For this purpose it is necessary to connect gravitation with the properties of electrical charges. The fundamental, purely relativistic property of the charge is its spin (See work [20]), that exists in the thinnest material environment, gravitational field. As megamasses consist of charges then the question of gravitational interaction of megamasses is narrowed down to the question of spin interaction of charges. To solve this question let us suppose that 2 spinors are located at the distance r so that their axes are parallel. Spinning each of them winds lines of force of this field on the own axis.  As the result a two-sided link is formed, the link of the type that enables the impulse carried by the link to change its direction into the opposite one during the period of the circulating there and back equaling to 2r/c. On this basis we can write down that , i.е. . It follows that the force developed by the spinor , where mc² – link energy, that equals to the work of the spinor in time t = r/c of its displacement. Having multiplied the link energy by this time we get that the product w×t equals to the proper moment of link impulse, that is:

.

It follows that

                                                                   mc² = h_o c/r.                                                        (2.1)

Hence, the force developed by the spinor is

                                                                    .                                                       (2.2)

As the first body has N₁ links with the spinors of the second body and the second body – N₂ links with the spinors of the first body, then the total interaction force of two bodies is,  

                                  ,  т.е.    ,                                    (2.3)

where N₁ N₂ h_o = h_og – full proper impulse moment of two bodies. Let's suppose that each spinor has one generalized link, that's why the number of potential links in the body will correspond to the number of spinors. 

Having denoted the average mass of one spinor as m_ō, we can find that

N₁ = m₁/m_ō, а    N₂ = m₂/m_ō,

where  m₁ и m₂ – masses of the first and the second bodies. As a result we get that:

.

Having denoted , we finally get that

                                                                ,                                                   (2.4)

where G  is a gravitational constant.

Its dimension is

.

This is the very Newton's law of gravitation. From the above mentioned there are no reasons according to which the law of gravitation would be unfair in the relativistic area. It is also seen that the gravitation itself has a relativistic origin.

Placing G = 6,6726×10^–11  and h_о = 7,6957×10^–37 J×s in the formula  we determine that the mass of the spinor equals to

m_ō = 1,85954×10^–9 kg.

Having solved the question of the link between the charge spin and gravitation let's start solving the direct question of the theory: finding out the reasons for a particular stability of a circular megaorbit. For this purpose let's represent the interaction force between two megamasses m_o1 иМ_о2 by the formula

                                                  ,                                           (2.5)

where f (b) – the function, considering relativistic effects appearing during the movement of a trial mass m_o1 in the central symmetrical field of the masses М_о2 with the velocity v, with the mass m_o1 << М_о2, b = v/c, where с – the velocity of light in the vacuum.

Considering the formula (2.3), if the right part of this formula is multiplied and divided by c then the proper impulse moment of the interacting pair is pointed out in the form of the following formula 

                                        ,                         (2.6)

where  – gravitational radius of the mass М_о2, then the formula (2.5) gets the form

                                                            .                                               (2.7)

On the other hand having multiplied and divided the right part of the formula (2.5) by с², we get that

                                                  .                                    (2.8)

But since, and r/r_og г = e, then

                                                             .                                               (2.9)

Comparing  (2.7) and (2.9), we get that .                                                 (2.10)

In the right part of this equation the quantity h_оg с/r determines the potential energy of the interaction U. Hence, in the left part there is a corresponding to it the link energy W_c of the mass m_о1.

The link energy of the mass m_о1 can be determined from the formula

                                             ,                             (2.11)

where Т – kinetic energy of the mass m_о1 in the filed of the mass М_о2.

The total energy of the system consisting of two interacting megamasses equals to the sum of the kinetic energy T of the circulating mass m_о1 and the potential energy of the interaction U, i.е.

                                                           W = Т + U = – ,                                              (2.12)

where r – the distance between interacting masses.

Then . Solving  (2.11) and (2.12) we find that

                                                              .                                               (2.13)

It follows that

                                                         ,                                          (2.14)

where e = r/r_og.

This formula is proved by the empirical data in the whole range of velocities [8].

