r-PROCESS PRODUCTION SITES AS INFERRED FROM Eu ABUNDANCES IN DWARF GALAXIES

The origin of about half of the elements heavier than Fe, produced via the rapid neutron capture process (r-process), is still a mystery (Cowan et al. 1991; Wanajo & Ishimaru 2006; Arnould et al. 2007; Qian & Wasserburg 2007; see Thielemann et al. 2016 for a recent review). Core-collapse supernovae (CCSNe) have long been thought of as the production sites (Burbidge et al. 1957; Cameron 1957). This scenario requires either a neutrino-driven wind with high entropy (Woosley et al. 1994; Qian & Woosley 1996; Hoffman et al. 1997; Goriely et al. 2015), a highly magnetized outflow (Suzuki & Nagataki 2005; Metzger et al. 2008; Winteler et al. 2012; Vlasov et al. 2014; Mösta et al. 2015; Nishimura et al. 2015), wind from a black hole torus (Wanajo & Janka 2012), or fallback SNe (Fryer et al. 2006). It is unclear whether any one of those occurs frequently enough (Qian 2000). With this realization, the alternative scenario of compact binary mergers (neutron star–neutron star or neutron star–black hole binaries) that eject highly neutron-rich material in which r-process nucleosynthesis takes place (Lattimer & Schramm 1976; Symbalisty & Schramm 1982; Eichler et al. 1989; Freiburghaus et al. 1999) has been attracting more and more attention. Nucleosynthesis studies have shown that all r-process nuclides can be produced in the neutron star merger outflows (Wanajo et al. 2014; Wu et al. 2016). The discovery of macronova/kilonova candidates after short gamma-ray bursts (GRBs; Berger et al. 2013; Tanvir et al. 2013; Yang et al. 2015; Jin et al. 2016) has provided recent support to this idea.

Dwarf galaxies are composed of old stellar populations (e.g., Weisz et al. 2015). The chemical abundances of dwarf galaxies have been frozen since shortly after those galaxies formed. These chemical abundances have unique implications for the r-process production scenario (e.g., Tsujimoto & Shigeyama 2014). The r-process elements have been detected in classical dwarf galaxies (Shetrone et al. 2001, 2003; Venn et al. 2004; Letarte et al. 2010; Tsujimoto et al. 2015). In addition to stable r-process elements, thorium is found in a star of Ursa Minor, and its abundance ratio to stable r-process elements is lower than solar (Aoki et al. 2007), suggesting that events producing heavy r-process elements took place in this galaxy in the early universe ∼10 Gyr ago.

The Sloan Digital Sky Survey discovered several ultrafaint dwarf galaxies (UFDs; e.g., Willman et al. 2005; Belokurov et al. 2007). Remarkably, these UFDs contain only ∼10³–10⁵ stars, and they are composed of an old and extremely metal-poor stellar population, suggesting that the UFDs are the closest objects to the first galaxies (see Bromm & Yoshida 2011 for a review). For some of them, the majority of the stars were likely to have formed before reionization, corresponding to a universe age of ∼13 Gyr (Brown et al. 2014). Recently, Ji et al. (2016a) and Roederer et al. (2016) reported the first discovery of highly r-process-enriched stars in a UFD, Reticulum II, while there are only strong upper limits on the abundances of r-process elements for some other UFDs (Frebel et al. 2010, 2014; Roederer & Kirby 2014). These measurements that show large fluctuations in the r-process abundance imply that only a single r-process event took place in Reticulum II. Following these observations of r-process material in the classical dwarfs and UFDs, a question that naturally arises is what are the rate and amount of matter produced in r-process events that are consistent with these observations.

We estimate here the most likely mass ejection of Fe and Eu and rate of CCSNe per total luminosity and of r-process production events relative to CCSNe. We use a likelihood analysis, based on Poisson statistics. The formulation is independent of the specific nature of the sources of r-process elements. We use the given measured Fe and Eu abundances in dwarf galaxies and their initial gas masses (to which those metals were injected). We assume that all dwarf galaxies (or alternatively, only UFDs) share the same mechanism for r-process production. The results in both cases (of all dwarf galaxies and just UFDs) strongly suggest that the r-process event rate is considerably smaller than the CCSN rate. They are comparable to the expectations from the neutron star merger scenario.

