CIRCUMSTELLAR DUST AROUND AGB STARS AND IMPLICATIONS FOR INFRARED EMISSION FROM GALAXIES

We use the radiative transfer code DUSTY (Ivezic & Elitzur 1997; Ivezic et al. 1999) to model dusty circumstellar shells around AGB stars. DUSTY solves the 1D radiative transfer equation for a source embedded in a dusty region assuming spherical symmetry. The input to DUSTY is straightforward—it requires an input spectrum, the dust properties (chemical composition, grain size distribution, the dust temperature at the inner boundary), a radial density profile, and the optical depth of the envelope at a reference wavelength. We altered the wavelength grid to be finer than the default (1968 wavelength points instead of 105). In this section we describe our choice of inputs and summarize those choices in Table 1.

Table 1.  Summary of the Parameters Chosen as Input for the DUSTY Models

Parameter	Carbon-rich	Oxygen-rich
Photosphere Properties
Photosphere Model	Aringer	BaSeL
${{T}_{{\rm eff}}}$	2400–4000 K	2000–4000 K
${\rm log} \;g$	0.0	0.0
Metallicity	${{Z}_{\odot }}$	${{Z}_{\odot }}$
Envelope Properties
${{R}_{{\rm in}}}$ Temperature (Tc)	1100 K	700 K
Density Profile	${{r}^{-2}}$	${{r}^{-2}}$
Dust Grain Properties
κ	3200 $\mu {{{\rm m}}^{2}}{{g}^{-1}}$	3000 $\mu {{{\rm m}}^{2}}{{g}^{-1}}$
${{Q}_{{\rm ext}}}$	0.1	0.1
Grain Size (a)	0.1 $\mu {\rm m}$	0.1 $\mu {\rm m}$
Density (${{\rho }_{d}})$	2.26 ${\rm g}\;{\rm c}{{{\rm m}}^{-3}}$	2.5 ${\rm g}\;{\rm c}{{{\rm m}}^{-3}}$

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We created two separate sets of models, one each for the carbon-rich and oxygen-rich AGB stars. In addition to composition, the models are a function of the optical depth, ${{\tau }_{{\rm AGB}}}$ (herein this refers to optical depth at 1 $\mu {\rm m}$), and the effective temperature of the star, ${{T}_{{\rm eff}}}$. The model DUSTY spectra are implemented in the stellar population synthesis code differentially, in that only the ratio of the output to input spectra are stored and used within the code.

In Figure 1 we show the behavior of the oxygen-rich and carbon-rich grids as a function of optical depth at a constant effective temperature for the carbon-rich and oxygen-rich grids. In this figure, we can see the characteristic features of the dust grains used in each set of models. For example, the $11\;\mu {\rm m}$ feature caused by SiC in the carbon-rich grid (right) and the $10\;\mu {\rm m}$ silicate feature in the oxygen-rich grid (left). The $10\;\mu {\rm m}$ silicate feature varies from an emission to an absorption feature as a function of ${{\tau }_{{\rm AGB}}}$.

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The differential implementation of the models allows us to keep the stellar parameters beyond effective temperature, such as stellar metallicity and surface gravity, constant. In Figure 2 we show the sensitivity of the differential spectra to surface gravity (middle panel) and metallicity (bottom panel). As can been seen, variations of these parameters do not have a significant impact on the differential spectrum. As such, surface gravity is kept constant at log(g) = 0.0 and the metallicity at $Z={{Z}_{\odot }}$ for both the carbon-rich and oxygen-rich differential spectra.

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For the oxygen-rich input stellar spectra we use the BaSeL (Lejeune et al. 1997) spectral library with effective temperatures ranging from 2000 to 4000 K. For the carbon-rich models we use the Aringer (Aringer et al. 2009) spectral library with effective temperatures ranging from 2400 to 4000 K. Stars hotter than ∼4000 K are not expected to contain significant circumstellar dust shells.

A key characteristic of the dust is quantified through the value of the extinction coefficient, κ,

$$\begin{eqnarray}&&\kappa =\frac{{{n}_{d}}\pi {{a}^{2}}{{Q}_{{\rm ext}}}}{{{\rho }_{d}}},\end{eqnarray} \tag{ 1 }$$

where ${{Q}_{{\rm ext}}}$ is the extinction efficiency factor, a is the grain size, ${{\rho }_{d}}$ is the internal grain density, and n_d is the spatial number density of a particular grain species in a dust mixture. The final κ for each grain mixture is the sum of the κ values for each component in that mixture. We fix ${{Q}_{{\rm ext}}}=0.1$ throughout, motivated by Suh (1999), and the grain densities are adopted from Bressan et al. (1998; see Table 1).

