The Velocity Curves and Three-Dimensional Kinematics of the Milky Way from Gaia DR3

Abstract

We investigate the fundamental kinematic parameters of stellar motions in the Milky Way using a statistical sample of 27,485,720 stars with measured three-dimensional velocities from the Gaia DR3 catalogue. From these data, we determine the local solar velocity $(U_{\odot },V_{\odot },W_{\odot })=\left(10.5\pm 0.7,14.5\pm 0.9,7.33\pm 0.1\right)\mathrm{km}/\mathrm{s}$, the average rotation velocity ${\mathrm{V}}_0=\left(228\pm 4\right)$ km/s, and the circular velocity ${\mathrm{V}}_c=\left(234\pm 4\right)$ km/s, both at the radius $R_{\odot }$. Next, within the range of $3-14$ kpc, we derive the Galactic circular velocity curve ${\mathrm{V}}_c\left(R\right)$ and determine the average spherical components of the stellar velocities as functions of both Galactocentric radius R and distance from the Galactic plane $\left|Z\right|$. Two-dimensional $R-Z$ maps of these components reveal a complex three-dimensional kinematic structure. In particular, the pattern observed at $R\lesssim 5$ kpc can be associated with the influence of the Galactic bar. To facilitate the description of the observations, we employ a simplified axisymmetric Monte Carlo model with several free parameters that reproduces the main kinematic trends.

1. Introduction

The Milky Way (MW) represents a unique laboratory for studying gravity and for understanding the formation and evolution of galaxies. In recent years, the Gaia space observatory has provided an unprecedented amount of highly precise astrometric, photometric, and spectroscopic data for millions of stars in the MW [1]. The scientific exploitation of these data has led to thousands of studies devoted to Galactic structure, stellar populations, and Galactic dynamics.

In particular, the Gaia DR3 catalogue provides exceptionally rich input data for constructing a detailed kinematic map of the MW, far beyond what was previously achievable. These data reveal structures on a broad range of spatial and dynamical scales, from gravitationally bound binaries and multiple stellar systems [2,3] to streaming motions associated with spiral arms [4,5], local velocity fluctuations and turbulence, and the global rotation of the Galactic disk and halo.

The observed Galactic rotation strongly suggests the presence of dark matter, which is believed to constitute a substantial fraction of the total mass of the MW (see [6,7,8,9,10,11,12,13,14,15,16,17] and the overview in [18]). Recent analyses of MW kinematics have provided increasingly accurate determinations of the Galactic circular velocity curve, defined as the radial dependence of the galaxy circular velocity [7,15,19,20,21,22]. Circular velocity curves represent one of the principal observational constraints for models of Galactic mass distribution and dark matter halos.

In addition to gravity, the formation and evolution of stars are governed by the forces of the microworld, namely the strong and electroweak interactions described by the Standard Model of particle physics. However, despite extensive observational and theoretical efforts, the microscopic nature and origin of dark matter remain unresolved.

Another important area of current research concerns the detailed mapping of kinematic substructures and deviations from axial symmetry within the MW, including spiral-arm perturbations, resonances induced by the Galactic bar, phase-space streams, and vertical oscillatory motions  [20,23,24,25,26,27,28,29,30,31]. A comprehensive overview of the integrated, structural, and kinematic properties of the Galaxy is provided in [32].

In general, the kinematic structure of galaxies also provides important clues to their evolutionary history. Observations have revealed galaxies with unusual kinematic properties, suggesting that their disks may have undergone significant changes over time. Such evolutionary processes could result in an atypical orientation of the galactic disk with respect to the dark matter halo. Previous studies have generally indicated an alignment between the angular momentum of dark matter and that of the baryonic component of galaxies [33,34,35,36,37]. Departures from this general picture may arise from interactions with smaller galaxies, including mergers or accretion events involving a more massive host galaxy. The MW itself may have experienced such interactions, and certain perturbations in its kinematic structure could provide observational evidence of these past events.

The primary objective of this paper is to derive the circular velocity curves of the MW using data from the Gaia observatory, specifically a subset of sources with measured radial velocities from the DR3 catalogue [38]. These data enable a complete 3D reconstruction of stellar kinematics in arbitrary coordinate systems. Our analysis focuses on the distribution of stellar velocity components in spherical coordinates and on their dependence on spatial position. This work extends our previous 2D kinematic analysis based on the full DR3 catalogue, in which sources lacking radial velocity measurements constitute the dominant fraction of the dataset [39].

Section 2 introduces the basic concepts and variables used throughout the study. Section 3 describes the criteria used for selecting stars and defines the angular sectors of the Galactic system that are suitable for separate analysis. A useful tool for modelling derived velocity distributions is a simple Monte Carlo (MC) simulation with several free parameters, the algorithm for which is described in Section 4. Essentially, the same model was employed in our earlier 2D analysis; however, the present 3D approach enables a more precise determination of the free parameters.

The results are presented in Section 5. In addition to circular velocity curves, our analysis also provides more general velocity maps in $R-Z$ coordinates. The accuracy of these results depends crucially on the precise determination of the Sun’s motion, which coincides with the motion of the Gaia observatory. The independent calculation of the Sun’s local velocity and the Galaxy’s mean rotational velocity at the radius of the Sun’s orbit is an important part of our analysis. In Section 6, the results obtained are discussed and, where appropriate, compared with the results of previous studies.

