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

Deokkeun An  Department of Science Education, Ewha Womans University, Seoul
03760, Republic of Korea  Timothy C. Beers  Department of Physics and JINA
Center for the Evolution of the Elements, University of Notre Dame, Notre
Dame, IN 46556, USA

(Accepted for publication in the Astrophysical Journal)

###### Abstract

We improve the identification and isolation of individual stellar populations
in the Galactic halo based on an updated set of empirically calibrated stellar
isochrones in the Sloan Digital Sky Survey (SDSS) and Pan-STARRS 1 (PS1)
photometric systems. Along the Galactic prime meridian ($l=0^{\circ}$ and
$180^{\circ}$), where proper motions and parallaxes from Gaia DR2 can be used
to compute rotational velocities of stars in the rest frame of the Milky Way,
we use the observed double color-magnitude sequences of stars having large
transverse motions, which are attributed to groups of stars in the metal-poor
halo and the thick disk with halo-like kinematics, respectively. The Gaia
sequences directly constrain color-magnitude relations of model colors, and
help to improve our previous calibration using Galactic star clusters. Based
on these updated sets of stellar isochrones, we confirm earlier results on the
presence of distinct groups of stars in the metallicity versus rotational-
velocity plane, and find that the distribution of the most metal-poor ([Fe/H]
$<-2$) stars in our sample can be modeled using two separate groups on
prograde and retrograde orbits, respectively. At $4$–$6$ kpc from the Galactic
plane, we find approximately equal proportions of the Splashed Disk, and the
metal-rich ($\langle{\rm[Fe/H]}\rangle\sim-1.6$) and metal-poor
($\langle{\rm[Fe/H]}\rangle\sim-2.5$) halos on prograde orbits. The Gaia-
Sausage-Enceladus, the metal-weak thick disk, and the retrograde halo
structure(s) ($\langle{\rm[Fe/H]}\rangle\sim-2.2$) constitute approximately
$10\%$ of the rest of the stellar populations at these distances.

Milky Way dynamics (1051), Milky Way stellar halo (1060), Stellar abundances
(1577), Stellar atmospheres (1584), Stellar evolutionary models (2046)

##  1 Introduction

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
($T_{\rm eff}$) 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, semi-empirical 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 $\sim 2\%$–$3\%$ 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.

Secondly, 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
$M_{r}\sim 7$ for the calibration in $g\,-\,r$ and $g\,-\,i$. For metal-poor
halo stars, this limiting magnitude corresponds to masses of $\sim 0.6\
M_{\odot}$. When the color indices include the $u$ or $z$ passbands, the
magnitude limit for the model calibration becomes brighter ($M_{r}\sim 6$ or
$0.7\ M_{\odot}$). This inhomogeneous coverage in stellar mass inevitably
limits the application of the isochrones, which in many cases requires multi-
band photometry to constrain stellar parameters.

Thirdly, 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 $\sim 5\%$–$7\%$ 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]111Throughout this paper, we use
[Fe/H] to indicate a logarithmic iron abundance with respect to the solar
value. Since we employ models with a specific relation between [Fe/H] and
$\alpha$-elemental abundance ([$\alpha$/Fe]) of stars (see § 3), an overall
metallicity ([M/H]) is higher than [Fe/H] for metal-poor stars with elevated
[$\alpha$/Fe].) and accurate crowded-field photometry are limited to those
with [Fe/H] $\lesssim-1.2$, while the open cluster samples have [Fe/H] $\geq
0$. There are metal-rich globular clusters that might be considered, such as
47 Tuc or M71 ([Fe/H] $\sim-0.7$), 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, b). 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
$\langle{\rm[Fe/H]}\rangle\sim-1.3$, and is likely a collection of stars from
the so-called inner halo (IH; Carollo et al., 2007, 2010; Beers et al., 2012)
and the debris of Gaia-Sausage-Enceladus (GSE; Belokurov et al., 2018; Helmi
et al., 2018). On the other hand, the red sequence has
$\langle{\rm[Fe/H]}\rangle\sim-0.4$, and belongs to the Splashed Disk (SD;
Bonaca et al., 2017; Belokurov et al., 2019).

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
cover 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 § 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 § 3, followed by a description of the procedures
used for the revised empirical corrections in § 4. Updated “blueprint” maps
showing the metallicity versus rotational-velocity distribution of stars based
on these new models and PS1 photometry are presented in § 5.

Figure 1: The $u\,-\,g$ color-magnitude diagrams of stars with precise Gaia
parallaxes ($\sigma_{\pi}/\pi<0.1$) along the Galactic prime meridian (within
$\pm 30\arcdeg$ from $l=0\arcdeg$ and $180\arcdeg$). Rotational velocities in
the Galactocentric cylindrical coordinate system are computed using proper-
motion and parallax measurements from Gaia DR2. Left: The number density of
all stars with $-150<v_{\phi}\leq 150$ km s-1. Middle: Only stars with large
retrograde motions ($-150<v_{\phi}\leq-50$ km s-1). Right: Stars with moderate
prograde rotation ($120<v_{\phi}\leq 150$ km s-1). In all panels, the blue and
red solid lines indicate fiducial sequences for the retrograde (middle panel)
and the prograde (right panel) populations, respectively.

##  2 Gaia’s Double Sequences

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 ($v_{\phi}$) 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 $v_{\phi}$ 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 $v_{\phi}$ along the Galactic prime meridian ($l=0\arcdeg$ or
$180\arcdeg$), where the $v_{\phi}$ vector essentially depends only on the
proper motions and distances (see Paper I for more information on the
$v_{\phi}$ estimation procedure).

For the calibration sample, only stars with good parallaxes
($\sigma_{\pi}/\pi<0.2$) and proper motions
[$(\mu_{\alpha}/\sigma_{\mu_{\alpha}})^{2}+(\mu_{\delta}/\sigma_{\mu_{\delta}})^{2}>(1.0/0.3)^{2}$]
are used. 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
$|b|>20\arcdeg$ 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.

