CHEMICAL EVOLUTION OF THE LARGE MAGELLANIC CLOUD

We adopt one-zone chemical evolution models that are essentially the same as those adopted in our previous studies on the chemical evolution of the Galaxy (Tsujimoto et al. 2010; TB12). Accordingly, we briefly describe the adopted models in the present study. The present model is improved in comparison with previous ones (Tsujimoto et al. 2010) because it includes the s-process elements and prompt SN Ia self-consistently in the models. The LMC disk is assumed to form through a continuous gas infall from outside the disk region (e.g., halo) for the last 13 Gyr, as in previous chemical evolution models (e.g., Tsujimoto et al. 2010). Some fraction of gas with metals can be expelled from the LMC as stellar winds through energetic feedback effects of SNe in some models of the present study (e.g., "wind models," as described later). The star formation can be suddenly and significantly enhanced (referred to as "bursts of star formation" or "starburst"). As demonstrated by theoretical studies, the Galaxy–LMC–SMC tidal interaction is important both for the formation of the Magellanic Stream and for reproducing the observed peaks of star formation in the LMC (e.g., Bekki & Chiba 2005; DB12). Therefore, the adopted assumption of a starburst is quite reasonable. Although the infall of metal-poor gas from the SMC and other dwarf galaxies can be important for the LMC chemical evolution, we discuss this later in Section 4.

We investigate the time evolution of the gas mass fraction (f_g(t)), the star formation rate (ψ(t)), and the abundance of the ith heavy element (Z_i(t)) for a given accretion rate (A(t)), IMF, and the ejection rate of ISM (w(t)). The basic equations for the adopted one-zone chemical evolution models are described as follows:

$$\begin{equation} \frac{df_g}{dt}=-\alpha _{\rm l}\psi (t)+A(t)-w(t) \end{equation} \tag{ 1 }$$

$$\begin{eqnarray} \frac{d(Z_if_g)}{dt} &=& {-}\alpha _{\rm l} Z_i(t)\psi (t)+Z_{A,i}(t)A(t)+y_{{\rm II},i}\psi (t) \nonumber \\ && +y_{{\rm Ia},i}\int ^t_0 \psi (t-t_{\rm Ia})g(t_{\rm Ia})dt_{\rm Ia} \nonumber \\ && +\int ^t_0 y_{{\rm agb},i}(m_{\rm agb}) \psi (t-t_{\rm agb})h(t_{\rm agb})dt_{\rm agb} -W_i(t) , \nonumber\\ \end{eqnarray} \tag{ 2 }$$

where α_l is the mass fraction locked up in dead stellar remnants and long-lived stars, y_{Ia, i}, y_{II, i}, and y_{agb, i} are the chemical yields for the ith element from Type II supernovae (SN II), SN Ia and AGB stars, respectively, Z_{A, i} is the abundance of heavy elements contained in the infalling gas, and W_i is the wind rate for each element. The quantities t_Ia and t_agb represent the time delay between star formation and SN Ia explosion, and between star formation and the onset of AGB phase, respectively. The terms g(t_Ia) and h_agb are the distribution functions of SNe Ia and AGB stars, respectively, the details of which are described later in this section. The term h_agb controls how much AGB ejecta can be returned into the ISM per unit mass for a given time in Equation (2). The total gas masses ejected from AGB stars depends on the original masses of the AGB stars (e.g., Weidemann 2000). Therefore, this term h_agb depends on the adopted IMF and the age–mass relation of the stars (later described). Thus, Equation (1) describes the time evolution of the gas due to star formation, gas accretion, and stellar wind. Equation (2) describes the time evolution of the chemical abundances due to chemical enrichment by supernovae and AGB stars.

The star formation rate ψ(t) is assumed to be proportional to the gas fraction with a constant star formation coefficient and thus is described as follows:

$$\begin{equation} \psi (t)=C_{\rm sf}f_{\rm g}(t). \end{equation} \tag{ 3 }$$

We assume that C_sf is different in (1) the "quiescent phase," when the LMC shows an almost steady star formation; and (2) the "burst phase," when the LMC experiences a burst of star formation. In the present study, we investigate models with no burst (referred to as "standard models") and those in which a starburst can occur only once ("burst models") or twice ("double-burst models"). The SFR is assumed to be constantly higher for t_{sb1, s} ⩽ t ⩽ t_{sb1, e} in the first starburst and t_{sb2, s} ⩽ t ⩽ t_{sb2, e} in the second one. The star formation coefficient (C_sf) is thus described as follows:

$$\begin{equation} C_{\rm sf}= \left\lbrace \begin{array}{@{}ll}C_q & \mbox{for quiescent phase} \\ C_{\rm sb1} & \mbox{for first starburst} \\ C_{\rm sb2} & \mbox{for second starburst} \\ \end{array}. \right. \end{equation} \tag{ 4 }$$

In the present study, a non-dimensional value of C_sf is given for each model.

For the accretion rate, we adopt the formula in which A(t) = C_aexp (− t/t_a) and t_a is a free parameter controlling the timescale of the gas accretion. The normalization factor, C_a, is determined such that the total gas mass accreted onto the LMC can be one for a given t_a. Although we investigated models with different t_a, we finally adopted the models with t_a = 0.3 Gyr for the LMC. This is mainly because the models with t_a = 0.3 Gyr can better explain the observations. The present results do not depend strongly on the parameter t_a for a reasonable parameter range. The initial [Fe/H] of the infalling gas is set to be −2 and we assume an SN II like enhanced [α/Fe] ratio (e.g., [Mg/Fe] ≈ 0.4) and [Ba/Fe] for the gas. This adoption of initial [Fe/H] is reasonable for the present study, which discusses stars with [Fe/H] as low as −2. Also, similar adoptions have been done in other chemical evolution models (e.g., Tsujimoto et al. 2010). We discuss our parameter study for the above variables in Section 2.5.

We investigate models in which metals from AGB stars and SNe can be removed from the LMC through energetic stellar winds so that they cannot be used for chemical enrichment of the LMC ISM. These models are referred to as "wind models" for convenience. The wind models are further divided into two categories: "selective wind models" and "non-selective wind models." Only SNe ejecta (not already existing ISM) can be expelled from the LMC in the selective wind model, whereas both ejecta from SNe and AGB stars, and already existing ISM can be expelled from the LMC in the non-selective wind model. For comparison, we also investigate models in which there is no stellar wind and thus gaseous ejecta from all stars can be fully mixed with the ISM ("non-wind models").

In the selective wind models, only a fraction, 1 − f_ej, of gaseous ejecta from SNe can be mixed with ISM for chemical enrichment processes, whereas all AGB ejecta can be mixed with ISM. Therefore, the wind rate w(t) is estimated only from the total mass of the gaseous ejecta from SNe M_{ej, sn} at each time step. The total mass, M_w, of ISM that is removed from the LMC at each time step in the selective wind models is

$$\begin{equation} M_{\rm w}=f_{\rm ej}M_{\rm ej, sn}. \end{equation} \tag{ 5 }$$

In the non-selective wind models, we adopt the same model that was used in PT98 in which M_ej is determined solely by the SFR:

$$\begin{equation} M_{\rm w}= C_{\rm ej} \psi (t), \end{equation} \tag{ 6 }$$

where C_ej is a parameter (in simulation units) that can control how efficiently the ISM with AGB and SN ejecta can be removed from the LMC. We investigate non-wind (i.e., standard, burst, and non-burst), selective wind, and non-selective wind models. We discuss our parameter study for the above variables in Section 2.5.