2. Definition of Variables

The positions of sources in Gaia data are expressed in galactic angular coordinates: longitude (l) and latitude (b). Using parallax, we can also define the distance ($r)$ of the source from the Sun. The proper motion of the stars in Gaia data is represented by the vector

$${\mu }_{ICRS}=\left({\mu }_{\alpha }^{*},{\mu }_{\delta }\right);{\mu }_{\alpha }^{*}\equiv {\mu }_{\alpha }\cos \delta ,$$

(1)

whose components are angular velocities in the directions of right ascension and declination in the ICRS. For our analysis, we prefer the representation of proper motion in the galactic reference frame:

$${\mu }_{gal}=\left({\mu }_l^{*},{\mu }_b\right),{\mu }_l^{*}\equiv {\mu }_l\cos b,$$

(2)

which is obtained by the transformation given in [40]. Then, the transverse 2D velocity ${\mathbf{v}}_T$ defined as

$${\mathbf{v}}_T=\left(v_l,v_b\right)[\mathrm{km}/\mathrm{s}]=4.7406r\left[\mathrm{kpc}\right]\left({\mu }_l^{*},{\mu }_b\right)[\mathrm{mas}/\mathrm{year}],$$

(3)

where $v_l,v_b$ are velocity components in directions of increasing latitude $\left(l\right)$ and longitude $\left(b\right)$, and the distance r is obtained from the parallax p:

$$r\left[\mathrm{kpc}\right]={\displaystyle \frac{1}{p\left[\mathrm{mas}\right]-\mathrm{ZP}}}$$

(4)

where ZP is the value of the parallax zero-point for Gaia EDR3 and Gaia DR3 returned by the Python gaiadr3_zeropoint package [41] based on the functions described in [42]. We work with data that also include radial velocity ($v_r$), so we have complete 3D data on position and velocity relative to the Sun’s position. For our analysis, it is convenient to express spherical components $v_l,v_b,v_r$ in the local standard of rest (LSR) frame. In fact, there are two LSR approaches:

(i)

Kinematic—based on the average motion of nearby stars. The key parameter is $V_0$, the average velocity of the stars along the circular path of radius $R_{\odot }$.

(ii)

Dynamic—a hypothetical point that would orbit the Galaxy along a perfectly circular path at a given distance from the Galactic centre. The key parameter is $V_c$, the circular velocity of this point along the path of radius $R_{\odot }$.

In the first step, the analysis is conducted within a kinematic framework. Subsequently, in Section 5, selected results are appropriately corrected to account for the effects associated with the dynamic formulation.

If $\Delta {\mathbf{V}}_{\odot }$ is the Sun’s velocity relative to the average velocity of the surrounding stars, then the corresponding LSR components take the form

$$w_{\alpha }=v_{\alpha }+\Delta {\mathbf{V}}_{\odot }.{\mathbf{n}}_{\alpha }$$

(5)

and satisfy

$$⟨\mathbf{w}⟩\to \left(0,0,0\right)\mathrm{for}r\to 0$$

(6)

independently in various sectors around the Sun. The practical application of this definition is demonstrated in Section 5. The resulting $\Delta {\mathbf{V}}_{\odot }$ may, of course, depend to some extent on the choice of these sectors. This inevitable uncertainty must be included in the measurement error. The vectors ${\mathbf{n}}_{\alpha }$ define directions of increasing coordinates $\alpha =l,b,r$. In terms of Cartesian coordinates, we have

$$\mathbf{w}=\left({\mathrm{w}}_x,{\mathrm{w}}_y,{\mathrm{w}}_z\right)=w_l{\mathbf{n}}_l+w_b{\mathbf{n}}_b+w_r{\mathbf{n}}_r.$$

(7)

In the next step, we transform spherical galactic LSR coordinates and velocities to the Galactocentric representation:

$$l,b,r,w_l,w_b,w_r\to \mathsf{\Phi },\mathsf{\Theta },R,V_{\mathsf{\Phi }},V_{\mathsf{\Theta }},V_R,$$

(8)

where

$$V_{\alpha }=\mathbf{w}.{\mathbf{N}}_{\alpha };\alpha =\mathsf{\Phi },\mathsf{\Theta },R.$$

(9)

The explicit forms of the unit vectors ${\mathbf{n}}_{\alpha }$ and ${\mathbf{N}}_{\alpha }$ are given in Appendix A, where the relationship between galactic and Galactocentric coordinates is defined. For clarity, we use the following convention throughout the paper: positions, velocities, and their coordinates are denoted in lowercase (uppercase) in the galactic (Galactocentric) reference frame. The LSR frame rotates around the Galactic centre, so the total Galactocentric velocity reads

$$\mathbf{W}\left(\mathbf{R}\right)=\mathbf{w}\left(\mathbf{R}\right)+{\mathbf{V}}_{LSR};W_{\alpha }=\mathbf{W}.{\mathbf{N}}_{\alpha },{\mathbf{V}}_{LSR}=\left(0,{\mathrm{V}}_0,0\right),$$

(10)

where velocity ${\mathrm{V}}_0>0$ is defined as the average rotation velocity of stars in the Galactic plane and at the radius of the Sun’s orbit $R_{\odot }$. For our analysis, we use the value $R_{\odot }\approx 8.178$ kpc [43]. The rotation velocity of a star reads

$$W_{\mathsf{\Phi }}=V_{\mathsf{\Phi }}-{\mathrm{V}}_0.$$

(11)

The direction of increasing $\mathsf{\Phi }$ (or ${\mathbf{N}}_{\mathsf{\Phi }}$) is opposite to the direction of the Galaxy’s rotation, which explains the negative value of $W_{\mathsf{\Phi }},$ see Figure A1 in Appendix A.

The velocities $V_{\mathsf{\Phi }},W_{\mathsf{\Phi }},V_{\mathsf{\Theta }},V_R$ are the subject of statistical analysis. The function

$$\mathrm{V}\left(R\right)=-⟨W_{\mathsf{\Phi }}\left(\mathbf{R}\right)⟩$$

(12)

averaged in the galactic plane at fixed R is the input for the calculation of the circular velocity curve ${\mathrm{V}}_c\left(R\right)$. Velocity ${\mathrm{V}}_0$ defined above is a special case: ${\mathrm{V}}_0=$$\mathrm{V}\left(R_{\odot }\right).$ The rotation curve $\mathrm{V}\left(R\right)$ reflects the collective rotation velocity rather than the velocity of a test particle [11]. We also study the functions $\mathrm{V}(R_c,Z)=-⟨W_{\mathsf{\Phi }}\left(\mathbf{R}\right)⟩$ and $⟨V_R(R_c,Z)⟩$ depending on two cylindrical coordinates (A5). The average $⟨\dots ⟩$ denotes the average value in the corresponding bin centred at the point $(R_c,Z)$.

