Prediction of changing of dynamic characteristics of solair system’s planets up to 10 milliard years forward.
By the use of known TitiusBode’s, Kepler’s and Newton’s laws there is a new, unknown before distribution pattern of planets from the central celestial body (the Sun), and satellites from the planet obtained. This pattern allows defining the Solar System planets’ dynamic characteristics’ temporal changes.
The dependence of Solar System planets’ removal from the Sun velocities look as follows:
v_{n} = v_{1} + 4(v_{n1} – v_{n2}) .
(1)
This formula is right for the Solar System formation’s every time point, because the planets’ removal velocities have constant values. Taking this into account we conclude, that the planets’ distances’ from the Sun dependence looks as follows:
v_{n} = A_{n} r_{n} ,
(2)
where v_{n} is the planets’ removal velocity, A_{n} is the planets’ removal velocities’ constant:
A_{n} = 1/t_{n} ,
(3)
where t_{n} is the time on the planet’s evolution given moment, r_{n} is the planet’s distance from the Sun in the n1 time point. Substituting (2) and (3) in the formula (1) we get:
r_{n} = r_{1} + 4(r_{n1} – r_{n2}) ,
(4)
so, the planet’s distance from the Sun (the central celestial body) is equal to the first planet’s distance from the Sun plus the quadruple distance between two preceding to the n1 one planets, where n is the given planet’s counted from the Sun number, r_{n} is the given planet’s distance from the Sun, r_{1} is the first planet’s distance from the Sun, r_{n2} is the preceding the given one planet’s distance from the Sun, r_{n1} is the preceding the last one planet’s distance from the Sun.
The equation (4) can be produced as follows:
r_{n1} – r_{n2} = (r_{n} – r_{1})/4 ,
(5)
so, the distance between placed one after other planets of a Solar System is equal to the quarter of the difference between the next and first planets’ distances from the Sun.
The equations (1) and (3) are showing the planet distance and removal velocity dependence of four revolving round the Sun bodies, taking into consideration the disturbing forces, arising in the neighbour planets’ gravitation fields.
Resolving the equation (4) together with Kepler’s 3rd law we get:
[Изображение не найдено] ,
(6)
where T_{1}, T_{2}, m_{1}, m_{2}, r_{1}, r_{2} are orbital periods, masses, and major orbit semiaxes correspondingly of the first and second planets, which are revolving round the Sun with the mass M. After transformations we get the value of the sought planet’s major orbit semiaxis:
[Изображение не найдено] .
(7)
The equation (7) represents the general form of the planet and satellite distances’ pattern, taken into consideration the disturbances experienced by every planet from gravitation forces, caused by surrounding planets and the Sun.
The known, obtained experimentally in accordance with classic celestial mechanics, planet distances and round the Sun revolution periods as well as obtained by authors calculating by TitiusBode formula, planet and satellite distance pattern are shown in the table 1.
Table 1. The Solar System planets characteristics.
Planet

