THE ORIGIN OF DUST EXTINCTION CURVES WITH OR WITHOUT THE 2175 Å BUMP IN GALAXIES: THE CASE OF THE MAGELLANIC CLOUDS

We adopt the following two steps to derive the dust extinction curves of the MCs. First, we investigate the time evolution of dust grains by using one-zone chemical evolution models of the MCs that have been developed by BT12. Second, we derive the extinction curves based on the abundances of dust grains by using the "two-size approximation model" developed by Hirashita (2015; H15). In this two-size approximation, the whole grain population is represented by small grains (radius a less than 0.03 μm) and large grains (a > 0.03 μm) because the major grain processing mechanisms work differently on small and large grains (H15). The original purpose of this two-size approximation method is to use it in statistical (theoretical) predictions and Nbody+hydrodynamical simulations of galaxy formation and evolution and thereby to dramatically reduce the total amount of calculation time for estimating the dust extinction curve for each galaxy and each particle of a simulated galaxy. We use the method in the present study because we need to run numerous models and the method enables us to complete such calculations of extinction curves much faster.

In the first step, each element (e.g., C, O, and Si) is divided into gas-phase metals and dust, and dust is further divided into "small" and "large" grains. We accordingly investigate the time evolution of the three components, gas-phase metals and small and large grains, separately for each element (e.g., C, O, and Si) in a model. The time evolution of small and large grains can be used for the derivation of the extinction curve in the second step. As demonstrated by H15, the two-size approximation for a dust size distribution is very useful and convenient, because a full calculation of the dust size distribution in a galaxy is complicated and time-consuming (e.g., Asano et al. 2013). Accordingly, we adopt the revised version of the two-size approximation model (H15) in order to derive the extinction curve of a galaxy in the second step. In the revised version of the two-size approximation model, the dust in the ISM of a galaxy is divided into four grains: small and large silicate (referred to as small and large "Si-grains," respectively) and small and large carbonaceous grains (small and large "C-grains," respectively). This two-size, two-component dust grain model enables us to derive the extinction curve of a galaxy based on the chemical evolution of the galaxy, as described later.

SFHs of galaxies can control their chemical evolution histories, because chemical evolution depends on the time evolution of metals ejected from SNe and AGB stars, the formation rates of which depend on star formation rates (SFRs). Therefore, the dust evolution histories of galaxies, which depend on their chemical evolution histories, can be controlled by their SFHs. Since we mainly discuss the extinction curves of the MCs, we need to choose carefully the model parameters for star formation of the MCs. Many observations suggest that the MCs have experienced at least a few starburst (SB) events in the past 6 Gyr (e.g., Harris & Zaritsky 2006, 2009; Rubele et al. 2012, 2015). Furthermore, previous and recent numerical simulations of the MCs have demonstrated that highly enhanced star formation is inevitable during the past LMC–SMC-Galaxy tidal interaction (e.g., Yoshizawa & Noguchi 2003; Bekki & Chiba 2005; Yozin & Bekki 2014). We consider these observational and theoretical results in constructing the SFHs of the MCs.

Since the importance and possibility of dust removal from galaxies through energetic radiation-driven dust wind in one-zone chemical evolution models have been discussed extensively by Bekki & Tsujimoto (2014, BT14), we do not discuss this issue in detail in this paper. The present model is more sophisticated than BT14 in the sense that it investigates the evolution of gas-phase metals and the formation and evolution of different dust grains in a self-consistent manner. Furthermore, the present model newly includes the accretion of gas-phase metals onto already existing dust grains, dependence on the cold gas fraction in the ISM, grain growth by coagulation, and grain disruption by shattering. Thanks to this, we can provide more reliable predictions of the time evolution of dust grains in a galaxy.

We adopt a revised version of our one-zone chemical evolution models used in BT12 in order to investigate the time evolution of gas, metals, and dust in a fully self-consistent manner. Although there are only two basic equations in BT12, there are four in the present study, which reflects the fact that the revised version includes the evolution of small and large dust grains. We investigate the time evolution of the total masses of gas (M_g) and that of the total masses of metal (M_z) and dust (M_d) for each element (e.g., C and O) in a model. Since the total metal mass includes gas-phase metals and dust, the total mass of gas-phase metals that is not depleted onto dust grains (M_z,g) can be estimated as ${M}_{{\rm{z}}}-{M}_{{\rm{d}}}$ in the present study. The total gas mass (M_g,t) accreted onto a galaxy is set to be 1.0 (i.e., normalized) in all models of the present study. The basic equations for the time evolution of gas and metals in the adopted one-zone chemical evolution models with dust physics are described as follows:

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{{\rm{g}}}}{{dt}}=-{\alpha }_{{\rm{l}}}\psi (t)+A(t)-w(t)-{w}_{{\rm{d}}}(t)\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\displaystyle \frac{{{dM}}_{{\rm{z}},i}}{{dt}} & = & -{\alpha }_{{\rm{l}}}{Z}_{i}(t)\psi (t)+{Z}_{A,i}(t)A(t)+{y}_{\mathrm{II},i}\psi (t)\\ & & +{y}_{\mathrm{Ia},i}{\displaystyle \int }_{0}^{t}\psi (t-{t}_{\mathrm{Ia}})g({t}_{\mathrm{Ia}}){{dt}}_{\mathrm{Ia}}\\ & & +{\displaystyle \int }_{0}^{t}{y}_{\mathrm{agb},i}({m}_{\mathrm{agb}})\psi (t-{t}_{\mathrm{agb}})h({t}_{\mathrm{agb}}){{dt}}_{\mathrm{agb}}\\ & & -{W}_{i}(t)-{W}_{{\rm{d}},i}(t),\end{eqnarray} \tag{ 2 }$$

where α_l is the mass fraction locked up in dead stellar remnants and long-lived stars, ψ(t) is the SFR, Z_i is the metallicity for ith element, y_II,i, y_Ia,i, and y_agb,i are the chemical yields for the ith element from type II supernovae (SNe II), from Type Ia supernovae (SNe Ia), and from 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 that between star formation and the onset of the AGB phase, respectively. The terms g(t_Ia) and h_agb are the distribution functions of SNe Ia and AGB stars, respectively, and 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 depend on the original masses of the AGB stars (e.g., Weidemann 2000). Therefore, the term h_agb depends on the adopted IMF and the relation between the mass and the lifetime in stars (later described). The term W_d,i is the dust wind rate for each element (i.e., w_d is the sum of W_d,i for all elements).

The basic equations for the time evolution of the total masses of small (M_d,s) and large (M_d,l) grains in the models are described as follows.

$$\begin{eqnarray}\displaystyle \frac{{{dM}}_{{\rm{d}},{\rm{s}},i}}{{dt}} & = & -{\alpha }_{{\rm{l}}}{D}_{{\rm{s}},i}\psi (t)+{D}_{{\rm{s}},i}(0)A(t)\\ & & -\displaystyle \frac{{M}_{{\rm{d}},{\rm{s}},i}}{{\tau }_{\mathrm{SN},{\rm{s}},i}}+\displaystyle \frac{{M}_{{\rm{d}},{\rm{l}},i}}{{\tau }_{\mathrm{sh},i}}-\displaystyle \frac{{M}_{{\rm{d}},{\rm{s}},i}}{{\tau }_{\mathrm{co},i}}+\displaystyle \frac{{M}_{{\rm{d}},{\rm{s}},i}}{{\tau }_{\mathrm{acc},{\rm{s}},i}}-{W}_{{\rm{d}},{\rm{s}},{\rm{i}}}(t)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}\displaystyle \frac{{{dM}}_{{\rm{d}},{\rm{l}},i}}{{dt}} & = & -{\alpha }_{{\rm{l}}}{D}_{{\rm{l}},i}\psi (t)+{D}_{{\rm{l}},i}(0)A(t)+{f}_{{\rm{c}},i}{E}_{{\rm{z}},i}\\ & & -\displaystyle \frac{{M}_{{\rm{d}},{\rm{l}},i}}{{\tau }_{\mathrm{SN},{\rm{l}},i}}-\displaystyle \frac{{M}_{{\rm{d}},{\rm{l}},i}}{{\tau }_{\mathrm{sh},i}}+\displaystyle \frac{{M}_{{\rm{d}},{\rm{s}},i}}{{\tau }_{\mathrm{co},i}}+\displaystyle \frac{{M}_{{\rm{d}},{\rm{l}},i}}{{\tau }_{\mathrm{acc},{\rm{l}},i}}-{W}_{{\rm{d}},{\rm{l}},i}(t)\end{eqnarray} \tag{ 4 }$$

where M_d,s,i and M_d,l,i are the masses of small and large grains for the ith element, respectively, D_s,i and D_l,i are the dust-to-gas ratios for small and large grains, respectively, D_s,i(0) and D_l,i(0) are the initial dust-to-gas ratios for small and large grains (in infalling gas), respectively, f_c,i and E_z,i are the condensation efficiency of dust and the ejection rate of a metal element, respectively, τ_SN,s,i and τ_SN,l,i are the timescales of dust destruction by SNe for small and large grains, respectively, τ_sh,i is the timescale for the shattering of large grains, τ_co,i is the timescale for coagulation of small grains, and W_d,s,i and ${W}_{{\rm{d}},{\rm{l}},i}$ are the dust wind rates for small and large grains, respectively. The dust wind term W_d,i in Equation (2) is therefore equivalent to ${W}_{{\rm{d}},{\rm{s}}i}+{W}_{{\rm{d}},{\rm{l}},i}$.