Placing the full mass impulse m_о1

                                                                                                                    (2.15)

and  r according to the formula (2.13)  into the formula L_zj = p_j×r  we find that the impulse moment of  a circular orbit relatively to the axis z equals to    

                                                        .                                         (2.16)

Multiplying formulae (2.11) and (2.16) and omitting intermediate transformations we get as e result the following formula

                                                     .                                       (2.17)

The formula (2.17) is easily transformed into an undulation equation. In particular it can be represented in the following form:

Е – Е_о = (h_og / L_z) р×с,

where Е – full relativistic energy of the mass m_o1, L_z – its full impulse moment in the projection on the axis z.

The full mass energy m_o1 and the full system energy W, stipulated by the movement of the mass m_o1 and its interaction with the mass М_о2 are linked by the formula Е = W – U. Placing this E into the previous formula we get the following equation

W – U – Е_о1 – (h_og / L_z) р×с=0,

where  h_og / L_z = a represents a non-dimensional but in a general case a vectorial number. Hence we can write down that 

                                                      .                                       (2.18)

In this equation a vector  has the meaning of a  full impulse, in particular

,

where  – orbit impulse (without a spin)  – spin addition to it,  – sum of additions stipulated by the unaccounted effects. Placing this  into the equation (1.18), we get that

                                         .                              (1.19)

According to the physical sense in this equation  – energy of the orbit movement of the mass m_o1;  – energy brought by the proper rotation of the mass m_o1;  – energy stipulated by the unaccounted effects expressed through the constant a.

Hence

                                                  W – U – m_o1 c² – T_l ± T_s – T_l s = 0.                                      (2.20)

To solve this equation (2.19) at the beginning we will simplify it casting away everything that is connected with the unknown effects, and then we will pass over into the spherical system of axes. In this system in the operators of undulation mechanics

,

where k = ± (j + ½) = ±1; ±2 – quantum number expressed by the quantum number j of the full impulse moment, i – imaginary unit. Having placed this operator into the equation (2.18), we will get

                                .                  (2.21)

This operational equation comprises all the possible states of the system stipulated by the quantums of the impulse moment h_оg. To chose out of them states stipulated by the quantums of the impulse moments of the basis state of the mass m_o1, let's divide  by the absolute a of the vector . Then , where  – unitary vector base (matrix), and  L_i = h_og/a.  Since | U | = h_og c/r, so finally the equation  (2.21) has the following form

                              .                (2.22)

For the circular orbits the present equation shows that

.

The solutions of the equation (2.22) are the undulation functions of the type

y = y_о е ^i j.

For the mass m_o1 moving relatively to the mass М_о2 along the arc S of the radius r undulation function y has the form

                                                               y = y_о е ^{i (kS ± wt ± j}_o⁾,                                               (2.23)

where  k = 2p/l – undulation number, l – wave length of the mass m_o1, j_о – initial phase.

This function can be simplified. Let us consider that with t = 0 the initial phase j_о = 0. And since for the circular orbit S = 2pr, then having assumed that D = h_og/p, where D = l/(2p), and impulse р = h_оg k, we will get that the phase j = k S = 2pр×r/h_og.

But р×r = L_z – is the projection of the full impulse moment on the axis z, hence, j = 2p (L_z/h_og). As the result we get that y/y_o = cos 2p n_j, where n_j = L_z/h_оg.

The square of this function determines the probability density w of the presence of the megamass in a certain place relatively to the axis of rotation depending on the impulse moment, i.е. w = | y/y_o | ² = cos² 2p n_j. From this formula it is seen that the probability density is extreme in the cases when n_j takes integers and half-integers. Therefore, the impulse moment

                                                                    L_z = h_og × n_j.                                                        (2.24)

where  n_j – integers and half-integers.

To determine the most possible distances of the mass m_o1, let us connect the number n_j with the parameter e, determining the orbit sizes. Such link is found in the first part of the present paper in the form of the following formula

                                                                                                                            (2.25)

It follows that with

 

nj =	1 2	3 2	5 2	7 2	9 2	11 2	13 2	15 2	17 2	19 2	21 2	…
e =	0	2	6	12	20	30	42	56	72	90	110	… .