Figure 2: Same as in the left panel of Figure 1, but displaying CMDs in other
color indices. Figure 3: Fiducial sequences in bins of different rotational
velocities. Each of the lines is obtained based on the same methodology as
shown in Figures 1–2.

The left panel of Figure 1 shows the $u\,-\,g$ CMD for stars with
$-150<v_{\phi}<150$ km s-1, along the Galactic prime meridian within a region
of $\pm 30\arcdeg$. The computed $v_{\phi}$ is corrected for a small
inclination with respect to the prime meridian (sometimes denoted as the
“projected” $v_{\phi}$). 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.

Figure 4: Same as in the left panel of Figure 1, but displaying CMDs in the
PS1 photometric system. The $u$-band data are taken from SDSS.

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
($-150<v_{\phi}<-50$ km s-1), a subset of the stars shown in the left panel,
and the right panel shows stars with prograde motions ($120<v_{\phi}<150$ km
s-1). The exact ranges of $v_{\phi}$ are adjusted until the sequences are most
clearly separated from each other, while keeping large numbers of stars in
each $v_{\phi}$ 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 $\Delta(u\,-\,g)=0.4$ mag (middle panel). Similarly, the rMS
is obtained after rejecting stars bluer than the bMS by $\Delta(u\,-\,g)=0.4$
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.