The ejection rate of metals from SNe and AGB stars (E_z,i) is described as follows.

$$\begin{eqnarray}{E}_{{\rm{z}},i} & = & {y}_{\mathrm{II},i}\psi (t)+{y}_{\mathrm{Ia},i}{\displaystyle \int }_{0}^{t}\psi (t-{t}_{\mathrm{Ia}})g({t}_{\mathrm{Ia}}){{dt}}_{\mathrm{Ia}}\\ & & +{\displaystyle \int }_{0}^{t}{y}_{\mathrm{agb},i}({m}_{\mathrm{agb}})\psi (t-{t}_{\mathrm{agb}})h({t}_{\mathrm{agb}}){{dt}}_{\mathrm{agb}}.\end{eqnarray} \tag{ 5 }$$

The formation rate of dust grains from SNe and AGB ejecta at each time step is estimated from E_z,i and f_c,i. It is assumed here that the dust produced by stars is in the large-grain regime (H15). It should also be stressed that the dust condensation efficiency for each ith element is different between SN Ia, SN II, and AGB ejecta and this difference is properly included in the present one-zone models: the way to describe the dust production term in Equation (4) is not very accurate, and such a description is adopted only for convenience. In the implementation of the dust production process, the dust production term is calculated in our code as follows.

$$\begin{eqnarray}{f}_{{\rm{c}},i}{E}_{{\rm{z}},i} & = & {f}_{{\rm{c}},\mathrm{SN}\;\mathrm{Ia},i}{E}_{{\rm{z}},\mathrm{SN}\;\mathrm{Ia},i}\\ & & +{f}_{{\rm{c}},\mathrm{SN}\;\mathrm{II},i}{E}_{{\rm{z}},\mathrm{SN}\;\mathrm{II},i}+{f}_{{\rm{c}},\mathrm{AGB},i}{E}_{{\rm{z}},\mathrm{AGB},i},\end{eqnarray} \tag{ 6 }$$

where f_{c,SN Ia,i} f_{c,SN II,i}, and f_c,AGB,i are the dust condensation efficiencies for the ith element and E_{z,SN Ia,i} E_{z,SN II,i}, and E_z,AGB,i are the metal ejection rates for the ith element.

It should be noted here that the way to describe the evolution of gas, metals, and dust in the above equations is slightly different from those adopted in BT12 and H15. For example, we investigate M_g and M_z instead of the gas mass fraction (f_g in BT12) and metallicity (Z_i) in the present study. Also, the timescales for dust accretion, coagulation, and shattering are defined for each element in the present study: they are assumed to be constant for all elements in H15. The dust wind for small and large grains is separately considered, for the first time, in the present study. The details of the models for star formation, chemical evolution, and dust physics in the ISM and the model parameters are described in the following subsections.

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

$$\begin{eqnarray}&&\psi (t)={C}_{\mathrm{sf}}{M}_{{\rm{g}}}(t).\end{eqnarray} \tag{ 7 }$$

The normalization factor C_sf is chosen appropriately so that the final gas mass fraction (and metallicity) can be consistent with the observed one. We assume that C_sf is different between (1) the "quiescent phase," when a galaxy shows an almost steady-star formation and (2) the "SB phase," when the galaxy experiences a bursty star formation. Many observational studies showed that the LMC and the SMC have experienced a few SB events in the past 6 Gyr (e.g., Harris & Zaritsky 2006, 2009; Rubele et al. 2012). Guided by these observations, we investigate models in which an SB can occur three times ("SB models"), and the first, second, and third SBs are denoted as SB1, SB2, and SB3, respectively. The star formation rate is assumed to be constantly higher for t_sb1,s ≤ t ≤ t_sb1,e in SB1, t_sb2,s ≤ t ≤ t_sb2,e in SB2, and t_sb3,s ≤ t ≤ t_sb3,e in SB3. The star formation coefficient (C_sf) is thus described as follows.

$$\begin{eqnarray}{C}_{\mathrm{sf}}=\{\begin{array}{ll}{C}_{{\rm{q}}} & {\rm{for}}\;{\rm{quiescent}}\;{\rm{phase}}\\ {C}_{\mathrm{sb}1} & {\rm{for}}\;{\rm{SB1}}\\ {C}_{\mathrm{sb}2} & {\rm{for}}\;{\rm{SB2}}\\ {C}_{\mathrm{sb}3} & {\rm{for}}\;{\rm{SB3}}.\end{array}\end{eqnarray} \tag{ 8 }$$

In the present study, C_sf is given for each model (e.g., SMC, LMC, and MW).

We adopt the following formula for the rate of gas accretion onto the disk of a galaxy.

$$\begin{eqnarray}&&A(t)={C}_{{\rm{a}}}\mathrm{exp}(-t/{t}_{{\rm{a}}}),\end{eqnarray} \tag{ 9 }$$

where ${t}_{{\rm{a}}}$ is a free parameter controlling the timescale of the gas accretion and C_a is the normalization factor and determined such that the total gas mass (M_g,t) accreted onto the galaxy can be 1 for a given t_a. This M_g,t is determined such that M_g,t is the total baryonic mass of a (present) galaxy (i.e., gas and stars, including gas, metals, and dust ejected from the galaxy through outflow). The parameter t_a can therefore control the time evolution of the total gas mass and the gas mass fraction of a galaxy. We mainly show the results of the models with t_a = 5 Gyr, though we investigated models with different t_a. The main purpose is to discuss how dust wind plays a key role in the evolution of extinction curves of galaxies in the present study. Such roles of dust wind do not depend on the adopted t_a.

Although we show only the results of the models with t_a = 5 Gyr, the final (i.e., present) gas masses and gas mass fractions of the three galaxies are roughly consistent with the observed values. For example, if we adopt M_g = 3 × 10⁹ M_⊙ (BT12) for the LMC, then the final gas mass is 1.1 × 10⁹ M_⊙ and the gas fraction is 0.37. These numbers are roughly consistent with the observed values (e.g., van den Bergh 2000), suggesting that the adopted t_a is reasonable. Furthermore, the typical star formation rate at non-SB phases in the LMC models is 0.15 M_⊙ yr⁻¹, which is similar to the observed present SFR of the LMC (0.26 M_⊙ yr⁻¹; Kennicutt et al. 1995). The final gas mass fractions of the SMC and the MW are 0.53 and 0.05 in the present models, respectively, which are also roughly consistent with the observed values (e.g., van den Bergh 2000). The evolution of SFRs and gas mass fractions are given in Figure 1 and later discussed.

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The assumption of slow gas accretion onto the SMC implies that the SMC has a disk component. Almost all previous theoretical models of the Magellanic Stream (MS) formation needed to assume the gas and stellar disks (e.g., Gardiner & Noguchi 1995; Diaz & Bekki 2012) in order to reproduce the observed properties of the MS. Furthermore, recent observations have confirmed that the SMC has a stellar disk (e.g., Dobbie et al. 2014), and it is a well known fact that the SMC has a rotating HI disk (e.g., Stanimirovic et al. 2004). Therefore, it is reasonable for the present study to assume that the SMC has a disk onto which gas can be accreted.

