A NEW CHEMICAL EVOLUTION MODEL FOR DWARF SPHEROIDAL GALAXIES BASED ON OBSERVED LONG STAR FORMATION HISTORIES

Here we describe our new chemical evolution model to reproduce simultaneously both the observed SFHs and MDFs of the following four local dSphs: Fornax, Sculptor, Leo II, and Sextans. We adopt the following assumptions in our model, which are the same as those in K11, except items 2, 6, and 8.

• 1.  
  The interstellar medium (ISM) of a dSph is treated as one zone, that is, we do not consider any spatial variations of gas density and metallicity in the ISM at any time.
• 2.  
  The observed SFH derived from the analysis of the CMD is used in our model. Therefore, the SFR at time t, Ψ(t), is fixed to the observed one. Namely, our model is calculated along with the SFH.
• 3.  
  We assume that there is a positive correlation between the gas mass and SFR like the Schmidt–Kennicutt law (Schmidt 1959; Kennicutt 1998). Since the SFR is given as input, the gas mass at a certain time is completely determined according to the SFR at the time.
• 4.  
  The interstellar gas can be expelled from a dSph (i.e., gas outflow) while supplied from its outside (i.e., gas infall), that is, the dSph in our model is not a closed box.
• 5.  
  Gas outflow is assumed to be induced via a superwind (a collective effect of a large number of SNe), so that its rate is proportional to the total number of SNe Ia and II at each time. Metallicity of the outflow gas is assumed to be the same as that of the system at that time.
• 6.  
  The number of SNe Ia is determined by the past SFH and the delay time distribution (DTD) function. The DTD function used in our model is the observationally derived one of Maoz et al. (2010). The minimum delay time of SNe Ia, t_{delay, min}, is adopted to be one of the model parameters.
• 7.  
  The number of SNe II is determined by the past SFH and the stellar lifetime given by Timmes et al. (1995). We assume that stars with m ⩾ 10 M_☉ explode as SNe II.
• 8.  
  Gas infall rate is adjusted to reproduce the observed SFH. Since we assume that the SFR is fixed (item 2) and the gas mass of the ISM correlates with the SFR (item 3), the gas mass at time t, M_gas(t), is determined by the observed SFH. Moreover, the time derivative of the gas mass, $\dot{M}_\mathrm{gas}(t)$, depends on the SFR, the ejected gas mass from evolved stars (i.e., AGBs and SNe Ia and II), the gas outflow via a superwind (item 5), and the gas infall. Therefore, in order to reproduce the gas mass, we set the gas infall rate as an adjustable amount (see Section 2.1.3). The metallicity of the infall gas is assumed to be primordial.
• 9.  
  The yield of SNe Ia and II is adopted from Iwamoto et al. (1999) and Nomoto et al. (2006), respectively. For the AGB yields, we adopt those of Karakas (2010). We assume that all stars with m < 10 M_☉ evolve into AGBs after their lifetimes. All heavy elements are assumed to stay in gas phase during the entire time of calculation.
• 10.  
  The initial mass function (IMF) used in our model, ϕ(m), is that derived by Kroupa et al. (1993) with a mass range of m = 0.08–100 M_☉.

We calculate the chemical evolution for 119 elements and isotopes listed in Iwamoto et al. (1999), Nomoto et al. (2006), and Karakas (2010) under the assumptions listed above. The equations of the mass evolution and the chemical evolution for an element i at time t are represented by

$$\begin{eqnarray} \dot{M}_\mathrm{gas}(t) &=& -\Psi (t) + E(t) + \dot{M}_\mathrm{in}(t) - \dot{M}_\mathrm{out}(t) \end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray} \dot{M}_{\mathrm{gas},i}(t) &=& -\Psi (t)X_i(t) + E_i(t) + \dot{M}_{\mathrm{in}}(t)X_{\mathrm{in},i}(t) \nonumber \\ && - \dot{M}_\mathrm{out}(t)X_i(t), \end{eqnarray} \tag{ 2 }$$

where M_{gas, i}(t) is the gas-phase mass of an element i in the ISM at time t, X_i(t) is the gas-phase mass fraction for the element to the total mass of interstellar gas (X_i(t) ≡ M_{gas, i}(t)/M_gas(t) and ∑_iX_i(t) = 1), E_i(t) is the rate of gas mass for the element ejected from evolved stars to the ISM and E(t) is its total (E(t) = ∑_iE_i(t)), and $\dot{M}_{\mathrm{in}}(t)$ and $\dot{M}_{\mathrm{out}}(t)$ are its infall and outflow rates, respectively. We assume that both the initial abundance of the ISM and the infall gas are primordial (i.e., X_H(t = 0) = X_{in, H} = 0.75 and X_He(t = 0) = X_{in, He} =0.25).

The ejected gas mass E_i(t) is given by the sum of three independent terms as follows:

$$\begin{equation} E_i(t) = \dot{\xi }_{i,\mathrm{Ia}}(t) + \dot{\xi }_{i,\mathrm{II}}(t) + \dot{\xi }_{i,\mathrm{AGB}}(t), \end{equation} \tag{ 3 }$$

where $\dot{\xi }_{i,\mathrm{Ia}} (t)$, $\dot{\xi }_{i,\mathrm{II}} (t)$, and $\dot{\xi }_{i,\mathrm{AGB}} (t)$ represent the rates of gas masses ejected from SNe Ia (Iwamoto et al. 1999), SNe II (Nomoto et al. 2006), and AGBs (Karakas 2010), respectively. The term $\dot{M}_{\mathrm{out}}(t)$ can be written by using the number of SNe II and Ia, $\dot{N}_\mathrm{II}(t)$ and $\dot{N}_\mathrm{Ia}(t)$, respectively, as

$$\begin{eqnarray} \dot{M}_\mathrm{out}(t) &=& A_\mathrm{out} [\dot{N}_\mathrm{Ia}(t) + \dot{N}_\mathrm{II}(t)], \end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray} \dot{N}_\mathrm{Ia}(t) &=& \int ^{t}_{0} {{\rm DTD}}(t-t^{\prime }) \Psi (t^{\prime })\ dt^{\prime }, \end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray} \dot{N}_\mathrm{II}(t) &=& \int ^{100\, M_\odot }_{10\, M_\odot } \Psi (t - \tau (m)) \phi (m)\ dm. \end{eqnarray} \tag{ 6 }$$

Here the outflow coefficient A_out (M_☉ SN⁻¹) controls the outflow gas mass per unit SN. For the DTD function, we adopt the following, which are observationally estimated by Maoz et al. (2010):

$$\begin{equation} {{\rm DTD}}(t_\mathrm{delay}) = \left\lbrace \begin{array}{@{}l@{\quad }l@{}} 1 \times 10^{-3}\ \mathrm{SN\ Gyr}^{-1}\ M_{\odot }^{-1}\\ \quad \times (t_\mathrm{delay} / \mathrm{Gyr})^{-1.1} & \mathrm{for}\ t_\mathrm{delay} \ge t_\mathrm{delay,min}\\ 0 & \mathrm{otherwise} \end{array}\right., \end{equation} \tag{ 7 }$$

where t_delay indicates an elapsed time since the progenitor star of an SN Ia is formed. The variable τ(m) in Equation (6) is the stellar lifetime of the star with mass of m.

