Photometric recalibration of the SDSS Stripe 82 to a few milimagnitude precision with the stellar color regression method and Gaia EDR3

Wide-field photometric surveys play a leading role in the discovery of new celestial objects, new phenomena, and new laws in modern astronomical research. The potential discovery space and amount of information that can be extracted from a given photometric survey is closely related to the uniformity of its photometric calibration. Hence, a uniform and accurate photometric calibration plays a central role in the current and next-generation wide-field imaging surveys, such as the Sloan Digital Sky Survey (SDSS; York et al. 2000), the Dark Energy Survey (DES; The Dark Energy Collaboration et al. 2005), the Pan-STARRS1 survey (PS1; Chambers et al. 2016), the SkyMapper Southern Survey (SMSS; Wolf et al. 2018), the Stellar Abundance and Galactic Evolution survey (SAGE; Zheng et al. 2018), the Javalambre Photometric Local Universe Survey (J-PLUS; Cenarro et al. 2019), the Southern Photometric Local Universe Survey (S-PLUS; Mendes de Oliveira et al. 2019), the Javalambre Physics of the Accelerating Universe Astrophysical Survey (J-PAS; Benitez et al. 2014). the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et al. 2019), the Chinese Space Station Telescope (CSST; Zhan 2021), the Wide Field Survey Telescope (WFST; Lou et al. 2016), and the Multi-channel Photometric Survey Telescope (Mephisto; Er et al. 2021, in preparation).

The calibration of astronomical photometric measurements are traditionally based on sets of standard stars (e.g., Landolt 1992, 2009, 2013; Stetson 2000; Clem & Landolt 2013). However, due to the very limited numbers of standard stars available, it is very challenging to achieve calibration precision better than 1 percent for ground-based observations (e.g. Stubbs & Tonry 2006). The challenges are mainly caused by the temporal and spatial variations of the Earth atmospheric transmission, and the difficulties in correcting for instrumental effects such as flat fielding and electronics of CCDs. In recent years, a number of approaches have been developed for the precise calibration of wide field imaging surveys. These approaches can be roughly divided into two categories: “hardware-driven” approaches and “software-driven” approaches. Note that all these approaches perform “relative” calibration (i.e., establishing an internally consistent photometric calibration across the whole survey region) rather than “absolute” calibration.

The “hardware-driven” approaches are based on better understanding of wide field imaging observations, including the ubercalibration method (Padmanabhan et al. 2008), the Forward Global Calibration Method (FGCM; Burke et al. 2018), and the hypercalibration method (Finkbeiner et al. 2016). The ubercalibration method is originally developed for the SDSS but has been widely used (e.g., Schlafly et al. 2012; Liu et al. 2014; Zhou et al. 2018; Gaia Collaboration et al. 2016). It requires a significant amount of over-lapping observations. The FGCM is developed for the DES and LSST. In addition to repeated observations, it requires data taken with auxiliary instrumentation at the observatory, and models of the instrument and atmosphere to estimate the spatial and time variations of the passbands. The hypercalibration method assumes that systematic errors of different surveys are independent. Therefore, different surveys can be used to calibrate each other.

The “software-driven” approaches are based on better understanding of stellar colors, including the Stellar Locus Regression method (SLR; High et al. 2009), the stellar locus method (SL; López-Sanjuan et al. 2019), and the Stellar Color Regression method (SCR; Yuan et al. 2015a). The position of the stellar locus was firstly used by Ivezić et al. (2004) to estimate the accuracy of photometric zeropoints of the SDSS. Assuming a universal stellar color-color locus, the SLR method can correct for effects including variations in instrumental sensitivity, in the Earth atmospheric transmission, and in the Galactic interstellar and reddening, producing real-time well calibrated colors for both stars and galaxies. However, the precision is limited to a few per cent, due to the variations of stellar populations and the interstellar extinction, especially in the low Galactic latitude region. The method also requires a blue filter in addition to at least two other filters to break possible degeneracy. The SLR method can only perform color calibration. To overcome this limitation, by using an existing well-calibrated data-set as anchors, the SL method can perform photometric calibration with the help of the de-reddened stellar locus. The method can be further improved by including the metallicity effect in colours via the metallicity-dependent stellar locus (Yuan et al. 2015b; López-Sanjuan et al. 2021).

