THE STELLAR METALLICITY DISTRIBUTION FUNCTION OF THE GALACTIC HALO FROM SDSS PHOTOMETRY

In the following analysis, we employ the same set of underlying stellar interior models and the same α-element enhancement scheme as in An et al. (2009b), motivated by the observed behavior of these elements among field dwarfs and cluster stars from spectroscopic studies (e.g., Venn et al. 2004; Kirby et al. 2008): [α/Fe] = +0.4 at [Fe/H] = −3.0, [α/Fe] = +0.3 at [Fe/H] = −2.0, −1.5, −1.0, [α/Fe] = +0.2 at [Fe/H] = −0.5, and [α/Fe] = +0.0 at [Fe/H] = −0.3, −0.2, −0.1, +0.0, +0.1, +0.2, +0.4. A linear interpolation was made in this metallicity grid to obtain isochrones at intermediate [Fe/H] values. We adopted an age of 13 Gyr for −3.0 ⩽ [Fe/H] ⩽ −1.2, and 4 Gyr at −0.3 ⩽ [Fe/H] ⩽ +0.4, with a linear interpolation between these two metallicity ranges. As described below, our strategy minimizes the application of our calibrated stellar models to main-sequence turnoff stars, for which colors are most sensitive to the adopted age. For this reason, our main results are insensitive to the age assumption, but readers are cautioned if they apply our models to stellar populations with different ages. The above age–metallicity–[α/Fe] relations are adopted throughout this paper.

Colors and magnitudes predicted by theoretical models typically do not agree with those observed from star clusters. Therefore, we defined color–T_eff–[Fe/H] corrections to better match the data. In An et al. (2009b), we used M67 cluster data to define color–T_eff relations in u − g, g − r, g − i, and g − z, at the cluster's metal abundance ([Fe/H] = 0.0). In An et al. (2009a), we adopted these correction factors in gri color indices at solar abundance, and used a linear ramp between [Fe/H] = −0.8 and [Fe/H] = 0.0. In this weighting scheme, empirical corrections become zero at [Fe/H] = −0.8 and below, so metallicity estimation of metal-poor stars are essentially those obtained from pure theoretical calculations, and are not affected by the empirical color corrections. This choice was motivated by the fact that the models are in satisfactorily agreement with data in gri color indices for globular clusters (An et al. 2009b). Above solar abundance, we applied the M67-based color corrections to all models, assuming that offsets in colors between the model and data are independent of the metal abundance of a star.

In principle, cluster sequences alone can be used to directly estimate stellar distances and metallicities, as long as there exist well-measured cluster-star samples over a wide range of [Fe/H]. However, cluster data are often noisy (e.g., some cluster fiducials are found to be essentially superimposed, even if their [Fe/H] values are different), so we choose instead to be guided by theory, in order to infer stellar colors at any given metallicity (and/or age; see also An et al. 2007a, 2007b; Pinsonneault et al. 2004).

In particular, updates on the empirical calibrations of models were necessary to adjust model u − g colors, which have a profound impact on the accuracy of photometric metallicity estimates, to better match observed cluster fiducial sequences over a wide range of metal abundances. Below we describe our adopted methodology used to generate a new set of calibrated isochrones for all of the ugriz color indices.

We expanded upon the strategy of An et al. (2009b) for the color–T_eff calibration by the addition of cluster fiducial sequences in An et al. (2008) over a wide range of metal abundances. Although our pure theoretical models predict colors that are consistent with observed fiducial sequences within the total systematic and random errors (An et al. 2009b), there still remains a small, but suggestive systematic residual pattern of the color offsets over T_eff and [Fe/H]. In the updated calibration set, we have attempted to minimize these effects to better constrain estimates of the stellar parameters.

In the color–T_eff–[Fe/H] calibration, which is described in detail below, we used cluster fiducial sequences for several globular and open clusters in An et al. (2008). These clusters are listed in Table 2, along with our adopted values for the cluster parameters, which are the same as those used in our earlier model comparisons (An et al. 2008). The [Fe/H] and E(B − V) estimates for globular clusters are from Kraft & Ivans (2003), who used Fe ii lines from high-resolution spectra to compare colors derived from high-resolution spectroscopic determinations of T_eff with the observed colors of the same stars. Distances to the globular clusters are all Hipparcos-based subdwarf fitting distances. We adopted the Carretta et al. (2000) distance estimates whenever they are available, since they employed the same metallicity scale for both subdwarfs and cluster stars in the subdwarf-fitting technique; otherwise, we adopted distances in Kraft & Ivans (2003). For M67, we took the average reddening and metallicity estimates from high-resolution spectroscopy in the literature (An et al. 2007b and references therein), and adopted a cluster distance estimated from an empirically calibrated set of isochrones in the Johnson–Cousins–2MASS system (An et al. 2007b). For NGC 6791, we adopted the average [Fe/H] from high-resolution spectroscopic studies (see references in An et al. 2009b). The cluster's reddening and distance estimates are based on the application of our calibrated isochrones in the Johnson–Cousins–2MASS system (D. An et al., in preparation).

We employed UberCal (Uber-calibration) magnitudes (Padmanabhan et al. 2008) for calibrating the cluster systems, instead of the "Photometric Telescope (PT)"-calibrated magnitudes (hereafter Photo magnitudes).¹³ Our original cluster sequences in An et al. (2008) were on the Photo system, where the standard SDSS photometric pipeline (Lupton et al. 2002) was used to define stellar colors and magnitudes in SDSS. However, Padmanabhan et al. (2008) later devised a method of improving a relative photometric calibration error by using repeat measurements in the overlapping fields of the survey. Since DR7, SDSS takes the UberCal magnitudes as the default magnitudes.

We transformed the fiducial sequences in the Photo system onto the UberCal system, by applying zero-point differences between the two systems in the cluster flanking fields, where we derived photometric zero points for the cluster fiducials (An et al. 2008). The differences between the two calibrations are not alarmingly large for the fiducial sequences used in the current study. The mean differences are +0.003 ± 0.014 mag, +0.003 ± 0.015 mag, −0.001 ± 0.010 mag, −0.005 ± 0.028 mag, and −0.003 ± 0.013 mag in ugriz, respectively, in the sense of UberCal minus Photo magnitudes, for all the cluster fields considered in An et al. (2008), except from the few imaging stripes from which we could not retrieve UberCal magnitudes. The zero-point differences between UberCal and Photo systems for several cluster fields used in this work are listed in Table 1.

Gray cross points in Figures 1–4 represent the color differences between cluster fiducial sequences and pure theoretical stellar models in g − r, g − i, g − z, and u − g, respectively. In each of the color indices, we have arranged the comparisons such that comparisons for metal-poor clusters are shown in the upper panels, and those for metal-rich clusters are shown in the lower panels. A gray strip represents a ±1σ range of a total systematic error in the comparison, including errors from the distance, reddening, age, and an assumed photometric zero-point error on the fiducial sequences (see also An et al. 2008 for more details).

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Solid lines in Figures 1–4 show empirical color corrections derived using cluster fiducial sequences. We based our calibration using cluster fiducial sequences for both M92 and M67. In other words, color–T_eff relations were defined with the observed main sequences of M92 at [Fe/H] = −2.4 (Kraft & Ivans 2003), and those at [Fe/H] = 0.0 were defined with respect to M67 (see An et al. 2007b). The choice of M92 and M67 for the base case was due to the fact that these clusters are well studied and have reliably determined distances, foreground reddening, and metallicity estimates (An et al. 2009b, see also Table 2). They both also have wide T_eff coverage (see Figures 1–4).