Classic celestial mechanics

TitiusBode law

Planet and satellite distance pattern

r, astronomical units

T, years

r, astronomical units

T, years

r, astronomical units

T, years

Mercury

0.387

0.24

0.4

0.26

0.378

0.24

Venus

0.723

0.62

0.7

0.588

0.723

0.62

Earth

1.0

1.0

1.0

1.0

1.0

1.0

Mars

1.524

1.88

1.6

2.03

1.528

1.889

The asteroid belt





2.8

4.69

2.91

4.49

Jupiter

5.203

11.66

5.2

11.8

5.2

11.96

Saturn

10.539

29.46

10.0

31.65

9.785

30.61

Uranus

19.182

84.01

19.6

86.83

19.64

86.45

Neptune

30.058

164.8

38.8

241.72

29.65

161.4

Pluto

39.439

248.4

77.2

676.59

39.084

244.5

Comparing the Solar System planet characteristics, obtained in accordance with classic celestial mechanics’ laws (table 1, columns 2 and 3), with the planet characteristics, calculated in accordance with the planet distance law we can see, that the results are similar.
The planet revolution period can be obtained by RadoDarwin formula:
[Изображение не найдено] ,
(8)
where ρ_{0} is the planet’s average density. This formula allows estimating t_{J2} for the epoch, when the balanced figure of the planet was “fixed”, and the remained to present day value J^{*}. Taking for the moment of inertia of Venus the value J^{*} = 0.334, obtained by model calculations, V. N. Zharkov gives certain Venus’ time paleoperiod:
[Изображение не найдено] .
(9)
This indicates that Venus was revolving more quickly some time. The “young” Venus’ revolution period, as it is mentioned by V. N. Zharkov, was shorter and equal to 10 hours, however the planet’s nonequilibrium, corresponding to that quick revolution, apparently is erased long ago from Venus’ “memory”, because of its mantle’s and nucleus’ “fluidity”.
In the same way, being aware of Mercury’s J^{*} value, there is in the same work estimated the planet’s revolution period in its silicate mantle’s cooling and hardening epoch. Supposing in the same way as for Venus, that in that epoch the value corresponded to the hydrostatic equilibrium, then supposing for Mercury J^{*} = 0.324 (the value, obtained on basis of model calculations), by RadoDarwin formula the planet’s paleoperiod is defined as [Изображение не найдено].
At the observations of Mercury’s revolution’s tidal decelerations is shown that the characteristic deceleration time is 10^{9} years. This estimation is not conflicting with “young” Mercury’s known revolution period, which is equal to 8 hours. Moreover, it is possible to consider, that Mercury’s mantle is “hardened” and is distinctly stronger now, than after about 10^{9} years its formation.
Assumed for Earth, that by the time of 0.6·10^{9} years after its formation the main gravitational differentiation processes were mainly completed and moments of inertia were stabilized, V. N. Zharkov adduces Earth’s revolution period T_{0} for that epoch:
T_{0} = (89) hours T = 0.6·10^{9} years.
(10)
Studying the specific angular momentum (the impulse per the mass unit) of planets there is an empiric pattern achieved, according to that this unchanged by tidal frictions momentum is proportional to M^{3/4}, where M is the planet’s mass. On basis of this pattern there are estimated Earth’s revolution period’s limits in its postformation early epoch as 13 hours and 10 hours.
There is the resonance correlations’ full system by A. M. Molchanov found for 9 known big planets.
Let’s using the results, obtained by the authors of abovementioned researches on Solar System planets’ dynamic parameters’ temporal changes, study the planets’ round own axes revolution’s changes, starting from their formation moment.
After the processing of Mercury and Venus planets’ radiolocation data there were obtained new facts about their revolution. There was shown, that Mercury’s revolution period T_{ω} relates to the revolution period T_{0} as 3/2, so there is the following resonance correlation:
[Изображение не найдено] .
(11)
The table 2 below, in which are represented Mercury’s dynamic characteristics, calculated by the formula (7), indicates, that Mercury’s paleoperiod, calculated by V. N. Zharkov, and, as it was mentioned above, equal to [Изображение не найдено] days, corresponds to the time period T – (0.70.8)·10^{9} years from the planet formation moment and the distance from the Sun, equal to r – (8.010.0) ·106 km, so 3.88.2 days.
Table 2. Mercury’s dynamic characteristics.
The time from Mercury’s formation, T, 10^{9 }years