3. Dataset

The data sectors of the sky used for analysis are defined in Table 1.

Table 1. Sectors of the sky and the number of stars in them. The sectors N₀, Q₀ and S₀ cover the entire range of l: $⟨\mathbf{0},\mathbf{360}⟩$.

b[deg]∖l[deg]	$⟨-45,45⟩$	$⟨45,135⟩$	$⟨135,225⟩$	$⟨225,315⟩$
$⟨20,90⟩$	N0:3,743,420
$⟨5,20⟩$	N1: 1,779,692	N2: 1,445,772	N3: 1,003,565	N4: 1,431,995
$⟨-5,5⟩$	Q0:7,300,540
$⟨-5,5⟩$	Q1: 2,589,645	Q2: 1,705,027	Q3: 973,140	Q4: 2,032,728
$⟨-20,-5⟩$	S1: 2,480,036	S2:1,626,841	S3: 902,638	S4: 1,586,298
$⟨-90,-20⟩$	S0:4,184,923

From the set of all Gaia DR3 sources with a defined radial velocity (33,812,183 stars) [38], we only accept sources with a positive parallax p and limited distance $r\le 6$ kpc that satisfy the following condition:

$$\Delta v_T\le 15\mathrm{km}/\mathrm{s},\Delta v_r\le 10\mathrm{km}/\mathrm{s},$$

(13)

where we use the relation [39]

$${\displaystyle \frac{\Delta v_T}{v_T}}\approx \sqrt{{\left({\displaystyle \frac{\Delta p}{p}}\right)}^2+{\displaystyle \frac{{\left({\mu }_{\alpha }^{*}\Delta {\mu }_{\alpha }^{*}\right)}^2+{\left({\mu }_{\delta }\Delta {\mu }_{\delta }\right)}^2}{{\mu }^4}}};\mu =\sqrt{{\mu }_{\alpha }^{*2}+{\mu }_{\delta }^2}.$$

(14)

All parameters, including errors, are included in the Gaia DR3 data. The resulting star counts are listed in the table. After these cuts, we are left with a total of 27,485,720 stars for further analysis. The distributions of their distances from the Sun and from the galactic plane over all sectors are shown in Figure 1.

Figure 1. Distribution of distances from the Sun (left) and from the galactic plane (right). Units: $r,z$ [kpc]. Bin widths: 0.024 and 0.008 kpc.

Note that the cuts (13) simultaneously select sources with higher parallax precision. Figure 2 show the corresponding distribution of relative parallax errors. The selected sources meet the condition $\Delta p/p<0.2$ in all sectors, which is the standard criterion for selecting “good” sources. The first cut on $\Delta v_T$ in Equation (13) should be regarded as approximate because covariance terms are neglected in (14). We have verified that the narrowing of the cuts does not systematically affect the parameters of the resulting curves in Section 5. Most of our calculations focus on mean values, which means that the resulting errors are much smaller than the errors of individual entries. The error bars in the figures in Section 5 represent the statistical errors of the average values in the corresponding bins. These errors are calculated using the formula

$$\Delta v={\displaystyle \frac{{\sigma }_b}{\sqrt{n_b}}};{\sigma }_b=\sqrt{⟨v^2⟩-{⟨v⟩}^2},$$

(15)

where ${\sigma }_b$ is the standard deviation of the velocities and $n_b$ is their number in the given bin.

Figure 2. Distribution of relative parallax errors across different sectors. Bin width: 0.002.

4. Algorithm of Monte Carlo Simulations

The velocity distributions are compared with a simple MC probabilistic model proposed in our previous study [39]. This model represents an axially symmetric approximation corresponding to a simplified scenario in which local velocity fluctuations are averaged out. The simulated velocity of a star in the Galactocentric reference frame is given by the relation

$${\mathbf{W}}^{gen}\left(\mathbf{R}\right)=\left(V_{\mathsf{\Phi }}-\mathrm{V}\left(R\right)\right){\mathbf{N}}_{\mathsf{\Phi }}+V_{\mathsf{\Theta }}{\mathbf{N}}_{\mathsf{\Theta }}+V_R{\mathbf{N}}_R,$$

(16)

where the velocity $V\left(R\right)>0$ is defined by Equation (12). The components $V_{\mathsf{\Phi }},V_{\mathsf{\Theta }}$, and $V_R$ are generated in the local reference frame defined by the basis (A6). The model is based on the multinormal distribution

$$P(V_{\mathsf{\Phi }},V_{\mathsf{\Theta }},V_R;R,Z)\sim \exp \left(-{\displaystyle \frac{V_{\mathsf{\Phi }}^2}{2{\sigma }_{\mathsf{\Phi }}^2}}-{\displaystyle \frac{V_{\mathsf{\Theta }}^2}{2{\sigma }_{\mathsf{\Theta }}^2}}-{\displaystyle \frac{V_R^2}{2{\sigma }_R^2}}\right),$$

(17)

with partial asymmetry (different ${\sigma }_{\mathsf{\Phi }}$ for positive and negative $V_{\mathsf{\Phi }}$). The dependence on R and Z is absorbed in the standard deviations ${\sigma }_{\alpha }(R,Z)$. As a first approximation, we assume

$${\sigma }_{\alpha }={\sigma }_{\alpha 0}+{\sigma }_{\alpha 1}\left|Z\right|;\alpha =\mathsf{\Phi },\mathsf{\Theta },R,$$

(18)

where ${\sigma }_{\alpha 0}$ and ${\sigma }_{\alpha 1}$ are free parameters. We have found that the optimal shape of the distribution $V_{\mathsf{\Phi }}$ is slightly asymmetric, with different ${\sigma }_{\mathsf{\Phi }}^{\pm }$ for the two opposite orientations, where $+(-)$ denotes in (against) the direction of rotation:

$${\sigma }_{\mathsf{\Phi }}^{+}={\sigma }_{\mathsf{\Phi }0},{\sigma }_{\mathsf{\Phi }}^{-}={\sigma }_{\mathsf{\Phi }0}+{\sigma }_{\mathsf{\Phi }1}^{-}\left|Z\right|.$$

