RECONSTRUCTING THE STELLAR MASS DISTRIBUTIONS OF GALAXIES USING S⁴G IRAC 3.6 AND 4.5 μm IMAGES. II. THE CONVERSION FROM LIGHT TO MASS

As described in Paper I, we assume that emission in each pixel mapped at each of two frequencies ν_i is a linear combination of the emission from N = 2 different sources (stellar and nonstellar) such that, for P samples (i.e., image pixels) in M = 2 frequency channels,

$$\begin{equation} \mathbf {x}=\mathbf {A}\cdot \mathbf {s}, \end{equation} \tag{ 1 }$$

where x is the M × P measurement set, A is an M × N mixing matrix containing the coefficients of the combination, and the N × P matrix s represents the source signals, under the assumption that the sources S are separable functions of location and frequency (i.e., s_j = S_j(x, y) and A_{i, j}∝S_j(ν)).

Given our two input channel 1 and 2 images x_i, we can apply ICA to extract measurements of at most two signals s_j (each representing either contaminants or emission from the true old stellar disk) in each of P pixels while simultaneously solving for the separation coefficients $A^{-1}_{ij}$.

Like principal component analysis, ICA extracts solutions via transformation of the input data, but the sources are statistically independent rather than orthogonal (with zero covariance). This grants greater uniqueness to ICA solutions, promising improved descriptions of sources that are the result of distinct physical processes.

Perhaps the strongest advantage of ICA is that it allows us to retain and exploit information about the different spectral shapes of the stars and dust between 3.6 and 4.5 μm, rather than requiring that a priori assumptions about the properties of either the stars or the contaminating emission (e.g., imposing a particular set of separation coefficients $A^{-1}_{ij}$ given some empirical recipe). This is a particularly favorable alternative to (1) subtracting scaled versions of the 8 μm image (which requires prior knowledge of the relation between the 3.3 μm PAH emission feature and the 8 μm and how, in particular, this varies depending on the size and ionization state of the dust) or (2) a simple scaling and subtraction of the 3.6 and 4.5 μm images. Given the difference in SED shapes for the old stars and dust, there is not a single scaling between the emission at 3.6 μm and at 4.5 μm that can be applied uniformly to all galaxies throughout the Hubble sequence. Combining the two images would furthermore require the loss of the [3.6]–[4.5] color, a key diagnostic of the properties of the old stellar population. By avoiding such measures, we gain a unique tracer of the nonstellar emission (and the thermal and nonthermal sources of dust heating; paper I) as well as a measure of the color of the old stars, which we leverage in this paper to assign the M/L_3.6.

In addition, ICA in principle allows much more information in individual structural components, like spiral arms, to be maintained in contaminant-cleaned maps than is supplied by other commonly employed mass estimation methods. Alternative schemes for the identification or correction of these features in IRAC images (Kendall et al. 2008; Hunter et al. 2006; Bolatto et al. 2007; de Blok et al. 2008) either require the loss of 2D information or employ auxiliary data (e.g., 8 μm, UV, optical images) that is not uniformly available for the full S⁴G sample. The ICA uses only the information from the first two IRAC bands and thus provides a correction for nonstellar emission that can be made uniformly throughout the sample.

As outlined in Paper I (and described in detail by M. Querejeta et al., in preparation), our application of ICA with additional postprocessing yields two maps: a corrected map of the old stellar light (a clean, smoothed version of the uncorrected image) and a map of significant contaminating nonstellar emission (containing most, if not all, of the bright and knotty features tracing, e.g., star formation in spiral arms).

2.2.1. Nonstellar Emission

2.2.2. Old Stellar Light Maps

As in other NIR bands, the light at 3.6 μm is dominated by the old stars that populate the red giant branch (RGB). This furnishes the powerful advantage that the stellar M/L, which must reflect the mix of these giants stars with the underlying mass-dominant dwarf stars, varies much less with age/SFH and metallicity than at optical wavelengths (Bell & de Jong 2001).

Consider that, in a simple stellar population, as stars evolve off the main sequence (and the number of stars left on the main sequence decreases) the RGB is populated by later and later spectral types with decreasing M/L. The M/L of the stars remaining on the main sequence meanwhile increases, weighted more toward the higher M/L of the late M spectral class. At the same time, the fraction of giants to dwarfs increases because the average time spent on the RGB is longer than their average main-sequence lifetime. These factors keep the NIR M/L for the oldest stars free of the relatively high degree of variation characteristic of optical wavelengths as the population evolves. Together with the reduced sensitivity to dust extinction, this is the primary reason these bands are favored as direct stellar mass tracers.

