# Kinematics of the Milky Way from the Gaia EDR3 red giants and sub-giants.

P. N. Fedorov,1 V. S. Akhmetov,1 A. B. Velichko,1 A. M. Dmytrenko1 and S. I.
Denischenko1
1Institute of astronomy of V. N. Karazin Kharkiv national university, Svobody
sq. 4, 61022, Kharkiv, Ukraine
E-mail<EMAIL_ADDRESS>(PNF)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the results of the kinematic investigations carried out with the
use of spatial velocities of red giants and sub-giants containing in the
$Gaia$ EDR3 catalogue. The twelve kinematic parameters of the
Ogorodnikov–Milne model have been derived for stellar systems with radii 0.5
and 1.0 kpc, located along the direction the Galactic center – the Sun – the
Galactic anticenter within the range of Galactocentric distances $R$ 0–8–16
kpc. By combining some of the local parameters the information related to the
Galaxy as a whole has been received in the distance range 4–12 kpc, in
particular the Galactic rotational curve, its slope, velocity gradients. We
show that when using this approach, there is an alternative possibility to
infer the behaviour of the Galactic rotational curve and its slope without
using the Galactocentric distance $R_{\odot}$. The kinematic parameters
derived within the Solar vicinity of 1 kpc radius are in good agreement with
those given in literature.

###### keywords:

stars: kinematics and dynamics–Galaxy: kinematics and dynamics–solar
neighborhood–methods: data analysis–proper motions

††pubyear: 2021††pagerange: Kinematics of the Milky Way from the Gaia EDR3 red
giants and sub-giants.–2

##  1 Introduction

Investigation of stellar kinematics is an important instrument for the
Galactic study. It is based on the analysis of stellar proper motions and
radial velocities within various physical and mathematical models. The
Ogorodnikov–Milne (O–M, Ogorodnikov (1932); Milne (1935); Ogorodnikov (1965))
model is most commonly used for the stellar kinematics analysis. The model
provides the ability to investigate the velocity field in a deformable stellar
system. The analysis of the stellar velocity field is usually confined to
solving the basic kinematic equations to compute the O–M model parameters in
the local coordinate system, which moves together with the Sun about the
Galactic centre.

In the last time, some new models are often used to analyse the stellar
kinematics. Since stellar proper motions and radial velocities are the
components of the velocity vector field, it would be reasonable to use the
methods of decomposition of the corresponding stellar velocity field on a set
of the vector spherical harmonics (VSH). The approach allows to detect all the
systematic constituents present in the stellar velocity field under study, as
well as make the mathematically complete kinematic model. Comparison of the
decomposition coefficients with model parameters shows whether the models are
complete as well as allows to reveal all significant systematics which is not
included into the models. This approach is effective, as demonstrated in a
number of papers, for instance Vityazev & Shuksto (2004); Vityazev & Tsvetkov
(2005); Makarov & Murphy (2007); Mignard & Klioner (2012); Velichko et al.
(2020).

##  2 The O–M model equations and forming the local coordinate systems

To write mathematically the basic kinematic equations of various models, the
Cartesian coordinate system $x,y,z$ is usually applied. The origin of the
system coincides with the Solar system barycenter. The $x$ axis points to the
Galactic center, $y$ axis coincides with the direction of the Galactic
rotation, while $z$ axis is perpendicular to the Galactic plane and
complements the right-handed Cartesian coordinate system (see Fig. 1). The
given system is called the local rectangular Galactic coordinate system.

Figure 1: The local rectangular Galactic $x,y,z$ and cylindrical
Galactocentric $R,\theta,z$ coordinate systems. $S$ is an arbitrary star with
the heliocentric distance $r$.

In this paper, we use the O–M model which describes the systematic
differential stellar velocity field within the vicinity of a chosen point.
According to the Helmholtz theorem, the stellar motion may be represented by
the sum of velocities, namely the translational motion of the centroid of the
corresponding stellar system, as well as its rigid-body rotation and
deformation velocities. By the term "centroid" we mean a point that coincides
with that moving at a velocity equal to the mean velocity of stars belonging
to the given stellar system. This approach has been used, for instance, in the
papers by Clube (1972); du Mont (1977); Miyamoto & Soma (1993); Miyamoto & Zhu
(1998); Bobylev & Khovrichev (2011).

In general, representation of the velocity field of a deformable stellar
system is based on the general form of expanding the continuous vector
function $\bf V(r)$ which is defined in the vicinity of any point $\bf r_{0}$
($x_{0},y_{0},z_{0}$). In the Cartesian Galactic coordinate system $x,y,z$ the
general form of the stellar velocity field expansion $\bf V(\bf r)$ is as
follows:

${\bf V}({\bf r})={\bf V}({\bf r}_{0}+d{\bf r})={\bf V}({\bf
r}_{0})+\left(\frac{\partial V}{\partial
x_{k}}\right)_{0}dx_{k}+\frac{1}{2!}\left(\frac{\partial^{2}V}{\partial
x_{k}\partial x_{m}}\right)_{0}dx_{k}dx_{m}+...$ (1)

where the indexes $k,m=$ 1,2,3, and the partial derivatives are evaluated at
the point ${\bf r}_{0}(x_{0},y_{0},z_{0})$.

When analysing the solar vicinity, we restrict the expansion 1 to the first-
order terms and derive the components $v_{i}$ of the linear velocity field
$\bf V$ at the point located at the heliocentric distance from the Sun ${\bf
r}={\bf r}_{0}+d{\bf r}$ as follows:

$\displaystyle v_{i}({\bf r}_{0}+d{\bf r})=v_{i}({\bf
r}_{0})+\frac{1}{2}\left(\frac{\partial v_{i}}{\partial x_{k}}-\frac{\partial
v_{k}}{\partial x_{i}}\right)_{0}dx_{k}+$ (2)
$\displaystyle+\frac{1}{2}\left(\frac{\partial v_{i}}{\partial
x_{k}}+\frac{\partial v_{k}}{\partial x_{i}}\right)_{0}dx_{k}=v_{i}({\bf
r}_{0})+M^{-}dx_{k}+M^{+}dx_{k}$

where $i,k$ = 1, 2, 3.

In this equation, $v_{i}({\bf r}_{0})$ is usually interpreted as the mean
velocity of the stellar system relative to the Sun, or on the contrary, as the
Solar velocity relative to the chosen centroid with the opposite signs, where
the components are $v_{i}({\bf r}_{0})=-X_{\odot}{\bf e}_{x}-Y_{\odot}{\bf
e}_{y}-Z_{\odot}{\bf e}_{z}$. Thus, finally one can write:

$\displaystyle v_{i}(r_{0}+dr)=-X_{\odot}{\bf e}_{x}-Y_{\odot}{\bf
e}_{y}-Z_{\odot}{\bf e}_{z}+M^{-}dx_{k}+M^{+}dx_{k}$ (3)

The matrix elements $M^{-}$ and $M^{+}$ are the partial derivatives of the
projections $(v_{1},v_{2},v_{3})$ of the velocity vector on the axes of the
rectangular Galactic coordinate system, and they are usually called kinematic
parameters.

Components of the antisymmetric matrix

$M^{-}=\frac{1}{2}\left(\frac{\partial v_{i}}{\partial x_{k}}-\frac{\partial
v_{k}}{\partial x_{i}}\right)_{0}=\omega_{ik}=-\omega_{ki}$ (4)

constitute antisymmetric tensor which is referred to as the tensor of local
rotation velocities. It describes the rigid-body rotation of the stellar
system under examination about the axis which passes through the stellar
centroid with the instant angular velocity ${\bf\omega}$, and
$\omega_{1},\omega_{2},\omega_{3}$ are projections of ${\bf\omega}$ on the
Galactic axes $x,y,z$ respectively.

The matrix

$M^{+}=\frac{1}{2}\left(\frac{\partial v_{i}}{\partial x_{k}}+\frac{\partial
v_{k}}{\partial x_{i}}\right)_{0}=u_{ik}$ (5)

is symmetric, and its components $v_{ik}=v_{ki}$ form the symmetric second-
rank tensor which is referred to as the tensor of local deformation
velocities. It defines the velocity of deformation motion in the stellar
system under study. Usually, it is these components of the tensor that
correspond to the forces inducing deformations in the dynamical investigations
(Tarapov (2002); Sedov (1970)).

The diagonal components of the symmetric matrix $M^{+}_{11}$, $M^{+}_{22}$ and
$M^{+}_{33}$ characterize velocities of relative contractions/expansions of
the stellar system along $x,y,z$ axes, while the components
$M^{+}_{12}=M^{+}_{21}$, $M^{+}_{23}=M^{+}_{32}$, $M^{+}_{13}=M^{+}_{31}$ do
velocities of the angular deformations in the $(x,y)$, $(y,z)$ and $(x,z)$
planes, respectively. Here the velocity of angular deformation means the
velocity of changing angles between segments in the planes.

