MEASURING REDDENING WITH SLOAN DIGITAL SKY SURVEY STELLAR SPECTRA AND RECALIBRATING SFD

The eighth data release of the SDSS provides uniform, contiguous imaging of about one-third of the sky, mostly at high Galactic latitudes (Aihara et al. 2011). The SDSS imaging is performed nearly simultaneously 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 the bandpasses u, g, r, i, and z, which have central wavelengths of about 3600, 4700, 6200, 7500, and 8900 Å. We use SDSS data that have been photometrically calibrated according to the "ubercalibration" procedure of Padmanabhan et al. (2008).

The technique presented here is sensitive to the SDSS photometric calibration, and so we briefly describe the basic units in which the SDSS data are calibrated: the run and camera column ("camcol"). The SDSS takes imaging data in runs that usually last for several hours. These runs observe regions of sky that are long in right ascension and narrow in declination. Each run is composed of observations from six camcols: the six sets of five CCDs (one for each of the photometric bands) that fill the SDSS focal plane.

The SDSS photometry is used to select targets for follow-up SDSS spectroscopy. Spectroscopy is performed with two multiobject double spectrographs, which are fed by 640 fibers. The spectra cover a wavelength range of 3800–9200 Å at a resolution of λ/Δλ ≃ 2000. The stars used in this work were targeted for spectroscopy either as part of the main SDSS survey or as part of the Sloan Extension for Galactic Understanding and Exploration (SEGUE; Yanny et al. 2009) and its successor, SEGUE-2. Targets from the main survey are observed until they reach a target signal-to-noise ratio (S/N) of about $(\mathrm{S}/\mathrm{N})^2 > 15 \,\mathrm{{\rm pixel}^{-1}}$ for stars with magnitudes g = 20.2, r = 20.25, i = 19.9. SEGUE targets are exposed long enough to achieve (S/N)² > 100 at the same depth, so that stellar parameters can be measured (Abazajian et al. 2009).

The SDSS has a number of programs targeting different stellar types for spectra. Figure 1 shows an ugr color–color plot of the different types of stars used in this work, together with the overall stellar locus. Yanny et al. (2009) gives a detailed description of the criteria that define these targets. The stars used in this work cover most of the stellar locus, except for the reddest stars, for which stellar parameters are poorly measured. Table 1 gives the names of each of the programs targeting stars used in this analysis, as well as the number of stars used. In addition to stars targeted for specific science goals, we also use the reddening and spectrophotometric standard star target types. These stars are F stars used for calibration purposes in the SDSS spectroscopic pipeline.

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The SEGUE is a spectroscopic and photometric extension to the SDSS designed to provide insight into the structure and history of the Galaxy (Yanny et al. 2009). As part of this program, the SSPP was developed to uniformly process the stellar spectra from the SDSS (Lee et al. 2008a). This work uses the eighth data release of the SDSS, which includes an updated version of the SSPP, and spectra from SEGUE-2 in addition to SEGUE. The SSPP provides multiple estimates of the spectral type, metallicity, gravity, and radial velocity for all of the stars with SDSS spectra. The accuracy of these parameters has been repeatedly tested by comparison with high-resolution spectroscopy and by using measurements from globular and open clusters (Lee et al. 2008b; Allende Prieto et al. 2008; Smolinski et al. 2011).

The SSPP provides a number of different estimates for the temperature, metallicity, and gravity of each star, and a single composite estimate. We cannot use the composite estimates produced by the SSPP because some of the SSPP parameter estimates rely on the photometry of the stars, which is affected by the dust. Instead we rely only on methods that do not make use of any photometry. These are the NGS1, ki13, ANNRR, and ANNSR methods (Lee et al. 2008a). For this work we have used the ANNRR estimator (Re Fiorentin et al. 2007), which uses a neural net trained on continuum-normalized SDSS spectra to make parameter estimates. However, the four methods produce compatible stellar parameters and the results of this work are insensitive to the method used.

