The Blue Tip of the Stellar Locus: Measuring Reddening with the SDSS
Edward F. Schlafly, Douglas P. Finkbeiner, David J. Schlegel, Mario Jurić, Željko Ivezić, Robert R. Gibson, Gillian R. Knapp, Benjamin A. Weaver
aa r X i v : . [ a s t r o - ph . GA ] S e p Draft version October 3, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
THE BLUE TIP OF THE STELLAR LOCUS:MEASURING REDDENING WITH THE SDSS
Edward F. Schlafly , Douglas P. Finkbeiner , David J. Schlegel , Mario Juri´c , ˇZeljko Ivezi´c , Robert R.Gibson , Gillian R. Knapp , Benjamin A. Weaver Draft version October 3, 2018
ABSTRACTWe present measurements of reddening due to dust using the colors of stars in the Sloan Digital SkySurvey (SDSS). We measure the color of main sequence turn-off stars by finding the “blue tip” of thestellar locus: the prominent blue edge in the distribution of stellar colors. The method is sensitive tocolor changes of order 18, 12, 7, and 8 mmag of reddening in the colors u − g , g − r , r − i , and i − z ,respectively, in regions measuring 90 ′ by 14 ′ . We present maps of the blue tip colors in each of thesebands over the entire SDSS footprint, including the new dusty southern Galactic cap data provided bythe SDSS-III. The results disfavor the best fit O’Donnell (1994) and Cardelli et al. (1989) reddeninglaws, but are well described by a Fitzpatrick (1999) reddening law with R V = 3 .
1. The Schlegel et al. (1998, SFD) dust map is found to trace the dust well, but overestimates reddening by factors of 1.4,1.0, 1.2, and 1.4 in u − g , g − r , r − i , and i − z , largely due to the adopted reddening law. In selectdusty regions of the sky, we find evidence for problems in the SFD temperature correction. A dustmap normalization difference of 15% between the Galactic north and south sky may be due to thesedust temperature errors. Subject headings: dust, extinction — ISM: clouds — Galaxy: stellar content INTRODUCTION
Most astronomical observations are affected by Galac-tic interstellar dust, whether as a source of foregroundlight in the microwave, far-infrared and gamma-ray wave-length regions or as a cause of extinction in the infraredthrough ultraviolet (Draine 2003). Characterizing theproperties of the dust and accounting for its effects onobservations is then a central problem in astronomy.Dust is formed as stars burn nuclear fuel to heavy ele-ments and emit these elements in stellar winds or in moreviolent eruptions, and these elements are reprocessed inthe interstellar medium (Draine 2009). The distributionof dust is correspondingly correlated with the hydrogenin the interstellar medium (ISM), though the dust-to-gasratio varies. Burstein and Heiles (1978, BH) took advan-tage of the correlation between H I and dust column den-sity to make the first widely used map of dust columndensity, combining H I emission and galaxy counts.The BH dust map was superseded by the Schlegel et al. (1998, SFD) dust map, which took advantage of thefull-sky far-infrared (FIR) data provided by IRAS andDIRBE, which are dominated by thermal emission fromthe Galactic dust at wavelengths of 100 µ m and longer.After correction for dust temperature using FIR color ra-tios, these maps trace dust column density. Calibration Department of Physics, Harvard University, 17 OxfordStreet, Cambridge, MA 02138, USA Harvard-Smithsonian Center for Astrophysics, 60 GardenStreet, Cambridge, MA 02138, USA Lawrence Berkeley National Lab, 1 Cyclotron Road, MS50R5032, Berkeley, CA 94720, USA Hubble Fellow Department of Astronomy, University of Washington, Box351580, Seattle, WA 98195, USA Department of Astrophysical Sciences, Princeton University,Peyton Hall, Princeton, NJ 08544, USA Department of Physics, New York University, 4 WashingtonPlace, New York, New York 10003, USA of dust column density to color excess E ( B − V ) was per-formed using the colors of a sample of 389 galaxies withMg indices (Faber et al. et al. et al. in the southern Galactic sky.This allows us to more tightly constrain the SFD normal-ization and the dust extinction spectrum, or “reddeninglaw,” over the SDSS bands, as well as to provide a map ofcolor residuals that point to problems with the SFD dustmap, and particularly with the temperature correction.Previous tests of SFD have usually found thatSFD overpredicts extinction in high-extinction regions.Shortly after its introduction, Arce and Goodman (1999)found that SFD extinction was too high by 40% in Tau-rus, using star counts, colors, and FIR emission. Like-wise, studies using globular cluster photometry, galaxycounts, and NIR colors have found that SFD overpre-dicts extinction by a similar fraction in other dusty re-gions (Stanek 1998; Chen et al. et al. et al. (2001) ex-plore the link between the SFD overestimation and dusttemperature using star counts in the Polaris Flare. At | b | < ◦ , Dobashi et al. (2005) use optical star countsto conclude that SFD overpredicts extinction by a fac-tor of two or more. We perform similar tests to thesein regions of lower extinction than had been previouslypossible, taking advantage of the high quality and depthof the SDSS stellar photometry, and complementing therecent work of Peek and Graves (2010), who use SDSSgalaxy spectra. In these regions, we do not find thatSFD overpredicts reddening by a large factor; rather, wefind that SFD overpredicts reddening by about 14% in B − V , though, because of the reddening law adopted bySFD, this varies from color to color.The colors of stars vary substantially with stellar type,and with location in the Galaxy due to the effect ofmetallicity on color. Nevertheless, we find that the colorsof the most blue main sequence stars in old populations—the main sequence turn-off (MSTO) stars—are remark-ably stable over the sky, and that we can empiricallymodel their slow spatial variation. We therefore presentmeasurements of the colors of the “blue tip” of the stel-lar locus, and use them to constrain the SFD dust map.This work is akin to that of High et al. (2009), in whichthe color of stellar populations is also used as a sort ofcolor standard.In § §
3, wepresent our method for measuring the blue tip of the stel-lar locus, and the corresponding maps of blue tip colorson the sky. In §
4, we present fits of the SFD dust map tothe blue tip colors. In § §
6, we discuss these resultsand conclude. The blue tip maps and measurements canbe found at the web site skymaps.info/bluetip . DATA
The SDSS
The SDSS is a digital spectroscopic and photomet-ric survey, that, with the additional south Galac-tic cap (SGC) data provided by the SDSS-III, coversjust over one third of the sky, mostly at high lati-tudes (Abazajian et al. u , g , r , i ,and z (Gunn et al. et al. stars. The SDSS is 95% complete up to magnitudes22.1, 22.4, 22.1, 21.2, and 20.3 in u , g , r , i , and z . TheSDSS imaging is performed in a drift-scanning mode inwhich the shutter is left open while the telescope movesat constant angular speed across the sky. The result-ing imaging “runs” tend to cover strips of sky that arelong in right ascension and narrow in declination. Theseruns are divided into fields, which are 13 . ′ by 9 ′ in size.We use SDSS data that have been photometrically cal-ibrated according to the “ubercalibration” procedure ofPadmanabhan et al. (2008). The SFD Dust Map
The SFD dust map is a map of thermal emission fromdust based on the
IRAS µ m maps. The IRAS µ mmap underwent three major processing steps before beingused in the SFD dust map: it was destriped, zodiacal-light subtracted, and calibrated to match COBE/DIRBEat degree scales. The first and third of these steps werenecessary because the zero point of the IRAS µ mdetector varied over time, imprinting the IRAS scan pat-tern on the data in the form of stripes, and making theoverall zero point of the data uncertain. The zodiacal-light subtraction is needed to remove the signature of thehot, bright, local interplanetary dust from the
IRAS andDIRBE data, which, though bright, contributes negligi-bly to the reddening. The DIRBE 100 µ m to 240 µ mflux ratio is used to constrain the dust temperature andto transform the 100 µ m flux map to a map propor-tional to dust column density. The destriped IRAS data have resolution of 6 arcminutes, while the temperaturecorrection is smoothed to degree scales. The destriped,temperature-corrected
IRAS µ m map is normalizedto a map of E ( B − V ) by using a sample of 389 galax-ies with Mg indices and B − V colors, for use as colorstandards (Faber et al. THE BLUE TIP OF THE STELLAR LOCUS
Main sequence stars are confined to a tight locus incolor-color space, as is apparent from a typical SDSScolor-color diagram (Fig. 1). The locus has a sharp blueedge at the color of the MSTO stars, blueward of whichthere are only rare blue stragglers, white dwarfs, andquasars. The intrinsic color of the MSTO is set by theproperties of the stellar population: primarily, metallic-ity and age. The observed colors of the MSTO are theintrinsic colors, altered by reddening due to dust andby systematic color shifts due to imperfect calibration ofthe observations. Empirically, the intrinsic color of theMSTO is slowly varying in space; the observed color ofMSTO stars is therefore a good probe of the reddeningdue to dust.Population effects cause the color of main sequencestars to vary as a function of the magnitude of starsprobed. At faint magnitudes and high latitudes, metal-poor halo stars dominate, rendering the MSTO starsbluer than at bright magnitudes and lower latitudes,where redder disk stars dominate (Fig. 2). Moreover, theshape of the blue edge changes with magnitude: owing tophotometric errors, the edge is sharp for bright stars andbecomes blurred at fainter magnitudes. Accordingly, thetypical color of MSTO stars depends on magnitude.Stars of similar metallicities may clump together in thehalo, changing the observed color of the blue tip. Becausewe do not have a good way to model such effects, wecannot distinguish such clumps from reddening due todust in the blue tip maps.
