###### Abstract

Detailed analysis of kinematics of the Milky Way disk in the solar
neighborhood based on the GAIA DR3 catalog reveals the existence of
peculiarities in the stellar velocity distribution perpendicular to the
galactic plane. We study the influence of resonances—the outer Lindblad
resonance and the outer vertical Lindblad resonance—of a rotating bar with
stellar oscillations perpendicular to the plane of the disk, and their role in
shaping the spatial and the velocity distributions of stars. We find that the
$Z$ and $V_{Z}$ distributions of stars with respect to $L_{Z}$ are affected by
the outer Lindblad resonance. The existence of bar resonance with stellar
oscillations perpendicular to the plane of the disk is demonstrated for a long
(large semi-axis 5 kpc)and fast rotating bar with $\Omega_{\rm
b}=60.0~{}\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$. We show also that, in the
model with the long and fast rotating bar, some stars in the 2:1 OLR region
deviate far from their original places, entering the bar region. A combination
of resonance excitation of stellar motions at the 2:1 OLR region together with
strong interaction of the stars with the bar potential leads to the formation
of the group of ’escapees’, i.e., stars that deviate in $R$ and $Z$—directions
at large distances from the resonance region. Simulations, however, do not
demonstrate any noticeable effect on $V_{Z}$-distribution of stars in the
solar neighborhood.

###### keywords:

Milky Way disk; kinematics; dynamics

1 1 0 Academic Editor: Yosuke Mizuno 31 May 2023 1 September 2023 4 September
2023 https://doi.org/ false Resonant Effects of a Bar on the Galactic Disk
Kinematics Perpendicular to Its Plane Resonant Effects of a Bar on the
Galactic Disk Kinematics Perpendicular to Its Plane Vladimir Korchagin 1,*,
Artem Lutsenko 2 , Roman Tkachenko 1, Giovanni Carraro 2 and Katherine Vieira
3V. Korchagin, A. Lutsenko, R. Tkachenko, G. Carraro and K. Vieira Korchagin,
V.; Lutsenko, A.; Tkachenko, R.; Carraro G.; Vieira, K. Correspondence:
<EMAIL_ADDRESS>
##  1 Introduction

Since the HIPPARCOS era it has been known that the solar neighborhood
manifests complex dynamical structures such as streams and moving groups. The
Gaia mission (Perryman et al., 2001; Lindegren et al., 2016; Fabricius et al.,
2016; Evans et al., 2018), with its accurate observational data, shed new
light the on properties of kinematical and structural peculiarities, both in
the solar neighborhood and beyond. Examples of such structures are the ridges
(diagonal distributions in the $(R,V_{\phi})$ plane) and arches, phase spirals
in the $(Z,V_{Z})$ plane, bimodality in $(V_{R},V_{\phi})$ plane including the
Hercules stream (Antoja et al., 2018; Kawata et al., 2018), and vertical waves
(Khanna et al., 2019).

Many of the abovementioned kinematical and structural features are related to
the resonant phenomena occurring within the Galactic disk. Resonances cause
the radial migration of the stars that in turn affect the disk’s density
distribution and its kinematical properties. Schönrich and Binney (2009)
distinguished two types of radial migration caused by resonance phenomena. In
the first type, the angular momentum of a star changes, so the star’s orbit
moves inwards or outwards, depending on whether the angular momentum is lost
or gained. This type of resonance occurring in stellar disks was studied by
Sellwood and Binney (2002) and Roškar et al. (2008), who showed that the
resonant interaction of stars with a spiral density wave churns the stars
around corotation causing their re-distribution within the Galactic disk. As
was stressed by Sellwood and Binney (2002), such behavior has profound
consequences for the chemical evolution of galaxies. Similar ’churning’ of
matter within corotation resonance in gaseous gravitating disks was shown by
Laughlin and Korchagin (1996) who found that an unstable spiral density wave
depletes mass around the corotation, leading eventually to an amplitude
saturation of an unstable spiral. If a star-periodic gravitational field
interaction increases a star’s epicycle amplitude without changing its angular
momentum, the star contributes to the density over a wider range of radii.
Schönrich and Binney (2009), in modification of the terminology introduced by
Sellwood and Binney (2002), called theses changes in the epicycle amplitude
’blurring’.

Dehnen (2000) demonstrated that a fast rotating bar with corotation resonance
located at 3.5–5 kpc causes a bimodal distribution of stars in the
$(V_{R},V_{\phi})$ plane at the outer Lindblad resonance (OLR). This idea was
further confirmed by Antoja et al. (2014) and Monari et al. (2017) who
explained the bimodality in $(V_{R},V_{\phi})$ as a result of the bar
interaction with the stars at the 2:1 OLR.

Hunt and Bovy (2018) discussed another possibility to account for the observed
kinematical peculiarity in the solar neighborhood—the Hercules stream, by the
4:1 OLR resonance with a slowly rotating bar. Hunt et al. (2018) demonstrated
that the Hercules stream and the observed ridges in the $(R,V_{\phi})$ plane
can be explained by transient winding spiral arms, independently or in
combination with a long and slow bar. It is also known that ridges can be
produced by a barred potential and its resonances (Antoja et al., 2018).

Antoja et al. (2018) demonstrated recently the existence of a phase-spiral
pattern seen in the solar neighborhood in the $(Z,V_{Z})$—plane. The plausible
explanation of the appearance of the Gaia phase-space spiral is a passage
through the Milky Way disk of the Sagittarius dwarf galaxy about 400 $\pm$ 150
Myr ago (see Antoja et al. (2018); Binney and Schönrich (2018) and Tepper-
García et al. (2022)). Alternatively, the observed feature can be explained by
disk perturbations from the subhalos (Chequers et al., 2018), a perturbation
generated by a buckling bar (Khoperskov et al., 2019), and a misaligned gas
accretion (Khachaturyants et al., 2022). In addition to the influence of
resonances on dynamic structures and phase-space mixing (Antoja et al., 2018;
Hunt et al., 2018; Trick et al., 2021), the influence of the resonance effects
on chemical evolution of the Galactic disk is also well known (Mishurov and
Tkachenko, 2018, 2019a, 2019b; Khoperskov and Gerhard, 2022).