This paper is organized as follows. In Section 2 we describe the sample of UFD galaxies. A critical factor in the analysis is the initial gas masses in UFDs. We estimate the initial gas masses in UFDs in Section 3 using several different methods. We turn to the initial masses of Fe and r-process materials in Section 4. We apply a likelihood analysis in order to examine the Fe and r-process abundances in dwarf galaxies and estimate the rates and ejecta mass of CCSNe and r-process events needed in Section 5. We then compare the implied rate and r-process yields with those expected in the double neutron star (DNS) merger scenario in Section 6.1 and peculiar CCSNe in Section 6.2. Finally, we summarize our results in Section 7.

We consider dwarf spheroidal galaxies and focus in particular on UFDs that are composed of old and metal-poor stars (e.g., Simon & Geha 2007). We consider galaxies for which there are reported detections or upper limits on the Fe and Eu abundances. Our sample includes five UFDs: Reticulum II, Segue I, Segue II, Coma Berenices, and Ursa Major II. In addition, we consider six more massive classical dwarf galaxies: Carina, Sculptor, Draco, Leo I, Ursa Minor, and Fornax. We do not consider Sagitarius, since this galaxy has been shown to be considerably tidally disrupted (Ibata et al. 1995; Mateo et al. 1996; Majewski et al. 2003), and its mass is therefore very uncertain. Table 1 shows the V-band luminosities, L_V, the velocity dispersions, σ, and the half-light radii, r_1/2, taken from Walker et al. (2009, 2015), and the average Fe and Eu abundances (Shetrone et al. 2001, 2003; Frebel et al. 2010, 2014; Letarte et al. 2010; Roederer & Kirby 2014; Tsujimoto et al. 2015; Ji et al. 2016a, 2016b; Roederer et al. 2016). The errors of the average abundances shown in Table 1 include statistical and systematic uncertainties. If the systematic errors of the measurements for a galaxy are not available in the literature, we simply assume a systematic error of 0.2 dex.

Table 1.  Summary of Observed Properties of Dwarf Galaxies

Type	Object	r1/2 (pc)	σ (km s−1)	Mh(<r1/2) (107 M⊙)	LV (LV,⊙)	$\langle [\mathrm{Fe}/{\rm{H}}]\rangle$	$\langle [\mathrm{Eu}/{\rm{H}}]\rangle$
UFDs	Reticulum II	${32}_{-1.1}^{+1.9}$	${3.6}_{-0.7}^{+1.0}$	${0.024}_{-0.008}^{+0.014}$	1.0 ± 0.09 · 103	−2.83 ± 0.23	−0.97 ± 0.19
	Segue I	29 ± 7	4.3 ± 1.2	0.031 ± 0.019	3.3 ± 2.1 · 102	−2.68 ± 0.21	<−2.0
	Segue II	34 ± 5	3.4 ± 1.8	0.023 ± 0.023	8.5 ± 1.7 · 102	−2.96 ± 0.19	<−3.0
	Ursa Mayjor II	140 ± 25	6.7 ± 1.4	0.36 ± 0.16	4.0 ± 1.9 · 103	−2.89 ± 0.22	<−3.1
	Coma Berenices	77 ± 10	4.6 ± 0.8	0.094 ± 0.035	3.7 ± 1.7 · 103	−2.57 ± 0.22	<−2.4
Classical dwarfs	Draco	196 ± 12	9.1 ± 1.2	0.94 ± 0.25	2.7 ± 0.4 · 105	−2.00 ± 0.21	−1.26 ± 0.26
	Leo I	246 ± 19	9.2 ± 1.4	1.2 ± 0.4	3.4 ± 1.1 · 106	−1.29 ± 0.20	−0.75 ± 0.24
	Fornax	668 ± 34	11.7 ± 0.9	5.3 ± 0.9	1.4 ± 0.4 · 107	−0.90 ± 0.15	−0.33 ± 0.22
	Carina	241 ± 23	6.6 ± 1.2	0.61 ± 0.23	2.4 ± 1.0 · 105	−1.64 ± 0.20	−1.34 ± 0.21
	Ursa Minor	280 ± 39	9.5 ± 1.2	1.5 ± 0.4	2.0 ± 0.9 · 105	−1.90 ± 0.11	−1.11 ± 0.23
	Sculptor	260 ± 39	9.2 ± 1.1	1.3 ± 0.4	1.4 ± 0.6 · 106	−1.64 ± 0.20	−1.16 ± 0.22

