A MONTE CARLO METHOD FOR MAKING THE SDSS u-BAND MAGNITUDE MORE ACCURATE

It is an increasing perception that the Galactic halo system comprises at least two spatially overlapping components with different kinematics, metallicity and spatial distribution (Carollo et al. 2007, 2010; An et al. 2013, 2015). Chemical abundance is the direct observational ingredient in investigating the dual nature of the Galactic halo. Since the chemical abundance of stars has a strong effect on the emergent flux, especially at blue end, the natural endeavor is to recover the metal information from large photometric surveys such as the Sloan Digital Sky Survey (SDSS; York et al. 2000). The advantage of photometric metallicity estimates is that information on the metallicity of large numbers of stars can be obtained.

Based on the SDSS ugriz photometry, Ivezić et al. (2008) used a polynomial-fitting method from spectroscopic calibration of de-reddened u − g and g − r colors to derive the photometric metallicity (see also Peng et al. 2012). However, due to the relatively large error of the SDSS u-band magnitude, only the metallicities [Fe/H] of stars brighter than g = 19.5 were obtained. Combining the more accurate SCUSS (Zhou et al. 2016) u-band photometry, SDSS g and r photometry, Gu et al. (2015) developed a three-order polynomial photometric metallicity estimator, in which the u-band magnitude can be used for faint magnitudes of g = 21. However, both the estimators developed by Ivezić et al. (2008) and Gu et al. (2015) based on polynomial-fitting have the intrinsic drawback that they cannot be extended to the metal-poor end. In order to solve this problem, Gu et al. (2016) (hereafter denoted as Paper I) devised a Monte Carlo method to estimate the stellar metallicity distribution function (MDF) which appears particularly accurate at both metal-rich and metal-poor ends. The natural step forward is to combine the SCUSS u and SDSS $g,r$ photometry with the method introduced in Paper I to investigate the MDF of the Galactic halo stars. But only those stars in the south Galactic cap are surveyed by the SCUSS. How can we estimate the photometric metallicity distribution of faint stars (deep in the Galactic halo) in both the south and north hemisphere? This paper provides a new method to achieve this goal. Due to the fact that the SCUSS u is more accurate than the SDSS u, we convert the latter to the former using a Monte Carlo method, through which the converted u magnitude becomes as accurate as the SCUSS u magnitude.

We organize this paper as follows. In Section 2, we give a brief overview of the SDSS and SCUSS. The technical details for converting the SDSS u to the SCUSS u are presented in Section 3. Section 4 evaluates the effectiveness of this conversion. A discussion of the potential application of the conversion is given in Section 5.

The SDSS is a digital multi-filter imaging and spectroscopic redshift survey using a dedicated 2.5 m wide-angle optical telescope at Apache Point Observatory in New Mexico, USA (Gunn et al. 2006). It began operation in 2000, and now covers over 35% of the sky, with about 500 million photometrically surveyed objects and more than 3 million spectroscopically surveyed objects. Five bands (u, g, r, i, and z) are used to simultaneously measure the objects' magnitude, with effective wavelengths of 3551, 4686, 6165, 7481, and 8931 Å respectively. The limit magnitudes of u, g, r, i, and z are 22.0, 22.2, 22.2, 21.3, and 20.5, respectively (Abazajian et al. 2004). The relative photometric calibration accuracy values for u, g, r, i, and z are 2%, 1%, 1%, 1% and 1%, respectively (Padmanabhan et al. 2008). Other technical details about the SDSS can be found on the SDSS website http://www.sdss3.org/, which also provides an interface for public data access.