The stellar parameters derived from the SSPP are transformed into predicted ugriz colors using the MARCS grid of model atmospheres (Gustafsson et al. 2008). Each model includes estimates of the surface flux at a resolution of λ/Δλ of 20,000, and covers the range 1300–200,000 Å. The grid covers an extensive range of stellar parameters: 2500 K ⩽ T_eff ⩽ 8000 K, −5 ⩽ [Fe/H] ⩽ 1, and −1 ⩽ log g ⩽ 5.5, as well as additional parameter combinations involving α-enhancement and microturbulence. Edvardsson (2008) and Plez (2008) have verified the accuracy of synthetic broadband colors predicted from this grid. For log g < 3, we use the MARCS spherical stellar atmosphere models, while for log g ⩾ 3, we use the plane parallel models. We have experimented with using other grids: the Munari et al. (2005) grid, the Castelli & Kurucz (2004) grid, and the NGS1 grid used internally in the SSPP. The Munari et al. (2005) grid predicts bluer colors than the other grids owing to the grid's not including "predicted lines," faint lines which have not been individually detected (Munari et al. 2005). Nevertheless, because we calibrate the predicted colors to match the observed colors (Section 4.2), our final results are insensitive to the choice of grid, even when the raw predicted colors are substantially discrepant from the observed colors.

The SDSS targets a wide variety of stars for spectroscopy, including exotic types for which the spectral parameters are difficult to measure. We are interested in the dust, and so want to select stars for which the stellar parameters are well understood and accurately measured. Accordingly, we exclude from this analysis stars targeted to have unusual colors and very red stars with dereddened g − r > 1 mag. Likewise, we exclude any white dwarf targets. Finally, we exclude any objects with stellar parameters marked as unreliable by the SSPP (according to the parameters' indicator variables).

The remaining target types and number of stars used in each type are listed in Table 1. We find it convenient to occasionally divide these target types into five classes. These classes are the standard stars, which were targeted as reddening standards or spectrophotometric standards; the FG stars, targeted to have spectral types F or G; the BHB stars; the K stars, targeted to have spectral near K; and the other stars, which include the low-metallicity and low-latitude targets.

We have chosen not to impose a cut on the S/N of the spectra. We have varied cuts on S/N from 0 to 50; the resulting best-fit reddening coefficients presented in this work vary by only 0.5% in that range.

The total number of stars used following these cuts is 261,496.

We test the quality of the Δ by restricting to the region of sky with |b| > 50° and E(B − V)_SFD < 0.04. In this region of low extinction, excursions of Δ from zero should be due primarily to the statistical uncertainties in the photometry and stellar parameters. However, additional systematic effects can cause Δ ≠ 0. These include any mismatch between the synthetic and real spectra and biases in the SSPP-derived stellar parameters.

The performance of Δ for this sample of stars is given in Table 2, for several classes of star. The rows marked "raw" give the means and standard deviations of the uncalibrated Δ relevant here. In all colors, the mean Δ is within about 30 mmag of 0, comparable with the 10–20 mmag uncertainty in the SDSS zero points (Abazajian et al. 2004). In r − i and i − z, the scatter in Δ is 20–30 mmag, comparable to the SDSS photometric uncertainty.

Table 2. Bias in Predicted Magnitudes

Target Class	$\overline{\Delta _{u-g}}$	σu − g	$\overline{\Delta _{g-r}}$	σg − r	$\overline{\Delta _{r-i}}$	σr − i	$\overline{\Delta _{i-z}}$	σi − z
Standards (raw)	22	51	3	33	28	22	17	25
Standards (cal)	−3	46	−2	31	−1	22	−1	24
FG (raw)	0	82	−2	42	30	28	20	35
FG (cal)	1	72	−2	37	0	27	0	34
BHB (raw)	64	83	−1	50	36	32	33	42
BHB (cal)	24	90	−13	46	0	31	−2	40
K (raw)	10	114	−24	57	29	32	14	32
K (cal)	5	73	4	35	0	27	0	28
Other (raw)	13	88	20	47	34	28	18	31
Other (cal)	−16	68	8	38	3	27	0	30
All (raw)	12	92	−6	51	30	29	18	33
All (cal)	1	71	0	37	0	27	0	31

Notes. The mean ($\overline{\Delta _{}}$) and standard deviation (σ) of Δ in mmag for different types of stars (Section 2.2), before and after calibration (Section 4.2), in each of the SDSS colors. Rows marked "(raw)" or "(cal)" indicate whether the means and standard deviations are given before or after calibration, respectively. After calibration, the mean of Δ is close to zero, and the scatter in Δ is reduced, especially in the color u − g.

