# M Subdwarf Research<EMAIL_ADDRESS>Atmospheric Parameters and Kinematics

Shuo Zhang CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, China  University of Chinese Academy of
Sciences, Beijing 100049, China  Department of Astronomy, School of Physics,
Peking University, Beijing 100871, P. R. China  Kavli institute of Astronomy
and Astrophysics, Peking University, Beijing 100871, P. R. China  A-Li Luo CAS
Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Beijing 100101, China  University of Chinese Academy of Sciences, Beijing
100049, China  School of Information Management & Institute for Astronomical
Science, Dezhou University, Dezhou 253023, China  Department of Physics and
Astronomy, University of Delaware, Newark, DE 19716, USA  Georges Comte  Aix-
Marseille Univ, CNRS, CNES, LAM, Laboratoire d’Astrophysique de Marseille,
Marseille, France  Rui Wang CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Beijing 100101, China  University of Chinese
Academy of Sciences, Beijing 100049, China  Yinbi Li  CAS Key Laboratory of
Optical Astronomy, National Astronomical Observatories, Beijing 100101, China
Bing Du  CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, China  Wen Hou  CAS Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Beijing 100101, China  Li Qin
CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Beijing 100101, China  School of Information Management & Institute for
Astronomical Science, Dezhou University, Dezhou 253023, China  John Gizis
Department of Physics and Astronomy, University of Delaware, Newark, DE 19716,
USA  Jian-Jun Chen  CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Beijing 100101, China  Xiang-Lei Chen  CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Beijing
100101, China  University of Chinese Academy of Sciences, Beijing 100049,
China  Yan Lu  CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, China  School of Information Management &
Institute for Astronomical Science, Dezhou University, Dezhou 253023, China
Yi-Han Song  CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Beijing 100101, China  Hua-Wei Zhang  Department of Astronomy,
School of Physics, Peking University, Beijing 100871, P. R. China  Kavli
institute of Astronomy and Astrophysics, Peking University, Beijing 100871, P.
R. China  Fang Zuo  CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Beijing 100101, China  University of Chinese
Academy of Sciences, Beijing 100049, China

###### Abstract

Applying the revised M subdwarf classification criteria discussed in Paper i@
to LAMOST DR7, combining the M subdwarf sample from Savcheva et al, a new M
subdwarf sample was constructed for further study. The atmospheric parameters
for each object were derived fitting with the PHOENIX grid, combining with
Gaia DR2, the relationship between the gravity and metallicity were explored
according to the locus both in the color-absolute magnitude diagram and the
reduced proper motion diagram. Objects that have both the largest gravity and
the lowest metallicity are located away from the main-sequence cloud and may
be considered as the intrinsic M subdwarfs, which can be classified as
luminosity class<EMAIL_ADDRESS>Another group of objects whose spectra show typical M
subdwarf characters have lower gravity and relatively moderate metal
deficiency and occupy part of the ordinary M dwarf region in both diagrams.
The Galactic $U$, $V$, $W$ space velocity components and their dispersion show
that the local Galactic halo population sampled in the solar neighborhood is
represented by objects of high gravity and an inconspicuous bimodal
metallicity distribution, with a fraction of prograde orbits. The other M
subdwarfs seem to partly belong to the thick disk component with a significant
fraction of thin disk moderately metal-poor objects intricately mixed with
them. However, the selection effects, especially the favored anti-center
direction of investigation in the LAMOST sub-sample, but also contamination by
multiplicity and parameter coupling could play important roles and need to be
further investigated.

Stellar astronomy — Stellar types — Late-type stars— M stars, Stellar
kinematics, Atmospheric parameters

††journal: ApJ††software: astropy (Astropy Collaboration et al., 2013),
matplotlib (Barrett et al., 2005), pandas(McKinney et al., 2010), numpy
(Oliphant, 2007), scipy (Virtanen et al., 2019)

##  1 Introduction

M subdwarfs are usually recognized as the oldest members of the low-mass
stellar populations and metal-poor counterparts of the late-type M dwarfs.
With their extremely long nuclear-burning lifetimes, subdwarfs are supposed to
represent early generations of star formation and are thus potential tracers
of Galactic dynamic and chemical evolution history (Savcheva et al., 2014).
Despite their scarcity in the solar neighborhood, M subdwarfs are supposed to
comprise the majority of stars in the Milky Way stellar halo (Bochanski et
al., 2013), while a number of them also populate the thick disk, whose
population is supposed to be older than that of the thin disk. Additionally,
the cool atmospheres of M subdwarfs provide conditions for studying molecule
and dust formation as well as radiative transfer in metal-poor environments
(Allard et al., 1997).

In many early investigations, M subdwarfs are considered as metal-poor main-
sequence dwarfs (e.g. Lépine et al. 2007; Rajpurohit et al. 2014) solely with
some noteworthy kinematic and spectral properties: red subdwarfs, exhibiting
large space velocities relative to the Sun, kinematically associated with the
Galactic halo (Gizis, 1997) and thick disk, display shallower TiO absorption
bands in the optical spectra than the ordinary red dwarfs dominating the
Galactic thin disk. Historically, they were usually searched based on their
low luminosity combined with large proper motion (Sandage, 1969; Jones, 1972;
Hartwick et al., 1984; Sandage & Kowal, 1986; Gizis, 1997; Lépine et al.,
2003, 2007; Smith et al., 2009; Jao et al., 2011) and characteristic spectral
features (Mould, 1976; Mould & McElroy, 1978; Ake & Greenstein, 1980; Lépine &
Scholz, 2008; Savcheva et al., 2014; Bai et al., 2016; Zhang et al., 2019;
Hejazi et al., 2020). The research of metal-poor subdwarfs also extends to the
late M-type and L-type in recent years (e.g. Zhang et al. 2017b, a, 2018;
Zhang 2019).

On the other hand, some studies have suggested that subdwarfs should have a
luminosity classification independent of dwarfs—vi@ because of 1$\sim$2
magnitude less luminous than the main sequence dwarfs in the H-R diagram,
forming an independent sequence, and more or less showing larger surface
gravity than the dwarfs (Jao et al., 2008). Results from Kesseli et al. (2019)
show that ultra subdwarfs can be as much as five times smaller than their
solar-metallicity counterparts for a given temperature. However, subdwarf
identification in most of the researches depends much more on the metallicity
rather than the gravity, and whether gravity constrain for subdwarfs is
necessary still remains an open question.

Characterizing M subdwarfs in terms of atmospheric parameters has long been a
challenge. Like M dwarfs, the opacity sources are dominated by the molecular
absorption bands such as TiO, VO, MgH, FeH and CaH in the optical, as well as
H${}_{\text{2}}$O and CO in the infrared, each of which has millions of
spectral lines. These complex band structures leave no window to access the
true continuum and only allow the strongest atomic lines such as Cai@, Nai@,
Ki@ to show out at low spectral resolution. In addition, dust clouds form with
decreasing temperature, the stellar atmosphere becomes more complicated, and
the spectra show increasingly diatomic and triatomic molecules (Rajpurohit et
al., 2016). Thus, the traditional technique of estimating the effective
temperature ($T_{\text{eff}}$) of a star based on black body approximation and
broadband photometry is not applicable to cool M subdwarfs in which the true
continuum is covered by a wide variety of molecular absorbers(Allard et al.,
1997). It is also difficult to disentangle the effect of reduced metal
abundance from that of the increasing surface gravity or the temperature
decrease because these three parameters affect the pressure structure of the
photosphere in a similar way (Jao et al., 2008).

Since accurate metallicities for most M subdwarfs are difficult to measure,
proxies for metallicity have been introduced. The most widely used
metallicity-tracer parameter $\zeta$, (a parameter derived from CaH2, CaH3,
and TiO5 molecular bandhead indices originally used by Gizis 1997), was
introduced by Lépine et al. (2007) and revised by Dhital et al. (2012) and
Zhang et al. (2019). With the parameter $\zeta$ or revised definition, two
large samples of M subdwarfs are spectroscopically identified by Savcheva et
al. (2014) and Zhang et al. (2019) using Sloan Digital Sky Survey (SDSS; York
et al. 2000) and the Large Sky Area Multi-Object Fiber Spectroscopic Telescope
(LAMOST; Cui et al. 2012) respectively. According to the decreasing order of
$\zeta$, the M subdwarfs can be divided into three metallicity subclasses: M
subdwarfs (sdM), extreme metal-poor M subdwarfs (esdM), and ultra metal-poor M
subdwarfs (usdM).

The first model grid based on the PHOENIX radiative transfer code (Allard &
Hauschildt, 1995) has been used by several authors to compare synthetic
spectra to observed ones and to the broad-band colors: Dahn et al. (1995);
Leggett et al. (1996); Jones et al. (1996). Gizis (1997) was thus able to
constrain the metallicity scale of subdwarfs from $-$2 dex for esdM to $-$1 to
$-$1.5 for sdM, while the stars not classified as subdwarfs were spanning the
range $-$1 dex to 0. However, this pioneering model grid was far from being
optimized because of the lack of theoretical molecular opacities, especially
for the very important species CaH.

The NextGen model grid (Hauschildt et al., 1999) brought a number of
improvements and was in turn used by e.g. Woolf & Wallerstein (2005) who
demonstrated the $\alpha$ enrichment of halo metal-poor M stars. Several
papers using NextGen models to interpret spectra of cool M dwarfs and
subdwarfs were subsequently published (e.g.Leggett et al. 2000; Lépine et al.
2004) while Burgasser & Kirkpatrick (2006) compared the performances of
NextGen models and AmesCond models on the spectrum of an esdM star. Progress
in modeling led the PHOENIX-based computations to the BT-Settl series of
grids, the last generation of which proved highly successful in reproducing
the features of the observed spectra in detail. Comparisons of low mass stars
between a few high-resolution spectroscopic spectra and theoretical synthetic
ones show that the latest PHOENIX models have already reached a
milestone.(Rajpurohit et al., 2014, 2018a, 2018b).

