THE BLUE TIP OF THE STELLAR LOCUS: MEASURING REDDENING WITH THE SLOAN DIGITAL SKY SURVEY

The SDSS is a digital spectroscopic and photometric survey, that, with the additional south Galactic cap (SGC) data provided by the SDSS-III, covers just over one-third of the sky, mostly at high latitudes (Abazajian et al. 2009). The SDSS provides near-simultaneous imaging in five optical filters: u, g, r, i, and z (Gunn et al. 1998; Fukugita et al. 1996). The photometric pipeline has uniformly reduced data for about 10⁸ stars. The SDSS is 95% complete up to magnitudes 22.1, 22.4, 22.1, 21.2, and 20.3 in u, g, r, i, and z. The SDSS imaging is performed in a drift-scanning mode in which the shutter is left open while the telescope moves at a constant angular speed across the sky. The resulting imaging "runs" tend to cover strips of sky that are long in right ascension and narrow in declination. These runs are divided into fields, which are 135 by 9' in size. We use SDSS data that have been photometrically calibrated according to the "ubercalibration" procedure of Padmanabhan et al. (2008).

The SFD dust map is a map of thermal emission from dust based on the IRAS 100 μm maps. The IRAS 100 μm map underwent three major processing steps before being used in the SFD dust map: it was destriped, zodiacal-light-subtracted, and calibrated to match COBE/DIRBE at degree scales. The first and third of these steps were necessary because the zero point of the IRAS 100 μm detector varied over time, imprinting the IRAS scan pattern on the data in the form of stripes, and making the overall zero point of the data uncertain. The zodiacal-light subtraction is needed to remove the signature of the hot, bright, local interplanetary dust from the IRAS and DIRBE data, which, though bright, contributes negligibly to the reddening. The DIRBE 100–240 μm flux ratio is used to constrain the dust temperature and to transform the 100 μm flux map to a map proportional to dust column density. The destriped IRAS data have resolution of 6 arcmin, while the temperature correction is smoothed to degree scales. The destriped, temperature-corrected IRAS 100 μm map is normalized to a map of E(B − V) by using a sample of 389 galaxies with Mg₂ indices and B − V colors, for use as color standards (Faber et al. 1989).

We perform fits to the location of the blue tip of the stellar locus in each SDSS field (or group of SDSS fields). To avoid biasing the fits, we select stars in each field for analysis in a reddening-independent way, and then find the location of the blue tip in the resulting sample.

Care must be taken when imposing cuts on the stars in each field to avoid biasing the resulting blue tip color. Because extinction changes the observed magnitudes of stars, a flat cut on magnitude would bias the measurement, as the distance range probed would change with the dust column. Instead, we perform cuts on D, the g magnitude of a star projected along the reddening vector to zero color:

$$\begin{equation} D = g - \frac{A_g}{A_g-A_r}(g-r). \end{equation} \tag{ 1 }$$

Provided A_g/(A_g − A_r) is accurate, the set of stars in a particular range of D is independent of the dust column toward the stars. D is related to the distance modulus of the MSTO stars (though not redder stars). Using the Jurić et al. (2008) "bright" photometric parallax relation, MSTO stars with r − i = 0.1 and g − r = 0.2 have M_g = 5.3. Accounting for the projection from g − r = 0.2 to zero color along the reddening vector, D ≈ g − M_g + 4.6. In this work, we frequently use the range 10 < D < 19. Including the effect of saturation on the bright end, this restriction on D selects stars with distances between about 1 and 8 kpc.

With a reddening-independent population of stars in hand, we measure the color of the blue edge of these stars in each SDSS color. The stellar colors are modeled as drawn from a probability distribution P(x), taken to be a step function convolved with a Gaussian to reflect the photometric uncertainty and the intrinsic width of the blue edge. The maximum-likelihood location for the step is denoted by the "blue tip" of the locus. Specifically, the form of the probability distribution is taken to be

$$\begin{equation} P= \frac{1}{2}\left[1+{\rm erf}\left(\frac{x-x_0}{\sqrt{2} \sigma _P} \right)\right] + F, \end{equation} \tag{ 2 }$$

where erf is the Gaussian error function, σ_P corresponds to the width of the edge, and F sets a floor to the probability distribution, to render the fit insensitive to the occasional white dwarf or blue straggler. In this work, we use F = 0.05. The variable x₀ is the only free parameter in the fits. We maximize the likelihood $\mathcal {L}=\prod _x P(x-x_0)$, where the product is taken over the color of all the stars satisfying the cuts in the field. The maximum likelihood location of x₀ gives the color of the blue tip. The final results of this work are insensitive to substantial changes in the floor F, from 0.01 to 0.05. The formal statistical uncertainty in the blue tip color is computed for each measurement by fitting $\mathcal {L}$ near the maximum likelihood to a Gaussian.