We assume that some fractions of metals from SNe can be removed from a galaxy thorough energetic outflows so that they do not contribute to the chemical enrichment of the galaxy. In the present study, only SNe ejecta can be expelled from a galaxy ("selective wind model"). We do not consider that the ejection efficiency depends on (1) chemical elements (e.g., C and O) and (2) whether the ejecta is from SN Ia or SN II just for simplicity in the present study. Some fraction, $1-{f}_{\mathrm{ej}}$, of gaseous ejecta from SNe can be mixed with the ISM for chemical enrichment processes whereas all AGB ejecta can be mixed with the ISM in the selective wind models.

The SN wind rate (w(t)) is estimated only from the total mass of gaseous ejecta from SNe (M_ej,sn) at each time step. The total mass, M_w, of the ISM that is removed from a galaxy at each time step is as follows.

$$\begin{eqnarray}&&{M}_{{\rm{w}}}={f}_{\mathrm{ej}}{M}_{\mathrm{ej},\mathrm{sn}}.\end{eqnarray} \tag{ 10 }$$

Therefore, the SN wind rate at each time step is simply

$$\begin{eqnarray}&&w(t)=\displaystyle \frac{{{dM}}_{{\rm{w}}}}{{dt}}.\end{eqnarray} \tag{ 11 }$$

We show the results of the models with f_ej = 0.4 in the present study, because BT12 have already investigated the SN wind models with f_ej = 0.2 and 0.4 and confirmed that such models can better reproduce the observed properties of chemical abundances of the LMC (e.g., Figure 16 of BT12). We also confirm that the roles of dust wind in the evolution of extinction curves do not depend strongly on f_ej, though the models with lower f_ej end up with higher gaseous metallicities.

Multiple SN events can strongly influence the chemical and dynamical evolution of star-forming regions, and one of the most dramatical influences is the formation of super shells and bubbles in the H i disks of the LMC and the SMC (e.g., Kim et al. 1999; Stanimirovic et al. 1999). Such SN events can cause more efficient dust destruction, shattering, and coagulation in shells and bubbles of MCs, which implies that the models of dust evolution would need to include such dust-related physical processes depending on the time evolution of SN rates. In the present study, we do not intend to model such dust-related physical processes dependent on SN rates, mainly because such complicated models require a few additional model parameters and thus would possibly prevent us from clarifying the key parameters for the evolution of dust extinction curves. Thus it would be our future study to include the dust-related physical processes dependent on SN rates of galaxies in dust evolution models.

We assume that there is a time delay (t_Ia) between the star formation and the metal ejection from SNe Ia. We here adopt the following, "DTD," (g(t_Ia)) for 0.1 Gyr ≤ t_Ia ≤ 10 Gyr, which is deduced from recent observational studies on the SN Ia rate in extra-galaxies (e.g., Maoz et al. 2010);

$$\begin{eqnarray}&&{g}_{\mathrm{Ia}}({t}_{\mathrm{Ia}})={C}_{{\rm{g}}}{t}_{\mathrm{Ia}}^{-1},\end{eqnarray} \tag{ 12 }$$

where C_g is a normalization constant that is determined by the number of SNe 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). We mainly investigate models with the above DTD (rather than those with a typical time delay of SN Ia being ∼1 Gyr). 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 investigate models with f_b = 0.05, because such models can better explain the observed chemical properties of the MCs and the Galaxy (e.g., Tsujimoto et al. 2010; BT12).

We adopt the nucleosynthesis yields of SNe II and Ia from Tsujimoto et al. (1995) to deduce y_II,i and y_Ia,i for a given IMF. For AGB yields, we calculate y_agb,i and h(t_agb) for a given IMF based on the yield tables from van den Hoek & Groenewegen (1997). Since our recent chemodynamical simulations of disk galaxy formation can reproduce the basic dust properties of galaxies (e.g., dust-to-gas ratios) reasonably well (Bekki 2013), we adopt the dust yields (corresponding to f_c,i) used in the simulations for the present study. The adopted ${f}_{{\rm{c}},i}$ is originally from Dwek (1998), which explained the dust properties of the Galaxy quite well. The adopted values are different between different elements and between SNe and AGB stars. We investigate the time evolution of dust and metals for the selected eight elements, C, O, Mg, Fe, Ca, SI, Ti, and S.

We adopt a standard power-law IMF;

$$\begin{eqnarray}&&{\rm{\Psi }}({m}_{{\rm{I}}})={M}_{s,0}{{m}_{{\rm{I}}}}^{-\alpha },\end{eqnarray} \tag{ 13 }$$

where m_I is the initial mass of each individual star and the slope α is set to be 2.35, which corresponds to the Salpeter IMF (Salpeter 1955). The normalization factor M_s,0 is a function of α, m_l (lower mass cut-off), and m_u (upper mass cut-off). These m_l and m_u are set to be 0.1 M_⊙ and 50 M_⊙, respectively. As discussed in BT12, the IMF slope α can influence the chemical evolution history of the LMC. However, we did not discuss such an influence in the present study, first, because the purpose of this paper is not to clarify the roles of IMF in galactic chemical evolution and, second, because the models with the standard IMF (with SN wind) can explain the observed abundances of the MCs (e.g., Tsujimoto & Bekki 2009; BT12).

The IMF slope can possibly influence the evolution of dust and metals of galaxies because it can influence the total amount of energy ejected from massive stars and SNe. If we adopt a top-heavy IMF (e.g., α = 2.1 instead of 2.35), then dust and metals can be more efficiently ejected from the MCs so that dust-to-gas ratios can be lower in the MCs. The time evolution of extinction curves of the MCs might be changed to some extent owing to the more efficient removal of dust. Although this is an important issue, we do not investigate how the IMF can influence the evolution of dust and extinction curves of the MCs in the present study: we will clarify these IMF influences in a future study.

Although we investigate the time evolution of small and large grains for each element (e.g., C, O, and Mg) separately, we divide these grains into the following two categories: C-grains and Si-grains. Dust grains formed from the element C are referred to as C-grains whereas those from elements other than C are referred to as Si-grains. Asano et al. (2013) also adopted this treatment of Si-grains, expecting that solid materials other than carbonaceous dust compose silicate materials after dust processing in the ISM. Accordingly, there are four different types of grains (four-component dust model), for each of which the total mass and dust-to-gas ratios (D) are investigated in the present study. For example, the dust-to-gas ratio for small Si-grains is estimated as follows.

$$\begin{eqnarray}&&{D}_{{\rm{s}},\mathrm{Si}}=\displaystyle \sum _{i}{D}_{{\rm{s}},{\rm{i}}},\end{eqnarray} \tag{ 14 }$$

where the summation is done for elements other than C (i.e., only for i = O, Fe, Mg, Ca, Si, Ti, and S). The dust-to-gas ratio for small C-grains is simply D_s,C. Thus, the total dust-to-gas ratio (D) is given as

$$\begin{eqnarray}&&D=\displaystyle \sum _{i}\displaystyle \sum _{j}{D}_{i,j},\end{eqnarray} \tag{ 15 }$$

where summation is done over i = s (small) and l (large) and j = C, O, Fe, Mg, Ca, Si, Ti, and S.