Measuring the Blue Tip
We perform fits to the location of the blue tip of thestellar locus in each SDSS field (or group of SDSS fields).To avoid biasing the fits, we select stars in each field foranalysis in a reddening-independent way, and then findthe location of the blue tip in the resulting sample.Care must be taken when imposing cuts on the starsin each field to avoid biasing the resulting blue tip color.Because extinction changes the observed magnitudes ofstars, a flat cut on magnitude would bias the measure-ment, as the distance range probed would change withthe dust column. Instead, we perform cuts on D , the g magnitude of a star projected along the reddening vectorto zero color: D = g − A g A g − A r ( g − r ) . (1)Provided A g / ( A g − A r ) is accurate, the set of stars in aparticular range of D is independent of the dust columntowards the stars. D is related to the distance modu-lus of the MSTO stars (though not redder stars). Usingthe Juri´c et al. (2008) “bright” photometric parallax re-lation, MSTO stars with r − i = 0 . g − r = 0 . M g = 5 .
3. Accounting for the projection from g − r = 0 . g -r g -r -0.5 0.0 0.5 1.0 1.5 2.0g-r-0.50.00.51.01.52.0 r- i -0.5 0.0 0.5 1.0 1.5 2.0g-r-0.50.00.51.01.52.0 r- i -0.5 0.0 0.5 1.0 1.5 2.0r-i-0.50.00.51.01.5 i - z -0.5 0.0 0.5 1.0 1.5 2.0r-i-0.50.00.51.01.5 i - z Figure 1.
Color-color diagrams for point sources from the SDSS, within two degrees of the north Galactic pole. The sharp cutoff instellar density blueward of the “blue tip” is prominent. In u − g , the points blueward of the blue tip are mostly quasars and white dwarfs.The blue tip is at about 0.8, 0.2, 0.1, and 0.0 mags in u − g , g − r , r − i , and i − z , respectively. u u g g r r i i Figure 2.
Color-magnitude diagrams from the SDSS, for the same stars as in Fig. 1. The location of the blue edge of the stellar locusdepends on magnitude, especially in the bluer bands, where bright, nearby, more metal-rich stars are redder than more distant halo stars.In u − g , blueward of the blue edge lie mostly quasars and white dwarfs. The location of the blue tip is as in Fig. 1. D ≈ g − M g + 4 .
6. In this work, we frequently use therange 10 < D <
19. Including the effect of saturationon the bright end, this restriction on D selects stars withdistances between about 1 and 8 kpc.With a reddening-independent population of stars inhand, we measure the color of the blue edge of thesestars in each SDSS color (Fig. 3). The stellar colors aremodeled as drawn from a probability distribution P ( x ),taken to be a step function convolved with a Gaussianto reflect the photometric uncertainty and the intrinsicwidth of the blue edge. The maximum-likelihood loca-tion for the step is denoted the “blue tip” of the locus.Specifically, the form of the probability distribution istaken to be P = 12 (cid:20) (cid:18) x − x √ σ P (cid:19)(cid:21) + F, (2)where erf is the Gaussian error function, σ P correspondsto the width of the edge and F sets a floor to the prob-ability distribution, to render the fit insensitive to theoccasional white dwarf or blue straggler. In this work weuse F = 0 .
05. The variable x is the only free parameterin the fits, and gives the color of the blue tip. The finalresults of this work are insensitive to substantial changesin the floor F , from 0.01 to 0.05. The formal statisti-cal uncertainty in the blue tip color is computed for eachmeasurement by fitting the likelihood near the maximumlikelihood to a Gaussian.The width of the blue edge is set by a combinationof the intrinsic width of the edge and the typical mea-surement error in star colors in the SDSS. We adopt thevalues 0.05, 0.05, 0.025 and 0.025 for the width parame-ter σ P in the colors u − g , g − r , r − i and i − z , respectively. Because the photometric uncertainty becomes larger atfaint magnitudes, at high extinctions the observed blueedge is expected to broaden. Despite this, σ P is keptconstant in the fits, introducing the possibility of biasin the measurements. Nevertheless, as we look only atrelatively bright ranges of D and low extinction, this is aminor effect. Simulations with a mock star catalog ( § < D <
19 used in this work,the introduced bias in the recovered ratio of blue tip colorto E ( B − V ) is less than 2% in u − g and smaller in theother colors.The statistical uncertainty of the fit depends on thenumber of stars included in the fit, and so on the lo-cal stellar density, the sky area and range of D used. Athigh latitudes, binning together ten SDSS fields, for starswith 10 < D <
19 mag, typical statistical uncertaintiesare 20, 15, 6 and 6 mmag for u − g , g − r , r − i , and i − z , respectively. Finding the standard deviations ofmeasurements taken near the north Galactic pole, wherethe expected reddening is smooth and small, we find un-certainties of 24, 17, 10, and 12 mmag, similar to the fitresults.The presence of dust reddens stars, pushing the blueedge redward (Fig. 4). In low-extinction regions whereall of the dust is closer than about 1 kpc, the observedblue tip is uniformly shifted. It is this shift that wetrack in our measurements. In high-extinction regions orregions with dust beyond about 1 kpc, we can in principlelearn about the change in dust column with distance byvarying D , though we have not done so in this work (see § P ( x, x ) chosen to fit the blueedge of the distribution is usually a close fit only near -1 0 1 2 3u-g242220181614 u -1 0 1 2 3u-g242220181614 u g -1 0 1 2 3g-r242220181614 g r -1 0 1 2 3r-i242220181614 r i -1 0 1 2 3i-z242220181614 i Figure 3.