We adopt the nucleosynthesis yields of SNe II and Ia from T95 to deduce y_{II, i} and y_{Ia, i} for a given IMF. Stars with masses larger than 8 M_☉ explode as SNe II soon after their formation and eject their metals into the ISM. In contrast, there is a time delay (t_Ia) between the star formation and the metal ejection for SNe Ia. We here adopt the following DTD (g(t_Ia)) for 0.1 Gyr ⩽t_Ia ⩽ 10 Gyr, which is consistent with recent observational studies on the SN Ia rate in extra galaxies (e.g., Totani et al. 2008; Maoz et al. 2010, 2011):

$$\begin{equation} g_{\rm Ia} (t_{\rm Ia}) = C_{\rm g}t_{\rm Ia}^{-1}, \end{equation} \tag{ 7 }$$

where C_g is a normalization constant that is determined by the number of SN Ia per unit mass (which is controlled by the IMF and the binary fraction for intermediate-mass stars for the adopted power-law slope of −1). Maoz & Badenes (2010) detected a population of prompt SN Ia in the LMC and showed that the number of prompt SN Ia per stellar mass formed is 2.7–11.0 × 10⁻³ M⁻¹_☉. Although we mainly investigate the "prompt SN Ia models" with the above DTD, we also investigate the "classical SN Ia models" with 1 Gyr ⩽t_Ia ⩽ 3 Gyr (e.g., Yoshii et al. 1996). This SN Ia lifetime in Yoshii et al. (1996) was deduced by using their Galactic chemical evolution models that can be consistent with the observed [O/Fe]–[Fe/H] relations of the Galactic stars in the solar neighborhood. The fraction of the stars that eventually produce SNe Ia for 3–8 M_☉ has not been observationally determined and thus is regarded as a free parameter, f_b. We mainly investigate models with f_b = 0.03, because such models can better explain the observed chemical properties of the LMC. We briefly compare the results of models with f_b = 0.03 and 0.06. For the site of r-process, we adopt the mass range of 8–10 M_☉ for SNe II (Mathews et al. 1992; Ishimaru & Wanajo 1999) as identified with the collapsing O–Ne–Mg core (Wheeler et al. 1998). The yield of Ba from r-process is 1.45 × 10⁻⁶ M_☉ for the adopted range of SNe II.

Low-mass AGB stars (<3 M_☉) release the s-process elements during the thermally pulsing AGB phase (e.g., Gallino et al. 1998). As a consequence of large uncertainties in convective mixing and ¹³C-pocket efficiencies, the s-process nucleosynthesis allows for a wide range of possible production levels. On the other hand, the observed abundances for the AGB stars can be directly compared with the theoretical nucleosynthesis results. Here, we investigate the element Ba by adopting the best empirical metallicity-dependent Ba-yield derived by TB12. The adopted metallicity-dependent Ba-yield y_Ba is

$$\begin{equation} y_{\rm Ba}= \left\lbrace \begin{array}{@{}ll}y_{\rm Ba,0}(m_{\rm agb}) \times 10^{1.5 {\rm [Fe/H]} +1.5} & -2 \le {\rm [Fe/H]} < -1 \\ y_{\rm Ba,0}(m_{\rm agb}) & -1 \le {\rm [Fe/H]}. \\ \end{array} \right. \end{equation} \tag{ 8 }$$

The value of y_{Ba, 0}(m_agb) depends on the mass of the stars m_agb that can finally become AGB stars. The adopted y_{Ba, 0}(m_agb) are 2.6 × 10⁻⁷ M_☉, 4.0 × 10⁻⁷ M_☉, and 5.2 × 10⁻⁷ M_☉ for m_agb = 1.5 M_☉, 2.0 M_☉, and 3.0 M_☉, respectively.

An AGB star with initial mass m_I and final mass m_F can eject its envelope with a total mass of m_ej and the gas can be mixed with the surrounding ISM to chemically pollute the ISM. The initial-to-final relationship for AGB stars, from which we can deduce m_ej, has been extensively discussed in Weidemann (2000). We derive an analytic form for m_ej (=m_I − m_F) from the observational data by Weidemann (2000) by using the least-squares fitting method, and find

$$\begin{equation} m_{\rm ej} =0.916 M_{\rm I}-0.444. \end{equation} \tag{ 9 }$$

The coefficient of determination (R-value) is 0.995 for the above fitting. In order to calculate the main-sequence turnoff mass m_TO, we use the following formula (Renzini & Buzzoni 1986):

$$\begin{equation} \log m_{\rm TO}(t_{\rm s}) = 0.0558 (\log t_{\rm s})^2 - 1.338 \log t_{\rm s} + 7.764, \end{equation} \tag{ 10 }$$

where m_TO is in solar units and time t_s is in years. By using the adopted IMF and Equations (9) and (10), we numerically estimate h_agb (i.e., how much AGB ejecta can be returned into ISM per unit mass) in Equation (2) at each time step.

The adopted IMF is defined as Ψ(m_I) = M_{s, 0}m_I^−α, where m_I is the initial mass of each individual star and the slope α = 2.35 corresponds to the Salpeter IMF (Salpeter 1955). The normalization factor M_{s, 0} is a function of α, m_l (lower mass cutoff), and m_u (upper mass cutoff). These m_l and m_u are set to be 0.1 M_☉ and 50 M_☉, respectively (so that the normalization factor M_{s, 0} is dependent simply on α). We investigate models with different α to find the model(s) that can best explain the observed abundance patterns of stars in the LMC. We do not discuss models with different m_u, because the effects of changing m_u on the LMC chemical evolution are similar to those of changing α.

We mainly investigate the standard (labeled as "S"), burst ("B"), and double-burst ("DB") models in which stellar wind effects are not included at all (i.e., "non-wind models"). By using these non-wind models, we demonstrate how the IMF slope (α) and the epochs of starburst (t_{sb1, s} and t_{sb2, s}) can influence the chemical evolution of the LMC. These t_{sb1, s} and t_{sb2, s} are given in units of Gyr. We then investigate the wind models ("W") so that we can discuss whether or not the removal of AGB and SN ejecta from the LMC is important in the chemical evolution of the LMC. In the present study, the time t is the time that has elapsed since the model calculation started. Therefore, t = 0 Gyr (13 Gyr) represents the time when the calculation starts (ends). Previous observational and theoretical studies have suggested that there could be at least two epochs of enhanced star formation (starburst) about ∼2 Gyr and 0.2 Gyr ago (e.g., HZ09; DB12). We therefore mainly investigate the burst and double-burst models with t_{sb1, s} = 11 Gyr and t_{sb2, s} = 12.8 Gyr. Table 1 summarizes the representative 26 models investigated in the present study.

Figure 1 illustrates the time evolution of SFRs and [Fe/H] of stars in the five representative models (S1, B1, B6, DB1, and DB6) with different parameters controlling the LMC star formation history. No star formation burst is assumed in S1, whereas only one burst is assumed in B1 and B6. Two bursts of star formation at different epochs are assumed in DB1 and DB6. These models are chosen so that they can, in combination, show a wide range of star formation histories in Figure 1. They are just examples of possible star formation histories of the LMC. We investigate how the final chemical abundances depend on the LMC star formation history by using the results of models with different star formation histories. Figure 2 shows the time evolution of [Mg/Fe] and the [Mg/Fe]–[Fe/H] relation in the models with prompt and classical SNe Ia. It is clear from this figure that the models with prompt SN Ia show no plateau in the [Mg/Fe]–[Fe/H] relation owing to the earlier chemical enrichment by SNe Ia that can eject an Fe-rich gas. Figure 2 also shows that the time evolution of [Mg/Fe] depends on f_b in such a way that [Mg/Fe] can steeply decrease with time in the model with a larger f_b. We discuss these in more detail for different models in the following sections.