The south Galactic Cap u-band Sky Survey (SCUSS) is an international cooperative project that is jointly undertaken by the National Astronomical Observatories of China and Steward Observatory of University of Arizona. It utilizes the 2.3 m Bok telescope located on Kitt Peak to photometrically survey the stars in the south Galactic Cap in the u band with an effective wavelength of 3538 Å. This project started in the summer of 2009, began its observation in the fall of 2010, completed in the fall of 2013, and finally about 5000 deg² area ($30^\circ \lt l\lt 210^\circ ,\,-80^\circ \lt b\lt -20^\circ$) were surveyed. Its main goal is to provide the essential input data to the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) project (Zhao et al. 2006). Figure 1 shows the similarity in response curve between the u band filter of the SCUSS and that of the SDSS. The limit magnitude for point sources is about 23.2 mag with a 5 minute exposure time, and is about 1.5 mag deeper than that of the SDSS u band magnitude (Jia et al. 2014; Peng et al. 2015). In Table 1, we provide a brief summary of the SCUSS. More detailed information and data reduction about SCUSS can be found in Zhou et al. (2016), Zou et al. (2015, 2016), and the SCUSS website http://batc.bao.ac.cn/Uband/, which also provides an interface for public data access.

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In Figure 2, the average errors of the SCUSS u and SDSS u of numerous main-sequence stars with $0.2\lt g-r\lt 0.8$ are plotted as functions of g-band magnitude. The figure clearly shows that the error of the SDSS u is much larger than that of the SCUSS u on the whole, especially at the faint end. The spectroscopically surveyed stars have limiting magnitude of g = 19.5. Coincidentally, the error of the SDSS u limits the application of photometric metallicity estimates in the range of $g\lt 19.5$. From Figure 2, we find that the error of theSDSS u is about 0.11 when g = 19.5. So we set 0.11 as the maximum error. Beneath this value, the SCUSS u corresponds to the range of $g\lt 20.5$. However, the SDSS u error is as high as 0.22 when g = 20.5. In the following, we will convert the SDSS u to the SCUSS u for stars brighter than g = 20.5 so that the error of the converted u does not exceed 0.11. Here, we only convert the SDSS u with $18.5\lt g\lt 20.5$ for main-sequence stars. Since g-, r-band magnitudes are much more accurate than those of u, we assume that they are absolutely precise, at least in the considered g-band magnitude range. So the error of u − g is the direct consequence of the error of u.

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For each object surveyed by the SCUSS, we can identify the same object from the SDSS catalog by matching their positions. So in the merged catalog, each star has the following information: position (R.A. and decl.), SCUSS u-band magnitude and its error, SDSS $u,g,r,i,z$-band magnitudes and their error and extinction. Here, the extinction for the SDSS u-band magnitude is also used by the SCUSS u-band magnitude. Throughout this paper, magnitudes and colors are understood to have been corrected for extinction and reddening following Schlegel et al. (1998). We select the stars from the SCUSS catalog using the following criteria:

• 1.  
  $18.5\lt g\lt 20.5;$
• 2.  
  $0.2\lt g-r\lt 0.8;$
• 3.  
  $0.6\lt {(u-g)}_{\mathrm{SDSS}}\lt 2.2;$
• 4.  
  $0.6\lt {(u-g)}_{\mathrm{SCUSS}}\lt 2.2;$
• 5.  
  main-sequence stars are selected by only including those objects at distances smaller than 0.15 mag from the stellar locus described by the following equation (Jurić et al. 2008):

  $$\begin{eqnarray*}(g-r) & = & 1.39\{1-\exp [-4.9{(r-i)}^{3}-2.45{(r-i)}^{2}\\ & & -1.68(r-i)-0.05]\}\end{eqnarray*}$$
• 6.  
  we further refine the selection of main-sequence stars by only including those objects at distances smaller than 0.3 mag from the stellar locus described by the following equation (Jia et al. 2014):

  $$\begin{eqnarray*}&&{(u-g)}_{\mathrm{SDSS}}=\exp [-{(g-r)}^{2}+2.8(g-r)-1]\end{eqnarray*}$$

