Physical NatuRe of the Phenomenon
of delogarithmic Periodicity
Part 2. Astronomical
Systems
Empirically the phenomenon of delogarithmic periodicity in
astronomical systems is described by the formula of the type (1.36). This means
that the theory describing this phenomenon in astronomical systems must be both
relativistic and undulatory. The current existing theories of gravitational
interaction are not of this type. In this connection our task is to find in the
gravitational interaction the undulatory equation of the type (1.21). Since the
mentioned phenomenon is observed not only in nonrelativistic but also in
relativistic areas of Metagalaxy, where the red shift z > 1, it is
relevant and even necessary to start with the analysis of the law of
gravitation, in particular with the finding the reasons of its universality.
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 twosided 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 mc^{2}
– 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
mc^{2} = h_{o }c/r. (2.1)
Hence, the force developed by the
spinor is
_{}. (2.2)
As the first body has N_{1} links with the spinors of the
second body and the second body – N_{2} links with the spinors of the
first body, then the total interaction force of two bodies is,
_{},
т.е. _{}, (2.3)
where N_{1 }N_{2 }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
N_{1} = m_{1}/m_{ō}, а N_{2} = m_{2}/m_{ō},
where m_{1} и m_{2} –
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}
и М_{о}_{2}
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_{о}_{2} with the velocity v, with
the mass m_{o1} << М_{о}_{2},
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_{о}_{2},
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 с^{2}, 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_{о}_{g }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_{о}_{1}.
The link energy of the mass m_{о}_{1}
can be determined from the formula
_{}, (2.11)
where Т – kinetic energy of the mass m_{о}_{1} in the filed of the mass М_{о}_{2}.
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_{о}_{1} 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_{о}_{1}
_{} (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_{о}_{2} 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 nondimensional 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 }c^{2} – 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_{о}_{g}.
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_{о}_{2} along the arc S of the
radius r undulation function y
has the form
y = y_{о} е^{ i (kS }^{±}^{ }^{w}^{t }^{±}^{ }^{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_{о}_{g }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_{о}_{g}.
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} = cos^{2} 2p n_{j}. From this formula
it is seen that the probability density is extreme in the cases when n_{j}
takes integers and halfintegers. Therefore, the impulse moment
L_{z} = h_{og} × n_{j}. (2.24)
where n_{j} – integers and halfintegers.
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
n_{j} =

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

n_{i} + n_{i + 1} = 2n^{2}

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_{о}_{1} = 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_{о}_{1}.
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
n^{2} – k_{g} – ¼
= 0. (2.33)
Its solution has the following form
_{}.
It follows that having denoted k_{g} = 2,
we find that n_{1} = 2,118034, n_{2} = – 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 n_{1} = 1,118034, n_{2} = – 0,118034;
k_{g} = 1, n_{1} = 1,618034, n_{2} = – 0,618034;
k_{g} = 2, n_{1} = 2,118034, n_{2} = – 0,118034; (2.35)
k_{g} = 3, n_{1} = (1,618034)^{2}, n_{2} =
(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
planetsatellite 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 TitziusBose 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 quantumundulation 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 energylike.
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 [1821],
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