Example fits for the blue tip of the stellar locus in u − g , g − r , r − i , and i − z for a one degree region around l = 150 ◦ and b = 20 ◦ . A grayscale of the color-magnitude diagram is plotted, with features of the fit superimposed. The diagonal solid lines in the g − r plot give the magnitude range within which stars are fitted, corresponding to 10 < D <
19. The histograms give the distribution of stellarcolors satisfying this cut. The solid error function gives the best fit to this histogram, within the gray dashed vertical lines. The blackdotted vertical line gives the derived location of the blue tip. The arrow in the upper left of the diagram gives the magnitude and directionof the reddening vector in this field, according to SFD. Finally, the right hand axis gives the value of the pdf P ; its integral between theouter dashed lines is unity. -1 0 1 2 3g-r242220181614 g -1 0 1 2 3g-r242220181614 g g -1 0 1 2 3g-r242220181614 g g -1 0 1 2 3g-r242220181614 g g -1 0 1 2 3g-r242220181614 g Figure 4.
Example fits for the blue tip of the stellar locus in regions of increasing extinction, at l = 150 ◦ and b = 80 ◦ , ◦ , ◦ and 10 ◦ ,from left to right. The SFD extinction prediction is indicated by the length of the arrow in the upper left of the diagram. As we approachthe plane, the stellar density increases. At b = 10 ◦ , the brightest stars are not behind all of the dust, and so the color of the blue tipbecomes redder at fainter magnitudes. the blue edge itself, and is rather bad over most of thecolor range. However, experiments with other functionalforms and attempts to adaptively shrink the range overwhich the fit is performed to a narrow region around theedge tend to render the fit less robust. Moreover, be-cause 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 lo-cation of the edge around slightly (mmags), but not in away that correlates with the dust.The blue tip fit can occasionally fail to find the bluetip of the stellar locus when too few stars are used. Insuch cases it might identify a single blue star or quasaras the blue tip, or latch on to the M-dwarf peak in colormagnitude diagrams. In the maps we present, with 10 19 and using 10 SDSS fields worth of stars for eachblue tip fit, fewer than 1% of measurements are affected. Changing Extinction with Distance In this work we treat all MSTO stars observed by theSDSS with 10 < D < 19 as behind all of the dust. Inprinciple, however, by comparing the color of the blue tipfor nearby stars and distant stars, distant clouds of dustcould be detected. Such clouds certainly exist, as dusthas been detected in the halos of other galaxies and inH I clouds outside the disk in our Galaxy (M´enard et al. et al. D and distant rangesof D reveal no readily identifiable structures larger thanthe noise in the halo. We note that if such structuresvary slowly spatially, then they will be difficult to distin-guish from variation in blue tip colors due to changing stellar populations. We defer to later work the attemptto identify such clouds.On most sight lines the great majority of the dust col-umn comes from the Galactic disk and is associated withthe H I disk, which has a scale height of about 150 pc(Kalberla and Kerp 2009). Accordingly, as the brightestunsaturated MSTO stars are approximately a kiloparsecaway, at high Galactic latitudes even the nearest MSTOstars observed by the SDSS are behind this dust.At low Galactic latitudes, extinction increases rapidlyand the color of the blue tip of the stellar locus becomesdramatically redder as fainter magnitudes are probed(Fig. 5). In these dusty regions, reddened blue tip starsmay be redder than intrinsically red foreground stars.Moreover, blue tip stars in these regions are spread outalong the reddening vector, making it hard to isolatestars of a given reddening and distance using D . Weattempt to identify and exclude such regions when ana-lyzing blue tip maps. Blue Tip Maps We measure the blue tip color over the entire SDSSfootprint, in each color and for a variety of ranges of D (Fig. 6). Here we present maps using blue tip fits to starsin a broad range of magnitudes: 10 < D < 19 mag, usingfits to the stars in 10 adjacent SDSS fields.The maps show unmistakable signatures of dust (darkclouds in left panels of Fig. 6) that are almost entirelyremoved when stellar colors are corrected according tothe SFD map. The residuals at high Galactic latitudeafter extinction correction are dominated by problemswith the survey calibration—striping in the SDSS scan 10 < D < 15 -1 0 1 2 3g-r242220181614 g 10 < D < 15 -1 0 1 2 3g-r242220181614 g 15 < D < 17 -1 0 1 2 3g-r242220181614 g 15 < D < 17 -1 0 1 2 3g-r242220181614 g 17 < D < 19 -1 0 1 2 3g-r242220181614 g 17 < D < 19 -1 0 1 2 3g-r242220181614 g 19 < D < 21 -1 0 1 2 3g-r242220181614 g 19 < D < 21 -1 0 1 2 3g-r242220181614 g Figure 5. Example fits for the blue tip of the stellar locus for different magnitude ranges D at l = 150 ◦ and b = 10 ◦ . The blue tipshifts redward as the range of D probed goes fainter, until we run out of stars. Additionally, intrinsically red foreground stars increasinglycontaminate the blue tip as we go fainter in dusty regions. -20020406080 -20020406080 NGP SGPGC ACplane -20020406080 -20020406080 u-g u-g corr -20020406080 -20020406080 g-r g-r corr -20020406080 -20020406080 r-i r-i corr 300 200 100 0-20020406080 300 200 100 0-20020406080 i-z 300 200 100 0 300 200 100 0 -0.10-0.050.000.05-0.10-0.050.000.05 i-z corr Figure 6. Blue tip colors over the SDSS footprint, for each SDSS color. The left hand panels give the observed blue tip color, while theright hand panels give blue tip colors after correction for dust extinction according to SFD and the coefficients ( A a − A b ) / (SFD E ( B − V ))presented in this paper. The upper row of panels gives SFD and the locations of important points in the sky. direction and a few runs with bad zero points in u − g (Fig. 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 shiftsvery blue.The former effect is caused by the various stellar popu-lation sampled in different parts of the sky. In the plane,the higher metallicity, redder disk population becomesincreasingly dominant over the halo population. Thelatter effect is a result of the violation of the assump-tion that all of the stars are behind all of the dust. Starsin front of the dust are dereddened with the full SFDdust column, and are therefore rendered extremely blue.These stars are then seen by the blue tip algorithm asthe signature of the blue tip, resulting in a spuriouslyblue color. Even in less dusty regions where at leastsome MSTO stars are behind all of the dust, the methodcan fail if an insufficiently faint range of D is used. Inthis case, MSTO stars behind different amounts of dustare grouped together in finding the best fit, blurring theblue edge of the stellar locus. Because σ P for the fit isfixed, this will result in a bluer than average blue tipcolor, while SFD would track the color of the MSTOstars behind all of the dust—the reddest stars in thegroup. These effects, however, are only noticeable in thedustiest region of the sky, where | b | . ◦ .The Monoceros stream is clearly visible as a feature at( α, δ ) = (110 ◦ , − ◦ ) – (110 ◦ , ◦ ) in the blue tip maps,because its MSTO is bluer than that of the thick andthin disks (Newberg et al. u − g .There are two types of survey-systematic inducedresiduals in the maps, both of which are seen as stripingalong the SDSS scan direction. The less important arethe few, large, ∼ ◦ wide stripes that are most noticeableon the eastern edge of the u − g blue tip maps. These re-gions correspond to the footprints of SDSS runs that haveslight zero point offsets from the rest of the survey. Themore troubling residual is the universal small-scale strip-ing along the scan direction. This corresponds to the 13arcminute scale of the SDSS camera columns. We havenot fully examined this residual, but attribute it to differ-ent filter responses between the 6 SDSS camera columns(Doi et al. et