We consider (1) that dust grains can grow quite efficiently through the accretion of gas-phase metals onto the grains in the cold molecular clouds, (2) that the timescale is shorter for clouds with higher metal-abundance, and (3) that the dust accretion timescale (τ_acc,s,i and τ_acc,l,i) depends on the metallicity and the molecular gas density in the ISM. The accretion timescale for the small grain of ith element is therefore

$$\begin{eqnarray}&&{\tau }_{\mathrm{acc},{\rm{s}},i}=\displaystyle \frac{{\tau }_{\mathrm{acc},{\rm{s}},i,\odot }}{{C}_{\mathrm{acc},i}},\end{eqnarray} \tag{ 16 }$$

where τ_acc,s,i,⊙ is the dust accretion timescale at the solar metallicity (0.014) and the total dust abundance (0.0064) at the solar neighborhood (in the present MW), and the correction factor C_acc,i is as follows:

$$\begin{eqnarray}&&{C}_{\mathrm{acc},i}=F({Z}_{{\rm{z}},{\rm{g}}},{f}_{{{\rm{H}}}_{2}})=\displaystyle \frac{{Z}_{{\rm{z}},{\rm{g}}}}{{Z}_{{\rm{z}},{\rm{g}},\odot }}{(\displaystyle \frac{D}{{D}_{\odot }})}^{\beta },\end{eqnarray} \tag{ 17 }$$

where D, Z_z,g, ${f}_{{{\rm{H}}}_{2}}$ are the dust-to-gas ratio, the gas-phase metallicity, and the cold gas (molecular gas) fraction, respectively, and Z_z,g,⊙ and D_⊙ are the gas-phase metallicity and the dust-to-gas ratio at the solar neighborhood, respectively. Here, D_⊙ is the total dust-to-gas ratio and thus is set to be 0.0064, and β relates the dust-to-gas ratio to the molecular fraction of the ISM. In this formula, we adopt a very reasonable assumption that the dust accretion timescale is shorter in higher metallicity and has a higher cold gas fraction. The cold gas fraction is assumed to be a function of D as follows:

$$\begin{eqnarray}&&{f}_{{{\rm{H}}}_{2}}\propto {(\displaystyle \frac{D}{{D}_{\odot }})}^{\beta },\end{eqnarray} \tag{ 18 }$$

where we estimate ${f}_{{{\rm{H}}}_{2}}$ by using D and gas mass at each time step in a model. We choose this β of 0.2 for ${f}_{{{\rm{H}}}_{2}}$, because such a choice is consistent with recent observational results by Corbelli (2012) who show a correlation between the total masses of molecular hydrogen and dust in galaxies.

Although we adopt exactly the same formula for the dust accretion timescale (τ_acc,l,i) of large grains, we consider that τ_acc,l,i,⊙ should be significantly longer than τ_acc,s,i,⊙ owing to the possible mass growth timescale proportional to dust sizes (H15). We therefore assume that τ_acc,l,i,⊙ is 10τ_acc,s,i,⊙ for all models. We show the results of the models with τ_acc,s,i,⊙ = 0.02 Gyr τ_acc,l,i,⊙ = 0.2 Gyr for all elements in the present study. The dust destruction timescale (τ_SN,s,i and τ_SN,l,i) is set to be 0.2 Gyr for small and large grains of all elements, and these values are similar to those adopted in previous studies (e.g., Dwek 1998; H15).

As discussed by H15, the timescales of coagulation and shattering depend on D (the dust-to-gas ratio) in the ISM of galaxies. We therefore adopt the following formula,

$$\begin{eqnarray}&&{\tau }_{\mathrm{sh},i}={\tau }_{\mathrm{sh},i,\odot }{(\displaystyle \frac{{D}_{{\rm{l}}}i}{{D}_{{\rm{l}},i,\odot }})}^{-1},\end{eqnarray} \tag{ 19 }$$

where τ_sh,i,⊙ and D_l,i,⊙ are τ_sh,i and D_l,i (the dust-to-gas ratio for large grains) at the solar neighborhood, respectively. Since C (carbon) becomes carbon dust whereas metals other than C become silicate, the shattering timescale for large C-grains is as follows:

$$\begin{eqnarray}&&{\tau }_{\mathrm{sh},i}={\tau }_{\mathrm{sh},{\rm{C}},\odot }{(\displaystyle \frac{{D}_{{\rm{l}},{\rm{C}}}}{{D}_{{\rm{l}},\odot }})}^{-1},\end{eqnarray} \tag{ 20 }$$

where D_l,C is the dust-to-gas ratio of large C-grains and D_l,⊙ is the dust-to-gas ratio for large grains (0.0045) at the solar neighborhood. The timescale for elements other than C is as follows:

$$\begin{eqnarray}&&{\tau }_{\mathrm{sh},i}={\tau }_{\mathrm{sh},\mathrm{Si},\odot }{(\displaystyle \frac{{D}_{{\rm{l}},\mathrm{Si}}}{{D}_{{\rm{l}},\odot }})}^{-1},\end{eqnarray} \tag{ 21 }$$

where D_l,Si is the dust-to-gas ratio of large C-grains. We choose τ_sh,C,⊙ = τ_sh,Si = 0.1 Gyr, which is similar to those adopted in previous studies (e.g., H15).

Similarly, the coagulation timescale for C is described as follows:

$$\begin{eqnarray}&&{\tau }_{\mathrm{co},i}={\tau }_{\mathrm{co},{\rm{C}},\odot }{(\displaystyle \frac{{D}_{{\rm{s}},{\rm{C}}}}{{D}_{{\rm{s}},\odot }})}^{-1},\end{eqnarray} \tag{ 22 }$$

where D_l,C is the dust-to-gas ratio of small C-grains and D_l,⊙ is the dust-to-gas ratio for small grains (0.0019) at the solar neighborhood. The coagulation timescale for elements other C is as follows:

$$\begin{eqnarray}&&{\tau }_{\mathrm{co},i}={\tau }_{\mathrm{co},\mathrm{Si},\odot }{(\displaystyle \frac{{D}_{{\rm{s}},\mathrm{Si}}}{{D}_{{\rm{s}},\odot }})}^{-1},\end{eqnarray} \tag{ 23 }$$

where ${D}_{{\rm{l}},\mathrm{Si}}$ is the dust-to-gas ratio of small Si-grains. The values of τ_co,C,⊙ and τ_co,Si,⊙ are set to be 0.02 Gyr, which is similar to those adopted by other studies (e.g., H15).

The new model for dust wind rate W_d(t) in the present study is different to some extent from those adopted in our previous study (BT14). The wind rate is assumed to be different not only between C- and Si-grains but also between small and large grains in the new model. Therefore, the wind rate W_d(t) is described as

$$\begin{eqnarray}&&{W}_{{\rm{d}}}(t)={C}_{{\rm{w}}}S(t){M}_{{\rm{d}}}(t),\end{eqnarray} \tag{ 24 }$$

where C_w is a parameter that controls the removal efficiency of dust through dust wind, S(t) is the total luminosity (proportional to the strength of radiation pressure of stars), and M_d is the total dust mass. The physical meaning of C_w, which is the most important parameter in the present study, is clearly explained and discussed in Appendix A. About 0.4% of dust can be lost through radiation-driven stellar wind within 10⁶ years in a model with C_w = 0.01.

The value of C_w can be different between small and large C-grains and Si-grains. For example, the wind rate for small C-grains (C_w,s,C; i = C) is given as

$$\begin{eqnarray}&&{W}_{{\rm{d}},{\rm{s}},{\rm{C}}}(t)={C}_{{\rm{w}},{\rm{s}},{\rm{C}}}S(t){M}_{{\rm{d}},{\rm{s}},{\rm{C}}},\end{eqnarray} \tag{ 25 }$$

where C_w,s,C is the value of C_w for small C-grains. As demonstrated by recent numerical simulations of dust wind in galaxies (Bekki 2015), it depends on a number of factors whether dust can be ejected and removed from galaxies beyond their halo regions. For example, gas-dust interaction through gaseous drag can prevent dust from being ejected from gas disks of galaxies. Since this gas-dust interaction cannot be modeled in the present one-zone models, we regard C_w as a free parameter. Thus, we investigate the effects of dust wind in the evolution of dust by changing the C_w parameters for different grains.

In the present study, we assume that dust wind can occur only in strong SB phases (SB1, SB2, and SB3): no dust can be removed from a galaxy in the quiescent star formation. This assumption is quite reasonable and realistic, because dust wind is observed in SB galaxies like M82 (e.g., Kaneda et al. 2010). The strength of radiation pressure of stars in a galaxy is assumed to be proportional to the total luminosity of the galaxy, and the total luminosity is calculated as

$$\begin{eqnarray}&&S(t)={\displaystyle \int }_{0}^{t}{f}_{M/L}^{-1}(t-T){M}_{\mathrm{ns}}(T){dT},\end{eqnarray} \tag{ 26 }$$

where f_M/L is the mass-to-light ratio (M/L) of a single stellar population (SSP) and M_ns(T) is the total mass of stars formed at t = T. Since the M/L of an SSP is a function of age and metallicity, we need to use a stellar population synthesis code in order to properly calculate f_M/L at each time step. Accordingly, we use the code MILES (Vazdekis et al. 2010) for the M/L estimation of all models in the present study. We adopt the SSP table for the Salpeter IMF and [Z/H] = −0.4 from the MILES and estimate M/L for stars with different ages in all models.