To set a relation between the velocity components ${\bf V}({\bf r})$ and the
observational data, namely proper motions $\mu_{l}$ and $\mu_{b}$ on the
Galactic longitude and latitude, respectively, as well as radial velocity
$V_{r}$ of a star, we project the vector ${\bf V(r)}$ on the unit vectors of
the spherical Galactic coordinate system ${\bf e}_{l}$, ${\bf e}_{b}$, ${\bf
e}_{r}$ and introduce the factor $k=4.74057$ to transform the dimension of
stellar proper motions from $\rm mas\,yr^{-1}$ to $\rm km\,s^{-1}\,kpc^{-1}$.
Then, the conditional equations for proper motions and radial velocities in
the local Galactic coordinate system have the following form:

$\displaystyle\ k\,\mu_{l}\,{\rm cos}\,b$ $\displaystyle=X_{\odot}/r\,{\rm
sin}\,l-Y_{\odot}/r\,{\rm cos}\,l-\omega_{1}\,{\rm sin}\,b\,{\rm
cos}\,l-\omega_{2}\,{\rm sin}\,b\,{\rm sin}\,l+$
$\displaystyle+\omega_{3}\,{\rm cos}\,b\,+M^{+}_{12}\,{\rm cos}\,b\,{\rm
cos}\,2l-M^{+}_{13}\,{\rm sin}\,b\,{\rm sin}\,l+$
$\displaystyle+M^{+}_{23}\,{\rm sin}\,b\,{\rm cos}\,l-0.5\,M^{+}_{11}{\rm
cos}\,b\,{\rm sin}\,2l+$ $\displaystyle+0.5\,M^{+}_{22}{\rm cos}\,b\,{\rm
sin}\,2l$ (6) $\displaystyle\ k\,\mu_{b}$ $\displaystyle=X_{\odot}/r\,{\rm
cos}\,l\,{\rm sin}\,b+Y_{\odot}/r\,{\rm sin}\,l\,{\rm
sin}\,b\,-Z_{\odot}/r\,{\rm cos}\,b+\omega_{1}\,{\rm sin}\,l-$
$\displaystyle-\omega_{2}\,{\rm cos}\,l-0.5\,M^{+}_{12}\,{\rm sin}\,2b\,{\rm
sin}\,2l+M^{+}_{13}\,{\rm cos}\,2b\,{\rm cos}\,l+$
$\displaystyle+M^{+}_{23}\,{\rm cos}\,2b\,{\rm sin}\,l\,-0.5\,M^{+}_{11}\,{\rm
sin}\,2b\,{\rm cos}^{2}\,l-$ $\displaystyle-0.5\,M^{+}_{22}\,{\rm
sin}\,2b\,{\rm sin}^{2}\,l+0.5\,M^{+}_{33}\,{\rm sin}\,2b$ (7) $\displaystyle
V_{r}/r$ $\displaystyle=-X_{\odot}/r\,{\rm cos}\,l{\rm
cos}\,b-Y_{\odot}/r\,{\rm sin}\,l\,{\rm cos}\,b-Z_{\odot}/r\,{\rm sin}\,b+$
$\displaystyle+M^{+}_{13}\,{\rm sin}\,2b\,{\rm cos}\,l+M^{+}_{23}\,{\rm
sin}\,2b\,{\rm sin}\,l+M^{+}_{12}\,{\rm cos}^{2}\,b\,{\rm sin}\,2l+$
$\displaystyle+M^{+}_{11}\,{\rm cos}^{2}\,b\,{\rm cos}^{2}\,l+M^{+}_{22}\,{\rm
cos}^{2}\,b\,{\rm sin}^{2}\,l+M^{+}_{33}\,{\rm sin}^{2}\,b$ (8)

The equations given above define the differential stellar velocity field and
contain 12 unknown parameters.

The local Galactic coordinate system can be introduced at any arbitrary point
of the Galactic plane provided that one know coordinates $x,y,z$, spacial
velocities $V_{x},V_{y},V_{z}$ of the point (for instance, a star) and for all
stars located within the corresponding vicinity. Taking the Galactocentric
distance of the Sun to be equal to $R_{\odot}=8.0$ kpc (Valleé (2017)), one
can make the transition from the local Galactic coordinate system with the
origin at the barycenter of the Solar system to one with the origin at the
chosen point. The corresponding transformation takes into account, in general,
transposing the origin of the Galactic coordinate system and turning the
coordinate axes. The $x^{\prime}$–axis of the new coordinate system always
points to the Galactic center, the $y^{\prime}$–axis coincides with the
direction of the Galactic rotation, and the $z^{\prime}$ one is perpendicular
to the Galactic plane. The procedure would translocate the fictitious observer
from the barycenter of the Solar system to the point given by the origin of
the new coordinate system.

Thus, one can write the conditional equations of the O–M model in the chosen
coordinate system and estimate the kinematic parameters in the vicinity of
each chosen point. It is evident that these kinematic parameters are local,
since they characterize the stellar kinematics within a small spacial volume,
for instance within a spherical area with a given radius $r^{\prime}$.
Nevertheless, such an analysis of stellar kinematics in various Galactic parts
provides insight into behaviour of some global kinematic parameters, i.e.
those related to the entire Galaxy.

In this paper, we present estimations within the O–M model of the kinematic
parameters for stellar systems located along the direction the Galactic center
– the Sun – the Galactic anticenter. In this case, the origin of the
coordinate system is shifting along the $X$-axis relative to the Sun with the
step of 250 pc, without any rotations of the coordinate system.

##  3 Solving the problem

For the analysis, we use the $Gaia$ EDR3 data (Gaia collaboration et al.
(2016b, 2021a)) of $\sim$7,21 million stars, for which, in addition to proper
motions, radial velocities were also derived within the second release of the
$Gaia$ mission (Katz et al. (2019). The diagram color $BP-RP$ – absolute
magnitude $M_{G}$ for 10% of these stars is shown in Fig. 2. The diagram is
rough because extinction was not be taken into account. Using the straight
lines in Fig. 2, the stars were split into the main sequence stars (located
below the lines), and higher luminosity stars, such as sub-giants and giants
(located above the lines)

The Fig. 3 shows the distribution of number of stars with the heliocentric
distance $r$. $r$ were computed from parallaxes as $1/\pi$. As one can see
from the Fig. 3, the distance range corresponding to the main sequence stars
occupies the relatively narrow range of the Galactocentric distances 7 kpc
$<R<$9 kpc. On the contrary, sub-giants and red giants cover a much wider
Galactocentric distance range, 0 kpc $<R<$ 16 kpc. Therefore, the latter
stellar sample is better to trace the Galactic kinematics. The total amount of
sub-giants and red giants is 4.5 million.

Figure 2: Selection of red giants and sub-giants.

Figure 3: The number of main sequence stars and red giants plus sub-giants
depending on the heliocentric distance.

Each point, from which the observations would be made by a fictitious
observer, is placed on the $x$ axis. The centers of the spheres, restricting
spacial volumes that contain stars around each point, are located every 250 pc
from $R=$0 kpc to 16 kpc . The radii $r^{\prime}$ of the spheres have been set
equal to 500 pc or 1000 pc.

To improve the accuracy of the derived results, in the specific coordinate
system the special procedure has been carried out consisting in exclusion
stars whose velocities deviate from the average value by more than 3$\sigma$.
The rejection procedure has been carried out in each pixel on the celestial
sphere that would be visible from the specific point. The pixelization has
been made according to the HEALPix scheme (The Hierarchical Equal Area iso-
Latitude Pixalization, Gorsky et al. (2005)) with N${}_{\rm side}=$ 10 that
allows to divide the sphere onto 1200 equal areas (pixels). The pixelization
has been applied only for excluding stars.

Once the rejection procedure has been applied, the individual values of
stellar positions, parallaxes, proper motions, and radial velocities has been
used to solve by the linear least square method (LSM) the O–M model equations
written for each local Galactic coordinate system. As a result, the derived
kinematic parameters characterize the stellar velocity field within vicinity
of each given point.

The O–M model parameters allow to estimate a number of kinematic and physical
parameters related to the Galaxy as a whole. Within the simplified form of the
O–M model is usually called as the Oort-Lindblad one Ogorodnikov (1965), the
stellar velocity field is assumed to be axisymmetric. Herewith, the system
rotates in the Galactic plane only, i.e. $\omega_{3}$ = $|\bf\omega|$ while
$\omega_{1}$ and $\omega_{2}$ are equal to zero. In this case the value of
$M^{-}_{21}$ = $\omega_{3}$ corresponds to the Oort constant $B$. The
deformation also exists in the Galactic plane only, i.e. it is characterized
by the Oort constant $A=M^{+}_{12}$, while $M^{+}_{13}$ and $M^{+}_{23}$ are
equal to zero. Contractions/expansions $M^{+}_{11},M^{+}_{22},M^{+}_{33}$
along the corresponding axes are not taken into account. The Oort constants
are usually used in papers on kinematics to characterize the Galactic rotation
(for instance, Bovy (2017); Vityazev et al. (2018); Chengdong et al. (2019);
Tsvetkov (2019)). Thus, the angular rotation velocity of the Galaxy in the
Solar vicinity is equal to

$\Omega=A-B=M^{+}_{12}-\omega_{3}$ (9)

In general, if not to restrict the stellar velocity field, taking into account
the fact that the pairs ($\omega_{1}$, $M^{+}_{23}$) and ($\omega_{2}$,
$M^{+}_{13}$) in the corresponding planes are analogous to the Oort constants
$B$ and $A$ respectively, for an arbitrary centroid located at the
Galactocentric distance $R$ one can compute the angular rotation of the Galaxy
velocity $\Omega_{R}$:

$\Omega_{R}=\sqrt{(M^{+}_{12}-\omega_{3})^{2}+(M^{+}_{13}-\omega_{2})^{2}+(M^{+}_{23}-\omega_{1})^{2}}$
(10)

It is obvious that, knowing values of the Solar velocity components
$X_{\odot},Y_{\odot}$ and $Z_{\odot}$, or velocity of an arbitrary point where
the origin of the coordinate system is placed, relative to the centroids, one
can compute the velocity module $V_{\odot}$ and the apex coordinates
$L_{\odot}$ and $B_{\odot}$:

$V_{\odot}=\sqrt{X^{2}_{\odot}+Y^{2}_{\odot}+Z^{2}_{\odot}}$ (11)

${\rm tg}\,L_{\odot}=\frac{Y_{\odot}}{X_{\odot}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
tg}\,B_{\odot}=\frac{Z_{\odot}}{\sqrt{X^{2}_{\odot}+Y^{2}_{\odot}}}$ (12)

##  4 Kinematic parameters of the O–M model

It is worth noting that, at this stage of our kinematic analysis the O–M model
parameters derived out of the range of Galactocentric distances 4–12 kpc, are
not assumed to be reliable due to a variety of reasons. This is, first of all,
significant decreasing the amount of stars (the number of stars containing in
each stellar system is given in tables 1,2). As a result, the uncertainties of
the derived parameters increase. Moreover, the error bars grow since the
astrometric parameters become less accurate with the heliocentric distance to
the sources. In third place, since the O–M model is linear, it can be
confidently applied if the radius of the stellar system under examination
$r^{\prime}$ is much less than the Galactocentric distance to the stellar
system $R$, i.e. $r^{\prime}<<R$. Unfortunately, this condition is not
fulfilled when analysing the stellar systems which are close to the Galactic
center ($R<$ 4 kpc), although toward the Galactic anticenter the condition if
well fulfilled.