Throughout this work we adopt a grain size distribution that is a delta function at 0.1 $\mu {\rm m}$. This is a common, though rather simplistic assumption (e.g., Suh 1999, 2002; Piovan et al. 2003). We tested the impact of adopting different grain size distributions in the DUSTY models and found that we can obtain similar emergent SEDs for different grain size distributions by varying ${{\tau }_{{\rm AGB}}}$ (see the top panel of Figure 2).

In our model the parameters κ, the dust-to-gas ratio, and ${{R}_{{\rm in}}}$ depend on whether the star is oxygen-rich or carbon-rich. To compute κ we need to assume a certain dust composition. It is not currently possible to compute κ from first principles owing to various uncertainties such as the detailed properties and evolution of circumstellar dust shells and dust grain formation and destruction processes, although efforts along these lines are currently ongoing (e.g., Jones et al. 2012; Nanni et al. 2013; Dell'Agli et al. 2014; Schneider et al. 2014; Ventura et al. 2014).

As Cassarà et al. (2013) discuss in detail, we expect the dust composition to change as a function of ${{\tau }_{{\rm AGB}}}$. In principle, this would require us to compute κ and ${{\tau }_{{\rm AGB}}}$ iteratively (Piovan et al. 2003). However, AGB stars can have diverse features from one another and it is unlikely that any one set of uniform models will fit all AGB stars equally. Since we have no a priori model for how the grain properties should vary with stellar properties, we choose to adopt a relatively simple, observationally motivated scheme.

Sargent et al. (2010) and Srinivasan et al. (2010) fit models to multi-wavelength broadband photometry of AGB stars with grain composition as a free parameter of the model. For oxygen-rich stars, Sargent et al. (2010) found that a grain composition of 100% oxygen-deficient silicates was sufficient to fit the data. However, as an oxygen-rich shell becomes more optically thick we expect the silicates to become colder (Suh 1999, 2002). Therefore, in our grid we adopt a grain type that varies with ${{\tau }_{{\rm AGB}}}$. In our grid, for oxygen-rich stars we use warm silicates for the dust composition up until ${{\tau }_{{\rm AGB}}}$ = 3 when we start to include cold silicates in the mixture. The grain densities, sizes, and Q_ext parameters are all kept constant between the warm and cold silicates. For this reason, the extinction coefficient (Equation (1)) remains the same as the grid transitions from cold to warm silicates.

For carbon-rich stars, Srinivasan et al. (2010) found that a mixture of amorphous carbon (amC) and 10% silicon carbide (SiC) was sufficient. These dust compositions were used for their entire grid of AGB dust models, GRAMS (Sargent et al. 2011 and Srinivasan et al. 2011). Furthermore, we found that changing the amC and SiC ratio made only a modest difference in our results so we decided to keep the grain mixture for carbon-rich stars constant at 90% amC and 10% SiC for all values of ${{\tau }_{{\rm AGB}}}$.

Despite the simplicity of our dust compositions we find that they are sufficient for modeling individual AGB stars. We demonstrate this in Figure 3 where we show how the DUSTY models compare to the SEDs of well-studied AGB stars HV 5715, SSTISAGE1C J052206.92, LPV 28579, and CW Leo. For the first three stars the data is as presented in Sargent et al. (2010) and Srinivasan et al. (2010) with UBVI (Zaritsky et al. 1997, Magellanic Clouds Photometric Survey), JHK (Skrutskie et al. 2006, 2MASS), Spitzer IRAC and MIPS bandpasses (Meixner et al. 2006, SAGE), and Spitzer IRS spectroscopy (Kemper et al. 2010, SAGE-Spec). CW Leo is an extensively observed object. Here we fit optical to IR photometry (as presented in Groenewegen et al. 2012)—gri (Ahn et al. 2012, Sloan Digital Sky Survey), VRI (Le Bertre 1987), JHKLM (Le Bertre 1992), IRAS (Beichman et al. 1988), AKARI (Ita et al. 2008, AKARI IRC Survey of the Large Magellanic Cloud), and SPIRE and PACS bandpasses (Groenewegen et al. 2011, Mass-loss of Evolved StarS).

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HV 5715, SSTISAGE1C J052206.92, LPV 28579, and CW Leo are all oxygen-rich AGB stars and LPV 28579 is carbon-rich. Sargent et al. (2011) presented detailed models of HV 5715 and SSTISAGE1C J052206.92, Srinivasan et al. (2011) modeled LPV 28579, and Groenewegen et al. (2012) modeled CW Leo. The models used in Figure 3 were made using the default parameters of our model grid and only allowing ${{T}_{{\rm eff}}}$ and ${{\tau }_{{\rm AGB}}}$ to vary. The shape of the observed SEDs and the observed features are fit well by the models. This comparison provides a test of our model grid both as a function of ${{T}_{{\rm eff}}}$, ${{\tau }_{{\rm AGB}}}$, and whether the star is oxygen- or carbon-rich. We emphasize however that these fits are not unique and there are many degeneracies between e.g., the shell density profile, ${{\tau }_{{\rm AGB}}}$, grain size distribution, etc.