The width of the blue edge is set by a combination of the intrinsic width of the edge and the typical measurement error in star colors in the SDSS. We adopt the values 0.05, 0.05, 0.025, and 0.025 for the width parameter σ_P in the colors u − g, g − r, r − i, and i − z, respectively. Because the photometric uncertainty becomes larger at faint magnitudes, at high extinctions the observed blue edge is expected to broaden. Despite this, σ_P is kept constant in the fits, introducing the possibility of bias in the measurements. Nevertheless, as we look only at relatively bright ranges of D and low extinction, this is a minor effect. Simulations with a mock star catalog (Section 4.4) verify that for the range 10 < D < 19 used in this work, the introduced bias in the recovered ratio of blue tip color to E(B − V) is less than 2% in u − g and smaller in the other colors.

Some example measurements of the blue tip of the stellar locus are illustrated in Figure 3. In these plots, features of the blue tip fit are overplotted on a gray scale giving the color–magnitude diagram of a one degree region around l = 150° and b = 20°. The measurement is performed using stars with 10 < D < 19, which corresponds to stars falling between the diagonal lines in the g − r panel. Solid curves give the probability distribution function (pdf) P for the maximum likelihood location of the blue tip color x₀, and the number of stars as a function of color satisfying 10 < D < 19. The blue tip color x₀ is given by the dotted line. In order that P can be normalized, the fit is performed only on stars in a finite range of colors. We first make a rough estimate of the blue tip color by using a very broad range of colors, and then perform the fit a second time using only stars within 0.8 (0.4) mag of the estimated blue tip in u − g and g − r (r − i and i − z). This color range is given by the dashed lines.

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The statistical uncertainty of the fit depends on the number of stars included in the fit, and so on the local stellar density, the sky area, and range of D used. At high latitudes, binning together 10 SDSS fields, for stars with 10mag < D < 19 mag, typical statistical uncertainties are 20, 15, 6, and 6 mmag for u − g, g − r, r − i, and i − z, respectively. Finding the standard deviations of measurements taken near the north Galactic pole, where the expected reddening is smooth and small, we find uncertainties of 24, 17, 10, and 12 mmag, similar to the fit results.

The presence of dust reddens stars, pushing the blue edge redward (Figure 4). In low-extinction regions where all of the dust is closer than about 1 kpc, the observed blue tip is uniformly shifted. It is this shift that we track in our measurements. In high-extinction regions or regions with dust beyond about 1 kpc, we can in principle learn about the change in dust column with distance by varying D, though we have not done so in this work (see Section 3.2).

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The functional form P(x, x₀) chosen to fit the blue edge of the distribution is usually a close fit only near the blue edge itself, and is rather bad over most of the color range. However, experiments with other functional forms and attempts to adaptively shrink the range over which the fit is performed to a narrow region around the edge tend to render the fit less robust. Moreover, because P is approximately constant away from the edge, the fit results are robust to outliers far from the edge. Alternative functional forms systematically shift the location of the edge around slightly (mmag), but not in a way that correlates with the dust.

The blue tip fit can occasionally fail to find the blue tip of the stellar locus when too few stars are used. In such cases, it might identify a single blue star or quasar as the blue tip, or latch on to the M-dwarf peak in color–magnitude diagrams. In the maps we present, with 10 < D < 19 and using 10 SDSS fields worth of stars for each blue tip fit, fewer than 1% of measurements are affected.

In this work, we treat all MSTO stars observed by the SDSS with 10 < D < 19 as behind all of the dust. In principle, however, by comparing the color of the blue tip for nearby stars and distant stars, distant clouds of dust could be detected. Such clouds certainly exist, as dust has been detected in the halos of other galaxies and in H i clouds outside the disk in our Galaxy (Ménard et al. 2010; Wakker et al. 1996). However, comparing blue tip maps made for nearby ranges of D and distant ranges of D reveals no readily identifiable structures larger than the noise in the halo. We note that if such structures vary slowly spatially, they will be difficult to distinguish from variation in blue tip colors due to changing stellar populations. We defer to later work the attempt to identify such clouds.

On most sight lines the great majority of the dust column comes from the Galactic disk and is associated with the H i disk, which has a scale height of about 150 pc (Kalberla & Kerp 2009). Accordingly, as the brightest unsaturated MSTO stars are approximately a kiloparsec away, at high Galactic latitudes even the nearest MSTO stars observed by the SDSS are behind this dust.