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We mainly investigate (1) the present gas mass fraction f_{g, 0}; (2) the stellar metallicity of the youngest stellar population, [Fe/H]₀; (3) AMR; (4) [Mg/Fe]–[Fe/H] relation (as an example of [α/Fe]–[Fe/H] relations); (5) [Ba/Fe]–[Fe/H] relation (as an example of the time evolution of s-process elements); and (6) MDF. We compare these results with the corresponding observations for the LMC. In order to demonstrate clearly how different chemical properties of the LMC are compared with those of the Galaxy, we also show the observational results for the Galaxy. We estimate the total stellar mass (M_s) of the LMC by using the observed V-band luminosity (≈3 × 10⁹ L_☉) and the reasonable mass-to-light ratio of 0.9 ± 0.2 for the observed B−V color (Bell & de Jong 2001). By using the observed total mass (M_g) of the LMC ISM (Bernard et al. 2008) and M_s, we can estimate f_{g, 0}. The observational error bar of f_{g, 0} is due largely to the uncertainty of the stellar mass-to-light ratio for the LMC. We adopt [Fe/H]₀ ≈ −0.3 as derived by Luck et al. (1998) for Cepheid variables that are 10–60 Myr, because these youngest populations can have the present-day stellar metallicity of the LMC. Each of the observed AMRs shows a wide range in each age bin (represented by an error bar), owing to the presence of stars with different [Fe/H] at each age bin. Although this could make it difficult for us to derive the best model, we try to give at least a few constraints on some of the model parameters. We mainly investigate the time evolution of [Mg/Fe] and [Ba/Fe], because these abundances are more reliably derived from observations. The [Mg/Fe]–[Fe/H] and [Ba/Fe]–[Fe/H] relations are examples of [α/Fe]–[Fe/H] and [s-process/Fe] relations, respectively, in this study.

3.1.1. AMR

First, we describe the five standard models with different IMFs in order to demonstrate the importance of the IMF slopes in the LMC chemical evolution. In these models, C_q is chosen such that the final stellar metallicity ([Fe/H]₀) can be −0.3 (i.e., the observed value). Figure 3 shows the AMRs for the five models as well as the observed [Fe/H] of the LMC field stars at different age bins and individual GCs. The AMR in each model is simply a plot of [Fe/H] at each time step (i.e., not average [Fe/H] over a given age-bin-like observation). Although the observational errors are not small for old GCs in the LMC, their data points are included to compare stars with the lowest [Fe/H] in the models with the corresponding observations.

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The five models with no star formation burst can reproduce the overall trend of the observed AMR reasonably well, which implies that the AMR alone cannot be used for discriminating burst and non-burst models for the LMC. These models, however, appear to be less consistent with the observed [Fe/H] around the ages of 2–3 Gyr (i.e., t =10–11 Gyr) owing to the presence of stars with significantly lower metallicities (−1 < [Fe/H] < − 0.5). The models with shallower IMFs show higher [Fe/H] at a given age, which reflects the fact that chemical evolution can proceed more rapidly owing to a larger amount of metals produced by a larger number of SNe.

It should be stressed that these standard models cannot reproduce the AMR as well around the ages of 3–9 Gyr (i.e., t = 4–10 Gyr) derived by HZ09 in which the AMR shows systematically lower [Fe/H] for a given age in comparison with other observations by C05 and Carrera et al. (2008, C08). The reason for the apparent variance in the observed AMRs between different observations could be related to different methods to determine ages and metallicities of stars in the observations. If the results by HZ09 are closer to the true AMR of the LMC, then the standard models with no burst can be regarded as less realistic models for the LMC evolution. Although the large [Fe/H] dispersion around the age of 2 Gyr (i.e., t = 11 Gyr) could be simply due to star formation from gas with different metallicities (i.e., inner higher and outer lower metallicities of the LMC ISM), it could also be caused by a sudden and rapid infall of metal-poor gas from outside the LMC disk. If the observed dispersion is due to an external gas infall, then it would have profound implications for the LMC evolution. We will later discuss the implications in Section 4.

3.1.2. [Mg/Fe]

Figure 4 shows that [Mg/Fe] slowly and monotonically decreases with time regardless of the adopted IMFs. This time dependence of [Mg/Fe] is simply a result of a later contribution of SN Ia to the chemical evolution of the LMC. The more significant decrease of [Mg/Fe] with time can be seen in the models with steeper IMFs owing to the greater contribution of SN Ia to the chemical evolution in these models. The [Mg/Fe]–[Fe/H] relations do not show a plateau at a low [Fe/H]. The lack of a plateau in the [Mg/Fe]–[Fe/H] relations reflects the fact that SNe Ia can chemically pollute the LMC ISM from a very early stage (t ∼ 10⁸ yr) of its evolution. The lack of a plateau appears to be seen in the observed [Mg/Fe]–[Fe/H] relation for [Fe/H] < −1.5, though it is not as clear. The [Mg/Fe]–[Fe/H] relations in the models with α ⩽ 2.75 can reproduce the overall trend of the observed [Mg/Fe] relation reasonably well, though the observation shows a large [Mg/Fe] dispersion at a given [Fe/H].

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The model with a very steep IMF (α = 2.95) shows systematically lower [Mg/Fe] and therefore cannot reproduce the locations of stars in the [Mg/Fe]–[Fe/H] relation as well. The models with shallower IMFs show systematically larger [Mg/Fe] in the present standard models. Owing to the observed large dispersion of [Mg/Fe], it is currently difficult to determine which IMFs (α = 2.35 or 2.55 or 2.75) can better explain the observed [Mg/Fe]–[Fe/H] relation. The observed high [Mg/Fe] (>0.4) for higher [Fe/H] (>  − 0.7) cannot be explained by any of the standard models in the present study.

3.1.3. [Ba/Fe]

Figure 5 shows that [Ba/Fe] starts to slowly increase with time about 2 Gyr after the commencement of the active star formation in the LMC disk. This slow [Ba/Fe] increase is due to the ejection of s-process elements from low-mass (<3 M_☉) AGB stars. The observed systematically high [Ba/Fe] (>0 for most stars) at [Fe/H] < − 1.5 seems to be better reproduced by the models with steeper IMFs, though the very high [Ba/Fe] (>0.2) in some stars at such low metallicities cannot be reproduced at all by any of the standard models in the present study. The reason for the higher [Ba/Fe] at [Fe/H] < −1 in steeper IMFs is that the numerical ratio of SNe II with masses of ∼8 M_☉ to those with ∼10 M_☉ (and thus the contribution of these SNe to the chemical evolution) can change in the steeper IMF and consequently [Ba/Fe] can increase more significantly in the early history of the chemical evolution of the LMC. The maximum [Ba/Fe] in the model with the Salpeter IMF is at most [Ba/Fe] ∼ 0.4, whereas most of the LMC stars at [Fe/H] > −0.6 show [Ba/Fe] ⩾0.4. These results strongly suggest that the LMC could have had a steeper IMF (at least steeper than the Salpeter IMF with α = 2.35) in its star formation history (if the LMC has not experienced starbursts). The observed very high [Ba/Fe] (>0.8) of some stars at [Fe/H] ∼ −0.6 cannot be explained at all by any of the standard models in the present study.