The interval of each time step in our calculation is Δt = 25 Myr.³ The calculation is terminated when the SFR reaches zero or a certain condition, which is described in detail in Section 2.2, is satisfied.

The assumptions adopted in our model listed above are the same as K11 except the assumptions for the SFH (item 2), minimum delay time of SNe Ia (item 6), and gas infall rate (item 8). It should be stated that, while the stellar lifetime adopted in our model (Timmes et al. 1995) is also different from that in K11 (i.e., Padovani & Matteucci 1993; Kodama 1997), the difference is found to not affect the resultant chemical evolution. In the following subsections we describe these major differences in the model assumptions between K11 and ours.

2.1.1. Star Formation History and Gas Mass

In our model, we use the SFHs estimated from the photometric observations for our sample dSphs (i.e., Fornax, Sculptor, Leo II, and Sextans) as the most fundamental inputs. On the other hand, in K11, SFH is not an input quantity but one of the output quantities, which is determined to reproduce the observed data of the MDF and abundance ratios for each one of their samples of the local dSphs, including our sample dSphs. In Figure 1 we show the SFHs of our sample dSphs derived in the literature (Fornax: de Boer et al. 2012b; Sculptor: de Boer et al. 2012a; Leo II: Dolphin 2002; Sextans: Lee et al. 2009). It should be emphasized that the observed SFHs have much longer timescales of about 8–13 Gyr than the timescales of the SFHs derived in K11 of ∼0.8–1.3 Gyr.

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It should be noted that the SFHs of both Leo II and Sextans start from 15 Gyr ago, which is longer than the age of the universe derived from the current cosmology (Komatsu et al. 2009). As for this discrepancy, Lee et al. (2009) mentioned that the current level of discrepancy is within the range of various plausible adjustments for the stellar models. However, for safety, we have compared the best-fit parameters, whose details are described in Section 2.2, of Sextans for the SFHs starting from 15 Gyr ago and 14 Gyr ago, and we have confirmed that there is no difference. Namely, the discrepancy is not critical to the chemical evolution of the dSphs. Therefore, we have decided to let it stand.

For each dSph in our sample, we derive its SFR at each time step via linear interpolation of the discrete data points of its observed SFH. Note that the observational estimates for SFH used here contain uncertainties, as shown in Figure 1, originating from the model that decodes the SFH of a dSph from its CMD, the so-called CMD model. However, we do not incorporate such uncertainties in the process to obtain Ψ(t) via linear interpolation; therefore, SFH used in our calculation does not include uncertainties of the CMD model.

We then calculate the gas-phase mass in the ISM of the dSph at time t, M_gas(t), as

$$\begin{equation} \frac{M_{\mathrm{gas}}(t)}{10^6\, M_\odot } = \left[\frac{1}{A_\ast }\left(\frac{\Psi (t)}{10^6\, M_\odot \,\mathrm{Gyr}^{-1}}\right)\right]^{1/\alpha }, \end{equation} \tag{ 8 }$$

where A_* and α are the same parameters as in the K11 model. While K11 defined the unit of A_* as 10⁶ M_☉ Gyr⁻¹, we set A_* as the nondimensional parameter. This formulation explains the relation between total mass of interstellar gas M_gas(t) and Ψ(t) (i.e., $\Psi \propto M_\mathrm{gas}^\alpha$, the generalization of a Kennicutt–Schmidt law). While K11 treated α as one of their model parameters and found that its best-fit value varies among their sample of dSphs (see Table 2 in K11), we fix it to unity in our model for simplicity. When α = 1, the parameter A_* takes the same value of the SFE (i.e., =Ψ(t)/M_gas(t)) in units of Gyr⁻¹. We investigate the effect of using a fixed α on other model parameters through a comparison with the K11 model results in Section 4.3.

2.1.2. Type Ia Supernovae

2.1.3. Gas Infall

While the gas infall rate at a certain time t, $\dot{M}_\mathrm{in}(t)$, is represented by $\dot{M}_\mathrm{in} (t) \propto t e^{-t / \tau _\mathrm{in}}$ in K11, where τ_in is one of their model parameters, it is determined in our model so that Equation (1) is satisfied at any time in the calculation. This difference comes from the different assumption for the SFH described in Section 2.1.1. In our model, once the SFH of a certain dSph is given, M_gas(t) at any t is determined via Equation (8) with an adopted parameter of A_*, and hence $\dot{M}_{\mathrm{gas}} (t)$ is also fixed. Moreover, $\dot{\xi }_{i,\mathrm{II}} (t)$ and $\dot{\xi }_{i,\mathrm{AGB}} (t)$ in Equation (3) and $\dot{N}_\mathrm{II} (t)$ in Equation (4) are completely determined because the stellar lifetime and both yields of SNe II and AGBs used in our model are all fixed. The terms $\dot{\xi }_{i,\mathrm{Ia}} (t)$ and $\dot{N}_\mathrm{Ia} (t)$ in these equations also depend on SFH, as well as one of the parameters in our model, t_{delay, min}. Therefore, all variables in Equation (1) except for $\dot{M}_\mathrm{in}(t)$ are determined for a given SFH and set of parameters in our model, that is, A_*, A_out, and t_{delay, min}. In a more qualitative sense, our prescription for gas infall rate means that, while the SN blows out part of the gas from the ISM of the dSph, star formation can take place according to the observed SFH so that the primordial gas flows into the system at the same time.

