In the structures shown in Figure 3 , the layers were successively rotated about the x-axis; as a result, these structures became asymmetric. In this connection, the MS05с99e.dat structure was created with the same parameters as the MS05с99с.dat structure, but without the rotation of layers, that is, with the parameter I_nx = 0. As evident from Figure 4(a) , the former structure is more symmetrical. It has N = 1488 bodies, and the layer parameters in it are the same as in the MS05c99c.dat structure. After 196 revolutions of first-layer bodies, that is, at T = 1.96, the structure under consideration (see Figure 4(b) ) also proved to be more symmetrical than the previous configuration ( Figure 3(e) ). However, the spread of body distances r_r1 in it ( Figure 4(c) ) is somewhat greater than in the previous structure ( Figure 3(f) ). The greatest distance of a body from the center of mass is r_r1max = 15.7 in comparison with r_r1max = 13.1 in the previous structure. It also exhibits slightly more collisions, namely, 46, compared with the previous structure with 44 collisions. Note that body 1434 exhibits the greatest distance from the center of mass r_r1max = 15.7.

The dynamics of inter-body collisions in the structure under consideration is shown in Figure 4(d) . By the time T = 1.96, 40 bodies with a double mass and 3 bodies with a triple mass have formed in this structure. As it is seen from Figure 4(d) , collisions occur more frequently during the initial time interval of T < 0.5, and less frequently during the interval T > 0.7. Initially, the rate of collisions is equal to v_imp = 64 collisions per 100 revolutions of first-layer bodies and, then, v_imp = 8. That is, the rate of collisions has decreased by eight times. If we relate the rate of collisions to the number of bodies, then in the last section it will be v_imp1 = v_imp/N = 5.4 × 10⁻³.

From the visual analysis of the structures, it follows that the remains of their initial organization shown in Figure 4(a) still persist by the time T = 0.1, whereas by the time T = 0.2 they are no longer observed. That is why we can assume that, from the latter moment, a structure with a disordered arrangement of bodies has been created, with its appearance changing little in the future. When the rate v_imp reaches the second stage of its change, we can assume that the dynamics of the structure has passed into the stable phase of its existence.

The trajectories of bodies in different layers were studied: that of first-layer body 2, that of third-layer body 401, and those of bodies 1434 and 1488 in the last layer. The study was carried out for two time intervals, initial T = 0 - 0.15 ( Figure 5(a) , Figure 5(b) ) and final T = 1.81 - 2.11 ( Figures 5(c)-(e) ).

Over the initial time interval, body 2 first moves in a circle (see Figure 5(a) ) with a period P₁ = 0.01. Its orbit lies in the xy-plane. Over time, the orbit becomes elliptical, and the period of revolution increases slightly. Over a finite time interval ( Figure 3(c) ), the orbital eccentricity of body 2 increases to e = 0.385, but the dimensions of the orbit show a decrease, and the period also decreases and becomes shorter than 0.01. In the 3D graph of Figure 5(e) , it is seen that the orbit of body 2 rotates in space, and with each revolution of this body

around the center of mass, this body departs farther and farther from the xy-plane.

The orbit of body 401 over the initial time interval ( Figure 5(a) ) during the first revolution of the body is approximately a circle with a period of P₂ = 0.028. Then, the trajectory becomes elliptical, and the period of revolution increases slightly. Over a finite time interval ( Figure 5(c) ), the eccentricity reaches a value e = 0.373, the orbit size increases, and the period reaches 1.25P₂. As it is seen from Figure 5(e) , here the orbit rotates in space with each body revolution.

The orbit of body 1488, the last body in the fifth layer, over the initial time interval ( Figure 5(b) ) is a circle with a period of revolution P₅ = 0.1046. Over the final time interval ( Figure 5(d) ), the orbit is already elliptical with an eccentricity e = 0.349, and the period has increased by a factor of 1.6. From Figure 5(e) , it is evident that the orbit of body 1488 changes in space little during this time.

Figure 5(b) shows that during the first revolution the orbit of the most distant body 1434 exhibits notable changes. Over the final time interval ( Figure 5(d) ), the orbit is already an ellipse with an eccentricity e = 0.491 and a period of 0.30, while the initial period of revolution of five-layer bodies is 0.1046. That is, here the period has increased three-fold.

It should be noted that the orbits are exactly ellipses, parabolas and hyperbolas only in the two cases: 1) the interaction involves two bodies, and 2) the interaction involves N bodies located axisymmetrically on one plane [18] . That is why under an elliptic trajectory we mean a trajectory bounded in space, while under hyperbolic and parabolic trajectories we understand infinite trajectories. In the latter case, the velocity of a body on a parabolic trajectory tends to zero at infinity.

Changes in the orbit of body 1434 occur when this body approaches other bodies. The approaches of body 1434 over the initial time interval are shown in Figure 6. Plotted on the vertical axis are the numbers of bodies k to which the body i = 1434 comes in proximity. Such bodies are marked with points or with

horizontal segments if the points overlap. Also indicated on the vertical axis is the distance R_ik at the moment of approach of body 1434 to body k. The distance R_ik is increased by the factor f. The distance R_ik is defined as the minimum distance of a body i from other bodies over the number of integration steps Kl3, where Kl3 is a parameter of the Galactica system. In the case under consideration, the integration was performed with a step dT = 1 × 10⁻⁷.

As it follows from Figure 6, at the initial time T = 0 body 1434 passes at a distance of 2.40 × 10⁻³ from body 1062, which is at the intersection of the initial location line. In this case, the distance between the bodies on this line is 3.17 × 10⁻³, that is, of the same order.

Then, body 1434 moves with an almost constant distance to the neighboring body 1435 until the time T = 2.16 × 10⁻²; and then at time T = 2.32 × 10⁻² it approaches body 1311 to a distance R_ik = 1.66 × 10⁻⁴. The latter distance corresponds to 90 radii of the body. Then, at time T = 3.44 × 10⁻² the body approaches body 1186 to a distance of R_ik = 5.84 × 10⁻⁴. These two approaches lead to a significant change in the trajectory of the body of interest, with its orbit becoming elliptical. The body moves away from all bodies to a distance R_ik = 3.83 × 10⁻² from the nearest body 1421 at time T = 0.12. This is the most distant point of its orbit, after which body 1434 starts approaching the center of the structure again. At this point, the distance of body 1434 from the center is r = 0.109.

As a result of subsequent interactions, the orbital eccentricity increases in magnitude, and at the apocenter by the time T = 1.96 ( Figure 5(d) ) the body moves away from the center to a distance r = 0.205, which value is 3.14 times greater than the initial size of the structure.

When considering the results of calculations for the motion of bodies in the structures shown in Figure 3(b) , Figure 3(d) , and Figure 4(b) , it becomes necessary to determine the type of motion of remote bodies. This allows one to determine whether such a body is a body of this structure or it is a body ejected out of it. For solving this problem, it is necessary to perform a special study of the trajectory of such body. Consider a method based on the results of the two-body problem, which will give an answer to the question of interest without performing special studies.

For a body remote from the structure, we can assume that it is affected by the entire structure with mass m_ss located in its center of mass. Then, similarly to (5), the parameter of the body trajectory will be

${\alpha }_1={\mu }_1/\left(R_p\cdot v_p\right)$(29)

where R_p and v_p are the pericenter radius of the body and its velocity in pericenter, and the interaction parameter is ${\mu }_1=-G{m}_{ss}$. As a result of calculations made using the Galactica program, we obtain the coordinates x, y, z and the velocities v_x, v_y, v_z, of the body in the center-of-mass system. From the definition of the scalar product of the radius vector r of the body and its velocity vector v, we can write

$x{v}_x+y{v}_y+z{v}_z=r\cdot v\cdot \cos {\beta }_1$,(30)

where β₁ is the angle between the vectors r and v;

$r=\sqrt{x^2+y^2+z^2}$; $v=\sqrt{v_x^2+v_y^2+v_z^2}$.

Then, from formula (30) we obtain the following expression for the angle β₁:

$\cos {\beta }_1=\frac{x{v}_x+y{v}_y+z{v}_z}{r\cdot v}$(31)

The radial velocity v_r is directed along the radius vector r, and the transversal velocity v_t, in the direction perpendicular to the latter velocity, so that

$v_r=v\cdot \cos {\beta }_1$; $v_t=v\cdot \sin {\beta }_1$.(32)

On the other hand, according to the two-body problem [33] [34] , the radial and transversal velocities, similarly to (12) and (13), can be written as follows:

$v_r=\pm v_p\sqrt{{\left({\alpha }_1+1\right)}^2-{\left({\alpha }_1+R_p/r\right)}^2}$; $v_t=v_p\cdot R_p/r$.(33)

Equating the right-hand sides in the velocity v_t from formulas (32) and (33), we obtain the pericentric velocity

$v_p=\frac{v\cdot r\cdot \sin {\beta }_1}{R_p}$,(34)

and excluding the velocity v_r from formulas (32) and (33) with taking into account formula (34), we obtain the expression

$R_p\cos {\beta }_1=r\sin {\beta }_1\sqrt{{\left({\alpha }_1+1\right)}^2-{\left({\alpha }_1+R_p/r\right)}^2}$.(35)

Three Equations (29), (34) and (35) include three parameters α₁, R_p, and v_p. As a result of successive substitutions and solutions of quadratic equations, the pericenter radius is obtained in the following form:

$R_p=\frac{{\mu }_v\pm \sqrt{{\mu }_v^2+r^2\sin {\beta }_1\left(1+2{\mu }_v/r\right)}}{1+2{\mu }_v/r}$(36)

where the designation μ_v = μ₁/v² is introduced. The quantity μ_v, which is measured in meters, is negative, μ_v < 0. Expression (36) gives two values of R_p: with the “-” sign in the case of an elliptical orbit we obtain the apocenter radius R_a, and in the case of “+”, the pericenter radius R_p.

