A Blueprint for the Milky Way's Stellar Populations. II. Improved Isochrone Calibration in the SDSS and Pan-STARRS Photometric Systems

Color–magnitude relations of stars are basic tools for studying the fundamental properties of stellar systems. Nevertheless, theoretical approaches exhibit limitations in matching model predictions to observed sequences of well-studied star clusters. This mismatch largely arises from an inaccurate conversion from theoretically predicted effective temperatures (T_eff) to observable colors, which reflects the complex nature and interplay of stellar interior and atmosphere models (e.g., Pinsonneault et al. 2003). To overcome this situation, semiempirical modeling has been suggested as a practical means for correcting theoretically predicted quantities (e.g., Lejeune et al. 1997, 1998; Westera et al. 2002; VandenBerg & Clem 2003; Pinsonneault et al. 2004; An et al. 2009b, 2013, 2015b). Stellar isochrones calibrated in his way have been successfully used for deriving accurate distances and metallicities of individual stars and star clusters (e.g., An et al. 2007, 2009a, 2013, 2015a; An 2019; An & Beers 2020).

Ideally, samples for the calibration of stellar models should comprise a single stellar population with well-defined physical parameters such as distance and chemical compositions. For metal-poor stars, the traditional approach is to employ subdwarf stars in the solar neighborhood with accurate parallaxes (e.g., Hipparcos; ESA 1997), but this approach is limited by the small size of the calibration sample. Calibration based on star clusters is also useful, because they provide well-defined sequences for a wide range of stellar masses and luminosity. Some of them also place useful constraints on the physical parameters, although such clusters with good distance measurements are limited to relatively few, nearby systems.

An et al. (2009b) provided an empirically calibrated set of isochrones in the ugriz photometric system (Fukugita et al. 1996) of the Sloan Digital Sky Survey (SDSS; Abolfathi et al. 2018), based on observations of Galactic star clusters. However, there remain some limitations in this study. First, the original cluster photometry in An et al. (2008), which is employed in the above study, is limited to ∼2%–3% errors, because putting the crowded-field photometry onto the global scale of the SDSS system can inevitably introduce systematic zero-point errors. In an effort to reduce the size of these errors, An et al. (2013) updated models by matching the An et al. (2008) photometry with the uber-calibrated SDSS photometry (Padmanabhan et al. 2008) in the cluster's (low-density) flanking fields. Nevertheless, the exact size of the systematic error in the cluster photometry is still poorly constrained.

Second, the SDSS imaging data do not cover faint main-sequence (MS) stars in the bright cluster samples, due to the relatively bright completeness limit in the survey—the 95% completeness limits for point sources are u = 22.0, g = 22.2, r = 22.2, i = 21.3, and z = 20.5. Useful photometry is limited to those of brighter stars, resulting in an effective absolute magnitude limit of M_r ∼ 7 for the calibration in g − r and g − i. For metal-poor halo stars, this limiting magnitude corresponds to masses of ∼0.6 M_⊙. When the color indices include the u or z passbands, the magnitude limit for the model calibration becomes brighter (M_r ∼ 6 or 0.7 M_⊙). This inhomogeneous coverage in stellar mass inevitably limits the application of the isochrones, which in many cases requires multiband photometry to constrain stellar parameters.

Third, the cluster-based calibration relies on subdwarf-fitting distances to globular clusters based on Hipparcos parallaxes of nearby subdwarfs (e.g., Reid 1997; Carretta et al. 2000; Kraft & Ivans 2003). Such distance determinations typically have ∼5%–7% errors. If the luminosity of a star is taken as an independent variable to compute color differences between observation and the model (Pinsonneault et al. 2004; An et al. 2009b), the current uncertainty in the cluster distances translates into a moderate systematic offset in colors, on the order of 0.02–0.04 mag.