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To evaluate the number of events that supplied Eu or Fe in each galaxy, the total amount of the elements produced in it is needed. We assume that the elemental abundances of stars in dwarf galaxies represent the amount of the elements in the gas from which the stars have been formed. The total amount of Fe and Eu produced in a galaxy during active star formation is estimated using the elemental abundances listed in Table 1, together with the gas mass into which metals are injected. The relatively low stellar luminosities as compared with the total halo masses suggest that UFDs have lost most of their initial gas due to either SN feedback or intergalactic medium reionization (Bovill & Ricotti 2009). Thus, the observed upper limits (∼10³–10⁴ M_⊙) on their current gas mass are not representative of the amount of material into which the r-process elements have been injected at the early stages of their evolution. Therefore, we estimate the amount of the gas based on the assumption that the initial gas mass is proportional to the halo mass within the size of the stellar spheroidal.

When considering the mass loss from the dwarf galaxies, we assume that the gas is chemically homogeneous with the average abundances, as observed in the remaining stars. However, of course, the chemical mixing process in the gas is inhomogeneous, and a fraction of the produced material may escape from the galaxies, or alternatively, gas with a primordial composition may escape. Such effects should be taken into account when considering the abundance distribution of stars in each dwarf galaxy. These issues should be addressed with hydrodynamics simulations (see e.g., Pallottini et al. 2014; Hirai et al. 2015; Ritter et al. 2015) and are beyond the scope of this work.

The initial gas mass, M_g, should be between the total stellar mass of the system and 1/6 of the total halo mass of the system (we take here the typical fraction of 1/6 for the ratio of baryonic to total mass in a halo). As mentioned above, however, the stellar mass of UFDs is significantly smaller than the halo mass due to significant gas loss throughout the formation and evolution of the galaxy. We estimate M_g using the halo mass within a given radius, M_h(<r). This can be calculated directly from measurements of the half-light radius r_1/2 and the velocity dispersion σ by using the Jeans equation (e.g., Walker et al. 2009):

$$\begin{eqnarray}{M}_{h}(r) & = & \displaystyle \frac{5{r}_{1/2}{\sigma }^{2}{(r/{r}_{1/2})}^{3}}{G[1+{(r/{r}_{1/2})}^{2}]}\\ & \approx & 1.2\times {10}^{3}\,{M}_{\odot }\left(\displaystyle \frac{r}{1\,\mathrm{pc}}\right){\left(\displaystyle \frac{\sigma }{1\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}\displaystyle \frac{{(r/{r}_{1/2})}^{2}}{1+{(r/{r}_{1/2})}^{2}},\end{eqnarray} \tag{ 1 }$$

where the Plummer model for the stellar distribution with an isotropic and constant velocity dispersion is assumed. The halo masses within the half-light radius estimated with this formula by ${M}_{h}(\lt {r}_{1/2})=5{r}_{1/2}{\sigma }^{2}/2G$ are shown in Table 1. Of course, this formula can only be used up to the radius where the above assumptions are valid. As a fiducial model, we use M_g ≈ M_h(<r₉₀)/6, where ${r}_{90}\approx 3.71{r}_{1/2}$. According to the Plummer distribution, this choice corresponds to assuming that the material within the radius r₉₀, where 90% of the stellar mass is included, contributes to the chemical mixing of the injected metals. For comparison, we consider also initial halo masses that are smaller, M_h(<r₅₀), or larger than M_h(<r₉₀) by a factor of 3.

As the assumptions made in Equation (1) may be invalid at r₉₀ for some large dwarf galaxies, we also employ an alternative way to estimate the halo masses within r₉₀: using the Navarro–Frenk–White (NFW) mass profile with the model parameters derived in Walker et al. (2009), and integrating the distribution up to r₉₀. This method can only be carried out for the classical dwarfs. For the UFDs there are insufficient observations of stellar kinematics at large enough radii that would constrain the model parameters. In all but one case (Fornax), this estimate results in similar values to those obtained with the first method. Table 2 provides a summary of the different masses.