Since the parameter C_w is already introduced in Equation (23), the above Equation (25) is slightly different from that (with a parameter C_r) used in BT14. The radiation-driven wind can remove dust from a galaxy, and thus C_w is non-zero only for t_sb1,s ≤ t ≤ t_sb1,e, t_sb2,s ≤ t ≤ t_sb2,e, and t_sb3,s ≤ t ≤ t_sb3,e: C_w = 0 in the quiescent SF phase. We have run a large number of models with different values of C_w,s,C, C_w,l,C, C_w,s,Si, and C_w,l,Si so that we can find models that can explain the observed extinction curves of the LMC and the SMC. We mainly show the results of three "fiducial" models with particular combinations of these four parameters for the SMC, the LMC, and the MW, because they can reproduce the observed extinction curves.

Based on the total masses (or fractions) of small and large C-grains and Si-grains in a model, we can draw the extinction curve by using the two-size approximation method by H15. Since the details of the method to derive extinction curves from dust masses and compositions are already given in H15, we briefly describe it here. We adopt the "modified" log-normal model for the grain size distribution of small and large grains, and the functional form (n_i,j(a)) for the distribution is given as

$$\begin{eqnarray}&&{n}_{i,j}(a)=\displaystyle \frac{{C}_{i,j}}{{a}^{4}}\mathrm{exp}\{-\displaystyle \frac{{[\mathrm{ln}(a/{a}_{0,i,j})]}^{2}}{2{\sigma }^{2}}\},\end{eqnarray} \tag{ 27 }$$

where a is the grain size, subscript i describes the small (i = s) or large (i = l) grains, subscript j describes carbonaceous (j = C) or silicate (j = Si) grains, C_i,j is the normalization constant, and ${a}_{0,i,j}$ and σ are the central grain radius and the standard deviation of the log-normal distribution, respectively. It should be noted here that the mass distribution ($\propto {a}^{3}{n}_{i,j}(a)$) is log-normal in the adopted model.

We adopt a_0,,s,C = a_0,,s,Si = 0.005 μm, a_0,,l,C = a_0,,l,Si = 0.1 μm, and σ = 0.75 so that the size distribution function can roughly cover the small and large grain size ranges. The normalizing constant for small and large C-grains and Si-grains are determined by the following equation,

$$\begin{eqnarray}&&\mu {{\rm{m}}}_{{\rm{H}}}{D}_{i,j}={\displaystyle \int }_{0}^{\infty }\displaystyle \frac{4}{3}\pi {a}^{3}{{sn}}_{i,j}(a){da},\end{eqnarray} \tag{ 28 }$$

where μ = 1.4 is the gas mass per hydrogen nucleus, m_H is the hydrogen atom mass, and s is the material density of a dust grain. Accordingly, we can derive ${C}_{i,j}$ for D_i,j derived from one-zone chemical evolution models by using the above equation.

Based on n_i,j, estimated from D_i,j at each time step in a chemical evolution model, we can derive the dust extinction curve at the time step. The extinction at wavelength λ in units of magnitude (A_λ) normalized to the column density of hydrogen nuclei (N_H) is described as follows (H15):

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{\lambda }}{{N}_{{\rm{H}}}}=2.5\mathrm{log}\;{\rm{e}}\displaystyle \sum _{i}\displaystyle \sum _{j}{\displaystyle \int }_{0}^{\infty }{n}_{i,j}\pi {a}^{2}{Q}_{\mathrm{ext}}(a,\lambda ),\end{eqnarray} \tag{ 29 }$$

where Q_ext(a,λ) is the extinction efficiency factor. The values of this Q_ext factor at each wavelength and for each grain are evaluated by using the Mie theory (Bohren & Huffman 1983) and the same optical constants for silicate and carbonaceous dust in Weingartner & Draine (2001). We adopt s = 3.5 g cm⁻³ for small and large Si-grains and s = 2.24 g cm⁻³ for small and large C-grains. As shown in H15, this extinction model can reproduce the observed MW extinction curve reasonably well.

We first investigated many models with different model parameters for dust growth, shattering, and coagulation and found that no models can reproduce the observed extinction curves without the 2175 Å bump (SMC) and the very weak one (LMC). We then considered the effects of dust wind on the evolution of dust extinction curves by incorporating the dust ejection/removal process associated with dust wind into the one-zone models. K. C. Hou & H. Hirashita (2015, in preparation) have adopted one-zone models without dust wind for deriving dust extinction curves of galaxies and found that the models with a wide range of parameters for dust-related physical processes (e.g., coagulation) cannot reproduce the observed extinction curves without the 2175 Å bump. Therefore, it seems that standard chemical evolution models without dust wind cannot reproduce the extinction curves of the MCs.

After extensive investigation of models with dust-related physical processes (e.g., selective destruction of C-grains by SNe and dust wind), we finally find that models with dust wind can reproduce better the observed extinction curves of the MCs. Therefore, we focus on the results of the models with dust wind in the present study. Although it is possible that physical processes other than dust wind can be responsible for the origin of the extinction curves of the MCs, we focus exclusively on the effects of dust wind on the evolution of extinction curves of the MCs.

We consider that the efficient ejection/removal of dust grains from the SMC and the LMC can be the key physical process for the origin of their dust extinction curves. We therefore (1) change the parameters of dust wind and (2) adopt reasonable values for other model parameters (e.g., for dust growth) so that we can more clearly elucidate the key factor for the presence or absence of the 2175 Å bump. The model parameters for star formation (e.g., burst epoch) and dust-related physical processes (e.g., dust accretion timescale) are summarized in Tables 1 and 2, respectively, for the MCs and the MW. BT12 have already discussed how the chemical evolution (thus dust evolution) histories of the MCs depend on star formation parameters, and H15 has discussed how the dust evolution of a galaxy depends on the model parameters for dust growth, destruction, shattering, and coagulation. Furthermore, we have confirmed that only some models with dust wind can explain the observed extinction curves of the MCs. Therefore, we focus exclusively on how dust wind plays a key role in controlling the extinction curves of the MCs.

In determining the three burst epochs, we consider both recent observational and theoretical results on the star formation and chemical evolution histories of the MCs. Harris & Zaritsky (2006) investigated the ages of stars in the SMC based on UBVI photometry from their MCs Photometric Survey and found significant rises in the mean star formation rate around 0.06, 0.4, and 2.5 Gyr ago. Using gas dynamical simulations of the SMC interacting with the LMC and the MW, Yoshizawa & Noguchi (2003) predicted that the SMC experienced SB events about 0.2 and 1.5 Gyr ago. Rubele et al. (2015) have recently detected the SB populations with ages of 1.5 and 5 Gyr in the deep images of the SMC from the VISTA survey of the MCs in the YJK_s bands. Guided by these observations, we assume that the SMC experienced three enhanced star formation episodes at t_sb,1,s = 8.5, 11.5, and 12.85 Gyr (assuming a current age of 13 Gyr for the LMC).

The LMC also appears to have experienced at least three major epochs of significantly enhanced star formation. Rubele et al. (2012) investigated the AMR of the LMC stellar populations and found that the SFR peaked around 2 Gyr ago, though the AMR appears to be different between different local regions. Based on the results of the gas dynamical simulations of the LMC over the past 6 Gyr, Bekki et al. (2004) suggested that an SB, which can form globular clusters, could have occurred in the LMC about 3 Gyr ago owing to the LMC–SMC-Galaxy tidal interaction. Harris & Zaritsky (2009) found a very strong enhancement of mean star formation from 10⁸ years ago to the present in the blue arm, Constellation III, and 30 Doradus region of the LMC. Thus, we assume that the LMC experienced three enhanced star formations at t_sb,1,s = 10.0, 11.5, and 12.85 Gyr (corresponding to 3, 1.5, and 0.15 Gyr ago, respectively).

Table 3 summarizes the model parameters of dust wind in the three galaxies. Although we investigated much more models than those listed in this table, we show only these representative ones, because they grasp some essential ingredients of the dust wind effects on the extinction curves of galaxies. We mainly describe the time evolution of the extinction curve in the fiducial model for the SMC (S1). This fiducial model is chosen because it can best reproduce the observed extinction curve of the SMC. As a comparative experiment, the model without dust wind (yet with SB; S2) is investigated for the SMC. These models with and without dust wind enable us to demonstrate more clearly how the dust wind can change the shapes of the extinction curves of the SMC. The models with different C_w,s,C (S3, S4, and S5) are investigated so that the importance of ejection/removal of small C-grains in the evolution of the extinction curve of the SMC can be more clearly described. The models without SBs (S6–S9) enable us to understand the importance of SB dust wind in the evolution of the extinction curve.