Gaia Data Release 2 (Gaia Collaboration et al., 2018b) has provided astrometry
and photometry on more than one billion stars. As a result, a large number of
subdwarfs that were out of the reach of parallax studies now have accurate
available trigonometric parallaxes permitting distance determination and hence
their placement in the Hertzsprung-Russell diagram, while their 3D kinematics
can be studied thanks to high accuracy proper motions from Gaia and radial
velocity from spectroscopy.

M subdwarfs have been found both at high Galactic latitudes, as expected for
halo population members, and at much lower latitudes in fields located towards
the Galactic anti-center, suggesting that one could use them to study the mix
of various population components in the Solar neighborhood. It becomes thus
especially interesting to explore the links between their atmospheric
parameters, especially gravity and metallicity, and their kinematic
properties. For this, we need a reasonably accurate estimation of the
atmospheric parameters of a large sample of subdwarfs for which we have
accurate photometry and astrometry and the same data on a control sample of
ordinary M dwarfs. The present work, based only on low-resolution
spectroscopy, has its limitations and must be considered as a preliminary
attempt towards this goal.

The paper is organized as follows. In Section 2, we briefly summarize the
observational material that will be used in Sections 4 and 5. In Section 3, we
analyze the properties of the PHOENIX-based BT-Settl CIFIST model grid to
evaluate its performance by studying spectral indices of BT-Settl synthetic
spectra compared with those measured on real-world template spectra of M
dwarfs and subdwarfs. Then, we fit the template spectra with the synthetic
grid to get the atmospheric parameter ranges for each template spectra. In
section 4, we derive the atmospheric parameters for spectra of a sample of M
dwarf and subdwarf from the best-fit PHOENIX BT-Settl synthetic spectra. This
leads to the construction of a cleaned genuine subdwarf sample, a normal dwarf
sample and a third population whose properties point towards non-genuine
subdwarfs but rather metal-deficient dwarfs. In section 5, we explore the
links between the distribution of the atmospheric parameters of the stars
belonging to these three samples and their photometric and kinematic
properties as provided by the Gaia Data Release 2, using color-absolute
magnitude diagram, reduced proper motion diagram and 3D Galactic motion
distributions. Finally, conclusions and summary are provided in Section 6.

##  2 Observational data

The low-resolution spectra of M dwarfs and subdwarfs are from LAMOST DR7 and
the SDSS DR7, and the astrometric data is from Gaia DR2.

LAMOST is a reflecting Schmidt telescope located in Xinglong Station of
National Astronomical Observatory, China (40∘N, 105∘E) with a mean aperture of
4.3 meters and a field of view of 5∘. The observable sky area extends from
$-$10∘ to +90∘ declination. 4000 optical fibers positioned on the focal plane
yield a high spectrum acquisition rate per night. Until March 2020, DR7
published more than 10 million low-resolution spectra (R$\sim$1800) covering
3800$-$9000$\rm\AA$ (Luo et al., 2015) and 2.17 million medium resolution
spectra (R$\sim$7500) covering 3700$-$5900$\rm\AA$ and 5700$-$9000$\rm\AA$
(Liu et al., 2020).

The SDSS began regular survey operations in 2000, and has progressed through
several phases until now: SDSS-i@ (2000$-$2005), SDSS-ii@ (2005$-$2008), SDSS-
iii@ (2008$-$2014), and SDSS-iv@ (2014$-$2020). The SDSS DR7 contains over 1.6
million low-resolution spectra in total, including 930,000 galaxies, 120,000
quasars, and 460,000 stars. It was collected by a dedicated wide-field 2.5 m
telescope at Apache Point Observatory (Gunn et al., 2006). The telescope
employed a drift-scan technique, imaging the sky in $u$,$g$,$r$,$i$,$z$ wide
bands along five camera columns (Gunn et al., 1998). The fiber-fed
spectrographs acquired 640 spectra (R$\sim$2000) simultaneously, and the
spectral range is 3800$-$9200 $\rm\AA$.

The proper motions, trigonometric parallaxes, and photometry used in this work
come from Gaia Data Release 2 made available in April 2018 by European Space
Agency. Gaia DR2 contains the position and brightness information of more than
1.69 billion stars, the parallax and proper-motion measurements for 1.33
billion stars (Gaia Collaboration et al., 2018a; Lindegren et al., 2018; Luri
et al., 2018), the color information for 1.38 billion stars (Gaia
Collaboration et al., 2018a), the radial velocities for more than 7 million
stars (Katz et al., 2019), etc. The mean parallax error is up to 0.7 mas for
sources having G=20 mag, and the proper motion uncertainty is up to 0.2 and
1.2 mas yr-1 for G = 17 mag and G = 20 mag respectively.

##  3 PHOENIX BT-Settl Model Analysis

###  3.1 PHOENIX BT-Settl Model Grid

Classical theoretical model calculation codes include ATLAS code from Kurucz
(1973) and Castelli & Kurucz (2004), MARCS code from Gustafsson et al. (1975,
2008), PHOENIX code from Allard & Hauschildt (1995), and so on. In the
atmosphere of very low mass stars, various molecular absorption (each with
hundreds of thousands to millions of spectral lines) and the presence of
numerous condensates make accurate modeling extremely complicated; the
convection zone extends to the most outer photosphere layer, which means that
the evolution of the model strictly depends on the precise treatment of
surface boundary (Allard et al., 2013). It is worth celebrating that in recent
years, the PHOENIX code differs from previous methods by computing the
opacities during the model execution (or on-the-fly), considering non-local
thermodynamic equilibrium effects, etc., and combining with the latest solar
abundance (Asplund et al., 2009; Caffau et al., 2011). The atmospheric model,
especially for modeling very low mass stars such as M-type stars, brown
dwarfs, and even Jupiter-like planets, has made breakthrough progress.

The latest version PHOENIX BT-Settl models represent decisive progress
compared to the previous model grids (Allard et al., 2001) in different
aspects, including updated molecular line lists, better solar abundance
estimates and treatment of convection, though residual incompleteness of
opacities and a limitation of the mixing length theory formalism still exist.
Further studies on a complete, comprehensive and uniform grid of models are
addressing these shortcomings and will better reproduce observational
constraints (Allard, 2016).

The synthetic spectra generated by BT-Settl model atmospheres (Allard et al.,
2012, 2013; Baraffe et al., 2015) have been compared with observed spectra and
their consistency verified in a number of recent M dwarf/subdwarf studies
(e.g., Rajpurohit et al. 2014, 2016, 2018a, 2018b).

In this work, we use the version BT-Settl CIFIST2011 model atmosphere which is
valid across the entire parameter range (Allard et al., 2011, 2012). A part of
pre-calculated grid as list in 1 is used in the analysis and the following
parameter measurement process.

The models are calculated with effective temperatures from $T_{\text{eff}}$ =
400 to 8000 K in 100 K steps, surface grativity ranging from log $g$ = 0.0 to
6.0 in steps of 0.5 dex, and metallicity from [M/H]=$-$2.5 to +0.5 in steps of
0.5 dex (M stands for all elements heavier than H and He; assuming [X/H] =
[Fe/H] for most elements and so [Fe/H] denotes the overall metallicity). Alpha
element (O, Ne, Mg, Si, S, Ar, Ca, and Ti) enhancement is taken into account
as follows: for [M/H] = 0.0, no enhancement; for [M/H]=$-$0.5, a value of
[$\alpha$/M]=+0.2 has been used; for [M/H] $\leq-$1.0, [$\alpha$/M]=+0.4. Note
that in PHOENIX the [$\alpha$/M] is similar to [$\alpha$/Fe] since it is
computed as the change in dex of the abundance of alpha elements relative to
the non-alpha average, and the heaviest is Fe.

Table 1: Parameter Space of the Grid Used in this Work  Variable | Range | Step size
---|---|---
T${}_{\text{eff}}$ | 2700 $-$ 4500 | 100 K
log g | 0 $-$ 6.0 | 0.5 dex
$[\text{M/H}]$ |  $-$2.5 $-$ +0.5 | 0.5 dex
$[\alpha/\text{M}]$ | 0 $-$ +0.4 | 0.2 dex

###  3.2 Comparison of Models with Real-world Template Spectra in the
“canonical” diagram

In this section, we firstly compare M-type template spectra from Zhong et al.
(2015) with BT-Settl models. The templates, assembled from SDSS low-resolution
(R$\sim$2000) spectra and classified using Lépine et al. (2007, 2013)
calibration, are spanning the spectral subtypes K7.0 to M8.5 and covering
every half-subtype across the range.

The relevant pre-calculated synthetic spectra111https://phoenix.ens-
lyon.fr/Grids/BT-Settl/CIFIST2011/SPECTRA/ generated by PHOENIX code have been
convolved down to the spectral resolution of SDSS, using a series of Gaussian
kernels with parameters sigma and corresponding FWHM listed in Table 2. To
determine the sigma value of a convolution kernel, we fit gaussians profiles
to night sky lines (assuming they have a natural width much smaller than the
spectral psf). In the red range, clean lines of OH molecule are numerous, in
the blue the 5577$\rm\AA$ is also good, as well as some mercury lines, except
5461$\rm\AA$ which has fine structure. Na D lines of city lights are avoided
because they have a very large width (high pressure lamps).