(19)

Thus, only ${\sigma }_{\mathsf{\Phi }}^{-}$ depends on $\left|Z\right|$. This asymmetry also reflects the effective deceleration $⟨\Delta V_{\mathsf{\Phi }}⟩$ of the collective rotation for larger $\left|Z\right|$:

$$-⟨W_{\mathsf{\Phi }}\left(\left|Z\right|\right)⟩={\mathrm{V}}_0-⟨\Delta V_{\mathsf{\Phi }}⟩\approx {\mathrm{V}}_0-\varkappa \left|Z\right|;\varkappa =\sqrt{{\displaystyle \frac{2}{\pi }}}{\sigma }_{\mathsf{\Phi }1}^{-}.$$

(20)

It should be noted that the parameter $\varkappa$$\left(\mathrm{or}{\sigma }_{\mathsf{\Phi }1}^{-}\right)$ represents only an average over the sectors $N_{\alpha }$ and $S_{\alpha }$. In Section 5, we show that the actual situation is somewhat more complex. The Wolfram Mathematica code of the generator is available at https://www.fzu.cz/~piska/Catalogue/genkinJUN26.nb (accessed on 8 October 2026). Figure 3 shows examples of the distribution of simulated velocities $V_{\mathsf{\Phi }},V_{\mathsf{\Theta }}$ and $V_R$. The distributions $V_{\mathsf{\Theta }}$ and $V_R$ are symmetric for any $\left|Z\right|$, whereas the distribution $V_{\mathsf{\Phi }}$ is asymmetric for $\left|Z\right|>0$.

Figure 3. Distribution of simulated velocities $V_{\mathsf{\Phi }},V_{\mathsf{\Theta }}$, and $V_R$ at $Z=0$ (top row) and $Z=1.5$ kpc (bottom row). Units: $V_{\alpha }$ [km/s].

The simulated distributions at different positions are obtained as follows. The position $\left(l,b,r\right)$ of the source in the galactic reference frame is transformed using Equation (A3) into the Galactocentric spherical coordinate system $\left(\mathsf{\Phi },\mathsf{\Theta },R\right)$. Then, at this position, the velocity ${\mathbf{W}}^{gen}\left(\mathbf{R}\right)$ is generated according to the distribution (16) using the parameters from Table 2. The optimal parameters ${\sigma }_{\alpha }$ listed in the table are obtained through comparison with the real data presented in the following section. The generated velocity can be transformed back to the galactic reference frame:

$$w_{\alpha }^{gen}\left(\mathbf{r}\right)=\left({\mathbf{W}}^{gen}\left(\mathbf{R}\right)-{\mathbf{V}}_{LSR}\right).{\mathbf{n}}_{\alpha };\alpha =l,b,r.$$

(21)

Table 2. Optimal parameter values in the MC simulation. Units: ${\sigma }_{\alpha 0}\left[\mathrm{km}{\mathrm{s}}^{-1}\right]$, ${\sigma }_{\alpha 1}\left[\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{kpc}}^{-1}\right]$.

${\mathit{\sigma }}_{\mathbf{\Theta }0}$	${\mathit{\sigma }}_{\mathbf{\Phi }0}$	${\mathit{\sigma }}_{\mathit{R}0}$	${\mathit{\sigma }}_{\mathbf{\Theta }1}$	${\mathit{\sigma }}_{\mathbf{\Phi }1}^{-}$	${\mathit{\sigma }}_{\mathit{R}1}$	Ref.
12	20	30	23	32	32	this work
$18.03\pm 0.03$	$24.35\pm 0.04$	$36.81\pm 0.07$	x	x	x	[44]
11	20	31	x	x	x	[45]

In this way, for any source defined by input $\left(l,b,r,w_l,w_b,w_r\right)$ or $\left(\mathsf{\Phi },\mathsf{\Theta },R,V_{\mathsf{\Phi }},V_{\mathsf{\Theta }},V_R\right)$, we generate the vector $\left(l,b,r,w_l^{gen},w_b^{gen},w_r^{gen}\right)$ or $\left(\mathsf{\Phi },\mathsf{\Theta },R,V_{\mathsf{\Phi }}^{gen},V_{\mathsf{\Theta }}^{gen},V_R^{gen}\right)$. To compare the model simulations with the measured data, we also need to know the local velocity of the Sun $\Delta {\mathbf{V}}_{\odot }$ and the rotation velocity ${\mathrm{V}}_0$, which we also determine in the following section.

It goes without saying that different methods used to analyse different data samples can lead to slightly different parameters. Since the 3D analysis is more consistent, we give preference to the parameters ${\sigma }_{\alpha },{\mathrm{V}}_0$, and $\Delta {\mathbf{V}}_{\odot }$ obtained in this work.

5. Results

In Figure 4, we show the dependence of average galactic velocities on the distance r. We compare the set of galactic velocities related to the Sun ($v_{\alpha }$) with those related to the LSR ($w_{\alpha }$), defined in Equation (5). Once the Sun’s local velocity $\Delta {\mathbf{V}}_{\odot }$ has been set correctly, the average velocities $⟨w_{\alpha }⟩$ should satisfy the condition (6). This condition is well satisfied for the values used in the figure:

$$\Delta {\mathbf{V}}_{\odot }=\left(\Delta V_{x\odot },\Delta V_{y\odot },\Delta V_{z\odot }\right)=\left(10.5\pm 0.7,20.5\pm 0.9,7.33\pm 0.1\right)\mathrm{km}/\mathrm{s}.$$

(22)

Figure 4. Dependence of mean velocity ($w_l-$blue; $w_b-$green; $w_r-$black; $v_l-$red; $v_b-$purple; and $v_r-$orange) on distance r in sectors $Q_1$–$Q_4$. Units: r [kpc], v [km/s].