At low Galactic latitudes, extinction increases rapidly and the color of the blue tip of the stellar locus becomes dramatically redder as fainter magnitudes are probed (Figure 5). In these dusty regions, reddened blue tip stars may be redder than intrinsically red foreground stars. Moreover, blue tip stars in these regions are spread out along the reddening vector, making it hard to isolate stars of a given reddening and distance using D. We attempt to identify and exclude such regions when analyzing blue tip maps.

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We measure the blue tip color over the entire SDSS footprint, in each color, and for a variety of ranges of D (Figure 6). Here we present maps using blue tip fits to stars in a broad range of magnitudes: 10mag < D < 19 mag, using fits to the stars in 10 adjacent SDSS fields.

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The maps show unmistakable signatures of dust (dark clouds in the left panels of Figure 6) that are almost entirely removed when stellar colors are corrected according to the SFD map. The residuals at high Galactic latitude after extinction correction are dominated by problems with the survey calibration—striping in the SDSS scan direction and a few runs with bad zero points in u − g (Figure 7). There is also a slowly varying residual in which blue tip color becomes redder near the Galactic plane, until deep in the Galactic plane when the color shifts very blue.

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The former effect is caused by the various stellar populations sampled in different parts of the sky. In the plane, the higher metallicity, redder disk population becomes increasingly dominant over the halo population. The latter effect is a result of the violation of the assumption that all of the stars are behind all of the dust. Stars in front of the dust are dereddened with the full SFD dust column, and are therefore rendered extremely blue. These stars are then seen by the blue tip algorithm as the signature of the blue tip, resulting in a spuriously blue color. Even in less dusty regions where at least some MSTO stars are behind all of the dust, the method can fail if an insufficiently faint range of D is used. In this case, MSTO stars behind different amounts of dust are grouped together in finding the best fit, blurring the blue edge of the stellar locus. Because σ_P for the fit is fixed, this will result in a bluer than average blue tip color, while SFD would track the color of the MSTO stars behind all of the dust—the reddest stars in the group. These effects, however, are only noticeable in the dustiest region of the sky, where |b| ≲ 15°.

The Monoceros stream is clearly visible as a feature at (α, δ) = (110°, − 10°)–(110°, 60°) in the blue tip maps, because its MSTO is bluer than that of the thick and thin disks (Newberg et al. 2002), except in u − g.

There are two types of survey-systematic induced residuals in the maps, both of which are seen as striping along the SDSS scan direction. The less important are the few, large, ∼3° wide stripes that are most noticeable on the eastern edge of the u − g blue tip maps. These regions correspond to the footprints of SDSS runs that have slight zero point offsets from the rest of the survey. The more troubling residual is the universal small-scale striping along the scan direction. This corresponds to the 13 arcmin scale of the SDSS camera columns. We have not fully examined this residual, but attribute it to different filter responses between the six SDSS camera columns (Doi et al. 2010). The SDSS ubercalibration algorithm ties stars of some mean color together, but does not account for color terms between the SDSS camera columns (Padmanabhan et al. 2008). As the MSTO stars are substantially more blue than the typical SDSS star, they are particularly vulnerable to color terms. Complicating this explanation of the camera column striping is the observation that the best-fit camera column offsets derived in Section 4.2 are different for the brightest range of D used than for fainter ranges of D, which are mutually compatible. We interpret this as a saturation or nonlinearity effect that depends on camera column, though we remove any stars flagged as saturated from the analysis. These systematics need to be addressed in fits to maps of the blue tip.

In some cases, a single run is long enough and dusty enough to individually constrain the dust reddening coefficients R_a−b in each color at the 10% level, even when fitting for camera column offsets and a quadratic. The fits show substantial variation in R_a−b (Figure 9). Most of this variation takes the form of an overall difference in dust absorption normalization, rather than as variation in the shape of the dust reddening spectrum. Some low, outlying values of R_u−g suggest that the run is so extinguished that substantial numbers of stars are not detected in the u band, ruining the fit. The fits remove nearly any trace of dust in these runs (Figure 10).

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Fits to the runs individually well constrained by the dust map indicate significant variation in normalization (∼20%) as well as, to a lesser extent, variation in dust extinction spectrum. Accordingly, in finding a global fit to the dust properties, we remove from the fit regions that we deem discrepant from the "average" dust properties. To do this, first we individually fit each SDSS run. Runs with best-fit R_a−b more than 5σ from the inverse variance weighted average of R_a−b for all the SDSS runs are excluded from the global fit. Additionally, the global fit is iterated and σ-clipped field-by-field at 5σ.

