# Age-metallicity dependent stellar kinematics of the Milky Way disc from
LAMOST and Gaia

Yaqian Wu1,2, Maosheng Xiang3,1, Yuqin Chen1,4, Gang Zhao1,4, Shaolan Bi2,
Chengdong Li 5, Yaguang Li6,7, Yang Huang8,9
1CAS Key Laboratory of Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences Beijing 100101, P. R. China;
2Department of Astronomy, Beijing Normal University, Beijing 100875, P. R.
China;
3Max-Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg,
Germany;
4School of Astronomy and Space Science, University of Chinese Academy of
Sciences Beijing, 101408, P. R. China;
5Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Parks
Road, Oxford, OX1 3PU, UK ;
6Sydney Institute for Astronomy (SIfA), School of Physics, University of
Sydney, NSW 2006, Australia ;
7Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark;
8South West Institute For Astronomy Research, Yunnan University Kunming
650500, P. R. China;
9Department of Astronomy, Peking University, Beijing 100871, P. R. China;
E-mail: wuyaqian@nao.cas.cnE-mail:mxiang<EMAIL_ADDRESS>
###### Abstract

We investigate the stellar kinematics of the Galactic disc in 7 $<$ $R$ $<$ 13
kpc using a sample of 118 945 red giant branch (RGB) stars from LAMOST and
Gaia. We characterize the median, dispersion and skewness of the distributions
of the 3D stellar velocities, actions and orbital parameters across the age-
metallicity and the disc $R$ – $Z$ plane. Our results reveal abundant but
clear stellar kinematic patterns and structures in the age – metallicity and
the disc $R$ – $Z$ plane. The most prominent feature is the strong variations
of the velocity, action, and orbital parameter distributions from the young,
metal-rich thin disc to the old, metal-poor thick disc, a number of smaller-
scale structures — such as velocity streams, north-south asymmetries, and
kinematic features of spiral arms — are clearly revealed. Particularly, the
skewness of $V_{\phi}$ and $J_{\phi}$ reveals a new substructure at $R\simeq
12$ kpc and $Z\simeq 0$ kpc, possibly related to dynamical effects of spiral
arms in the outer disc. We further study the stellar migration through
analysing the stellar orbital parameters and stellar birth radii. The results
suggest that the thick disc stars near the solar radii and beyond are mostly
migrated from the inner disc of $R\sim 4-6$ kpc due to their highly
eccentrical orbits. Stellar migration due to dynamical processes with angular
momentum transfer (churning) are prominent for both the old, metal-rich stars
(outward migrators) and the young metal-poor stars (inward migrators). The
spatial distribution in the $R$ – $Z$ plane for the inward migrators born at a
Galactocentric radius of $>$12 kpc show clear age stratifications, possibly an
evidence that these inward migrators are consequences of splashes triggered by
merger events of satellite galaxies that have been lasted in the past few giga
years.

###### keywords:

catalogues – Galaxy: kinematic – Galaxy: fundamental parameters – stars:

††pagerange: Age-metallicity dependent stellar kinematics of the Milky Way
disc from LAMOST and Gaia–18††pubyear: 2017

##  1 Introduction

Understanding the formation and evolution of the Galactic disc is an
interesting issue of contemporary astrophysics. Modern large-scale sky
surveys, such as the Gaia astrometry survey (e.g. Gaia Collaboration et al.,
2016b, a, 2018; Lindegren et al., 2018), and a number of Galactic spectrocopic
surveys, such as the RAdial Velocity Experiment (RAVE; Steinmetz et al.,
2006), SDSS (Yanny et al., 2009), LAMOST (Zhao et al., 2006, 2012), GALAH (De
Silva et al., 2015) and APOGEE (Majewski et al., 2017), have revealed
unprecedented details and phenomena of our Milky Way disc through the study of
stellar distributions (e.g. Bovy et al., 2012a; Widrow et al., 2012; Mackereth
et al., 2017; Xiang et al., 2018; Wang et al., 2018b; Dobbie & Warren, 2020),
chemistry (Cheng et al., 2012; Hayden et al., 2014; Xiang et al., 2015a;
Kawata et al., 2017; Frankel et al., 2018; Ness et al., 2018, 2019; Wang et
al., 2019a; Feuillet et al., 2019; Lian et al., 2020) and kinematics (Lee et
al., 2011; Bovy et al., 2012b; Carlin et al., 2013; Sun et al., 2015; Aumer et
al., 2017; Antoja et al., 2018; Katz et al., 2018; Kawata et al., 2018; Tian
et al., 2018; Wang et al., 2018a; Bland-Hawthorn et al., 2019; Katz et al.,
2019; Wang et al., 2019b; Yan et al., 2019; Coronado et al., 2020; Li & Shen,
2020) of millions of stars. These studies have provided great insights on the
structure, formation and chemo-dynamical evolution of the Galactic disc(s)
(e.g. Minchev et al., 2015; Aumer & Binney, 2017; Ting & Rix, 2019; Spitoni et
al., 2019).

The Milky Way is suggested to have two major components, the thin and the
thick disc, which differ in various ways, such as spatial distribution, age,
metallicity and kinematics (e.g. Gilmore & Reid, 1983; Bensby et al., 2003;
Spagna et al., 2010; Lee et al., 2011; Haywood et al., 2013; Hayden et al.,
2014; Xiang et al., 2015a, 2017a). On kinematics, the thin disc contains stars
that show larger rotation velocities and smaller velocity dispersion than the
thick disc (e.g. Chiba & Beers, 2000). Stars in the thick disc at the solar
neighbourhood are found to exhibit a positive relation between rotation
velocity and metallicity, while a negative relation is found in thin disc
stars (Spagna et al., 2010; Lee et al., 2011; Adibekyan et al., 2013; Recio-
Blanco et al., 2014; Grieves et al., 2018; Peng et al., 2018; Wojno et al.,
2018). Using these differences in stellar kinematics, a star could be
categorized into the thin or the thick disc population (e.g. Bensby et al.,
2003; Li & Zhao, 2017).

The stellar velocities are found to exhibit significant variations in both the
radial and the vertical directions of the Galactic disc. In the radial
direction, Siebert et al. (2011) detected a radial velocity gradient in the
local disc with the RAVE survey data. Sun et al. (2015) revealed a complex
radial velocity pattern with significant radial variations in the 8$<$$R$$<$12
kpc with the LAMOST spectroscopic survey of the Galactic anti-center (LSS-GAC;
Liu et al., 2014; Yuan et al., 2015). Utilizing the LAMOST data, Tian et al.
(2017) also revealed the radial velocity variations in the outer disc clearly.
With the Gaia DR2, Katz et al. (2018) revealed the complexity of the velocity
field of the Galactic disc. They also observed streaming motions in $V_{R}$,
$V_{\phi}$ and $V_{Z}$. With APOGEE red giant stars, Eilers et al. (2020)
found clear radial velocity patterns across the whole disc of 0$<$$R$$<$20
kpc. These radial patterns were suggested to be related to the perturbation
induced by both the rotating bar and the spiral arms.

In the vertical direction, the velocities of stars in the outer disc show
strong north-south asymmetry (e.g. Carlin et al., 2013; Sun et al., 2015).
Williams et al. (2013) and Sun et al. (2015) investigated the stellar bulk
motions of the nearby disc in detail, and revealed interesting patterns such
as bending, breathing, and ripples. With LAMOST red clump stars, Wang et al.
(2018a) suggested that the outer disc exhibits complex kinematic structures.
For example, there is a high radial velocity peak at $Z$ $\approx$ 0.5 kpc,
$R$ $\approx$ 10–11 kpc. The complex structures are likely associated with
over-density in the stellar number density (e.g. Xu et al., 2015) and stellar
mass distribution (e.g. Xiang et al., 2018). The outermost disc shows a strong
warp structure (e.g. Skowron et al., 2019; Chen et al., 2019a), which exhibits
significant features in stellar velocities (Huang et al., 2018; Poggio et al.,
2018; Romero-Gómez et al., 2019; Wang et al., 2020).