Lastly, the globular cluster samples with both good cluster parameters (distance, reddening, and metallicity [Fe/H]³ ) and accurate crowded-field photometry are limited to those with [Fe/H] ≲ −1.2, while the open cluster samples have [Fe/H] ≥ 0. There are metal-rich globular clusters that might be considered, such as 47 Tuc or M71 ([Fe/H] ∼ −0.7), but they are either too distant or attenuated by a large amount of foreground dust. None of the other clusters within the metallicity gap have well-constrained parameters and/or accurate ugriz photometry.

To overcome the above limitations, we turn our attention to a dynamical family of stars in the Milky Way as a calibration sample. In particular, we focus on the double sequences in a color–magnitude diagram (CMD) of stars with large transverse motions, discovered from the analysis of Gaia DR2 (Gaia Collaboration et al. 2018a, 2018b). These sequences are ascribed to metal-poor halo stars and thick-disk (TD) stars with halo-like kinematics in the Galaxy, respectively. More clear identification of stars belonging to each sequence can be seen in An & Beers (2020, hereafter Paper I). According to the chemo-dynamical mapping in Paper I, the blue sequence is composed of stars with 〈[Fe/H]〉 ∼ −1.3, and is likely a collection of stars from the debris of Gaia–Sausage–Enceladus (GSE; Belokurov et al. 2018; Helmi et al. 2018) and, to a lesser extent, the so-called inner halo (IH; Carollo et al. 2007, 2010; Beers et al. 2012). On the other hand, the red sequence has 〈[Fe/H]〉 ∼ −0.4, and belongs to the TD and the Splashed Disk (SD; Bonaca et al. 2017; Belokurov et al. 2020).

The use of Gaia's double sequences can help avoid problems of imprecise distance estimates for the Galactic clusters and the global zero-point variations in the survey data. Stars that constitute Gaia's sequences are relatively nearby, and therefore have good proper motion and parallax measurements, in addition to low foreground reddening. Because of their proximity to the Sun, the observed MS extends far below the faint limit set by the former calibrating samples of Galactic globular clusters, and therefore covers a wider range of stellar mass. Since these stars are spread over the celestial sphere, the observed sequences represent the average color–magnitude relation of each dynamical family, and are less affected by local photometric zero-point errors. In addition, the red sequence has an intermediate metallicity between the globular and open clusters, so it can fill in the missing metallicity baseline in the model calibration.

In this paper, we employ Gaia's double sequences to improve our previous calibration of isochrone colors. With the same methodology, we also construct an empirically calibrated set of models in the Pan-STARRS 1 (PS1) photometric system (Tonry et al. 2012; Chambers et al. 2016). This paper is organized as follows. In Section 2, we describe how we extract Gaia's double sequences from stars with good astrometric quantities in Gaia DR2. A summary on the model construction is provided in Section 3, followed by a description of the procedures used for the revised empirical corrections in Section 4. Updated "blueprint" maps showing the metallicity versus rotational-velocity distribution of stars based on these new models and PS1 photometry are presented in Section 5.

Gaia's double sequences are two distinct fiducial lines observed from stars with high transverse motions. In the original work by the Gaia team, the double sequences are presented in its native photometric filter system, B_G and R_G (Gaia Collaboration et al. 2018a). To utilize these in the isochrone calibration, we reproduce the double sequences in the SDSS (Abolfathi et al. 2018) and PS1 (Chambers et al. 2016) photometric systems by cross-matching photometric catalogs with Gaia DR2.

To refine the sample of stars with high transverse motions, a rotational velocity (v_ϕ) component in the Galactocentric cylindrical coordinate system is employed, instead of a transverse velocity on the celestial sphere, as used by the Gaia team. This is physically motivated by our previous work in Paper I, in which individual stellar components are clearly distinguished from each other in the [Fe/H] versus v_ϕ plane. This leads to a more precise isolation of stars that are associated with each of Gaia's double sequences. In the absence of radial-velocity measurements for a large number of stars, we compute v_ϕ along the Galactic prime meridian (l = 0° or 180°), where the v_ϕ vector essentially depends only on the proper motions and distances (see Paper I for more information on the v_ϕ estimation procedure).