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However, in the colors u − g and g − r, the scatter in Δ is somewhat larger than expected. The expected uncertainty in Δ_{u − g} and Δ_{g − r} is about 40 and 30 mmag, respectively, including both the photometric uncertainty and the uncertainty in the stellar parameters. However, the scatter in Δ in this sample of stars is 92 and 51 mmag in the colors u − g and g − r. Figure 2 (left panels) shows that additionally there are trends in Δ with temperature, metallicity, and gravity, especially in the color u − g. Because we expect that typical stellar metallicity is correlated with the dust column to that star, these trends could bias this analysis and must be removed. Moreover, these trends contribute to the scatter in Δ_{u − g} and Δ_{g − r}.

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We do not fully understand the source of these trends. The two most likely candidates are mismatches between the synthetic stellar spectra and real stellar spectra, and biases in the SSPP stellar parameters. We have tested for mismatch between the synthetic and observed spectra by comparing with other synthetic grids. Comparing the grids of Gustafsson et al. (2008), Castelli & Kurucz (2004), and Lee et al. (2008a) indicates that the grids can disagree with one another by about 50 mmag in u − g and g − r. The larger effect, however, is probably due to biases in the SSPP stellar parameters. The data release 8 version of the SSPP introduced substantial changes to metallicity estimates for stars around solar metallicity, by about 0.4 dex (Smolinski et al. 2011). Colors predicted using parameters from the data release 7 version of the SSPP were too blue by as much as 200 mmag in u − g.

However, we calibrate the predicted colors to remove the trends and to render our results insensitive to these sources of systematic errors. We have repeated the full analysis using four different grids (Gustafsson et al. 2008; Castelli & Kurucz 2004; Lee et al. 2008a; Munari et al. 2005). The final results of this work were unchanged at the 1% level. We have also repeated the analysis using the data release 7 and 8 versions of the SSPP, which changed the final results by at most 2%.

We remove these trends by calibrating the synthetic photometry so that Δ ≈ 0 on the high-latitude sample of stars where reddening is unimportant. This procedure has been done before; for instance, Fitzpatrick & Massa (2005) also calibrate synthetic photometry to match observed photometry. In this work, we perform the calibration by fitting an empirical curve to Δ as a function of temperature, metallicity, and gravity, and adjusting the synthetic photometry to remove this curve from Δ.

To properly model the trends in Δ, we should find the function f of temperature T_eff, metallicity log Z, and gravity log g that most accurately predicts the colors. To be more precise, f should be chosen to minimize the distance between Δ and f in color space, considering the covariance matrix for the observed colors and for the predicted colors. We however deemed this computationally intractable and unnecessary, given the wide range of temperatures (4000 K < T_eff < 8000 K), metallicities (−3 < log Z < 0), and gravities (2 < log g < 5) available and the small estimated uncertainties in the stellar parameters for the objects we look at (σ_T ≲ 150 K, σ_log Z ≲ 0.1 dex, σ_log g ≲ 0.1 dex). We therefore ignore the covariance between the stellar parameters and Δ, and rely on the fact that the stars cover a broad range of parameters (much broader than the uncertainties in those parameters) to guarantee that the bias introduced by this is small.

To maximize the effective range of stellar parameters, we need a set of stars for calibration that cover parameter space evenly without overweighting stars of a particular type. Accordingly we choose stars from the ugr color plane, capping the maximum number of stars chosen from any location in the plane. This guarantees that all of the available stars are used in areas of color space where few stars were targeted, but limits the number of stars used in areas where many stars were targeted. The primary consequence of this selection is to reduce the importance of the many SEGUE1_GD targets chosen from a narrow range in temperature.