Table 2.  Summary of Derived Quantities for the Dwarf Galaxies in the Sample

Type	Object	r90 (pc)	Mg(<r90)a (106 M⊙)	Mg(<r90)b (106 M⊙)
UFDs	Reticulum II	${120}_{-4}^{+7}$	${0.28}_{-0.12}^{+0.15}$	...
	Segue I	110 ± 3	0.37 ± 0.21	...
	Segue II	130 ± 20	0.27 ± 0.28	...
	Ursa Major II	520 ± 90	4.2 ± 1.8	...
	Coma Berenices	290 ± 40	1.1 ± 0.8	...
Other dwarfs	Draco	730 ± 50	11 ± 3	${9.8}_{-2.7}^{+3.5}$
	Leo I	920 ± 70	14 ± 4	13 ± 4
	Fornax	2500 ± 100	61 ± 10	30 ± 4
	Carina	900 ± 90	7.0 ± 2.7	5.3 ± 1.5
	Ursa Minor	1000 ± 200	17 ± 5	14 ± 6
	Sculptor	970 ± 150	15 ± 4	13 ± 5

Notes.

^aThe initial gas mass estimated with the assumption of constant velocity dispersion up to r₉₀. ^bThe initial gas mass estimated with the NFW model.

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We compare the gas masses estimated here with those from the literature. Using a cosmological simulation of the first galaxies, Greif et al. (2010) show that Population II stars are formed from cold gas of ∼10⁵ M_⊙ at the center of a galaxy with a total mass of ∼10⁸ M_⊙. The smallest gas masses in our sample, (2–3) × 10⁵ M_⊙ for Reticulum II, Segue I, and Sugue II, are consistent with the result of Greif et al. (2010).

The total masses of Fe and Eu produced in the system are estimated as

$$\begin{eqnarray}&&{M}_{\mathrm{Fe}}\approx 120\cdot {10}^{[\mathrm{Fe}/{\rm{H}}]}\,{M}_{\odot }\left(\displaystyle \frac{{M}_{g}}{{10}^{5}\,{M}_{\odot }}\right),\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{M}_{\mathrm{Eu}}\approx 3.7\cdot {10}^{-5+[\mathrm{Eu}/{\rm{H}}]}\,{M}_{\odot }\left(\displaystyle \frac{{M}_{g}}{{10}^{5}\,{M}_{\odot }}\right),\end{eqnarray} \tag{ 3 }$$

where we assume that metals with an Eu (Fe) abundance of [Eu/H] ([Fe/H]) are injected into a gas with an initial mass M_g and used the solar abundances from Lodders (2003).

While only the abundances of Eu and some other r-process elements are measured, the total mass of r-process elements produced in a dwarf galaxy can be estimated by extrapolating the Eu abundance to other r-process ones using the solar r-process abundance pattern (Goriely 1999):

$$\begin{eqnarray}{M}_{r}\sim \left\{\begin{array}{ll}3.3\cdot {10}^{-2+[\mathrm{Eu}/{\rm{H}}]}\,{M}_{\odot }\left(\displaystyle \frac{{M}_{g}}{{10}^{5}\,{M}_{\odot }}\right) & ({A}_{\min }=70)\\ 0.7\cdot {10}^{-2+[\mathrm{Eu}/{\rm{H}}]}\,{M}_{\odot }\left(\displaystyle \frac{{M}_{g}}{{10}^{5}\,{M}_{\odot }}\right) & ({A}_{\min }=90)\\ 0.6\cdot {10}^{-2+[\mathrm{Eu}/{\rm{H}}]}\,{M}_{\odot }\left(\displaystyle \frac{{M}_{g}}{{10}^{5}\,{M}_{\odot }}\right) & ({A}_{\min }=110)\end{array}\right.\end{eqnarray} \tag{ 4 }$$

where A_min is the minimum mass number of r-process elements. It is worthwhile noting that the abundances of the r-process elements with atomic numbers of Z < 40, corresponding to A < 90, in Reticulum II are depleted compared to solar (Ji et al. 2016b) and similar to those of an extreme metal-poor star CS 22892–052 (Sneden et al. 2003). This suggests that there are two types of events, light and heavy r-process events, and only the latter took place in Reticulum II.

Figure 1 depicts the estimated total Fe and Eu masses for each dwarf galaxy as a function of the total stellar luminosity. Note that the Fe mass is proportional to the luminosity within uncertainties, implying that the number of SNe producing Fe is proportional to the number of stars. For the Eu masses, while a similar trend can be found, the dispersion is larger for the smaller galaxies. We find M_Eu ≈ 10⁻⁵ M_⊙ for Reticulum II. For the other UFDs there are strong upper limits, in some cases as small as 10⁻⁷ M_⊙. This led Ji et al. (2016a) to suggest that there was only a single r-process production event in Reticulum II and none in the other UFDs.

