Formation of Globular Clusters with Internal Abundance Spreads in r-Process Elements: Strong Evidence for Prolonged Star Formation

Recent photometric and spectroscopic investigation of the Galactic globular clusters (GCs) has confirmed that they exhibit multiple stellar populations (see, e.g., Gratton et al. 2012 for a recent review). Individual GCs show internal abundance spreads in different elements: light elements (e.g., Carretta et al. 2009, hereafter C09), s-process (e.g., Marino et al. 2009), C+N+O (e.g., Yong et al. 2015), r-process (e.g., Sneden et al. 1997; Roederer 2011,  hereafter R11; Sobeck et al. 2011, hereafter S11; Worley et al. 2013, hereafter W13), helium (e.g., Piotto et al. 2005), and Fe (e.g., Freeman & Rodgers 1975). Furthermore, the vast majority of the Galactic GCs investigated so far possess clear anticorrelations between light elements (e.g., CNO; C09). Such internal abundance spreads have also been observed for GCs in the Large and Small Magellanic Clouds (e.g., Mucciarelli et al. 2009; Niederhofer et al. 2017) and the Galactic dwarf satellites (e.g., Larsen et al. 2014). A key question related to GC formation is why GCs show both (i) ubiquitous anticorrelations between light elements (e.g., the Na–O anticorrelation) and (ii) a diversity in the degree of chemical abundance inhomogeneity (e.g., only some GCs with [Fe/H] spreads).

Previous theoretical models of GC formation tried to reproduce the apparently universal anticorrelations between O and Na and between Mg and Al observed in the Galactic GCs (e.g., Fenner et al. 2004; Bekki et al. 2007; D'Ercole et al. 2008, 2010; Ventura et al. 2016). D'Antona & Caloi (2004) discussed the origin of Y (helium abundance spread) in NGC 2808 based on self-enrichment by massive asymptotic giant branch (AGB) stars. The origin of GCs with abundance spreads in Fe and s-process elements has been recently discussed in the context of GC merging in their host dwarf galaxies (Bekki & Tsujimoto 2016). However, the origin of internal abundance spreads in r-process elements has not been discussed extensively so far, though not only M15 and M92 (e.g., Sneden et al. 1997; Roederer & Sneden 2011) but also several others (e.g., M5 and NGC 3201) are now observed to have such abundance spreads (e.g., R11). Tsujimoto & Shigeyama (2014, hereafter TS14) suggested that efficient accretion of r-process elements ejected from neutron star merging (NSM) onto some fraction of (i.e., not all of) the stars in a GC can introduce a large star-to-star variation in the abundances of r-process elements within the GC.

If NSM ejecta can be mixed with an intracluster medium (ICM) and finally used for secondary star formation in a GC, then large star-to-star abundance spreads in r-process elements (e.g., [Eu/H]) can be expected. A key question here is whether the NSM ejecta with very high ejection speeds (10%–30% of the speed of light) can be stopped by the ICM within the forming GC (R < 10 pc). Recently, Komiya & Shigeyama (2016) showed that ejecta from NSMs can be stopped through interaction with the interstellar medium (ISM), because the ejecta can lose kinetic energy and momentum through this interaction. Recent numerical simulations have shown that a large amount of AGB ejecta can be accumulated in the central regions of forming GCs to form very high-density gaseous regions with ρ_g = 10⁵ cm⁻³ (Bekki 2017a, 2017b, hereafter B17a, B17b). If a single NSM occurs in such a high-density gaseous region, then the ejecta is highly likely to be trapped and mixed with the AGB ejecta. New stars formed from such mixed gas can have chemical abundances of r-process elements that are quite different from those of stars formed in the initial starburst at GC formation. Thus, it is possible that secondary star formation from AGB ejecta mixed with gas from a single NSM can explain the large internal abundance variations observed in several Galactic GCs (e.g., M5 and M15). This NSM scenario has not been explored by previous theoretical works.

