LIFTING THE DUSTY VEIL WITH NEAR- AND MID-INFRARED PHOTOMETRY. II. A LARGE-SCALE STUDY OF THE GALACTIC INFRARED EXTINCTION LAW

To ensure the most homogeneous sample of stars for study, we choose to use G- and K-type red clump (RC) stars, a population that is relatively luminous (M_Ks ∼ −1.54; Groenewegen 2008), homogeneous in color and absolute magnitude ((J − K_s) ∼0.65 ± 0.10, $\sigma _{M_{K_{s}}}$ ∼0.04; Girardi et al. 2002; Groenewegen 2008), and common in disk populations across a wide range of metallicities and ages. In the color–magnitude diagram (CMD) of an unreddened region of sky, the RC appears as a prominent vertical stripe, with the counts per magnitude tracing the density of the stellar population along the line of sight. When dust is present along the line of sight, its extinction acts to both dim the RC stars and redden them, resulting in a diagonal shift of their CMD to the right at fainter magnitudes (Figure 2(a)). However, even in highly reddened disk midplane fields, the RC forms a distinctive locus, which we utilize to identify the stars for study.

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For our analysis, we selected only those stars detected in all three of the 2MASS JHK_s bands with σ ⩽ 0.4 mag; this requirement removes many intrinsically red sources (i.e., those not detected in the bluer 2MASS bands) with low line-of-sight reddening from potentially contaminating the highly reddened part of the RC sample. For the four Spitzer-IRAC bands ([3.6μ], [4.5μ], [5.8μ], [8μ]), we follow Indebetouw et al. (2005) in requiring signal to noise ≳10 and paring the catalog to those stars with σ ≲ 0.2 mag. To remove sources with possible intrinsic IR-excesses (e.g., YSOs or evolved stars with circumstellar shells), we adopt color restrictions of ([3.6μ]−[4.5μ]) ⩽ 0.6 mag and ([5.8μ]−[8μ]) ⩽ 0.2 mag (Flaherty et al. 2007). Figure 2(b) shows the stars remaining after these photometric quality and color limits are imposed.

To ensure the purest possible sample of RC stars, we require that RC sample candidates have the characteristics of RC stars in both observed and intrinsic color space. To that end, we deredden the remaining stars with a new technique, detailed elsewhere (S. R. Majewski et al. 2010, in preparation), and then select only those stars whose dereddened colors fall within the expected range of RC colors. In brief, this dereddening technique takes advantage of the fact that infrared photometry (λ ≳ 1.5 μm) measures primarily the Rayleigh–Jeans portion of stellar spectral energy distributions, which has a constant shape only minimally dependent on intrinsic stellar atmospheric variations; this constancy implies a homogeneous set of infrared colors, particularly at the longer wavelengths sampled by the Spitzer-IRAC bands. Thus, any departure from the nominal infrared colors measures the amount of reddening and extinction toward each star individually. We apply this technique to the sample of stars remaining after the photometric quality and color cuts imposed above, and we then retain only stars with 0.55 mag ⩽ (J − K_s)₀ ⩽ 0.9 mag (Figure 2(c)). This step is invoked to help remove marginally reddened RGB stars or highly reddened bluer dwarfs that overlap the RC in the observed CMD. We note that the potential circularity of selecting stars dereddened with an assumed extinction law is not a source of concern, because the generous Δ(J − K_s)₀ = 0.35 mag RC selection bin will still conservatively include very nearly all RC stars anyway.

Next, we divide the catalog into ∼25 × 2° blocks (in Galactic longitude and latitude, respectively) and plot a CMD for each block; straggler stars lying outside the obvious RC locus, which are primarily redder giant stars, as well as the brightest (unreddened) RC stars are then removed manually (Figure 2(d)). Because many fainter stars unable to be positively associated with the RC locus are removed, this step has the effect of reducing the overall range of 2MASS photometric uncertainties for stars ultimately used in the sample analysis. Although high uncertainties in the PSC, particularly in the midplane, are potential indicators of a flux overestimation bias,⁷ Figure 3 shows that our final distribution of NIR photometric uncertainties is well below the limit where this bias is a concern.

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Finally, in order to make reliable color–color linear fits (Section 3.2), we require an average RC reddening of at least 0.35 mag and a minimum spread in RC reddening of at least 0.15 mag (i.e., $\overline{(J-K_s)}$ ⩾ 1 and ΔE(J − K_s) ⩾ 0.15 mag) in each block's CMD. We performed the color–color fits described below using a variety of photometric quality cuts, and the set of restrictions detailed here is the most lenient one (i.e., includes the most stars to provide better statistics) yielding the same fit results as stricter requirements.