Table 1: Gaia Double Sequences in the SDSS System | bMS | rMS
---|---|---
$M_{r}$ | $u\,-\,g$ | $g\,-\,r$ | $g\,-\,i$ | $g\,-\,z$ | $u\,-\,g$ | $g\,-\,r$ | $g\,-\,i$ | $g\,-\,z$
$3.300$ | $0.955$ | $0.290$ | $0.380$ | $0.419$ | $1.248$ | $0.443$ | $0.580$ | $0.632$
$3.500$ | $0.941$ | $0.282$ | $0.371$ | $0.398$ | $1.209$ | $0.420$ | $0.551$ | $0.595$
$3.700$ | $0.929$ | $0.275$ | $0.364$ | $0.382$ | $1.174$ | $0.401$ | $0.527$ | $0.563$
$3.900$ | $0.917$ | $0.272$ | $0.362$ | $0.373$ | $1.148$ | $0.387$ | $0.510$ | $0.542$
$4.100$ | $0.908$ | $0.273$ | $0.366$ | $0.375$ | $1.133$ | $0.381$ | $0.502$ | $0.532$
$4.300$ | $0.903$ | $0.280$ | $0.378$ | $0.388$ | $1.131$ | $0.382$ | $0.506$ | $0.536$
$4.500$ | $0.902$ | $0.292$ | $0.397$ | $0.410$ | $1.143$ | $0.392$ | $0.520$ | $0.553$
$4.700$ | $0.908$ | $0.309$ | $0.422$ | $0.442$ | $1.167$ | $0.408$ | $0.543$ | $0.581$
$4.900$ | $0.922$ | $0.331$ | $0.453$ | $0.481$ | $1.206$ | $0.429$ | $0.575$ | $0.620$
$5.100$ | $0.946$ | $0.358$ | $0.489$ | $0.527$ | $1.258$ | $0.456$ | $0.613$ | $0.667$
$5.300$ | $0.981$ | $0.388$ | $0.531$ | $0.579$ | $1.324$ | $0.488$ | $0.658$ | $0.722$
$5.500$ | $1.028$ | $0.422$ | $0.578$ | $0.637$ | $1.405$ | $0.524$ | $0.709$ | $0.786$
$5.700$ | $1.089$ | $0.458$ | $0.630$ | $0.700$ | $1.498$ | $0.567$ | $0.768$ | $0.857$
$5.900$ | $1.164$ | $0.497$ | $0.685$ | $0.768$ | $1.605$ | $0.615$ | $0.835$ | $0.939$
$6.100$ | $1.254$ | $0.539$ | $0.744$ | $0.841$ | $1.724$ | $0.672$ | $0.913$ | $1.032$
$6.300$ | $1.361$ | $0.585$ | $0.807$ | $0.920$ | $1.851$ | $0.737$ | $1.001$ | $1.137$
$6.500$ | $1.482$ | $0.637$ | $0.878$ | $1.007$ | $1.981$ | $0.808$ | $1.100$ | $1.255$
$6.700$ | $1.614$ | $0.696$ | $0.960$ | $1.106$ | $2.107$ | $0.885$ | $1.208$ | $1.383$
$6.900$ | $1.752$ | $0.763$ | $1.055$ | $1.218$ | $2.220$ | $0.963$ | $1.320$ | $1.517$
$7.100$ | $1.890$ | $0.837$ | $1.162$ | $1.341$ | $2.312$ | $1.038$ | $1.433$ | $1.654$
$7.300$ | $2.021$ | $0.913$ | $1.276$ | $1.472$ | $2.380$ | $1.107$ | $1.541$ | $1.787$
$7.500$ | $2.137$ | $0.988$ | $1.387$ | $1.600$ | $2.424$ | $1.166$ | $1.643$ | $1.914$
$7.700$ | $2.232$ | $1.055$ | $1.486$ | $1.717$ | $2.447$ | $1.215$ | $1.735$ | $2.032$
$7.900$ | $2.301$ | $1.111$ | $1.569$ | $1.818$ | $2.456$ | $1.253$ | $1.819$ | $2.140$
$8.100$ | $2.345$ | $1.159$ | $1.635$ | $1.900$ | $2.458$ | $1.283$ | $1.895$ | $2.241$
$8.300$ | $2.366$ | $1.202$ | $1.689$ | $1.969$ | $2.457$ | $1.304$ | $1.965$ | $2.335$
$8.500$ | $2.373$ | $1.243$ | $1.736$ | $2.030$ | $2.456$ | $1.320$ | $2.030$ | $2.424$
$8.700$ | $2.371$ | $1.284$ | $1.782$ | $2.087$ | $2.456$ | $1.331$ | $2.091$ | $2.509$
$8.900$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.455$ | $1.338$ | $2.149$ | $2.592$
$9.100$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.453$ | $1.343$ | $2.204$ | $2.670$
$9.300$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.450$ | $1.347$ | $2.254$ | $2.744$
$9.500$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.448$ | $1.350$ | $2.300$ | $2.814$
$9.700$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.447$ | $1.353$ | $2.344$ | $2.880$
$9.900$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.449$ | $1.356$ | $2.387$ | $2.945$
$10.100$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.453$ | $1.359$ | $2.430$ | $3.010$
$10.300$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.459$ | $1.364$ | $2.473$ | $3.074$
$10.500$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.465$ | $1.369$ | $2.517$ | $3.138$
$10.700$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.470$ | $1.375$ | $2.562$ | $3.204$
$10.900$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.473$ | $1.380$ | $2.607$ | $3.270$
$11.100$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.475$ | $1.386$ | $2.652$ | $3.337$
Table 2: Gaia Double Sequences in the PS1 System | bMS | rMS
---|---|---
$M_{r}(PS1)$ | $u_{SDSS}\,-\,g_{PS1}$ | $(g\,-\,r)_{PS1}$ | $(g\,-\,i)_{PS1}$ | $(g\,-\,z)_{PS1}$ | $u_{SDSS}\,-\,g_{PS1}$ | $(g\,-\,r)_{PS1}$ | $(g\,-\,i)_{PS1}$ | $(g\,-\,z)_{PS1}$
$3.300$ | $0.970$ | $0.265$ | $0.330$ | $0.330$ | $1.290$ | $0.389$ | $0.526$ | $0.545$
$3.500$ | $0.964$ | $0.253$ | $0.322$ | $0.325$ | $1.251$ | $0.368$ | $0.494$ | $0.512$
$3.700$ | $0.958$ | $0.245$ | $0.317$ | $0.322$ | $1.217$ | $0.349$ | $0.467$ | $0.484$
$3.900$ | $0.951$ | $0.240$ | $0.317$ | $0.321$ | $1.192$ | $0.337$ | $0.448$ | $0.464$
$4.100$ | $0.944$ | $0.242$ | $0.323$ | $0.326$ | $1.177$ | $0.331$ | $0.440$ | $0.455$
$4.300$ | $0.938$ | $0.250$ | $0.335$ | $0.337$ | $1.176$ | $0.333$ | $0.443$ | $0.457$
$4.500$ | $0.936$ | $0.264$ | $0.353$ | $0.355$ | $1.188$ | $0.341$ | $0.456$ | $0.472$
$4.700$ | $0.940$ | $0.281$ | $0.376$ | $0.381$ | $1.213$ | $0.355$ | $0.477$ | $0.496$
$4.900$ | $0.954$ | $0.301$ | $0.403$ | $0.413$ | $1.253$ | $0.373$ | $0.505$ | $0.529$
$5.100$ | $0.981$ | $0.324$ | $0.435$ | $0.451$ | $1.307$ | $0.396$ | $0.540$ | $0.570$
$5.300$ | $1.020$ | $0.349$ | $0.472$ | $0.495$ | $1.377$ | $0.424$ | $0.581$ | $0.618$
$5.500$ | $1.072$ | $0.375$ | $0.512$ | $0.543$ | $1.463$ | $0.456$ | $0.627$ | $0.674$
$5.700$ | $1.137$ | $0.403$ | $0.555$ | $0.596$ | $1.563$ | $0.493$ | $0.681$ | $0.737$
$5.900$ | $1.218$ | $0.433$ | $0.604$ | $0.655$ | $1.678$ | $0.535$ | $0.742$ | $0.810$
$6.100$ | $1.316$ | $0.468$ | $0.661$ | $0.722$ | $1.804$ | $0.583$ | $0.812$ | $0.892$
$6.300$ | $1.436$ | $0.510$ | $0.727$ | $0.799$ | $1.937$ | $0.637$ | $0.889$ | $0.984$
$6.500$ | $1.575$ | $0.560$ | $0.802$ | $0.886$ | $2.072$ | $0.695$ | $0.973$ | $1.085$
$6.700$ | $1.725$ | $0.615$ | $0.884$ | $0.982$ | $2.202$ | $0.755$ | $1.063$ | $1.192$
$6.900$ | $1.871$ | $0.671$ | $0.967$ | $1.078$ | $2.318$ | $0.816$ | $1.157$ | $1.303$
$7.100$ | $1.993$ | $0.721$ | $1.040$ | $1.165$ | $2.415$ | $0.875$ | $1.251$ | $1.414$
$7.300$ | $2.077$ | $0.760$ | $1.097$ | $1.234$ | $2.489$ | $0.930$ | $1.343$ | $1.524$
$7.500$ | $2.120$ | $0.788$ | $1.138$ | $1.282$ | $2.540$ | $0.979$ | $1.432$ | $1.630$
$7.700$ | $2.137$ | $0.808$ | $1.167$ | $1.317$ | $2.572$ | $1.021$ | $1.517$ | $1.733$
$7.900$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.593$ | $1.056$ | $1.598$ | $1.833$
$8.100$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.607$ | $1.085$ | $1.674$ | $1.929$
$8.300$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.620$ | $1.107$ | $1.744$ | $2.021$
$8.500$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.632$ | $1.124$ | $1.808$ | $2.109$
$8.700$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.643$ | $1.135$ | $1.866$ | $2.194$
$8.900$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.653$ | $1.141$ | $1.920$ | $2.276$
$9.100$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.659$ | $1.141$ | $1.976$ | $2.358$
$9.300$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.656$ | $1.137$ | $2.037$ | $2.442$
$9.500$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.642$ | $1.128$ | $2.108$ | $2.530$
$9.700$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.617$ | $1.116$ | $2.190$ | $2.623$
$9.900$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.585$ | $1.102$ | $2.278$ | $2.718$

The rMS covers a wide range of $M_{r}$, from the MS turn-off to $M_{r}\sim
11$, 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 $\sim 3$ 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 $v_{\phi}$ cuts. The fiducial line for stars with
$-300<v_{\phi}<-150$ 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 the IH and GSE. The contribution
of stars from the outer halo (OH; Carollo et al., 2007, 2010) is negligible in
local volumes, 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 SD. For stars
with larger $v_{\phi}$ 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.