To be consistent with our earlier color calibration with M67, we used the same color corrections as in An et al. (2009b). These are polynomial fits to the points over 4000 ⩽ T_eff(K) ⩽ 6000, and are expressed as follows:

$$\begin{eqnarray} \Delta (g - r) &=& 7.610 - 19.910 \theta + 17.279 \theta ^2 - 4.983 \theta ^3, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \Delta (g - i) &=& 6.541 - 16.313 \theta + 13.461 \theta ^2 - 3.675 \theta ^3, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \Delta (g - z) &=& 3.755 - 7.595 \theta + 4.535 \theta ^2 - 0.672 \theta ^3, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \Delta (u - g) &=&-7.923 + 24.131 \theta - 23.727 \theta ^2 + 7.627 \theta ^3, \quad \end{eqnarray} \tag{ 4 }$$

where θ ≡ T_eff/5040 K. These corrections are shown as solid lines in Figures 1–4 (second panels from the bottom) and are in the sense that the above values should be added to the model colors. For M92, we used the color differences between the model and the cluster fiducial sequences as the baseline of the model calibration at the cluster's metal abundance (see second panels from the top in Figures 1–4).

A quadratic relation in [Fe/H] was then used to parameterize the metallicity dependence of the color–temperature corrections, assuming the following functional form:

$$\begin{equation} \Delta _i (T_{\rm eff},{\rm [Fe/H]}) = \delta _i (T_{\rm eff}) + \zeta _i {\rm [Fe/H]} + \xi _i {\rm [Fe/H]}^2, \end{equation} \tag{ 5 }$$

where Δ_i represents color corrections in each of g − r, g − i, g − z, or u − g, while δ_i represents color–T_eff corrections in Equations (1)–(4) in the ith color index at solar metallicity. Here, ζ_i and ξ_i are coefficients to be derived from the fit, where we used fiducial sequences for M15, M13, M3, M5, and NGC 6791. In fact, the problem is reduced to a single parameter fit, since our models were defined to match the observed main sequences of M67 and M92 at their metal abundances. The solid curves in Figures 1–4 show the resulting color corrections on the color–T_eff–[Fe/H] space. We extrapolated color corrections below [Fe/H] = −2.4 using the best-fitting parameters obtained from the above equation.

The T_eff versus [Fe/H] range covered by our calibrating sample clusters is sparse and non-uniform. In particular, hot stars (T_eff ≳ 6000 K) are not covered at or near solar abundance, while cool, metal-poor stars (T_eff ≲ 5000 K) are outside of our calibration range. This is mainly because the clusters are found at varying distances from the Sun, while all of the cluster images were taken in drift-scan or time-delay-and-integrate (TDI) mode, with the same effective exposure time of 54.1 s per band. As a result, turnoff stars in M67 were too bright and lower main-sequence stars in M92 were too faint for SDSS to obtain reliable magnitudes. The cluster age is an additional factor that affects the non-uniform coverage in the stellar parameter space.

In order to obtain full coverage of the color corrections in the T_eff versus [Fe/H] plane, we assumed that the color difference from the model for M67/M92 at the high/low T_eff end extends and remains constant to T_eff = 7000 K/4000 K, respectively. We note that this reasonable, but arbitrary, extrapolation of the color–T_eff relations has minimal impact on the results of our analysis, because our calibrating cluster sample is in fact representative of the majority of stars detected in SDSS (in terms of T_eff and [Fe/H], among other parameters). The purpose of the extrapolation is to obtain stable color–T_eff–[Fe/H] relations going forward. Note that our model calibration is valid for main-sequence stars only.

Our adopted age of 13 Gyr at −3.0 ⩽ [Fe/H] ⩽ −1.2 in the color calibration (Section 2.1) is justified by our earlier result that the main-sequence turnoff ages of our calibrating clusters are approximately 13 Gyr, when pure theoretical YREC models are directly used in the age estimation (see Table 9 in An et al. 2009b). To a first approximation, a systematic error in the adopted age in the calibration results in a scale error in our color–T_eff corrections. Therefore, special attention should be paid when applying the models to stellar populations with different age–metallicity relations.

We applied calibrated stellar isochrones to the observed ugriz magnitudes, and searched for the best-fitting stellar parameters—T_eff (or stellar mass), [Fe/H], and an absolute magnitude (or distance)—by minimizing the χ² of the fit, defined as

$$\begin{equation} \chi ^2 = \sum _i \frac{(X_{{\rm obs},i} - X_{{\rm model},i})^2}{\sigma _i^2}, \end{equation} \tag{ 6 }$$

for each star over −3 ⩽ [Fe/H] ⩽ +0.4. Here, X_{obs, i} and X_{model, i} are the observed and model magnitudes, respectively, in the ith bandpass. Stellar mass, [Fe/H], and distances were set as free parameters (i.e., three parameters and five data points for each star). This is equivalent to fitting a model to the observed spectral energy distribution in the wavelength versus flux space.

We adopted foreground dust estimates by Schlegel et al. (1998), with theoretical extinction coefficients given by An et al. (2009b):

$$\begin{eqnarray} R_\lambda {\rm (YREC)} &=& \frac{A_\lambda }{E(B-V)} \nonumber\\ & = & [4.858,3.708,2.709,2.083,1.513], \end{eqnarray} \tag{ 7 }$$

where λ = u, g, r, i, z, respectively. These extinction coefficients were computed using theoretical spectral energy distributions and the standard Cardelli et al. (1989) extinction curve. Our values are in good agreement with those provided in Girardi et al. (2004). The above values lie in between the default extinction coefficients in SDSS (Schlegel et al. 1998, SFD98) and those in Schlafly & Finkbeiner (2011, SF11):

$$\begin{equation} R_\lambda {\rm (SFD98)} = [5.155,3.793,2.751,2.086,1.479], \end{equation} \tag{ 8 }$$

$$\begin{equation} R_\lambda {\rm (SF11)}= [4.239,3.303,2.285,1.698,1.263]. \end{equation} \tag{ 9 }$$

In terms of the u − g colors, which have a significant impact on photometric metallicity estimates, our E(u − g) ≡ A_u − A_g coefficient is 18% smaller than the SFD98 value, but 19% larger than the Schlafly & Finkbeiner (2011) value. These differences are clearly important and we discuss their effect on our derived photometric MDFs in Section 4 below.

Photometric temperatures and metallicities in our approach are derived simultaneously, based on the observed ugriz photometry. Each of these parameters are primarily constrained by different portions of the stellar spectral energy distribution; u − g colors are mostly responsible for the photometric metallicities, while griz colors are sensitive to temperatures. Nevertheless, there is a moderate level of correlation between them, so checking a photometric temperature scale is an important step toward obtaining accurate photometric metal abundances of stars.

In Pinsonneault et al. (2012), we verified the accuracy of our photometric T_eff estimation procedure using the most recent temperature scale from the infrared flux method (IRFM) by Casagrande et al. (2010). Photometric temperatures were estimated from griz photometry in the Kepler Input Catalog (Brown et al. 2011) using a calibrated set of isochrones—the same set of models used in the current work—then they were compared to the IRFM temperatures derived from 2MASS J − K_s colors. Overall, we found good agreement between these two fundamental temperature scales (〈ΔT_eff〉 ≲ ±40 K) at 4400 K <T_eff < 6000 K, where the IRFM scale is well defined; see Pinsonneault et al. (2012) for detailed discussions on this comparison.

Here we compare our photometric temperature estimates with the spectroscopic estimates obtained from medium-resolution SDSS/SEGUE spectra (Yanny et al. 2009); comparisons are shown in Figure 5. In this comparison, we included 51, 999 stars from the initial sample of 162, 645 objects that satisfy the following selection criteria:

• 1.  
  Sources are detected in all five bandpasses.
• 2.  
  χ²_min/ν < 3.0, where χ²_min is a minimum χ² defined in Equation (6) and ν (=2) is a degree of freedom.
• 3.  
  σ_u < 0.03 mag.
• 4.  
  log g_(YREC) ⩾ 4.15.
• 5.  
  S/N_(SSPP) > 20/1.
• 6.  
  log g_(SSPP) ⩾ 3.5.

where the subscripts YREC and SSPP indicate parameters estimated from the model isochrones and SSPP, respectively.