With the rapid development of multi-fiber spectroscopic surveys, e.g., SDSS/SEGUE (Yanny et al. 2009), LAMOST (Deng et al. 2012; Liu et al. 2014), APOGEE (Jönsson et al. 2020), and GALAH (De Silva et al. 2015) surveys, we have entered into the era of millions of stellar spectra. On the other hand, thanks to the modern template-matching based as well as data-driven stellar parameter pipelines (e.g., Lee et al. 2008a,b; Wu et al. 2011; Xiang et al. 2015, 2017), stellar atmospheric parameters ($T_{\rm eff}$, log $g$, and [Fe/H]) can be determined to a very high internal precision (e.g., Niu et al. 2021a). Therefore, stellar colors can be accurately predicted based on the large-scale spectroscopic surveys with the star-pair technique (Yuan et al. 2013). Taking advantage of the above fact and using millions of spectroscopically observed stars as color standards, Yuan et al. (2015a) have proposed the spectroscopy-based SCR method to perform precise color calibrations. Compared to the SLR and SL methods, the SCR method fully accounts the effects of metallicity, surface gravity, and dust reddening on stellar colors. Applying the method to the SDSS Stripe 82 standard stars catalog, we have achieved a precision of about 5 mmag in $u-g$, 3 mmag in $g-r$, and 2 mmag in $r-i$ and $i-z$, an improvement by a factor of two to three. The method has also been applied to the Gaia DR2 and EDR3 (Niu et al. 2021a, b) to correct for magnitude/color-dependent systematic errors in the Gaia colors, with a precision of about 1 millimagnitude.

With the data releases of the Gaia DR2 and EDR3 (Gaia Collaboration et al. 2016, 2018, 2021), accurate and homogeneous photometric data of the whole sky and with an exquisite quality are now accessible, reaching down to the unprecedented millimagnitude level for the G, BP, and RP passbands. With the help from the Gaia photometry, the SCR method can accurately predict the magnitudes of stars in various bands, providing a great opportunity to break the 1 percent precision barrier of ground-based photometric surveys. Such approach has been applied to the second data release from the SkyMapper Southern Survey (SMSS DR2). Large zero-point offsets are detected, particularly for the gravity- and metallicity-sensitive $uv$ bands (Huang et al. 2021a).

In this work, by combining spectroscopic data from the LAMOST DR7, SDSS DR12 and corrected photometric data from the Gaia EDR3, we apply the SCR method to recalibrate the SDSS Stripe 82 standard stars catalog. The paper is organized as follows. In Section 2, we introduce our data. In Section 3, we apply the SCR method to the SDSS Stripe 82 region to recalibrate the data and check the precision of our method. In Section 4, we apply the method to the latest version of the catalog (V4.2, Thanjavur et al. 2021). The discussions and conclusions are given in Section 5.

The repeatedly scanned Stripe 82 ($|{\rm Dec.}|<1.266\arcdeg$, 20^h34^m $<$ R.A. $<$ 4^h00^m) is a contiguous equatorial region of 300 deg² in the SDSS. Ivezić et al (2007; I07 hereafter) delivered an accurate photometric catalog of about one million stars in $u,g,r,i,z$ bands, with an internal calibration consistency of one percent, thus providing a practical definition of the SDSS photometric system. The random photometric errors of the I07 catalog are below 0.01 mag for stars brighter than 19.5, 20.5, 20.5, 20.0, and 18.5 in $u$, $g$, $r$, $i$, and $z$ bands, respectively.

A new version (V4.2) of SDSS Stripe 82 standard stars catalog is recently released (Thanjavur et al. 2021). Compared to the original version, the new version delivers averaged SDSS $ugriz$ photometry for nearly one million stars brighter than $\sim$22. Thanks to 2–3 times more measurements per star, their random errors are 1.4–1.7 times smaller than those in the original catalog. The new catalog is calibrated against Gaia EDR3, firstly using the $G$ photometry to derive grey photometric zeropoint corrections as functions of R.A. and Dec., then using the $BP-RP$ color to derive relative corrections in the $ugiz$ bands to the $r$ band. In this section, we apply our method to the new catalog with the same procedures. The $\delta_{m}^{ext}({\rm R.A.})$ and $\delta_{c}^{ff}({\rm Dec.})$ values obtained for the V4.2 catalog are displayed in Figure 20 and listed in Tables A4–A6 in the Appendix. Same to Tables A1–A3, the entries in the tables should be added to cataloged photometry.