Given the pericenter radius R_p, the pericenter velocity v_p can be found using expression (34). In the case of v_p > 0, the orbit is passed counterclockwise. Then, formula (29) can be used to calculate the trajectory parameter α₁, and formula (10) yields the orbital eccentricity e. In accordance with formula (4), these parameters make it possible to determine the trajectory of the body, as well as the time of motion along it [33] [34] .

In the case of a parabolic or hyperbolic orbit, the velocity at infinity can be calculated as follows:

$v_{\infty }=v_p\sqrt{2{\alpha }_1+1}$(37)

For a hyperbolic orbit, the apocenter radius R_a calculated using expression (36) with the “-” sign turns out to be negative.

For body 1434, the parameters calculated by this algorithm for time T = 1.96 have the following values: R_p = 0.0694, v_p = 4.638, α₁ = –0.669, e = 0.494, R_a = 0.2049, and v_a = 1.572. Since for an elliptical orbit we have: –0.5 > α₁ > –1, the trajectory of body 1434 is an ellipse. At time T = 1.96, the body is at a distance r = 0.2049 and has a velocity v = 1.572. These parameters coincide with the parameters of the apocenter. As it is seen from Figure 5(d) , body 1434 shown with a circle is indeed located at the most distant point of the trajectory from the center, that is, at its apocenter. Therefore, algorithm (29)-(37) presented above can be used to estimate the trajectory of a remote body from its coordinates and velocity at some point in time.

Due to the fact that in the structure with layers rotated around the x-axis ( Figure 3(d) ) there were slightly fewer collisions than in the structure without rotations ( Figure 4(a) ), we have decided to create 10- and 15-layer structures with rotated layers. Such a ten-layer structure is shown in Figure 7(a) . The initial parameters of this structure in the MLSpStr2.dat file are the same as those of the structure in Figure 3(d) , except for the number of layers, which is set to N₂ = 10. All close encounters at the self-crossings of the formation line are eliminated. The number of bodies in the layers varies from 99 in the 1st layer to 991 in the 10^th layer, the semi-axis varies from a₁ = 0.0118 to a₁₀ = 0.1179, and the period, from P₁ = 0.0100 to P₁₀ = 0.2536. The total number of bodies is N = 5451.

In the previous cases, the calculation of bodies’ motion implemented with the help of the Galactica program was carried out with a time step dT = 1 × 10⁻⁷. At this step, the calculation time for the interval ∆T = 0.01, which is equivalent to one revolution of first-layer bodies lasted for 7 hours. For a ten-layer structure with N = 5451, the calculation time of this interval took 97 hours. For solving the problem over the required time interval, the duration of calculation will exceed one year. Therefore, the solution of problems with such a number of bodies was performed with a step of dT = 10⁻⁶ in automatic step selection mode, which can be launched in the Galactica system using the key Kl4 = 3. Since the results of calculation are issued following a certain number of integration steps, in the latter case the time interval between these results may be different.

It should be noted that the Galactica system also automatically modifies the step when the bodies approach each other up to a distance of the order of their diameters. These circumstances must be taken into account when analyzing points on the graphs of parameter variation over time.

After 81.5 revolutions of first-layer bodies, i.e. at T = 0.815, the structure is shown in Figure 7(b) . By the time T = 0.1, elements of the initial organization of

the structure are still preserved, and by the time T = 0.2 they completely disappear. By the time T = 0.4, the structure acquires a form later showing almost no changes. The scatter of body distances in Figure 7(c) is limited to 40a₁. About 30 bodies were ejected over a large distance, the largest of which is r_r1 = 600 for body 1575.

By the time T = 0.815, there were 306 collisions, one of which was with the central body. During this period, 202 double-mass bodies, 34 triple-mass bodies, 9 quadruple-mass bodies, and 2 bodies with mass 5m₁ were formed. The dynamics of collisions is shown in Figure 7(d) . Before the time T = 0.061, there were no collisions. Then frequent collisions began, with the rate which we will also consider in two regions. On the first interval ∆T = 0.0688 - 0.142 the rate of collisions is v_imp = 2158 collisions per 100 revolutions of first-layer bodies, and on the second interval ∆T = 0.339 - 0.815 the average rate of collisions is v_imp = 69.3. At the same time, over this time interval there is a tendency towards a further decrease of velocity, possibly to zero at infinity. In this structure, the rate of collisions in the second trajectory section compared with the first section has decreased by 31 times. In this case, the rate of collisions per one body is v_imp1 = 1.27 × 10⁻², which value is 2.4 times higher than that in the five-layer structure.

Figure 8 shows the trajectories of the central body 1, first-layer body 57, second-layer body 170, sixth-layer body 1563, and tenth-layer body 5301. These calculations were performed over the interval T = 0.663 - 1.021 with a step dT = 1 × 10⁻⁷. The trajectory of the central body 1 around the center of mass is irregular ( Figure 8(a) ). The three-dimensional appearance of the trajectory is shown in Figure 8(d) . Body 1 does not move away from the center of mass to a distance greater than 0.003, which value is 0.23 of the radius of the inner layer.

The orbit of body 57 ( Figure 8(b) ) continuously rotates in space ( Figure 8(e) ). The period of revolution of body 57 fluctuates within small limits relative to the period 1.34 P₁, where P₁ is the initial period. The semi-axis of the orbit also oscillates around a value of 0.015. The orbital eccentricity reaches 0.7.

The orbit of body 170 is an ellipse with a large eccentricity reaching 0.9 ( Figure 8(b) ). This orbit also varies in space ( Figure 8(e) ). The orbital period fluctuates around 1.16P₂, where P₂ = 0.028 is the initial period.

It should be noted that at the pericenters of the orbits in Figure 8(b) and Figure 8(d) there are trajectory sections with straight segments. This is due to the fact that during the intervals of outputting the trajectory coordinates (with the Galactica system parameter Kli = 2000), the orbits change more substantially than in other regions.

The trajectory of body 1563 is almost a circle, its eccentricity being e = 0.01 ( Figure 8(c) ). The orbit slightly changes in space ( Figure 8(e) ). The orbital period is 1.2P₆, where P₆ = 0.134. The semi-axis of the orbit has also changed little compared to the initial one and is equal to 1.06a₆.

The trajectory of body 5301 is an ellipse with eccentricity e = 0.23 ( Figure 8(c) ). Its plane is almost perpendicular to the xy-plane ( Figure 8(e) ). That is why on the xy-plane this trajectory is depicted as a flattened ellipse ( Figure 8(c) ).

As already noted, the distances r_r1 of about 30 bodies were outside the graph in Figure 7(c) . For some of these bodies, according to algorithm (29)-(36), the types of trajectories and their parameters were determined. At time T = 0.815, the most remote body 1575 had a distance r = 7.076 and a velocity v = 9.735. Calculations by algorithm (29)-(36) gave the following results: R_p = 0.0560, v_p = 11.41; α₁ = −0.1371; e = 6.293; and v_∞ = 9.720. With this value of α₁, the trajectory is a hyperbola. Thus, the body was ejected from the structure along a hyperbolic orbit. At this time, its velocity already approached the velocity at infinity.

For body 1575, the calculations were repeated for the time T = 0.385, when the distance and velocity were r = 2.883 and v = 9.766. The parameters of the hyperbolic trajectory were confirmed up to seven digits. These calculations were also performed for the time T = 0.1035, when the parameters of the body were r = 0.1907 and v = 10.450. The parameters of the hyperbolic orbit were confirmed to within 4 digits. At this time, the distance r is close to the radius of the outer layer. It follows from here that algorithm (29)-(36) can be used for determining the trajectories of all bodies outside their main cluster.

These studies were carried out for all remote bodies. It was found that 14 bodies with mass m₁ and 3 bodies with mass 2m₁ were ejected from the structure. Two bodies have strongly elongated elliptical orbits, whose apocenter radii R_a are equal to 0.5646 for body 4601 and 3.024 for body 3032.

As already noted, in this structure there are two bodies with a mass of 5m₁. Figure 9 shows the encounters of one of these bodies, namely body 1951, with other bodies over the interval 0.069 < T < 0.075. The calculations were performed in step correction mode with key Kl4 = 3 and key Kl3 = 300 steps for outputting results. From the time T = 0.069, body 1951 approaches body 1952, and at time T = 7.153 × 10⁻² the bodies merge together. Then, the approach to body 1950 begins and at time T = 7.153 × 10⁻² the two bodies merge together.