The first two criteria above require a good fit to the data, and the limit on the u-band measurement errors exclude data points with less well-defined photometric metallicity estimates. The log g limit restricts the analysis to main-sequence dwarfs. The last two criteria select dwarfs with reliable spectroscopic abundance measurements. We applied further cuts, based on colors (g − r > 0.2) and magnitudes (r > 14 mag), in order to remove stars with extremely blue colors or saturated brightness measurements. Most of the stars in the original SSPP sample were rejected due to their large u-band errors. Even if a more conservative cut on χ²_min/ν < 10 were used, there would still remain a total of 58, 335 stars in the comparison.

As shown in Figure 5, there is a constant T_eff offset between the photometric and SSPP temperature scales over 5000 K <T_eff < 6500 K, where the maximum deviation is less than 100 K in this temperature range. However, the good agreement between our temperature scale and the IRFM scale (Pinsonneault et al. 2012) ensures that our temperature estimates are more or less closer to the fundamental temperature scale.¹⁴ The random scatter seen in Figure 5 is ∼100 K for individual T_eff estimates.

Figure 6 shows a comparison between the spectroscopic (SSPP) metallicities from SDSS/SEGUE and our photometric estimates for the same set of stars, as in Figure 5. The central line in the top panel shows a median photometric metallicity for each 0.2 dex bin in the spectroscopic metallicity, and the other two lines represent the first and the third quartiles. An extended metal-poor tail at a given SSPP [Fe/H] appears in Figure 6, in particular at low [Fe/H] values. This is because the metallicity sensitivity of stellar colors degrades at lower metal abundances, due to the non-zero photometric errors. In the next section, we model these profiles based on simulations of artificial stars. As will be shown, artificial star tests indicate that the median photometric metallicities (central line in Figure 6) are insensitive to photometric errors and the lower limit in the model grid at [Fe/H] = −3 and exhibit a maximum deviation of only Δ[Fe/H] ∼ 0.1 dex from the true [Fe/H] down to [Fe/H] = −2.8. The overall good agreement between our photometric metallicities and the SSPP values shown in Figure 6, with a maximum deviation of less than 0.2 dex, confirms that our photometric metallicity scale is not far from the spectroscopic scale down to [Fe/H] ∼ −2.5, perhaps as low as ∼ − 3. However, in order to perform a more stringent test of the accuracy of our photometric metallicity estimates, we require more comparison star samples below [Fe/H] ∼ −2.5.

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Because we included u-band measurements, which are more sensitive to metal abundances than the other SDSS bandpasses, the observed dispersion in Figure 6 is smaller than the equivalent Figure 2 in An et al. (2009a), which is based only upon gri. The 1σ random scatter (estimated from the interquartile range divided by 1.349, assuming a normal distribution) is about 0.2–0.4 dex for individual stars. The [Fe/H] estimates in SDSS/SEGUE are precise to ∼0.2 dex for individual stars (Lee et al. 2008a, 2008b; Allende Prieto et al. 2008; Smolinski et al. 2011), so this comparison indicates that the photometric metallicities can be as precise as ∼0.3 dex per star, when good photometry is available. The downturn of the mean trend at [Fe/H] ∼ −0.8 in Figure 6 is likely caused by our ad hoc assumptions on the [α/Fe] and age as a function of [Fe/H] in the model.

An analysis of likely member stars of Galactic open and globular clusters and high-resolution spectra of SDSS/SEGUE stars indicates that the SSPP exhibits a tendency to estimate [Fe/H] higher by about 0.25 dex for stars with [Fe/H] < −3.0, in particular for cool giants. Although the SSPP underestimates metallicities for stars with super solar abundances, the effect is less than 0.1 dex. Nevertheless, it should be kept in mind that the reliability of measuring photometric metallicities eventually requires a test against high-resolution spectroscopy.

In An et al. (2009a), we tested the accuracy of photometric metallicities from the application of gri photometry and found that the photometric technique systematically underestimates [Fe/H]_phot in the low-metallicity range (Δ[Fe/H] ∼ 0.3 dex at [Fe/H] ∼ −1.6. We speculated that the mismatch between the SSPP and photometric metallicities were likely caused by either unresolved binaries in the sample or a small zero-point offset in the model colors. Although the effect of unresolved binaries in the photometric metallicities cannot be completely ignored (see Section 2.5 below), small systematic color differences can actually induce a systematic trend in [Fe/H]_phot, especially when photometric metallicities are derived based on color indices that are weakly dependent on [Fe/H], such as the application of gri photometry in An et al. (2009a). The accuracy of the photometric technique (Figure 6) has improved as more metallicity-sensitive color indices (ugriz) are employed in the calculation of [Fe/H]_phot.

Figure 7 is the same as Figures 5 and 6, but based only on ugr colors. The scatter in the comparison is larger than those seen in Figures 5 and 6, because of the weaker constraints on these parameters. The downturn of the mean trend at [Fe/H] ∼ −0.8 is stronger than seen in Figure 6. Clearly, the inclusion of all ugriz passbands is preferred, in order to strongly constrain both photometric temperatures and metal abundances.

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We performed artificial star tests in order to evaluate the effects of photometric errors and unresolved binaries and/or blends on our determinations of photometric [Fe/H] estimates. In particular, we use these simulation results to construct metallicity kernels, which are employed to assess the broadening of the [Fe/H] profiles resulting from the above effects. These kernels are then used in Section 4 to deconvolve the observed MDF of the halo.

Figure 8 shows the artificial star test results for a single stellar population without binaries. In this exercise, we used the same set of models as in Section 2.2, which include the main sequence and a portion of the subgiant branch. For each of the various metallicity bins, we generated 20,000 artificial stars with masses above 0.65 M_☉ from a model isochrone. The magnitudes were then convolved with a Gaussian having a standard deviation of 0.02 mag in gri, and 0.03 mag in the u and z passbands. We derived photometric metallicities and distance estimates as described in Section 2.3; the resulting MDFs are shown as solid black histograms in the left panels, and the distributions in distance modulus are in the right panels. We used the following selection criteria.

• 1.  
  χ²_min/ν < 3.0, where χ²_min is a minimum χ² defined in Equation (6) and ν (=2) is a degree of freedom.
• 2.  
  log g_(YREC) ⩾ 4.15.
• 3.  
  0.65 < M_*/M_sun < 0.75.

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The last criterion is specified in order to be consistent with the analysis of the actual data sets described in Section 4 and assumes that stars are on the main sequence.

The solid red histograms shown in Figure 8 are those resulting from the simulations, but assuming photometric errors twice the size as above (0.04 mag error in gri, 0.06 mag error in u and z, and assuming no correlation between the errors in different bandpasses), illustrating the effect of the size of photometric errors on the photometric metallicity estimates. For a given photometric error, the resulting dispersion is higher at lower metallicity, due to the weaker dependence of broadband photometric colors on metallicity at lower abundances, e.g., see Figure 1 in An et al. (2009a). At solar abundance, the standard deviation of the [Fe/H] distribution in Figure 8 is 0.1 dex, but increases to 0.3 dex at [Fe/H] = −1.6 and 0.5 dex at [Fe/H] = −2.4. At the lowest [Fe/H], the distribution becomes asymmetric, with an extended low-metallicity tail. On the other hand, the distribution in distance modulus becomes progressively more symmetric at the lower input metallicity, because of the weaker metallicity sensitivity of colors and magnitudes at lower metal abundances. Our model set does not extend below [Fe/H] = −3 or above [Fe/H] = +0.4, which results in a piling up of stars at these metallicities.