We divide the color range of $0.2\lt g-r\lt 0.8$ into six equal bins, and also divide the magnitude range of $18.5\lt g\lt 20.5$ into 20 bins. Thus, we totally get 120 $0.1\times 0.1$ mag² bins, and designate each square bin by an index computed in the following manner:

$$\begin{eqnarray*}&&\mathrm{index}=\mathrm{int}((u-g-0.2)/0.1)\ast 20+\mathrm{int}((g-18.5)/0.1)\end{eqnarray*}$$

where the symbol int stands for the integer portion. In this way the index takes values from 0 to 119. Main-sequence stars whose colors and magnitudes match a position specified by index will be used to construct a "convertor." Thus, we will obtain a total of 120 convertors, and each convertor is denoted as convertor[index]. In the following, each convertor has the form of a 16 × 16 array in which each element is further denoted as $\mathrm{convertor}[\mathrm{index}][i][j]$, where i, j range from 0 to 15. Each main-sequence star that is associated with one convertor is further classified with two labels of integer number, i and j, which can be computed in the following manner:

$$\begin{eqnarray*}i & = & {\rm{int}}(({(u-g)}_{\mathrm{SDSS}}-0.6)/0.1)\\ j & = & {\rm{int}}(({(u-g)}_{\mathrm{SCUSS}}-0.6)/0.1),\end{eqnarray*}$$

where the symbol int also stands for the integer portion. Each element in each convertor array records the number of stars whose ${(u-g)}_{\mathrm{SDSS}}$ and ${(u-g)}_{\mathrm{SCUSS}}$ colors match its position. We use convertor[index][i][:] to denote the set of 16 numbers of convertor[index][i][j] for j taking integer values from 0 to 15. The maximum value of the convertor[index][i][:] is further denoted as max[index][i].

Figure 3 shows the two-color diagrams for ${(u-g)}_{\mathrm{SCUSS}}$ versus ${(u-g)}_{\mathrm{SDSS}}$. Main-sequence stars in different magnitude and color ranges are selected. Stars in panels from top row to bottom row have $18.5\lt g\lt 18.6$, $19.4\lt g\lt 19.5$ and $20.4\lt g\lt 20.5$, respectively. Stars in panels from left column to right column have $0.2\lt g-r\lt 0.3$, $0.5\lt g-r\lt 0.6$ and $0.7\lt g-r\lt 0.8$, respectively. Each panel corresponds to one convertor array. The more scattered the points in each panel, the larger error in u-band magnitude is implied. As shown in Figure 3, the error of ${(u-g)}_{\mathrm{SDSS}}$ is larger than that of ${(u-g)}_{\mathrm{SCUSS}}$, especially for the faint stars. The comparison of error for each panel is quantized by the displayed ratio of standard deviation between ${(u-g)}_{\mathrm{SDSS}}$ and ${(u-g)}_{\mathrm{SCUSS}}$. In the bottom left panel of the diagram, the ratio has a maximum value of 1.536 compared with the others. This panel corresponds to the fainter and bluer stars that reasonably belong to the Galactic halo.

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The central idea for converting the SDSS u to SCUSS u is that we obtain the color distribution of converted u − g according to the inputting distribution of ${(u-g)}_{\mathrm{SDSS}}$ and the scatter diagram in each panel of Figure 3. The scatter diagram of ${(u-g)}_{\mathrm{SCUSS}}$ versus ${(u-g)}_{\mathrm{SDSS}}$ is now explained as the consequence of probability. More points in a small region imply that a star has a higher probability of being located within it. We reproduce a given distribution using the Monte Carlo method. The converted u − g should be considered the same as ${(u-g)}_{\mathrm{SCUSS}}$. Here, in order to distinguish the converted u − g from the original ${(u-g)}_{\mathrm{SDSS}}$ and ${(u-g)}_{\mathrm{SCUSS}}$, we denote the converted u − g with a subscript, namely ${(u-g)}_{\mathrm{CONV}}$.