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5.1. The Statistical Method

We aim to constrain the event rate and mass per event for both SNe and r-process events. We begin with a description of the general statistical method that we use in both cases. We define the likelihood function in terms of two unknown parameters: R, the average rate of events, and ${\tilde{m}}_{X}$, the mass of an element X produced by a single event.

Using Poisson statistics, we calculate the probability of obtaining N events in the ith galaxy, given an expected number of events ${N}_{R,i}$. We assume that the mass of X produced per event follows a normal distribution with an average of ${\tilde{m}}_{X}$ and a standard deviation of $0.5{\tilde{m}}_{X}$. For each N the associated probability is then the Poisson probability of obtaining N events times the probability that N events would lead to an accumulated mass of element X in the ith galaxy, ${M}_{X,i}$ (calculated from the observed values of [X/H] and M_g). Summing these probabilities over all N values, we obtain the final probability: ${P}_{i}({M}_{X,i}| R,{\tilde{m}}_{X})$. Finally, we write the likelihood function as

$$\begin{eqnarray}&&L(R,{M}_{X})={{\rm{\Pi }}}_{i}{P}_{i}({M}_{X}| R,{\tilde{m}}_{X}).\end{eqnarray} \tag{ 5 }$$

We maximize, of course, this function over its free parameters.

5.2. The SN Rate and Iron Production in Dwarf Galaxies

We begin by applying the method described in Section 5.1 to constrain the rate of Fe-producing SNe and the amount of Fe produced per event in these dwarf galaxies. The resulting likelihood function is seen in Figure 2. The main constraint is on the product of the two parameters. This reflects the fact that the average Fe mass per solar luminosity should match the observed ratio. The actual peak of the likelihood function is not strongly constrained, although interestingly, it coincides with an Fe mass per event of ${\tilde{m}}_{\mathrm{Fe}}=0.1\,{M}_{\odot }$ and a rate of ≈16 SNe per 10³ L_V,⊙ (approximately 10³ stars).

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These values are in fact expected from SN studies. In the local universe, two-thirds of CCSNe are Type II SNe that typically produce 0.02 M_⊙ of Fe and Type Ibc SNe that produce 0.2 M_⊙ (Drout et al. 2011; Kushnir 2015), and one-third of CCSNe are Type Ibc (Li et al. 2011). Therefore, the Fe production is dominated by SNe Ibc. In addition, since the star formation epoch in dwarf galaxies had lasted for a short timescale of ∼1 Gyr, we assume that SNe Ia do not contribute to the Fe production. Thus, 0.1 M_⊙ per CCSN is expected for dwarf galaxies. In addition, the rate of 16 SNe per 10³ L_V,⊙ is consistent with observations suggesting one SN per 100M_⊙ of star formation (e.g., Horiuchi et al. 2011; Li et al. 2011). The observational constraints on the rate and Fe mass per event from the local universe are depicted by the red cross in Figure 2. This result suggests that the Fe production in each CCSN and their rate in metal-poor environments are similar to those in the local universe. In addition, it demonstrates the validity of this method.

5.3. Results and Implications for r-process Production in Dwarf Galaxies

We turn now to the production of Eu in r-process events. We calculate the rates relative to the CCSN rate, R_rp/SN. Using the results of the previous section (and independent analyses; see Section 5.2), we assume an average production of 0.1 M_⊙ of Fe per CCSN. The average number of r-process events in the ith galaxy is then N_R,i = R_rp/SN × (M_Fe,i/0.1 M_⊙), where M_Fe,i is the total amount of Fe in the same galaxy.

Figure 3 shows the results for two cases: all dwarf galaxies (left) and only UFDs (right). The likelihood function peaks at a rate relative to the SN rate of R_rp/SN = 6 × 10⁻⁴ (R_rp/SN = 3 × 10⁻⁴), and at a mass per event of ${\tilde{m}}_{\mathrm{Eu}}=9\times {10}^{-5}\,{M}_{\odot }$ (${\tilde{m}}_{\mathrm{Eu}}=7\times {10}^{-6}\,{M}_{\odot }$), corresponding to a total mass of r-process matter produced per event of roughly ∼1.7 × 10⁻² M_⊙ (1.3 × 10⁻³ M_⊙). The strong upper limits on the Eu abundance of some galaxies, coupled with the detection of a significant amount of Eu in galaxies of similar masses, imply that the Eu mass produced per event should be approximately the smallest amount detected in any given galaxy. Notice that this situation is very different than the case for Fe production, where the same statistical method results in no preference for a small number of events per galaxy. This leads to a relative rate of r-process to CCSN events $\ll 1$. Thus, we can rule out the possibility that normal CCSNe dominate the production of r-process materials in dwarf spheroidal galaxies.