Resonances of an external perturbing potential with the stellar motions in the
direction perpendicular to the Galactic plane were first considered by Binney
(1981). These resonances, however, were not studied in much detail. Combes et
al. (1990) demonstrated that vILR-inner resonance of the bar with the vertical
motions of the stars can be responsible for the formation of the bar’s peanut
shape. Further, the influence of vertical inner Lindblad resonances on the
formation of X- or peanut-shaped Galactic bulges was discussed by Quillen
(2002) and Quillen et al. (2014). The resonant thickening of self-gravitating
disks was analytically studied by Fouvry et al. (2017). Recently, Vieira et
al. (2022) presented a detailed analysis of kinematics of the Milky Way disk
in the solar neighborhood using the GAIA DR3 catalogue. The authors found an
excess of stars in the $V_{Z}$-distribution of red giants in the solar
neighborhood at $-40<V_{Z}<-20~{}\mbox{km~{}s}^{-1}$ and a dearth of stars in
the $V_{Z}$ distribution at $25<V_{Z}<50~{}\mbox{km~{}s}^{-1}$.

The discovery of dips and bumps in the $V_{Z}$-velocity distribution of stars
in the solar neighborhood (Vieira et al. (2022)) posed the question if the
development of such features can be caused by the resonance of the rotating
bar with the stellar motions perpendicular to the galactic disk in the solar
neighborhood. The resonances of a rotating bar with stellar motions in the
plane of the disk lead to a considerable reshuffling of matter within the
plane of the disk and can also influence the kinematical and density
distributions of stars in a Galactic disk in the direction perpendicular to
the disk’s plane.

We study in this paper how the resonances of the stellar motions with a
rotating bar, occurring in the solar neighborhood, namely, 2:1 outer Lindblad
resonance, 1:1 outer Lindblad resonance, and the vertical outer Lindblad
resonance can influence the disk’s kinematical properties, in particular, the
peculiar kinematical features reported by Vieira et al. (2022).

Section 2 describes the method we chose to study the resonance dynamics of the
thin disk stars under non-axisymmetric time-dependent bar potentials. Section
3 presents the results of our study. Section 4 gives a brief summary of our
results.

##  2 The Model

We study the influence of the bar on the dynamics of a thin collisionless
disk, aiming at possible detection of resonance effects, both in the plane of
the disk and perpendicular to disk’s plane. The behaviour of a thin
collisionless disk is simulated with help of the astrophysical package Galpy
(Bovy, 2015), by integrating the particle dynamics using Dormand–Prince method
(dop853) (Alexander, 1990)—the eighth order Runge–Kutta family method with the
adaptive timesteps. To generate the equilibrium collisionless disk, we
followed the dynamics of the initial velocity and density distribution of
particles in the Milky Way-like potential during three Gyr until the
equilibrium distribution of stars is achieved. We then study the dynamics of
the resulting equilibrium configuration for another three Gyr, while
considering the influence of the bar potential. We assume that the bar
smoothly switches on over a period of one Gyr after the axisymmetric
equilibrium of the disk is settled. We find that the integration time of three
Gyr is sufficient to study the influence of the bar—further increase of
integration time does not lead to the qualitative change of disk dynamics.

###  2.1 Initial Distribution

As it was mentioned above, we discuss the dynamics of a thin Galactic disk,
taking into account the possible influence of a bar. We are interested in the
influence of resonances with a bar outside the bar region; therefore, the
stellar spatial distribution was generated outside four kpc using $10^{6}$
particles and assuming the commonly accepted exponential density distribution
law $\propto\exp\\{-r/r_{d}\\}$, where $r_{d}$ is the radial scale length of
the Galactic thin disk. There is a fairly large spread of the measured values
of $r_{d}$ ranging from two to four kpc (Drimmel and Spergel, 2001; Gerhard,
2001; Piffl et al., 2014; Khrapov et al., 2021). Following Jurić et al.
(2008), we choose in this paper the value of $r_{d}=2.5$ kpc. For the vertical
distribution of particles, we use the exponential law
$\propto\exp\\{-z/h_{z}\\}$ with the adopted value for vertical scale height
of $h_{z}=300$ pc (Jurić et al., 2008; Kordopatis et al., 2011). Average
velocities of the particles in the radial direction and in the direction
perpendicular to the disk were chosen to be zero ${<V_{R}>}=0$ and
${<V_{Z}>}=0~{}\mbox{km~{}s}^{-1}$. To generate mean velocities of the
particles in the azimuthal direction ${<V_{\phi}>}$, we use the disk’s
rotation curve in the axisymmetric Galactic potential McMillan17 (McMillan,
2017) (from now on MC17), described in detail in next subsection. We adopt the
values of the velocity dispersions of particles in the solar neighborhood
$\sigma_{R},\sigma_{\phi},\sigma_{Z}$ equal to 31, 20 and 11 km s-1 Vieira et
al. (2022). The dependence of the velocity dispersion of particles on the
radius was adopted in the form
$\sigma_{R}=\sigma_{R,0}/\exp(r/7.4\,[\,\mathrm{kpc}])$, taken from Khrapov et
al. (2021); Tiede and Terndrup (1999) where
$\sigma_{R,0}=93~{}\mbox{km~{}s}^{-1}$ to satisfy the observed value of the
velocity dispersion in the solar neighborhood of
$\sigma_{R}=31~{}\mbox{km~{}s}^{-1}$. The azimuthal velocity dispersion was
set using the equilibrium condition for the collisionless disk (Khrapov et
al., 2021):

$\sigma_{\varphi}=\sigma_{R}\,\frac{\varkappa}{2\Omega}\,,$ (1)

where $\Omega$ and $\varkappa$ are the angular velocity and the epicyclic
frequency of the disk calculated from the equilibrium rotation curve in the
MC17 potential. For setting the velocity dispersion perpendicular to the disk
direction, we use the ration $\sigma_{Z}/\sigma_{R}=0.36$ to satisfy the
observed value of $\sigma_{Z}$ velocity dispersion in the solar neighborhood
$\sigma_{Z}=11~{}\mbox{km}~{}\mbox{s}^{-1}$.