With this set of stars in hand, we find the function f that minimizes

$$\begin{equation} \sum _i \left(\frac{\Delta _{c,i}-f_c(x_i)}{\sigma _{c,i}} \right)^2 \end{equation} \tag{ 2 }$$

for each color c, where i indexes over stars, Δ gives the measured color minus the predicted color, and x gives the stellar parameters for the stars. The uncertainty σ is computed from the photometric uncertainty and the uncertainty in the predicted colors derived from the uncertainty in the stellar parameters. The values Δ here are corrected for reddening according to S10, though as E(B − V)_SFD is less than 0.04 for this sample this is of little importance. We use for f a fifth-order polynomial, though we have also used a second-order polynomial with negligible effect on the final results of this work.

Figure 2 (right panels) shows that the remaining trends in Δ are small. The rows in Table 2 marked "cal" show the means and standard deviations of Δ after calibration for several classes of star. The calibration brings the mean of Δ to zero. The calibration has the additional benefit of reducing the scatter in Δ to be much closer to the expected statistical uncertainty of about 40, 30, 25, and 30 mmag in u − g, g − r, r − i, and i − z.

To verify that the final results are insensitive to the details of the calibration, we have varied the order of the polynomial used and the sampling method used to select calibration objects. As long as the order of the polynomial is at least two, we achieve consistent results at the 1% level. The sampling method is similarly unimportant except in extreme cases where the calibration stars do not cover a satisfactory range of stellar parameters.

The calibrated measurements of Δ immediately permit the construction of a reddening map, which is shown in Figure 3 (left panels). The right panels show the map after correction for dust according to SFD, using the prescription of S10. The derived reddening map agrees well with that found in S10, clearly identifying the same runs with bad zero points in u − g and the SFD-underpredicted region in the northwest of the north Galactic cap. In the Galactic plane, the stars used for the reddening measurements may not be behind the entire dust column, and so it is unsurprising that their reddening is less than that predicted by SFD. Over most of the observed sky, however, the residuals after correction for dust are within the uncertainties.

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As pointed out in S10, we again find that the dereddened south is somewhat redder than the dereddened north (Section 5.5). Apart from this trend and the cloud in the northwest, the residual maps are reassuringly flat and reveal no trends in color with Galactic latitude or longitude.

Figure 4 shows the residual g − r map of Peek & Graves (2010). This map is compared with our residual g − r map, smoothed to match the 45 resolution of the Peek & Graves (2010) map. The two maps show several consistent large-scale features. Both maps detect the SFD underprediction in the northwest, and additionally find that the southwest of the North Galactic Cap is somewhat redder than the southeast, by about 10 mmag.

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We use the reddenings Δ to constrain R_{a − b}, the reddening in the color a − b divided by E(B − V)_SFD. We measure the R_{a − b} by fitting Δ as a linear function of E(B − V)_SFD, the slope of which gives R_{a − b}. We find that we can detect calibration errors in the SDSS using this technique. Accordingly, we perform a second similar linear fit, where additionally a constant zero-point offset in each SDSS run is also fit.

The R_{a − b} are given by the equation

$$\begin{equation} \Delta _{a-b} = R_{a-b} E(B-V)_\mathrm{SFD}+ C, \end{equation} \tag{ 3 }$$

where C gives a constant offset which should give zero if the calibration of the reddening values in the high Galactic latitude, low reddening is accurate when extended over the entire footprint. We find R_{a − b} through least-squares minimization of

$$\begin{equation} \chi ^2 = \sum _i \left(\frac{\Delta _{a-b,i} - R_{a-b} E(B-V)_{\mathrm{SFD},i} - C}{\sigma _i} \right)^2, \end{equation} \tag{ 4 }$$

where i indexes over stars, R_{a − b} and C are the fit parameters, E(B − V)_{SFD, i} is E(B − V)_SFD in the direction of star i, and σ_i is the uncertainty in Δ_{a − b, i}, considering both the uncertainty in the measured color and the uncertainty in the predicted color, propagated from the uncertainty in the stellar parameters.