Mercury’s revolution round the Sun period, T, years

Mercury’s distance from the Sun, r, 10^{6} km

Mercury’s revolution period, T_{ω}, hours

0.7

0.0138

8.60

3.42

1.0

0.029

10.084

7.20

1.7

0.0525

20.941

13.00

2.7

0.1045

33.259

25.90

3.7

0.168

45.58

41.60

4.7

0.24

57.90

59.44

5.7

0.32

70.20

79.25

6.7

0.41

82.50

101.50

7.7

0.505

94.85

125.10

8.7

0.606

107.17

150.20

9.7

0.721

120.35

178.20

10.7

0.827

131.80

204.80

11.7

0.946

144.00

234.30

12.7

1.07

156.40

285.00

13.7

1.195

168.80

296.00

14.7

1.3315

187.00

329.70

At this time Mercury was already in hydrostatic equilibrium and, as it was already mentioned above, was experiencing the cooling and hardening stage, thanks to its silicate mantle.
The result, obtained for “young” Mercury’s revolution time was equal to t = 8 hours, which in the table 2 corresponds to the time T = 0.15·10^{9} years from the planet formation and it was at the distance r – 1.82·10^{6} km from the Sun.
Table 2 indicates, that Mercury’s revolution period is increasing linearly from the formation moment till now and further (in the table are shown Mercury’s dynamic characteristics for up to 14.7·10^{9} years), and will increasing with planet’s removal in the time.
According to radiolocation measurements, Venus’ revolution period is T^{V} = 234.24 ± 0.1, which in measurement accuracy limits coincides with the value 243.16 days, at which Venus has to face Earth by its same side at every lower connection. The connection period is t = 583.02 days.
The resonance takes place at the following relationship:
[Изображение не найдено] .
(12)
From here we get the following dependence for Venus’ revolution period:
[Изображение не найдено] .
(13)
Supposing that the resonance dependence has remained unchanged, as it is noticed by A. M. Molchanov, from the moment “…of system’s evolutional maturity…”, we can calculate Venus’ revolution periods starting from its formation moment till its 14.7·10^{9} years age. In the table 3 are shown planet’s dynamic characteristics for time points starting from 0.7·10^{9} years from Venus’ formation. The calculation results are shown in table 3.
Table 3. Venus’ dynamic characteristics.
The time from Venus’ formation, T, 10^{9 }years

Venus’ revolution round the Sun period, T, years

Venus’ distance from the Sun, r, 10^{6} km

Venus’ revolution period, T_{ω}, hours

0.7

0.035

16.10

13.76

1.0

0.06

23.01

23.4

1.7

0.113

39.10

52.782

2.7

0.267

62.14

104.85

3.7

0.43

85.15

170.93

4.7

0.62

108.16

243.25

5.7

0.81

130.20

308.30

6.7

1.047

154.20

414.20

7.7

1.29

177.20

507.90

8.7

1.55

200.20

616.9

9.7

1.823

232.20

719.96

10.7

2.114

246.20

841.60

11.7

2.42

269.30

961.5

12.7

2.73

292.30

1078.25

13.7

3.055

315.30

10205.50

14.7




From the formula (12) we get Earth’s revolution period’s dependence, expressed by Venus’ dynamic parameters, when [Изображение не найдено] expressed by T^{E}. Since [Изображение не найдено] , then
[Изображение не найдено] .
(14)
By the use of this formula are obtained Earth’s revolution periods starting from the time point T = 0.7·10^{9} years from its formation. The numerical values of Earth’s revolution periods in different time points from its formation moment are shown in the table 4.
Table 4. Earth’s dynamic characteristics.
The time from Earth’s formation, T, 10^{9 }years

Earth’s revolution round the Sun period, T, years

Earth’s distance from the Sun, r, 10^{6} km

Earth’s revolution period, T_{ω}, hours

0.7

0.057 = 21 days

22.29

1.55

1.0

35.8 days

31.83

2.35

1.7

0.216 = 78.9 days

54.11

5.2

2.7

0.435

85.94

10.44

3.7

0.698

117.77

16.75

4.7

1.0 = 365.26 days

149.60

24 = 1.0 day

5.7

1.3317

187.43

32

6.7

1.702

213.26

40.9

7.7

2.10

245.10

50.4

8.7

2.515

276.92

60.4

9.7

2.965

308.75

71.2

10.7

3.43

340.58

82.33

11.7

3.93

372.4

94.3

12.7

4.44

404.24

106.6

13.7

4.97

436.10

119.3

14.7

5.55

467.90

133.2

The table 4, in which are shown Earth’s dynamic characteristics starting from its formation moment till our days and 10 milliard years after, indicates, that Earth’s revolution periods are increased linearly increasing the time.
The mentioned above astronomic and paleontologic research results are indicating Earth’s revolution’s deceleration as 2 sec per 100000 years, in the same time the observations during last 200 years are indicating 1 sec.
The numerical values of the revolution’s deceleration per 100000 years, obtained by authors, are shown in the table 5.
Table 5. Earth’s revolution’s deceleration.
Time from Earth’s formation, years

