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Gravitation is one of the central forces playing an important role in formation of natural systems like galaxies and planets. Gravitational forces between particles of a gaseous cloud transform the cloud into spherical shells and disks of higher density during gravitational contraction. The density can reach that of a solid body. The theoretical model was tested to model the formation of a spiral galaxy and Saturn. The formations of a spiral galaxy and Saturn and its disk are simulated using a novel N-body self-gravitational model. It is demonstrated that the formation of the spirals of the galaxy and disk of the planet is the result of gravitational contraction of a slowly rotated particle cloud that has a shape of slightly deformed sphere for Saturn and ellipsoid for the spiral galaxy. For Saturn, the sphere was flattened by a coefficient of 0.8 along the axis of rotation. During the gravitational contraction, the major part of the cloud transformed into a planet and a minor part transformed into a disk. The thin structured disk is a result of the electromagnetic interaction in which the magnetic forces acting on charged particles of the cloud originate from the core of the planet.

We simulate the formation of spiral galaxies and disk planets and their inner structures using a novel N-body model. Particularly, the ringed planets are especially interesting. In our numerical approach, the dynamics of selfgravitating N-body systems is considered using Newtonian gravitational interaction [_{eff} = 570 K using the initial radius 1 × 10^{9} m and the mass of a proton. The corresponding velocity of a proton is about 4000 m/sec. One can suppose that the local clouds are expanding while the whole system is contracting. The temperature of the local cloud will decrease and proton’s velocity will decrease as well. The contraction time obtained from the simulation is of the order of 1 × 10^{5} sec. During this time, if the proton moves with average velocity of 1000 m/sec., it will travel the distance of 1 × 10^{8} m that is about one tenth of the initial radius of the N-body sphere. Due to random directions of local thermal movement, not many protons escape from the system.

This assumption allows considering systems much lighter than the Sun. It is supposed that the i^{th} body of the N-body cloud had a mass m_{i} and coordinated x_{i}, y_{i}, z_{i}. Then the total mass is given by

Although one can associate the mass M with the average density M = r_{av}V, this density used in the Jeans condition is different from the density obtained in the present model

The stability of the system is revealed during the simulation when the relative positions between the bodies change very slowly at the beginning of the process during almost one third of the contraction time.

A new simulation program is designed for the modeling of the gravitational contraction of the self-gravitating N-body system. The system has a shape of the sphere slightly flattened along the axis of rotation. Because the mass of the cloud is given by 3N d-functions, the traditional codes are difficult to apply to this system. We use a direct implementation of equations of motion combining two mechanisms: the Newtonian gravitational interaction and conservation of angular momentum. The algorithm is based on calculations of the gravitational accelerations of all bodies and their displacements using the gravitational law in its original differential form

. In calculations of the gravitational masses, positions of all bodies are used. The rotational velocities w_{i} are obtained by applying the Kepler’s second law for small angles given by. During gravitational attraction, the velocities of the bodies increase towards the center of mass. The total velocity of each body is a vectorial sum of the radial and tangential (orbital) velocities. An increase of the radial velocity results in an increase of the angular velocity calculated from the Kepler’s formula. As a result, the remote particles will rotate slower than those situated closer to the center. Due to non-spherical geometry of the cloud, the disk will develop at the edges of the cloud in the rotation plane. In addition, rotational movement of the system will lead to its shrinking along z-axis and flattening of the planet. The charged particles of the faster rotating core of the planet will establish the magnetic field. This magnetic field will interact with the charged particles of the disk and, finally, this interaction results in formation of Saturn’s structured disk.

It is assumed that prior to its present form the Saturnlike planet had consisted of a slightly deformed spherical cloud of particles. The number of particles in gravitational systems may be very large and ill-defined and, therefore, to simplify the model we combine neighboring particles into clusters and consider the gravitational interaction between these clusters resulting in self-contraction of the cloud. One can imagine that two particles situated near to each other will have the same gravitational aceleration due to interaction with all the other particles. Therefore, by combining them into cluster means that we consider the movement of their center of mass. The gravitational law for a cluster can be written in the form

where G is the universal gravitational constant , M the gravitational mass, r_{av} an average density, r the relative position between the cluster and the center of mass of the whole system, P is the internal pressure, S_{eff} an effective cross section and m the mass of the cluster. The initial condition comprises randomly distributed positions, r, of the clusters. Their speeds are zero and the average density r_{av}(t = 0) = r_{0}. Suppose that the time steps are small enough so that during one time step the average density does not change. Then the solution of the differential Eq.3 is given by

At the beginning of the process the cloud contracts without changing its structure. Suppose that the internal pressure is very small at the beginning of the process. Then Eq.3 is a simple harmonic oscillator with the frequency given by. The speed for, being a linear function of r at a particular time. At small internal pressure the average density r_{av} is given by

This density does not depend on the position, r, and is therefore constant within the volume of the cloud.