We find the best-fit (least-squares sense) combination of the SFD dust map, constant offsets for each camera column other than the first, and quadratics in field number for each run to the blue tip map via singular value decomposition. We do not fit for the offset for the first camera column because changing the offsets for all of the camera columns is degenerate with changing the constant offsets in the quadratics for each run. Specifically, we find the parameter x satisfying

$$\begin{equation} \mathbf {A}^{\intercal }\mathbf {C}^{-1}\mathbf {A}\mathbf {x} = \mathbf {A}^{\intercal }\mathbf {C}^{-1}\mathbf {b} \end{equation} \tag{ 5 }$$

which provides the least-squares solution to Ax = b. Here, C is a diagonal covariance matrix, with the variances determined by the formal statistical uncertainties in the blue tip fit to each field (or binned fields). The blue tip color for each field (or binned fields) is in b. The design matrix A has dimensions n_field × n_param. It contains one column for the SFD dust map values in each field, five columns for the offsets for the SDSS camera columns two through six, and three columns for each of the 792 runs composing the SDSS-III, for the three terms in the quadratics for each run.

The SDSS-III contains imaging data on 1,147,506 fields. We require that the SDSS score⁸ of each run be greater than 0.5 and the point-spread function (PSF) full width at half-maximum (FWHM) be less than 1.8 arcsec in r, which reduces the number of fields to 686,554. Excluded runs typically are unphotometric, have bad seeing, or are Apache Wheel calibration runs. The blue tip method is run on all of the remaining fields, or binned sets of these fields. It occasionally fails, due to an insufficient number of stars or apparent detection of the blue tip blueward of −0.75 mag or redward of 2.4, 2.4, 1.8, and 1.6 mag in u − g, g − r, r − i, and i − z, respectively. The number of binned fields that successfully make it through the blue tip fit depends on the range of D used and the color, but in the ranges of D examined here it ranges from 684,000 to 677,000. Unbinned fields fail for lack of stars at high Galactic latitudes, and so fewer fields are successful: in ranges of D of interest, between 683,000 and 620,000 fields pass. The median number of stars contributing to the fits in each unbinned field at 10 < D < 19 is 100. About 10⁸ stars enter into the fits.

The global fit produces the coefficients R_a−b in Table 1 and at high latitudes leaves few noticeable coherent residuals (Figure 11). Nevertheless, a few dust clouds are clearly imperfectly subtracted. Most prominently, in the north, a few undersubtracted clouds in g − r stand out on the eastern side of the north Galactic cap (NGC). Undersubtracted clouds in the northwest and at (α, δ) = (145°, 0°) cause the runs including them to be excluded from the fit. The former feature is the largest residual in the Peek & Graves (2010) maps. Subtraction in the south is remarkably clean, though at right ascensions and declinations of (340°, 20°) and (60°, 0°) there are slightly oversubtracted clouds. A cloud at (45°, 20°) barely makes it into the SDSS footprint, and is badly oversubtracted, causing the exclusion of its run from the global fit. In g − r especially, a residual associated with the Monoceros stream stands out at the western edge of the NGC. We have tried masking this region and repeating the fit, but as the Monoceros stream is not correlated with the dust, the fit results were unchanged.

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The blue tip color residuals as a function of E(B − V)_SFD are generally flat (Figure 12). In r − i and i − z, any trend with E(B − V) is at less than the 2% level. In u − g there is no discernible trend, though the scatter is large. In g − r there seem to be disturbing 5%–10% trends with E(B − V) but they may stabilize around E(B − V) = 0.4 mag, with reddenings off by only 15 mmag there.

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The blue tip maps have residuals with distributions given in Figure 13. The core of the distribution is Gaussian, but the wings are non-Gaussian, falling off more slowly than would be expected. The blue tip method gives median estimated uncertainties as 17.5, 12.5, 5.7, and 5.8 mmag, closely reproducing the Gaussian fits to the residual distribution, which give 18.1, 12.3, 6.9, and 7.8 mmag in u − g, g − r, r − i, and i − z, respectively.

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The polynomial terms from the global fit are intended to model the slow, intrinsic variation in the blue tip color and per-run calibration offsets, and so are of independent interest (Figure 14). These maps are clearly heavily striped. Some of this striping is clearly removing calibration problems with the SDSS, as in the eastern side of the NGC in u − g. The striping at high latitudes in the north in other colors is probably substantially an artifact of the low signal in these regions, but the typical stripe-to-stripe difference there is only ∼5 mmag, and at worst 10 mmag.

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The SDSS imaging contains a number of runs crossing the Galactic plane, as part of SEGUE (Yanny et al. 2009). All of these runs and a few other low Galactic latitude runs get identified as discrepant by the fitting algorithm and discarded from the final fit. In a run-by-run treatment, this is inevitable, as at very high extinction the blue tip method will fail because the stars are not behind all the dust. However, for |b|>10°, inspection of the blue tip residual maps suggests that the SEGUE runs could be well fit also, although we have not tried that here. The signal from the dust in the SEGUE runs is sufficiently rich to merit treatment separate from the bulk of the high Galactic latitude sky, as is done in M. J. Berry et al. (2011, in preparation).