(0.71.7)·10^{9}

(1.72.7)·10^{9}

(2.73.7)·10^{9}

(3.74.7)·10^{9}

(4.75.7)·10^{9}

Deceleration time per 100000 years, t, sec

1.83

1.34

1.12

1.1

0.96

The table 5 indicates, that in the time period T = (0.71.7)·10^{9} Earth’s revolution’s deceleration was really t = 1.83 or about 2 sec per 100000 years, but in present time the revolution’s deceleration is t = 0.96 or about 1 sec per 100000 years.
This way can be explained the abovementioned inconsistency.
In the table 6 are shown Mars’ temporal dynamic characteristics, obtained at the supposition, that Mars’ and Earth’s revolution periods’ initial ratio [Изображение не найдено] is remained.
Mars’ revolution has not decelerated by the tidal friction, and, so, the planet has kept its initial revolution angular velocity and the corresponding angular momentum. The lunar and solar tides have decelerated Earth’s revolution, and about two times prolonged terrestrial days in the last two milliards years. This data coincides with the results, obtained by V. N. Zharkov. So, when excluded the lunar and solar tides’ influence, then Earth’s revolution period today would be equal to Earth’s and Mars’ revolution period 2.0·10^{9} years ago, so T_{ω} = 10.44 – 10.8 hours, or about 11 hours.
Furthermore, the paleontologists have recently shown, that the warm sea corals are growing, forming daily a fascia, the thickness of which depends on the light amount received during the day.
Table 6. Mars’ dynamic characteristics.
The time from Mars’ formation, T, 10^{9 }years

Mars’ revolution round the Sun period, T, years

Mars’ distance from the Sun, r, 10^{6} km

Mars’ revolution period, T_{ω}, hours

0.7

0.11

33.96

1.6

1.0

0.18

51.40

2.4

1.7

0.41

82.47

5.4

2.7

0.82

130.97

10.8

3.7

1.35

179.48

17.3

4.7

1.88

227.99

24.7

5.7

2.52

276.50

39.2

6.7

3.24

325.00

42.1

7.7

3.94

373.50

51.88

8.7

4.75

422.00

62.13.

9.7

5.60

470.50

73.25

10.7

6.46

519.00

84.73

11.7

7.39

567.60

97.10

12.7

8.35

616.10

109.70

13.7

9.36

664.60

122.80

14.7

10.40

713.10

137.10

The results of the day duration determination by coral fossils researches are shown in the table 7.
Table 7. Earth’s revolution’s deceleration according to paleontologic data.
Period

Time, million years

Year duration, days

Day duration, hours

Relative day duration

Present time

0

365.25

24

1

Cretaceous

72

370.33

23.67

0.986

Permian

270

384.10

22.82

0.951

Carboniferous

298

387.50

22.62

0.943

Devonian

380

398.75

21.98

0.916

Silurian

440

407.10

21.53

0.897

The table indicates, that the day duration is noticeably increasing with the time. So, e.g. in Devonian period the day duration was equal 0.916·84164 = 78926 sec., so less 7238 sec. in 380 million years, so less 1.9 sec. in 100000 years.
It is quite obvious that these numbers cannot be considered exact, however they give unambiguously the order of values.
Before this time there was apparently only the solar tides’ influence on Earth and Mars, and their evolution followed the same laws.
The revolution period was equal to 11 hours. This way, if supposed, that the lunar tides’ influence has started affecting Earth’s revolution 4·10^{9} years ago then really Earth’s revolution is decelerated two times. This conclusion allows concluding, that the Moon was separately formed and by Earth’s gravitation field caught 4·10^{9} years ago.
The adduced calculations by formula 7 are confirming many authors’ opinion, that in their formation’s early period the planets Mercury, Venus, Earth and Mars were revolving very quickly about their axes, like untwisted tops. With the time, moving away from the Sun, they have decelerated their movement.
The revolution velocities of planets and their distances from the Sun in the early period of their formation can testify their “heat” origination.
In the present the Earth is moving nonuniformly on its orbit, and we have to take it into account while resolving earthquakes’ accurate prediction problem. 