###  2.2 Galactic Potential Models

We adopt the axisymmetric model of the Galactic potential from (McMillan,
2017), which includes potentials of the bulge, the thin and the thick disks,
the gravitational potentials of the gaseous ($H_{1}$ and $H_{2}$) disks, and
the gravitational potential of the dark matter halo.

To model the time-dependent potential of a rotating bar, we add to the MC17
potential model a potential of a smoothly growing for one Gyr bar, using the
prescription taken from Dehnen (2000):

$A(t)=\left(\frac{3}{16}\xi^{5}-\frac{5}{8}\xi^{3}+\frac{15}{16}\xi+\frac{1}{2}\right),\quad$
(2)

where $\xi=2(\frac{t-t_{form}}{t_{steady}})-1$, when $t_{form}\leq t\leq
t_{form}+t_{steady}$; $\xi=-1$, when $t\leq t_{form}$ and $\xi=1$, when $t\geq
t_{form}+t_{steady}$. The amplitude multiplier function $A(t)$ smoothly
changes from 0 to 1 for one Gyr, so the bar potential is smoothly turned on
after the equilibrium axisymmetric distribution is settled.

The bar potential is modeled by the equation taken from Long and Murali
(1992):

$\Phi_{r}(x,y,z)=\frac{GM_{b}}{2a}\ln\left(\frac{x-a+T_{-}}{x+a+T_{+}}\right).$
(3)

Here, $T_{\pm}=[(a\pm x)^{2}+y^{2}+(b+\sqrt{c^{2}+z^{2}})^{2})]^{1/2}$.
$a,b,c$ are characteristic bar parameters and $x,y,z$ are Cartesian
coordinates. We adopt values of bar semi-axes equal to $b=1$ kpc, $c=0.3$ kpc.
For the major semi-axis of the bar, $a$, we adopt two values: $a=5$ kpc—the
long bar, (LB) and $a=3.5$ kpc—the short bar (SB). In accordance with Portail
et al. (2017) and Kent (1992), we choose the bar mass equal to
$M_{\mathrm{b}}=1.88\times 10^{10}\,\,{\rm M}_{\odot}$, which is also in a
agreement with the estimate of Zhao et al. (1994).

A number of studies have been undertaken to measure the bar angular velocity
(Debattista et al., 2002; Sanders et al., 2019; Clarke et al., 2019; Shen and
Zheng, 2020; Sormani et al., 2015). Debattista et al. (2002) estimated the
angular velocity of the Galactic bar to be $\Omega_{\rm b}=59\pm
5\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$. Recent data based on Gaia proper motion
measurements allowed for a more accurate estimation of the pattern speed of
the Galactic bar. However, there is still disagreement between the results.
For example, Sanders et al. (2019) used the VVV Infrared Astrometric Catalogue
(VIRAC) and Gaia DR2 proper motions and estimated the angular velocity of the
bar to be $\Omega_{\rm b}=41\pm 3\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$, Clarke
et al. (2019) obtained the value of $\Omega_{\rm b}=37.5\,\mathrm{{km\
s}^{-1}\ {kpc}^{-1}}$ (Shen and Zheng, 2020). Dehnen (2000) suggested that, to
explain some kinematical features observed in the solar neighborhood, a bar
with the angular velocity of $\Omega_{\rm b}=55.5~{}\,\mathrm{{km\ s}^{-1}\
{kpc}^{-1}}$ is required. The fast bar is also supported by some other studies
(Gardner and Flynn, 2010; Antoja et al., 2014; Chakrabarty, 2007; Bissantz et
al., 2003) with the bar angular velocity varying within $\Omega_{\rm
b}=\mbox{55--60}~{}\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$. We use in this paper
two values of the angular velocity of the bar: $\Omega_{\rm b}$ = 40
$\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$(slow bar) and $60~{}\,\mathrm{{km\
s}^{-1}\ {kpc}^{-1}}$ (fast bar). Thus, we explore two values of the angular
velocity of a bar and the two values of the bar’s major semi-axis. We use the
following designations for our models. The model with the fast long bar
($\Omega_{\rm b}=60\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$, a = 5 kpc) is
designated as “LB60”, and the slow short bar ($\Omega_{\rm b}=40\,\mathrm{{km\
s}^{-1}\ {kpc}^{-1}}$, a = 3.5 kpc) is designated as “SB40”. The models we use
are listed in Table 1.

Table 1: List of different bar models.

Name |  $\Omega_{\rm b}~{}[\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}]$ |  $a~{}[\mbox{{kpc}}]$
---|---|---
SB40 |  40 |  3.0
SB60 |  60 |  3.0
LB40 |  40 |  5.0
LB60 |  60 |  5.0

Here $\Omega_{\rm b}$ is angular bar velocity and $a$ is the bar major semi-
axis.

###  2.3 Rotation Curves

The rotation curves of our barred, and non-barred models are shown in Figure
1. In the bar potential, the rotational curve is systematically higher
compared to that in the axisymmetric model MC17 due to the fact that a bar
with mass $M_{\mathrm{b}}=1.88\times 10^{10}\,\,{\rm M}_{\odot}$ is added to
the axisymmetric MC17 model. The rotation curves were determined from the
equation $v_{rot}=(rd\Phi/dr)^{1/2}$, where $\Phi$ represents the total
potential in the plane Z = 0. With the bar potential the disk rotation is not
circular in its central regions, so the “rotation curve” depends now on the
direction relative to the bar major axis. Figure 1 presents the rotation
curves calculated along a few directions relative to the bar major axis. As
one can see from Figure 1, with the adopted mass of the bar of 1.88 $\times$
1010 M⊙, which is comparable to that of the unperturbed axisymmetric disk, the
bar changes the disk’s rotation, which shifts the position of corotation
resonance from $\approx$ 3.8 kiloparsecs in the MC17 model to about 5 kps. The
long bar model satisfies the criterion of Contopoulos (1980), for which
$\mathbb{R}=R_{CR}/R_{bar}\geq 1$. We also notice that the position of
corotation is important for studying the particle dynamics at corotation
resonance, but is not of much importance in our study focused on disk dynamics
at the outer and the vertical outer Lindblad resonances.