Given the presence of dust-correlated residuals in different parts of the sky, one wonders whether the fit is actually "global" in a useful sense, or is primarily a product of the footprint we have decided to look at. Accordingly, we have cut the sky into a number of subregions and separately found the best-fit R_a−b in each. To test for large-scale variations in R_a−b, we divide the sky into northern (b>25°) and southern (b < −25°) Galactic regions, as well as octants of the sky divided by lines of constant Galactic longitude (Figure 15). To try to test variation in dust properties as a function of extinction or temperature, we divide the sky into regions of dust of different extinctions and temperatures. Finally, to verify that we have satisfactorily accounted for the camera column offsets, we divide the SDSS survey into its six camera columns.

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We restrict to the main global fit region and apply the globally derived camera column offsets to the blue tip colors. We remove best-fit polynomials from the filtered SFD dust map and from the blue tip colors, as in Section 4, using the global fit region. We then find the best fit of the filtered, polynomial-subtracted dust map to the polynomial-subtracted blue tip map, restricted to various subsets of the full region.

The results are encouragingly consistent, given that we have already seen that dust reddening normalization can vary substantially, along with, to a lesser extent, its spectrum (Figure 9). We find the dust extinction normalization and spectrum to be consistent at the 10% level. The north/south normalization discrepancy is the most surprising, with the north preferring a normalization about 15% larger. We also find that dust with 0.6mag < E(B − V) < 1.0 mag prefers a ∼15% smaller normalization than dust with 0mag < E(B − V) < 0.05 mag. Dust at different temperatures likewise has similar extinction normalizations and spectra.

The reliability of the fitting procedure can be tested by running the fit on a star catalog from a mock galaxy model. The mock catalog was generated with the galfast catalog generation code (M. Jurić 2011, in preparation). Given a model of stellar number density, metallicity, a three-dimensional extinction map, the photometric system, and instrumental errors, galfast generates realistic mock photometric survey catalogs.

As inputs, we used the number density distribution parameters from Jurić et al. (2008) and the metallicity distribution from Ivezić et al. (2008). The three-dimensional dust distribution map was generated using the model of Amôres & Lépine (2005). The generated u, g, r, i, and z magnitudes were convolved with magnitude-dependent errors representative of SDSS, and the final catalog was flux limited at r < 22.5.

The three-dimensional dust map was modified from Amôres & Lépine (2005) to contain the small-scale clouds seen in SFD. This was performed by scaling each line of sight from the model by the ratio E(B − V)_SFD/E(B − V)_{model,100 kpc}, so that the model matches SFD at 100 kpc. The resulting map contains the expected average three-dimensional distribution of the dust combined with the angular structures present in SFD.

We construct a mock galaxy using galfast and then construct catalogs of observations of these stars for each SDSS run. The resulting catalogs are processed by the blue tip analysis code in exactly the way that the actual SDSS catalogs are processed. As for the actual catalogs, the median number of stars per field is ∼100.

Blue tip fits performed on the mock galaxy recover the R_a−b used to within 3%. This verifies that we properly account for shifts in the blue tip color due to metallicity. However, for these tests we have assumed that SFD correctly predicts the line of sight column density and that the reddening law is the same everywhere on the sky, at least at high Galactic latitudes and in the optical. Insofar as the final fit residuals are largely flat (Figure 11), this assumption seems justified. Moreover, we can verify that the ratios of the R_a−b derived independently from SFD agree with the ratios of the R_a−b derived here.

The preceding analysis has assumed that SFD is a good template for the dust. However, we can relax our dependence on SFD if we restrict our attention to ratios between different R_a−b.

In the absence of changing stellar populations and calibration errors, blue tip colors will fall along a single line given by the reddening vector. By removing a quadratic in field number and accounting for camera column offsets as in Section 4.2, we remove variations in blue tip color not associated with reddening. By fitting a line to the resulting polynomial-subtracted blue tip color–color diagrams, we measure the ratio of the R_a−b. Some striping along the SDSS scan direction is evident in the blue tip residuals despite our attempts to remove it, and this striping will be deleterious to the fit results because we cannot rely on its being uncorrelated with the dust template we fit. Accordingly, we fit lines for each camera column, eliminating the effect of striping, and average the results (Figure 16).