For the calibration sample, only stars with good parallaxes (σ_π/π < 0.2) and proper motions $[{({\mu }_{\alpha }/{\sigma }_{{\mu }_{\alpha }})}^{2}+{({\mu }_{\delta }/{\sigma }_{{\mu }_{\delta }})}^{2}\,\gt {(1.0/0.3)}^{2}]$ are used; as a result, more than 90% of stars are found within ∣Z∣ < 2.3 kpc. Having good u-band photometry (u < 20.5) is another important constraint for obtaining accurate fiducial sequences in u − g. The foreground extinction estimates in Schlegel et al. (1998) are adopted, along with the extinction coefficients of Schlafly & Finkbeiner (2011) in the above filter sets. Since extinction values in Schlegel et al. (1998) represent integrated dust absorption along each line of sight, the sample is limited to ∣b∣ > 20° to minimize the impact of uncertainties in the extinction correction. About 90% of stars in the calibration sample are located above 500 pc, which is approximately five times the scale height of the Galactic dust layer (Drimmel & Spergel 2001; Li et al. 2018). A small fraction of stars with E(B − V) > 0.1 are further rejected from the sample. The E(B − V) distribution of the remaining stars is peaked at 0.02 mag, with a median value of 0.03 mag.

The left panel of Figure 1 shows the u − g CMD for stars with −150 < v_ϕ < 150 km s⁻¹, along the Galactic prime meridian within a region of ± 30°. The computed v_ϕ is corrected for a small inclination with respect to the prime meridian (sometimes denoted as the "projected" v_ϕ). The CMDs for the same stars in other color indices (g − r, g − i, and g − z) are displayed in Figure 2. As shown in Figures 1 and 2, the double sequences are most clearly separated in the u − g CMD. The large separation between the blue and the red MS (hereafter bMS and rMS, respectively) is because u − g has the strongest sensitivity to metallicity, and the metallicity difference between the two populations constituting the double sequences is sufficiently large to exhibit a clear division.

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The blue and red lines in Figures 1 and 2 indicate fiducial lines for the bMS and rMS, respectively, which trace the median colors of stars in bins of M_r. The middle panel in Figure 1 includes stars with retrograde motions (−150 < v_ϕ < −50 km s⁻¹), a subset of the stars shown in the left panel, and the right panel shows stars with prograde motions (120 < v_ϕ < 150 km s⁻¹). The exact ranges of v_ϕ are adjusted until the sequences are most clearly separated from each other, while keeping large numbers of stars in each v_ϕ bin. To improve the isolation of stars belonging to each MS, the bMS is constructed after rejecting outliers as those having redder colors than the rMS by Δ(u − g) = 0.4 mag (middle panel). Similarly, the rMS is obtained after rejecting stars bluer than the bMS by Δ(u − g) = 0.4 mag (right panel). This process requires a few iterations; the same groups of stars are used in the derivation of fiducial lines in the other color indices.

The rMS covers a wide range of M_r, from the MS turn-off to M_r ∼ 11, which is significantly deeper than the limit set in the best available case of M67. The bMS is shorter than the rMS, due to the smaller number of metal-poor stars in the survey volume. Nevertheless, it is still ∼3 mag deeper in M_r than for the bright globular cluster samples in the SDSS photometric catalog (An et al. 2008).

Figure 3 shows a comparison between the bMS and the rMS with fiducial lines obtained using other v_ϕ cuts. The fiducial line for stars with −300 < v_ϕ < −150 km s⁻¹ is similar to the bMS, indicating a similar mean metallicity of these groups of stars. Our previous result in Paper I indicates that the bMS is mainly composed of stars in GSE and, to a lesser extent, the IH. The contribution of stars from the outer halo (OH; Carollo et al. 2007, 2010) is negligible in local volumes (∣Z∣ < 2 kpc), and therefore the inclusion of stars with higher retrograde velocities makes little change in the derived fiducial line. Regarding metal-rich populations, the rMS includes mostly stars belonging to the TD and the SD. For stars with larger v_ϕ than 150 km s⁻¹, their fiducial lines become slightly redder, and the MSTO becomes bluer (younger) than the rMS. The metallicity difference between the two populations is approximately 1 dex (see below), which helps to bridge the gap between the metal-rich and the metal-poor systems in the model calibration.