Table 2: The gaussian kernels with the following sigmas and corresponding fwhm used for convolving synthetic spectra.  Central $\rm\lambda(\rm\AA)$ | $\rm\sigma$ | Pixels of 0.5 $\rm\AA$ | FWHM
---|---|---|---
4200 | 1.05 | 2.10 | 4.935
5100 | 1.08 | 2.16 | 5.075
6000 | 1.25 | 2.50 | 5.875
6900 | 1.43 | 2.86 | 6.720
7800 | 1.40 | 2.80 | 6.580
8700 | 1.39 | 2.78 | 6.535
Figure 1: The “canonical” diagram classically used to identify subdwarfs from
dwarfs: spectral composite index CaH2+CaH3 versus TiO5. Models are plotted
with the following codes : continuous lines: log $g$ = 4.5 models, dashed
lines: log $g$ = 5.0 models, and dotted lines: log $g$ = 5.5 models. Black :
solar metallicity, blue: $-$1.0 dex, and magenta: $-$2.0 dex. The small
2-digit labels indicate $T_{\text{eff}}$ points (e.g. “32” means 3200 K).
Observed template spectra: black dots represent normal main-sequence dwarfs,
subdwarf templates colored in blue are for “sdM” subclass, green for “esdM”
and magenta for “usdM” subclass. The M0v@ to M7v@ “standard dwarf” spectral
type sequence is labeled on all graphs.

Figure 1 is the “canonical” [TiO5, CaH2+CaH3] diagram classically used by many
authors to identify and spectroscopically separate the subdwarfs from the
normal dwarfs. The spectral indices were defined initially by Reid et al.
(1995). The boundaries of the d/sd/esd/usd metallicity classification scheme
are also plotted. This diagram may represent more or less a metallicity-
gravity diagram, since TiO5 is a metallicity (and alpha) indicator while
CaH2+CaH3 is rather a gravity indicator, and both are naturally strongly
$T_{\text{eff}}$ dependent. The subdwarfs are supposed to order along
temperature sequences, while the fan-shape of their distribution reflects
their atmospheric metal deficiencies.

The normal dwarf template sequence is well-ordered and fits quite well with
the model family {log $g$ = 4.5, [Fe/H] = 0.0} track, although with the
coolest types (M6$\sim$M7) being a bit offset, as it would be expected from an
insufficient value of the model gravity.This is an important point and needs
to be further discussed. All late-type M dwarfs are expected to have higher
surface gravity than early M dwarfs due to the near 1-to-1 mass radius
relation for low-mass stars. Note that between metallicities of 0.0 and
$-$1.0, the displacement of the model track is quite small even at the lowest
$T_{\text{eff}}$, making the estimates of moderate metal deficiencies rather
difficult on this diagram.

The subdwarf templates also draw quite well-defined and well separated
sequences with increasing slopes from sd to usd, the latter one being almost
vertical at a constant TiO5 index of almost null absorption. The coolest types
of dwarfs and sd are somehow confused, this questions the validity of the
templates themselves which are built from spectra either in insufficient
number or too noisy to be really representative of the genuine subtype
definition.

The metallicity of the sd subclass does not comply with a $-$1.0 value, it
must be more deficient, or affected by a higher gravity, or both. The model
family {log $g$ = 5.0, [Fe/H] = $-$2.0} could provide a reasonable fit for
usdM0$\sim$3, but the track moves towards significant values of the TiO5
opacity for later subtypes, thus the real stars must be more deficient. The
model family {log $g$ = 5.5, [Fe/H] = $-$2.0} track yields more or less an
outer envelope for the subdwarfs, implying that most of the right-hand
distribution of their template representative points is driven more by large
metal deficiencies than by high gravity.

###  3.3 Fitting the dM and sdM Templates with the Model

Using template spectra rather than hundreds of individual spectra to compare
with the models enables to smooth accidental differences due to noise and
uncertainties in individual classification, and in principle gives access to
high signal-to-noise data. However, the final quality of a template depends on
the number of individual spectra co-added to build it. We shall hereafter use
the COMPLETE collection of templates of Zhong et al. (2015), i.e. the
interpolated 221 spectra in total spanning 12 subclasses (from dMr to usdMp)
and 18 subtypes (from M0 to M8.5).

The parameter grid of the synthetic spectra is shown in Table 1. Note that the
grid is not uniform: the value log $g$ = 6.0 only pertains for
$T_{\text{eff}}\geq$ 3000 K, and the [$\alpha$/M] ratio is dependent on the
metallicity.

The spectral fitting region extends from 6000$\rm\AA$ to 8800$\rm\AA$ with a
step size of 1$\rm\AA$ which covers most prominent spectral features used for
classification. Before the fitting process, the synthetic spectra were
smoothed at the resolution of the observed spectra.

In the fitting process, we set the spectral flux as a function of
$T_{\text{eff}}$, log $g$, and [M/H], and assume:

$F_{\text{o},\lambda}=F_{\text{m},\lambda}\times
P_{\text{n},\lambda}+\epsilon_{\lambda},$ (1)

where $F_{\text{o},\lambda}$ is the flux of the observed spectrum at
wavelength $\lambda$, $F_{\text{m},\lambda}$ is the flux of the synthetic
spectrum generated by PHOENIX model wavelength $\lambda$,
P${}_{\text{n},\lambda}$ is an n-order polynomial factor used to correct the
inaccuracies of the continuum of the observed spectrum (here we set n = 4
based on the optimal experimental results), $\epsilon_{\lambda}$ is the
difference between the synthetic spectrum.

The equation above can be solved by minimizing the loss function
L=$\sum_{\lambda=1}^{\text{N}}(F_{\text{m},\lambda}\times$
P${}_{\text{n},\lambda}-F_{\text{o},\lambda})^{2}$ using the least square
method to obtain the coefficient matrix solution of P${}_{\text{n},\lambda}$.
Then we obtain the best-fit synthetic spectrum by minimizing the $\chi^{2}$
distance,

$\chi^{2}\ =\
\frac{\sum_{i\text{=1}}^{N}[(F_{\text{o}_{i}}-F^{\prime}_{\text{m}_{i}})^{2}/\epsilon_{i}^{2}]}{N}$
(2)

where $F^{\prime}_{\text{m}_{i}}$ is the flux of the $i^{\text{th}}$ pixel of
the synthetic spectrum corrected by polynomial, $\epsilon_{i}$ is the flux
difference of the $i^{\text{th}}$ pixel, and $N$ is the number of pixels
involved in the calculation.

For each given template spectrum, we calculate the $\chi^{2}$ values when
matching all the synthetic spectra in the grid and then interpolate between
the parameters by synthesizing linear combinations of top-five best-fit
synthetic spectra, as the same way in Yee et al. (2017).

Figure 2: Template spectra of subtype dM0, sdM2, esdM4 and usdM6 with their
corresponding synthetic spectra and fitting residuals.  Figure 3: The
residuals between template spectra of different spectral subtypes and their
best-fit synthetic spectra. Black, blue, purple, and red represent subclass
dM, sdM, esdM, and usdM respectively. The error bar is the standard deviation
of the difference between the flux of the template spectrum and the flux of
the synthetic spectrum. The number beside each node represents the number of
spectra used to construct the template spectrum.  Figure 4: The distributions
of estimated atmospheric parameters of the templates spectra of different
subclasses. The colors represent the same as in Figure 3.

The fitting results of four template spectra of different spectral subtypes
and metallicity subclasses (dM0, sdM2, esdM4 and usdM6) with their
corresponding estimated parameters are shown in Figure 2 as examples. The
shape of the residual spectra indicate that the errors mainly come from the
region of 6000$-$7000Å, where there remain some incompleteness in line lists
and/or missing/incomplete opacities on some molecules used in the modeling
(CaOH, CN, CaH, etc.). Besides, the far red region of the observed spectra
could be contaminated by telluric absorption.

Figure 3 shows the residual levels of the fitting results of all the
templates. The residual level of the fitting results gradually increases from
$\sim$5% for early type M stars to 10% for the late types: both spectral
quality and the number of objects for template building decrease dramatically
for the latest types.

Finally, the distributions of the estimated results of log $g$ and [Fe/H] of
the templates above are shown in Figure 4. The dispersions are large for both
the two parameters, with the basic trend being a clear decrease of the average
metallicity from sd to usd accompanied by a more confuse increase of gravity.
Note that when using the complete set of templates (221 in total), we can meet
spectra of slightly atypical characteristics, as for instance illustrated in
the top panel of Figure 2 in which a dM0 is metal-poor and 100 K cooler than a
standard dM0 of solar metallicity.

##  4 Sample Construction and Atmospheric Parameter Estimates

To estimate the atmospheric parameters $T_{\text{eff}}$, log $g$ and [M/H] of
M subdwarfs from low-resolution spectra and investigate the trends of these
parameters with the kinematic properties of the stars, we need a sufficiently
large sample of subdwarfs spanning the broadest possible range of parameters.

In Paper i@, we had explored LAMOST DR4 in which 2,791 M subdwarf were
identified. Extending the search to LAMOST DR7, a total of 4,767 visually
selected candidates are retrieved including the previously obtained DR4 ones.
Combining this sample with SDSS objects published by Savcheva et al. (2014),
we submitted these spectra to a least-squares based fitting process with
PHOENIX synthetic spectra. The determination of atmospheric parameters is not
always possible with a reasonable accuracy because the error budget of the
fitting process depends on the signal-to-noise (S/N) of the spectra. Thus, the
final combined LAMOST + SDSS sample of stars with known atmospheric parameters
will be reduced with respect to the initial numbers.