Figure 9 shows the effects of unresolved binaries and/or blends for three metallicity bins, [Fe/H]_input = −1.6 (left panels), −2.0 (middle panels), and −2.4 (right panels), respectively. Photometric errors of 0.02 mag in gri and 0.03 mag in u and z passbands were assumed. To reduce the effects of interpolation errors in the isochrones, which sometimes produce a non-continuous distribution in [Fe/H]_phot, we convolved the derived photometric metallicities with Gaussians having the grid size of the model (σ_[Fe/H] = 0.1 dex). The top panels apply to stellar populations comprised entirely of single stars. The middle two panels show the results when unresolved binaries are included in the sample. In total, 10,000 primary stars were generated from a single [Fe/H] population, and the same number of secondary stars was generated, based either on a flat mass function, or adopting the M35 mass function (Barrado y Navascués et al. 2001). These simulated stars were merged with 10,000 single stars with no secondary components to simulate a 50% unresolved binary fraction. The resulting magnitudes in ugriz were convolved with Gaussians having the specified photometric errors.

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As can be appreciated by inspection of Figure 9, the presence of unresolved binaries has little impact on the derived photometric [Fe/H] distribution, because the dispersion is dominated by the effects of photometric errors. Binaries, due to the modified flux distributions due to a secondary component, typically have lower photometric metal abundances than those of the primaries alone, but these effects are mostly buried in the extended [Fe/H] distribution produced by the presence of non-zero photometric errors. The bottom panels in Figure 9 show photometric metallicity distributions of the SDSS/SEGUE spectroscopic sample in Figures 5 and 6 in the same metallicity bins: −1.7 ⩽ [Fe/H]_SSPP < −1.5 (bottom left), −2.1 ⩽ [Fe/H]_SSPP < −1.9 (bottom middle), −2.5 ⩽ [Fe/H]_SSPP < −2.3 (bottom right). Except for a small offset in the metallicity scale between the SSPP and photometric metallicities (median [Fe/H]_phot values are shown in the panels), which has already been pointed out in Section 2.4, the overall [Fe/H]_phot distributions appear quite similar to those from the artificial star tests.

Our strategy in this study is to deconvolve the observed MDF of halo stars using simulated profiles and infer the underlying [Fe/H] distribution (Section 4). In order to carry out the deconvolution in a straightforward manner, we parameterized simulated [Fe/H] kernels by simultaneously fitting two Gaussian functions, as shown by the two green curves in each panel of Figure 9. The red curve is a sum of these two distributions. As discussed above, the precise binary fractions and the exact form of the adopted mass functions are not dominant factors in determining the overall shape of a given MDF. Therefore, we opted to choose the M35 mass function with a 50% binary and/or blending fraction when comparing the observed MDF of the halo with simulated results.

The top panel in Figure 10 displays photometric metallicity kernels for various input metallicity values ([Fe/H]_input), normalized to the peak value of each profile. These simulated profiles were generated from the same set of artificial star tests as above, assuming a M35-type mass function and by taking 0.02 mag error in gri and 0.03 mag error in u and z. The input [Fe/H]_input values are from −1.2 to −2.8 in intervals of 0.1 dex; for clarity, the simulated profiles are alternatively shown in thick and thin curves. Again, note that the resulting photometric distribution becomes broader at lower metallicities.

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The bottom panel in Figure 10 shows the median differences between photometric metallicities and input [Fe/H] values for the same data set used to construct the kernels in the top panel. Longer error bars represent the interquartile ranges, while the shorter error bars represent ±1σ errors, where we computed the 1σ error by dividing the interquartile ranges by 1.349, assuming a normal distribution.

An unbiased sample of stars is, of course, an important ingredient for obtaining a representative MDF of the Milky Way's halo population(s). Although photometric samples are less susceptible to sample biases than spectroscopic studies that make use of metallicity or color in their sample selection, a bias still exists that needs to be taken into account, as discussed below.

In this work we adopted a sample selection based on stellar mass, as estimated using our isochrones. Figure 11 shows our color-calibrated models at [Fe/H] = −2.4, −1.6, −1.2, and −0.8; the thick solid lines indicate where 0.65 < M_*/M_☉ < 0.75. This range of stellar mass is similar to what is adopted in this work (see below). Our choice for the mass-based sample selection was motivated by our reasoning that, in order to obtain a representative sample of the halo, stars at different metallicities should be sampled in identical mass ranges. We consider this to be a superior choice to the more commonly adopted color-based selections, because of the strong relationship between color, metallicity, and mass; narrow color cuts in a stellar sample would produce a mix of stars, including less massive, lower metallicity stars and more massive, higher metallicity stars. Although colors can be used as a surrogate for temperatures, they have only a limited applicability for stellar masses, hence we believe our mass-based selection should produce a less biased sample of stars for the assembly of a valid halo MDF.

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Figure 12 illustrates metallicity–luminosity relations as a function of stellar mass, where the luminosity is expressed in terms of a maximum heliocentric distance that can be reached by a star at a specific magnitude limit. Note that we adopted an age of 13 Gyr for models at −3.0 ⩽ [Fe/H] ⩽ −1.2, and 4 Gyr at −0.3 ⩽ [Fe/H] ⩽ +0.4, with a linear interpolation between these two metallicity ranges (Section 2.1). The u-band magnitude limit was used in Figure 12, because of the strong sensitivity of this band on metal abundances. The u_max = 20.6 (top panel) and u_max = 21.0 (bottom panel) correspond to a median photometry error of σ ≈ 0.03 mag in the calibration and the co-added catalogs, respectively. The error size is similar to those adopted in the artificial star tests (Section 2.5).

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At a given u-band magnitude limit, we computed a maximum distance to which each star can be observed at a given stellar mass and metal abundance. At a fixed mass, metal-poor stars are brighter than metal-rich stars, thus they can be observed at greater distances than metal-rich stars (see Figure 11). Note that a color-based selection would have the opposite consequence on the luminosity of stars that would be included; at a fixed color (or temperature) stars are brighter at higher metallicity. It follows that, in a magnitude-limited survey such as SDSS, color-based selection would produce samples that are biased against more metal-poor stars at greater distances from the Sun.

The gray shaded region in Figure 12 indicates the mass–distance limit set in our halo sample. In both panels, these areas are surrounded by an [Fe/H] = −1.2 model to insure that the sample is unbiased at [Fe/H] ⩽ −1.2. Our photometric halo MDFs should be less affected by this choice, since we generally expect to obtain relatively few halo stars outside this metallicity range. An additional constraint on the stellar mass has been applied to the sample, because main-sequence turnoff masses (the high-mass end points of each curve in Figure 12) are varying at different metallicities. Therefore, we imposed an upper mass limit, M_* < 0.75 M_☉, which is close to the main-sequence turnoff mass from an [Fe/H] = −3 model; turnoff masses occur at higher masses for [Fe/H] > −3. The lower limit to the heliocentric distance (5 kpc) was set to exclude possible thick-disk interlopers in the sample; this naturally results in a lower mass cut-off at ∼0.65 M_☉. We further imposed a cut based on Galactic latitude (see discussion below in Section 4.1). The application of these constraints results in a relatively narrow parameter space (gray areas in Figure 12) in the stellar mass versus distance plane (5 kpc ⩽d_helio ≲ 8 kpc for the calibration catalog; 5 kpc ⩽d_helio ≲ 9 kpc for the co-added catalog). In short, our sample selection (delineated by the gray regions in Figure 12) collects all of the stars at [Fe/H] ≲ −1.2 in a limited volume of the halo, given the magnitude limit set in a photometric catalog (σ_u < 0.03 mag in this work). The resulting distribution of our sample in Galactocentric distances is shown in Figure 13.