The photometric properties of stellar populations, which progressively redden as the RGB becomes dominant,²⁰ encode this difference in stellar M/L with wavelength. By construction, the mass-to-light ratio at 3.6 μm is

$$\begin{equation} \Upsilon _{3.6}=\Upsilon _{\lambda }\frac{10^{-0.4(m^\lambda -m^{3.6})}}{10^{-0.4(M_{\odot }^{\lambda }-M_{\odot }^{3.6})}}, \end{equation} \tag{ 2 }$$

where, at any wavelength λ, ϒ_λ is the mass-to-light ratio, m^λ is apparent magnitude, and $M_{\odot }^\lambda$ is the absolute magnitude of the sun. The prominence of the RGB that leads to red optical–NIR colors ensures a smaller ϒ_3.6 than in the optical. For old stellar populations (redder than the Sun), the variation in ϒ_3.6 compared to ϒ_λ will also tend to be smaller because

$$\begin{equation} \frac{d\Upsilon _{3.6}}{d\Upsilon _{\lambda }}=\frac{10^{-0.4(m_\lambda -m_{3.6})}}{10^{-0.4(M_{\odot }^{\lambda }-M_{\odot }^{3.6})}}\left(1-0.4\frac{\partial (m_\lambda -m_{3.6})}{\partial \beta }\frac{d\beta }{d\log {\Upsilon _{\lambda }}}\right), \end{equation} \tag{ 3 }$$

where variation in parameter β (representing, e.g., age or metallicity) is responsible for the change in the properties of the population, both color and M/L. Even progressing from the H band to 3.6 μm, the variation in the stellar M/L is intrinsically reduced; assuming a color H−[3.6] = 0.4 typical of an old, dust-free population, Δϒ_3.6 will be at most 0.75Δϒ_H.

In the next section, we explore the variation in M/L in more detail using population synthesis models to follow the evolution in the fraction of giant to dwarf stars, for a given (Chabrier) initial mass function (IMF), as well as the dependence of their combined properties on metallicity and SFH. We compare ϒ_3.6 with ϒ_B to underscore the contrast between tracing the RGB and tracing young, main-sequence stars at optical wavelengths, although a more symbolic comparison may be with the I-band M/L. The I band is preferred to the shorter wavelength B band for tracing the stellar mass for the same reasons we prefer 3.6 μm imaging of old stars (Bell & de Jong 2001) (reduced sensitivity to young stars and dust extinction). It has even been argued that the I band is favorable to NIR (and 3.6 μm) bands for estimating stellar mass, given the uncertain contribution from stars in the late (i.e., AGB) stages of stellar evolution (Rix & Rieke 1993; Rhoads 1998; Portinari et al. 2004; along with the contribution from nonstellar emission at 3.6 μm). Here we emphasize that, for old stellar populations, we can expect Δϒ_3.6  ∼ 0.3Δϒ_I, given their typical colors I-[3.6] = 2.2 (I − H = 1.8).

Population synthesis models provide a clean way to explore the dependencies of the observed properties of a stellar population on age/SFH and metallicity and relate these to the stellar M/L. Figure 1 shows the stellar M/L in the optical (ϒ_B) and at 3.6 μm (ϒ_3.6) as a function of two different colors for a set of dust-free synthesized stellar populations assuming a Chabrier IMF. Each population is constructed with an exponentially declining SFH with e-folding time τ from GALAXEV models (Bruzual & Charlot 2003; hereafter BC03) and observed after 13 Gyr or when the population has a mean age A = 13 Gyr–τ.

Zoom In Zoom Out Reset image size

Note first that ϒ_3.6 is characterized by much less variation (a factor of nine less) than ϒ_B over this range of ages and metallicities (see also de Denus-Baillargeon et al. 2013). Second, ϒ_3.6 and ϒ_B show opposite dependencies on metallicity (clearer in the right panel). The young stars that dominate in the optical are bluer and brighter at lower metallicities (due to a combination of reduced line blanketing and opacity), reducing ϒ_B compared to higher metallicities. The metallicity dependence of ϒ_3.6 is reversed because the RGB traced in the NIR is brighter and redder at higher metallicities.