###  4.1 Sample Construction and Individual Spectra Fitting

Because of magnitude limitation, most M dwarfs of LAMOST range in spectral
type before M3, and only a few cooler and fainter spectra of low S/N are
available. The data of SDSS DR7 covers a more extended parameter space, with
cooler and lower metallicity objects. The strategies of two surveys point to
different fields, i.e. SDSS at high Galactic latitude favoring halo and thick
disk objects detection, while LAMOST survey has in priority been targeted at
regions of the Galactic anti-center. Therefore, in this study we combine the
subdwarfs from SDSS DR7 (SEGUE-I/II) and LAMOST DR7 to construct a consistent
sample of M subdwarfs, more representative of the total population of these
stars.

First, it is useful to discuss three points on which the processing of the
LAMOST and SDSS spectra sets will differ: (1) selecting criteria, (2)
spectroscopic resolution, and (3) spectral data reduction.

(1) Selecting criteria.

Several works have been conducted to identify M subdwarfs from SDSS and LAMOST
low resolution spectral data sets (West et al., 2004; Lépine & Scholz, 2008;
Bochanski et al., 2013; Savcheva et al., 2014; Zhang et al., 2016; Bai et al.,
2016), Paper i@, and the two largest samples published were from Savcheva et
al. (2014) and Paper i@ respectively.

The selecting conditions differ for these two latter samples: the principal
separator used is the standard parameter $\zeta$ initially introduced by
Lépine et al. (2007) based on a combination of CaH2, CaH3 and TiO5 spectral
indices:

$\zeta_{\text{CaH}/\text{TiO}}=\frac{1-\text{TiO5}}{1-[\text{TiO5}]_{Z_{\odot}}}$
(3)

in which the [TiO5]${}_{Z_{\odot}}$ is a third order polynomial fit of the
TiO5 spectral index as a function of the CaH2+CaH3 index as:

$\begin{split}[\text{TiO5}]_{Z_{\odot}}\ &=\ a\\\ +&b\times(\text{CaH2}\ +\
\text{CaH3})\\\ +&c\times(\text{CaH2}\ +\ \text{CaH3})^{2}\\\
+&d\times(\text{CaH2}\ +\ \text{CaH3})^{3}\end{split}$ (4)

so that $\zeta$ = 1 for solar metallicity and $\zeta$ = 0 for the most metal-
poor objects where TiO5 absorption becomes undetectable. The coefficients $a$,
$b$, $c$, $d$ derived for SDSS objects by Lépine et al. (2007) led to the
calibration $\zeta_{\text{L07}}$ which was later revised by Lépine et al.
(2013) (hereafter $\zeta_{\text{L13}}$). The condition for a star to belong to
the subdwarf population was set as $\zeta_{\text{L07}}<$0.825 and
$\zeta_{\text{L13}}<$0.825

Since $\zeta$ was found to be dependent on the M dwarfs reference sample,
instrumental setup and various factors influential in the reduction process
(Lépine et al., 2013), we proposed a new calibration adapted to LAMOST spectra
in Paper i@ (hereafter $\zeta_{\text{Z19}}$), and we suggested the condition
for LAMOST subdwarfs being set as $\zeta_{\text{Z19}}<$0.75 with an additional
screening condition based on the CaH1 index.

(2) Spectroscopic resolution.

Along the wavelength of a spectrum from SDSS or LAMOST, the physical
resolution (depending on the grating$\times$focal length of
camera$\times$pixel size of the detector) varies in a complicated way. For
LAMOST, when spectra are extracted and rebinned in constant log-wavelength
steps by the 1D pipeline, this variation is +/- “dissolved” into another
variation. Additionally, though in theory, all spectra obtained from the same
instrument are supposed to have an identical resolving power, in practice for
a large-scale survey project, the resolution changes slightly at different
observation dates, due to imperfect instrument stability.

In the actual calculation, assuming a gaussian instrumental profile, the
resolution R of a spectrum at wavelength $\lambda$ can be derived from
$\lambda$/$\Delta\lambda$, where $\Delta\lambda$ is the full width at half
maximum (FWHM) of the spectral line broadening at that wavelength. The
Gaussian kernel $\sigma$ = FWHM/2.355 used to convolve the synthetic spectrum
is derived from $\sqrt{{\sigma_{\text{o}}}^{2}-{\sigma_{\text{m}}}^{2}}$,
where $\sigma_{\text{m}}$ corresponds to the synthetic spectrum, and
$\sigma_{\text{o}}$ corresponds to the observed spectrum. Since the resolution
of the synthetic spectra is extremely high, we set $\sigma_{\text{m}}$ as 0
here.

Prior to the fitting process, the observed spectra are shifted to rest-frame
based on the measured radial velocities. The fitting range was set to
6000$-$8800Å, in which the pixels marked by “bad” flags and the spectral
intervals heavily contaminated by telluric absorption (7210$-$7350Å,
7560$-$7720Å, and 8105$-$8240Å) were masked. We divided the region
6000$-$8800$\rm\AA$ into 7 bands evenly and used a dedicated convolution
kernel for each 400$\rm\AA$ band.

For SDSS and LAMOST spectra, we adopted different convolution kernels
calculated as follows. For an SDSS spectrum, we computed the mean value of the
$\sigma$ in each band by the “dispersion” value of each pixel stored in the
fits file. For a LAMOST spectrum, we calculated the mean value of the FWHM of
the arc lamp spectrum in each band, obtained on the same day for wavelength
calibration purpose.

(3) Data reduction.

In the fitting process described above, we first correct the continuum of an
observed spectrum using a fourth-order polynomial and then choose the best-fit
synthetic spectrum based on the chi-square minimum principle. We find that the
continuum slopes of some observed spectra of LAMOST set differ largely from
their corresponding corrected spectra. This deformation is mainly caused by
flux calibration problems (Du et al., 2016) and would affect the accuracy of
the spectral indices. To control this unwanted bias, we measure the spectral
indices of each polynomial-corrected observed spectrum, and calculate the
difference between them and the original indices measured on the uncorrected
spectrum. We then construct the distribution of each index differences, as
shown in Figure 5. Although the offset of a single index is generally small,
the error on the $\zeta$ parameter composed of multiple indices may be
considerably enlarged. Hence, targets with at least one spectral index
difference beyond 3$\sigma$ of the corresponding difference distribution are
removed from the final subdwarf sample.

Figure 5: Each subdiagram shows a spectral index error distribution for the
LAMOST subdwarfs. The errors are the differences between the values measured
from the original spectra minus the values measured from the continuum-
corrected spectra. They reflect the deformation level of the local
pseudocontinuum, due to imperfect flux calibration. The bottom right panel
shows the distribution of the derived parameter $\zeta$ values with a scatter
0.133 although the scatters of each of the indices (CaH2, CaH3 and TiO5) used
to calculate $\zeta$ are quite smaller.

We summarize the specific solution to build the combined LAMOST + SDSS
subdwarf sample in seven steps as follows:

(1) Select 4,767 visually inspected subdwarfs from LAMOST DR7.

(2) Recalculate the indices of the SDSS subdwarfs from Savcheva et al. (2014)
with the radial velocities recorded in the catalog (see Paper i@ for more
comments).

(3) Calculate the resolution of each spectrum using the profile of
corresponding arc-lamp spectrum (LAMOST) or the “dispersion” value stored in
the fits file (SDSS) and get the gaussian convolution kernels.

(4) Smooth the synthetic spectra to fit every observed spectrum and estimate
the atmospheric parameters. Figure 6 shows the fitting result of a LAMOST
extreme subdwarf in the upper panel and an SDSS subdwarf spectrum showing
H$\alpha$ emission in the lower panel. The spectral bands contaminated by
telluric absorptions are masked.

(5) For each index, measure its value on each observed spectrum and on its
corresponding continuum-corrected spectrum, and get their difference.

(6) Exclude the objects with one or more spectral index differences beyond
3$\sigma$ of the corresponding difference distribution.

(7) Combine the two samples above, and apply the selecting cuts by removing
the objects of $\zeta_{\text{L07}}\geq$0.825, $\zeta_{\text{L13}}\geq$0.825,
and $\zeta_{\text{Z19}}\geq$0.75.

Finally, a sample of 3,131 subdwarfs is assembled, 1,722 of which have
reliable kinematic properties. To provide better understanding, we summarize
in Table 3 the various subsamples used along this work.

Table 3: The various stellar samples used in the present work.

(1) |  LAMOST M dwarfs (2) |  LAMOST M subdwarfs initial sample |  SDSS M subdwarfs initial sample (3) |  LAMOST+SDSS combined subdwarf sample with atmospheric param. before CaOH/CaH1 cut |  LAMOST+SDSS green-marked “subdwarfs” |  LAMOST+SDSS “genuine” subdwarfs after CaOH/CaH1 cut (4) |  LAMOST M giants (5)
---|---|---|---|---|---|---|---
N |  $\sim$70,000 |  4,767 |  2,780 |  3,131 (LAMOST: 1,852) (SDSS: 1,279) |  855 |  2,276 |  7,200
N${}_{\text{Gaia}}$ |  35,382 |  - |  - |  1,722 (LAMOST: 1,160) (SDSS: 562) |  526 |  1,196 |  -

  * •
  * •

Notes to Table 3:

  * •

(1) The numbers corresponding to N${}_{\text{Gaia}}$ means the objects (N)
have available data from Gaia DR2 and can be used for kinematic analysis. The
flag is set as 1 if all of the following conditions are true:
a. phot_bp_mean_flux_over_error $>$ 10
b. phot_rp_mean_flux_over_error $>$ 10
c. phot_g_mean_flux_over_error $>$ 10
d. ResFlag = 1 $\&$ ModFlag = 1 (from Bailer-Jones et al. 2018)
e. astrometric_excess_noise_sig $\leq$ 2
f. ruwe $\leq$ 1.4

  * •

(2) A visually identified sample from Zhang et al. (2019) for comparison.

