About the tendency to synchronization of the Earth and the Moon rotary movements
The Hierarchical structure of the Earth–Moon system provides process of self-organizing. The principle of synchronization and symmetry of initial structure of the Earth–Moon system and the processes, breaking symmetry form the basis of self-organizing model. The theory of a nonlinear parametrical re...
Saved in:
| Published in: | Кинематика и физика небесных тел |
|---|---|
| Date: | 2005 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Головна астрономічна обсерваторія НАН України
2005
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/79671 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | About the tendency to synchronization of the Earth and the Moon rotary movements / G.S. Kurbasova, L.V. Rykhlova // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 343-346. — Бібліогр.: 6 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859719557647695872 |
|---|---|
| author | Kurbasova, G.S. Rykhlova, L.V. |
| author_facet | Kurbasova, G.S. Rykhlova, L.V. |
| citation_txt | About the tendency to synchronization of the Earth and the Moon rotary movements / G.S. Kurbasova, L.V. Rykhlova // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 343-346. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Кинематика и физика небесных тел |
| description | The Hierarchical structure of the Earth–Moon system provides process of self-organizing. The principle of synchronization and symmetry of initial structure of the Earth–Moon system and the processes, breaking symmetry form the basis of self-organizing model. The theory of a nonlinear parametrical resonance allows finding the features as results of perturbation effects. In the present paper some ratio which determine stationary structure of the Earth–Moon system is given.
|
| first_indexed | 2025-12-01T09:08:58Z |
| format | Article |
| fulltext |
ABOUT THE TENDENCY TO SYNCHRONIZATION
OF THE EARTH AND THE MOON ROTARY MOVEMENTS
G. S. Kurbasova1, L. V. Rykhlova2
1Scientific–Research Institute “Crimean Astrophysical Observatory”
Nauchny, 98409 Crimea, Ukraine
e-mail: gsk@simeiz.ylt.crimea.com
2Institute of Astronomy, Russian Academy of Sciences
e-mail: rykhlova@inasan.rssi.ru
The Hierarchical structure of the Earth–Moon system provides process of self-organizing. The prin-
ciple of synchronization and symmetry of initial structure of the Earth–Moon system and the pro-
cesses, breaking symmetry form the basis of self-organizing model. The theory of a nonlinear
parametrical resonance allows finding the features as results of perturbation effects. In the present
paper some ratio which determine stationary structure of the Earth–Moon system is given.
INTRODUCTION
Piphagor and J. Kepler ideas about world harmony have deep physical sense. The hypothesis about full
resonance Solar System was put forward A. M. Molchanov [6].
E. A. Grebenikov has come to a conclusion [3]: “The Numerical analysis and theoretical generalizations have
allowed stating a principle of the least interaction (or a principle of synchronism). According to this principle
any planetary system, irrespective of an initial condition, sooner or later evolves to resonant condition where
resonant ratio prevails between the basic frequencies systems.
Some calculations specify that the Solar System now is near to the first resonant condition and to this
condition she has come for a time interval in some billions years”.
Time of evolution of satellite systems to the first resonant condition is much less. It depends on initial
conditions in parameters of system, i.e., weights, forms of bodies, elements of orbits during formation of system
as dynamic structure.
Classical example of resonant movement is the indignant movement of terrestrial axis. An attraction the Sun
and the Moon brings the comparable contribution in this movement.
The movement of the Moon is described by empirical Cassini laws and has double 1:1 resonance: firstly,
between axial and orbital rotations and, secondly, between precession orbits and the Moon’s axes of rotation.
Thus, one axis of the Moon inertia in the movement “traces” the radius-vector of an orbit, i.e., takes place
the law “a constant phase”: at each passage of a pericenter of an orbit one of the main axes of inertia, normal
to an axis of rotation, and a radius-vector of a pericenter have equal distances up to a line of knot [2].
Marked the ancients general harmony of the universe and existing tendencies to synchronization in the Earth–
Moon system are shown in simple quantitative ratio between the sizes of forms and the orbits of both bodies.
Result of synchronization is the coordination of parameters at formation of structure of the Earth–Moon system
as single whole.