A star's color excess E(λ₁ − λ₂) is simply the difference between the intrinsic color (λ₁ − λ₂)₀ and the observed color (λ₁ − λ₂); this excess depends on both the column density of dust along the line of sight to the star and the difference in extinction (for the same amount of dust) between λ₁ and λ₂. However, the color excess ratio removes the total column density dependence (see the Appendix) and reflects the differential behavior of extinction as a function of wavelength. An example of a CER_λ for a source is

$$\begin{equation} \mbox{CER}_\lambda =\frac{E(H-\lambda)}{E(H-K_s)}=\frac{(H-\lambda)-(H-\lambda)_{\rm 0}}{(H-K_s)-(H-K_s)_{\rm 0}}, \end{equation} \tag{ 1 }$$

where (H − λ)₀ and (H − K_s)₀ are the intrinsic colors of the source. Here we have chosen E(H − K_s) to be the common reference denominator for our CER_λ analysis because (H − K_s)₀ is a well-defined color for RC giants, more well measured in our sample than MIR colors, and less susceptible to color variations from metallicity, temperature, or non-RC contamination than the other NIR colors (J − K_s)₀ and (J − H)₀.

Equation (1) can be rearranged as

$$\begin{equation} (H-\lambda)=\frac{E(H-\lambda)}{E(H-K_s)}[(H-K_s)\, -\, (H-K_s)_0]\, +\, (H-\lambda)_0,\nonumber\\ \end{equation} \tag{ 2 }$$

where (H − λ) and (H − K_s) are the observed stellar colors. For this work, we have adopted intrinsic colors for RC stars from the Padova suite of stellar models, where (H − K_s)₀ = 0.10 nearly independently of RC metallicity (e.g., Girardi et al. 2002). When pairs of observed colors ([H − K_s] and [H − λ], for λ = J, [3.6μ], [4.5μ], [5.8μ], [8μ]) are plotted against one another for stars spanning a wide range of extinctions, we find extremely linear correlations; a linear fit to these distributions gives estimates of the CER_λ values as the slope, and the intrinsic (H − λ)₀ colors as the y-intercept when (H − λ) is on the ordinate. Figure 4 shows an example of these linear correlations and fits.

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Our goal is to measure CER_λ values using RC stars in many different parts of the Galactic plane and to look for variations. Before fitting, the data are binned by Galactic longitude into the same ∼25× 2° bins used in the RC sample selection. By adopting only fields meeting the average-reddening and differential-reddening requirements (as before: $\overline{E(J-K_s)}$ ⩾ 0.35 and ΔE(J − K_s) ⩾ 0.15 mag), we fit 65 longitude bins, spanning from 10° to 100° from the Galactic center (i.e., 10° ⩽ l ⩽ 65°, l = 90°, and 260° ⩽ l ⩽ 350°). For each bin, we fit a set of five CER_λ (i.e., [H − K_s] versus each of [H − λ], λ ≠ K_s) using a linear weighted least-squares algorithm. We take the uncertainties in the fits to be 1σ, where σ is the standard deviation of the fitted parameters. Because of the intrinsic stellar density gradient in the disk, the number of stars in each bin varies (from 820 to >6 × 10⁴) across our longitude range. We check for any effects this variation in sample size has on the fits by "normalizing" to the number of stars in the least-populated longitude bin (820); that is, for every other bin, we apply the fitting algorithm to 820 randomly selected stars a total of 25 times. Though we find no noticeable difference with the parameters derived by fitting all the stars in each bin, the results described below are the mean of the 25 fits for each bin, and the quoted uncertainties are either the average 1σ or the standard deviation of the 25 fit parameters, whichever is larger.

3.3.1. Derived CER_λ

Figure 5 shows the derived CER_λ values as a function of angle from the Galactic center. This figure straightforwardly demonstrates the existence of a large-scale longitudinal variation in the infrared extinction law, symmetric about the Galactic center, and that the slopes of the CER_λ trends increase as λ increases (i.e., more drastic CER_λ variations at longer MIR wavelengths). Because an increased CER_λ for λ > H (and decreased CER_λ for λ < H) results from relatively greater reddening E(H − λ), these results indicate a steeper extinction curve (A_λ/A_Ks; Section 5.1) in the outer Galaxy. The CER_λ values, averaged in 5° bins, are also presented in Table 2.