 

If the numbers e of this case are summed in a strict sequence, e.g. adding the following number to the each previous one, then as the result we will get a number of states suitable for the occupation by the masses m_o1 with the given n, and to be exact:

 

n	e
1	0 + 2 = 2
2	2 + 6 = 8
3	6 + 12 = 18
××××××××××××××××××
n	ni + ni + 1 = 2n2

 

This result is also secondary. The more essential is the coordination of the numbers e and n_j of the most possible orbits with the wave length of the mass m_o1. According to the determination of its wave length

                                                                     .                                                        (2.26)

But since

                                                                   ,                                                   

and  h_о1 = m_o1 with r_og2 = p_o1× r_og2, then as a result

                                                                                                                (2.27)

Let us transform this formula to the form . Then instead of b let us place the quantity n_j according to the formula

.

After certain transformations we will get that 

                                                               ,                                                (2.28)

but  = e, therefore  D = r_og2×e/n_j. It follows that D n_j = r_og2×e, i.е.   

                                                              D n_j = 2p r_og2 e = 2p r.                                             (2.29)

Formula (2.29) shows that the circular megaorbit will be stable in the case when the circle length contains a multiple of the wave lengths of the circulating mass m_о1.

If we choose the quantum of the mass action m_o1 of its basic state as the prototype of the measurement of the impulse moment in the system of interacting megamasses , then l = h_i/р. It follows that knowing that h_I = h_og/a, we will get that

                                                                                                               (2.30)

or that, i.e. it is the same

                                                                   ,

or that , i.e. even simpler

                                                                  ,                                                   

where  r_og эф = r_og/a –scale prototype of the initial state. As the result we get that 

                                                                  l × n_j = 2p r_эф.                                                      (2.31)

This formula represents the general condition of the stationarity of a circular megaorbit. To determine the number of the lengths l, with which the circular orbit will be the most stable, let us represent the formula (2.30) in the form of the equation

                                                              .                                              (2.32)

Having denoted D/r_o эф = k_g, where k_g – number of harmonic, quantizing the wave length D,we will get an equation

                                                                  n² – k_g – ¼ = 0.                                                    (2.33)

Its solution has the following form

                                                           .

It follows that having denoted k_g = 2, we find that n₁ = 2,118034, n₂ = – 0,118034. If 0,5 is regularly added to or subtracted from these solutions, we will get a whole set of similar numbers that appeared to be described by the equation of the type

                                                               .                                      (2.34)

where k_g = 0, 1, 2, 3, 4,… . In this equation out of all the solutions the most attention should be paid to the solutions with k_g = 0, 1, 2 and 3. In particular, if

 

                   k_g = 0,     then                n₁ = 1,118034,                    n₂ = – 0,118034;

                   k_g = 1,                           n₁ = 1,618034,                    n₂ = – 0,618034;

                   k_g = 2,                           n₁ = 2,118034,                    n₂ = – 0,118034;                (2.35)

                   k_g = 3,                           n₁ = (1,618034)²,                n₂ =    (0,618034)².

 

The equation (2.34) is completely identical to the equation (1.40). The analysis of its solutions is represented in the first part of this paper. It follows that the parameter e of the link energy W_c can change either according to the law

                                                                   ,                                                    (2.36)

where with m = 1, 2, 3… k = 0, 1, 2, 3…, or according to the law

                                                                ,                                              (2.37)

where with m¢ = 1, 2, 3… k¢ = 0, 1, 2, 3…, or according to the law

                                                      ,                                         (2.38)

where k_g = 1, 2, 3.

The formula (2.36) is proved by the bulk of empirical data in the planet-satellite system , in the galactic system and in the Metagalaxy [1, 7, 8].

The formula (2.37) with the integers k¢/m¢ transforms into the law of Titzius-Bose in its modern form. In the general form it describes all the empirical data obtained in the paper [17]. As far as the formula (2.38) is concerned it as well as the formula (2.37) was verified by the concrete calculation of the planet location in the solar system and satellites in the system of Jupiter, Saturn and Uranus. The results are represented in the tables of comparison. The best accordance with the empirical data was shown by the formula (2.38): the divergence with empirical data doesn't exceed 0,5%, for the formula (2.37) it reaches 1%. The both results are acceptable but there is one fundamental divergence. The formula (2.37) couldn't allow the harmonic of the positions VI and VII, VIII and IX of the satellites of Jupiter in the multiplicity of the 8^th and 12^th while the formula (2.38) managed to do this.