Then, body 1951 approaches body 1953 and at time T = 7.172 × 10⁻² to merge with it. Then, the approach to body 1954 begins, and at time T = 7.321 × 10⁻² there occurs merging with this body. As a result of these four events, the mass of 1951 became equal to 5m₁. Further, at time T = 7.321 × 10⁻² the body 1951 approaches body 1949 to a distance of 14 peripheral-body radii R₁ = 1.676 × 10⁻⁶, and the bodies diverge. Then, at time T = 7.456 × 10⁻² there occurs an encounter with body 1955 to a distance of 48R₁.

Figure 10(a) shows the trajectories of all bodies approaching and colliding with body 1951. These calculations were performed in mode Kl4 = 3 with an initial step dT = 0.5 × 10⁻⁶ ( Figure 10(b) ) over the interval of output steps Kli =

300. The first approach of body 1951 occurs with body 1952. On the section of approach ( Figure 10(b) ), the calculations were carried out with the interval of output steps Kli = 3. Body 1952 moves ahead of body 1951 and, as the latter body approaches the former, it begins to go around it. On a larger scale ( Figure 10(c) ), the diameters of the bodies are marked with circles. The bodies are almost at the same distance from the xy-plane. The merging of the two bodies occurs when their surfaces come in contact. The condensation of points on the trajectory indicates the inclusion of step correction mode before the collision. The length of the condensation section is proportional to the body velocity. It is seen that the velocity of body 1951 is greater than the velocity of body 1952. This difference may be due to the interaction between these bodies. In this case, the body 1951 is accelerating, and the body 1952, decelerating.

It should be noted that all images in the graphs are drawn to scale, and the values of the limits y₁, y₂, x₁, x₁, etc. are presented in Table 1 .

The second approach of body 1951 occurs with body 1950 ( Figure 10(a) ). Figure 10(d) shows that the body 1950 catches up with the body 1951, crosses its trajectory, and due to the attraction of this body, the trajectory of body 1950 bends. The merging of the two bodies occurs at the moment of contact. In this case, the z-coordinate of body 1951 is greater, so the contact occurs below the visible contour of body 1951. As a result of the merging with body 1952, the radius of body 1951 increases to 1.26R₁. Before the collision, the body 1950 was slowing down as a result of the interaction with body 1951, while the body 1951 experienced acceleration.

The third approach of body 1951 occurs with body 1953 ( Figure 10(a) ). It is seen from Figure 10(e) that body 1953 moves ahead of body 1951. That is why body 1953 goes around body 1951 and collides with it on the side opposite to the side of approach. In this case, the radius of body 1951 was already 1.44R₁.

As it is evident from Figure 10(f) , the fourth approach to body 1954 occurs similarly, yet with the enveloping motion of body 1954 being more pronounced than that of body 1953. This is due to the greater mass of body 1951. This mass is 4m₁, and the radius of 1951 is 1.59R₁. After the merging of body 1951 with body 1954, its mass became 5m₁, and its radius, 1.71R₁. In these two cases (see Figure 10(e) , and Figure 10(f) ,), body 1951 more closely approaches the xy-plane than body 1954; therefore, this plane is partially shaded by this body.

In the four collisions considered above, three bodies approached the target body from the left side and collided with it on the right side ( Figure 10(c) , Figure 10(e) , Figure 10(f) ). On the other hand, the body approaching the target body from the right side collided with it on the left side ( Figure 10(d) ). At the same time, the greater the mass of body 1951, the greater is the angle through which the approaching body envelopes body 1951.

The approach of body 1949 to body 1951 is shown in Figure 10(g) . These calculations were performed with the output interval Kli = 30. As noted above, in this case the body 1949 approached body 1951 to a distance of 14R₁. At the moment of approach, the velocity of body 1949 was almost three times greater than the velocity of body 1951. That is why the body 1949 overtook body 1951 with a further increase in velocity and went around it by almost 90˚. After approach, the velocity of body 1949 began to decrease.

Despite the above collisions, body 1951 kept moving within the structure. By the end of the studied time interval T = 0.815, the distance of body 1951 from the center of mass was r_r1 = 7.17. Body 1949 was also inside the structure with a distance of r_r1 = 4.28.

When bodies collide, the body formed from them acquires its own angular momentum, which we call spin S_p, and thermal energy E_t. The algorithm for calculating these quantities was reported in [33] [34] , and its software implementation was described in [24] . Figure 11(a) shows the variation of spin projections S_px, S_py, S_pz and thermal energy E_t of body 1951 during its collision with four bodies. As a result of each collision, these quantities changed. The circles in the graphs mark the moments T of issuing output files by the Galactica program; those files were used to determine the values of interest.

The smallest change in spins occurs during the first collision (point 2m₁) with body 1952 ( Figure 10(c) ). This collision is close to a frontal impact. As is seen from Figure 11(a) , the thermal energy E_t changes significantly in this case. During the second collision (point 3m₁) with body 1950, the spin projection S_p has increased more significantly, and the thermal energy E_t changed three times less than it did so during the first collision. As it is seen from Figure 10(d) , this collision occurred along a tangent line. Other collisions with bodies 1953 and 1954 also occurred in nearly tangential directions. But body 1951 became more

massive, and the velocity of the approaching bodies increased. Therefore, the increment of spins and thermal energy were significant. As a result, body 1951 acquired thermal energy E_t = 2.87 × 10⁻³ and spin modulus S_p = 6.385 × 10⁻⁹. The spin vector makes an angle β₂ = −23.8˚ with the z-axis. Since the angle is negative, body 1951 rotates clockwise.

For comparison, Figure 11(b) shows the rotational and thermal characteristics of body 3685, which also has a mass of 5m₁. During the first collision at point 2m₁, the body has acquired a small thermal energy E_t. During the second collision (3m₁), the spin projections S_px and S_py have slightly decreased, while the thermal energy E_t increased significantly. The third collision at point 5m₁ occurred with a body of double mass. In this case, the spin projection S_py dominates. The spin value is S_p = 3.48 × 10⁻⁹, and its vector makes an angle β₂ = −78.1˚ with the z-axis, i.e. the axis of rotation of body 3685 is close to the xy-plane. Its thermal energy is E_t = 6.45 × 10⁻³. Thus, with the same masses of these bodies, body 1951 has a 1.8 times greater spin and 2.2 times lower thermal energy.

Due to the fact that with an increase in the number of bodies, the time required for calculating the evolution of a structure of interest becomes excessively large, a 15-layer structure was created with a smaller initial number of bodies in the first layer, namely N_3,1 = 49. In this case, the absolute peripheral-body mass was 2.02 times greater than that in the previous structures. This structure, contained in the MS15c49b.dat file, is shown in Figure 12(a) . All close encounters at self-crossings of the formation line were eliminated by changing the number of bodies in accord with (21). The number of bodies in the layers varies from 47 in the first layer to 733 in the 15^th layer; the semi-axes, from a₁ = 0.0118 - 0.0105 to a₁₅ = 0.1573; and the periods, from P₁ = 0.0100 to P₁₀ = 0.3916. The total number of bodies was N = 5866.

After 77.9 revolutions of first-layer bodies, i.e. at T = 0.779, the structure is shown in Figure 12(b) . By the time T = 0.2, elements of the initial organization of the structure are still preserved, whereas by the time T = 0.3 they completely disappear. By the time T = 0.4, the structure acquires a form that shows almost no subsequent changes. The scatter of body distances in Figure 12(c) is limited to 50 radii of first-layer bodies. About 20 bodies were ejected over a greater distance, the largest of which is r_r1 = 168 for body 5336.

By the time T = 0.779, there occurred 164 collisions. In this case, 120 bodies of double mass, 17 bodies of triple mass and 2 bodies of quadruple mass were formed. In addition, there occurred four collisions with the central body.

The dynamics of collisions is illustrated in Figure 12(d) . For the interval T = 0.085 - 0.286, the rate of collisions was v_imp = 547 collisions per 100 revolutions of first-layer bodies, and during the second interval T = 0.378 - 0.779, the average rate of collisions was v_imp = 85. Thus, over the second time interval the collision rate has decreased by 6.4 times.

When compared with the 10-layer structure, the velocity over the second time interval of the 15-layer structure was 1.23 times higher. The velocity per one body, v_imp1 = 1.45 × 10⁻², is also 1.14 times higher.

Figure 13 shows the trajectories of the central body 1, first-layer body 25, seventh-layer body 1325, and 15^th-layer body 5660. These calculations were carried out over the interval T = 0.597 - 0.853 with a step dT = 1 × 10⁻⁷. The trajectory of the central body 1 ( Figure 13(a) ) is irregular and lies in the range of distances r < 0.003 from the center of mass.

The orbit of body 25 is continuously changing in space. Its eccentricity also changes: it decreases up to the time T = 0.77 and then increases. The average value of the eccentricity is e = 0.69. The period undergoes similar changes, and on average it equals 0.83P₁, where P₁ = 0.01 is the initial period. After T = 0.77, a significant change in the plane of the orbit occurs ( Figure 13(c) ), both its eccentricity and period increase, and its change in space also becomes more pronounced.