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Table 2. Derived CER_λ Values for Use in Equation (3)

Galactic Longitude, l	CERJ	CER[3.6μ]	CER[4.5μ]	CER[5.8μ]	CER[8μ]
10–15	−1.97	1.79	1.92	2.08	2.00
15–20	−1.95	1.78	1.92	2.07	1.99
20–25	−1.98	1.80	1.93	2.08	2.01
25–30	−2.01	1.80	1.94	2.09	2.05
30–35	−1.99	1.82	2.01	2.16	2.10
35–40	−1.97	1.83	2.02	2.19	2.15
40–45	−2.00	1.83	1.99	2.19	2.11
45–50	−1.96	1.81	2.00	2.21	2.13
50–55	−2.01	1.83	2.03	2.30	2.20
55–60	−2.06	1.83	2.02	2.30	2.20
60–65	−2.07	1.82	2.00	2.30	2.19
90	−2.18	1.94	2.38	2.78	2.49
260–265	−2.16	1.83	2.11	2.54	2.42
265–270	−2.10	1.77	2.05	2.39	2.29
270–275	−2.14	1.77	2.01	2.43	2.34
275–280	−2.13	1.80	2.06	2.51	2.35
280–285	−2.02	1.81	2.06	2.40	2.29
285–290	−2.11	1.82	2.01	2.43	2.42
290–295	−2.06	1.80	1.98	2.36	2.31
295–300	−2.13	1.85	2.05	2.38	2.28
300–305	−2.05	1.81	2.00	2.27	2.21
305–310	−2.03	1.82	2.02	2.21	2.12
310–315	−2.02	1.81	2.02	2.20	2.08
315–320	−2.04	1.82	2.02	2.22	2.09
320–325	−2.05	1.84	2.03	2.24	2.12
325–330	−2.00	1.81	2.01	2.18	2.07
330–335	−2.00	1.81	1.99	2.15	2.05
335–340	−2.01	1.80	1.95	2.11	2.02
340–345	−1.98	1.79	1.97	2.12	2.00
345–350	−2.00	1.80	1.93	2.10	2.01

Notes. For the contiguous data, the results have been averaged over 5°.

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CER_J appears nearly constant (with large scatter) until the angle ∼55°, when it becomes increasingly negative, while the MIR CER_λ values have a somewhat flat distribution only out to ∼25°. Of all the IR colors under consideration, (H − J) is the most sensitive to spectral variations due to differences (spatially dependent or random) in RC metallicity and temperature. Thus, the behavior of CER_J is consistent with the presence of global extinction law trends, but because it is also the most susceptible to unaccounted-for RC population variations, we do not evaluate the significance of the different Galactocentric angle of the trend inflection at this time.

3.3.2. Checks on (H − λ)₀

In Figure 6, we plot the derived intrinsic RC colors (H − λ)₀ against the field Galactocentric angle. Exploring the y-intercept values of our fits serves as a sanity check on our results—if our method is viable, then (H − λ)₀ from our fits should uniformly correspond to the intrinsic (H − λ)₀ colors of RC stars. Figure 6 shows that we indeed find (H − λ)₀ colors that are not only consistent across our survey but also in good agreement with the predictions of the Padova stellar models (Girardi et al. 2002). Some slight differences (⩽0.025 mag) compared to the theoretical colors may be attributable to inconsistencies in bolometric corrections or inaccuracies in the filter transmission profiles applied to the stellar models.

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It is possible, of course, that the true intrinsic colors of the RC stars in our sample do vary slightly, since the derived (H − λ)₀ values depend on our assumption of a constant (H − K_s)₀. We validate the latter assumption using the recent Milky Way disk abundance gradient of Pedicelli et al. (2009) and the relationship between [Fe/H] and (H − K_s)₀ in the Padova isochrones. For metallicities spanning −1.22 ⩽ [Fe/H] ⩽ +0.18, the intrinsic (H − K_s)₀ color of RC stars changes steadily by ≲0.06 mag. Pedicelli et al. (2009) find an [Fe/H] gradient of ∼⁻0.13 dex kpc⁻¹ inside the solar circle and ∼⁻0.042 dex kpc⁻¹ outside of it. Given the approximate range of R_GC of our RC sample (see Section 5.5), this corresponds to a maximum Δ[Fe/H] of 0.46 dex along any single line of sight, resulting in a maximum intrinsic (H − K_s)₀ scatter of ≪0.06 mag in any single fitted field. Since the measured E(H − K_s) values are so much larger (usually ⩾ 0.8 mag), this possible scatter does not affect the derived CER_λ values. This conclusion is further supported by the very tight linear color–color correlations observed (Figure 4), indicating no significant intrinsic color variations within any single field.