Nevertheless, the gas infall rate determined in this way would be negative when the time derivative of gas mass, $\dot{M}_{\mathrm{gas}} (t)$, is negative and the outflow rate, $\dot{M}_\mathrm{out} (t)$, is low. Such a negative infall rate acts effectively as outflow. However, this implies that the system would blow out not only enriched gas by SNe but also extra primordial gas simultaneously. This is contrary to one of the assumptions in our model, that the system is chemically homogeneous. To avoid such physically unlike situations, we artificially add an extra outflow rate $\dot{M}_\mathrm{out}^\mathrm{ex}$ to Equation (1) so that the resulting infall rate becomes zero during such a situation. Therefore, the gas infall rate is determined via the following equation rather than Equation (1):

$$\begin{equation} \dot{M}_\mathrm{in}(t) = \dot{M}_\mathrm{gas}(t) + \Psi (t) - E(t) + \dot{M}_\mathrm{out}(t) + \dot{M}^\mathrm{ex}_\mathrm{out}(t). \end{equation} \tag{ 9 }$$

Since the metallicity of outflow gas is assumed to be the same as that of the system at that time in our model, introducing such extra outflow gas solves the physically unlike situation described above. Although the introduction of such an extra outflow rate is completely artificial, it should be stated that external processes to remove gas from dSphs such as tidal stripping can act as this extra outflow rate. This is because the outflow rate in our model is simply assumed to be linearly proportional to the SN rates as represented by Equation (4).

As a consequence of introducing the extra outflow rate, the net outflow rate ($= \dot{M}_\mathrm{out} (t) + \dot{M}_\mathrm{out}^\mathrm{ex} (t)$) is not proportional to the SN rate transiently when $\dot{M}_\mathrm{out}^\mathrm{ex}$ is not equal to zero. In order to evaluate the importance of the extra outflow rate quantitatively, we define F_ex as the total mass ratio of the extra outflow to the net outflow in the entire SFH of a dSph, which can be represented by

$$\begin{equation} F_\mathrm{ex} = \int ^{t_\mathrm{fin}}_0 \dot{M}^\mathrm{ex}_\mathrm{out} (t) dt \left / \right.\int ^{t_\mathrm{fin}}_0 (\dot{M}_\mathrm{out} (t) + \dot{M}_\mathrm{out}^\mathrm{ex} (t))\, dt, \end{equation} \tag{ 10 }$$

where t_fin is the time at which our calculation ends (see Section 2.2 for its detailed definition). If F_ex = 0, there is no extra outflow gas required for the dSph during the model calculation. If 0 < F_ex ⩽ 1, the model needs extra gas outflow during the model calculation.

Although this modification is artificial, the derived best-fit model parameters are consistent within their 1σ uncertainties whether or not this modification is adopted for our sample. The detailed definition of the best-fit model and 1σ uncertainties are described in the next section and Section 3.1, respectively.

In our model, there are three free parameters to fit the observed MDF as follows: SFE (A_*), outflow coefficient (A_out), and minimum delay time for SNe Ia (t_{delay, min}). The best-fit parameters are determined through the following two steps. First, for each dSph in our sample, we calculate the evolution history of iron abundance, ζ_Fe(t) ≡ [Fe/H](t), using the model provided with its SFH. When we calculate the abundance, we adopt the solar composition of Sneden et al. (1992) for Fe: 12 + log (n_Fe/n_H)_☉ = 7.52. The parameter ranges explored are A_* = 10⁻³–10 and A_out = 10²–10⁵ M_☉ SN⁻¹, equally spaced in logarithmic scale with a step of 0.01 dex. For t_{delay, min}, we examine the following two cases: t_{delay, min} = 0.1 Gyr and 0.5 Gyr. Therefore, the total number of the parameter sets is 120,000 for each t_{delay, min}. Then, the best-fit model for each t_{delay, min} is individually determined by using a maximum likelihood estimation, which maximizes the following likelihood L:

$$\begin{eqnarray} L &=& \prod _j \int ^{t_\mathrm{fin}}_0 \frac{1}{\sqrt{2\pi }\ \Delta \zeta _{\mathrm{Fe},j}}\ \exp {\left[-\frac{(\zeta _{\mathrm{Fe},j} - \zeta _\mathrm{Fe}(t))^2}{2(\Delta \zeta _{\mathrm{Fe},j})^2}\right]} \nonumber \\ && \times \ \frac{\dot{M}_*(t)}{M_*}\ dt, \end{eqnarray} \tag{ 11 }$$

where ζ_{Fe, j} = [Fe/H]_j and Δζ_{Fe, j} = Δ[Fe/H]_j are the observed iron abundance and its uncertainty of the jth star in the dSph, respectively. The variable $\dot{M}_\ast (t) / M_\ast$ is evaluated in our model according to the SFH of the dSph, where M_* is the stellar mass of the dSph at present and $\dot{M}_*(t)$ is the mass of the stars that are newly formed in the dSph at time t and survive until the present. We note that, since both M_* and $\dot{M}_\ast (t)$ are completely determined from the input SFH, the factor $\dot{M}_\ast (t) / M_\ast$ does not depend on our model parameters and simply acts as a weight in the likelihood calculation. Therefore, the best-fit parameters can be different among our sample dSphs.

For the observed iron abundance of our sample dSphs, we use the data provided in Kirby et al. (2010), in which the iron abundance [Fe/H] of nearly 3000 stars in eight dSphs, including our sample dSphs, is spectroscopically measured by using Fe i absorption lines. It should be noted that these sample stars are different from those used to derive the SFHs. In general, the area in a dSph spectroscopically observed is different from that photometrically observed, as shown in Table 1; the area observed spectroscopically is typically smaller than that observed photometrically and concentrates on the center of each dSph. However, since we assume that the dSph in our model is chemically homogeneous at any time, we simply assume that the stars sampled in the photometric and spectroscopic observations both well represent the entire population in the dSph. We will discuss the influences of such differences in the observed area on the resultant chemical evolution histories in Section 4.2.

Table 1. Half-light Radii and Observed Area of the dSphs

dSph	Half-light Radius	Observed Area		Reference
		Spectroscopy	Photometry
Fornax	∼16'	∼20' × 20'	∼15 × 10	1, 2, 3
Sculptor	∼11'	∼20' × 20'	∼17 × 17	1, 2, 4
Leo II	∼26	∼20' × 15'	∼13 × 13	1, 2, 5
Sextans	∼28'	∼35' × 20'	∼42' × 28'	1, 2, 6

References. (1) McConnachie 2012; (2) Kirby et al. 2010; (3) de Boer et al. 2012b; (4) de Boer et al. 2012a; (5) Mighell & Rich 1996; (6) Lee et al. 2009.