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3.1.4. f_g–[Fe/H] Relation

In the present study, the C_q for each model is chosen such that the final [Fe/H] (i.e., [Fe/H]₀) is consistent with the observed one (−0.3). A smaller amount of metals can be ejected from SNe in the models with steeper IMFs owing to the smaller numbers of SNe. Therefore, the ISM in the models with steeper IMFs needs to be more rapidly consumed by star formation and chemically enriched by SN ejecta to reach [Fe/H]₀ = −0.3. Figure 6 shows that (1) the models with steeper IMFs show smaller f_{g, 0} and (2) the model with α = 2.55 is the most consistent with the observed f_{g, 0}. These results suggest that the observed f_{g, 0} and [Fe/H]₀ combine to support the steeper IMF (α ≈ 2.55) of the LMC. However, this suggestion depends on the adopted assumption that no ISM can be expelled from the LMC in the standard models. If a significant fraction of cold ISM can be stripped from the LMC, even the models with shallower IMFs (e.g., α = 2.15) may be able to explain both f_{g, 0} and [Fe/H]₀. Likewise, if a significant fraction of cold ISM can be recently accreted onto the LMC, the models with rather steep IMFs (e.g., α > 2.75) may explain both f_{g, 0} and [Fe/H]₀ as well.

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3.2.1. AMR

The star formation burst and the subsequent efficient production of metals can rapidly and significantly increase [Fe/H] in the burst models. Therefore, the chemical evolution in the quiescent phase of star formation in the burst models needs to proceed more slowly (i.e., smaller C_q) in comparison with the standard models so that the final [Fe/H] can be as low as −0.3. Figure 7 shows the AMRs in the five burst models with different IMFs and C_q. The burst model B3 cannot reproduce the observed AMR owing to the systematically low [Fe/H] in the quiescent phase of the star formation. The model B1 with the Salpeter IMF can better explain the AMR of C05 and C08, whereas the models with steeper IMFs (B2 and B5) can better explain the AMR by HZ09. The observed AMR thus cannot give strong constraints on the IMF of the LMC. Owing to the observed large [Fe/H] dispersion at an age of ∼2 Gyr (t = 11 Gyr), the AMR alone cannot allow us to make a robust conclusion as to whether the LMC experienced a burst of star formation at an age of 2 Gyr (i.e., t = 11 Gyr).

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3.2.2. [Mg/Fe]

As shown in Figure 8, the overall trends of the [Mg/Fe] evolution in the quiescent phase for the burst models are similar to those for the standard ones. The [Mg/Fe]–[Fe/H] relations in the burst models show "bumps" (a sharp increase followed by a decrease in the time evolution of [Mg/Fe]), whose magnitudes depend on C_s1 and α. The observed locations of the LMC stars in the [Mg/Fe]–[Fe/H] plane do not clearly show such a bump. However, if the two stars with [Mg/Fe] ∼ 0.5 at [Fe/H] ∼ −1.2 and −1.0 are removed from Figure 8, then the remaining stars appear to show a bump around −0.7 < [Fe/H] < −0.6. Furthermore, the apparent lack of stars with [Mg/Fe] > 0.3 at [Fe/H] ∼ −0.4 suggests that [Mg/Fe] has been decreasing since [Fe/H] = −0.5. Thus the higher [Mg/Fe] at [Fe/H] ∼ −0.6 could be evidence of a starburst around t = 11 Gyr (i.e., 2 Gyr ago). However, the observed [Mg/Fe] dispersion at [Fe/H] ∼ −0.6 is so large that we cannot make a robust conclusion as to whether a starburst occurred in the LMC at about t = 11 Gyr (i.e., 2 Gyr ago). Also, it should be noted that the smaller number of stars at [Fe/H] > −0.5 could be responsible for the apparent lack of stars with higher (>0.3) [Mg/Fe].

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3.2.3. [Ba/Fe]

Figure 9 shows that irrespective of the model parameters, [Ba/Fe] rapidly decreases soon after the starbursts owing to the ejection of Fe-rich gas from prompt SNe Ia in the time evolution of [Ba/Fe]. The temporary [Ba/Fe] decrease is subsequently followed by its rapid and sharp increase due to chemical pollution by AGB ejecta. Owing to the rapid [Ba/Fe] increase, the burst models can show systematically higher final [Ba/Fe] in comparison with the standard models with no burst. Therefore, the observed presence of stars with higher [Ba/Fe] (>0.5) at [Fe/H] ≈ −0.3 could be more consistent with the burst models. The burst models with steeper IMFs (α = 2.75) show [Ba/Fe] ∼ 0.7 at [Fe/H] = −0.3, which suggests that the combination of a steeper IMF and a secondary starburst could be closely associated with the origin of stars with [Ba/Fe] as large as 0.7. However, very high [Ba/Fe] (>0.9) at [Fe/H] ∼ −0.6 cannot be explained by the present starburst models. These stars with very high [Ba/Fe] could have been formed directly from AGB ejecta that did not mix well with the ISM.

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3.2.4. f_g–[Fe/H] Relation

Figure 10 clearly shows that the burst model with the Salpeter IMF cannot simultaneously reproduce the observed f_{g, 0} and [Fe/H]₀ owing to efficient chemical enrichment. On the other hand, a larger amount of gas needs to be consumed for the ISM to be chemically enriched to [Fe/H] ∼ −0.3 in the models with steep IMFs (α = 2.75) so that f_{g, 0} can become significantly smaller in the model. The models with α = 2.55 and different star formation histories in the quiescent phases can best reproduce both the observed f_{g, 0} and [Fe/H]₀. These results on the IMF dependences are essentially the same as those derived for the standard model.

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3.3.1. AMRs

Figure 11 shows the dependence of AMRs on IMFs and star formation histories in the quiescent phase for double-burst models. The derived IMF dependence of AMRs are very similar to those in the burst models. The model DB3, with a steep IMF (α = 2.75) and a less rapid star formation in the quiescent phase, is inconsistent with any observational results on the AMR. The model DB1 with the Salpeter IMF can better reproduce the observed AMRs of C05 and C08, whereas the model DB2 with moderately steep IMF (α = 2.55) can be better fitted to the AMR by HZ09. Although the burst at t = 12.8 Gyr is as strong as that at t = 11 Gyr, the signature of the burst in the AMR is less clear in the double-burst models. The observed AMR shows a large [Fe/H] dispersion in the youngest stellar population (i.e., [Fe/H] ranging from ∼ − 0.6 to ∼0), and the presence of such relatively metal-poor ([Fe/H] ∼ −0.6) stars at the present time is puzzling. The recent accretion of metal-poor gas and the resultant active star formation (before the burst at t = 12.8 Gyr) could result in the formation of these metal-poor stars and thus introduce a larger scatter in [Fe/H] in young populations.

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3.3.2. [Mg/Fe]

Figure 12 shows that the time evolution of [Mg/Fe] in the double-burst models is characterized by two occurrences of a sharp [Mg/Fe] increase/decrease after starbursts, which results in two bumps. The double-burst models show two bumps in the [Mg/Fe]–[Fe/H] plane, though such bumps are not clearly seen in the observations. The final [Mg/Fe] is larger than 0 for all double-burst models owing to the last starburst at t = 12.8 Gyr, which is a feature that discriminates between the burst and double-burst models. The models (DB3, 4, and 5) with steeper IMFs show a smaller final [Mg/Fe] and the final [Mg/Fe] in the model DB1 with α = 2.35 is more consistent with the observed ratio for the LMC field stars. The observed [Mg/Fe] of the field stars with −0.5 < [Fe/H] < −0.3 appears to increase from ∼0 to 0.2, which may be regarded as the second bump due to the starburst at t = 12.8 Gyr. The second bump can be shown more clearly in the models DB3 and DB5 with steeper IMFs (α = 2.75). As noted in Section 3.2, the observed large [Mg/Fe] dispersion at a given [Fe/H] makes it difficult for us to confirm the presence (or absence) of the first and second starbursts in the observed [Mg/Fe]–[Fe/H] relation.