  * •

(3) Savcheva et al. (2014), the spectral indices are recalculated using the
radial velocities published in the catalog.

  * •

(4) i.e. using the Equation 5 in the present paper.

  * •

(5) A visually identified sample from Zhang et al. (2019) for comparison.

Figure 6: Upper panel: a LAMOST extreme subdwarf spectrum and its best-fit
synthetic spectrum. The spectral bands contaminated by telluric absorptions
are masked. Lower panel: an SDSS subdwarf spectrum showing H$\alpha$ emission
and its best-fit synthetic spectrum.  Figure 7: The M subdwarf sample
constructed in this work, combining the low-resolution spectra from SDSS (blue
dots) and LAMOST (red dots) data sets. The M dwarfs used for comparison (black
dots) are visually confirmed from LAMOST data (Zhang et al., 2019). The bottom
right internal subdiagram shows the spectral type distributions of the two
sub-samples.

We must underline that we haven’t used the CaH1 index as a filter until now.
For comparison, we also estimate the atmospheric parameters of a M dwarf
sample which contains around 70,000 visually inspected spectra from LAMOST
(Zhang et al., 2019). Figure 7 shows the combined LAMOST+SDSS sample of
subdwarfs together with the comparison sample of LAMOST M dwarfs. The LAMOST
sample contains a large number of early-type and medium metal-poor stars,
while the SDSS sample covers a broader range of spectral types and metal
abundances.

###  4.2 Results of Parameter Estimation

####  4.2.1 Signal-to-Noise impact on quality

Statistically, the fit residuals between the observed spectra and the
synthetic spectra vary with signal-to noise ratio (S/N). Figure 8 shows the
distribution of fitting residuals for subdwarfs from the two surveys. The left
panel shows the LAMOST subdwarfs mainly covering M0$-$M3 while the SDSS
subdwarfs on the right panel cover M0 to M8. The residual levels stabilize
below 10% after the S/N becomes larger than $\sim$30\. The later-type
subdwarfs from both samples show higher residual levels because of an average
lower spectrum quality.

Figure 8: The spectral fitting residual of subdwarfs as a function of the
signal-to-noise in $i$ band. Left panel: LAMOST subdwarfs, right panel: SDSS
subdwarfs.

####  4.2.2 Comparison with literature results

In order to estimate the accuracy of the derived parameters, we cross-match
the entire LAMOST DR7 M-type stars with objects having known parameters from
literature. As a result, we find a total of 269 observations for 167 stars
studied with high or medium spectral resolution in 9 papers, and the
comparison is shown in Figure 9. Although a systematic bias in the effective
temperature appears from around 3700K to the cool end, it generally follows a
1-to-1 trend with the literature values. The metallicity also seems to follow
a 1-to-1 trend, while the gravity values show a larger range than the
literature ones. Note that we have masked several telluric absorption bands
where gravity-sensative K and Na lines are located in the fitting process, the
accuracy of our estimated gravity values is therefore affected somehow. The
distributions of the differences (measurements from this work $-$ literature
values) for the three parameters are shown in Figure 9. The bias and scatter
are 165 K and 170 K for $T_{\text{eff}}$, 0.03 dex and 0.32 dex for log $g$,
and $-$0.17 dex and 0.41 dex for [Fe/H], respectively.

Figure 9: The upper panels show the comparison of atmospheric parameters of
167 M stars estimated from 269 LAMOST spectra (including multi-observations)
with the ones compiled from literature. Parameter values from different
literature are shown in different colors as listed in legend. Error estimates
provided are shown as corresponding error bars. The uncertainties for the
values from LAMOST spectra are the upper values of internal errors determined
from signal-to-ratio and $\sqrt{\chi^{2}}$. The bottom panels show the
distribution of differences along with the corresponding mean value and
standard deviation.

##  5 Sample Properties Analysis

In this section, we analyze the properties of the subdwarf sample by comparing
them with the dwarf sample on spectral index diagrams and Gaia DR2 H-R
diagram, and provide a preliminary account of their kinematic properties from
the reduced proper motion diagram and in the 3D galactic motion system.

In Paper i@, we suggested a screening condition associated with the CaH1 index
be combined with the classical $\zeta$ condition to identify a subdwarf via
its low-resolution spectrum. To show the importance of gravity and gravity-
dependent index—CaH1 for identifying a “genuine” subdwarf, we divide the
subdwarf sample selected above into two groups by the separator curve
(equation 9 in Paper i@) on the [CaOH, CaH1] index diagram:

CaH1 $\displaystyle<0.4562\times\text{CaOH}^{3}-0.1977\times\text{CaOH}^{2}$
(5) $\displaystyle-0.01899\times\text{CaOH}-0.7631$

The objects that do not meet Equation (5) are marked in green in the following
figures.

For further analysis, we retrieve proper motions, and photometric magnitudes
in $G$, $BP$, and $RP$ bands of these stars from Gaia Data Release 2, and
estimated distance from Bailer-Jones et al. (2018). To ensure reliability, we
set a series of filter conditions as listed in the notes to Table 3, and
finally 1,722 objects among the 3,131 selected in Section 4 meet the
conditions, 526 of which are green-marked “subdwarfs”.

Figure 10: The gravity and metallicity distributions of the green-marked
“subdwarfs” vs. the other sdMs. The two upper panels show the distribution of
spectral indices of four groups of subdwarfs: green-marked “subdwarfs” (the
subdwarfs that don’t meet Equation 5), sdMs, esdMs, and usdMs (the remaining
subdwarfs classified into to categories according to the $\zeta$ values). The
sdMs, esdMs and usdMs are colored in blue, purple and magenta respectively.
The bottom left and middle panels show the comparison of green-marked
“subdwarfs” with the sdMs. The bottom right panel compares the total proper-
motion distribution of the green-marked “subdwarfs” with that of the other
sdMs.

###  5.1 Spectral Index Diagram

As shown in Figure 10, the upper left panel shows the green marked
“subdwarfs”. According to the $\zeta$ values, we also divide the remaining
subdwarfs into sdM/esdM/usdM subclasses on the [TiO5, CaH2+CaH3] diagram, as
can be seen in the upper right panel, and the objects in the three subclasses
are plotted as blue, purple and magenta points respectively. Most of the green
marked “subdwarfs” appear located in the “sdM” region, which is colored in
blue. Therefore, we compare the gravity and metallicity of the two groups and
display those distributions in the bottom left and middle panels. As shown in
the bottom middle panel, the green objects share similar metal abundance
distribution with the blue labeled ones, however, their gravities are
systematically lower than the latter ones by some 0.2 dex.

With the available proper-motions of these targets from Gaia DR2, we also
compare the total proper motions of the two groups in bottom right panel of
Figure 10. The proper-motions of green marked “subdwarfs” are overall smaller
than those of the sdMs. Generally speaking, large proper motion is a classical
property of subdwarfs, though some objects may have relatively small proper
motion due to the 3D motion direction. Proper motions of esdMs and usdMs are
larger than sdMs (not shown in the figure), which means that the green ones do
behave more like dwarfs than subdwarfs.

Figure 11: M-type stars with different gravity shown on the spectral index
diagram CaH1 vs. CaOH. Color coding is the same as in the upper left panel of
Figure 10. To illustrate the sensitivity to gravity of this diagram, an M
giant sample (7,200 objects visually identified from Zhang et al. 2019 and
colored in yellow) and the M dwarf sample are added.

The gravity differences can be seen more clearly in Figure 11, in which we add
the LAMOST M dwarfs and a new sample of M giants onto the [CaOH, CaH1] index
diagram. The giants were also spectroscopically identified by visual
inspection (Zhang et al., 2019). Giants, and dwarfs subdwarfs are classified
as luminosity class iii@, v@ respectively and Jao et al. (2008) have proposed
that subdwarfs be classified as vi@, this sequence tracing the gravity
variation trend. Therefore, Figure 11 clearly illustrates the gravity
segregation between the luminosity classes of giants/dwarfs/subdwarfs thanks
to the sensitivity of CaH1 index to log $g$ especially when compared to CaOH
index.

###  5.2 H-R Diagram

Using estimated distance by Bailer-Jones et al. (2018), we derive the absolute
magnitudes M${}_{\text{G}}$, M${}_{\text{BP}}$ and M${}_{\text{RP}}$ of the
dwarfs and subdwarfs. Since most of the M-type stars we analyze here are
located within 1 kpc, we choose to ignore the effect of extinction in the
calculation. Figure 12 shows the stars on Gaia color-absolute magnitude
diagram [BP$-$RP, M${}_{\text{RP}}$]. On this diagram, which is an
observational Hertzsprung-Russel diagram, the gray dots represent the dwarf
sample, the green dots represent the “subdwarfs” that do not meet Equation
(5), and the red dots represent the remaining subdwarfs (all subclasses
together). The green-marked “subdwarfs” fall almost completely in the dwarf
region, and thus once again verify the conclusion that they behave more like
dwarfs than subdwarfs.