The purpose of this paper is to investigate the origin of GCs with internal abundance spreads in r-process elements based on the NSM scenario. We particularly investigate the following questions: (1) under what physical conditions NSM ejecta can be trapped within forming GCs, (2) what fraction of NSM ejecta can be trapped within GCs and used for secondary star formation, and (3) whether the degrees of internal abundance spreads depend on the physical properties of GCs (e.g., initial total masses). To do so, we use both analytical models and numerical simulations of GC formation from star-forming molecular clouds (MCs). We provide predictions of (i) the number of GCs with such internal abundance spreads (e.g., Δ[Eu/H]), (ii) the bimodal distributions of [Eu/H] in GCs, and (iii) the possible correlations between [Na/Fe] and [Eu/H].

The plan of the paper is as follows. We discuss the possibility of NSM ejecta being trapped by the ICM of forming GCs using analytical models in Section 2. We derive the possible internal spreads of [Eu/H] among GC stars by assuming secondary star formation from NSM ejecta mixed with the ICM in Section 3. We present the results of numerical simulations of GC formation to discuss whether the ICM of forming GCs can be as high as the required density of the ICM for trapping the NSM ejecta in Section 4. Based on these results, we provide several predictions of the NSM scenario in Section 5. We summarize our conclusions in Section 6. In order to discuss the present results, we show some observational results (W13) in  the Appendix: the distribution of [Eu/H] and a correlation between [Eu/H] and [Na/Fe] for M15.

2.1. Ruling Out Supernovae as a Source of r-Process Elements in GCs

2.2. Secondary Star Formation in Mixed Gas from AGB Stars and NSMs

Figure 1 describes the new NSM scenario of GCs with internal abundance spreads in r-process elements (e.g., [Eu/H]). In the new scenario, the first-generation (FG) stars are formed from a GC-forming MC. After all massive stars with masses (m) larger than 9 M_⊙ explode as SNe, the gas ejected from AGB stars begins to be accumulated into the central region of the FG stellar system. The formation of second-generation (SG) stars in the central high-density gaseous region is possible due to accretion of AGB ejecta. One NSM occurs during this accretion phase when the density of the ICM in the central FG system becomes quite high (ρ_icm > 10⁴ M_⊙). As a result, the ejecta from the NSM can be trapped by the ICM very efficiently, owing to interaction between the NSM ejecta and hydrogen atoms. New stars are then formed from the ICM mixed with the NSM ejecta, so that the [Eu/H] of the new stars can be much higher than that of FG stars. It should be stressed here that SG stars formed before the NSM can have an [Eu/H] that is almost identical to that of FG stars.

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If the NSM occurs after the AGB ejecta are removed from a GC by some physical process, such as expulsion of the gas by SNe Ia or ram pressure stripping of the gas by the Galactic hot halo gas or the warm gas of the GC host dwarf galaxy, then such a GC cannot show internal spreads in r-process elements. This is because there is no gas that can stop the r-process elements ejected from an NSM in the forming GC. Accordingly, the timing of an NSM within a forming GC is quite important as to whether the GC can finally have abundance spreads in r-process elements. Since neutron stars are the outcome of the deaths of massive stars (>10 M_⊙, for which stars finally become SNe), NSMs occur only after SNe in forming GCs. Therefore, gas ejected from SNe should be entirely removed before the formation of SG stars. Otherwise, this scenario cannot explain the observed lack of abundance spreads in [Fe/H] among GCs with internal spreads in r-process elements. Given the wide range of delay-time distributions for NSMs (e.g., Dominik et al. 2012), it is possible that NSM events can occur more than 1 Gyr after the formation of FG stars in GCs. This could explain why only a fraction of old GCs show significant abundance spreads in r-process elements (e.g., R11, W13).

In this scenario, FG and SG stars formed before NSMs should show smaller [Eu/H] than SG stars formed after NSMs. As discussed later in this paper, high [Eu/H] can be seen in SG stars with higher [Na/Fe] in M15, which is consistent with the new scenario. SG stars with very high [Na/Fe] formed from gas ejected by massive AGB stars (m ∼ 8−9 M_⊙) might be less likely to have high [Eu/H], because the time interval between the onset of the AGB phase for such intermediate-mass (m ∼ 8−9 M_⊙) stars and the last (lowest-mass) SNe is very short: fine-tuning is required for the epoch of an NSM. The wide spread of [Eu/H] in SG stars with different [Na/Fe] is expected in this scenario, if one NSM occurs when intermediate-mass stars (3−9 M_⊙) enter into the AGB evolutionary stage.