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In our calculation for chemical evolution, the metallicity of the system can change violently with time. Specifically, such a violent change of metallicity happens just before the end of the SFH, where the mass of interstellar gas becomes very low and comparable with that of outflowing gas. In this condition, a small amount of yield is sufficient to change the metallicity of the interstellar gas, which is supplied via infall of gas with primordial composition. Therefore, we set the following two criteria to determine the condition for the violent change of metallicity:

$$\begin{eqnarray} (1)\;\; && \vert \zeta _\mathrm{Fe}(t - \Delta t) - \zeta _\mathrm{Fe}(t) \vert > 0.3 \quad \mathrm{and} \nonumber \\ && \quad \left[\zeta _\mathrm{Fe}(t - \Delta t) - \zeta _\mathrm{Fe}(t)\right] \left[\zeta _\mathrm{Fe}(t) - \zeta _\mathrm{Fe}(t + \Delta t)\right] < 0, \quad\quad\;\;\; \end{eqnarray} \tag{ 12 }$$

or

$$\begin{eqnarray} (2)\;\; &&\vert \zeta _\mathrm{Fe}(t - \Delta t) - \zeta _\mathrm{Fe}(t) \vert > 0.3 \quad \mathrm{and} \quad t > 5\, \mathrm{Gyr},\hspace{45pt} \end{eqnarray} \tag{ 13 }$$

where Δt is the interval of individual time steps (=25 Myr). If either of these criteria is satisfied, we terminate the calculation and set t_fin at the time, which is used to calculate the likelihood via Equation (11).

We first compare the MDFs and chemical abundance ratios of the best-fit models for our sample dSphs with those of the observed data in Figure 2. For the observed data in both the left and right panels of Figure 2, we plot only the stars with the uncertainties in both metallicity and abundance ratio less than or equal to 0.3 dex, while we do not adopt any cutoff for the uncertainties during the likelihood calculation. We also show the abundance ratios of some of the other elements calculated in our model, that is, [Si/Fe], [Ca/Fe], and [Ti/Fe], in Appendix B.

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It is found that the resultant MDFs of the best-fit models reasonably reproduce the observed data. Specifically, the peak metallicities at which the observed MDFs have the largest frequencies of stars are well fitted by the best-fit models for all of our sample dSphs. Moreover, the tails in the observed MDFs elongated into metal-poor sides from the peak metallicities, which are different among the dSphs, are also reproduced by the best-fit models. On the other hand, although the observed MDF for Sextans shows a broader metal-rich tail with Δ[Fe/H] ∼ 1 dex than those for the other three dSphs of Δ[Fe/H] ∼ 0.5 dex, the model MDFs have similar metal-rich tails with Δ[Fe/H] ∼ 0.5 dex in all dSphs despite the difference in their SFHs. It is also found that there are bumps in the model MDFs with t_{delay, min} = 0.5 Gyr at [Fe/H] ∼ −2.3 in Sculptor and [Fe/H] ∼ −2.5 in Sextans, while such bumps are not evident in the observed MDFs.

The peak metallicities in the MDFs are found to correlate with A_out; the model MDF with smaller A_out has a peak at higher metallicity. It is also found that the metal-poor tail is related to SFH; the dSph forming stars more intensively in the past tends to have a more elongated tail. On the other hand, the significance of the bump seen in the metal-poor tail is found to correlate with the model parameters A_* and t_{delay, min}, as well as the SFH. We discuss the effects of our model parameters of A_out, A_*, and t_{delay, min} on the MDF in detail in Section 4.1.

The best-fit model parameters with their 1σ uncertainties are listed in Table 2 and shown in Figure 3. It is important to note again that these uncertainties do not include the uncertainties in SFHs shown in Figure 1, as described in Section 2.1.1. If the uncertainties in the SFH are included, the derived uncertainties of the best-fit model parameters will be larger.

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Table 2. The Best-fit Model Parameters with Their 1σ Uncertainties, the Maximum Likelihoods, and the Fractions of Extra Outflow Gas Mass

dSph	tdelay, min = 0.1 Gyr				tdelay, min = 0.5 Gyr
	log A*	log [Aout/(M☉ SN−1)]	−log Lmax	Fex (%)	log A*	log [Aout/(M☉ SN−1)]	−log Lmax	Fex (%)
Fornax	$-1.50^{+0.05}_{-0.03}$	$3.26^{+0.02}_{-0.02}$	175	4.4	$-1.01^{+0.04}_{-0.06}$	$3.16^{+0.01}_{-0.02}$	170	0.54
Sculptor	$-1.91^{+0.07}_{-0.06}$	$3.88^{+0.02}_{-0.02}$	252	0.00	$-1.01^{+0.11}_{-0.11}$	$3.72^{+0.02}_{-0.02}$	243	0.00
Leo II	$-2.43^{+0.16}_{-0.11}$	$3.72^{+0.13}_{-0.24}$	110	21	$-1.75^{+0.11}_{-0.13}$	$3.80^{+0.02}_{-0.03}$	113	0.00
Sextans	$-2.05^{+0.13}_{-0.13}$	$4.13^{+0.07}_{-0.09}$	111	0.00	$-1.09^{+0.30}_{-0.26}$	$4.04^{+0.04}_{-0.06}$	108	0.00

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As shown in Figure 3, both of the model parameters A_* and A_out are well determined using the observed MDFs, except those with t_{delay, min} = 0.1 Gyr for Leo II and Sextans, whose contours are apparently elongated. Such elongated contours of constant confidence levels are attributed to large F_ex. Large F_ex implies that the extra gas outflow contributes significantly to the outflow from the system rather than the superwind; therefore, the model results depend weakly on A_out.

For each dSph, the likelihoods of the best-fit models with t_{delay, min} = 0.1 Gyr and 0.5 Gyr are found to be similar to each other, as shown in Table 2. This result implies that the best-fit value of t_{delay, min} for a certain dSph cannot be evaluated by analyzing its MDF alone, while A_out and A_* can be determined by the metallicity at which the MDF has a peak and the significance of the metal-poor tail in the MDF, respectively. However, this difficulty for t_{delay, min} can be solved through a comparison of the model with observed data in the [Mg/Fe]–[Fe/H] plane, as shown in the right panels of Figure 2. It is clearly shown that, for all of the sample dSphs, the evolution tracks of the model with t_{delay, min} = 0.5 Gyr in the plane provide better fits to the observed data than those with t_{delay, min} = 0.1 Gyr. This difference of the evolution track in the [Mg/Fe]–[Fe/H] plane originates from the difference between the SN Ia and SN II yields. Since the SN Ia yields have a smaller abundance ratio of [α/Fe] than the SN II yields, [α/Fe] of the system starts to decrease when SN Ia starts to explode, that is, when the time of t_{delay, min} is elapsed since the onset of star formation.