PROBLEM OF SYNCHRONIZATION. MODEL PARAMETRICAL OSCILLATOR
The Earth–Moon represents an open self-organizing subsystem of the Solar System. She models external
influence and her parts.
For example, barycenter of the Earth–Moon system goes on an elliptic orbit around the Sun as a result of
self-organizing system.
Displacement of the lunar orbit perigee under action of the Sun is responsible for the reduction of dura-
tion astral period of the Moon rotation and changes oscillatory energy in system. This process concerns to
management at more high level.
The hierarchical structure allows to consider modelling of dynamic processes in the Earth–Moon system as
the act of compression of the entrance information for reception of qualitative and quantitative estimations of
considered process at various hierarchical levels. Thus, the question is not discussed, how might be carried out
c© G. S. Kurbasova, L. V. Rykhlova, 2004
343
dynamically (at least at a theoretical level) procedure of self-organizing, or compression of the information. Such
approach to modelling provides opportunity of the reduction of degrees of freedom quantity and application of
the unified descriptions of the investigated process models.
The Earth–Moon system represents complex nonlinear oscillator. The mathematical decision of a problem of
synchronization of movements of two connected nonlinear oscillators that are the Earth and the Moon includes,
at least, two requirements:
1. opportunity of drawing up of two van der Pol equations with various managing parameters and mathe-
matical description of their connections;
2. the description of the dissipation.
Performance of these conditions is the difficult problem and not solved till now.
Nevertheless, universal character of process of synchronization in a nature, in various mechanical and electric
devices is reflected in a generality of existing mathematical descriptions.
Opportunities of application of these models for the description process of synchronization of oscillators with
close frequencies in complex oscillatory systems are based on the assumption of sufficiency of weak interaction.
A simple model can described a parametrical resonance in the Earth–Moon system, concerning to paramet-
rical excitation of the Chandler oscillation [5].
The equation of oscillation looks like
Θ̈ + ω2
0 [1 + e cos(2ωt)] · f(Θ) = 0, (1)
where Θ is the angular displacement of the Earth, ω0 is the frequency of own oscillation of the Earth in
the Earth–Moon system, 2ω is the frequency of parameter change ρ, ρ is the distances between the centers
of masses, e is the eccentricity of the Moon’s orbit, f(Θ) = Θ + 1
2γΘ3 is the function describing nonlinearity
of system, γ = ±√
1 + (4λ)2, 2λ =
1 + μ
1 − μ
, μ =
mM
mE
(mM is the mass of the Moon; mE is the mass of
the Earth).
The description of model (1) has universal character and represents a special case of the Hill’s-equation –
the Mathieu equation, which has the decision in the first area of instability (ω0 = ω).
The parameter ρ in the Equation (1) changes under the periodic law:
ρ =
ρ0
[1 + e cos(2ωt)]
(2)
that to similarly periodic change of a gravitational field as
gt = g0 + g1 cos(2ωt). (3)
We can discussed the qualitative analysis of process of the symmetry infringement in case of a parametrical
resonance.
The equation (1) for small deviations from equilibrium movement (x = Θ − Θ0) is given by
ẍ + ω2
0(1 + α)f(x) = 0, (4)
where 1 > α ≥ 0 and f(x) = x + 1
2γx3.
The equation of fluctuation of system (4) is the elementary movement with one degree of freedom.
In this case, F = −ω2(1 + α)f(x) there is a restoring force which acts upon the point-mass. The force
F = −∂V
∂x
, where V (x) is the potential. If α = 0 and f(x) = x the point-mass (system) is in a potential hole.
V0(x) =
1
2
ω2
0x
2. (5)
This case is shown in Fig. 1: there is one stationary condition
dx
dt
= 0, x = 0, and it is steady.