Radial migration is expected to significantly alter the position of stars in
the Galactic disc (Sellwood & Binney, 2002; Roškar et al., 2008; Loebman et
al., 2011; Minchev et al., 2013; Grand & Kawata, 2016; Anders et al., 2018).
Generally, there are two widely suggested processes responsible for the radial
migration: one is the so-called blurring, which changes the stellar radial
position due to epicycle motion without changing the angular momentum; the
other is the so-called churning, which alters the Galactocentric position of
the stars due to interactions with the Galactic bar or the spiral arms without
significantly changing their orbital eccentricities (e.g. Schönrich & Binney,
2009). Thick disc stars may have more blurring than thin disc stars due to
their highly eccentrical motions. Thin disc stars, however, may suffer strong
churning effect. At the solar neighborhood, previous works have revealed the
existence of super metal-rich stars (SMR), which have metallicities exceeding
the present-day interstellar medium. These stars are likely a consequence of
the radial migration due to the churning process (e.g. Nordström et al., 2004;
Boeche et al., 2013; Kordopatis et al., 2013; Xiang et al., 2015b; Anders et
al., 2017; Chen et al., 2019b; Wang et al., 2019a).

In this work, we present a study on stellar kinematics of the Galactic disc in
7 $<$ $R$ $<$ 13 kpc with a large sample of red giant branch (RGB) stars from
the LAMOST Galactic surveys. We characterize the median, dispersion and
skewness of the distributions of the 3D stellar velocities, actions and
orbital parameters across the age – metallicity and the disc $R$ – $Z$ plane.
We further study the blurring and churning effects through analysing the
stellar orbital parameters and stellar birth radii. The paper is organized as
follows. Section 2 introduces the data we adopted. Section 3 presents the
velocities and actions across the age – metallicity and $R$ – $Z$ planes. The
orbital parameters and stellar migration are investigated in Section 4,
followed by conclusions in Section 5.

##  2 Data

We make use of the LAMOST RGB stars from Wu et al. (2019), who estimated
stellar ages and masses for 640 986 RGB stars selected from the forth data
release of the LAMOST Galactic surveys (Deng et al., 2012; Zhao et al., 2012).
The RGB stars are selected with criteria of $T_{\rm eff}$ $<$ 5500 K and log
$g$ $<$ 3.8 dex, and they are further distinguished from red clump stars using
the g-mode period spacing ($\Delta$ $P$) derived from the LAMOST spectra with
a data-driven method based on kernal principal component analysis (KPCA),
taking the asterosesimic measurements from the $Kepler$ (Vrard et al., 2016)
as the training set. The stellar ages and masses are also estimated with the
KPCA method, using a sample of 5376 RGB stars with asteroseismic ages from Wu
et al. (2018) as the training set. The typical age uncertainty is 20 per cent
for stars with the spectral signal-to-noise ratios (SNR) higher than 50 per
pixel. Note that, the stellar ages for metal-poor stars are underestimated due
to bias inherited from the training set. Here we opt to update the age
estimates utilizing the same method as Wu et al. (2019) but removing the
metal-poor stars with artificially young ages from the training set. As a
result, we find that, although not perfectly, this reduces the systematics of
the age estimates by about 1 – 2 Gyr at the metal-poor end ([Fe/H]$\lesssim$
$-$0.6 dex).

Stellar parameters of the RGB stars, including the radial velocity $V_{\rm
r}$, effective temperature $T_{\rm eff}$, surface gravity log $g$, metallicity
[Fe/H], absolute magnitudes ${\rm M}_{V}$ and ${\rm M}_{K_{\rm s}}$, alpha-
element to iron abundance ratio [$\alpha$/Fe], C and N abundances [C/H] and
[N/H], as well as extinction $E_{B-V}$ and spectroscopic distance, are
estimated with the LAMOST Stellar Parameter Pipeline at Peking University
(LSP3; Xiang et al., 2015b, 2017a), publicly available as the LAMOST value-
added catalogue (Xiang et al., 2017b)111http://dr4.lamost.org/doc/vac. Given a
spectral SNR higher than 50, the stellar parameters yielded by LSP3 have a
typical uncertainty of 4 km/s in $V_{\rm r}$,100 K for $T_{\rm eff}$, 0.1 dex
for log $g$, 0.3 mag for $M_{V}$ and $M_{Ks}$, 0.1 dex for [Fe/${\rm H}$],
[C/H] and [N/H], and better than 0.05 dex for [$\alpha$/Fe] (Xiang et al.,
2017b).

We cross-match the LAMOST RGB sample stars with Gaia DR2 for astrometric
parameters. Furthermore, instead of using the spectroscopic distance in the
LAMOST value-added catalogue, we make use of the Gaia distances from Bailer-
Jones et al. (2018) (here after BJ18). Given Gaia DR2 coordinates, proper
motions, the BJ18 distances and the LAMOST radial velocities, we compute the
Galactocentric 3D velocities ($V_{R}$, $V_{\phi}$, $V_{Z}$), actions ($J_{R}$,
$J_{\phi}$, $J_{Z}$), defined as the integral of the canonical momentum along
the orbit

$J_{i}=1/2\pi\oint{p_{i}dx_{i}},$ (1)

where $p_{i}$ are the conjugate momenta, the orbital eccentricity ($e$),
epicentre ($R_{apo}$) and pericentre radii ($R_{peri}$), minimal and maximal
vertical distances from the disc ($Z_{max}$ and $Z_{min}$), and guiding-centre
radii ($R_{g}$) with the Galpy (Bovy, 2015), using the MWPotential2014. We
adopt a right-hand Cartesian coordinate to calculate the space motions of our
sample stars in cylindrical coordinates. $V_{R}$ is positive for motion away
from the Galactic Centre, $V_{\phi}$ is positive in the direction of Galactic
rotation (note that we have changed the direction of $V_{\phi}$, in this
reference frame a retrograde rotation is indicated by $V_{\phi}$ $<$ 0 km
s-1), and $V_{Z}$ is positive towards the north Galactic pole. The Sun is
assumed to be located at (X, Y, Z)= ($-$8, 0, 0) kpc, and the solar motion
w.r.t. the local standard of rest is ($U_{\odot}$, $V_{\odot}$, $W_{\odot}$) =
(7.01, 10.13, 4.95) km s-1 (Huang et al., 2015). As a result, we obtained 568
590 LAMOST-Gaia RGB stars with $X$, $Y$, $Z$, $V_{R}$, $V_{\phi}$, $V_{Z}$,
$J_{R}$, $J_{\phi}$, $J_{Z}$, $e$, $Z_{max}$, $Z_{min}$, $R_{apo}$,
$R_{peri}$, $R_{g}$.

Figure 1: Colour-coded stellar number density distributions of the LAMOST RGB
sample stars in the age – [Fe/H] (left) and the Galactic $R$ – $Z$ (right)
planes. The adopted bin size is 0.25 Gyr $\times$ 0.05 dex for the age –
[Fe/H] plane, and 0.1 kpc $\times$ 0.1 kpc for the $R$ – $Z$ plane.