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We have carried out comparisons of our samples obtained by both the calibration and the co-added catalogs, using a search radius of 1''. As recommended in Ivezić et al. (2007), we only included sources in the calibration catalog with at least four repeated observations in each bandpass, for both the photometric comparisons and the following analysis, and proceed with the mean magnitudes and their standard errors. After carrying out iterative 3σ rejections, we found zero-point differences of −0.007 ± 0.022 mag, −0.001 ± 0.011 mag, +0.002 ± 0.013 mag, and +0.003 ± 0.015 mag, in u − g, g − r, g − i, and g − z, respectively, for stars with u < 20.6 and fainter than 16th magnitude in each filter bandpass; see also photometric comparisons in Annis et al. (2011). The sense of these differences is that the u − g color measurements in the co-added catalog are redder than those in the calibration catalog. We restricted our comparison to only include those stars with Galactic latitudes at |b| > 35°, to be consistent with the analysis of the halo MDF carried out below. The listed errors indicate the derived dispersions in color difference (rather than the errors in the mean), using ∼100, 000 objects in each comparison. Because fainter u-band measurements yield less strong UV excesses, the use of photometry from the co-added catalog would lead to systematically higher photometric metallicity estimates than in the calibration catalog (this is confirmed in the next section). The rms deviations of the color differences between the two catalogs are stable (σ_color < 0.003 mag) along the 110° length of Stripe 82.

Although a detailed study is beyond the scope of the present work, we identified a strong systematic deviation in the u passband (Δu ∼ 0.2 mag at u ∼ 22 mag) at the faint end, beyond the magnitude limit set in our sample (u < 20.6). One likely cause of the systematic offset is a Malmquist-type bias in the calibration catalog, because faint sources near the detection limit can either be detected or missed, due to the existence of large Poisson errors. Since the calibration catalog took the average of the source magnitudes from individual images, the mean magnitude could therefore be biased toward brighter source measurements. We avoided the photometry zero-point issue by requiring σ_u < 0.03 mag; small photometric errors also make the photometric metallicity estimates more reliable.

We have not made use of the main SDSS photometric database in this work. The 95% completeness limit in the u bandpass of the main survey is 22.0 mag, but the σ_u = 0.03 mag limit corresponds to u ≈ 18.7 mag, on the order of 2 mag shallower than for the Stripe 82 catalog. At this limiting magnitude, use of the main SDSS database would allow exploration of heliocentric distances only up to ∼2.2 kpc for the construction of an unbiased [Fe/H] sample at [Fe/H] < −1.2, which is an insufficient volume to probe the halo MDF.

To summarize, in the following analysis we used the selection criteria listed below.

• 1.  
  Sources are detected in all five bandpasses.
• 2.  
  χ²_min/ν < 3.0, where χ²_min is a minimum χ² defined in Equation (6) and ν (=2) is a degree of freedom.
• 3.  
  log g_(YREC) ⩾ 4.15.
• 4.  
  0.65 M_☉ < M_* < 0.75 M_☉.
• 5.  
  Mass–metallicity–luminosity limits (gray region in Figure 12).
• 6.  
  d_helio > 5 kpc.
• 7.  
  |b| > 35°.

The above selection restricts the photometric sample to 0.2 ≲ g − r ≲ 0.5 (see Figure 11). The total numbers of objects that passed the above selection criteria are 2523 from the calibration catalog and 2626 from the co-added catalog. The median r magnitudes are 18.7 mag and 19.1 mag for the calibration and the co-added catalogs, respectively, with a ∼0.3 mag dispersion. The standard SDSS star–galaxy separation based on the difference between point-spread function (PSF) and model magnitudes is robust to r ∼ 21.5 (Lupton et al. 2002; Scranton et al. 2002), so the contamination by galaxies in our sample should be negligible (see also Annis et al. 2011; Bovy et al. 2012a).

Our calibration is valid for main-sequence stars only, and we have explicitly assumed that the great majority of stars in the sample are in their main-sequence phase of evolution. However, there certainly exist distant background giants and subgiants along each line of sight, and their distances can be greatly underestimated if a photometric parallax relation for main-sequence stars is blindly applied. Unfortunately, ugriz photometry alone is not sufficient to reliably discriminate giants from dwarfs, unlike horizontal-branch stars (Ivezić et al. 2007), and therefore some level of contamination by giants in our sample is unavoidable.

We evaluated the level of expected giant and subgiant contamination using cluster photometry for both M13 ([Fe/H] = −1.6; see Figure 11) and M92 ([Fe/H] = −2.4) (An et al. 2008). In order to carry this out, we applied cuts based on χ and sharp parameters in DAOPHOT (Stetson 1987) as in An et al. (2008). Based on the large extent of these clusters on the SDSS CCD chips, we opted not to apply cuts based on the distance from the cluster centers.

We assumed that the number density of the halo follows a power-law profile with an index of −2.8 and a halo ellipticity of 0.6 (e.g., Jurić et al. 2008). Using this model, along with the photometry for each cluster, we simulated color–magnitude diagrams for each line of sight in Stripe 82, and applied the same mass–metallicity–luminosity cuts to the sample as in the previous section. We integrated the number of giants out to 30 kpc from the Sun, beyond which the giant contamination is negligible, because either those distant giants are too faint to be included in our sample or they do not satisfy our selection criteria. Each object was tagged as either a (sub)giant or a dwarf, depending on the location on the original cluster color–magnitude diagram, and the number of giants included in the sample was counted. From the above calculation we found that the contamination rates are at about the 10%–15% level from the M13 photometry, and the 15%–20% level when using the M92 photometry. M92 is more metal-poor than M13, so more cluster giants fall in the color range (0.2 ≲ g − r ≲ 0.5), which is implicitly set by our sample selection criteria. This results in a slightly higher contamination rate from the M92 photometry.

More than any other parameters (e.g., shape parameters of the halo) in the model, we found that the assumed fraction of giants is the dominating factor that determines the total contamination rate in our sample. Note that our estimate for giant contamination in the sample is higher than what Jurić et al. (2008) estimated (∼15% versus ∼4%). Jurić et al. (2008) obtained an estimate of the fraction of giants to be about 5% from M13, using the SDSS pipeline Photo values in this crowded region, over a color range similar to that in our analysis, and concluded that the bias in the number density of the halo is about 4% arising from the misidentification of giants as main-sequence dwarfs.

Giant contamination in our sample produces an overall shift of the photometric metallicity estimates toward higher values. Figure 14 shows the fiducial sequences on the u − g versus g − r diagram for a number of clusters that were used in our color calibration. The black lines shown are fiducial sequences, with the dotted lines indicating a giant sequence. For M67, only a main sequence was detected in SDSS (giants are too bright to be included; An et al. 2008). The overlaid gray lines in each panel show our calibrated models at [Fe/H] = −3.0 (left most), −2.4, −1.6, −0.8, and 0.0 (right most), respectively. At a given g − r, giants have redder u − g colors than the main sequence, which leads to an overestimated photometric metallicity if they are misidentified as dwarfs. The size of a bias in the photometric metallicity can be as large as 0.5–1 dex, but fortunately this only applies for a limited number of stars along each line of sight.

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On the other hand, contamination by thick-disk stars in our sample is negligible. We performed a set of Galactic simulations to check the overall fraction of thick-disk interlopers in our sample along various lines of sight, using artificial star test results in Section 2.5. To simulate a dispersion in the underlying [Fe/H] distribution, we combined artificial stars at the central [Fe/H] (≈−0.7 for the thick disk and ≈ − 1.6 for the halo) with those at ±1σ values (±0.2 and ±0.4 dex for the thick disk and the halo, respectively), taking normalizations from a Gaussian distribution. We adopted the Galactic structural parameters in Jurić et al. (2008).