The temperature of the dust at the inner radius, ${{R}_{{\rm in}}}$, of the shell, T_c, is an important source of uncertainty in the modeling. Table 2 summarizes the different values for T_c adopted in the literature and demonstrates the range of values considered by various authors. Our choice for T_c was based on fitting the data from Martini et al. (2013) (as presented in Section 4).

We compared the dust models of individual stars for the different T_c values to the stars shown in Figure 3 to check that on an individual basis the values gave reasonable agreement to AGB stars. In Figure 3 we compare the different models for the range of T_c values to CW Leo. Since CW Leo is the most dust enshrouded AGB star of our sample any effects due to changes to the input parameters will be amplified compared to the other stars in the sample. These comparisons lead us to adopt ${{T}_{c}}=700$ K for the oxygen-rich grid. We emphasize that T_c could in principle vary with stellar type (i.e., with ${{T}_{{\rm eff}}}$), metallicity, etc., and our choice of a constant value for T_c may introduce systematic uncertainties in the final model results.

The grid and the code used to generate the grid are publicly available.³

With grids of circumstellar AGB dust emission as a function of optical depth, ${{T}_{{\rm eff}}}$, and C/O taken from the dustless isochrones, our goal now is to connect the grids to stellar isochrones by computing the value of ${{\tau }_{{\rm AGB}}}$ at each isochrone point.

The optical depth, ${{\tau }_{{\rm AGB}}}$, is the key quantity that connects the stellar parameters provided by the isochrones to the models of the dust shells. Computing ${{\tau }_{{\rm AGB}}}$ from stellar parameters has been discussed extensively in the literature beginning with the work of Vassiliadis & Wood (1993) and Habing et al. (1994). Subsequent work by Bressan et al. (1998), Piovan et al. (2003), and Cassarà et al. (2013) refined this technique to incorporate dusty envelopes into their SPS codes. In this work, we largely follow these previous efforts in coupling circumstellar dust models to stellar isochrones. We will be brief in our description of computing ${{\tau }_{{\rm AGB}}}$ as the former three papers provide extensive details of the derivations of the following equations.

We start with an initial equation for ${{\tau }_{{\rm AGB}}}$ that is derived assuming spherical symmetry and by integrating over the thickness of the shell while assuming that ${{R}_{{\rm out}}}\gg {{R}_{{\rm in}}}$, where ${{R}_{{\rm out}}}$ and ${{R}_{{\rm in}}}$ are the outer and inner radii of the dust envelope:

$$\begin{eqnarray}&&{{\tau }_{{\rm AGB}}}=\frac{\delta \dot{M}\kappa }{4\pi {{v}_{{\rm exp} }}}\frac{1}{{{R}_{{\rm in}}}}.\end{eqnarray} \tag{ 2 }$$

In Vassiliadis & Wood (1993) they used observationally estimated mass-loss rates of Galactic Mira variables and OH/IR stars to empirically determine equations for $\dot{M}$, pulsation period (P), and v_exp:

$$\begin{eqnarray}&&\frac{{{v}_{{\rm exp} }}}{{\rm km}\;{{{\rm s}}^{-1}}}=-13.5+0.056\frac{P}{{\rm days}},\end{eqnarray} \tag{ 3 }$$

with an additional condition that v_exp lie in the range of 3–15 km s⁻¹, and:

$$\begin{eqnarray}&&{\rm log} \frac{P}{{\rm days}}=-2.07+1.94\;{\rm log} \frac{R}{{{R}_{\odot }}}-0.9\;{\rm log} \frac{M}{{{M}_{\odot }}}.\end{eqnarray} \tag{ 4 }$$

The equations for $\dot{M}$ are a function of initial mass of the star and whether the star is in a super-wind phase, defined by Vassiliadis & Wood (1993) as stars with $P\gt 500$ days. When not in the super-wind phase the mass-loss rate for a star with an initial mass $\lt 2.5\;{{M}_{\odot }}$ is described by a simple relation with pulsation period,

$$\begin{eqnarray}&&{\rm log} \dot{M}=-11.4+0.0123\;P,\end{eqnarray} \tag{ 5 }$$

where P is in days, and $\dot{M}$ is in ${{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}$, and for stars with initial mass $\gt 2.5\;{{M}_{\odot }}$,

$$\begin{eqnarray}&&{\rm log} \dot{M}=-11.4+0.0125\left[ P-100\left( \frac{M}{{{M}_{\odot }}}-2.5 \right) \right].\end{eqnarray} \tag{ 6 }$$

In the super-wind phase the mass-loss rate is described as,

$$\begin{eqnarray}&&\dot{M}=\frac{1}{c\;{{v}_{{\rm exp} }}}\frac{L}{{{L}_{\odot }}}.\end{eqnarray} \tag{ 7 }$$

where c and v_exp are in units of km s⁻¹.

