies also predict short-lived mergers ( 1 Myr), the fraction of such binaries is estimated to be less than several percent (Dominik et al. 2012). In our model, we adopt the bimodal NSM coalescence timescales from IWP15, where the proportions of long-lived (100 Myr) and short-lived (1 Myr) NSMs are set to 95% and 5%, respectively. The average frequency of NSM to CCSN is assumed to be f NSM /f CCSN = 1/1000 according to the population synthesis model by Dominik et al. (2012).
The number of CCSN events occurring within a given time interval ∆t is where τ m is the lifetime of the star with initial mass m adopted from Schaller et al. (1992). The number of NSM events occurring within the time interval ∆t is then where t NSM is the NSM coalescence timescale. Equations (6) and (7) account for the average numbers of CCSNe and NSMs, respectively, occurring in a subhalo with mass M sub in the corresponding time interval. In our model, we first calculate the average number of NSMs occurring in each time interval (taken to be a few percent of t) for a subhalo of M sub . The total average number of NSMs occurring in a group of subhalos with masses between M 1 and M 2 is then obtained as ∆N NSM N sub in the corresponding time interval. Using a Monte Carlo method, we randomly choose the subhalos in which NSMs occurred. In Table 1 the resulting average, minimum, and maximum numbers of NSMs (for Table 2. NSM-related Model Parameters Short-lived 1.00 5 5.00 × 10 −5 4 × 10 −5 8 yEu Long-lived 100 95 9.50 × 10 −4 4 × 10 −5 8 yEu case 1) in the subhalos of masses between M 1 and M 2 after 2 Gyr are presented in the 4th, 5th, and 6th columns, respectively. The yield of Eu ejected by a single NSM event is assumed to be y Eu = 4 × 10 −5 M ⊙ so that the observed average [Eu/Fe] values are reproduced (Figures 4). This value is about a factor of 2 greater than that obtained by the nucleosynthesis calculation in Wanajo et al. (2014), ∼ 2 × 10 −5 M ⊙ , which is, however, dependent on the still uncertain total ejecta mass from an NSM (e.g., Shibata et al. 2017). Besides Eu, we also calculate the chemical evolution of Ba as well, its yield being y Ba = 8 y Eu , which matches the solar r-process abundance ratio (Burris et al. 2000). The parameters concerning NSMs are summarized in Table 2  In Figure 1, we present the chemical evolutions of five selected subhalos with different stellar masses for case 1 (see Sec. 2.1). Their stellar masses are 10 4 , 10 5 , 10 6 , 10 7 , and 10 8 M ⊙ corresponding to those from the thinnest to thickest curves in different colors. Figure 1a shows the age-metallicity relations of the five subhalos. The Fe yields are those produced by CC-SNe. Since CCSNe occur constantly and sufficiently even for low-mass subhalos (see IWP15), [Fe/H] in each subhalo increases monotonically with time. For case 1, the star formation efficiency k SF is a function of the subhalo mass, in which heavier subhalos have higher k SF values. Therefore, a heavier subhalo shows a higher increasing rate of metallicity. Regarding case 2 (not shown here), k SF is constant regardless of the subhalo mass and hence the age-metallicity relations are the same for all subhalo models. These results are consistent with IWP15. Figure 1b shows the cumulative number of NSMs computed stochastically, which occur in each subhalo. In the subhalos with masses above 10 7 M ⊙ , the first NSM occurs at 0.01-0.02 Gyr. This is a consequence of the fact that these subhalos contain a large enough number of massive stars for triggering even rare (short-lived) NSMs. After 0.1 Gyr, the cumulative N NSM quickly increases owing to the long-lived NSMs. For the heaviest subhalo, the cumulative N NSM increases smoothly unlike the others because the NSM occurrence rate is high. For the subhalos with masses below 10 6 M ⊙ , the numbers of stars are too small to experience short-lived NSMs. The NSMs occurring at 0.1-0.3 Gyr are due to the long-lived NSMs. In any case, NSMs do not necessarily occur at t = 0.1 Gyr because of the stochastic nature. The probabilistic effects appear more clearly in the subhalos with lower masses. The total number of N NSM that occur in these subhalos with masses of 10 5 M ⊙ and 10 4 M ⊙ are only 3 and 1, respectively. Figure 1c shows the time evolutions of [Ba/H] in the five subhalos. We compare our results for these subhalos with the average evolutions that are calculated from the one-zone models (the same as those in IWP15   In general, once a single NSM occurs in a given subhalo, the evolution of [r/Fe] begins with a high value that decreases monotonically with increasing metallicity until the next NSM event, as discussed in section 3.1. As clearly seen in the average abundance ratios, the long-lived NSMs occur at lower metallicities for less massive subhalos. The stars in lower-mass subhalos have lower metallicities on average. Massive subhalos also have stars at low metallicities, but their r-process elements (Ba) originate from the short-lived NSMs. Figure 3 shows the chemical evolutions of Ba for case 2.