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 $1\arcsec$
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.

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.

##  3 Theoretical Stellar Isochrones

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 $T_{\rm eff}$ 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
$\alpha$-element abundance ([$\alpha$/Fe]) in these models, motivated by
spectroscopic observations of stars in the Milky Way (e.g., Venn et al., 2004;
Hayden et al., 2015): [$\alpha$/Fe]$=+0.4$ at [Fe/H] $=-3.0$,
[$\alpha$/Fe]$=+0.3$ at $-2.0\leq{\rm[Fe/H]}\leq-1.0$, [$\alpha$/Fe]$=+0.25$
at [Fe/H] $=-0.75$, [$\alpha$/Fe]$=+0.2$ at [Fe/H] $=-0.5$, and the solar
abundance ratio at [Fe/H] $\geq-0.3$. 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.

##  4 Derivation of Empirical Color Corrections

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
$\Delta(g\,-\,r)/\Delta{\rm[Fe/H]}\sim 0.2$ mag dex-1, the metallicity of a
calibrating system should be determined to a precision of $\sim 0.1$ 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
$\langle{\rm[Fe/H]}\rangle\sim-1.3$ and $\langle{\rm[Fe/H]}\rangle\sim-0.5$
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: $\langle{\rm[Fe/H]}\rangle\sim-1.4$
for the bMS and $\langle{\rm[Fe/H]}\rangle\sim-0.7$ for the rMS.

Figure 5: Color differences between observed fiducial lines and YREC+MARCS
theoretical models (observation minus model). Dotted and solid lines are those
derived from cluster fiducial sequences over a wide range of metallicity. The
open circles with error bars show color differences of Gaia’s double sequences
(see Figures 1–2) . Figure 6: Derivation of empirical color corrections in the
SDSS photometric system. The blue open circles are color differences obtained
from the key globular and open clusters. The red diamond symbols are those
from Gaia’s double sequences. The adopted empirical correction function is
shown by a red solid line. For comparison, the original correction is
displayed by the green dashed line. The case at $T_{\rm eff}=5800$ K is shown
in this example.

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 $T_{\rm
eff}$ range. Figure 6 shows mean color offsets at a fixed $T_{\rm eff}$
($5800$ K in this example; see 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 $T_{\rm eff}$. 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).

On the other hand, the observed trend of the metal-rich systems is more
dramatic, with a strong curvature in the color-$T_{\rm eff}$ 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 $\Delta T_{\rm eff}=100$ 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 double
sequences in Figure 6.

Figure 7: Same as in Figure 6, but in the PS1 photometric system.

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 $T_{\rm eff}=5800$ 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 $T_{\rm eff}$ and/or incorrect line-blanketing in the models.

Figure 8: Comparison of photometric metallicities from $ugriz$, based on the
revised calibration in this study with those from Paper I. All of the stars in
the SDSS data set are included in the above comparison. The crosses and error
bars show mean differences and standard deviations in bins of $0.2$ dex in
[Fe/H]. Contour levels include 20%, 60%, 85%, and 95% of the sample.

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 $T_{\rm
eff}$ becomes higher at a given color. A higher $T_{\rm eff}$ 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 $-1\lesssim{\rm[Fe/H]}\lesssim 0$ 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 $\sim
0.1$ dex at intermediate metallicities
($-1.4\lesssim{\rm[Fe/H]}\lesssim-0.2$), and are $\sim 0.2$ 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 $T_{\rm eff}$, [Fe/H], and distance. Therefore, we construct
stellar distributions in the [Fe/H] versus $v_{\phi}$ plane in the following
section, in order to judge the consequence of the revised calibration.

##  5 Chemo-dynamical Distribution of the Galactic Halo

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]-$v_{\phi}$ plane with a proper normalization scheme (Paper I), or after
a masking of dominant stellar components (this study).

Figure 9: Decomposition of stellar populations in the Milky Way at $4$–$6$ kpc
from the Galactic plane. Rotational velocities in the Galactocentric
cylindrical coordinate system are computed using Gaia’s proper motions and
photometric distances along the Galactic prime meridian. Photometric distances
and metallicities are derived using SDSS photometry. (a) A logarithmic number-
density distribution of the stars. (b) Same as in (a), but showing a linear
density distribution, after correcting for bias in stellar mass (see text).
(c) A number-density distribution after subtracting three major components –
the prograde metal-poor halo (MPH), the metal-rich halo (MRH), and the SD –
modeled using Gaussian distributions (red solid ellipses). The metal-weak
thick disk (MWTD) is seen in this residual distribution, which is fit using a
Gaussian ellipse (blue dotted ellipse). (d) A residual distribution after
subtracting the MWTD in (c). The remaining stars are fit using two Gaussian
functions (red dotted ellipses), each of which represents GSE and the
retrograde halo structure(s) (RHS), respectively. (e) A residual distribution
after subtracting all sub-components (solid ellipses) in (c) and (d). (f) The
significance of the residual map in (e). Note that the color scales are
different in all panels.

Figure 9 shows the observed distribution of stars in the [Fe/H]-$v_{\phi}$
plane extracted using the revised isochrones, along with its decomposition
into sub-populations. Panel (a) shows a raw, unweighted stellar distribution
of stars at $4<|Z|\leq 6$ 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 under-
estimation 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 ($<0.1\ M_{\odot}$) 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 ($<0.1\
M_{\odot}$).