Consider Eq.3 when speeds are not zero. By substituting r = Acos(wt) + Bsin(wt)+ pressure term into Eq.3 we get the solution at A = r_{1}, B = –v_{1}/w and w^{2} = 4p/3Gr_{av}, here r_{1} and v_{1} are position and the speed of the cluster at time t_{1}. The following phenomenon can be deduced from this result: Consider any two clusters labelled i and j situated close to each other. Then there may exist a time d t_{ij} after which their positions from the center become equal i.e.

dt_{ij} depends on neither r_{1} nor r_{2} , but only on the derivative dv/dr. This means that more than two clusters can reach the same radius and this process will take place in a whole volume resulting in increasing density at places in which dv/dr satisfies the equation

Then the local density will vary leading to formation of structured spherical layers in accordance with the equation

where v is the speed and r the position of the layer from the center of mass. One can expect that the amplitude of the oscillation of the local density would increase towards the center of mass because there are fewer clusters there and their speeds are smaller.

The rotational movement is described using Kepler’s second law. Due to small time intervals after which all velocities are recalculated, the Kepler’s law can be applied to small angles

where w_{i}_{1} and w_{i}_{2} are rotational velocities separated by one time step. The associated additional displacement is calculated adding to a full displacement of the cluster.

There arises a question how angular momenta of the clusters relate to each other? One can expect that clusters situated closer to the center of mass will have different angular velocities due to strong dissipation process. For remote clusters, the gravitational masses will differ very little because most of the clusters are concentrated closer to the center. Therefore, for a remote cluster one can rewrite Eq.9 using Eq.4 as

where Dt is the travelling time from r_{i}_{1} to r_{i}_{2}. This ratio does not depend on r_{i}. For the same travelling times other remote clusters will have the same relation leading us to a conclusion that

The meaning of this equation is that during the gravitational contraction the angular momenta of the remote clusters equalize. Then there exists a curve w(r) which fit all the values w_{i}(r_{i}) of remote clusters. One can expect that this curve terminates on the surface of the planet. Then Eq.11 can be written as

here r_{s} is the planet’s radius and w_{S} its angular velocity. These parameters are known for Saturn and, therefore, this equation can be used to get angular velocities of the remote clusters during planet’s formation. We would notice that the final angular velocities will change dramatically from Eq.12 due to the electromagnetic interaction mechanism described below. In addition to gravitational contraction of the cloud, the disk itself shrinks along the z-axis. This gravitational shrinking is almost uniform resulting in a very thin disk. First estimations of the vertical thickness of the disk based on the photometric measurements yielded a thickness of one km [5,6]. Then photo-polarimeter experiments revealed that the thickness is 200 m [

The magnetic forces play a very important role in creation of the structured disk. This mechanism takes place at final stage of the planet’s formation. The magnetic field is established by the rotating planet’s core. In the equatorial plane, the plane of the disk, the angle between the velocity vector and magnetic field is p/2 and therefore the magnetic force will be given simply by F_{m} = qvB where q is the charge of a particle. The additional increase of the orbital velocity is given by

where a is a turning angle. The magnetic field needed for turning the particle onto the orbit, corresponding to the angle alpha of p/2, is obtained from Eq.13

