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4. Planets         

4.2. Quantization of Orbits of Planets            

           On the analogy of the Edied's theory let us assume, that the Sun in the initial moment was a giant star and admit the presence of a cavity inside it. Then we shall also assume that under the action of the gravitational forces the hollow Sun was in the state of the elastic contraction. (For more details see the chapter "The Sun".) Then such system can be described by the differential second-kind equation by analogy with elastic hollow spheres considered in classical mechanics under external regular pressure. If we presume that the external regular pressure functions as the gravitational interaction between the particles of the shell of the Sun, then the differential equation has the solutions as the regular motion of a body, its rotation about the axis and the motion of waves in the shell. All these motions are proper to the Sun. In the last case the potential energy of the elastic gravitational contraction passes into the kinetic energy of the continuous motion of waves along the shell, which, in turn, is transmitted to planets by means of gravitational interactions. In this case any disturbance on the internal equatorial surface of the Sun can generate a soliton – a drop of a substance, which will be a germ of a future planet. The birth of planets by the Sun is shown in Fig.4.2.



Fig.4.2. Birth of Planets by the Sun 
At contraction of the Sun's shell this germ saves the impulse of motion and remains on the fixed orbit. In this case the gravitational potential of the planet in relation to the internal surface of the Sun's shell undergoes the jump of 4, similarly to the jump of the potential at transition through a double electrostatic or a double gravitational layer.
The kinetic energy of the Sun shown as a rotating sphere with the mass concentrated mainly in the shell, is derived by the following conventional way for a hollow sphere:

           Wk = ( M /  3 = 2.64 10 36 J,
            wherein is the equatorial rotation rate of the surface of the Sun.
The kinetic energy of motion of all planets of the solar system without taking into account the kinetic energies of rotation of planets around their own axes and the energy of motion of Pluto is:

          

J,
            wherein  Mn is the mass of the n-planet,   vn is the orbital velocity of the n-planet.
Let's admit that at the birth of a planet and the transition of its germ through the solar shell there occurs a jump of the gravitational potential of 4. Equating the above-stated equations in view of it jump, we obtain the following expression for the law of conservation of the kinetic energy in the solar system:


          

.
At calculation under this formula the error is 5.4 %.

 
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  > Quantization of Orbits of Planets  (continuation) Page 2

The law of conservation energy of the solar system shows that our solar system could not collide with other stars after the moment of its birth and it has no other substantially large planets except for the known eight planets. Further it will be shown that the ninth planet the Pluto is a small fraction of the eighth planet the Neptune and it can be neglected.
The law of conservation of the full moment of momentum in the solar system without any arguments follows from the following expression:

          

,
             wherein   n = 1, 2, 3, 4, 5, 6, 7, 8,   m = 0, 0, 0, 0, 1, 2, 3, 4.

The calculation according to this formula gives an error of 1.6 %.
Omitting the intermediate calculations, we shall determine the main parameters of orbits of the planets. We shall present the Sun as the plasma sphere consisting of rotating protons and electrons. We shall assume that their electrostatic field rotates together with them. We shall determine the limit distances (relative to the center of the Sun), at which the velocity of the end of the position the radius-vector of the electrostatic field surrounding the proton will be equal to the speed of light. We shall assume, that the minimum velocity of motion of a proton relative plasma cannot be less than vp 4c . Then the limit frequency of the precession of a proton about its axis is:

          

Hz ,

            wherein 


 is a maximum accessible de Brogile wavelength of a proton,
             h  is the Plank's constant,   m p is the mass of a proton,   с  is the speed of light,
             = 1 /  137.036  is the fine structure constant.

From here we shall define a limiting radius-vector of the Coulombian field of a proton:

          

.
The maximum value of the major semiaxis of the orbit of the first planet from the Sun – the Mercury gives R1* = 5.79110 10 m. Fig.4.3. shows the orbit of Mercury defined by the radius-vector of the Coulombian field of a proton. The difference between the experimental data and the calculated value gives 0.086 %. Hence, we shall take this value as the first point – the beginning of the scale of measurement of the orbits of planets.



Fig.4.3. The Orbit of Mercury .
The scale of mean orbital velocities of planets also begins with the value of the velocity of the Mercury and makes:
           v1 = 3  2c  = 47.89307 km/s.
This value differs from the experimentally measured value of the mean orbital velocity of the Mercury of v1* = 47.89 km/s only by 0.015 %, that it is considered the absolute coincidence for astronomical measurements.

 
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Now we shall proceed to the electronic component of the Sun. It we assume that the electron also rotates together with the electrostatic field and as it is more light, than the proton, its speed limit does not exceed  ve =  3c ,  the limiting frequency of precession about the axis will make:

          

 Hz ,

             wherein 


 is generalized extreme accessible value of De Brogile wavelengths
             of an electron,   m e is the mass of an electron.