These variations in metallicity are well tracked by the optical–NIR color, which exhibits a weaker dependence on age than the B–R color. In the left panel, we see that age and Z contribute more equally to the B–R color and introduce comparable variation in ϒ_3.6, while their individual influences on ϒ_B are degenerate (Bell & de Jong 2001). The R–H color breaks this degeneracy, revealing similar degrees of variation in ϒ_B due to age and metallicity over the ranges in these parameters considered here. The nearly orthogonal dependencies of ϒ_3.6 on age and metallicity are likewise responsible for similar degrees of variation in ϒ_3.6. But, again, these are about one-tenth as large as in the B band.

These well-known dependencies of optical and optical/NIR colors on population age and metallicity offer the opportunity to estimate the stellar M/L: the B–R color provides a good constraint on ϒ_B (Bell & de Jong 2001), whereas the combination of R-H and B–R provide the potential to estimate ϒ_3.6 (by constraining the metallicity and the age, respectively; as used by Zibetti et al. 2009). But such determinations are at risk of the uncertainties introduced by dust (explored below), to which colors tied to optical bands, in particular, are sensitive.

The task of assigning M/L requires finding and using a color (or colors) to optimally trace these age and metallicity variations. Unfortunately, this task is complicated by the presence of dust, which can reduce the diagnostic power of colors tied to optical bands in disk galaxies, in particular (Bell & de Jong 2001; MacArthur et al. 2004; Z09).

The impact of dust at optical wavelengths is demonstrated in Figure 1 by the extinction vector A_V = 0.3 at the top left, assuming a Galactic extinction law (Cardelli et al. 1989; Indebetouw et al. 2005). Dust both reddens and dims (optical) light, increasing B–R and ϒ_B. But because the dust vector is not exactly parallel to the locus of dust-free stellar population models, the broken age/metallicity–dust degeneracy leads to underestimated masses for young, dust-reddened populations relative to old, dust-free populations. As a result of dust, typical uncertainties on total masses are 0.1–0.2 dex using global colors to assign the stellar M/L (Bell & de Jong 2001), although colors are expected to remain reliable in one dimensions for all but the most massive (and therefore dustiest; e.g., Dalcanton & Bernstein 2002) disks (MacArthur et al. 2004). Because our main concern is not only the total mass but also the 2D mass distribution within the disk, dust extinction can pose a real problem. This is especially true because we are interested in mapping the stellar mass in dusty, star-forming disks with a wide range of masses. In star-forming disk galaxies A_V can locally be as high as 1–3 mag (e.g., Prescott et al. 2007; Kreckel et al. 2013; Ganda et al. 2009) and even higher in edge-on disks (Jansen et al. 1994; Ganda et al. 2009). For a Galactic dust extinction law, even A_V = 1 mag introduces a color excess of 1.5 mag in B–R color, exceeding by far the total amount of variation exhibited by stellar populations with ages between 1 and 12 Gyr.

Although we can expect optical colors to retain their usefulness, e.g., in early-type E/S0 galaxies characterized by lower dust content, for dusty, star-forming galaxies (the bulk of the S⁴G sample) the use of optical colors clearly requires incorporating a realistic model of the dust (MacArthur et al. 2004; Zibetti et al. 2009). So far, the effect of dust on UV-to-optical emission from late-type spiral disks has been well modeled assuming a mix of foreground screen and birth-cloud extinction (e.g., Popescu et al. 2011; da Cunha et al. 2008). Given the variety of dust geometries, which likely vary from position to position in the disk, models can be difficult to implement realistically in two dimensions without knowledge of the actual dust content and properties measured from far-IR data sampling the dust SED. The first implementation of a dust model for the purposes of estimating the stellar M/L by Zibetti et al. (2009) may not be optimal as recent work suggests that stellar reddening is a poor tracer of the overall dust distribution (Kreckel et al. 2013).

In this section, we explore the specific age, SFH, and metallicity dependencies of the [3.6]–[4.5] colors of old stellar populations. The colors of RGB stars may not provide the leverage on age/SFH that the (optical) colors of main-sequence stars can. Even so, between 3.6 and 4.5 μm these colors are less susceptible to the systematic uncertainties plaguing optical colors (e.g., dust extinction/reddening) otherwise used to constrain the M/L. Plus, any variations in M/L due to age will be intrinsically much less than at shorter wavelengths. In this case, the [3.6]–[4.5] colors of old stars may alone provide suitable constraints on the M/L.