The galactocentric distance of the Sun is chosen to be $R_{\odot}=8.122$ kpc
(GRAVITY Collaboration et al. (2018)).

Figure 1: Rotation curves of axisymmetric (orange line) and non-axisymmetric
models. For the models with bars, the rotation curves are shown for different
directions with respect to the bar major semi-axis. Short bar—green lines.
Long bar—purple lines. Grey lines—the linear rotation curves corresponding to
the angular velocities of a bar. The vertical lines mark the positions of
corotation in MC17 model and in the models with the bar.

###  2.4 Resonances

####  2.4.1 Resonances in the Plane of the Disk

Resonances in the Galactic plane of the disk are determined by the condition:

$m(\Omega_{b}-\Omega)=l\kappa\,,$ (4)

where $\kappa$ is the epicyclic frequency, $l$ and $m$ are integer numbers
equal to $(l=0,~{}\pm 1,~{}\pm 2,~{}...)$ and $(m=1,2,~{}...)$. We use the
following notation for resonances: if $(m,l)=~{}(2,-1),~{}(2,1),\\\ ~{}(1,1)$,
this determines the 2:1 inner Lindblad resonance (2:1 ILR) and 2:1 outer
Lindblad resonance (2:1 OLR) and 1:1 outer Lindblad resonance (1:1 OLR). We
consider the dominating m = 2 and m = 1 modes in the bar potential, neglecting
higher order modes. It should be noticed, however, that m = 4 Fourier
component of the bar potential can influence the disk dynamics as shown by
Hunt and Bovy (2018). The corotation resonance (CR) is determined by the
conditions $l=0$, when the rotation frequency of the stars is equal to the bar
pattern frequency.

Figure 2 shows the positions of ILR, OLR and CR resonances in the plane of the
disk in our models for bar pattern speeds $\Omega_{\rm b}=40\,\mathrm{{km\
s}^{-1}\ {kpc}^{-1}}$ and 60 $\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$.

Our focus is on the study of the resonance effects in the solar neighborhood
and beyond, so we exclude from consideration the disk dynamics in the central
regions of the disk—namely, the inner Linblad resonance and corotation.

Figure 2: The positions of resonances in the plane of the disk (a) and in
perpendicular to the disk direction (b). (a) Black solid and dotted horizontal
lines are the angular velocities of the bar. Purple, green and orange lines
show the radial dependence of the quantities $\Omega-\kappa/2$, $\Omega$,
$\Omega+\kappa/2$ and $\Omega+\kappa$, respectively. The vertical lines mark
the positions of 2:1 ILR, CR, 2:1 OLR and 1:1 OLR resonances. (b) Frequencies
of oscillations of particles (colored dots) perpendicular to the disk as a
function of galactocentric radius. Black solid and dotted lines show the
radial dependence of frequency of vertical oscillations, $\nu_{Z}$ in the left
hand side of the equation 5. Crossings correspond to the positions of the 2:1
vertical outer Lindblad resonance (2:1 vOLR) in the disk.

####  2.4.2 Vertical Resonances

Resonance between oscillation of a star in the direction perpendicular to the
plane of the disk and rotating perturbing potential that has m-fold rotational
symmetry occurs when the following condition is met (Binney, 1981; Sellwood,
2013; Sellwood and Wilkinson, 1993):

$m(\Omega_{b}-\Omega)=l\Omega_{Z}\,,$ (5)

where _l_ is an integer number.

We will be interested in resonance condition when $l=+1$ because, for larger
values of $l$, resonance conditions are achieved far from the solar
neighborhood. The value of $m$ we assume to be equal to $m=2$. We study thus
an influence of the vertical outer 2:1 Lindblad resonance (2:1 vOLR).
$\Omega_{Z}$ is the frequency of the vertical oscillations, which we will use
further in the form of $\nu_{Z}=\Omega_{Z}/2\pi$, where $\nu_{Z}$ is the
linear frequency of oscillations.

In the direction perpendicular to the disk, its density and potential are
changing rapidly so the periods of oscillations of the particles depend on
their velocities in the Z-direction at $Z=0$ and have to be determined
separately. To find the positions of the vertical resonance, we measure the
frequencies of the vertical oscillations of particles by direct integration of
the orbits of particles in the axisymmetric potential. To do this, we generate
at different radii of the disk the particles that have velocities
perpendicular to the disk’s plane equal to $V_{Z}=1,5,10,20,30,40$
$~{}\mbox{km~{}s}^{-1}$, and the azimuthal velocities corresponding to the
equilibrium rotational velocity at a given radius. We also assume that
$V_{R}=0~{}\mbox{km~{}s}^{-1}$. To estimate the frequencies, we followed the
dynamics of the particles during a fixed period of time and calculated the
number of intersections of the particles with the mid-plane of the disk.
Results of simulations, covering the range of $V_{Z}$-velocities in the solar
neighborhood, are shown in Figure 2b. We notice that frequencies of particle
oscillations are in agreement with the results of Antoja et al. (2018).
Beraldo e Silva et al. (2023) used the python package naif for the frequency
analysis of orbits. In this paper, we determined the frequencies of the non-
linear oscillations from the numerical simulations that we deem sufficient for
our purpose.