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These color–color fit results are in good agreement with the results of Section 4.2 (Table 2). The results agree to about 1σ with the SFD fit results, where the uncertainties σ are given by the standard deviations of the fit results between the camera columns. The original SFD prescription is seen to clearly overestimate R_u−g/R_g−r (see Section 5.1.3), while both the original SFD prescription and the updated R_V = 3.1 O'Donnell reddening law are found to underestimate R_r−i/R_i−z.

5.1.1. Connecting Reddening Laws to R_a−b

A reddening law gives the extinction A(λ) over a range of wavelengths λ and so makes a prediction for the R_a−b that we have fit. The extinction in band b, in the limit that the variation in the extinction over that band is small, is given by Δm_b = A(λ_eff,b,S)E(B − V), with

$$\begin{equation} \lambda _{\mathrm{eff},b,S} = \frac{\int {d\lambda \lambda S(\lambda) W_b(\lambda)}}{\int {d\lambda S(\lambda) W_b(\lambda)}}. \end{equation} \tag{ 6 }$$

Here the system response for band b is given by W_b(λ), S(λ) is the source spectrum in photons s⁻¹ Å⁻¹, and A(λ) is the reddening law, normalized to give A(λ_eff,B,S) − A(λ_eff,V,S) = 1. The reddening in the color a−b is then given by (A(λ_eff,a,S) − A(λ_eff,b,S))E(B − V), from which it follows that R_a−b = (A(λ_eff,a,S) − A(λ_eff,b,S))E(B − V)/E(B − V)_SFD. Here we have assumed that the variation in the extinction over the band pass is small; for E(B − V) < 1, this assumption changes the predicted R_a−b by less than 1% in r − i and i − z, and less than 5% in g − r and u − g.

Accordingly, we evaluate the extinctions in each SDSS pass band using the SDSS system throughputs and an MSTO source spectrum from a Kurucz model with T_eff = 7000 K, for a variety of reddening laws (Gunn et al. 1998; Kurucz 1993). We examine in particular the CCM, O'Donnell, and Fitzpatrick (1999, F99) reddening laws, which are parameterized by R_V = A_V/E(B − V). We find the best-fit factors R_V to our measured R_a−b for each of these reddening laws.

The other free parameter in these fits is the best-fit normalization N' = E(B − V)/E(B − V)_SFD. We do not report this parameter directly. The SFD dust map is based on a map of thermal emission from dust and is correspondingly proportional to τ_100 μm, the optical depth of dust at 100 μm. Empirically, the ratio between τ_100 μm and E(B − V) is itself a function of R_V, and so N' is covariant with R_V. However, the ratio between τ_1 μm and τ_100 μm depends less on R_V, so we report N = A_{1 μm,predicted}/A_1 μm,SFD. Here, A_{1 μm,predicted} is the extinction at 1 μm implied by the best-fit reddening law and A_1 μm,SFD is the SFD-predicted extinction, extrapolated to 1 μm following the R_V = 3.1 O'Donnell reddening law assumed by SFD. The quantity A_1 μm,SFD is simply 1.32 · E(B − V)_SFD. We find that the F99 reddening law provides the best fit to the R_a−b we measure, so we additionally mention for reference that the F99 reddening law ratio E(B − V)/A_1 μm is 11% larger than the O'Donnell ratio E(B − V)/A_1 μm, though this depends on the adopted R_V and source spectrum.

5.1.2. R_V and N for Individual Runs

We have found that the dust extinction spectrum normalization and shape vary over the sky (Figure 9). We can quantify this effect in terms of variation in N and R_V by fitting reddening laws to the runs that individually well constrain R_a−b (Section 4.1). We use the F99 reddening law here rather than the traditional CCM or O'Donnell reddening law because the latter two reddening laws (especially the O'Donnell reddening law) predict that R_r−i ≈ R_i−z, which is a poor fit to the data. The derived N and R_V vary from run to run, especially as a ∼20% scatter in normalization (Figure 17). Almost all of the runs are consistent with 2.8 < R_V < 3.2. The distribution of R_V and N validates our use of 1 μm as a reference wavelength for N, as the two parameters appear uncorrelated.

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5.1.3. R_V and N for the Global Fit

In order to compute best-fit reddening laws for the global fit R_a−b, we need to account for the covariance in the R_a−b over the sky in addition to the formal statistical uncertainties in the fit. These uncertainties are about a part in one thousand. However, the best-fit normalization can vary at the 10% level from field to field, and the part in a thousand uncertainty reflects averaging these fluctuations over many fields. While the dust extinction spectrum seems relatively constant, it would be surprising if the average reddening reflected the reddening of any particular dust cloud at the part in a thousand level, so another method for estimating the uncertainties is needed. The uncertainties reported in Table 1 are the standard deviations in R_a−b for eight octants in Galactic longitude, and do not account for the covariance of the measurements in the different colors owing to the changing best-fit normalization.