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The methodology described above is also applied to the PS1 system (except in the y-band). Only the primary detections in the stacked imaging catalog are taken, with a sample cut on i-band photometry from a point-spread function being less than the i-band Kron magnitude plus 0.05 mag to select point-like sources. Because PS1 does not include u-band observations, its griz measurements are cross-matched with the SDSS u-band data using a 1'' search radius and the same magnitude cut in the u-band as above. Figure 4 shows the bMS and the rMS overlaid on CMDs from PS1 (along with the SDSS u-band) constructed in this way.

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The bMS and rMS in both systems look alike, because of their similar filter-response functions. Nonetheless, they are not identical, and therefore can serve as accurate "fiducial" sequences in the native filter systems. The fiducial lines are tabulated in Tables 1 and 2 for the SDSS and PS1 systems, respectively.

The same suite of underlying stellar isochrones as in our previous work (An et al. 2009b, 2013) is adopted, which is based on the YREC evolutionary models (Sills et al. 2000) and MARCS atmosphere models (Gustafsson et al. 2008). YREC models are generated over a wide range of stellar ages, but the base set consists of 13 Gyr old models below [Fe/H] = −1.2, 4 Gyr old models above [Fe/H] = −0.3, and models at linearly interpolated ages between the two metallicity ranges. The T_eff and luminosity predicted by the YREC models are converted into synthetic colors and magnitudes using MARCS atmospheric models. We assume a one-to-one relation between [Fe/H] and α-element abundance ([α/Fe]) in these models, motivated by spectroscopic observations of stars in the Milky Way (e.g., Venn et al. 2004; Hayden et al. 2015): [α/Fe] = +0.4 at [Fe/H] = −3.0, [α/Fe] = +0.3 at −2.0 ≤ [Fe/H] ≤ −1.0, [α/Fe] = +0.25 at [Fe/H] = −0.75, [α/Fe] = +0.2 at [Fe/H] = −0.5, and the solar abundance ratio at [Fe/H] ≥ −0.3. In this study, a new set of synthetic magnitudes in the PS1 photometric bands are computed using filter transmission curves in Tonry et al. (2012). The AB corrections in Abolfathi et al. (2018) and Scolnic et al. (2015) are adopted in each of the native filter systems.

In An et al. (2013), fiducial sequences of M67 are used on the metal-rich side, and those of M92 are taken on the metal-poor side, as a baseline of the calibration. They are partly replaced and reinforced by Gaia's double sequences in this revised work. The models are adjusted to match the bMS, along with other globular cluster sequences (M3, M5, M13, M15, and M92), from which color corrections are defined in the metal-poor regime. Similarly, the rMS is used to calibrate models at an intermediate metallicity between the bMS and the solar-metallicity cluster, M67. NGC 6791 is used as a fiducial case at a super-solar metallicity. These fiducial sequences are collectively used to model the metallicity dependence of the color corrections.

The cluster fiducial sequences in An et al. (2008) and Bernard et al. (2014) are taken in the SDSS and PS1 systems, respectively. To employ the improved photometric calibration for the cluster photometry, SDSS DR6 photometry in each cluster's flanking fields is compared to DR14 photometry that is based on the uber-calibration (Finkbeiner et al. 2016), and the mean differences are applied to the original sequences in An et al. (2008) to be consistent with the DR14 scale. The size of the corrections amount to a few hundredths of magnitudes. The adopted distance, metallicity, and reddening for the globular cluster sample (M15, M92, M13, M3, and M5, in increasing order of [Fe/H]) are the same as in An et al. (2009b, 2013), and those for the open clusters (M67 and NGC 6791) are taken from An et al. (2019). For globular clusters, uniform errors in the color differences (0.02–0.04 mag) are adopted in each color index, which are propagated through errors in the adopted cluster parameters (see An et al. 2009b, for a detailed error analysis). The errors for the open cluster samples are those propagated from errors in An et al. (2019).