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3.3.3. [Ba/Fe]

Figure 13 shows that [Ba/Fe] rapidly decreases soon after the first starburst at t = 11 Gyr and then increases until the commencement of the second starburst at t = 12.8 Gyr for the double-burst models with different IMFs (in the time evolution of [Ba/Fe]). The increase of [Ba/Fe] after the second starburst cannot be seen, because the 0.2 Gyr difference between the present and the last starburst is not long enough for AGB stars to have chemically polluted the LMC ISM. Owing to the first starburst, [Ba/Fe] can become significantly high (up to ∼0.7) in the double-burst models. However, the maximum value across all models is still significantly lower than those (>0.9) observed in some stars with [Fe/H] = −0.6 ∼ −0.5. The models with steeper IMFs can show a higher final [Ba/Fe] and they can better explain the clear differences in the locations of field stars in the [Ba/Fe]–[Fe/H] plane between the LMC and the Galaxy.

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3.3.4. f_g–[Fe/H] Relation

Figure 14 shows that the best model to explain both f_{g, 0} and [Fe/H]₀ simultaneously is the one with α = 2.55 among the double-burst models (i.e., DB2 and DB4). This result combined with those in the standard and burst models strongly suggests that the IMF of the LMC should be moderately steeper (α ≈ 2.55) in its long-term star formation history. As demonstrated, the models with steeper IMF can better explain the presence of the LMC field stars with [Ba/Fe] > 0.5 and higher [Ba/Fe] of GCs at low metallicities ([Fe/H] < −1.5). It should be noted here that all ejecta from AGB stars and SNe are retained in the LMC for these non-wind models. Thus, it can be concluded that a steeper IMF (α ∼ 2.55) can better explain the star formation and chemical evolution histories of the LMC, as long as the LMC has not lost a significant amount of its chemically enriched ISM through stellar winds.

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3.4.1. AMR

Figure 15 shows that (1) the AMRs are not so different between selective wind models (W1−W4) with different α, C_q, and f_ej, and (2) the chemical evolution proceeds significantly more slowly in the non-selective wind model (W5) than in the selective ones until recently (t ∼ 8 Gyr) so that the AMR in the non-selective wind model shows systematically lower [Fe/H] for a given age. Although the non-selective wind model can explain the observed [Fe/H]₀, the total gas mass ejected from the LMC (M_{ej, t}) for 13 Gyr is about 3.4 times larger than the final stellar mass (i.e., M_{ej, t} = 9.1 × 10⁹ M_☉ for M_s = 2.7 × 10⁹ M_☉) and thus appears to be too large. Both already existing ISM and newly synthesized metals can be efficiently removed from the LMC in the non-selective wind models so that the chemical evolution of the LMC can proceed much more slowly. As a result of this, a much larger amount of gas can be removed from the LMC until the stellar metallicity finally becomes [Fe/H] ≈ 0.3 in the non-selective wind model (W5). The AMRs in the selective wind models are very similar to the standard models: they are more consistent with the one observed by C05 and C08 than that by HZ09. This result implies that the observed AMR alone does not enable us to discuss the effects of stellar winds in the LMC chemical evolution.

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3.4.2. [Mg/Fe]

Figure 16 shows that the monolithic decrease of [Mg/Fe] with time and the [Mg/Fe]–[Fe/H] relation in the five wind models are similar to those derived in the standard models with no wind. The selective wind models with larger f_ej show slightly higher [Mg/Fe] for a given IMF, whereas those with steeper IMFs show lower [Mg/Fe] for a given f_ej. The five wind models cannot explain the observed higher [Mg/Fe] (>0.2) for [Fe/H] >−0.6 in the LMC field stars. There is no remarkable difference in the [Mg/Fe] evolution between the selective and non-selective wind models.

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3.4.3. [Ba/Fe]

Figure 17 shows that the selective wind models can show higher [Ba/Fe] (>0.5) for [Fe/H] > −0.6 for a range of α and f_ej. The model W2 with α = 2.55 and f_ej = 0.4 shows [Ba/Fe] > 0.6 without secondary starbursts, because Fe-rich gas is ejected from the LMC disk, whereas the AGB ejecta containing s-process elements (e.g., Ba) can be retained and thus used for chemical evolution. These results imply that the efficient removal of SNe ejecta from the LMC disk is a way to significantly increase [Ba/Fe] without starburst(s). The models with steeper IMFs show higher [Ba/Fe], which is essentially the same as the results derived for the standard models. The final (thus maximum) [Ba/Fe] in the non-selective wind model at [Fe/H] = −0.3 is lower than 0.4, and therefore inconsistent with the observed value, which implies that the non-selective wind model is less promising than the selective wind as a reasonable model for LMC chemical evolution.

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3.4.4. f_g–[Fe/H] Relation

Figure 18 shows that the selective wind model W1 with the Salpeter IMF appears to better fit the observed f_{g, 0} and [Fe/H]₀ than the models with a steeper IMF (α = 2.55) for f_ej = 0.4 (W2). However, the model with a steeper IMF and f_ej = 0.2 (W4) can also explain the two observed quantities better than the model with the Salpeter IMF and f_ej = 0.2 (W3). It should be noted that the model W5 can also explain f_{g, 0} but does not show the observed high [Ba/Fe] at [Fe/H] ∼ −0.6. These results mean that if we carefully choose the two parameters (α and f_ej), the observation can be well reproduced. The non-selective wind model with the Salpeter IMF can also reproduce both f_{g, 0} and [Fe/H]₀ reasonably well. We do not intend to discuss the wind models with starburst(s), because the effects of starbursts on the LMC chemical evolution are essentially the same as those already described in the burst and double-burst models.

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In this section, we briefly summarize the advantages and disadvantages of the four different types of models in reproducing recent observational results (see Table 2 for a brief summary). The observed AMR can be consistent with most models with a reasonable set of parameters. The models with steeper IMFs (α ⩾ 2.55) can better explain the observed f_{g, 0}–[Fe/H] relation and [Ba/Fe] > −0.2 at [Fe/H] > −1.5 in a self-consistent manner. The apparent bump in the observed [Mg/Fe]–[Fe/H] relation at [Fe/H] ∼ −0.6 can be better reproduced in models with starbursts. The higher [Ba/Fe] ∼ 0.5 at [Fe/H] ∼ −0.3 can be better explained by non-starburst models (i.e., standard ones) with steeper IMFs and starbursts. Very high [Ba/Fe] (>0.9) at [Fe/H] ∼ −0.6 cannot be explained by any of the models in the present study. The significantly higher (>0.2) [Ba/Fe] observed in some old, metal-poor GCs ([Fe/H] < −1.5) cannot be explained by any of these models, either.

The selective wind models with the Salpeter IMF can better explain observations than the non-wind models with the Salpeter IMF, though the selective wind models need to assume an apparently large f_ej = 0.4 and cannot accurately reproduce the [Ba/Fe] of the stars in the metal-poor GCs with [Fe/H] < −1.5. This leads us to suggest that a steeper IMF is required for explaining the different observed chemical properties of the LMC in a self-consistent manner. The removal of metals from SNe can also significantly increase the [Ba/Fe] (up to ∼0.8) after a starburst if a large f_ej > 0.5 is chosen in the selective wind models. This implies that the selective removal of SNe ejecta could be partly responsible for the observed rather high [Ba/Fe] (∼0.8) in the LMC field stars. Thus, there are two possible ways to explain the high [Ba/Fe]: one is star formation directly from Ba-rich AGB ejecta and the other is a significantly efficient selective removal of SN ejecta.

Thus, as summarized in Table 2, the four different types of models can explain most of the observed chemical properties of the LMC reasonably well (except the unusually high [Ba/Fe]), as long as the model parameters are carefully chosen. The observed AMR and overall trends of the [Mg/Fe]–[Fe/H] (and equally the [α/Fe]–[Fe/H]) and the [Ba/Fe]–[Fe/H] relations can give less strong constraints on the IMF and the star formation history of the LMC. Table 3 describes how the four key chemical properties of the LMC constrain the IMF and the presence or absence of past starbursts.