However, the main reason of unreliability is due to our estimations of the
parameters of rotation $\omega_{1},\omega_{2}$ and deformation in $yz$ and
$xz$ planes are actually the values averaged over the northern and southern
Galactic hemispheres. In the papers by Vityazev & Tsvetkov (2014); Velichko et
al. (2020) the stellar kinematics was analysed using the vector spherical
harmonics. The authors have shown that the values of the deformation velocity
component $M^{+}_{12}$ in the $xy$ plane and rotation velocity $\omega_{3}$
are equal in the northern and southern Galactic hemispheres. The values of the
rotation velocity components $\omega_{1}$, $\omega_{2}$ as well as deformation
velocity components $M^{+}_{23}$ and $M^{+}_{13}$ have different signs in the
northern and southern hemispheres while their modules are virtually equal.
Therefore, it is not surprising that we see in Fig. 4 the near zero averaged
values of these parameters within the range 4 $<R<$ 12 kpc, since the centers
of the spheres with radii $r^{\prime}$ 500 pc and 1000 pc are located in the
Galactic plane and encompass the stars of both the northern and southern
Galactic hemispheres. However, it would be wrongly to consider as actual any
deviations of the values from zero out of the Galactocentric range 4–12 kpc,
since the deviations can be caused, for instance, by shifting the LSM
solutions due to the celestial sphere is not covered uniformly (distribution
of objects through the whole celestial sphere is not uniform). Nevertheless,
we show in figures the Galactocentric distance range from 0 kpc to 16 kpc to
have at least some understanding of behaviour of the kinematic parameters near
the Galactic center as well as in its outer part.

###  4.1 Components of the rotational tensor

As one can see from Fig. 4 (left panel), in the range 4–12 kpc the parameters
$\omega_{1}$ and $\omega_{2}$ are virtually equal to zero, while $\omega_{3}$
gradually changes from –30 $\rm km\,s^{-1}\,kpc^{-1}$to –10 $\rm
km\,s^{-1}\,kpc^{-1}$and at the Solar distance has the value
$\omega_{3}=-13.5\pm 0.08$ $\rm km\,s^{-1}\,kpc^{-1}$which is found to be in
good agreement with those given in numerous papers (for instance, Vityazev &
Tsvetkov (2012); Bovy (2017); Vityazev et al. (2018); Chengdong et al. (2019);
Tsvetkov (2019); Velichko et al. (2020)). The behaviour of the components
$\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ indicate that the vectors of
instantaneous angular rotational velocity of the corresponding stellar systems
are virtually perpendicular to the Galactic plane, and in the modulus almost
coincide with the value $\omega_{3}$. Out of range 4–12 kpc the components of
the rotation vector $\omega_{1}$ and $\omega_{2}$ slightly change but we
cannot guarantee the behaviour to be realistic, for the reasons mentioned
above.

Figure 4: The components of local rotation velocities tensor $\omega_{3}$
(Oort B), $\omega_{2},\omega_{1}$ (left panel), and the components of
deformation velocity tensor $M^{+}_{12}$ (Oort A), $M^{+}_{13},M^{+}_{23}$
(right panel) as a function of the Galactocentric distance.

Figure 5: The diagonal components of deformation velocity tensor (left panel)
and the Solar motion components (right panel) as a function of the
Galactocentric distance.

###  4.2 Components of the deformation velocity tensor

The right panel of Fig. 4 demonstrates the deformation components
$M^{+}_{12},M^{+}_{23},M^{+}_{13}$ in the corresponding planes. One can
clearly see that in the range 4–8 kpc $M^{+}_{12}$ gradually increases from 0
to about 16 $\rm km\,s^{-1}\,kpc^{-1}$that followed by a decrease up to 0 at
$R=$ 12 kpc. The value of $M^{+}_{12}$ = 14.92$\pm$ 0.08 $\rm
km\,s^{-1}\,kpc^{-1}$at the Solar distance also is in good agreement with
estimations derived in numerous papers (the same ones as in subsection 4.1).
As in the case of rotational components, in the range 4–8 kpc the deformation
tensor components $M^{+}_{13}$ and $M^{+}_{23}$ are virtually equal to zero,
while within the entire distance range 0–16 kpc their behaviour is very
similar to that for $\omega_{2}$ and $\omega_{1}$ but with opposite sign.

###  4.3 Diagonal components of the deformation velocity tensor

Left panel of Fig. 5 shows dependencies on Galactocentric distance of the
kinematic parameters $M^{+}_{11}$, $M^{+}_{22}$ and $M^{+}_{33}$ which
characterize contractions/expansions of the stellar systems under study along
$x$, $y$, and $z$ axes, respectively. As one can see in left panel of Fig. 5,
the behaviour of $M^{+}_{11}$ shows pronounced peculiarity in the Solar
vicinity. At the same time, in range 4–12 kpc $M^{+}_{22}$ and $M^{+}_{33}$ do
not show such drastic changes in their behavior but demonstrate near zero
variations. This indicates relatively small contractions/expansions along the
$y$ and $z$ axes within the specified distance range. The $x$-axis direction
to the Galactic center (vertex), given inaccurate, may be one of the reasons
of the peculiar behaviour of the parameter $M^{+}_{11}$. However, to
unambiguously establish the real reasons of the behaviour requires additional
investigations which will be present in our future papers.

Figure 6: Coordinates of the Solar apex and the Solar velocity modulus as a
function of the Galactocentric distance.

Figure 7: The modulus of the angular Galactic rotational velocity $|\Omega|$
as a function of the Galactocentric distance.

###  4.4 Components of the Solar velocity relative to different centroids and
coordinates of the Solar apex

Right panel of Fig. 5 shows the components of the linear Solar velocity, while
its modulus and coordinates of the Solar apex are given in Fig, 6 as functions
of the Galactocentric distance.

As one can see from the right panel of Fig. 5, the component of the Solar
velocity vector $X_{\odot}$ relative to centroids located within the range
from 7 to 10 kpc undergoes noticeable changes, and at $R=8$ kpc has minimal
value equal to $8.34\pm 0.06$ $\rm km\,s^{-1}\,kpc^{-1}$at $R=8.75$ kpc. It
worth noting that within the specified distance range the parameter
$M^{+}_{11}$ also changes significantly. It is obvious that the changes also
affect the values of the Solar apex coordinates $L$ and $B$. In this distance
range, one can see that there are less noticeable but still obvious variations
of the parameters $M^{+}_{12}$ and $\omega_{3}$. The values of
$X_{\odot},Y_{\odot}$ and $Z_{\odot}$ relative to the centroid located at the
Solar distance, as well as the Solar apex coordinates are in good agreement
with those given in other papers (for instance, Vityazev & Tsvetkov (2014);
Velichko et al. (2020)), that indicates the behaviour of these Solar velocity
components is realistic.

Thus, we have a very interesting result. We know that the Solar velocity
relative to the centroids taken with opposite sign is equal to the velocity of
centroids relative to the Sun. On the other hand, the Solar velocity modulus
relative to the Galactic center is constant. Hence, the behaviour of the
centroids’ linear velocity components relative to the Galactic center taken
with opposite sign has to follow that of the $X_{\odot},Y_{\odot}$ and
$Z_{\odot}$ components. Since the absolute values of the Solar velocity
components relative to the Galactic center are not known, it is impossible to
establish the offsets on axes for the functions $X_{\odot},Y_{\odot}$ and
$Z_{\odot}$, but their behaviour is determined reliably. In particular, the
behaviour of the function $Y_{\odot}(R)$ have to reproduce that of the
Galactic rotational curve $V_{\rm rot}(R)$. Moreover, since the values of the
Solar peculiar velocity component $Y_{\odot}$ have been derived only from the
$Gaia$ data, not using Galactocentric distances, we derive the behaviour of
the Galactic rotational curve more accurately compared to the traditional
estimation present in subsection 5.1.

Irregular behaviour of the component $X_{\odot}$ may be explained by influence
of the Gould Belt’s stellar velocity field. The stellar kinematics of this
local system with the characteristic radius of about 0.5 kpc is known to
differ from that of stars do not belonging to it (see, for instance, Bobylev
(2004, 2014); Alves and all. (2020)).