al. § D used thanfor fainter ranges of D , which are mutually compatible.We interpret this as a saturation or nonlinearity effectthat depends on camera column, though we remove anystars flagged as saturated from the analysis. These sys-tematics need to be addressed in fits to maps of the bluetip. FITTING THE BLUE TIP MAP The sensitivity of the blue tip colors to dust redden-ing makes for a natural test of the SFD map. We carryout the simplest tests of SFD permitted by the blue tipmaps here, checking the dust map normalization and the R V = 3 . B − V . The reddeninglaw used in SFD is from O’Donnell (1994), which is sim-ilar to the Cardelli et al. (1989, CCM) reddening law.The SFD dust map is often used as a map of redden- ing E ( B − V ), but is based fundamentally on thermalemission from dust. In SFD, the thermal emission isconverted to reddening based on a single dust map nor-malization constant. This constant was derived by com-paring the observed reddenings E ( B − V ) of a sampleof elliptical galaxies to the SFD temperature-correctedemission from dust at their locations (Faber et al. B − V color of galax-ies. Because of the five band photometry provided by theSDSS, we are able to extend the SFD test by addition-ally checking the assumed O’Donnell R V = 3 . E ( B − V ) SFD ,as these are proportional) to reddening. We extend thistest to multiple colors and check for variation over thesky.To perform this test each blue tip map would be fit,ideally, as( m a − m b ) bluetip = R a − b E ( B − V ) SFD + C, (3)over a part of the sky, where ( m a − m b ) bluetip is the bluetip map for the color a − b , R a − b is the normalizationconstant to be measured, and C is the intrinsic blue tipcolor. However, because the blue tip of the stellar locusis not a universal color standard, but rather varies withposition in a way likely substantially covariant with thedust map, this fit is impractical. Moreover, the surveystriping artifacts discussed in § m a − m b ) bluetip = R a − b E ( B − V ) SFD + Q r ( f ) + C i (4)where Q r ( f ) is a quadratic for run r in SDSS field num-ber f and C i is the camera column offset for cameracolumn i . We fix C to be zero to remove the degen-eracy with the constant term in Q r ( f ). The quadratic Q r ( f ) simultaneously accounts for zero point errors inthe SDSS calibration and slow intrinsic variation in bluetip color.Because we bin 10 SDSS fields together for each bluetip measurement, variations in the dust map on scalesmuch smaller than 1.5 ◦ in the scan direction will not becaptured in the blue tip measurements. The blue tipmethod is not a linear operator on the underlying stellarcolors, so the measured color of the blue tip is not linearlyrelated to the mean of E ( B − V ) SFD in the fields. In thiswork, for each set of fields contributing to a blue tipmeasurement, we use the median E ( B − V ) SFD in thosefields as proportional to the expected reddening of thatmeasurement. This is not strictly correct, but insofar asthe dust does not vary too quickly, this filtering seems toapproximate the correct behavior.After finding the best fit R a − b in each color, we re-peat the blue tip measurements on the SDSS stars, thistime with individual stellar colors corrected according tothe derived fit parameters R a − b , at the full resolutionof the dust map. This process converges in a few itera-tions. The iteration renders the final fits insensitive to -20020406080 -20020406080 u-g corr g-r corr 300 200 100 0-20020406080 300 200 100 0-20020406080 r-i corr 300 200 100 0 300 200 100 0 i-z corr Figure 7. Stretched version of Fig. 6, highlighting residuals. The Monoceros stream is prominent at the right hand side of the NGPregion of the footprint. White to black spans 0.225, 0.11, 0.05, and 0.06 mag. the details of the dust map filtering. We have verified theinsensitivity to the filtering by noting that fits to the un-binned blue tip maps using an unfiltered SFD map givefit coefficients that are compatible with the fits to thebinned blue tip maps.We model the striping in camera columns by constantterms in the fit for each camera column. Empiricallythis correction is satisfactory, but we note that if ourinterpretation of the camera column offsets as derivedfrom color terms is correct, the response of each cameracolumn to dust is slightly biased. However, we detectcamera column offsets of ∼ § Fits to Individual SDSS Runs In some cases, a single run is long enough and dustyenough to individually constrain the dust reddening co-efficients R a − b in each color at the 10% level, even whenfitting for camera column offsets and a quadratic. Thefits show substantial variation in R a − b (Fig. 9). Mostof this variation takes the form of an overall differencein dust absorption normalization, rather than as varia-tion in the shape of the dust reddening spectrum. Somelow, outlying values of R u − g suggest that the run is soextinguished that substantial numbers of stars are notdetected in the u band, ruining the fit. The fits removenearly any trace of dust in these runs (Fig. 10).Fits to the runs individually well constrained by thedust map indicate significant variation in normalization( ∼ R a − b more than 5 σ from the inversevariance weighted average of R a − b for all the SDSS runsare excluded from the global fit. Additionally, the global u-g u-g g-r g-r -10-50510-10-50510 r-i r-i i-z i-z Figure 8. The camera column offsets used to destripe the SDSSblue tip maps, derived from the global blue tip map fits ( § D to which the offsets apply.The right hand labels give the value of the grayscale in mmag.The offsets are compatible for different ranges of D , except at thebrightest magnitudes (10 < D < fit is iterated and σ -clipped field-by-field at 5 σ . Global Fit to All SDSS Runs We find the best fit (least squares sense) combinationof the SFD dust map, constant offsets for each cameracolumn other than the first, and quadratics in field num-ber for each run to the blue tip map via singular valuedecomposition. We do not fit for the offset for the firstcamera column because changing the offsets for all of thecamera columns is degenerate with changing the constantoffsets in the quadratics for each run. Specifically, we findthe parameters x satisfying A ⊺ C − Ax = A ⊺ C − b (5) u-g g-r r-i i-z 0.20.40.60.81.01.21.41.6 R a - b u-g g-r r-i i-z Figure 9. R a − b for the four SDSS colors, for a selection of indi-vidually well constrained SDSS runs. The reddening spectrum hasa similar shape among the different runs, but there is substantialscatter in the overall normalization (left). After forcing the meanof R g − r , R r − i , and R i − z to match for each run, there is closeagreement among the various dust extinction spectra (right). Er-ror bars account only for the formal statistical uncertainties. Thethick gray line in each plot gives the global best fit R a − b derivedin this work. which provides the least squares solution to Ax = b .Here C is a diagonal covariance matrix, with the vari-ances determined by the formal statistical uncertaintiesin the blue tip fit to each field (or binned fields). Theblue tip color for each field (or binned fields) is in b . Thedesign matrix A has dimensions n field × n param . It con-tains one column for the SFD dust map values in eachfield, five columns for the offsets for the SDSS cameracolumns two through six, and three columns for each ofthe 792 runs composing the SDSS-III, for the three termsin the quadratics for each run.The SDSS-III contains imaging data on 1,147,506fields. We require that the SDSS score of each runbe greater than 0.5 and the PSF FWHM be less than1.8 arcseconds in r , which reduces the number of fieldsto 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, dueto an insufficient number of stars or apparent detectionof the blue tip blueward of − . 