The fraction of thick-disk stars, of course, is varying at different Galactic latitudes, so we computed the fraction of thick-disk stars along Stripe 82. We found that the average fraction is negligible (0.4% of the entire sample) below photometric [Fe/H] = −1.0 (see also Bovy et al. 2012b). This is because our sample selection, based on mass, metallicity, and distance, is strongly biased against stars with [Fe/H] > −1.2. If all the stars below solar metallicity are included in this estimate, the fraction becomes ∼3.4%. However, only stars with photometric metallicities less than [Fe/H] = −1 are concerned in the following discussions, and we can safely assume that the thick-disk contamination is negligible.

Figure 15 shows our in situ observed MDFs from the calibration (top panel) and the co-added catalogs (bottom panel), respectively, including only stars satisfying the selection criteria described above. Jurić et al. (2008) found that both thick-disk and halo stars have approximately the same number density at 3 kpc above the Galactic plane. In order to minimize the contribution from thick-disk interlopers in our halo sample, we imposed a heliocentric distance limit of greater than 5 kpc and Galactic latitudes |b| > 35°, which correspond to a minimum vertical distance |Z| = 2.9 kpc from the Galactic plane at the low-latitude limit. As noted above, we imposed a sample cut at ${\rm [Fe/H]_{\rm true}} = -1.2$, where ${\rm [Fe/H]_{\rm true}}$ indicates a true metallicity value. The great majority of thick-disk stars having metallicities above this limit would have been excluded from our photometric sample selection. Nevertheless, it remains possible that some metal-weak thick-disk stars (Chiba & Beers 2000; Carollo et al. 2010, and references therein) have entered our sample, at least near the low-latitude limit. Note that some stars near ${\rm [Fe/H]_{\rm true}} = -1.2$ could have photometric metallicities greater than ${\rm [Fe/H]_{\rm phot}} = -1.2$, because of non-zero photometric errors.

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The thick solid histograms in Figure 15 show our derived photometric MDFs, constructed using the YREC-based extinction coefficients in Section 2.3; the dotted lines represent the case with 10% smaller extinction values than in Schlegel et al. (1998). The thin solid histograms are those based on extinction coefficients from Schlafly & Finkbeiner (2011). Although their extinction coefficients are ∼20% smaller than our default YREC-based values, the difference has only a moderate effect on the resulting MDFs; the overall shape of the MDFs does not change significantly, because of the small level of foreground dust toward these high Galactic latitude stars.

The photometric MDFs are insensitive to our adopted [α/Fe] and age scheme (Section 2.1). Even if we had taken a different [α/Fe] or age relation as a function of [Fe/H], model colors would still be forced to match the same set of cluster data from our color calibration procedure (Section 2.2). The resulting color–[Fe/H] relation would then be essentially unaffected by this change. Although intrinsically broad distributions in [α/Fe] and/or age at each [Fe/H] could result in additional spreading of the observed MDFs, we neglect their effects in the following discussion.

Figure 16 compares our photometric MDFs with the spectroscopic MDF obtained by Ryan & Norris (1991b, gray shaded), which is based on a kinematically selected sample of 372 local halo subdwarfs (d_helio ≲ 400 pc). We have multiplied their sample by a factor of eight to directly compare the shapes of MDFs. The mean [Fe/H] value in the Ryan & Norris (1991b) sample is −1.80 over −3 < [Fe/H] < −1, with a standard deviation of 0.46 dex. Note that, according to these authors, their sample is incomplete above [Fe/H] = −1.

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The abundance measurements in Ryan & Norris (1991b) are accurate to σ ∼ 0.2 dex, while our photometric metallicity estimates have larger dispersions and asymmetric error distributions. In order to perform an "apples-to-apples" comparison with our photometric MDFs, we convolved the MDF of Ryan & Norris (1991b) with the convolution kernels in Figure 10. We binned their data at [Fe/H] < −1 with intervals of Δ[Fe/H] = 0.1 dex, and applied the convolution kernels to each bin, ignoring spectroscopic [Fe/H] errors in Ryan & Norris. The resulting MDF is shown (with an arbitrary normalization) as a dotted line in Figure 16, which includes the effects of both photometric errors and unresolved binaries and/or blending, as in our photometric MDFs. The Ryan & Norris (1991b) MDFs are similar in their overall shape, both before and after the convolution is applied, and both match our in situ MDFs rather well.

As shown in Figure 10, there is a systematic shift in the metallicity scale in the co-added catalog. A Kolmogorov–Smirnov (K-S) test rejects the null hypothesis that the MDF of Ryan & Norris (1991b) and that of the co-added catalog (bottom panel) are drawn from the same parent population at significant levels (p < 0.001). On the other hand, the null hypothesis that the Ryan & Norris MDF and that of the calibration catalog (top panel) are drawn from the same parent population cannot be rejected (p = 0.364); in other words, their MDFs are statistically similar with each other. Photometric zero-point shifts may be responsible for the systematically higher photometric [Fe/H] estimates in the co-added catalog (Section 3.1 and see below in Section 4.2.2).

Perhaps the overall good agreement between our photometric MDFs and the spectroscopic Ryan & Norris (1991b) MDFs should not be so surprising, given the similar range of Galactocentric distances of these two samples. The Ryan & Norris (1991b) stars occupy a local volume (although their orbits extend to much larger distances), at roughly 8 kpc away from the Galactic center. Our sample stars in Stripe 82 are found at present distances within 7–14 kpc from the Galactic center, as shown in Figure 13, although they are much farther away from the Galactic disk than the Ryan & Norris (1991b) sample stars. It is worth keeping in mind that, although we do not presently have full space motions for the stars in our photometric MDFs, they surely contain many stars with orbits that take them far outside the local volume. Both samples of stars can be thought of as local "snapshots" of a volume in the relatively nearby halo.

We now describe our attempts to recover the "true" halo MDF, using experiments that consider different ways for understanding the nature of the underlying [Fe/H] distributions. This requires confronting the possibility that more than one parent stellar population may be required, as discussed below.

We began by deconvolving the observed MDFs of the halo using a single [Fe/H]_phot profile (one of the kernels in Figure 10), where a single [Fe/H]_true is assumed, without an intrinsic dispersion. We found that none of these profiles are able to match the overall shape of the MDFs, immediately suggesting that either the hypothesis of a small dispersion in [Fe/H]_true should be abandoned, and/or at least two sub-components are required to properly account for the overall shape of the MDF. We consider each of these two cases in turn below.

4.2.1. A One-component Model

Figure 17 shows the fitting results for the observed photometric MDFs, assuming that the metallicity of the underlying halo population can be well described by a single Gaussian distribution with a dispersion in metallicity. The solid red line in Figure 17 shows the resulting fit, obtained using the nonlinear least-squares fitting routine MPFIT (Markwardt 2009) over $-2.8 < {\rm [Fe/H]_{\rm phot}} < -1.0$, after application of the convolution kernels (Figure 10). As described earlier, each of these simulated profiles includes the effects of photometric errors (σ_{g, r, i} = 0.02 mag, σ_{u, z} = 0.03 mag) and a 50% unresolved binary fraction and/or blends with the M35 mass function for secondaries. Note that these kernels are not Gaussian functions, nor are they the products of a chemical evolution model.

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For the calibration catalog (top panel), we found that a best-fitting Gaussian has a peak at [Fe/H]_true = −1.80 ± 0.02 with σ_[Fe/H] = 0.41 ± 0.03 dex. The reduced χ² value (χ²_ν) of the fit is 2.0 for 15 degrees of freedom, assuming Poisson errors. The errors in the above parameters were scaled based on the χ²_ν of the fit. The estimated peak [Fe/H] and the dispersion are similar to those obtained for the Ryan & Norris (1991b) sample, as expected from the similar MDF shape from these two studies. For the co-added catalog (bottom panel), we found a best-fitting Gaussian with a peak at [Fe/H]_true = −1.55 ± 0.03 and σ_[Fe/H] = 0.43 ± 0.04 dex, with χ²_ν = 2.7.