It is remarkable that the magnetic field does not depend on the velocity. It means that all similar charged particles of equal masses situated at the same position from the center will turn on their orbits during the same time. A typical gravitational contraction time of the final stage when the system is already separated into two subsystems, planet and disk-forming cloud, is about 10,000 seconds. One can estimate values of the magnetic field needed to turn particles on to their orbits. For electron the magnetic field is 0.9 × 10^{–15} T. For ionized hydrogen atom it will be 1.5 × 10^{–12} T. How many charged particles have to rotate to produce that magnetic field? To simplify the problem, consider only one circular loop of radius R inside the planet. Then the magnetic field B outside the planet at equatorial plane is given by

where m_{L} is the magnetic dipole moment, r is the position from the center, q is the charge of a particle, N number of the particles and m_{0} permeability of free space m_{0} = 4p × 10^{7} Henry/m. For an orbit of radius R = 2 × 10^{7} m, the angular velocity estimated from Eq.12 w =1 × 10^{–3} rad/sec. Using the value of elementary charge q = 1.6 × 10^{–19} C, the magnetic field B = 1 × 10^{–10} T and mass of hydrogen atom m_{H} = 1.66 × 10^{–27} kg, the calculated number N is 2 × 10^{29} and the total mass of N particles is as small as around 300 kg. This estimation suggests that during the gravitational contraction, a slow rotation of a large number of charged particles produces a sufficient magnetic field to enable formation of the disk. Probably, the magnetic field at that stage is much stronger.

One can conclude that electrons will form a ring at earlier times and much further from the planet than hydrogen atoms. The electron’s belt should rotate in opposite direction to the main disk. The ring’s structure observed on Saturn’s disk can be related with different atomic structure. The magnetic force results in separate different rings associated with different masses of the particles and different positions at the time when the magnetic field “switched on”. The disk is decomposed into light and heavy elements during electro-magnetic interaction. The heavier particles will establish rings closer to the planet. Very heavy particles will be not affected by the magnetic field. Therefore, these particles will continue their movement towards the planet remaining spokes-like structures on the disk. When approaching the planet the particles interact with each other and the effective cross section increases resulting in widening the spokes towards the planet’s surface. This is clearly seen in the images of the B-ring. The gaps between the rings indicate the differences in the atomic masses. This can be possible if chemical composition of the particles is simple and particles consist of few atoms. The mass of the disk, in accordance with this mechanism, is much smaller than the mass of the planet. This correlates well with the observational data. The elliptical orbits observed on the disk support this model because the elliptical orbits may originate only if the particles change their trajectories from free fall.

To estimate the temperature of the system we consider each cluster as a volume of particles and correlate the thermal energy with the distribution of the kinetic energy of the system [

where m is the mass of a particle, is the average density, r is the local density, is the average squared velocity of the layer.

To estimate the internal pressure and its influence on the disk structure one can write an equation for the work of the force F_{P} associated with the pressure acting on a particle. The work of this force will be equal to the change of the kinetic energy of the particle.

where S_{eff} is a cross section of the particle, v_{i}_{+} the velocity of the particle at position r_{i} without pressure and v_{i} the velocity of the particle at the same position under pressure. The adiabatic process is described by the equation or

The values P_{0}, V_{0} and r_{0} will be referred to the planet’s surface. Suppose that at a position r_{A} the pressure and gravitation compensate each other. Then

being M a mass of the planet and m particle mass. Then using Eqs.18 and 19 the surface pressure can be written as

The integral Eq.17 can be transformed as follows

Then the actual velocity v_{i} is given

The velocity at the equilibrium state

The value of g is calculated using a well-known relation g = C_{P}/C_{V}. One can expect complex molecular structures with a big effective number of degrees of freedom including vibration. Then a model of the bent triatomic molecule can be applied. For this molecule type; C_{P} = 14R/2; g = 7/6. Some examples of this type of molecules are H_{2}O, ozone O_{3}, SO_{2}, H_{2}S etc. If the number of degrees of freedom is very large or infinite then

If there are different types of particles then the different positions are obtained using the condition Eq.20

This equation implies the formation of structured disk. The radius depends on both parameters particle mass m_{i} and its effective cross section S_{eff i}. During the formation of the planet the adiabatic coefficient g can change also starting form 1.7 for one-atom particles down to 1 for many atom particles. The structure of the disk associated with the variation of g depends on the number of degrees of freedom. Then using g =L_{1}/L_{2} where L_{1} and L_{2} are integers one can write Eq.25 as

This equation explains the formation of numerous ringlets in the disk of Saturn due to variations of L_{1} and L_{2}. Because these values are integers, the discrete structure is obtained. Comparison of the contributions of the gravitational attraction force and the force associated with the pressure reveals dramatic decrease of the pressure at large positions due to its dependence F_{p}µ r^{–}^{3}^{g} whiles the gravitational force F µ r^{–}^{2}.