Figure 2b shows the positions of the vertical resonances in our mock Galactic
disk, determined in this way. Black solid and dotted lines demonstrate the
radial dependence of the theoretical frequency $\nu_{Z}$ from Equation (5) as
a function of radius for two values of the bar angular velocity $\Omega_{\rm
b}=40\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$ and 60 $\,\mathrm{{km\ s}^{-1}\
{kpc}^{-1}}$. The positions of the vertical resonances are marked in Figure 2b
by the grey zones.

As one can see from Figure 2b, the resonance of stellar motions with a bar
that has an angular velocity of $\Omega_{\rm b}=40\,\mathrm{{km\ s}^{-1}\
{kpc}^{-1}}$ occurs at $\approx 11-13$ kpc, while resonance with the fast
rotating bar that has $\Omega_{\rm b}=60\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$
occurs at $\approx\mbox{8--9}$ kpc.

When analyzing the resonance regions, it is important to know variations of
the vertical force $K_{Z}$ in the models with the Galactic bar
$K_{Z}=-{\partial\Phi}/{\partial Z}$. Figure 3 shows the radial dependence of
the Z-component of the gravitational force taken at Z = 200 pc and Z = 500 pc
for the axisymmetric MC17 potential and for two models with bar major semi-
axis a = 3.5 (SB) and a = 5 (LB) kpc. With the bar potential, the calculations
are carried out in three different directions with respect to the large semi-
axis: $0^{\circ},45^{\circ}$ and $90^{\circ}$. The amplification of
oscillations of the particles perpendicular to the plane of the disk is caused
by the periodic force from the non-axisymmetric rotating bar potential. The
efficiency of such amplification of particle oscillations is determined by the
difference of the bar gravitational force acting perpendicular to the disk in
the direction of the bar’s major and minor axes.

As one can see from the Figure 3, for the long bar (LB) model the variation of
the force at the R = 9 kpc is about 10%, while at R = 13kpc the force
variation is about 4%. For the short bar (SB), variation of gravitational
force caused by the bar is about 3% at the R = 9 kpc, and at R = 13 kpc.

Figure 3: Radial dependence of the vertical component of the force for the
axisymmetric MC17 potential (orange) and potentials with the bar with major
semi-axis a = 3.5 kpc (green) and a = 5 kpc (purple). For models with the bar,
the force dependence is shown for different angles with respect to the bar
major axis. The vertical gray line marks the position of the Sun.

##  3 Results and Discussion

###  3.1 Equilibrium in the Axisymmetric Potential

To construct the axisymmetric equilibrium distribution in the MC17 potential
model, we integrate the dynamics of the initially generated distribution of
particles for five Gyr. Figure 4 shows the result of such integration. After
$\approx$ three Gyr of integration the disk rebuilds itself and reaches
equilibrium, remaining unchanged during further integration, with the radial
scale length of the disk beyond four kpc approximately equal to its initial
value of $r_{d}=2.5~{}\mbox{kpc}$.

Figure 4: Evolution of the particle distribution in the MC17 model from the
initial distribution (left) to the equilibrium one after three Gyr of
evolution. The size of the radial bins is 0.1 kpc.

Figure 5 shows the distributions along the radius of the particles
Z-coordinate (first column), velocity component perpendicular to the disk
$V_{Z}$ (second column), azimuthal velocity $V_{\phi}$ (third column), and the
radial velocity of the particles $V_{R}$ (fourth column) of the initially
generated distribution of particles (first row) and the equilibrium
distribution reached after three Gyr integration in MC17 potential (second
row). There is a redistribution of particles both in the coordinate and in the
velocity space. As one can see, the re-arrangement of the azimuthal velocity
of the particles $V_{\phi}$ is noticeable. After three Gyr, there is seen the
appearance of thin diagonal ridges. These ridges, seen also in simulations by
Antoja et al. (2018) and Fux (2001); Minchev et al. (2009); Gómez et al.
(2012), result from the phase-space mixing in the plane of the disk that has
initial distribution out of equilibrium (Antoja et al., 2018). The equilibrium
distribution established in the axisymmetric potential after three Gyr of
integration was taken as the initial distribution to simulate the dynamics of
the disk dynamics in the models with a bar.

###  3.2 Dynamics in a Barred Potential

####  3.2.1 Resonances in the Plane of the Disk

Figure 6 demonstrates distribution of stars along the radius in four models
with bar potential after three Gyr of evolution from the equilibrium state.
The positions of corotation, OLR and 1:1 resonances are marked on the figure
with the vertical lines. One can see the correlation of the dips on the
distribution of particles along the radius with the position of the outer
Lindblad resonance, especially noticeable in the models with the fast rotating
bar. This result is known from previous studies (see, e.g., Melnik (2019)).
Stars near the OLR resonance change their angular momenta, which affects the
orbits, eventually causing the radial migration of the stars. The process can
move the stars radially by kiloparsecs (Mikkola et al., 2020; Sellwood and
Binney, 2002).

Figure 5: Rearrangement of the initially generated distribution in the MC17
potential (top frames) after three Gyr of evolution (bottom frames). First
column—the distribution of stars along the radius, second, third, and fourth
columns distributions of $V_{Z}$, $V_{\phi}$ and $V_{R}$ velocity components
along the radius. Particles’ density distributions are encoded in the colour
bars.

Figure 6: Radial distributions of particles in four barred potentials after
three Gyr of evolution (SB40 top left; SB60 top right; LB40 bottom left; LB60
bottom right). The size of the radial bins is 0.1 kpc. The vertical dashed,
solid and dotted lines show the positions of CR, 2:1 OLR and 1:1 OLR,
respectively.