To obtain a clean sample of disc stars with high quality of age estimates, we
further make a few quality cuts. First of all, to ensure good age estimates,
we discard stars with low spectral quality, for those of spectral SNR $<$ 50
or stars with $d_{g}$ $<$0.8, where $d_{g}$ is an indicator of the similarity
between the training and the target spectra defined in Wu et al. (2019). This
cut removes 65 per cent stars in our sample. Further we discard stars with log
$g$ $<$ 2.0 dex, as well as stars with [Fe/H] $<$$-$1 dex, because in these
parameter spaces, there is a lack of calibration stars with good seismic ages.
In this work, we focus on a characterization of the observed kinematical
properties of the disc stars. To minimize contamination from the halo stars,
we remove stars with $V_{\phi}$ $<$ 50 km s-1 from our sample. We further
discard stars with parallax error larger than 20 per cent to ensure good
quality of the kinematic data. Ultimately, our final sample contains 118 945
stars. Fig. 1 shows the finally adopted RGB stars in the age – [Fe/H] space
and the disc $R$ – $Z$ plane. The left panel illustrates that the sample stars
cover a wide age range, from 0 to $\sim$ 14 Gyr. For all ages, the [Fe/H]
exhibits a broad distribution. For stars older than 8 Gyr, the bulk stars
exhibit a negative trend between age and [Fe/H]. The right panel illustrates
that the sample covers a Galactocentric radial distance of 5–14 kpc, and a
vertical distance of $-$3 to 3.5 kpc from the disc mid-plane. About 84 per
cent of the stars are located with 2 kpc from the Sun.

Fig. 2 shows the distribution of $R$ and $Z$ for our RGB stars in the age –
[Fe/H] plane. It shows that the metal-poor young stars have a median
Galactocentric radius larger than 10 kpc, while the other stars are mainly
located at a Galactocentric radius not far from the solar value. In the
vertical direction, as expected, the young, metal-rich thin disc stars are
located close to the disc mid-plane, while the old, metal-poor thick disc
stars are distributed at larger heights in general.

Figure 2: [Fe/H] versus age diagram for the LAMOST-Gaia RGB stars, colour-
coded by median values of $R$ and $Z$ for stars in each 0.25 Gyr $\times$ 0.05
dex bin.

##  3 Velocities and actions

In this section, we present the distributions of velocities and actions on the
age – metallicity plane in section 3.1 and on the $R$ – $Z$ plane in section
3.2.

###  3.1 Velocities and actions across the age – metallicity plane

We investigate the distributions of $V_{R}$, $J_{R}$, $V_{\phi}$, $J_{\phi}$,
$V_{Z}$ and $J_{Z}$ for mono-age and mono-metallicity stars. In steady state,
the actions are invariant quantities describing the orbit of the star, while
the velocities change along the orbit. Therefore a comparison of the
distribution patterns between velocities and actions for mono-age and mono-
metallicity stars tells us information about the impact of stellar motion on
the morphology of stellar distribution in the age–metallicity plane. A
comparison of the two also reflects the impact of dynamical evolution due to
angular momentum transfer, as the resultant alteration may differ between the
velocity distribution and action distribution. For example, ′churning′ process
may significantly increase the angular momentum, but may not significantly
alter the rotation. difference caused by orbital motion.

We divide the age-[Fe/H] plane into individual bins, and characterize the
distribution of the velocities and actions in each individual bin using three
parameters – median, dispersion and skewness. Here the dispersion is defined
as the standard deviation, and the skewness is defined as

$\alpha_{i}=(Ei^{3}-3\mu\sigma^{2}-\mu^{3})/\sigma^{3},$ (2)

where $\mu$ represents the median and $\sigma$ represents the standard
deviation. To calculate the median, dispersion and skewness of the
distribution, we adopt a bin size of 1 Gyr $\times$ 0.1 dex. We require the
minimum number of stars in each bin to be at least 100. Bins with fewer stars
are neglected.

####  3.1.1 the radial kinematics

Fig. 3 plots the results for $V_{R}$ and $J_{R}$. It shows that the median
values of $V_{R}$ in individual bins are small ($-10\lesssim V_{R}\lesssim 10$
km/s) across the whole age – [Fe/H] plane. There are no strong variations
across the plane except for the young, metal-poor stars, which exhibit larger
mean radial motion than the other stars. These young, metal-poor stars are
found to be thin disc stars in the outer disc, and they also exhibit strong
stream motions in the disc $R$ – $Z$ plane (Fig. 6). The mean $J_{R}$ shows a
clear separation in the age – [Fe/H] plane: the younger, more metal-rich stars
exhibit small $J_{R}$, while the older, metal-poor stars exhibit large
$J_{R}$. This suggests that the younger, more metal-rich stars indeed have
small radial motion. While the old, metal-poor stars have strong radial
motion, and their small mean $V_{R}$ value is simply due to an average effect.
Although the young, metal-poor stars have relatively large mean $V_{R}$, they
have small $J_{R}$, which may be an evidence that these stars have experienced
perturbations. The sharp border is a good distinction between the thin and
thick disc populations — the more metal-poor, older stars with larger $J_{R}$
are dominated by the thick disc stars, which have larger orbit eccentricities
than the more metal-rich younger thin disc stars, which are dominated by
circular orbits (thus small $J_{R}$), (e.g. Jing et al., 2016; Mackereth et
al., 2017; Feuillet et al., 2019; Yan et al., 2019). This can also be seen in
Fig. 6, in which the stars with large $J_{R}$ are distributed at large heights
above the disc plane. For convenience, hereafter, we will use the terms ‘thin
disc’ and ‘thick disc’ to qualitatively represent the younger, more metal-rich
and older, more metal-poor stars, respectively, as separated by the border.
Note that the old ($\gtrsim 10$ Gyr), metal-rich (${\rm[Fe/H]}\gtrsim 0$)
stars also exhibit small $V_{R}$ and $J_{R}$ values, implying that these thin
disc stars are co-formed the metal-poor thick disc stars since they have
comparably old ages. Finally, we note that the stellar age for thick disc
stars at the metal-poor end might have been underestimated due to a lack of
seismic calibration stars. While this does not affect the main conclusions in
the current paper as the difference of kinematics between the thin and thick
disc stars are obvious.

The dispersion of both $V_{R}$ and $J_{R}$ exhibit clear increasing trend from
the young metal-rich stars to the old, metal-poor stars. For populations with
$\tau\lesssim 8$ Gyr and ${\rm[Fe/H]}>0$ dex, which are mostly composed of the
′thin′ disc stars, the dispersion is mainly a function of stellar age, while
the trend with [Fe/H] is not obvious. Since there are strong radial
metallicity gradients for these relatively young stellar populations (e.g.
Xiang et al., 2015a; Anders et al., 2017), a lack of trend with [Fe/H] implies
that the dispersion for mono-age and mono-metallicity stellar population does
not change significantly with Galactocentric radius. This conclusion can be
also reached by combining Fig. 3 with Fig. 2, which shows that the stars with
age $<$ 5 Gyr and [Fe/H] $<$ $-$0.2 dex have a mean Galactocentric radius
beyond 10 kpc, while the stars with solar metallicity at the same age mainly
locate at $\sim$ 8 kpc. Such a lack of radial variations in $V_{R}$ and
$J_{R}$ is probably related to the bifurcation structure in the $V_{R}$
distribution of the outer-disc stars(Liu et al., 2012; Sun et al., 2015), as
these bifurcation structures will increase the measured dispersions at the
outer disc thus the metal-poor bins of a given age.