A key question in this scenario is how much gas (ICM) is required to stop r-process elements ejected from NSMs, so that the NSM ejecta can be trapped in the central regions of GCs. Komiya & Shigeyama (2016) analytically investigated how r-process elements from an NSM can lose their initial kinetic energy through interaction with neutral hydrogen. They estimated how long the r-process elements can travel before they lose all of their kinetic energy through Coulomb scattering and found that the "stopping length" (l_s) is described as follows:

$$\begin{eqnarray}&&{l}_{{\rm{s}}}=2.6{\left(\displaystyle \frac{{n}_{{\rm{H}}{\rm{I}}}}{1{\mathrm{cm}}^{-3}}\right)}^{-1}\,\mathrm{kpc},\end{eqnarray} \tag{ 1 }$$

where n_{H i} is the number density of hydrogen atoms. This l_s should be as small as the core sizes of forming GCs (∼2−3 pc) where secondary star formation should occur. It should be noted here that Tsujimoto et al. (2017) derived l_s using the observed abundances of ²⁴⁴Pu of presolar grains and pointed out that the value of l_s can be significantly smaller than that derived by Komiya & Shigeyama (2016). Therefore, l_s in the above equation is an upper limit for l_s.

Figure 2 shows l_s as a function of the total mass of AGB ejecta (M_gas) for a given size of gas sphere (R_gas). Here, the AGB ejecta is assumed to form a uniform gaseous sphere just for simplicity of discussion. Clearly, if M_gas > 10⁴ M_⊙ and R_gas < 3 pc, then l_s can be smaller than 3 pc. This required M_gas is reasonable for GCs with initial total masses (M_gc,0) of ∼10⁵ M_⊙, because ∼10% of the masses in AGB stars can be ejected through stellar winds (e.g., Bekki 2011). If M_gas ∼ 10^5.5 M_⊙, then r-process elements can be trapped by the gas that is more diffusely distributed (i.e., R_gas ∼ 10 pc). These results imply that retaining NSM ejecta in the central regions of forming GCs is highly likely after gas ejected from AGB stars is accumulated in the central regions.

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If an NSM occurs within the gaseous sphere of an FG stellar system with a mass of M_FG ≥ 10⁵ M_⊙ and R_gas < 3 pc, then almost all of the NSM ejecta can be retained by the gas. If an NSM occurs outside the gaseous sphere of the FG stellar system, then only a fraction of the NSM ejecta can be trapped by the gas (i.e., AGB ejecta). Figure 3 shows the fraction of NSM ejecta retained by a GC (f_ret) as a function of the distance of the NSM from the center of the GC (r_gc) for R_gas = 1, 3, and 10 pc. Since the flux of the mass ejected from an NSM falls as ${r}_{{\rm{g}}}^{-2}$, f_ret can be rather small outside R_gas. It is likely that only a fraction of NSM ejecta can be retained in GCs, because NSMs are likely to occur in the outer regions of GCs (R > 3 pc, where more stars can exist), rather than the central regions.

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We can estimate (i) the [Eu/H] of stars formed from NSM ejecta mixed with AGB ejecta in a GC and (ii) the internal abundance spreads of [Eu/H] among FG and SG stars (Δ[Eu/H]) in the GC for a given f_ret by using the following simple model for the mixing of AGB and NSM ejecta. We assume that the AGB and NSM ejecta can be mixed uniformly so that the final abundance of Eu depends on the total mass of AGB ejecta for a given yield of Eu from NSM events. The total mass of gas ejected from one NSM (M_nsm) is ∼0.01 M_⊙, and the mass fraction of Eu in the ejecta (f_Eu,0) is 10⁻² (TS14). Using these numbers, we can calculate the chemical abundance of Eu (f_Eu) as follows:

$$\begin{eqnarray}&&{f}_{\mathrm{Eu}}=\displaystyle \frac{{f}_{\mathrm{Eu},0}{M}_{\mathrm{nsm}}+{f}_{\mathrm{Eu},\mathrm{agb}}{M}_{\mathrm{gas}}}{{M}_{\mathrm{agb}}},\end{eqnarray} \tag{ 2 }$$