The orbit of body 1325 ( Figure 13(b) ) is an ellipse with eccentricity e = 0.408. The semi-axis of the orbit is 1.1 times greater than a₇ = 0.073, where a₇ is the semi-axis of the initial orbit, and the period is 0.95 times less than P₇ = 0.1645. As it follows from Figure 13(b) and Figure 13(d) , the orbit changes significantly during one revolution.

The orbit of body 5600 in Figure 13(b) and Figure 13(e) is represented by its part smaller than half the body’s range of revolution about the center of mass. The distance of the body from the center is r = 0.238 and practically does not change. The slight ellipticity in Figure 13(b) is due to the inclination of the orbit to the xy-plane (see Figure 13(d) ). The distance r is 1.5 a₁₅, and the period of revolution of body 5600 is therefore longer than the initial period P₁₅ = 0.3916.

As already noted, the distances r_r1 of approximately 20 bodies were beyond the bounds of Figure 12(c) . For most of these bodies, algorithm (29)-(36) was used to determine their trajectory types. For bodies 5336, 543, 264, and 5146, remote to distances r_r1 = 168, 125, 115, and 107, respectively, the trajectories were hyperbolas, i.e. the bodies were ejected out of the structure. In total, six bodies were ejected. In this case, for two bodies, 5146 and 264, the eccentricities were 1.045 and 1.0098, respectively, i.e. they are close to unity, which value characterizes a parabolic orbit. For body 5594, remote to r_r1 = 94, the orbit is an ellipse with eccentricity e = 0.894. Other distant bodies also have elliptical orbits.

Encounters of the most distant body 5336 with other bodies were studied ( Figure 14 ). In the first time interval T = 0.184 - 0.193 (see part a of Figure 14 ), this body first passes close to body 5335, and then its approach to body 5333 begins. At time T = 0.1854, the distance between the bodies R_ik becomes equal to 4.05 × 10⁻⁶, this value amounting to 2.15 radii of the body R₁, i.e. the distance between the surfaces of the bodies is 0.15∙R₁. In this case, the velocities of the bodies increase to v = 5.53 for body 5336 and v = 5.37 for body 5333. At the initial time T = 0, the velocities of both bodies were equal to 2.53. After the approach, the bodies move away from each other and their velocities at the final time T = 0.779 decrease to v = 2.77 and 1.87, respectively.

In the final time interval T = 0.75 - 0.85, it is seen that body 5336 is continuously moving away from body 5146, which is also moving away from the structure along a hyperbolic orbit. The distance between the bodies increases almost linearly with an average velocity of 1.38. The bodies move in approximately the same direction: the angle between their velocities is 8.27º.

As for body 5333, after moving away from body 5336, its velocity, as already noted, decreased more significantly, and its motion proceeded along an elliptical orbit with an eccentricity e = 0.597. At time T = 0.779, the body is at a distance of r_r1 = 13.87. Thus, the ejection of body 5336 has occurred due to its acceleration when approaching body 5333 to a distance r = 2.15R₁.

In the considered 5, 10, and 15-layer models of globular clusters, the arrangement of the layers relative to each other, the number of bodies in the first layer, and the number of layers changed. All of them are stable and do not destroy. Therefore, with the variations considered, it is possible to create models of globular clusters with any number of layers in them.

In Figure 15(a) and Figure 15(b) , the points show the velocity profiles of the five-layer MS05c99e.dat structure at the final time T = 1.96, and the lines show the same profiles at T = 0. In Figure 15(a) , the velocities v are shown depending on the body numbers k, and in Figure 15(b) , depending on the relative radius r_rN2 = r/r_N2. At the initial time T = 0, the line segments mark the velocities of bodies in five layers ( Figure 15(a) ). They decrease stepwise from the first layer to the last fifth layer. After the interaction of bodies proceeding during 196 revolutions of first-layer bodies, the distribution of velocities no longer shows the initial stepping behavior. The velocities were distributed around the initial velocity profile. At the same time, most of the velocities retain the initial trend of decreasing the velocity magnitude with increasing the body number k.

The zero velocities on the horizontal axis mark the bodies that have merged with other bodies, and their velocities were therefore assigned zero values. Body 76, which is at a distance r_rN2 = 0.082, or r_r1 = 0.41 from the center of mass, has the highest velocity v = 14.33; i.e. it is located near the central body. All in all, there are eight bodies whose velocities exceed the limit of the graph v = 12.

The distribution of velocities over body distances r_rN2 ( Figure 15(b) ) has a more compact form. In the trajectory section r_rN2 ≤ 1, the velocities of bodies are distributed uniformly around the initial profile represented by segments. It should be noted that the initial velocity profile is discrete. It is represented by points at which the line breaks.

Consider the reasons for the deviation of velocities from the initial profile. At first the orbits are circles, and then they become ellipses. In the pericenters of the orbits, the bodies have high velocities, and in the apocenters they have lower velocities. In addition, for some bodies the semi-axes of their orbits become smaller, and the velocities become greater. For the other part of the bodies, on the contrary, the semi-axes increase, and the velocities decrease. For these two reasons, there appears a velocity spread relative to the initial profile. Evidently, the action of these two factors is symmetrical with respect to this profile.

According to formulas (12)-(13), the velocity of the bodies at the initial time is equal to

$v=\sqrt{v_r^2+v_t^2}=v_p\sqrt{2{\alpha }_1\left(1-R_p/r\right)+1}$, (38)

where the values of α₁, v_p, and R_p can be calculated using dependencies (4)-(9). For circular orbits with r = R_p, formula (38) gives the velocities of bodies at the initial time, and for arbitrary distances r it gives such velocities for all bodies. The above parameters depend on the mass of the bodies within the radius r, as well as on the number of bodies and their mass at the radius r (see formulas (6) and (7)). Therefore, expression (38) can be used to estimate the masses of stars in globular clusters from their velocities.

Outside the distances of the graph in Figure 15(b) , there is only one body 1434 with r_rN2 = 3.14. The velocity of this body, v = 1.572, falls onto the same dependence v(r_rN2) as for other bodies at distances r_rN2 close to 3.

The velocity distributions of the ten-layer structure over body numbers ( Figure 15(c) ) are more compact than those for the five-layer structure. The velocities are distributed more uniformly about the initial profile for layers 1 to 8. The distribution of velocities over distances ( Figure 15(d) ) is also more compact compared to the distribution over body numbers. The highest velocity v = 17.7 is exhibited by the body located at a distance of r_rN2 = 0.0227, or r_r1 = 0.227, so that this body is almost twice as close to the center as in the five-layer structure. A total of four bodies exceed the velocity bounds on the graph. The bounds of the graph over distances are exceeded for 30 bodies, with the largest distance r_rN2 = 60 belonging to body 1575. The velocity v = 9.735 of this body does not fall on the dependence v(r_rN2) for distances close to 3. Starting from r_rN2 = 6, the bodies have an ascending velocity profile. This profile can be represented as an average dependence

$v=-0.526+0.171\cdot r_{rN2}$ for $r_{rN2}>6$.(39)

These bodies leave the structure by moving along hyperbolic orbits.

The 15-layer structure has an even more compact distribution of bodies over their numbers ( Figure 15(e) ). It lasts up to the 14^th layer. The velocities here are distributed more evenly around the initial velocity profile.

The distribution of velocities over distances is also more compact ( Figure 15(f) ). However, the middle of this distribution noticeably deviates from the initial velocity profile in the region of the last layers. This difference is observed already for the 10-layer structure in Figure 15(d) . In the 15-layer structure, the highest velocity v = 16.9 is exhibited by a body located at a distance of r_rN2 = 0.022, or r_r1 = 0.33. The velocities of only two bodies exceed the velocity bounds in the graph. The bounds of the graph in terms of body distances are exceeded for about 30 bodies with the largest distance r_rN2 = 11.2 found for body 5336. Its velocity is v = 2.763. Starting from r_rN2 = 4.4, an increasing velocity profile is set:

$v=-0.964+0.333\cdot r_{rN2}$ for $r_{rN2}>4.4$(40)

The velocity profiles presented in Figure 15 for different structures depending on the numbers of bodies and on their distances are similar, but they differ in quantitative characteristics. These quantitative differences are seen from formulas (39)-(40) for the ascending sections of velocity profiles. From the analysis of velocity profiles over body numbers at the initial time ( Figure 15(a) , Figure 15(c) , Figure 15(e) ), it can be inferred that with an increase in the number of layers, the velocity difference between the layers decreases, and the v-profile asymptotically approaches a horizontal line. Moreover, in the 14^th layer ( Figure 15(e) ) the velocity becomes minimal, v = 2.524, and in the 15^th layer it already increases. This increase in body velocities can also be traced in the velocity profile over distances ( Figure 15(f) ) at r_rN2≈1 for the final time T = 0.779.

In order to elucidate the behavior of velocity in structures with a large number of layers, structures MS24c99b.dat with 24 layers ( Figure 16(c) ) and MS34c49b.dat with 34 layers ( Figure 16(d) ) were created. They have approximately the same number of bodies: N = 29701 and 29156, respectively. In the first structure, the number of bodies in the first layer is N_3,1 = 99, and in the second, N_3,1 = 49. That is why the second structure contains ten extra-layers at an approximately the same number of bodies N.