The skewness of $V_{R}$ is $\sim$ 0 and it uniformly distributes across the
age – [Fe/H] plane, indicating that for all mono-age and mono-abundance bins,
the $V_{R}$ follows nearly Gaussian distributions (See Appendix, Fig. A1).
However, the skewness in $J_{R}$ shows strong patterns, as the young, metal-
rich stars show much larger positive skewness than the old, metal-poor stars.
Note that given the definition of the $J_{R}$, the skewness is always positive
(Fig. A1). Interestingly, Fig. 3 shows that the skewness of the old, metal-
rich stars are different from the young metal-rich stars but similar to the
old, metal-poor stars, in contrast to the case of mean $J_{R}$. We find that
the $J_{R}$ distribution for the old, metal-rich stars is more dispersed than
the young, metal-rich stars. It makes the skewness of the old, metal-rich
stars lower than the young metal-rich stars.

Figure 3: Distributions of the LAMOST-Gaia RGB stars in the age – [Fe/H]
plane, colour-coded by the values of median (left), dispersion (middle), and
skewness (right) of the distribution functions of the radial velocity
($V_{R}$) and action ($J_{R}$) for stars in each 1.0 Gyr $\times$ 0.1 dex age
– [Fe/H] bin.

####  3.1.2 azimuthal kinematics

Fig. 4 plots the results of $V_{\phi}$ and $J_{\phi}$. It shows that the
median $V_{\phi}$ and $J_{\phi}$ are clearly different between the younger,
more metal-rich thin disc stars and the older, more metal-poor thick disc
stars. As the latter has lower rotation velocities and angular momentum. The
sharp border between the two overall populations (i.e., the thin disc and the
thick disc) is consistent with Fig. 3. For the thin disc stars, the values of
$V_{\phi}$ and $J_{\phi}$ increase with decreasing [Fe/H]. In contrast, the
values for $V_{\phi}$ and $J_{\phi}$ of the thick disc stars exhibit a
positive trend with [Fe/H]. This is consistent with previous results (Spagna
et al., 2010; Lee et al., 2011; Adibekyan et al., 2013; Recio-Blanco et al.,
2014; Allende Prieto et al., 2016; Peng et al., 2018; Yan et al., 2019). The
figure shows such a negative trend for thin disc stars of all mono-age stellar
populations of $\tau\lesssim 7$ Gyr. The reason of such a negative trend
between $V_{\phi}$ and [Fe/H] is likely a combined consequence of the stellar
migration due to epicycle motion and the negative radial metallicity gradients
of the Galactic disc. That is to say, the outer, metal-poor stars are mainly
inward migrators, at their peri-center phase of their orbits (see Section 4)
and with large $V_{\phi}$, while the inner/solar neighbourhood, metal-rich
stars are mainly outward migrators, at their apo-center phase of their orbits
and with small $V_{\phi}$ (see e.g. Loebman et al., 2011; Curir et al., 2012;
Minchev et al., 2015; Kawata et al., 2018). For the thick disc stars, the
positive trend between $V_{\phi}$ and [Fe/H] is likely a consequence of a
negative vertical metallicity gradient (e.g. Curir et al., 2012; Xiang et al.,
2015a; Anders et al., 2017; Kawata et al., 2018) and the stellar migration.
The more metal-poor thick disc stars are generally at larger heights and have
smaller $V_{\phi}$ either because they were born hotter so that they have
larger eccentricity and/or because the thick disc was born with a positive
radial metallicity gradient (e.g. Curir et al., 2004). For a given [Fe/H],
younger stars have generally larger $V_{\phi}$ or $J_{\phi}$, which is
consistent with the fact that older stars exhibit more dynamical evolution
(e.g. Ting & Rix, 2019). Note that at the metal-poor and $\tau<5$ Gyr region,
the data present a sharp boundary that exhibits a negative trend between
[Fe/H] and age. This is likely a consequence of Galactic chemical enrichment
with time (e.g. Schönrich & Binney, 2009). This effect also causes a steeper
$V_{\phi}$ – [Fe/H] gradient for the younger population.

The dispersions of $V_{\phi}$ and $J_{\phi}$ also exhibit clear increasing
trend from the younger, more metal-rich stars to the older, more metal-poor
stars. For the thin disc population, the dispersion of $V_{\phi}$ does not
change significantly with [Fe/H], indicating that the values for
$\sigma_{V_{\phi}}$ is nearly constant along the Galactocentric radius for
mono-age stellar population. However the values of $\sigma_{J_{\phi}}$ at a
given age increases with decreasing [Fe/H]. This is simply a consequence of
constant $\sigma_{V_{\phi}}$ and the fact that the more metal-poor stars are
located at larger Galactocentric radius.

The skewness of $V_{\phi}$ is mostly negative for thin disc stars and $\sim$ 0
for thick disc stars. This means that the $V_{\phi}$ distribution for the thin
disc stars has a negative tail, which is largely contributed by stars in the
outer disc (i.e., with larger $R$). While the $V_{\phi}$ distribution for the
thick disc stars is nearly Gaussian. The skewness of $J_{\phi}$ is close to
zero in the majority of the age–[Fe/H] bins, even for many of which the
$V_{\phi}$ skewness is negative. This is not surprising, because some of stars
in the negative tail of $V_{\phi}$ are located at larger $R$, so that the
$J_{\phi}$ ($=V_{\phi}$ $\times$ $R$) distribution will not exhibit a similar
negative tail to the $V_{\phi}$. Nonetheless, for some age – [Fe/H] bins of
$\tau$ $<$ 5 Gyr, ${\rm[Fe/H]}\sim-0.6$ dex, the stars exhibit strong negative
skewness in $J_{\phi}$, possibly due to spiral arm dynamics (Wu et al. in
preparation, see also Kawata et al. 2018).

Figure 4: Same as Fig. 2, but for the azimuthal component of the stellar
velocities and actions.

####  3.1.3 vertical kinematics

Fig. 5 shows that the median value of $V_{Z}$ is generally small ($-$5 km/s
$\lesssim V_{Z}\,\lesssim 5$ km/s), and largely across the whole age – [Fe/H]
plane expect for the young, metal-poor stars, which exhibit systematically
positive mean motion. The $J_{Z}$ exhibits a clear increasing trend from the
younger, metal-rich thin disc stars to the older, metal-poor thick disc stars,
and there is a clear border between the thin and thick stars.

The dispersion of $V_{Z}$ increases smoothly with both age and [Fe/H]. For
stars with the same age, the velocity dispersion increases as [Fe/H]
decreases. At a given [Fe/H], the dispersion of $V_{Z}$ increases with age for
the ‘thin’ disc stars. This relation is known as the age – velocity dispersion
relation (AVR), the results are consistent with previous works (e.g. Holmberg
et al., 2009; Tian et al., 2015; Hayden et al., 2017; Hayden et al., 2020).
The observed AVR of the thin disc stars suggested to be a consequence of
gradual heating through scattering process, and the stars were born with
roughly constant velocity dispersion of smaller than 10 km s-1 (Aumer &
Binney, 2009; Ting & Rix, 2019). Similar to the case of mean $J_{Z}$, the
dispersion of $J_{Z}$ shows a sharp border between the younger, more metal-
rich stars and the older, more metal-poor stars.

Similar to the case of mean $V_{Z}$, the skewness of $V_{Z}$ is also $\sim$ 0
and uniformly distributes across the age – [Fe/H] plane, indicating that for
all mono-age and mono-abundance bins, the $V_{Z}$ follows nearly Gaussian
distributions (Fig. A1). While the skewness of $J_{Z}$ shows strong patterns,
as the young metal-rich populations show larger positive skewness than the
old, metal-poor populations. Note that given the definition of $J_{Z}$, the
skewness is also always positive (Fig. A1).