Then a limited radius-vector for an electrostatic field of an electron is:

          

.
The maximum value of a major semiaxis of orbit of the fifth planet from the Sun – the Jupiter is R 5* = 7.78310 11 m. Fig.4.4. shows the orbit of the Jupiter, defined by the position vector of a Coulombian field of an electron. The difference between the experimental data and the calculated value is 0.23 %.



Fig.4.4. The Orbit of the Jupiter .
The increase of the error for the Jupiter in comparison with the Mercury in 2.7 times is explained by the 13 times increase of the distance from the Sun to the Jupiter in comparison with distance from the Sun to the Mercury resulting in the systematical error of the measurement of the distances from the Earth. The increase of an error can be also explained by the disturbances caused by Saturn.
Let's take the given value as the second point of the scale of measurements of distances in the solar system.
It follows from our suppositions that the whole solar system is stratified into two subspaces, genetically connected with a proton and an electron.
Let's unite these subspaces by forming the scale in the form of distances of the major semiaxises of the orbits of planets:

          

 .
Similarly let's construct a scale for the mean orbital velocities:

          

 ,
            wherein   n  = 1, 2, 3, 4, 5, 6, 7, 8, 9,   m  = 0, 0, 0, 0, 1, 2, 3, 4, 5.

In this case distances and velocities are the functions of two variables.

 
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  > Quantization of Orbits of Planets  (continuation) Page 4

If gravitational fields are presented as a medium with a self action, then it follows from the differential geometry that a torus is the only stable topological formation in this medium. Hence, such space can be decomposed only into enclosed toruses or in special case - into enclosed spheres. Following this idea it is possible to assume that every hydrogen atom of the solar substance forms around itself a toroidal gravitational field. A random distribution of atoms in the space results in superposition of these fields. As a consequence, the aggregate gravitational field has the spherical symmetry. However, on the account of the rotation of the Sun, there is formed a certain anisotropy of the total gravitational field of the Sun, as a result of which the shape of the gravitational field of the Sun acquires a toroidal character. Hence, in the space around the Sun there appears the allocated direction, relative to which there can be performed the quantization of the orbits of planets and the slope of the planes of the orbits.
We shall present the expansion of the space around of the Sun in the form of enclosed toruses. In this case each envelope of a torus coincides with the above calculated scale of orbits. This implies that the cross-sections of a torus form a circular or an elliptical orbits of the planets depending on the angle of the intersecting plane. The topology of the solar system is shown in Fig.4.5.



Fig.4.5. The Topology of a Solar System 
In this case, distances and velocities are functions of two variables. Repeated attempts in the past to find the laws of distances between the planets of the Solar system result in large errors because they are generalized to functions with one variable as for example in the well known Titius-Bode law RnR3 ( 0.4 + 0.32 n ), which also is a function of one variable giving the 29 % error for the Neptune and the 96 % error for the Pluto.
Let's calculate the average orbital velocities and the values of the major semiaxes of the orbits and compare them with the experiment. The obtained data are set forth in Table A.

Table A. Orbital Dimensions for Planets

Planet number Experimental average orbit velocity,
vn*, km/s
Theoretical average orbit velocity,
vn , km/s
Error,

, %
Experimental value of major semiaxis of orbits,
Rn*, (x106 km)
Theoretical
value of major semiaxis of orbits,
Rn , (x106 km)
Error,

, %
1
2
3
4
5
6
7
8
9
47.89
35.03
29.79
24.13
13.06
9.64
6.81
5.43
4.74
47.893
35.919
28.74
23.95
13.0617
8.980
6.841
5.526
4.635
+0.0064
+2.50
-3.60
-0.75
+0.013
-7.30
+0.46
+1.77
-2.20
57.90
108.20
149.6
227.9
778.3
1427.0
2869.6
4496.6
5900.0
57.95
103.02
160.97
231.80
779.11
1648.36
2839.57
4352.71
6187.81
+ 0.10
- 5.0
+10.76
+1.73
+ 0.120
+15.50
-1.0
-3.30
+4.90

(*) – the average heliocentric distance.
In June 2002 a new small planet was discovered – the Quaoar (2002LM60) having the radius of the orbit of 6488,62 million km. This distance varies by 4.86 % from the theoretical value of the radius of the 9 th planet. The radius of the Quaoar is 1250 km.
The analysis of the above Table shows that the orbit parameters of the planets the Solar system are strictly fixed at the quantum level and, hence, they are generically connected with the Sun, i.e. they were "born" together with the Sun or from the proper Sun. This implies that no big planets should exist between the orbits the of the Mars and the Jupiter. A minor deviation of parameters of some orbits from their experimental value reveals the fact of the exchange additional impulses between some of them. Such situation is possible only in case of their joint contraction /expansion and, consequently, the change of the temperature or by means of the interaction or exchange by their satellites.

 
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