11institutetext: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China;
<EMAIL_ADDRESS>
22institutetext: School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China
33institutetext: Key Laboratory of Space Astronomy and Technology, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101,
People’s Republic of China<EMAIL_ADDRESS>
44institutetext: Department of Astronomy, Beijing Normal University, Beijing
100875, China

# Search for the metal-weak thick disk from the LAMOST DR5

T.-S. Yan 1122 J.-R. Shi 1122 H. Tian 33 W. Zhang 11 B. Zhang 44

###### Abstract

Based on the data release of the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope survey (LAMOST DR5) and the _Gaia_ Early Data Release
3 (_Gaia_ EDR3), we construct a sample containing 46,109 giant (log g
$\leqslant$ 3.5 dex) stars with heliocentric distance d $\leqslant$ 4 kpc, and
the sample is further divided into two groups of the inner ($\mathrm{R_{GC}}$
$<$ 8.34 kpc) and outer region ($\mathrm{R_{GC}}$ $>$ 8.34 kpc). The
$\mathrm{L_{Z}}$ distributions of our program stars in the panels with
different [Fe/H] and [$\alpha$/Fe] suggest that the thick-disk consists of two
distinct components with different chemical compositions and kinematic
properties. For the inner region, the metal-weak thick disk (MWTD) contributes
significantly when [$\alpha$/Fe] $>+$0.2 dex and [Fe/H] $<-$0.8 dex, while the
canonical thick-disk (TD) dominates when [Fe/H]$>-$0.8 dex. However, MWTD
clearly appears only when [$\alpha$/Fe] $>+$0.2 dex and [Fe/H] $<-$1.2 dex for
the outer region, and its proportion is lower than that of the inner region
within the same metallicity. Similar result can be obtained from the
$\mathrm{V_{\phi}}$ distribution. The high fraction of MWTD in the inner than
that in the outer region imply that MWTD may form in the inner disk, and is an
observational evidence about the inside-out disk formation scenario.

###### keywords:

stars: abundances – stars: atmospheres – Galaxy: structure – Galaxy:
kinematics and dynamics – methods: data analysis

##  1 Introduction

It is well known that the Milky Way is composed of multiple components, such
as the thin-disk, the thick-disk and the halo (An & Beers, 2020; Chiappini et
al., 1997; van der Kruit & Freeman, 2011), and it provides a unique chance for
studying the formation and evolution of galaxies in detail by analysing the
full location, kinematics characteristic and chemical compositions of these
stellar components.

As an important component of our Galaxy, the thick-disk has been studied by
numerous works since its discovery (Beraldo e Silva et al., 2021; Di Matteo et
al., 2011, 2019; Franchini et al., 2021; Gilmore & Reid, 1983; Girard et al.,
2006; Helmi et al., 2018; Recio-Blanco et al., 2014; Ruchti et al., 2011). The
thick-disk is composed primarily of older stars, resulting in different
chemical composition from the young thin-disk stars. In the metallicity
distribution, the thick-disk exhibits a peak aound [Fe/H]$\sim$ $-$0.6 dex,
and most of the thick-disk stars fall within the interval of [Fe/H] from
$-$1.0 dex to $-$0.3 dex (Reddy & Lambert, 2008). At the mooment, there is no
uniform conclusion on the upper and lower bounds of the metallicity
distribution for the thick-disk stars (Reddy & Lambert, 2008), and some disk-
like kinematics stars with very low metallicity (down to [Fe/H] = $-1.7$ dex,
or even lower) have been found (Carollo et al., 2010, 2019; Chiba & Beers,
2000; Norris et al., 1985). These metal-poor tail stars are commonly
considered belong to the metal-weak thick-disk (MWTD, Beers et al., 2014;
Carollo et al., 2010, 2019; Carollo & Chiba, 2020; Ivezić et al., 2008;
Morrison et al., 1990; Reddy & Lambert, 2008; Ruchti et al., 2011), while the
remaining component after removing MWTD from the thick-disk is called as the
canonical thick-disk (TD, Carollo et al., 2010).