The images of the structures, obtained using the Galactica program, are shown in Figure 16 . They are projections onto the xy-plane of the structures

pre-rotated around the x-axis through an angle of 20˚. The bodies in the structures are shown as points, and the velocity vectors, as segments. In order not to obscure the images of bodies in the 24-layer and 34-layer structures, the velocity vectors are reduced. Condensation of points in these structures occurs in layers whose mid-planes coincide with the line of sight. In the 15-layer structure at the final time T = 0.779 ( Figure 16(b) ), the enlarged points represent bodies with large masses. Since the graphs show projections of velocities, the small size of segments can be for bodies whose velocity is close to the perpendicular to the xy-plane. Sets of bodies with velocities directed similarly can be constellations.

All structures in Figure 16 were obtained with Inx = 1, i.e. all layers in them were uniformly rotated around the x-axis. As it is evident from the images of the structures, with an increase in the number of layers their shape approaches the spherical shape: the last structure ( Figure 16(d) ) shows no difference from a sphere.

When analyzing the velocity profiles of these structures, it turned out that those profiles are similar, and they can be generalized as shown in Figure 17 . The velocity of bodies v_r1 is plotted along the vertical axis. This is the velocity normalized by the maximum velocity, or the body velocity in the first layer v₁. Plotted along the horizontal axis is the body distance r_r1 normalized by the semi-axis a₁ of the first layer. Numbers 1, 2, and 3 mark the velocity profiles of the 15-, 24-, and 36-layer structures. All those profiles form a general dependence. This dependence also determines the velocity profiles of the 5-layer and 10-layer structures. The latter velocity profiles are not shown in Figure 17 .

The orbits of all bodies in the structures under consideration are circles; that is why their velocities can be calculated using expression (9) with r = R_p and v = v_p. From formula (9), it follows that the velocity v is inversely proportional to $\sqrt{r}$ and, according to formulas (6) and (16), they vary in proportion to $\sqrt{m}$, where m is the mass of bodies inside the sphere of radius r. Therefore, with an increase in r, the velocity decreases unless the mass of the bodies increases so much that its influence becomes predominant.

In the ascending section of the graph in Figure 17 , the velocity varies non-linearly, and its average change obeys the law

$v_{r1}=0.333+3.582\cdot {10}^{-3}\cdot r_{r1}$ for $r_{r1}>14$.(41)

For structures 1, 2, and 3 shown in Figure 17 we have: v₁ = 6.590, 5.396, and 4.485, respectively. These velocities are given in dimensionless form. They change due to the fact that the dimensional velocities are multiplied by the velocity coefficient k_v, which, in accordance with formulas (26)-(27), can be determined as follows:

$k_v=1/\left(A_m{k}_t\right)$.(42)

The presented values of v₁ vary in proportion to k_v. It follows from here that the generalized velocity profile in Figure 17 is also valid in dimensional form.

The angular momenta of five structures, from the 5-layer MS05c99b structure to the 34-layer MS34c49b structure, are given in Table 2 for the initial time T = 0. This quantity is also called the kinetic momentum. The table shows the projections M_x, M_y and M_z of the angular momentum onto the coordinate axes x, y, z, as well as the absolute value of M. Based on these projections, the angle β₃ between the angular momentum vector M and the z-axis is calculated as follows:

${\beta }_3=arctg\frac{\sqrt{M_x^2+M_y^2}}{M}$. (43)

In addition, the total orbital momentum of all bodies M_us is given on the condition that the orbits of the bodies lie in the same plane. For circular orbits, the total orbital momentum is defined as follows:

$M_{us}={\displaystyle \underset{i=1}{\overset{N}{\sum }}{m}_i{v}_i{r}_i}$.(44)

The orbital moments of bodies vary from layer to layer. For example, for the

34-layer structure, the angular momentum of the central body is 2.19 × 10⁻³³, for the first-layer bodies it is 9.41 × 10⁻⁷, and for the 34^th-layer bodies, 1.46 × 10⁻⁵. It should be noted that the presence of the kinetic moment of the central body is due to its deviation from the center of mass of the structure.

As evident from Table 2 , the values of kinetic moments M and M_us increase with the number of layers in the structures. For example, on going from the 5-layer structure to the 34-layer structure the value of M_us increases by almost 10 times. The projections of moments M_x, M_y, and M_z also vary in a wide range, and they can be either positive or negative. The angular momentum vector M is least inclined to the z-axis for the MS05c99b structure: β₃ = 19.5º. In this structure, there are no rotations of layers about the x-axis. Therefore, the angular momentum of the structure M = 0.02608 differs little from the orbital momentum of all bodies M_us: their ratio is 0.9428.

In other structures, the layers are uniformly rotated around the x-axis. This has led to a significant change in the vector M. For instance, for the MS10c99b structure we have: β₃ = 94˚, i.e. the vector M lies practically in the xy-plane. In addition, it is located in the southern hemisphere. Therefore, the total rotation of bodies in the structure proceeds in clockwise direction. In other structures, the total rotation of bodies proceeds counterclockwise.

For the 15-layer structure, we have: β₃ = 87˚, i.e. the angular momentum vector lies even closer to the xy-plane, but it is located in the northern hemisphere. Therefore, the total rotation of bodies proceeds counterclockwise. In these two structures, the momentum ratio M/M_us is close to 0.5. In structures with a large number of layers, this ratio slightly increases, and the angle β₃ decreases, i.e. the vector M moves away from the xy-plane.

In Table 2 , the values of the kinetic momentum are given for the initial time. The largest relative change in the angular momentum at the final time, δM_z = 5.8 × 10⁻⁵, was found to occur in the MS15c49b structure. This means that the magnitudes of the momentum remain unchanged to four significant digits. This means that the values of the angular momentum given in Table 2 are also valid for other moments of time.

In Figure 11 , the change in the spins of a body and its thermal energy was considered when other bodies were successively attached to it. Consider now the spins and thermal energies of all bodies at the final stage of structure formation.

Figure 18(a) shows the spin projections S_px, S_py, S_pz and thermal energy E_t for all merged bodies of the 5-layer structure at moment T = 1.96. The numbers of merged bodies i_u are plotted along the horizontal axis. The spin projections S_px, S_py, and S_pz are marked with a cross, a rhombus, and a square, respectively, and the thermal energies are marked with straight segments of a thin line that breaks at body numbers. The break points and the symbols of spin projections falling onto one and the same vertical line with number i_u, belong to the body i_u.

All in all, there are N_u = 43 merged bodies. The spin projections ( Figure 18(a) ) vary over wide ranges, and the plots show no dominant direction. The spin modulus S_p changes from a minimum value of 2.86 × 10⁻¹⁰ to a maximum value of 2.71 × 10⁻⁹, those values differing by one order of magnitude. The average value of spin modulus S_ppm = 1.21 × 10⁻⁹ in the graph is shown with a dashed line.

The total value of the angular momentum of the entire 5-layer structure, i.e. its spin, is given in Table 2 as projections S_sx, S_sy, and S_sz. The total spin modulus is S_s = 1.21 × 10⁻⁸, i.e. it is exactly ten times greater than the average spin of an individual body S_sm. However, there are N_u bodies in total, and the ratio of the total spin to the sum of the spins of individual bodies is S_s/(N_u∙S_ppm) = 0.23. Thus,

the spins of individual bodies compensate each other to 77%.

As evident from Table 2 , the total spin of the system is inclined to the z-axis by an angle β₃ = 120.7˚, whereas the kinetic momentum of the system M has an inclination angle β₃ = 19.5˚. The ratio of these momenta is 4.66 × 10⁻⁷, i.e. the rotational momentum is a small fraction of the orbital angular momentum.

The thermal energies E_t of bodies in Figure 18 are reduced by a factor f = 1 × 10⁻⁶. For the 5-layer structure, these energies vary from a minimum value of 5.85 × 10⁻⁵ to a maximum value of 3.24 × 10⁻³, these values differing by 55 times. The average value of the thermal energy of one body is E_tm = 1.08 × 10⁻³, and the thermal energy of all bodies is E_ts = 0.0466.

In the 10-layer structure ( Figure 18(b) ), there are N_u = 248 merged bodies in total. In this system, there was one merging with the central body, so we will consider the parameters of collisions of peripheral bodies without a central body. The spins of these bodies vary from

2.80 × 10⁻¹¹ to 6.39 × 10⁻⁹ at the mean value of S_ppm = 8.287 × 10⁻¹⁰. In Figure 18(b) , the mean value is shown with a dashed line. Here again, the dominating spin direction cannot be identified on the spin projection graphs.

The spin of the central body, S_p1 = 2.13 × 10⁻⁷, is 257 times greater than the average spin of the peripheral body. Simultaneously, the total spin of the system ( Table 2 ) is S_s = 2.29 × 10⁻⁷. Therefore, the projection of the spin of the central body determines the direction of the entire spin of the system. The spin makes an angle β₃ = 56˚ with the z-axis, while the orbital momentum vector is inclined to this axis by an angle β₃ = 94˚. In this case, if one look from the end of the z-axis onto the xy-plane, then the total orbital motion will proceed in clockwise direction, and the total rotational motion, counterclockwise.