For n stars corresponding to index = a, i = b, we generate n random numbers according to the distribution exhibited by the 15 numbers from convertor[a][b][0] to convertor[a][b][15]. The obtained n random numbers are all real numbers from 0 to 15. Then, they are converted to ${(u-g)}_{\mathrm{CONV}}$ values using ${(u-g)}_{\mathrm{CONV}}=0.1\ast r+0.6$, where r is one random number. How can we generate a sequence of random numbers that complies with a given distribution? This is explained as follows. Suppose that there are two stochastic variables, X and Y, which can be assigned a random number generating function $X={\mathrm{rand}}_{X}()$ and $Y={\mathrm{rand}}_{Y}()$, respectively. In each trial, we obtain a random number pair (X, Y), where X is modulated to take the uniform probability distributed random real number from 0 to 15. For any star, whose index and i has been determined, Y is modulated to take the uniform probability distributed random real number from 0 to max[index][i]. When $Y\leqslant \mathrm{convertor}[\mathrm{index}][i][\mathrm{int}(X)]$ (int(X), the integer portion of X), we record X as a useful value, and otherwise discard it. After numerous trials, we obtain a sequence of random numbers $\{{X}_{1},{X}_{2},{X}_{3},\cdots \}$ that follow the same probability distribution as those recorded in convertor[index][i][:]. Here, because convertor[index][i][j] can be equal to zero for some j values, we can discard them and record the non-zero elements and their positions in a new array. Through this method, the sampling efficiency can be greatly improved.

From the top three two-color diagrams of Figure 3, we find that ${(u-g)}_{\mathrm{SDSS}}$ may be expressed as a linear function of ${(u-g)}_{\mathrm{SCUSS}}$. For selected stars with $18.5\lt g\lt 18.6$, the error of u plays a minor role for the distribution of points in these diagrams. If the u-band (both SDSS and SCUSS) magnitudes were absolutely precise , the resulting transformation relation would be as follows:

$$\begin{eqnarray*}&&{(u-g)}_{\mathrm{SDSS}}=k\ast {(u-g)}_{\mathrm{SCUSS}}+h,\end{eqnarray*}$$

where k is the slope and h represents a constant.

In evaluating which color (either ${(u-g)}_{\mathrm{SDSS}}$ or ${(u-g)}_{\mathrm{SCUSS}}$) has the greater error, the reliability of the standard deviation ratio shown in Figure 3 depends on the assumption  $k\approx 1$. In Figure 4, we plot a two-color diagram of ${(u-g)}_{\mathrm{SCUSS}}$ versus ${(u-g)}_{\mathrm{SDSS}}$ for main-sequence stars with $0.2\lt g-r\lt 0.8$ and $16.99\lt g\lt 17.01$. We notice that the error of the u-band magnitude at the bright magnitude g = 17 is small, and therefore its effect on the color distribution in Figure 4 can be neglected. The trend of ${(u-g)}_{\mathrm{SDSS}}$ versus ${(u-g)}_{\mathrm{SCUSS}}$ is fitted by a line with the expression shown in the figure. The slope k = 0.9955 is almost equal to 1. The assumption of $k\approx 1$ holds. We can evaluate which u has the greater error by the dispersion degree of the points in Figure 3. In addition, for convenience we may also approximately assume that the SDSS u and SCUSS u are from the same photometric system, since they are almost similar after neglecting the error, as shown in Figure 4.

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In order to evaluate the effect of this conversion, we plot histograms of the distribution of ${(u-g)}_{\mathrm{SDSS}}$, ${(u-g)}_{\mathrm{SCUSS}}$ and ${(u-g)}_{\mathrm{CONV}}$ for main-sequence stars with different magnitude and color ranges in Figure 5. The top three panels show color distributions of stars with $18.5\lt g\lt 19$, the middle three with $19.3\lt g\lt 19.7$, and the bottom three with $20\lt g\lt 20.5$. Corresponding to the color range, stars for panels from left column to right column have $0.2\lt g-r\lt 0.3$, $0.5\lt g-r\lt 0.6$ and $0.7\lt g-r\lt 0.8$, respectively. The histograms in each panel are normalized to the maximum, with the actual peak values labeled. It is clear that the profiles of the histograms of ${(u-g)}_{\mathrm{CONV}}$ in each panel are almost the same as those of ${(u-g)}_{\mathrm{SCUSS}}$. This effect indicates that the conversion has the ability to reduce the error of u_SDSS, to be as small as that of u_SCUSS. Actually, the distribution of ${(u-g)}_{\mathrm{CONV}}$ will completely coincide with that of ${(u-g)}_{\mathrm{SCUSS}}$ as long as the number of stars selected is large enough for the histogram, since the convertor arrays are constructed using the SCUSS u data. Thus, for a larger sky area in which there is no SCUSS u, the convertor array could be used to reduce the error of SDSS u. As a result, the error of the converted u magnitude when g = 20.5 is equal to the original error of SDSS u when g = 19.5. However, we are still aware that the extent to which this conversion method can reduce the u magnitude error cannot be fully tested until a deeper survey is available.