Figure 7 shows the distribution of stars in $(R,Z)$, $(R,V_{Z})$ and
$(R,V_{\phi})$ planes after three Gyr of orbits evolution in the barred
potential. Diagrams $(R,Z)$, $(R,V_{Z})$ do not show a noticeable effect of
the bar potential on the distribution of the particles at OLR resonances. The
role of the 2:1 outer Lindblad resonance is revealed, however, as a decreasing
of density of particles at OLR resonance in models with the fast rotating bar
($\Omega_{b}=60\,\mathrm{{km\ s}^{-1}\ {kpc}^{-1}}$) due to proximity in these
models of resonance to the bar region as shown in Figure 6. The influence of
the OLR resonances on the distribution of particles is also illustrated in the
$(R,V_{\phi})$-diagram of Figure 7, which shows the appearance of noticeable
diagonal ridges seen on the figure. Similar ridges, confirmed observationally
(Kawata et al., 2018), are connected to the resonant orbital structure in the
barred potential that forms regions in the phase space with stable and
unstable orbits, and hence the regions with overdensities and gaps
(Michtchenko et al., 2018; Antoja et al., 2018; Hunt et al., 2019).

Figure 7:  Distribution of particles shown on $(R,Z)$—plane (first row),
$(R,V_{Z})$—plane (second row), and $(R,V_{\phi})$—plane (third row) after
three Gyr of evolution in models SB40 (first column), SB60 (second column),
LB40 (third column), and LB60 (fourth column). The vertical dashed, solid and
dotted lines show positions of CR, 2:1 OLR and 1:1 OLR, respectively.

To demonstrate the influence of resonances occurring in the plane of the disk
on the disk’s vertical structure, we follow Trick et al. (2021) and describe
the dynamics of the disk in $(L_{Z},<|V_{Z}|>)$ and $(L_{Z},<|Z|>)$ space.
Figure 8 shows $(L_{Z},<|Z|>)$ and $(L_{Z},<|V_{Z}|>)$ distributions after
three Gyr of evolution of the models for moving averages of the values $|Z|$
and $|L_{Z}|$. As is seen from Figure 8, the position of 2:1 OLR resonance
correlates with the bumps on the $|Z|$ distribution of particles in all
models. The dependence of the velocity component $V_{Z}$ on radius is also
affected by bar at the 2:1 OLR resonance region. The upper right panel of
Figure 8 shows an elevation of $<|V_{Z}|>$ velocity in the vicinity of 2:1
outer Lindblad resonance in the model with the fast rotating long bar. A
similar result was reported by Trick et al. (2021) (see their Figure 9c), who
also demonstrated that the 2:1 OLR resonance with the bar can contribute to
the peculiarities in distribution of $V_{Z}$ along the $L_{Z}$. To find the
location of the resonances in $L_{Z}$ space in Figure 8, we use the radial
positions corresponding to the axisymmetric potential and the velocities
corresponding to the rotation curve of MC17.

Figure 8: Green lines—average coordinate $<|Z|>$ (left column) and average
velocity component $<|V_{Z}|>$ (right column) as functions of the angular
momentum $L_{Z}$ after three Gyr of evolution in barred potentials (SB60-first
row, LB60-second row, SB40-third row and LB40 fourth row). Orange lines are
the same as for green ones, but for the model MC17.

Figure 9: Evolution of resonant (blue) and non-resonant (orange) particles in
model SB60 shown on $(x,y)$, $(R,Z)$, $(t,-L_{Z})$ and $(t,R)$—planes at 2:1
(top row) and 1:1 (bottom row) OLR.

In models with a slowly rotating bar (SB40 and LB40), the influence of the 2:1
OLR on disk dynamics in the perpendicular direction is less noticeable due to
its further location from the galactic center. The $Z$-distribution of the
particles along the radius is perturbed by a few percent around the 2:1 outer
Lindblad resonance. The influence of the 2:1 OLR on the particles $V_{Z}$
distribution alone is negligible.

To clarify in more detail the influence of resonances on $(-L_{Z},<|Z|>)$ and
($-L_{Z},<|V_{Z}|>$) distributions, we trace the dynamics of the resonant and
non-resonant particles in the model with the short fast rotating bar SB60.
Figure 9 shows the evolution of resonant (blue) and non-resonant (orange)
particles at 2:1 (top row) and 1:1 (bottom row) outer Lindblad resonances. As
one can see from the $L_{Z}-t$ plot in Figure 9, the angular momentum of
resonant particles shows an oscillational behavior atop of the smaller time-
scale oscillations, corresponding to the periodic disturbances from the bar,
i.e., there is an oscillational behavior with a larger period. Similar
behavior of the particles at the 2:1 outer Lindblad resonance was reported by
Melnik (2019), who gave an explanation for such behavior. Such behavior of the
resonant particles influences the properties of the disk in the Z-direction.
As one can see from Figure 9, the resonant particles can deviate up to 10 kpc
from the resonance region, located at $\approx$ 6.5 kpc, bringing the
particles with higher velocity dispersion $\sigma_{z}$ to the regions with
smaller values of z-component of the velocity dispersion of the disk, which
leads to the appearance of a bump on $<Z>$–$L_{Z}$ distribution in Figure 8.

Let us discuss in more detail the $(-L_{Z},<|Z|>)$-distribution of the
particles in LB60 model. Figure 8 shows that particles with angular momenta of
$\approx$ 2200–2800 $~{}\mbox{km}~{}\mbox{s}^{-1}~{}\mbox{kpc}$ considerably
increase the thickness of the disk with an average growth of the Z-coordinate
of one hundred percent. The appearance of a bump on $(-L_{Z},<|Z|>)$-diagram
is related to the ’escapers’—the particles at the 2:1 OLR region that have
amplitudes of oscillations large enough so the particles reach the bar region,
and get an additional “kick” from a bar. Figure 10 supports this picture. As
one can see from $(t,L_{Z})$ diagram of this figure, most of the resonant
particles have a ’typical’ behavior at the resonant region with angular
momenta nonlinearly oscillating around the equilibrium value. A few percent of
the particles that closely approach the bar region increase their angular
momenta from from $-L_{Z}\approx$ 1600–1800 to 2200–3000
$~{}\mbox{km~{}s}^{-1}~{}\mbox{kpc}$, reaching as a consequence large
distances, both in radius and in the direction perpendicular to the disk.
Khoperskov et al. (2020) reported about the formation of the escaping
particles from the OLR-region due to an interaction with a slowing bar. Our
simulations show that the formation of escapers occurs in a constantly
rotating bar potential as well.