###  4.5 Components of the angular Galactic rotational velocity

Fig. 10 shows the components of the angular Galactic rotational velocity
vector $\bf\Omega$ derived from the analysis of the stellar systems under
examination. Fig. 7 shows the vector’s modulus as a function of the
Galactocentric distance $R$. One can clearly see from the figures that in the
range 4–12 kpc the components of the angular Galactic rotational velocity are
nearly zero, and therefore the Galactic rotation vector within the range are
virtually perpendicular to the Galactic plane. The value of $\Omega$ at the
Solar distance is about 30 $\rm km\,s^{-1}\,kpc^{-1}$.

Finally, We emphasize that the kinematic parameters derived in this work for
the Solar centroid are very close to those published in earlier works
mentioned above. This indicates that all the values derived for other
centroids located along the $x$-axis are adequate.

##  5 Interpretation of the kinematic parameters given in the cylindrical
coordinate system

To interpret the derived results and get some information about the Galaxy as
a whole we move to the Galactocentric coordinate system (Miyamoto & Soma
(1993); Bobylev (2006); Vityazev & Tsvetkov (2012), Fig.1). As before, here we
have in mind that the components of the rotational and deformation velocity
tensors defined in the vertical planes $xz$ and $yz$ have opposite signs in
the northern and southern Galactic hemispheres. When solving the equations by
the LSM this regularity may be broken due to stellar distribution over the
sphere is not uniform, especially in spatial areas which are distant from the
Sun. This may result in biases in the least squares solutions.

The elements of the rotational and deformation tensors in the local Galactic
coordinate system are related to the stellar velocity field components
$V_{R}$, $V_{\theta}$ and $V_{z}$ in the cylindrical Galactocentric coordinate
system ($R,\theta,z$) in the following way:

Figure 8: Circular velocity of Galaxy as a function of the Galactocentric
distance ( Galactic rotation curves).

Figure 9: Comparison of the behaviours of the Galactic rotational curves
derived from the equation 15 and using the component $Y_{\odot}$.

$\displaystyle
M=0.5\left(\begin{matrix}2\,M^{+}_{11}&M^{+}_{12}+M^{-}_{12}&M^{+}_{13}+M^{-}_{13}\\\\[6.0pt]
M^{+}_{21}+M^{-}_{21}&2\,M^{+}_{22}&M^{+}_{23}+M^{-}_{23}\\\\[6.0pt]
M^{+}_{31}+M^{-}_{31}&M^{+}_{32}+M^{-}_{32}&2\,M^{+}_{33}\end{matrix}\right)=$

$\displaystyle=\left(\begin{matrix}\frac{\partial V_{R}}{\partial
R}&\frac{1}{R}\frac{\partial
V_{R}}{\partial\theta}-\frac{V_{\theta}}{R}&\frac{\partial V_{R}}{\partial
z}\\\\[10.0pt] \frac{\partial V_{\theta}}{\partial
R}&\frac{1}{R}\frac{\partial
V_{\theta}}{\partial\theta}+\frac{V_{R}}{R}&\frac{\partial
V_{\theta}}{\partial z}\\\\[10.0pt] \frac{\partial V_{z}}{\partial
R}&\frac{1}{R}\frac{\partial V_{z}}{\partial\theta}&\frac{\partial
V_{z}}{\partial z}\end{matrix}\right)$ (13)

Figure 10: The component of the angular Galactic rotational velocity vector
$\bf\Omega$ and gradients of components Galactocentric velocity as a function
of the Galactocentric distance.

###  5.1 Galactic rotational velocity in the centroids’ vicinity

The components of the Galactocentric velocities $V_{R}$, $V_{\theta}$ and
$V_{z}$ as well as their derivatives can be expressed via the combinations of
the O–M model parameters. Thus,

$\frac{V_{\theta}}{R}+\frac{1}{R}\frac{\partial
V_{R}}{\partial\theta}=(\omega_{3}-M^{+}_{12}).$ (14)

Since the stellar system’s angular rotational velocity components in the $xz$
and $yz$ planes (Fig. 4) are nearly zero within the distance range 4–12 kpc,
the radical expression 10 contains only the first term. As a result, the
stellar velocity field within the specified distance range may be considered
as axisymmetric. Therefore, we assume that

$\frac{\partial V_{R}}{\partial\theta}=0.$

Then, given the Galactic rotation is opposite to the direction of reference of
the azimuth angle $\theta$, we can compute the circular velocity of the local
coordinate system $V_{\rm rot}$ (i.e. Galactic rotational velocity) as
follows:

$V_{\rm rot}=-V_{\theta}=(M^{+}_{12}-\omega_{3})_{R}R=\Omega_{R}R$ (15)

Fig. 8 shows the linear circular Galactic rotational velocity $V_{\rm
rot}=\Omega_{R}R$ as a function of the Galactocentric distance $R$. As one can
see from the figure, the values of $V_{\rm rot}$ determined in the Solar
vicinity is in good agreement with those given in numerous papers (see above),
and it is equal to about 227.36$\pm$0.11 $\rm km\,s^{-1}$. The dependency,
commonly referred to as the Galactic rotational curve, is reliable only within
the distance range 4–12 kpc. Unfortunately, out of the specified range the
values of $V_{\rm rot}$ should not be trusted. Nevertheless, as mentioned
above, Fig. 8 demonstrates the entire distance range from 0 to 16 kpc to get
insight into values and behaviour of $V_{\rm rot}$ near the Galactic center
and at its outer part.

For comparison, Fig. 9 also shows the dependency of $Y_{\odot}$ on $R$,
derived only from the peculiar Solar velocity relative to the centroids given
with opposite sign. These two curves have been align by ordinate at $R=$ 0
kpc.

###  5.2 Radial gradient of the Galactic rotational velocity

The radial gradient of the Galactic rotational velocity, or the slope of the
Galactic rotational curve, is defined by the relation:

$\frac{\partial V_{\rm rot}}{\partial R}=-(M^{+}_{12}+\omega_{3})$ (16)

The top panel of Fig.10 shows $\partial V_{\rm rot}/\partial R$ as a function
of $R$. It is clearly seen that the values and behaviour of the slope within
the distance range from 0 to about 8 kpc are noticeably different from those
in the range 8–16 kpc. In the latter case, the curve’s slope decreases, and
its magnitude oscillates around $\partial V_{\rm rot}/\partial R$= –2 - –5
$\rm km\,s^{-1}\,kpc^{-1}$, that, in principle, does not contradict to the
behaviour of the Galactic rotational curve.

At the same time, we can derive the slope of the Galactic rotational curve via
differentiation of $Y_{\odot}$ by $R$. For comparison, Fig. 11 demonstrates
the numerical derivative of the function $Y_{\odot}$ by $R$ which is the slope
of the Galactic rotational velocity derived only from the analysis of the
peculiar Solar velocity relative to the centroids. Its values are in good
agreement with those computed from the relation 16.

Figure 11: Comparison of the slope of the Galactic rotational curve
$-(M^{+}_{12}+\omega_{3})$ and the numerical derivative $\partial
Y_{\odot}/\partial R$.

###  5.3 Vertical gradient of the Galactic rotational curve

The vertical gradient of the Galactic rotational velocity $\partial V_{\rm
rot}/\partial z$ is defined as follows:

$\frac{\partial V_{\rm rot}}{\partial z}=M^{+}_{23}-\omega_{1}$ (17)

The analysis of the kinematic parameter using vector spherical harmonics was
carefully carried out in the papers by Vityazev & Tsvetkov (2014); Velichko et
al. (2020). It was shown that the gradients derived from the $Gaia$ DR2 data
in the northern and southern hemispheres have opposite signs and their modules
are nearly equal. Therefore, when calculating over the entire sphere, within
the distance range 4–12 kpc $\partial V_{\rm rot}/\partial z$ is nearly zero
but still non-zero. The fact is given in the left panel of Fig. 10 (in the
middle) that shows the dependency of $\partial V_{\rm rot}/\partial z$ on $R$.

###  5.4 Galactic warp

The relation

$\Sigma=-\frac{1}{R}\frac{\partial
V_{z}}{\partial\theta}=\omega_{1}+M^{+}_{23}$ (18)

is usually interpreted as kinematic manifestation of the local Galactic warp.
Its values were derived for the Solar vicinity by Miyamoto & Soma (1993); Zhu
(2000); Mignard (2000); Vityazev & Tsvetkov (2012). In the present paper the
value is found to be nearly zero within the entire distance range (the right
panel of Fig. 10, in the middle).

###  5.5 Gradient of the vertical velocity component along the radius-vector

The gradient of the vertical component of the stellar velocity field along the
radius-vector is defined as:

$\frac{\partial V_{z}}{\partial R}=\omega_{2}-M^{+}_{13}$ (19)

It is shown in the left panel of Fig. 10 (bottom raw). One can see that its
values are nearly zero within the distance range 2–14 kpc.

###  5.6 Vertical gradient of the Galactic expansion velocity

The vertical gradient of the Galactic expansion velocity can be found from the
equation:

$\frac{\partial V_{R}}{\partial z}=-(\omega_{2}+M^{+}_{13})$ (20)

Its behaviour is given in the right panel of Fig. 10 (bottom raw). One can see
that its numerical values are small but statistically significant. Besides,
its values grow slowly with distance up to about 11.5 kpc. It is obvious that
more accurate estimations of the given gradients may be established only
solving similar tasks for the northern and southern hemispheres separately.