Due to lack of large-scale sample stars of high-precision parameters, the
existence of MWTD remains controversial (Ruchti et al., 2010). Norris et al.
(1985) suggested that these stars with [Fe/H]$\leqslant-$1.0 dex and
$e\leqslant$ 0.4 should belong to a population, and Morrison et al. (1990)
refered to these stars of $-$1.6 dex$<$[Fe/H]$<-$1.0 dex with kinematical and
spatial properties similar to the thick-disk ones as MWTD stars. It need to be
pointed out that the metallicity derived by Norris et al. (1985) and Morrison
et al. (1990) is besed on the information from the David Dunlap Observatory
(DDO) photometry, and Twarog & Anthony-Twarog (1994) found that the
photometric metallicity used by Norris et al. (1985) has an $\sim$0.5 dex
offset from the spectral ones for giants near [Fe/H]$\sim-$1.2 dex. In
addition, by comparing the metallicity obtained from the high-resolution
spectra with these from the DDO photometry for giants of Norris et al. (1985)
and Morrison et al. (1990), Ryan & Lambert (1995) concluded that most stars
with the DDO photometric metallicity of [Fe/H]$<-$1.0 dex have spectral
metallicities of [Fe/H]$>-$1.0 dex. As a result, these stars identified as
belonging to MWTD by Norris et al. (1985) and Morrison et al. (1990),
therefore should belong to TD. Although the conclusions from Twarog & Anthony-
Twarog (1994) and Ryan & Lambert (1995) cannot rule out the existence of MWTD,
the evidences of MWTD suggested by Norris et al. (1985) and Morrison et al.
(1990) become weaker.

Fortunately, the advent of large-scale surveys, such as the Large Sky Area
Multi-Object Fibre Spectroscopic Telescope (LAMOST) survey (Cui et al., 2012),
the Sloan Digital Sky Survey (SDSS, York et al., 2000), the Radial Velocity
Experiment spectroscopic survey (RAVE, Steinmetz et al., 2020a, b), and the
_Gaia_ mission (Gaia Collaboration et al., 2021), make it possible to perform
accurate chemodynamical studies on the stellar populations in our Galaxy
(Carollo et al., 2019). The spectra from LAMOST, SDSS and RAVE provide
estimation of the stellar atmospheric parameters (effective temperature,
surface gravity and metallicity) and radial velocities ($\mathrm{RV}$), and
the _Gaia_ satellite delivers high-precision astrometric data (position,
trigonometric parallax, and proper motions). The combination of parameters
obtained from spectrum and astrometry is a powerful tool for studying the
properties of stellar component related to the Milky Way by investigating the
perspectives of stellar spatial distribution, chemical composition and
kinematics.

In this paper, based on a sample combining spectral information from LAMOST
and astrometric data from _Gaia_ , we examine MWTD. The paper is organized as
follows. In Sect. 2, we describe the selection of our sample stars. Sect. 3
presents the search for MWTD in the kinematic space, and the discussion and
conclusions are given in Sect. 4.

##  2 Sample

###  2.1 Selection of the Sample Stars

The sample stars are selected from LAMOST DR5 with a range of the effective
temperature from 4,500 K to 7,000 K, and only the giants (log g $\leqslant$
3.5 dex) with signal-to-noise (S/N) ratio of their spectra higher than 50 have
been considered. This removes all the dwarf stars. The stellar atmosphere
parameters and [$\alpha$/Fe] ratios are taken from the recommended values of
Xiang et al. (2019) with internal uncertainties of [Fe/H] and [$\alpha$/Fe]
less than 0.07 and 0.05 dex, respectively, while, the radial velocities
($\mathrm{RV}$s) are adopted from LAMOST DR5 (Luo et al., 2015), and only the
objects with a $\mathrm{RV}$ uncertainty less than 10 km s-1 have been
selected. Moreover, to further ensure the accuracy of the metallicity, only
the objects with a difference of metallicity between LAMOST DR5 and that of
Xiang et al. (2019) less than 0.1 dex are selected.