Figure 12: All the subdwarfs and dwarfs on the Gaia color-magnitude diagram.
The absolute magnitudes in RP band are derived from the Gaia DR2 apparent
magnitudes and estimated distances from Bailer-Jones et al. (2018). Most of
the green-marked “subdwarfs” fall into the main sequence region populated by
the M dwarfs plotted as black dots, and the remaining subdwarfs plotted as red
dots lie in part below the main sequence with a large dispersion, but some are
also distributed across the main sequence region.  Figure 13: After the
removal of the green-marked “subdwarfs”, the remaining samples are shown on
the Gaia color-absolute magnitude diagram again. On the left panel, subdwarf
are color-coded for different metallicities, while the same ones on the right
panel are color-coded for different gravities. Note that the subdwarfs with
log $g>$ 5.5 or log $g<$ 4.5 (38 objects in total) are not plotted on the
right panel to make the color bar more tight.

Besides, we find that a number of Equation (5) selected subdwarfs (red dots)
also overlap with dwarfs on this diagram. This point will be developed below.

Figure 13 shows the metal abundance and gravity variation of these stars on
the Gaia color-magnitude diagram. The subdwarfs falling in the dwarf region
are the ones with the highest metal abundances or the smallest surface
gravities. In other words, stars that are away from the main sequence have
both large gravity and low metal abundance, and they should be the “genuine”
subdwarfs.

###  5.3 Kinematics

####  5.3.1 Reduced Proper Motion diagram

Aside from the H-R diagram, another useful tool for effectively separating
different stellar populations is the reduced proper motion (RPM; Jones 1972)
diagram (e.g. Salim & Gould 2002; Subasavage et al. 2005; Lépine et al. 2007;
Faherty et al. 2009), which uses photometric information (color) and proper
motions of the stars. Even without knowledge of the distance to the stars, the
RPM diagram can be used to constrain kinematic properties of large stellar
samples, because a given stellar population whose members are orbiting in the
Galactic potential has a characteristic space velocity distribution reflected
in a combination of luminosity and transverse motion. The definition of the
reduced proper motion $H$ is

$H=m+5log\ \mu+5=M+5log\ T$ (6)

where $m$ is the apparent magnitude of a star in a given photometric band,
$\mu$ is the annual total proper motion (in arcsec yr-1), $M$ is the absolute
magnitude, and $T$ is the tangential velocity (in a.u. yr-1).

Since the dwarfs belong to the thin disk population and the subdwarfs are
claimed to belong to the thick disk and halo populations, the comparison
between blue to red or optical to near-infrared colors and the reduced proper
motions can in principle relatively clearly distinguish the Population i@
dwarfs and Population ii@ subdwarfs : (1) given an absolute magnitude,
subdwarfs are bluer in color; (2) subdwarfs being members of a population
spanning a spatial volume more extended out of the Galactic plane with more
eccentric orbits, they have higher average tangential velocities (Salim &
Gould, 2002; Lépine et al., 2007). The RPM diagram combines these two features
so that the subdwarfs and main sequence stars fall into two separate regions.
This method has long been used to separate high-velocity subdwarfs from dwarfs
(Jones, 1972; Yong & Lambert, 2003a, b).

Figure 14: After the removal of the green-marked “subdwarfs”, the remaining
samples are shown on the reduced proper motion diagram. The color coding is
the same as in Figure 12. The fiducial separation line between Cloud A and
Cloud B objects is drawn (see text).

Figure 14 shows the RPM diagram for our stars, using Gaia DR2 color and proper
motions. A clear similarity with the information contained in the H-R diagram
of Figure 14 can be noted. Although the selection biases of the combined
LAMOST+SDSS subdwarf sample are poorly known (mix of pointing directions,
different limiting depth of surveys, velocity ellipsoid sampling, density
distribution sampling, spatial inhomogeneities, etc.), basic trends can be
outlined. First, the subdwarfs falling below the region occupied by the main-
sequence dwarfs have low metal abundances and higher gravities. Second,
remaining subdwarfs appear clearly divided in two clouds (noted A and B in the
following) with a quite clean separation: Cloud A is not distinct from the
lower part of the main-sequence dwarf cloud, while Cloud B is centered around
a half magnitude below the center of Cloud A. The fact that Cloud A and B are
especially distinct in the RPM diagram imply that these two clouds sample two
distinct populations whose kinematic behaviors are different.

####  5.3.2 Galactic velocity components

Using the usual textbook transformation formula, we compute the three-
dimensional space velocity components $U$, $V$, $W$ in the right-handed
Galactic frame: the $U$ and $V$ are contained in the Galactic plane, $U$
points toward the Galactic center, while $V$ is the velocity component in the
direction of the solar motion. The $W$ component is perpendicular to the
Galactic plane and points towards the North Galactic pole. $U$, $V$, $W$ for
our targets are determined using the radial velocities, coordinates, proper
motions and distances derived from the parallaxes. The components are reduced
to the Local Standard of Rest (LSR; Fuchs et al. 2009) by correcting from the
solar motion assumed to be ($U_{\odot}$, $V_{\odot}$, $W_{\odot}$)=[11.1,
12.24, 7.25] km s-1 (Schönrich et al., 2010).

Figure 15: Velocity component distribution of dwarfs and subdwarfs with
different gravities in the Galactic 3D motion system ($U$, $V$, $W$). From
left to right, the abscissa represents the velocity component of $U$, $V$, and
$W$ respectively. The ordinate represents the distance from Galactic plane.
Except for the green-marked “subdwarfs”, the remaining subdwarfs are divided
into groups with different gravities and distinguished by different colors.
Each dwarf bin contains 7,000 stars and each subdwarf bin contains 100 stars.
The node and the error bar show the mean value and the standard deviation of
each bin respectively.  Figure 16: Velocity component distribution of dwarfs
and subdwarfs with different metallicities in the Galactic 3D motion system
($U$, $V$, $W$). Except for the green-marked “subdwarfs”, the remaining
subdwarfs are divided into groups with different metallicities and
distinguished by different colors. The meaning of the axes and the number of
stars contained in each bin are the same as in Figure 15.

In Figures 15 and 16, we divide our remaining subdwarfs into different gravity
and metallicity groups and compare them with the green-marked “subdwarfs” and
dwarfs. Each dwarf bin contains 7,000 objects and each subdwarf bin contains
100 objects. As expected, the green-marked “subdwarfs” behave mostly like
field dwarfs with only a slightly larger dispersion in all the velocity
components. In addition, the yellow bins in Figure 15 representing the
subdwarf group with the lowest gravity and the light blue bins in Figure 16
representing the subdwarf group with the highest metallicity have nearly the
same behaviors in $V$ as the dwarfs and the green-marked “subdwarfs”.

To further explore the kinematic behavior of the full subsamples, we draw
several histograms in Figure 17, in which the dwarfs, green-marked
“subdwarfs”, Cloud A and Cloud B subdwarfs are plotted separately. The mean
values and standard deviations are also shown on each panel, except for
$|$z$|$, the absolute value of the distance to the Galactic plane, for which
the quartiles are computed. The separation between Cloud A and Cloud B objects
is defined by the fiducial line drawn on Figure 14. Several points emerge:

Figure 17: Histograms of some characteristics for the various sub-populations
of dwarfs and subdwarfs for which Gaia astrometry is available. From left to
right, log $g$, [Fe/H], $U$, $V$, $W$ space velocity components and $|$Z$|$.
From top to bottom: (a) all spectroscopically identified M dwarfs, (b) green-
marked “subdwarfs”, (c) Cloud A subdwarfs (see Figure 14 for definition), (d)
Cloud B subdwarfs, (e) Extended Cloud A (see text for definition), (f) Disk
outliers (spectroscopically identified M dwarfs whose total space velocity
w.r.t. the LSR is larger than that of 99 % of the population, see Figure 19
for their location on Toomre diagram). The $\mu$ and $\sigma$ on each panel
are indicated for the mean value and standard deviation respectively. For the
$|$Z$|$ distributions, the quartiles are indicated.

(a) The gravities increase systematically from dwarfs (mean close to 5.0, in
agreement with the results from Section 3) to Cloud B subdwarfs (mean around
5.2).

(b) Metallicity decreases systematically from dwarfs (mean around $-$0.4 dex)
to Cloud B subdwarfs. The metallicity of the dwarfs, quite surprisingly,
appears systematically poorer than solar, with a distribution tail down to
$-$1\. dex. Although the error assessment and control is difficult in the
spectra fitting process, we recall that the comparison in Subsection 4.2 with
the sample of 167 stars having a metallicity measurement in the literature did
not reveal any major systematic deviation.

Figure 18: The bimodal distribution of metallicity of the halo candidates,
which are selected from Cloud B subdwarfs with $v_{tot}\geq$220 km s-1. The
mean values and standard deviations of each Gaussian fit are also shown on the
diagram.

(c) As shown in Figure 18, the halo candidates in Cloud B subdwarfs exhibit an
inconspicuous bimodal distribution of metallicity, with two sub-populations,
one around [Fe/H] = $-$1 dex and the other around $-1.5$ dex.

(d) The mean values and dispersions of the $U$, $V$, and $W$ components agree
quite well with the dwarf sample belonging to the thin disk, with a velocity
lag of $V$ = $-$10 km s-1. The green-marked and Cloud A objects have more
velocity dispersion suggesting possible membership of the thick disk, but
their mean $V$ value significantly deviates from the usually adopted $-$60 km
s-1. Cloud B objects are obviously dominated by halo members, with a mean $V$
largely retrograde and considerable dispersion.