These values were obtained after a minor correction (through visual fitting) of our previously measured values in [39], which also includes a comparison with other studies. Since the method for determining $\Delta {\mathbf{V}}_{\odot }$ is virtually the same in both studies, we adopt the magnitude of systematic errors from the previous study. Statistical errors, on the other hand, are insignificant.

The slope of the curves $⟨w_l⟩$ in sectors $Q_2$ and $Q_4,$ as we discussed in the cited paper, is a direct consequence of MW rotation. In Figure 5, we show these curves on a larger scale together with their simulated counterparts given by Equation (21). Consistency with the simulation further confirms the condition (6) for $⟨w_l⟩$. The slope can be explained as follows. In the plane of the MW disc $\left(b,z=0\right)$, the rotation is defined by the function $\mathrm{V}\left(R\right)$ in (12), so the average velocity $w_{\alpha }$ at the position $\mathbf{r}$ observed from the LSR frame reads

$$⟨w_{\alpha }\left(\mathbf{r}\right)⟩=-\left(\mathrm{V}\left(R\right){\mathbf{N}}_{\mathsf{\Phi }}\left(\mathbf{R}\right)+{\mathbf{V}}_{LSR}\right).{\mathbf{n}}_{\alpha }\left(\mathbf{r}\right);\alpha =l,b,r.$$

(23)

Figure 5. The curves $⟨w_l⟩$ in sectors $Q_2$ and $Q_4$: data (blue); simulation model (red). Units: r [kpc], $w_l$ [km/s].

For the curves $⟨w_l⟩$, we then obtain

$$⟨w_l⟩={\mathrm{V}}_0\left({\displaystyle \frac{R_{\odot }\cos l-r}{\sqrt{R_{\odot }^2+r^2-2R_{\odot }r\cos l}}}-\cos l\right).$$

(24)

In Figure 6, we show the set of these curves for different values of the parameters ${\mathrm{V}}_0$ and l. The curves in Figure 5 correspond to a superposition of curves over l from the sectors $Q_2$ and $Q_4$. The MC simulation is based on the value obtained in the previous study:

$${\mathrm{V}}_0=\left(228\pm 4\right)\mathrm{km}/\mathrm{s}.$$

(25)

Figure 6. Dependence of velocity $⟨w_l⟩$ on distance r for various rotation velocities $V_0$ (at fixed $l=90$ deg) and various angles l (at fixed $V_0=228$ km/s) in the galactic plane ($b=0$). Units: r [kpc], $w_l$ [km/s].

It is clear that $\mathrm{V}\left(R\right)\approx {\mathrm{V}}_0$ agrees perfectly with the measured values in the range $r\lesssim 3.5$ kpc, which confirms the previous result.

For consistent comparison with analyses resulting from the dynamic definition of the LSR (see Section 2), the velocities $\Delta V_{y\odot }$ and ${\mathrm{V}}_0$ require a correction based on the calculation of asymmetric drift

$${\mathrm{V}}_a={\mathrm{V}}_c-{\mathrm{V}}_0,$$

(26)

where $V_c$ is the MW circular velocity at $R_{\odot }$. The Sun’s rotation velocity $V_{\odot \mathsf{\Phi }}$ in the Galactocentric frame can be decomposed alternatively as

$$V_{\odot \mathsf{\Phi }}={\mathrm{V}}_0+\Delta V_{y\odot }={\mathrm{V}}_c+\Delta V_{y\odot }^{corr},$$

(27)

so

$$\Delta V_{y\odot }^{corr}=\Delta V_{y\odot }-{\mathrm{V}}_a,{\mathrm{V}}_c={\mathrm{V}}_0+{\mathrm{V}}_a.$$

(28)

In [46], the asymmetric drift $V_a$ around the Sun’s position was estimated to be about 6 km/s. A similar value can be extracted from Figure 3 in [47], where the parameter $⟨v_R^2⟩$ is replaced with our value of ${\sigma }_{R0}^2=900$ km²s⁻², as shown in Table 2. In the standard notation, we can identify

$$U_{\odot }=\Delta V_{x\odot },V_{\odot }=\Delta V_{y\odot }^{corr},W_{\odot }=\Delta V_{z\odot }.$$

(29)

These results are summarised in Table 3.

Table 3. Sun’s local velocity, mean rotation velocity ${\mathrm{V}}_0$, and circular velocity ${\mathrm{V}}_c$. Units: $\left[\mathrm{km}{\mathrm{s}}^{-1}\right]$.

${\mathit{U}}_{\odot }$	${\mathit{V}}_{\odot }$	${\mathit{W}}_{\odot }$	${\mathbf{V}}_0$	${\mathbf{V}}_{\mathit{c}}$
$10.5\pm 0.7$	$14.5\pm 0.9$	$7.33\pm 0.1$	$228\pm 4$	$234\pm 4$

The Galactocentric velocities $V_{\mathsf{\Phi }},V_{\mathsf{\Theta }},V_R$ are calculated using Equation (9). The distributions $P\left(V_{\mathsf{\Phi }}\right),$$P\left(V_{\mathsf{\Theta }}\right)$, and $P\left(V_R\right)$ in the sectors $Q_1$–$Q_4$ covering the galactic plane are shown in Figure 7, Figure 8 and Figure 9.

Figure 7. Distributions of $V_{\mathsf{\Phi }}$ in sectors $Q_{\alpha }$ of the galactic plane defined in Table 1 and in Equation (30): data (light blue); simulation model (red). Unit: $V_{\mathsf{\Phi }}$ [km/s]. Bin width: 1.6 km/s.

Figure 8. Distributions of $V_{\mathsf{\Theta }}$ in sectors $Q_{\alpha }$ of the galactic plane defined in Table 1 and in Equation (30): data (light blue); simulation model (red). Unit: $V_{\mathsf{\Theta }}$ [km/s]. Bin width: 1.6 km/s.

Figure 9. Distributions of $V_R$ in sectors $Q_{\alpha }$ of the galactic plane defined in Table 1 and in Equation (30): data (light blue); simulation model (red). Unit: $V_R$ [km/s]. Bin width: 1.6 km/s.