Clustering of Subhalos
According to the hierarchical structure formation scenario, our Galactic halo has been formed from the clustering of subhalos (e.g., Prantzos 2008). In this context, one can examine the chemical evolution of the Galactic halo as the ensemble of the subhalos discussed in section 3.2.    Figure 3. Same as Figure 2, but for case 2.  By comparing Figures 2 and 4a, it is clear that the highly r-process-enhanced stars in the Galactic halo originate from the low-mass subhalos (M sub 10 6 M ⊙ ). On the other hand, the Galactic halo stars with subsolar [r/Fe] values mostly come from the heavier subhalos with masses greater than 10 6 M ⊙ . The presence of such a large dispersion in [r/Fe] at [Fe/H] ∼ −3 cannot be reproduced by the evolutions of averaged abundance ratios (red lines in Figure 2; the same as the models in IWP15), which is obviously due to a stochastic nature of NSM events in subhalos. However, the large standard deviation between [Fe/H] ∼ −3 and −2.5 (Figure 4) is mainly due to the mass-dependent star formation efficiency for case 1, k SF ∝ M sub +0.3 , not to the stochastic occurrences of NSMs. As shown in IWP15, the subhalos evolve at different rates, reaching different metallicities at a given time (or the same metallicity at different times, thus with different NSM rates). As a consequence, the stars of a given [Ba/Fe] are distributed in different metallicity regions when all subhalos are displayed (see the red lines in Figure 2). Figure 5 shows the results for case 2. The calculated stellar [r/Fe] values with large star-to-star scatters appear at the metallicity [Fe/H] −2.5, which is substantially higher than [Fe/H] ∼ −3 in the observations. The mean value of our result shows a sudden increase at [Fe/H] ∼ −2.5; however, the observed [r/Fe] values gradually increase from [Fe/H] ∼ −3 to −2. Unlike case 1, the standard deviation overall is small despite its large star-to-star scatter. The reason is the constant star formation efficiency for case 2, k SF = 0.20 Gyr −1 , which keeps all subhalos on the same evolutionary path of [r/Fe] (see the red lines in Figure 3), resulting in a small standard deviation.
We find that our model of case 1 is successful in explaining the observational aspects of the r-process abundances in Galactic halo stars: the large [r/Fe] dispersion with highly r-process-enhanced stars at [Fe/H] −2.5 and the mean value of [r/Fe], which gradually increases with metallicity from [Fe/H] ∼ −3. Therefore, the model of subhalos with k SF ∝ M sub +0.3 appears suitable to describe the chemical evolution of the Galactic halo, and thus we focus on case 1 in the subsequent discussion.

NSM Coalescence Timescales
In the previous subsections, the NSM coalescence timescale t NSM was assumed as 100 Myr for 95% of the NSMs and 1 Myr for the rest. In this subsection, we test other choices of t NSM .  In order to examine the case of little contribution from the short-lived NSMs, we first calculate a model using only the long t NSM , that is, t NSM = 100 Myr for all NSMs. Figure 6 shows the calculated [Ba/Fe] evolution in the Galactic halo for case 1. The Galactic halo has the metal-poor stars with subsolar [Ba/Fe] values at metallicities less than −3, which cannot be explained solely with t NSM = 100 Myr in our model.
Since population synthesis studies as well as the observation of binary neutron stars infer the NSM coalescence timescales ≥ 100 Myr, we next test the case of a longer t NSM . Figure 7  The abundance ratios of r-process elements in the metal-poor halo stars show large dispersions. In contrast, the abundance ratios of α-elements such as Mg show a small spread of ∼ 1 dex (see the gray dots in Figure 8). Moreover, a well-selected homogeneous sample shows little intrinsic scatter in [Mg/Fe] at low metallicity (within a total range of 0.2 dex with a standard deviation of 0.06 dex for 23 stars; Arnone et al. 2005).