The present study has shown that the standard, burst, and double-burst models with steeper IMFs (α ∼ 2.55) can explain well the observed f_g and [Fe/H]₀ in a self-consistent manner. Furthermore such non-wind models can not only better explain the observed higher [Ba/Fe] at [Fe/H] < −1.5 in the LMC (due to the steeper IMFs which raise the r-process/Fe ratio), but can also reproduce well the overall dependence of [Ba/Fe] on [Fe/H]. Although the selective wind models with the Salpeter IMFs (α = 2.35) can explain both f_g and [Fe/H]₀ in a self-consistent manner for a larger f_ej (∼0.4), they cannot explain as well the observed [Ba/Fe] at [Fe/H] ∼ −1.5. Therefore, the non-wind models with steeper IMFs seem to be slightly better models for the adopted stellar yields.

Although it seems premature to determine observationally whether the IMF of the LMC is clearly steeper than the Salpeter one, a number of previous observations have suggested a steeper IMF in the LMC. For example, Mateo (1988) investigated the IMFs of stars with masses ranging from 0.9 M_☉ to 10.5 M_☉ in the six clusters of the LMC and found that the slopes are typically α ∼ 3.5. Hill et al. (1994) also found a steeper IMF (α ∼ 3) for young stars with masses larger than 9 M_☉ in the LMC and also suggested that the IMF can be different below and above 9 M_☉. Holtzman et al. (1997) investigated the IMF for stars on the main sequence that are fainter than the oldest turnoff, and discussed the possibility of a steep IMF with α = 2.75 for the LMC dominated by young populations. As shown in Figures 4 and 6 of this paper, rather steep IMFs (e.g., α ⩾ 2.95) cannot be consistent with the observed chemical properties and AMR of the LMC.

The only difference in the non-wind models with steeper IMFs and the selective wind models with the Salpeter ones is the [Fe/H] dependence of [Ba/Fe] at lower [Fe/H] (< − 1.5). Most of the observed field stars and GCs with [Fe/H] < −1.5 in the LMC show [Ba/Fe] > −0.2, which is more consistent with the non-wind models with steeper IMFs. However, the total number of data points in the observed [Ba/Fe]–[Fe/H] diagram is sufficiently small enough that we cannot make a robust conclusion as to whether the non-wind or the selective wind models are better at explaining the observed [Ba/Fe] trend with [Fe/H]. Thus, it is undoubtedly worthwhile for future spectroscopic observations to investigate [Ba/Fe] and other s-process elements of the metal-poor stellar populations of the LMC with [Fe/H] < −1.5 to obtain better constraints on the IMF and the ejection processes of SN ejecta in the LMC.

The present study first investigated how a prompt SNe Ia influences the chemical evolution of the LMC and thereby predicted that [α/Fe] decreases monotonically (until a secondary starburst occurs) with no remarkable plateau in the [α/Fe]–[Fe/H] relation for low [Fe/H]. We should first investigate [Mg/Fe]–[Fe/H] relations, first because this element is most reliably determined by the observations among α-elements, and second because its yield is most reliably predicted by nucleosynthesis calculations. The second-best α-element for this investigation is Ca. On the other hand, Ti and Si are not as good since Ti is not predicted well by SN II nucleosynthesis, and the observed determination of the Si abundance involves a large uncertainty.

The observed [Mg/Fe]–[Fe/H] relation does not show the predicted trends as clearly, partly because of the large dispersion in [Mg/Fe] at each [Fe/H]. Although the physical reasons for the large [Mg/Fe] dispersion are not as clear, one of the possible explanations is described as follows. If different local regions have different star formation histories (e.g., due to different local gas densities and gas infall rates) in the LMC, as observed in recent observations (e.g., HZ09; R12), then they can have different [Mg/Fe] owing to different chemical enrichment histories. Therefore, the observed large scatter could be due to different evolutions of [Mg/Fe] in different local regions of the LMC.

The observed [Ca/Fe]–[Fe/H] relation can also be used for discussing whether there is evidence for the prompt SNe Ia playing a significant role in the chemical evolution of the LMC. Figure 19 shows the locations of the LMC stars on the [Ca/Fe]–[Fe/H] plane as well as the predicted [Ca/Fe]–[Fe/H] relations for three different models. These three models are chosen because they, in combination, show widely different [Ca/Fe]–[Fe/H] relations of the LMC stars. It is clear that [Ca/Fe] decreases as [Fe/H] increases without showing a clear plateau for [Fe/H] < −1 in the LMC stars, which is consistent with the predictions of the present models with prompt SNe Ia. In addition, the observed locations of the LMC stars on the [Ti/Fe]–[Fe/H] plane clearly show a trend for a wide range of metallicity similar to [Ca/Fe] (e.g., C12). However, the [Si/Fe]–[Fe/H] relation does not show such a clear trend owing to a larger [Si/Fe] dispersion for stars with [Fe/H] < −1.5 (e.g., Figure 4 in C12). These results on the [Ca/Fe]–[Fe/H] and [Ti/Fe]–[Fe/H] relations are supporting evidence that demonstrate that SNe Ia have influenced the chemical enrichment history of the LMC.

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Figure 19 also shows that there are significant differences in the distribution of field stars on the [Ca/Fe]–[Fe/H] plane between the LMC and the Galaxy. The LMC stars with [Fe/H] > −1 have systematically lower [Ca/Fe] in comparison with their Galactic counterparts with similar metallicities. The significantly lower [Ca/Fe] in the LMC stars cannot be easily explained by the models, even those with very steep IMFs with α = 2.95, though the models with steeper IMFs can show a lower [Ca/Fe]. On the other hand, the observations do not show significant differences in the locations of field stars on the [Mg/Fe]–[Fe/H] plane between the LMC and the Galaxy. Given that the models with rather steep IMFs (α > 2.75) show significantly low [Mg/Fe] (< − 0.2), the models with unusually steep IMFs (α > 2.95) are unable to explain both the observed distributions of the LMC field stars on the [Ca/Fe]–[Fe/H] and [Mg/Fe]–[Fe/H] planes. So how can we explain these observations?

If Ca can be expelled more effectively from the LMC ISM during explosions of SNe II in comparison with other α-elements, then [Ca/Fe] becomes significantly lower than [Mg/Fe] as the chemical evolution proceeds. We suggest that the nucleosynthesis of jet-induced SNe II could be responsible for the origin of the rather low [Ca/Fe] as follows. Shigeyama et al. (2010) have recently investigated hydrodynamical processes and nucleosynthesis in jet-induced SNe and derived aspherical distributions of chemical yields of the SNe. They have found that both the [Ca/Fe] and the [Mg/Fe] in the ejecta of a jet-induced SN II can depend strongly on azimuthal angles θ (where θ = 0 corresponds to the direction of the jet). They have also found that [Mg/Ca] can be lower in the lower θ in their A3 model in which the total explosion energy of an aspherical supernova is 10⁵² erg and the chemical abundances of O, Mg, Fe, Ca, Cr, Mn, and Zn for each θ are investigated (see their Figure 1 for the θ dependences). If the jet-induced SNe II eject with lower θ (where a larger amount of Ca-rich gas exists) can be more efficiently expelled from the LMC, then [Ca/Fe] in the LMC becomes rapidly lower (in comparison with [Mg/Fe]) as the chemical enrichment proceeds. Although it is not clear at this stage whether or not this more efficient removal of Ca from the LMC is really possible, here we discuss how much more efficiently Ca should be removed in order to explain the observed [Ca/Fe]–[Fe/H] relation.