Table 3.  The Model Parameters for Radiation-driven Dust Wind in Each Model

Model Namea	Cw,s,Si b	Cw,l,Si	Cw,s,C	${C}_{{\rm{w}},{\rm{l}},{\rm{C}}}$	Comments
S1	0.01	0.002	0.05	0.01	fiducial SMC
S2	0.0	0.0	0.0	0.0	no dust wind
S3	0.01	0.002	0.01	0.01	⋯
S4	0.01	0.002	0.03	0.01	⋯
S5	0.01	0.002	0.1	0.01	⋯
S6	⋯	⋯	⋯	⋯	no SB
S7	0.01	0.002	0.05	0.01	no SB1
S8	0.01	0.002	0.05	0.01	no SB2
S9	0.01	0.002	0.05	0.01	no SB3
L1	0.01	0.002	0.03	0.01	fiducial LMC
L2	0.0	0.0	0.0	0.0	no dust wind
M1	⋯	⋯	⋯	⋯	fiducial MW

Notes.

^aThe first capital letters "S," "L," and "M" indicate the SMC, LMC, and MW models, respectively. The mark "-" means no dust wind owing to no starburst events whereas the zero value (0.0) means no dust wind in spite of starburst. ^bThe dust removal coefficient for small and large Si-grains and small and large C-grains is represented by C_w,s,Si, C_w,l,Si, C_w,s,C, and C_w,l,C, respectively.

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We show the results of the LMC models with and without dust wind also in order to clarify the physical reasons for the observed weak 2175 Å extinction bump in the LMC. We show the results of the MW model (M1) without SB (thus without dust wind) only, because there is little observational evidence for recent strong SB in the MW. This MW model and those for the MCs (S1 and M1) can combine to demonstrate that the present models can explain not only the extinction curve of the MW but also those of the MCs. Figure 1 shows the time evolution of M_g (normalized), f_g (gas mass fraction), and SFR (normalized) for the fiducial SMC, LMC, and MW models.

Figures 2 and 3 show the time evolution of the extinction curve and the relative fractions of dust grains, respectively, in the standard SMC model (S1) in which the SMC can experience three strong SB events. Before the first SB event (SB1), the star formation rate of the SMC can be kept low so that chemical evolution can proceed very slowly (T < 8.5 Gyr). In this low-metallicity phase (pre-SB1 phase), stellar dust production dominates the dust content (i.e., still little dust growth) since the dust abundance is too low for other dust processing mechanisms (in particular, shattering) to work efficiently. As a result of this, the ISM dust of the SMC is dominated by large grains. The SMC thus can show a rather flat extinction curve without the 2175 Å bump in this pre-SB1 phase.

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As the gas-phase metallicity becomes higher through more rapid chemical evolution during SB1, the mass fraction of small grains becomes larger owing to the interplay between shattering and accretion (T = 9.8 Gyr). The extinction curve can therefore show a slightly steeper rise in the FUV regime and the 2175 Å bump can be appreciably seen at this post-SB1 phase. The extinction curve becomes progressively steeper and the 2175 Å bump becomes more prominent after the SB1 (T = 11.5 Gyr). The fraction of cold gas can become higher at this epoch, because D, which can determine the gas fraction, is higher. The fraction of small Si-grains can be as large as that of large Si-grains at this pre-SB2 phase, because the higher metallicity and the higher cold gas fraction combine to allow the more rapid growth of the small grain abundance. These behaviors in the evolution of Si grains are apparently similar to those reported in Asano et al. (2014).

Although the fraction of small Si-grains can be temporarily lower owing to the efficient destruction of dust by SNe during the second SB phase (SB2), it can start to rise again soon after SB2 (T = 12.5 Gyr). The total mass of small Si-grains becomes 1.6 times larger than that of large Si-grains at T = 12.5 Gyr so that the mass fraction of small Si-grains can be the largest in the SMC history. Consequently, the extinction curve of the SMC in this model at T = 12.5 Gyr shows a very steep rise that is even steeper than the observed SMC extinction curve and thus inconsistent with the observed curve. Although small C-grains can also suffer a significant loss of their mass owing to the grain destruction by SNe during SB2, their mass can steadily and rapidly increase through the accretion of gas-phase metals after SB2. This increase of small C-grain abundance is responsible for the strong 2175 Å bump in the extinction curve at T = 12.5 Gyr.

The fraction of small Si-grains can become smaller owing to their destruction by SNe and ejection from the SMC through radiation-driven dust wind in the third SB phase (SB3) so that the extinction curve at T = 12.9 Gyr can be slightly less steeper than that at T = 12.5 Gyr. The dust wind effects are more significant in this SB3, mainly because the total luminosity of stars in the SMC takes its maximum. Although a significant fraction of small C-grains can be removed from the SMC through dust wind, the extinction curve still clearly shows the 2175 Å bump in the early phase of this SB3. The 2175 Å bump can completely disappear only after most of the small C-grains can be removed from the SMC through dust wind at T = 12.9 Gyr. The final extinction curve at T = 13 Gyr is very similar to the one observed both in the lack of the 2175 Å bump and in the steep linear rise at the FUV wavelength. Thus the present standard SMC model with dust wind can reproduce the observed extinction curve quite well.

As is clearly shown in Figure 3, the time evolution of C-grains is different between the SMC models with and without dust wind (S1 and S2, respectively) after SB2: the fractions of small and large C-grains and small Si-grains (relative to large Si-grains) are not so different between the two models during and after SB1 and SB2. This suggests that the SMC cannot show the extinction curve without the 2175 Å bump until quite recently. This also suggests (1) that star-bursting dwarf galaxies do not necessarily show extinction curves without the 2175 Å bump and (2) that dust wind needs to be strong enough to remove most (90%) of the small C-grains in star-bursting dwarfs with no 2175 Å bump in their extinction curves. It is thus highly likely that SB dwarfs have different 2175 Å bump strength.

Figure 4 shows that the final extinction curve in the SMC model with three SB events, yet no dust wind, has the conspicuous 2175 Å bump. The SMC models with lower dust wind efficiency (C_w,s,C = 0.01 and 0.03 for S4 and S5, respectively) also show the 2175 Å bump and the FUV rise that is less consistent with the observed extinction curve of the SMC. The SMC model with higher dust wind efficiency (C_w,s,C = 0.1, S6) can reproduce the observed extinction curve of the SMC as well as the standard SMC model. These results confirm that the removal of small C-grains from the SMC through dust wind is a viable mechanism for the observed lack of the 2175 Å bump in the extinction curve of the SMC.

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The model with no SB, and thus no dust wind (S7), shows the conspicuous 2175 Å bump and a steeper FUV rise, which clearly demonstrates that dust wind can significantly influence the shape of the extinction curve of the SMC. As shown in models S8 and S9, the presence or absence of the 2175 Å bump does not depend on whether SB1 or SB2 occurs in the SMC. However, the 2175 Å bump of the extinction curve is too strong in model S10 without SB3, which suggests that the strength of the 2175 Å bump can depend on whether the SMC experiences SB3 with dust wind. These results thus confirm that the last SB with strong dust wind about 0.2 Gyr ago is a key physical process for the observed characteristic extinction curve of the SMC. The results of the SMC models with different strengths of SBs are discussed in Appendix B. The metallicity evolution of the SMC (LMC and MW) in the present new models with dust evolution is discussed briefly in Appendix C.

Figure 5 presents the time evolution of the extinction curve of the LMC in the model L1 with three SB events. Gas can be more rapidly converted into new stars in this L1 than in the SMC models (owing to the adopted higher SF efficiency) so that chemical abundances can be higher even well before the first SB event (T = 8.4 Gyr). As shown in Figure 6, small C- and Si-grains can start to grow earlier in this L1, and their mass fraction can take their peak values before SB1. Consequently, the extinction curve can have a rather conspicuous 2175 Å bump and a very steep FUV rise at T = 8.4 and 9.8 Gyr. Since SNe can efficiently destroy dust grains during SB1 owing to high SF rates (i.e., high SN rates), the 2175 Å bump becomes weakened after SB1 (T = 11.5 Gyr). However, such destruction of small grains cannot severely weaken the 2175 Å bump after SB1 and even after SB2 (T = 12.5 Gyr).