[Mg/Fe] appears uniform throughout the whole metallicity range until type Ia supernovae start to increase the Fe abundance after [Fe/H] ∼ −1. This contradicts the inhomogeneous chemical evolution models with inefficient ISM mixing, in which the variation of Mg/Fe in CCSNe yields remains at low metallicity (see, e.g., Argast et al. 2002). Recent hydrodynamical studies show that the implementation of substantial ISM mixing is fundamental to reproduce the observed little scatter in [Mg/Fe] (e.g., Shen et al. 2015;. Each of the one-zone homogeneous subhalos in our model can be regarded as a limiting case of such efficient ISM mixing. In this subsection, we test whether our model is  Figure  11). Such a trend cannot be explained from a single r-process site that produces both Sr and Ba with the solar r-process ratio. This fact implies the presence of a "weak" r-process ) that makes Sr but little Ba, in addition to the "main" r-process. HD 122563 ([Fe/H] = −2.7) is such a star with the signature of a possible weak r-process (Honda et al. 2006), exhibiting a descending trend of abundances toward heavier neutron-capture elements (see also Aoki et al. 2017). In this section, we examine our model (case 1) for the evolution of Sr by taking into account the contributions from both the main and weak r-processes. The assumed site of the main r-process is NSMs as in the previous sections. Note that such a modification of our model does not affect the evolutions of Ba or Eu in section 3. For the weak r-process we assume the stars with 10 -11 M ⊙ , at the low-mass end of CCSN range, which are suggested as the primary sources of light trans-iron elements including Sr (Wanajo et al. 2011(Wanajo et al. , 2018. The solar r-process abundance of Sr by mass is 16 times greater than that of Eu. Thus, we define y Sr,m and y Sr,w , the yields of Sr from the main and weak r-processes, respectively, which satisfy where N w is the number of CCSNe from the range 10 -11 M ⊙ . Observations show that the highly r-processenhanced stars such as CS 22892-052 and CS 31082-001 have somewhat smaller light-to-heavy r-process abundance ratios than that of the solar r-process abundances (Sneden et al. 2008;Siqueira Mello et al. 2014; see also Ji & Frebel 2018 for the stars in Ret II). For simplicity, we assume that half of the Sr abundances come from the main r-process and the other half from the weak r-process, i.e., N NSM y Sr,m = N w y Sr,w .
The yields of Sr from the main and weak r-processes are thus derived from Equations (8) and (9). The parameters related to Sr yields are summarized in Table 4.
We present the chemical evolutions of Sr relative to Fe as functions of [Fe/H] (case 1) for different mass subhalos in Figure 9. The average enhancement of [Sr/Fe] occurs first by the short-lived NSMs up to [Sr/Fe] ∼ −2.4 along with an increase of metallicity to [Fe/H] ∼ −3. Afterward, the weak r-process increases the Sr abundance ratio up to [Sr/Fe]∼ −1, followed by the enrichment due to the long-lived NSMs. The weak r-process (low-mass CCSNe) is responsible for the enrichment of the stars with [Sr/Fe] −1, whereas the higher [Sr/Fe] stars are entirely due to the main r-process (NSMs). As discussed in sections 3.1 and 3.2, NSMs highly enhance the [Sr/Fe] values, especially in a fraction of low-mass subhalos. In contrast to NSMs, low-mass CCSNe occur uniformly in all subhalos, making no star-to-star scatters or large enhancements of [Sr/Fe].
The chemical evolution of Sr as an ensemble of all subhalos is shown in Figure 10, which is compared with the abundances of Galactic halo stars. The observed trend of [Sr/Fe] such as the large star-to-star scatter at [Fe/H] ∼ −3 with highly Sr enhanced stars at [Sr/Fe] ∼ 1 is well reproduced as those of Ba and Eu discussed in section 3. The mean calculated value of [Sr/Fe] at [Fe/H] −2.5 is lower than the solar value because we exclude the contributions from the s-process.
We also present the evolution of [Sr/Ba] in Figure 11. As found in the evolution of [Sr/Fe], the average enhancement of [Sr/Ba] to a constant value of −0.6 occurs first by the short-lived NSMs. The rise of [Sr/Ba] at [Fe/H] −3 with a large dispersion is due to the production of Sr (without Ba) by the weak r-process in low-mass CCSNe. Afterwards, the long-lived NSMs start producing both Sr and Ba, which leads to a convergence of [Sr/Ba] values at [Fe/H] −3. Note that the [Sr/Ba] value is constant at −0.54 with our adopted yields for NSMs, meaning the higher values are purely due to the weak r-process. We also find an overall agreement of our model with the observed trend of [Sr/Ba], that is, the large star-to-star scatter of the abundances at [Fe/H] ∼ −3 with few stars below [Sr/Ba] ∼ −0.6. It is emphasized that the reasonable agreement here is due to the additional sources of Sr, low-mass CCSNe in our model, in addition to NSMs.