(e) The quartiles of the $|$z$|$ distributions show that the dwarf sample is
the one which is the most concentrated around the Galactic plane, while the
Cloud B contains the most diffuse population out of this plane. These
histograms also show that the LAMOST+SDSS surveys sample the M dwarf
population inside scale height of the thin disk above the Galactic plane (600
pc) (Jurić et al., 2008), while the subdwarf one is sampled only up to less
than 1 kpc (one scale height of the thick disk or less) (Gilmore & Reid, 1983;
Veltz et al., 2008; Carollo et al., 2010).

(f) The striking similarity of all the indicators (except for a marginal
difference in log $g$ ) between the green-marked objects and the Cloud A ones
imply that they are members of the same stellar population as defined
kinematically, only differing by the CaH1 and CaOH indices. In the following,
these two groups will be considered as a unique subsample which will be
designated as Extended Cloud A.

####  5.3.3 The Toomre diagram

Trying to push forward the analysis, we have plotted in Figure 19 the Toomre
diagram for all our stars.

The dwarf cloud appears slightly asymetric, with an extension towards negative
$V$ velocity component. This is not unexpected since 2 or 3 - component disk
(thin, thick and intermediate) model populations (Schönrich & Binney 2009 and
references herein) predict such an asymmetry. The Extended Cloud A group seems
to be centered more or less coincident with the dwarf cloud, while Cloud B
objects are scattered across a large area of the diagram.

  1. 1.

Halo objects.

To select halo objects, a stringent condition on their total velocity (Bonaca
et al., 2017) which would avoid pollution by possibly numerous thick disk
stragglers (Smith et al., 2009), is:

$|V-V_{\text{LSR}}|\geq 220\ \text{km s}^{-1}$ (7)

which translates, which the adopted velocity components of the LSR, ($U$ = 0,
$V$ = 220, $W$ = 0 km s-1) into:

$v_{\text{tot}}=\sqrt{U^{2}+V^{2}+W^{2}}\geq 220\ \text{km s}^{-1}$ (8)

where $v_{\text{tot}}$ is the total space velocity with respect to the LSR.

Figure 19: Left panel: Toomre diagram for all stars in the present study.
Black points: all spectroscopically identified M dwarfs, green circles: the
green-marked “subdwarfs”, blue circles: Cloud A subdwarfs, red circles: Cloud
B subdwarfs. The black dashed curve encloses 99 % of the M dwarf population:
all black points outside this curve are considered as representative of
“outliers”. The dashed red circle is our adopted limit $v_{\text{tot}}$ = 220
km s-1 for halo population assignment. The dashed magenta vertical line
separates retrograde orbits on its left from prograde orbits on its right.
Right panel: the M dwarfs with thin disk kinematics have been removed. The
green-marked “subdwarfs” and Cloud A subdwarfs are merged into a single group:
Extended Cloud A. The M dwarfs outliers have been separated based on their
total space velocity $v_{\text{tot}}$ w.r.t. the LSR. The Cloud B are color-
coded in four categories based on their metallicity and $v_{\text{tot}}$.

This selection provides a kinematic sample of the local halo haunting the
Solar Neighbourhood, which contains essentially stars from Cloud B, (220
objects), with a handful from Extended Cloud A, (9 objects) accompanied by
targets considered as outliers from the dwarf cloud (71 objects)(see below).
Its kinematic characteristics are in fair agreement with those of already
identified halo population stars selected across widely more extended volumes
and using more luminous targets (Smith et al., 2009; Bond et al., 2010;
Bochanski et al., 2013): $<$$U$$>$ = 5 km s-1 and $\sigma_{U}$ = 173 km s-1,
$<$$V$$>$ = $-$248 km s-1 and $\sigma_{V}$ = 77 km s-1, and $<$$W$$>$ = 6 km
s-1 and $\sigma_{W}$ = 89 km s-1. With the adopted value of the LSR velocity,
the fiducial line at $V$ = $-$220 km s-1 separates the stars which have
retrograde orbits from those who have prograde ones. A striking fact is the
quite high fraction of prograde objects (258 stars or 40%) observed in the
present local halo sample, thus the strongly negative value of $<$$V$$>$
quoted above does not account completely for a complex orbital distribution.

The metallicity of this halo sample, in spite of the kinematical cut applied,
remains bimodal. Previous studies, mostly based on F-G-K stars in which
metallicity is derived from atomic lines have already found a substantial
metal-rich component in the halo population, (and the metallicity distribution
of globular clusters is one historic and well-documented example (see e.g.
Zinn 1985; Harris 1996) but the metal-rich peak is generally found around
[Fe/H] = $-$0.6 dex: Bonaca et al. (2017). The prograde objects are also
reputedly more metal-rich than the retrograde ones, implying that they could
belong to different subpopulations, the metal-rich one containing inner halo
objects while the metal-poor one is made of outer halo objects, these two
subpopulations having different formation histories and different space
distributions. In the present local halo sample, we do not find such a
segregation, the number of prograde metal-poor objects remaining important
while many higher metallicity stars are on obviously retrograde orbits, as
shown in the right panel of Figure 19.

  2. 2.

Thin disk dwarfs:

Since some spectroscopically identified dwarfs are found to kinematically
belong to the local halo, we could expect some continuity in the dwarf high
velocity distribution tail, and therefore a number of them would also be
kinematically mixed with the Extended Cloud A population and need to be
disentangled.

To select thin disk dwarfs, a condition of type $v_{\text{tot}}$ = constant is
not completely satisfactory because of the asymmetry of the dwarf cloud in the
Toomre diagram. A correct way to proceed would be to compute for each star its
probability of membership to one of the three populations - halo, thick disk
and thin disk as proposed e.g. by Bensby et al. (2003) but we prefer to
postpone such a calculation because the selection effects in the present
sample are not correctly assessed. We choose a simplified and empirical
procedure as follows: we assume that the distribution of the dwarf stars in
the Toomre diagram is consistent with an incomplete ellipse, and that the
isodensity contours of this ellipse trace constant probability level of a star
to belong to its parent population. We select the isodensity contour
containing 99% of the dwarf cloud and fit an ellipse to this contour. All
stars found outside are considered as “outliers” from the thin disk dwarf
population.

The kinematically “purified” dwarf sample shows typical velocity component
values of the thin disk population (Edvardsson et al., 1993; Reddy et al.,
2003): $<$$U$$>$ = 0 km s-1 and $\sigma_{U}$ = 44 km s-1, $<$$V$$>$ = $-$13 km
s-1 and $\sigma_{V}$ = 30 km s-1, and $<$$W$$>$ = 0 km s-1 and $\sigma_{W}$ =
26 km s-1, and its stars have exclusively prograde orbits as expected. The
metallicity, as outlined above, is in average slightly deficient, the
histogram showing three peaks, one solar or slightly super-solar, the
principal one around $-$0.4 dex and the third one around $-$1 dex. Since the
selection effects are not under control, we doubt that this three-component
would be significant.

  3. 3.

Thick disk population and its elusive limits

The Extended Cloud A occupies a large area, somewhat fan-shaped vertically, in
the Toomre diagram, broadly covering the dwarf cloud and merging beyond it
until the halo region, with practically all orbits prograde. The shape of the
$|$Z$|$ histogram, with its peak displaced from that of the thin disk dwarfs
and its quartiles systematically larger shows that this population is clearly
thicker in $z$. The average metal content is poorer than that of the thin disk
dwarfs at [Fe/H] = $-$0.74 dex with a [Fe/H] distribution having a long metal-
poor tail extending to $-$2.5 dex. We suspect that the Extended Cloud A
population is an intricate mix of objects really belonging to a thick disk
component with pollution by an important contribution of thin disk “mild”
subdwarfs very difficult to disentangle, with unknown selection effects
playing in background. A possible way to separate these contributions is the
detailed computation of individual orbits and study of the $z_{\text{max}}$
and eccentricities, but such a work is far beyond the scope of the present
study.

In the Toomre diagram, the Extended Cloud A population is mixed with a
fraction of the Cloud B population, corresponding to stars that preferentially
belong to the high metallicity peak of the bimodal [Fe/H] distribution of
Cloud B in Figure 17. These stars likely belong to the halo for a part,
because our halo selection kinematic cut is very stringent, and to the thick
disk for another part, but their respective assessment to the two components
would probably also need a detailed orbital calculation.

The [Fe/H] distribution of the outliers from the dwarf cloud show that they
span the same broad metallicity range as the principal dwarf population,
peaking at $-$0.6 dex, only slightly more metal-poor than the ordinary disk
dwarfs. Their distribution in $|$Z$|$ is clearly more extended, making them
probable members of the thick disk. Their velocity dispersions $\sigma_{U}$ =
119 km s-1, $\sigma_{V}$ = 90 km s-1, $\sigma_{W}$ = 75 km s-1 suggest that
they contain a majority of objects belonging to the thick disk with some
pollution by an inner halo fraction.

To conclude, the combined subdwarf sample extracted from LAMOST DR7 and SDSS
DR7 contains a mix of objects belonging to various components: clear halo
members with an implicit bimodal metallicity and considerable orbital mixing
between retrograde and prograde cases, and an important thick disk fraction
merging both into halo and thin disk. The spectroscopically selected M dwarfs
themselves, whose quasi-totality belong to the standard thin disk, contain a
small fraction (less than 1% of the total) of outliers belonging both to the
thick disk and the halo. It is quite unclear if the present situation is
dependent or not on the selection effects, especially the favoured directions
of investigation towards the Galactic anticenter in the LAMOST survey.

