The sample for ${13}^{\circ }>b>-{13}^{\circ }$ consists of 92 GC ( Table 1 ) In the original table, [7] the following GC parameters are given:

• RA: Right ascension

• DEC: Declination

• l: Galactic longitude

• b: Galactic latitude

• $R_{\odot }$: Distance from Sun

• RGC: Distance from the Galactic center

• RV: Radial velocity

• $\mu \alpha \text{cos}\delta$: Proper motion in right ascension

• $\mu \delta$: Proper motion in declination

• $\rho \mu \alpha \mu \delta$: Correlation coefficient

• X: Distance from the Gal. center in direction of Sun

• Y: Distance from the Gal. center in direction of Solar motion

• Z: Distance above/below galactic plane

• U: Velocity in X direction

• V: Velocity in Y direction

• W: Velocity in Z direction

• RPeri: Average perigalactic distance

• RApo: Average apogalactic distance

As stated in the article [7] distances $R_{\odot }$ to GC are determined either by using eclipsing binaries [8] [9] or through fits of their color-magnitude diagrams (CMDs) with theoretical isochrones e.g. [10] - [13] , by using variable stars that follow known relations between their periods and absolute luminosities like RR Lyrae stars e.g. [14] [15] Type II Cepheid [16] or Mira type variables [17] . Finally, it is possible to determine distances by comparing the magnitudes of main-sequence stars with stars of similar metallicity in the solar neighborhood, the so-called subdwarf method e.g. [18] [19] or kinematically by comparing line-of-sight and proper motion velocity dispersion profiles in GC e.g. [20] - [22] . The latter method has the advantage that the derived distance is not influenced by the reddening of the cluster. For GC with accurately measured radial velocities and dispersion of proper motions, the distance to the GC can be determined by achieving a better fit with a theoretical cluster model e.g. [23] .

RV radial velocities were calculated by [24] by combining literature data with ESO and Keck data archives. [25] also added radial velocities from Gaia DR2 and the Anglo-Australian Observatory (AAO) to the table. The radial velocities given in [26] were also added based on MUSE data. Own motions were added to the table from Gaia EDR3 or HST.

For this work, we used the following table parameters: RA, DEC, l, b, RGC, $\mu \alpha \text{cos}\delta$, $\mu \delta$, Z and W.

The method for determining the velocity of the GC movement relative to the plane and center of the Galaxy is as follows: we find the position of the GC today (or rather at the time of observation) and in 1 year. The difference between these GC positions gives the velocity of movement for 1 year. Next, we convert this result into km/s. Although the parameters l, b, RGC, Z and W are given in the original table, in order to use the same calculation tools for the two GC positions, we recalculate them using the corresponding formulas.

We obtain l and b ourselves by converting coordinates using the formulas:

$\sin \left(b\right)=\sin \left({\delta }_{NGP}\right)\ast \sin \left(\delta \right)+\cos \left({\delta }_{NGP}\right)\ast \cos \left(\delta \right)\ast \cos \left(\alpha -{\alpha }_{NGP}\right)$ (1)

$\tan \left(l_{NCP}-l\right)=\frac{\cos \left(\delta \right)\ast \sin \left(\alpha -{\alpha }_{NGP}\right)}{\sin \left(\delta \right)\ast \cos \left({\delta }_{NGP}\right)-\cos \left(\delta \right)\ast \sin \left({\delta }_{NGP}\right)\ast \cos \left(\alpha -{\alpha }_{NGP}\right)}$ (2)

where:

• $\alpha$ and $\delta$: equatorial coordinates of the GC,

• ${\delta }_{NGP}$ and ${\alpha }_{NGP}$ are the equatorial coordinates of the north galactic pole,

• $l_{NCP}$: Galactic longitude of the North celestial pole,

• $l$ and $b$ are the galactic coordinates of the GC.

The proper motions $\mu l$, $\mu b$ were calculated based on $\mu \alpha$, $\mu \delta$ as follows: first, $l$ and $b$ are calculated based on the above formulas and then $l^{\prime }$ and $b^{\prime }$ (GC galactic coordinates in a year) using ${\alpha }^{\prime }=\alpha +\mu \alpha$ and ${\delta }^{\prime }=\delta +\mu \delta$. Next we get $\mu l=l^{\prime }-l$, $\mu b=b^{\prime }-b$.