Recent studies of dark molecular clouds have shown that the MIR extinction curve becomes increasingly shallow with increasing local dust density (e.g., Flaherty et al. 2007; Román-Zúñiga et al. 2007; McClure 2009; Chapman et al. 2009). Theoretical models predict this behavior as an effect of grain growth, from either the coagulation of grains or the accumulation of refractory ice mantles in dense clouds (Li & Greenberg 2003; Draine 2003). In either case, the growth of grains serves both to remove small particles (including PAHs) and to add larger ones, leading to an overall flatter ("grayer") extinction. In addition, the presence of certain ices may lead to enhanced extinction around their solid-state resonance bands (e.g., CO at 4.7 μm and H₂O at 3.1 and 6.0 μm). However, neither the specific environmental conditions producing this grain growth, nor the detailed behavior of A_λ/A_Ks as a function of grain size, is yet well constrained. Chapman et al. (2009) approach these questions by using A_V as a proxy for density to probe specific molecular clouds and find a steady progression toward shallower 3.6–8 μm extinction curves as 〈A_V〉 increases in the cloud. The 25-wide longitude bins and the large total spatial extent of our stellar sample mean that any density variations potentially driving the extinction law variations seen here are likely dominated by the large-scale density gradient of the Galactic disk, not the small-scale filamentary structure of dark clouds.

This density gradient may be understood as either a change in the density of the mean ISM or a changing fractional contribution of molecular cloud material. One of the stated goals of this study is to test the commonly used description of a simple, bimodal ISM comprising individually homogeneous "dense" regions (dark clouds and cores) and "diffuse" regions (everything in between). Thus, we must examine what we mean hereafter in this analysis when we refer to "diffuse" material and attempt to distinguish its reddening effects from those of denser clouds.

Here, we explore the diffuse ISM using two separate methods to identify those stars tracing it. Each of these—the "RPD" and "¹³CO" methods—lead to their own implicit definition of interstellar diffusivity, and we will discuss the relative merits of each method and definition.

4.1.1. The RPD Method

4.1.2. The ¹³CO Method

To assess the effects of high-density clouds on our average ISM extinction curves, we sort our data set into rough divisions of "dense" and "diffuse" lines of sight, separately for each of the two methods described above. For the ¹³CO-emission method, we use the integrated-intensity maps of the Boston University-FCRAO Galactic Ring Survey (GRS; spanning 18° ⩽ l ⩽ 54°; Jackson et al. 2006) as tracers of high ISM density. These maps have resolution high enough (∼075) to distinguish filamentary structures of potentially dense ISM at the median distances of our stellar sample (where distances are estimated using the technique detailed in Section 5.5). As the most conservative selection of "diffuse" material, we consider only those stars with no measured ¹³CO emission at their projected position on the sky (comprising ∼55% of the RC stars in the longitudes covered by the GRS). As described above, this should eliminate any typically sized molecular clouds with n(H) ≳ 1–5 cm⁻³ from affecting our derived extinction law. For the RPD method, we consider two choices of maximum reddening per distance for "diffuse" ISM—RPD ⩽ 0.22 and RPD ⩽ 0.15 mag kpc⁻¹—to assess the effects of the particular limiting value. These cuts retain ∼79% and ∼57%, respectively, of the RC stars in the longitudes covered by the GRS.

We apply the CER_λ-fitting process as before (Section 3.2) to these various "diffuse" subsamples. The GRS coverage limits the ¹³CO-selection method to 18° ⩽ l ⩽ 54°, and we show results for the RPD method both with and without this longitude limitation. The number of randomly selected stars for the 25 CER_λ fits per field (i.e., the minimum starcount in any longitude bin) changes from 820 (Section 3.2) to 14,000 for RPD ⩽ 0.22 mag kpc⁻¹, 5700 for RPD ⩽ 0.15 mag kpc⁻¹, and 8000 for W(¹³CO) = 0 K; this increase in the number of fitted stars, compared to the original fits, occurs because we are now considering only fields no further than 54° from the Galactic center. The increased counts are responsible for the smaller error bars throughout Figure 7 (compared to Figure 5).

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In the case where CER_λ values are derived from stars with no ¹³CO emission detected at their positions (Figure 7(a)), we observe very nearly the same behavior with Galactic angle as seen in the case where no distinction between "dense" and "diffuse" lines of sight is made (i.e., Figure 5). That is, even with the likeliest sites of larger dust grains explicitly removed, the ISM extinction behavior continues to depend on the Galactocentric angle, with the reddening curve becoming steeper toward the outer Galaxy.