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Interestingly, this rate of Eu production is consistent with the constraints from our own, much more massive Galaxy. Assuming one CCSN per 100 M_⊙ of star formation, and given a total stellar mass of 6.4 × 10¹⁰ M_⊙ (McMillan 2011), we obtain 6.4 × 10⁸ such SNe in the history of the Milky Way (Venn et al. 2004). Combining this with a total Eu mass of $\approx 18\,{M}_{\odot }$ leads to ${R}_{\mathrm{rp}/\mathrm{SN}}=2.8\times {10}^{-3}{({\tilde{m}}_{\mathrm{Eu}}/{10}^{-5}{M}_{\odot })}^{-1}$, with a systematic uncertainty of a factor of ∼3. The range permitted by these observations is depicted as the area between the dashed lines in Figure 3. As seen in that figure, this range is consistent with the best-fit parameters found for the dwarf galaxies. Furthermore, the low rate is consistent with ${R}_{\mathrm{rp}/\mathrm{SN}}\lt 5\cdot {10}^{-3}$, which Hotokezaka et al. (2015) find in order to reproduce the ²⁴⁴Pu abundances of both the early solar system material (Turner et al. 2007) and the present-day deep-sea archives (Wallner et al. 2015). These suggest that the r-process production in the Milky Way is dominated by the same events as in dwarf galaxies. Therefore, these results for the rate and mass per event can be generalized, and the same process could be the dominant source of r-process production in the whole universe.

5.4. The Robustness of the Method and Dependence on Parameters

We turn now to exploring the dependence of the results on the assumed halo masses. As mentioned in Section 3, there is some uncertainty in the values of these masses. We repeat the calculation (for all dwarf galaxies) with halo masses of ${M}_{h}(\lt {r}_{50})$ (smaller by about an order of magnitude as compared with M_h(<r₉₀)) and $3{M}_{h}(\lt {r}_{90})$. For ${M}_{h}(\lt {r}_{50})$ ($3{M}_{h}(\lt {r}_{90})$), the likelihood peaks at an Eu mass per event of ${\tilde{m}}_{\mathrm{Eu}}=3\times {10}^{-5}\,{M}_{\odot }$ (${\tilde{m}}_{\mathrm{Eu}}=2.5\times {10}^{-4}\,{M}_{\odot }$) and at a rate relative to the SN rate of ${R}_{\mathrm{rp}/\mathrm{SN}}=2\times {10}^{-3}$ (${R}_{\mathrm{rp}/\mathrm{SN}}=3\times {10}^{-4}$). As expected, the amount of Eu mass per event is approximately the minimal Eu mass observed in any galaxy and thus tracks linearly the halo mass. In addition, since R_rp/SN is roughly

$$\begin{eqnarray}&&{R}_{\mathrm{rp}/\mathrm{SN}}=\displaystyle \frac{{M}_{\mathrm{Eu}}{\tilde{m}}_{\mathrm{Fe}}}{{M}_{\mathrm{Fe}}{\tilde{m}}_{\mathrm{Eu}}},\end{eqnarray} \tag{ 6 }$$

and since ${M}_{\mathrm{Eu}},{M}_{\mathrm{Fe}},{\tilde{m}}_{\mathrm{Eu}}$ are linear in the halo mass, it follows that R_rp/SN is inversely proportional to this quantity. Most importantly, the conclusion that ${R}_{\mathrm{rp}/\mathrm{SN}}\ll 1$ holds in all of these cases.

6.1. DNS Mergers

The r-process mass per event that we find in the likelihood analysis described in Section 5 is roughly consistent with expectations for DNS mergers from simulations. These simulations typically find ejecta masses of 10⁻³–10⁻² M_⊙, corresponding to Eu masses of 5 × 10⁻⁶ to 5 × 10⁻⁵ M_⊙ (Korobkin et al. 2012; Bauswein et al. 2013; Perego et al. 2014; Just et al. 2015; Sekiguchi et al. 2015; Radice et al. 2016; Wu et al. 2016). The results are also consistent with macronova estimates (Berger et al. 2013; Hotokezaka et al. 2013; Piran et al. 2014; Jin et al. 2016).