In an attempt to characterize the observed distribution and isolate individual
Galactic subpopulations, we fit mixtures of elliptical Gaussians to the
[Fe/H]-$v_{\phi}$ 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 (An et al., in preparation).

Figure 10: Same as in Figure 9, but based on the PS1 photometry.

We begin by modeling major structures visible in panel (b): the metal-poor
halo over a wide range of metallicity ($-2.6<{\rm[Fe/H]}<-1.6$) and the metal-
rich clump at [Fe/H] $\sim-0.6$, 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] $\sim-1.6$ 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 under-estimates 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 solid red 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
$-1.6<{\rm[Fe/H]}<-0.8$ on prograde orbits ($v_{\phi}\sim 130$ km s-1). In
Paper I, the clump near the same $v_{\phi}$ 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 also occupied by
stars associated with the MWTD from spectroscopic data (Carollo et al.,
2019).222This parameter space also overlaps with the Helmi streams (Helmi et
al., 1999), which is characterized by high vertical speeds with respect to the
Galactic plane. However, the current data set, based on $|Z|-v_{\phi}-$[Fe/H],
is not sufficient to discriminate such objects from the MWTD members. 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]-$v_{\phi}$ distribution and its decomposition
into sub-populations as in Figure 9, but based on the PS1 photometry and the
newly calibrated set of isochrones. The locations and extents of the sub-
populations 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.

Table 3: Component Analysis at $4<|Z|<6$ kpc | $\langle{\rm[Fe/H]}\rangle$ | $\langle v_{\phi}\rangle$ | $\sigma({\rm[Fe/H]})$ | $\sigma(v_{\phi})$ | Contribution
---|---|---|---|---|---
Stellar Population | (dex) | (km s-1) | (dex) | (km s-1) | (%)
Splash Disk (SD) | $-0.57\pm 0.03$ | $90.1\pm 11.2$ | $0.31\pm 0.01$ | $91.4\pm 16.5$ | $24.8\pm 4.9$
Metal Weak Thick Disk (MWTD) | $-0.90\pm 0.17$ | $129.3\pm 1.7$ | $0.40\pm 0.02$ | $31.8\pm 1.2$ | $5.7\pm 1.1$
Gaia-Sausage-Enceladus (GSE) | $-1.16\pm 0.04$ | $-60.3\pm 1.3$ | $0.20\pm 0.04$ | $32.3\pm 0.5$ | $0.5\pm 0.1$
Prograde Metal-rich Halo (MRH) | $-1.58\pm 0.05$ | $64.6\pm 5.3$ | $0.46\pm 0.08$ | $76.8\pm 1.2$ | $30.3\pm 11.8$
Prograde Metal-poor Halo (MPH) | $-2.50\pm 0.00$ | $58.6\pm 9.9$ | $0.54\pm 0.05$ | $73.7\pm 0.8$ | $35.2\pm 5.7$
Retrograde Halo Structure(s) (RHS) | $-2.28\pm 0.14$ | $-142.0\pm 11.4$ | $0.98\pm 0.05$ | $54.7\pm 1.5$ | $3.5\pm 0.1$

Table 3 shows the mean central positions and standard deviations of each
component in [Fe/H] and $v_{\phi}$ 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 Summary and Discussion

###  6.1 Empirically Calibrated Isochrones

Previous and ongoing imaging surveys can provide useful information on the
properties of stars to a significant depth, with a high volume-filling factor,
and therefore can complement the relatively smaller spectroscopic survey data.
Extracting fundamental stellar parameters from the photometry, however,
requires a well-calibrated relation between a limited number of observables
and physical quantities. As one of such approaches, our previous exercise
relied on observations of well-studied Galactic star clusters to calibrate
theoretically predicted quantities in stellar isochrones. This semi-empirical
approach has been successfully applied to the SDSS data in Paper I to map the
distribution of halo stars on the metallicity versus rotational-velocity
space, leading to clear identification of a number of distinct stellar
populations.

Our previous work is ultimately tied to the Hipparcos (ESA, 1997) distance
scale, since cluster distances are adopted from the subdwarf-fitting to
globular clusters and the parallaxes of individual stars in open clusters.
With the advent of Gaia, however, it becomes possible to utilize its exquisite
parallaxes and other astrometric information in the isochrone calibration.
This study provides the first look at the consequence of the revised
calibration, which incorporates Gaia’s double sequences from stars with large
rotational velocities.

Gaia’s double sequences open up a new possibility for sampling a single
stellar population from a dynamical family of stars in the Milky Way, and
utilizing it to define a fiducial color-magnitude relation. The observed
double sequences are sufficiently narrow to define purely empirical color-
magnitude relations in the native SDSS and PS1 photometric systems. In
particular, the rMS covers a wide range of mass, down to $\sim 0.3\
M_{\odot}$. Since these stars are found across a large survey area, the
observed sequences are essentially free from small-scale fluctuations in the
photometric and astrometric zero-point levels, which are the real advantages
over using star clusters.

Our previous mapping result in Paper I suggests that the bMS belongs to a
combination of the IH and GSE, while the rMS is composed of the SD population.
The metallicity difference between the two sequences is approximately $1$ dex
in [Fe/H], providing a useful leverage on the metallicity dependence of
stellar colors. When compared to theoretical stellar isochrones, we find that
bMS consistently exhibits redder colors; the size of the color offset is
consistent with those found from globular clusters. Because the metallicity of
the bMS population is comparable to that of metal-rich globular clusters in
the halo, and the colors change mildly in the metal-poor regime, the agreement
confirms our adopted distance scale for the globular clusters and the accuracy
of the cluster photometry.