where M_gas is the total mass of AGB ejecta and f_Eu,0 and f_Eu,agb are the mass fractions of Eu in the NSM and AGB ejecta, respectively. We ignore the total mass of the NSM ejecta in the denominator, because it is too small in comparison with M_gas. We can convert this f_Eu into the final [Eu/H] of the mixed gas ([Eu/H]_f) by assuming the solar abundance of Eu (3.8 × 10⁻¹⁰) for a given initial [Eu/H] ([Eu/H]_i) of the AGB ejecta. The [Eu/H] spread is thus estimated as follows:

$$\begin{eqnarray}&&{\rm{\Delta }}{[\mathrm{Eu}/{\rm{H}}]={[\mathrm{Eu}/{\rm{H}}]}_{{\rm{f}}}-[\mathrm{Eu}/{\rm{H}}]}_{{\rm{i}}}.\end{eqnarray} \tag{ 3 }$$

Figure 4 shows Δ[Eu/H] for a metal-poor GC with [Eu/H]_i  = −2 as a function of M_gas for three different f_ret. Clearly, the Δ[Eu/H] for this metal-poor GC can be quite large (≥1 dex) even for f_ret = 0.01, if M_gas ≤ 10^4.4 M_⊙. The GC with M_gas ≤ 10⁵ M_⊙ has a Δ[Eu/H] that is much larger than the one observed for M15 (∼−1). A particular combination of M_gas and f_ret is required for the observed Δ[Eu/H] to be reproduced. Since GCs with M_gc,0 can have M_gas ∼ 10⁴ M_⊙, these results suggest that only a small fraction of NSM ejecta should be mixed with the ICM and retained in the GCs. If M_gc,0 is significantly larger than 2 × 10⁵ M_⊙ (typical present-day GC mass), then f_ret can be larger. These results, combined with those in Figure 4, imply that GCs with internal abundance spreads of r-process elements experience only one NSM event in their outer parts in their early phases of formation. A future numerical study will investigate where NSM events can occur in forming GCs with AGB ejecta.

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Observational study of [Ba/H] distributions of GCs with internal [Ba/H] spreads shows a bimodal distribution of [Ba/H] in M15 (W13), and the [Eu/H] distribution in the data by W13 and S11 shows such a bimodality in the [Eu/H] distribution for M15 (see Appendix). It is thus important to discuss whether and how the observed bimodality can be achieved in the present scenario of GC formation. Guided by recent results from hydrodynamical simulations of turbulent diffusion (Greif et al. 2009), we consider that r-process elements ejected from NSMs can be spread over the ICM of a forming GC through diffusion processes within a short timescale (<10⁶ yr). We adopt a working hypothesis that the chemical abundances of Eu can be different in different regions of the ICM owing to turbulent diffusion. Although gaseous regions close to an NSM event can have very high [Eu/H] initially, the Eu abundances can become progressively lower as time passes owing to diffusion. We accordingly consider the following functional form (the Green function; described in Greif et al. 2009) for the distribution of [Eu/H] for stars formed from NSM ejecta mixed with AGB ejecta:

$$\begin{eqnarray}&&N([\mathrm{Eu}/{\rm{H}}])=\displaystyle \frac{{N}_{0}}{{(2\pi {\sigma }^{2})}^{3/2}}\exp \left(\displaystyle \frac{-{(Z-{Z}_{{\rm{m}}})}^{2}}{2{\sigma }^{2}}\right),\end{eqnarray} \tag{ 4 }$$

where N₀ is the normalization constant, σ is the dispersion of [Eu/H], Z represents [Eu/H] (just for convenience), and Z_m is the mean value of [Eu/H].