The strong upper limits of $\lesssim {10}^{-6}\,{M}_{\odot }$ for the other UFDs imply that such events did not take place in these UFDs. We turn now to estimating the expected rates of DNS mergers in these galaxies and comparing them to the results from the likelihood analysis in Section 5. The expected number of r-process production events by DNS inside a UFD during its star formation epoch is

$$\begin{eqnarray}&&{R}_{\mathrm{rp}/\mathrm{SN}}=f({v}_{\mathrm{CM}}\lt {v}_{\mathrm{esc}},{t}_{{\rm{m}}}\lt {t}_{\mathrm{SF}})\displaystyle \frac{\mathrm{Number}\,\mathrm{of}\,\mathrm{mergers}}{\mathrm{Number}\,\mathrm{of}\,\mathrm{ccSNe}},\end{eqnarray} \tag{ 7 }$$

where $f({v}_{\mathrm{CM}}\lt {v}_{\mathrm{esc}},{t}_{{\rm{m}}}\lt {t}_{\mathrm{SF}})$ is the fraction of mergers with a center-of-mass velocity, v_CM, smaller than the escape velocity of a UFD, ${v}_{\mathrm{esc}}$, and with a merger time, ${t}_{{\rm{m}}}$, shorter than the duration of the star formation phase, ${t}_{\mathrm{SF}}$. The small escape velocities from UFDs (down to $\approx 15\,\mathrm{km}\,{{\rm{s}}}^{-1}$ in some cases) and the relatively short durations of their star formation, $\lesssim 1\,\mathrm{Gyr}$, reduce the number of mergers providing r-process elements in the star-forming gas. However, in Beniamini et al. (2016) we have shown that since many DNS systems receive small kicks at birth (Beniamini & Piran 2016), a large fraction of them would have v_CM < 15 km s⁻¹. Furthermore, given limits on the separation of the double pulsar system before its second collapse, a significant fraction of systems are expected to both remain confined in UFDs and merge within less than a gigayear. All together, we estimate that $6\times {10}^{-2}\lesssim f({v}_{\mathrm{CM}}\lt {v}_{\mathrm{esc}},{t}_{{\rm{m}}}\lt {t}_{\mathrm{SF}})\lesssim 0.6$. Therefore, using our result in Section 5, the number of mergers per CCSN is estimated as 5 × 10⁻⁴ to 2 × 10⁻².

We compare the results also with the merger rate estimated using short GRB observations. The local short GRB rate is estimated as ${4.1}_{-1.9}^{+2.3}{f}_{b}^{-1}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ (Wanderman & Piran 2015), where ${f}_{b}^{-1}$ is a beaming factor that is in the range $1\lt {f}_{b}^{-1}\lesssim 100$ (Fong et al. 2015). The rate of local CCSNe is ${7.1}_{-1.3}^{+1.4}\times {10}^{4}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$. Therefore, the number of short GRBs per CCSN is ${5.8}_{-2.8}^{+3.2}\times {10}^{-5}{f}_{b}^{-1}$, consistent with our estimates above.

6.2. Peculiar CCSNe

We have explored the question of iron and r-process production in dwarf spheroidal galaxies in a model-independent fashion. We assume that all dwarf galaxies (or alternatively, only UFDs) share the same mechanism for production of these elements. We then use the element abundances and apply a likelihood method to estimate the most likely rates and element mass produced per such event. Using the mass within the radius that contains 90% of the galaxies' light, M_h(<r₉₀), as a proxy for the halo masses (which in turn provides the initial gas mass), we find that on average 0.1 M_⊙ of Fe is produced per CCSN and that the rate is approximately 16 CCSNe per ${10}^{3}{L}_{\odot }$, verifying the estimated values of these parameters from independent studies using different methods (Horiuchi et al. 2011).