The inner radius, ${{R}_{{\rm in}}}$, is derived by equating the stellar luminosity with the luminosity at the inner radius, remembering that the inner radius is defined as the radius where the temperature is equal to the dust condensation temperature, T_c,

$$\begin{eqnarray}&&{{R}_{{\rm in}}}={{\left[ \frac{L}{4\pi T_{{\rm c}}^{4}} \right]}^{\frac{1}{2}}}.\end{eqnarray} \tag{ 8 }$$

We adopt our T_c values from Table 2 to obtain a final relation for carbon-rich stars,

$$\begin{eqnarray}&&{{R}_{{\rm in}}}=1.92\times {{10}^{12}}{{\left( \frac{L}{{{L}_{\odot }}} \right)}^{\frac{1}{2}}}\;{\rm cm},\end{eqnarray} \tag{ 9 }$$

and for oxygen-rich stars,

$$\begin{eqnarray}&&{{R}_{{\rm in}}}=4.74\times {{10}^{12}}{{\left( \frac{L}{{{L}_{\odot }}} \right)}^{\frac{1}{2}}}\;{\rm cm}.\end{eqnarray} \tag{ 10 }$$

The dust-to-gas ratio, δ, is obtained by inverting the observed correlation between it and v_exp found in Habing et al. (1994),

$$\begin{eqnarray}&&\delta =\frac{{{\delta }_{{\rm AGB}}}\;v_{{\rm exp} }^{2}}{225}{{\left( \frac{L}{{{10}^{4}}{{L}_{\odot }}} \right)}^{-0.06}},\end{eqnarray} \tag{ 11 }$$

where we adopt ${{\delta }_{{\rm AGB}}}$ = 0.01 for oxygen-rich stars (Suh 1999) and ${{\delta }_{{\rm AGB}}}$ = 0.0025 for carbon-rich stars (Blanco et al. 1998). Note that in the following we assume that δ does not depend on metallicity. See Section 2.3 for details.

With the above equations we can now explore the behavior of ${{\tau }_{{\rm AGB}}}$ along the isochrones as a function of metallicity and age. This is shown in Figure 4 where we show the number of AGB stars as a function of luminosity (top) and their corresponding ${{\tau }_{{\rm AGB}}}$ values (bottom). In this figure we see that the knee of the luminosity functions (LFs) coincide with the upturn of the ${{\tau }_{{\rm AGB}}}$ values. This indicates that brighter stars are both rarer and have larger values of ${{\tau }_{{\rm AGB}}}$. Only for these rare stars is there an appreciable level of dust obscuration.

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The models are now included in the FSPS population synthesis code (Conroy & Gunn 2010) as of v2.5.⁴ FSPS computes ${{\tau }_{{\rm AGB}}}$ as per the equations above for identified AGB stars in the isochrones. With ${{\tau }_{{\rm AGB}}}$, ${{T}_{{\rm eff}}}$, and the composition of the stellar envelope, FSPS interpolates within the grid of DUSTY models and grafts the interpolated model onto the stellar SED.

We briefly review here the salient characteristics of the FSPS stellar population synthesis model. FSPS takes as input a set of stellar isochrones (here we use the Padova models detailed in Marigo et al. 2008), stellar spectral libraries (in our case the BaSeL library), allows the user to specify a stellar IMF, and outputs simple stellar populations (SSPs), and, if a SFH is specified, composite stellar populations. FSPS includes models for diffuse dust absorption and emission. The model has been extensively calibrated and tested against observations, see Conroy & Gunn (2010) for details.

When AGB dust is turned on in FSPS, ${{\tau }_{{\rm AGB}}}$ is computed at each isochrone point using the method described above. Fundamentally, ${{\tau }_{{\rm AGB}}}$ depends on the mass and age of the star.

To first order, the seed elements relevant for grain formation in oxygen-rich environments, such as O, Si, and Fe, will vary in proportion to the metallicity of the star. It therefore might seem reasonable for the dust-to-gas ratio to increase linearly with the metallicity of the star (e.g., Habing et al. 1994).