##  6 Summary and Conclusion

In the most widely used classification system, M-type main sequence stars are
classified into dM/sdM/esdM/usdM subclasses corresponding to different metal
abundances according to the CaH2, CaH3 and TiO5 molecular bands in the low-
resolution spectrum. The metal-poor subclasses are named as subdwarfs, which
have typical kinematic characteristics of thick disk and halo population.
However, in the present work, we suggest the “genuine” subdwarfs belonging to
population ii@ be classified as luminosity class vi@ on the H-R diagram. For
these stellar objects, having larger surface gravity is a necessary condition
as important as “being metal deficient”, and the three spectral indices in
classical classification system can not be fully competent for selecting these
subdwarfs. This conclusion is drawn by a series of experiments including the
comparison of synthetic spectra with the templates, measuring atmospheric
parameters for a subdwarf sample and combine with kinematics analysis, etc).
According to the spectral index CaOH-CaH1 relationship and the empirical
relationship on RPM diagram, we divide a spectroscopically identified
subdwarfs into multiple groups and analyze the multi-population properties in
this sample from their multi-dimensional parameters.

The sample including subdwarfs from both LAMOST and SDSS combined with a
control group of dwarfs selected from LAMOST, and estimate their atmospheric
parameters fitting with the PHOENIX grid. The technical details of the whole
fitting procedure of individual spectra is developed, including the use of
different convolution kernels to smooth the synthetic spectra and the
pseudocontinuum flux correction specifically implemented for the LAMOST data.
The residuals of the fitting procedure can be used to determine the precision
which depend on the S/N of the observed spectra. The accuracy of the process
is evaluated from 167 M-type stars with measured atmospheric parameters from
literature which have LAMOST spectra: and the bias and scatter of the
comparison of parameters are 165$\pm$170 K for $T_{\text{eff}}$, 0.03$\pm$0.32
dex for gravity, and -0.17$\pm$0.41 dex for metallicity. A systematic bias in
effective temperature exists for $T_{\text{eff}}$ lower than 3700 K while the
gravity and metallicity both appear in fair agreement. The effective
temperature scale produced by these detailed fits is reasonably consistent
with those published by Rajpurohit et al. (2014) for the subdwarfs and
Rajpurohit et al. (2018a) for the dwarfs. The following analysis are mainly
dependent on the latter two parameters.

For the sample of subdwarfs and dwarfs, apparent magnitudes, broad-band
colors, and proper-motions are obtained from cross-matching these two sets
with Gaia DR2, and the distances are from Bailer-Jones et al. (2018). The
objects are selected so as to have acceptable quality flags in photometry,
distance and accuracy. The spectroscopic “subdwarfs” do not entirely lie below
the main-sequence but in an extended region which overlaps that occupied by
the M dwarfs in the color-absolute magnitude diagram, while the objects that
lie away (and below) from the main sequence are the ones with both larger
gravity and lower metal abundance. They are the “genuine” subdwarfs, for which
the luminosity class vi@ ((Jao et al., 2008)) is justified. Alternatively,
some “subdwarfs” with gravity similar as that of the dwarfs or only slightly
higher also behave like dwarfs in many aspects although they are medium metal-
deficient through the definition using the CaH and TiO spectral indices, we
have been able to sort out a significant subsample of objects which, in spite
of average low metallicity, appear intricately mixed with the ordinary main-
sequence dwarfs and also sharing their location in a reduced proper motion
diagram. These “subdwarfs” should perhaps better be classified as “metal-poor
dwarfs”.

A provisional kinematic analysis of these samples show that the sdM population
lying below the main-sequence cloud is dominated by halo stars, in which an
implicit bimodal metallicity distribution appears, with a peak around $-$1.0
dex and a second peak around $-$1.5 dex. This group, which samples the
galactic halo only in a very local volume (typically less than 1 kpc radius),
contains a mix of objects of prograde and retrograde orbits, as revealed by
their location in the Toomre diagram, without clear trend with the
metallicity. The other subdwarfs, including those which we prefer to consider
as “metal-poor dwarfs” appear as a mixed population of thick disk and thin
disk members. If “lying below the main sequence” and “associated with halo
kinematics” (large negative $V$ component and large dispersions in $U$, $V$
and $W$) define intrinsic features of the genuine subdwarfs belonging to
Population ii@ stars, then both large gravity (average log $g$ = 5.2 to 5.4)
$and$ low metal abundance should be essential attributes of them. The fact
that the two properties must exist simultaneously needs to be underlined: we
have only found a handful of halo objects among the other subdwarfs, even
those which have a metallicity lower than $-$1.0 dex.

The ordinary dwarfs populate in their vast majority the Galactic thin disk, as
expected, with an average metallicity slightly lower than solar. A small
fraction (less than 1%) form a set of kinematic outliers, halo and thick disk
probable members. Finally, CaH1 index is a good indicator of gravity
experimentally, which behaves better in separating subdwarfs from dwarfs than
the composite index CaH2+CaH3.

Though the contamination by low ratio of multiplicity (12.5$\pm$1.9%: Ziegler
et al. 2015) and the parameter coupling effect would influence the accuracy of
parameters measurement, a preliminary indicative conclusion can still be drawn
from the analysis results above. Besides, accurate discrimination of subdwarfs
has to be revisited with more information such as high-resolution spectra and
multi-band data, and the estimates of parameters also need a more complete
model grid (an independent alpha enrichment such as the analysis in Hejazi et
al. 2020, more complete opacity and line lists, etc). The sub-population
membership assessment of the subdwarfs found in numbers in the large area
spectroscopic surveys, in turn needs careful selection effects control and
detailed orbital calculation.

ZS thanks Dr. Yan. H.L. for the thoughtful discussion about smoothing the
synthetic spectra, and Dr. Allard, F. for her helpful private communication
about PHOENIX models. This work is supported by National Key R&D Program of
China(No. 2019YFA0405502), National Science Foundation of China (No.
U1931209). Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built
by the Chinese Academy of Sciences. Funding for the project has been provided
by the National Development and Reform Commission. LAMOST is operated and
managed by the National Astronomical Observatories, Chinese Academy of
Sciences.

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Table 4: Data description of the subdwarf catalog.

Column Name | Unit | Description
---|---|---
ID | - | SpecObjID for SDSS objects and obsid for LAMOST objects
specname | - | Name of spectrum
ra | degrees | Right Ascension (J2000)
dec | degrees | Declination (J2000)
l | degrees | Galactic longitude
b | degrees | Galactic latitude
rv |  km s-1 | Heliocentric radial velocity
Teff | K | Estimated effective temperature
logg | dex | Estimated surface gravity
FeH | dex | Estimated overall metallicity
snri | - | Average spectrum signal to noise ratio in i band
spt | - | Spectral subtype
CaOH | - | Spectral index of CaOH band
CaH1 | - | Spectral index of CaH1 band
CaH2 | - | Spectral index of CaH2 band
CaH3 | - | Spectral index of CaH3 band
TiO5 | - | Spectral index of TiO5 band
ZetaL07 | - |  The value of $\zeta$ parameter calibrated by Lépine et al. (2007)
ZetaD12 | - |  The value of $\zeta$ parameter calibrated by Dhital et al. (2012)
ZetaL13 | - |  The value of $\zeta$ parameter calibrated by Lépine et al. (2013)
ZetaZ19 | - |  The value of $\zeta$ parameter calibrated by Zhang et al. (2019)
source | - | The source of spectrum (lamost/sdss)
parallax | mas | Gaia DR2 parallax
parallax_error | mas | Standard error of Gaia DR2 parallax
pmra |  mas yr-1 | Gaia DR2 proper motion in right ascension direction
pmra_error |  mas yr-1 | Standard error of Gaia DR2 proper motion in right ascension direction
pmdec |  mas yr-1 | Gaia DR2 proper motion in declination direction
pmdec_error |  mas yr-1 | Standard error of Gaia DR2 proper motion in declination direction
astrometric_gof_al | - | Goodness of fit statistic of model wrt along-scan observations
astrometric_excess_noise_sig | - | Significance of excess noise
phot_g_mean_flux_over_error | - | G-band mean flux divided by its error
phot_g_mean_mag | mag | Gaia DR2 G-band mean magnitude
phot_bp_mean_flux_over_error | - | Integrated BP mean flux divided by its error
phot_bp_mean_mag | mag | Gaia DR2 integrated BP mean magnitude
phot_rp_mean_flux_over_error | - | Integrated RP mean flux divided by its error
phot_rp_mean_mag | mag | Gaia DR2 integrated RP mean magnitude
bp_rp | mag | Gaia DR2 BP-RP colour.
ruwe | - | Renormalized Unit Weight Error associated to each Gaia source
rest | pc |  Estimated distance from Bailer-Jones et al. (2018), based on Gaia DR2
ResFlag | - |  The Result Flag from Bailer-Jones et al. (2018)
ModFlag | - |  The Modality Flag from Bailer-Jones et al. (2018)
flag | - |  Set as 1 for the final sample used in kinematic part, see table 3 for details
$V_{total}$ |  km s-1 | Total spacial velocity
$Z$ | pc | Vertical distance away from Galactic plane
$U$ |  km s-1 | Velocity component points towards the galactic center
$V$ |  km s-1 | The tangential velocity component rotating around galactic center
$W$ |  km s-1 | Velocity component points towards the north galactic pole

Note. — The complete catalog can be accessed online222http://paperdata.china-
vo.org/work/subdwarf-catalog.csv.