The consequence of adopting uncertainties based on the field-to-field variation in the dust extinction spectrum is that the resulting uncertainties on N and R_V will describe the range of N and R_V among the clouds within the footprint we analyze. They do not give the uncertainties on the best-fit N and R_V within our sample, which are more tightly constrained. As mentioned, the formal statistical uncertainties are about one part in a thousand. Uncertainties estimated from using different ranges of D for selecting the stars give uncertainties of 2%, 2%, 1%, and 2% in R_a−b for the colors u − g, g − r, r − i, i − z, respectively.

The field-to-field variation in R_a−b can be estimated from the sample covariance matrix of best-fit R_a−b in different sky regions. We find the best-fit R_a−b at each point in the footprint. Because over most of the footprint the signal-to-noise ratio (S/N) in the filtered blue tip maps is too low to make a reliable determination of R_a−b, we include nearby pixels in the fit with weights given by a 7° smoothed Gaussian. The best-fit R_a−b is then given by

$$\begin{equation} R_{a-b, n} = \frac{\sum _i D_i C^{-1}_i W_{i, n} b_i}{\sum _i D_i^2 C^{-1}_i W_{i, n}}, \end{equation} \tag{ 7 }$$

where n is the pixel on the sky, i indexes over blue tip measurements, D gives the filtered dust map, b and C⁻¹ give the blue tip measurement and its inverse variance, and W_i,n is the weight matrix, corresponding to Gaussian smoothing with an FWHM of 7°.

The sample covariance matrix of the resulting maps of R_a−b is used as the covariance matrix for fits of reddening laws to R_a−b (Table 3). Only pixels with estimated uncertainty in R_i−z less than 0.01 are used for computing the sample covariance matrix, to avoid overestimating the intrinsic variance due to the uncertainty in the estimates. The largest eigenvalue of the covariance matrix (2.4 × 10⁻²) is 94 times larger than the smallest eigenvalue (2.6 × 10⁻⁴). The eigenvector with the largest eigenvalue is roughly parallel to a vector corresponding to changing the normalization of the spectrum, and that with the smallest eigenvalue corresponds to changing the relative amount of reddening in r − i to i − z. This quantitatively agrees with our claim that the normalization of the reddening law is poorly constrained relative to its shape.

Table 3. Eigenvalues and Eigenvectors of the Reddening Covariance Matrix

$\sqrt{\lambda }$	vu−g	vg−r	vr−i	vi−z
0.155	0.88	1.21	0.45	0.36
0.079	1.23	−0.51	−0.41	−0.79
0.031	0.51	−0.91	0.61	1.05
0.016	−0.15	−0.15	1.35	−0.83

Notes. Eigenvalues λ and eigenvectors v for the sample covariance matrix. Eigenvectors are normalized so that |v| = |R_globalfit|, to facilitate comparison between R_a−b and the first, least well-constrained eigenvector, which is roughly parallel to R_a−b.

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With this covariance matrix, the best-fit reddening laws have R_V = 3.9 ± 0.4, N = 0.98 ± 0.10 for the O'Donnell reddening law, R_V = 3.0 ± 0.4, N = 0.75 ± 0.10 for the CCM reddening law, and R_V = 3.1 ± 0.2, N = 0.78 ± 0.06 for the F99 reddening law (Figure 18). The χ²/dof of the O'Donnell, CCM, and F99 reddening laws are 7.8, 3.5, and 2.1, respectively. The O'Donnell and CCM reddening laws are disfavored because they predict smaller reddening differences between r − i and i − z relative to the data and the F99 reddening law. The F99 reddening law fits all of the points reasonably well.

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The O'Donnell reddening law in particular and CCM reddening law to a lesser extent provide poor fits to the blue tip data, and the best-fit parameters are seriously driven by the poorly matching r − i and i − z data. Accordingly, the derived R_V and N for these reddening laws are unphysical.

All of the fits give R_u−g substantially less than the value given in SFD (Figure 18). The values given in the SFD appendix were based on preliminary estimates of the system response of the SDSS, which varied somewhat from the actual system response. Using the SDSS system response from Gunn et al. (1998) for the R_V = 3.1 O'Donnell reddening law gives values different from those in SFD by 20% in u − g and 10% in g − r and i − z.

The predicted R_a−b depends somewhat on the source spectrum used (Figure 19). We illustrate the effect of changing the source spectrum by plotting the derived reddening laws for the O'Donnell, CCM, and F99 reddening laws for Kurucz models (Kurucz 1993) with T = 6500, 5500, and 4500 from the stellar spectra grid of Munari et al. (2005). The blue tip best-fit reddening law is relatively insensitive to T over this range.