A special N-body simulation computer program has been designed for continuous calculation of the position, speed and acceleration of each cluster during gravitational contraction. The particles are randomly distributed within a deformed spherical volume at the initial time. The deformed shape is obtained by flattening the sphere by a coefficient 0.8 along z-axis, the axis of rotation.

The whole time range consists of short periods. During each period of time the clusters have constant accelerations that are different for different clusters. At the end of each period the acceleration of every cluster is re-calculated taking into account the new positions of the other clusters. In addition, rotational velocities are calculated using Eq.9 and its contribution is added into final displacements. In this model we use hydrogen atoms because it is believed that Saturn consists mostly of hydrogen. The total number of clusters used is 2500, having equal masses. These clusters have a total mass of 5.68 × 10^{26} kg that is the round mass of Saturn.

The process of gravitational contraction can be divided in three stages.

1) The First stage is the gravitational contraction of the slowly rotating slightly flattened large spherical cloud. This process takes about two thirds of the total contraction time from the beginning during approximately 1 × 10^{5} seconds. The cloud decreases in size without significant change of its geometry. The dynamics of the cloud at this stage are revealed in

2) During second stage the cloud separates into the planet’s cloud and the disk. During this stage the structure of the cloud changes and the disk starts to form. The second stage is lasting from 1 × 10^{5} to 1.45 × 10^{5} seconds demonstrated in

3) ^{5} to 1.5 × 10^{5} seconds. At this stage, the planet approaches its present geometry and dynamics. The disk has still a relatively large mass of the order of 1 × 10^{24} kg and contracts as well. But, the processes in the planet’s cloud occur much faster than the contraction of the disk. The magnetic field is established from the rotating core of the planet. The radiation coming from the planet ionizes atoms in the disk’s cloud. The magnetic forces acting on charged particles turn the particles onto round orbits. The difference in masses and charges of different ionized particles will result in formation of structured rings. Heavy particles continue their movement further towards the planet remaining traces on the disk known as spokes. The mass of the disk decreases dramatically because heavy particles move towards the planet. The stripes on the surface of the planet may be

associated with this process as well. Finally, the disk becomes thinner and lighter. There is a region of very low density between the planet and the disk where the internal pressure dominates the gravitation. Comparison of the velocities of the simulation and observed orbital velocities of the disk reveal a perfect correlation. The compared values are collected in

The inner structure of the planet is a complex composition of spherical shells separated by regions of very low density. Figures 4 and 5 reveal how randomly distributed clusters combine into spherical shells during gravitational contraction. The outer shell of Saturn has a spherical shape but is open at the top and bottom. The equatorial radius is about 6 × 10^{7} m and that along the z-axis 5 × 10^{7} m. This agrees very well with the observed values. At final stage, around 5000 seconds, the dynamics of the formation process is most dramatic. The central region of the planet is almost empty at 1.45 × 10^{5} seconds and during the last 5000 seconds it became filled with very fast moving bodies. Some of the bodies go through the center of mass of the planet and move away. Most of them will be trapped in the outer shells but some of them can escape from the planet through the equatorial strip and form the orbiting satellites. Therefore, the information about Saturn’s satellites may reveal the information about the inner structure of the planet.

The surface of the planet is very rough, hundreds of km in height variations. The relief of the planet changes dramatically from middle altitudes to the equator. At the equatorial strip the height variations are as large as 4000 km and there are pillar-like structures of enormous height here. The width of this strip is around 12000 km.

The time development of the density of the planet is plotted in Figures 6-8. The initial density is very low, about 0.5 kg/m^{3}. It starts to grow during gravitational contraction and a number of peaks can be seen after 1 × 10^{5} seconds corresponding to appearing of spherical shells. The density at peaks grows significantly to few thousands of kg/m^{3} at final stage and varies between 1000 kg/m^{3} and 4000 kg/m^{3} and increases towards the center of the planet reaching almost 4 × 10^{4} kg/m^{3} at around 1.0 × 10^{7} m from the center. The density of the outer shell is around 3000 kg/m^{3}. This suggests that the surface of Saturn is a solid-state material. However, the central region of the planet has a very low density. There are also regions of very low density between the shells. There is a post-formation process. This is a long-term process of electro-magnetic interaction between charged particles. The electrostatic interaction between positively charged particles and electrons may result in neutralization of the particles.