For comparison, in the case where stars are filtered using the RPD method, the distribution of CER_λ shows significant variation with longitude (Figure 7(b)), though no global trends are apparent. In addition, we find the counterintuitive result that the departure from canonical "diffuse ISM" values (e.g., Indebetouw et al. 2005) actually increases as the restriction on maximum RPD is lowered from 0.22 to 0.15 mag kpc⁻¹. Moreover, the extinction curves that derive from these "pure" diffuse-only samples are difficult to explain physically, as they are steeper than even a simple power law or R_V = 3.1 dust model (e.g., Section 5.3). At least part of the larger scatter in CER_λ is due to the uncertainties inherent to fitting a straight line (as in Figure 4) to the increasingly incoherent color–color distributions that are obtained when the high-reddening stars are preferentially removed and the color–color plots become dominated by the photometric errors of the stars. If one considers the lower CER_λ values near l ∼ 28° and 50° to be troughs, rather than attributing the values at l ∼ 37° to a peak, these depressions do correspond roughly with the expected positions of the tangencies of the Scutum and Sagittarius spiral arms (e.g., Englmaier & Gerhard 1999). This might be one possible explanation for the troughs since decreased CER_λ values indicate a shallower extinction curve (see Section 5.1), consistent with what would be expected in a dense spiral arm.

However, ascribing this behavior to the spiral arms has many problems. First, because the removal of stars with high RPD estimates is explicitly intended to filter out all stars in or behind molecular clouds, it is unlikely that features caused by molecular clouds would become increasingly prominent as the RPD limit is lowered (in theory producing an even purer diffuse-only stellar sample). Second, while the precise heliocentric distances of these spiral arm tangent points are not tightly constrained, recent models place them near or beyond the range of our RC stars (e.g., Hou et al. 2009), making it impossible for the stars with low RPD values in our data set to probe spiral arm tangencies. Third, we can check for the corresponding spiral arm tangencies in the fourth Galactic quadrant (Carina, l ∼284° and Crux, l ∼310°), since the RPD method is not limited to the longitude range imposed by the ¹³CO maps. Figure 7(c) shows CER_λ values derived from stars with RPD < 0.22 mag kpc⁻¹ over the entire longitude span of our data—no dips corresponding to the Crux and Carina arms are visible. In light of these inconsistencies, we cannot ascribe the CER_λ behavior for 18° ⩽ l ⩽ 54° to any known Galactic structure. The physical reality of the observed features themselves (Figures 7(b) and (c)) are called into question by (1) the number and type of assumptions that must be made to select a diffuse RPD limit from a wide (and potentially varying) range of possibilities, (2) the increasing departure from realistic CER_λ values as the RPD limit is tightened, and (3) the strong dependence of the calculated RPD values on the stellar distance estimates, which themselves require an assumed extinction law to calculate.

Given this wide range of complications, we conclude that the ¹³CO selection filter, while not perfect, has fewer systematic uncertainties than does the RPD filter, and we therefore adopt the lack of ¹³CO emission as the preferred tracer of diffuse, intra-cloud interstellar material. The similarity in behavior between this "diffuse" extinction and the "all stars" extinction derived in Section 3 indicates that the environments probed by our full set of RC stars are dominated by ISM too thin to harbor ¹³CO molecules; the following analyses, therefore, are not limited to the available longitudes of the ¹³CO data and are able to explore the extinction behavior across the largest possible span of the Galactic disk. The persistence of the trends between relative extinction and longitude in the diffuse ISM, particularly at 5.8 and 8 μm, indicates effects on the reddening law of either the mean Galactic density gradient (beyond the known growth effects in very dense clouds) or of secondary factors that vary throughout the Galactic disk, such as the chemical composition, size, or crystalline fraction of the dust grains. These secondary factors, particularly grain size and crystallinity, may of course be at least partially due to the overall Galactic ISM density gradient (see the beginning of Section 4.1).

For now, we adopt a working definition of "diffuse ISM" to refer to interstellar matter with a hydrogen number density too low to shield ¹³CO molecules from dissociation (i.e., n(H) ≲ 1–5 cm⁻³). We believe that this definition is not only close to what is normally envisioned as "diffuse ISM" but also that which is most useful in terms of practical and immediate applicability of our results to common dereddening problems. In the four-phase ISM model (e.g., Whittet 2003), material defined thusly corresponds to a mixture of the "warm" and "cool (atomic)" phases. We recognize that this thin matter is likely to have a large-scale radial density gradient throughout the disk, since it probably contains the outskirts of molecular clouds, which have a varying filling factor throughout the disk. However, we emphasize that by our (or nearly any) definition this ISM would not be considered "dense cloud" material and so, prior to this study, would have been assigned a constant "diffuse" extinction law, regardless of its location in the Galaxy. Our results here show that this constant extinction law does not accurately describe the extinction behavior along all lines of sight, and any reddening corrections applied without considering the varying nature of the diffuse extinction law could be potentially subject to severe systematic errors.