Using $\mu b$, we obtain the velocity of movement $V\uparrow$ of the GC in the northern region relative to the Galaxy plane according to the formula:

$V\uparrow =\left(R_{\odot }+RV\ast SY/K\right)\ast \sin \left(b+\mu b\right)-R_{\odot }\ast \sin \left(b\right)$ where:

• $R_{\odot }$: Distance from the Sun

• RV: Radial velocity of GC

• SY: seconds in a year

• K: km in kpc

Thus, we obtained the velocities V↑ of the GC motion in the direction orthogonal to the galactic plane. We, then sorted them in Table 1 in descending order of b. For the southern region of the Galaxy, we will do the same with negative values of the indicated ranges b and denote the velocities V↓. Figure 1 and Figure 2 show the GC velocities V↑ and V↓ in the northern and southern regions of the Galaxy.

Thus, it can be argued that the GCs of the northern region in the range $0^{\circ }<b<{13}^{\circ }$ on average move away from the Galactic plane. The average velocity of the GC’s receding from the Galactic plane in the northern region is 30.36 ± 15.29 km/s (N = 42). Similarly, GCs in the southern region of the Galaxy in the range $-{13}^{\circ }<b<0^{\circ }$ on average move away from the Galactic plane at a velocity of 17.85 ± 13.76 km/s (N = 50).

Applying Student’s t-test shows that the GC velocities V↑ and V↓ of the northern and southern regions of the Galaxy in the ranges $0^{\circ }<b<{13}^{\circ }$ and $-{13}^{\circ }<b<0^{\circ }$ differ at the level of 0.0205.

In the previous section, we obtained the GC movement velocities depending on the galactic latitude b. Now let’s move on to determining the GC velocities relative to the distance to the Galactic plane (Dpl). The GC sample from the general table for the range of distances to the plane from −1.4 to 1.4 kpc contains 77 GCs ( Table 2 ). GC velocities V↑ and V↓ for the northern and southern regions of the Galaxy are shown in Figure 3 and Figure 4 .

On average, the GCs of the northern and southern regions of the Galaxy move away from the Galactic plane. The average velocity of the GCs receding from the plane for the northern region of the Galaxy is 27.74 ± 17.13 km/s (N = 35) and for the southern region -17.41 ± 14.51 km/s (N = 42). Applying Student’s t-test shows that the GC velocities V↑ and V↓ of the northern and southern regions of the Galaxy for the distance ranges from the plane $0<Dpl<1.4$ kpc and $-1.4<Dpl<0$ kpc differ at the level of 0.0496.

It should be noted that similar results are obtained if we use the data from the original table: Z (Distance to the plane) and W (Velocity in Z direction) ( Figure 5 and Figure 6 ). The difference in velocities V↑ and V↓ from the tabular data W is approximately 7.24 km/s, which can be attributed to the oscillation of the solar orbit ( [27] - [32] ).

The dependence of the GC velocities V↑ and V↓ over the entire range $-1.4<Dpl<1.4$ kpc is shown in Figure 7 . As can be seen from the figure, there is an obvious dependence between the parameters discussed. Let us evaluate the significance of the correlation. To do this, we use the value

$t=R\sqrt{\frac{n-2}{1-R^2}}$ (3)

which obeys the Student distribution. Here,

$R\left(V,Dpl\right)=\frac{{\displaystyle \sum \left(V_i-\langle V\rangle \right)\left(D_{pli}-\langle D_{pl}\rangle \right)}}{\left(n-1\right){\sigma }_V{\sigma }_{Dpl}}$ (4)

- assessment of the correlation between the values of V and Dpl, ${\sigma }_V$ and ${\sigma }_{Dpl}$: standard deviations of the corresponding values.