However, recent models attempting to follow the formation and dust grains self-consistently as a function of AGB mass and metallicity have revealed a complicated relation between the dust-to-gas ratio and metallicity (e.g., Ferrarotti & Gail 2006). Schneider et al. (2014) detailed the complicated factors that contribute to the dust production of AGB stars by comparing the observed dust production rates of AGB stars in the Large and Small Magellanic Clouds to theoretical models. Schneider et al. (2014) noted that efficiency of the Hot Bottom Burning (HBB), the TDU, and the mass of the star all contribute to the efficiency of the dust production for AGB stars. And, as Ventura et al. (2014) noted, the efficiency of these mechanisms are influenced by the modeling of convection in the star. Discrepancies between the theoretical dust yields and observed dust yields found in Schneider et al. (2014) highlight that we do not have a solid predictive model for dust formation in AGB winds. Moreover, current models suggest that dust production could be a complex, non-monotonic function of metallicity. Recent observational studies are consistent with this viewpoint (e.g., Gerke & Kochanek 2013). For these reasons we decided to adopt a simple approach and keep the dust-to-gas ratio fixed for all oxygen-rich stars, independently of metallicity.

For carbon-rich stars the seed elements for the grains (carbon) are made in situ in the star, so one would expect that the dust-to-gas ratio to have little dependence on the original metallicity of the star. Hence also for carbon-rich stars we keep the dust-to-gas ratio fixed, independent of metallicity.

In this section we first describe the general conditions in which circumstellar dust may affect the integrated light of simple and complex stellar populations (Section 4.1), then move to discussing two examples in which there is evidence for the impact of circumstellar dust (early-type galaxies in Section 4.2 and low-metallicity dwarf galaxies in Section 4.3), and one example in which circumstellar dust appears to play no important role (actively star-forming galaxies, in Section 4.4).

With the new circumstellar dust models we can calculate revised SEDs and compare them to models without circumstellar dust. In Figure 8 we show model SEDs with (solid lines) and without (dashed lines) dust shells around AGB stars for ages of 0.2, 2, and 10 Gyr for $Z=0.2\;{{Z}_{\odot }}$ and $Z={{Z}_{\odot }}$. The circumstellar dust substantially increases the model flux red-ward of $\sim 4\;\mu {\rm m}$. At old ages the influence of the AGB dust is sharply diminished, having little discernible impact on the SED. We note here that the models in this figure contain no diffuse dust associated with the interstellar medium (see below).

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The trends with age and metallicity in Figure 8 can at least partially be understood by considering separately the effects of carbon-rich and oxygen-rich stars on the integrated SED. We explore this in Figure 9, which shows the fractional contribution of oxygen-rich and carbon-rich stars to ${{L}_{{\rm bol}}}$ (top panels) and the relative contribution of oxygen-rich and carbon-rich stars to the $24\;\mu {\rm m}$ flux (bottom panels). Note that we are using Padova stellar models here.

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We see the well-known fact that the appearance of carbon-rich stars is both metallicity and age dependent. In the top row of Figure 9 we see that for the lowest metallicity model, $Z=0.1\;{{Z}_{\odot }}$, carbon-rich stars dominate ${{L}_{{\rm bol}}}$ for a longer period of time than the higher metallicity models. Moreover, carbon-rich stars do not exist at old ages, $9.3\lesssim {\rm log} t/{\rm yr}\lesssim 10$ for high metallicities. The age dependence is complicated, depending on processes such as the TDU and HBB. Detailing when and under what conditions carbon-rich stars are produced is outside the scope of this work but it is a much discussed subject in the literature (e.g., Iben & Renzini 1983; Di Criscienzo et al. 2013, and the references therein). It is important to understand because at certain ages and metallicities carbon-rich stars are dustier and therefore redder than oxygen-rich stars, as demonstrated in the lower panels of Figure 9. We see in Figure 9 that carbon-rich stars have a disproportionate influence on the $24\;\mu {\rm m}$ flux relative to their overall ${{L}_{{\rm bol}}}$ contribution. The lack of carbon-rich stars at old ages, combined with the generally lower AGB luminosities at late times (see Figure 4) explains why circumstellar dust emission from AGB stars has a relatively modest effect on the integrated light SED at old ages.

In Figure 10 we show how AGB dust affects the NIR and mid-IR integrated light of composite stellar population models as a function of age for different metallicities (note that this refers to populations as a whole, rather than the metallicity of individual stars discussed in Section 2.1, which will have an effect on the SPS models), SFHs (represented by exponential models with an e-folding time given by ${{\tau }_{{\rm SF}}}$), and diffuse dust content (the contribution of dust around young stars and dust in the ISM). In this figure four wavelength ranges are shown and in each panel we compute the ratio of luminosities between models with and without circumstellar AGB dust.