The thermal energy of peripheral bodies varies from 5.46 × 10⁻⁶ to 6.45 × 10⁻³, i.e. by 1180 times, with the average thermal energy of one body being equal to E_tpm = 9.44 × 10⁻⁴. The thermal energy of the central body is E_t1 = 0.762, and that of all bodies, E_ts = 0.995. Thus, the central body makes the main contribution to the thermal energy of the system.

The number of merged bodies in the 15-layer structure is N_u = 140 ( Figure 18(c) ). Here, four bodies have merged with the central body. The spin modules of peripheral bodies vary from 1.73 × 10⁻¹⁰ to 3.75 × 10⁻⁹ at an average value of S_ppm = 1.29 × 10⁻⁹. The S_ppm-value is shown in Figure 18(c) with a dashed line.

The spin of the central body is S_p1 = 8.99 × 10⁻⁷, i.e. it exceeds the spin of the peripheral body by 696 times. Simultaneously, the total spin of the system (see Table 2 ) is S_s = 9.07 × 10⁻⁷, i.e. it is almost completely due to the central body. The total spin vector and, consequently, the spin of the central body makes an angle β₃ = 120.7˚ with the z-axis. Thus, the total rotational motion with respect to the z-axis proceeds in clockwise direction, and the total orbital motion with angle β₃ = 87 proceeds in an almost vertical plane. The ratio of the total spin to the total angular momentum is 1.15 × 10⁻⁵ (see Table 2 ).

The thermal energy of peripheral bodies varies from 2.25 × 10⁻⁵ to 6.46 × 10⁻³ with an average value of E_tpm = 1.56 × 10⁻³. The thermal energy of the central body is E_t1 = 0.149, and that of all bodies is E_ts = 0.366. The energy E_t1 is five times lower than the thermal energy of the central body in the 10-layer structure, despite the fact that there occurred four collisions with the central body. This fact indicates that the collision in the 10-layer structure has occurred at a higher velocity of the peripheral body. In this structure, the spin of the central body is four times less than that in the 15-layer structure. This indicates that the collision in the former structure was more frontal, while the collisions in the latter structure were closer to those occurring along tangential lines.

In the images of structures at the final time ( Figure 4 , Figure 7 and Figure 12 ), we observe close arrangements of bodies to each other involving two, three or more bodies. Some of these arrangements may be due to the coincidence of their visual lines at a sufficiently large distance between the bodies. Another fraction may be due to the short-term convergence of bodies in visual images. However, there may be cases when the bodies form stable associations, that is, constellations. In order to identify constellations in the output files of the Galactica system, a Constns.for program was developed. The output file of Galactica contains the coordinates of bodies at time T. The output files are generated after a set number of integration steps Kl3. The Constns.for program determines the number of bodies with numbers i for which there are bodies with numbers k located at a distance R_ik ≤ d_max from body i. For each body i, one can find up to six bodies numbered k with their distances R_ik. The total number of all nearby bodies k is determined without any restriction, but their numbers and distances R_ik are not memorized.

Figure 19(a) shows the results of calculations performed using this program for the ten-layer structure at the final time T = 0.815 for d_max = 0.003. This value of d_max corresponds to the distance between the bodies on the line of their location at time T = 0 ( Figure 7(a) ). For choosing this distance, the distances between closely spaced bodies in Figure 7(b) were measured. This distance turned out to be close to 0.003.

Along the horizontal axis, Figure 19(a) shows the numbers of bodies i, and along the vertical line, the numbers of bodies k. All in all, N_is = 1032 bodies i were found among the total number of bodies in the structure, N = 5451. If body i has several close bodies k, this number of bodies is marked in the graph by the number of points along the vertical line of body i. The largest number of bodies i had one close body each, and the smallest, five bodies. There were two such bodies.

It should be noted that the number N_is = 1032 includes all bodies that have a close body. Therefore, if each body has one close body, then there will be 1032/2 = 516 constellations, and if each body has two close bodies, then there will be 1032/3 = 344 constellations, etc.

From Figure 19(a) , it is seen that for all bodies, starting from the body i = 1000, the number of close bodies k is concentrated on the diagonal line. In

Figure 19(b) , for bodies i = 4600 - 4800 the graph of nearby bodies is shown on an enlarged scale. The points show the location of bodies at time T = 0.815. It is seen from the graph that on the diagonal line the bodies k have numbers close to those of bodies i. As already noted, the close positions of the bodies determined at the time T = 0.815 may appear occasional. To make sure that these close positions are not occasional, but form constellations, we have to repeat such calculations for another point in time. The results of the calculation for the previous time T = 0.770 are shown with symbols “x”. Most of them coincide with the points for T = 0.815. Out of 41 cases, there are no matches in only three cases. Therefore, we can assume that in 38 cases of close locations, constellations appear. Of the 38 cases, three are different. Body 4746 has a nearby body 4750; body 4747 has two nearby bodies, 4750 and 5409; and body 4750 also has two nearby bodies, 4746 and 4747. This situation was also confirmed for T = 0.770. Therefore, the bodies 4746, 4747, 4750 and 5409 form a constellation of four bodies. It follows from here that in the region of bodies i = 4600 - 4800 there are 17 constellations formed by two bodies and one constellation formed by four bodies. In total, out of 200 bodies i in Figure 8(b) there are 18 constellations. Thus, the total number of constellations can be estimated as 18/200 = 0.09 of the total number of bodies. Then, there must be 0.09∙N = 491 constellations in the ten-layer structure. This estimate is close to the number 516 that was estimated previously from the number of encounters with one body. Therefore, we can assume that up to 9% of all bodies in the structure can form constellations.

The above-mentioned constellation 4746 - 5409 involving four bodies 4746, 4747, 4750 and 5409 is shown in Figure 20 . Its position at time T = 0.770 is marked with the velocity vectors of bodies 4746 and 5409. The position of this constellation at the final time T = 0.815 is also shown in Figure 7(b) . Evidently, during this time the constellation has moved on the sphere by about 90˚.

In this constellation, body 4747 has double mass, and the rest of the bodies have one mass m₁ each. When viewed on a larger scale, body 4747 is in the center, and the rest of the bodies surround it.

Bodies with a mass of 5m₁ do not form constellations. Most constellations are created with masses m₁. In Figure 19(a) , there are four cases of bodies i with mass 4m₁, 13 cases with mass 3m₁, and 76 cases with mass 2m₁. The number of constellations with these bodies will be slightly less, since several bodies with increased masses can enter one constellation.

In the upper right corner of Figure 19(b) , four bodies, 4776, 4781, 4785 and 4789, attract attention; one of those bodies, namely, body 4789, has a twice increased mass. It turned out that these bodies form two pairs: 4776 - 4789 and 4781 - 4785. As evident from Figure 20 , at time T = 0.770 these pairs occupy position in different places of the 10-layer structure. Based on 20 points, starting from the time T = 0.7167, relative trajectories of the bodies of interest were analyzed: that of body 4776 relative to body 4789 and that of body 4781 relative to body 4785. Figure 19(c) shows the relative trajectories of these bodies, i.e. 4776 and 4781, as projected onto the xy-plane. Relative coordinates are denoted as x_r and y_r. From the initial point I to the final point F, body 4781 makes a little more than three revolutions around body 4789. The average interval between the points is ΔT = 5.195 × 10⁻³. The orbit of body 4781 is an ellipse with semi-axis a = 1.69 × 10⁻³ and eccentricity e = 0.255. The orbital period is P = 0.0325.

The orbit of body 4776, from the initial point I to the final point F, is represented with intersecting segments, since the period of revolution of this

body is comparable with the interval ΔT between the points. The orbit of the body is an ellipse with axis a = 3.12 × 10⁻⁴ and eccentricity e = 0.513. The period of revolution of body 4776 is given by formula (8). It equals P = 3.15 × 10⁻³, a value ten times shorter than the period of revolution of body 4781. During this time interval, body 4776 makes 31 revolutions about body 4789. The position of these two constellations at time T = 0.770 is shown in Figure 20 , and at the final time T = 0.815, in Figure 7(b) . During this time interval, the motion of these constellations over the sphere has occurred through an angle of not more than 45˚.

Calculations were performed using the Constns.for program with a doubled distance d_max = 6 × 10⁻³. For time T = 0.815, 2650 bodies i that approached other bodies k to a distance d ≤ d_max were identified. In this case, the number of bodies k reached 19 for body i = 93. For time T = 0.770, there were 2715 bodies i with the largest number of bodies k equal to 17 for body i = 84. For this case, the selected first few bodies were not confirmed for the time T = 0.815. Over the interval of change of bodies i = 4600 - 4800 ( Figure 19(b) ), the cases of discrepancy noted earlier have coincided in this case. Thus, a twofold increase in the distance d_max makes it possible to identify constellations with more distant bodies and with a large number of such bodies.

Similar calculations were also performed for the 5- and 15-layer structures. For the five-layer structure, at the final time T = 1.96, at d_max = 3.16 × 10⁻³, 367 bodies i approaching other bodies k to a distance R_ik ≤ d_max were identified. The number of bodies k reached 5. The distribution of bodies on a graph similar to that of Figure 19(a) was more uniform, and there was no concentration of bodies along the diagonal line.