The stellar proper motions (PMs) are obtained from _Gaia_ EDR3 (Gaia
Collaboration et al., 2021) with errors both in right ascension and in
declination direction less than 0.2 mas yr-1, while the geometric distances
($\mathrm{\emph{r}_{med}}$) from Bailer-Jones et al. (2021) have been adopted.
Our sample stars are confined to d $\leqslant$ 4 kpc with an uncertainty of
($\mathrm{\emph{r}_{hi}}$$-$$\mathrm{\emph{r}_{lo}}$)/($\mathrm{2\times\emph{r}_{med}}$)
smaller than 20%. Here, $\mathrm{\emph{r}_{lo}}$ and $\mathrm{\emph{r}_{hi}}$
are the 16th and 84th percentiles of the posterior (see Bailer-Jones et al.,
2021, for details).

After these considerations included, the number of stars is reduced to
211,046, and the distribution of these stars in the [$\alpha$/Fe] - [Fe/H]
plane is shown in Fig. 1. Obviously, most of them are belong to the thin-disk,
and these thin-disk stars are not important for this study. Therefore,
according to the criteria of Hayden et al. (2014) (see the black yellow
polyline in Fig. 1) we remove them from our sample. Finally, there are 46,109
stars in our sample.

Figure 1: Distribution of giant stars in the [$\alpha$/Fe] - [Fe/ H] plane.
The black yellow polyline is the criteria for dividing the thin- and thick-
disk stars from Hayden et al. (2014).

To avoid the systematic offset of LAMOST RVs (Anguiano et al., 2018; Tian et
al., 2020), we compare the LAMOST RVs with those from _Gaia_ EDR3 for the
common stars, only stars with Gaia RV errors less than 1 km s-1 have been
selected. The offset of LAMOST RV is around $-$4.85 km s-1 (Fig. 2), which is
very close to those of the previous works ($\sim-$5 km s-1, Anguiano et al.,
2018; Tian et al., 2020). Therefore, a value of 4.85 km s-1 is added to the
LAMOST RV to compensate for the offset.

Figure 2: Comparison of the LAMOST RVs with those from _Gaia_ EDR3, only stars
with Gaia RV errors less than 1 km s-1 have been used.

###  2.2 Stellar Kinematic Parameters

The full phase-space information, positions (including $\alpha$, $\delta$ and
l, b), PMs and distances, combined with $\mathrm{RV}$s, provide the parameters
required to calculate the kinematics. Following Kordopatis et al. (2011), the
Galactic-centerd Cartesian coordinates ($\mathrm{X_{GC}}$, $\mathrm{Y_{GC}}$,
Z) of stars are obtained through their distances and Galactic coordinates (l,
b), and $\mathrm{R_{GC}}$ = $\mathrm{\sqrt{X^{2}_{GC}+Y^{2}_{GC}}}$
representes the distance to the Galactic center in planar radial coordinate.
In our calculation, the solar position of (X⊙, Y⊙, Z${}_{\odot}=($8.34 kpc, 0,
0) is adopted (Reid et al., 2014). The galactocentric velocities
$\mathrm{V_{R}}$ and $\mathrm{V_{\phi}}$ are also calculated, and
$\mathrm{V_{R}}$ and $\mathrm{V_{\phi}}$ are defined as positive with
increasing $\mathrm{R_{GC}}$ and $\phi$, respectively. The Galactic rotation
velocity of the Local Standard of Rest (LSR) is adopted as Vc = 240 km s-1
(Reid et al., 2014), and the solar motion with respected to LSR of
($\mathrm{U_{\odot},V_{\odot},W_{\odot}})=$ (11.1, 12.24, 7.25) km s-1, is
taken from Schönrich et al. (2010). The Z-axis angular momentum is derived
using the relationship of $\mathrm{L_{Z}}$ =
$\mathrm{R_{GC}}$$\times$$\mathrm{V_{\phi}}$.

##  3 Results

Although MWTD overlaps TD and the halo in parameter space, it is still
possible to trace the footprint of MWTD with the help of their distribution in
the metallicities, kinematics and dynamics (An & Beers, 2020, 2021; Carollo et
al., 2010, 2019; Carollo & Chiba, 2020; Cordoni et al., 2021; Kordopatis et
al., 2013).