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 , re-spectively. The number of binned fields that successfullymake it through the blue tip fit depends on the rangeof D used and the color, but in the ranges of D exam-ined here it ranges from 684,000 to 677,000. Unbinnedfields fail for lack of stars at high Galactic latitudes, andso fewer fields are successful: in ranges of D of interest,between 683,000 and 620,000 fields pass. The mediannumber of stars contributing to the fits in each unbinnedfield at 10 < D < 19 is 100. About 10 stars enter intothe fits.The global fit produces the coefficients R a − b in Ta-ble 1 and at high latitudes leaves few noticeable coher-ent residuals (Fig. 11). Nevertheless, a few dust cloudsclearly are imperfectly subtracted. Most prominently, inthe north, a few undersubtracted clouds in g − r standout on the eastern side of the north Galactic cap (NGC). The SDSS score of a field reflects sensitivity to point sourcesand is derived from sky brightness and seeing. It further is cappedat 0.5 for fields deemed unphotometric or runs fields using binnedpixels (i.e., Apache Wheel fields). Table 1 R a − b for the SDSS colorsSDSS color SFD R V = 3 . u − g . 362 1 . 138 1 . ± . g − r . 042 1 . 141 1 . ± . r − i . 665 0 . 616 0 . ± . i − z . 607 0 . 624 0 . ± . Note . — R a − b according to the original SFD prescrip-tion, an R V = 3 . § Undersubtracted clouds in the northwest and at ( α , δ ) =(145 ◦ , 0 ◦ ) cause the runs including them to be excludedfrom the fit. The former feature is the largest resid-ual in the Peek and Graves (2010) maps. Subtractionin the south is remarkably clean, though at right ascen-sions and declinations of (340 ◦ , 20 ◦ ) and (60 ◦ , 0 ◦ ) thereare slightly oversubtracted clouds. A cloud at (45 ◦ , 20 ◦ )barely makes it into the SDSS footprint, and is badlyoversubtracted, causing the exclusion of its run from theglobal fit. In g − r especially, a residual associated withthe Monoceros stream stands out at the western edge ofthe NGC. We have tried masking this region and repeat-ing the fit, but as the Monoceros stream is not correlatedwith the dust, the fit results were unchanged.The blue tip color residuals as a function of E ( B − V ) SFD are generally flat (Fig. 12). In r − i and i − z , anytrend 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 . u − g , g − r , r − i , and i − z , respectively.The polynomial terms from the global fit are intendedto model the slow, intrinsic variation in the blue tip colorand per-run calibration offsets, and so are of indepen-dent interest (Fig. 14). These maps are clearly heavilystriped. Some of this striping is clearly removing cali-bration problems with the SDSS, as in the eastern sideof the NGC in u − g . The striping at high latitudes inthe north in other colors is probably substantially an ar-tifact of the low signal in these regions, but the typicalstripe-to-stripe difference there is only ∼ et al. u-g g-r r-i -80-4004080-80-4004080 i-z -200-1000100200-200-1000100200 u-g -160-80080160-160-80080160 g-r -80-4004080-80-4004080 r-i -80-4004080-80-4004080 i-z Figure 10. Blue tip colors (left) and residuals (right) for the runs in Fig. 9, in mmag. Coherent signals from the dust are manifest (left).After fitting according to § 4, the residuals show little structure (right). behind all the dust. However, for | b | > ◦ , inspectionof the blue tip residual maps suggests that the SEGUEruns could be well fit also, although we have not triedthat here. The signal from the dust in the SEGUE runsis sufficiently rich to merit treatment separate from thebulk of the high Galactic latitude sky, as is done in Berry et al. (in prep). Fits to different sky regions Given the presence of dust-correlated residuals in dif-ferent parts of the sky, one wonders whether the fit isactually “global” in a useful sense, or is primarily a prod-uct of the footprint we have decided to look at. Accord-ingly, we have cut the sky into a number of subregionsand separately found the best fit R a − b in each. To testfor large scale variations in R a − b , we divide the sky intonorthern ( b > ◦ ) and southern ( b < − ◦ ) Galacticregions, as well as octants of the sky divided by linesof constant Galactic longitude (Fig. 15). To try to testvariation in dust properties as a function of extinctionor temperature, we divide the sky into regions of dust ofdifferent extinctions and temperatures. Finally, to ver-ify that we have satisfactorily accounted for the cameracolumn offsets, we divide the SDSS survey into its sixcamera columns.We restrict to the main global fit region and apply theglobally derived camera column offsets to the blue tipcolors. We remove best fit polynomials from the filteredSFD dust map and from the blue tip colors, as in § . < E ( B − V ) < . ∼ 15% smaller nor-malization than dust with 0 < E ( B − V ) < . 05 mag.Dust at different temperatures likewise has similar ex-tinction normalizations and spectra. Fits to a Mock Star Catalog The reliability of the fitting procedure can be testedby running the fit on a star catalog from a mock galaxymodel. The mock catalog was generated with the galfast catalog generation code (Juri´c, in prep). Given a modelof stellar number density, metallicity, a 3D extinctionmap, the photometric system, and instrumental errors, galfast generates realistic mock photometric survey cat-alogs.As inputs, we used the number density distributionparameters from Juri´c et al. (2008) and the metallic-ity distribution from Ivezi´c et al. (2008). The three-dimensional dust distribution map was generated usingthe model of Amˆores and L´epine (2005). The gener-ated u , g , r , i , and z magnitudes were convolved withmagnitude-dependent errors representative of SDSS, andthe final catalog was flux limited at r < . -20020406080 -20020406080 u-g σ =17.6 g-r σ =13.2 300 200 100 0-20020406080 300 200 100 0-20020406080 r-i σ = 6.1 300 200 100 0 300 200 100 0 i-z σ = 6.3 Figure 11. Blue tip color residuals after removing best fit linear combination of dust map and polynomials, as described in § σ , while black corresponds to − σ . -30-20-100102030 R e s i d u-g g-r R e s i d r-i i-z Figure 12. Blue tip color residuals with E ( B − V ) SFD . Theresiduals in u − g , r − i , and i − z reassuringly show no trendwith E ( B − V ). In g − r , the residuals have some structure, butthe positive slope at E ( B − V ) < . . < E ( B − V ) < . E ( B − V ). Amˆores and L´epine (2005) to contain the small-scaleclouds seen in SFD. This was performed by scalingeach line of sight from the model by the ratio E ( B − V ) SFD /E ( B − V ) model , , so that the model matchesSFD at 100 kpc. The resulting map contains the ex-pected average three dimensional distribution of the dustcombined with the angular structures present in SFD.We construct a mock galaxy using galfast and then o f m ea s u r e m en t s u-g σ = 18.1 g-r σ = 12.3-50 -25 0 25 50Resid (mmag) o f m ea s u r e m en t s r-i σ = 6.9 -50 -25 0 25 50Resid (mmag) i-z σ = 7.8 Figure 13. Fit residuals, in mmag. The core of the distributionis well fit by a Gaussian, but the wings fall off more slowly than in aGaussian. The colors r − i and i − z have the tightest distributions,likely because the blue tip is less dependent on metallicity in thesecolors. construct catalogs of observations of these stars for eachSDSS run. The resulting catalogs are proccesed by theblue tip analysis code in exactly the way that the actualSDSS catalogs are processed. As for the actual catalogs,the median number of stars per field is ∼ R a − b used to within 3%. This verifies that we properlyaccount for shifts in the blue tip color due to metallicity.However, for these tests we have assumed that SFD cor-1 -20020406080 -20020406080 u-g poly g-r poly 300 200 100 0-20020406080 300 200 100 0-20020406080 r-i poly 300 200 100 0 300 200 100 0 i-z poly Figure 14. Map of the polynomial terms from the global fits. The scale is the same as in Fig. 7. These maps model the intrinsic variationof the blue tip color on the sky as well as per-run SDSS calibration offsets. North/South u-g g-r r-i i-z0.40.60.81.01.2 R edden i ng / S F D E ( B - V ) NS Longitude slices Extinction slices u-g g-r r-i i-z