Figure 5: Same as Fig. 2, but for the vertical component of the stellar
velocities and actions.

###  3.2 Kinematics across the R – Z plane

####  3.2.1 Radial velocity $V_{R}$ & action $J_{R}$

Fig. 6 plots the results of $V_{R}$ and $J_{R}$ across the $R$ – $Z$ plane. To
calculate the median, dispersion and skewness of the distribution, we adopt a
bin size of 0.5 $\times$ 0.5 kpc. We further require a minimal number of 100
in each bin, and discard bins with fewer stars. The mean $V_{R}$ shows
irregular substructures. At the solar radius of 7 kpc $<$ $R$ $<$ 9 kpc, and
$|Z|$ $<$ 0.5 kpc, stars with $R$ $<$ 8 kpc move outwards (with positive
velocity), while stars with $R$ $>$ 8 kpc move inwards (with negative
velocity). It shows a negative radial velocity gradients in the solar radius.
Utilizing the RAVE data, Siebert et al. (2011) measured a negative radial
velocity gradient from about 2 kpc inside of the Sun to about 1 kpc outside.
Williams et al. (2013) confirmed the negative radial velocity gradients using
RAVE red-clump stars. With LAMOST red-clump stars, Tian et al. (2017) and Wang
et al. (2018b) found that the radial velocity is oscillating along $R$, as the
value is negative at $R$ $<$ 9 kpc but becomes positive beyond $R$ $\sim$ 9
kpc. These patterns also revealed by Katz et al. (2018). In the outer disc,
the most prominent features are those with positive $V_{R}$ in $10<R<14$ kpc.
There are two positive velocity peaks, one at $10\lesssim R\lesssim 12$ kpc
and $Z\sim 0.5$ kpc, the other at $R\sim 13$ kpc and $Z\sim 0$ kpc. These
outer-disc substructures correspond to those presented in Fig. 3 for the
young, metal-poor stars. In Fig. 7, we slice the stars into age bins to show
how these $V_{R}$ structures vary with age. The figure illustrates that the
substructure with positive $V_{R}$ at the outer disc of $Z\sim 0.5$ kpc is
strong for young stellar populations, and is also present in old populations.
This structure is likely related to the so-called ‘north-near’
overdensity/stream presented in previous works (Xu et al., 2015; Wang et al.,
2018b; Xiang et al., 2018). The peak at $R\sim 13$ kpc and $Z\sim 0$ kpc
occurs in the age bin of 3-6 Gyr, but it is hard to tell if they are also
present in other age bins due to the limited spatial coverage of the data (but
see Wang et al. 2018 for velocity structure in the outermost disc of $R>12$
kpc). The $J_{R}$ distribution exhibits a flaring structure in the outer disc,
and the $J_{R}$ values are small near the Galactic plane across the whole
radii. In contrast, the stars with $R<10$ kpc and $|Z|\gtrsim 1$ kpc exhibit
large $J_{R}$ values, suggesting that stars at those high Galactic latitude,
mostly thick disc stars, have strong radial motions.

The dispersion of $V_{R}$ and $J_{R}$ exhibits a strong flaring structure
across the whole radial range from $R$ $\sim$ 6 kpc to $R$ $\sim$ 13 kpc, with
larger dispersion at larger heights. At $|Z|$ $<$ 1 kpc, the dispersion of
$V_{R}$ decreases significantly with increasing Galactic radius, which is
likely driven by the age of the stellar populations — the outer disc is
dominated by younger stars.

The skewness of $V_{R}$ is close to 0 across the $R$ – $Z$ plane, except for
the outer disc region of $R$$\gtrsim$11 kpc, which shows negative skewness,
possibly caused by the strong substructures with positive $V_{R}$. The $J_{R}$
for stars at low Galactic heights or outer disc, mostly thin disc stars,
exhibits strong positive skewness due to the sharp distributions (see also
Fig. 3 and the according explanations in section 3.1.1).

Figure 6: Distributions of the LAMOST RGB stars in the Galactic $R$–$Z$ plane.
Colours represent the median (left), dispersion (middle), and skewness (right)
of the radial velocities and action distribution functions for stars in each
0.5 kpc $\times$ 0.5 kpc bin. The substructures (A & B) with high $V_{R}$
values are marked in the top left panel (see text).  Figure 7: Colour-coded
distributions of median $V_{R}$ across the distance $R$–$Z$ plane for stars in
different age bins. Also shown are the contours indicating 50,70,90, and 99
per cent of the sample stars.

####  3.2.2 Azimuthal velocity $V_{\phi}$ & action $J_{\phi}$

Fig. 8 plots the results of $V_{\phi}$ and $J_{\phi}$ across the $R$ – $Z$
plane. It shows a clear vertical gradient on both sides of the disc that
represents the transition from thin to thick disc. The $V_{\phi}$ of the thin
disc exhibits a flaring structure. In the outer disc of $R$ $\geq$ 10 kpc and
$|Z|$ $\geq$ 0.5 kpc, stars in the south are rotating faster than those at the
symmetric positions in the north, which is consistent with previous findings
of Wang et al. (2018b). The $J_{\phi}$ shows regularized flaring patterns
across the whole disc covered by our data, a consequence of the increase of
Galactocentric radius and the vertical $V_{\phi}$ gradient, as the former
causes larger $J_{\phi}$ at outer disc while the latter causes smaller
$J_{\phi}$ at larger height.

The dispersion of $V_{\phi}$ and $J_{\phi}$ also exhibits a clear increasing
trend from the smaller heights to larger heights. For the thin disc stars, the
dispersion of $V_{\phi}$ exhibits a clear flaring structure — stars at larger
$|Z|$ or smaller $R$ have larger $V_{\phi}$ dispersion. As a consequence, the
dispersion of $J_{\phi}$ is largely invariant with $R$. However, for the thick
disc stars, the dispersion of $V_{\phi}$ is largely invariant with $R$. As a
consequence, the dispersion of $J_{\phi}$ increases significantly with $R$ in
the thick disc.

The skewness of $V_{\phi}$ and $J_{\phi}$ is generally negative for thin disc
stars and $\sim$ 0 for thick disc stars. Interestingly, in 11 $<$ $R$ $<$ 13
kpc, $-$0.25 $\lesssim$ $Z$ $\lesssim$ 0.25 kpc, the skewness of $V_{\phi}$
and $J_{\phi}$ is clearly different from the other regions nearby. We find
that there are bifurcations in the $J_{\phi}$ distribution that is probably an
effect of spiral arms. The detailed discussions on this structure will be
presented in Wu et al. (in preparation).

Figure 8: Same as Fig. 6, but for the azimuthal component of the stellar
velocities and actions.

####  3.2.3 Vertical velocity $V_{Z}$ & Action $J_{Z}$

Fig. 9 plots the results of $V_{Z}$ and $J_{Z}$ across the $R$ – $Z$ plane. It
shows that stars in the north and south sides of the plane at $R$ $\sim$ 8.5
kpc exhibit net upward vertical motions. This is bending mode (Widrow et al.,
2012; Williams et al., 2013; Widrow et al., 2014; Sun et al., 2015; Carrillo
et al., 2018; Chequers et al., 2018), and the results are similar to the
results shown by Williams et al. (2013); Carrillo et al. (2018); Wang et al.
(2019b). At the outer disc $R$ $\sim$ 13 kpc, It is revealed that the vertical
velocities in southern disc stars have larger vertical velocities than the
northern disc stars, due to the effect of the warp structure (Skowron et al.,
2019; Chen et al., 2019b). The values for $J_{Z}$ show a strong dependence on
$Z$ while they are only marginally dependent on $R$. This is determined by the
nature of $J_{Z}$. As $J_{Z}$ quantifies the vertical excursions of an orbit,
and vanishes for orbits lying entirely in the Galactic plane, stars at large
heights have large vertical excursions, while stars near the plane have large
probabilities to be with small vertical excursions.