###  5.7 Relationships between the diagonal components of the deformation
tensor and the stellar velocity field components $V_{R}$, $V_{\theta}$ and
$V_{z}$

$\frac{\partial V_{R}}{\partial R}=M^{+}_{11}$ (21)

$\frac{V_{R}}{R}+\frac{1}{R}\frac{\partial
V_{\theta}}{\partial\theta}=M^{+}_{22}$ (22)

$\frac{\partial V_{z}}{\partial z}=M^{+}_{33}$ (23)

As mentioned above, the behaviour of the Solar velocity components
$X_{\odot},Y_{\odot}$ and $Z_{\odot}$ reflects that of centroids’ linear
velocity components relative the Galactic center. This obviously implies that
the centroid’s velocity vector has not only the $Y_{\odot}$ component related
to $V_{\theta}$ but $X_{\odot}$ and $Z_{\odot}$ components related to $V_{R}$
and $V_{Z}$, respectively. The parameters $X_{\odot}$ and $M^{+}_{11}$ shown
in the top panel of Fig. 5 have been computed as independent unknowns by
solving the O–M model equations. Therefore, the excellent agreement between
the $M^{+}_{11}$ dependency and that derived by numerical differentiation of
the function $X_{\odot}$ by $R$ is an evidence of both actuality and
reliability of their estimations (see Fig. 12).

Figure 12: Comparison of the component of deformation velocity $M^{+}_{11}$
along axis $x$ and the numerical derivative $\partial X_{\odot}/\partial R$.

##  6 Summary and conclusions

The method applied in the present paper for deriving the O–M model kinematic
parameters with the use of stellar spatial velocities, allowed both to
estimate the parameters in a specific local Galactic region and to compute a
number of global characteristics related to much larger Galactic volumes.
Besides, this approach have demonstrated appearance of new, alternative
possibilities, in particular, to trace the behaviour of the Galactic
rotational curve and its slope. The present results demonstrate the ability of
the method to check consistency and reliability of the derived parameters.
Numerical estimations of the kinematic parameters as well as their errors are
given in Tables 1, 2 of the Appendix A.

Since there are some processes in the Galaxy which have opposite directions in
the northern and southern hemispheres, when analysing the stellar velocity
field over the entire sphere, they compensate each other and become
undetectable. Therefore, the final interpretation of the derived results
requires additional investigations of the stellar velocity field in the
northern and southern Galactic hemispheres separately. In the near future, we
plan to publish the results of further research aimed at eliminating the
uncertainties appeared at large and small Galactocentric distances as well as
enlarging the volume of the Galactic spatial area under study.

##  7 Acknowledgements

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.

## Data availability

The used catalogue data is available in a standardised format for readers via
the CDS (https://cds.u-strasbg.fr). The software code used in this paper can
be made available upon request by emailing the corresponding author.

## References

  * Alves and all. (2020) Alves J., Zucker C., Goodman A.A., Speagle J., Meingast S., Robitaille T., Finkbeiner D., Schlafly T., Grnee G. 2020, Nature, 578, 237–239
  * Bobylev (2004) Bobylev V. V. 2004, Astron. Let., 30, No. 11, 785-796
  * Bobylev (2006) Bobylev V. V. 2006, Astron. Let., 32, No. 9, 676-690
  * Bobylev & Khovrichev (2011) Bobylev V. V., Khovritchev, M. Yu. 2011, MNRAS, 417, 1952-1963
  * Bobylev (2014) Bobylev V. V. 2014, Astrophysics, 57, 4
  * Bovy (2017) Bovy J. 2017, MNRAS, 468, L63
  * Clube (1972) Clube S. V. M. 1972, MNRAS, 159, 289-314
  * Chengdong et al. (2019) Chengdong L., Gang Zh., Chengqun Y. 2019, ApJ, 872:205
  * Gaia collaboration et al. (2016b) Gaia collaboration et al. 2016, A&A, 595, A2
  * Gaia collaboration et al. (2021a) Gaia collaboration et al. 2021, A&A, 649, A1
  * Gorsky et al. (2005) Gorski K. M., Hivon E., Banday A. J., Wandelt B. D., Hansen F. K., Reinecke M., Bartelmann M. 2005, AJ, 622, pp. 759-771
  * Katz et al. (2019) Katz D., Sartoretti P., Cropper M., et al. 2019, A& A, 622, A205
  * Makarov & Murphy (2007) Makarov V. V. & Murphy D. W. 2007, AJ 134, 367-375
  * Mignard (2000) Mignard F. 2000, A&A, 354, 522–536
  * Mignard & Klioner (2012) Mignard F. & Klioner, S. 2012, A&A, 547, A59
  * Milne (1935) Milne E. A. 1935, MNRAS, 95, 560.
  * Miyamoto & Zhu (1998) Miyamoto M. & Zhu Z. 1998, AJ, 115, 1483.
  * Miyamoto & Soma (1993) Miyamoto M. & Soma M. 1993, AJ, 105, 691-701.
  * du Mont (1977) du Mont B., 1977, A& A, 61, 127–132
  * Ogorodnikov (1965) Ogorodnikov K. F. 1965, Dynamics of stellar systems, Fizmatgiz, Moscow [in russian]
  * Ogorodnikov (1932) Ogorodnikov K. F. 1932, Z. Astrophys., 4, 190
  * Sedov (1970) Sedov L. I. 1970, Continuum mechanics, V. 1., Science, Moscow, 492 pp. [in russian]
  * Tsvetkov (2019) Tsvetkov A., Amosov F. 2019, Astron. Let., 45, No. 7, 517-528
  * Tarapov (2002) Tarapov I. E. 2002, Continuum mechanics. V. 2: General laws of kinematics and dynamics. 2002, Gould pages, Kharkov, 516 pp. [in russian]
  * Valleé (2017) Valleé J. P. 2017, Astrophys Space Sci, 362, 7971
  * Vityazev et al. (2018) Vityazev, V. V., Popov, A. V., Tsvetkov, A. S., Petrov, S. D., Trofimov, D. A., Kiyaev, V. I. 2018, Astron. Lett., 44, 4, pp. 236-247
  * Vityazev & Tsvetkov (2014) Vityazev V. V., Tsvetkov A. S. 2014, MNRAS, 442, 1249–1264
  * Velichko et al. (2020) Velichko A. B., Fedorov P. N., Akhmetov V. S. 2020, MNRAS, 494, pp. 1430-1447
  * Vityazev & Tsvetkov (2012) Vityazev V. V., Tsvetkov A. S. 2012, Astron. Lett., 38, 7, pp. 411-427
  * Vityazev & Tsvetkov (2005) Vityazev V. V., Tsvetkov A. S. 2005, Vestn. Spb. Univ., Ser.1, 1
  * Vityazev & Shuksto (2004) Vityazev V. V., Shuksto A. K. 2004, in ASP Conference Series, Vol. 316, Order and Chaos on Stellar and Planetary Systems, ed. G. Byrd, K. Kholshevnikov, A. Mylläri, I. Nikiforov & V. Orlov, 230
  * Zhu (2000) Zhu Z. 2000, Publ. Astron. Soc. Japan, 52, 1133.