In case of nonlinear function f(x) = x +
1
2
γx3 and α = 0 the potential V (x) looks like
V (x) =
1
2
(ω2
0x
2 +
1
4
ω2
0γx4) (6)
or
V (x) = V0(x)(1 +
1
4
γx2). (7)
344
Figure 1. Potential V0(x) = 1
2
ω0
2x2 Figure 2. Potential V (x) = V0(x)(1 + 1
4
γx2)
Now the number of special points is three. If γ > 0, the form of new potential is former (Fig. 1), i.e.,
the unique stationary condition x0 = 0 is steady. In a case γ < 0 the form of potential V (x) changes, but
symmetrically as it is shown in Fig. 2. Former steady condition x0 = 0 becomes unstable and the varying
particle (system) moves in one of equiprobable conditions x1 or x2.
The periodic potential (α �= 0) in case of nonlinear function f(x) has difficult character.
From the analysis a conclusion was made that for occurrence of uncertainty concerning the following condition
of system has not necessarily big number of degrees of freedom: only one nonlinearity does makes unpredictable
behaviour of system.
The resonance arising under action of external periodic forces on nonlinear oscillator, results in increase of
amplitude oscillations and, hence, to the exit of frequency of the oscillator from a resonance. In the Earth–Moon
system (according to Gamilton) the asymptotic steady conditions or asymptotic steady cycles are absent [1].
Therefore, a bit later the system again comes back to a vicinity of a resonance.
Synchronization in system of nonlinear Earth–Moon oscillator is based on exchange of energy between
amplitudes and the frequencies of fluctuations on all hierarchical structures.
The analytical description of changes of parameters of the Chandler oscillation is a description of process of
self-organizing of the Earth–Moon system on various hierarchical levels under influence of external management.
The model of self-organizing is based on symmetry of initial structure of the Earth–Moon system. Nonlinear
processes break this symmetry and influence on evolution of system. Special methods find out the new features
caused by perturbations.
One of these methods is the theory of a parametrical resonance.
CHARACTERISTICS OF STATIONARY STRUCTURE
As consequence of action of a principle of synchronism into the Solar System as a result of long evolution was
defined the coordinated behaviour of planets and their satellites.
The steady structure of the Solar System and its parts was determined as a result of this. The Earth–Moon
system is open, i.e., the exchange of energy with an environment takes place. Due to this the homogeneous
stationary condition transfers in non-uniform stationary condition which is steady concerning small perturba-
tions. A formation of structures of the Earth–Moon system, most likely, had wave character. It has determined
independence its characters from initial conditions.
Stationary connections between average parameters of figures and orbits of both bodies determine structure
of the Earth–Moon system. Dependence between squares of own frequencies before and after an establishment
of connection is: ∣
∣
∣
∣
ω2
1 − ω2
2
ν2
1 − ν2
2
∣
∣
∣
∣ = γ, (8)
where
∣
∣ν2
1 − ν2
2
∣
∣ is the difference of squares of own frequencies of the Earth and the Moon up to establishments
of connection,
∣
∣ω2
1 − ω2
2
∣
∣ is the difference of squares of own frequencies of the Earth and the Moon in the Earth–
Moon system, γ = 2.280730667 is the dimensionless constant determined through the relation of the Moon and
the Earth masses.
345
Characteristics of constant structure were derived as ratios:
a) Between the Earth and the Moon radiuses
4 · 1 − ρ ′
1 + ρ ′ = γ, (9)
where ρ ′ = ρ0kR, ρ0 is the relation of the Moon average radius RM to the Earth equatorial radius ae. Dimen-
sionless factor kR looks like
kR =
1 − 1
2
√
5 − 2PM
1
2
√
5 − 2P�
, (10)
where PM = 1+sinπ′
M +cosπ′
M , P� = 1+sinπ�+cosπ�, π′
M is the corner under which is seen an average radius
of the Moon RM on average distance between the centers of masses, π� is the constant of the Sun parallax.