CHEMICAL EVOLUTION OF ULTRA-FAINT DWARF GALAXIES
Ultra-faint dwarfs (UFDs) are small galaxies in mass and size with low luminosities. Their faintness makes their discovery as well as their spectroscopic study difficult. Up to date, about 10 UFDs have been discovered around our Galaxy. They are metal-poor (e.g., Kirby et al. 2013), mostly showing very low [r/Fe] (e.g., François et al. 2007;Koch et al. 2013;Frebel et al. 2014;François et al. 2016). However, one of the UFDs, Reticulum II (Ret II), has highly r-process-enhanced stars (Ji et al. 2016a;Roederer et al. 2016;Ji et al. 2016b;Ji & Frebel 2018). The fact that Ret II is one such peculiar galaxy out of about 10 UFDs is reminiscent of our result for low-mass subhalos discussed in section 3. This similarity motivates us to explore the chemical evolutions of UFDs by adopting our models of low-mass subhalos. In this subsection, we apply our subhalo model to the chemical evolution of Ret II. As discussed in section 3.1 and 3.2, the relative abundance ratio in a low-mass subhalo sizably deviates from the average evolutionary curve. This is due to the small number of NSMs occurring in each subhalo. As a result, the value of maximum N NSM divided by the mean N NSM for a given mass range (see Table 1) is greater for lower-mass subhalos.
The stellar mass in a typical UFD galaxy is ∼ 10 3 -10 4 M ⊙ . Provided that UFDs evolve the same as subhalos, we consider the lowest-mass range in our model (case 1), 10 4.0 -10 4.1 M ⊙ , in which only 5 out of 138 subhalos experience (only single) NSMs (Table 5).
In Figure 12 we compare the chemical evolutions of these subhalos with the spectroscopic data of Ret II stars. Each band made by successive colored circles represents the evolution of a single galaxy, which is guided by a thin line. For Mg (Figure 12a), the evolutions of all 138 galaxies are overlapped because of their common origins for Mg and Fe, i.e., CCSNe, and cannot be distinguished. For Sr (Figure 12b), the early evolutions at [Fe/H] < −3.6 are due to the weak r-process from lowmass CCSNe and thus are overlapped as well. However, at [Fe/H] > −3.6, five UFD models (see the first column in   ter the (single) NSM event, respectively. The fraction of r-process-enhanced stars in model SH2 is 0.750 (the fourth column in Table 5), which is in good agreement with that in Ret II (7/9 = 0.778). However, the models considered here reach the end of evolution at [Fe/H] = −2.4 and thus do not account for the presence of two stars at [Fe/H] = −2.2 and −2.1.
Given that the evolutions of UFD galaxies are represented by our models of the lowest-mass subhalos, the fact that 5 out of 138 subhalos experience NSMs implies a chance of several percent to discover Ret II-like galaxies with respect to all observed UFDs. Our result also suggests that UFDs with various fractions of r-processenhanced stars will be discovered in future observations, e.g., 2 out of 10 stars with the enrichment of r-process elements (model SH3; see the fourth column in Table 5).
One should keep in mind, however, that a trial here is for a qualitative purposes only. The least massive subhalo in our model has 10 4 M ⊙ , which corresponds to intermediate-mass UFDs. It is currently unclear if the scaling of the mass-metallicity relation extends to ∼ 10 3 M ⊙ for the majority of UFDs (Kirby et al. 2013). It also is unknown if the chemical evolution of such a lowmass system, probably with only a few episodes of star formation, can be approximated with our simple picture of a homogeneous and continuous chemical evolution.

SUMMARY AND CONCLUSION
We have revisited the study of Galactic chemical evolution by IWP15 in order to investigate the role of NSMs as the dominant sources of r-process elements in the Galaxy. Our chemical evolution model was constructed on the basis of the hierarchical structure formation scenario in IWP15, in which the different mass (M sub ) subhalos that formed the Galactic halo had different star formation histories. The number of NSMs occurring in each subhalo was obtained with the Monte Carlo method. The star formation histories of the subhalos were determined from the observed mass-metallicity relation of the local dwarf galaxies, assuming the same correlation for both systems. In the framework of our simple model of galactic chemical evolution, this massmetallicity relation leads to OFR/SFR ∝ M sub −0.3 , where OFR and SFR are the outflow rate and the star formation rate, respectively. We examined two extreme cases as in IWP15, in which either of the coefficients for SFR and OFR (Equations (1) and (2)) were kept constant such that k OF = 1.0 Gyr −1 and k SF ∝ M sub +0.3 for case 1 or k SF = 0.20 Gyr −1 and k OF ∝ M sub −0.3 for case 2.