The accordance of the rated empirical data in various systems of the megaworld allows to draw a number of general and particular conclusions.

                            1. Independently of their sizes all megamasses possess undulation properties in reality, the properties being revealed in the process of interaction.    

2. Gravitational interaction also belongs to undulation interactions.

3. All undulation interactions are described by an undulation relativistic equation that is universal for all systems independently of the interaction nature, see the equation (1.19) and (2.21).

              4. The phenomenon of delogarithmic periodicity is a universal phenomenon, it depicts the reflection of undulation properties and relativism by the interacting objects, including the dependence of the mass on the velocity and spins.

              5. The formulae describing the phenomenon of delogarithmic periodicity show that in dynamically stable systems there exist quantum-undulation states  organized in a particular manner:

·        the peculiarity of their organization is the division of a characteristic circle of the state by the wave length of the circulating mass into 10 equal parts, the wave length of the circulating object being directly connected with the proportions of a golden section that emerge in the state circle -- it equals to the mean proportional between the biggest and the smallest parts of the circle radius, divided in the extreme and average rate.

·        the increase of the circle radius of such state in several times stipulated by the undulation multiplicity k/m does not upset the proportions of the golden section, but the link energy changes. Such states become dimensional and energy-like.

Delogarithmic periodicity reflects this likeness (through the excitation resonances.

6. In Metagalaxy the phenomenon of delogarithmic periodicity is observed (in the distribution of quasars) till z = 5 not less. And since the golden section is the property of Euclidean geometry it follows that Metagalaxy space (observed universe) is euclidean (till z = 5 not less). It is an established scientific fact that should be seriously taken into consideration.

In the conclusion it should be mentioned that the attempts to understand the physical essence of the phenomenon of delogarithmic periodicity (PDP) have been mad earlier and some results have been obtained, see [18-21], but they were the necessary stages of the process that helped to reach the necessary clearness. In this connection special attention should be paid to the paper [22], where one of the physical applications of the phenomenon of delogarithmic periodicity is considered. Then it will come the turn of the realization of an energy project based on this phenomenon. 

Tables comparing empirical and rated results

 

Planet system.

For the Sun r_og = 1477 m, 

 

Formula (2.38)							Formula (2.37)
№	Planet	r emp, m	rrated, м	d, %	k/m	k¢/m¢	Wrated, м	d, %	k¢/m¢
1	Mercury	5,719×1010	5,807×1010 5,782×1010	0,28 0,16	58/8 175/24	3/16 0	5,739×1010	0,89	279/8
2	Venus	1,082×1011	1,086×1011	0,37	60/8	5/16	1,068×1011	1,29	434/12
3	Earth	1,496×1011	1,492×1011	0,18	61/8	3/8	1,509×1011	0,47	295/8
4	Мars	2,279×1011	2,283×1011	0,18	63/8	1/16	2,289×1011	0,44	302/8 453/12
5	Jupiter	7,783×1011	7,710×1011	0,94	67/8	2/8	7,963×1011	1,97	484/12
6	Saturn	1,427×1012	1,427×1012	0,00	104/12	1/12	1,448×1012	1,47	499/12
7	Uranus	2,8696×1012	2,8625×1012	0,25	107/12	8/24	2,8634×1012	0,22	344/8 516/12
8	Neptune	4,4966×1012	4,5133×1012	0,37	110/12	1/12	4,451×1012	1,01	527/12
9	Pluto	5,912×1012	5,925×1012	0,22	74/8	1/8	5,893×1012	0,32	356/8 534/12
				0,30				0,90

 

Jupiter, r_og = 1,410 m

 