These means and dispersions in [Eu/H] are different in the stars formed from the original gas of a GC-forming MC (FG) than in those from NSM ejecta mixed with AGB ejecta (SG). If SG stars are formed from AGB ejecta before NSMs occur, then the [Eu/H] should be the same as that of FG stars. Accordingly, there could be a significant difference in [Eu/H] even in SG stars. Therefore, we use the two terms "polluted" (p) and "nonpolluted" (np) to discriminate between stars that are formed from gas polluted by NSM ejecta and those without such pollution. For example, N_p and σ_p are the number of polluted stars and their [Eu/H] dispersion, respectively. A basic parameter for the entire distribution of [Eu/H] is σ_p, σ_np, Z_p, Z_np, and R_p, which is the number ratio of polluted to nonpolluted stars:

$$\begin{eqnarray}&&{R}_{{\rm{p}}}=\displaystyle \frac{{N}_{{\rm{p}}}}{{N}_{\mathrm{np}}}.\end{eqnarray} \tag{ 5 }$$

Figure 5 shows the normalized N_p and N_np in the four models with different R_p, σ_p, and σ_np. For consistency with observations by W13, we adopt Z_m = −2.0 and −1.6 for nonpolluted and polluted stars, respectively. Clearly, the [Eu/H] distributions depend strongly on the three parameters, with the model with R_p being the most similar to the observed distribution. The relatively large [Eu/H] dispersion (0.1 dex) in the best model with R_p = 3.4 implies that diffusion of r-process elements can proceed efficiently within the ICM of the GC. The present study cannot discuss whether such a large dispersion of 0.1 dex can be achieved in the ICM through turbulent diffusion. The larger R_p implies that 77% of the present stars in M15 can be the polluted population. The large fraction of the polluted population in M15 implies that the original mass of FG (nonpolluted) stars, from which AGB ejecta originates, should be at least a factor of ∼10 larger than the present-day mass of the FG. This is a classic mass-budget problem discussed by many previous works (e.g., Smith & Norris 1982). The selective stripping of FG stars in the early phase of GC formation is a promising explanation for solving this mass-budget problem (Bekki 2011).

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In order to discuss whether the density of the ICM of forming GCs can become as high as ρ_icm = 10⁴ cm⁻³, we perform smooth particle hydrodynamics simulations of GC formation within massive MCs. We have already investigated the general trends of GC formation within MCs (B17b), and we use the same numerical methods used in B17b in the present study. Since the details of the numerical methods are given in B17b, we briefly describe them in this paper. We use our original simulation code that can be run on a cluster of graphics processing unit machines (Bekki 2013, 2015). An MC with a total mass of M_mc and a size of R_mc is assumed to have (i) a fractal mass distribution with a three-dimensional fractal dimension of 2.6 and (ii) a power-law radial density distribution with a slope of −1. The initial virial ratio of a GC-forming MC is set to 0.35, which ensures rapid gravitational collapse of the MC. The initial global rotation of fractal MCs is not included in the present study.

Feedback effects of SNe with different masses are separately implemented in the simulations, and the gas ejection of individual AGB stars is also self-consistently included. Star formation is assumed to occur if (i) the gas density of a particle (ρ_g) exceeds the threshold gas density of star formation (ρ_th) and (ii) $\mathrm{div}{\boldsymbol{v}}\lt 0$, where v is the velocity of the gas particle. We adopt ρ_th = 10⁵ cm⁻³ in the present study. We consider two models for star formation from AGB ejecta. In one model (M1), star formation cannot occur from AGB ejecta even if ρ_g ≥ 10⁵ cm⁻³. This model is constructed to obtain a better understanding of the density enhancement achieved through accumulation of AGB ejecta in the central regions of forming GCs. In the other model (M2), star formation occurs if the above two physical conditions are met for AGB ejecta.

Here, we describe the results of the models (M1 and M2) with M_mc = 3 × 10⁶ M_⊙ and R_mc = 100 pc for which GCs with a final stellar mass (M_gc) of ∼10⁶ M_⊙ can be formed. The mass and spatial resolutions of the models are 2.9 M_⊙ and 0.2 pc in the present study. The total number (N) of a simulation significantly increases from the initial N = 1,048,911 owing to the new addition of "AGB particles," which represent gas ejected from AGB stars with different masses.

Figures 6–8 describe how the AGB ejecta can be accreted into the central region of a forming GC within a giant MC in the fiducial model M1 without secondary star formation from AGB ejecta. FG stars are formed from numerous small gas clumps that are developed from local gravitational instabilities within the fractal MC, because the ρ_g of the small clump can become higher than 10⁵ cm⁻³. These numerous groups of FG stars merge with one another to form a larger stellar system with a stellar halo within a timescale of 10⁷ yr (at T = 12 Myr). The gas density of the GC-forming MC can dramatically decrease owing to gas consumption by the FG formation. The remaining cold gas of the MC can be rapidly expelled by multiple SN explosions (T = 12 and 24 Myr) because the lifetimes of massive stars with m > 30 M_⊙ are quite short (<10 Myr). The cold gas gradually disappears from the inner region of the forming GC (T = 36 Myr), and almost all of the gas can be expelled from the GC by T = 110 Myr.