##  Appendix A Kinematic parameters of the O–M model

Table 1: The O–M model parameters for stellar systems with $r^{\prime}$ = 1.0 kpc. $R$ is the Galactocentric distance to centroids, in kpc; Nstars is the number of stars containing in each stellar system; the Solar motion components are in $\rm km\,s^{-1}$; the other O–M model parameters are in $\rm km\,s^{-1}\,kpc^{-1}$. $R$ | Nstars | $X_{\odot}$ | $Y_{\odot}$ | $Z_{\odot}$ | $\omega_{1}$ | $\omega_{2}$ | $\omega_{3}$
---|---|---|---|---|---|---|---
16.00 | 202 | 8.01$\pm$2.02 | 44.98$\pm$2.02 | 5.75$\pm$2.02 | 1.76$\pm$3.24 | -3.47$\pm$3.19 | -2.36$\pm$3.00
15.75 | 242 | 9.68$\pm$1.78 | 43.85$\pm$1.78 | 7.83$\pm$1.78 | 0.32$\pm$2.79 | -6.32$\pm$2.77 | -6.30$\pm$2.63
15.50 | 296 | 12.00$\pm$1.57 | 42.16$\pm$1.57 | 6.39$\pm$1.57 | 0.89$\pm$2.50 | -3.02$\pm$2.47 | -5.53$\pm$2.32
15.25 | 372 | 12.35$\pm$1.33 | 41.30$\pm$1.33 | 4.11$\pm$1.33 | 4.43$\pm$2.09 | 0.10$\pm$2.04 | -3.51$\pm$1.96
15.00 | 469 | 13.92$\pm$1.30 | 41.00$\pm$1.30 | 4.09$\pm$1.30 | 3.58$\pm$2.01 | 2.59$\pm$2.01 | -7.68$\pm$1.91
14.75 | 619 | 14.95$\pm$1.18 | 40.19$\pm$1.18 | 4.37$\pm$1.18 | 3.71$\pm$1.78 | 1.37$\pm$1.82 | -7.30$\pm$1.72
14.50 | 809 | 14.97$\pm$1.04 | 39.41$\pm$1.04 | 3.90$\pm$1.04 | 5.00$\pm$1.58 | -1.09$\pm$1.59 | -4.61$\pm$1.49
14.25 | 1055 | 16.32$\pm$0.88 | 38.40$\pm$0.88 | 3.82$\pm$0.88 | 4.53$\pm$1.33 | -2.51$\pm$1.34 | -1.81$\pm$1.29
14.00 | 1383 | 16.56$\pm$0.74 | 36.25$\pm$0.74 | 3.72$\pm$0.74 | 4.38$\pm$1.13 | -1.84$\pm$1.13 | -1.99$\pm$1.07
13.75 | 1815 | 17.05$\pm$0.69 | 35.10$\pm$0.69 | 3.64$\pm$0.69 | 5.53$\pm$1.04 | -1.37$\pm$1.04 | -1.46$\pm$0.98
13.50 | 2396 | 17.08$\pm$0.63 | 33.42$\pm$0.63 | 3.47$\pm$0.63 | 6.57$\pm$0.94 | -1.31$\pm$0.94 | -4.09$\pm$0.88
13.25 | 3160 | 16.84$\pm$0.53 | 32.23$\pm$0.53 | 3.18$\pm$0.53 | 6.99$\pm$0.80 | -0.73$\pm$0.80 | -3.47$\pm$0.75
13.00 | 4183 | 16.70$\pm$0.45 | 30.97$\pm$0.45 | 2.96$\pm$0.45 | 5.76$\pm$0.69 | -0.71$\pm$0.69 | -4.92$\pm$0.64
12.75 | 5438 | 16.00$\pm$0.39 | 29.79$\pm$0.39 | 2.88$\pm$0.39 | 5.30$\pm$0.59 | -0.33$\pm$0.59 | -6.11$\pm$0.55
12.50 | 7209 | 15.87$\pm$0.32 | 29.07$\pm$0.32 | 3.09$\pm$0.32 | 4.26$\pm$0.49 | -0.30$\pm$0.50 | -7.40$\pm$0.45
12.25 | 9410 | 15.50$\pm$0.28 | 28.77$\pm$0.28 | 3.71$\pm$0.28 | 2.82$\pm$0.43 | -0.21$\pm$0.43 | -7.83$\pm$0.39
12.00 | 12527 | 15.33$\pm$0.25 | 28.96$\pm$0.25 | 4.41$\pm$0.25 | 2.13$\pm$0.38 | -0.86$\pm$0.38 | -8.42$\pm$0.34
11.75 | 16842 | 15.06$\pm$0.22 | 28.72$\pm$0.22 | 4.75$\pm$0.22 | 1.39$\pm$0.32 | -1.33$\pm$0.33 | -8.30$\pm$0.29
11.50 | 22936 | 14.85$\pm$0.19 | 28.57$\pm$0.19 | 5.15$\pm$0.19 | 0.69$\pm$0.28 | -1.88$\pm$0.28 | -9.41$\pm$0.26
11.25 | 30955 | 14.49$\pm$0.17 | 28.65$\pm$0.17 | 5.48$\pm$0.17 | 0.54$\pm$0.25 | -2.15$\pm$0.25 | -9.73$\pm$0.22
11.00 | 42149 | 14.29$\pm$0.14 | 28.50$\pm$0.14 | 5.73$\pm$0.14 | 0.51$\pm$0.21 | -2.03$\pm$0.22 | -9.56$\pm$0.19
10.75 | 56686 | 13.98$\pm$0.13 | 28.11$\pm$0.13 | 5.92$\pm$0.13 | 0.65$\pm$0.19 | -1.96$\pm$0.20 | -9.49$\pm$0.17
10.50 | 74710 | 13.54$\pm$0.11 | 27.48$\pm$0.11 | 5.99$\pm$0.11 | 0.76$\pm$0.17 | -1.38$\pm$0.17 | -9.85$\pm$0.15
10.25 | 96055 | 12.90$\pm$0.10 | 26.98$\pm$0.10 | 6.22$\pm$0.10 | 0.79$\pm$0.15 | -0.85$\pm$0.16 | -10.51$\pm$0.13
10.00 | 119880 | 12.04$\pm$0.08 | 26.58$\pm$0.08 | 6.56$\pm$0.08 | 0.50$\pm$0.14 | -0.58$\pm$0.14 | -11.22$\pm$0.12
9.75 | 145752 | 11.12$\pm$0.07 | 26.27$\pm$0.07 | 6.83$\pm$0.07 | 0.33$\pm$0.13 | -0.55$\pm$0.13 | -11.49$\pm$0.11
9.50 | 172728 | 9.98$\pm$0.07 | 26.01$\pm$0.07 | 7.06$\pm$0.07 | 0.18$\pm$0.12 | -0.60$\pm$0.13 | -11.43$\pm$0.10
9.25 | 199244 | 9.09$\pm$0.06 | 25.76$\pm$0.06 | 7.26$\pm$0.06 | -0.00$\pm$0.12 | -0.76$\pm$0.12 | -11.22$\pm$0.09
9.00 | 225976 | 8.51$\pm$0.06 | 25.45$\pm$0.06 | 7.43$\pm$0.06 | -0.21$\pm$0.12 | -0.83$\pm$0.12 | -10.94$\pm$0.09
8.75 | 250562 | 8.34$\pm$0.06 | 25.00$\pm$0.06 | 7.52$\pm$0.06 | -0.24$\pm$0.11 | -0.85$\pm$0.11 | -11.06$\pm$0.08
8.50 | 273889 | 8.65$\pm$0.06 | 24.47$\pm$0.06 | 7.63$\pm$0.06 | -0.14$\pm$0.11 | -0.90$\pm$0.11 | -11.47$\pm$0.08
8.25 | 296300 | 9.43$\pm$0.05 | 23.98$\pm$0.05 | 7.67$\pm$0.05 | -0.10$\pm$0.11 | -0.93$\pm$0.11 | -12.39$\pm$0.08
8.00 | 320733 | 10.47$\pm$0.05 | 23.59$\pm$0.05 | 7.72$\pm$0.05 | -0.04$\pm$0.11 | -0.91$\pm$0.11 | -13.50$\pm$0.08
7.75 | 347258 | 11.62$\pm$0.05 | 23.46$\pm$0.05 | 7.79$\pm$0.05 | 0.06$\pm$0.11 | -0.72$\pm$0.11 | -14.66$\pm$0.08
7.50 | 375367 | 12.73$\pm$0.05 | 23.58$\pm$0.05 | 7.88$\pm$0.05 | 0.07$\pm$0.11 | -0.57$\pm$0.11 | -15.56$\pm$0.08
7.25 | 401070 | 13.70$\pm$0.05 | 23.88$\pm$0.05 | 7.97$\pm$0.05 | 0.13$\pm$0.10 | -0.61$\pm$0.10 | -16.11$\pm$0.07
7.00 | 417071 | 14.45$\pm$0.05 | 24.57$\pm$0.05 | 8.09$\pm$0.05 | 0.37$\pm$0.10 | -0.28$\pm$0.10 | -16.83$\pm$0.08
6.75 | 418459 | 14.95$\pm$0.05 | 25.48$\pm$0.05 | 8.20$\pm$0.05 | 0.61$\pm$0.10 | -0.28$\pm$0.10 | -17.46$\pm$0.08
6.50 | 405412 | 15.20$\pm$0.05 | 26.58$\pm$0.05 | 8.34$\pm$0.05 | 0.72$\pm$0.11 | -0.10$\pm$0.11 | -18.18$\pm$0.08
6.25 | 376332 | 15.21$\pm$0.06 | 27.93$\pm$0.06 | 8.48$\pm$0.06 | 0.75$\pm$0.11 | 0.02$\pm$0.11 | -18.99$\pm$0.09
6.00 | 332420 | 15.10$\pm$0.07 | 29.60$\pm$0.07 | 8.68$\pm$0.07 | 0.69$\pm$0.12 | 0.10$\pm$0.12 | -20.20$\pm$0.10