It is noted that the fraction of metal-poor stars is lower (see Fig. 8 of
Miranda et al., 2016), and Carollo et al. (2010) pointed out that the
significant contribution of MWTD is within $-$1.8 dex $<$ [Fe/H] $<-$0.8 dex,
and possibly up to $\sim-$0.7 dex. Therefore, we divide our sample stars into
four metallicity intervals of [Fe/H] $<-$1.2 dex, $-$1.2 dex $<$ [Fe/H]
$<-$0.8 dex, $-$0.8 dex $<$ [Fe/H] $<-$0.4 dex and [Fe/H] $>-$0.4 dex.
Moreover, Carollo et al. (2019) suggested that the MWTD stars tend to have
higher [$\alpha$/Fe] ratios than those of TD, which means it will improve the
search efficiency of MWTD when the sample is decomposed in [$\alpha$/Fe]
space. In Fig. 3 we plot the [$\alpha$/Fe] distribution of our sample stars of
[Fe/H] $<-$0.8 dex, and the distribution can be well fitted with two Gaussian
functions. According to their [$\alpha$/Fe] distribution, the sample is
further divided into high ([$\alpha$/Fe] $>+$0.2 dex) and low $\alpha$
([$\alpha$/Fe] $<+$0.2 dex) intervals.

Figure 3: [$\alpha$/Fe] distribution of the sample stars with [Fe/H] $<-$0.8
dex. The distribution can be well fitted with two Gaussian components. The
vertical dotted line is labeled as [$\alpha$/Fe] $=+$0.2 dex.

In order to compare the differences due to different $\mathrm{R_{GC}}$ (e.g.,
An & Beers, 2021), we present our discussions on our program stars with the
inner ($\mathrm{R_{GC}}$ $<$ 8.34, 23,664 stars) and outer ($\mathrm{R_{GC}}$
$>$ 8.34 kpc, 22,445 stars) regions.

###  3.1 The $\mathrm{L_{Z}}$ distributions

Following Carollo et al. (2019), we focus on the $\mathrm{L_{Z}}$
distribution. In Fig. 4, we plot the $\mathrm{L_{Z}}$ distribution for the
inner region stars ($\mathrm{R_{GC}}$ $<$ 8.34) in slices of [Fe/H] and
[$\alpha$/Fe]. We apply multi-Gaussians to fit the $\mathrm{L_{Z}}$
distribution in each panel, and the fitting parameters of each component
including the fraction, $\mathrm{L_{Z}}$ and scatter ($\mathrm{\sigma_{LZ}}$)
are derived according to the Bayesian method, which can effectively avoid the
contamination between different components (Tian et al., 2019). To perform
this fitting, we apply _emcee_ (Foreman-Mackey et al., 2013) to run a Markov
Chain Monte Carlo (MCMC) simulation. We use 100 walkers for 2000 iterations
with 1000 burn-in have been used. The median value for each parameter is
adopted, and the difference between the median value and the 16% and 84%
values are used as the upper and lower uncertainties, respectively. Fig 5
shows the MCMC result of panel b in Fig. 4.

The best-fit parameters and uncertainties of the MCMC simulations for all the
subsamples are listed in Table 1. When the fraction is not a free parameter
during the MCMC simulation (one in each panel), its uncertainty is marked as 0
(Table 1). The fitted distributions are represented with brown, red, green and
blue lines for the halo, MWTD, TD and total, respectively, in Fig. 4.