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Massive AGB stars (m = 6−9 M_⊙) start to eject (Na-rich) gas after the removal of gas chemically polluted by the SN explosion (T ∼ 50 Myr). The AGB ejecta can be gravitationally trapped in the FG stellar system with a total mass of ∼2 × 10⁶ M_⊙ because of the relatively slow wind velocity (v_wind = 10 km s⁻¹). The gas can be gradually accumulated onto the central regions (R < 2 pc) of the FG stellar system (T = 110 Myr), so that ρ_g can become very high. The gas can efficiently lose its kinetic energy owing to energy dissipation during its accretion process within the GC. Finally, a very compact gaseous sphere in the central region of the FG stellar system can be formed 149 Myr after the start of the gravitational collapse of the MC in this model. These basic formation processes of a GC can be seen in M2 with secondary star formation from AGB ejecta.

Figure 9 shows the radial distributions of ρ_icm at T = 149 Myr for M1 and M2. Since star formation is not included in M2, ρ_icm can become rather high, >10⁵ cm⁻³, in the center of the FG stellar system (R < 0.4 pc). Clearly, most of the gas particles within R < 1 pc have the required log ρ_icm ≥ 3.4 for l_s ≤ 3 pc (for stopping r-process elements and trapping them within GCs). This means that if NSMs occur around this timestep (T = 149 Myr), then the ejecta can be easily trapped by the ICM in this model. It is confirmed that ρ_icm can also be quite high (ρ_icm > 10⁴ cm⁻³) at T = 110 Myr, which means that fine-tuning in the epoch of an NSM is not required for the ejecta to be trapped by the ICM.

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However, secondary star formation from AGB ejecta can continue to decrease the total mass of the ICM so that ρ_icm can be significantly lower. Figure 9 shows that although the radial gradient of ρ_icm is very steep, ρ_icm in the inner region of the simulated GC can still be high (>10⁴ cm⁻³) in M2 with secondary star formation. This confirms that, as long as ρ_th is similar to the gas densities of the molecular cores of star-forming MCs (10⁵ cm⁻³), AGB ejecta (ICM) can continue to have ρ_icm ≥ 10⁴ cm⁻³ over several Myr. We thus conclude that AGB ejecta can stop the r-process elements ejected from an NSM so that the NSM ejecta can be used for secondary star formation in the central region of a forming GC.

5.1. The Epoch of an NSM in a Forming M15

5.2. Why Only Some GCs Show Abundance Spreads in r-Process Elements

R11 investigated the chemical abundances of r-process elements for 17 Galactic GCs and found that only four of them have large internal spreads of [Eu/H]. So far, only these GCs have clear evidence of internal abundance spreads in r-process elements among all Galactic GCs. This observation raised a question as to why only a fraction of GCs show such clear abundance spreads of [Eu/H] in the Galaxy. In the present NSM scenario, NSMs need to occur when forming GCs still retain enough AGB ejecta (>10³ M_⊙). Forming GCs can lose their AGB ejecta by some physical process such as (i) ram pressure stripping by the warm/hot gas of their GC host galaxies and (ii) prompt Type Ia SNe. Accordingly, NSMs need to occur before the stripping of AGB ejecta for the ejecta to be converted into new stars. Given the observed wide range of the delay-time (t_delay) distribution of NSMs (10⁷–10¹⁰ yr), NSMs do not so easily synchronize with when GCs still retain AGB ejecta. Therefore, only GCs that happen to experience NSMs during secondary star formation from AGB ejecta can have internal abundance spreads in r-process elements.