Two phenomena are found. One is that the magnitude offsets in the Dec. direction are well corrected in the V4.2 catalog, particularly in the small scales within a given CCD camera. However, there are small (a few mmag) but significant global offsets for certain CCD cameras, e.g., the fourth camera in the south strip. The offsets for the fourth camera in the south strip are very similar in the $ugriz$ bands, probably due to the fact that the calibration of the $ugiz$ bands are based on the $r$ band and the offset comes from the $r$ band. The cause of such global offsets needs further investigation.

The other is that the magnitude offsets in the R.A. direction still suffer certain systematic, up to 0.01 – 0.02 mag in the $ugriz$ bands. The errors are caused by ignoring the effects of varying reddening and metallicity along the R.A. direction in the calibration process of Thanjavur et al. (2021). Both reddening and metallicity increase toward low Galactic latitude regions. The variations of reddening along the Dec. direction are very weak. Therefore, the calibration of Thanjavur et al. (2021) along the Dec. direction works well. For the $griz$ bands whose metallicity sensitivities are weak (about 0.02 mag/dex, see Yuan et al. 2015b), their errors are mainly caused by the effect of varying reddening. Consequently, their errors show a trend similar to extinction (see Figure 1). For the $u$ band whose metallicity sensitivity is strong (about 0.2 mag/dex, see Yuan et al. 2015b), its errors are caused by the combined effects of varying reddening and metallicity.

The variations of magnitude offset as a function of magnitude and color $g-i$ for the six SDSS CCD columns after corrections for V4.2 are similar to Figures 14–15. Fit coefficients for the $z$ magnitude offsets as a function of $z$ magnitude are listed in Table 3. The corrections should be added to the cataloged photometry.

In this work, by combining spectroscopic data from the LAMOST DR7, SDSS DR12 and corrected photometric data from the Gaia EDR3, we apply the Stellar Color Regression method to recalibrate the SDSS Stripe 82 standard stars catalog. With a total number of about 30,000 spectroscopically targeted stars, we have mapped out the relatively large and strongly correlated photometric zero-point errors present in the catalog, $\sim$2.5 per cent in the $u$ band and $\sim$ 1 per cent in the $griz$ bands. Our study also confirms some small but significant magnitude dependence errors in the $z$ band for some CCDs. Various tests show that we have achieved an internal precision of about 5 mmag in the $u$ band and about 2 mmag in the $griz$ bands, which is about 5 times better than previous results. We also apply the method to the latest V4.2 version of the catalog, and find modest systematic calibration errors along the R.A. direction and smaller errors along the Dec. direction. The updated catalogs are publicly available¹¹1https://faculty.washington.edu/ivezic/sdss/catalogs/
stripe82.html²²2http://paperdata.china-vo.org/Huang.Bowen/2022/ApJS/
PRoStripe82³³3There are two catalogs in the link of the third footnote: the stripe82calibStars_v4.2_corrected.dat for V4.2 and the stripe82calibStars_v2.6_corrected.dat for I07..

The results demonstrate the power of the SCR method when combining spectroscopic data and Gaia photometry in breaking the 1 percent precision barrier of ground-based photometric surveys. Our work paves the way for the re-calibration of existing surveys, such as the whole SDSS photometric survey, and has important implications for the calibration of future surveys, such as the LSST, CSST, and Mephisto.

The key idea behind the SCR method is that stellar colors are intrinsically simple, which can be fully determined by a small number of parameters including $T_{\rm eff}$, [Fe/H], log $g$, and other elemental abundances. To make the method flexible under different situations, different forms can be adopted to predict intrinsic colors of the selected target stars:

1. 1.