The dispersion of $V_{Z}$ exhibits a clear flaring structure, which shows a
negative radial gradient and a positive vertical gradient. The negative radial
gradient of $\sigma_{V_{Z}}$ is expected since $\sigma_{V_{Z}}$ associated
with the surface mass density (e.g. Freeman & Bland-Hawthorn, 2002; Binney &
Tremaine, 2008), which decreases with increasing Galactocentric radius (e.g.
Xiang et al., 2018). The positive vertical gradient mainly reflects the
increase of surface mass density enclosed by increasing height above the disc.
The dispersion of $J_{Z}$, however, has less dependence with $R$. Again, this
is likely due to the nature of $J_{Z}$, as discussed above.

The skewness of $V_{Z}$ is close to 0 and uniformly distributes across the $R$
– $Z$ plane, except for the outermost region ($R>$11kpc), whereas the skewness
in $J_{Z}$ shows strong patterns. These characteristics are similar to the
case of skewness for $V_{R}$ and $J_{R}$, and they share similar physical
reasons (see Section 3.2.1).

Figure 9: Same as Fig. 6, but for the vertical component of the stellar
velocities and actions.

##  4 Orbital parameters and stellar migration

In this section, we present the orbital parameters of the RGB stars across the
age – metallicity plane and the disc $R$ – $Z$ plane, and discuss the impact
of non-circular orbital motion on the stellar distribution by comparing the
current position of the stars with the guiding-centre radii of their orbits.
We further estimate the birth radii of the stars based on their age and
metallicity in the same way as Chen et al. (2019b), initially proposed by
Minchev et al. (2018), and study the effect of stellar migration (churning).
Note that here we refer to the stellar migration as the churning process, in
which stars change their orbital radii through angular momentum transfer via
scattering or perturbation (e.g. Sellwood & Binney, 2002; Minchev et al.,
2013).

###  4.1 Orbital parameters

As mentioned in Section 2, the orbital parameters of our sample stars are
derived using positions and proper motions from Gaia DR2, distances from BJ18,
radial velocities from LAMOST DR4. The orbit calculation is implemented with
the Galpy, assuming the MWpotential2014 (Bovy, 2015).

Fig. 10 plots the median, dispersion, and skewness of the stellar eccentricity
distributions in the age – [Fe/H] plane and the disc $R$ – $Z$ plane. The
figure illustrates that the young or old but metal-rich thin disc stars have
mean orbital eccentricities smaller than 0.2, and have small ($<$0.1)
dispersion among stars in each mono-age and mono-[Fe/H] bin, which is
consistent with previous work (e.g. Kordopatis et al., 2015). While the old,
metal-poor thick disc stars have larger eccentricities and broader
distribution, the mean eccentricities vary from 0.2 to larger than 0.5. For
the thin disc stars, the eccentricity steady increases with stellar age and
exhibits little change with [Fe/H] at a fixed age, possibly suggesting that
the steady disc heating processes has played a decisive role to determine the
orbital evolution of these stars (e.g. Ting & Rix, 2019). At a given age, the
[Fe/H] of a star is an indicator of its Galactocentric radius because of the
existence of radial [Fe/H] gradient for mono-age populations (e.g. Boeche et
al., 2013; Xiang et al., 2015a, 2017a; Anders et al., 2017; Wang et al.,
2019a). The lack of variations with [Fe/H] suggests that the overall
eccentricity for stars with the same age has little radial variations in the
range of our sample ($7<R<13$ kpc). The spatial patterns of eccentricity in
the $R$ – $Z$ plane reflect mainly the superposition of stellar populations of
different ages. The young, outer disc stars exhibit positive skewness. The
patterns in both the age – [Fe/H] and $R$ – $Z$ plane are similar to the
inverse cases for $V_{\phi}$ (see Figs. 4 and 8). This is expected since the
eccentricity is largely determined by $V_{\phi}$. Interestingly, in 11 $<$ $R$
$<$ 13 kpc, $-$0.25 $\lesssim$ $Z$ $\lesssim$ 0.25 kpc, the dispersion and
skewness of eccentricity exhibit peculiar behaviours, similar to the case of
$V_{\phi}$. This is might be an effect caused by spiral arms (Wu et al. in
preparation).

Figure 10: Top panels: Colour-coded distributions of median (left), dispersion
(middle), and skewness (right) of the eccentricity distribution functions for
stars in individual age–[Fe/H] bins of 1 Gyr $\times$ 0.1 dex. Bottom panels:
Colour-coded distributions of median (left), dispersion (middle), and skewness
(right) of the eccentricity distribution functions for stars in individual
$R$–$Z$ bins of 0.5 kpc $\times$ 0.5 kpc.

Fig. 11 plots the distributions of $Z_{max}$, $R_{g}$, $R_{peri}$, and
$R_{apo}$. The figure clearly shows that the younger, more metal-rich stars
have maximal orbital heights smaller than 1 kpc, while the older, metal-poor
stars have maximal orbital heights larger than 1 kpc, giving credence to our
definition of ‘thin’ and ‘thick’ disc populations in Section 3. The thick disc
stars have orbital guiding-centre radii of 4 – 6 kpc, peri-centre radii of 3 –
6 kpc, and apo-centre radii of $<10$ kpc. This is consistent with the
convention that thick disc stars are all formed in the inner disc (e.g.
Haywood et al., 2013). It is clear that the thick disc stars exhibit smaller
guiding-centre radii than those thin disc stars. The figure also illustrates
that, except for the young, metal-poor ones, stars in all age and [Fe/H] bins
have apo-centre radii $\simeq$10 kpc. In other words, the ‘inner’ ($R_{\rm
apo}<10$ kpc) disc has complex stellar populations, which dominate most of age
and [Fe/H] space, while the ‘outer’ ($R_{\rm apo}>10$ kpc) disc has a
relatively simple formation history, as the stars dominate only a small,
specific part of the age - [Fe/H] space.

Figure 11: Distributions of the maximal height $Z_{\rm max}$, guiding center
radius $R_{g}$, pericenter radius $R_{\rm peri}$, and apocenter radius $R_{\rm
apo}$ of the stellar orbits across the age–[Fe/H] plane. Colours represent the
median values of the orbital parameters for stars in individual age–[Fe/H]
bins of 0.25 Gyr $\times$ 0.05 dex.

###  4.2 Non-circular motion

Due to non-circular nature of the stellar orbits, stars change their radial
positions along the orbits without changing the angular momentum. Here we
study the impact of non-circular effect on the stellar distribution by
comparing the current radial position ($R$) of the star with their orbital
guiding-centre radii ($R_{g}$). Fig. 12 shows the distributions of median
value of $R-R_{g}$ for stars across the age – metallicity and $R$ – $Z$
planes. The figure shows a sharp border between the younger, metal-rich thin
disc stars and the older, metal-poor thick disc stars: the thick disc stars
show positive values of 1 – 4 kpc, indicating that they are currently at their
apo-centre side. This is a strong non-circular effect considering the fact
that 4 kpc is comparable to their guiding-centre radii (Fig. 11). However, the
thin disc stars exhibit only small such effect, since the $R-R_{g}$ has a
value of only $-$1 kpc to $\sim$ 1 kpc, depending on the location in the
age–[Fe/H] and $R$ – $Z$ planes. The young, metal-poor stars at the outermost
disc ($R>12$ kpc, $Z<0$ kpc) exhibit a strongest negative $R-R_{g}$ of about
$-1$ kpc. Note that these stars belong to the warp structure of the outer disc
in the third quadrant. The negative $R-R_{g}$ values suggest that these stars
are currently at the peri-centre side of their orbits.