Our result shows that the observed properties of the enrichment histories of [r/Fe] in the Galactic halo can be explained by assuming that NSMs are the sources of r-process elements. The adopted low SFR in lowmass subhalos for case 1 makes the occurrence of NSMs at low metallicities ([Fe/H] ∼ −3) possible as shown by IWP15. In addition, the presence of the r-processenhanced metal-poor stars in the Galactic halo is accounted for as a result of the large enhancement of [r/Fe] by a single or a few NSMs in a small fraction of low-mass subhalos. However, our case 2 adopting a constant star formation efficiency (k SF ) results in the enrichment of r-process elements at higher metallicities, [Fe/H] > −2.5, as found in previous studies. We conclude, therefore, that the reality is closer to our case 1, i.e., star formation is less efficient in lower-mass subhalos while gas outflow only weakly depends on the subhalo masses, provided that NSMs are the dominant contributors of r-process elements in the Galaxy.
The observed trend of [r/Fe] in the Galactic halo can be mostly reproduced solely by long-lived NSMs with a coalescence timescale of 100 Myr. However, a small fraction of short-lived (1 Myr in our model) NSMs appear to be necessary, which are responsible for explaining the presence of stars with subsolar [r/Fe] values at [Fe/H] −3. A test shows that a long coalescence timescale appreciably greater than 100 Myr, such as 500 Myr, has difficulty in reproducing the enhancement of r-process elements at low metallicities. This implies either that in reality the distribution of coalescence timescales (for the long-lived NSMs) has a sharp peak at ∼ 100 Myr or the NSMs with longer timescales escape from subhalos because of neutron star kicks and do not contribute to Galactic chemical evolution Safarzadeh & Scannapieco 2017). In the future, localizations of NSMs in galaxies by identification of electromagnetic counterparts (kilonovae) of gravitational waves will provide us with information on the distribution of coalescence timescales.
It is important to note that our model naturally reproduces a large dispersion of abundance ratios (relative to Fe) for r-process elements, but a small one for intermediate-mass elements such as Mg; the latter are produced by the same sources and on the same timescales as Fe, while the former result from different sources, operating on very different timescales. This feature, which is in good agreement with spectroscopic results, is another important aspect of NSMs as sources of r-process elements.
Our model is also successful in explaining the spectroscopic abundances of light neutron-capture elements such as Sr when assuming an additional contribution (a weak r-process) from low-mass CCSNe. The resulting evolution of [Sr/Ba] as a function of metallicity reasonably reproduces the observed star-to-star scatters with higher [Sr/Ba] values than those predicted solely by the enrichment from NSMs. This supports the idea that there are extra sources (weak r-process) of light neutroncapture elements in addition to the (main) r-process.
Finally, our models of the least massive subhalos with M sub ∼ 10 4 M ⊙ well account for the observed nature of UFDs, namely, only 1 (Ret II) out of about 10 such galaxies shows enhancement of r-process elements. Moreover, the fact that seven out of observed nine stars in Ret II show enrichment of r-process elements can be reasonably reproduced by such a lowest-mass subhalo model. This supports the idea that the UFDs are the leftovers of the building blocks that made the Galaxy. Our models also predict the presence of Ret II-like UFDs with various fractions of r-process-enhanced stars. This will be tested by future spectroscopic explorations of UFD galaxies. However, it is currently unclear if the mass-metallicity relation can be applied to the bulk of UFDs. It also is cautioned that we applied our homogeneous and continuous chemical evolution model to such a small system that probably had only a few episodes of star formation. Obviously, further studies of UFDs will be necessary from both observational and theoretical sides.
It should be noted that our study neglected a spatial inhomogeneity of ISM in each subhalo as well as a merging process of subhalos during the evolutionary time of the Galactic halo (2 Gyr). The contribution of s-process elements also was excluded, which could be important for the evolutions of Ba and Sr at [Fe/H] −2.5. Nevertheless, our simplified approach enabled us to disentangle the different sources of dispersion: the subhalomass-dependent SFRs that lead to a spread in [Fe/H] for the same [r/Fe] (as shown by IWP15) and the stochastic NSM events that lead to a spread in [r/Fe] for the same [Fe/H]. However, our models failed to explain the moderate dispersion of the observed [r/Fe] ratios at [Fe/H] ∼ −1.5 (Figure 4), which may be due to the combined effects of ISM inhomogeneity and s-process contribution. Such effects will be explored in our forthcoming paper.