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As is known, u-band measurements are very important in deriving the photometric metallicity and therefore for constructing a precise MDF. Because of the relatively shallow survey limit ($u\sim 22$) and the relatively large error in the SDSS u-band near the faint end, the application of the photometric metallicity estimates is greatly restricted in the range of $g\lt 19.5$, an insufficient depth to explore the distant halo and substructures. However, the SCUSS u is 1.5 mag deeper than the SDSS u, and its error is smaller than the latter's error on the whole. The potential application of the conversion from the SDSS u to the SCUSS u is very important in deriving relatively accurate photometric metallicities of distant stars. In Paper I, we developed a new method to estimate the photometric metallicity distribution of large numbers of stars. Compared with other photometric calibration methods, this effectively reduces the error induced by the method itself, and therefore enables a more reliable determination of the photometric MDF. However, there is another error source: that of the SDSS u-band magnitude. This error behavior limits the application of the method in the range of $g\lt 19.5$ in Paper I, which is same as that of Ivezić et al.'s (2008) photometric metallicity estimator. The more accurate SCUSS u-band measurements guarantee the accuracy of the stellar distribution in the u − g versus g − r panel, and it extends the application of the method in Paper I to even fainter stars. Thus, the photometric MDF of distant stars such as halo stars or some stream stars can be estimated.

However, only the stars in south Galactic Cap are surveyed by the SCUSS, and these have a more accurate u band magnitude; how can we derive the photometric metallicity of stars in the north Galactic hemisphere? The conversion from the SDSS u to SCUSS u statistically reduces the error of the u-band magnitude, which make it possible to estimate the photometric MDF of stars in the whole sky. In this study, we have made the conversion for stars in the range $18.5\lt g\lt 20.5$. This conversion combined with the method introduced in Paper I enables us to estimate the photometric metallicity distribution function for stars at least in the range of $g\lt 20.5$, which is 1 mag deeper than that of spectroscopically surveyed stars. So we can study the chemical structure of the Galactic halo in more detail. Besides the application described above, the more accurate u band magnitude from the conversion can be applied to address other scientific issues.

This work was supported by joint fund of Astronomy of the National Natural Science Foundation of China and the Chinese Academy of Science, under Grants U1231113. This work was also by supported by the Special Funds of Cooperation between the Institute and the University of the Chinese Academy of Sciences. In addition, this work was supported by the National Natural Foundation of China (NSFC, No.11373033, No.11373035), and by the National Basic Research Program of China (973 Program) (No. 2014CB845702, No.2014CB845704, No.2013CB834902).

We would like to thank all those who participated in observations and data reduction of the SCUSS for their hard work and kind cooperation. The SCUSS is funded by the Main Direction Program of Knowledge Innovation of Chinese Academy of Sciences (No. KJCX2-EW-T06). It is also an international cooperative project between the National Astronomical Observatories, Chinese Academy of Sciences and Steward Observatory, University of Arizona, USA. Technical support and observational assistances of the Bok telescope are provided by Steward Observatory. The project is managed by the National Astronomical Observatory of China and Shanghai Astronomical Observatory.

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 National Laboratory, 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.