To convert the derived CER_λ into the more familiar IR extinction law format A_λ/A_Ks, we use the relation

$$\begin{equation} \frac{A_\lambda }{A_{K_s}}=\frac{A_H}{A_{K_s}}-\left(\frac{A_H}{A_{K_s}}-1\right)\cdot \mbox{CER}_\lambda. \end{equation} \tag{ 3 }$$

Deriving the true, longitude-dependent extinction law requires an independent determination of A_H/A_Ks along each line of sight for which we have measured CER_λ. Unfortunately, this is an extremely challenging measurement to make. Nishiyama et al. (2006, 2009) have done it toward the Galactic center by assuming that all RC stars are at the same distance; we can make no such assumption for our lines of sight. Indebetouw et al. (2005) fit the RC locus in NIR CMDs to directly extract A_H/A_Ks, leaving distance as a free parameter and assuming that the amount of extinction in those bands is constant per unit distance along the line of sight, the equivalent of assuming a smooth, homogeneous dust distribution. This assumption is not generally applicable, and in our more extensive disk survey we see many lines of sight with CMDs containing shifts and kinks in the RC locus that provide definitive evidence for a nonhomogeneous extinction and dust distribution. Thus, given no currently reliable method for determining A_H/A_Ks around the disk, we cannot provide an absolute, independently calibrated set of A_λ/A_Ks curves at this time.

Nevertheless, to facilitate comparison of A_λ/A_Ks behavior with other studies, we use the commonly adopted value A_H/A_Ks = 1.55 as determined by Indebetouw et al. (2005), which corresponds to an A_λ ∝ λ^−β power law with β = 1.66. The range of typical β values (1.6–1.8) gives A_H/A_Ks = 1.52–1.6; when applied to our observed CER_λ with Equation (3), this range adds a ∼3%–15% uncertainty to the calculated A_λ/A_Ks values. This uncertainty is several times smaller than the percentage change in A_λ/A_Ks with the Galactic angle due to the intrinsic CER_λ trends we observe here. In addition, we note that the choice of A_H/A_Ks only sets the absolute value scale for the relative extinction law; it is not sensitive to variations in extinction law shape.

In Figure 8, we compare our mean A_λ/A_Ks curve, derived using the total RC sample at all Galactocentric angles, with the results of Lutz et al. (1996), Indebetouw et al. (2005, fit from their Equation (4)), Jiang et al. (2006), Flaherty et al. (2007), and Chapman et al. (2009). We find very similar values and overall curve shape to the extinction laws found by many of these authors, but a closer inspection reveals that our data yield slightly lower relative extinction values in the MIR (i.e., a steeper MIR extinction curve). This is entirely in accordance with the difference in environments probed by these various studies: the shallowest curve results (Lutz et al. 1996; Jiang et al. 2006; Flaherty et al. 2007) are derived using stars in the Galactic Center and behind dense star-forming regions, where dust grains are expected to be significantly different than in the disk's diffuse ISM (Section 4).

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Our results more closely match the curve of Indebetouw et al. (2005), which was derived using three lines of sight comprising both diffuse ISM and an H II region, with Galactocentric angles ∼42° and 76°. The fact that our mean curve lies below these data is entirely consistent with our relatively larger inclusion of diffuse and outer-Galaxy regions. Another comparable match to our data comes from the two lowest A_Ks bins of Chapman et al. (2009), probing regions where A_Ks ⩽ 1. The least-extinguished stars in this dark-cloud survey are likely to be either foreground dwarfs unaffected by the molecular cloud material or stars observed through the low-density cloud edges.

Because our study finds evidence for variations in the extinction law with longitude, it is useful to compare extinction curves from different sight lines with curves extracted from theoretical dust models to attempt explanation of the observed differences. Figure 9 shows such a comparison between the most current suite of dust models of Weingartner & Draine (2001)⁸ and our extinction curves sampled from various Galactocentric angles, ranging from the innermost GLIMPSE data (10°–15° from Galactic center) to dust entirely beyond the solar circle (>90°). The Case A and Case B models of Weingartner & Draine (2001) differ slightly in their carbon abundances (see below) and grain size restrictions—Case A contains grains only up to 1 μm, while Case B grains can be as large as 10 μm.