For the 15-layer structure with d_max = 5.3 × 10⁻³, at the final time T = 0.780, 1843 bodies i that approached other bodies k to a distance R_ik ≤ d_max were identified. The largest number of bodies k was 7. The distribution of bodies in a graph similar to Figure 19(a) was also more uniform, but with lower concentrations of bodies near the diagonal line. The diagonal line could also be traced, but it started from i = 2000.

As noted above, in the MLSpStr2.for program, files of structures with initial conditions are created in dimensionless form using parameters m_ss, A_m, and k_t [25] . The dimensionless parameters of the structures are indicated in Table 3 . Here, the number of bodies N varies from 1488 to 29,701; the mass of the central body m₀, from 0.8683 to 0.1427; the mass of peripheral bodies m₁, from 9.332 × 10⁻⁵ to 2.532 × 10⁻⁵; the radius of the central body Ra₀, from 2.171 × 10⁻⁵ to 3.965 × 10⁻⁵; the radii of peripheral bodies Ra₁, from 1.220 × 10⁻⁶ to 1.884 × 10⁻⁶; the semi-axis of the orbit of first-layer bodies a₁, from 0.0149 to 0.008587; and the semi-axis of the orbit of last-layer bodies a_N2, from 0.0652 to 0.2427. All structures have a dimensionless mass m_ss = 1 and the same period of revolution of first-layer bodies, P₁ = 0.01. The period of revolution of last-layer bodies P_N2 varies from 0.1046 to 0.7475.

As already noted, the dimensional semi-axes of these structures are identical, and the dimensional masses of peripheral bodies in the 5-, 10-, and 24-layer structures are also identical. In the 15- and 34-layer structures, the masses are twice as large. Yet, the dimensionless semi-axes and the masses of bodies are different in all structures. The passage to dimensionless quantities is related with the masses of structures, and those masses are different. Thus, the identical dimensional parameters in dimensionless form also become different.

The last column of Table 3 indicate the relative outer radius r_95rN2 of the structures whose evolution was studied. This radius was determined at the final time as the largest distance from the center of mass of 95% of the bodies, i.e. this is the size of the structure with 95% confidence. Evidently, for all structures the value of r_95rN2 is approximately the same, with an average value r_95rN2m = 1.662. Only 5% of all bodies are located at a distance from the center larger than r_95rN2m.

Based on the dimensionless parameters summarized in Table 3 , we calculate the dimensional parameters of five models of globular star clusters. We set the peripheral-body mass to half the mass of the Sun: m_1,m = 0.5 m_S. In the center of globular constellations, the density of stars can reach 100 to 1000 stars per cubic parsec [7] . With this in mind, for all models in dimensional form, we set the semi-axis of the orbit of first-layer bodies to a_1,m = 0.2 pc, where one parsec (pc) is 3.0856776 × 10¹⁶ m.

Taking into account the values of the dimensionless masses in Table 3 , the dimensional central-body mass will be

$m_{0,m}=m_{1,m}\cdot m_0/m_1$,(45)

and the dimensional mass of the entire structure,

$m_{ss,m}=m_{0,m}+\left(N-1\right)\cdot m_{1,m}$.(46)

Given the semi-axis of the orbit of first-layer bodies a_1,m, the geometric scale is

$A_{m,m}=a_{1,m}/a_1$.(47)

With the help of this scale, we find the semi-axis of the orbit of last-layer bodies a_N2,m = A_m,m·a_N2. Given the values of A_m,m and m_ss,m, formula (27) can be used to determine the time factor:

$k_{t.m}=\sqrt{G\cdot m_{ss,m}/A_{m,m}^3}$.(48)

The velocity coefficient k_v,m is calculated by formula (42). Using these coefficients, converting the angular momentum and energy to dimensionless form, we obtain expressions for their scale factors:

$k_M=\frac{k_{v,m}}{A_{m,m}{m}_{ss,m}}$; $k_E=\frac{{\left(k_{v,m}\right)}^2}{m_{ss,m}}$.(49)

With the help of scale factors, the transition from dimensionless quantities to dimensional ones is carried out. For example, the dimensional period of revolution of outer-layer bodies is calculated as follows: P_N2,m = P_N2/k_t,m.

Table 4 indicates the scale factors and dimensional values of globular clusters modeled with five multilayer structures. The body masses are given in solar masses; the semi-axes, in parsecs; and periods are in years. The last column gives the number of stars in a cubic parsec at the initial time. This value is defined as follows:

$n_{pk}=\frac{3N}{4\pi {\left(a_{N2,m}\right)}^3}$.(50)

The density of stars n_pk in Table 4 changes from 355 for the 5-layer structure to 22 for the 34-layer structure. Since these are average densities, these values correspond to the observed densities of stars in the center of globular star clusters.

The revolution periods of stars in the first layer P_1,m vary from 120 to 170 thousand years ( Table 4 ), and the revolution periods of stars in the last layer P_N2,m, from 1.2 to 12.7 million years. The periods of revolution of stars in constellations are much shorter than in the outer layer. For instance, in the ten-layer structure the period of revolution of body 4781 relative to body 4785 is 388 thousand years ( Figure 19(c) ), and that of body 4776 about body 4788 is 37.6 thousand years.

Taking into account the average value of r_95rN2m, the diameter of these globular star clusters D_g = 2 × 1.622 × r_95rN2m varies from 3.2 pc for the 5-layer structure to 22 pc for the 34-layer structure. The diameter of observed globular star clusters varies from 20 to 100 pc, and the number of stars in them, from 10⁴ to 10⁵ [7] . Evidently, the globular cluster models based on the 24- and 34-layer structures satisfy these parameters.

The shape of the observed globular star clusters may differ from spherical, so such clusters are characterized by ellipticity [2] . Ellipticity is defined as the difference between the largest and smallest diameters, divided by the largest diameter. In our Galaxy, the clusters NGC 7492, NGC 6144 and NGC 6273 have ellipticity of 0.24, 0.25 and 0.27, respectively [2] . The highest ellipticity of the simulated clusters, equal to 0.6, was exhibited by the 5-layer cluster. For other models, with an increase in the number of layers, the ellipticity approaches zero.

It is also noted [7] that deformation tails were found in the shape of all globular clusters for which high-quality optical images were obtained; in other words, there are deviations from spherical symmetry. Therefore, we may assume that the asymmetry of the 10-layer and 15-layer models of globular clusters in Figure 7(b) and Figure 12(b) is their quite acceptable shape.

In order to verify the results of the scale transition, the period of revolution of first-layer bodies was calculated from the dimensional parameters according to formula (8). In formula (8) for a circular orbit we have α₁ = −1, the pericenter radius is R_p = a₁, the pericenter velocity v_p can be calculated by formula (9), and the interaction parameter µ₁, by formula (6). Thus, the periods of revolution calculated up to the units of the fourth decimal place proved to be coincident with the periods calculated using the scaling approach.

The presented parameters of globular cluster models are of interest to scientists who develop cluster models based on an analysis of their brightness. These models are based on the Virial Theorem -U/T_kin = 2, where U is the potential energy of the cluster and T_kin is its kinetic energy. They are defined in a known manner; see for example equation (1.2) in [13] . Dimensionless values of T_kin and -U are given in Table 3 , from which it follows that the ratio -U/T_kin varies from 1.960 to 1.986, i.e. is close to 2.

The total mechanical energy E = T_kin + U in the process of evolution of the presented models towards the end slightly decreases in absolute value in the 5-layer structure by 0.02%, in the 10-layer structure by 1.2%, and in the 15-layer structure by 0.87%. In statistical modeling of clusters, one important scale is the virial radius $R_v=G\cdot m_{ss}^2/\left(\text{2}\left|U\right|\right)$ [5] . As can be seen from Table 3 , the value of R_v almost completely coincides with the outer radius a_N2 for a 10-layer structure; in other cases, R_v is greater or less than a_N2.

When considering the approaches and collisions of bodies in the three studied structures, the maximum spins and thermal energies were reported. In addition, the maximum values of velocities v_mx were indicated. With the help of scale factors, we find the dimensional values of the velocities, the periods of rotation of merged bodies P_rt, and the increase in their temperature Δt. For determining the radii of stars in globular star clusters, we specify their average density ρ_gk = ρ_S = 1.406 × 10³ kg/m³, where ρ_S is the average density of the Sun. Then, the dimensional radii of the bodies can be determined from their dimensional masses m_m:

$R{a}_m={\left(\frac{3m_m}{4\pi {\rho }_{gk}}\right)}^{1/3}$.(51)

The moments of inertia of bodies are defined as for a homogeneous globe:

$J_m=0.4m_m\cdot R{a}_m^2$.(52)

The dimensional spins and thermal energies are calculated using the scale factors

$S_{p,m}=S_p/k_M$; $E_{t,m}=E_t/k_E$.(53)

Then, the dimensional periods of rotation of stars are defined as follows:

$P_{rt,m}=2\pi J_m/S_{p,m}$.(54)

For calculating the heating temperature Δt of merged stars, we specify their specific heat equal to the specific heat of water C_t = 1.183 × 10³ J/(kg·deg). Then, the increase in the temperature of merged bodies will be:

$\Delta t=E_{t,m}/\left(m_m\cdot C_t\right)$.(55)

The results of calculations performed according to these formulas are summarized in Table 5 . The first five columns indicate the dimensionless spins, the thermal energies, and the maximum velocities. The values with subscript “0” refer to the central body, and those with subscript “1”, to peripheral bodies. For a peripheral body, the maximum values of spins and thermal energies are given. These bodies with the maximum parameters in the 5- and 15-layer structures have doubled masses, and in the 10-layer structure, the body mass is 5m₁. As already noted, bodies located near the center of structures have a maximum dimensionless velocity v_mx.