5.75 | 276790 | 14.77$\pm$0.08 | 31.62$\pm$0.08 | 8.85$\pm$0.08 | 0.88$\pm$0.13 | 0.24$\pm$0.14 | -21.42$\pm$0.11
5.50 | 218552 | 14.45$\pm$0.10 | 34.09$\pm$0.10 | 8.92$\pm$0.10 | 0.95$\pm$0.15 | 0.25$\pm$0.16 | -22.72$\pm$0.13
5.25 | 166997 | 14.06$\pm$0.11 | 36.85$\pm$0.11 | 9.04$\pm$0.11 | 0.39$\pm$0.18 | 0.22$\pm$0.18 | -24.05$\pm$0.16
5.00 | 126335 | 13.75$\pm$0.14 | 40.31$\pm$0.14 | 9.08$\pm$0.14 | 0.29$\pm$0.21 | -0.17$\pm$0.21 | -25.49$\pm$0.19
4.75 | 94575 | 13.59$\pm$0.17 | 44.52$\pm$0.17 | 9.14$\pm$0.17 | 0.13$\pm$0.25 | 0.36$\pm$0.25 | -27.51$\pm$0.23
4.50 | 70530 | 13.68$\pm$0.20 | 49.18$\pm$0.20 | 9.37$\pm$0.20 | -0.20$\pm$0.30 | 0.67$\pm$0.30 | -29.60$\pm$0.28
4.25 | 53084 | 13.93$\pm$0.25 | 54.24$\pm$0.25 | 9.50$\pm$0.25 | 0.05$\pm$0.35 | 0.98$\pm$0.36 | -31.28$\pm$0.35
4.00 | 40550 | 14.63$\pm$0.29 | 60.15$\pm$0.29 | 9.45$\pm$0.29 | 0.46$\pm$0.42 | 0.70$\pm$0.43 | -33.29$\pm$0.42
3.75 | 31430 | 15.48$\pm$0.35 | 66.16$\pm$0.35 | 9.42$\pm$0.35 | -0.07$\pm$0.49 | -0.05$\pm$0.51 | -34.84$\pm$0.51
3.50 | 24730 | 16.20$\pm$0.41 | 72.37$\pm$0.41 | 9.80$\pm$0.41 | -0.91$\pm$0.58 | -0.29$\pm$0.60 | -36.12$\pm$0.61
3.25 | 19871 | 17.51$\pm$0.48 | 78.65$\pm$0.48 | 9.61$\pm$0.48 | -1.11$\pm$0.68 | -0.96$\pm$0.70 | -37.04$\pm$0.73
3.00 | 16167 | 18.35$\pm$0.56 | 85.82$\pm$0.56 | 9.16$\pm$0.56 | -1.07$\pm$0.80 | -1.06$\pm$0.81 | -39.66$\pm$0.86
2.75 | 13375 | 19.74$\pm$0.66 | 93.26$\pm$0.66 | 8.84$\pm$0.66 | -1.17$\pm$0.93 | -1.05$\pm$0.94 | -40.33$\pm$1.02
2.50 | 11135 | 20.30$\pm$0.76 | 100.65$\pm$0.76 | 9.12$\pm$0.76 | -2.37$\pm$1.07 | 0.10$\pm$1.08 | -41.04$\pm$1.19
2.25 | 9481 | 21.55$\pm$0.87 | 108.27$\pm$0.87 | 10.47$\pm$0.87 | -2.93$\pm$1.22 | 1.16$\pm$1.23 | -42.33$\pm$1.36
2.00 | 8118 | 21.78$\pm$0.99 | 116.56$\pm$0.99 | 10.80$\pm$0.99 | -4.87$\pm$1.39 | 0.23$\pm$1.40 | -43.44$\pm$1.55
1.75 | 7121 | 22.64$\pm$1.11 | 123.98$\pm$1.11 | 9.62$\pm$1.11 | -6.11$\pm$1.54 | -2.77$\pm$1.55 | -44.74$\pm$1.74
1.50 | 6084 | 22.53$\pm$1.26 | 132.80$\pm$1.26 | 9.33$\pm$1.26 | -6.41$\pm$1.73 | -3.58$\pm$1.75 | -44.18$\pm$1.97
1.25 | 5172 | 22.59$\pm$1.46 | 141.34$\pm$1.46 | 9.56$\pm$1.46 | -7.76$\pm$1.99 | -3.27$\pm$1.99 | -45.37$\pm$2.26
1.00 | 4463 | 22.68$\pm$1.66 | 147.17$\pm$1.66 | 7.37$\pm$1.66 | -10.12$\pm$2.24 | -3.08$\pm$2.21 | -44.29$\pm$2.51
0.75 | 3869 | 22.22$\pm$1.89 | 154.05$\pm$1.89 | 7.85$\pm$1.89 | -10.87$\pm$2.50 | -0.64$\pm$2.46 | -46.63$\pm$2.80
0.50 | 3399 | 22.08$\pm$2.09 | 161.86$\pm$2.09 | 10.04$\pm$2.09 | -8.25$\pm$2.75 | 5.02$\pm$2.69 | -46.09$\pm$3.07
0.25 | 2978 | 22.99$\pm$2.31 | 171.72$\pm$2.31 | 13.92$\pm$2.31 | -6.41$\pm$3.02 | 4.24$\pm$3.01 | -50.13$\pm$3.41
0.00 | 2663 | 21.57$\pm$2.52 | 180.78$\pm$2.52 | 13.87$\pm$2.52 | -7.30$\pm$3.30 | -0.52$\pm$3.25 | -53.29$\pm$3.71
Table 2: The O–M model parameters for stellar systems with $r^{\prime}$ = 1.0 kpc. $R$ is the Galactocentric distance to centroids, in kpc; Nstars is the number of stars containing in each stellar system; the O–M model parameters are in $\rm km\,s^{-1}\,kpc^{-1}$. $R$ | Nstars | $M^{+}_{23}$ | $M^{+}_{13}$ | $M^{+}_{12}$ | $M^{+}_{11}$ | $M^{+}_{22}$ | $M^{+}_{33}$
---|---|---|---|---|---|---|---
16.00 | 202 | -6.92$\pm$3.24 | -5.18$\pm$3.19 | 10.08$\pm$3.00 | -10.29$\pm$4.18 | -3.88$\pm$4.32 | -0.28$\pm$4.82
15.75 | 242 | -7.93$\pm$2.79 | 1.40$\pm$2.77 | 8.17$\pm$2.63 | -4.49$\pm$3.67 | -3.11$\pm$3.76 | -1.38$\pm$4.13
15.50 | 296 | -5.45$\pm$2.50 | 2.03$\pm$2.47 | 7.80$\pm$2.32 | -1.95$\pm$3.25 | -4.07$\pm$3.32 | -2.92$\pm$3.74
15.25 | 372 | -2.99$\pm$2.09 | 1.28$\pm$2.04 | 10.69$\pm$1.96 | -0.90$\pm$2.71 | -2.61$\pm$2.83 | -7.29$\pm$3.06
15.00 | 469 | -3.37$\pm$2.01 | -0.89$\pm$2.01 | 9.37$\pm$1.91 | -2.16$\pm$2.70 | 1.64$\pm$2.70 | -8.96$\pm$2.97
14.75 | 619 | -3.54$\pm$1.78 | -0.67$\pm$1.82 | 9.94$\pm$1.72 | -4.18$\pm$2.50 | -0.23$\pm$2.37 | -6.18$\pm$2.64
14.50 | 809 | -4.93$\pm$1.58 | -0.33$\pm$1.59 | 9.38$\pm$1.49 | -4.28$\pm$2.14 | -1.40$\pm$2.08 | -4.31$\pm$2.37
14.25 | 1055 | -4.33$\pm$1.33 | 0.02$\pm$1.34 | 9.66$\pm$1.29 | -2.85$\pm$1.84 | 0.17$\pm$1.80 | -2.48$\pm$1.95
14.00 | 1383 | -3.97$\pm$1.13 | 0.15$\pm$1.13 | 10.28$\pm$1.07 | -0.09$\pm$1.51 | 0.52$\pm$1.53 | -3.71$\pm$1.67
13.75 | 1815 | -5.96$\pm$1.04 | 0.11$\pm$1.04 | 9.99$\pm$0.98 | 0.10$\pm$1.37 | -0.07$\pm$1.39 | -4.23$\pm$1.56
13.50 | 2396 | -5.97$\pm$0.94 | -0.68$\pm$0.94 | 8.72$\pm$0.88 | 1.47$\pm$1.25 | -1.60$\pm$1.25 | -3.65$\pm$1.40
13.25 | 3160 | -5.69$\pm$0.80 | -0.70$\pm$0.80 | 9.92$\pm$0.75 | 1.43$\pm$1.06 | -1.16$\pm$1.06 | -3.70$\pm$1.20
13.00 | 4183 | -5.07$\pm$0.69 | -1.02$\pm$0.69 | 9.79$\pm$0.64 | 1.63$\pm$0.91 | -0.99$\pm$0.91 | -4.58$\pm$1.03
12.75 | 5438 | -4.96$\pm$0.59 | -2.15$\pm$0.59 | 9.64$\pm$0.55 | 1.07$\pm$0.78 | -0.01$\pm$0.77 | -3.79$\pm$0.90
12.50 | 7209 | -4.10$\pm$0.49 | -2.35$\pm$0.50 | 9.20$\pm$0.45 | 0.72$\pm$0.65 | 0.16$\pm$0.64 | -3.89$\pm$0.75
12.25 | 9410 | -2.94$\pm$0.43 | -2.36$\pm$0.43 | 8.77$\pm$0.39 | 0.44$\pm$0.56 | 0.56$\pm$0.55 | -3.38$\pm$0.65
12.00 | 12527 | -2.77$\pm$0.38 | -2.76$\pm$0.38 | 8.82$\pm$0.34 | 1.23$\pm$0.49 | 0.87$\pm$0.48 | -2.69$\pm$0.58
11.75 | 16842 | -1.93$\pm$0.32 | -3.05$\pm$0.33 | 9.20$\pm$0.29 | 1.41$\pm$0.42 | 0.80$\pm$0.41 | -2.03$\pm$0.50
11.50 | 22936 | -1.63$\pm$0.28 | -2.86$\pm$0.28 | 9.60$\pm$0.26 | 1.19$\pm$0.37 | 0.94$\pm$0.35 | -1.28$\pm$0.43