Figure 10: Orbit evolution of escaping particles in the model LB60 shown on
$(x,y)$, $(R,Z)$, $(t,-L_{Z})$ and $(t,R)$-planes.

The dynamics of resonant particles in the model LB60 with the fast elongated
bar has some other features not seen in the models with a shorter (SB60) or a
slowly rotating bar (SB40 and LB40). Figure 11 shows evolution of the
particles in the LB60 model shown in $(L_{Z},Z)$, $(R,V_{\phi})$ and
$(R,V_{R})$-planes over three Gyr of evolution. The $(L_{Z},Z)$-plane of this
figure demonstrates growth of the resonant structure for $L_{Z}\approx$
2000–3000 $~{}\mbox{km}~{}\mbox{s}^{-1}~{}\mbox{kpc}$. We already discussed
this feature also seen in Figure 8.

Another manifestation of escaping particles, shown in Figure 10, is the growth
with time of high-velocity and low-velocity “tails”, seen on the
$(R,V_{\phi})$ distributions of Figure 11. The origin of the tails is related
to the escaping particles, which, due to the resonant set-up at the 2:1 outer
Lindblad resonance, come close enough to the bar and deviate far from their
original locations. The orbits of such resonant particles, occupying a
relatively narrow band of angular momenta $L_{Z}\approx\mbox{2200--
2800}$$~{}\mbox{km}~{}\mbox{s}^{-1}~{}\mbox{kpc}$, become more elongated with
time, with growing apocentric distances and pericentric velocities of the
particles seen on the $(R,V_{\phi})$ diagram of Figure 11 as high- and low-
velocity tails.

The bottom row of Figure 11 shows the formation of chevron-like structures in
the $(R,V_{R})$ distribution of the particles interacting with the long fast
bar. Chevron-like structures are observed in the velocity distribution of the
Milky Way halo stars (Davies et al., 2023; Belokurov et al., 2023), and were
recently discovered in the Andromeda galaxy by Dey et al. (2023). Davies et
al. (2023), Belokurov et al. (2023) and Wenbo Wu WenboWu et al. (2023) suggest
that such structures can result from accretion of a few satellite galaxies in
the past of the Milky Way’s history. Interaction of stars with a bar at
corotation resonance can also lead to the formation of chevron-like
structures, as shown recently by Dillamore et al. (2023). Our simulations
demonstrate that chevrons can also be formed by the particles escaping from
the 2:1 outer Lindblad resonance due to a strong interaction with the fast
long bar. We find that about five percent of the resonant stars are involved
in the escaping process, and the formation of the chevron-like structures and
tails. The formation of chevron-like structures by the escapees from the OLR
region, requires, however, a more detailed study.

Figure 11: Evolution of $(Z)$ (first row), $(V_{\phi})$ (second row), and
$V_{R})$ (third row) distributions of particles in LB60 model.

It should be noted that such effects appear only in the LB60 model, for which
a fast rotation provides a 2:1 OLR resonance position closer to the centre
(near 7 kpc), unlike the slow bar (10 kpc). The parameters of the Milky Way
bar, such as the length of the semi-major axis and the angular velocity of
rotation, are still largely uncertain and it is not possible to draw an
unambiguous conclusion about whether the LB60 model is acceptable for our
galaxy. But it should also be noted that in the observational data of other
galaxies there are both very rapidly rotating bars and long bars (Lee et al.,
2022), in which such resonant effects may take place. However, the observation
of structures such as chevrons, or high and low-velocity “tails” remain
difficult or impossible tasks. However, such bar configurations (fast rotation
and long major semi-axis) are quite rare for Milky Way Analogue galaxies
(Garma-Oehmichen et al., 2022).

####  3.2.2 Resonance with Motions Perpendicular to the Disk Direction

Let us discuss the influence of the 2:1 vertical outer Lindblad resonance
occurring in the solar neighborhood on the disk’s vertical structure if the
angular velocity of the bar is equal to 60
$~{}\mbox{km}~{}\mbox{s}^{-1}~{}\mbox{kpc}$. For resonant buildup of
oscillations, it is necessary that the azimuthal variation of the vertical
force from the perturbing potential of a bar be considerable. We find that the
model LB60, with its fast rotating and elongated bar, satisfies this
criterion. To demonstrate the influence of 2:1 vOLR, we follow the dynamics of
two test particles in the resonance region, and integrate their orbits in the
model LB60 with the fast rotating elongated bar. These particles have
initially circular orbits with initial $R=8~{}\mbox{kpc}$, but differ by
frequencies of oscillations in the direction perpendicular to the plane of the
disk. The first particle has a frequency of oscillations equal to
$\nu_{Z}\approx 9.7~{}\mbox{Gyr}^{-1}$, corresponding to the resonant
frequency of oscillations at eight kpc (see Figure 2b). Another particle has a
non-resonant frequency perpendicular to the disk direction equal to
$\nu_{Z}\approx 12~{}\mbox{Gyr}^{-1}$. Figure 12 demonstrates the dynamics of
the resonant and the non-resonant particles. As can be seen from the figure,
in the plane of the disk, both particles have close to circular orbits, and
oscillate along the radius with approximately the same amplitude. However, the
particles demonstrate different behavior when perpendicular to the disk
direction. Non-resonant particles (orange line) oscillate with constant
amplitudes in both the coordinate $Z$ and velocity $V_{Z}$. The resonant
particles (blue line) oscillate perpendicular to the disk with periodically
increasing and decreasing amplitudes of oscillations of $Z$ and $V_{Z}$
values.

Figure 12: Dynamics of resonant (blue) and non-resonant (orange) particles
shown in the model LB60 on $(x,y)$, $(R,Z)$, $(t,Z)$ and $(t,V_{Z})$ planes at
2:1 vOLR.