For an additional test, we assumed that the photometric MDF is shaped primarily by large photometric errors, even though the underlying [Fe/H] distribution is single-peaked at around [Fe/H] = −1.6. We found marginal agreement with the observed MDFs only if the size of photometric errors were underestimated by a factor of two (i.e., σ_{g, r, i} ≈ 0.04 mag, σ_{u, z} ≈ 0.06 mag) for both of the Stripe 82 catalogs. However, we consider it unlikely that the errors have been underestimated by this much. This hypothesis is also inconsistent with the intrinsically wide range of spectroscopic [Fe/H] determinations from Ryan & Norris (1991b).

The dashed blue line in Figure 17 shows the best-fitting simple chemical evolution model (Hartwick 1976). We used the mass-loss modified version as in Ryan & Norris (1991b), who found an excellent match of this model to their MDF. The model MDF is characterized by a single parameter, the effective yield (y_eff), which relates to the mass of ejected metals relative to the mass locked in stars. The simple mass-loss modified model adopts instantaneous recycling and mixing of metal products in a leaky box, and further assumes a zero initial metallicity, constant initial mass function, and a fixed effective yield. We utilized [Fe/H] in this model as a surrogate for the metallicity. After convolving the model MDF with the deconvolution kernels (Figure 10), we found log₁₀y_eff = −1.65 ± 0.02 (χ²_ν = 2.6 for 15 degrees of freedom) for the calibration catalog, and log₁₀y_eff = −1.37 ± 0.04 (χ²_ν = 5.0) for the co-added catalog, which simply correspond to the peak of the MDF. The goodness of the fit to the calibration catalog data is comparable to that using the single Gaussian fit, and the resulting effective yield is close to what Ryan & Norris (1991b) obtained from their MDF (log₁₀y_eff = −1.6).

4.2.2. A Two-component Model

Although a single-peak Gaussian [Fe/H] distribution describes the observed photometric MDF rather well, it is not a unique solution. Here we consider a two-peak Gaussian [Fe/H] distribution fit to the Stripe 82 MDFs.

The red and blue curves in the top panel of Figure 18 show the best-matching pair of simulated profiles to the calibration catalog MDF, searched using the MPFIT routine over $-2.8 < {\rm [Fe/H]_{\rm phot}} < -1.0$. In this fitting exercise, we fixed the dispersion of the underlying [Fe/H] distributions to σ_[Fe/H] = 0.30 dex for both Gaussians. The green curve is the sum of these individual components, which exhibits an excellent fit to the observed profile (χ²_ν = 1.9 for 14 degrees of freedom). The underlying true [Fe/H] distribution for each of these curves is shown in the bottom panel, which exhibits peaks at [Fe/H]_true = −1.67 ± 0.08 and −2.33 ± 0.30, respectively.

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The bottom panel in Figure 18 also shows the spectroscopic MDF from Ryan & Norris (1991b, gray histogram). Our deconvolved [Fe/H] distribution matches their observed MDF well on the metal-poor side. Their sample is contaminated by disk stars above [Fe/H] ≈ −1, while our photometry sample selection excludes metal-rich stars with [Fe/H]_true > −1.2 (Section 3.1).

The fractional contribution of the low-metallicity component with a peak at [Fe/H]_true = −2.33 (the area under the blue curve in the top panel of Figure 18) to the entire halo sample (the area under the green curve in the top panel) is 24%, with a clear strong dependence on metallicity. Below [Fe/H] = −2.0, the contribution from the low-metallicity component is 52% of the total numbers of halo stars in this metallicity regime. Above [Fe/H] = −1.5, the high-metallicity component (the red curve with a peak at [Fe/H]_true = −1.67) contributes 96% of the total numbers of halo stars.

The relative fraction of the low-metallicity component depends on our adopted value for the dispersion of the underlying [Fe/H] distribution. In the above exercise, we assumed σ_[Fe/H] = 0.30 dex for each component. However, the contribution from the low [Fe/H] component becomes as high as 42% if the standard deviation is set to σ_[Fe/H] = 0.20 dex. On the other hand, fixing the value to σ_[Fe/H] = 0.4 dex leads to a similar result as in the previous section, which requires only one component in the fit. If σ_[Fe/H] = 0.2 dex is a reasonable lower limit to the dispersion of each of the underlying [Fe/H] components, we can only set an upper limit on the fractional contribution of the low-metallicity component, at ∼40%–50%.

To investigate this further, we have also performed two-component deconvolutions leaving each component's dispersion as a free parameter. For this, we use the extreme deconvolution (Bovy et al. 2011) technique, which fits all parameters of arbitrary numbers of Gaussian components to data with individual uncertainties, and which has been extended to handle non-Gaussian uncertainty kernels, such as those in Section 2.5. The best two-component fit (with Δχ² = 6.5 compared to the fixed σ_[Fe/H] = 0.3 dex fit above) consists of a 69% contribution from a low-metallicity component ([Fe/H] = −2) with σ_[Fe/H] = 0.35 dex and a higher-metallicity component ([Fe/H] = −1.4). However, the latter has an unrealistically small dispersion of only σ_[Fe/H] = 0.07 dex. Fitting instead with a weak prior on the dispersion (a χ² distribution with 2 degrees of freedom and a mean of 0.3 dex) yields a slightly worse fit (Δχ² = 2 with respect to the best fit), with almost equal contributions from a low-metallicity (53%, mean [Fe/H] = −2.14, σ_[Fe/H] = 0.3 dex) and a higher-metallicity (mean [Fe/H] = −1.5, σ_[Fe/H] = 0.2 dex) component. These results further show that the parameters of a two-component fit are only poorly constrained on the basis of the photometric-metallicity data alone.

An even stronger constraint on the fraction of the low-metallicity halo component can be obtained if our deconvolved MDF is forced to match the shape of the Ryan & Norris (1991b) MDF. As shown in the bottom panel of Figure 18, our assumed [Fe/H] dispersion of 0.3 dex results in a good agreement of our MDF with that of the Ryan & Norris sample (gray shaded). However, the agreement breaks down when a smaller (or larger) value is assumed for a dispersion of the individual underlying [Fe/H] distributions; a smaller dispersion leads to an enhanced contribution from the metal-poor component, resulting in an overestimated number of metal-poor stars with respect to the Ryan & Norris MDF. From the above considerations, we find that individual components of the underlying MDF with ∼0.25–0.32 dex dispersion in [Fe/H] provide a reasonably good fit to the Ryan & Norris (1991b) MDF; in other words, the Ryan & Norris MDF is also well fit by double Gaussians with these dispersion values. The fractions of the low-metallicity component of the halo corresponding to these values are ∼20%–35% of the entire halo population.

In the case of the co-added catalog (Figure 19), the overall metallicities are shifted toward higher [Fe/H] values; peak [Fe/H]_true values of the two components are [Fe/H] = −1.39 ± 0.12 and −1.99 ± 0.23, respectively, with a total χ²_ν value of 2.6. The shift in [Fe/H] is expected from the photometric zero-point difference between the two Stripe 82 catalogs (Section 3.1), where the larger (fainter) u-band magnitudes in the co-added catalog produce higher photometric metallicities than in the calibration catalog. The contribution of the metal-poor component is comparable (30%) to that derived from the calibration catalog (24%).

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A strong disagreement with the Ryan & Norris (1991b) MDF is evident in the lower panel, where the deconvolved MDF shows systematically higher metallicities. If we expect both MDFs to be the same at similar Galactocentric distances, this would indicate that the MDF in Figure 19 has been shifted toward higher [Fe/H] due to photometric zero-point errors in the co-added catalog (Section 3.1). Some of the stars in Stripe 82 have spectroscopic estimates from the SSPP and were included in the comparison of the spectroscopic metallicities with the photometric determinations (Section 2.4). However, there were not a sufficient number of stars satisfying both the spectroscopic (Section 2.4) and photometric (Section 3.1) selection criteria for a meaningful comparison to verify which photometric catalog provides a more consistent result with the SSPP estimates.