${\sigma }_V=\sqrt{\frac{{{\displaystyle \sum }}_{i=0}^n{\left(V_i-\langle V\rangle \right)}^2}{n-1}},{\sigma }_{Dpl}=\sqrt{\frac{{{\displaystyle \sum }}_{i=0}^n{\left(Dp{l}_i-\langle Dpl\rangle \right)}^2}{n-1}},$ (5)

We get t = 2.55, from which it follows that the significance of the correlation is high $\alpha =1-P\approx 0.01$. Thus, we can conclude that GCs far from the Galactic plane, on average, have relatively high speeds of receeding from the Galactic plane.

Figure 8 shows the plot of GC by distance Z from the galactic plane versus the velocities W at a distance of up to 4 kpc. As can be seen from the figure, GCs are distributed quite symmetrically along the Z axis. Visually, it seems that the GCs are also distributed symmetrically by velocities W. In fact, the velocities of the northern and southern GCs differ significantly. The GC of the northern region is 23.02 km/s, and of the southern region - 10.48 km/s. The asymmetry value Asm = 12.54 km/s. We assume that this is due to the oscillation of the solar orbit (about 7 km/s relative to the plane). It is logical to accept that in order to obtain spatial symmetry, it is necessary to have symmetry by velocities as well, otherwise, over the past billions of years, spatial symmetry would have been broken.

The result obtained in the previous sections that GCs on average move away from the Galactic plane suggests that it is possible that GCs on average can also move away from the Galactic center, so this work attempts to test this assumption.

Using the table data from 151 GCs, the distances of the GC from the center of the Galaxy $Rgal$ (Galactocentric distances) and the velocity of movement of the GC (VR) relative to the center of the Galaxy were calculated (it should be noted that this data is not in the original table). To do this, we calculated the distance of each GC from the center of the Galaxy for two points of location of these GCs: the “current” (or rather at the moment of observation) distance of the GC from the center $Rgal$ (Galactocentric distance) and the distance calculated after a 1 year $R^{\prime }gal$. The difference between these distances gives us the velocity of movement of the GC relative to the center, i.e. $VR=R^{\prime }gal-Rgal$. In the table, the current distance of the GC from the center of the Galaxy RGC is given rounded to the second decimal number, so we considered it appropriate to calculate these values ourselves. Using simple trigonometry, the velocities of globular clusters relative to the center of the Galaxy are calculated. The current distance of the GC from the Galactic center is calculated using the following formula:

$Rgal=\sqrt{{\left(D_{CL}\ast \sin \left(b\right)\right)}^2+{\left(D_{GC}\ast \sin \left(l\right)\right)}^2+{\left(D_{CL}\ast \cos \left(b\right)-D_{GC}\ast \cos \left(l\right)\right)}^2}$ (6)

where:

• $D_{GC}$ (kpc): distance from the Sun to the center of the Galaxy;

• $b$: galactic latitude of the Globular Cluster (GC);

• $l$: galactic longitude of the Globular Cluster (GC);

• $D_{Cl}$: distance from the Sun to the center of the GC;

• $Rgal$: distance from the center of the Galaxy to the center of the GC.

The distance of the GC from the Galactic center in 1 year is calculated using the following formula:

$R^{\prime }gal=\sqrt{{\left({D^{\prime }}_{CL}\ast \sin \left(b^{\prime }\right)\right)}^2+{\left(D_{GC}\ast \sin \left(l^{\prime }\right)\right)}^2+{\left({D^{\prime }}_{CL}\ast \cos \left(b^{\prime }\right)-D_{GC}\ast \cos \left(l^{\prime }\right)\right)}^2}$ (7)

where:

• ${D^{\prime }}_{Cl}=D_{Cl}+RV$ distance from the Sun to the GC in a year;

• $b^{\prime }=b+\mu b$: galactic latitude in a year;

• $l^{\prime }=l+\mu l$: galactic longitude in a year;

• RV: radial velocity of the GC relative to the Sun.

The average velocity of the GC relative to the Galactic center for the range 0 to 12 kpc is 31.70 ± 14.74 km/s (N = 119). At the same time, with increasing distance this speed decreases and for a distance of 90 kpc it reaches about 9 km/s, which can be attributed to the gravitational influence of the galaxy on the GC. Based on the calculations obtained, it can be assumed that GCs, on average, are probably also moving away from the Galactic center.