It should be noticed that the resonance with the particle motions
perpendicular to the plane of the disk occurs at eight kpc if the velocities
of the particles in that direction are about $V_{Z}\approx
33~{}\mbox{km}~{}\mbox{s}^{-1}$ at Z = 0. Taking into account that the
velocity dispersion of the thin disk in the solar neighborhood is equal to
$\sigma_{Z}=11~{}\mbox{km}~{}\mbox{s}^{-1}$, there is a small number of such
stars. Moreover, we find that 2:1 vOLR resonance occurs if stars are in nearly
circular orbits. Also, the build up of resonance oscillations depends on the
phase of a star’s motion perpendicular to the disk relative to the phase of a
rotating bar. Taking these factors into account, we conclude that it is
unfeasible to observationally confirm a manifestation of the 2:1 vertical
resonance in the solar neighborhood.

##  4 Summary

The results of our study can be summarized as follows.

In agreement with previous studies, we confirm that 2:1 OLR resonance of
particles orbiting in the plane of the disk with the rotating bar leads to
their spatial redistribution. The angular momenta of the resonant stars change
under the influence of the bar. The maximum of such momentum exchange occurs
at 2:1 outer Lindblad resonance. The momentum exchange has oscillational
behavior when atop of the smaller time-scale oscillations corresponding to the
periodic disturbances from the rotating bar, i.e., there are oscillations with
a larger period.

Spatial redistribution of the particles caused by their interaction with the
potential of a bar is more prominent for a larger bar’s semi-axis and its
angular velocity is larger due to the fact that resonances in this case are
closer to the perturbing potential. In addition to the redistribution of
particles at OLR we find—in agreement with previous studies (Antoja et al.
(2018))—that resonance leads to the formation of the ridges seen in
$(R,V_{\phi})$ particles distribution.

The 2:1 and 1:1 OLR do not cause any noticeable influence on the distribution
of particles and their velocities when perpendicular to the plane of the disk
direction as a function of the radius. Such influence is seen, however, in the
distributions of the average coordinate of the particles,$<|Z|>$ and their
average velocity components,$<|V_{Z}|>$, as a function of the angular momentum
$L_{Z}$. The effect manifests itself as bumps on $L_{Z}$—$<|Z|>$ and
$L_{Z}$–$<|V_{Z}|>$\- distributions developing outside the resonance region.

Simulations show an appearance of prominent bump of the average coordinate of
the particles,$<|Z|>$ as a function of the momentum $L_{Z}$ at $\approx$
2200–2800 $~{}\mbox{km}~{}\mbox{s}^{-1}~{}\mbox{kpc}$ in the model with a
long, fast bar. Appearance of the bump on $(-L_{Z},<|Z|>)$—diagram is related
to the ’escapers’—the particles at 2:1 OLR region that have the amplitudes of
oscillations grown large enough so they reach the bar region. Strong
interaction with the bar potential of such resonant particles provides an
additional “kick” from a bar. We find that a few percent of the particles in
the vicinity of the 2:1 OLR can closely approach the bar region and increase
their angular momenta from $-L_{Z}\approx$ 1600–1800 to 2200–3000
$~{}\mbox{km}~{}\mbox{s}^{-1}~{}\mbox{kpc}$, propagating, as a consequence, at
large distances both in radius and in the direction perpendicular to the disk.

The escaping stars in the model with a long fast bar generate and strengthen
with time the high- and low-velocity “tails” seen on the
$(R,V_{\phi})$—particle distribution.

Another manifestation of the 2:1 resonance in the LB60 model is the appearance
of the chevron-like structures seen on the $R,V_{R}$ diagram. Such structures
are observed in the Milky Way halo and assumed to be the result of an
accretion of a massive satellite galaxy into the Milky Way. Our simulations
demonstrate that chevrons can also be formed by the particles escaping from
the 2:1 outer Lindblad resonance due to a strong interaction with the fast
long bar.

The resonance of the particle motions in the direction perpendicular to the
plane of the disk with the rotating bar (vOLR–resonance) can also affect the
disk’s kinematical properties in the direction perpendicular to its plane. We
find that the “vertical resonance” can amplify the amplitude of oscillations
of resonance particles by about a hundred percent. The condition for the
vertical resonance in the solar neighborhood is not, however, satisfied for
all particles. Only particles, that have at the solar neighborhood the
velocity of $V_{Z}\approx 33~{}\mbox{km s}^{-1}$ at Z = 0 satisfy the
resonance condition. Moreover, we find that 2:1 vOLR resonance occurs if stars
have nearly circular orbits. In addition, the build-up of resonance
oscillations depends on the phase of a star’s motion perpendicular to the disk
relative to the phase of a rotating bar. Taking these factors into account, we
conclude that it is unfeasible to observationally confirm a manifestation of
the 2:1 vertical resonance in the solar neighborhood.

The resonances 2:1 OLR and vOLR do not have any noticeable effect on
$V_{Z}$-distribution of stars in the solar neghborhood. We conclude therefore
that peculiarities of $V_{Z}$-velocity distribution in the solar neighborhood
reported by Vieira et al. (2022) cannot be caused by the influence of
resonances with a rotating bar.

Conceptualization, V.K., G.C., K.V. and R.T.; methodology, V.K., G.C. and
R.T.; software, R.T. and A.L.; writing original draft preparation, V.K., R.T.
and G.C.; writing review and editing, V.K., R.T., A.L., G.C. and K.V. All
authors have read and agreed to the published version of the manuscript.

The research by V.K. and R.T. was carried out at Southern Federal University
with the financial support of the Ministry of Science and Higher Education of
the Russian Federation (State contract GZ0110/23-10-IF).

All data used in this paper are taken from the open sources and the references
are given.

###### Acknowledgements.

We thank the anonymous reviewers for the careful reading of the manuscript and
valuable comments. The authors declare no conflict of interest. Abbreviations
The following abbreviations are used in this manuscript:
Corotation Resonance | CR
---|---
inner Lindblad resonance | ILR
outer Lindblad resonance | OLR
vertical outer Lindblad resonance | vOLR
Long bar | LB
Short bar | SB
MC17 | McMillan17
References

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