Sizes PM and P� have geometrical analogue – the dimensionless perimeters of appropriate rectangular triangles.
b) Between constant inclinations equator the Earth ξ and the Moon IM
2 · 1 + h0
1 − h0
= γ, (11)
where h0 =
2IM
εPM
.
c) Between the main moments of inertia of the Earth CE and the Moon CM
4 · 1 − 2Z
√
C0
1 + 2Z
√
C0
= γ, (12)
where Z =
1 +
√
sinΔ
1 −√
sinΔ
, Δ is the difference of the coordinated inclinations equator of the Earth and the Moon
to the ecliptic, C0 = CM/CE .
d) Between constants general of the Earth’s precession p and of the corner in top of internal cone of
the Cassini 2κ [4]
2p
κ
· PM = γ. (13)
CONCLUSIONS
The hierarchical device of the Solar System, the features self-organizing provide stability of structure and of
dynamic regimes in the Earth–Moon system. The self-organizing on various hierarchical levels is based on
the principle of synchronization.
The theory of a nonlinear parametrical resonance allows finding the features as results of effects of pertur-
bation.
Structure of the Earth–Moon system was generated as a result of long evolution of the Solar System.
Characteristics of this structure are constant ratio between average parameters of the figures and orbits of
the Earth and the Moon.
Acknowledgements. Researches were supported by grant RFFI (04–02–16633).
[1] Arnol’d V. I. Small denominators and problems of stability of movement in the classical and heavenly mechanics //
Successes Mat. Sci.–1963.–6.–114 p.
[2] Beletsky V. V. Sketches about movement of space bodies.–Moscow: Nauka, 1977.–432 p. (in Russian).
[3] Grebenikov E. A. Introduction in the theory of resonant systems.–Moscow State University, 1976.–176 p.
(in Russian).
[4] Kulikov K. A., Gurevich V. B. Bases lunar astrometry.–Moscow: Nauka, 1972.–392 p. (in Russian).
[5] Kurbasova G. S., Rykhlova L. V. // Astron. J.–2001.–78, N 11.–P. 1049–1056.
[6] Molchanov A. M. The resonant structure of the Solar System // Icarus.–1968.–8, N 2.
346
|
| id | nasplib_isofts_kiev_ua-123456789-79671 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0233-7665 |
| language | English |
| last_indexed | 2025-12-01T09:08:58Z |
| publishDate | 2005 |
| publisher | Головна астрономічна обсерваторія НАН України |
| record_format | dspace |
| spelling | Kurbasova, G.S. Rykhlova, L.V. 2015-04-03T18:27:05Z 2015-04-03T18:27:05Z 2005 About the tendency to synchronization of the Earth and the Moon rotary movements / G.S. Kurbasova, L.V. Rykhlova // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 343-346. — Бібліогр.: 6 назв. — англ. 0233-7665 https://nasplib.isofts.kiev.ua/handle/123456789/79671 The Hierarchical structure of the Earth–Moon system provides process of self-organizing. The principle of synchronization and symmetry of initial structure of the Earth–Moon system and the processes, breaking symmetry form the basis of self-organizing model. The theory of a nonlinear parametrical resonance allows finding the features as results of perturbation effects. In the present paper some ratio which determine stationary structure of the Earth–Moon system is given. Researches were supported by grant RFFI (04–02–16633). en Головна астрономічна обсерваторія НАН України Кинематика и физика небесных тел MS4: Positional Astronomy and Global Geodynamics About the tendency to synchronization of the Earth and the Moon rotary movements Article published earlier |
| spellingShingle | About the tendency to synchronization of the Earth and the Moon rotary movements Kurbasova, G.S. Rykhlova, L.V. MS4: Positional Astronomy and Global Geodynamics |
| title | About the tendency to synchronization of the Earth and the Moon rotary movements |
| title_full | About the tendency to synchronization of the Earth and the Moon rotary movements |
| title_fullStr | About the tendency to synchronization of the Earth and the Moon rotary movements |
| title_full_unstemmed | About the tendency to synchronization of the Earth and the Moon rotary movements |
| title_short | About the tendency to synchronization of the Earth and the Moon rotary movements |
| title_sort | about the tendency to synchronization of the earth and the moon rotary movements |
| topic | MS4: Positional Astronomy and Global Geodynamics |
| topic_facet | MS4: Positional Astronomy and Global Geodynamics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79671 |
| work_keys_str_mv | AT kurbasovags aboutthetendencytosynchronizationoftheearthandthemoonrotarymovements AT rykhlovalv aboutthetendencytosynchronizationoftheearthandthemoonrotarymovements |