The advantage of Gaia's double sequences over the cluster fiducial sequences is that they cover a large dynamic range of stellar mass based on an accurate distance scale. However, the mean metallicities of the bMS and rMS are not as precisely known as those of the calibrating cluster systems. Given that the observed colors depend on metallicity by Δ(g − r)/Δ[Fe/H] ∼ 0.2 mag dex⁻¹, the metallicity of a calibrating system should be determined to a precision of ∼0.1 dex, in order to make calibrated isochrones useful. By cross-matching sources with SDSS/APOGEE (Majewski et al. 2017), Gaia Collaboration et al. (2018a) found 〈[Fe/H]〉 ∼ −1.3 and 〈[Fe/H]〉 ∼ −0.5 for the bMS and rMS, respectively. Sahlholdt et al. (2019) used photometric metallicities of red-giant stars, based on SkyMapper (Wolf et al. 2018) DR1, and found slightly lower metallicities: 〈[Fe/H]〉 ∼ −1.4 for the bMS and 〈[Fe/H]〉 ∼ −0.7 for the rMS.

As described below, metallicities of the bMS and rMS are searched by minimizing changes of the color difference with respect to metallicity, and are adopted throughout this study. In Figure 5, the color offsets from the five globular clusters and two open clusters are shown over the entire T_eff range. Figure 6 shows mean color offsets at a fixed T_eff (5800 K in this example; see the Appendix for more cases) as a function of [Fe/H]. For the globular cluster samples, there is no strong trend in the color residual over [Fe/H] at a given T_eff. Since the bMS is on the metal-rich end of the globular clusters in our sample (more or less similar to that of M5), this leads to a plausible assumption that the bMS should have a similar color offset from the models as other globular clusters. The most satisfying color offsets in all color indices are found at [Fe/H] = −1.3 for the bMS. If a higher (lower) metallicity is adopted by 0.1 dex, the bMS becomes too blue (red) with respect to the color residuals from the globular clusters. We adopt an age of 13 Gyr for the bMS, based on an eyeball fit to the MS turn-off. The same age is adopted for other globular clusters, except M5 (12 Gyr).

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On the other hand, the observed trend of the metal-rich systems is more dramatic, with a strong curvature in the color–T_eff relations for cool stars. The metallicity of the rMS, which is found between those of globular clusters (and the bMS) and M67, is determined by exploring different [Fe/H], while inspecting the resulting mean color offsets. Because colors in the theoretical models are smoothly varying with metallicity, second-order terms (i.e., color corrections) should not exceed the underlying color changes of the models. If a higher (lower) [Fe/H] is taken, the rMS becomes too blue (or red) with respect to M67 (as well as the globular clusters), resulting in a strong curvature in the color corrections, and therefore model colors. If the metallicity is set to [Fe/H] = −0.4, as shown in Figure 6, a smooth transition from M67 to globular clusters can be achieved. An age of 12 Gyr is found for the rMS, based on fitting of the original isochrone models to its MS turn-off.

The red line in Figure 6 shows an empirical correction function in each color index. As discussed above, we adopt a constant color correction at [Fe/H] < −1, and make a smooth transition to the color corrections from the rMS, M67, and NGC 6791 using a quadratic interpolation. The correction functions are computed at ΔT_eff = 100 K intervals. The impact of the Bayesian parallaxes in Bailer-Jones et al. (2018) is negligible in the color calibration, as the color offsets change little for both the bMS (0.01–0.02 mag) and rMS (<0.01 mag), when Bayesian parallaxes are employed. These differences are included in the error bars for the two sequences in Figure 6.