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The diffuse dust model is a combination of the Charlot & Fall (2000) model for dust attenuation, with a power-law attenuation curve (with ${{\tau }_{{\rm diff}}}$ quoted at 5500 Å and the power-law index, $\alpha =-0.7$ unless stated otherwise) and the Draine & Li (2007) model for diffuse dust emission. The overall amount of diffuse dust emission is determined by energy conservation. In this paper, models that include diffuse dust also include dust around young stars, the so-called "birth cloud component", where ${{\tau }_{{\rm bc}}}=3{{\tau }_{{\rm diff}}}$. Note that the Draine & Li (2007) dust models include both thermal emission from cold dust and PAH emission. The models are described by three parameters ${{q}_{{\rm PAH}}}$, ${{U}_{{\rm min} }}$, γ. Unless otherwise stated we adopt 3.5%, 1.0, 0.01 respectively for these values.

We see in all models that only after ∼0.1 Gyr does AGB dust become influential, peaking at about 1 Gyr, with a significant reduction by the time the galaxy reaches 10 Gyr in age. The influence of AGB dust on the SED varies in interesting ways as a function of wavelength, metallicity, and SFH, but by far the most influential parameter is the amount of diffuse dust.

For galaxies where there is no diffuse dust the impact of the dust around AGB stars is significant and extends to wavelengths as short as $\sim 4\;\mu {\rm m}$. However, even a modest amount of diffuse dust significantly diminishes the relative influence of AGB dust at $\gtrsim 8\;\mu {\rm m}$. At shorter wavelengths the effect of AGB dust dominates over that of diffuse dust, since in our model the diffuse dust component is generally not hot enough to emit significantly at $\lt 8\;\mu {\rm m}$. We investigate this in more detail in Figure 11 where we show the effect of AGB dust at 4.5 and 24 $\mu {\rm m}$ as a function of the amount of diffuse dust for several ages. At 4.5 $\mu {\rm m}$ the AGB flux is relatively insensitive to the diffuse dust content but at 24 $\mu {\rm m}$ the diffuse dust greatly reduces the influence of the emission from the AGB dust shells. From ∼1.0 to 10.0 Gyr the AGB dust has a significant influence of the mid-IR light only for low diffuse dust values and, consistent with the previous figures, drops off as the diffuse dust becomes more prominent. We therefore conclude that circumstellar dust around AGB stars will be important in regimes where the diffuse dust optical depth is roughly $\lesssim 0.1$.

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We also investigate the effect of AGB dust on optical–NIR and NIR–mid-IR colors in Figure 12. The difference is not large for $V-{{K}_{s}}$, but this color is commonly employed in high-precision SED modeling of galaxies and so even a small change could potentially induce large differences in derived parameters. The models that include AGB dust tend to be bluer in $V-{{K}_{s}}$ than models that do not for a range of metallicities and ages. This is because AGB stars, with or without dust shells, contribute essentially zero flux in the V band and significant flux in ${{K}_{s}}$. Dusty AGB stars can be optically thick even in the ${{K}_{s}}$ band, resulting in a lower flux and hence bluer $V-{{K}_{s}}$ color. In the middle and bottom panels of Figure 12 we see that the models with AGB dust are significantly redder than models without for optical–mid-IR and mid-IR–mid-IR colors.

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Early-type galaxies are potential candidate systems where the effect of AGB dust may be important, owing to their generally low diffuse dust content. They are known to host intermediate and old age populations, $1\lt t\lt 10$ Gyr, based on optical absorption line spectroscopy (e.g., Trager et al. 1998, 2005; Thomas et al. 2005; Jimenez et al. 2007; Conroy et al. 2014). While many early-type galaxies do show evidence for diffuse dust at low levels (e.g., Lauer et al. 2005), the origins of which are actively debated, there exists galaxies with only upper limits in the FIR, suggesting that they have very little diffuse dust (e.g., Jura et al. 1987; Knapp et al. 1989; Temi et al. 2009; Martini et al. 2013).

Athey et al. (2002) presented a mid-IR spectrum from the Infrared Space Observatory (ISO) for the early-type galaxy NGC 1404. These authors found a clear excess in the data at 8–14 μm compared to a blackbody extrapolation. They attributed this excess to O-rich AGB features. In Figure 13 we show their data for NGC 1404 compared to our models with no dust (blue line) and three models with AGB dust for different ages (red lines). All models assume the galaxy has a metallicity of ${{Z}_{\odot }}$.