Through this comparison for our sample dSphs, it is found that t_{delay, min} = 0.1 Gyr is too short to fit the observed distribution of the stars in the [Mg/Fe]–[Fe/H] plane. However, this result is clearly against the observational estimates for the minimum delay time of SNe Ia, which provides t_{delay, min} ≲ 0.1 Gyr almost certainly (Totani et al. 2008; Maoz et al. 2010). This discrepancy between our model prediction and the observational estimate to t_{delay, min} is discussed in Section 4.4.

We show the time evolution of stellar mass M_*(t) and total mass of interstellar gas M_gas(t), as well as the total mass of the system M_*(t) + M_gas(t) calculated from the best-fit models for our sample dSphs in Figure 4. Note that since the observed SFH is adopted as the input in our model, the time evolution of stellar mass is completely independent of model parameters; therefore, both of the shapes and absolute values of M_*(t) for a dSph calculated from the best-fit models with t_{delay, min} = 0.1 Gyr and 0.5 Gyr are identical to each other. Moreover, since gas mass at a given time is assumed to be linearly proportional to the SFR at the time, Ψ(t), according to Equation (8), the time evolution of gas mass for a certain dSph has exactly the same shape as that of its SFR by definition. Therefore, the shape of M_gas(t) is independent of the model parameters, as shown in Figure 4. However, the ratio of M_gas(t)/Ψ(t) is controlled by one of the model parameters A_* in the sense that smaller A_* results in larger gas mass for a given SFR. For the dSphs for which A_* of the best-fit models are very low (i.e., log A_* ≲ −2), that is, Sculptor, Leo II, and Sextans with t_{delay, min} = 0.1 Gyr, gas mass dominates the total mass of the dSph in almost the entire time during the calculation.

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The time evolution of the outflow and infall rates of the best-fit models are shown in Figure 5. It is found that, while the outflow rate evolves smoothly, the shape of the time evolution of the infall rate is like a step function, except for Leo II. This step-function-like shape of the infall rate originates from the prescription for the infall rate in our model; that is, the infall rate is adjusted to satisfy the condition of Equation (9) for a given SFH. Thus, the infall rate can suddenly change at the time when the slope of SFR changes (see Figure 1). In this sense, such step-function-like evolution of the infall rate is a rather artificial result caused by using the observationally estimated SFH with coarse time resolution. On the other hand, since the outflow rate is determined by the number of SNe Ia and SNe II as described in Equation (4) and hence the SFR, the time evolution of the outflow rate looks similar to the SFH and smoother than the infall rate.

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For the best-fit models for Leo II with t_{delay, min} = 0.1 Gyr and for Fornax with both t_{delay, min}, bumps are also seen in the outflow rates at the late stages of evolution, that is, the look-back time of ∼1–6 Gyr for Leo II and ∼2 Gyr and ∼0 Gyr for Fornax. These bumps originate from the contributions of the extra outflow gas masses to avoid the negative infall rate explained in Section 2.1.3. The fractions of extra outflow gas masses F_ex of these models are F_ex = 4.4%, 0.54%, and 21% for the best-fit models for Fornax with t_{delay, min} = 0.1Gyr and 0.5 Gyr and for Leo II with t_{delay, min} = 0.1Gyr, respectively. This artificial prescription for the extra outflow tends to appear in the late stages of evolution, where the increasing rate of metallicity of the system is smaller than that in the early stage, as shown in Figure 6. Therefore, the effect of this extra gas outflow on the MDF is small.

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The gas flow rates of the models with lower A_* are larger since the gas masses are larger. Total outflow gas masses are ∼10–100 times larger than the present stellar masses. It is worth noting that the outflow and infall rates are comparable. This result implies that a large amount of infalling gas, which is comparable to outflowing gas associated with star formation, is inevitably required to reproduce both low average metallicity and relatively long SFH of dSph simultaneously. Specifically, in order to explain the observed low metallicities of dSphs, a large amount of metals is required to be blown away by superwinds. However, since a large amount of gas outflow terminates the star formation in dSphs rapidly, a comparable gas infall is necessary to reproduce relatively long SFHs estimated observationally for dSphs. We will provide a discussion for the chemical evolution of dSphs and comparison with the results of previous works in Section 4.5.

In Section 3.1, we show that the observed MDFs of our sample dSphs are reasonably reproduced by our best-fit models. Here we investigate the detailed dependence of MDF on our model parameters of A_*, A_out, and t_{delay, min}. Considering that the MDF is roughly determined by the SFH and the AMR as

$$\begin{equation} \frac{{dN}_\mathrm{stars}}{d\mathrm{[Fe/H]}} \propto \frac{d\Psi }{dt} \left / \frac{d\mathrm{[Fe/H]}}{dt},\right. \end{equation} \tag{ 14 }$$

and that the SFH is fixed in our model, the MDF will depend only on the AMR. Therefore, through investigating the effects of the model parameters of A_*, A_out, and t_{delay, min} on the AMR, which are shown in Figure 7, we can examine their effects on the MDF. Note that while the illustrated models in Figure 7 are only the cases for Sculptor, the AMRs for the other dSphs show a similar dependence on the model parameters.

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As shown in Figure 7, compared with the AMR of the best-fit model with t_{delay, min} = 0.5 Gyr (black solid curve), the AMR of the model with larger A_out (red dotted curve) results in a slower metal evolution, and its deviation from the best-fit model increases monotonically with time. This is because, for a given SFR, a larger A_out leads to a larger amount of metal elements expelled from the system as a more intense outflow, resulting in a smaller growth rate of metallicity. Since the deviation from the best-fit model seems to be evident throughout the late stages of chemical evolution, A_out affects the mean metallicity of the system; larger A_out leads to smaller mean metallicity. In fact, the best-fit models for our sample dSphs show this trend (see Table 2 and Figure 2). In other words, our model predicts that the observed low metallicities of dSphs are the consequence of intense gas outflows accompanied by a large amount of metal.

On the other hand, the effect of changing t_{delay, min} from the value in the best-fit model of t_{delay, min} = 0.5 Gyr into 0.2 Gyr appears in the early phase of chemical evolution, as shown by the blue dot-dashed curve in Figure 7. Since the SN Ia supplies much abundant Fe to the ISM compared to SNe II and AGBs, metallicity of the system increases rapidly just after the SN Ia starts to explode, that is, the minimum delay time t_{delay, min} is elapsed since the onset of star formation. Therefore, the AMRs with different t_{delay, min} are clearly different in the early stages of chemical evolution. However, the growth rate of metallicity decreases with time, and hence the difference between the models with different t_{delay, min} becomes negligible in the late phases of chemical evolution. Therefore, the effect of t_{delay, min} is important in the early phase of chemical evolution, in particular around t ∼ t_{delay, min}.