Figure 4: The $\mathrm{L_{Z}}$ distributions of the inner region stars in slices of [Fe/H] and [$\alpha$/Fe]. The distribution can be best-fitted by one to three Gaussian(s), and each represents one component: the halo (brown), MWTD (red) and TD (green). The black histogram is the sum of all stars, while the light blue solid line represents the sum of the fit components.  Figure 5: The MCMC simulation results of the panel b of Fig. 4. The adopted median values of the parameters are represented by the solid blue lines. Table 1: The parameters of the best-fit components in the $\mathrm{L_{Z}}$ distributions for the inner region stars in slices of [Fe/H] and [$\alpha$/Fe]. The units of the $\mathrm{L_{Z}}$ peak and scatter ($\sigma_{\mathrm{LZ}}$) are in kpc km s-1. [$\alpha$/Fe] | Component |  [Fe/H]$<$$-$1.2 dex |  $-$1.2 dex$<$[Fe/H]$<$$-$0.8 dex |  $-$0.8 dex$<$[Fe/H]$<$$-$0.4 dex |  [Fe/H]$>$$-$0.4 dex
---|---|---|---|---|---
fraction |  $\mathrm{L_{Z}}$ peak | $\sigma_{\mathrm{LZ}}$ | fraction |  $\mathrm{L_{Z}}$ peak | $\sigma_{\mathrm{LZ}}$ | fraction |  $\mathrm{L_{Z}}$ peak | $\sigma_{\mathrm{LZ}}$ | fraction |  $\mathrm{L_{Z}}$ peak | $\sigma_{\mathrm{LZ}}$
$>0.2$ | TD |  |  |  |  0.40$\pm$0 |  1253.5$\pm$11.3 |  391.6$\pm$26.4 |  0.89$\pm$0 |  1328.2$\pm$7.1 |  346.1$\pm$5.0 |  1.00$\pm$0 |  1412.0$\pm$8.0 |  341.3$\pm$5.9
MWTD |  0.35$\pm$0 |  950.7$\pm$7.0 |  444.9$\pm$35.4 |  0.39$\pm$0.08 |  948.5$\pm$11.2 |  527.5$\pm$37.0 |  |  |  |  |  |
the halo |  0.65$\pm$0.03 |  -0.2$\pm$7.1 |  733.6$\pm$33.6 |  0.21$\pm$0.02 |  1.8$\pm$11.2 |  508.8$\pm$31.5 |  0.11$\pm$0.01 |  391.6$\pm$41.5 |  405.2$\pm$18.8 |  |  |
$<0.2$ | TD |  |  |  |  0.67$\pm$0 |  1265.3$\pm$40.8 |  390.0$\pm$33.7 |  0.94$\pm$0 |  1383.1$\pm$7.2 |  356.2$\pm$5.5 |  1.00$\pm$0 |  1464.0$\pm$4.5 |  331.6$\pm$3.3
the halo |  |  |  |  0.33$\pm$0.05 |  91.2$\pm$61.4 |  490.6$\pm$50.1 |  0.06$\pm$0.01 |  329.6$\pm$48.1 |  384.9$\pm$23.8 |  |  |

It can be seen from Fig. 4 that the three low $\alpha$ ([$\alpha$/Fe] $<+$0.20
dex) panels (f, g and h) are dominated by the TD stars, and their
$\mathrm{L_{Z}}$ peaks gradually decrease with decreasing metallicity. When
[Fe/H] $<-$0.4 dex, the halo exists (panels f and g), and it accounts for
about one third when $-$1.2 dex $<$ [Fe/H] $<-$0.8 dex (panel f).

For the four high $\alpha$ panels, the $\mathrm{L_{Z}}$ distributions vary
regularly with the metallicity. The TD stars dominate both in panels c and d,
and the $\mathrm{L_{Z}}$ distribution of these two panels are very similar to
those of panels g and h. The typical $\mathrm{L_{Z}}$ peak value is around
1350 kpc km s-1.

We note that there are three components in panel b, and one of them has a
$\mathrm{L_{Z}}$ peak of 948.5$\pm$11.2 kpc km s-1 with a scatter of
527.5$\pm$37.0 kpc km s-1, this value is lower than that of TD
(1253.5$\pm$11.3 kpc km s-1), while, it is higher than that of the halo
(1.8$\pm$11.2 kpc km s-1). Which indicates that this component is dynamically
different from those of the halo and TD, and it is most likely to be MWTD (An
& Beers, 2020; Carollo et al., 2010; Naidu et al., 2020). The difference of
the $\mathrm{L_{Z}}$ peak between MWTD and TD is around 300 kpc km s-1, which
is larger than that of the fitting error. Our results present that the
fraction of MWTD in this panel is around 0.39, while the fractions of the halo
and TD are $\sim$0.21 and $\sim$0.40, respectively.

For the more metal-poor panel a, although there are only two components, we
notice that there is a component with a $\mathrm{L_{Z}}$ peak of 950.7$\pm$7.0
kpc km s-1 with a scatter of 444.9$\pm$35.4 kpc km s-1. This $\mathrm{L_{Z}}$
peak value is very close to that of MWTD in panel b, which indicates this
component is also MWTD. The contribution of MWTD in this panel is about one
third (0.35). The remaining contribution in the panel accounts for about two-
thirds, and they comes from the halo of a $\mathrm{L_{Z}}$ peak of
$-$0.2$\pm$7.1 kpc km s-1 with a sctter of 733.6$\pm$33.6 kpc km s-1.