The same distributions for other sectors (outside the galactic plane) are shown in Figure 10, Figure 11 and Figure 12. In the figures, we merge the sectors, which, due to rotational symmetry, give very similar results:

$$\begin{array}{cc}\hfill & Q_5=Q_2+Q_4{Q}_6=S_1+N_1{Q}_7=S_3+N_3\hfill \\ \hfill & Q_8=S_2+N_2+S_4+N_4{Q}_9=S_0+N_0\hfill \end{array}$$

(30)

Figure 10. Distributions of $V_{\mathsf{\Phi }}$ in sectors $Q_{\alpha }$ outside the galactic plane defined in Equation (30): data (blue); simulation model (red). Unit: $V_{\mathsf{\Phi }}$ [km/s]. Bin width: 1.6 km/s.

Figure 11. Distributions of $V_{\mathsf{\Theta }}$ in sectors $Q_{\alpha }$ outside the galactic plane defined in Equation (30): data (light blue) and the simulation model (red). Unit: $V_{\mathsf{\Theta }}$ [km/s]. Bin width: 1.6 km/s.

Figure 12. Distributions of $V_R$ in sectors $Q_{\alpha }$ outside the galactic plane defined in Equation (30): data (light blue); simulation model (red). Unit: $V_R$ [km/s]. Bin width: 1.6 km/s.

Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 are used to determine the parameters of the simulated curves in Table 2.

Since these parameters generate 24 different images, we optimised them through visual fitting. The distributions in Figure 7, Figure 8 and Figure 9 correspond to the galactic plane ($Z=0$). Their simulated shapes are controlled solely by the parameters ${\sigma }_{\alpha 0}$, which are fixed based on these figures. The remaining parameters ${\sigma }_{\alpha 1}$ are then fixed so that the simulation best matches the data in Figure 10, Figure 11 and Figure 12, which correspond to the sectors outside the galactic plane ($\left|Z\right|>0$). In all figures except for sectors $Q_1$ and $Q_6$ oriented towards the MW centre ($-{45}^0<l<{45}^0$), we observe very good agreement between the MC simulation and the data. A slight asymmetry of the $V_{\mathsf{\Phi }}$ distribution (both data and simulation) is observed outside the galactic plane in Figure 10. A slight widening with increasing $\left|Z\right|$ is most clearly evident in Figure 9 and Figure 12 when comparing the $V_R$ distributions in sectors $Q_0$ and $Q_9$. All parameters are close to the values we obtained previously [39] and in other studies listed in the table. In the cited studies, the velocity dispersions are related to the cylindrical coordinate system, whereas ours are related to the spherical one. However, since small $\left|Z\right|$ values dominate the data, it can be assumed that $V_{\mathsf{\Theta }}\approx V_Z,V_R\approx V_{R_c}$ and ${\sigma }_{\mathsf{\Theta }0}\approx {\sigma }_Z,{\sigma }_{R0}\approx {\sigma }_{R_c}$ correspondingly.

In the left panel of Figure 13, we show the velocity curves $⟨V_{\mathsf{\Phi }}⟩,⟨V_{\mathsf{\Theta }}⟩$, and $⟨V_R⟩$ in sectors covering the galactic plane depending on the radius R in this plane. All curves give zero at the point $R_{\odot }$, which once again confirms the correct setting of $\Delta {\mathbf{V}}_{\odot }$ in our calculation. The curves fluctuate around zero, while the simulations yield a net zero (not shown in the figure). Local kinematic substructures generate deviations $\Delta v\lesssim 5$ km/s. A more pronounced increase in $⟨V_{\mathsf{\Phi }}⟩$ (which means a decrease in rotation velocity) is evident for $R\lesssim 5$ kpc. The apparent increase in $⟨V_R⟩$ for $R\lesssim 4$ kpc may indicate that a slight centrifugal component accompanies the lower rotation velocity in this region. The resulting rotation velocity curve $\mathrm{V}\left(R\right)={\mathrm{V}}_0-⟨V_{\mathsf{\Phi }}⟩$ is shown in the right panel, where we used an additional cut $b<2$ deg (see note in the following section). The high number of stars in sector $Q_0$ makes it possible to obtain curves with very good resolution of local kinematic substructures.

Figure 13. Dependence of $⟨V_{\mathsf{\Phi }}⟩$ (blue)$,⟨V_{\mathsf{\Theta }}⟩$ (green), and $⟨V_R⟩$(black) on radius R in the galactic plane (left) in sector Q₀ and rotation curve V(R) (right) in the same sector but with an additional cut $\left|b\right|<2$ deg. Units: V [km/s], R [kpc].

Table 4 contains the corresponding circular velocity curve

$${\mathrm{V}}_c\left(R\right)=\mathrm{V}\left(R\right)+{\mathrm{V}}_a,$$

(31)

where, in accordance with [46], the asymmetric drift in the galactic plane $Z\approx 0$ is assumed to be ${\mathrm{V}}_a\approx 6$ km/s regardless of radius R.

Table 4. Circular velocity curve ${\mathrm{V}}_c\left(R\right)$ with corresponding errors in the defined $\Delta R$ bin. N is the number of entries in a given bin. Units: ${\mathrm{V}}_c$ [km/s], R [kpc].