Unfortunately, SPS models are so far not well trusted at IRAC wavelengths, where molecular line opacities and the TP-AGB phase of evolution remain weakly constrained in template spectra. Peletier et al. (2012) show the degree to which predictions vary from model to model. We note that the modeled [3.6]–[4.5] colors do not match the observed global colors of galaxies, either E/S0 (Peletier et al. 2012) or later-type (Pahre et al. 2004), which do not exceed [3.6]–[4.5] = 0 and strictly fall in the range −0.15 < [3.5]–[4.5] <−0.02. Models that do not properly include CO absorption in the 4.5 μm wavelength range tend to predict redder [3.6]–[4.5] colors (BC03 and CB07; see 6) or a weaker dependence on metallicity (Marigo et al. 2008; Maraston 2005) than observed.²¹ Even at ages long after the AGB phase, the models have very little predictive power at mid-IR wavelengths. To avoid this issue, in the Appendix we calibrate a relation between the NIR and [3.6]–[4.5] colors of GLIMPSE giants in order to relate the NIR colors of SPS modelswhich successfully reproduce the colors of real galaxiesto realistic [3.6]–[4.5] colors. These better match expectations (based on observation) for the mid-IR colors of old stellar populations.

Figure 2 shows the same BC03 tracks of metallicity and age as shown in the previous section, but now using the relation defined in the Appendix. Notably, the trend in ϒ_3.6 with [3.6]–[4.5] is the reverse of the trend with R–H color in Figure 1; due to CO absorption in the stellar photospheres of giant stars in the 4.5 μm band, the [3.6]–[4.5] color is bluer at higher metallicity. In addition, only the influence of metallicity can account for the full range in [3.6]–[4.5] colors observed in real galaxies (as measured, e.g., by Pahre et al. 2004). Peletier et al. (2012) find this same dependence of [3.6]–[4.5] on metallicity in early-type galaxies, linking the relation between [3.6]–[4.5] and galaxy velocity dispersions in the SAURON sample to a mass–metallicity relation, and also linking [3.6]–[4.5] directly to absorption lines like Mgb and Fe. With our GLIMPSE-based calibration to relate [3.6]–[4.5] color to M/L it is now clear that metallicity is in fact the primary driver of [3.6]–[4.5], the age dependence of [3.6]–[4.5] being weak. Indeed, age variation introduces a limited span of only ∼0.002 mag in [3.6]–[4.5] color from τ = 2–12 Gyr at fixed metallicity (see Figure 2). Variation in metallicity across the range 0.0004 < Z/Z_☉ < 0.05, on the other hand, can account for colors spanning −0.08 ≲ [3.6]–[4.5] < −0.02.

Zoom In Zoom Out Reset image size

Over the full range of [3.6]–[4.5] probed here, metallicity introduces a change in M/L by 0.2 dex. At fixed metallicity, age variations are responsible for a similar degree of variation, but the corresponding change in [3.6]–[4.5] color of only 0.002 mag is below the typical photometric accuracy of these bands (Reach et al. 2005). As is the case for NIR colors to which our IRAC color is tied, age variations are not well probed. Still, as discussed in Section 3.1, age variations themselves do not lead to greatly varying ϒ_3.6. In fact, Figure 2 suggests that these are at most ±0.1 dex at fixed metallicity. Considering the range of ages and metallicities exhibited by the galaxies studied by MacArthur et al. (2004), the oldest age (lowest τ) considered here may even be unrealistic. More realistic SFHs, characterized by at least one additional burst of star formation during the decline of initial star formation, correspond to younger mean ages and, as demonstrated in the next section, tend to tighten the spread in ϒ_3.6 at fixed metallicity.

Here we explore the impact of bursts on ϒ_3.6. For each of the solar metallicity (Z = 0.02) exponentially declining SFHs we add one burst of star formation with 0.5 Gyr duration. Each burst contributes 10% to the total stellar mass formed over the lifetime of the population and is chosen to occur between 1 and 6 Gyr from the present "observation" epoch. Following MacArthur et al. (2004), we assume that more recent and more massive bursts are not likely in our sample, according to the Kauffmann et al. (2003) study of burst fraction in the Sloan Digital Sky Survey (SDSS). Note that this simple model does not account for bursts that produce stars with metallicities other than the metallicity characteristic at earlier times.

As can be seen in Figure 3, the [3.6]–[4.5] color is less sensitive to bursty SFHs than the B–R color, as expected (i.e., the [3.6]–[4.5] color itself is not very sensitive to age; see the previous section). However, older bursts introduce a slightly larger fractional change to the [3.6]–[4.5] color than to the B–R color. This change in [3.6]–[4.5] stays in roughly fixed proportion to the change in ϒ_3.6 for bursts of all ages. As a result, the fixed metallicity age track in the right panel remains robust; with the addition of bursts the population slides down the right-hand-side age track, while on the left the age track itself is changed.