Fig. 6 presents the $\mathrm{L_{Z}}$ distributions of the outer region stars
($\mathrm{R_{GC}}$ $>$ 8.34 kpc, black histogram) in slices of [Fe/H] and
[$\alpha$/Fe]. Similar to the inner region, we apply _emcee_ to perform a MCMC
simulation to determine the components contained of each panel for the outer
region stars. The best-fit components obtained through the simulation are
superimposed on the histogram of Fig. 6 with color curves, and the best-fit
parameters are listed in Table 2.

Figure 6: Similar to Fig. 4 but for the $\mathrm{L_{Z}}$ Distribution of outer region stars. Table 2: Similar to Table 1, but for the outer region stars. [$\alpha$/Fe] | Component |  [Fe/H]$<$$-$1.2 |  $-$1.2$<$[Fe/H]$<$$-$0.8 |  $-$0.8$<$[Fe/H]$<$$-$0.4 |  [Fe/H]$>$$-$0.4
---|---|---|---|---|---
fraction |  $\mathrm{L_{Z}}$ peak | $\sigma_{\mathrm{LZ}}$ | fraction |  $\mathrm{L_{Z}}$ peak | $\sigma_{\mathrm{LZ}}$ | fraction |  $\mathrm{L_{Z}}$ peak | $\sigma_{\mathrm{LZ}}$ | fraction |  $\mathrm{L_{Z}}$ peak | $\sigma_{\mathrm{LZ}}$
$>0.2$ | TD |  |  |  |  0.72$\pm$0 |  1432.5$\pm$12.8 |  475.6$\pm$15.0 |  0.91$\pm$0 |  1600.4$\pm$10.5 |  386.8$\pm$6.4 |  1.00$\pm$0 |  1697.4$\pm$10.2 |  378.7$\pm$7.3
MWTD |  0.24$\pm$0 |  996.9$\pm$22.0 |  687.8$\pm$62.1 |  |  |  |  |  |  |  |  |
the halo |  0.76$\pm$0.04 |  60.8$\pm$21.9 |  701.4$\pm$30.2 |  0.28$\pm$0.02 |  19.1$\pm$15.2 |  555.4$\pm$28.1 |  0.09$\pm$0.02 |  625.3$\pm$136.6 |  494.0$\pm$55.2 |  |  |
$<0.2$ | TD |  |  |  |  0.59$\pm$0 |  1736.4$\pm$44.1 |  427.2$\pm$30.1 |  0.96$\pm$0 |  1779.6$\pm$49.8 |  427.9$\pm$14.2 |  1.00$\pm$0 |  1808.0$\pm$4.8 |  350.2$\pm$3.4
the halo |  |  |  |  0.41$\pm$0.04 |  211.3$\pm$86.7 |  525.5$\pm$62.6 |  0.04$\pm$0.01 |  450.4$\pm$30.1 |  485.2$\pm$35.8 |  |  |

MWTD has obvious contribution only in panel a of Fig. 6. The fraction of MWTD
in this panel is 0.24, which is lower than that in the same panel of the inner
region. The $\mathrm{L_{Z}}$ peak of MWTD in this panel is 996.9$\pm$22.0 kpc
km s-1 with a scatter of 687.8$\pm$62.1 kpc km s-1, and we note that this
scatter is larger than that of the inner ones ($\sim$500 kpc km s-1). The
reason is most likely due to the fraction of MWTD in the outer region is lower
than that in the inner one.

###  3.2 The $\mathrm{V_{\phi}}$ distributions

It is noted by An & Beers (2021), Carollo et al. (2010) and Kordopatis et al.
(2013) that MWTD has kinematic properties different from those of the halo and
TD, thus, we display the $\mathrm{V_{\phi}}$ distribution of the inner and
outer region stars in Figs. 7 and 8 in slices of [Fe/H] and [$\alpha$/Fe],
respectively. The best-fit result obtained with the MCMC simulation are
superimposed on the histogram with color curves, and the best-fit parameters
are listed in Tables 3 and 4.