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The 2175 Å bump of the LMC becomes less conspicuous during SB3 owing to the combination effect of dust destruction and radiation-driven dust wind (T = 12.9 Gyr). The shape of the extinction curve of the LMC cannot become quite similar to the observed one until the end of SB3 (T = 13 Gyr). The rather weak 2175 Å bump at T = 13 Gyr in this model is due to the more preferential removal of small C-grains from the LMC through radiation-driven dust wind during the longer SB (SB3). The major difference in the dust abundances between this LMC model L1 and the SMC model S1 is that the final fractions of small and large C-grains are both slightly larger in L1 than in S1 (See Figures 4 and 6). These differences can cause a difference in the extinction curves between the LMC and the SMC in the present study.

Figure 7 compares the extinction curve of L1 with dust wind and that of L2 without dust wind. Clearly, L1 with dust wind can better reproduce the observed extinction curve of the LMC, which confirms that the efficient removal of C-grains through radiation driven dust wind is essential for explaining the origin of the extinction curve of the LMC. Previous numerical simulations of the MCs (e.g., Yoshizawa & Noguchi 2003; Bekki & Chiba 2005) suggested that the observed enhanced SFRs in the MCs (e.g., Harris & Zaritsky 2006, 2009) can be due largely to the LMC–SMC-Galaxy tidal interaction in the past 3 Gyr. Thus the results in Figures 2–7 suggest that the origin of the observed extinction curves of the LMC and the SMC can be closely associated with the past active star formation triggered by tidal interaction among the three galaxies.

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It is instructive for the present study to show whether the new models for dust evolution of galaxies can reproduce the observed extinction curve of the MW just by changing the model parameters. Figure 8 describes the time evolution of the MW model M1 without SB events (thus without dust wind). In the very early chemical evolution history of model M1 (T = 1 Gyr), the extinction curve is rather flat without the clear 2175 Å bump owing to the very small fraction of small C-grains. The extinction curve of the model M1 evolves to have the 2175 Å bump and a steeper rise at 1/λ > 6 μm⁻¹ only 2 Gyr after the commencement of its star formation (i.e., disk formation). The overall feature of the extinction curve becomes quite similar to the observed one for the MW at T = 5 Gyr, which implies that MW-like galaxies at higher redshifts might already have extinction curves similar to that of the present MW.

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The 2175 Å bump is a bit too conspicuous at T = 5 Gyr in comparison with the observed one, and the bump does not evolve significantly with time after T = 5 Gyr. This is mainly because the relative mass fractions of small Si-grains and small and large C-grains (with respect to large Si-grains) do not change significantly after T = 5 Gyr. The overall shape of the extinction curve at T = 13 Gyr becomes closer to that of the observed one. However, the final shape of the 2175 Å bump is still slightly different from the observe one (e.g., more remarkable bump), which implies that the present models do not reproduce the dust abundances of the MW completely well. In a future study, we will reproduce the observed extinction curve of the MW better by using a more sophisticated dust evolution model.

These results imply that luminous disk galaxies like the MW are likely to show clear 2175 Å bumps in their extinction curves, because a significant amount of dust is unlikely to be removed from the galaxies owing to their deep gravitational potential. Although a small amount of dust and metals could be removed/ejected from the galaxies through SN and dust wind, such a less significant removal/ejection process cannot influence the shapes of the extinction curves to a large extent. The present study therefore predicts that the shapes of extinction curves depend on the masses of galaxies in such a way that more massive galaxies are likely to have the 2175 Å bump. It is an interesting observational question how the strength of the 2175 Å bump in a massive galaxy depends on its physical parameter (e.g., total mass and luminosity). It is worth noting here that Asano et al. (2014) and Nozawa et al. (2015) attempted to reproduce the 2175 Å bump of the MW by freely changing the mass fractions of various ISM phases.

Finally, Figure 9 summarizes the time evolution of mass fractions of the key three dust grains with respect to large Si-grains for the MCs and the MW. The differences in the evolution between the three galaxies can be clearly seen, which reflects on the time evolution of the extinction curves of the three. SFHs with or without SBs and capabilities to retain dust within galaxies are key determinants for the time evolution of the extinction curves of galaxies. Since the present study has investigated only three different types of galaxies (i.e., SMC, LMC, and MW), we will need to investigate models with different star formation and dust evolution histories to fully understand the origin of various extinction curves observed in galaxies with different masses and Hubble types.

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We have, for the first time, demonstrated that the observed extinction curve without the 2175 Å bump in the SMC can be reproduced, if the small C-grains can be more preferentially lost through radiation-driven wind during the last SB around 0.2 Gyr ago. In this "lost small C-grain" scenario, the LMC can have a weak 2175 Å bump in its dust extinction curve, because it can retain a larger amount of small C-grains owing to its deeper gravitational potential in comparison with the SMC. This "lost small C-grain" scenario predicts that more massive galaxies (like the MW) are likely to show more distinctive 2175 Å bumps in their dust extinction curves. It should be stressed, however, that the present low-mass galaxies can show such 2175 Å bumps if they have not experienced recent strong SBs and the resultant loss of small C-grains.

The key physical process required for the lack of the 2175 Å bump in a galaxy is the more efficient removal of C-grains than Si-grains from the galaxy. The removal (or stripping) process of dust grains in galaxies should be rather complicated, because gravitational dynamics, gas-dust drag, and radiation pressure of stars can all be involved in the removal process. Therefore, it would be a formidable task to accurately estimate the dust removal efficiency in a galaxy for the physical parameters of the galaxy. Thus we investigate whether and in what physical conditions the above two requirements can be met in a galaxy by using a simplified yet reasonable model.

The first question is whether these C-grains can be more efficiently removed from a galaxy through radiation-driven wind than Si-grains. The key parameter for this removal process in a galaxy is the frequency-averaged radiation pressure coefficient (${Q}_{\mathrm{pr}}^{*}$), as demonstrated in previous theoretical models and recent numerical simulations of dust wind in galaxies (e.g., Ferrara et al. 1991; Bekki 2015; Bianchi & Ferrara 2005). We have therefore investigated ${Q}_{\mathrm{pr}}^{*}$ for C and Si grains by using the publicly available data for optical properties of dust (Draine & Lee 1984; Laor & Draine 1993) and the stellar population synthesis code ("MILES") developed by Vazdekis et al. (2010). The radiation pressure coefficient (Q_pr) for a grain with a radius is given as follows.

$$\begin{eqnarray}&&{Q}_{\mathrm{pr}}={Q}_{\mathrm{abs}}+(1-g(\theta )){Q}_{\mathrm{sca}},\end{eqnarray} \tag{ 30 }$$

where Q_abs and Q_sca are the absorption and scattering coefficient, respectively, and g(θ) is the scattering asymmetry parameter.

Using Q_pr and the flux of a SED (f(λ)), we can estimate ${Q}_{\mathrm{pr}}^{*}$ as follows.

$$\begin{eqnarray}&&{Q}_{\mathrm{pr}}^{*}={\displaystyle \int }_{{\lambda }_{\mathrm{min}}}^{{\lambda }_{\mathrm{max}}}{Q}_{\mathrm{pr}}f(\lambda )d\lambda ,\end{eqnarray} \tag{ 31 }$$

where λ_min and λ_max are set to be 3340.5 and 7409.6 Å, respectively, in MILES. The above wavelength range is publicly available one in MILES and f(λ) is the normalized flux (i.e., ${\displaystyle \int }_{{\lambda }_{\mathrm{min}}}^{{\lambda }_{\mathrm{max}}}f(\lambda )=1$). Figure 10 shows the wavelength-dependence of the radiation pressure coefficient (Q_pr) for silicate and graphite with the radii of 0.01 and 0.1 μm and the SED of a galaxy for a metallicity ([Z/H]) of −0.7 dex and an age of 63 Myr (which is the youngest stellar population that can be adopted in MILES). We use this SED for an SSP to mimic the SMC with young SB components. We discuss just this model with two representative dust radii as an example, though, ideally speaking, we should show the results for all dust radii.