Zoom In Zoom Out Reset image size

In either the left or right panels of Figure 3, it is clear that later bursts have the largest impact on color and ϒ_3.6, and this increases for shorter τ. Imposing a burst of star formation on an exponentially declining SFH primarily reduces the mean age of the stellar population (MacArthur et al. 2004) and thereby also effectively reduces the spread in ϒ_3.6 compared to the no-burst SFHs. Note that the [3.6]–[4.5] color for populations with the youngest age (corresponding to a burst 1 Gyr ago) is the most likely to be inaccurate because it does not account for AGB stars (only giant stars). At ∼1 Gyr, the contribution from AGB stars in the NIR is predicted to reach its peak of ∼30% or as high as ∼70%, depending on the model (see BC03 and Maraston 2005). The trends in Figure 3 described above remain secure for all other SFHs.

Given that bursty SFHs reduce the mean population age and hence tighten the spread in ϒ_3.6, metallicity is arguably the source of largest variation in ϒ_3.6. This is especially clear if we focus only on populations with ages younger than ∼10 Gyr (τ = 4–12 Gyr) in Figure 2, even discounting the lowest metallicity models. Such a reduced range of ages and metallicities would be representative of the large majority of nearby S0–Irr galaxies studied by MacArthur et al. (2004).

Because the [3.6]–[4.5] color is itself primarily driven by metallicity, it serves as a natural constraint on ϒ_3.6. In Figure 4 we show our parameterization of the relation between ϒ_3.6 and [3.6]–[4.5]. Assuming that all regions of the parameter space in Figure 2 are equally likely, we bin our set of models by metallicity, letting the spread in ϒ_3.6 and [3.6]–[4.5] due to age set the dispersion around the mean. On average, unconstrained age variations introduce an rms uncertainty of 0.06 dex on ϒ_3.6, shown as the vertical error bar at fixed metallicity in Figure 4. The horizontal error bar reflects age-induced variation in [3.6]–[4.5] color at fixed metallicity (∼0.002 mag) together with the uncertainty in our GLIMPSE-based calibration between J − K and [3.6]–[4.5], with which we can estimate the [3.6]–[4.5] color to within ∼0.009 mag (the Appendix).

Zoom In Zoom Out Reset image size

Taking a linear least-squares fit to the points in Figure 4, we find

$$\begin{equation} \log {\Upsilon _{3.6}}=3.98(\pm 0.98) ([3.6]-[4.5]) + 0.13(\pm 0.08). \end{equation} \tag{ 4 }$$

We adopt this fit as our optimal color–M/L relation. As demonstrated by Table 1, this relation is not especially sensitive to the particular range of age and metallicity probed by our models. Even with restricted metallicity and age ranges, errors in the slope and intercept primarily quantify the uncertainty introduced by age/SFH.

Considering that the ages of either early- or late-type galaxies (ETGs and LTGs) are expected to each fall within narrower ranges than the full range considered at each Z, the errors on the parameters of our optimal relation between ϒ_3.6 and [3.6]–[4.5] color are even conservative. Fits only to models in Figure 2 with ages >7 Gyr (<7 Gyr), as would be representative of ETGs (LTGs), yield best-fit parameters a = 3.86 ± 0.82 and b = 0.18 ± 0.07 (a = 4.07 ± 0.67 and b = 0.09 ± 0.06).

For completeness we note that the spread in ϒ_3.6 with age at fixed metallicity is larger the nearer the population is observed to the start of star formation. Also, this relation between ϒ_3.6 and [3.6]–[4.5] does not account for dust extinction.

In addition, our best-fit relation predicts a change in ϒ_3.6 by 0.2 dex for every 0.05 mag difference in color, slightly larger than typically emerges from standard single (optical)–color relations (see Z09; Bell & de Jong 2001). We emphasize, though, that by passing from optical colors to NIR-only colors, the stellar tracer changes from main-sequence stars to red giants, and we can expect the latter to show much less variation than the former for the same evolution in a given population.

Over the color range −0.14 < [3.6]–[4.5] < −0.04, the 3.6 μm mass-to-light ratio for the oldest stars spans ∼0.3 dex, driven primarily by metallicity. Using the [3.6]–[4.5] color to constrain the metallicity, the only lingering uncertainty in ϒ_3.6 is due to variation in the population age, which contributes an rms of 0.06–0.08 dex uncertainty at a given metallicity (see Figure 4). To reduce the uncertainty in ϒ_3.6 to any greater degree would require better constraints on the population age. According to the trends described in Section 4.1, this would be possible using an optical color, but only by also incorporating a realistic model of the dust to account for extinction and reddening (as well as the scattering of light by dust).