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The maps of R_a−b over the sky can be combined to make maps of R_V and reddening law normalization (Figure 20). We have combined the measured values with a prior of R_V = 3.1 ± 0.2 and N = 0.78 ± 0.06 to reduce the scatter in regions of low S/N.

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One feature of the maps of R_V and N is that the two maps are substantially uncorrelated. This again confirms the expectation that the ratio of τ_1 μm/τ_100 μm does not depend on R_V.

We can directly compare our predicted E(B − V) with E(B − V)_SFD by using our best-fit reddening law and accounting for the difference in source spectrum between the galaxies that SFD analyzed and the MSTO stars we analyze. We get E(B − V)_bluetip = 0.86 · E(B − V)_SFD, suggesting SFD overpredicts reddening by about 14%. The estimated normalization uncertainty in SFD was 4%, while we claim 8% uncertainty in our normalization. However, the estimated fractional uncertainty of SFD was 16%, and we are now in a position to attribute most of this uncertainty to varying best-fit normalization. Accordingly, it is unsurprising that the SFD normalization differs from ours by 14%, particularly given that the footprint we analyze is different from that used in SFD, and that we have found north/south normalization differences of about 15%.

The final E(B − V)_SFD takes the form E(B − V)_SFD = pI_corrX, where p represents a normalization coefficient, I_corr represents the destriped, zodiacal-light subtracted IRAS 100 μm flux and X represents a temperature correction factor. An error in determining the temperature correction factor will lead to dust with a different best-fit normalization, but with the same reddening spectrum, very much like what we see in the blue tip maps.

Accordingly, the blue tip fit residuals can plausibly be attributed to errors in the SFD temperature correction factor. In Section 4.3, we attempted to test the temperature correction by verifying that the best-fit dust normalization was consistent for regions of different temperature according to SFD. However, this procedure assumes that the SFD temperature correction adequately distinguishes hot and cold dust. At high latitudes, the S/N of the DIRBE 100 μm and especially the 240 μm map is too low to construct a 100 μm/240 μm ratio map without substantial filtering. The SFD ratio map was constructed by first smoothing the 100 and 240 μm maps to 1° and then further weighting low S/N pixels to a high |b| average ratio. Insofar as this procedure mixes dust of different temperatures together to a single reported SFD average temperature, it makes it difficult for us to test the SFD temperature correction using the SFD temperature map in this way.

One way to test the accuracy of the temperature correction at high |b| is to compare the SFD temperature correction with the temperature correction used for the FIRAS dust fits (Finkbeiner et al. 1999). The SFD temperature correction is weighted to constants in the north and south in regions of low S/N, while the FIRAS temperature correction is weighted to a 7° smoothed map in these regions. In high S/N regions the two corrections agree. Letting X_SFD be the SFD temperature correction and X_FIRAS be the FIRAS temperature correction, we can plot blue tip color residuals versus the change in predicted dust column density from switching from the SFD to the FIRAS temperature correction: Δ = E(B − V)_SFD(X_FIRAS/X_SFD − 1) (Figure 21).

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Were the FIRAS temperature correction a much better predictor of dust temperature than SFD, we would hope for a linear trend between dust correction and blue tip residual, with slope equal to the best-fit dust coefficients. Instead we find slopes of approximately half the best-fit coefficients, suggesting that the true temperature map is between the SFD and F99 temperature maps. The specific values of the coefficients of the linear fits shown are not reliable because we have not accounted for the uncertainties in Δ in performing the fit, and so the slopes may be underestimated. Regardless, the fact that a clear linear trend with significant slope exists provides strong evidence that the temperature correction at high |b| is unreliable.

The FIRAS temperature correction may also explain the ∼15% normalization difference in best-fit dust extinction spectrum observed between the north and the south in Section 4.3. The ratio X_FIRAS/X_SFD in regions with E(B − V) < 0.05 is about 10% higher in the south than in the north (Figure 22).

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The color of the blue tip changes because of both dust and changing stellar populations. If we deredden the blue tip according to SFD and the best-fit R_a−b of this work, we expect the remaining variation in blue tip color to be due to changing stellar populations (Figure 7).

A surprising feature of the dereddened blue tip maps is that the SGC appears redder than the NGC in g − r, r − i, and i − z, especially outside a region in the southeast of the SGC (Table 4). This may be a genuine structure in the stellar population in the south, but we are unable to distinguish that possibility from calibration errors or errors in the dust map. However, because the south tends to prefer a smaller SFD normalization than the north, the dereddened south would have been expected to be too blue rather than too red (Figure 15). Moreover, the shape of the extra reddening in the different SDSS bands does not look like any plausible reddening law.