The effect of changing A_* appears also in the early phase of chemical evolution, as shown by the green dashed curve in Figure 7. Since a higher A_* means that the system has smaller gas mass for a given SFR according to Equation (8), the interstellar gas of the system is easily enriched by the yields from the evolved stars, which are fixed from the past SFH. Therefore, the model with a higher A_* results in a more rapid chemical enrichment in the early phase. However, the growth rate of metallicity decreases with time similar to the model with changing t_{delay, min}. As a result, changing A_* affects the metal evolution mainly in the early phase.

Specifically, a lower A_* results in a larger number of metal-poor stars, producing a broader metal-poor tail in the MDF. In other words, the dSph whose MDF has a broad metal-poor tail is expected to prefer a low A_*. However, the best-fit parameters of A_* show no relation with the broadness of the metal-poor tails in the MDFs against the expectation from our model; while Sculptor has a broader metal-poor tail than Fornax and Leo II, the best-fit parameter of A_* for Sculptor is similar to that for Fornax and larger than that for Leo II (see Table 2). This contradiction comes from the fact that the effect of A_* on the metal-poor tail in MDF is not primary but secondary. The broadness of the metal-poor tail in MDF is primarily determined by the SFH, or more specifically, the fraction of the stars formed in the early phase of chemical evolution. As shown in Figure 1, the fraction of the stars formed in the early phase is large in Sculptor but small in Fornax and Leo II. When we calculate the chemical evolution for Fornax, Sculptor, and Leo II with the same A_*, the metal-poor tails in the resultant MDFs would be broader in Sculptor and narrower in Fornax and Leo II. In other words, the value of A_* is determined not only by the shape of the metal-poor tail in MDF but also by the shape of the SFH.

Moreover, the similarity between the effects of changing t_{delay, min} and A_* on the AMR leads to the degeneracy between these parameters in our chemical evolution model. As shown in Figure 2 and Table 2, the observed MDFs can be reproduced regardless of t_{delay, min} since the model MDF with short t_{delay, min} and low A_* is similar to the model with long t_{delay, min} and high A_*. This degeneracy can be resolved by comparing the model results with the observed [Mg/Fe]–[Fe/H] diagrams simultaneously since different t_{delay, min} results in different abundance ratios of α elements as described in Section 3.2. Therefore, both the observed MDF and abundance ratio are required to determine t_{delay, min} and A_* accurately.

In summary, it is difficult to understand the chemical evolution of dSphs only by analyzing the MDF and abundance ratio. It is important to examine the MDF, abundance ratio, and SFH simultaneously for investigating the chemical evolution of dSphs.

As already stated in Section 2.2, the observed area for a dSph in which its MDF is evaluated is typically different from that to obtain SFH. In our model, the difference in the observed area is completely neglected since dSph is assumed to be chemically homogeneous. However, photometric observations for dSphs have reported that some dSphs have a radial gradient with age and metallicity; young stars are more concentrated than old stars, and metal-rich stars are more concentrated than metal-poor stars (de Boer et al. 2012a, 2012b; Lee et al. 2009). Therefore, the dSphs are not chemically homogeneous, and the difference in the observed area for the SFH and MDF may affect our results.

In order to investigate the effect of this difference on the model results, we calculate the chemical evolution for Fornax and Sculptor by using their SFHs and MDFs evaluated from the stars in the same area, that is, the central region (r_ell < 02) obtained by de Boer et al. (2012a, 2012b), where r_ell is the elliptical radius of the dSphs used in de Boer et al. (2012a, 2012b). As shown in Figures 8 and 9, while the number of metal-poor stars in the observed MDFs is slightly underestimated, our model can reproduce the observed MDFs for Fornax and Sculptor with almost the same parameters, except A_out for Fornax.⁴ A large amount of extra gas outflow (i.e., F_ex > 0.8) in the past several gigayears.

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We also calculate the chemical evolution for Fornax by using the MDF evaluated from the 562 stars in a wide area (∼10 × 15) of Fornax (Battaglia et al. 2006), which is comparable to the observed area for SFH by de Boer et al. (2012b). Although the observed MDF in the wide area shows a larger number of metal-poor stars than the MDF shown in Figure 2, our model cannot reproduce it as shown in Figure 10. Moreover, a significant amount of extra outflow (i.e., F_ex > 0.8) is required for the past ∼8 Gyr, as shown in the bottom panels of Figure 10.

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The results of the above comparison indicate that our model for chemical evolution cannot explain both the SFH and MDF simultaneously in the central and/or the whole regions of Fornax. This difficulty can be relaxed if the SFE (i.e., A_* with the units of Gyr⁻¹) is adopted to be time variant, which is suggested from observational studies (e.g., Combes et al. 2013). Since the SFE controls the growth rate of metallicity in the early phase as described in Section 4.1, the model with a time-variable SFE can form a larger number of metal-poor stars.

A multizone model is also expected to resolve the difficulty in our model for the number of metal-poor stars. Some studies presented a signature of multicomponent structure in dSphs. It is reasonable that the metal-poor population in a dSph has experienced different chemical evolution from the metal-rich population. Hence, it is important to gather many observational results of dSphs and construct a more realistic multizone chemical evolution model.

As described in Section 2.1, our model is basically the same as that of K11. The best-fit parameters in K11 for our sample dSphs are listed in Table 3. Since the unit of A_* in our model is different from that in the K11 model ($A_\ast ^\mathrm{K11}$), we show converted values of A_* from $A_\ast ^\mathrm{K11}$ in Table 3 by $A_\ast = A_\ast ^\mathrm{K11} / (10^6\, M_\odot \, \mathrm{Gyr}^{-1}$). Then we can directly compare the best-fit parameters derived from the K11 model with ours.