11.25 | 30955 | -1.37$\pm$0.25 | -2.63$\pm$0.25 | 10.48$\pm$0.22 | 1.25$\pm$0.33 | 0.85$\pm$0.31 | -0.65$\pm$0.39
11.00 | 42149 | -1.08$\pm$0.21 | -2.66$\pm$0.22 | 11.71$\pm$0.19 | 1.03$\pm$0.28 | 0.59$\pm$0.26 | -0.14$\pm$0.34
10.75 | 56686 | -0.86$\pm$0.19 | -2.49$\pm$0.20 | 12.56$\pm$0.17 | 1.48$\pm$0.25 | 0.42$\pm$0.23 | 0.22$\pm$0.30
10.50 | 74710 | -0.70$\pm$0.17 | -2.36$\pm$0.17 | 12.66$\pm$0.15 | 2.46$\pm$0.22 | 0.45$\pm$0.20 | 0.26$\pm$0.27
10.25 | 96055 | -0.65$\pm$0.15 | -2.11$\pm$0.16 | 12.53$\pm$0.13 | 3.34$\pm$0.19 | 0.18$\pm$0.18 | -0.02$\pm$0.25
10.00 | 119880 | -0.40$\pm$0.14 | -2.00$\pm$0.14 | 12.44$\pm$0.12 | 4.30$\pm$0.17 | -0.09$\pm$0.16 | -0.10$\pm$0.23
9.75 | 145752 | -0.27$\pm$0.13 | -1.60$\pm$0.13 | 12.44$\pm$0.11 | 4.94$\pm$0.16 | -0.60$\pm$0.15 | -0.48$\pm$0.22
9.50 | 172728 | -0.10$\pm$0.12 | -1.42$\pm$0.13 | 12.41$\pm$0.10 | 4.70$\pm$0.14 | -0.81$\pm$0.14 | -0.59$\pm$0.21
9.25 | 199244 | 0.08$\pm$0.12 | -1.41$\pm$0.12 | 12.76$\pm$0.09 | 3.54$\pm$0.13 | -1.06$\pm$0.13 | -0.78$\pm$0.20
9.00 | 225976 | 0.24$\pm$0.12 | -1.23$\pm$0.12 | 13.15$\pm$0.09 | 1.68$\pm$0.13 | -1.26$\pm$0.12 | -0.85$\pm$0.20
8.75 | 250562 | 0.33$\pm$0.11 | -1.06$\pm$0.11 | 13.82$\pm$0.08 | -0.59$\pm$0.12 | -1.11$\pm$0.12 | -0.84$\pm$0.19
8.50 | 273889 | 0.25$\pm$0.11 | -1.08$\pm$0.11 | 14.45$\pm$0.08 | -2.81$\pm$0.12 | -0.82$\pm$0.11 | -0.68$\pm$0.19
8.25 | 296300 | 0.31$\pm$0.11 | -1.07$\pm$0.11 | 14.90$\pm$0.08 | -4.54$\pm$0.11 | -0.37$\pm$0.11 | -0.60$\pm$0.19
8.00 | 320733 | 0.20$\pm$0.11 | -0.99$\pm$0.11 | 14.92$\pm$0.08 | -5.22$\pm$0.11 | -0.23$\pm$0.11 | -0.38$\pm$0.19
7.75 | 347258 | -0.04$\pm$0.11 | -0.95$\pm$0.11 | 14.73$\pm$0.08 | -5.33$\pm$0.11 | -0.11$\pm$0.11 | -0.39$\pm$0.19
7.50 | 375367 | -0.19$\pm$0.11 | -0.88$\pm$0.11 | 14.52$\pm$0.08 | -4.50$\pm$0.11 | -0.12$\pm$0.11 | -0.13$\pm$0.18
7.25 | 401070 | -0.26$\pm$0.10 | -1.00$\pm$0.10 | 14.50$\pm$0.07 | -3.29$\pm$0.11 | -0.14$\pm$0.11 | 0.21$\pm$0.18
7.00 | 417071 | -0.47$\pm$0.10 | -0.79$\pm$0.10 | 14.20$\pm$0.08 | -2.11$\pm$0.11 | 0.07$\pm$0.11 | 0.27$\pm$0.18
6.75 | 418459 | -0.64$\pm$0.10 | -0.84$\pm$0.10 | 13.95$\pm$0.08 | -1.21$\pm$0.11 | 0.36$\pm$0.11 | 0.40$\pm$0.18
6.50 | 405412 | -0.69$\pm$0.11 | -0.84$\pm$0.11 | 13.64$\pm$0.08 | -0.46$\pm$0.12 | 0.79$\pm$0.11 | 0.37$\pm$0.18
6.25 | 376332 | -0.67$\pm$0.11 | -0.69$\pm$0.11 | 13.02$\pm$0.09 | 0.12$\pm$0.13 | 1.52$\pm$0.12 | 0.53$\pm$0.19
6.00 | 332420 | -0.55$\pm$0.12 | -0.71$\pm$0.12 | 12.55$\pm$0.10 | 0.61$\pm$0.14 | 2.35$\pm$0.13 | 0.70$\pm$0.20
5.75 | 276790 | -0.67$\pm$0.13 | -0.67$\pm$0.14 | 12.11$\pm$0.11 | 1.02$\pm$0.16 | 3.28$\pm$0.15 | 0.98$\pm$0.22
5.50 | 218552 | -0.81$\pm$0.15 | -0.42$\pm$0.16 | 11.79$\pm$0.13 | 0.98$\pm$0.19 | 3.84$\pm$0.18 | 1.11$\pm$0.25
5.25 | 166997 | -0.53$\pm$0.18 | -0.22$\pm$0.18 | 11.69$\pm$0.16 | 1.44$\pm$0.22 | 4.19$\pm$0.22 | 1.55$\pm$0.28
5.00 | 126335 | -0.58$\pm$0.21 | -0.12$\pm$0.21 | 11.06$\pm$0.19 | 1.06$\pm$0.27 | 4.70$\pm$0.27 | 1.55$\pm$0.33
4.75 | 94575 | -0.80$\pm$0.25 | 0.08$\pm$0.25 | 10.57$\pm$0.23 | 0.27$\pm$0.33 | 4.26$\pm$0.33 | 1.67$\pm$0.38
4.50 | 70530 | -0.82$\pm$0.30 | 0.46$\pm$0.30 | 9.86$\pm$0.28 | -0.13$\pm$0.40 | 3.30$\pm$0.40 | 1.40$\pm$0.44
4.25 | 53084 | -0.76$\pm$0.35 | 0.44$\pm$0.36 | 9.34$\pm$0.35 | -0.62$\pm$0.50 | 2.34$\pm$0.49 | 0.65$\pm$0.51
4.00 | 40550 | -1.07$\pm$0.42 | 0.22$\pm$0.43 | 9.35$\pm$0.42 | -2.50$\pm$0.61 | 0.57$\pm$0.59 | 0.06$\pm$0.60
3.75 | 31430 | -0.75$\pm$0.49 | 0.26$\pm$0.51 | 8.79$\pm$0.51 | -3.71$\pm$0.74 | -0.50$\pm$0.70 | -1.20$\pm$0.69
3.50 | 24730 | 0.24$\pm$0.58 | 0.02$\pm$0.60 | 10.10$\pm$0.61 | -4.69$\pm$0.89 | -1.53$\pm$0.85 | -1.67$\pm$0.80
3.25 | 19871 | 0.32$\pm$0.68 | -0.85$\pm$0.70 | 10.14$\pm$0.73 | -4.62$\pm$1.05 | -1.52$\pm$1.01 | -0.37$\pm$0.91
3.00 | 16167 | 2.00$\pm$0.80 | -1.52$\pm$0.81 | 8.70$\pm$0.86 | -3.03$\pm$1.24 | -2.84$\pm$1.20 | 0.65$\pm$1.05
2.75 | 13375 | 3.67$\pm$0.93 | -0.42$\pm$0.94 | 9.19$\pm$1.02 | -3.70$\pm$1.45 | -4.84$\pm$1.42 | 0.50$\pm$1.19
2.50 | 11135 | 4.57$\pm$1.07 | -0.08$\pm$1.08 | 7.76$\pm$1.19 | -3.57$\pm$1.69 | -5.06$\pm$1.66 | 0.85$\pm$1.35
2.25 | 9481 | 5.87$\pm$1.22 | -1.78$\pm$1.23 | 7.85$\pm$1.36 | -3.04$\pm$1.93 | -6.05$\pm$1.91 | 1.34$\pm$1.52
2.00 | 8118 | 6.80$\pm$1.39 | -2.56$\pm$1.40 | 8.61$\pm$1.55 | -2.18$\pm$2.21 | -5.08$\pm$2.18 | 1.82$\pm$1.71
1.75 | 7121 | 7.40$\pm$1.54 | -0.91$\pm$1.55 | 10.17$\pm$1.74 | -2.72$\pm$2.48 | -7.38$\pm$2.45 | 2.96$\pm$1.87
1.50 | 6084 | 6.47$\pm$1.73 | -0.13$\pm$1.75 | 7.19$\pm$1.97 | 3.56$\pm$2.80 | -6.93$\pm$2.77 | 1.86$\pm$2.09
1.25 | 5172 | 7.30$\pm$1.99 | 0.35$\pm$1.99 | 7.31$\pm$2.26 | -2.66$\pm$3.20 | -4.45$\pm$3.20 | 3.91$\pm$2.37
1.00 | 4463 | 6.80$\pm$2.24 | 2.52$\pm$2.21 | 11.92$\pm$2.51 | 4.67$\pm$3.52 | -9.54$\pm$3.59 | 5.17$\pm$2.67
0.75 | 3869 | 5.53$\pm$2.50 | -0.96$\pm$2.46 | 12.49$\pm$2.80 | 3.77$\pm$3.92 | -9.24$\pm$4.01 | 8.42$\pm$2.98
0.50 | 3399 | 5.19$\pm$2.75 | -6.60$\pm$2.69 | 12.22$\pm$3.07 | 0.38$\pm$4.26 | -8.55$\pm$4.43 | 7.43$\pm$3.28
0.25 | 2978 | 10.07$\pm$3.02 | -6.11$\pm$3.01 | 10.28$\pm$3.41 | -1.12$\pm$4.82 | -2.53$\pm$4.83 | 6.17$\pm$3.62
0.00 | 2663 | 10.66$\pm$3.30 | -2.32$\pm$3.25 | 9.39$\pm$3.71 | -4.27$\pm$5.18 | -2.13$\pm$5.31 | 5.78$\pm$3.92