In this work, we construct a set of isochrones in the SDSS and PS1 passbands
with the revised color corrections, and use it to map the [Fe/H]-$v_{\phi}$
distribution of stars at $4$–$6$ kpc from the Galactic plane. Because a number
of populations reveal themselves in this distance bin, we choose this sample
of stars for the purpose of verifying the accuracy of the newly calibrated
models. We find that both sets of isochrones in SDSS and PS1 (along with the
SDSS $u$-band) produce qualitatively similar results to those in Paper I. We
also refine positions in the [Fe/H]-$v_{\phi}$ plane, and estimate the rough
contributions of each population. Among various populations, the position and
the extent of the MWTD are particularly well-captured in the revised map,
because colors in the intermediate-metallicity range are better handled by the
metal-rich sequence. This result strengthens the conclusion reached in Paper
I, that the MWTD is a separate component from the canonical TD (see also
Carollo et al., 2019).

The photometric metallicity scale may differ from the spectroscopic
metallicity scale. Unfortunately, there are a limited number of metal-poor MS
stars with both homogeneous, high-resolution spectroscopic abundance analyses
and good $ugriz$ photometry in the native filter systems. For relatively
metal-rich stars ([Fe/H] $>-1$), the comparison with the GALAH survey (Buder
et al., 2018) indicates that the systematic difference does not exceed $0.1$
dex (see also Figure 2 in Paper I). The scale error may be as large as
$0.1$–$0.3$ dex in the lower metallicities, judged from the comparisons to
medium-resolution spectroscopic data from SDSS/SEGUE (Yanny et al., 2009) and
high-resolution follow-up observations (C. M. Rockosi et al., in preparation).
Inhomogeneous abundance ratios of Milky Way stars can also affect the
photometric metallicity scale, since enhancement of $\alpha$-element
abundances has a net effect of increasing the overall metallicity of a star
(e.g., Kim et al., 2002). Therefore, distinct [Fe/H]-[$\alpha$/Fe] sequences
discovered in the Milky Way (e.g., Nissen & Schuster, 2010; Hayden et al.,
2015; Matsuno et al., 2019) can exhibit shifted [Fe/H] values in our mapping,
unless specific [$\alpha$/Fe] are taken in the isochrone calibration.

Based on comparisons to SDSS/SEGUE spectroscopic metallicities, it can be seen
that the precision of the models in the PS1 system is lower than the
calibrated models in the SDSS $ugriz$ bands. The comparison of the PS1-based
photometric metallicity with the spectroscopic data indicates that standard
deviations of the metallicity difference ($\Delta{\rm[Fe/H]}=0.6$, $0.4$, and
$0.2$ dex at [Fe/H]=$-2$, $-1$, and $0$, respectively) are $10\%$–$15\%$
larger for metal-poor stars ($-2\lesssim{\rm[Fe/H]}\lesssim-1$), and are
$50\%$ larger at solar metallicities than the case with the revised
calibration in the SDSS $ugriz$. This mild deterioration in the models may
originate from inaccurate cluster fiducial sequences and/or extinction
corrections in the PS1 filter passbands, among other sources of possible
systematic errors.

In the current data release of Gaia (DR2), good parallaxes are limited to
relatively nearby stars, and do not provide statistically meaningful results
for the majority of globular clusters, because of a non-negligible systematic
error in the measurements (Gaia Collaboration et al., 2018c). The
color-$T_{\rm eff}$ relations can be further improved, once more accurate
distances to the key open and globular clusters become available. Deep
crowded-field photometry of the globular clusters will also be useful to
accurately determine color-$T_{\rm eff}$ relations for low-mass metal-poor
stars. On the contrary, the calibration will benefit from relatively ‘shallow’
photometry of nearby clusters, because the upper MS in these well-studied
clusters are often saturated in the broadband imaging surveys.

###  6.2 Halo Dichotomy

There is confusion in the literature on the relationship of the dual halos
(Carollo et al., 2007, 2010; Beers et al., 2012) to recently discovered
structures in the halo. Helmi (2020) proposed a connection between the halo
dichotomy and Gaia’s double sequences, according to which the IH is, by and
large, the same component as the SD, and the OH is likely GSE. However, the
dual halos were originally suggested based on a sample of metal-poor ([Fe/H]
$\lesssim-1$) stars, which probably does not include the red-sequence stars.
The peak metallicity of each component is also significantly different from
each other ($\Delta{\rm[Fe/H]}\sim 1$ dex), which is too large to be explained
by a scale error in the spectroscopic metallicity measurements.

According to Figure 9 (see also Paper I), the metal-rich
($\langle{\rm[Fe/H]}\rangle\sim-1.6$) halo component in prograde rotation
($\langle v_{\phi}\rangle\sim 60$ km s-1) can plausibly be assigned to the IH.
In addition, the original IH sample in Carollo et al. (2007, 2010) may include
a large number of stars in GSE, due to their similar metallicity ranges. Since
GSE has a small scale height, the inclusion of the GSE would have led them to
conclude that the IH has an oblate distribution, in contrast with a spherical
distribution of the OH.

Meanwhile, the mean metallicities of the metal-poor halo component and the
retrograde halo structure(s) are comparable to that of the OH
($\langle{\rm[Fe/H]}\rangle\sim-2.2$) in Carollo et al. (2007, 2010).
According to our mapping, the former has a prograde rotation with $\langle
v_{\phi}\rangle\sim+60$ km s-1, while the latter structures have $\langle
v_{\phi}\rangle\sim-140$ km s-1. Indeed, the asymmetric $v_{\phi}$
distribution of stars with [Fe/H] $<-2$ has previously been noted (Carollo et
al., 2010; Beers et al., 2012; An et al., 2013, 2015a), according to which at
least two Gaussian functions are required to model the observed $v_{\phi}$
distribution. Carollo et al. (2007, 2010) argue that the OH has a net zero or
a small retrograde rotation, which can be understood if the OH is composed of
the prograde metal-poor halo and the retrograde structures, as our map shows.
Recent re-analysis of the spectroscopic sample also supports this picture
(Carollo & Chiba, 2020).