Figure 7: Similar to Fig. 4 but for the $\mathrm{V_{\phi}}$ distribution. Table 3: The parameters of the best-fit components in the $\mathrm{V_{\phi}}$ distributions for inner region stars ($\mathrm{R_{GC}}$ $<$ 8.34 kpc). The units of $\left\langle{\mathrm{V_{\phi}}}\right\rangle$ and $\sigma_{\mathrm{V\phi}}$ are in km s-1. [$\alpha$/Fe] | Component |  [Fe/H]$<$$-$1.2 |  $-$1.2$<$[Fe/H]$<$$-$0.8 |  $-$0.8$<$[Fe/H]$<$$-$0.4 |  [Fe/H]$>$$-$0.4
---|---|---|---|---|---
fraction |  $\mathrm{V_{\phi}}$ peak | $\sigma_{\mathrm{V\phi}}$ | fraction |  $\mathrm{V_{\phi}}$ peak | $\sigma_{\mathrm{V\phi}}$ | fraction |  $\mathrm{V_{\phi}}$ peak | $\sigma_{\mathrm{V\phi}}$ | fraction |  $\mathrm{V_{\phi}}$ peak | $\sigma_{\mathrm{V\phi}}$
$>0.2$ | TD |  |  |  |  0.38$\pm$0 |  177.4$\pm$7.2 |  47.6$\pm$3.9 |  0.88$\pm$0 |  176.5$\pm$0.9 |  43.6$\pm$0.6 |  1.00$\pm$0 |  183.7$\pm$1.0 |  42.9$\pm$0.7
MWTD |  0.35$\pm$0 |  124.9$\pm$6.3 |  63.4$\pm$6.5 |  0.40$\pm$0.12 |  117.8$\pm$10.7 |  64.6$\pm$8.8 |  |  |  |  |  |
the halo |  0.65$\pm$0.05 |  1.4$\pm$6.8 |  101.6$\pm$5.5 |  0.22$\pm$0.05 |  6.2$\pm$10.6 |  71.2$\pm$6.2 |  0.12$\pm$0.01 |  57.8$\pm$6.0 |  55.9$\pm$2.6 |  |  |
$<0.2$ | TD |  |  |  |  0.54$\pm$0 |  181.5$\pm$6.4 |  42.0$\pm$5.3 |  0.92$\pm$0 |  185.0$\pm$0.9 |  43.0$\pm$0.7 |  1.00$\pm$0 |  190.2$\pm$0.5 |  40.4$\pm$0.4
the halo |  |  |  |  0.46$\pm$0.08 |  43.3$\pm$14.9 |  77.4$\pm$8.2 |  0.08$\pm$0.01 |  54.6$\pm$6.4 |  53.7$\pm$3.1 |  |  |
Figure 8: Similar to Fig. 7 but for the outer stars ($\mathrm{R_{GC}}$ $>$ 8.34 kpc). Table 4: Similar to Table 3 but for the $\mathrm{V_{\phi}}$ distributions of outer region stars ($\mathrm{R_{GC}}$ $>$ 8.34 kpc). [$\alpha$/Fe] | Component |  [Fe/H]$<$$-$1.2 |  $-$1.2$<$[Fe/H]$<$$-$0.8 |  $-$0.8$<$[Fe/H]$<$$-$0.4 |  [Fe/H]$>$$-$0.4
---|---|---|---|---|---
fraction |  $\mathrm{V_{\phi}}$ peak | $\sigma_{\mathrm{V\phi}}$ | fraction |  $\mathrm{V_{\phi}}$ peak | $\sigma_{\mathrm{V\phi}}$ | fraction |  $\mathrm{V_{\phi}}$ peak | $\sigma_{\mathrm{V\phi}}$ | fraction |  $\mathrm{V_{\phi}}$ peak | $\sigma_{\mathrm{V\phi}}$
$>0.2$ | TD |  |  |  |  0.70$\pm$0 |  157.7$\pm$1.9 |  49.8$\pm$1.7 |  0.94$\pm$0 |  172.1$\pm$0.7 |  41.6$\pm$0.5 |  1.00$\pm$0 |  186.2$\pm$1.1 |  39.6$\pm$0.8
MWTD |  0.21$\pm$0 |  107.4$\pm$7.1 |  74.3$\pm$20.7 |  |  |  |  |  |  |  |  |
the halo |  0.79$\pm$0.07 |  10.0$\pm$6.9 |  77.2$\pm$4.0 |  0.30$\pm$0.02 |  7.0$\pm$2.8 |  59.5$\pm$3.0 |  0.06$\pm$0.01 |  43.0$\pm$5.5 |  43.0$\pm$2.9 |  |  |
$<0.2$ | TD |  |  |  |  0.56$\pm$0 |  185.6$\pm$4.2 |  40.9$\pm$2.9 |  0.96$\pm$0 |  191.9$\pm$0.6 |  40.9$\pm$0.5 |  1.00$\pm$0 |  196.4$\pm$0.5 |  36.1$\pm$0.4
the halo |  |  |  |  0.44$\pm$0.04 |  27.2$\pm$8.5 |  59.2$\pm$6.4 |  0.04$\pm$0 |  41.7$\pm$5.1 |  48.9$\pm$3.2 |  |  |