To render the fit insensitive to calibration errors in the SDSS, we also fit

$$\begin{equation} \Delta _{a-b} = R_{a-b} E(B-V)_\mathrm{SFD}+ C_r \end{equation} \tag{ 5 }$$

in an analogous least-squares sense, where r indexes over SDSS run number, which absorbs any zero-point calibration errors in the SDSS runs. We call this the fit with zero-point offsets. The results of this fit are similar to those given by Equation (3). Each fit is iterated and clipped at 3σ in each color, until finally all clipped points are removed and the fit is repeated with the same clipped set of stars in each color.

The results from the fits are given in Table 3. The fits with and without zero-point offsets agree to within 4%. Both sets of fits give values consistently slightly smaller than the S10 values (∼4%). In S10, we found that the best-fit normalization of SFD varied over the sky by about 10%, and so this 4% normalization difference may be due to the different sky footprint available to the S10 and spectrum-based analyses. However, when we tested this at high Galactic latitudes in the north where the two footprints overlapped, the results were inconclusive because the uncertainties were similar in size to the 4% effect we observe. The χ² per degree of freedom (χ²/dof) for the fits are quite good, especially in the redder colors; without including zero-point offsets, we achieve 1.45, 1.19, 1.04, and 1.02 χ²/dof in the colors u − g, g − r, r − i, and i − z, respectively.

These results confirm the preference for an R_V = 3.1 F99 reddening law and a 14% recalibration of SFD, first reported in S10. The predicted R_{a − b} according to SFD and an O'Donnell (1994) reddening law are in disagreement with the measurements.

Figure 5 shows the residuals to the fit without zero-point offsets. The residuals are largely Gaussian, with σ of 56, 34, 25, and 29 mmag in u − g, g − r, r − i, and i − z. This scatter is substantially better than the scatter we achieved in the calibration sample of stars (Table 2), because those stars were chosen to be widely distributed over the ugr color–color plane and therefore many of those stars had outlying photometry.

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The residuals as a function of E(B − V)_SFD display a striking trend, shown in Figure 6. The residuals increase with E(B − V) for E(B − V) ≲ 0.2, and then decrease thereafter. This result is consistent with the S10 result that the best-fit normalization of the dust was larger for E(B − V) < 0.2 than it was for E(B − V) > 0.2 by about 15%, which is about the size of the trend seen here. That said, the entire E(B − V) ≳ 0.3 footprint for this analysis occurs in a relatively small area near the Galactic anticenter and a few SEGUE low-latitude fields, and may not apply more generally.

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The formal statistical uncertainties in the fit results are small (∼2 parts in a thousand). These uncertainties dramatically underestimate the true uncertainty in the fit, stemming from biases in the photometry (Section 5.3.1), unmodeled calibration errors in the SDSS, and problems with using SFD as a template (S10). The reported uncertainties in Table 3 are derived by a Monte Carlo simulation with simple models for the calibration errors and SFD template errors. Calibration errors in the SDSS are modeled as normally distributed run zero-point errors, with standard deviations of 20 mmag in u and 10 mmag in g, r, i, and z, according to the estimated calibration uncertainties of Padmanabhan et al. (2008). We additionally allow the true zero point to change by $\dot{a}t$ over the course of the night, where $\dot{a}$ gives the rate of change of zero point and t is the time of night. We choose $\dot{a}$ to be normally distributed with standard deviations of 20 mmag per 6 hr in u and 10 mmag per 6 hr in g, r, i, and z. For the SFD template errors, we multiply SFD by a normalization which varies over the sky on 1° scales, with a mean of 1 and standard deviation of 0.1. This is very roughly intended to simulate the uncertainty in the SFD dust temperature correction, but does not include the covariance between temperature and extinction that may be important.

The resulting uncertainties are dominated by uncertainties coming from the calibration effects, with the zero point and change in zero point with time effects contributing similarly.

We can free the analysis from a dependence on SFD and its normalization by looking only at ratios of R_{a − b}. If Δ has been appropriately calibrated, then in the absence of dust Δ should be consistent with zero. The presence of dust will spread Δ along a line in color–color space. The slope of this line in a pair of colors a − b and c − d gives the ratio R_{a − b}/R_{c − d} and is determined by the reddening law.