Table 3. The Best-fit Model Parameters with t_{delay, min} = 0.1 Gyr of K11, Our Model, and Comparison Models for Our Sample dSphs

dSph	Parameter	K11	Comparison Model				Our Model
			Case 1	Case 2	Case 3	Case 4
	α	0.98$^{+0.15}_{-0.04}$	0.98 (fixed)	0.98 (fixed)	1.0 (fixed)	1.0 (fixed)	1.0 (fixed)
Fornax	log A*	0.70$^{+0.16}_{-0.10}$	1.14$^{+0.06}_{-0.12}$	1.14$^{+0.11}_{-0.18}$	1.14$^{+0.12}_{-0.19}$	−1.30$^{+0.04}_{-0.04}$	−1.50$^{+0.05}_{-0.03}$
	log [Aout/(M☉ SN−1)]	3.18$^{+0.01}_{-0.02}$	3.18 (fixed)	3.16$^{+0.01}_{-0.01}$	3.16$^{+0.01}_{-0.01}$	3.32$^{+0.01}_{-0.02}$	3.25$^{+0.02}_{-0.02}$
	α	0.83$^{+0.14}_{-0.08}$	0.83 (fixed)	0.83 (fixed)	1.0 (fixed)	1.0 (fixed)	1.0 (fixed)
Sculptor	log A*	−0.33$^{+0.08}_{-0.13}$	−0.32$^{+0.02}_{-0.02}$	−0.30$^{+0.05}_{-0.04}$	−0.41$^{+0.06}_{-0.06}$	−2.21$^{+0.02}_{-0.03}$	−1.91$^{+0.07}_{-0.06}$
	log [Aout/(M☉ SN−1)]	3.73$^{+0.01}_{-0.01}$	3.73 (fixed)	3.70$^{+0.02}_{-0.03}$	3.68$^{+0.02}_{-0.03}$	2.95$^{+0.45}_{-0.95}$	3.88$^{+0.02}_{-0.02}$
	α	0.66$^{+0.17}_{-0.40}$	0.66 (fixed)	0.66 (fixed)	1.0 (fixed)	1.0 (fixed)	1.0 (fixed)
Leo II	log A*	−0.37$^{+0.50}_{-0.11}$	−0.26$^{+0.02}_{-0.03}$	−0.25$^{+0.06}_{-0.06}$	−0.48$^{+0.12}_{-0.10}$	−2.08$^{+0.07}_{-0.07}$	−2.43$^{+0.16}_{-0.11}$
	log [Aout/(M☉ SN−1)]	3.82$^{+0.02}_{-0.02}$	3.82 (fixed)	3.80$^{+0.02}_{-0.02}$	3.78$^{+0.03}_{-0.02}$	3.91$^{+0.03}_{-0.03}$	3.72$^{+0.13}_{-0.24}$
	α	0.50$^{+0.20}_{-0.25}$	0.50 (fixed)	0.50 (fixed)	1.0 (fixed)	1.0 (fixed)	1.0 (fixed)
Sextans	log A*	−0.28$^{+0.27}_{-0.18}$	−0.29$^{+0.03}_{-0.03}$	−0.30$^{+0.08}_{-0.06}$	−0.74$^{+0.14}_{-0.13}$	−2.47$^{+0.14}_{-0.12}$	−2.05$^{+0.13}_{-0.13}$
	log [Aout/(M☉ SN−1)]	3.98$^{+0.04}_{-0.03}$	3.98 (fixed)	3.91$^{+0.09}_{-0.11}$	3.91$^{+0.07}_{-0.11}$	3.97$^{+0.15}_{-0.64}$	4.13$^{+0.07}_{-0.09}$

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It is found that while the values of A_out in K11 are similar to those in our best-fit models, A_* of our best-fit models with t_{delay, min} = 0.1 Gyr are much smaller than those of K11. To investigate the origin of this difference, we explore the best-fit models for the following four cases: (Case 1) the model with the fixed SFH, α, and A_out that are the same as those in K11 and with only one adjustable parameter of A_*; (Case 2) the model with the same SFH and α as those in K11 and with adjustable parameters of A_* and A_out; (Case 3) the model with the same SFH as that in K11 and with three adjustable parameters α (fixed to unity), A_*, and A_out; and (Case 4) the model with the same shape of the SFH as K11, but its timescale is extended to fit the timescale of our SFH. The resultant best-fit parameters for the four cases are listed in Table 3.

In the Case 1 model, in which only one parameter of A_* can be used to fit the observed MDF, the best-fit values of A_* for our sample dSphs are similar to those in K11, as shown in Table 3. This result indicates that, if the same SFH and parameters are adopted, the resultant MDFs from the K11 and our models are similar to each other. Although the best-fit A_* is significantly different from that in K11 for Fornax, the discrepancy of A_* between the K11 and our model is attributed to the different prescription for gas infall rate. Since the infall rate at t = 0 is adopted to be exactly equal to zero in the K11 model, high A_* results in rapid consumption of interstellar gas, terminating star formation in a very short timescale. Since the best-fit SFH of Fornax in the K11 model decreases rapidly in the first short epoch (i.e., ≲ 0.1 Gyr), A_* is considered to be a maximum in their model. On the other hand, since the infall rate in our model is adjusted to keep the interstellar gas mass satisfying the required condition of Equation (8) for a given SFR, A_* can be explored to fit the observed MDF.

The best-fit parameters of A_* and A_out in the Case 2 model are similar to those in the Case 1 model. In the Case 3 model, in which α is changed from the best-fit values in K11 to unity, the best-fit parameters of A_out are similar to those of Case 2, but those of A_* become lower. In other words, A_* and α are degenerate.

The best-fit values of A_* change dramatically in the Case 4 model, in which the SFH timescales are modified into those used in our model of ∼10 Gyr; A_* decreases more than one order of magnitude, while A_out is similar to that in the Case 3 model, except for Sculptor.⁵ The significant decrease of A_* can be interpreted as follows. In order to reproduce a given MDF, a longer star formation timescale requires a slower chemical evolution, that is, a lower value of A_*. As a consequence, the Case 4 model provides a set of the best-fit parameters that is quite similar to that of our model, as shown in Table 3. The tiny differences between the two sets of the best-fit parameters are caused by differences in the shapes of SFH.

Therefore, we can conclude that the difference in the best-fit parameters of A_* between the K11 and our model originates from the difference in the star formation timescale for dSph. Furthermore, although Fenner et al. (2006) made their model fitting for the observed chemical abundance of Sculptor and found that the [α/Fe] is degenerated between the duration of the SFH and the SN feedback, our model fitting for the observed MDFs shows no degeneracy between the duration of the SFHs and the M_out (see Cases 3 and 4 in Table 3). We emphasize that, since the parameters in the chemical evolution model are sensitive not only to the MDF but also to the SFH, it is important to fit the MDF and SFH simultaneously in order to investigate the chemical evolution of dSphs.