Temperature slices T<17.617.6-18.2T>18.2 Camcol slices u-g g-r r-i i-z Figure 15. R a − b for the SDSS colors over different subsets of the SDSS footprint. The first panel gives the coefficients for the north( b > ◦ ) and south ( b < − ◦ ). The next panel gives the best fit coefficients for eight slices in Galactic longitude. The third panel givescoefficients in regions of sky with increasing extinction. The fourth panel divides the sky into regions of three different temperatures, andthe fifth panel divides the survey by SDSS camera column. The thick gray line in each panel gives the global best fit R a − b derived in thiswork. rectly predicts the line of sight column density and thatthe reddening law is the same everywhere on the sky, atleast at high Galactic latitudes and in the optical. Inso-far as the final fit residuals are largely flat (Fig. 11), thisassumption seems justified. Moreover we can verify thatthe ratios of the R a − b derived independently from SFDagree with the ratios of the R a − b derived here. Fitting Ratios of R a − b without SFD The preceding analysis has assumed that SFD is a goodtemplate for the dust. However, we can relax our de-pendence on SFD if we restrict our attention to ratiosbetween different R a − b .In the absence of changing stellar populations and cal-ibration errors, blue tip colors will fall along a single linegiven by the reddening vector. By removing a quadraticin field number and accounting for camera column off-sets as in § R a − b . Some striping alongthe SDSS scan direction is evident in the blue tip resid-uals despite our attempts to remove it, and this striping will be deleterious to the fit results because we cannotrely on its being uncorrelated with the dust template wefit. Accordingly, we fit lines for each camera column,eliminating the effect of striping, and average the results(Fig. 16).These color-color fit results are in good agreement withthe results of § σ with the SFD fit results, where the uncertainties σ are given by the standard deviations of the fit resultsbetween the camera columns. The original SFD pre-scription is seen to clearly overestimate R u − g /R g − r (see § R V = 3 . R r − i /R i − z . DISCUSSION In this section, we use the blue tip fits presented aboveto constrain reddening laws and to test the SFD nor-malization and temperature correction. We then brieflypoint out a discrepancy in the color of the dereddenedblue tip between the north and south, which is plausi-bly a consequence of a different stellar population in thenorth than in the south.2 -0.2 0.0 0.2 0.4 0.6 0.8u-g0.00.20.40.60.8 g -r blue tipSFDR V =3.1 O’Dcolor color fit r- i -0.1 0.0 0.1 0.2 0.3 0.4r-i-0.20.00.20.4 i - z Figure 16. Plots of the colors of the blue tip in different colors, for camera column one. Spatially slowly varying terms have beenremoved, so points should fall along the line R a − b /R c − d . The reddening coefficients presented here best trace the distribution of points. Table 2 Blue Tip Color-Color Fit ResultsColor ratio SFD O’Donnell blue tip color color A u − g /A g − r . ± . A g − r /A r − i . ± . A r − i /A i − z . ± . Note . — Blue tip color-color fit results. The results are inclose agreement with the global blue tip fits, while excludingthe original SFD reddening law and an R V = 3 . Constraining Reddening Laws Connecting Reddening Laws to R a − b A reddening law gives the extinction A ( λ ) over a rangeof wavelengths λ and so makes a prediction for the R a − b that we have fit. The extinction in band b , in the limitthat the variation in the extinction over that band insmall, is given by ∆ m b = A ( λ eff ,b,S ) E ( B − V ), with λ eff ,b,S = R dλλS ( λ ) W b ( λ ) R dλS ( λ ) W b ( λ ) (6)Here the system response for band b is given by W b ( λ ), S ( λ ) is the source spectrum in photons/s/˚A, 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 thengiven by ( A ( λ eff ,a,S ) − A ( λ eff ,b,S )) E ( B − V ), from whichit 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 vari-ation 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 than5% in g − r and u − g .Accordingly, we evaluate the extinctions in each SDSSpass band using the SDSS system throughputs and aMSTO source spectrum from a Kurucz model with T eff =7000K, for a variety of reddening laws (Gunn et al. R V = A V /E ( B − V ). Wefind 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 fitnormalization N ′ = E ( B − V ) /E ( B − V ) SFD . We donot report this parameter directly. The SFD dust mapis based on a map of thermal emission from dust andis correspondingly proportional to τ µ m , the opticaldepth of dust at 100 µ m. Empirically, the ratio between τ µ m and E ( B − V ) is itself a function of R V , and so N ′ is covariant with R V . However, the ratio between τ µ m and τ µ m depends less on R V , so we report N = A µ m , predicted /A µ m , SFD . Here A µ m , predicted is the extinction at 1 µ m implied by the best fit redden-ing law and A µ m , SFD is the SFD-predicted extinction,extrapolated to 1 µ m following the R V = 3 . A µ m , SFD is simply 1 . · E ( B − V ) SFD . We find that the F99 red-dening law provides the best fit to the R a − b we measure,so we additionally mention for reference that the F99 red-dening law ratio E ( B − V ) /A µ m is 11% larger than theO’Donnell ratio E ( B − V ) /A µ m , though this dependson the adopted R V and source spectrum. R V and N for Individual Runs We have found that the dust extinction spectrum nor-malization and shape vary over the sky (Fig. 9). Wecan quantify this effect in terms of variation in N and R V by fitting reddening laws to the runs that individ-ually well constrain R a − b ( § R r − i ≈ R i − z , which is a poor fit to the data. Thederived N and R V vary from run to run, especially asa ∼ 20% scatter in normalization (Fig. 17). Almost allof the runs are consistent with 2 . < R V < . 2. Thedistribution of R V and N validates our use of 1 µ m asa refererence wavelength for N , as the two parametersappear uncorrelated. R V and N for the Global Fit In order to compute best-fit reddening laws for theglobal fit R a − b , we need to account for the covariancein the R a − b over the sky in addition to the formal sta-tistical uncertainties in the fit. These uncertainties are3 V N Figure 17. Best fit R V and dust map normalization N for SDSSruns that individually well constrain R a − b , examples of which arein Fig. 9. Dust map normalizations vary by as much as a factor oftwo, though most are consistent at the 20% level. The large red“X” indicates the location of the best fit R V and normalizationto the global fit R a − b . Error bars account only for the formalstatistical uncertainties. about a part in one thousand. However, the best fit nor-malization can vary at the 10% level from field to field,and the part in a thousand uncertainty reflects averagingthese fluctuations over many fields. While the dust ex-tinction spectrum seems relatively constant, it would besurprising if the average reddening reflected the redden-ing of any particular dust cloud at the part in a thousandlevel, so another method for estimating the uncertaintiesis needed. The uncertainties reported in Table 1 are thestandard deviations in R a − b for 8 octants in Galacticlongitude, and do not account for the covariance of themeasurements in the different colors owing to the chang-ing best fit normalization.The consequence of adopting uncertainties based onthe field-to-field variation in the dust extinction spec-trum is that the resulting uncertainties on N and R V will describe the range of N and R V among the cloudswithin the footprint we analyze. They do not give theuncertainties on the best fit N and R V within our sam-ple, which are more tightly constrained. As mentioned,the formal statistical uncertainties are about one part ina thousand. Uncertainties estimated from using differentranges of D for selecting the stars give uncertainties of2%, 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 estimatedfrom the sample covariance matrix of best fit R a − b indifferent sky regions. We find the best fit R a − b at eachpoint in the footprint. Because over most of the foot-print the signal-to-noise in the filtered blue tip maps istoo low to make a reliable determination of R a − b , we in-clude nearby pixels in the fit with weights given by a 7 ◦ smoothed Gaussian. The best fit R a − b is then given by R a − b,n = P i D i C − i W i,n b i P i D i C − i W i,n (7) Table 3 Eigenvalues and eigenvectors of thereddening covariance matrix √ λ v u − g v g − r v r − i v i − z . 