Turning now to r-process events, we find that the Eu mass per event is between $3\times {10}^{-5}\,{M}_{\odot }\lt {\tilde{m}}_{\mathrm{Eu}}\lt 2\times {10}^{-4}\,{M}_{\odot }$ ($2.5\times {10}^{-6}\,{M}_{\odot }\lt {\tilde{m}}_{\mathrm{Eu}}\lt 3.5\times {10}^{-5}\,{M}_{\odot }$ for UFDs only). The very strong upper limits on the Eu mass in some galaxies, combined with detections of significantly larger masses in other dwarfs, imply that Eu production is dominated by rare events and that the minimal Eu mass observed in any galaxy is approximately the amount of Eu mass produced per event. This corresponds to a total r-process mass of $6\times {10}^{-3}\,{M}_{\odot }\lt {\tilde{m}}_{r \mbox{-} \mathrm{process}}\lt 4\times {10}^{-2}\ {M}_{\odot }$ ($5\times {10}^{-4}\,{M}_{\odot }\,\lt {\tilde{m}}_{r \mbox{-} \mathrm{process}}\lt 7\times {10}^{-3}\,{M}_{\odot }$ for UFDs only), assuming that r-process events dominate the production of elements with mass number $A\geqslant 90$. The rate of these events is $2.5\times {10}^{-4}\lt {R}_{\mathrm{rp}/\mathrm{SN}}\lt 1.4\times {10}^{-3}$ as compared with the CCSN rate ($1.6\times {10}^{-5}\lt {R}_{\mathrm{rp}/\mathrm{SN}}\lt 1.4\times {10}^{-3}$ for UFDs only). We conclusively rule out the possibility that the rates of r-process events could be comparable to that of CCSNe. Interestingly, both the r-process mass per event and the rate are, however, consistent with expectations for DNS mergers. However, rare magnetic-driven SNe, long GRBs, superluminos SNe, or even low-luminosity GRBs are all possible candidates. Furthermore, these values are consistent with the total Eu mass observed in our own Galaxy. This suggests that the same mechanism is behind the production of r-process events in both UFDs and the Milky Way, and that it may be the dominant mechanism for production of r-process elements as a whole in the universe. Association of a gravitational-wave event from a DNS merger and a clear macronova from which the amount of r-process production can be well estimated could therefore confirm or reject this scenario.

A large scatter in the abundances of r-process elements is also seen in metal-poor stars in our own Galaxy, which again supports rare events (e.g., Cescutti & Chiappini 2014; Tsujimoto & Shigeyama 2014; Wehmeyer et al. 2015). In addition, Hotokezaka et al. (2015) have shown that the rarity of r-process events is broadly consistent with ²⁴⁴Pu abundances of both the early solar system material (Turner et al. 2007) and the present-day deep-sea archives (Wallner et al. 2015), which are difficult to explain otherwise. The r-process element enrichment in halo stars with very low metallicities, proposed as a problem for this scenario (Argast et al. 2004), can be resolved by taking into account the turbulent mixing of material in the galaxy (Piran et al. 2014), the assembly of subhalos during the formation of the Galaxy (Komiya et al. 2014; Ishimaru et al. 2015; Shen et al. 2015; van de Voort et al. 2015), the large typical distances between the locations of the stellar collapses and the eventual mergers (Beniamini et al. 2016), and the likely possibilities of rapid mergers and stellar collapses with small amounts of mass ejecta (Beniamini & Piran 2016; Beniamini et al. 2016).

Given the orbits of the observed DNS systems in the Galaxy, a significant amount of DNS systems are expected to both remain confined within UFDs and merge within a relatively short time since formation (Beniamini et al. 2016). Using these results and assuming that r-process production is dominated by DNS mergers, we estimate the DNS merger rate to be 5 × 10⁻⁴ to 2 × 10⁻³ per CCSN. This is comparable to the rates estimated using other methods such as from short GRBs or from population synthesis.

There are two major differences between the DNS and core-collapse scenarios. First, DNS mergers have delay times relative to the star formation, but the core-collapse events do not. Moreover, since the rare core-collapse events, discussed above, may take place preferably in low-metallicity environments, they may even precede the average star formation and produce r-process elements in very low metallicity environments. However, the region will be "polluted" by mass outflow from the progenitor stars, oxygen and carbon from winds, and Fe from the associated SN. Second, during the formation of the second neutron star in a DNS, the center of mass receives a velocity of ∼15 km s⁻¹ (Beniamini et al. 2016). This is slower than the typical escape velocity from UFDs but fast enough to run away from the star-forming regions where they have been formed. Thus, the mergers can take place in low-metallicity environments that are "unpolluted" by the progenitors. This depends, of course, on the pace of turbulent mixing within the UFD. These differences should be reflected in the distribution of Eu and Fe within the galaxies and in the chemical history of the Galaxy. Abundance measurements for a larger number of stars in Reticulum 2 and other dwarf galaxies may provide vital information that may enable us to distinguish between the two scenarios.

We thank Alex Ji and Kohei Inayoshi for useful discussions. This work was supported in part by an Israel Space Agency (SELA) grant, the Templeton Foundation, and the I-Core center for excellence "Origins" of the Israel Science Foundation (ISF).