In Section 3.2, we show that our model disfavors the minimum delay time of t_{delay, min} ∼ 0.1 Gyr since it is too short to reproduce the observed distribution of the stars in our sample dSphs in the [Mg/Fe]–[Fe/H] plane (see right panels of Figure 2). This is against the recent observational and/or theoretical estimates for t_{delay, min}, which show that t_{delay, min} ≲ 0.1 Gyr (e.g., Totani et al. 2008; Cooper et al. 2009; Maoz & Mannucci 2012; Kistler et al. 2013). While the K11 model with t_{delay, min}  =  0.1 Gyr successfully reproduces the observed distributions in the [Mg/Fe]–[Fe/H] plane, it results in too short star formation timescales compared to those estimated observationally for the dSphs. Therefore, to explain both the SFH and the chemical abundance of dwarf galaxies simultaneously, a longer t_{delay, min} than the observationally favorable value of ∼0.1 Gyr seems to be required inevitably. What are the origins of this discrepancy in t_{delay, min} between our chemical evolution model and the observational and theoretical estimates?

One of the origins of this discrepancy can be the dependence of SN Ia rate on metallicity. Taking account of the suggestion from some theoretical works that the SN Ia rate in metal-poor galaxies is smaller compared to metal-rich galaxies (Kobayashi et al. 1998; Kobayashi & Nomoto 2009; Meng et al. 2011), we consider that the effect of the SN Ia yields on the abundance ratio of [Mg/Fe] will be smaller in the early metal-poor stages of chemical evolution than in the late metal-rich stages. Incorporating such a metal-dependent SN Ia rate into our model with short t_{delay, min} will result in the same abundance ratios as the model with long t_{delay, min} because, in addition to the onset time of the first SN Ia, extra metal enrichment time is required. Therefore, the discrepancy can be relaxed if such a metallicity dependence of the SN Ia rate is taken into account. However, the opposite dependence on metallicity has also been suggested (e.g., Cooper et al. 2009; Kistler et al. 2013). Therefore, it is still unclear how t_{delay, min} depends on the metallicity of the system.

Another possible origin of this discrepancy can be the one-zone approximation adopted in our model; it is an apparently oversimplified approximation for the dSphs because the age and metallicity gradients have been observed in them as described in Section 4.2. If we adopt a multizone approximation instead of the one-zone one, the discrepancy can be relaxed. While the one-zone model assumes that the system is chemically homogeneous, multizone models probably can describe these complicated structures. In the multizone approximation, zones can have different SFHs from each other, and the ejecta of SNe II and SNe Ia can flow from one zone to another. If a zone has a short (<0.1 Gyr) star formation event and its star formation ceases before the SN Ia ejecta from nearby zones flows into the zone, the chemical abundance of stars formed there reflects only the SN II ejecta. In other words, the chemical evolution model with a multizone approximation can increase the metallicity of the stars without the effect of SNe Ia.

The assumption of the time-independent IMF is also one of the origins of the discrepancy, since some theoretical works predicted a different IMF when the metallicity of the system is low. Tumlinson (2006) suggested that low-mass star formation was inhibited when the metallicity of the system was below a critical metallicity (i.e., Z_cr ≲ 10⁻⁴ Z_☉). If the system forms only massive stars when the metallicity of the ISM is low, the ISM is chemically enriched without the contribution of SNe Ia. Therefore, the metallicity-dependent IMF can relax the discrepancy about t_{delay, min}.

Some previous studies argue that dSphs have a bottom-heavy IMF and/or lack very massive (⩾25 M_☉) stars (Tsujimoto 2011; Li et al. 2013; Weidner et al. 2013). Li et al. (2013) analyzed the chemical abundance of the stars in Fornax dSph and concluded that a bottom-heavy IMF is necessary to explain the low [α/Fe]. However, the low [α/Fe] is also explained by the large contribution of the SNe Ia (Ikuta & Arimoto 2002; Lanfranchi & Matteucci 2004; Tolstoy et al. 2009; K11), and a bottom-heavy IMF cannot explain the high [α/Fe] (e.g., [Mg/Fe] > 0.5), such as observed in our samples. To confirm the effect of the bottom-heavy IMF on our model results, we calculate the chemical evolution of the samples by changing the stellar mass upper limit from 100 M_☉ to 25 M_☉. However, we have found that the model MDFs have no changes and the model [Mg/Fe] decrease with decreasing mass upper limit at the same [Fe/H] about 0.2 dex. Therefore, the model with a bottom-heavy IMF and t_{delay, min} = 0.1 Gyr results in worse fitting to the observed data.

4.5.1. The Star Formation Efficiencies of Dwarf Galaxies

As shown in Table 2, the best-fit values of the SFE (A_*) for our sample dSphs are 10^−2.4 to 10^−1.0 Gyr⁻¹. Here we compare these model results with the observational estimates for the SFEs of dwarf galaxies. Note that the SFE derived in our model is constant during the entire history of the galaxy, while the SFE estimated from a star-forming dwarf galaxy reveals its current value.

Figure 11 shows such a comparison of the SFE as a function of the stellar mass between our model results and the observational estimates. In this plot, the observational estimates for the SFEs of the star-forming dwarf galaxies are shown instead of dSphs since the star-forming dwarf galaxies have enough interstellar gas and SFR to be detected by observations, which are inevitable to evaluate current SFE. From 48 Im galaxies whose H i gas masses and Hα luminosities are obtained by McCall et al. (2012) and Kennicutt et al. (2008), respectively, the mean SFE is evaluated as 〈log [SFE/Gyr⁻¹]〉 = −1.4 ± 0.7, which is consistent with our best-fit SFEs. The mean SFEs of Im and Sm galaxies in Hunter & Elmegreen (2004) are 〈log [SFE/Gyr⁻¹]〉 = −1.4 ± 0.5 and −1.1 ± 0.5, respectively, which are also comparable to those of our best-fit models. The mean SFE of blue compact dwarfs (BCDs), 〈log [SFE/Gyr⁻¹]〉 = −0.6 ± 0.5, is found to be higher by almost 1 mag than our best-fit SFEs.

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The result of this comparison of the SFE is consistent with the observed result that the mean SFHs of the different morphological types of dwarf galaxies, including dSph and dIrr, are indistinguishable from each other over most of cosmic time (Weisz et al. 2011). Therefore, the dIrrs may experience the same chemical evolution histories as the dSphs essentially. It is indicated that if dIrrs exhaust all their interstellar gas, they can evolve into dSphs (Kormendy 1985; Peeples et al. 2008; Zahid et al. 2012).

4.5.2. The Gas Outflow from dSphs