155 0 . 88 1 . 21 0 . 45 0 . . 079 1 . − . − . − . . 031 0 . − . 91 0 . 61 1 . . − . − . 15 1 . − . Note . — Eigenvalues λ and eigen-vectors v for the sample covariance ma-trix. Eigenvectors are normalized so that | v | = | R global fit | , to facilitate compari-son between R a − b and the first, least wellconstrained eigenvector, which is roughlyparallel to R a − b . where n is the pixel on the sky, i indexes over blue tipmeasurements, D gives the filtered dust map, b and C − give the blue tip measurement and its inverse variance,and W i,n is the weight matrix, corresponding to Gaussiansmoothing with a FWHM of 7 ◦ .The sample covariance matrix of the resulting maps of R a − b is used as the covariance matrix for fits of reddeninglaws to R a − b (Table 3). Only pixels with estimated un-certainty in R i − z less than 0 . 01 are used for computingthe sample covariance matrix, to avoid overestimatingthe intrinsic variance due to the uncertainty in the esti-mates. The largest eigenvalue of the covariance matrix(2 . · − ) is 94 times larger than the smallest eigenvalue(2 . · − ). The eigenvector with the largest eigenvalueis roughly parallel to a vector corresponding to chang-ing the normalization of the spectrum, and that with thesmallest eigenvalue corresponds to changing the relativeamount of reddening in r − i to i − z . This quantitativelyagrees with our claim that the normalization of the red-dening law is poorly constrained relative to its shape.With this covariance matrix, the best fit reddeninglaws have R V = 3 . ± . N = 0 . ± . 10 for theO’Donnell reddening law, R V = 3 . ± . N = 0 . ± . R V = 3 . ± . N =0 . ± . 06 for the F99 reddening law (Fig. 18). The χ / dof of the O’Donnell, CCM, and F99 reddening lawsare 7.8, 3.5, and 2.1 respectively. The O’Donnell andCCM reddening laws are disfavored because they predictsmaller reddening differences between r − i and i − z rel-ative to the data and the F99 reddening law. The F99reddening law fits all of the points reasonably well.The O’Donnell reddening law in particular and CCMreddening law to a lesser extent provide poor fits to theblue tip data, and the best-fit parameters are seriouslydriven by the poorly-matching r − i and i − z data. Ac-cordingly, the derived R V and N for these reddening lawsare unphysical.All of the fits give R u − g substantially less than thevalue given in SFD (Fig. 18). The values given in theSFD appendix were based on preliminary estimates ofthe system response of the SDSS, which varied somewhatfrom the actual system response. Using the SDSS sys-tem response from Gunn et al. (1998) for the R V = 3 . u − g and 10% in g − r and i − z .The predicted R a − b depend somewhat on the sourcespectrum used (Fig. 19). We illustrate the effect of4 u-g g-r r-i i-z0.20.40.60.81.01.21.4 R edden i ng / S F D E ( B - V ) SFDR V =3.1 O ′ Donnellblue tipBest fit F99Best fit CCMBest fit O’Donnell Figure 18. The original SFD reddening coefficients (solid line),updated coefficients according to the actual SDSS filters (crosses),and the best fit global coefficients found in this work (stars). Wefit our best fit coefficients with reddening laws according to F99(diamonds), CCM (triangles) and O’Donnell (squares). The F99reddening law seems a good fit to each SDSS band ( § changing the source spectrum by plotting the derivedreddening laws for the O’Donnell, CCM, and F99 red-dening laws for Kurucz models (Kurucz 1993) with T =6500, 5500, and 4500 from the stellar spectra grid ofMunari et al. (2005). The blue tip best fit reddeninglaw is relatively insensitive to T over this range.The maps of R a − b over the sky can be combined tomake maps of R V and reddening law normalization overthe sky (Fig. 20). To suppress noise in the measurement,we have combined the measured values with a prior of R V = 3 . ± . N = 0 . ± . 06 to reduce the scatterin regions of low signal-to-noise. One feature of the mapsof R V and N is that the two maps are substantially un-correlated. This again confirms the expectation that theratio of τ µ m /τ µ m does not depend on R V . Comparison with SFD Normalization We can directly compare our predicted E ( B − V ) with E ( B − V ) SFD by using our best fit reddening law andaccounting for the difference in source spectrum betweenthe galaxies that SFD analyzed and the MSTO stars weanalyze. We get E ( B − V ) blue tip = 0 . · E ( B − V ) SFD ,suggesting SFD overpredicts reddening by about 14%.The estimated normalization uncertainty in SFD was4%, while we claim 8% uncertainty in our normaliza-tion. However, the estimated fractional uncertainty ofSFD was 16%, and we are now in a position to attributemost of this uncertainty to varying best-fit normaliza-tion. Accordingly, it is unsurprising that the SFD nor-malization differs from ours by 14%, particularly giventhat the footprint we analyze is different from that usedin SFD, and that we have found north/south normaliza-tion differences of about 15%. The SFD Temperature Correction u -A g )/(A g -A r )1.01.21.41.61.82.02.2 ( A g - A r ) / ( A r - A i ) CCM T=6500CCM T=5500CCM T=4500F99 T=6500F99 T=5500F99 T=4500O’D T=6500O’D T=5500O’D T=4500 g -A r )/(A r -A i )0.81.01.21.41.6 ( A r - A i ) / ( A i - A z ) Figure 19. O’Donnell, CCM, and F99 reddening laws for dif-ferent stars of different temperatures. R V = 2 . 6, 3 . 1, and 3 . R V increases to the left. Ellipses are forthe best fit blue tip values, with uncertainties computed from thecovariance matrix of Table 3. 300 200 100 0-20020406080 300 200 100 0-20020406080 2.62.83.03.23.43.62.62.83.03.23.43.6300 200 100 0-20020406080 300 200 100 0-20020406080 0.60.70.80.91.00.60.70.80.91.0 Figure 20. Maps of R V (top) and dust map normalization (bot-tom) over the SDSS footprint. The map has been combined with aprior ( § The final E ( B − V ) SFD takes the form E ( B − V ) SFD = p I corr X , where p represents a normalization coefficient, I corr represents the destriped, zodiacal-light subtracted IRAS µ m flux and X represents a temperature cor-rection factor. An error in determining the temperaturecorrection factor will lead to dust with a different bestfit 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 be5 -20-1001020 R e s i d ( mm ag ) y = 0.32x + -0.63 u-g y = 0.41x + -0.37 g-r -15 0 15 ∆ (mmag)-20-1001020 R e s i d ( mm ag ) y = 0.22x + -0.55 r-i -15 0 15 ∆ (mmag) y = 0.28x + -0.61 i-z Figure 21. Blue tip color residuals versus change in filtered dustmap E ( B − V ) from using the FIRAS temperature correction ratherthan the SFD temperature correction (both in mmag). The lineartrend suggests that the temperature correction may be at fault. attributed to errors in the SFD temperature correctionfactor. In § µ m and especially the 240 µ m mapare too low to construct a 100 µ m / µ m ratio mapwithout substantial filtering. The SFD ratio map wasconstructed by first smoothing the 100 µ m and 240 µ mmaps to 1 ◦ and then further weighting low S/N pixelsto a high | b | average ratio. Insofar as this proceduremixes dust of different temperatures together to a singlereported SFD average temperature, it makes it difficultfor us to test the SFD temperature correction using theSFD temperature map in this way.One way to test the accuracy of the temperature cor-rection at high | b | is to compare the SFD tempera-ture correction with the temperature correction used forthe FIRAS dust fits (Finkbeiner et al. ◦ smoothed map in these regions. In high signal-to-noiseregions the two corrections