Figure 12: Distributions of the $R-R_{\rm g}$ across the age–[Fe/H] (left) and
the Galactic $R$–$Z$ (right) planes. Colours represent the median values of
$R-R_{\rm g}$ for stars in individual age–[Fe/H] and $R$–$Z$ bins.

###  4.3 Churning

Determining the stellar churning effect relies on knowledge about the stellar
birth radii. This is a tough task, as the stellar kinematics themselves tell
little about the birth position of a migrator. It has been suggested that, for
some particular stellar populations, the stellar metallicity, as a fossil
record for metallicity of the interstellar medium from which the star was born
from, can be an indicator of the stellar birth place. For example, old
($\gtrsim 8$ Gyr), metal-rich (${\rm[Fe/H]}>0$) stars are generally expected
to be formed at the inner part of the thin disc. So that if such stars occur
at the solar neighbourhood or the outer disc with near circular orbits, they
are likely arrived their current position via churning (e.g. Nordström et al.,
2004; Schönrich & Binney, 2009; Kordopatis et al., 2013; Boeche et al., 2013;
Chen et al., 2019b; Wang et al., 2019a).

With the justifiable assumption that a negative radial metallicity gradient in
the interstellar medium (ISM) is existed for most of the disc lifetime, and
that the dispersion of the ISM metallicity at a given age and radius is small,
Minchev et al. (2018) proposed a method to determine the stellar birth radii
based on their age and metallicity. Briefly, the method assumes that the ISM
has a radial metallicity gradients of $-0.15$ dex/kpc at the beginning of the
disc formation, and monotonically flats to $-0.07$ dex/kpc at present. The ISM
metallicity at solar radius is assumed to vary from ${\rm[Fe/H]}\sim-1.0$ dex
at 13 Gyr ago to 0.1 dex at present. With these prior knowledge, the birth
radius of a star then can be derived from their age and [Fe/H].

Furthermore, the effectiveness of this method has been verified by Chen et al.
(2019b) via applying to the LAMOST stars. Similar to Chen et al. (2019b), here
we adopt the Minchev et al. (2018) method to estimate the birth radii of our
RGB stars. Note that the thick disc stars are widely suggested to exhibit an
insignificant radial metallicity gradient, which is explained as an evidence
that the disc stars are born from radially well-mixed gas (e.g. Cheng et al.,
2012; Boeche et al., 2013; Haywood et al., 2013; Xiang et al., 2015b; Wang et
al., 2019a). The birth radii inferred from their age and metallicity might be
problematic. Therefore we focus only on the thin disc stars, for which the
method applies.

We select the thin disc stars through the [Fe/H]–[$\alpha$/Fe] plane. The
distribution of our RGB stars in the [Fe/H] – [$\alpha$/Fe] plane is shown in
the left panel of Fig. 13. The figure clearly displays two prominent sequences
of stars that correspond to the high-$\alpha$ thick disc sequence and
low-$\alpha$ thin disc sequence, respectively. We adopt the broken line shown
in the Fig. 13 as a criterion to distinguish the thin or thick disc stars. The
right panel of Fig. 13 suggests that the selected thin disc stars are located
in the regions of the age – [Fe/H] plane where the stars have small $J_{R}$
values, verifying the effectiveness of the selection criteria.

In Fig. 14, we plot the distribution of the difference between the guiding-
centre radii and the birth radii $R_{g}-R_{b}$ of the thin disc stars in age –
metallicity and $R$ – $Z$ planes. The $R_{g}-R_{b}$ is a direct measurement of
the effection of stellar redistribution due to radial migration. The figure
suggests that the relatively old stars with super-solar metallicities show
positive $R_{g}-R_{b}$ values, which means that they are outward migrators
from the inner disc. While the younger, metal-poor stars show negative
$R_{g}-R_{b}$ values, indicating that they are inward migrators from the outer
disc. At a given age from 4 to 12 Gyr, the maximal difference between $R_{g}$
and $R_{b}$ could reach 5 kpc for the most metal-rich stars, indicating that
these stars may have a minimal average migration speed of about 0.5 to 1
kpc/Gyr, if we assume that the stars start to migrate since they are born.
This is consistent with Quillen et al. (2018) and Chen et al. (2019b), who
suggest that the stellar migration due to the excitation of the Galactic bar
have a speed of 0.5–1 kpc/Gyr, depending on the pitch angle of the bar.
Interestingly, the data exhibits a lack of stars with ${\rm[Fe/H]}\gtrsim 0.3$
dex and age younger than 4 Gyr. For the most metal-poor stars at age younger
than 2 Gyr, the distance of inward migration could reach 4–5 kpc, suggesting a
fast migration speed of larger than 2 kpc/Gyr. We do not fully understand such
a large migration speed. It is possible that such a fast migration speed might
be related to some violent processes, such as perturbations of galaxy mergers.
On the other hand, it is also possible that the $R_{b}$ estimates for these
outer disc stars suffer large uncertainties.

Fig. 14 further shows the median distance alteration due to the radial
migration for stars at different positions across the disc $R$–$Z$ plane. It
suggests that, at around the solar radius ($7<R<9$ kpc), the median distance
alteration is only 0 – 1 kpc. For stars older than 8 Gyr, the redistribution
effect due to radial migration is much stronger, with a median value of 3 kpc.
In the outer disc of 9$<$$R$$<$12 kpc, the median distance alteration due to
the stellar migration for the overall sample stars is small, from $-1$ to 0
kpc, but the young stars shows a larger overall inward migration of $-$3 kpc.

However, a small alteration in the radial position for the overall stars does
not mean a weak migration effect. In fact, Fig. 15 shows that, the stars with
common birth radius show strong migrations. Particularly, for stars born at 12
kpc, very few of them remain at their birth place, but most of them have
migrated inward by a large distance ($>2$ kpc). Such a net inward migration
effect might be driven by splashes triggered by merger events of satellite
galaxies at the outer disc. Particularly, the Figure shows an age
stratification for these inward migrators, possibly an evidence suggesting
that the migration in the outer disc was triggered by merger events that has
been lasted in the past few giga years. This provides constraints on
understanding the merger history of our Galaxy (e.g. Minchev et al., 2016).

The birth radius determination depends on assumptions on the time-dependent
radial metallicity gradient of the ISM, in order to understand possible
uncertainties raised from this assumption, we test a different ISM gradients
by adopting the mono-age stellar metallicity gradients of Xiang et al.
(2015a)( see their Fig. 15) as an approximation of the ISM gradients. The
comparison of the ISM metallicity adopted in this work for the birth radius
calculation (i.e., from Minchev et al. 2018) and that of the Xiang et al.
(2015a) is shown in Fig. 16. The figure also shows the comparison of the birth
radius derived from these metallicity gradients, as well as the current
position of $R_{b}$ $=$ 12 kpc stars according to the Xiang et al. (2015a)
metallicity gradients. The results suggest that our conclusion about the
migration is robust, i.e., the results are not significantly impacted by the
adopted metallicity gradients. This is consistent with the findings of
Feltzing et al. (2020), who showed that the birth radii of stars with
different ages depends only marginally on the adopted ISM metallicity
gradients, confirming the effectiveness of the Minchev et al. (2018) method
for estimating stellar birth radii.