Our results illustrate a simple limiting case, where one could imagine that the stellar halo formed out of individual subhalos which, in the mean, were pre-enriched in metals that peak roughly at the observed locations. Nevertheless, the fact that the observed photometric [Fe/H] distribution is well described by primarily two metallicity peaks at [Fe/H] ∼ −1.7 and −2.3 is consistent with the argument by Carollo et al. (2007, 2010) that our Milky Way stellar halo is a superposition of at least two components, the inner and outer halos, that are distinct in metallicity, kinematics, and spatial distributions. The apparent duality of the Galactic halo system is further supported by a limited kinematic analysis of stars in our photometric sample, as described below (Section 4.3). It should be kept in mind that our description of these two populations, following Carollo et al., assumes that they are "smooth" distributions, even though it is recognized that there exist numerous, possibly even locally dominant, examples of substructure (such as stellar streams and overdensities) being present as well.

From the above tests, we conclude that we do not have sufficiently strong constraints, from the observed photometric MDFs alone, in order to discriminate between the hypothesis of the duality of the halo (Figures 18 and 19) and the single stellar population (Figure 17) model. We also cannot preclude the possibility that the halo comprises more than two sub-components, because of the limited number of constraints presented by the observed MDFs to perform multi-component fitting. However, we show below that the degeneracy in this solution can be at least partially removed by combining independent information from the available kinematics of our sample stars, as well as for those in the Ryan & Norris (1991b) sample.

Most of the stars in our Stripe 82 photometric sample do not have full velocity information available. However, at high Galactic latitudes, proper-motion measurements alone can provide useful constraints on the U (toward the Galactic center) and V (in the direction of Galactic disk rotation) velocity components, which are sufficient to constrain the motions of stars parallel to the Galactic plane.

We searched for proper-motion measurements in SDSS (Munn et al. 2004), with the proper-motion flags set to the recommended values: ${\tt match} = 1$, ${\tt nFit} = 6$ and ${\tt dist22} > 7$, ${\tt sigRa} < 525$, and ${\tt sigDec} < 525$ (taking into account the correction described by Munn et al. 2008). We then combined proper-motion measurements with our distance estimates to compute approximate rotation velocities¹⁶ (v_ϕ) in the Galactocentric cylindrical system (see Bond et al. 2010).

Figure 20 shows the approximate velocity distribution in v_ϕ for the calibration and co-added catalogs, respectively. The top panels show the full velocity distribution of all stars in each catalog, while the middle and bottom panels show subsets of these in the metal-rich (−1.5 < [Fe/H] ⩽ −1.0; middle panel) and in the metal-poor (−3.0 < [Fe/H] ⩽ −2.0; bottom panel) regimes, respectively. Inspection of the distributions in these two metallicity ranges indicates that they are not identical, as might be expected for a single halo population. The thick black histograms show the v_ϕ distributions of stars at |b| > 45°; red histograms show the more restricted case with |b| > 60° (the restriction to higher latitude provides a better approximation for the rotational velocity). Comparisons between the black and red histograms show that the latitudinal restrictions only mildly change the overall distributions in all metallicity bins. For the calibration catalog, a K-S test rejects the null hypothesis that the samples in the two different metallicity ranges are drawn from the same parent population at significant levels (p = 0.019 for |b| > 45°; p = 0.003 for |b| > 60°). For the co-added catalog, the K-S test is unable to reject this hypothesis (p = 0.167 for |b| > 45°; p = 0.297 for |b| > 60°). Although not a strong effect, there exist greater fractions of more metal-poor halo stars in retrograde, rather than prograde, rotation for both catalogs.

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A more striking difference is obtained when the MDFs are considered for different v_ϕ ranges, as shown in Figure 21. As before, the black and red histograms indicate the two samples at different Galactic latitude ranges. The middle panels show the MDFs for stars with high prograde rotation (v_ϕ > 80 km s⁻¹); bottom panels show those of stars with extreme retrograde rotation (v_ϕ < −80 km s⁻¹). Note that the ±80 km s⁻¹ limit was set in order to maximize the numbers of stars in each split, so as to enable statistically meaningful interpretations.

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Inspection of Figure 21 indicates that the MDFs for stars in retrograde motion are shifted toward lower metallicities than the MDFs for those in prograde rotation. For the calibration catalog, whether the samples are restricted to |b| > 45° or |b| > 60°, a K-S test rejects the null hypothesis that the two samples for different ranges of v_ϕ are drawn from the same parent population at highly significant levels (p = 0.0001 for |b| > 45°; p = 0.0002 for |b| > 60°). For the co-added catalog, the lower-latitude cut results in a significant rejection (p = 0.026 for |b| > 45°); no rejection can be made for |b| > 60° (p = 0.257). We conclude that greater fractions of low-metallicity halo stars exist in retrograde, rather than prograde, rotation.

Inspired by these results, we decided to check how the MDFs of the spectroscopic Ryan & Norris (1991b) sample vary for prograde versus retrograde rotation, and how the V_ϕ distributions vary for higher- and lower-metallicity divisions. Note that we distinguish V_ϕ from v_ϕ to indicate the fact that the rotation component for the Ryan & Norris sample is obtained from full space motions (Ryan & Norris 1991a). Figure 22 shows the [Fe/H] distributions (left panels) and V_ϕ distributions (right panels) of their local kinematically selected sample, with the same limits as on v_ϕ and [Fe/H] in Figures 20 and 21, except in the middle left panel, where we set V_ϕ > 50 km s⁻¹ to include a sufficient number of stars. Clear differences exist in these distributions. A K-S test indicates that the null hypothesis that the samples are drawn from the same parent population is rejected at significant levels: p = 0.019 for the different ranges in V_ϕ and p = 0.021 for the different ranges in [Fe/H]. Both divisions indicate that greater fractions of metal-poor halo stars are found in retrograde, rather than prograde, rotation.

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Table 3 summarizes the average [Fe/H] values (and errors in the mean) for the kinematically divided samples considered above. For the calibration and coadded catalogs, the mean [Fe/H] is between −1.7 and −1.5 for stars in highly prograde motion (v_ϕ > 80 km s⁻¹). For the calibration catalog, the mean [Fe/H] for the highly retrograde components (v_ϕ < −80 km s⁻¹) is ∼0.3–0.4 dex lower than for the metal-rich counterparts; the difference is ∼0.2 dex for the co-added catalog. The difference in the mean [Fe/H] between the highly retrograde component and the highly prograde component in the Ryan & Norris (1991b) sample is 0.3 dex. Taken as a whole, these simple kinematic divisions indicate that a single stellar population is inadequate to describe the behavior of the higher and lower-metallicity halo stars for all three samples.

We caution that the observed kinematic–metallicity correlation can also be produced by (sub)giant contamination in our sample (e.g., Ryan 1989). Unrecognized giants and subgiants are brighter than dwarfs, so their distances and velocities will be underestimated, while their photometric metallicities are higher than those for dwarfs (Figure 14). We used the photometry of M13 (An et al. 2008) to perform the same suite of simple Galactic halo models as in Section 3.2, and found that (sub)giants have systematically higher photometric metallicities, and higher v_ϕ, than dwarfs. The overall contamination rate of (sub)giants in the sample is estimated around 15%, but the fraction increases to ∼40% when high-metallicity stars (−1.5 < [Fe/H] < −1.0) are considered. However, it is less likely that the observed kinematic–metallicity correlation is caused entirely by the (sub)giant contamination, because the Ryan & Norris sample, which is independent from our photometric sample and has different selection criteria, also exhibits the same behavior (Figure 22). Further investigation is needed on the effect of (sub)giant contamination.