r_{rated},
м

d, %

k/m

k¢/m¢

W_{rated},
м

d, %

k¢/m¢

1

Mercury

5,719×10^{10}

5,807×10^{10}
5,782×10^{10}

0,28
0,16

58/8
175/24

3/16
0

5,739×10^{10}

0,89

279/8

2

Venus

1,082×10^{11}

1,086×10^{11}

0,37

60/8

5/16

1,068×10^{11}

1,29

434/12

3

Earth

1,496×10^{11}

1,492×10^{11}

0,18

61/8

3/8

1,509×10^{11}

0,47

295/8

4

Мars

2,279×10^{11}

2,283×10^{11}

0,18

63/8

1/16

2,289×10^{11}

0,44

302/8
453/12

5

Jupiter

7,783×10^{11}

7,710×10^{11}

0,94

67/8

2/8

7,963×10^{11}

1,97

484/12

6

Saturn

1,427×10^{12}

1,427×10^{12}

0,00

104/12

1/12

1,448×10^{12}

1,47

499/12

7

Uranus

2,8696×10^{12}

2,8625×10^{12}

0,25

107/12

8/24

2,8634×10^{12}

0,22

344/8
516/12

8

Neptune

4,4966×10^{12}

4,5133×10^{12}

0,37

110/12

1/12

4,451×10^{12}

1,01

527/12

9

Pluto

5,912×10^{12}

5,925×10^{12}

0,22

74/8

1/8

5,893×10^{12}

0,32

356/8
534/12





0,30




0,90


Jupiter, r_{og }= 1,410 m
№

Satellite

r_{emp},
m

r_{rated},
m

d, %

k/m

k¢/m¢

r_{rtaed},
м

d, %

k¢/m¢

1

Almateus

1,812×10^{8}

1,779×10^{8}

1,82

62/8

2/8

1,825×10^{8}

0,72

299/8

2

Io

4,212×10^{8}

4,241×10^{8}

0,69

65/8

2/8

4,235×10^{8}

0,14

313/8

3

Europa

6,703×10^{8}

6,687×10^{8}

0,24

67/8

0

6,717^{×}10^{8}

0,21

481/12

4

Ganymede

1,068×10^{9}

1,069×10^{9}

0,094

68/8

3/8

1,087×10^{9}

1,78

493/12

5

Callisto

1,881×10^{9}

1,891×10^{9}

0,53

211/24

4/24

1,905×10^{9}

1,28

338/8
507/12

6

XIII

1,111×10^{10}

1,100×10^{10}

0,99

76/8

7/16

1,112×10^{9}

0,090

551/12

7

VI

1,1445×10^{10}

1,1474×10^{10}

0,25

115/12

1/8

1,1579×10^{10}

1,17

368/8 552/12

8

VII

1,1732×10^{10}

1,1693×10^{10}

0,33

115/12

2/12

1,1579×10^{10}

1,30

368/8 552/12

9

X

1,1825×10^{10}

1,1824×10^{10}

0,0085

115/12

3/16

1,2053×10^{10}

1,93

553/12

10

XII

2,1206×10^{10}

2,1238×10^{10}
2,1147×10^{10}

0,15
0,27

118/12
79/8

5/24
0

2,1131×10^{10}

0,35

378/8 567/12

11

XI

2,259×10^{10}

2,2555×10^{10}
2,2458×10^{10}

0,15
0,58

118/12
79/8

4/12
1/8

2,244×10^{10}

0,66

379/8

12

VIII

2,349×10^{10}

2,327×10^{10}

0,94

119/12

0

2,383×10^{10}

1,45

380/8 570/12

13

IX

2,394×10^{10}

2,3857×10^{10}

0,38

79/8

2/8

2,383×10^{10}

0,46

380/8 570/12





0,50




0,89


Saturn, r_{og }= 0,422 m
Formula (2.38)

Formula (2.37)

№

Satellite

r_{emp},
m

r_{rated},
m

d, %

k/m

k¢/m¢

r_{rated},
m

d, %

k¢/m¢

1

Janus

1,59×10^{8}

1,594×10^{8}

0,25

66/8

1/8

1,580×10^{8}

0,63

475/12

2

Mimus

1,855×10^{8}

1,855×10^{8}

0

100/12

1/24

1,8554×10^{8}

0,022

479/12

3

Enteladus

2,383×10^{8}

2,383×10^{8}

0

67/8 101/12

1/24 2/12

2,360×10^{8}

0,97

485/12

4

Теоfius

2,947×10^{8}

2,950×10^{8}

0,10

102/12

5/24

2,942×10^{8}

0,17

327/8

5

Diona

3,774×10^{8}

3,780×10^{8}

0,16

96/8

1/8

3,743×10^{8}

0,82

331/8

6

Rhea

5,271×10^{8}

5,247×10^{8}

0,46

105/12

5/24

5,263×10^{8}

0,15

505/12

7

Titan

1,221×10^{9}

1,225×10^{9}

0,33

109/12

9/24

1,7217×10^{9}

0,057

526/12

8

Triton

1,481×10^{9}

1,484×10^{9}

0,20

110/12

9/24

1,493×10^{9}

0,81

531/12

9

Japetus

3,561×10^{9}

3,559×10^{9}
3,574×10^{9}

0,056

231/24
115/12

0
5/24

3,607×10^{9}

1,29

553/12

10

Phoebe

1,295×10^{10}

1,290×10^{10}

0,39

122/12

1/12

1,302×10^{10}

0,54

390/8 585/12




<d> = 0,20


<d> = 0,55

Uranus, r_{og }= 0,0644 m
№

Satellite

r_{emp},
m

r_{rated},
m

d, %

k/m

k¢/m¢

r_{rated},
m

d, %

k¢/m¢

1

Miranda

1,298×10^{8}

1,288×10^{8}

0,77

72/8

0

1,2996×10^{8}

0,12

517/12

2

Ariel

1,908×10^{8}

1,891×10^{8}

0,89

110/12

0

1,902×10^{8}

0,31

351/8

3

Umbriel

2,659×10^{8}

2,662×10^{8}

0,11

74/8

5/16

2,675×10^{8}

0,60

535/12

4

Titania

4,376×10^{8}

4,413×10^{8}

0,85

114/12

1/8

4,415×10^{8}

0,89

365/8

5

Oberonus

5,857×10^{8}

5,885×10^{8}

0,48

77/8 231/24

2/12 4/24

5,965×10^{8}

1,84

370/8 555/12




<d> = 0,62


<d> = 0,75

Literature
1.
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