We find the best-fit slope of the line in Δ in color space in a least-squares sense, considering the covariance between the Δ_{a − b} induced by the photometry and the color predictions from the stellar parameters. Figure 7 shows the result of this fit, together with the predicted line from S10 and from an O'Donnell (1994) reddening law. Table 4 tabulates the best-fit parameters for the ratios of R_{a − b}. The fits do a good job at reproducing the S10 ratios for R_{a − b}, but disagree with the O'Donnell (1994) reddening law predictions. We achieve χ²/dof of 1.4 on average over all the stars we fit. On the more limited subset of standard stars we get χ²/dof of 1.1.

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An important caveat applies to this analysis. The Δ_{a − b} for a star are correlated. We track the correlation induced by the uncertainty in the photometry and the uncertainty in the stellar parameters. However, any unknown effect that changes the predicted or measured colors of stars will additionally spread out Δ in color–color space, and there is a danger of interpreting that signal as signal from the dust. When instead we fit Δ as a function of E(B − V)_SFD, we have the advantage of using a template that is uncorrelated with the uncertainties in Δ and that is known to be tightly correlated with the dust.

To verify the robustness of these results we have separately fit the standard stars, the F and G stars, the BHB stars, the K stars, and all of the target types we look at. The results were consistent at the 3% level.

5.3.1. The Photometric Bias

5.3.2. The Three-dimensional Footprint

The Δ are sufficiently numerous and cover a sufficiently large region of the sky that they can be usefully cut into different subsets to study the variation in R_{a − b} over the sky. Figure 8 shows the results of such an analysis. The panels show R_{a − b} for different ranges in Galactic latitude, longitude, E(B − V)_SFD, and dust temperature according to SFD. The rightmost panel shows R_{a − b} as derived for stars observed with each of the SDSS camcols. All R_{a − b} are fit with zero-point offsets to minimize sensitivity to calibration errors in the SDSS.

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The fits' results show smaller scatter between the various sky regions. We recover the S10 result that stars with E(B − V)_SFD < 0.1 prefer a larger normalization than other stars by ∼10%; likewise in the north relative to the south, and possibly for the same reason, as the north has less dust than the south. Fits to the dust in different ranges of Galactic longitude and at different temperatures according to SFD are reasonably consistent.

We can get some idea as to whether or not the filter differences between the SDSS camcols are important to this analysis by splitting the survey by camera column. We find very close (2%) agreement between the various camcols. This also serves to place an upper bound on the statistical uncertainty in the measurements.

These results on the universality of the R_{a − b} are consistent with those found in S10. In that work, we did a more detailed analysis of the variation in R_{a − b} over the sky and found the dominant mode of variation in R_{a − b} took the form of a 10% variation in the overall normalization of the R_{a − b}. We have not repeated that analysis here, but we try to account for the effects of a 10% normalization variation in our uncertainties (Section 5.1).

In S10, we noted an asymmetry between the north and south in the dereddened color of the main-sequence turnoff, which we called the blue tip color. We were unable to tell whether this color difference was a calibration effect or a result of the south having an intrinsically different blue tip color than the north. Because the spectrum-based reddening measurements are sensitive to calibration problems and not to the intrinsic color of the blue tip, we can use them to break this degeneracy.

Table 5 shows the differences in Δ between the Galactic north and south, compared with the blue tip color differences found in S10. We find the offset in Δ for each target class, and report the mean of these offsets in Table 5. For the uncertainty in the means, we report the standard deviation of the offsets. The color differences are comparable in r − i and i − z, suggesting that calibration errors in the SDSS are the source of the discrepancy in these colors. In g − r, however, the blue tip color differences are larger than the spectrum-based differences. In u − g, the uncertainty is too large to make any definitive statement. The difference in these colors could then be caused by a fundamentally different blue tip color in the north than in the south. This result is heartening because the color of the blue tip varies due to age and metallicity changes in stellar populations, which should primarily affect the u − g and g − r colors while leaving the r − i and i − z colors unchanged.