Figure 13: Left: [$\alpha$/Fe] versus [Fe/H] diagram for our RGB sample. The
blue solid line delineates the demarcation of the thin and thick disc
sequences adopted by this work. Right: [Fe/H] versus Age diagram for our RGB
sample stars, colour-coded by values of $J_{R}$. Also shown are the contours
indicating 50,70,90, and 99 per cent of the selected thin (red) and thick disc
(black) stars. Figure 14: Panel a), Colour-coded distribution of $R_{\rm
g}-R_{\rm b}$ for the thin disc stars in the age–[Fe/H] plane. Panel b):
Colour-coded distribution of $R_{\rm g}-R_{\rm b}$ for the thin disc stars in
the Galactic $R$–$Z$ plane. Panel c): Same as Panel b), but for stars younger
than 4 Gyr. Panel d): Same as Panel c), but for stars older than 8.  Figure
15: Distributions in the $R$ – $Z$ plane for stars born at 12 kpc ($R_{\rm
b}=12$ kpc), as indicated by the red box. The color represents stellar age.
Stratification effect in stellar age is visible.  Figure 16: Left panel:
Comparison of the fits to the evolution functions of the metallicity gradient
of Minchev et al. (2018) with Xiang et al. (2015a)(see their Fig. 15). In the
middle panel: Comparison of birth radius estimates of this work with
metallicity gradient using of Xiang et al. (2015a) (see their Fig. 15). The
black solid line represents the 1:1 line. The colour bar represent the number
of stars. Right panel: Distributions in the $R$ – $Z$ plane for stars born at
12 kpc (birth radii are calculated with metallicity gradient using of Xiang et
al. (2015a)), as indicated by the red box. The colour bar presents stellar
age.

##  5 Summary

We have investigated the stellar kinematics of the Galactic disc in 7 $<$ $R$
$<$ 13 kpc with a large sample of RGB stars combining LAMOST and Gaia. We
characterize the median, dispersion and skewness of the distributions of the
3D stellar velocities, actions and orbital parameters across the age-
metallicity and the disc $R$ – $Z$ planes, and discussed the impact of stellar
migration for the thin disc stars by comparing the guiding-centre radii and
the birth radii, which are derived based on their age and [Fe/H]. The results
reveal abundant kinematic trends and structures. Particularly, we find that:

i) The most prominent feature is the clear difference in the velocity and
action distributions between the thin and thick disc stars. In the age–[Fe/H]
plane, the kinematics exhibit a sharp border between the young, metal-rich
thin disc stars and the old, metal-poor thick disc stars. This is particularly
clear in the distributions of the actions, and presents in the median,
dispersion, and skewness for all the 3D actions $J_{R}$, $J_{\phi}$, and
$J_{Z}$. The kinematic distinction between the thin and thick stars is
consistent with the chemical distinction in the [Fe/H]–[$\alpha$/Fe] plane.
These results support the widely accepted conception that the Milky Way disc
is composed of two major components which related to different formation and
evolution history.

ii) The velocity distributions reveal several known kinematic substructures in
the outer disc. Among of them, we confirm the existence of the north-south
asymmetries for $V_{R}$, which present for stars of all ages and exhibit
complicated patterns in the $R$ – $Z$ planes. As a part of these complicated
patterns, the high-velocity stream at $R\sim 11$ kpc and $Z\sim 0.5$ kpc is
also shown in stars of all ages, and is particularly strong for young
populations. The data also reveals the north-south asymmetries in $V_{Z}$
(also $V_{\phi}$) for $R>10$ kpc, which is the kinematic feature of the disc
warp in the outer disc.

iii) The skewness of the distributions of both $V_{\phi}$ and $J_{\phi}$
reveals a new substructure at $R\sim 12$ kpc and $Z\sim 0$ kpc, which is
related to stars in between the spiral arms (the perseus arm and the outer
arm) in the outer disc. The nature of this substructure deserves to be further
studied.

iv) The thick disc stars have a typical guiding-centre radii smaller than 6
kpc, indicating that they are mostly born in the inner disc of $R<6$ kpc and
reach their current positions via highly eccentric orbital motions. The thin
disc stars, however, have much smaller eccentricities, which lead to typically
only $\sim$1 kpc difference between the guiding-centre radii and their current
positions. These results are well consistent with former works (e.g. Haywood
et al., 2013; Kordopatis et al., 2015; Jia et al., 2018).

v) The old ($>4$ Gyr), metal-rich stars in our sample exhibit strong outward
migration. At a given age, the most metal-rich stars are outward migrators
born at 5 kpc inner than their current positions, indicating a mean migration
speed of about 0.5–1.0 kpc/Gyr. While the metal-poor thin disc stars exhibit
strong inward radial migration. At a given age, the most metal-poor stars
might have been born at 3–5 kpc further away from their current positions. For
young stars, this suggests a fast mean migration speed, probably triggered by
merge events.

vi) All stars born at the outer disc of $R\gtrsim 12$ kpc may have experienced
strong inward migration. The spatial distribution in the $R$ – $Z$ plane for
the inward migrators born at a given distance (e.g., 12 kpc) shows an age
stratification, possibly an evidence that these inward migrators are
consequences of splashes triggered by merger events of satellite galaxies that
have lasted in the past few giga years.

##  6 Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## Acknowledgments

It is a pleasure to thank the anonymous referee and editior for their helpful
suggestions. This work is supported by the National Natural Science Foundation
of China under grant No.11988101, 11903044, 11890694, 11625313, the Joint
Research Fund in Astronomy (U2031203), National Key R&D Program of China No.
2019YFA0405502, M.-S. Xiang and Y. Huang acknowledge supports from NSFC Grant
No. 11703035. Cultivation Project for LAMOST Scientific Payoff and Research
Achievement of CAMS-CAS. The LAMOST FELLOWSHIP is supported by Special Funding
for Advanced Users, budgeted and administrated by Center for Astronomical
Mega-Science, Chinese Academy of Sciences (CAMS). 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 also made use of
data from the European Space Agency (ESA) mission Gaia
222https://www.cosmos.esa.int/gaia, processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
333https://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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## Appendix A

Fig A1 shows an example of the 1D distribution of $V_{R}$, $J_{R}$,
$V_{\phi}$, $J_{\phi}$, $V_{Z}$ and $J_{Z}$ for different age and metallicity
bins. In particular, we select two age and metallicity bins, and mark the
value of skewness of these distribution in the figure.

Figure 17: 1D distribution of $V_{R}$, $J_{R}$, $V_{\phi}$, $J_{\phi}$,
$V_{Z}$ and $J_{Z}$ for different age bins and metallicity bins, black line
represents 4 $<$ $\tau$ $<$ 6 Gyr, $-$0.2 $<$ [Fe/H] $<$ 0, red line
represents 9 $<$ $\tau$ $<$ 11 Gyr, $-$0.9 $<$ [Fe/H] $<$ $-$0.7 dex.
Distributions with positive tails exhibit positive values of skewness, while
distributions with negative tails exhibit negative values of skewness.

Fig. A2 shows the distributions at different birth radius in the $R$ – $Z$
plane. We select three regions in birth radii to validate the median distance
alteration due to the radial migration shown in Fig. 15. From Fig. A2, we can
see that for stars born beyond 10 kpc, the results are consistent with Fig.
15, whereas stars in the outer disc show much stronger inward migration
effect. This may be caused by the stronger interactions with satellite
galaxies at the outer disc.

Figure 18: Distributions in the $R$ – $Z$ plane for stars born at 9
kpc,(left), 10 kpc (middle), 14 kpc (right), as indicated by the red box. The
color represents stellar age.