In each panel of the two figures (Figs. 7 and 8), the components determined
from the $\mathrm{V_{\phi}}$ distribution are very similar to those from the
$\mathrm{L_{Z}}$ ones. The $\mathrm{V_{\phi}}$ peak of MWTD in the panels a
and b of the inner region are 124.9$\pm$6.3 km s-1 and 117.8$\pm$10.7 km s-1,
respectively, while it is 107.4$\pm$7.1 km s-1 in the panel a of the outer
region, which are consistent with those of Carollo et al. (2010) and
Kordopatis et al. (2013), however, they are around 15 km s-1 lower than those
of An & Beers (2021). We note that the matallicites for the sapmle stars of An
& Beers (2021) were estimated from the photometric data, while the
$\mathrm{V_{\phi}}$ of their individual stars were computed using photometric
distances and proper-motion measurements from Gaia EDR3, and their sample
restrict to stars within $\pm 30^{\circ}$ from the Galactic prime meridian.
Therefore, the difference in $\mathrm{V_{\phi}}$ may due to the different in
sample stars.

##  4 Discussion and Conclusions

Based on the combined dataset of LAMOST DR5 and Gaia EDR3, we investigate the
kinematic properties of the giant stars in the solar neighbourhood. Our
results show that MWTD occupies a significantly fraction with metallicity
lower than -0.8 and [$\alpha$/Fe]$>+$0.2 dex. It is found that the fraction of
MWTD is significantly higher within the solar orbit radius, while the Z-axis
angular momentum is similar for both subsamples divided by the solar orbit
radius, 950 and 1000 kpc km s-1 for inner and outer region, respectively. The
high fraction of MWTD in the inner solar orbit radius and higher $\alpha$
element abundances may suggest that the MWTD stars were born in the inner
region of the primordial disk, and some of them migrated to larger radii, or
MWTD is the result of inside out scenario. However, the number of the MWTD
stars is relatively small compared to the TD and halo ones, thus, it is hard
to make an ascertained estimation of the origin of MWTD under the current
situations. To figure out the origin of MWTD, additional constraints from both
observations (such as the abundances of C, N, O, Mg, Si, Ca and Ti, etc) and
numerical simulations are necessary.

## ACKNOWLEDGEMENTS

Our research is supported by National Key R&D Program of China
No.2019YFA0405500, the National Natural Science Foundation of China under
grant Nos. 12090040, 12090044, 11833006, 12022304, 11973052, 11973042,
1214028, U2031143 and U1931102. This work is supported by the Astronomical Big
Data Joint Research Center, co-founded by the National Astronomical
Observatories, Chinese Academy of Sciences and Alibaba Cloud. This work is
also partially supported by the Open Project Program of the Key Laboratory of
Optical Astronomy, National Astronomical Observatories, Chinese Academy of
Sciences. 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. This work has made use of data from the European Space Agency (ESA)
mission Gaia (https://www. cosmos.esa.int/gaia), processed by the Gaia Data
Processing and Analysis Consortium (DPAC, https://www.
cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been
provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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