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As in the case with the observational studies listed above, our results clearly show a shallower extinction law than either the R_V = 3.1 theoretical model or any power law that could be extrapolated from the three NIR measurements alone (e.g., the solid line in Figure 9). The Case B models with R_V = 4.0 and R_V = 5.5 provide the closest overall matches to the "inner" Galactic sight lines, though these are nearly indistinguishable from each other blueward of ∼3 μm; we note that if we had scaled the theoretical curves using A_H/A_Ks = 1.55, as we did for our derived extinction curves, the MIR range of the Case B models would shift downward and align more closely with the |l| ∼ 30°–60° data points. The original carbon abundances of the Weingartner & Draine (2001) models have been called into question by observational constraints (see discussion in Draine 2003), but only those with R_V = 3.1 or a Case A size distribution have subsequently been modified (with a change in C/H on the order of ≲15%). These Case A models more closely follow the R_V = 3.1 curve, but all three yield MIR extinction curves too steep to explain our inner Galaxy observations.

Thus, there is a clear relationship between the Galactocentric angle and best-matched dust model: though the inner fields correspond to the higher R_V values, the outer fields are increasingly similar to the highly diffuse R_V = 3.1 and the small-grain Case A models. The diffuse nature of outer disk Galactic dust is one of the reasons a large-scale extinction law survey of this type has not before been conducted—a diffuse ISM produces small reddenings that are difficult to study, particularly in the relatively transparent MIR portion of the curve (compared to shorter, more dust-sensitive wavelengths); our extensive disk coverage and high-quality MIR photometry allow us to make headway on overcoming these obstacles. After determining extinction behavior in a consistent way across a wide swath of the Galactic disk and comparing the results to theoretical dust models, we interpret the variation in extinction law as a consequence of a decrease in mean dust grain size toward the outer Galaxy.

The presence of increased relative extinction in the IRAC-4 channel at ∼8 μm is generally attributed to the broad 9.7 μm resonance of amorphous silicates. Indebetouw et al. (2005) estimate that this absorption band has a ≲20% effect on the expected flux through the [8μ] filter (but a negligible effect on the [5.8μ] channel). Though the exact behavior of the absorption band is loosely constrained (Whittet 2003), generally the band strength appears to decrease with both increasing grain size and increasing crystallinity (Figure 9; models of Weingartner & Draine 2001). Thus, one may expect a general trend of more dramatic 8 μm inflections with decreasing ISM density (loosely, with increasing Galactic angle), and as shown in Figure 10, we see just that. This figure shows the mean A_[8μ]/A_[5.8μ] from our data for each 10° bin in Galactic angle, and this ratio clearly increases toward the outer Galaxy. Also included for comparison are the approximate corresponding ratios for the five dust models in Figure 9 (Weingartner & Draine 2001). These latter ratios are approximate because they do not take into account the true filter transmission profiles of the two IRAC channels; we use the FWHM and isophotal wavelength of each bandpass (convolved with a K2 giant star) to calculate the relative extinction values. This analysis again suggests decreasing ISM grain size toward the outer Galaxy via the observed increasing 8 μm relative extinction and is further evidence for the variable nature of the ISM constituents on Galactic scales.

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Because each line of sight analyzed here thus far probes a range of Galactocentric radii (R_GC), the observed longitudinal variations in CER_λ demonstrate that the assumption of a universal infrared extinction law, even for diffuse material, is incorrect–if it were otherwise, then the same extinction curve would be measured in all directions in the Galactic plane and at all distances. Given that RC stars are good standard candles and that the factors affecting A_λ/A_Ks (e.g., dust composition, grain size) are likely to have a radial dependence, the possibility of measuring the IR extinction law as a function of Galactocentric radius is a tantalizing one. The difficulty, of course, lies in the fact that the reddening we observe toward each star is caused by dust spread along the line of sight to the star, so that the observed extinction behavior cannot be assigned to the dust at one particular R_GC.