agree. Letting X SFD be theSFD temperature correction and X FIRAS be the FIRAStemperature correction, we can plot blue tip color resid-uals versus the change in predicted dust column densityfrom switching from the SFD to the FIRAS tempera-ture correction: ∆ = E ( B − V ) SFD ( X FIRAS / X SFD − FIRAS /X SFD NorthSouth Figure 22. The ratio X FIRAS / X SFD in the north (solid) and inthe south (dashed). The temperature correction is systematicallyhigher in the south than in the north by ∼ § for a linear trend between dust correction and blue tipresidual, with slope equal to the best fit dust coefficients.Instead we find slopes of approximately half the best fitcoefficients, suggesting that the true temperature mapis between the SFD and F99 temperature maps. Thespecific values of the coefficients of the linear fits shownare not reliable because we have not accounted for theuncertainties in ∆ in performing the fit, and so the slopesmay be underestimated. Regardless, the fact that a clearlinear trend with significant slope exists provides strongevidence that the temperature correction at high | b | isunreliable.The FIRAS temperature correction may also explainthe ∼ 15% normalization difference in best fit dust ex-tinction spectrum observed between the north and thesouth in § X FIRAS / X SFD in regions with E ( B − V ) < . 05 is about 10% higher in the south thanin the north (Fig. 22). The Dereddened Blue Tip in the North and South The color of the blue tip changes because of both dustand because of changing stellar populations. If we dered-den the blue tip according to SFD and the best fit R a − b ofthis work, we expect the remaining variation in blue tipcolor to be due to changing stellar populations (Fig. 7).A surprising feature of the dereddened blue tip mapsis that the SGC appears redder than the NGC in g − r , r − i , and i − z , especially outside a region in the southeastof the SGC (Table 4). This may be a genuine structurein the stellar population in the south, but we are unableto distinguish that possibility from calibration errors orerrors in the dust map. However, because the south tendsto prefer a smaller SFD normalization than the north, thedereddened south would have been expected to be too6 Table 4 North/South color asymmetryRegion u − g g − r r − i i − z North (mag) 0.817 0.228 0.064 -0.033South (mag) 0.824 0.250 0.071 -0.020Difference (mmag) 7.6 21.8 7.2 12.4 Note . — Median blue tip color in the north andsouth, with 40 < | b | < 70. The south is system-atically redder than the north. This may be a signof an interesting stellar structure in the south, or acalibration problem or SFD error. blue rather than too red (Fig. 15). Moreover, the shapeof the extra reddening in the different SDSS bands doesnot look like any plausible reddening law. CONCLUSION The blue tip of the stellar locus provides a viable colorstandard for testing reddening. It is also a sensitive probeof systematics in survey photometry: we detect strip-ing artifacts from the SDSS camera columns as well asoccasional runs with bad zero points. Reddening mea-surements based on the stellar locus are limited becausethe stellar locus varies with position on the sky, due tochanging stellar populations. However, we find that weare able to overcome this limitation by looking for small-scale fluctuations that correlate with those in the SFDdust map. By removing the best fit quadratic from theblue tip colors in each SDSS run, both survey systematicsand sky-varying stellar populations seem adequately ac-counted for: Gaussian fits to the blue tip residuals have σ of 18.1, 12.3, 6.9, and 7.8 mmag, compared with themedian formal statistical uncertainties σ of 17.5, 12.5,5.7, and 5.8 mmag in u − g , g − r , r − i , and i − z , respec-tively. Because our errors are not Gaussian, in terms of χ / dof we do worse; the fits have χ / dof of 1.49, 1.30,2.16, and 2.28 in the four colors.Using these reddening measurements over the SDSSfootprint, including, especially, the new, dustier, south-ern data, we can sensitively constrain the SFD dust mapnormalization in each SDSS color. The original SFD val-ues for A b /E ( B − V ) were in error because the filterresponses used in the SFD appendix did not match theeventual filter responses used in the SDSS. After takingthis into account, our measurements and an R V = 3 . i − z ,where the O’Donnell reddening law overpredicts redden-ing dramatically. We find an F99 reddening law providesa good fit to the data, with R V = 3 . N = 0 . 78. Werecommend the use of this reddening law and normaliza-tion for use with SFD and dereddening optical data, or,for SDSS data, the use of the constants R a − b presentedin Table 1.This result largely vindicates the SFD normalization athigh Galactic latitudes and low column densities, whichhad been called into question by earlier studies in dustierregions. Except in u − g , where the SDSS system responseassumed by SFD was in error, and in i − z , where theO’Donnell reddening law overpredicts reddening, we donot find that SFD overpredicts extinction by 40%. In g − r , the closest band to the SFD B − V calibration, ournormalization agrees with the SFD normalization within4%. Extrapolating from the SDSS bands to B − V , we find that SFD overpredicts E ( B − V ) by 14%. However,because the best fit normalization varies over the sky,this result depends on the footprint that we have ana-lyzed, and will be different in other areas. The fit resultsindicate that the normalization varies between clouds byabout 10%.We have also made maps of the R V and N over theSDSS footprint. However, we do not yet recommend theuse of these maps except in regions where the signal-to-noise is high, where we use these values to computethe R a − b covariance matrix. We are actively investigat-ing incorporating similar maps of R V and N into futuredust extinction maps, properly combining them with theavailable FIR data.The fact that the best fit normalization of the dust mapvaries over the sky points to problems with the SFD tem-perature correction. However, the dust extinction spec-trum is relatively stable, indicating that at least in re-gions with E ( B − V ) . . 5, objects can be dereddened inthe optical assuming a universal extinction law to withina few percent accuracy.The blue tip colors, dereddened according to the SFDdust map with the coefficients from this work, show nocoherent residuals greater than about 30, 30, 10, and10 mmag in the footprint we analyze, which covers mostof the high-latitude sky. Over much of the sky, residualsare within the statistical uncertainties.These reddening measurements permit detailed tests ofdust maps over a large sky area and over a range of dusttemperatures and column densities. We have been ableto find clear signs of shortcomings in the SFD tempera-ture correction. In future work, we plan to construct newdust maps based on dust emission, constrained to best fitthese measurements. Future surveys like Pan-STARRS(Kaiser et al. APPENDIX USING SFD TO PREDICT EXTINCTION IN SPECIFIC CLOUDS Frequently it is useful to estimate the reddening through a particular cloud for which the reddening law is expectedto be different from the Galactic average, as when the cloud has R V substantially different from three. The correctprocedure in such cases is not to use A V = R V E ( B − V ) SFD , because in such clouds E ( B − V ) = E ( B − V ) SFD , asSFD is ultimately a map of optical depth at 100 µ m, and the ratio of the 100 µ m optical depth to E ( B − V ) is itselfa function of R V .If the cloud in question has SDSS coverage, the surest footing is to use the measured blue tip colors for that cloud toderive the extinction law in that cloud. If SDSS coverage is not available, extrapolation from SFD or some alternativemethod is required.In order to extrapolate from SFD, we have found that the best procedure is to take the SFD prediction for E ( B − V )and transform this to a prediction for extinction at 1 µ m. 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