The above procedure is repeated for the PS1 system (with SDSS u-band photometry), and the resulting color-correction functions are shown in Figure 7 at T_eff = 5800 K. The overall trend of the color correction is quite similar to the case for SDSS, implying that the observed systematic departures from the models are unlikely caused by systematic errors in the photometry and/or filter-response functions; rather, they may be a sign of a scale error in T_eff and/or incorrect treatment of line-blanketing in the models.

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Compared to the old calibration (green dashed line in Figure 6), the revised calibration for metal-poor ([Fe/H] < −1) stars makes isochrones redder in g − r, g − i, and g − z, because mean color offsets from all globular cluster samples are taken in this study, instead of taking M92 as a baseline. The sense of this change is that the photometrically derived T_eff becomes higher at a given color. A higher T_eff can further lead to a longer distance and a higher photometric metallicity. The newer calibration, which is designed to directly follow M67 and NGC 6791, also has a sharper dip at −1 ≲ [Fe/H] ≲ 0 in all color indices.

The net difference in metallicity between the original and the revised calibration is shown in Figure 8. The mean differences in [Fe/H] are ∼0.1 dex at intermediate metallicities (−1.4 ≲ [Fe/H] ≲ −0.2), and are ∼0.2 dex on the metal-poor side. However, it is not trivial to determine the impact of the revised calibration, because of the interplay of the color corrections in the derivation of T_eff, [Fe/H], and distance. Therefore, we construct stellar distributions in the [Fe/H] versus v_ϕ plane in the following section, in order to judge the consequence of the revised calibration.

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Based on the revised set of isochrones, metallicities and distances of individual stars are computed, employing the same procedure as in Paper I. Gaia's proper-motion data are used along the Galactic prime meridian, in combination with photometric distances, to estimate the rotational velocities of stars. The following analysis is restricted to 4–6 kpc from the Galactic plane, to verify the accuracy of the current models, where a number of stellar populations in the Milky Way can reveal themselves on the [Fe/H]–v_ϕ plane with a proper normalization scheme (Paper I), or after a masking of dominant stellar components (this study).

Figure 9 shows the observed distribution of stars in the [Fe/H]–v_ϕ plane extracted using the revised isochrones, along with its decomposition into subpopulations. Panel (a) shows a raw, unweighted stellar distribution of stars at 4 < ∣Z∣ ≤ 6 kpc on a logarithmic scale. In our sample, there is a luminosity bias for MS stars, according to which the observed metal-rich MS stars tend to be more massive than their metal-poor counterparts in a given distance bin. As more-massive MS stars are outnumbered by less-massive stars in a standard mass function of a stellar population, this implies an underestimation of the number of metal-rich stars in our sample. This problem is mitigated by multiplying the raw star count by ${({M}_{* }/0.7{M}_{\odot })}^{2.35}$, based on the Salpeter (1955) mass function. Here, M_* is the median value of stellar mass in each bin, which usually span a narrow range of mass (<0.1 M_⊙) at a given metallicity. The scaled distribution based on this simple correction is shown in panel (b). The impact of the bias correction is minimal, however, because the mass difference between the metal-poor and metal-rich star samples is small (<0.1 M_⊙).

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In an attempt to characterize the observed distribution and isolate individual Galactic subpopulations, we fit mixtures of elliptical Gaussians to the [Fe/H]–v_ϕ plane. This analysis is done mostly for the purpose of demonstration; a more detailed modeling will be presented in the subsequent paper of this series (D. An et al. 2021, in preparation).

We begin by modeling major structures visible in panel (b): the metal-poor halo over a wide range of metallicity (−2.6 <[Fe/H] < −1.6) and the metal-rich clump at [Fe/H] ∼ −0.6, which is attributed to the SD according to Paper I. They comprise approximately 90% of the stars in this data set. The former exhibits double peaks at [Fe/H] ∼ −1.6 and −2.5, reminiscent of the central values of the IH and OH populations in Carollo et al. (2007, 2010), respectively. Indeed, a fit using a single Gaussian underestimates the number of stars at both ends of the metallicity distribution, and at least two Gaussians are required to capture the observed distribution of the metal-poor stars. The red solid ellipses in panel (c) show the positions and the extents of the three best-fitting Gaussians to the major structures—the SD, the metal-rich, and the metal-poor halos on prograde orbits, respectively. The residual distribution after subtracting these three components is shown by the pixel colors. This step requires a few iterations, with fitting other components, including the metal-weak thick disk (MWTD; Beers et al. 2014; Carollo et al. 2019, see below).