On the other hand, we can leverage the age–metallicity relation (AMR) characteristic of old stellar populations to place a constraint on the age at a given metallicity. With the [3.6]–[4.5] color tracing both the metallicity and the age, the uncertainty in ϒ_3.6 can be reduced. The AMR even suggests that a constant ϒ_3.6 may be fairly reasonable, given that the relation between ϒ_3.6 and Z is almost orthogonal to the relation between ϒ_3.6 and τ. As such, old stellar populations across the Hubble sequence seem to be well described with a constant ϒ_3.6. Indeed, according to Figure 2, old, metal-rich populations have comparable ϒ_3.6 to younger, more metal-poor populations (compare ϒ_3.6 at Z = 0.05, τ = 4 Gyr, with Z = 0.004, τ = 12 Gyr).

We confirm the implied independence of ϒ_3.6 on color by adopting the relation between the ages and metallicities of elliptical galaxies measured by Gallazi et al. (2005) across the stellar mass range 8.91 <log M_*/M_☉ < 11.91. This supplies an estimate of the age at a given assumed metallicity, which we then translate back to the quantities ϒ_3.6 and [3.6]–[4.5] color using metallicity-dependent linear fits of the relation between each quantity and population age from the set of models in Figure 2. Over the range of Gallazi et al. (2005) measured metallicities (from 0.002 ≲ Z  ≲  0.02), ϒ_3.6 exhibits much less variation with color than implied in Figure 2. The AMR is consistent with a constant $\Upsilon _{3.6}^{{\rm AMR}}$ = 0.6 for a Chabrier IMF²², good to within 0.06 dex. The uncertainty stems from the dispersion around the mean value over the adopted metallicity range, together with propagated errors due to the uncertainty in the fitted AMR represented by the intrinsic 1σ spread in the Gallazi et al. (2005) measurements.

The relation between old stellar age and metallicity allows us to know ϒ_3.6 even better than suggested by the full grid of models in Figure 2. Uniformly weighted, those models are consistent with an average constant (color-independent) ϒ_3.6 = 0.6 (Chabrier) and an estimated uncertainty 0.11 dex. But while our $\Upsilon _{3.6}^{{\rm AMR}}$ incorporates a realistic intrinsic spread in the relation between age and metallicity as measured by Gallazi et al. (2005), it should be used with caution.

The existence of an AMR in stellar populations depends on the history of star formation in these systems. We have assumed that the AMR measured for the old stellar populations in elliptical galaxies can be applied to the old stellar populations in late-type galaxies (isolated from younger populations at 3.6 μm). This implicitly assumes that late-type disks had similar formation and evolutionary histories as present-day ellipticals, up to the point when the latter underwent quenching (and an associated morphological change), whereas the former continued to form stars.

For systems in which an AMR in the stellar populations is not likely (e.g., due to an atypical evolutionary history), we therefore recommend either adopting the more conservative uncertainty of 0.11 dex on the constant ϒ_3.6 = 0.6 or using the [3.6]–[4.5] color-dependent relation in Section 4.3, which is preferred when color information is available.

The same applies locally within stellar systems, given indications that a link between age and metallicity persists within disks (e.g., MacArthur et al. 2004). This implies that ϒ_3.6 will be relatively flat within a galaxy with population gradients (as well as that a single ϒ_3.6 can be applied to galaxies of all types). The observed 2D [3.6]–[4.5] color distribution can be used to assess which ϒ_3.6 and uncertainty may be most appropriate. An implied smooth, monotonic variation in metallicity ([3.6]–[4.5] color) throughout the system may indicate that $\Upsilon _{3.6}^{{\rm AMR}}$ can be reliably applied; otherwise, the color-dependent relation should be adopted. In cases where color information is compromised (either because of contamination from nonstellar emission or data quality), we advocate the use of the most conservative estimate, log ϒ_3.6 = −0.22 ± 0.11 dex.

The remarkable single, color-independent ϒ_3.6 implied by the existence of an AMR within (and among) galaxies turns out to be a particular advantage for generating mass maps from our ICA maps of the old stellar light. As noted in Section 2.2, it may not be possible to retain full 2D color information tracking age or metallicity gradients in galaxies with pervasive nonstellar emission (Meidt et al. 2012a). The old stellar light map can nevertheless be converted to stellar mass with ∼15% accuracy (adopting the single, global [3.6]–[4.5] color that is retained; M. Querejeta et al., in preparation).