In the last five columns of Table 5 , the dimensional values are given. The periods of rotation of the central body P_rt0,m vary from 2 to 17 days, while the period of rotation of the peripheral body (P_rt1,m) reach thousandths of a day, which is equivalent to several minutes. The heating temperature of the central body Δt₀ varies from several hundred to two thousand degrees, while that of the peripheral body (Δt₁) reaches several tens of thousands of degrees. In the central region of the clusters, the highest velocity of star motion is equal to two tens of kilometers per second.

With the masses of bodies used in the models of globular star clusters, their relative radii do not coincide with the relative radii at which the problem of interaction of bodies (28) was solved. Their rotation periods and temperatures Δt depend on the body radii. To make sure that the values in Table 5 are reliable, the periods of rotation and temperatures were calculated for the initial dimensional radii of the bodies.

On average, the temperatures proved to be ten times higher. The temperatures are proportional to the squares of body velocities. At the initial parameters, the velocities are three times higher, which fact explains one order of magnitude higher temperatures.

At the initial parameters, the rotation periods were 60 times longer. The periods of rotation are inversely proportional to body velocities, and they vary in proportion to the squares of body radii. The radii of the bodies with the initial parameters are 15 times smaller. This explains the 60-fold increase in periods.

Thus, in order of magnitude, the increase in the temperature of stars and their periods of rotation in the models of globular star clusters presented in Table 5 adequately reflect these properties of stars.

The processes of merging of bodies depend on their radii. The connection of radii with body masses differs from the connection of distances with masses in the interaction Equation (28). That is why for more accurate modeling of body parameters during the merging of bodies, it is necessary to specify the radii of bodies corresponding to the simulated star system.

The rotation periods and temperatures of peripheral bodies presented in Table 5 are extreme. As it was shown in Section 8.3, the minimum values of spins and thermal energies are one or two orders of magnitude smaller. Therefore, on average, the temperatures Δt₁ for peripheral bodies can be expected to have several times lower, and the rotation periods P_rt1,m, several times larger values.

One of the main problems concerning such stellar associations as globular star clusters and galaxies is the central-body mass: how big should it be? For the problem of interaction of bodies of a plane axisymmetric structure [18] [33] , there is an exact solution in the absence of a central body, that is, for m₀ = 0. However, such a structure is unstable. As a result of numerical experiments with a single-layer spherical structure [20] [21] , it was found that such structures can be created with a relative central-body mass p_m0 < 0.95. Therefore, multilayer structures were created with certain margins at p_m0 = 0.99.

In a multilayer structure, the central-body mass, related to the mass of the entire structure, decreases with an increase in the number of layers from m₀ = 0.87 for the five-layer structure to m₀ = 0.14 for the 34-layer structure ( Table 3 ). However, the dimensional central-body mass m_0,m remains equal to thousands of solar masses ( Table 4 ). When modeling globular clusters, other researchers also obtain such masses of the central bodies [9] . However, in order to reduce the central-body mass, we will replace it with a model in the form of a multilayer structure similar to the above-considered structures. Let the mass of such a structure be equal to the central-body mass

$m_{ss3,m}=m_{0,m}$,(56)

and the outer semi-axis of the orbit is the fraction, equal to k_a1, of the semi-axis of the inner layer of the globular cluster:

$a_{N23,m}=k_{a1}\cdot a_{1,m}$(57)

Here, the subscript “3” denotes the parameters of the model of the first central body. Subsequently, there will be more models of other central bodies. Two conditions (56)-(57) uniquely determine the parameters of this model.

Based on the parameters of the multilayer structures considered, central-body models were calculated. For a globular cluster model with N₂ layers, a central-body model with the same number of layers was created. Table 6 summarizes the parameters of the models of the 1st central body with k_a1 = 0.5, i.e. the outer size of the central-body model is equal to half the inner size of the multilayer structure. Given in Table 6 are the central-body mass m_03,m and the peripheral-body mass m_13,m (in solar masses), and the period of revolution of inner-layer bodies P_13,m and the period of revolution of outer-layer bodies P_N23,m (in sidereal years). For the five-layer model, the central-body mass m_0,m has decreased from 4901 ( Table 4 ) down to 4255 solar masses ( Table 6 ), i.e., this is an insignificant decrease. For the 34-layer structure, the central-body mass has decreased from 2426 to 346.1 solar masses, that is, by a factor of 7. In this structure, the peripheral bodies have a mass of m_1,m = 0.5 solar masses, and in the central-body model, the peripheral-body mass is m_13,m = 0.0713 solar masses. That is, also 7 times less. The period of revolution of bodies on the inner layer is P_13,m = 800.6 years, and on the outer layer, P_N23,m = 59,840 years. These periods are 212 times shorter than the corresponding periods in the globular cluster model ( Table 4 ).

This central-body model models the central body of a multilayer structure, i.e. this is the model of the first central body. The central-body model also has a central body, i.e. this is a second central body. This body can also be substituted with a multilayer model according to conditions similar to (56)-(57). Table 6 summarizes the parameters of the model of the 2^nd central body; those parameters are marked with the subscript “4”. The mass of the 2^nd central body m_04,m for its 34-layer model has decreased most significantly, namely, also by 7 times with respect to m_03,m. The peripheral-body mass m_14,m has decreased by the same factor. The periods of revolution P_14,m and P_N24,m have also decreased by 212 times compared to the corresponding periods of the model of the first central body.

In these central-body models with two 34-layer structures, the central-body mass is 49.37 solar masses. If we additionally use the model of a 3rd central body, then the central-body mass will be further decreased to seven solar masses. In this case, the central body will be surrounded by 34 layers of revolving bodies with a mass equal to one and a half Jupiter masses. On the inner layer, the period of revolution of bodies will be 6.5 days, and on the 34^th layer, 1.3 years. There will be 29,155 peripheral bodies in total. Then, the same number of bodies with a mass equal to 0.0102 solar mass will be located in 34 layers with periods of revolution of 3.78 years in the 35^th layer and 282.6 years in the 68^th layer. The next 29,155 bodies with a mass of 0.0713 solar mass are located in 34 layers with periods of revolution of 800.6 years in the 69^th layer and 59.84 thousand years in the 102nd layer. The 34 outer layers of this globular cluster contain 29,155 stars with a mass of 0.5 solar mass and periods of revolution of 169.6 thousand years in layer 103 and 12.68 million years in layer 136. All in all, this cluster contains 116,620 peripheral bodies revolving around the central body with a mass equal to 7 solar masses. The radius of the first layer is 0.13 AU, and that of the 136^th layer, 6.8 pc, or 2.06 × 10⁵ AU. As it was already noted, the period of revolution of first-layer bodies here is 6.5 days, and that of bodies in the last layer, 12.68 million years.

Bodies in the inner layer with a mass equal to 1.5 Jupiter masses are not the sources of radiation, so they will shield the central body from light. In layers, starting from the 35^th layer, the mass of bodies will increase, and their luminosity will appear first in the infrared range, then in the red, and then in other spectral ranges.

The scaling of the results of the performed studies makes it possible to foresee such a hierarchically folded multilayer structure of a globular star cluster with a moderate central-body mass. The high velocities of bodies and their close arrangement on the inner layers can contribute to their more frequent collisions than in the structures considered; this circumstance can facilitate the destruction of such structures. Therefore, in order to make sure that such star clusters are possible, it is necessary to create their models and study their evolution. Such models can be created based on the multilayer structures whose evolution from the initially organized state to the steady state has already taken place, for example, based on the 15-layer MS15c49b.dat structure at the time T = 0.779. Then, the study of the evolution of such globular clusters will take much less time.

Unlike in the 34-layer structure, for the 15-layer structure it will be necessary to create not three but several more central-body models. For example, with seven such models, the peripheral-body mass on the inner layer will be equal to 2.2 Jupiter masses. In this case, the period of revolution of such a body will be eight orders of magnitude shorter, which raises doubts about the stability of such a structure.

With three central-body models, the peripheral-body mass in the inner layer will be 51.4 Jupiter masses, its semi-axis will be 1.96 × 10⁻² pc, or 4047 AU, and the period will be 9.13 × 10⁻² years, or 33.4 days. In this case, the central-body mass will be equal to 238 solar masses, i.e. a value ten times less than without the central-body model ( Table 4 ). The globular cluster will contain 23,465 bodies, and the parameters of its 15 outer layers will be such as indicated in Table 4 for the 15-layer structure. Seemingly, such a hierarchical model of a globular cluster will be stable.