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First, we see that by including AGB dust in the model the fits to the data are greatly improved at 8–12 μm. Furthermore, we see from the three models with AGB dust the mid-IR can be a powerful age discriminator, as already discussed in Bressan et al. (1998), Bregman et al. (2006), and Athey et al. (2002), for simple systems with only one burst of star formation. We find that a model with AGB dust and an age of ∼7.5 Gyr provides the best fit to the $10\;\mu {\rm m}$ feature. This is in good agreement with the 9 Gyr estimated by Bregman et al. (2006), who used the AGB dust models from Piovan et al. (2003) to estimate galaxy ages based on mid-IR SEDs. This age is also in agreement with an optical spectroscopically estimated age for this galaxy of 7 Gyr (C. Conroy & P. van Dokkum 2015, in preparation).

Interpretation of the mid-IR data for NGC 1404 is complicated by the fact that this galaxy is apparently not entirely devoid of diffuse dust. It is detected by Herschel at 70 and $160\;\mu {\rm m}$ (Temi et al. 2009), indicating the presence of cold dust. However, we attempted to fit the data from Athey et al. (2002) with only diffuse dust (no AGB dust) and found that such a model could not simultaneously fit the mid-IR and FIR data (see also Figure 15). Therefore, we conclude that circumstellar AGB dust appears to be necessary to fit the mid-IR data for this galaxy.

Martini et al. (2013) recently published mid-IR photometry of early-type galaxies from Spitzer's IRAC and MIPS bands. Martini et al. (2013) divided the sample based on whether there were diffuse dust lanes detected in HST imaging and detected at 70 and 160 $\mu {\rm m}$. Rampazzo et al. (2013) also published an atlas of mid-IR spectroscopy from Spitzer-IRS of early-type galaxies that they sort into classifications based on the contribution of old stellar populations to the mid-IR continuum, where class 0 are galaxies where the old stellar populations are most significant.

In Figure 14 we compare the data for galaxies that did not have detected dust lanes and were classified as class 0 in Rampazzo et al. (2013) to models that include (solid-red) and do not include (dashed-blue) AGB dust. All models plotted were made for a solar metallicity, single-age stellar population. The data were normalized to the flux at $5.72\;\mu {\rm m}$, which should eliminate aperture size issues between IRAC and IRS. The ages of the galaxies were estimated by hand to provide a satisfactory fit to the 3–20 μm data. The model that includes AGB dust fits the 3.6–8.0 μm data just as well as the model that does not, only for $\gtrsim 8.0\;\mu {\rm m}$ are there significant differences between models with and without circumstellar dust (for these ages). The models provide a good fit to the mid-IR spectra from Spitzer-IRS. In particular, the overall shape, from 5 to 20 μm, is reproduced well and the relative strength and positions of the broad 10 and $18\;\mu {\rm m}$ dust features are generally well matched.

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In Figure 15 we compare the observations of NGC 2300 with both the AGB dust model and two models that includes only diffuse dust. The fundamental difference between the latter two models is the adopted value of ${{q}_{{\rm PAH}}}$. In one case we adopt the fiducial value while in the other we adopt an unreasonably high value of 10%.⁵ We added as much diffuse dust into the models as possible without exceeding the constraints of the FIR flux at 70 and $160\;\mu {\rm m}$. It is clear that the model with only diffuse dust cannot reproduce the 8–24 μm features seen in early-type galaxies with influential old stellar populations. Varying the diffuse dust parameters ${{q}_{{\rm PAH}}}$ and ${{U}_{{\rm min} }}$ does not qualitatively improve the fits.

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The slight over prediction of the strength of the $10\;\mu {\rm m}$ emission feature is likely due to our treatment in the dust grain properties. We reiterate that our grain scheme is relatively simplistic and there is an abundance of evidence that the actual grain mixtures in AGB stars, and their dependence on the properties of the star, are more complicated than represented in the models presented here (e.g., McDonald et al. 2010; Jones et al. 2012). Other groups producing similar dust shell models have incorporated more detailed approaches to the grain schemes. For example, Cassarà et al. (2013) detail a sophisticated approach to the grain mixtures that accounts for changes in ${{\tau }_{{\rm AGB}}}$ and $\dot{M}$ and Groenewegen (2006) include aluminum oxide dust along with the more standard silicate in oxygen-rich AGB stars. In future work we will explore the effect of different grain species on the mid-IR spectra of early-type galaxies.

Low-metallicity galaxies are another class of objects in which circumstellar dust may significantly contribute to the integrated SED owing to their generally low diffuse dust content. Johnson et al. (2013) compared observations for a subset of ANGST galaxies with SPS models. They found and discussed the possible causes of a 0.2 dex under-prediction by the models compared to Spitzer data from the LVL survey presented in Dale et al. (2009) (see Figure 5 in Johnson et al. 2013). They considered the possibility that circumstellar dust around AGB stars, not included in the SPS models, may be the source of the discrepancy.