The retrograde structure(s) are likely related to other known components in
the halo – for example, Sequoia (Myeong et al., 2019) and/or Thamnos
(Koppelman et al., 2019). They may also include stars in the arc-like feature
seen in the Toomre diagram (Helmi et al., 2018), which is proposed to be a
part of the tidal debris from GSE by these authors. In this regard, a long
chain of stars connecting GSE and the retrograde structure(s) in panel (d) of
Figure 9 may reflect a metallicity gradient of the progenitor(s) of GSE. On
the other hand, there appears no matching substructure to the prograde metal-
poor halo; although detailed dynamical analysis in Limberg et al. (2020)
reveals a number of cold stellar streams of metal-poor stars, their
contribution may not be enough to account for the entire population.

In the following work of this series, we will continue our effort to constrain
chemo-dynamical properties of the stellar populations in the local halo, by
applying the revised set of empirically calibrated isochrones presented in
this paper.

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 web
site 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 ($T_{\rm eff}=5800$ K) are
shown at $T_{\rm eff}=6200$ K (left) and $4500$ K (right).

Figure 11: Same as in Figure 6, but at $T_{\rm eff}=6200$ K (left) and $4500$
K (right).

## References

  * Abolfathi et al. (2018) Abolfathi, B., Aguado, D. S., Aguilar, G., et al. 2018, ApJS, 235, 42. doi:10.3847/1538-4365/aa9e8a
  * An (2019) An, D. 2019, ApJ, 878, L31. doi:10.3847/2041-8213/ab2467
  * An & Beers (2020) An, D. & Beers, T. C. 2020, ApJ, 897, 39. doi:10.3847/1538-4357/ab8d39 (Paper I)
  * An et al. (2013) An, D., Beers, T. C., Johnson, J. A., et al. 2013, ApJ, 763, 65. doi:10.1088/0004-637X/763/1/65
  * An et al. (2015a) An, D., Beers, T. C., Santucci, R. M., et al. 2015a, ApJ, 813, L28. doi:10.1088/2041-8205/813/2/L28
  * An et al. (2009a) An, D., Johnson, J. A., Beers, T. C., et al. 2009a, ApJ, 707, L64. doi:10.1088/0004-637X/707/1/L64
  * An et al. (2008) An, D., Johnson, J. A., Clem, J. L., et al. 2008, ApJS, 179, 326. doi:10.1086/592090
  * An et al. (2009b) An, D., Pinsonneault, M. H., Masseron, T., et al. 2009b, ApJ, 700, 523. doi:10.1088/0004-637X/700/1/523
  * An et al. (2019) An, D., Pinsonneault, M. H., Terndrup, D. M., et al. 2019, ApJ, 879, 81. doi:10.3847/1538-4357/ab23ed
  * An et al. (2007) An, D., Terndrup, D. M., Pinsonneault, M. H., et al. 2007, ApJ, 655, 233. doi:10.1086/509653
  * An et al. (2015b) An, D., Terndrup, D. M., Pinsonneault, M. H., et al. 2015b, ApJ, 811, 46. doi:10.1088/0004-637X/811/1/46
  * Bailer-Jones et al. (2018) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., et al. 2018, AJ, 156, 58. doi:10.3847/1538-3881/aacb21
  * Beers et al. (2012) Beers, T. C., Carollo, D., Ivezić, Ž., et al. 2012, ApJ, 746, 34. doi:10.1088/0004-637X/746/1/34
  * Beers et al. (2014) Beers, T. C., Norris, J. E., Placco, V. M., et al. 2014, ApJ, 794, 58. doi:10.1088/0004-637X/794/1/58
  * Belokurov et al. (2018) Belokurov, V., Erkal, D., Evans, N. W., et al. 2018, MNRAS, 478, 611. doi:10.1093/mnras/sty982
  * Belokurov et al. (2019) Belokurov, V., Sanders, J. L., Fattahi, A., et al. 2020, MNRAS, 494, 3880. doi:10.1093/mnras/staa876
  * Bernard et al. (2014) Bernard, E. J., Ferguson, A. M. N., Schlafly, E. F., et al. 2014, MNRAS, 442, 2999. doi:10.1093/mnras/stu1081
  * Bonaca et al. (2017) Bonaca, A., Conroy, C., Wetzel, A., et al. 2017, ApJ, 845, 101. doi:10.3847/1538-4357/aa7d0c
  * Buder et al. (2018) Buder, S., Asplund, M., Duong, L., et al. 2018, MNRAS, 478, 4513. doi:10.1093/mnras/sty1281
  * Carollo & Chiba (2020) Carollo, D. & Chiba, M. 2020, arXiv:2010.00235
  * Carollo et al. (2010) Carollo, D., Beers, T. C., Chiba, M., et al. 2010, ApJ, 712, 692. doi:10.1088/0004-637X/712/1/692
  * Carollo et al. (2007) Carollo, D., Beers, T. C., Lee, Y. S., et al. 2007, Nature, 450, 1020. doi:10.1038/nature06460
  * Carollo et al. (2019) Carollo, D., Chiba, M., Ishigaki, M., et al. 2019, ApJ, 887, 22. doi:10.3847/1538-4357/ab517c
  * Carretta et al. (2000) Carretta, E., Gratton, R. G., Clementini, G., et al. 2000, ApJ, 533, 215. doi:10.1086/308629
  * Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560
  * Drimmel & Spergel (2001) Drimmel, R. & Spergel, D. N. 2001, ApJ, 556, 181. doi:10.1086/321556
  * ESA (1997) ESA 1997, ESA Special Publication, 1200