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Clearly, ${Q}_{\mathrm{pr}}^{*}$ is higher in graphite than in silicate for the adopted SED and dust radii (0.01 and 0.1 μm), which means that graphite is more likely to be influenced by the radiation pressure of stars and thus removed from a galaxy if the gravitational potential is sufficiently shallow. Furthermore, the difference in ${Q}_{\mathrm{pr}}^{*}$ is significantly larger in the small grains (a = 0.01 μm) than in the large ones (a = 0.1 μm), which implies that small C-grains can be much more efficiently removed than small Si-grains. Although the real SED of an SMC-type galaxy might be different from the adopted SED in this investigation, it is unlikely that this result changes qualitatively if a more realistic SED is adopted, given Q_pr of graphite always higher than those of silicate for the relevant wavelength range. These results thus suggest that graphite is more likely to be lost from an SMC-type galaxy than silicate, though a full investigation on this dust removal process for different dust grains and more realistic galaxy spectra needs to be done in our future studies.

If a rather small mass fraction of small C-grains is responsible for the observed lack of the 2175 Å bump in the SMC, then is there any observational evidence for that? Using ISOPHOTO and HiRes IRAS data for dust emission and ATCA data for HI column gas density, Bot et al. (2004) investigated the gas-to-dust ratios of polycyclic aromatic hydrocarbon (PAH) molecules or small grains (VSGs) for different regions of the SMC. They found that the derived difference in the gas-to-dust ratio of PAHs and very small grains between the Galaxy and the SMC is three times larger than the metallicity difference between the two galaxies. This observational result implies that the dust depletion levels for VSG and PAHs are low in the SMC for some physical reasons (e.g., destruction by SNe; Bot et al. 2004). Since this observational study cannot distinguish small C-grains from small Si-grains, it is not clear whether the observed low depletion level of VSGs implies that the mass fraction of small C-grains is significantly low owing to the efficient removal or destruction of such grains. It appears to be difficult for photometric study of the SMC alone to estimate the contributions of small C- and Si-grains to the observed dusty SED separately.

The present study predicts that the star-forming regions of the SMC for which there is no 2175 Å bump can be dominated by Si-grains (i.e., small fractions of C-grains). However, Welty et al. (2001) found that the Si appears to be undepleted in a local region of the SMC toward the SMC star Sk 155, which means that the local region should contain little Si dust. If this is not just for the Sk 155 region but for other star-forming regions generally in the SMC, then the present scenario and other dust models for the SMC like Pei (1992) appears to be ruled out. Thus it would be observationally important to make a robust estimation of the dust depletion level of Si in many star-forming regions with and without the 2175 Å bump for the SMC.

A number of possible carriers of the 2175 Å bump have been extensively discussed by many authors so far (e.g., Whittet 2003). Mishra & Li (2015) have recently found a strong correlation between the strength of the 2175 Å bump and the carbon depletion level ([C/H]_dust) in the MW and thus suggested that the carrier of the observed 2175 Å bump is graphite or PAH molecules. Their results are consistent with the present "lost small C-grains" scenario, though they did not discuss small and large C-grains separately in the context of the origin of the 2175 Å bump. The results imply that the time evolution of C-grains in a galaxy can be reflected on the strength of the 2175 Å bump in the extinction curve of the galaxy.

The severely deficient small C-grain of the SMC ISM discussed in the present study would be just one of the promising ways that explain the observed lack of the 2175 Å bump in the SMC. Nozawa et al. (2015) have recently reproduced the observed dust extinction curve without the 2175 Å bump in SDSS J1048+4637 by adopting the optical constant of amorphous carbon for carbonaceous grains (See Figure 4 in their paper). Although their dust extinction curve (Nozawa et al. 2015) is quite different from that of the SMC, it would be possible that amorphous carbon dust might be responsible for the lack of the 2175 Å bump in the SMC extinction curve. A recent observational study of dust in the LMC by Herschel (Meixner et al. 2010) has shown that their dust model with amorphous carbon can better explain their observations for a more reasonable dust-to-gas ratio. It appears to be currently difficult to model the evolution of amorphous carbon by considering the possible physical effects (e.g., radiation fields and gaseous shock) related to the transformation of carbon grain properties.

As demonstrated in the present study, a way to reproduce the observed lack of the 2175 Å bump in the SMC extinction curve is to reduce the relative contribution of small C-grains. Therefore, it does not make much difference whether small C-grains can be lost either through energetic dust wind or through some destruction process of dust grains. If small C-grains can be more preferentially destroyed or transformed by SNe or gaseous shock or a strong radiation field in the ISM of the SMC, the 2175 Å bump might not be observed in the SMC. It would be possible that the last strong LMC–SMC collision/interaction about 0.2 Gyr ago (e.g., Diaz & Bekki for the latest model), which can strongly compress the ISM of the SMC (e.g., Bekki & Chiba 2007), could preferentially destroy the small C-grains. In a future study, we will investigate whether small C-grains can be more preferentially destroyed by these physical processes in the ISM by using parsec-scale numerical simulations of the ISM.

Interstellar dust can have significant influences on the physical processes of galaxy formation and evolution, such as the formation of molecular hydrogen in the ISM of galaxies (e.g., Gould & Salpeter 1963) and photoelectric heating of the ISM (e.g., Watson 1972). Although these influences have been recently included in Nbody+hydrodynamical simulations of galaxy formation and evolution in a self-consistent manner (Bekki 2013, 2014, 2015), the evolution of dust sizes, which can control dust-related physical processes of ISM (e.g., Yamasawa et al. 2011), has not been incorporated properly into numerical simulations of galaxy formation and evolution. The present study has demonstrated that the evolution of dust sizes and compositions is a key factor for the evolution of extinction curves in galaxies. We thus suggest that if future numerical simulations of galaxy formation and evolution can include the time evolution of dust sizes and compositions, such numerical studies can have the following improvements in their predictive power.

First, such simulations with the evolution of dust sizes and compositions can predict the SEDs of galaxies in a more consistent manner. The SEDs of dusty galaxies are strongly influenced by the adopted extinction curves, and the shapes of the extinction curves of galaxies depend on their SFHs that control their dust evolution histories. Therefore, previous SED models of galaxies with a fixed extinction curve (e.g., an adoption of the MW dust extinction curve) might be much less self-consistent and realistic. Construction of SEDs from age and metallicity distributions of stars and spatial distributions of stars in numerical simulations has been done by many authors (e.g., Bekki et al. 1999; Bekki & Shioya 2001; Jonsson 2006; Yajima et al. 2014), though they had to assume a constant dust-to-metal ratio and a fixed dust extinction curve. It is doubtlessly worthwhile for these simulations to include the evolution of dust sizes and compositions for a more consistent prediction of galactic SEDs.

Second, future simulations with dust size evolution can provide more accurate predictions of molecular hydrogen abundances in the ISM of galaxies. As shown in sophisticated one-zone models of galaxy formation with dust size evolution by Yamasawa et al. (2011), the formation efficiency of molecular hydrogen on dust grains in a forming galaxy depends on the evolution of the dust size distribution of the galaxy: since the molecular hydrogen formation rate on the surface of a dust grain depends on the surface area, the net formation rate of molecular hydrogen on dust grains in a galaxy relates to the dust extinction curve. In order for numerical simulations to predict the dust size distribution of a galaxy, they need to follow the time evolution of dust grains with different sizes for each gas particle. This is a very formidable task, because such dust grain evolution is computationally heavy even in one-zone chemical evolution models (e.g., Asano et al. 2014). A wise way to implement the dust size evolution in numerical simulations of galaxy formation would be to adopt a two-size approximation, as done in this study.

Third, the effects of photoelectric heating of dust on the thermal evolution of the ISM in galaxies can be more self-consistently investigated in numerical simulations with dust size evolution. The photoelectric heating rate depends on the dust size distribution (not just dust abundances), and the photoelectric heating has been demonstrated to influence galaxy-wide star formation through its influences on the thermal history of galactic ISM (e.g., Bekki 2015). Therefore, photoelectric heating would be one of the key ingredients that future numerical simulations, in particular, those for the evolution of galaxy-wide star formation, should include. Thus, we suggest that incorporation of dust size evolution into numerical simulations of galaxy formation and evolution is essential for understanding various aspects of galaxy formation and evolution.