Recently, Eskew et al. (2012) assembled an estimate of ϒ_3.6 by using resolved measurements of the SFH of the LMC (Harris & Zaritsky 2009) to measure the mass in regions sampling throughout the map of 3.6 μm brightness. They find that mass is well traced by 3.6 μm flux and, based on the relation between the two, estimate ϒ_3.6 = 0.53, with an uncertainty of ∼30%. Then, finding that the scatter about the relation tends to correlate with the [3.6]–[4.5] color, they fit the additional relation

$$\begin{equation} \log {\Upsilon _{3.6}}=-0.74([3.6]-[4.5]) -0.23 \end{equation} \tag{ 5 }$$

attributing some part of the trend with [3.6]–[4.5] to the presence of either younger stellar populations (<1 Gyr) or dust emission (traced at 8 μm).

These estimates are reassuringly consistent with our own, although the second, in terms of [3.6]–[4.5] color, suggests a different (much weaker) color dependence than we found in Section 4.1. This may be expected, given that the fairly uniform metallicity of the LMC would restrict the range in [3.6]–[4.5] color. But if all of the color variation stems from age variation in this case (ages range from 12 Gyr to <1 Gyr in the LMC; Harris & Zaritsky 2009), then the shallower color dependence may suggest either that ϒ_3.6 is weakly dependent on young populations, in particular, or that dust emission acts to remove the steeper trend we find.

There are several techniques to estimate the 3.6 μm M/L in the presence of nonstellar emission, $\Upsilon _{3.6}^d$, based on global colors and total masses. Zhu et al. (2010) link SDSS masses with Spitzer luminosities and colors in SWIRE. Jarrett et al. (2013) use the Zhu et al. (2010) Two Micron Sky Survey (2MASS) relations to derive K-band masses and link these to WISE luminosities and colors for a small sample. Both relations extend redder than [3.6]–[4.5] = 0 and parameterize a dependence on [3.6]–[4.5] that is nearly the reverse of what we find. Zhu et al. (2010) also suggest that $\Upsilon _{3.6}^d$ is much more sensitive to SFH than metallicity, contrary to our findings.

These calibrations primarily reflect the fact that, as the nonstellar emission tracing young stellar populations makes an increased fractional contribution to the integrated 3.6 and 4.5 μm, the global [3.6]–[4.5] color reddens (even independent of the dust properties) and the total luminosity increases, thus reducing the mass-to-light ratio.

The fractional contribution of nonstellar emission to the integrated light of a galaxy can be increased due to, e.g., an enhanced star formation rate (SFR) (at fixed mass) or an intrinsically less massive stellar component (at fixed SFR). Both of these cases would be associated with redder [3.6]–[4.5] colors and thus a lower M/L. But at the local level, the [3.6]–[4.5] color in galaxies also reflects real changes in the dust properties (PAH versus larger grains and the physical state of the dust; see Paper I), rendering the above estimates ineffective. For mapping the 2D stellar mass distribution, we advocate first removing the nonstellar emission and then applying the ϒ_3.6 derived in this work.

The relation between ϒ_3.6 and [3.6]–[4.5] color presented here, while robust, still hinges on the assumed stellar population model and IMF. The J − H colors of old stellar populations, to which our [3.6]–[4.5] colors are tied, are fairly consistent from model to model (Bruzual 2007), even as predictions for optical–NIR colors like V − K vary by ∼0.3 mag at fixed age. In contrast, discrepancies in the NIR M/L of populations with (exponentially) declining SFHs emerge at all ages, as a result of differences in the treatment of stars in the AGB phase that occurs at Simple Stellar Population ages ∼1 Gyr. Depending on the model, predictions for the NIR contribution of AGB stars at this intermediate age are as low as 30% (BC03) or as high as 70% (Maraston 2005; and see the 2007 version of the BC03 models, hereafter CB07). As a result, ϒ_H predicted by CB07 and BC03 differ by ∼0.1 dex at all ages (Zibetti et al. 2009), with CB07 predicting a lower M/L.²³ The larger NIR contribution of AGB stars at intermediate ages predicted by the Maraston (2005) models would imply a similarly lowered M/L. In these cases, ϒ_3.6 would be reduced by ∼0.1 dex across all [3.6]–[4.5] colors.