  * Finkbeiner et al. (2016) Finkbeiner, D. P., Schlafly, E. F., Schlegel, D. J., et al. 2016, ApJ, 822, 66. doi:10.3847/0004-637X/822/2/66
  * Fukugita et al. (1996) Fukugita, M., Ichikawa, T., Gunn, J. E., et al. 1996, AJ, 111, 1748. doi:10.1086/117915
  * Gaia Collaboration et al. (2018a) Gaia Collaboration, Babusiaux, C., van Leeuwen, F., et al. 2018, A&A, 616, A10. doi:10.1051/0004-6361/201832843
  * Gaia Collaboration et al. (2018b) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1. doi:10.1051/0004-6361/201833051
  * Gaia Collaboration et al. (2018c) Gaia Collaboration, Helmi, A., van Leeuwen, F., et al. 2018, A&A, 616, A12. doi:10.1051/0004-6361/201832698
  * Gustafsson et al. (2008) Gustafsson, B., Edvardsson, B., Eriksson, K., et al. 2008, A&A, 486, 951. doi:10.1051/0004-6361:200809724
  * Hayden et al. (2015) Hayden, M. R., Bovy, J., Holtzman, J. A., et al. 2015, ApJ, 808, 132. doi:10.1088/0004-637X/808/2/132
  * Helmi (2020) Helmi, A. 2020, ARA&A, 58, 205. doi:10.1146/annurev-astro-032620-021917
  * Helmi et al. (2018) Helmi, A., Babusiaux, C., Koppelman, H. H., et al. 2018, Nature, 563, 85. doi:10.1038/s41586-018-0625-x
  * Helmi et al. (1999) Helmi, A., White, S. D. M., de Zeeuw, P. T., et al. 1999, Nature, 402, 53. doi:10.1038/46980
  * Kim et al. (2002) Kim, Y.-C., Demarque, P., Yi, S. K., et al. 2002, ApJS, 143, 499. doi:10.1086/343041
  * Koppelman et al. (2019) Koppelman, H. H., Helmi, A., Massari, D., et al. 2019, A&A, 631, L9. doi:10.1051/0004-6361/201936738
  * Kraft & Ivans (2003) Kraft, R. P. & Ivans, I. I. 2003, PASP, 115, 143. doi:10.1086/345914
  * Lejeune et al. (1997) Lejeune, T., Cuisinier, F., & Buser, R. 1997, A&AS, 125, 229. doi:10.1051/aas:1997373
  * Lejeune et al. (1998) Lejeune, T., Cuisinier, F., & Buser, R. 1998, A&AS, 130, 65. doi:10.1051/aas:1998405
  * Li et al. (2018) Li, L., Shen, S., Hou, J., et al. 2018, ApJ, 858, 75. doi:10.3847/1538-4357/aabaef
  * Limberg et al. (2020) Limberg, G., Rossi, S., Beers, T. C., et al. 2020, arXiv:2011.08305
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94. doi:10.3847/1538-3881/aa784d
  * Matsuno et al. (2019) Matsuno, T., Aoki, W., & Suda, T. 2019, ApJ, 874, L35. doi:10.3847/2041-8213/ab0ec0
  * Myeong et al. (2019) Myeong, G. C., Vasiliev, E., Iorio, G., et al. 2019, MNRAS, 488, 1235. doi:10.1093/mnras/stz1770
  * Naidu et al. (2020) Naidu, R. P., Conroy, C., Bonaca, A., et al. 2020, ApJ, 901, 48. doi:10.3847/1538-4357/abaef4
  * Nissen & Schuster (2010) Nissen, P. E. & Schuster, W. J. 2010, A&A, 511, L10. doi:10.1051/0004-6361/200913877
  * Padmanabhan et al. (2008) Padmanabhan, N., Schlegel, D. J., Finkbeiner, D. P., et al. 2008, ApJ, 674, 1217. doi:10.1086/524677
  * Pinsonneault et al. (2003) Pinsonneault, M. H., Terndrup, D. M., Hanson, R. B., et al. 2003, ApJ, 598, 588. doi:10.1086/378854
  * Pinsonneault et al. (2004) Pinsonneault, M. H., Terndrup, D. M., Hanson, R. B., et al. 2004, ApJ, 600, 946. doi:10.1086/379925
  * Reid (1997) Reid, I. N. 1997, AJ, 114, 161. doi:10.1086/118462
  * Sahlholdt et al. (2019) Sahlholdt, C. L., Casagrande, L., & Feltzing, S. 2019, ApJ, 881, L10. doi:10.3847/2041-8213/ab321e
  * Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161. doi:10.1086/145971
  * Schlafly & Finkbeiner (2011) Schlafly, E. F. & Finkbeiner, D. P. 2011, ApJ, 737, 103. doi:10.1088/0004-637X/737/2/103
  * Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525. doi:10.1086/305772
  * Scolnic et al. (2015) Scolnic, D., Casertano, S., Riess, A., et al. 2015, ApJ, 815, 117. doi:10.1088/0004-637X/815/2/117
  * Sills et al. (2000) Sills, A., Pinsonneault, M. H., & Terndrup, D. M. 2000, ApJ, 534, 335. doi:10.1086/308739
  * Tonry et al. (2012) Tonry, J. L., Stubbs, C. W., Lykke, K. R., et al. 2012, ApJ, 750, 99. doi:10.1088/0004-637X/750/2/99
  * VandenBerg & Clem (2003) VandenBerg, D. A. & Clem, J. L. 2003, AJ, 126, 778. doi:10.1086/376840
  * Venn et al. (2004) Venn, K. A., Irwin, M., Shetrone, M. D., et al. 2004, AJ, 128, 1177. doi:10.1086/422734
  * Westera et al. (2002) Westera, P., Lejeune, T., Buser, R., et al. 2002, A&A, 381, 524. doi:10.1051/0004-6361:20011493
  * Wolf et al. (2018) Wolf, C., Onken, C. A., Luvaul, L. C., et al. 2018, PASA, 35, e010. doi:10.1017/pasa.2018.5
  * Yanny et al. (2009) Yanny, B., Rockosi, C., Newberg, H. J., et al. 2009, AJ, 137, 4377. doi:10.1088/0004-6256/137/5/4377