Milutin Marjanov

Resonant motions are possible only in an inhomogeneous gravitational field. That’s the reason why in the beginning of the work we elucidate the principal differences between a homogeneous and an inhomogeneous field. We show that the resonant motions in closed orbits are consequences of the kinematical extrema conditions at perihelion and/or aphelion. Resonance ratios are determined and it is shown that resonance may take place after one, two and four revolutions. Finally, in the Appendix, we propose a simple criterion for ranking of the gravitational fields.

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Stable motions of one large and two small bodies under mutual gravitational interactions in closed, stable orbits are numerically investigated in this work.

As known, this is one of the oldest problems of the classical mechanics posed by Newton himself in his attempt to find laws of the Moon movements.

Orbits of the bodies were obtained in this work combining the perturbation technique and the method of the successive approximations. One example was given as an illustration of the exposed procedure. A somewhat intriguing result was obtained: the outcome of the arbitrary chosen initial conditions was the resonant motion of three bodies. Was this a pure chance?

Gravitational forces acting on the bodies meet at one point called center of the gravitational attraction. All the “exact” solutions of the three – body problem are given for the cases when center of attraction and center of mass coincide. However, excepting the restricted three – body problem, all such motions are unstable [2] and of theoretical interest, only.

Generally, center moves in the invariant plane describing the locus of these points. It was shown in the second part of this work how to find this curve if one large and two small bodies are in question. We propose the short name attractide for this locus.

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Three bodies moving in the closed orbits, exposed to the mutual gravitational interactions only, enter gradually, as a rule, into the gravitational resonances: periods of their rotations become related as the rational fractions.

In long term, this phenomenon may lead to either and for the most part, stabilization of the orbits or, for some period ratios, be the cause of the chaotic motions, also.

Investigation of that phenomenon in this work was based on the, so cold, Newton’s three-body system consisting of one massive and 2 considerably smaller bodies ( , ) turning around it. The same procedure may be used for examination of the correspondent restricted three-body problem ( , 0).

Numerical analysis of the problem was carried out in the following way. After discretizing the time sub-space, the differential equations of motions were replaced by the corresponding difference equations. Their approximate solutions were procured by use of the perturbation technique.

It was revealed that the chaotic motions zones are situated around the : ~ 1:3 and 3:1 resonances. This outcome is in accordance with the Wisdom’s result for the Sun-Jupiter-asteroid triad /9/, as well as with the observational data. Extensions of the chaotic motions zone depend on the mass ratios and .

Neglecting all the other gravitational influences, the adopted model may be applied to the Sun, together with any pair of the heavenly bodies, provided that they do not form the binary system. Also, it is not necessary that the small bodies form the mutually closest pair.

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What is period of the Earth’s rotation around its axis?

As a rule, answer to this question is that that one turn around the axis of the Earth requires, approximately, 24 hours and that this period is cold the mean solar day. Either, more precisely, some four minutes shorter stellar day.

In fact, neither solar, nor the stellar day stands for the Earth’s full spin period. Earth turns around its axis considerably more slowly. The only frame of reference, in which its intrinsic rotation is clearly defined, while its orbital motions are excluded, has to be the one related with the Moon.

The full spin period of our planet is about 50 minutes longer lunar day – the time between one lunar zenith to the next. The term tidal day is also in use in oceanography.

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Besides translation, spin around its axis and rotation around center of the Milky Way, the Sun performs relative motion in the solar system Laplacian plane, also. This motion was anticipated by Newton himself, in his Principia.

The form of the Sun’s orbit is substantially different from the other solar system bodies’ orbits. Namely, the Sun moves along the path composed of the chain of large and small loops [1, 2, 6, 9]. This chain is situated within the circular outline with the diameter approximately twice as large as the Sun’s is. Under supposition that the solar system is stable, the Sun is going to move along it, in the same region, for eternity, never reitereiting the same path.

It was also shown in this work that velocity and acceleration of the Sun’s center of mass are completely defined by the relative velocities and accelerations of the planets with respect to the Sun.

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1. Introduction

If three bodies move in closed orbits, exposed only to the mutual gravitational interactions, they, as a rule, gradually enter into a gravitational resonance: periods of their rotations become related as the rational fractions [12]. This proportionality is the consequence of the fact that, over a time, such a system assumes dynamical configuration which provides minimal potential energy to every individual body. That phenomenon may lead to, either and for the most part, stabilization of the orbits or, for some period ratios, may also generate chaotic motions.

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