We have opted to use the BC03 models for this work, given recent evidence suggesting that the assumed AGB prescription may be the most realistic. The NIR spectra of post-starburst galaxies studied by Zibetti et al. (2012) do not exhibit the strong molecular absorption predicted by Maraston (2005) models. A lower NIR contribution from AGB stars, as in the BC03 models, seems to provide the best fit to the SEDs of post-starburst galaxies analyzed by Kriek et al. (2010). A similarly low contribution is inferred for AGB stars in ∼1 Gyr old clusters in M100, arguably due to the extinction of the AGB by the circumstellar dust shell produced during this phase (Meidt et al. 2012b; which leads to reprocessed emission detected at mid-IR wavelengths). We therefore argue that the relation we derive based on BC03 models is optimal and that the uncertainties stemming from the properties of stellar populations are representative.

Only the uncertainty arising with the assumed IMF will dominate the lingering uncertainty in ϒ_3.6 due to age or SFH. The Chabrier IMF, for one, entails an increased fraction of giants to dwarfs at low turnoff mass M_TO, effectively decreasing the total M/L below the traditional Salpeter-based value (although the heavier main sequence will partially compensate in this case). Generally, the choice of the IMF will introduce systematic variation in log ϒ_3.6 by 0.05 dex to 0.3 dex (see Bell & de Jong 2001; Z09; Bernardi et al. 2010). Given that the uncertainties on ϒ_3.6 associated with the IMF exceed those arising with the stellar population, it should be possible to use ϒ_3.6 to explore variation in the IMF, e.g., with global galaxy parameters.

The presence of nonstellar emission is another potential source of discrepancy. We can expect the (positive) trend between ϒ_3.6 and [3.6]–[4.5] color to weaken or even reverse (Jarrett et al. 2013) when dust contributes to the [3.6]–[4.5] color, and the mass-to-light ratio is therefore $\Upsilon _{3.6}^{d}$ = $M/(L_{3.6}^{{\rm dust}}+L_{3.6}^{{\rm stars}})$. In such cases, our strictly stellar log ϒ_3.6 = −0.23 ± 0.1 dex can still be useful and can even provide an estimate for the fractional contamination by dust emission since

$$\begin{equation} \frac{L_{3.6}^{{\rm dust}}}{L_{3.6}^{{\rm dust}}+L_{3.6}^{{\rm stars}}}=1-\frac{\Upsilon _{3.6}^{d}}{\Upsilon _{3.6}}, \end{equation} \tag{ 6 }$$

where, e.g., $\Upsilon _{3.6}^{d}$ is estimated with the observed dust reddened [3.6]–[4.5] following Jarrett et al. (2013).

We emphasize, again, that our estimate for ϒ_3.6 here is most appropriate for 2D maps where the nonstellar emission has been removed and which thus arguably trace a uniformly old stellar population. Uncorrected 3.6 μm maps would most certainly require a more highly color-dependent M/L (at least locally) given the contribution from nonstellar emission. Without removing such emission, which can locally contribute as much as 30% (Paper I), then applying our color-independent stellar ϒ_3.6 = 0.6 will result in significant overdensities, e.g., at the locations of dusty spiral arms. The overestimation in 2D stellar mass surface density exceeds the uncertainty associated with our optimal estimate of ϒ_3.6.²⁴ The local effect of the contaminants can be quite large and should be kept in mind, e.g., when estimating the stellar mass in individual structural components mapped at 3.6 μm. The global contribution from dust will be smaller, introducing ∼10% overestimation in the total mass (Paper I).

In Section 4.4 we discussed the use of the AMR in old stellar populations to reduce the uncertainty in ϒ_3.6 due to age variations left unconstrained by the [3.6]–[4.5] color, as in Figure 4. As an alternative, age constraints are possible using optical colors dominated by younger stars, with the incorporation of a realistic dust model to account for extinction and reddening (as well as the scattering of light by dust; e.g., Zibetti et al. 2009). Ideally, some part of the constraint on the dust model would come from the far-IR dust emission itself. SED fitting with MAGPHYS (da Cunha et al. 2008), which samples throughout the combined dust and stellar SED, can provide a dust-free view of the properties of the stellar population and is potentially very powerful when applied pixel by pixel.

On the other hand, it might be possible to use the information on the dust emission contained in the IRAC bands (which can be isolated with ICA). Because this is primarily emission from the smallest grains (PAH and the hottest dust), it would be necessary to first define a relation between these grains and the bulk of the dust.