In Figure 16 we plot the offset between observed IRAC 8 $\mu {\rm m}$ and MIPS $24\;\mu {\rm m}$ magnitudes and predicted magnitudes for models with (blue squares) and without (red circles) AGB dust as a function of metallicity. For about half the galaxies, the metallicities are well constrained by nebular emission lines (Berg et al. 2012). For the rest of the sample, metallicities are estimated the mass–metallicity relationship (Lee et al. 2006; Berg et al. 2012); the resulting metallicities that are broadly consistent with those estimated from HST CMDs (Weisz et al. 2011). For each galaxy, the corresponding model has the same SFH and metallicity. The model photometry for each galaxy is based on the estimated SFH for each galaxy derived from CMDs; see Johnson et al. (2013) for details. The default models do not contain any diffuse dust.

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In both panels we include linear fits to both sets of offsets; in general the offset is reduced with the inclusion of AGB dust. For the $8.0\;\mu {\rm m}$ flux, the average difference between observations and models is close to zero. However, for the $24\;\mu {\rm m}$ flux, while there is some improvement in the offset in all the galaxies, a significant discrepancy remains even after including AGB dust.

We see a very strong correlation between metallicity and the size of the offsets at $24\;\mu {\rm m}$. The origin of the correlation is unclear to us. In Figure 17 we compare the residuals between the observations and models that only include circumstellar AGB dust and models that include AGB dust and diffuse dust at $24\;\mu {\rm m}$ (${{q}_{{\rm PAH}}}=0.05$ and dust attenuation index $\alpha =-1.3$). We see that by including diffuse dust we are able to better capture the $24\;\mu {\rm m}$ flux. However, the correlation with metallicity remains. This indicates that joint modeling of UV–IR SEDs is necessary for these galaxies to fully understand their characteristics.

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Finally, we consider the effect of circumstellar dust on the SEDs of actively star-forming galaxies.

Kelson & Holden (2010) suggested that AGB stars may be responsible for a substantial fraction of the mid-IR luminosities in galaxies from z = 0 to z = 2. In Figure 18 we show color–color diagrams of galaxies at $z\sim 1$ from the FIREWORKS catalog (see Wuyts et al. 2008 for details; also see Figure 2 from Kelson & Holden 2010). In each panel we include three models: a model with the standard amount of AGB dust ("$1\;\times$ AGB dust") where the amount of AGB dust included in the model is as described in Section 2.2; a model with a much larger amount AGB dust ("$100\;\times$ AGB dust") where we multiplied the standard dust optical depth by 100, and a model with no AGB dust but a modest amount of diffuse dust (${{\tau }_{{\rm diff}}}=0.05$). Each model assumes $Z={{Z}_{\odot }}$, ${{\tau }_{{\rm SF}}}=2.0$ Gyr, and uses a delayed τ-model to be as similar as possible with the models from Kelson & Holden (2010). In the first panel we see no difference among the different models. This is expected since only optical colors are plotted and no dust will effect the flux at these wavelengths. This plot serves as a check that the models we have adopted are a reasonable description of the data and that they span the full range of optical–optical colors.

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The second panel is an optical-mid-IR color–color plot and we see significant differences among the models. The standard AGB model is able to reproduce only a small fraction of the galaxy colors but fails to span the whole extent of the color range. Even when boosting the AGB dust optical depth by a factor of 100 (far beyond what is plausible) we are unable to reproduce the full color range of the data. Furthermore, reducing the metallicity of the model to $Z=0.4\;{{Z}_{\odot }}$ and thereby increasing the number of carbon stars (see Figure 9), and hence the MIR flux, only modestly improves the fits of the model to the data. However, a modest amount of diffuse dust can readily explain the full range of mid-IR colors. It therefore seems unlikely that AGB dust alone can explain the full range of the mid-IR colors.

This issue has also been discussed in Melbourne & Boyer (2013) where they compared the mid-IR colors of AGB stars in the SMC and LMC with the colors of AGB stars adopted for the models presented in Kelson & Holden (2010). Melbourne & Boyer (2013) concluded that the average AGB color adopted by Kelson & Holden (2010) were significantly redder than the observed average AGB colors. This is due to the fact that Kelson & Holden (2010) used heavily dust enshrouded AGB stars as representative of more typical AGB stars. Furthermore, Kelson & Holden (2010) used the Maraston (2005) stellar population models, which appear to assign too much weight to the TP-AGB phase (Kriek et al. 2010; Zibetti et al. 2013).

The results of this section are consistent with our findings from Section 4.1 as well as the findings of Melbourne & Boyer (2013). Actively star-forming galaxies generally contain significant quantities of diffuse dust that tends to overwhelm the effect of circumstellar dust in the mid-IR.