№	Satellite	remp, m	rrated, m	d, %	k/m	k¢/m¢	rrtaed, м	d, %	k¢/m¢
1	Almateus	1,812×108	1,779×108	1,82	62/8	2/8	1,825×108	0,72	299/8
2	Io	4,212×108	4,241×108	0,69	65/8	2/8	4,235×108	0,14	313/8
3	Europa	6,703×108	6,687×108	0,24	67/8	0	6,717×108	0,21	481/12
4	Ganymede	1,068×109	1,069×109	0,094	68/8	3/8	1,087×109	1,78	493/12
5	Callisto	1,881×109	1,891×109	0,53	211/24	4/24	1,905×109	1,28	338/8 507/12
6	XIII	1,111×1010	1,100×1010	0,99	76/8	7/16	1,112×109	0,090	551/12
7	VI	1,1445×1010	1,1474×1010	0,25	115/12	1/8	1,1579×1010	1,17	368/8 552/12
8	VII	1,1732×1010	1,1693×1010	0,33	115/12	2/12	1,1579×1010	1,30	368/8 552/12
9	X	1,1825×1010	1,1824×1010	0,0085	115/12	3/16	1,2053×1010	1,93	553/12
10	XII	2,1206×1010	2,1238×1010 2,1147×1010	0,15 0,27	118/12 79/8	5/24 0	2,1131×1010	0,35	378/8 567/12
11	XI	2,259×1010	2,2555×1010 2,2458×1010	0,15 0,58	118/12 79/8	4/12 1/8	2,244×1010	0,66	379/8
12	VIII	2,349×1010	2,327×1010	0,94	119/12	0	2,383×1010	1,45	380/8 570/12
13	IX	2,394×1010	2,3857×1010	0,38	79/8	2/8	2,383×1010	0,46	380/8 570/12
				0,50				0,89

 

 

 

 

 

Saturn, r_og = 0,422 m

 

Formula (2.38)							Formula (2.37)
№	Satellite	remp, m	rrated, m	d, %	k/m	k¢/m¢	rrated, m	d, %	k¢/m¢
1	Janus	1,59×108	1,594×108	0,25	66/8	1/8	1,580×108	0,63	475/12
2	Mimus	1,855×108	1,855×108	0	100/12	1/24	1,8554×108	0,022	479/12
3	Enteladus	2,383×108	2,383×108	0	67/8 101/12	1/24 2/12	2,360×108	0,97	485/12
4	Теоfius	2,947×108	2,950×108	0,10	102/12	5/24	2,942×108	0,17	327/8
5	Diona	3,774×108	3,780×108	0,16	96/8	1/8	3,743×108	0,82	331/8
6	Rhea	5,271×108	5,247×108	0,46	105/12	5/24	5,263×108	0,15	505/12
7	Titan	1,221×109	1,225×109	0,33	109/12	9/24	1,7217×109	0,057	526/12
8	Triton	1,481×109	1,484×109	0,20	110/12	9/24	1,493×109	0,81	531/12
9	Japetus	3,561×109	3,559×109 3,574×109	0,056	231/24 115/12	0 5/24	3,607×109	1,29	553/12
10	Phoebe	1,295×1010	1,290×1010	0,39	122/12	1/12	1,302×1010	0,54	390/8 585/12
			<d> = 0,20				<d> = 0,55

 

 

Uranus, r_og = 0,0644 m

 

№	Satellite	remp, m	rrated, m	d, %	k/m	k¢/m¢	rrated, m	d, %	k¢/m¢
1	Miranda	1,298×108	1,288×108	0,77	72/8	0	1,2996×108	0,12	517/12
2	Ariel	1,908×108	1,891×108	0,89	110/12	0	1,902×108	0,31	351/8
3	Umbriel	2,659×108	2,662×108	0,11	74/8	5/16	2,675×108	0,60	535/12
4	Titania	4,376×108	4,413×108	0,85	114/12	1/8	4,415×108	0,89	365/8
5	Oberonus	5,857×108	5,885×108	0,48	77/8 231/24	2/12 4/24	5,965×108	1,84	370/8 555/12
			<d> = 0,62				<d> = 0,75

 

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21.   Petrunenko V.V. Chapter 18. The phenomenon of delogarithmic periodicity in astronomical systems.// Cycles as the basis of the universe. Stavropol: Press of North Caucasian state technical university. Stavropol, 2001. P. 327-356.

22.   Petrunenko V.V. Physical parameters of the Metagalaxy. // Cycles. 2002. Part 3. P. 46‑59.

 

 

Petrunenko V.V.