R	3.1	3.3	3.5	3.7	3.9	4.1	4.3	4.5	4.7	4.9	5.1
${\mathrm{V}}_c\left(R\right)$	227.33	228.41	224.4	223.35	222.99	221.22	222.09	221.41	222.1	223.38	224.95
${\Delta }_{stat}$	1.41	1.26	1.07	0.85	0.73	0.6	0.48	0.39	0.31	0.25	0.2
${\Delta }_{tot}$	7.41	7.26	7.07	6.85	6.73	6.6	6.48	6.39	6.31	6.25	6.2
N	489	647	984	1411	2103	3085	4872	7370	11,050	16,730	24,148
R	5.3	5.5	5.7	5.9	6.1	6.3	6.5	6.7	6.9	7.1	7.3
${\mathrm{V}}_c\left(R\right)$	226.66	228.7	230.1	230.95	232.09	233.23	233.59	233.93	234.45	234.84	235.32
${\Delta }_{stat}$	0.16	0.13	0.11	0.1	0.1	0.09	0.09	0.08	0.08	0.07	0.07
${\Delta }_{tot}$	6.16	6.13	6.11	6.1	6.1	6.09	6.09	6.08	6.08	6.07	6.07
N	34,811	48,698	64,079	78,064	88,319	90,507	97,609	112,446	114,642	125,791	146,645
R	7.5	7.7	7.9	8.1	8.3	8.5	8.7	8.9	9.1	9.3	9.5
${\mathrm{V}}_c\left(R\right)$	235.68	235.73	235.4	234.96	233.57	232.63	232.2	231.43	230.6	229.59	228.78
${\Delta }_{stat}$	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.07	0.08	0.09	0.1
${\Delta }_{tot}$	6.06	6.06	6.06	6.06	6.06	6.06	6.06	6.07	6.08	6.09	6.1
N	166,802	186,689	201,779	169,888	153,551	150,941	131,385	105,659	79,150	59,109	47,036
R	9.7	9.9	10.1	10.3	10.5	10.7	10.9	11.1	11.3	11.5	11.7
${\mathrm{V}}_c\left(R\right)$	228.23	228	227.38	227.05	226.3	226.22	226.55	226.4	226.85	227.16	227.22
${\Delta }_{stat}$	0.11	0.12	0.12	0.12	0.13	0.13	0.14	0.15	0.15	0.17	0.18
${\Delta }_{tot}$	6.11	6.12	6.12	6.12	6.13	6.13	6.14	6.15	6.15	6.17	6.18
N	38,934	34,222	30,928	29,118	27,059	24,620	21,992	19,205	16,344	13,962	11,977
R	11.9	12.1	12.3	12.5	12.7	12.9	13.1	13.3	13.5	13.7	13.9
${\mathrm{V}}_c\left(R\right)$	227.5	226.78	226.74	226.3	226.21	225.29	225.55	225.35	225.4	224.75	223.59
${\Delta }_{stat}$	0.19	0.22	0.23	0.26	0.29	0.34	0.39	0.45	0.54	0.61	0.93
${\Delta }_{tot}$	6.19	6.22	6.23	6.26	6.29	6.34	6.39	6.45	6.54	6.61	6.93
N	9787	7815	6125	4860	3716	2750	1993	1446	965	696	435

In Figure 14, we show the circular velocity curve ${\mathrm{V}}_c\left(R\right)$ together with the results of some other studies. For better clarity, we have illustrated the errors in a separate part.

Figure 14. Circular velocity curve together with the curves from the studies [7,15,20,21] (top panel) and corresponding errors (bottom panel). Units: ${\mathrm{V}}_c$ [km/s], R [kpc].

In Figure 15, we show the velocity curves in the sectors outside the galactic plane as functions of distance $\left|Z\right|$. Local fluctuations of $⟨V_{\mathsf{\Theta }}⟩$ and $⟨V_R⟩$ in all analysed sectors do not exceed 5 km/s. On the other hand, a clear increase in $⟨V_{\mathsf{\Phi }}⟩$ is observed in all sectors. The corresponding simulation, represented by a linear curve controlled by Equation (20), which is common to all sectors, is clearly only an approximation. The differences between the measured data and the simulation vary across different sectors. For $⟨V_{\mathsf{\Theta }}⟩$ and $⟨V_R⟩$, the simulations yield a net zero (not shown in the figure).

Figure 15. Dependence $⟨V_{\mathsf{\Phi }}⟩$ (blue, simulation red)$,⟨V_{\mathsf{\Theta }}⟩$ (green), and $⟨V_R⟩$ (black) on distance $\left|Z\right|$ from the galactic plane in sectors Q_α. Units: V [km/s], Z [kpc].

Diagrams of the rotation velocity $\mathrm{V}(R_c,Z)$, radial velocity $⟨V_R(R_c,Z)⟩$, polar velocity $⟨V_{\mathsf{\Theta }}(R_c,Z)⟩$, and simulated rotation velocity ${\mathrm{V}}_{sim}(R_c,Z)$ obtained from all sectors are shown in Figure 16.

Figure 16. The $R-Z$ diagrams. Top: rotation velocity $\mathrm{V}(R_c,Z)$ and simulated rotation velocity ${\mathrm{V}}_{sim}(R_c,Z)$. Bottom: radial velocity $⟨V_R(R_c,Z)⟩$ and polar velocity $⟨V_{\mathsf{\Theta }}(R_c,Z)⟩$. Units: V [km/s], $R,Z$ [kpc].

The plots provide additional information about the kinematics of the MW in the depicted region. Consistent with the left panel of Figure 13, we again observe that the decrease in rotation velocity $\mathrm{V}(R_c,Z)$ in the region $R_c\lesssim 5$ kpc is accompanied by a centrifugal component in $⟨V_R(R_c,Z)⟩$. We also observe a slight asymmetry in $⟨V_{\mathsf{\Theta }}(R_c,Z)⟩$ at smaller R. For $Z<0$, $V_{\mathsf{\Theta }}>0$ is dominant; conversely, for $Z>0$, $V_{\mathsf{\Theta }}<0$ dominates.

The rotation velocity $\mathrm{V}(R_c,Z)$ decreases as $\left|Z\right|$ increases. This decrease is more pronounced for smaller $R_c$ (sector $Q_6$) and less pronounced for larger $R_c$ (sector $Q_7$). This pattern is more complex than the pattern from the simulation V_sim(R_c,Z), which assumes a constant slope $\varkappa$ throughout the entire region (see Equation (20)).

6. Discussion and Conclusions

Using a subset of the DR3 data from the Gaia space observatory, which includes information on radial velocities, we studied the 3D kinematics of the MW. This builds on our earlier 2D analysis, which relied on high-statistics data on stars without defined radial velocities. Although the two approaches differ in both method and input dataset, they yield identical or very close results, for example, concerning the Sun’s local velocity $\Delta {\mathbf{V}}_{\odot }$ and the mean rotation velocity ${\mathrm{V}}_0$ at radius $R_{\odot }$.