We have investigated the selective wind model (W6) in which Ca is removed more efficiently by a factor of 1.75 from the LMC in comparison with other elements of SNe II ejecta. Therefore, M_w in Equation (5) is rewritten in the model W6 as follows:

$$\begin{equation} M_{\rm w}= \left\lbrace \begin{array}{@{}ll}1.75f_{\rm ej}M_{\rm ej, sn} & \mbox{for Ca} \\ f_{\rm ej}M_{\rm ej, sn} & \mbox{for other elements}. \\ \end{array} \right. \end{equation} \tag{ 11 }$$

The parameter values of α, C_sq, and f_ej in model W6 are the same as those adopted in model W2 (i.e., α = 2.55, C_q = 0.015, and f_ej = 0.4). Figure 19 demonstrates that model W6 shows a steeper [Ca/Fe] decrease with [Fe/H] and a rather low final [Ca/Fe] (∼ − 0.3) at [Fe/H] ∼−0.3. Furthermore, it is confirmed that the [Mg/Fe]–[Fe/H] in the model is also consistent with observations. Although the results of the model are broadly consistent with observations, it is not clear why some of the intermediate-age LMC GCs with [Fe/H] ∼ −0.5 have higher [Ca/Fe] (∼0) than the field stars with similar metallicities. The possible difference in [Ca/Fe] between the GCs and the field stars in the LMC could be related to the differences in the formation processes between field stars and GCs and thus is beyond the scope of this paper. We will discuss the origin of this intriguing difference in our forthcoming papers based on chemodynamical numerical simulations of the LMC.

Recent observational studies have investigated the AMR for different local regions in the LMC and thereby discussed the spatially resolved star formation and chemical evolution histories of the LMC (e.g., HZ09; R12). HZ09 have found that there are peaks of star formation roughly 2 Gyr, 500 Myr, 100 Myr, and 12 Myr ago in the LMC. R12 also have revealed the presence of peaks in SFRs 2.0 Gyr and 250 Myr ago. Recent numerical simulations on the formation of the Magellanic Stream have shown that the LMC and the SMC could have experienced strong tidal interactions about 2 Gyr and 250 Myr ago, and suggested that the two interactions could have significantly enhanced star formation in the LMC (DB12). These recent observational and theoretical studies imply that the LMC might have experienced a burst of star formation at least twice, though the epoch and the strength of each burst have not been precisely determined yet. In the following discussion, we focus on the possible starbursts ∼2 Gyr and ∼200 Myr ago.

As shown in previous theoretical models, past starburst events can be imprinted on the chemical abundances of the stellar and gaseous components of the LMC (e.g., Russell & Dopita 1992; T95). In particular, [α/Fe] as a function of [Fe/H] can significantly change during starburst events (e.g., T95) and thus can be used to give strong constraints on the epochs of the events. PT98 clearly showed that (1) the non-selective wind model with a secondary starburst about 2 Gyr ago can better explain the observed AMR, and (2) [α/Fe] can significantly and rapidly increase during the starburst and then slowly decrease to the solar value (e.g., [Mg/Fe] ∼ 0). The present study has predicted that the chemical signatures of the starburst about 2 Gyr ago include (1) a rapid increase of [α/Fe] followed by a rapid decrease, (2) a rapid decrease of [Ba/Fe] followed by a rapid increase, and (3) a time delay between the [α/Fe] and [Ba/Fe] peaks.

However, as pointed out in previous sections, these three chemical signatures of the past starburst event around 2 Gyr ago cannot be clearly seen in the observational results. Figure 20 shows [Mg/Fe]–[Fe/H] and [Ba/Fe]–[Fe/H] relations for the five burst models (B4−B9) with different epochs and strengths of starburst. The results shown in this figure (and Figure 8) suggest that although the observed higher [Mg/Fe] (>0.2) at [Fe/H] ∼ −0.6 can be consistent with the rapid increase of [Mg/Fe] due to the starburst event, the large scatter in [Mg/Fe] around [Fe/H] ∼ −0.6 does not allow us to make a robust conclusion on the origin of the higher [Mg/Fe]. Similarly, the observed larger [Ba/Fe] (>0.5) for [Fe/H] > −0.4, which is consistent with the present starburst models, cannot be regarded as strong evidence for the presence of a starburst about 2 Gyr ago owing to the small number of stars with known [Ba/Fe] for [Fe/H] >−0.4. More observational data sets are necessary to investigate how the secondary starburst about 2 Gyr ago might have changed the chemical evolution history of the LMC. The AMRs derived by different observational studies are significantly different so that the comparison between the observed AMRs and the simulated ones in the present study cannot provide strong constraints on the nature of past starburst events either.

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Concerning a possible starburst about 0.2 Gyr ago, a chemical signature of the starburst is the sudden increase of [Mg/Fe] around [Fe/H] ∼ −0.4, as shown in the present double-burst models. There is a hint of such a [Mg/Fe] increase in the observational data for the LMC field stars by P08, though the number of the stars is very small (only three). If the starburst is strong enough, then it can be imprinted on the MDF of the LMC. Figure 21 describes the MDFs for [Fe/H] and [Mg/Fe] in the five double-burst models. The MDFs here are relative frequency histograms that are binned with a 0.1 dex bin width and normalized to the most populated bin. Clearly the MDF of [Fe/H] show the double peaks for the five models and such double peaks are not observed in the previous observation by C08. The observed apparent lack of a bimodal MDF in C08 could be due to the small number of young and metal-rich samples in C08. If the observational result is real, then it means that the starburst about 0.2 Gyr ago is much weaker than modeled in the present study (so that it cannot be detected in the observed MDF).

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The AMR derived for the LMC by HZ09 (in their Figure 20) appears to show a sudden and significant decrease of [Fe/H] around 1.5 Gyr ago followed by a rapid increase of [Fe/H], though HZ09 did not discuss the origin of the possible "dip" in the AMR. The AMRs for some local regions of the LMC derived by R12 also appear to show the dips, whereas the AMR by C08 does not clearly show any. If the observed dip around 1.5 Gyr ago (HZ09) is real, it has profound implications on the gas accretion history of the LMC. One likely explanation for the possible dip is that a large amount of external metal-poor gas ([Fe/H] < − 1.0, significantly smaller than the gaseous metallicity of the LMC about 2 Gyr ago) was accreted onto the LMC from outside the LMC disk, and consequently [Fe/H] rapidly and significantly decreased. The observed apparently rapid decrease of [Fe/H] by almost 0.2 dex at [Fe/H] ∼ −0.7 (HZ09) can give some constraints on the amount of gas accreted onto the LMC for a given metallicity of the metal-poor gas.

In order to discuss how the AMR of the LMC can change owing to the infall of metal-poor gas from outside the LMC disk, we investigated the AMRs of models with gas infall, but without any starburst before the infall. The purpose of this investigation is to clearly show how the dip of the AMR can be formed during the infall of metal-poor gas (thus not to reproduce the observed AMR completely self-consistently). Therefore, we think that the adopted, somewhat idealized models would be enough to clearly show the formation of the dip in the AMR due to the infall of metal-poor gas. We have mainly investigated how much gas needs to be accreted onto the LMC by using the results of the "infall models" in which metal-poor gas with [Fe/H] = −1.6 can be accreted onto the LMC 1.5 Gyr ago. Here, the metallicity of [Fe/H] = −1.6 is chosen just for a representative case to show clearly the formation of the dip in the AMR (we also investigated the infall models with [Fe/H] = −1.0 for comparison).