On the other hand, for certain lines of sight, we can make simple assumptions regarding the dust distribution that allow reasonable estimates of trends in A_λ/A_Ks with R_GC. First, we assume that the large-scale Galactic dust distribution has a strong radial density gradient (scalelength ∼2–3 kpc; Drimmel & Spergel 2001; López-Corredoira et al. 2002; Misiriotis et al. 2006) from the Galactic center to the outer disk. In this case, for sight lines toward the Galactic center (|l| ⩽ 13°), we can assign the majority of the dust to the approximate R_GC of the farthest concentration of stars in our sample, where the dust is presumably thickest (for our RC sample, R_GC ∼ 5.5 kpc, assuming R_GC,☉ = 8 kpc). Likewise, for large-angle sight lines (⩾150°, including the Galactic anti-center), we estimate most of the dust to lie in the foreground of the survey stars nearest the sun (for our sample, R_GC ∼10.4 kpc), where we expect the dust density to be highest. Note that we discard the minimum reddening requirements (Section 3.1) in order to use these outer Galaxy fields; the small stellar reddenings at these longitudes are responsible for the increased uncertainties in the linear color-color fit parameters. Finally, there exists an optimal Galactic longitude at which the majority of the observed stars, at varying heliocentric distances, fall along the same Galactocentric ring (with a constant, well-defined R_GC). To calculate this, we recognize that each line of sight (with |l| ⩽ 90°) intersects the tangent point of a Galactocentric ring, where a concentration of stars (and the intervening dust) have the same R_GC; generally there will also be stars and dust with larger R_GC on the near and far sides of the tangent point (as seen from the Sun's position). But as Figure 11 demonstrates, there are longitudes at which other factors serve to remove stars in front of and behind the tangent point—at these longitudes, nearer stars are either too bright for 2MASS/Spitzer or are blue main sequence stars, and farther stars are actually extinguished out of our database altogether. For our particular sample's heliocentric distance distribution, this critical angle is 58° (i.e., l = 58° and 302°), corresponding to the ring at R_GC ∼ 6.8 kpc and indicated by the dotted line in Figure 11.

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Distances to the RC stars are estimated using the extinction-measurement technique mentioned in Section 3.1 (to be detailed in S. R. Majewski et al. 2010, in preparation); the law of cosines is applied to convert the midplane heliocentric distances (d) into R_GC:

$$\begin{equation} R_{\rm GC} = \sqrt{d^2+R_{{\rm GC},\odot }^2-2dR_{{\rm GC},\odot }\cos (l)}, \end{equation} \tag{ 4 }$$

where l is a star's Galactic longitude and R_GC,☉ is assumed to be 8 kpc. In converting reddenings to extinction, we must assume an absolute magnitude M_Ks for red clump stars (M_Ks ∼ −1.54; Groenewegen 2008) and a NIR+MIR extinction law, for which we choose that of Indebetouw et al. (2005) for all three lines of sight described above. We realize that this introduces a significant systematic uncertainty into our calculated R_GC values, which varies depending on the relative molecular cloud filling factor along the line of sight (Section 4.1.1); however, we note that in choosing a consistent law for all three cases, the systematic differences in CER_λ among them are still highly informative. The approximate R_GC values themselves are sufficient for our illustrative goals here, as we are not attempting to "fit" a global relationship but rather to show relative trends.

Figure 12 shows the fitted MIR CER_λ values (i.e., no A_H/A_Ks assumed) for the three ranges of Galactic angle described above. The doubled points for the inner angles correspond to the sight lines on either side of the Galactic center. We clearly see a trend with Galactic radius, evidence for an intrinsic A_λ/A_Ks radial dependence underlying the apparent dependence of the infrared extinction law on the Galactic angle. The behavior with R_GC is as expected from both the analyses in this section and theoretical predictions: the presumably more diffuse ISM in the outer disk produces higher CER_λ values (which appear as a steeper extinction curve) relative to the inner disk. Comparison to theoretical models suggests consistency with the presence of larger grains and a higher selective-to-relative extinction ratio in the inner Galaxy, decreasing to almost exclusively sub-micron particles in the outer disk.

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Interstellar dust grain size is a function of both the initial size distribution at formation and the ongoing processes that further affect grain size, crystallinity, and even composition. The formation size distribution of Galactic dust, and indeed the formation mechanism(s) itself, is not yet definitively understood (e.g., Speck et al. 2009; Zhukovska & Gail 2009), and layered on top of these are multiple complex processes, including grain coagulation, gas accretion, thermal annealing, collisional shattering, and gas–grain sputtering, which depend on local ISM phase and conditions. Our initial conclusions from the various analyses presented here regarding the mean grain size spatial distribution—larger (at least micron-sized) grains at small Galactocentric radii, decreasing steadily to sub-micron ones past the solar circle—provide an observational constraint on the convolved patterns of grain size formation distribution and the relative importance of local conditions and processing mechanisms at different locations in the disk.