As for $\Delta {\mathbf{V}}_{\odot }$, the component perpendicular to the galactic disk plane remains completely unchanged. The velocity ${\mathrm{V}}_0$ also remains unchanged. It is important to emphasise that both velocities $\Delta {\mathbf{V}}_{\odot }$ and ${\mathrm{V}}_0$ are obtained completely independently of each other by analysing the stellar velocities in sectors $Q_1-Q_4$. For ${\mathrm{V}}_0$ measurements, sectors $Q_2$ and $Q_4$ are optimal, with a total sample of 3,737,755 stars. It follows from Equations (27)–(29) and Table 3 that the Sun’s total velocity ${\mathrm{V}}_0+\Delta V_{y\odot }={\mathrm{V}}_c+V_{\odot }=248.5\pm 5$ km/s (that is, it does not contain the parameter ${\mathrm{V}}_a$). Our result is consistent with a very reliable estimate of $251.55\pm 0.98$ km/s based on the longitudinal speed of the radio source Sgr A*[48] and with the total velocity of $247.4\pm 1.4$ km/s obtained in [43]. Very good agreement with other measurements also holds for the average rotation velocity ${\mathrm{V}}_0=228\pm 5$ km/s and the Sun’s local velocities $(U_{\odot },V_{\odot },W_{\odot })=\left(10.5\pm 0.7,14.5\pm 0.9,7.33\pm 0.1\right)$$\mathrm{km}/\mathrm{s}$, as shown in our previous 2D analysis.

Important results are the curves $⟨V_{\alpha }\left(R\right)⟩,\mathrm{V}\left(R\right),$ and $⟨V_{\alpha }\left(\left|Z\right|\right)⟩$ and the corresponding $R-Z$ diagrams in Figure 13, Figure 15, and Figure 16. In the region $R_c\gtrsim 5$ kpc in the figures, we observe fluctuations of local kinematic substructures in the range up to 10 km/s. The behaviour of the curves in the region $R_c\lesssim 5$ kpc may be related to the proximity of the galactic bar.

Our velocity curve ${\mathrm{V}}_c\left(R\right)$ in Figure 14 agrees very well (in the overlap region) with the velocity curves presented in recent studies [7,15,20,21], which were also obtained using the Gaia DR3 catalogue. Small differences may be due to the parameters for the selection of stars, including the shape of the selection region.

As can be seen in Figure 15 and Figure 16, the average rotation velocity of stars slightly decreases with increasing $\left|Z\right|$. Therefore, to determine the curves $\mathrm{V}\left(R\right)$ and ${\mathrm{V}}_c\left(R\right)$ accurately, the selection region around the galactic plane should be as narrow as possible. That is why we used a very narrow range $\left|b\right|<2\mathrm{deg}$ to generate the $\mathrm{V}\left(R\right)$ curve. This strict cut affects the curve particularly in the region $R_c\lesssim 5$ kpc.

The decrease in rotation velocity with increasing $\left|Z\right|$ has also been described in [20,44]. This effect is consistent with the Besançon Galaxy model [49], which provides a prediction for the region $R_c\approx R_{\odot }$. This prediction agrees well with our curves in the sectors $Q_8$ and $Q_9$ of Figure 15, where $⟨R_c⟩\approx R_{\odot }$. Asymmetric drift and a mixed population of thin and thick disks most likely contribute to the negative vertical gradient.

A useful tool in the analysis of stellar velocities is a simple MC simulation model described in Section 4. The simulation corresponds to an averaged axisymmetric approximation of the MW, so it may be useful in validating axisymmetric dynamical models or determining the scale of local kinematical substructures out of axial symmetry. The simulation model describes the distribution of velocities $V_{\mathsf{\Phi }},V_{\mathsf{\Theta }}$, and $V_R$ in the region $R_c\gtrsim 5$ kpc. Some discrepancy between the simulated $V_{\mathsf{\Phi }},V_R$ and the data in sectors $Q_1$ and $Q_6$ was mentioned in the previous section (Figure 7, Figure 9, Figure 10, Figure 12, and Figure 15). The simulations play an important role in determining the parameter ${\mathrm{V}}_0$ in Figure 5. In the relevant sectors $Q_2$ and $Q_4$, the agreement with the simulation is perfect in the region $r\lesssim 3$ kpc. In these sectors, we also observe very good agreement between the measured data and the simulation for all velocity distributions, as can be seen in Figure 7, Figure 8 and Figure 9, sector $Q_5=Q_2+Q_4$. In other sectors where simulations yield poorer results, introducing additional free parameters into the MC generator would improve the fit to the data.

The goal was to present a methodology for the kinematic analysis of the MW. We have shown that our approach makes it possible—using only two external parameters ($R_{\odot }$ and ${\mathrm{V}}_a$) taken from other studies—to derive, with reasonable accuracy, valuable results that are fully consistent with the findings of other studies. It should be noted that the parameter ${\mathrm{V}}_a$, which characterises asymmetric drift, is a quantity derived from Jeans modelling [50], which necessarily assumes an axially symmetric gravitational potential. It would certainly be interesting to attempt to calculate the function ${\mathrm{V}}_a(R_c,Z)$ for subsamples of different ages within the global and, presumably, time-dependent gravitational potential. It is clear, however, that solving such a problem would require further independent research.

In conclusion, we believe that the proposed methods for calculating the local velocity of the Sun, the average rotation velocity $V_0$, the velocity curve ${\mathrm{V}}_c\left(R\right)$, and, more generally, the velocity maps $\mathrm{V}(R_c,Z),$$⟨V_R(R_c,Z)⟩$, and $⟨V_{\mathsf{\Theta }}(R_c,Z)⟩$ represent a useful contribution to the analysis, description, and understanding of the global kinematics of the MW. The MC simulation model proved to be an effective tool in our study.