The dynamics of the formation process can be seen in Figures 9 and 10 where velocities of all clusters versus position are shown for different times. The observed orbital velocities of Saturn’s rings and satellites are also included combined in ^{4} m/sec after 1.5 × 10^{5} seconds that is very close to the observed value of 3.5 × 10^{4} m/sec. The small difference is present because some of the clusters of the disk did not approach the planet yet. When those clusters fall onto the planet the mass of the planet increases and the escape velocity increases a little as well. In

It is interesting that during the final stage, around 5000 seconds, of the gravitational contraction the central region of the planet is filled with bodies whose velocities vary from near to zero up to 25 km/s. The bodies have a high density and their velocities are enough to escape from the planet. There are natural obstacles for them, the outer shells. During this escape the bodies can be trapped within the shells but some of them can go through. Those bodies that came through the shells became the Saturn’s satellites. This also explains why the satellites are different in chemical composition, density etc. They initially formed in different shells inside Saturn. Therefore, the knowledge about the satellites may help to understand the atomic structure of the interior shells of Saturn.

The rotational dynamics is presented in ^{–7} rad/s, the clusters of the disk approach the curve given by Eq.12 during the gravitational attraction, revealing the equalizing of their angular momenta. The clusters of the planet’s cloud situated closer to the center have a variety of angular velocities revealing a strong dissipation of the kinetic energy and a rise of the temperature.

For the calculation of the temperature each cluster is considered as a volume of separate atoms. The process of

the gravitational contraction is an adiabatic one. The temperature of the contracting cloud is calculated using Eq.16 and the results are shown in ^{6} K in the central region of the planet at final stage.

For simulation of formation of a spiral galaxy, we assume that initially the elliptical E-type shape of the galaxy is obtained by proportional shrinking the spheroid along yand z-axes by 0.8, the most frequently observed apparent axis ratio [15-18]. Initially the particles are randomly distributed within this volume. The radial velocities of all particles are zero. The galaxy has slow rotation on z-axis. The number of the particles in the galaxy N = 2000. In addition, 300 particles are created outside the body of the galaxy to study the dynamics of the process in more detail. The galaxy has the total mass of 1 × 10^{43} kg, a typical galaxy’s mass. The particles have equal masses m = 5 × 10^{39} kg. The starting angular velocity was of the order of 1 × 10^{–15} rad/s and the corresponding initial period of rotation T = 2p/w was about

two hundred million years.

^{15} seconds which is a little smaller than the initial period of rotation. The formation of the spiral can be divided in three stages. The first stage lasts from the beginning of the contraction until the formation of two different geometrical shapes, the central part and two regions apart of it, after 3.35 × 10^{15} sec. (^{15} sec. and 3.90 × 10^{15}. At the end of this stage the spiral is formed (

The comparison of the photo images of these type spiral galaxies such as NGC 2997, NGC 157, NGC1566, NGC 300 and others, reveals an excellent similarity of the shape of the arms and the core shown in

The formation of a spiral galaxy, Saturn, its disk and interior, along with dynamical properties is simulated using a novel gravitational N-body model for system of particles and the computer simulation program developed by the author. The self-gravitating cloud of particles contracts in accordance with Newtonian mechanics where interaction between all particles is considered. Particularly, the model considers an extremely large number of particles comprising the gaseous cloud. The rotation of the cloud is described by Kepler’s second law applied to a short periods of time as a result of the conservation of angular momentum. The simulation results show that gravitational contraction produces complex structures inside planets and stars. For disk planets like Saturn, the model predicts the formation of structured disk. Both disk and interior are complex structures. The formation of the structured disk is incorporated with the interaction between the magnetic field and the charged

particles. The planet consists of several spherical shells having high density and the outer shell has the density of a solid-state material.

The theoretical model and computer simulation predict the shape of spirals of the spiral galaxies. The dynamics of rotating elliptical galaxy and its transformation to a Seyfert spiral galaxy is modeled. The two arms and core of the spiral are the result of the gravitational interaction and differential rotation. The results of the simulation show that the type of the spiral galaxy is determined mainly by the initial shape of the galaxy and its initial rotation. The computer simulation reveals that a spiral shaped galaxy is a result of the transformation of a cluster of non-spherical symmetry. In addition, the spiral can form if the contraction time is larger than the period of rotation of the galaxy.