Once these major structures are subtracted off, the residual distribution shown in panel (c) is dominated by an elongated structure of stars with −1.6 < [Fe/H] < −0.8 on prograde orbits (v_ϕ ∼ 130 km s⁻¹). In Paper I, the clump near the same v_ϕ is attributed to the MWTD. The central position and the extent of the clump are somewhat sensitively affected by how the major components are fit by elliptical Gaussians. In this regard, it is reassuring that this volume of the parameter space is occupied by stars associated with the MWTD (Carollo et al. 2019).⁴ The revised calibration makes the metallicity of MWTD slightly lower than in Paper I, since the color–metallicity relations are adjusted by employing the rMS at the intermediate metallicity near the MWTD. The result of fitting an elliptical Gaussian is shown in panel (c) by a blue dashed line, and the residual after subtraction is shown in panel (d).

As shown in Panel (d) of Figure 9, the retrograde halo substructure(s) and GSE begin to reveal themselves after removing the MWTD and narrowing down the range in the density plot. The blue dashed curves show the best-fitting elliptical Gaussians to these components, assuming that metal-poor halo stars in retrograde orbits can be modeled as a single population. The fitting residuals after subtracting these components are shown in panel (e). The significance of the residual is computed assuming Poisson statistics, and is displayed in panel (f). There are some weak signals left in panels (e)–(f), some of which may be related to other substructures in the halo (e.g., Helmi 2020; Naidu et al. 2020, and references therein).

Figure 10 shows the same [Fe/H]–v_ϕ distribution and its decomposition into subpopulations as in Figure 9, but based on the PS1 photometry and the newly calibrated set of isochrones. The locations and extents of the subpopulations from PS1 are quite similar to those from SDSS, indicating that the models in the PS1 griz colors are accurately calibrated, although the same SDSS u-band data are used in both cases. The two systems are at least on a self-consistent metallicity and distance scale with each other.

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Table 3 shows the mean central positions and standard deviations of each component in [Fe/H] and v_ϕ from Figures 9–10. The errors indicate half of the differences. A fraction of each component is computed from a direct integration of the fitted profile. According to these estimates, about equal proportions of stars at 4 < ∣Z∣ < 6 kpc belong to the SD, the metal-rich, and the metal-poor prograde halos, respectively. Only the remaining 10% of stars can be found in the MWTD, GSE, and the retrograde structure(s). The fraction of GSE is small at this distance, because its main body primarily encompasses a more local volume.

6.1. Empirically Calibrated Isochrones

6.2. Halo Dichotomy

We thank Donald M. Terndrup for useful comments. D.A. thanks Coryn Bailer-Jones for kindly providing extensive data tables with Bayesian parallaxes. D.A. acknowledges support provided by the National Research Foundation of Korea to the Center for Galaxy Evolution Research (2017R1A5A1070354) and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1A02085433). T.C.B. acknowledges partial support from grant PHY 14-30152 (Physics Frontier Center/JINA-CEE), awarded by the U.S. National Science Foundation.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS website is www.sdss.org.

SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics ∣ Harvard & Smithsonian (CfA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

This work presents results from the European Space Agency (ESA) space mission Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is provided by national institutions, in particular the institutions participating in the Gaia MultiLateral Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia archive website is https://archives.esac.esa.int/gaia.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

In Figure 11, empirical color-correction functions other than the representative case in Figure 6 (T_eff = 5800 K) are shown at T_eff = 6200 K (left) and 4500 K (right).

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