In the infall models, the following infall rate of external gas ($\dot{M}_{\rm ext}$) is added to the right-hand side of Equation (1):

$$\begin{equation} \dot{M}_{\rm ext}= \frac{ M_{\rm ext} }{t_{\rm in,e} - t_{\rm in, s} }, \end{equation} \tag{ 12 }$$

where M_ext is the total mass of the external gas that can infall onto the LMC for t_{in, s} ⩽ t ⩽ t_{in, e}, where t_{in, s} and t_{in, e} are the epochs when the gas infall starts and ends, respectively. Therefore, the gas infall rate is assumed to be steady and constant in the present infall models. We investigate the models with t_{in, s} = 11.5 Gyr and t_{in, e} = 12.0 Gyr. The chemical abundance patterns of the gas is assumed to be the same as that of the LMC halo. Therefore, the term $Z_{I, i}\dot{M}_{\rm ext}$ (where Z_{I, i} is the chemical abundance of each element in the infalling gas) is added to the right-hand side of Equation (2) to calculate the evolution of Z_i. Owing to the infall of metal-poor gas, the mean metallicity of the youngest population becomes significantly lower. Therefore, a starburst needs to occur after the gas infall so that the final [Fe/H] can be −0.3, as observed. We thus assume that starbursts can occur before and after accretion of the metal-poor gas onto the LMC disk in the infall models because the models with this assumption can be consistent also with the observational results by HZ09 (i.e., a possible starburst about 2 Gyr ago). We have particularly investigated the four infall models (I1−I4) with α = 2.55, C_q = 0.004, C_sb1 = 0.8, t_{sb1, s} = 11.0 Gyr, t_{sb1, e} = 11.1 Gyr, t_{sb2, s} = 12.0, t_{sb2, e} = 13.0 Gyr, and [Fe/H] = −1.6 in external infalling gas, and different C_sb2 and M_ext. These models were chosen because they can together show different degrees of a sudden [Fe/H] drop (or different depths of the dip) in the AMRs. For comparison, we have investigated a model (I5) with [Fe/H] = −1.0 in external gas for comparison. The model parameters for these models are shown in Table 4.

Figure 22 shows the AMRs for the last ∼5 Gyr (i.e., t = 8–13 Gyr) in the four infall models in which M_ext ranges from 0.1 to 0.4 in simulation units. M_ext = 1 thus means that the total amount of external gas accretion is the same as the total amount of gas accreted onto the LMC from its own halo for the last 13 Gyr. The value of C_sb2 in each model is chosen such that the final [Fe/H] can be consistent with the observed one. For comparison, the AMR of the burst model B2 without gas infall is shown in this figure. Clearly stellar [Fe/H] can rapidly decrease soon after the metal-poor gas is accreted onto the LMC with the magnitude of the decrease depending on the amount of the accreted gas (M_ext). The model I3 with M_ext = 0.3 can show a dip of ∼0.2 dex, which means that ∼10⁹ M_☉ needs to be accreted onto the LMC in order to explain the observed dip. Other models with lower M_ext (i.e., I1 and I2) show less remarkable dips in the AMRs and thus are less consistent with the observations. If the metallicity of the accreted gas is higher than the adopted one ([Fe/H] = −1.6), then an even larger amount of gas needs to be accreted onto the LMC to form the remarkable dip in the AMR. For example, if the infalling gas has [Fe/H] = −1, then M_ext ≈ 0.4 is required to explain the dip of ∼0.2 dex observed in the AMR. We thus conclude that if the observed dip is real, then a massive accretion event of metal-poor gas with M_ext of at least ∼10⁹ M_☉ is required to explain the observed magnitude of the dip.

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Previous numerical simulations by BC07 and DB12 demonstrated that the gas of the SMC can be accreted onto the LMC after tidal stripping of the SMC gas caused by the strong LMC–SMC–Galaxy interaction. These simulations showed that there could be two accretion events around 1.5 Gyr and 0.2 Gyr ago, that means that the first accretion event about 1.5 Gyr ago is a promising candidate that can provide a large enough amount of gas to form the dip in the LMC AMR. However, the total amount of the SMC gas transferred to the LMC about 1.5 Gyr ago in these simulations is less than 10⁸ M_☉ for the total SMC mass of 3 × 10⁹ M_☉. The predicted gas mass is much smaller than the required mass (M_ext ∼ 10⁹ M_☉) for explaining the observed dip. This means which the SMC gas accreted onto the LMC is unlikely to explain the observed dip of 0.2 dex if the total mass of the SMC about 2 Gyr ago is similar to that of the present SMC (∼3 × 10⁹ M_☉). However, if the SMC was originally much more massive, the required amount of gas mass could be transferred from the SMC to the LMC.

Then where does such a large amount of metal-poor gas come from? One possible scenario is that a gas-rich dwarf galaxy with a total gas mass of 10⁹ M_☉ merged with the LMC about 1.5 Gyr ago and the mixing of the gas with the LMC ISM caused a significant decrease of [Fe/H]. Since the gas mass of the dwarf should be similar to that of the LMC in order to explain the observed dip, the dwarf would have to have had a total mass similar to the LMC. This means that the LMC stellar disk could have been severely damaged by the violent dynamical process of such a major merger event. The thick disk and counter-rotating stellar components (e.g., Subramaniam & Prabhu 2005) might have been formed from this major merger event that occurred in the LMC about 1.5 Gyr ago. A merged dwarf as massive as the LMC should have a much lower SFR than the LMC so that it can have a low metallicity. This discussion depends on the assumption that the observed dip of 0.2 dex is real and the dip was caused by a massive gas accretion event. Given that the amplitude of the dip provides information about the total mass of a gas-rich dwarf merging with the LMC (or the total mass of gas accretion onto the LMC), it would be quite important for future extensive observational studies to confirm the presence or the absence of the dip(s) in the AMR.

Although the present models can reproduce the overall trend of [Ba/Fe] with [Fe/H] and the higher [Ba/Fe] (∼0.5) for [Fe/H] < −0.6 in the LMC reasonably well, they cannot explain the stars with [Ba/Fe] > 0.9. We have confirmed that even the selective wind models with steep IMFs and efficient metal ejection (f_ej ∼ 0.6) can show at most [Ba/Fe] ∼ 0.8. This failure of the present one-zone chemical evolution models is related to the adopted assumption that AGB ejecta can be mixed with ISM soon after AGB stars eject their s-process elements. If the AGB ejecta does not mix well with the surrounding ISM and consequently can be converted into new stars, then the stars can show rather high [Ba/Fe]. Figure 1 in TB12 shows the rather high (>1) observed [Ba/Fe] in the envelope of AGB stars with different metallicities. We thus propose that the stars with unusually high [Ba/Fe] (>0.9) in the LMC were formed as a result of incomplete mixing of ISM and AGB ejecta. A key question here is how energetic stellar winds from AGB stars can cool down to become cold gas for star formation without mixing so well with the surrounding ISM.

Recent hydrodynamical simulations have shown that secondary star formation directly from AGB ejecta is possible in massive star clusters owing to the deeper gravitational potentials (e.g., Bekki 2011). Although this secondary star formation appears to be a convincing mechanism for the formation of stars with unusually high [Ba/Fe] in the LMC GCs, it cannot explain why some of the LMC field stars show such high [Ba/Fe]. One possibility for the field star formation from AGB ejecta is that AGB ejecta can assemble in the H i holes (e.g., Kim et al. 1999), where ISM can be almost completely blown away by SNe, and then can be converted into cold gas there without mixing efficiently with chemically enriched ISM. The new stars formed from AGB ejecta in the H i holes can naturally have very high [Ba/Fe]. Thus, it would be important for observational studies to confirm whether the locations of young stars with very high [Ba/Fe] are more likely to be in the present H i holes. Our future chemodynamical simulations will investigate whether this star formation from AGB ejecta in H i holes is really possible in the LMC.