We discuss the probability (R_r) of one GC having one NSM in a more quantitative way. The t_delay distribution of NSMs has the following power-law profile for t_delay > 3 × 10⁷ yr (Figure 8 in Dominik et al. 2012):

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dt}}\propto {t}_{\mathrm{delay}}^{-\beta },\end{eqnarray} \tag{ 6 }$$

where β can be approximated as 1.3. If the duration of secondary star formation from AGB ejecta is t_sf, then P_r is described as follows:

$$\begin{eqnarray}&&{P}_{{\rm{r}}}=\displaystyle \frac{{n}_{\mathrm{nsm}}{t}_{\mathrm{sf}}}{{t}_{\mathrm{delay},{\rm{m}}}},\end{eqnarray} \tag{ 7 }$$

where n_nsm is the number of NSMs that can occur in one GC and t_delay,m is the average of t_delay. The total number of GCs with internal abundance spreads in r-process elements in the Galaxy (N_gc,r) is as follows:

$$\begin{eqnarray}&&{N}_{\mathrm{gc},{\rm{r}}}={P}_{{\rm{r}}}{N}_{\mathrm{gc},0},\end{eqnarray} \tag{ 8 }$$

where N_gc,0 is the number of Galactic GCs. The total number of core-collapse SNe (CCSNe) that occur in one GC (with a total mass of ≈10⁵ M_⊙) is ≈500, and one NSM occurs for every 1500 CCSNe (Tsujimoto et al. 2017). Therefore, n_nsm = 500/1500 = 0.3 is a reasonable value. If we adopt reasonable numbers for t_delay,m (=10⁹ yr; Dominik et al. 2012) and t_sf = 10⁸ yr and n_nsm = 0.3, then N_gc,r = 4.5 for N_gc,0 = 150 (Harris 1996—2010 edition). This is roughly consistent with the number of GCs with internal abundance spreads in r-process elements.

5.3. Alternative Scenarios

5.4. Relation to the Galactic Halo Stars

In order to understand the origin of GCs with internal abundance spreads in r-process elements (e.g., [Eu/H]), we have investigated whether ejecta from NSMs can be retained in forming GCs and subsequently converted into new stars using analytical models and numerical simulations of GC formation. In the NSM scenario, stars with high [Eu/H] are formed from NSM ejecta mixed with the ICM (i.e., AGB ejecta) several tens of Myr after the initial starburst of GC formation. Therefore, prolonged star formation is essential in this GC formation scenario. We have also constructed a theoretical model that explains the observed bimodal distribution of [Eu/H] in GCs. The principal results are as follows:

• (1)  
  The high-speed gaseous ejecta (i.e., r-process elements) from NSMs can be stopped by the ICM in forming GCs if the total mass of the ICM within the central regions of GCs (R ∼ 1−3 pc) is as large as 10⁴ M_⊙. This means that AGB ejecta in GCs with the original stellar masses (M_s) of ∼10⁵ M_⊙ can stop the r-process elements from NSMs, because ∼10% of M_s can be AGB ejecta. It is concluded that AGB ejecta is essentially important for trapping NSM ejecta in the central regions of GCs. The required larger amount of AGB ejecta implies that the original GCs should be massive.
• (2)  
  NSMs are possibly more likely to occur outside the central region of a GC than within it. Therefore, only a fraction of NSM ejecta can interact with the ICM (i.e., AGB ejecta), so that the mass fraction of the ejecta trapped and retained by the ICM (f_ret) can be very small (<0.1). Our models suggest that a small f_ret (∼0.01) is required for explaining the observed spreads of [Eu/H] (e.g., Δ[Eu/H] ∼ −1 dex for M15). Therefore, one NSM in the outer part of a forming GC is not a problem in the present scenario.
• (3)  
  The observed apparent bimodal distribution of [Eu/H] in M15 can be explained if SG stars were formed from the ICM with the mean [Eu/H] being ∼1 dex higher than the initial [Eu/H] of the GC-forming MC. The observed large spread in [Eu/H] in the SG stars is due to turbulent diffusion of NSM ejecta in the ICM. The larger number of SG stars with enhanced [Eu/H] suggests that the original mass of the GC (M15) should be significantly larger than its present mass. This is the classic mass-budget problem in GC formation.
• (4)  
  Our new hydrodynamical simulations of GC formation from fractal MCs show that the required higher ICM density (ρ_icm > 10³–10⁴ cm⁻³) is possible ∼100 Myr after the initial burst of star formation in GCs. This result confirms the validity of the NSM scenario. Although secondary star formation can decrease ρ_icm, the ICM can keep its higher density owing to slow star formation in SG formation. We thus conclude that the NSM scenario is very promising in explaining the observed internal spreads of r-process elements in GCs.
• (5)  
  The scenario predicts that the number of GCs with such internal spreads of r-process elements should be small (≤5) owing to (i) the rarity of NSMs and (ii) the very wide distribution of delay times of NSMs. The scenario also predicts that there are two SG populations with low and high [Eu/Fe], which appears to be consistent with the observed [Eu/H]–[Na/Fe] relation for M15. Although the observed large number of stars with high [Eu/H] and [Na/Fe] in M15 supports the NSM scenario, other GCs need to be investigated to reach a more robust conclusion on the validity of the scenario.
• (6)  
  We rule out the possibility of unusual SNe (e.g., magnetorotational SNe; Nishimura et al. 2017) as the origin of internal spreads of r-process elements in M15 for two reasons. First, if the ICM of forming GCs is chemically polluted by SNe, then internal abundance spreads can be seen not only in [Eu/H] but also in [Fe/H], which is not seen in GCs. Second, if SNe are the site of r-process elements, they produce both light (Y, Sr, etc.) and heavy (Ba, Eu, etc.) r-process elements. Stars with enhanced Ba and Eu, however, do not exhibit any enhancement in Sr in GCs (e.g., M15). The observed large scatter seen only in Ba and Eu (but not in the light r-process elements) is inconsistent with the above (unusual) SN scenario.
• (7)  
  Since short gamma-ray bursts (GRBs) originate from NSMs (e.g., Grindlay et al. 2006), direct evidence for the NSM scenario would be the discovery of short GRBs located in very young and massive GCs with ages of 10⁷–10⁸ yr at high redshifts. Such young GCs can also have very low-level star formation, and the optical light curves of the GRBs could be significantly influenced by dust (AGB ejecta) within the GCs. Although young, massive GCs with short GRBs shrouded by dust would be very hard to identify with current telescopes owing to the low luminosities, the discovery of such GCs would provide irrefutable supporting evidence for the NSM scenario.

We are grateful to the referee, Timothy C. Beers, for his constructive and useful comments. TT is supported in part by JSPS KAKENHI grant number 15K05033.

A recent observational study of the Galactic GC M15 has revealed that the distributions of [Eu/H] and [Ba/H] (r-process elements) are bimodal with two distinct peaks (W13). We have reproduced the bimodal [Eu/H] distribution using the data by W13 in order to discuss the results of our theoretical models. Figure 10 shows that the [Eu/H] distribution has two peaks around [Eu/H] ∼ −1 and −1.6, with a large dispersion for the entire population (−2.2 ≤ [Eu/H] ≤ −1.2; Δ[Eu/H] ∼ 1.0 dex). If one star with a very Eu-rich abundance around [Eu/H] ∼ −1.2 is removed, then the dispersion is 0.8 dex. The two peaks strongly suggest that there were two major episodes of star formation in M15. The physical origin of the two major episodes of star formation is given in the main text.

Using the same data, we have investigated the distribution of stars in the [Na/Fe]–[Eu/H] diagram for M15. Figure 11 shows three trends for the GC stars. First, most of the stars with 0.3 ≤ [Na/Fe] ≤ 1.0 (corresponding to SG stars) have higher [Eu/H] (>−1.8, i.e., around the second peak of the [Eu/H] distribution). Second, four stars with very high [Na/Fe] (>1.0) do not show such high [Eu/H] (>−1.8). Third, most of the stars with [Na/Fe] < 0 have lower [Eu/H] (<−1.8); it should be noted that one star with [Na/Fe] ∼ −0.1 has a high [Eu/H] (∼−1.5). The absence of stars with very high [Na/Fe] and [Eu/H] appears to be remarkable, though the number of stars investigated is not very large. Gas ejected from metal-poor, massive AGB stars with m = 8 M_⊙ can have [Na/Fe] ∼ 1.0 (e.g., Ventura et al. 2011). Therefore, one possible interpretation for the absence of such high-Na and high-Eu stars is that one NSM occurred after the massive AGB stars died away. More details on this discussion are given in the main text.