   Intrinsic colors are functions of stellar atmospheric parameters. They can be computed via the star-pair technique or polynomial fitting relations. Depending on the color of interest, different atmospheric parameters can be included. For broadband filters as in this work, $T_{\rm eff}$, [Fe/H] and log $g$ are sufficient. For some narrow-band filters such as $J510$ of the J-PLUS survey, one may need to include the effect of [Mg/Fe] as well.

2. 2.

   Intrinsic colors are functions of normalized stellar spectra. One can use machine learning techniques to predict intrinsic colors of a given star from its normalized stellar spectrum. In this way, stellar atmospheric parameters are not needed anymore. This approach is pure data-driven and model-free.

3. 3.

   Intrinsic colors of main-sequence stars (or red giants if necessary) are functions of a given color (e.g., $BP-RP$) and metallicity via the tools of metallicity-dependent stellar locus (Yuan et al. 2015b; Huang et al. 2021a; López-Sanjuan et al. 2021; Zhang et al. 2021). The metallicities of target stars could be from spectroscopic measurements when they are available, or photometric ones delivered by well-calibrated photometric surveys (e.g., Yuan et al. 2015c; Huang et al. 2021b; Xu et al. 2021; Yang et al. 2022). It is worth mentioning that in the literature, stellar color-color relations are widely used as transformation relations between different photometric systems (e.g., Riello et al. 2021), ignoring the effects of metallicity and reddening. Although it is convenient, it will cause spatially dependent systematic errors up to a few per cent and should not be used when high-precision calibrations are required. However, using stars within a very limited sky area where variations of reddening and metallicity can be safely ignored, the color-color relations may serve as an excellent tool in correcting for small-scale effects (e.g., flat fielding).

4. 4.

   Intrinsic colors are functions of a given set of colors (e.g., $U-B$, $B-V$, $V-R$, and $R-I$) that are well-calibrated and sensitive to stellar atmospheric parameters (e.g., Yang et al. 2021). It will also be very interesting to explore predicting observed colors directly from the Gaia $BP$ and $RP$ spectrophotometry when the data is available.

Alternatively, with Gaia parallaxes accurate to a few per cent, and absolute magnitudes predicted from stellar spectra to 0.1 – 0.2 magnitude (e.g., Xiang et al. 2017; Wang et al. submitted), it is also possible to predict the observed magnitudes directly based on Gaia parallaxes and spectra, without the help from Gaia photometry. Such approach will be explored in future.

Precise reddening correction is another key ingredient of the SCR method. Given that mmag precision has been achieved with the Gaia photometry, reddening correction precise to mmag also should be pursued with efforts. Three factors have to be considered:

1. 1.

   Systematics of widely used 2D reddening maps. Using millions of LAMOST stars, Sun et al. (submitted) have investigated the SFD and Planck 2D extinction maps (Planck collaboration 2014; Irfan et al. 2019) in the middle and high Galactic latitude regions. Spatially dependent errors are found, which are correlated with the dust temperature, dust reddening, and spectral index of the dust emission. Sun et al. (submitted) have provided recalibrated SFD and Planck extinction maps within the LAMOST footprint, along with empirical relations for regions outside. Nevertheless, further improvements of the Galactic all-sky extinction maps are needed in the era of precision astronomy.

2. 2.

   Varying reddening coefficients for very broad or blue filters. For very broad (e.g., the Gaia passbands) or blue (e.g., $u$, $NUV$, and $FUV$) filters, their reddening coefficients relative to $E(B-V)$ show strong dependences on stellar types and reddening (e.g., Niu et al. 2021a,b, Zhang et al. in preparation), even for a given reddening curve. Such dependences should be carefully taken into account in future.

3. 3.

   Variations of reddening laws across the Galaxy, particularly in the Galactic disk. We will map the spatial variations of the reddening law across the Galactic disk in future (Zhang et al. in preparation).

Last but not least, the predicted magnitudes of the SCR method are for the “standard” passbands defined by the reference field. “Chromatic correction” (e.g., Burke et al. 2018) to the standard system is necessary in many cases to account for the variations of passbands caused by atmospheric extinction and other factors.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven NationalLaboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.